LMFDB - Embedded newform 144.5.m.a.91.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,5,Mod(19,144)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("144.19");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	144.m (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.8852746841$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} + 15 x^{12} - 34 x^{11} + 62 x^{10} - 312 x^{9} + 1432 x^{8} - 4960 x^{7} + \cdots + 2097152 $ x^14 - 4x^13 + 15x^12 - 34x^11 + 62x^10 - 312x^9 + 1432x^8 - 4960x^7 + 11456x^6 - 19968x^5 + 31744x^4 - 139264x^3 + 491520x^2 - 1048576*x + 2097152
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{21} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			91.2
Root			$0.336831 - 2.80830i$ of defining polynomial
Character	$\chi$	$=$	144.91
Dual form			144.5.m.a.19.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.47147 + 3.14513i) q^{2} +(-3.78368 - 15.5462i) q^{4} +(-27.2309 - 27.2309i) q^{5} +50.3097 q^{7} +(58.2460 + 26.5217i) q^{8} +O(q^{10})$	\(q+(-2.47147 + 3.14513i) q^{2} +(-3.78368 - 15.5462i) q^{4} +(-27.2309 - 27.2309i) q^{5} +50.3097 q^{7} +(58.2460 + 26.5217i) q^{8} +(152.945 - 18.3444i) q^{10} +(53.1047 - 53.1047i) q^{11} +(-125.128 + 125.128i) q^{13} +(-124.339 + 158.231i) q^{14} +(-227.367 + 117.644i) q^{16} -286.271 q^{17} +(-99.5010 - 99.5010i) q^{19} +(-320.304 + 526.370i) q^{20} +(35.7745 + 298.268i) q^{22} +100.505 q^{23} +858.049i q^{25} +(-84.2940 - 702.795i) q^{26} +(-190.356 - 782.124i) q^{28} +(-343.872 + 343.872i) q^{29} +208.400i q^{31} +(191.927 - 1005.85i) q^{32} +(707.510 - 900.360i) q^{34} +(-1369.98 - 1369.98i) q^{35} +(-1159.47 - 1159.47i) q^{37} +(558.857 - 67.0299i) q^{38} +(-863.882 - 2308.31i) q^{40} +2335.63i q^{41} +(-2079.41 + 2079.41i) q^{43} +(-1026.51 - 624.643i) q^{44} +(-248.396 + 316.103i) q^{46} +1054.04i q^{47} +130.069 q^{49} +(-2698.67 - 2120.64i) q^{50} +(2418.71 + 1471.82i) q^{52} +(-2136.46 - 2136.46i) q^{53} -2892.18 q^{55} +(2930.34 + 1334.30i) q^{56} +(-231.653 - 1931.39i) q^{58} +(-3721.44 + 3721.44i) q^{59} +(2496.46 - 2496.46i) q^{61} +(-655.446 - 515.055i) q^{62} +(2689.20 + 3089.57i) q^{64} +6814.72 q^{65} +(-329.116 - 329.116i) q^{67} +(1083.16 + 4450.42i) q^{68} +(7694.64 - 922.903i) q^{70} +1040.71 q^{71} +2673.24i q^{73} +(6512.27 - 781.089i) q^{74} +(-1170.38 + 1923.34i) q^{76} +(2671.68 - 2671.68i) q^{77} +4475.80i q^{79} +(9394.98 + 2987.88i) q^{80} +(-7345.86 - 5772.43i) q^{82} +(-1457.69 - 1457.69i) q^{83} +(7795.43 + 7795.43i) q^{85} +(-1400.82 - 11679.2i) q^{86} +(4501.56 - 1684.71i) q^{88} +1146.97i q^{89} +(-6295.17 + 6295.17i) q^{91} +(-380.281 - 1562.48i) q^{92} +(-3315.11 - 2605.04i) q^{94} +5419.01i q^{95} -13101.5 q^{97} +(-321.461 + 409.084i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{2} - 8 q^{4} + 2 q^{5} - 4 q^{7} + 92 q^{8}+O(q^{10}) $ 14 * q + 2 * q^2 - 8 * q^4 + 2 * q^5 - 4 * q^7 + 92 * q^8	$ 14 q + 2 q^{2} - 8 q^{4} + 2 q^{5} - 4 q^{7} + 92 q^{8} - 100 q^{10} - 94 q^{11} - 2 q^{13} - 44 q^{14} - 168 q^{16} + 4 q^{17} - 706 q^{19} - 1900 q^{20} + 900 q^{22} - 1148 q^{23} + 3416 q^{26} - 3784 q^{28} - 862 q^{29} - 3208 q^{32} + 7508 q^{34} - 1340 q^{35} - 1826 q^{37} - 3568 q^{38} - 5144 q^{40} + 1694 q^{43} + 14636 q^{44} - 5316 q^{46} + 682 q^{49} - 20070 q^{50} + 20452 q^{52} + 482 q^{53} - 11780 q^{55} + 6952 q^{56} - 20456 q^{58} + 2786 q^{59} - 3778 q^{61} + 11472 q^{62} + 15808 q^{64} + 2020 q^{65} + 7998 q^{67} - 18032 q^{68} + 15296 q^{70} - 19964 q^{71} + 23780 q^{74} - 23996 q^{76} + 9508 q^{77} - 1384 q^{80} + 16016 q^{82} + 17282 q^{83} + 9948 q^{85} + 4796 q^{86} + 7288 q^{88} - 28036 q^{91} + 14632 q^{92} + 432 q^{94} - 4 q^{97} + 12246 q^{98}+O(q^{100}) $ 14 * q + 2 * q^2 - 8 * q^4 + 2 * q^5 - 4 * q^7 + 92 * q^8 - 100 * q^10 - 94 * q^11 - 2 * q^13 - 44 * q^14 - 168 * q^16 + 4 * q^17 - 706 * q^19 - 1900 * q^20 + 900 * q^22 - 1148 * q^23 + 3416 * q^26 - 3784 * q^28 - 862 * q^29 - 3208 * q^32 + 7508 * q^34 - 1340 * q^35 - 1826 * q^37 - 3568 * q^38 - 5144 * q^40 + 1694 * q^43 + 14636 * q^44 - 5316 * q^46 + 682 * q^49 - 20070 * q^50 + 20452 * q^52 + 482 * q^53 - 11780 * q^55 + 6952 * q^56 - 20456 * q^58 + 2786 * q^59 - 3778 * q^61 + 11472 * q^62 + 15808 * q^64 + 2020 * q^65 + 7998 * q^67 - 18032 * q^68 + 15296 * q^70 - 19964 * q^71 + 23780 * q^74 - 23996 * q^76 + 9508 * q^77 - 1384 * q^80 + 16016 * q^82 + 17282 * q^83 + 9948 * q^85 + 4796 * q^86 + 7288 * q^88 - 28036 * q^91 + 14632 * q^92 + 432 * q^94 - 4 * q^97 + 12246 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−2.47147	+	3.14513i	−0.617867	+	0.786282i
$2$	−2.47147	+	3.14513i	−0.617867	+	0.786282i
$3$		0			0
$3$		0			0
$4$	−3.78368	−	15.5462i	−0.236480	−	0.971636i
$5$	−27.2309	−	27.2309i	−1.08924	−	1.08924i	−0.995607	−	0.0936308i	$-0.970153\pi$
$5$	−27.2309	−	27.2309i	−1.08924	−	1.08924i	−0.0936308	−	0.995607i	$-0.529847\pi$
$6$		0			0
$7$	50.3097			1.02673			0.513365	−	0.858171i	$-0.328399\pi$
$7$	50.3097			1.02673			0.513365	+	0.858171i	$0.328399\pi$
$8$	58.2460	+	26.5217i	0.910094	+	0.414402i
$9$		0			0
$10$	152.945	−	18.3444i	1.52945	−	0.183444i
$11$	53.1047	−	53.1047i	0.438881	−	0.438881i	−0.452754	−	0.891635i	$-0.649558\pi$
$11$	53.1047	−	53.1047i	0.438881	−	0.438881i	0.891635	+	0.452754i	$0.149558\pi$
$12$		0			0
$13$	−125.128	+	125.128i	−0.740404	+	0.740404i	−0.972656	−	0.232252i	$-0.925391\pi$
$13$	−125.128	+	125.128i	−0.740404	+	0.740404i	0.232252	+	0.972656i	$0.425391\pi$
$14$	−124.339	+	158.231i	−0.634382	+	0.807299i
$15$		0			0
$16$	−227.367	+	117.644i	−0.888154	+	0.459546i
$17$	−286.271			−0.990557			−0.495279	−	0.868734i	$-0.664934\pi$
$17$	−286.271			−0.990557			−0.495279	+	0.868734i	$0.664934\pi$
$18$		0			0
$19$	−99.5010	−	99.5010i	−0.275626	−	0.275626i	0.555734	−	0.831360i	$-0.312437\pi$
$19$	−99.5010	−	99.5010i	−0.275626	−	0.275626i	−0.831360	+	0.555734i	$0.812437\pi$
$20$	−320.304	+	526.370i	−0.800760	+	1.31593i
$21$		0			0
$22$	35.7745	+	298.268i	0.0739143	+	0.616255i
$23$	100.505			0.189991			0.0949957	−	0.995478i	$-0.469716\pi$
$23$	100.505			0.189991			0.0949957	+	0.995478i	$0.469716\pi$
$24$		0			0
$25$			858.049i			1.37288i
$26$	−84.2940	−	702.795i	−0.124695	−	1.03964i
$27$		0			0
$28$	−190.356	−	782.124i	−0.242801	−	0.997607i
$29$	−343.872	+	343.872i	−0.408885	+	0.408885i	−0.881350	−	0.472465i	$-0.843364\pi$
$29$	−343.872	+	343.872i	−0.408885	+	0.408885i	0.472465	+	0.881350i	$0.343364\pi$
$30$		0			0
$31$			208.400i			0.216858i	0.994104	+	0.108429i	$0.0345820\pi$
$31$			208.400i			0.216858i	−0.994104	+	0.108429i	$0.965418\pi$
$32$	191.927	−	1005.85i	0.187429	−	0.982278i
$33$		0			0
$34$	707.510	−	900.360i	0.612033	−	0.778858i
$35$	−1369.98	−	1369.98i	−1.11835	−	1.11835i
$36$		0			0
$37$	−1159.47	−	1159.47i	−0.846946	−	0.846946i	0.142805	−	0.989751i	$-0.454388\pi$
$37$	−1159.47	−	1159.47i	−0.846946	−	0.846946i	−0.989751	+	0.142805i	$0.954388\pi$
$38$	558.857	−	67.0299i	0.387020	−	0.0464196i
$39$		0			0
$40$	−863.882	−	2308.31i	−0.539927	−	1.44269i
$41$			2335.63i			1.38943i	0.719286	+	0.694714i	$0.244469\pi$
$41$			2335.63i			1.38943i	−0.719286	+	0.694714i	$0.755531\pi$
$42$		0			0
$43$	−2079.41	+	2079.41i	−1.12461	+	1.12461i	−0.133575	+	0.991039i	$0.542646\pi$
$43$	−2079.41	+	2079.41i	−1.12461	+	1.12461i	−0.991039	+	0.133575i	$0.957354\pi$
$44$	−1026.51	−	624.643i	−0.530220	−	0.322646i
$45$		0			0
$46$	−248.396	+	316.103i	−0.117389	+	0.149387i
$47$			1054.04i			0.477159i	0.971123	+	0.238580i	$0.0766818\pi$
$47$			1054.04i			0.477159i	−0.971123	+	0.238580i	$0.923318\pi$
$48$		0			0
$49$	130.069			0.0541728
$50$	−2698.67	−	2120.64i	−1.07947	−	0.848256i
$51$		0			0
$52$	2418.71	+	1471.82i	0.894494	+	0.544312i
$53$	−2136.46	−	2136.46i	−0.760576	−	0.760576i	0.215850	−	0.976426i	$-0.430748\pi$
$53$	−2136.46	−	2136.46i	−0.760576	−	0.760576i	−0.976426	+	0.215850i	$0.930748\pi$
$54$		0			0
$55$	−2892.18			−0.956092
$56$	2930.34	+	1334.30i	0.934420	+	0.425479i
$57$		0			0
$58$	−231.653	−	1931.39i	−0.0688625	−	0.574136i
$59$	−3721.44	+	3721.44i	−1.06907	+	1.06907i	−0.0716407	+	0.997431i	$0.522823\pi$
$59$	−3721.44	+	3721.44i	−1.06907	+	1.06907i	−0.997431	+	0.0716407i	$0.977177\pi$
$60$		0			0
$61$	2496.46	−	2496.46i	0.670912	−	0.670912i	−0.287014	−	0.957926i	$-0.592663\pi$
$61$	2496.46	−	2496.46i	0.670912	−	0.670912i	0.957926	+	0.287014i	$0.0926628\pi$
$62$	−655.446	−	515.055i	−0.170512	−	0.133989i
$63$		0			0
$64$	2689.20	+	3089.57i	0.656542	+	0.754289i
$65$	6814.72			1.61295
$66$		0			0
$67$	−329.116	−	329.116i	−0.0733162	−	0.0733162i	0.669498	−	0.742814i	$-0.266509\pi$
$67$	−329.116	−	329.116i	−0.0733162	−	0.0733162i	−0.742814	+	0.669498i	$0.766509\pi$
$68$	1083.16	+	4450.42i	0.234247	+	0.962461i
$69$		0			0
$70$	7694.64	−	922.903i	1.57033	−	0.188348i
$71$	1040.71			0.206449			0.103225	−	0.994658i	$-0.467084\pi$
$71$	1040.71			0.206449			0.103225	+	0.994658i	$0.467084\pi$
$72$		0			0
$73$			2673.24i			0.501639i	0.968034	+	0.250820i	$0.0807001\pi$
$73$			2673.24i			0.501639i	−0.968034	+	0.250820i	$0.919300\pi$
$74$	6512.27	−	781.089i	1.18924	−	0.142639i
$75$		0			0
$76$	−1170.38	+	1923.34i	−0.202628	+	0.332988i
$77$	2671.68	−	2671.68i	0.450612	−	0.450612i
$78$		0			0
$79$			4475.80i			0.717161i	0.933499	+	0.358580i	$0.116739\pi$
$79$			4475.80i			0.717161i	−0.933499	+	0.358580i	$0.883261\pi$
$80$	9394.98	+	2987.88i	1.46797	+	0.466857i
$81$		0			0
$82$	−7345.86	−	5772.43i	−1.09248	−	0.858482i
$83$	−1457.69	−	1457.69i	−0.211597	−	0.211597i	0.593349	−	0.804945i	$-0.297805\pi$
$83$	−1457.69	−	1457.69i	−0.211597	−	0.211597i	−0.804945	+	0.593349i	$0.797805\pi$
$84$		0			0
$85$	7795.43	+	7795.43i	1.07895	+	1.07895i
$86$	−1400.82	−	11679.2i	−0.189402	−	1.57913i
$87$		0			0
$88$	4501.56	−	1684.71i	0.581297	−	0.217550i
$89$			1146.97i			0.144801i	0.997376	+	0.0724003i	$0.0230659\pi$
$89$			1146.97i			0.144801i	−0.997376	+	0.0724003i	$0.976934\pi$
$90$		0			0
$91$	−6295.17	+	6295.17i	−0.760194	+	0.760194i
$92$	−380.281	−	1562.48i	−0.0449292	−	0.184603i
$93$		0			0
$94$	−3315.11	−	2605.04i	−0.375182	−	0.294821i
$95$			5419.01i			0.600444i
$96$		0			0
$97$	−13101.5			−1.39244			−0.696222	−	0.717826i	$-0.745137\pi$
$97$	−13101.5			−1.39244			−0.696222	+	0.717826i	$0.745137\pi$
$98$	−321.461	+	409.084i	−0.0334716	+	0.0425952i
$99$		0			0
$100$	13339.4	−	3246.59i	1.33394	−	0.324659i
$101$	7488.18	+	7488.18i	0.734063	+	0.734063i	0.971422	−	0.237359i	$-0.0762818\pi$
$101$	7488.18	+	7488.18i	0.734063	+	0.734063i	−0.237359	+	0.971422i	$0.576282\pi$
$102$		0			0
$103$	7141.23			0.673129			0.336565	−	0.941660i	$-0.390735\pi$
$103$	7141.23			0.673129			0.336565	+	0.941660i	$0.390735\pi$
$104$	−10606.8	+	3969.60i	−0.980662	+	0.367012i
$105$		0			0
$106$	11999.6	−	1439.25i	1.06796	−	0.128093i
$107$	−1794.26	+	1794.26i	−0.156718	+	0.156718i	−0.781111	−	0.624393i	$-0.785346\pi$
$107$	−1794.26	+	1794.26i	−0.156718	+	0.156718i	0.624393	+	0.781111i	$0.285346\pi$
$108$		0			0
$109$	5362.57	−	5362.57i	0.451357	−	0.451357i	−0.444448	−	0.895805i	$-0.646600\pi$
$109$	5362.57	−	5362.57i	0.451357	−	0.451357i	0.895805	+	0.444448i	$0.146600\pi$
$110$	7147.93	−	9096.28i	0.590738	−	0.751759i
$111$		0			0
$112$	−11438.8	+	5918.62i	−0.911894	+	0.471829i
$113$	5165.40			0.404527			0.202263	−	0.979331i	$-0.435170\pi$
$113$	5165.40			0.404527			0.202263	+	0.979331i	$0.435170\pi$
$114$		0			0
$115$	−2736.86	−	2736.86i	−0.206946	−	0.206946i
$116$	6647.00	+	4044.80i	0.493981	+	0.300594i
$117$		0			0
$118$	−2506.99	−	20901.8i	−0.180048	−	1.50114i
$119$	−14402.2			−1.01703
$120$		0			0
$121$			9000.79i			0.614766i
$122$	1681.77	+	14021.6i	0.112992	+	0.942061i
$123$		0			0
$124$	3239.83	−	788.521i	0.210707	−	0.0512826i
$125$	6346.13	−	6346.13i	0.406152	−	0.406152i
$126$		0			0
$127$		−	22886.9i		−	1.41899i	−0.704711	−	0.709495i	$-0.748923\pi$
$127$		−	22886.9i		−	1.41899i	0.704711	−	0.709495i	$-0.251077\pi$
$128$	−16363.4	+	822.099i	−0.998740	+	0.0501770i
$129$		0			0
$130$	−16842.4	+	21433.2i	−0.996590	+	1.26824i
$131$	−19202.2	−	19202.2i	−1.11894	−	1.11894i	−0.991897	−	0.127048i	$-0.959450\pi$
$131$	−19202.2	−	19202.2i	−1.11894	−	1.11894i	−0.127048	−	0.991897i	$-0.540550\pi$
$132$		0			0
$133$	−5005.87	−	5005.87i	−0.282993	−	0.282993i
$134$	1848.51	−	221.713i	0.102947	−	0.0123476i
$135$		0			0
$136$	−16674.1	−	7592.40i	−0.901500	−	0.410489i
$137$		−	33680.5i		−	1.79448i	−0.441547	−	0.897238i	$-0.645570\pi$
$137$		−	33680.5i		−	1.79448i	0.441547	−	0.897238i	$-0.354430\pi$
$138$		0			0
$139$	11747.9	−	11747.9i	0.608036	−	0.608036i	−0.334397	−	0.942432i	$-0.608532\pi$
$139$	11747.9	−	11747.9i	0.608036	−	0.608036i	0.942432	+	0.334397i	$0.108532\pi$
$140$	−16114.4	+	26481.6i	−0.822163	+	1.35110i
$141$		0			0
$142$	−2572.09	+	3273.17i	−0.127558	+	0.162328i
$143$			13289.8i			0.649899i
$144$		0			0
$145$	18727.9			0.890746
$146$	−8407.68	−	6606.82i	−0.394430	−	0.309947i
$147$		0			0
$148$	−13638.2	+	22412.4i	−0.622637	+	1.02321i
$149$	14877.7	+	14877.7i	0.670136	+	0.670136i	0.957747	−	0.287611i	$-0.0928611\pi$
$149$	14877.7	+	14877.7i	0.670136	+	0.670136i	−0.287611	+	0.957747i	$0.592861\pi$
$150$		0			0
$151$	−8005.74			−0.351114			−0.175557	−	0.984469i	$-0.556173\pi$
$151$	−8005.74			−0.351114			−0.175557	+	0.984469i	$0.556173\pi$
$152$	−3156.60	−	8434.47i	−0.136626	−	0.365065i
$153$		0			0
$154$	1799.81	+	15005.8i	0.0758900	+	0.632727i
$155$	5674.94	−	5674.94i	0.236210	−	0.236210i
$156$		0			0
$157$	12150.9	−	12150.9i	0.492958	−	0.492958i	−0.416279	−	0.909237i	$-0.636666\pi$
$157$	12150.9	−	12150.9i	0.492958	−	0.492958i	0.909237	+	0.416279i	$0.136666\pi$
$158$	−14077.0	−	11061.8i	−0.563891	−	0.443110i
$159$		0			0
$160$	−32616.7	+	22164.0i	−1.27409	+	0.865780i
$161$	5056.40			0.195070
$162$		0			0
$163$	−23646.5	−	23646.5i	−0.890002	−	0.890002i	0.104520	−	0.994523i	$-0.466669\pi$
$163$	−23646.5	−	23646.5i	−0.890002	−	0.890002i	−0.994523	+	0.104520i	$0.966669\pi$
$164$	36310.1	−	8837.28i	1.35002	−	0.328572i
$165$		0			0
$166$	8187.26	−	981.989i	0.297113	−	0.0356361i
$167$	−42493.7			−1.52367			−0.761836	−	0.647770i	$-0.775702\pi$
$167$	−42493.7			−1.52367			−0.761836	+	0.647770i	$0.775702\pi$
$168$		0			0
$169$		−	2753.14i		−	0.0963950i
$170$	−43783.8	+	5251.48i	−1.51501	+	0.181712i
$171$		0			0
$172$	40194.7	+	24459.1i	1.35866	+	0.826767i
$173$	16142.1	−	16142.1i	0.539347	−	0.539347i	−0.383990	−	0.923337i	$-0.625450\pi$
$173$	16142.1	−	16142.1i	0.539347	−	0.539347i	0.923337	+	0.383990i	$0.125450\pi$
$174$		0			0
$175$			43168.2i			1.40957i
$176$	−5826.84	+	18321.7i	−0.188108	+	0.591480i
$177$		0			0
$178$	−3607.36	−	2834.69i	−0.113854	−	0.0894676i
$179$	22442.0	+	22442.0i	0.700415	+	0.700415i	0.964500	−	0.264084i	$-0.0850698\pi$
$179$	22442.0	+	22442.0i	0.700415	+	0.700415i	−0.264084	+	0.964500i	$0.585070\pi$
$180$		0			0
$181$	−9891.06	−	9891.06i	−0.301916	−	0.301916i	0.539847	−	0.841763i	$-0.318482\pi$
$181$	−9891.06	−	9891.06i	−0.301916	−	0.301916i	−0.841763	+	0.539847i	$0.818482\pi$
$182$	−4240.81	−	35357.4i	−0.128028	−	1.06743i
$183$		0			0
$184$	5854.04	+	2665.58i	0.172910	+	0.0787328i
$185$			63146.9i			1.84505i
$186$		0			0
$187$	−15202.3	+	15202.3i	−0.434737	+	0.434737i
$188$	16386.4	−	3988.17i	0.463625	−	0.112839i
$189$		0			0
$190$	−17043.5	−	13392.9i	−0.472119	−	0.370995i
$191$		−	2033.60i		−	0.0557442i	−0.999611	−	0.0278721i	$-0.991127\pi$
$191$		−	2033.60i		−	0.0557442i	0.999611	−	0.0278721i	$-0.00887311\pi$
$192$		0			0
$193$	29257.4			0.785453			0.392727	−	0.919655i	$-0.371532\pi$
$193$	29257.4			0.785453			0.392727	+	0.919655i	$0.371532\pi$
$194$	32380.0	−	41206.0i	0.860346	−	1.09486i
$195$		0			0
$196$	−492.140	−	2022.08i	−0.0128108	−	0.0526363i
$197$	−28194.9	−	28194.9i	−0.726504	−	0.726504i	0.243417	−	0.969922i	$-0.421732\pi$
$197$	−28194.9	−	28194.9i	−0.726504	−	0.726504i	−0.969922	+	0.243417i	$0.921732\pi$
$198$		0			0
$199$	54100.9			1.36615			0.683075	−	0.730348i	$-0.260642\pi$
$199$	54100.9			1.36615			0.683075	+	0.730348i	$0.260642\pi$
$200$	−22756.9	+	49977.9i	−0.568923	+	1.24945i
$201$		0			0
$202$	−42058.1	+	5044.49i	−1.03073	+	0.123627i
$203$	−17300.1	+	17300.1i	−0.419814	+	0.419814i
$204$		0			0
$205$	63601.4	−	63601.4i	1.51342	−	1.51342i
$206$	−17649.3	+	22460.1i	−0.415905	+	0.529270i
$207$		0			0
$208$	13729.5	−	43170.6i	0.317343	−	0.997842i
$209$	−10567.9			−0.241934
$210$		0			0
$211$	−31994.1	−	31994.1i	−0.718630	−	0.718630i	0.249694	−	0.968325i	$-0.419670\pi$
$211$	−31994.1	−	31994.1i	−0.718630	−	0.718630i	−0.968325	+	0.249694i	$0.919670\pi$
$212$	−25130.1	+	41297.5i	−0.559142	+	0.918865i
$213$		0			0
$214$	−1208.72	−	10077.6i	−0.0263937	−	0.220055i
$215$	113249.			2.44994
$216$		0			0
$217$			10484.6i			0.222654i
$218$	3612.56	+	30119.4i	0.0760155	+	0.633773i
$219$		0			0
$220$	10943.1	+	44962.3i	0.226097	+	0.928974i
$221$	35820.6	−	35820.6i	0.733412	−	0.733412i
$222$		0			0
$223$			94185.8i			1.89398i	0.321261	+	0.946991i	$0.395893\pi$
$223$			94185.8i			1.89398i	−0.321261	+	0.946991i	$0.604107\pi$
$224$	9655.79	−	50604.2i	0.192438	−	1.00853i
$225$		0			0
$226$	−12766.1	+	16245.9i	−0.249944	+	0.318072i
$227$	−62683.9	−	62683.9i	−1.21648	−	1.21648i	−0.968857	−	0.247622i	$-0.920351\pi$
$227$	−62683.9	−	62683.9i	−1.21648	−	1.21648i	−0.247622	−	0.968857i	$-0.579649\pi$
$228$		0			0
$229$	−19781.9	−	19781.9i	−0.377221	−	0.377221i	0.492877	−	0.870099i	$-0.335945\pi$
$229$	−19781.9	−	19781.9i	−0.377221	−	0.377221i	−0.870099	+	0.492877i	$0.835945\pi$
$230$	15371.8	−	1843.72i	0.290583	−	0.0348528i
$231$		0			0
$232$	−29149.3	+	10909.1i	−0.541566	+	0.202681i
$233$			25062.1i			0.461642i	0.972996	+	0.230821i	$0.0741412\pi$
$233$			25062.1i			0.461642i	−0.972996	+	0.230821i	$0.925859\pi$
$234$		0			0
$235$	28702.6	−	28702.6i	0.519740	−	0.519740i
$236$	71934.9	+	43773.4i	1.29156	+	0.785934i
$237$		0			0
$238$	35594.6	−	45296.9i	0.628392	−	0.799676i
$239$		−	93041.8i		−	1.62885i	−0.580265	−	0.814427i	$-0.697051\pi$
$239$		−	93041.8i		−	1.62885i	0.580265	−	0.814427i	$-0.302949\pi$
$240$		0			0
$241$	−80981.9			−1.39429			−0.697146	−	0.716929i	$-0.745547\pi$
$241$	−80981.9			−1.39429			−0.697146	+	0.716929i	$0.745547\pi$
$242$	−28308.7	−	22245.2i	−0.483380	−	0.379844i
$243$		0			0
$244$	−48256.3	−	29364.6i	−0.810540	−	0.493225i
$245$	−3541.90	−	3541.90i	−0.0590071	−	0.0590071i
$246$		0			0
$247$	24900.8			0.408149
$248$	−5527.14	+	12138.5i	−0.0898663	+	0.197361i
$249$		0			0
$250$	4275.14	+	35643.7i	0.0684023	+	0.570299i
$251$	−25910.0	+	25910.0i	−0.411264	+	0.411264i	−0.882179	−	0.470915i	$-0.843924\pi$
$251$	−25910.0	+	25910.0i	−0.411264	+	0.411264i	0.470915	+	0.882179i	$0.343924\pi$
$252$		0			0
$253$	5337.31	−	5337.31i	0.0833837	−	0.0833837i
$254$	71982.2	+	56564.2i	1.11573	+	0.876747i
$255$		0			0
$256$	37855.9	−	53496.7i	0.577636	−	0.816295i
$257$	−15800.6			−0.239225			−0.119613	−	0.992821i	$-0.538165\pi$
$257$	−15800.6			−0.239225			−0.119613	+	0.992821i	$0.538165\pi$
$258$		0			0
$259$	−58332.6	−	58332.6i	−0.869584	−	0.869584i
$260$	−25784.7	−	105943.i	−0.381431	−	1.56720i
$261$		0			0
$262$	107851.	−	12935.8i	1.57117	−	0.188447i
$263$	82043.7			1.18613			0.593067	−	0.805153i	$-0.297917\pi$
$263$	82043.7			1.18613			0.593067	+	0.805153i	$0.297917\pi$
$264$		0			0
$265$			116356.i			1.65690i
$266$	28115.9	−	3372.26i	0.397365	−	0.0476604i
$267$		0			0
$268$	−3871.23	+	6361.78i	−0.0538988	+	0.0885745i
$269$	30820.2	−	30820.2i	0.425923	−	0.425923i	−0.461314	−	0.887237i	$-0.652622\pi$
$269$	30820.2	−	30820.2i	0.425923	−	0.425923i	0.887237	+	0.461314i	$0.152622\pi$
$270$		0			0
$271$		−	110808.i		−	1.50880i	−0.656412	−	0.754402i	$-0.727927\pi$
$271$		−	110808.i		−	1.50880i	0.656412	−	0.754402i	$-0.272073\pi$
$272$	65088.7	−	33678.0i	0.879768	−	0.455206i
$273$		0			0
$274$	105930.	+	83240.4i	1.41097	+	1.10875i
$275$	45566.4	+	45566.4i	0.602531	+	0.602531i
$276$		0			0
$277$	25634.0	+	25634.0i	0.334084	+	0.334084i	0.854135	−	0.520051i	$-0.174087\pi$
$277$	25634.0	+	25634.0i	0.334084	+	0.334084i	−0.520051	+	0.854135i	$0.674087\pi$
$278$	7914.08	+	65983.0i	0.102403	+	0.853773i
$279$		0			0
$280$	−43461.7	−	116130.i	−0.554358	−	1.48125i
$281$			48800.5i			0.618033i	0.951057	+	0.309017i	$0.0999999\pi$
$281$			48800.5i			0.618033i	−0.951057	+	0.309017i	$0.900000\pi$
$282$		0			0
$283$	−111466.	+	111466.i	−1.39178	+	1.39178i	−0.570439	+	0.821340i	$0.693227\pi$
$283$	−111466.	+	111466.i	−1.39178	+	1.39178i	−0.821340	+	0.570439i	$0.806773\pi$
$284$	−3937.72	−	16179.1i	−0.0488212	−	0.200594i
$285$		0			0
$286$	−41798.1	−	32845.3i	−0.511004	−	0.401551i
$287$			117505.i			1.42657i
$288$		0			0
$289$	−1569.88			−0.0187963
$290$	−46285.5	+	58901.8i	−0.550363	+	0.700378i
$291$		0			0
$292$	41558.6	−	10114.7i	0.487411	−	0.118628i
$293$	−13093.1	−	13093.1i	−0.152514	−	0.152514i	0.626726	−	0.779240i	$-0.284395\pi$
$293$	−13093.1	−	13093.1i	−0.152514	−	0.152514i	−0.779240	+	0.626726i	$0.784395\pi$
$294$		0			0
$295$	202676.			2.32895
$296$	−36783.3	−	98285.6i	−0.419824	−	1.12178i
$297$		0			0
$298$	−83562.0	+	10022.5i	−0.940971	+	0.112861i
$299$	−12576.1	+	12576.1i	−0.140670	+	0.140670i
$300$		0			0
$301$	−104615.	+	104615.i	−1.15467	+	1.15467i
$302$	19785.9	−	25179.1i	0.216942	−	0.276075i
$303$		0			0
$304$	34328.9	+	10917.6i	0.371461	+	0.118136i
$305$	−135962.			−1.46156
$306$		0			0
$307$	25274.9	+	25274.9i	0.268171	+	0.268171i	0.828363	−	0.560192i	$-0.189273\pi$
$307$	25274.9	+	25274.9i	0.268171	+	0.268171i	−0.560192	+	0.828363i	$0.689273\pi$
$308$	−51643.2	−	31425.6i	−0.544392	−	0.331270i
$309$		0			0
$310$	3822.99	+	31873.9i	0.0397813	+	0.331674i
$311$	−53808.0			−0.556322			−0.278161	−	0.960534i	$-0.589725\pi$
$311$	−53808.0			−0.556322			−0.278161	+	0.960534i	$0.589725\pi$
$312$		0			0
$313$			137345.i			1.40192i	0.713199	+	0.700961i	$0.247245\pi$
$313$			137345.i			1.40192i	−0.713199	+	0.700961i	$0.752755\pi$
$314$	8185.60	+	68246.8i	0.0830217	+	0.692187i
$315$		0			0
$316$	69581.6	−	16935.0i	0.696819	−	0.169594i
$317$	−115546.	+	115546.i	−1.14984	+	1.14984i	−0.163252	+	0.986584i	$0.552198\pi$
$317$	−115546.	+	115546.i	−1.14984	+	1.14984i	−0.986584	+	0.163252i	$0.947802\pi$
$318$		0			0
$319$			36522.4i			0.358904i
$320$	10902.5	−	157361.i	0.106470	−	1.53673i
$321$		0			0
$322$	−12496.7	+	15903.0i	−0.120527	+	0.153380i
$323$	28484.2	+	28484.2i	0.273023	+	0.273023i
$324$		0			0
$325$	−107366.	−	107366.i	−1.01648	−	1.01648i
$326$	132813.	−	15929.7i	1.24970	−	0.149890i
$327$		0			0
$328$	−61944.9	+	136041.i	−0.575782	+	1.26451i
$329$			53028.7i			0.489913i
$330$		0			0
$331$	68009.3	−	68009.3i	0.620744	−	0.620744i	−0.324978	−	0.945722i	$-0.605357\pi$
$331$	68009.3	−	68009.3i	0.620744	−	0.620744i	0.945722	+	0.324978i	$0.105357\pi$
$332$	−17146.1	+	28176.9i	−0.155557	+	0.255633i
$333$		0			0
$334$	105022.	−	133648.i	0.941427	−	1.19804i
$335$			17924.3i			0.159718i
$336$		0			0
$337$	−146703.			−1.29176			−0.645878	−	0.763440i	$-0.723509\pi$
$337$	−146703.			−1.29176			−0.645878	+	0.763440i	$0.723509\pi$
$338$	8658.97	+	6804.29i	0.0757937	+	0.0595593i
$339$		0			0
$340$	91693.7	−	150685.i	0.793198	−	1.30350i
$341$	11067.0	+	11067.0i	0.0951749	+	0.0951749i
$342$		0			0
$343$	−114250.			−0.971108
$344$	−176267.	+	65967.9i	−1.48955	+	0.557462i
$345$		0			0
$346$	10874.3	+	90663.8i	0.0908343	+	0.757324i
$347$	80120.9	−	80120.9i	0.665406	−	0.665406i	−0.291243	−	0.956649i	$-0.594069\pi$
$347$	80120.9	−	80120.9i	0.665406	−	0.665406i	0.956649	+	0.291243i	$0.0940688\pi$
$348$		0			0
$349$	100990.	−	100990.i	0.829143	−	0.829143i	−0.158255	−	0.987398i	$-0.550587\pi$
$349$	100990.	−	100990.i	0.829143	−	0.829143i	0.987398	+	0.158255i	$0.0505869\pi$
$350$	−135770.	−	106689.i	−1.10832	−	0.870929i
$351$		0			0
$352$	−43223.3	−	63607.7i	−0.348845	−	0.513363i
$353$	−129855.			−1.04210			−0.521052	−	0.853525i	$-0.674460\pi$
$353$	−129855.			−1.04210			−0.521052	+	0.853525i	$0.674460\pi$
$354$		0			0
$355$	−28339.6	−	28339.6i	−0.224873	−	0.224873i
$356$	17830.9	−	4339.76i	0.140694	−	0.0342425i
$357$		0			0
$358$	−126048.	+	15118.3i	−0.983488	+	0.117961i
$359$	55943.2			0.434068			0.217034	−	0.976164i	$-0.430362\pi$
$359$	55943.2			0.434068			0.217034	+	0.976164i	$0.430362\pi$
$360$		0			0
$361$		−	110520.i		−	0.848061i
$362$	55554.1	−	6663.23i	0.423935	−	0.0508472i
$363$		0			0
$364$	121685.	+	74046.9i	0.918403	+	0.558861i
$365$	72794.8	−	72794.8i	0.546405	−	0.546405i
$366$		0			0
$367$			144947.i			1.07616i	0.842892	+	0.538082i	$0.180851\pi$
$367$			144947.i			1.07616i	−0.842892	+	0.538082i	$0.819149\pi$
$368$	−22851.7	+	11823.8i	−0.168742	+	0.0873097i
$369$		0			0
$370$	−198605.	−	156066.i	−1.45073	−	1.14000i
$371$	−107485.	−	107485.i	−0.780906	−	0.780906i
$372$		0			0
$373$	24034.7	+	24034.7i	0.172751	+	0.172751i	0.788187	−	0.615436i	$-0.211020\pi$
$373$	24034.7	+	24034.7i	0.172751	+	0.172751i	−0.615436	+	0.788187i	$0.711020\pi$
$374$	−10241.2	−	85385.4i	−0.0732164	−	0.610436i
$375$		0			0
$376$	−27955.1	+	61393.9i	−0.197736	+	0.434260i
$377$		−	86056.2i		−	0.605480i
$378$		0			0
$379$	−27907.2	+	27907.2i	−0.194284	+	0.194284i	−0.797544	−	0.603260i	$-0.793868\pi$
$379$	−27907.2	+	27907.2i	−0.194284	+	0.194284i	0.603260	+	0.797544i	$0.293868\pi$
$380$	84244.9	−	20503.8i	0.583414	−	0.141993i
$381$		0			0
$382$	6395.95	+	5025.99i	0.0438307	+	0.0344425i
$383$			96652.7i			0.658895i	0.944174	+	0.329448i	$0.106863\pi$
$383$			96652.7i			0.658895i	−0.944174	+	0.329448i	$0.893137\pi$
$384$		0			0
$385$	−145505.			−0.981648
$386$	−72308.6	+	92018.2i	−0.485306	+	0.617588i
$387$		0			0
$388$	49572.0	+	203678.i	0.329286	+	1.35295i
$389$	−133129.	−	133129.i	−0.879777	−	0.879777i	0.113734	−	0.993511i	$-0.463719\pi$
$389$	−133129.	−	133129.i	−0.879777	−	0.879777i	−0.993511	+	0.113734i	$0.963719\pi$
$390$		0			0
$391$	−28771.8			−0.188197
$392$	7576.00	+	3449.65i	0.0493024	+	0.0224493i
$393$		0			0
$394$	158359.	−	18993.8i	1.02012	−	0.122354i
$395$	121880.	−	121880.i	0.781159	−	0.781159i
$396$		0			0
$397$	33406.1	−	33406.1i	0.211956	−	0.211956i	−0.593142	−	0.805098i	$-0.702113\pi$
$397$	33406.1	−	33406.1i	0.211956	−	0.211956i	0.805098	+	0.593142i	$0.202113\pi$
$398$	−133709.	+	170154.i	−0.844100	+	1.07418i
$399$		0			0
$400$	−100944.	−	195092.i	−0.630900	−	1.21933i
$401$	87329.7			0.543092			0.271546	−	0.962425i	$-0.412465\pi$
$401$	87329.7			0.543092			0.271546	+	0.962425i	$0.412465\pi$
$402$		0			0
$403$	−26076.8	−	26076.8i	−0.160562	−	0.160562i
$404$	88079.6	−	144745.i	0.539651	−	0.886834i
$405$		0			0
$406$	−11654.4	−	97167.8i	−0.0707031	−	0.589482i
$407$	−123146.			−0.743418
$408$		0			0
$409$			47133.9i			0.281765i	0.990026	+	0.140883i	$0.0449940\pi$
$409$			47133.9i			0.281765i	−0.990026	+	0.140883i	$0.955006\pi$
$410$	42845.8	+	357223.i	0.254883	+	2.12506i
$411$		0			0
$412$	−27020.2	−	111019.i	−0.159182	−	0.654037i
$413$	−187224.	+	187224.i	−1.09765	+	1.09765i
$414$		0			0
$415$			79388.5i			0.460958i
$416$	101845.	+	149876.i	0.588509	+	0.866055i
$417$		0			0
$418$	26118.3	−	33237.5i	0.149483	−	0.190229i
$419$	70487.1	+	70487.1i	0.401497	+	0.401497i	0.878760	−	0.477264i	$-0.158371\pi$
$419$	70487.1	+	70487.1i	0.401497	+	0.401497i	−0.477264	+	0.878760i	$0.658371\pi$
$420$		0			0
$421$	109929.	+	109929.i	0.620225	+	0.620225i	0.945589	−	0.325364i	$-0.105487\pi$
$421$	109929.	+	109929.i	0.620225	+	0.620225i	−0.325364	+	0.945589i	$0.605487\pi$
$422$	179698.	−	21553.2i	1.00906	−	0.121028i
$423$		0			0
$424$	−67777.6	−	181103.i	−0.377012	−	1.00738i
$425$		−	245634.i		−	1.35991i
$426$		0			0
$427$	125596.	−	125596.i	0.688845	−	0.688845i
$428$	34682.8	+	21105.0i	0.189333	+	0.115212i
$429$		0			0
$430$	−279891.	+	356182.i	−1.51374	+	1.92635i
$431$			8391.44i			0.0451733i	0.999745	+	0.0225867i	$0.00719017\pi$
$431$			8391.44i			0.0451733i	−0.999745	+	0.0225867i	$0.992810\pi$
$432$		0			0
$433$	112221.			0.598545			0.299272	−	0.954168i	$-0.403256\pi$
$433$	112221.			0.598545			0.299272	+	0.954168i	$0.403256\pi$
$434$	−32975.3	−	25912.3i	−0.175069	−	0.137571i
$435$		0			0
$436$	−103658.	−	63077.3i	−0.545292	−	0.331818i
$437$	−10000.4	−	10000.4i	−0.0523666	−	0.0523666i
$438$		0			0
$439$	−95834.1			−0.497269			−0.248634	−	0.968597i	$-0.579982\pi$
$439$	−95834.1			−0.497269			−0.248634	+	0.968597i	$0.579982\pi$
$440$	−168458.	−	76705.6i	−0.870134	−	0.396206i
$441$		0			0
$442$	24130.9	+	201190.i	0.123518	+	1.02982i
$443$	−48800.8	+	48800.8i	−0.248668	+	0.248668i	−0.820424	−	0.571756i	$-0.806262\pi$
$443$	−48800.8	+	48800.8i	−0.248668	+	0.248668i	0.571756	+	0.820424i	$0.306262\pi$
$444$		0			0
$445$	31233.0	−	31233.0i	0.157722	−	0.157722i
$446$	−296227.	−	232777.i	−1.48920	−	1.17023i
$447$		0			0
$448$	135293.	+	155435.i	0.674091	+	0.774451i
$449$	246669.			1.22355			0.611776	−	0.791031i	$-0.290455\pi$
$449$	246669.			1.22355			0.611776	+	0.791031i	$0.290455\pi$
$450$		0			0
$451$	124033.	+	124033.i	0.609794	+	0.609794i
$452$	−19544.3	−	80302.3i	−0.0956626	−	0.393053i
$453$		0			0
$454$	352070.	−	42227.7i	1.70812	−	0.204874i
$455$	342847.			1.65606
$456$		0			0
$457$		−	6030.04i		−	0.0288727i	−0.999896	−	0.0144364i	$-0.995405\pi$
$457$		−	6030.04i		−	0.0288727i	0.999896	−	0.0144364i	$-0.00459540\pi$
$458$	111107.	−	13326.3i	0.529675	−	0.0635298i
$459$		0			0
$460$	−32192.3	+	52903.1i	−0.152137	+	0.250015i
$461$	−13122.6	+	13122.6i	−0.0617473	+	0.0617473i	−0.737306	−	0.675559i	$-0.763902\pi$
$461$	−13122.6	+	13122.6i	−0.0617473	+	0.0617473i	0.675559	+	0.737306i	$0.263902\pi$
$462$		0			0
$463$		−	408077.i		−	1.90362i	−0.306689	−	0.951810i	$-0.599221\pi$
$463$		−	408077.i		−	1.90362i	0.306689	−	0.951810i	$-0.400779\pi$
$464$	37731.0	−	118640.i	0.175252	−	0.551054i
$465$		0			0
$466$	−78823.5	−	61940.1i	−0.362981	−	0.285233i
$467$	108240.	+	108240.i	0.496311	+	0.496311i	0.910288	−	0.413977i	$-0.135860\pi$
$467$	108240.	+	108240.i	0.496311	+	0.496311i	−0.413977	+	0.910288i	$0.635860\pi$
$468$		0			0
$469$	−16557.8	−	16557.8i	−0.0752759	−	0.0752759i
$470$	19335.8	+	161211.i	0.0875321	+	0.729792i
$471$		0			0
$472$	−315458.	+	118060.i	−1.41598	+	0.529930i
$473$			220853.i			0.987144i
$474$		0			0
$475$	85376.7	−	85376.7i	0.378401	−	0.378401i
$476$	54493.5	+	223900.i	0.240508	+	0.988187i
$477$		0			0
$478$	292629.	+	229950.i	1.28074	+	1.00642i
$479$		−	223150.i		−	0.972583i	−0.873797	−	0.486291i	$-0.838349\pi$
$479$		−	223150.i		−	0.972583i	0.873797	−	0.486291i	$-0.161651\pi$
$480$		0			0
$481$	290165.			1.25416
$482$	200144.	−	254699.i	0.861488	−	1.09631i
$483$		0			0
$484$	139928.	−	34056.2i	0.597329	−	0.145380i
$485$	356767.	+	356767.i	1.51670	+	1.51670i
$486$		0			0
$487$	225880.			0.952399			0.476200	−	0.879337i	$-0.342014\pi$
$487$	225880.			0.952399			0.476200	+	0.879337i	$0.342014\pi$
$488$	211620.	−	79198.5i	0.888620	−	0.332566i
$489$		0			0
$490$	19893.4	−	2386.04i	0.0828548	−	0.00993770i
$491$	101698.	−	101698.i	0.421842	−	0.421842i	−0.463996	−	0.885837i	$-0.653585\pi$
$491$	101698.	−	101698.i	0.421842	−	0.421842i	0.885837	+	0.463996i	$0.153585\pi$
$492$		0			0
$493$	98440.7	−	98440.7i	0.405024	−	0.405024i
$494$	−61541.4	+	78316.1i	−0.252182	+	0.320920i
$495$		0			0
$496$	−24517.0	−	47383.5i	−0.0996561	−	0.192603i
$497$	52357.9			0.211968
$498$		0			0
$499$	226481.	+	226481.i	0.909559	+	0.909559i	0.996236	−	0.0866770i	$-0.0276248\pi$
$499$	226481.	+	226481.i	0.909559	+	0.909559i	−0.0866770	+	0.996236i	$0.527625\pi$
$500$	−122670.	−	74646.4i	−0.490680	−	0.298585i
$501$		0			0
$502$	−17454.6	−	145526.i	−0.0692631	−	0.577476i
$503$	−125734.			−0.496956			−0.248478	−	0.968637i	$-0.579930\pi$
$503$	−125734.			−0.496956			−0.248478	+	0.968637i	$0.579930\pi$
$504$		0			0
$505$		−	407820.i		−	1.59914i
$506$	3595.54	+	29977.5i	0.0140431	+	0.117083i
$507$		0			0
$508$	−355804.	+	86596.7i	−1.37874	+	0.335563i
$509$	82499.2	−	82499.2i	0.318430	−	0.318430i	−0.529734	−	0.848164i	$-0.677708\pi$
$509$	82499.2	−	82499.2i	0.318430	−	0.318430i	0.848164	+	0.529734i	$0.177708\pi$
$510$		0			0
$511$			134490.i			0.515048i
$512$	74694.3	+	251277.i	0.284936	+	0.958546i
$513$		0			0
$514$	39050.7	−	49694.9i	0.147809	−	0.188099i
$515$	−194462.	−	194462.i	−0.733198	−	0.733198i
$516$		0			0
$517$	55974.7	+	55974.7i	0.209416	+	0.209416i
$518$	327631.	−	39296.4i	1.22103	−	0.146451i
$519$		0			0
$520$	396930.	+	180738.i	1.46794	+	0.668410i
$521$			225057.i			0.829120i	0.910022	+	0.414560i	$0.136064\pi$
$521$			225057.i			0.829120i	−0.910022	+	0.414560i	$0.863936\pi$
$522$		0			0
$523$	−230384.	+	230384.i	−0.842264	+	0.842264i	−0.989153	−	0.146889i	$-0.953074\pi$
$523$	−230384.	+	230384.i	−0.842264	+	0.842264i	0.146889	+	0.989153i	$0.453074\pi$
$524$	−225866.	+	371176.i	−0.822599	+	1.35182i
$525$		0			0
$526$	−202768.	+	258038.i	−0.732873	+	0.932636i
$527$		−	59659.0i		−	0.214810i
$528$		0			0
$529$	−269740.			−0.963903
$530$	−365953.	−	287569.i	−1.30279	−	1.02374i
$531$		0			0
$532$	−58881.5	+	96762.7i	−0.208044	+	0.341889i
$533$	−292253.	−	292253.i	−1.02874	−	1.02874i
$534$		0			0
$535$	97718.9			0.341406
$536$	−10441.0	−	27898.5i	−0.0363423	−	0.0971070i
$537$		0			0
$538$	20762.4	+	173105.i	0.0717319	+	0.598060i
$539$	6907.27	−	6907.27i	0.0237755	−	0.0237755i
$540$		0			0
$541$	−183995.	+	183995.i	−0.628655	+	0.628655i	−0.947730	−	0.319074i	$-0.896628\pi$
$541$	−183995.	+	183995.i	−0.628655	+	0.628655i	0.319074	+	0.947730i	$0.396628\pi$
$542$	348506.	+	273859.i	1.18635	+	0.932241i
$543$		0			0
$544$	−54943.1	+	287947.i	−0.185659	+	0.973003i
$545$	−292056.			−0.983271
$546$		0			0
$547$	−416851.	−	416851.i	−1.39318	−	1.39318i	−0.818100	−	0.575075i	$-0.804973\pi$
$547$	−416851.	−	416851.i	−1.39318	−	1.39318i	−0.575075	−	0.818100i	$-0.695027\pi$
$548$	−523604.	+	127436.i	−1.74358	+	0.424358i
$549$		0			0
$550$	−255928.	+	30696.3i	−0.846043	+	0.101475i
$551$	68431.2			0.225399
$552$		0			0
$553$			225176.i			0.736330i
$554$	−143976.	+	17268.6i	−0.469104	+	0.0562649i
$555$		0			0
$556$	−227085.	−	138184.i	−0.734578	−	0.447001i
$557$	233211.	−	233211.i	0.751691	−	0.751691i	−0.223104	−	0.974795i	$-0.571619\pi$
$557$	233211.	−	233211.i	0.751691	−	0.751691i	0.974795	+	0.223104i	$0.0716189\pi$
$558$		0			0
$559$		−	520386.i		−	1.66534i
$560$	472659.	+	150320.i	1.50720	+	0.479335i
$561$		0			0
$562$	−153484.	−	120609.i	−0.485949	−	0.381862i
$563$	82256.5	+	82256.5i	0.259510	+	0.259510i	0.824855	−	0.565345i	$-0.191257\pi$
$563$	82256.5	+	82256.5i	0.259510	+	0.259510i	−0.565345	+	0.824855i	$0.691257\pi$
$564$		0			0
$565$	−140659.	−	140659.i	−0.440626	−	0.440626i
$566$	−75090.4	−	626061.i	−0.234397	−	1.95427i
$567$		0			0
$568$	60617.3	+	27601.5i	0.187888	+	0.0855530i
$569$		−	130218.i		−	0.402203i	−0.979570	−	0.201102i	$-0.935548\pi$
$569$		−	130218.i		−	0.402203i	0.979570	−	0.201102i	$-0.0644522\pi$
$570$		0			0
$571$	62508.2	−	62508.2i	0.191719	−	0.191719i	−0.604720	−	0.796438i	$-0.706715\pi$
$571$	62508.2	−	62508.2i	0.191719	−	0.191719i	0.796438	+	0.604720i	$0.206715\pi$
$572$	206605.	−	50284.3i	0.631465	−	0.153688i
$573$		0			0
$574$	−369568.	−	290410.i	−1.12168	−	0.881429i
$575$			86238.6i			0.260835i
$576$		0			0
$577$	522256.			1.56867			0.784336	−	0.620336i	$-0.213004\pi$
$577$	522256.			1.56867			0.784336	+	0.620336i	$0.213004\pi$
$578$	3879.92	−	4937.49i	0.0116136	−	0.0147792i
$579$		0			0
$580$	−70860.6	−	291148.i	−0.210644	−	0.865481i
$581$	−73336.0	−	73336.0i	−0.217252	−	0.217252i
$582$		0			0
$583$	−226912.			−0.667606
$584$	−70898.8	+	155705.i	−0.207880	+	0.456539i
$585$		0			0
$586$	73538.9	−	8820.34i	0.214152	−	0.0256856i
$587$	−309702.	+	309702.i	−0.898811	+	0.898811i	−0.995331	−	0.0965202i	$-0.969229\pi$
$587$	−309702.	+	309702.i	−0.898811	+	0.898811i	0.0965202	+	0.995331i	$0.469229\pi$
$588$		0			0
$589$	20736.0	−	20736.0i	0.0597717	−	0.0597717i
$590$	−500909.	+	637444.i	−1.43898	+	1.83121i
$591$		0			0
$592$	400030.	+	127221.i	1.14143	+	0.363008i
$593$	447350.			1.27215			0.636075	−	0.771628i	$-0.280557\pi$
$593$	447350.			1.27215			0.636075	+	0.771628i	$0.280557\pi$
$594$		0			0
$595$	392186.	+	392186.i	1.10779	+	1.10779i
$596$	174999.	−	287584.i	0.492655	−	0.809603i
$597$		0			0
$598$	−8472.01	−	70634.7i	−0.0236910	−	0.197522i
$599$	−462149.			−1.28804			−0.644019	−	0.765009i	$-0.722734\pi$
$599$	−462149.			−1.28804			−0.644019	+	0.765009i	$0.722734\pi$
$600$		0			0
$601$		−	374481.i		−	1.03677i	−0.855149	−	0.518383i	$-0.826534\pi$
$601$		−	374481.i		−	1.03677i	0.855149	−	0.518383i	$-0.173466\pi$
$602$	−70474.8	−	587578.i	−0.194465	−	1.62134i
$603$		0			0
$604$	30291.2	+	124459.i	0.0830315	+	0.341155i
$605$	245100.	−	245100.i	0.669627	−	0.669627i
$606$		0			0
$607$		−	86755.6i		−	0.235462i	−0.993046	−	0.117731i	$-0.962438\pi$
$607$		−	86755.6i		−	0.235462i	0.993046	−	0.117731i	$-0.0375620\pi$
$608$	−119180.	+	80986.4i	−0.322402	+	0.219081i
$609$		0			0
$610$	336026.	−	427618.i	0.903053	−	1.14920i
$611$	−131891.	−	131891.i	−0.353290	−	0.353290i
$612$		0			0
$613$	−112325.	−	112325.i	−0.298920	−	0.298920i	0.541671	−	0.840591i	$-0.317792\pi$
$613$	−112325.	−	112325.i	−0.298920	−	0.298920i	−0.840591	+	0.541671i	$0.817792\pi$
$614$	−141959.	+	17026.7i	−0.376553	+	0.0451642i
$615$		0			0
$616$	226472.	−	84757.2i	0.596834	−	0.223365i
$617$		−	602706.i		−	1.58320i	−0.611041	−	0.791599i	$-0.709249\pi$
$617$		−	602706.i		−	1.58320i	0.611041	−	0.791599i	$-0.290751\pi$
$618$		0			0
$619$	−150969.	+	150969.i	−0.394010	+	0.394010i	−0.876114	−	0.482104i	$-0.839873\pi$
$619$	−150969.	+	150969.i	−0.394010	+	0.394010i	0.482104	+	0.876114i	$0.339873\pi$
$620$	−109696.	−	66751.5i	−0.285369	−	0.173651i
$621$		0			0
$622$	132985.	−	169233.i	0.343733	−	0.437426i
$623$			57703.6i			0.148671i
$624$		0			0
$625$	190658.			0.488084
$626$	−431968.	−	339444.i	−1.10231	−	0.866202i
$627$		0			0
$628$	−234876.	−	142925.i	−0.595551	−	0.362401i
$629$	331922.	+	331922.i	0.838948	+	0.838948i
$630$		0			0
$631$	−693714.			−1.74230			−0.871148	−	0.491020i	$-0.836624\pi$
$631$	−693714.			−1.74230			−0.871148	+	0.491020i	$0.836624\pi$
$632$	−118706.	+	260698.i	−0.297193	+	0.652684i
$633$		0			0
$634$	−77838.8	−	648975.i	−0.193650	−	1.61454i
$635$	−623231.	+	623231.i	−1.54562	+	1.54562i
$636$		0			0
$637$	−16275.3	+	16275.3i	−0.0401098	+	0.0401098i
$638$	−114868.	−	90264.0i	−0.282200	−	0.221755i
$639$		0			0
$640$	467976.	+	423203.i	1.14252	+	1.03321i
$641$	17843.0			0.0434261			0.0217131	−	0.999764i	$-0.493088\pi$
$641$	17843.0			0.0434261			0.0217131	+	0.999764i	$0.493088\pi$
$642$		0			0
$643$	−230136.	−	230136.i	−0.556626	−	0.556626i	0.371719	−	0.928345i	$-0.378768\pi$
$643$	−230136.	−	230136.i	−0.556626	−	0.556626i	−0.928345	+	0.371719i	$0.878768\pi$
$644$	−19131.8	−	78607.8i	−0.0461301	−	0.189537i
$645$		0			0
$646$	−159985.	+	19188.7i	−0.383366	+	0.0459813i
$647$	−568528.			−1.35814			−0.679068	−	0.734075i	$-0.737616\pi$
$647$	−568528.			−1.35814			−0.679068	+	0.734075i	$0.737616\pi$
$648$		0			0
$649$			395251.i			0.938391i
$650$	603032.	−	72328.4i	1.42730	−	0.171191i
$651$		0			0
$652$	−278142.	+	457083.i	−0.654291	+	1.07523i
$653$	−371799.	+	371799.i	−0.871932	+	0.871932i	−0.992683	−	0.120751i	$-0.961470\pi$
$653$	−371799.	+	371799.i	−0.871932	+	0.871932i	0.120751	+	0.992683i	$0.461470\pi$
$654$		0			0
$655$			1.04579e6i			2.43759i
$656$	−274772.	−	531046.i	−0.638506	−	1.23403i
$657$		0			0
$658$	−166782.	−	131059.i	−0.385210	−	0.302701i
$659$	71107.3	+	71107.3i	0.163736	+	0.163736i	0.784219	−	0.620484i	$-0.213064\pi$
$659$	71107.3	+	71107.3i	0.163736	+	0.163736i	−0.620484	+	0.784219i	$0.713064\pi$
$660$		0			0
$661$	570193.	+	570193.i	1.30502	+	1.30502i	0.924960	+	0.380065i	$0.124098\pi$
$661$	570193.	+	570193.i	1.30502	+	1.30502i	0.380065	+	0.924960i	$0.375902\pi$
$662$	45815.2	+	381981.i	0.104543	+	0.871617i
$663$		0			0
$664$	−46244.2	−	123565.i	−0.104887	−	0.280259i
$665$			272629.i			0.616494i
$666$		0			0
$667$	−34561.0	+	34561.0i	−0.0776846	+	0.0776846i
$668$	160783.	+	660615.i	0.360318	+	1.48046i
$669$		0			0
$670$	−56374.3	−	44299.4i	−0.125583	−	0.0986842i
$671$		−	265148.i		−	0.588901i
$672$		0			0
$673$	−57084.2			−0.126033			−0.0630167	−	0.998012i	$-0.520072\pi$
$673$	−57084.2			−0.126033			−0.0630167	+	0.998012i	$0.520072\pi$
$674$	362573.	−	461402.i	0.798134	−	1.01569i
$675$		0			0
$676$	−42800.8	+	10417.0i	−0.0936609	+	0.0227955i
$677$	107264.	+	107264.i	0.234032	+	0.234032i	0.814373	−	0.580341i	$-0.197081\pi$
$677$	107264.	+	107264.i	0.234032	+	0.234032i	−0.580341	+	0.814373i	$0.697081\pi$
$678$		0			0
$679$	−659134.			−1.42966
$680$	247305.	+	660801.i	0.534828	+	1.42907i
$681$		0			0
$682$	−62159.1	+	7455.43i	−0.133640	+	0.0160289i
$683$	−41763.8	+	41763.8i	−0.0895280	+	0.0895280i	−0.750452	−	0.660924i	$-0.770164\pi$
$683$	−41763.8	+	41763.8i	−0.0895280	+	0.0895280i	0.660924	+	0.750452i	$0.270164\pi$
$684$		0			0
$685$	−917153.	+	917153.i	−1.95461	+	1.95461i
$686$	282365.	−	359331.i	0.600016	−	0.763565i
$687$		0			0
$688$	228161.	−	717420.i	0.482019	−	1.51564i
$689$	534662.			1.12627
$690$		0			0
$691$	473605.	+	473605.i	0.991882	+	0.991882i	0.999967	−	0.00808568i	$-0.00257378\pi$
$691$	473605.	+	473605.i	0.991882	+	0.991882i	−0.00808568	+	0.999967i	$0.502574\pi$
$692$	−312025.	−	189872.i	−0.651594	−	0.396504i
$693$		0			0
$694$	53974.3	+	450007.i	0.112065	+	0.934330i
$695$	−639811.			−1.32459
$696$		0			0
$697$		−	668623.i		−	1.37631i
$698$	68033.3	+	567223.i	0.139640	+	1.16424i
$699$		0			0
$700$	671101.	−	163335.i	1.36959	−	0.333336i
$701$	139642.	−	139642.i	0.284170	−	0.284170i	−0.550599	−	0.834770i	$-0.685601\pi$
$701$	139642.	−	139642.i	0.284170	−	0.284170i	0.834770	+	0.550599i	$0.185601\pi$
$702$		0			0
$703$			230737.i			0.466880i
$704$	306879.	+	21261.6i	0.619188	+	0.0428994i
$705$		0			0
$706$	320934.	−	408412.i	0.643882	−	0.819388i
$707$	376728.	+	376728.i	0.753684	+	0.753684i
$708$		0			0
$709$	−161047.	−	161047.i	−0.320377	−	0.320377i	0.528535	−	0.848912i	$-0.322742\pi$
$709$	−161047.	−	161047.i	−0.320377	−	0.320377i	−0.848912	+	0.528535i	$0.822742\pi$
$710$	159172.	−	19091.3i	0.315755	−	0.0378720i
$711$		0			0
$712$	−30419.5	+	66806.2i	−0.0600057	+	0.131782i
$713$			20945.4i			0.0412011i
$714$		0			0
$715$	361893.	−	361893.i	0.707894	−	0.707894i
$716$	263974.	−	433801.i	0.514914	−	0.846183i
$717$		0			0
$718$	−138262.	+	175949.i	−0.268197	+	0.341300i
$719$			132212.i			0.255749i	0.991790	+	0.127874i	$0.0408154\pi$
$719$			132212.i			0.255749i	−0.991790	+	0.127874i	$0.959185\pi$
$720$		0			0
$721$	359273.			0.691122
$722$	347600.	+	273147.i	0.666815	+	0.523989i
$723$		0			0
$724$	−116344.	+	191193.i	−0.221955	+	0.364749i
$725$	−295059.	−	295059.i	−0.561349	−	0.561349i
$726$		0			0
$727$	98417.4			0.186210			0.0931050	−	0.995656i	$-0.470321\pi$
$727$	98417.4			0.186210			0.0931050	+	0.995656i	$0.470321\pi$
$728$	−533627.	+	199710.i	−1.00687	+	0.376822i
$729$		0			0
$730$	49039.0	+	408859.i	0.0920229	+	0.767234i
$731$	595275.	−	595275.i	1.11399	−	1.11399i
$732$		0			0
$733$	−369797.	+	369797.i	−0.688265	+	0.688265i	−0.961848	−	0.273584i	$-0.911791\pi$
$733$	−369797.	+	369797.i	−0.688265	+	0.688265i	0.273584	+	0.961848i	$0.411791\pi$
$734$	−455879.	−	358233.i	−0.846169	−	0.664927i
$735$		0			0
$736$	19289.7	−	101094.i	0.0356098	−	0.186624i
$737$	−34955.2			−0.0643542
$738$		0			0
$739$	117481.	+	117481.i	0.215120	+	0.215120i	0.806438	−	0.591318i	$-0.201392\pi$
$739$	117481.	+	117481.i	0.215120	+	0.215120i	−0.591318	+	0.806438i	$0.701392\pi$
$740$	981693.	−	238928.i	1.79272	−	0.436318i
$741$		0			0
$742$	603698.	−	72408.2i	1.09651	−	0.131516i
$743$	273733.			0.495849			0.247925	−	0.968779i	$-0.420251\pi$
$743$	273733.			0.495849			0.247925	+	0.968779i	$0.420251\pi$
$744$		0			0
$745$		−	810267.i		−	1.45988i
$746$	−134993.	+	16191.2i	−0.242568	+	0.0290939i
$747$		0			0
$748$	293859.	+	178817.i	0.525213	+	0.319600i
$749$	−90268.8	+	90268.8i	−0.160907	+	0.160907i
$750$		0			0
$751$		−	875863.i		−	1.55295i	−0.630150	−	0.776473i	$-0.717007\pi$
$751$		−	875863.i		−	1.55295i	0.630150	−	0.776473i	$-0.282993\pi$
$752$	−124002.	−	239655.i	−0.219276	−	0.423791i
$753$		0			0
$754$	270658.	+	212685.i	0.476078	+	0.374106i
$755$	218004.	+	218004.i	0.382446	+	0.382446i
$756$		0			0
$757$	66071.4	+	66071.4i	0.115298	+	0.115298i	0.762402	−	0.647104i	$-0.224020\pi$
$757$	66071.4	+	66071.4i	0.115298	+	0.115298i	−0.647104	+	0.762402i	$0.724020\pi$
$758$	−18800.0	−	156743.i	−0.0327204	−	0.272804i
$759$		0			0
$760$	−143721.	+	315636.i	−0.248825	+	0.546461i
$761$		−	333176.i		−	0.575314i	−0.957734	−	0.287657i	$-0.907124\pi$
$761$		−	333176.i		−	0.575314i	0.957734	−	0.287657i	$-0.0928763\pi$
$762$		0			0
$763$	269790.	−	269790.i	0.463422	−	0.463422i
$764$	−31614.8	+	7694.51i	−0.0541631	+	0.0131824i
$765$		0			0
$766$	−303985.	−	238874.i	−0.518078	−	0.407110i
$767$		−	931313.i		−	1.58309i
$768$		0			0
$769$	−110911.			−0.187552			−0.0937762	−	0.995593i	$-0.529894\pi$
$769$	−110911.			−0.187552			−0.0937762	+	0.995593i	$0.529894\pi$
$770$	359610.	−	457631.i	0.606528	−	0.771853i
$771$		0			0
$772$	−110701.	−	454840.i	−0.185744	−	0.763175i
$773$	800998.	+	800998.i	1.34052	+	1.34052i	0.895544	+	0.444972i	$0.146787\pi$
$773$	800998.	+	800998.i	1.34052	+	1.34052i	0.444972	+	0.895544i	$0.353213\pi$
$774$		0			0
$775$	−178818.			−0.297719
$776$	−763111.	−	347475.i	−1.26726	−	0.577032i
$777$		0			0
$778$	747730.	−	89683.6i	1.23534	−	0.148168i
$779$	232397.	−	232397.i	0.382962	−	0.382962i
$780$		0			0
$781$	55266.6	−	55266.6i	0.0906068	−	0.0906068i
$782$	71108.6	−	90491.1i	0.116281	−	0.147976i
$783$		0			0
$784$	−29573.5	+	15301.8i	−0.0481138	+	0.0248949i
$785$	−661762.			−1.07390
$786$		0			0
$787$	−528761.	−	528761.i	−0.853709	−	0.853709i	0.136879	−	0.990588i	$-0.456293\pi$
$787$	−528761.	−	528761.i	−0.853709	−	0.853709i	−0.990588	+	0.136879i	$0.956293\pi$
$788$	−331643.	+	545004.i	−0.534094	+	0.877702i
$789$		0			0
$790$	82106.0	+	684553.i	0.131559	+	1.09686i
$791$	259870.			0.415340
$792$		0			0
$793$			624756.i			0.993491i
$794$	22504.4	+	187629.i	0.0356966	+	0.297617i
$795$		0			0
$796$	−204701.	−	841063.i	−0.323068	−	1.32740i
$797$	−771661.	+	771661.i	−1.21481	+	1.21481i	−0.245390	+	0.969424i	$0.578916\pi$
$797$	−771661.	+	771661.i	−1.21481	+	1.21481i	−0.969424	+	0.245390i	$0.921084\pi$
$798$		0			0
$799$		−	301742.i		−	0.472653i
$800$	863071.	+	164683.i	1.34855	+	0.257317i
$801$		0			0
$802$	−215833.	+	274663.i	−0.335559	+	0.427024i
$803$	141961.	+	141961.i	0.220160	+	0.220160i
$804$		0			0
$805$	−137691.	−	137691.i	−0.212477	−	0.212477i
$806$	146463.	−	17566.9i	0.225454	−	0.0270412i
$807$		0			0
$808$	237557.	+	634756.i	0.363869	+	0.972263i
$809$		−	100893.i		−	0.154157i	−0.997025	−	0.0770783i	$-0.975441\pi$
$809$		−	100893.i		−	0.154157i	0.997025	−	0.0770783i	$-0.0245592\pi$
$810$		0			0
$811$	−51769.7	+	51769.7i	−0.0787107	+	0.0787107i	−0.745366	−	0.666655i	$-0.767725\pi$
$811$	−51769.7	+	51769.7i	−0.0787107	+	0.0787107i	0.666655	+	0.745366i	$0.267725\pi$
$812$	334409.	+	203493.i	0.507184	+	0.308629i
$813$		0			0
$814$	304352.	−	387311.i	0.459333	−	0.584536i
$815$			1.28783e6i			1.93885i
$816$		0			0
$817$	413807.			0.619946
$818$	−148242.	−	116490.i	−0.221547	−	0.174093i
$819$		0			0
$820$	−1.22941e6	−	748111.i	−1.82838	−	1.11260i
$821$	−48584.3	−	48584.3i	−0.0720791	−	0.0720791i	0.670148	−	0.742227i	$-0.266231\pi$
$821$	−48584.3	−	48584.3i	−0.0720791	−	0.0720791i	−0.742227	+	0.670148i	$0.766231\pi$
$822$		0			0
$823$	−338093.			−0.499157			−0.249578	−	0.968355i	$-0.580292\pi$
$823$	−338093.			−0.499157			−0.249578	+	0.968355i	$0.580292\pi$
$824$	415948.	+	189398.i	0.612611	+	0.278946i
$825$		0			0
$826$	−126126.	−	1.05156e6i	−0.184860	−	1.54126i
$827$	458054.	−	458054.i	0.669740	−	0.669740i	−0.287916	−	0.957656i	$-0.592962\pi$
$827$	458054.	−	458054.i	0.669740	−	0.669740i	0.957656	+	0.287916i	$0.0929624\pi$
$828$		0			0
$829$	495530.	−	495530.i	0.721042	−	0.721042i	−0.247775	−	0.968818i	$-0.579699\pi$
$829$	495530.	−	495530.i	0.721042	−	0.721042i	0.968818	+	0.247775i	$0.0796995\pi$
$830$	−249687.	−	196206.i	−0.362443	−	0.284811i
$831$		0			0
$832$	−723087.	−	50097.9i	−1.04458	−	0.0723724i
$833$	−37235.0			−0.0536613
$834$		0			0
$835$	1.15714e6	+	1.15714e6i	1.65964	+	1.65964i
$836$	39985.7	+	164291.i	0.0572127	+	0.235072i
$837$		0			0
$838$	−395898.	+	47484.5i	−0.563761	+	0.0676182i
$839$	−696514.			−0.989477			−0.494739	−	0.869042i	$-0.664736\pi$
$839$	−696514.			−0.989477			−0.494739	+	0.869042i	$0.664736\pi$
$840$		0			0
$841$			470785.i			0.665626i
$842$	−617429.	+	74055.1i	−0.870889	+	0.104455i
$843$		0			0
$844$	−376331.	+	618443.i	−0.528305	+	0.868189i
$845$	−74970.5	+	74970.5i	−0.104997	+	0.104997i
$846$		0			0
$847$			452827.i			0.631198i
$848$	737102.	+	234420.i	1.02503	+	0.325989i
$849$		0			0
$850$	772552.	+	607078.i	1.06928	+	0.840246i
$851$	−116533.	−	116533.i	−0.160912	−	0.160912i
$852$		0			0
$853$	78365.1	+	78365.1i	0.107702	+	0.107702i	0.758904	−	0.651202i	$-0.225735\pi$
$853$	78365.1	+	78365.1i	0.107702	+	0.107702i	−0.651202	+	0.758904i	$0.725735\pi$
$854$	84609.4	+	705424.i	0.116012	+	0.967241i
$855$		0			0
$856$	−152096.	+	56921.7i	−0.207572	+	0.0776838i
$857$		−	576243.i		−	0.784593i	−0.919839	−	0.392296i	$-0.871681\pi$
$857$		−	576243.i		−	0.784593i	0.919839	−	0.392296i	$-0.128319\pi$
$858$		0			0
$859$	74196.4	−	74196.4i	0.100553	−	0.100553i	−0.655040	−	0.755594i	$-0.727348\pi$
$859$	74196.4	−	74196.4i	0.100553	−	0.100553i	0.755594	+	0.655040i	$0.227348\pi$
$860$	−428497.	−	1.76058e6i	−0.579363	−	2.38045i
$861$		0			0
$862$	−26392.2	−	20739.2i	−0.0355190	−	0.0279111i
$863$			299840.i			0.402594i	0.979530	+	0.201297i	$0.0645157\pi$
$863$			299840.i			0.402594i	−0.979530	+	0.201297i	$0.935484\pi$
$864$		0			0
$865$	−879131.			−1.17495
$866$	−277350.	+	352948.i	−0.369821	+	0.470625i
$867$		0			0
$868$	162995.	−	39670.3i	0.216339	−	0.0526534i
$869$	237686.	+	237686.i	0.314749	+	0.314749i
$870$		0			0
$871$	82363.5			0.108567
$872$	454573.	−	170124.i	0.597821	−	0.223734i
$873$		0			0
$874$	56168.2	−	6736.87i	0.0735305	−	0.00881933i
$875$	319272.	−	319272.i	0.417009	−	0.417009i
$876$		0			0
$877$	−438353.	+	438353.i	−0.569934	+	0.569934i	−0.932110	−	0.362176i	$-0.882034\pi$
$877$	−438353.	+	438353.i	−0.569934	+	0.569934i	0.362176	+	0.932110i	$0.382034\pi$
$878$	236851.	−	301411.i	0.307246	−	0.390994i
$879$		0			0
$880$	657588.	−	340247.i	0.849157	−	0.439368i
$881$	1526.15			0.00196628			0.000983142	−	1.00000i	$-0.499687\pi$
$881$	1526.15			0.00196628			0.000983142		1.00000i	$0.499687\pi$
$882$		0			0
$883$	114980.	+	114980.i	0.147469	+	0.147469i	0.776986	−	0.629517i	$-0.216747\pi$
$883$	114980.	+	114980.i	0.147469	+	0.147469i	−0.629517	+	0.776986i	$0.716747\pi$
$884$	−692407.	−	421339.i	−0.886047	−	0.539172i
$885$		0			0
$886$	−32875.2	−	274095.i	−0.0418795	−	0.349167i
$887$	833028.			1.05880			0.529398	−	0.848373i	$-0.322418\pi$
$887$	833028.			1.05880			0.529398	+	0.848373i	$0.322418\pi$
$888$		0			0
$889$		−	1.15143e6i		−	1.45692i
$890$	21040.4	+	175423.i	0.0265629	+	0.221466i
$891$		0			0
$892$	1.46423e6	−	356369.i	1.84026	−	0.447889i
$893$	104878.	−	104878.i	0.131517	−	0.131517i
$894$		0			0
$895$		−	1.22223e6i		−	1.52584i
$896$	−823236.	+	41359.6i	−1.02544	+	0.0515182i
$897$		0			0
$898$	−609636.	+	775807.i	−0.755993	+	0.962058i
$899$	−71663.1	−	71663.1i	−0.0886699	−	0.0886699i
$900$		0			0
$901$	611606.	+	611606.i	0.753394	+	0.753394i
$902$	−696642.	+	83556.0i	−0.856242	+	0.102699i
$903$		0			0
$904$	300864.	+	136995.i	0.368157	+	0.167637i
$905$			538686.i			0.657716i
$906$		0			0
$907$	−1.13198e6	+	1.13198e6i	−1.37602	+	1.37602i	−0.524787	+	0.851234i	$0.675855\pi$
$907$	−1.13198e6	+	1.13198e6i	−1.37602	+	1.37602i	−0.851234	+	0.524787i	$0.824145\pi$
$908$	−737319.	+	1.21167e6i	−0.894302	+	1.46965i
$909$		0			0
$910$	−847335.	+	1.07830e6i	−1.02323	+	1.30213i
$911$		−	369952.i		−	0.445768i	−0.974845	−	0.222884i	$-0.928453\pi$
$911$		−	369952.i		−	0.445768i	0.974845	−	0.222884i	$-0.0715471\pi$
$912$		0			0
$913$	−154820.			−0.185732
$914$	18965.3	+	14903.1i	0.0227021	+	0.0178395i
$915$		0			0
$916$	−232684.	+	382381.i	−0.277316	+	0.455727i
$917$	−966058.	−	966058.i	−1.14885	−	1.14885i
$918$		0			0
$919$	815703.			0.965831			0.482915	−	0.875667i	$-0.339578\pi$
$919$	815703.			0.965831			0.482915	+	0.875667i	$0.339578\pi$
$920$	−86824.9	−	231997.i	−0.102581	−	0.274099i
$921$		0			0
$922$	−8840.19	−	73704.4i	−0.0103992	−	0.0867025i
$923$	−130222.	+	130222.i	−0.152856	+	0.152856i
$924$		0			0
$925$	994881.	−	994881.i	1.16275	−	1.16275i
$926$	1.28346e6	+	1.00855e6i	1.49678	+	1.17618i
$927$		0			0
$928$	279887.	+	411883.i	0.325002	+	0.478275i
$929$	−329861.			−0.382208			−0.191104	−	0.981570i	$-0.561207\pi$
$929$	−329861.			−0.382208			−0.191104	+	0.981570i	$0.561207\pi$
$930$		0			0
$931$	−12942.0	−	12942.0i	−0.0149314	−	0.0149314i
$932$	389620.	−	94827.0i	0.448548	−	0.109169i
$933$		0			0
$934$	−607940.	+	72917.1i	−0.696895	+	0.0835863i
$935$	827947.			0.947064
$936$		0			0
$937$		−	884572.i		−	1.00752i	−0.863844	−	0.503760i	$-0.831949\pi$
$937$		−	884572.i		−	1.00752i	0.863844	−	0.503760i	$-0.168051\pi$
$938$	92998.3	−	11154.3i	0.105699	−	0.0126776i
$939$		0			0
$940$	−554818.	−	337615.i	−0.627906	−	0.382090i
$941$	441727.	−	441727.i	0.498856	−	0.498856i	−0.412226	−	0.911082i	$-0.635249\pi$
$941$	441727.	−	441727.i	0.498856	−	0.498856i	0.911082	+	0.412226i	$0.135249\pi$
$942$		0			0
$943$			234743.i			0.263979i
$944$	408330.	−	1.28394e6i	0.458213	−	1.44079i
$945$		0			0
$946$	−694611.	−	545831.i	−0.776174	−	0.609924i
$947$	98614.4	+	98614.4i	0.109961	+	0.109961i	0.759947	−	0.649985i	$-0.225225\pi$
$947$	98614.4	+	98614.4i	0.109961	+	0.109961i	−0.649985	+	0.759947i	$0.725225\pi$
$948$		0			0
$949$	−334497.	−	334497.i	−0.371416	−	0.371416i
$950$	57514.9	+	479526.i	0.0637285	+	0.531331i
$951$		0			0
$952$	−838872.	−	381972.i	−0.925597	−	0.421461i
$953$		−	327664.i		−	0.360780i	−0.983595	−	0.180390i	$-0.942264\pi$
$953$		−	327664.i		−	0.360780i	0.983595	−	0.180390i	$-0.0577360\pi$
$954$		0			0
$955$	−55377.0	+	55377.0i	−0.0607187	+	0.0607187i
$956$	−1.44644e6	+	352041.i	−1.58265	+	0.385192i
$957$		0			0
$958$	701837.	+	551509.i	0.764725	+	0.600927i
$959$		−	1.69446e6i		−	1.84244i
$960$		0			0
$961$	880090.			0.952973
$962$	−717133.	+	912605.i	−0.774906	+	0.986127i
$963$		0			0
$964$	306410.	+	1.25896e6i	0.329723	+	1.35475i
$965$	−796705.	−	796705.i	−0.855545	−	0.855545i
$966$		0			0
$967$	386677.			0.413519			0.206760	−	0.978392i	$-0.433708\pi$
$967$	386677.			0.413519			0.206760	+	0.978392i	$0.433708\pi$
$968$	−238716.	+	524260.i	−0.254760	+	0.559495i
$969$		0			0
$970$	−2.00381e6	+	240340.i	−2.12968	+	0.255436i
$971$	−718740.	+	718740.i	−0.762313	+	0.762313i	−0.976740	−	0.214427i	$-0.931212\pi$
$971$	−718740.	+	718740.i	−0.762313	+	0.762313i	0.214427	+	0.976740i	$0.431212\pi$
$972$		0			0
$973$	591032.	−	591032.i	0.624288	−	0.624288i
$974$	−558254.	+	710421.i	−0.588456	+	0.748855i
$975$		0			0
$976$	−273921.	+	861308.i	−0.287559	+	0.904188i
$977$	1.27845e6			1.33935			0.669675	−	0.742654i	$-0.266433\pi$
$977$	1.27845e6			1.33935			0.669675	+	0.742654i	$0.266433\pi$
$978$		0			0
$979$	60909.2	+	60909.2i	0.0635503	+	0.0635503i
$980$	−41661.6	+	68464.5i	−0.0433794	+	0.0712875i
$981$		0			0
$982$	68510.0	+	571197.i	0.0710446	+	0.592329i
$983$	414529.			0.428991			0.214495	−	0.976725i	$-0.431189\pi$
$983$	414529.			0.428991			0.214495	+	0.976725i	$0.431189\pi$
$984$		0			0
$985$			1.53555e6i			1.58267i
$986$	66315.6	+	552902.i	0.0682122	+	0.568714i
$987$		0			0
$988$	−94216.6	−	387112.i	−0.0965192	−	0.396572i
$989$	−208992.	+	208992.i	−0.213667	+	0.213667i
$990$		0			0
$991$		−	176697.i		−	0.179921i	−0.995945	−	0.0899605i	$-0.971326\pi$
$991$		−	176697.i		−	0.179921i	0.995945	−	0.0899605i	$-0.0286741\pi$
$992$	209620.	+	39997.7i	0.213015	+	0.0406454i
$993$		0			0
$994$	−129401.	+	164672.i	−0.130968	+	0.166666i
$995$	−1.47322e6	−	1.47322e6i	−1.48806	−	1.48806i
$996$		0			0
$997$	−1.20434e6	−	1.20434e6i	−1.21160	−	1.21160i	−0.970500	−	0.241102i	$-0.922491\pi$
$997$	−1.20434e6	−	1.20434e6i	−1.21160	−	1.21160i	−0.241102	−	0.970500i	$-0.577509\pi$
$998$	−1.27205e6	+	152572.i	−1.27716	+	0.153184i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.5.m.a.91.2		14
3.2	odd	2		16.5.f.a.11.6	yes	14
4.3	odd	2		576.5.m.a.271.1		14
12.11	even	2		64.5.f.a.15.6		14
16.3	odd	4	inner	144.5.m.a.19.2		14
16.13	even	4		576.5.m.a.559.1		14
24.5	odd	2		128.5.f.b.31.6		14
24.11	even	2		128.5.f.a.31.2		14
48.5	odd	4		128.5.f.a.95.2		14
48.11	even	4		128.5.f.b.95.6		14
48.29	odd	4		64.5.f.a.47.6		14
48.35	even	4		16.5.f.a.3.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.5.f.a.3.6	✓	14	48.35	even	4
16.5.f.a.11.6	yes	14	3.2	odd	2
64.5.f.a.15.6		14	12.11	even	2
64.5.f.a.47.6		14	48.29	odd	4
128.5.f.a.31.2		14	24.11	even	2
128.5.f.a.95.2		14	48.5	odd	4
128.5.f.b.31.6		14	24.5	odd	2
128.5.f.b.95.6		14	48.11	even	4
144.5.m.a.19.2		14	16.3	odd	4	inner
144.5.m.a.91.2		14	1.1	even	1	trivial
576.5.m.a.271.1		14	4.3	odd	2
576.5.m.a.559.1		14	16.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	144.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.47147 + 3.14513i) q^{2} +(-3.78368 - 15.5462i) q^{4} +(-27.2309 - 27.2309i) q^{5} +50.3097 q^{7} +(58.2460 + 26.5217i) q^{8} +O(q^{10})\)	\(q+(-2.47147 + 3.14513i) q^{2} +(-3.78368 - 15.5462i) q^{4} +(-27.2309 - 27.2309i) q^{5} +50.3097 q^{7} +(58.2460 + 26.5217i) q^{8} +(152.945 - 18.3444i) q^{10} +(53.1047 - 53.1047i) q^{11} +(-125.128 + 125.128i) q^{13} +(-124.339 + 158.231i) q^{14} +(-227.367 + 117.644i) q^{16} -286.271 q^{17} +(-99.5010 - 99.5010i) q^{19} +(-320.304 + 526.370i) q^{20} +(35.7745 + 298.268i) q^{22} +100.505 q^{23} +858.049i q^{25} +(-84.2940 - 702.795i) q^{26} +(-190.356 - 782.124i) q^{28} +(-343.872 + 343.872i) q^{29} +208.400i q^{31} +(191.927 - 1005.85i) q^{32} +(707.510 - 900.360i) q^{34} +(-1369.98 - 1369.98i) q^{35} +(-1159.47 - 1159.47i) q^{37} +(558.857 - 67.0299i) q^{38} +(-863.882 - 2308.31i) q^{40} +2335.63i q^{41} +(-2079.41 + 2079.41i) q^{43} +(-1026.51 - 624.643i) q^{44} +(-248.396 + 316.103i) q^{46} +1054.04i q^{47} +130.069 q^{49} +(-2698.67 - 2120.64i) q^{50} +(2418.71 + 1471.82i) q^{52} +(-2136.46 - 2136.46i) q^{53} -2892.18 q^{55} +(2930.34 + 1334.30i) q^{56} +(-231.653 - 1931.39i) q^{58} +(-3721.44 + 3721.44i) q^{59} +(2496.46 - 2496.46i) q^{61} +(-655.446 - 515.055i) q^{62} +(2689.20 + 3089.57i) q^{64} +6814.72 q^{65} +(-329.116 - 329.116i) q^{67} +(1083.16 + 4450.42i) q^{68} +(7694.64 - 922.903i) q^{70} +1040.71 q^{71} +2673.24i q^{73} +(6512.27 - 781.089i) q^{74} +(-1170.38 + 1923.34i) q^{76} +(2671.68 - 2671.68i) q^{77} +4475.80i q^{79} +(9394.98 + 2987.88i) q^{80} +(-7345.86 - 5772.43i) q^{82} +(-1457.69 - 1457.69i) q^{83} +(7795.43 + 7795.43i) q^{85} +(-1400.82 - 11679.2i) q^{86} +(4501.56 - 1684.71i) q^{88} +1146.97i q^{89} +(-6295.17 + 6295.17i) q^{91} +(-380.281 - 1562.48i) q^{92} +(-3315.11 - 2605.04i) q^{94} +5419.01i q^{95} -13101.5 q^{97} +(-321.461 + 409.084i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} - 8 q^{4} + 2 q^{5} - 4 q^{7} + 92 q^{8}+O(q^{10}) \) 14 * q + 2 * q^2 - 8 * q^4 + 2 * q^5 - 4 * q^7 + 92 * q^8	\( 14 q + 2 q^{2} - 8 q^{4} + 2 q^{5} - 4 q^{7} + 92 q^{8} - 100 q^{10} - 94 q^{11} - 2 q^{13} - 44 q^{14} - 168 q^{16} + 4 q^{17} - 706 q^{19} - 1900 q^{20} + 900 q^{22} - 1148 q^{23} + 3416 q^{26} - 3784 q^{28} - 862 q^{29} - 3208 q^{32} + 7508 q^{34} - 1340 q^{35} - 1826 q^{37} - 3568 q^{38} - 5144 q^{40} + 1694 q^{43} + 14636 q^{44} - 5316 q^{46} + 682 q^{49} - 20070 q^{50} + 20452 q^{52} + 482 q^{53} - 11780 q^{55} + 6952 q^{56} - 20456 q^{58} + 2786 q^{59} - 3778 q^{61} + 11472 q^{62} + 15808 q^{64} + 2020 q^{65} + 7998 q^{67} - 18032 q^{68} + 15296 q^{70} - 19964 q^{71} + 23780 q^{74} - 23996 q^{76} + 9508 q^{77} - 1384 q^{80} + 16016 q^{82} + 17282 q^{83} + 9948 q^{85} + 4796 q^{86} + 7288 q^{88} - 28036 q^{91} + 14632 q^{92} + 432 q^{94} - 4 q^{97} + 12246 q^{98}+O(q^{100}) \) 14 * q + 2 * q^2 - 8 * q^4 + 2 * q^5 - 4 * q^7 + 92 * q^8 - 100 * q^10 - 94 * q^11 - 2 * q^13 - 44 * q^14 - 168 * q^16 + 4 * q^17 - 706 * q^19 - 1900 * q^20 + 900 * q^22 - 1148 * q^23 + 3416 * q^26 - 3784 * q^28 - 862 * q^29 - 3208 * q^32 + 7508 * q^34 - 1340 * q^35 - 1826 * q^37 - 3568 * q^38 - 5144 * q^40 + 1694 * q^43 + 14636 * q^44 - 5316 * q^46 + 682 * q^49 - 20070 * q^50 + 20452 * q^52 + 482 * q^53 - 11780 * q^55 + 6952 * q^56 - 20456 * q^58 + 2786 * q^59 - 3778 * q^61 + 11472 * q^62 + 15808 * q^64 + 2020 * q^65 + 7998 * q^67 - 18032 * q^68 + 15296 * q^70 - 19964 * q^71 + 23780 * q^74 - 23996 * q^76 + 9508 * q^77 - 1384 * q^80 + 16016 * q^82 + 17282 * q^83 + 9948 * q^85 + 4796 * q^86 + 7288 * q^88 - 28036 * q^91 + 14632 * q^92 + 432 * q^94 - 4 * q^97 + 12246 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more