LMFDB - Embedded newform 342.5.d.a.37.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,5,Mod(37,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("342.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	342.d (of order $2$, degree $1$, 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:	$35.3525273747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 $ x^8 + 450x^6 + 68229x^4 + 4001228*x^2 + 77475204
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.5
Root			$13.9305i$ of defining polynomial
Character	$\chi$	$=$	342.37
Dual form			342.5.d.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} -41.6240 q^{5} -62.4342 q^{7} -22.6274i q^{8} +O(q^{10})$	\(q+2.82843i q^{2} -8.00000 q^{4} -41.6240 q^{5} -62.4342 q^{7} -22.6274i q^{8} -117.730i q^{10} -122.938 q^{11} +68.9610i q^{13} -176.591i q^{14} +64.0000 q^{16} -297.375 q^{17} +(-195.642 - 303.390i) q^{19} +332.992 q^{20} -347.721i q^{22} -268.685 q^{23} +1107.56 q^{25} -195.051 q^{26} +499.474 q^{28} +561.405i q^{29} -252.423i q^{31} +181.019i q^{32} -841.104i q^{34} +2598.76 q^{35} +2407.33i q^{37} +(858.116 - 553.358i) q^{38} +941.843i q^{40} +690.246i q^{41} +218.905 q^{43} +983.502 q^{44} -759.956i q^{46} +83.1049 q^{47} +1497.03 q^{49} +3132.64i q^{50} -551.688i q^{52} -4389.68i q^{53} +5117.16 q^{55} +1412.73i q^{56} -1587.89 q^{58} +476.981i q^{59} +3965.09 q^{61} +713.960 q^{62} -512.000 q^{64} -2870.43i q^{65} +5111.96i q^{67} +2379.00 q^{68} +7350.41i q^{70} -8182.93i q^{71} -7345.23 q^{73} -6808.95 q^{74} +(1565.13 + 2427.12i) q^{76} +7675.53 q^{77} -1485.61i q^{79} -2663.94 q^{80} -1952.31 q^{82} -5347.30 q^{83} +12377.9 q^{85} +619.156i q^{86} +2781.76i q^{88} +11172.3i q^{89} -4305.53i q^{91} +2149.48 q^{92} +235.056i q^{94} +(8143.39 + 12628.3i) q^{95} +1078.13i q^{97} +4234.25i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7}+O(q^{10}) $ 8 * q - 64 * q^4 - 18 * q^5 - 162 * q^7	$ 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7} + 6 q^{11} + 512 q^{16} - 510 q^{17} - 12 q^{19} + 144 q^{20} + 396 q^{23} + 3458 q^{25} + 192 q^{26} + 1296 q^{28} - 1002 q^{35} + 3216 q^{38} - 8654 q^{43} - 48 q^{44} - 3210 q^{47} + 9222 q^{49} + 17146 q^{55} - 960 q^{58} + 1314 q^{61} + 15168 q^{62} - 4096 q^{64} + 4080 q^{68} + 23398 q^{73} - 13152 q^{74} + 96 q^{76} + 44622 q^{77} - 1152 q^{80} + 16512 q^{82} + 10440 q^{83} + 21274 q^{85} - 3168 q^{92} + 34686 q^{95}+O(q^{100}) $ 8 * q - 64 * q^4 - 18 * q^5 - 162 * q^7 + 6 * q^11 + 512 * q^16 - 510 * q^17 - 12 * q^19 + 144 * q^20 + 396 * q^23 + 3458 * q^25 + 192 * q^26 + 1296 * q^28 - 1002 * q^35 + 3216 * q^38 - 8654 * q^43 - 48 * q^44 - 3210 * q^47 + 9222 * q^49 + 17146 * q^55 - 960 * q^58 + 1314 * q^61 + 15168 * q^62 - 4096 * q^64 + 4080 * q^68 + 23398 * q^73 - 13152 * q^74 + 96 * q^76 + 44622 * q^77 - 1152 * q^80 + 16512 * q^82 + 10440 * q^83 + 21274 * q^85 - 3168 * q^92 + 34686 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$-1$

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 $

$2$			2.82843i			0.707107i
$2$			2.82843i			0.707107i
$3$		0			0
$3$		0			0
$4$	−8.00000			−0.500000
$5$	−41.6240			−1.66496			−0.832480	−	0.554055i	$-0.813080\pi$
$5$	−41.6240			−1.66496			−0.832480	+	0.554055i	$0.813080\pi$
$6$		0			0
$7$	−62.4342			−1.27417			−0.637084	−	0.770794i	$-0.719860\pi$
$7$	−62.4342			−1.27417			−0.637084	+	0.770794i	$0.719860\pi$
$8$		−	22.6274i		−	0.353553i
$9$		0			0
$10$		−	117.730i		−	1.17730i
$11$	−122.938			−1.01601			−0.508007	−	0.861353i	$-0.669618\pi$
$11$	−122.938			−1.01601			−0.508007	+	0.861353i	$0.669618\pi$
$12$		0			0
$13$			68.9610i			0.408053i	0.978965	+	0.204027i	$0.0654029\pi$
$13$			68.9610i			0.408053i	−0.978965	+	0.204027i	$0.934597\pi$
$14$		−	176.591i		−	0.900973i
$15$		0			0
$16$	64.0000			0.250000
$17$	−297.375			−1.02898			−0.514490	−	0.857496i	$-0.672019\pi$
$17$	−297.375			−1.02898			−0.514490	+	0.857496i	$0.672019\pi$
$18$		0			0
$19$	−195.642	−	303.390i	−0.541944	−	0.840415i
$19$	−195.642	−	303.390i	−0.541944	−	0.840415i
$20$	332.992			0.832480
$21$		0			0
$22$		−	347.721i		−	0.718431i
$23$	−268.685			−0.507911			−0.253956	−	0.967216i	$-0.581732\pi$
$23$	−268.685			−0.507911			−0.253956	+	0.967216i	$0.581732\pi$
$24$		0			0
$25$	1107.56			1.77209
$26$	−195.051			−0.288537
$27$		0			0
$28$	499.474			0.637084
$29$			561.405i			0.667545i	0.942654	+	0.333772i	$0.108322\pi$
$29$			561.405i			0.667545i	−0.942654	+	0.333772i	$0.891678\pi$
$30$		0			0
$31$		−	252.423i		−	0.262667i	−0.991338	−	0.131333i	$-0.958074\pi$
$31$		−	252.423i		−	0.262667i	0.991338	−	0.131333i	$-0.0419259\pi$
$32$			181.019i			0.176777i
$33$		0			0
$34$		−	841.104i		−	0.727599i
$35$	2598.76			2.12144
$36$		0			0
$37$			2407.33i			1.75846i	0.476400	+	0.879228i	$0.341941\pi$
$37$			2407.33i			1.75846i	−0.476400	+	0.879228i	$0.658059\pi$
$38$	858.116	−	553.358i	0.594263	−	0.383212i
$39$		0			0
$40$			941.843i			0.588652i
$41$			690.246i			0.410616i	0.978697	+	0.205308i	$0.0658197\pi$
$41$			690.246i			0.410616i	−0.978697	+	0.205308i	$0.934180\pi$
$42$		0			0
$43$	218.905			0.118391			0.0591954	−	0.998246i	$-0.481146\pi$
$43$	218.905			0.118391			0.0591954	+	0.998246i	$0.481146\pi$
$44$	983.502			0.508007
$45$		0			0
$46$		−	759.956i		−	0.359148i
$47$	83.1049			0.0376211			0.0188105	−	0.999823i	$-0.494012\pi$
$47$	83.1049			0.0376211			0.0188105	+	0.999823i	$0.494012\pi$
$48$		0			0
$49$	1497.03			0.623505
$50$			3132.64i			1.25306i
$51$		0			0
$52$		−	551.688i		−	0.204027i
$53$		−	4389.68i		−	1.56272i	−0.624080	−	0.781360i	$-0.714526\pi$
$53$		−	4389.68i		−	1.56272i	0.624080	−	0.781360i	$-0.285474\pi$
$54$		0			0
$55$	5117.16			1.69162
$56$			1412.73i			0.450486i
$57$		0			0
$58$	−1587.89			−0.472025
$59$			476.981i			0.137024i	0.997650	+	0.0685120i	$0.0218252\pi$
$59$			476.981i			0.137024i	−0.997650	+	0.0685120i	$0.978175\pi$
$60$		0			0
$61$	3965.09			1.06560			0.532799	−	0.846242i	$-0.321140\pi$
$61$	3965.09			1.06560			0.532799	+	0.846242i	$0.321140\pi$
$62$	713.960			0.185734
$63$		0			0
$64$	−512.000			−0.125000
$65$		−	2870.43i		−	0.679392i
$66$		0			0
$67$			5111.96i			1.13878i	0.822069	+	0.569388i	$0.192820\pi$
$67$			5111.96i			1.13878i	−0.822069	+	0.569388i	$0.807180\pi$
$68$	2379.00			0.514490
$69$		0			0
$70$			7350.41i			1.50008i
$71$		−	8182.93i		−	1.62327i	−0.584161	−	0.811637i	$-0.698577\pi$
$71$		−	8182.93i		−	1.62327i	0.584161	−	0.811637i	$-0.301423\pi$
$72$		0			0
$73$	−7345.23			−1.37835			−0.689175	−	0.724595i	$-0.742027\pi$
$73$	−7345.23			−1.37835			−0.689175	+	0.724595i	$0.742027\pi$
$74$	−6808.95			−1.24342
$75$		0			0
$76$	1565.13	+	2427.12i	0.270972	+	0.420207i
$77$	7675.53			1.29457
$78$		0			0
$79$		−	1485.61i		−	0.238041i	−0.992892	−	0.119021i	$-0.962025\pi$
$79$		−	1485.61i		−	0.238041i	0.992892	−	0.119021i	$-0.0379754\pi$
$80$	−2663.94			−0.416240
$81$		0			0
$82$	−1952.31			−0.290350
$83$	−5347.30			−0.776209			−0.388104	−	0.921615i	$-0.626870\pi$
$83$	−5347.30			−0.776209			−0.388104	+	0.921615i	$0.626870\pi$
$84$		0			0
$85$	12377.9			1.71321
$86$			619.156i			0.0837150i
$87$		0			0
$88$			2781.76i			0.359215i
$89$			11172.3i			1.41047i	0.708975	+	0.705234i	$0.249158\pi$
$89$			11172.3i			1.41047i	−0.708975	+	0.705234i	$0.750842\pi$
$90$		0			0
$91$		−	4305.53i		−	0.519928i
$92$	2149.48			0.253956
$93$		0			0
$94$			235.056i			0.0266021i
$95$	8143.39	+	12628.3i	0.902315	+	1.39926i
$96$		0			0
$97$			1078.13i			0.114585i	0.998357	+	0.0572927i	$0.0182468\pi$
$97$			1078.13i			0.114585i	−0.998357	+	0.0572927i	$0.981753\pi$
$98$			4234.25i			0.440884i
$99$		0			0
$100$	−8860.45			−0.886045
$101$	−1393.71			−0.136625			−0.0683126	−	0.997664i	$-0.521762\pi$
$101$	−1393.71			−0.136625			−0.0683126	+	0.997664i	$0.521762\pi$
$102$		0			0
$103$			11163.2i			1.05224i	0.850411	+	0.526119i	$0.176353\pi$
$103$			11163.2i			1.05224i	−0.850411	+	0.526119i	$0.823647\pi$
$104$	1560.41			0.144269
$105$		0			0
$106$	12415.9			1.10501
$107$		−	502.078i		−	0.0438534i	−0.999760	−	0.0219267i	$-0.993020\pi$
$107$		−	502.078i		−	0.0438534i	0.999760	−	0.0219267i	$-0.00698005\pi$
$108$		0			0
$109$		−	11387.7i		−	0.958477i	−0.877685	−	0.479238i	$-0.840913\pi$
$109$		−	11387.7i		−	0.958477i	0.877685	−	0.479238i	$-0.159087\pi$
$110$			14473.5i			1.19616i
$111$		0			0
$112$	−3995.79			−0.318542
$113$		−	4779.02i		−	0.374267i	−0.982334	−	0.187133i	$-0.940080\pi$
$113$		−	4779.02i		−	0.374267i	0.982334	−	0.187133i	$-0.0599197\pi$
$114$		0			0
$115$	11183.7			0.845652
$116$		−	4491.24i		−	0.333772i
$117$		0			0
$118$	−1349.11			−0.0968906
$119$	18566.4			1.31109
$120$		0			0
$121$	472.695			0.0322857
$122$			11215.0i			0.753491i
$123$		0			0
$124$			2019.38i			0.131333i
$125$	−20085.9			−1.28550
$126$		0			0
$127$		−	17552.6i		−	1.08826i	−0.839001	−	0.544130i	$-0.816860\pi$
$127$		−	17552.6i		−	1.08826i	0.839001	−	0.544130i	$-0.183140\pi$
$128$		−	1448.15i		−	0.0883883i
$129$		0			0
$130$	8118.80			0.480403
$131$	7401.84			0.431317			0.215659	−	0.976469i	$-0.430810\pi$
$131$	7401.84			0.431317			0.215659	+	0.976469i	$0.430810\pi$
$132$		0			0
$133$	12214.7	+	18941.9i	0.690528	+	1.07083i
$134$	−14458.8			−0.805236
$135$		0			0
$136$			6728.83i			0.363799i
$137$	−1692.77			−0.0901896			−0.0450948	−	0.998983i	$-0.514359\pi$
$137$	−1692.77			−0.0901896			−0.0450948	+	0.998983i	$0.514359\pi$
$138$		0			0
$139$	−37366.7			−1.93399			−0.966996	−	0.254791i	$-0.917993\pi$
$139$	−37366.7			−1.93399			−0.966996	+	0.254791i	$0.917993\pi$
$140$	−20790.1			−1.06072
$141$		0			0
$142$	23144.8			1.14783
$143$		−	8477.91i		−	0.414588i
$144$		0			0
$145$		−	23367.9i		−	1.11143i
$146$		−	20775.4i		−	0.974641i
$147$		0			0
$148$		−	19258.6i		−	0.879228i
$149$	6037.22			0.271935			0.135967	−	0.990713i	$-0.456586\pi$
$149$	6037.22			0.271935			0.135967	+	0.990713i	$0.456586\pi$
$150$		0			0
$151$			20619.1i			0.904307i	0.891940	+	0.452153i	$0.149344\pi$
$151$			20619.1i			0.904307i	−0.891940	+	0.452153i	$0.850656\pi$
$152$	−6864.92	+	4426.87i	−0.297131	+	0.191606i
$153$		0			0
$154$			21709.7i			0.915402i
$155$			10506.8i			0.437330i
$156$		0			0
$157$	−1043.55			−0.0423362			−0.0211681	−	0.999776i	$-0.506739\pi$
$157$	−1043.55			−0.0423362			−0.0211681	+	0.999776i	$0.506739\pi$
$158$	4201.95			0.168321
$159$		0			0
$160$		−	7534.75i		−	0.294326i
$161$	16775.2			0.647165
$162$		0			0
$163$	1587.92			0.0597658			0.0298829	−	0.999553i	$-0.490487\pi$
$163$	1587.92			0.0597658			0.0298829	+	0.999553i	$0.490487\pi$
$164$		−	5521.97i		−	0.205308i
$165$		0			0
$166$		−	15124.5i		−	0.548862i
$167$			21277.9i			0.762948i	0.924380	+	0.381474i	$0.124583\pi$
$167$			21277.9i			0.762948i	−0.924380	+	0.381474i	$0.875417\pi$
$168$		0			0
$169$	23805.4			0.833493
$170$			35010.1i			1.21142i
$171$		0			0
$172$	−1751.24			−0.0591954
$173$		−	23629.3i		−	0.789513i	−0.918786	−	0.394756i	$-0.870829\pi$
$173$		−	23629.3i		−	0.789513i	0.918786	−	0.394756i	$-0.129171\pi$
$174$		0			0
$175$	−69149.5			−2.25794
$176$	−7868.02			−0.254004
$177$		0			0
$178$	−31600.1			−0.997351
$179$		−	54580.1i		−	1.70345i	−0.523993	−	0.851723i	$-0.675558\pi$
$179$		−	54580.1i		−	1.70345i	0.523993	−	0.851723i	$-0.324442\pi$
$180$		0			0
$181$		−	48461.8i		−	1.47925i	−0.673017	−	0.739627i	$-0.735002\pi$
$181$		−	48461.8i		−	1.47925i	0.673017	−	0.739627i	$-0.264998\pi$
$182$	12177.9			0.367645
$183$		0			0
$184$			6079.65i			0.179574i
$185$		−	100203.i		−	2.92776i
$186$		0			0
$187$	36558.6			1.04546
$188$	−664.839			−0.0188105
$189$		0			0
$190$	−35718.2	+	23033.0i	−0.989424	+	0.638033i
$191$	−10718.8			−0.293819			−0.146909	−	0.989150i	$-0.546933\pi$
$191$	−10718.8			−0.293819			−0.146909	+	0.989150i	$0.546933\pi$
$192$		0			0
$193$		−	25358.2i		−	0.680776i	−0.940285	−	0.340388i	$-0.889442\pi$
$193$		−	25358.2i		−	0.680776i	0.940285	−	0.340388i	$-0.110558\pi$
$194$	−3049.42			−0.0810241
$195$		0			0
$196$	−11976.3			−0.311752
$197$	4536.94			0.116904			0.0584521	−	0.998290i	$-0.481383\pi$
$197$	4536.94			0.116904			0.0584521	+	0.998290i	$0.481383\pi$
$198$		0			0
$199$	−67698.1			−1.70951			−0.854753	−	0.519035i	$-0.826291\pi$
$199$	−67698.1			−1.70951			−0.854753	+	0.519035i	$0.826291\pi$
$200$		−	25061.1i		−	0.626529i
$201$		0			0
$202$		−	3942.01i		−	0.0966085i
$203$		−	35050.9i		−	0.850564i
$204$		0			0
$205$		−	28730.8i		−	0.683660i
$206$	−31574.2			−0.744044
$207$		0			0
$208$			4413.50i			0.102013i
$209$	24051.8	+	37298.1i	0.550623	+	0.853874i
$210$		0			0
$211$		−	2816.81i		−	0.0632691i	−0.999500	−	0.0316346i	$-0.989929\pi$
$211$		−	2816.81i		−	0.0632691i	0.999500	−	0.0316346i	$-0.0100713\pi$
$212$			35117.5i			0.781360i
$213$		0			0
$214$	1420.09			0.0310090
$215$	−9111.68			−0.197116
$216$		0			0
$217$			15759.8i			0.334682i
$218$	32209.2			0.677746
$219$		0			0
$220$	−40937.3			−0.845812
$221$		−	20507.3i		−	0.419878i
$222$		0			0
$223$		−	36870.4i		−	0.741426i	−0.928747	−	0.370713i	$-0.879113\pi$
$223$		−	36870.4i		−	0.741426i	0.928747	−	0.370713i	$-0.120887\pi$
$224$		−	11301.8i		−	0.225243i
$225$		0			0
$226$	13517.1			0.264647
$227$			56826.9i			1.10281i	0.834236	+	0.551407i	$0.185909\pi$
$227$			56826.9i			1.10281i	−0.834236	+	0.551407i	$0.814091\pi$
$228$		0			0
$229$	7632.64			0.145547			0.0727736	−	0.997348i	$-0.476815\pi$
$229$	7632.64			0.145547			0.0727736	+	0.997348i	$0.476815\pi$
$230$			31632.4i			0.597966i
$231$		0			0
$232$	12703.1			0.236013
$233$	−91249.2			−1.68080			−0.840402	−	0.541964i	$-0.817681\pi$
$233$	−91249.2			−1.68080			−0.840402	+	0.541964i	$0.817681\pi$
$234$		0			0
$235$	−3459.16			−0.0626375
$236$		−	3815.85i		−	0.0685120i
$237$		0			0
$238$			52513.7i			0.927083i
$239$	110161.			1.92856			0.964280	−	0.264884i	$-0.0853337\pi$
$239$	110161.			1.92856			0.964280	+	0.264884i	$0.0853337\pi$
$240$		0			0
$241$			58423.0i			1.00589i	0.864319	+	0.502944i	$0.167750\pi$
$241$			58423.0i			1.00589i	−0.864319	+	0.502944i	$0.832250\pi$
$242$			1336.98i			0.0228295i
$243$		0			0
$244$	−31720.7			−0.532799
$245$	−62312.6			−1.03811
$246$		0			0
$247$	20922.0	−	13491.6i	0.342934	−	0.221142i
$248$	−5711.68			−0.0928668
$249$		0			0
$250$		−	56811.6i		−	0.908985i
$251$	17258.3			0.273936			0.136968	−	0.990575i	$-0.456264\pi$
$251$	17258.3			0.273936			0.136968	+	0.990575i	$0.456264\pi$
$252$		0			0
$253$	33031.5			0.516045
$254$	49646.1			0.769517
$255$		0			0
$256$	4096.00			0.0625000
$257$		−	41195.7i		−	0.623714i	−0.950129	−	0.311857i	$-0.899049\pi$
$257$		−	41195.7i		−	0.623714i	0.950129	−	0.311857i	$-0.100951\pi$
$258$		0			0
$259$		−	150300.i		−	2.24057i
$260$			22963.4i			0.339696i
$261$		0			0
$262$			20935.6i			0.304987i
$263$	41179.7			0.595349			0.297675	−	0.954667i	$-0.403789\pi$
$263$	41179.7			0.595349			0.297675	+	0.954667i	$0.403789\pi$
$264$		0			0
$265$			182716.i			2.60187i
$266$	−53575.8	+	34548.5i	−0.757191	+	0.488277i
$267$		0			0
$268$		−	40895.7i		−	0.569388i
$269$			19858.8i			0.274440i	0.990541	+	0.137220i	$0.0438168\pi$
$269$			19858.8i			0.274440i	−0.990541	+	0.137220i	$0.956183\pi$
$270$		0			0
$271$	110976.			1.51110			0.755549	−	0.655093i	$-0.227370\pi$
$271$	110976.			1.51110			0.755549	+	0.655093i	$0.227370\pi$
$272$	−19032.0			−0.257245
$273$		0			0
$274$		−	4787.87i		−	0.0637737i
$275$	−136161.			−1.80047
$276$		0			0
$277$	−87534.8			−1.14083			−0.570415	−	0.821357i	$-0.693218\pi$
$277$	−87534.8			−1.14083			−0.570415	+	0.821357i	$0.693218\pi$
$278$		−	105689.i		−	1.36754i
$279$		0			0
$280$		−	58803.3i		−	0.750042i
$281$			133298.i			1.68815i	0.536228	+	0.844073i	$0.319849\pi$
$281$			133298.i			1.68815i	−0.536228	+	0.844073i	$0.680151\pi$
$282$		0			0
$283$	29905.8			0.373407			0.186704	−	0.982416i	$-0.440220\pi$
$283$	29905.8			0.373407			0.186704	+	0.982416i	$0.440220\pi$
$284$			65463.4i			0.811637i
$285$		0			0
$286$	23979.1			0.293158
$287$		−	43095.0i		−	0.523194i
$288$		0			0
$289$	4910.99			0.0587994
$290$	66094.5			0.785903
$291$		0			0
$292$	58761.8			0.689175
$293$			42625.4i			0.496516i	0.968694	+	0.248258i	$0.0798580\pi$
$293$			42625.4i			0.496516i	−0.968694	+	0.248258i	$0.920142\pi$
$294$		0			0
$295$		−	19853.8i		−	0.228140i
$296$	54471.6			0.621708
$297$		0			0
$298$			17075.8i			0.192287i
$299$		−	18528.8i		−	0.207255i
$300$		0			0
$301$	−13667.1			−0.150850
$302$	−58319.6			−0.639441
$303$		0			0
$304$	−12521.1	−	19416.9i	−0.135486	−	0.210104i
$305$	−165043.			−1.77418
$306$		0			0
$307$			39047.5i			0.414302i	0.978309	+	0.207151i	$0.0664191\pi$
$307$			39047.5i			0.414302i	−0.978309	+	0.207151i	$0.933581\pi$
$308$	−61404.2			−0.647287
$309$		0			0
$310$	−29717.9			−0.309239
$311$	−79232.0			−0.819180			−0.409590	−	0.912270i	$-0.634328\pi$
$311$	−79232.0			−0.819180			−0.409590	+	0.912270i	$0.634328\pi$
$312$		0			0
$313$	−107079.			−1.09298			−0.546492	−	0.837464i	$-0.684037\pi$
$313$	−107079.			−1.09298			−0.546492	+	0.837464i	$0.684037\pi$
$314$		−	2951.59i		−	0.0299362i
$315$		0			0
$316$			11884.9i			0.119021i
$317$			132924.i			1.32277i	0.750046	+	0.661385i	$0.230031\pi$
$317$			132924.i			1.32277i	−0.750046	+	0.661385i	$0.769969\pi$
$318$		0			0
$319$		−	69017.9i		−	0.678235i
$320$	21311.5			0.208120
$321$		0			0
$322$			47447.3i			0.457614i
$323$	58179.0	+	90220.6i	0.557649	+	0.864770i
$324$		0			0
$325$			76378.2i			0.723107i
$326$			4491.31i			0.0422608i
$327$		0			0
$328$	15618.5			0.145175
$329$	−5188.59			−0.0479356
$330$		0			0
$331$		−	44623.1i		−	0.407290i	−0.979045	−	0.203645i	$-0.934721\pi$
$331$		−	44623.1i		−	0.407290i	0.979045	−	0.203645i	$-0.0652788\pi$
$332$	42778.4			0.388104
$333$		0			0
$334$	−60182.9			−0.539486
$335$		−	212780.i		−	1.89602i
$336$		0			0
$337$			5142.51i			0.0452810i	0.999744	+	0.0226405i	$0.00720730\pi$
$337$			5142.51i			0.0452810i	−0.999744	+	0.0226405i	$0.992793\pi$
$338$			67331.8i			0.589368i
$339$		0			0
$340$	−99023.5			−0.856605
$341$			31032.3i			0.266873i
$342$		0			0
$343$	56438.4			0.479718
$344$		−	4953.25i		−	0.0418575i
$345$		0			0
$346$	66833.8			0.558270
$347$	175661.			1.45887			0.729436	−	0.684049i	$-0.239782\pi$
$347$	175661.			1.45887			0.729436	+	0.684049i	$0.239782\pi$
$348$		0			0
$349$	137125.			1.12581			0.562906	−	0.826521i	$-0.309683\pi$
$349$	137125.			1.12581			0.562906	+	0.826521i	$0.309683\pi$
$350$		−	195584.i		−	1.59661i
$351$		0			0
$352$		−	22254.1i		−	0.179608i
$353$	199391.			1.60013			0.800065	−	0.599913i	$-0.204798\pi$
$353$	199391.			1.60013			0.800065	+	0.599913i	$0.204798\pi$
$354$		0			0
$355$			340606.i			2.70269i
$356$		−	89378.5i		−	0.705234i
$357$		0			0
$358$	154376.			1.20452
$359$	−6307.95			−0.0489440			−0.0244720	−	0.999701i	$-0.507790\pi$
$359$	−6307.95			−0.0489440			−0.0244720	+	0.999701i	$0.507790\pi$
$360$		0			0
$361$	−53769.6	+	118711.i	−0.412594	+	0.910915i
$362$	137071.			1.04599
$363$		0			0
$364$			34444.2i			0.259964i
$365$	305738.			2.29490
$366$		0			0
$367$	52946.3			0.393100			0.196550	−	0.980494i	$-0.437026\pi$
$367$	52946.3			0.393100			0.196550	+	0.980494i	$0.437026\pi$
$368$	−17195.8			−0.126978
$369$		0			0
$370$	283416.			2.07024
$371$			274067.i			1.99117i
$372$		0			0
$373$		−	9312.99i		−	0.0669378i	−0.999440	−	0.0334689i	$-0.989345\pi$
$373$		−	9312.99i		−	0.0669378i	0.999440	−	0.0334689i	$-0.0106555\pi$
$374$			103403.i			0.739251i
$375$		0			0
$376$		−	1880.45i		−	0.0133011i
$377$	−38715.0			−0.272394
$378$		0			0
$379$		−	178291.i		−	1.24123i	−0.784117	−	0.620613i	$-0.786884\pi$
$379$		−	178291.i		−	1.24123i	0.784117	−	0.620613i	$-0.213116\pi$
$380$	−65147.1	−	101026.i	−0.451157	−	0.699628i
$381$		0			0
$382$		−	30317.3i		−	0.207761i
$383$		−	27764.8i		−	0.189276i	−0.995512	−	0.0946382i	$-0.969831\pi$
$383$		−	27764.8i		−	0.189276i	0.995512	−	0.0946382i	$-0.0301694\pi$
$384$		0			0
$385$	−319486.			−2.15541
$386$	71723.9			0.481381
$387$		0			0
$388$		−	8625.07i		−	0.0572927i
$389$	−241135.			−1.59353			−0.796765	−	0.604289i	$-0.793457\pi$
$389$	−241135.			−1.59353			−0.796765	+	0.604289i	$0.793457\pi$
$390$		0			0
$391$	79900.3			0.522631
$392$		−	33874.0i		−	0.220442i
$393$		0			0
$394$			12832.4i			0.0826638i
$395$			61837.2i			0.396329i
$396$		0			0
$397$	234920.			1.49052			0.745261	−	0.666772i	$-0.232325\pi$
$397$	234920.			1.49052			0.745261	+	0.666772i	$0.232325\pi$
$398$		−	191479.i		−	1.20880i
$399$		0			0
$400$	70883.6			0.443023
$401$			311500.i			1.93718i	0.248667	+	0.968589i	$0.420007\pi$
$401$			311500.i			1.93718i	−0.248667	+	0.968589i	$0.579993\pi$
$402$		0			0
$403$	17407.3			0.107182
$404$	11149.7			0.0683126
$405$		0			0
$406$	99138.9			0.601440
$407$		−	295951.i		−	1.78662i
$408$		0			0
$409$			243636.i			1.45645i	0.685338	+	0.728225i	$0.259655\pi$
$409$			243636.i			1.45645i	−0.685338	+	0.728225i	$0.740345\pi$
$410$	81263.0			0.483420
$411$		0			0
$412$		−	89305.5i		−	0.526119i
$413$		−	29779.9i		−	0.174592i
$414$		0			0
$415$	222576.			1.29236
$416$	−12483.3			−0.0721343
$417$		0			0
$418$	−105495.	+	68028.7i	−0.603780	+	0.389349i
$419$	193805.			1.10392			0.551960	−	0.833871i	$-0.313880\pi$
$419$	193805.			1.10392			0.551960	+	0.833871i	$0.313880\pi$
$420$		0			0
$421$			136536.i			0.770339i	0.922846	+	0.385169i	$0.125857\pi$
$421$			136536.i			0.770339i	−0.922846	+	0.385169i	$0.874143\pi$
$422$	7967.13			0.0447380
$423$		0			0
$424$	−99327.2			−0.552505
$425$	−329360.			−1.82345
$426$		0			0
$427$	−247557.			−1.35775
$428$			4016.62i			0.0219267i
$429$		0			0
$430$		−	25771.7i		−	0.139382i
$431$		−	323841.i		−	1.74332i	−0.490113	−	0.871659i	$-0.663044\pi$
$431$		−	323841.i		−	1.74332i	0.490113	−	0.871659i	$-0.336956\pi$
$432$		0			0
$433$			286129.i			1.52611i	0.646333	+	0.763056i	$0.276302\pi$
$433$			286129.i			1.52611i	−0.646333	+	0.763056i	$0.723698\pi$
$434$	−44575.5			−0.236656
$435$		0			0
$436$			91101.3i			0.479238i
$437$	52566.0	+	81516.3i	0.275259	+	0.426856i
$438$		0			0
$439$		−	26283.3i		−	0.136380i	−0.997672	−	0.0681901i	$-0.978278\pi$
$439$		−	26283.3i		−	0.136380i	0.997672	−	0.0681901i	$-0.0217224\pi$
$440$		−	115788.i		−	0.598079i
$441$		0			0
$442$	58003.3			0.296899
$443$	317887.			1.61981			0.809907	−	0.586558i	$-0.199518\pi$
$443$	317887.			1.61981			0.809907	+	0.586558i	$0.199518\pi$
$444$		0			0
$445$		−	465036.i		−	2.34837i
$446$	104285.			0.524267
$447$		0			0
$448$	31966.3			0.159271
$449$			68728.7i			0.340915i	0.985365	+	0.170457i	$0.0545245\pi$
$449$			68728.7i			0.340915i	−0.985365	+	0.170457i	$0.945476\pi$
$450$		0			0
$451$		−	84857.3i		−	0.417192i
$452$			38232.1i			0.187133i
$453$		0			0
$454$	−160731.			−0.779808
$455$			179213.i			0.865660i
$456$		0			0
$457$	182859.			0.875555			0.437778	−	0.899083i	$-0.355766\pi$
$457$	182859.			0.875555			0.437778	+	0.899083i	$0.355766\pi$
$458$			21588.4i			0.102917i
$459$		0			0
$460$	−89470.0			−0.422826
$461$	−142804.			−0.671951			−0.335976	−	0.941871i	$-0.609066\pi$
$461$	−142804.			−0.671951			−0.335976	+	0.941871i	$0.609066\pi$
$462$		0			0
$463$	96212.9			0.448819			0.224409	−	0.974495i	$-0.427955\pi$
$463$	96212.9			0.448819			0.224409	+	0.974495i	$0.427955\pi$
$464$			35929.9i			0.166886i
$465$		0			0
$466$		−	258092.i		−	1.18851i
$467$	−287321.			−1.31745			−0.658724	−	0.752384i	$-0.728904\pi$
$467$	−287321.			−1.31745			−0.658724	+	0.752384i	$0.728904\pi$
$468$		0			0
$469$		−	319162.i		−	1.45099i
$470$		−	9783.98i		−	0.0442914i
$471$		0			0
$472$	10792.8			0.0484453
$473$	−26911.6			−0.120287
$474$		0			0
$475$	−216684.	−	336021.i	−0.960374	−	1.48929i
$476$	−148531.			−0.655547
$477$		0			0
$478$			311583.i			1.36370i
$479$	−101492.			−0.442346			−0.221173	−	0.975235i	$-0.570989\pi$
$479$	−101492.			−0.442346			−0.221173	+	0.975235i	$0.570989\pi$
$480$		0			0
$481$	−166012.			−0.717544
$482$	−165245.			−0.711270
$483$		0			0
$484$	−3781.56			−0.0161429
$485$		−	44876.2i		−	0.190780i
$486$		0			0
$487$			203340.i			0.857363i	0.903456	+	0.428682i	$0.141022\pi$
$487$			203340.i			0.857363i	−0.903456	+	0.428682i	$0.858978\pi$
$488$		−	89719.7i		−	0.376745i
$489$		0			0
$490$		−	176247.i		−	0.734055i
$491$	227758.			0.944735			0.472368	−	0.881402i	$-0.343399\pi$
$491$	227758.			0.944735			0.472368	+	0.881402i	$0.343399\pi$
$492$		0			0
$493$		−	166948.i		−	0.686890i
$494$	38160.1	+	59176.5i	0.156371	+	0.242491i
$495$		0			0
$496$		−	16155.1i		−	0.0656667i
$497$			510895.i			2.06833i
$498$		0			0
$499$	−132829.			−0.533447			−0.266724	−	0.963773i	$-0.585941\pi$
$499$	−132829.			−0.533447			−0.266724	+	0.963773i	$0.585941\pi$
$500$	160687.			0.642750
$501$		0			0
$502$			48813.8i			0.193702i
$503$	−279294.			−1.10389			−0.551945	−	0.833880i	$-0.686114\pi$
$503$	−279294.			−1.10389			−0.551945	+	0.833880i	$0.686114\pi$
$504$		0			0
$505$	58011.9			0.227475
$506$			93427.3i			0.364899i
$507$		0			0
$508$			140420.i			0.544130i
$509$		−	304788.i		−	1.17642i	−0.808709	−	0.588209i	$-0.799833\pi$
$509$		−	304788.i		−	1.17642i	0.808709	−	0.588209i	$-0.200167\pi$
$510$		0			0
$511$	458594.			1.75625
$512$			11585.2i			0.0441942i
$513$		0			0
$514$	116519.			0.441032
$515$		−	464656.i		−	1.75193i
$516$		0			0
$517$	−10216.7			−0.0382235
$518$	425112.			1.58432
$519$		0			0
$520$	−64950.4			−0.240201
$521$			181301.i			0.667919i	0.942587	+	0.333959i	$0.108385\pi$
$521$			181301.i			0.667919i	−0.942587	+	0.333959i	$0.891615\pi$
$522$		0			0
$523$			5966.86i			0.0218144i	0.999941	+	0.0109072i	$0.00347193\pi$
$523$			5966.86i			0.0218144i	−0.999941	+	0.0109072i	$0.996528\pi$
$524$	−59214.7			−0.215659
$525$		0			0
$526$			116474.i			0.420976i
$527$			75064.3i			0.270279i
$528$		0			0
$529$	−207649.			−0.742026
$530$	−516799.			−1.83980
$531$		0			0
$532$	−97718.0	−	151535.i	−0.345264	−	0.535415i
$533$	−47600.1			−0.167553
$534$		0			0
$535$			20898.5i			0.0730142i
$536$	115671.			0.402618
$537$		0			0
$538$	−56169.1			−0.194058
$539$	−184042.			−0.633490
$540$		0			0
$541$	−121383.			−0.414727			−0.207364	−	0.978264i	$-0.566488\pi$
$541$	−121383.			−0.414727			−0.207364	+	0.978264i	$0.566488\pi$
$542$			313889.i			1.06851i
$543$		0			0
$544$		−	53830.7i		−	0.181900i
$545$			474000.i			1.59583i
$546$		0			0
$547$			297533.i			0.994399i	0.867636	+	0.497200i	$0.165638\pi$
$547$			297533.i			0.994399i	−0.867636	+	0.497200i	$0.834362\pi$
$548$	13542.2			0.0450948
$549$		0			0
$550$		−	385120.i		−	1.27312i
$551$	170325.	−	109834.i	0.561014	−	0.361772i
$552$		0			0
$553$			92753.2i			0.303304i
$554$		−	247586.i		−	0.806689i
$555$		0			0
$556$	298933.			0.966996
$557$	−192568.			−0.620687			−0.310344	−	0.950624i	$-0.600444\pi$
$557$	−192568.			−0.620687			−0.310344	+	0.950624i	$0.600444\pi$
$558$		0			0
$559$			15095.9i			0.0483097i
$560$	166321.			0.530360
$561$		0			0
$562$	−377023.			−1.19370
$563$		−	21470.7i		−	0.0677376i	−0.999426	−	0.0338688i	$-0.989217\pi$
$563$		−	21470.7i		−	0.0677376i	0.999426	−	0.0338688i	$-0.0107828\pi$
$564$		0			0
$565$			198922.i			0.623139i
$566$			84586.4i			0.264039i
$567$		0			0
$568$	−185159.			−0.573914
$569$		−	51249.7i		−	0.158295i	−0.996863	−	0.0791474i	$-0.974780\pi$
$569$		−	51249.7i		−	0.158295i	0.996863	−	0.0791474i	$-0.0252198\pi$
$570$		0			0
$571$	515089.			1.57983			0.789914	−	0.613218i	$-0.210125\pi$
$571$	515089.			1.57983			0.789914	+	0.613218i	$0.210125\pi$
$572$			67823.3i			0.207294i
$573$		0			0
$574$	121891.			0.369954
$575$	−297584.			−0.900065
$576$		0			0
$577$	−419849.			−1.26108			−0.630538	−	0.776159i	$-0.717166\pi$
$577$	−419849.			−1.26108			−0.630538	+	0.776159i	$0.717166\pi$
$578$			13890.4i			0.0415775i
$579$		0			0
$580$			186943.i			0.555717i
$581$	333855.			0.989020
$582$		0			0
$583$			539658.i			1.58775i
$584$			166204.i			0.487320i
$585$		0			0
$586$	−120563.			−0.351090
$587$	−311469.			−0.903939			−0.451970	−	0.892033i	$-0.649278\pi$
$587$	−311469.			−0.903939			−0.451970	+	0.892033i	$0.649278\pi$
$588$		0			0
$589$	−76582.5	+	49384.5i	−0.220749	+	0.142351i
$590$	56155.1			0.161319
$591$		0			0
$592$			154069.i			0.439614i
$593$	131604.			0.374247			0.187124	−	0.982336i	$-0.440083\pi$
$593$	131604.			0.374247			0.187124	+	0.982336i	$0.440083\pi$
$594$		0			0
$595$	−772807.			−2.18292
$596$	−48297.8			−0.135967
$597$		0			0
$598$	52407.3			0.146551
$599$		−	199776.i		−	0.556788i	−0.960467	−	0.278394i	$-0.910198\pi$
$599$		−	199776.i		−	0.556788i	0.960467	−	0.278394i	$-0.0898021\pi$
$600$		0			0
$601$		−	328476.i		−	0.909399i	−0.890645	−	0.454700i	$-0.849747\pi$
$601$		−	328476.i		−	0.909399i	0.890645	−	0.454700i	$-0.150253\pi$
$602$		−	38656.5i		−	0.106667i
$603$		0			0
$604$		−	164953.i		−	0.452153i
$605$	−19675.5			−0.0537544
$606$		0			0
$607$		−	275312.i		−	0.747219i	−0.927586	−	0.373610i	$-0.878120\pi$
$607$		−	275312.i		−	0.747219i	0.927586	−	0.373610i	$-0.121880\pi$
$608$	54919.4	−	35414.9i	0.148566	−	0.0958031i
$609$		0			0
$610$		−	466811.i		−	1.25453i
$611$			5731.00i			0.0153514i
$612$		0			0
$613$	−473140.			−1.25912			−0.629562	−	0.776950i	$-0.716766\pi$
$613$	−473140.			−1.25912			−0.629562	+	0.776950i	$0.716766\pi$
$614$	−110443.			−0.292956
$615$		0			0
$616$		−	173677.i		−	0.457701i
$617$	−10864.2			−0.0285382			−0.0142691	−	0.999898i	$-0.504542\pi$
$617$	−10864.2			−0.0285382			−0.0142691	+	0.999898i	$0.504542\pi$
$618$		0			0
$619$	8963.62			0.0233939			0.0116969	−	0.999932i	$-0.496277\pi$
$619$	8963.62			0.0233939			0.0116969	+	0.999932i	$0.496277\pi$
$620$		−	84054.8i		−	0.218665i
$621$		0			0
$622$		−	224102.i		−	0.579248i
$623$		−	697535.i		−	1.79717i
$624$		0			0
$625$	143834.			0.368214
$626$		−	302864.i		−	0.772857i
$627$		0			0
$628$	8348.36			0.0211681
$629$		−	715879.i		−	1.80942i
$630$		0			0
$631$	−439268.			−1.10324			−0.551621	−	0.834095i	$-0.685991\pi$
$631$	−439268.			−1.10324			−0.551621	+	0.834095i	$0.685991\pi$
$632$	−33615.6			−0.0841603
$633$		0			0
$634$	−375966.			−0.935340
$635$			730608.i			1.81191i
$636$		0			0
$637$			103237.i			0.254423i
$638$	195212.			0.479585
$639$		0			0
$640$			60278.0i			0.147163i
$641$			351470.i			0.855406i	0.903919	+	0.427703i	$0.140677\pi$
$641$			351470.i			0.855406i	−0.903919	+	0.427703i	$0.859323\pi$
$642$		0			0
$643$	−257336.			−0.622412			−0.311206	−	0.950342i	$-0.600733\pi$
$643$	−257336.			−0.622412			−0.311206	+	0.950342i	$0.600733\pi$
$644$	−134201.			−0.323582
$645$		0			0
$646$	−255182.	+	164555.i	−0.611485	+	0.394318i
$647$	−352433.			−0.841915			−0.420957	−	0.907080i	$-0.638306\pi$
$647$	−352433.			−0.841915			−0.420957	+	0.907080i	$0.638306\pi$
$648$		0			0
$649$		−	58639.0i		−	0.139218i
$650$	−216030.			−0.511314
$651$		0			0
$652$	−12703.3			−0.0298829
$653$	51680.8			0.121200			0.0606001	−	0.998162i	$-0.480699\pi$
$653$	51680.8			0.121200			0.0606001	+	0.998162i	$0.480699\pi$
$654$		0			0
$655$	−308094.			−0.718126
$656$			44175.8i			0.102654i
$657$		0			0
$658$		−	14675.6i		−	0.0338956i
$659$		−	144227.i		−	0.332105i	−0.986117	−	0.166053i	$-0.946898\pi$
$659$		−	144227.i		−	0.332105i	0.986117	−	0.166053i	$-0.0531021\pi$
$660$		0			0
$661$		−	615181.i		−	1.40799i	−0.710205	−	0.703995i	$-0.751398\pi$
$661$		−	615181.i		−	1.40799i	0.710205	−	0.703995i	$-0.248602\pi$
$662$	126213.			0.287997
$663$		0			0
$664$			120996.i			0.274431i
$665$	−508426.	−	788438.i	−1.14970	−	1.78289i
$666$		0			0
$667$		−	150841.i		−	0.339054i
$668$		−	170223.i		−	0.381474i
$669$		0			0
$670$	601834.			1.34069
$671$	−487459.			−1.08266
$672$		0			0
$673$			334936.i			0.739489i	0.929133	+	0.369745i	$0.120555\pi$
$673$			334936.i			0.739489i	−0.929133	+	0.369745i	$0.879445\pi$
$674$	−14545.2			−0.0320185
$675$		0			0
$676$	−190443.			−0.416746
$677$			504808.i			1.10141i	0.834700	+	0.550705i	$0.185641\pi$
$677$			504808.i			1.10141i	−0.834700	+	0.550705i	$0.814359\pi$
$678$		0			0
$679$		−	67312.5i		−	0.146001i
$680$		−	280081.i		−	0.605711i
$681$		0			0
$682$	−87772.6			−0.188708
$683$		−	690506.i		−	1.48022i	−0.672486	−	0.740110i	$-0.734773\pi$
$683$		−	690506.i		−	1.48022i	0.672486	−	0.740110i	$-0.265227\pi$
$684$		0			0
$685$	70459.8			0.150162
$686$			159632.i			0.339212i
$687$		0			0
$688$	14009.9			0.0295977
$689$	302717.			0.637673
$690$		0			0
$691$	461905.			0.967379			0.483690	−	0.875240i	$-0.339296\pi$
$691$	461905.			0.967379			0.483690	+	0.875240i	$0.339296\pi$
$692$			189035.i			0.394756i
$693$		0			0
$694$			496846.i			1.03158i
$695$	1.55535e6			3.22002
$696$		0			0
$697$		−	205262.i		−	0.422516i
$698$			387848.i			0.796069i
$699$		0			0
$700$	553196.			1.12897
$701$	410170.			0.834694			0.417347	−	0.908747i	$-0.362960\pi$
$701$	410170.			0.834694			0.417347	+	0.908747i	$0.362960\pi$
$702$		0			0
$703$	730358.	−	470974.i	1.47783	−	0.952985i
$704$	62944.1			0.127002
$705$		0			0
$706$			563962.i			1.13146i
$707$	87015.4			0.174083
$708$		0			0
$709$	368245.			0.732562			0.366281	−	0.930504i	$-0.380631\pi$
$709$	368245.			0.732562			0.366281	+	0.930504i	$0.380631\pi$
$710$	−963380.			−1.91109
$711$		0			0
$712$	252801.			0.498675
$713$			67822.3i			0.133411i
$714$		0			0
$715$			352884.i			0.690272i
$716$			436641.i			0.851723i
$717$		0			0
$718$		−	17841.6i		−	0.0346086i
$719$	460066.			0.889943			0.444971	−	0.895545i	$-0.353214\pi$
$719$	460066.			0.889943			0.444971	+	0.895545i	$0.353214\pi$
$720$		0			0
$721$		−	696965.i		−	1.34073i
$722$	−335766.	−	152083.i	−0.644114	−	0.291748i
$723$		0			0
$724$			387695.i			0.739627i
$725$			621788.i			1.18295i
$726$		0			0
$727$	196263.			0.371339			0.185670	−	0.982612i	$-0.440555\pi$
$727$	196263.			0.371339			0.185670	+	0.982612i	$0.440555\pi$
$728$	−97422.9			−0.183822
$729$		0			0
$730$			864757.i			1.62274i
$731$	−65096.8			−0.121822
$732$		0			0
$733$	578892.			1.07743			0.538716	−	0.842487i	$-0.318910\pi$
$733$	578892.			1.07743			0.538716	+	0.842487i	$0.318910\pi$
$734$			149755.i			0.277964i
$735$		0			0
$736$		−	48637.2i		−	0.0897869i
$737$		−	628453.i		−	1.15701i
$738$		0			0
$739$	721624.			1.32136			0.660681	−	0.750667i	$-0.270268\pi$
$739$	721624.			1.32136			0.660681	+	0.750667i	$0.270268\pi$
$740$			801621.i			1.46388i
$741$		0			0
$742$	−775177.			−1.40797
$743$			963975.i			1.74618i	0.487562	+	0.873088i	$0.337886\pi$
$743$			963975.i			1.74618i	−0.487562	+	0.873088i	$0.662114\pi$
$744$		0			0
$745$	−251293.			−0.452760
$746$	26341.1			0.0473322
$747$		0			0
$748$	−292469.			−0.522729
$749$			31346.8i			0.0558766i
$750$		0			0
$751$		−	632486.i		−	1.12143i	−0.828010	−	0.560713i	$-0.810527\pi$
$751$		−	632486.i		−	1.12143i	0.828010	−	0.560713i	$-0.189473\pi$
$752$	5318.71			0.00940526
$753$		0			0
$754$		−	109503.i		−	0.192611i
$755$		−	858249.i		−	1.50563i
$756$		0			0
$757$	684418.			1.19434			0.597172	−	0.802113i	$-0.296291\pi$
$757$	684418.			1.19434			0.597172	+	0.802113i	$0.296291\pi$
$758$	504283.			0.877680
$759$		0			0
$760$	285746.	−	184264.i	0.494712	−	0.319016i
$761$	201444.			0.347845			0.173923	−	0.984759i	$-0.444356\pi$
$761$	201444.			0.347845			0.173923	+	0.984759i	$0.444356\pi$
$762$		0			0
$763$			710980.i			1.22126i
$764$	85750.4			0.146909
$765$		0			0
$766$	78530.6			0.133839
$767$	−32893.1			−0.0559131
$768$		0			0
$769$	343992.			0.581695			0.290847	−	0.956769i	$-0.406063\pi$
$769$	343992.			0.581695			0.290847	+	0.956769i	$0.406063\pi$
$770$		−	903643.i		−	1.52411i
$771$		0			0
$772$			202866.i			0.340388i
$773$			323326.i			0.541105i	0.962705	+	0.270553i	$0.0872064\pi$
$773$			323326.i			0.541105i	−0.962705	+	0.270553i	$0.912794\pi$
$774$		0			0
$775$		−	279573.i		−	0.465469i
$776$	24395.4			0.0405121
$777$		0			0
$778$		−	682032.i		−	1.12680i
$779$	209414.	−	135041.i	0.345088	−	0.222531i
$780$		0			0
$781$			1.00599e6i			1.64927i
$782$			225992.i			0.369556i
$783$		0			0
$784$	95810.2			0.155876
$785$	43436.5			0.0704881
$786$		0			0
$787$		−	526771.i		−	0.850496i	−0.905077	−	0.425248i	$-0.860187\pi$
$787$		−	526771.i		−	0.850496i	0.905077	−	0.425248i	$-0.139813\pi$
$788$	−36295.5			−0.0584521
$789$		0			0
$790$	−174902.			−0.280247
$791$			298374.i			0.476879i
$792$		0			0
$793$			273436.i			0.434820i
$794$			664454.i			1.05396i
$795$		0			0
$796$	541585.			0.854753
$797$		−	937353.i		−	1.47566i	−0.674986	−	0.737831i	$-0.735850\pi$
$797$		−	937353.i		−	1.47566i	0.674986	−	0.737831i	$-0.264150\pi$
$798$		0			0
$799$	−24713.3			−0.0387113
$800$			200489.i			0.313264i
$801$		0			0
$802$	−881055.			−1.36979
$803$	903006.			1.40042
$804$		0			0
$805$	−698249.			−1.07750
$806$			49235.4i			0.0757891i
$807$		0			0
$808$			31536.1i			0.0483043i
$809$	339941.			0.519405			0.259702	−	0.965689i	$-0.416376\pi$
$809$	339941.			0.519405			0.259702	+	0.965689i	$0.416376\pi$
$810$		0			0
$811$		−	421572.i		−	0.640958i	−0.947256	−	0.320479i	$-0.896156\pi$
$811$		−	421572.i		−	0.640958i	0.947256	−	0.320479i	$-0.103844\pi$
$812$			280407.i			0.425282i
$813$		0			0
$814$	837077.			1.26333
$815$	−66095.4			−0.0995076
$816$		0			0
$817$	−42826.9	−	66413.4i	−0.0641612	−	0.0994974i
$818$	−689108.			−1.02987
$819$		0			0
$820$			229846.i			0.341830i
$821$	832671.			1.23534			0.617671	−	0.786437i	$-0.288077\pi$
$821$	832671.			1.23534			0.617671	+	0.786437i	$0.288077\pi$
$822$		0			0
$823$	193463.			0.285627			0.142813	−	0.989750i	$-0.454385\pi$
$823$	193463.			0.285627			0.142813	+	0.989750i	$0.454385\pi$
$824$	252594.			0.372022
$825$		0			0
$826$	84230.4			0.123455
$827$			414119.i			0.605500i	0.953070	+	0.302750i	$0.0979047\pi$
$827$			414119.i			0.605500i	−0.953070	+	0.302750i	$0.902095\pi$
$828$		0			0
$829$		−	1.10281e6i		−	1.60469i	−0.596858	−	0.802347i	$-0.703585\pi$
$829$		−	1.10281e6i		−	1.60469i	0.596858	−	0.802347i	$-0.296415\pi$
$830$			629540.i			0.913834i
$831$		0			0
$832$		−	35308.0i		−	0.0510066i
$833$	−445181.			−0.641574
$834$		0			0
$835$		−	885669.i		−	1.27028i
$836$	−192414.	−	298384.i	−0.275311	−	0.426937i
$837$		0			0
$838$			548164.i			0.780589i
$839$		−	420379.i		−	0.597196i	−0.954379	−	0.298598i	$-0.903481\pi$
$839$		−	420379.i		−	0.597196i	0.954379	−	0.298598i	$-0.0965189\pi$
$840$		0			0
$841$	392105.			0.554384
$842$	−386181.			−0.544712
$843$		0			0
$844$			22534.4i			0.0316346i
$845$	−990875.			−1.38773
$846$		0			0
$847$	−29512.4			−0.0411374
$848$		−	280940.i		−	0.390680i
$849$		0			0
$850$		−	931570.i		−	1.28937i
$851$		−	646813.i		−	0.893140i
$852$		0			0
$853$	−1.00746e6			−1.38462			−0.692310	−	0.721600i	$-0.743407\pi$
$853$	−1.00746e6			−1.38462			−0.692310	+	0.721600i	$0.743407\pi$
$854$		−	700198.i		−	0.960074i
$855$		0			0
$856$	−11360.7			−0.0155045
$857$			425334.i			0.579119i	0.957160	+	0.289560i	$0.0935089\pi$
$857$			425334.i			0.579119i	−0.957160	+	0.289560i	$0.906491\pi$
$858$		0			0
$859$	−101991.			−0.138221			−0.0691105	−	0.997609i	$-0.522016\pi$
$859$	−101991.			−0.138221			−0.0691105	+	0.997609i	$0.522016\pi$
$860$	72893.5			0.0985580
$861$		0			0
$862$	915959.			1.23271
$863$			295578.i			0.396872i	0.980114	+	0.198436i	$0.0635862\pi$
$863$			295578.i			0.396872i	−0.980114	+	0.198436i	$0.936414\pi$
$864$		0			0
$865$			983547.i			1.31451i
$866$	−809295.			−1.07912
$867$		0			0
$868$		−	126079.i		−	0.167341i
$869$			182638.i			0.241853i
$870$		0			0
$871$	−352526.			−0.464681
$872$	−257673.			−0.338873
$873$		0			0
$874$	−230563.	+	148679.i	−0.301833	+	0.194638i
$875$	1.25405e6			1.63794
$876$		0			0
$877$			267600.i			0.347926i	0.984752	+	0.173963i	$0.0556573\pi$
$877$			267600.i			0.347926i	−0.984752	+	0.173963i	$0.944343\pi$
$878$	74340.5			0.0964354
$879$		0			0
$880$	327498.			0.422906
$881$	−1.32998e6			−1.71353			−0.856765	−	0.515707i	$-0.827529\pi$
$881$	−1.32998e6			−1.71353			−0.856765	+	0.515707i	$0.827529\pi$
$882$		0			0
$883$	−229460.			−0.294296			−0.147148	−	0.989114i	$-0.547009\pi$
$883$	−229460.			−0.294296			−0.147148	+	0.989114i	$0.547009\pi$
$884$			164058.i			0.209939i
$885$		0			0
$886$			899120.i			1.14538i
$887$		−	161937.i		−	0.205825i	−0.994690	−	0.102913i	$-0.967184\pi$
$887$		−	161937.i		−	0.205825i	0.994690	−	0.102913i	$-0.0328162\pi$
$888$		0			0
$889$			1.09588e6i			1.38663i
$890$	1.31532e6			1.66055
$891$		0			0
$892$			294963.i			0.370713i
$893$	−16258.8	−	25213.2i	−0.0203885	−	0.0316173i
$894$		0			0
$895$			2.27184e6i			2.83617i
$896$			90414.4i			0.112622i
$897$		0			0
$898$	−194394.			−0.241063
$899$	141711.			0.175342
$900$		0			0
$901$			1.30538e6i			1.60801i
$902$	240013.			0.295000
$903$		0			0
$904$	−108137.			−0.132323
$905$			2.01717e6i			2.46290i
$906$		0			0
$907$		−	530320.i		−	0.644649i	−0.946629	−	0.322325i	$-0.895536\pi$
$907$		−	530320.i		−	0.644649i	0.946629	−	0.322325i	$-0.104464\pi$
$908$		−	454615.i		−	0.551407i
$909$		0			0
$910$	−506891.			−0.612114
$911$		−	969623.i		−	1.16833i	−0.811634	−	0.584166i	$-0.801422\pi$
$911$		−	969623.i		−	1.16833i	0.811634	−	0.584166i	$-0.198578\pi$
$912$		0			0
$913$	657385.			0.788639
$914$			517203.i			0.619111i
$915$		0			0
$916$	−61061.1			−0.0727736
$917$	−462128.			−0.549571
$918$		0			0
$919$	853445.			1.01052			0.505260	−	0.862967i	$-0.331397\pi$
$919$	853445.			1.01052			0.505260	+	0.862967i	$0.331397\pi$
$920$		−	253059.i		−	0.298983i
$921$		0			0
$922$		−	403910.i		−	0.475141i
$923$	564303.			0.662382
$924$		0			0
$925$			2.66625e6i			3.11614i
$926$			272131.i			0.317363i
$927$		0			0
$928$	−101625.			−0.118006
$929$	−580185.			−0.672257			−0.336128	−	0.941816i	$-0.609118\pi$
$929$	−580185.			−0.672257			−0.336128	+	0.941816i	$0.609118\pi$
$930$		0			0
$931$	−292882.	−	454185.i	−0.337905	−	0.524002i
$932$	729993.			0.840402
$933$		0			0
$934$		−	812667.i		−	0.931577i
$935$	−1.52172e6			−1.74065
$936$		0			0
$937$	−173996.			−0.198180			−0.0990901	−	0.995078i	$-0.531593\pi$
$937$	−173996.			−0.198180			−0.0990901	+	0.995078i	$0.531593\pi$
$938$	902725.			1.02601
$939$		0			0
$940$	27673.3			0.0313188
$941$		−	288487.i		−	0.325797i	−0.986643	−	0.162899i	$-0.947916\pi$
$941$		−	288487.i		−	0.325797i	0.986643	−	0.162899i	$-0.0520844\pi$
$942$		0			0
$943$		−	185459.i		−	0.208557i
$944$			30526.8i			0.0342560i
$945$		0			0
$946$		−	76117.6i		−	0.0850556i
$947$	−484204.			−0.539919			−0.269959	−	0.962872i	$-0.587010\pi$
$947$	−484204.			−0.539919			−0.269959	+	0.962872i	$0.587010\pi$
$948$		0			0
$949$		−	506534.i		−	0.562440i
$950$	950412.	−	612876.i	1.05309	−	0.679087i
$951$		0			0
$952$		−	420110.i		−	0.463542i
$953$		−	834271.i		−	0.918590i	−0.888284	−	0.459295i	$-0.848102\pi$
$953$		−	834271.i		−	0.918590i	0.888284	−	0.459295i	$-0.151898\pi$
$954$		0			0
$955$	446159.			0.489196
$956$	−881290.			−0.964280
$957$		0			0
$958$		−	287064.i		−	0.312786i
$959$	105687.			0.114917
$960$		0			0
$961$	859804.			0.931006
$962$		−	469552.i		−	0.507380i
$963$		0			0
$964$		−	467384.i		−	0.502944i
$965$			1.05551e6i			1.13346i
$966$		0			0
$967$	−268509.			−0.287148			−0.143574	−	0.989640i	$-0.545859\pi$
$967$	−268509.			−0.287148			−0.143574	+	0.989640i	$0.545859\pi$
$968$		−	10695.9i		−	0.0114147i
$969$		0			0
$970$	126929.			0.134902
$971$			513847.i			0.544999i	0.962156	+	0.272499i	$0.0878503\pi$
$971$			513847.i			0.544999i	−0.962156	+	0.272499i	$0.912150\pi$
$972$		0			0
$973$	2.33296e6			2.46423
$974$	−575132.			−0.606248
$975$		0			0
$976$	253766.			0.266399
$977$			880047.i			0.921970i	0.887408	+	0.460985i	$0.152504\pi$
$977$			880047.i			0.921970i	−0.887408	+	0.460985i	$0.847496\pi$
$978$		0			0
$979$		−	1.37350e6i		−	1.43306i
$980$	498500.			0.519055
$981$		0			0
$982$			644196.i			0.668029i
$983$		−	1.15115e6i		−	1.19131i	−0.803239	−	0.595656i	$-0.796892\pi$
$983$		−	1.15115e6i		−	1.19131i	0.803239	−	0.595656i	$-0.203108\pi$
$984$		0			0
$985$	−188845.			−0.194641
$986$	472200.			0.485705
$987$		0			0
$988$	−167376.	+	107933.i	−0.171467	+	0.110571i
$989$	−58816.4			−0.0601320
$990$		0			0
$991$		−	1.55344e6i		−	1.58178i	−0.611956	−	0.790892i	$-0.709617\pi$
$991$		−	1.55344e6i		−	1.58178i	0.611956	−	0.790892i	$-0.290383\pi$
$992$	45693.4			0.0464334
$993$		0			0
$994$	−1.44503e6			−1.46253
$995$	2.81787e6			2.84626
$996$		0			0
$997$	−1.19520e6			−1.20240			−0.601200	−	0.799099i	$-0.705310\pi$
$997$	−1.19520e6			−1.20240			−0.601200	+	0.799099i	$0.705310\pi$
$998$		−	375697.i		−	0.377204i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.5.d.a.37.5		8
3.2	odd	2		38.5.b.a.37.1	✓	8
12.11	even	2		304.5.e.e.113.8		8
19.18	odd	2	inner	342.5.d.a.37.1		8
57.56	even	2		38.5.b.a.37.8	yes	8
228.227	odd	2		304.5.e.e.113.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.b.a.37.1	✓	8	3.2	odd	2
38.5.b.a.37.8	yes	8	57.56	even	2
304.5.e.e.113.1		8	228.227	odd	2
304.5.e.e.113.8		8	12.11	even	2
342.5.d.a.37.1		8	19.18	odd	2	inner
342.5.d.a.37.5		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	342.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} -41.6240 q^{5} -62.4342 q^{7} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} -41.6240 q^{5} -62.4342 q^{7} -22.6274i q^{8} -117.730i q^{10} -122.938 q^{11} +68.9610i q^{13} -176.591i q^{14} +64.0000 q^{16} -297.375 q^{17} +(-195.642 - 303.390i) q^{19} +332.992 q^{20} -347.721i q^{22} -268.685 q^{23} +1107.56 q^{25} -195.051 q^{26} +499.474 q^{28} +561.405i q^{29} -252.423i q^{31} +181.019i q^{32} -841.104i q^{34} +2598.76 q^{35} +2407.33i q^{37} +(858.116 - 553.358i) q^{38} +941.843i q^{40} +690.246i q^{41} +218.905 q^{43} +983.502 q^{44} -759.956i q^{46} +83.1049 q^{47} +1497.03 q^{49} +3132.64i q^{50} -551.688i q^{52} -4389.68i q^{53} +5117.16 q^{55} +1412.73i q^{56} -1587.89 q^{58} +476.981i q^{59} +3965.09 q^{61} +713.960 q^{62} -512.000 q^{64} -2870.43i q^{65} +5111.96i q^{67} +2379.00 q^{68} +7350.41i q^{70} -8182.93i q^{71} -7345.23 q^{73} -6808.95 q^{74} +(1565.13 + 2427.12i) q^{76} +7675.53 q^{77} -1485.61i q^{79} -2663.94 q^{80} -1952.31 q^{82} -5347.30 q^{83} +12377.9 q^{85} +619.156i q^{86} +2781.76i q^{88} +11172.3i q^{89} -4305.53i q^{91} +2149.48 q^{92} +235.056i q^{94} +(8143.39 + 12628.3i) q^{95} +1078.13i q^{97} +4234.25i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7}+O(q^{10}) \) 8 * q - 64 * q^4 - 18 * q^5 - 162 * q^7	\( 8 q - 64 q^{4} - 18 q^{5} - 162 q^{7} + 6 q^{11} + 512 q^{16} - 510 q^{17} - 12 q^{19} + 144 q^{20} + 396 q^{23} + 3458 q^{25} + 192 q^{26} + 1296 q^{28} - 1002 q^{35} + 3216 q^{38} - 8654 q^{43} - 48 q^{44} - 3210 q^{47} + 9222 q^{49} + 17146 q^{55} - 960 q^{58} + 1314 q^{61} + 15168 q^{62} - 4096 q^{64} + 4080 q^{68} + 23398 q^{73} - 13152 q^{74} + 96 q^{76} + 44622 q^{77} - 1152 q^{80} + 16512 q^{82} + 10440 q^{83} + 21274 q^{85} - 3168 q^{92} + 34686 q^{95}+O(q^{100}) \) 8 * q - 64 * q^4 - 18 * q^5 - 162 * q^7 + 6 * q^11 + 512 * q^16 - 510 * q^17 - 12 * q^19 + 144 * q^20 + 396 * q^23 + 3458 * q^25 + 192 * q^26 + 1296 * q^28 - 1002 * q^35 + 3216 * q^38 - 8654 * q^43 - 48 * q^44 - 3210 * q^47 + 9222 * q^49 + 17146 * q^55 - 960 * q^58 + 1314 * q^61 + 15168 * q^62 - 4096 * q^64 + 4080 * q^68 + 23398 * q^73 - 13152 * q^74 + 96 * q^76 + 44622 * q^77 - 1152 * q^80 + 16512 * q^82 + 10440 * q^83 + 21274 * q^85 - 3168 * q^92 + 34686 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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