LMFDB - Embedded newform 208.4.w.c.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(17,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("208.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.w (of order $6$, degree $2$, not 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:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
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} + 51x^{6} - 224x^{5} + 2520x^{4} - 5712x^{3} + 16675x^{2} + 9072x + 6561 $ x^8 + 51x^6 - 224x^5 + 2520x^4 - 5712x^3 + 16675x^2 + 9072x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.3
Root			$2.54083 + 4.40084i$ of defining polynomial
Character	$\chi$	$=$	208.49
Dual form			208.4.w.c.17.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.643469 + 1.11452i) q^{3} -15.8968i q^{5} +(1.02552 + 0.592083i) q^{7} +(12.6719 - 21.9484i) q^{9} +O(q^{10})$	\(q+(0.643469 + 1.11452i) q^{3} -15.8968i q^{5} +(1.02552 + 0.592083i) q^{7} +(12.6719 - 21.9484i) q^{9} +(-33.6987 + 19.4560i) q^{11} +(-45.4488 + 11.4634i) q^{13} +(17.7173 - 10.2291i) q^{15} +(-18.4601 + 31.9739i) q^{17} +(-37.1887 - 21.4709i) q^{19} +1.52395i q^{21} +(-101.967 - 176.612i) q^{23} -127.707 q^{25} +67.3632 q^{27} +(-29.4958 - 51.0882i) q^{29} -77.3896i q^{31} +(-43.3681 - 25.0386i) q^{33} +(9.41221 - 16.3024i) q^{35} +(223.689 - 129.147i) q^{37} +(-42.0210 - 43.2773i) q^{39} +(146.884 - 84.8034i) q^{41} +(-183.168 + 317.255i) q^{43} +(-348.908 - 201.442i) q^{45} -249.834i q^{47} +(-170.799 - 295.832i) q^{49} -47.5141 q^{51} +157.459 q^{53} +(309.287 + 535.701i) q^{55} -55.2634i q^{57} +(-582.168 - 336.115i) q^{59} +(-290.125 + 502.512i) q^{61} +(25.9905 - 15.0056i) q^{63} +(182.231 + 722.489i) q^{65} +(-156.637 + 90.4343i) q^{67} +(131.225 - 227.289i) q^{69} +(449.996 + 259.805i) q^{71} +982.663i q^{73} +(-82.1757 - 142.332i) q^{75} -46.0782 q^{77} +1265.85 q^{79} +(-298.795 - 517.528i) q^{81} -1026.42i q^{83} +(508.282 + 293.457i) q^{85} +(37.9592 - 65.7473i) q^{87} +(1286.09 - 742.526i) q^{89} +(-53.3958 - 15.1535i) q^{91} +(86.2523 - 49.7978i) q^{93} +(-341.318 + 591.180i) q^{95} +(-303.313 - 175.118i) q^{97} +986.176i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{7} - 70 q^{9}+O(q^{10}) $ 8 * q + 36 * q^7 - 70 * q^9	$ 8 q + 36 q^{7} - 70 q^{9} - 72 q^{11} + 62 q^{13} - 96 q^{15} + 88 q^{17} + 144 q^{19} + 20 q^{23} - 84 q^{25} + 432 q^{27} - 484 q^{29} + 1038 q^{33} - 40 q^{35} + 996 q^{37} + 236 q^{39} + 156 q^{41} - 504 q^{43} - 1530 q^{45} + 922 q^{49} + 1808 q^{51} - 1164 q^{53} + 1128 q^{55} - 600 q^{59} - 1224 q^{61} - 6480 q^{63} + 670 q^{65} - 960 q^{67} + 1738 q^{69} + 2964 q^{71} - 1448 q^{75} - 3972 q^{77} + 3968 q^{79} - 4132 q^{81} + 3870 q^{85} + 1660 q^{87} + 5430 q^{89} + 1720 q^{91} + 3324 q^{93} - 2400 q^{95} - 3042 q^{97}+O(q^{100}) $ 8 * q + 36 * q^7 - 70 * q^9 - 72 * q^11 + 62 * q^13 - 96 * q^15 + 88 * q^17 + 144 * q^19 + 20 * q^23 - 84 * q^25 + 432 * q^27 - 484 * q^29 + 1038 * q^33 - 40 * q^35 + 996 * q^37 + 236 * q^39 + 156 * q^41 - 504 * q^43 - 1530 * q^45 + 922 * q^49 + 1808 * q^51 - 1164 * q^53 + 1128 * q^55 - 600 * q^59 - 1224 * q^61 - 6480 * q^63 + 670 * q^65 - 960 * q^67 + 1738 * q^69 + 2964 * q^71 - 1448 * q^75 - 3972 * q^77 + 3968 * q^79 - 4132 * q^81 + 3870 * q^85 + 1660 * q^87 + 5430 * q^89 + 1720 * q^91 + 3324 * q^93 - 2400 * q^95 - 3042 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	0.643469	+	1.11452i	0.123836	+	0.214490i	0.921277	−	0.388907i	$-0.127147\pi$
$3$	0.643469	+	1.11452i	0.123836	+	0.214490i	−0.797442	+	0.603396i	$0.793814\pi$
$4$		0			0
$5$		−	15.8968i		−	1.42185i	−0.703268	−	0.710925i	$-0.748276\pi$
$5$		−	15.8968i		−	1.42185i	0.703268	−	0.710925i	$-0.251724\pi$
$6$		0			0
$7$	1.02552	+	0.592083i	0.0553728	+	0.0319695i	0.527431	−	0.849598i	$-0.323155\pi$
$7$	1.02552	+	0.592083i	0.0553728	+	0.0319695i	−0.472058	+	0.881567i	$0.656489\pi$
$8$		0			0
$9$	12.6719	−	21.9484i	0.469329	−	0.812903i
$10$		0			0
$11$	−33.6987	+	19.4560i	−0.923686	+	0.533290i	−0.884809	−	0.465954i	$-0.845711\pi$
$11$	−33.6987	+	19.4560i	−0.923686	+	0.533290i	−0.0388769	+	0.999244i	$0.512378\pi$
$12$		0			0
$13$	−45.4488	+	11.4634i	−0.969632	+	0.244567i
$13$	−45.4488	+	11.4634i	−0.969632	+	0.244567i
$14$		0			0
$15$	17.7173	−	10.2291i	0.304972	−	0.176076i
$16$		0			0
$17$	−18.4601	+	31.9739i	−0.263367	+	0.456165i	−0.967135	−	0.254265i	$-0.918166\pi$
$17$	−18.4601	+	31.9739i	−0.263367	+	0.456165i	0.703767	+	0.710430i	$0.251500\pi$
$18$		0			0
$19$	−37.1887	−	21.4709i	−0.449035	−	0.259250i	0.258388	−	0.966041i	$-0.416809\pi$
$19$	−37.1887	−	21.4709i	−0.449035	−	0.259250i	−0.707423	+	0.706791i	$0.750142\pi$
$20$		0			0
$21$			1.52395i			0.0158358i
$22$		0			0
$23$	−101.967	−	176.612i	−0.924418	−	1.60114i	−0.792494	−	0.609879i	$-0.791218\pi$
$23$	−101.967	−	176.612i	−0.924418	−	1.60114i	−0.131924	−	0.991260i	$-0.542115\pi$
$24$		0			0
$25$	−127.707			−1.02166
$26$		0			0
$27$	67.3632			0.480150
$28$		0			0
$29$	−29.4958	−	51.0882i	−0.188870	−	0.327132i	0.756004	−	0.654567i	$-0.227149\pi$
$29$	−29.4958	−	51.0882i	−0.188870	−	0.327132i	−0.944874	+	0.327435i	$0.893816\pi$
$30$		0			0
$31$		−	77.3896i		−	0.448374i	−0.974546	−	0.224187i	$-0.928027\pi$
$31$		−	77.3896i		−	0.448374i	0.974546	−	0.224187i	$-0.0719726\pi$
$32$		0			0
$33$	−43.3681	−	25.0386i	−0.228770	−	0.132081i
$34$		0			0
$35$	9.41221	−	16.3024i	0.0454558	−	0.0787318i
$36$		0			0
$37$	223.689	−	129.147i	0.993898	−	0.573827i	0.0874610	−	0.996168i	$-0.472125\pi$
$37$	223.689	−	129.147i	0.993898	−	0.573827i	0.906437	+	0.422340i	$0.138791\pi$
$38$		0			0
$39$	−42.0210	−	43.2773i	−0.172532	−	0.177690i
$40$		0			0
$41$	146.884	−	84.8034i	0.559498	−	0.323026i	−0.193446	−	0.981111i	$-0.561966\pi$
$41$	146.884	−	84.8034i	0.559498	−	0.323026i	0.752944	+	0.658085i	$0.228633\pi$
$42$		0			0
$43$	−183.168	+	317.255i	−0.649600	+	1.12514i	0.333619	+	0.942708i	$0.391730\pi$
$43$	−183.168	+	317.255i	−0.649600	+	1.12514i	−0.983219	+	0.182432i	$0.941603\pi$
$44$		0			0
$45$	−348.908	−	201.442i	−1.15583	−	0.667316i
$46$		0			0
$47$		−	249.834i		−	0.775362i	−0.921794	−	0.387681i	$-0.873276\pi$
$47$		−	249.834i		−	0.775362i	0.921794	−	0.387681i	$-0.126724\pi$
$48$		0			0
$49$	−170.799	−	295.832i	−0.497956	−	0.862485i
$50$		0			0
$51$	−47.5141			−0.130457
$52$		0			0
$53$	157.459			0.408087			0.204044	−	0.978962i	$-0.434592\pi$
$53$	157.459			0.408087			0.204044	+	0.978962i	$0.434592\pi$
$54$		0			0
$55$	309.287	+	535.701i	0.758259	+	1.31334i
$56$		0			0
$57$		−	55.2634i		−	0.128418i
$58$		0			0
$59$	−582.168	−	336.115i	−1.28461	−	0.741668i	−0.306919	−	0.951736i	$-0.599298\pi$
$59$	−582.168	−	336.115i	−1.28461	−	0.741668i	−0.977687	+	0.210068i	$0.932632\pi$
$60$		0			0
$61$	−290.125	+	502.512i	−0.608963	+	1.05476i	0.382449	+	0.923977i	$0.375081\pi$
$61$	−290.125	+	502.512i	−0.608963	+	1.05476i	−0.991412	+	0.130778i	$0.958252\pi$
$62$		0			0
$63$	25.9905	−	15.0056i	0.0519761	−	0.0300084i
$64$		0			0
$65$	182.231	+	722.489i	0.347738	+	1.37867i
$66$		0			0
$67$	−156.637	+	90.4343i	−0.285615	+	0.164900i	−0.635963	−	0.771720i	$-0.719397\pi$
$67$	−156.637	+	90.4343i	−0.285615	+	0.164900i	0.350347	+	0.936620i	$0.386064\pi$
$68$		0			0
$69$	131.225	−	227.289i	0.228952	−	0.396556i
$70$		0			0
$71$	449.996	+	259.805i	0.752179	+	0.434271i	0.826481	−	0.562965i	$-0.190339\pi$
$71$	449.996	+	259.805i	0.752179	+	0.434271i	−0.0743017	+	0.997236i	$0.523673\pi$
$72$		0			0
$73$			982.663i			1.57551i	0.615991	+	0.787753i	$0.288756\pi$
$73$			982.663i			1.57551i	−0.615991	+	0.787753i	$0.711244\pi$
$74$		0			0
$75$	−82.1757	−	142.332i	−0.126518	−	0.219135i
$76$		0			0
$77$	−46.0782			−0.0681960
$78$		0			0
$79$	1265.85			1.80278			0.901388	−	0.433013i	$-0.142550\pi$
$79$	1265.85			1.80278			0.901388	+	0.433013i	$0.142550\pi$
$80$		0			0
$81$	−298.795	−	517.528i	−0.409870	−	0.709915i
$82$		0			0
$83$		−	1026.42i		−	1.35740i	−0.734414	−	0.678702i	$-0.762543\pi$
$83$		−	1026.42i		−	1.35740i	0.734414	−	0.678702i	$-0.237457\pi$
$84$		0			0
$85$	508.282	+	293.457i	0.648599	+	0.374469i
$86$		0			0
$87$	37.9592	−	65.7473i	0.0467777	−	0.0810213i
$88$		0			0
$89$	1286.09	−	742.526i	1.53175	−	0.884355i	0.532466	−	0.846452i	$-0.321266\pi$
$89$	1286.09	−	742.526i	1.53175	−	0.884355i	0.999281	−	0.0379029i	$-0.0120678\pi$
$90$		0			0
$91$	−53.3958	−	15.1535i	−0.0615099	−	0.0174563i
$92$		0			0
$93$	86.2523	−	49.7978i	0.0961714	−	0.0555246i
$94$		0			0
$95$	−341.318	+	591.180i	−0.368615	+	0.638460i
$96$		0			0
$97$	−303.313	−	175.118i	−0.317493	−	0.183305i	0.332782	−	0.943004i	$-0.392013\pi$
$97$	−303.313	−	175.118i	−0.317493	−	0.183305i	−0.650274	+	0.759699i	$0.725346\pi$
$98$		0			0
$99$			986.176i			1.00116i
$100$		0			0
$101$	715.853	+	1239.89i	0.705248	+	1.22153i	0.966602	+	0.256282i	$0.0824977\pi$
$101$	715.853	+	1239.89i	0.705248	+	1.22153i	−0.261354	+	0.965243i	$0.584169\pi$
$102$		0			0
$103$	1492.10			1.42739			0.713695	−	0.700456i	$-0.247020\pi$
$103$	1492.10			1.42739			0.713695	+	0.700456i	$0.247020\pi$
$104$		0			0
$105$	24.2258			0.0225162
$106$		0			0
$107$	580.909	+	1006.16i	0.524847	+	0.909062i	0.999581	+	0.0289326i	$0.00921082\pi$
$107$	580.909	+	1006.16i	0.524847	+	0.909062i	−0.474734	+	0.880129i	$0.657456\pi$
$108$		0			0
$109$		−	1599.05i		−	1.40515i	−0.711611	−	0.702574i	$-0.752034\pi$
$109$		−	1599.05i		−	1.40515i	0.711611	−	0.702574i	$-0.247966\pi$
$110$		0			0
$111$	287.874	+	166.204i	0.246160	+	0.142121i
$112$		0			0
$113$	−187.357	+	324.511i	−0.155974	+	0.270155i	−0.933413	−	0.358803i	$-0.883185\pi$
$113$	−187.357	+	324.511i	−0.155974	+	0.270155i	0.777439	+	0.628958i	$0.216518\pi$
$114$		0			0
$115$	−2807.56	+	1620.95i	−2.27658	+	1.31438i
$116$		0			0
$117$	−324.320	+	1142.79i	−0.256268	+	0.902999i
$118$		0			0
$119$	−37.8624	+	21.8599i	−0.0291667	+	0.0168394i
$120$		0			0
$121$	91.5691	−	158.602i	0.0687973	−	0.119160i
$122$		0			0
$123$	189.030	+	109.137i	0.138571	+	0.0800043i
$124$		0			0
$125$			43.0377i			0.0307952i
$126$		0			0
$127$	−324.648	−	562.306i	−0.226833	−	0.392887i	0.730035	−	0.683410i	$-0.239504\pi$
$127$	−324.648	−	562.306i	−0.226833	−	0.392887i	−0.956868	+	0.290523i	$0.906171\pi$
$128$		0			0
$129$	−471.450			−0.321774
$130$		0			0
$131$	612.966			0.408818			0.204409	−	0.978886i	$-0.434473\pi$
$131$	612.966			0.408818			0.204409	+	0.978886i	$0.434473\pi$
$132$		0			0
$133$	−25.4251	−	44.0375i	−0.0165762	−	0.0287108i
$134$		0			0
$135$		−	1070.86i		−	0.682702i
$136$		0			0
$137$	180.827	+	104.401i	0.112767	+	0.0651062i	0.555323	−	0.831635i	$-0.312595\pi$
$137$	180.827	+	104.401i	0.112767	+	0.0651062i	−0.442556	+	0.896741i	$0.645928\pi$
$138$		0			0
$139$	−184.795	+	320.074i	−0.112763	+	0.195312i	−0.916883	−	0.399155i	$-0.869303\pi$
$139$	−184.795	+	320.074i	−0.112763	+	0.195312i	0.804120	+	0.594467i	$0.202637\pi$
$140$		0			0
$141$	278.445	−	160.760i	0.166307	−	0.0960175i
$142$		0			0
$143$	1308.53	−	1270.55i	0.765211	−	0.742999i
$144$		0			0
$145$	−812.137	+	468.888i	−0.465133	+	0.268545i
$146$		0			0
$147$	219.807	−	380.718i	0.123329	−	0.213613i
$148$		0			0
$149$	−857.541	−	495.101i	−0.471493	−	0.272217i	0.245371	−	0.969429i	$-0.421090\pi$
$149$	−857.541	−	495.101i	−0.471493	−	0.272217i	−0.716865	+	0.697212i	$0.754423\pi$
$150$		0			0
$151$		−	2855.12i		−	1.53872i	−0.638816	−	0.769360i	$-0.720575\pi$
$151$		−	2855.12i		−	1.53872i	0.638816	−	0.769360i	$-0.279425\pi$
$152$		0			0
$153$	467.850	+	810.340i	0.247212	+	0.428184i
$154$		0			0
$155$	−1230.24			−0.637520
$156$		0			0
$157$	−947.845			−0.481823			−0.240912	−	0.970547i	$-0.577446\pi$
$157$	−947.845			−0.481823			−0.240912	+	0.970547i	$0.577446\pi$
$158$		0			0
$159$	101.320	+	175.491i	0.0505357	+	0.0875305i
$160$		0			0
$161$		−	241.492i		−	0.118213i
$162$		0			0
$163$	−2330.22	−	1345.35i	−1.11973	−	0.646479i	−0.178401	−	0.983958i	$-0.557092\pi$
$163$	−2330.22	−	1345.35i	−1.11973	−	0.646479i	−0.941333	+	0.337479i	$0.890426\pi$
$164$		0			0
$165$	−398.033	+	689.413i	−0.187799	+	0.325277i
$166$		0			0
$167$	1563.89	−	902.913i	0.724656	−	0.418380i	−0.0918081	−	0.995777i	$-0.529265\pi$
$167$	1563.89	−	902.913i	0.724656	−	0.418380i	0.816464	+	0.577397i	$0.195931\pi$
$168$		0			0
$169$	1934.18	−	1041.99i	0.880374	−	0.474280i
$170$		0			0
$171$	−942.502	+	544.154i	−0.421491	+	0.243348i
$172$		0			0
$173$	177.022	−	306.610i	0.0777959	−	0.134747i	−0.824503	−	0.565858i	$-0.808545\pi$
$173$	177.022	−	306.610i	0.0777959	−	0.134747i	0.902299	+	0.431111i	$0.141878\pi$
$174$		0			0
$175$	−130.966	−	75.6133i	−0.0565720	−	0.0326619i
$176$		0			0
$177$		−	865.117i		−	0.367380i
$178$		0			0
$179$	1297.09	+	2246.63i	0.541615	+	0.938104i	0.998812	+	0.0487386i	$0.0155201\pi$
$179$	1297.09	+	2246.63i	0.541615	+	0.938104i	−0.457197	+	0.889366i	$0.651147\pi$
$180$		0			0
$181$	−2338.97			−0.960522			−0.480261	−	0.877126i	$-0.659458\pi$
$181$	−2338.97			−0.960522			−0.480261	+	0.877126i	$0.659458\pi$
$182$		0			0
$183$	−746.747			−0.301645
$184$		0			0
$185$	−2053.02	−	3555.93i	−0.815897	−	1.41317i
$186$		0			0
$187$		−	1436.64i		−	0.561805i
$188$		0			0
$189$	69.0821	+	39.8846i	0.0265872	+	0.0153501i
$190$		0			0
$191$	15.3004	−	26.5011i	0.00579633	−	0.0100395i	−0.863113	−	0.505011i	$-0.831488\pi$
$191$	15.3004	−	26.5011i	0.00579633	−	0.0100395i	0.868909	+	0.494972i	$0.164822\pi$
$192$		0			0
$193$	−2634.49	+	1521.02i	−0.982563	+	0.567283i	−0.903043	−	0.429550i	$-0.858672\pi$
$193$	−2634.49	+	1521.02i	−0.982563	+	0.567283i	−0.0795202	+	0.996833i	$0.525339\pi$
$194$		0			0
$195$	−687.969	+	667.999i	−0.252649	+	0.245315i
$196$		0			0
$197$	3612.02	−	2085.40i	1.30632	−	0.754205i	0.324842	−	0.945768i	$-0.394689\pi$
$197$	3612.02	−	2085.40i	1.30632	−	0.754205i	0.981480	+	0.191563i	$0.0613556\pi$
$198$		0			0
$199$	−2315.80	+	4011.08i	−0.824938	+	1.42883i	0.0770283	+	0.997029i	$0.475457\pi$
$199$	−2315.80	+	4011.08i	−0.824938	+	1.42883i	−0.901967	+	0.431806i	$0.857877\pi$
$200$		0			0
$201$	−201.582	−	116.383i	−0.0707387	−	0.0408410i
$202$		0			0
$203$		−	69.8558i		−	0.0241523i
$204$		0			0
$205$	−1348.10	−	2334.98i	−0.459295	−	0.795522i
$206$		0			0
$207$	−5168.47			−1.73543
$208$		0			0
$209$	1670.95			0.553023
$210$		0			0
$211$	−2701.82	−	4679.69i	−0.881522	−	1.52684i	−0.849649	−	0.527349i	$-0.823186\pi$
$211$	−2701.82	−	4679.69i	−0.881522	−	1.52684i	−0.0318728	−	0.999492i	$-0.510147\pi$
$212$		0			0
$213$			668.707i			0.215113i
$214$		0			0
$215$	5043.34	+	2911.77i	1.59978	+	0.923634i
$216$		0			0
$217$	45.8210	−	79.3644i	0.0143343	−	0.0248277i
$218$		0			0
$219$	−1095.20	+	632.313i	−0.337930	+	0.195104i
$220$		0			0
$221$	472.462	−	1664.79i	0.143806	−	0.506724i
$222$		0			0
$223$	4319.82	−	2494.05i	1.29720	−	0.748941i	0.317283	−	0.948331i	$-0.397229\pi$
$223$	4319.82	−	2494.05i	1.29720	−	0.748941i	0.979920	+	0.199390i	$0.0638962\pi$
$224$		0			0
$225$	−1618.29	+	2802.97i	−0.479494	+	0.830509i
$226$		0			0
$227$	2196.74	+	1268.29i	0.642303	+	0.370834i	0.785501	−	0.618860i	$-0.212405\pi$
$227$	2196.74	+	1268.29i	0.642303	+	0.370834i	−0.143198	+	0.989694i	$0.545739\pi$
$228$		0			0
$229$		−	747.558i		−	0.215721i	−0.994166	−	0.107860i	$-0.965600\pi$
$229$		−	747.558i		−	0.215721i	0.994166	−	0.107860i	$-0.0343999\pi$
$230$		0			0
$231$	−29.6499	−	51.3551i	−0.00844510	−	0.0146273i
$232$		0			0
$233$	−1661.81			−0.467248			−0.233624	−	0.972327i	$-0.575058\pi$
$233$	−1661.81			−0.467248			−0.233624	+	0.972327i	$0.575058\pi$
$234$		0			0
$235$	−3971.55			−1.10245
$236$		0			0
$237$	814.535	+	1410.82i	0.223248	+	0.386677i
$238$		0			0
$239$			1995.23i			0.540002i	0.962860	+	0.270001i	$0.0870240\pi$
$239$			1995.23i			0.540002i	−0.962860	+	0.270001i	$0.912976\pi$
$240$		0			0
$241$	1129.49	+	652.109i	0.301895	+	0.174299i	0.643294	−	0.765619i	$-0.277567\pi$
$241$	1129.49	+	652.109i	0.301895	+	0.174299i	−0.341399	+	0.939918i	$0.610901\pi$
$242$		0			0
$243$	1293.93	−	2241.16i	0.341588	−	0.591648i
$244$		0			0
$245$	−4702.78	+	2715.15i	−1.22632	+	0.708019i
$246$		0			0
$247$	1936.31	+	549.517i	0.498803	+	0.141559i
$248$		0			0
$249$	1143.97	−	660.471i	0.291149	−	0.168095i
$250$		0			0
$251$	2528.81	−	4380.03i	0.635926	−	1.10146i	−0.350393	−	0.936603i	$-0.613952\pi$
$251$	2528.81	−	4380.03i	0.635926	−	1.10146i	0.986318	−	0.164852i	$-0.0527148\pi$
$252$		0			0
$253$	6872.32	+	3967.74i	1.70774	+	0.985967i
$254$		0			0
$255$			755.321i			0.185490i
$256$		0			0
$257$	2225.65	+	3854.95i	0.540204	+	0.935661i	0.998892	+	0.0470638i	$0.0149864\pi$
$257$	2225.65	+	3854.95i	0.540204	+	0.935661i	−0.458687	+	0.888598i	$0.651680\pi$
$258$		0			0
$259$	305.863			0.0733798
$260$		0			0
$261$	−1495.07			−0.354569
$262$		0			0
$263$	−2287.94	−	3962.83i	−0.536428	−	0.929120i	−0.999093	−	0.0425868i	$-0.986440\pi$
$263$	−2287.94	−	3962.83i	−0.536428	−	0.929120i	0.462665	−	0.886533i	$-0.346893\pi$
$264$		0			0
$265$		−	2503.09i		−	0.580239i
$266$		0			0
$267$	1655.12	+	955.584i	0.379370	+	0.219029i
$268$		0			0
$269$	2705.66	−	4686.34i	0.613260	−	1.06220i	−0.377427	−	0.926039i	$-0.623191\pi$
$269$	2705.66	−	4686.34i	0.613260	−	1.06220i	0.990687	−	0.136159i	$-0.0434757\pi$
$270$		0			0
$271$	322.418	−	186.148i	0.0722712	−	0.0417258i	−0.463429	−	0.886134i	$-0.653381\pi$
$271$	322.418	−	186.148i	0.0722712	−	0.0417258i	0.535700	+	0.844408i	$0.320048\pi$
$272$		0			0
$273$	−17.4696	−	69.2615i	−0.00387292	−	0.0153549i
$274$		0			0
$275$	4303.57	−	2484.67i	0.943692	−	0.544841i
$276$		0			0
$277$	−2953.92	+	5116.35i	−0.640737	+	1.10979i	0.344532	+	0.938775i	$0.388038\pi$
$277$	−2953.92	+	5116.35i	−0.640737	+	1.10979i	−0.985269	+	0.171014i	$0.945296\pi$
$278$		0			0
$279$	−1698.57	−	980.673i	−0.364484	−	0.210435i
$280$		0			0
$281$		−	3598.50i		−	0.763946i	−0.924174	−	0.381973i	$-0.875245\pi$
$281$		−	3598.50i		−	0.763946i	0.924174	−	0.381973i	$-0.124755\pi$
$282$		0			0
$283$	1706.65	+	2956.00i	0.358480	+	0.620905i	0.987707	−	0.156316i	$-0.0499619\pi$
$283$	1706.65	+	2956.00i	0.358480	+	0.620905i	−0.629227	+	0.777221i	$0.716629\pi$
$284$		0			0
$285$	−878.509			−0.182591
$286$		0			0
$287$	200.843			0.0413079
$288$		0			0
$289$	1774.95	+	3074.30i	0.361275	+	0.625748i
$290$		0			0
$291$		−	450.732i		−	0.0907985i
$292$		0			0
$293$	−976.718	−	563.908i	−0.194746	−	0.112436i	0.399457	−	0.916752i	$-0.369199\pi$
$293$	−976.718	−	563.908i	−0.194746	−	0.112436i	−0.594202	+	0.804316i	$0.702532\pi$
$294$		0			0
$295$	−5343.14	+	9254.58i	−1.05454	+	1.82652i
$296$		0			0
$297$	−2270.05	+	1310.62i	−0.443508	+	0.256059i
$298$		0			0
$299$	6658.85	+	6857.92i	1.28793	+	1.32643i
$300$		0			0
$301$	−375.683	+	216.901i	−0.0719402	+	0.0415347i
$302$		0			0
$303$	−921.258	+	1595.67i	−0.174670	+	0.302537i
$304$		0			0
$305$	7988.32	+	4612.06i	1.49970	+	0.865854i
$306$		0			0
$307$			7338.57i			1.36428i	0.731221	+	0.682141i	$0.238951\pi$
$307$			7338.57i			1.36428i	−0.731221	+	0.682141i	$0.761049\pi$
$308$		0			0
$309$	960.121	+	1662.98i	0.176762	+	0.306160i
$310$		0			0
$311$	−7550.86			−1.37675			−0.688376	−	0.725354i	$-0.741676\pi$
$311$	−7550.86			−1.37675			−0.688376	+	0.725354i	$0.741676\pi$
$312$		0			0
$313$	−1262.18			−0.227932			−0.113966	−	0.993485i	$-0.536356\pi$
$313$	−1262.18			−0.227932			−0.113966	+	0.993485i	$0.536356\pi$
$314$		0			0
$315$	−238.541	−	413.165i	−0.0426675	−	0.0739023i
$316$		0			0
$317$		−	4085.51i		−	0.723865i	−0.932204	−	0.361933i	$-0.882117\pi$
$317$		−	4085.51i		−	0.723865i	0.932204	−	0.361933i	$-0.117883\pi$
$318$		0			0
$319$	1987.94	+	1147.74i	0.348913	+	0.201445i
$320$		0			0
$321$	−747.594	+	1294.87i	−0.129990	+	0.225148i
$322$		0			0
$323$	1373.02	−	792.711i	0.236522	−	0.136556i
$324$		0			0
$325$	5804.14	−	1463.96i	0.990633	−	0.249864i
$326$		0			0
$327$	1782.17	−	1028.94i	0.301390	−	0.174007i
$328$		0			0
$329$	147.922	−	256.209i	0.0247879	−	0.0429339i
$330$		0			0
$331$	3910.68	+	2257.83i	0.649397	+	0.374930i	0.788225	−	0.615387i	$-0.211000\pi$
$331$	3910.68	+	2257.83i	0.649397	+	0.374930i	−0.138828	+	0.990316i	$0.544334\pi$
$332$		0			0
$333$		−	6546.14i		−	1.07726i
$334$		0			0
$335$	1437.61	+	2490.02i	0.234463	+	0.406102i
$336$		0			0
$337$	−123.448			−0.0199545			−0.00997723	−	0.999950i	$-0.503176\pi$
$337$	−123.448			−0.0199545			−0.00997723	+	0.999950i	$0.503176\pi$
$338$		0			0
$339$	−482.233			−0.0772605
$340$		0			0
$341$	1505.69	+	2607.93i	0.239113	+	0.414156i
$342$		0			0
$343$		−	810.677i		−	0.127616i
$344$		0			0
$345$	−3613.16	−	2086.06i	−0.563844	−	0.325535i
$346$		0			0
$347$	1226.45	−	2124.28i	0.189739	−	0.328637i	−0.755424	−	0.655236i	$-0.772569\pi$
$347$	1226.45	−	2124.28i	0.189739	−	0.328637i	0.945163	+	0.326599i	$0.105903\pi$
$348$		0			0
$349$	−7602.22	+	4389.14i	−1.16601	+	0.673196i	−0.952737	−	0.303797i	$-0.901746\pi$
$349$	−7602.22	+	4389.14i	−1.16601	+	0.673196i	−0.213273	+	0.976993i	$0.568412\pi$
$350$		0			0
$351$	−3061.57	+	772.210i	−0.465569	+	0.117429i
$352$		0			0
$353$	3395.57	−	1960.43i	0.511977	−	0.295590i	−0.221669	−	0.975122i	$-0.571150\pi$
$353$	3395.57	−	1960.43i	0.511977	−	0.295590i	0.733646	+	0.679532i	$0.237817\pi$
$354$		0			0
$355$	4130.07	−	7153.48i	0.617468	−	1.06949i
$356$		0			0
$357$	−48.7265	−	28.1323i	−0.00722376	−	0.00417064i
$358$		0			0
$359$		−	9182.02i		−	1.34988i	−0.737870	−	0.674942i	$-0.764168\pi$
$359$		−	9182.02i		−	1.34988i	0.737870	−	0.674942i	$-0.235832\pi$
$360$		0			0
$361$	−2507.50	−	4343.12i	−0.365578	−	0.633200i
$362$		0			0
$363$	235.688			0.0340782
$364$		0			0
$365$	15621.2			2.24013
$366$		0			0
$367$	−4207.89	−	7288.28i	−0.598502	−	1.03664i	−0.993042	−	0.117757i	$-0.962430\pi$
$367$	−4207.89	−	7288.28i	−0.598502	−	1.03664i	0.394541	−	0.918878i	$-0.370904\pi$
$368$		0			0
$369$		−	4298.48i		−	0.606423i
$370$		0			0
$371$	161.477	+	93.2286i	0.0225969	+	0.0130463i
$372$		0			0
$373$	−6212.22	+	10759.9i	−0.862350	+	1.49363i	0.00730479	+	0.999973i	$0.497675\pi$
$373$	−6212.22	+	10759.9i	−0.862350	+	1.49363i	−0.869655	+	0.493661i	$0.835659\pi$
$374$		0			0
$375$	−47.9664	+	27.6934i	−0.00660526	+	0.00381355i
$376$		0			0
$377$	1926.19	+	1983.77i	0.263140	+	0.271007i
$378$		0			0
$379$	−1036.75	+	598.565i	−0.140512	+	0.0811246i	−0.568608	−	0.822609i	$-0.692518\pi$
$379$	−1036.75	+	598.565i	−0.140512	+	0.0811246i	0.428096	+	0.903733i	$0.359185\pi$
$380$		0			0
$381$	417.801	−	723.653i	0.0561801	−	0.0973068i
$382$		0			0
$383$	−778.326	−	449.367i	−0.103840	−	0.0599519i	0.447181	−	0.894444i	$-0.352428\pi$
$383$	−778.326	−	449.367i	−0.103840	−	0.0599519i	−0.551020	+	0.834492i	$0.685761\pi$
$384$		0			0
$385$			732.494i			0.0969646i
$386$		0			0
$387$	4642.16	+	8040.46i	0.609753	+	1.05612i
$388$		0			0
$389$	−1927.09			−0.251175			−0.125588	−	0.992083i	$-0.540082\pi$
$389$	−1927.09			−0.251175			−0.125588	+	0.992083i	$0.540082\pi$
$390$		0			0
$391$	7529.31			0.973846
$392$		0			0
$393$	394.425	+	683.164i	0.0506262	+	0.0876872i
$394$		0			0
$395$		−	20122.9i		−	2.56328i
$396$		0			0
$397$	−1880.41	−	1085.66i	−0.237721	−	0.137248i	0.376408	−	0.926454i	$-0.377159\pi$
$397$	−1880.41	−	1085.66i	−0.237721	−	0.137248i	−0.614129	+	0.789206i	$0.710493\pi$
$398$		0			0
$399$	32.7205	−	56.6736i	0.00410545	−	0.00711084i
$400$		0			0
$401$	−9051.43	+	5225.85i	−1.12720	+	0.650789i	−0.943229	−	0.332143i	$-0.892228\pi$
$401$	−9051.43	+	5225.85i	−1.12720	+	0.650789i	−0.183970	+	0.982932i	$0.558895\pi$
$402$		0			0
$403$	887.146	+	3517.26i	0.109657	+	0.434757i
$404$		0			0
$405$	−8227.03	+	4749.88i	−1.00939	+	0.582774i
$406$		0			0
$407$	−5025.35	+	8704.17i	−0.612033	+	1.06007i
$408$		0			0
$409$	−3741.15	−	2159.95i	−0.452293	−	0.261132i	0.256505	−	0.966543i	$-0.417429\pi$
$409$	−3741.15	−	2159.95i	−0.452293	−	0.261132i	−0.708798	+	0.705411i	$0.750762\pi$
$410$		0			0
$411$			268.714i			0.0322499i
$412$		0			0
$413$	−398.015	−	689.383i	−0.0474214	−	0.0821364i
$414$		0			0
$415$	−16316.8			−1.93002
$416$		0			0
$417$	−475.639			−0.0558564
$418$		0			0
$419$	−1071.86	−	1856.51i	−0.124973	−	0.216460i	0.796749	−	0.604310i	$-0.206551\pi$
$419$	−1071.86	−	1856.51i	−0.124973	−	0.216460i	−0.921722	+	0.387850i	$0.873218\pi$
$420$		0			0
$421$		−	11308.5i		−	1.30912i	−0.756009	−	0.654561i	$-0.772853\pi$
$421$		−	11308.5i		−	1.30912i	0.756009	−	0.654561i	$-0.227147\pi$
$422$		0			0
$423$	−5483.45	−	3165.87i	−0.630294	−	0.363900i
$424$		0			0
$425$	2357.49	−	4083.30i	0.269071	−	0.466045i
$426$		0			0
$427$	−595.057	+	343.557i	−0.0674399	+	0.0389365i
$428$		0			0
$429$	2258.06	+	640.829i	0.254126	+	0.0721200i
$430$		0			0
$431$	−7167.83	+	4138.35i	−0.801072	+	0.462499i	−0.843846	−	0.536585i	$-0.819714\pi$
$431$	−7167.83	+	4138.35i	−0.801072	+	0.462499i	0.0427737	+	0.999085i	$0.486381\pi$
$432$		0			0
$433$	156.904	−	271.766i	0.0174142	−	0.0301622i	−0.857187	−	0.515005i	$-0.827790\pi$
$433$	156.904	−	271.766i	0.0174142	−	0.0301622i	0.874601	+	0.484843i	$0.161123\pi$
$434$		0			0
$435$	−1045.17	−	603.429i	−0.115200	−	0.0665108i
$436$		0			0
$437$			8757.30i			0.958623i
$438$		0			0
$439$	−2399.23	−	4155.59i	−0.260840	−	0.451789i	0.705625	−	0.708585i	$-0.250666\pi$
$439$	−2399.23	−	4155.59i	−0.260840	−	0.451789i	−0.966465	+	0.256797i	$0.917333\pi$
$440$		0			0
$441$	−8657.38			−0.934822
$442$		0			0
$443$	871.652			0.0934841			0.0467420	−	0.998907i	$-0.485116\pi$
$443$	871.652			0.0934841			0.0467420	+	0.998907i	$0.485116\pi$
$444$		0			0
$445$	−11803.8	−	20444.7i	−1.25742	−	2.17792i
$446$		0			0
$447$		−	1274.33i		−	0.134840i
$448$		0			0
$449$	6902.27	+	3985.03i	0.725475	+	0.418853i	0.816765	−	0.576971i	$-0.195765\pi$
$449$	6902.27	+	3985.03i	0.725475	+	0.418853i	−0.0912892	+	0.995824i	$0.529099\pi$
$450$		0			0
$451$	−3299.87	+	5715.53i	−0.344533	+	0.596749i
$452$		0			0
$453$	3182.09	−	1837.18i	0.330039	−	0.190548i
$454$		0			0
$455$	−240.892	+	848.821i	−0.0248202	+	0.0874579i
$456$		0			0
$457$	7830.03	−	4520.67i	0.801473	−	0.462731i	−0.0425127	−	0.999096i	$-0.513536\pi$
$457$	7830.03	−	4520.67i	0.801473	−	0.462731i	0.843986	+	0.536365i	$0.180203\pi$
$458$		0			0
$459$	−1243.53	+	2153.86i	−0.126456	+	0.219028i
$460$		0			0
$461$	−1989.15	−	1148.44i	−0.200963	−	0.116026i	0.396142	−	0.918189i	$-0.370349\pi$
$461$	−1989.15	−	1148.44i	−0.200963	−	0.116026i	−0.597104	+	0.802163i	$0.703682\pi$
$462$		0			0
$463$			10243.3i			1.02817i	0.857738	+	0.514087i	$0.171869\pi$
$463$			10243.3i			1.02817i	−0.857738	+	0.514087i	$0.828131\pi$
$464$		0			0
$465$	−791.624	−	1371.13i	−0.0789477	−	0.136741i
$466$		0			0
$467$	11561.2			1.14558			0.572792	−	0.819701i	$-0.305860\pi$
$467$	11561.2			1.14558			0.572792	+	0.819701i	$0.305860\pi$
$468$		0			0
$469$	−214.178			−0.0210871
$470$		0			0
$471$	−609.909	−	1056.39i	−0.0596669	−	0.103346i
$472$		0			0
$473$		−	14254.8i		−	1.38570i
$474$		0			0
$475$	4749.26	+	2741.99i	0.458760	+	0.264865i
$476$		0			0
$477$	1995.30	−	3455.96i	0.191527	−	0.331735i
$478$		0			0
$479$	−3471.88	+	2004.49i	−0.331178	+	0.191206i	−0.656364	−	0.754444i	$-0.727906\pi$
$479$	−3471.88	+	2004.49i	−0.331178	+	0.191206i	0.325186	+	0.945650i	$0.394573\pi$
$480$		0			0
$481$	−8685.93	+	8433.80i	−0.823377	+	0.799476i
$482$		0			0
$483$	269.148	−	155.393i	0.0253554	−	0.0146389i
$484$		0			0
$485$	−2783.81	+	4821.70i	−0.260632	+	0.451427i
$486$		0			0
$487$	−2566.87	−	1481.98i	−0.238842	−	0.137896i	0.375802	−	0.926700i	$-0.377367\pi$
$487$	−2566.87	−	1481.98i	−0.238842	−	0.137896i	−0.614644	+	0.788804i	$0.710701\pi$
$488$		0			0
$489$		−	3462.77i		−	0.320228i
$490$		0			0
$491$	−8977.60	−	15549.7i	−0.825160	−	1.42922i	−0.901797	−	0.432159i	$-0.857752\pi$
$491$	−8977.60	−	15549.7i	−0.825160	−	1.42922i	0.0766377	−	0.997059i	$-0.475582\pi$
$492$		0			0
$493$	2177.99			0.198969
$494$		0			0
$495$	15677.0			1.42349
$496$		0			0
$497$	307.653	+	532.870i	0.0277668	+	0.0480935i
$498$		0			0
$499$		−	15327.6i		−	1.37507i	−0.726153	−	0.687533i	$-0.758694\pi$
$499$		−	15327.6i		−	1.37507i	0.726153	−	0.687533i	$-0.241306\pi$
$500$		0			0
$501$	2012.63	+	1161.99i	0.179476	+	0.103621i
$502$		0			0
$503$	−6413.37	+	11108.3i	−0.568505	+	0.984679i	0.428209	+	0.903679i	$0.359145\pi$
$503$	−6413.37	+	11108.3i	−0.568505	+	0.984679i	−0.996714	+	0.0809994i	$0.974189\pi$
$504$		0			0
$505$	19710.3	−	11379.8i	1.73683	−	1.00276i
$506$		0			0
$507$	2405.91	+	1485.20i	0.210750	+	0.130098i
$508$		0			0
$509$	−2082.85	+	1202.54i	−0.181377	+	0.104718i	−0.587939	−	0.808905i	$-0.700061\pi$
$509$	−2082.85	+	1202.54i	−0.181377	+	0.104718i	0.406563	+	0.913623i	$0.366727\pi$
$510$		0			0
$511$	−581.818	+	1007.74i	−0.0503681	+	0.0872401i
$512$		0			0
$513$	−2505.15	−	1446.35i	−0.215604	−	0.124479i
$514$		0			0
$515$		−	23719.6i		−	2.02954i
$516$		0			0
$517$	4860.76	+	8419.08i	0.413493	+	0.716191i
$518$		0			0
$519$	455.631			0.0385356
$520$		0			0
$521$	13258.4			1.11490			0.557450	−	0.830210i	$-0.311780\pi$
$521$	13258.4			1.11490			0.557450	+	0.830210i	$0.311780\pi$
$522$		0			0
$523$	2257.47	+	3910.05i	0.188742	+	0.326911i	0.944831	−	0.327558i	$-0.106226\pi$
$523$	2257.47	+	3910.05i	0.188742	+	0.326911i	−0.756089	+	0.654469i	$0.772892\pi$
$524$		0			0
$525$		−	194.619i		−	0.0161788i
$526$		0			0
$527$	2474.45	+	1428.62i	0.204532	+	0.118087i
$528$		0			0
$529$	−14711.1	+	25480.4i	−1.20910	+	2.09422i
$530$		0			0
$531$	−14754.3	+	8518.42i	−1.20581	+	0.696173i
$532$		0			0
$533$	−5703.56	+	5538.00i	−0.463506	+	0.450051i
$534$		0			0
$535$	15994.8	−	9234.58i	1.29255	−	0.746254i
$536$		0			0
$537$	−1669.27	+	2891.27i	−0.134142	+	0.232341i
$538$		0			0
$539$	11511.4	+	6646.11i	0.919910	+	0.531110i
$540$		0			0
$541$		−	7456.65i		−	0.592581i	−0.955098	−	0.296291i	$-0.904250\pi$
$541$		−	7456.65i		−	0.592581i	0.955098	−	0.296291i	$-0.0957497\pi$
$542$		0			0
$543$	−1505.06	−	2606.83i	−0.118947	−	0.206022i
$544$		0			0
$545$	−25419.7			−1.99791
$546$		0			0
$547$	6132.43			0.479349			0.239674	−	0.970853i	$-0.422959\pi$
$547$	6132.43			0.479349			0.239674	+	0.970853i	$0.422959\pi$
$548$		0			0
$549$	7352.88	+	12735.6i	0.571609	+	0.990055i
$550$		0			0
$551$			2533.20i			0.195859i
$552$		0			0
$553$	1298.15	+	749.488i	0.0998246	+	0.0576338i
$554$		0			0
$555$	2642.11	−	4576.26i	0.202074	−	0.350003i
$556$		0			0
$557$	9031.62	−	5214.41i	0.687042	−	0.396664i	−0.115461	−	0.993312i	$-0.536835\pi$
$557$	9031.62	−	5214.41i	0.687042	−	0.396664i	0.802503	+	0.596648i	$0.203501\pi$
$558$		0			0
$559$	4687.92	−	16518.6i	0.354701	−	1.24984i
$560$		0			0
$561$	1601.16	−	924.433i	0.120501	−	0.0695714i
$562$		0			0
$563$	−11392.3	+	19732.0i	−0.852801	+	1.47709i	0.0258691	+	0.999665i	$0.491765\pi$
$563$	−11392.3	+	19732.0i	−0.852801	+	1.47709i	−0.878670	+	0.477429i	$0.841569\pi$
$564$		0			0
$565$	5158.68	+	2978.37i	0.384119	+	0.221771i
$566$		0			0
$567$		−	707.646i		−	0.0524133i
$568$		0			0
$569$	6439.75	+	11154.0i	0.474461	+	0.821791i	0.999572	−	0.0292427i	$-0.00930957\pi$
$569$	6439.75	+	11154.0i	0.474461	+	0.821791i	−0.525111	+	0.851034i	$0.675976\pi$
$570$		0			0
$571$	−6071.36			−0.444971			−0.222485	−	0.974936i	$-0.571417\pi$
$571$	−6071.36			−0.444971			−0.222485	+	0.974936i	$0.571417\pi$
$572$		0			0
$573$	39.3813			0.00287117
$574$		0			0
$575$	13022.0	+	22554.7i	0.944440	+	1.63582i
$576$		0			0
$577$			2429.06i			0.175257i	0.996153	+	0.0876283i	$0.0279288\pi$
$577$			2429.06i			0.175257i	−0.996153	+	0.0876283i	$0.972071\pi$
$578$		0			0
$579$	−3390.42	−	1957.46i	−0.243353	−	0.140500i
$580$		0			0
$581$	607.727	−	1052.61i	0.0433955	−	0.0751631i
$582$		0			0
$583$	−5306.16	+	3063.51i	−0.376944	+	0.217629i
$584$		0			0
$585$	18166.7	+	5155.63i	1.28393	+	0.364375i
$586$		0			0
$587$	−12664.9	+	7312.09i	−0.890523	+	0.514144i	−0.874114	−	0.485722i	$-0.838557\pi$
$587$	−12664.9	+	7312.09i	−0.890523	+	0.514144i	−0.0164096	+	0.999865i	$0.505224\pi$
$588$		0			0
$589$	−1661.62	+	2878.01i	−0.116241	+	0.201335i
$590$		0			0
$591$	4648.44	+	2683.78i	0.323538	+	0.186795i
$592$		0			0
$593$			15282.4i			1.05830i	0.848529	+	0.529149i	$0.177489\pi$
$593$			15282.4i			1.05830i	−0.848529	+	0.529149i	$0.822511\pi$
$594$		0			0
$595$	347.501	+	601.890i	0.0239431	+	0.0414707i
$596$		0			0
$597$	−5960.58			−0.408627
$598$		0			0
$599$	11067.6			0.754940			0.377470	−	0.926022i	$-0.376794\pi$
$599$	11067.6			0.754940			0.377470	+	0.926022i	$0.376794\pi$
$600$		0			0
$601$	−9143.59	−	15837.2i	−0.620590	−	1.07489i	−0.989376	−	0.145379i	$-0.953560\pi$
$601$	−9143.59	−	15837.2i	−0.620590	−	1.07489i	0.368786	−	0.929514i	$-0.379774\pi$
$602$		0			0
$603$			4583.90i			0.309570i
$604$		0			0
$605$	−2521.27	−	1455.65i	−0.169428	−	0.0978194i
$606$		0			0
$607$	1381.29	−	2392.46i	0.0923638	−	0.159979i	−0.816142	−	0.577852i	$-0.803891\pi$
$607$	1381.29	−	2392.46i	0.0923638	−	0.159979i	0.908505	+	0.417873i	$0.137224\pi$
$608$		0			0
$609$	77.8557	−	44.9500i	0.00518042	−	0.00299091i
$610$		0			0
$611$	2863.94	+	11354.6i	0.189628	+	0.751816i
$612$		0			0
$613$	22586.0	−	13040.0i	1.48816	−	0.859187i	0.488248	−	0.872705i	$-0.337636\pi$
$613$	22586.0	−	13040.0i	1.48816	−	0.859187i	0.999909	+	0.0135178i	$0.00430298\pi$
$614$		0			0
$615$	1734.92	−	3004.97i	0.113754	−	0.197028i
$616$		0			0
$617$	−7941.30	−	4584.91i	−0.518160	−	0.299160i	0.218021	−	0.975944i	$-0.430040\pi$
$617$	−7941.30	−	4584.91i	−0.518160	−	0.299160i	−0.736182	+	0.676784i	$0.763373\pi$
$618$		0			0
$619$			17765.0i			1.15353i	0.816909	+	0.576767i	$0.195686\pi$
$619$			17765.0i			1.15353i	−0.816909	+	0.576767i	$0.804314\pi$
$620$		0			0
$621$	−6868.83	−	11897.2i	−0.443859	−	0.768787i
$622$		0			0
$623$	1758.55			0.113089
$624$		0			0
$625$	−15279.3			−0.977872
$626$		0			0
$627$	1075.20	+	1862.30i	0.0684840	+	0.118618i
$628$		0			0
$629$			9536.28i			0.604509i
$630$		0			0
$631$	−25243.7	−	14574.4i	−1.59261	−	0.919492i	−0.992858	−	0.119303i	$-0.961934\pi$
$631$	−25243.7	−	14574.4i	−1.59261	−	0.919492i	−0.599748	−	0.800189i	$-0.704733\pi$
$632$		0			0
$633$	3477.08	−	6022.47i	0.218328	−	0.378154i
$634$		0			0
$635$	−8938.86	+	5160.85i	−0.558626	+	0.322523i
$636$		0			0
$637$	11153.8	+	11487.3i	0.693769	+	0.714510i
$638$		0			0
$639$	11404.6	−	6584.45i	0.706040	−	0.407632i
$640$		0			0
$641$	12306.7	−	21315.8i	0.758323	−	1.31345i	−0.185383	−	0.982666i	$-0.559352\pi$
$641$	12306.7	−	21315.8i	0.758323	−	1.31345i	0.943705	−	0.330787i	$-0.107314\pi$
$642$		0			0
$643$	23498.4	+	13566.8i	1.44119	+	0.832071i	0.997929	−	0.0643213i	$-0.0204882\pi$
$643$	23498.4	+	13566.8i	1.44119	+	0.832071i	0.443261	+	0.896393i	$0.353822\pi$
$644$		0			0
$645$			7494.54i			0.457515i
$646$		0			0
$647$	11867.2	+	20554.5i	0.721092	+	1.24897i	0.960562	+	0.278064i	$0.0896929\pi$
$647$	11867.2	+	20554.5i	0.721092	+	1.24897i	−0.239470	+	0.970904i	$0.576974\pi$
$648$		0			0
$649$	26157.7			1.58210
$650$		0			0
$651$	117.938			0.00710037
$652$		0			0
$653$	−9688.22	−	16780.5i	−0.580596	−	1.00562i	−0.995409	−	0.0957153i	$-0.969486\pi$
$653$	−9688.22	−	16780.5i	−0.580596	−	1.00562i	0.414813	−	0.909907i	$-0.363847\pi$
$654$		0			0
$655$		−	9744.19i		−	0.581278i
$656$		0			0
$657$	21567.8	+	12452.2i	1.28073	+	0.739432i
$658$		0			0
$659$	5635.78	−	9761.46i	0.333139	−	0.577014i	−0.649986	−	0.759946i	$-0.725225\pi$
$659$	5635.78	−	9761.46i	0.333139	−	0.577014i	0.983126	+	0.182932i	$0.0585587\pi$
$660$		0			0
$661$	−7260.87	+	4192.07i	−0.427254	+	0.246675i	−0.698176	−	0.715926i	$-0.746005\pi$
$661$	−7260.87	+	4192.07i	−0.427254	+	0.246675i	0.270922	+	0.962601i	$0.412672\pi$
$662$		0			0
$663$	2159.46	−	544.672i	0.126495	−	0.0319055i
$664$		0			0
$665$	−700.055	+	404.177i	−0.0408225	+	0.0235689i
$666$		0			0
$667$	−6015.20	+	10418.6i	−0.349190	+	0.604814i
$668$		0			0
$669$	5559.33	+	3209.68i	0.321280	+	0.185491i
$670$		0			0
$671$		−	22578.7i		−	1.29902i
$672$		0			0
$673$	2703.65	+	4682.86i	0.154856	+	0.268219i	0.933007	−	0.359859i	$-0.117175\pi$
$673$	2703.65	+	4682.86i	0.154856	+	0.268219i	−0.778151	+	0.628078i	$0.783842\pi$
$674$		0			0
$675$	−8602.77			−0.490549
$676$		0			0
$677$	−10090.8			−0.572854			−0.286427	−	0.958102i	$-0.592468\pi$
$677$	−10090.8			−0.572854			−0.286427	+	0.958102i	$0.592468\pi$
$678$		0			0
$679$	−207.369	−	359.173i	−0.0117203	−	0.0203002i
$680$		0			0
$681$			3264.41i			0.183690i
$682$		0			0
$683$	11461.5	+	6617.28i	0.642109	+	0.370722i	0.785427	−	0.618955i	$-0.212444\pi$
$683$	11461.5	+	6617.28i	0.642109	+	0.370722i	−0.143317	+	0.989677i	$0.545777\pi$
$684$		0			0
$685$	1659.63	−	2874.57i	0.0925713	−	0.160338i
$686$		0			0
$687$	833.169	−	481.030i	0.0462698	−	0.0267139i
$688$		0			0
$689$	−7156.31	+	1805.01i	−0.395695	+	0.0998046i
$690$		0			0
$691$	2406.47	−	1389.37i	0.132484	−	0.0764895i	−0.432293	−	0.901733i	$-0.642295\pi$
$691$	2406.47	−	1389.37i	0.132484	−	0.0764895i	0.564777	+	0.825244i	$0.308962\pi$
$692$		0			0
$693$	−583.898	+	1011.34i	−0.0320064	+	0.0554367i
$694$		0			0
$695$	5088.14	+	2937.64i	0.277704	+	0.160332i
$696$		0			0
$697$			6261.93i			0.340298i
$698$		0			0
$699$	−1069.32	−	1852.12i	−0.0578619	−	0.100220i
$700$		0			0
$701$	−19737.6			−1.06345			−0.531725	−	0.846917i	$-0.678456\pi$
$701$	−19737.6			−1.06345			−0.531725	+	0.846917i	$0.678456\pi$
$702$		0			0
$703$	−11091.6			−0.595060
$704$		0			0
$705$	−2555.57	−	4426.38i	−0.136522	−	0.236464i
$706$		0			0
$707$			1695.38i			0.0901856i
$708$		0			0
$709$	5242.69	+	3026.87i	0.277706	+	0.160333i	0.632384	−	0.774655i	$-0.282076\pi$
$709$	5242.69	+	3026.87i	0.277706	+	0.160333i	−0.354679	+	0.934988i	$0.615410\pi$
$710$		0			0
$711$	16040.7	−	27783.3i	0.846096	−	1.46548i
$712$		0			0
$713$	−13667.9	+	7891.19i	−0.717908	+	0.414485i
$714$		0			0
$715$	−20197.7	−	20801.5i	−1.05643	−	1.08802i
$716$		0			0
$717$	−2223.72	+	1283.87i	−0.115825	+	0.0668715i
$718$		0			0
$719$	6878.79	−	11914.4i	0.356795	−	0.617987i	−0.630628	−	0.776085i	$-0.717203\pi$
$719$	6878.79	−	11914.4i	0.356795	−	0.617987i	0.987423	+	0.158098i	$0.0505361\pi$
$720$		0			0
$721$	1530.18	+	883.448i	0.0790385	+	0.0456329i
$722$		0			0
$723$			1678.45i			0.0863377i
$724$		0			0
$725$	3766.83	+	6524.34i	0.192961	+	0.334218i
$726$		0			0
$727$	18384.8			0.937903			0.468951	−	0.883224i	$-0.344632\pi$
$727$	18384.8			0.937903			0.468951	+	0.883224i	$0.344632\pi$
$728$		0			0
$729$	−12804.5			−0.650537
$730$		0			0
$731$	−6762.60	−	11713.2i	−0.342166	−	0.592650i
$732$		0			0
$733$		−	7848.53i		−	0.395487i	−0.980254	−	0.197744i	$-0.936639\pi$
$733$		−	7848.53i		−	0.395487i	0.980254	−	0.197744i	$-0.0633613\pi$
$734$		0			0
$735$	−6052.18	−	3494.23i	−0.303725	−	0.175356i
$736$		0			0
$737$	3518.97	−	6095.04i	0.175879	−	0.304632i
$738$		0			0
$739$	−13925.8	+	8040.07i	−0.693193	+	0.400215i	−0.804807	−	0.593537i	$-0.797731\pi$
$739$	−13925.8	+	8040.07i	−0.693193	+	0.400215i	0.111614	+	0.993752i	$0.464398\pi$
$740$		0			0
$741$	633.505	+	2511.65i	0.0314067	+	0.124518i
$742$		0			0
$743$	−8126.52	+	4691.85i	−0.401256	+	0.231665i	−0.687026	−	0.726633i	$-0.741084\pi$
$743$	−8126.52	+	4691.85i	−0.401256	+	0.231665i	0.285770	+	0.958298i	$0.407751\pi$
$744$		0			0
$745$	−7870.51	+	13632.1i	−0.387051	+	0.670393i
$746$		0			0
$747$	−22528.3	−	13006.7i	−1.10344	−	0.637069i
$748$		0			0
$749$			1375.79i			0.0671163i
$750$		0			0
$751$	4055.29	+	7023.97i	0.197043	+	0.341289i	0.947568	−	0.319553i	$-0.103533\pi$
$751$	4055.29	+	7023.97i	0.197043	+	0.341289i	−0.750525	+	0.660842i	$0.770199\pi$
$752$		0			0
$753$	6508.85			0.315001
$754$		0			0
$755$	−45387.3			−2.18783
$756$		0			0
$757$	12990.3	+	22499.9i	0.623701	+	1.08028i	0.988790	+	0.149310i	$0.0477051\pi$
$757$	12990.3	+	22499.9i	0.623701	+	1.08028i	−0.365089	+	0.930973i	$0.618962\pi$
$758$		0			0
$759$			10212.5i			0.488391i
$760$		0			0
$761$	−26638.1	−	15379.5i	−1.26890	−	0.732598i	−0.294118	−	0.955769i	$-0.595026\pi$
$761$	−26638.1	−	15379.5i	−1.26890	−	0.732598i	−0.974779	+	0.223171i	$0.928359\pi$
$762$		0			0
$763$	946.770	−	1639.85i	0.0449219	−	0.0778069i
$764$		0			0
$765$	12881.8	−	7437.30i	0.608813	−	0.351498i
$766$		0			0
$767$	30311.8	+	8602.39i	1.42698	+	0.404973i
$768$		0			0
$769$	−16156.1	+	9327.72i	−0.757611	+	0.437407i	−0.828437	−	0.560082i	$-0.810770\pi$
$769$	−16156.1	+	9327.72i	−0.757611	+	0.437407i	0.0708261	+	0.997489i	$0.477436\pi$
$770$		0			0
$771$	−2864.28	+	4961.08i	−0.133793	+	0.231736i
$772$		0			0
$773$	−14683.6	−	8477.58i	−0.683225	−	0.394460i	0.117844	−	0.993032i	$-0.462402\pi$
$773$	−14683.6	−	8477.58i	−0.683225	−	0.394460i	−0.801069	+	0.598572i	$0.795735\pi$
$774$		0			0
$775$			9883.22i			0.458085i
$776$		0			0
$777$	196.813	+	340.890i	0.00908704	+	0.0157392i
$778$		0			0
$779$	−7283.22			−0.334979
$780$		0			0
$781$	−20219.1			−0.926370
$782$		0			0
$783$	−1986.93	−	3441.46i	−0.0906859	−	0.157073i
$784$		0			0
$785$			15067.7i			0.685081i
$786$		0			0
$787$	23675.8	+	13669.2i	1.07236	+	0.619129i	0.928826	−	0.370516i	$-0.120819\pi$
$787$	23675.8	+	13669.2i	1.07236	+	0.619129i	0.143537	+	0.989645i	$0.454152\pi$
$788$		0			0
$789$	2944.44	−	5099.91i	0.132858	−	0.230116i
$790$		0			0
$791$	−384.275	+	221.861i	−0.0172734	+	0.00997280i
$792$		0			0
$793$	7425.36	−	26164.4i	0.332512	−	1.17166i
$794$		0			0
$795$	2789.74	−	1610.66i	0.124455	−	0.0718543i
$796$		0			0
$797$	8686.11	−	15044.8i	0.386045	−	0.668649i	−0.605869	−	0.795565i	$-0.707174\pi$
$797$	8686.11	−	15044.8i	0.386045	−	0.668649i	0.991914	+	0.126915i	$0.0405077\pi$
$798$		0			0
$799$	7988.16	+	4611.97i	0.353693	+	0.204205i
$800$		0			0
$801$		−	37636.8i		−	1.66021i
$802$		0			0
$803$	−19118.7	−	33114.5i	−0.840203	−	1.45527i
$804$		0			0
$805$	−3838.94			−0.168081
$806$		0			0
$807$	6964.03			0.303774
$808$		0			0
$809$	−12746.3	−	22077.3i	−0.553939	−	0.959450i	−0.997985	−	0.0634464i	$-0.979791\pi$
$809$	−12746.3	−	22077.3i	−0.553939	−	0.959450i	0.444046	−	0.896004i	$-0.353543\pi$
$810$		0			0
$811$			7742.96i			0.335255i	0.985850	+	0.167628i	$0.0536106\pi$
$811$			7742.96i			0.335255i	−0.985850	+	0.167628i	$0.946389\pi$
$812$		0			0
$813$	414.932	+	239.561i	0.0178995	+	0.0103343i
$814$		0			0
$815$	−21386.7	+	37042.9i	−0.919196	+	1.59209i
$816$		0			0
$817$	13623.5	−	7865.54i	0.583386	−	0.336818i
$818$		0			0
$819$	−1009.22	+	979.926i	−0.0430587	+	0.0418088i
$820$		0			0
$821$	11683.2	−	6745.28i	0.496645	−	0.286738i	−0.230682	−	0.973029i	$-0.574096\pi$
$821$	11683.2	−	6745.28i	0.496645	−	0.286738i	0.727327	+	0.686291i	$0.240762\pi$
$822$		0			0
$823$	16550.2	−	28665.7i	0.700975	−	1.21412i	−0.267149	−	0.963655i	$-0.586082\pi$
$823$	16550.2	−	28665.7i	0.700975	−	1.21412i	0.968124	−	0.250470i	$-0.0805850\pi$
$824$		0			0
$825$	5538.43	+	3197.61i	0.233725	+	0.134941i
$826$		0			0
$827$		−	5211.34i		−	0.219125i	−0.993980	−	0.109562i	$-0.965055\pi$
$827$		−	5211.34i		−	0.219125i	0.993980	−	0.109562i	$-0.0349449\pi$
$828$		0			0
$829$	7049.97	+	12210.9i	0.295363	+	0.511583i	0.975069	−	0.221901i	$-0.0712262\pi$
$829$	7049.97	+	12210.9i	0.295363	+	0.511583i	−0.679707	+	0.733484i	$0.737893\pi$
$830$		0			0
$831$	−7603.03			−0.317384
$832$		0			0
$833$	12611.9			0.524581
$834$		0			0
$835$	−14353.4	−	24860.8i	−0.594874	−	1.03035i
$836$		0			0
$837$		−	5213.21i		−	0.215287i
$838$		0			0
$839$	−6335.36	−	3657.72i	−0.260693	−	0.150511i	0.363958	−	0.931415i	$-0.381425\pi$
$839$	−6335.36	−	3657.72i	−0.260693	−	0.150511i	−0.624650	+	0.780904i	$0.714759\pi$
$840$		0			0
$841$	10454.5	−	18107.7i	0.428656	−	0.742454i
$842$		0			0
$843$	4010.61	−	2315.52i	0.163858	−	0.0946037i
$844$		0			0
$845$	−16564.3	−	30747.2i	−0.674355	−	1.25176i
$846$		0			0
$847$	187.812	−	108.433i	0.00761899	−	0.00439882i
$848$		0			0
$849$	−2196.35	+	3804.19i	−0.0887851	+	0.153780i
$850$		0			0
$851$	−45617.8	−	26337.5i	−1.83756	−	1.06091i
$852$		0			0
$853$		−	24221.1i		−	0.972234i	−0.873894	−	0.486117i	$-0.838413\pi$
$853$		−	24221.1i		−	0.972234i	0.873894	−	0.486117i	$-0.161587\pi$
$854$		0			0
$855$	8650.28	+	14982.7i	0.346004	+	0.599297i
$856$		0			0
$857$	−9191.63			−0.366371			−0.183186	−	0.983078i	$-0.558641\pi$
$857$	−9191.63			−0.366371			−0.183186	+	0.983078i	$0.558641\pi$
$858$		0			0
$859$	1899.06			0.0754309			0.0377155	−	0.999289i	$-0.487992\pi$
$859$	1899.06			0.0754309			0.0377155	+	0.999289i	$0.487992\pi$
$860$		0			0
$861$	129.236	+	223.843i	0.00511539	+	0.00886011i
$862$		0			0
$863$			18688.4i			0.737149i	0.929598	+	0.368574i	$0.120154\pi$
$863$			18688.4i			0.737149i	−0.929598	+	0.368574i	$0.879846\pi$
$864$		0			0
$865$	−4874.11	−	2814.07i	−0.191589	−	0.110614i
$866$		0			0
$867$	−2284.25	+	3956.43i	−0.0894775	+	0.154980i
$868$		0			0
$869$	−42657.5	+	24628.3i	−1.66520	+	0.961403i
$870$		0			0
$871$	6082.27	−	5905.71i	0.236613	−	0.229745i
$872$		0			0
$873$	−7687.11	+	4438.15i	−0.298017	+	0.172060i
$874$		0			0
$875$	−25.4819	+	44.1359i	−0.000984508	+	0.00170522i
$876$		0			0
$877$	25996.8	+	15009.3i	1.00097	+	0.577910i	0.908535	−	0.417808i	$-0.137202\pi$
$877$	25996.8	+	15009.3i	1.00097	+	0.577910i	0.0924350	+	0.995719i	$0.470535\pi$
$878$		0			0
$879$		−	1451.43i		−	0.0556945i
$880$		0			0
$881$	5969.14	+	10338.9i	0.228270	+	0.395375i	0.957295	−	0.289112i	$-0.0933599\pi$
$881$	5969.14	+	10338.9i	0.228270	+	0.395375i	−0.729026	+	0.684486i	$0.760027\pi$
$882$		0			0
$883$	22169.9			0.844933			0.422467	−	0.906379i	$-0.361164\pi$
$883$	22169.9			0.844933			0.422467	+	0.906379i	$0.361164\pi$
$884$		0			0
$885$	−13752.6			−0.522359
$886$		0			0
$887$	−364.388	−	631.139i	−0.0137936	−	0.0238913i	0.859046	−	0.511898i	$-0.171057\pi$
$887$	−364.388	−	631.139i	−0.0137936	−	0.0238913i	−0.872840	+	0.488007i	$0.837724\pi$
$888$		0			0
$889$		−	768.874i		−	0.0290070i
$890$		0			0
$891$	20138.0	+	11626.7i	0.757182	+	0.437159i
$892$		0			0
$893$	−5364.15	+	9290.99i	−0.201013	+	0.348165i
$894$		0			0
$895$	35714.1	−	20619.5i	1.33384	−	0.770095i
$896$		0			0
$897$	−3358.53	+	11834.3i	−0.125015	+	0.440508i
$898$		0			0
$899$	−3953.69	+	2282.67i	−0.146678	+	0.0846843i
$900$		0			0
$901$	−2906.71	+	5034.57i	−0.107477	+	0.186155i
$902$		0			0
$903$	−483.481	−	279.138i	−0.0178175	−	0.0102870i
$904$		0			0
$905$			37182.1i			1.36572i
$906$		0			0
$907$	19565.6	+	33888.7i	0.716280	+	1.24063i	0.962464	+	0.271411i	$0.0874901\pi$
$907$	19565.6	+	33888.7i	0.716280	+	1.24063i	−0.246183	+	0.969223i	$0.579177\pi$
$908$		0			0
$909$	36284.9			1.32397
$910$		0			0
$911$	−40955.0			−1.48946			−0.744730	−	0.667366i	$-0.767422\pi$
$911$	−40955.0			−1.48946			−0.744730	+	0.667366i	$0.767422\pi$
$912$		0			0
$913$	19970.0	+	34589.1i	0.723890	+	1.25381i
$914$		0			0
$915$			11870.9i			0.428895i
$916$		0			0
$917$	628.608	+	362.927i	0.0226374	+	0.0130697i
$918$		0			0
$919$	−6283.90	+	10884.0i	−0.225557	+	0.390676i	−0.956486	−	0.291777i	$-0.905754\pi$
$919$	−6283.90	+	10884.0i	−0.225557	+	0.390676i	0.730929	+	0.682453i	$0.239087\pi$
$920$		0			0
$921$	−8178.99	+	4722.14i	−0.292624	+	0.168947i
$922$		0			0
$923$	−23430.0	−	6649.36i	−0.835545	−	0.237125i
$924$		0			0
$925$	−28566.7	+	16493.0i	−1.01542	+	0.586256i
$926$		0			0
$927$	18907.8	−	32749.2i	0.669916	−	1.16033i
$928$		0			0
$929$	−8538.00	−	4929.42i	−0.301532	−	0.174089i	0.341599	−	0.939846i	$-0.389031\pi$
$929$	−8538.00	−	4929.42i	−0.301532	−	0.174089i	−0.643131	+	0.765756i	$0.722365\pi$
$930$		0			0
$931$			14668.8i			0.516381i
$932$		0			0
$933$	−4858.74	−	8415.59i	−0.170491	−	0.295299i
$934$		0			0
$935$	−22837.9			−0.798802
$936$		0			0
$937$	14064.4			0.490355			0.245177	−	0.969478i	$-0.421154\pi$
$937$	14064.4			0.490355			0.245177	+	0.969478i	$0.421154\pi$
$938$		0			0
$939$	−812.175	−	1406.73i	−0.0282261	−	0.0488891i
$940$		0			0
$941$			37759.0i			1.30808i	0.756458	+	0.654042i	$0.226928\pi$
$941$			37759.0i			1.30808i	−0.756458	+	0.654042i	$0.773072\pi$
$942$		0			0
$943$	−29954.7	−	17294.3i	−1.03442	−	0.597222i
$944$		0			0
$945$	634.036	−	1098.18i	0.0218256	−	0.0378031i
$946$		0			0
$947$	37020.8	−	21374.0i	1.27034	−	0.733433i	0.295291	−	0.955407i	$-0.404583\pi$
$947$	37020.8	−	21374.0i	1.27034	−	0.733433i	0.975053	+	0.221974i	$0.0712500\pi$
$948$		0			0
$949$	−11264.6	−	44660.8i	−0.385317	−	1.52766i
$950$		0			0
$951$	4553.39	−	2628.90i	0.155262	−	0.0896403i
$952$		0			0
$953$	−14794.6	+	25625.0i	−0.502879	+	0.871012i	0.497115	+	0.867684i	$0.334393\pi$
$953$	−14794.6	+	25625.0i	−0.502879	+	0.871012i	−0.999994	+	0.00332777i	$0.998941\pi$
$954$		0			0
$955$	−421.282	−	243.227i	−0.0142747	−	0.00824151i
$956$		0			0
$957$			2954.13i			0.0997843i
$958$		0			0
$959$	123.628	+	214.129i	0.00416282	+	0.00721022i
$960$		0			0
$961$	23801.9			0.798961
$962$		0			0
$963$	29444.9			0.985305
$964$		0			0
$965$	24179.4	+	41879.9i	0.806592	+	1.39706i
$966$		0			0
$967$			7140.75i			0.237468i	0.992926	+	0.118734i	$0.0378835\pi$
$967$			7140.75i			0.237468i	−0.992926	+	0.118734i	$0.962117\pi$
$968$		0			0
$969$	1766.99	+	1020.17i	0.0585797	+	0.0338210i
$970$		0			0
$971$	17433.3	−	30195.3i	0.576169	−	0.997954i	−0.419745	−	0.907642i	$-0.637880\pi$
$971$	17433.3	−	30195.3i	0.576169	−	0.997954i	0.995914	−	0.0903116i	$-0.0287863\pi$
$972$		0			0
$973$	−379.021	+	218.828i	−0.0124880	+	0.00720996i
$974$		0			0
$975$	5366.39	+	5526.82i	0.176269	+	0.181538i
$976$		0			0
$977$	27921.3	−	16120.4i	0.914311	−	0.527878i	0.0324954	−	0.999472i	$-0.489655\pi$
$977$	27921.3	−	16120.4i	0.914311	−	0.527878i	0.881816	+	0.471594i	$0.156321\pi$
$978$		0			0
$979$	−28893.1	+	50044.3i	−0.943236	+	1.63373i
$980$		0			0
$981$	−35096.5	−	20263.0i	−1.14225	−	0.659478i
$982$		0			0
$983$		−	16467.0i		−	0.534298i	−0.963655	−	0.267149i	$-0.913918\pi$
$983$		−	16467.0i		−	0.534298i	0.963655	−	0.267149i	$-0.0860816\pi$
$984$		0			0
$985$	−33151.1	−	57419.4i	−1.07237	−	1.85739i
$986$		0			0
$987$	380.734			0.0122785
$988$		0			0
$989$	74708.3			2.40201
$990$		0			0
$991$	6174.36	+	10694.3i	0.197916	+	0.342801i	0.947853	−	0.318709i	$-0.103249\pi$
$991$	6174.36	+	10694.3i	0.197916	+	0.342801i	−0.749936	+	0.661510i	$0.769916\pi$
$992$		0			0
$993$			5811.38i			0.185719i
$994$		0			0
$995$	63763.3	+	36813.7i	2.03159	+	1.17294i
$996$		0			0
$997$	−1942.47	+	3364.46i	−0.0617038	+	0.106874i	−0.895227	−	0.445610i	$-0.852987\pi$
$997$	−1942.47	+	3364.46i	−0.0617038	+	0.106874i	0.833523	+	0.552484i	$0.186320\pi$
$998$		0			0
$999$	15068.4	−	8699.75i	0.477220	−	0.275523i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.w.c.49.3		8
4.3	odd	2		52.4.h.a.49.2	yes	8
12.11	even	2		468.4.t.g.361.4		8
13.4	even	6	inner	208.4.w.c.17.3		8
52.3	odd	6		676.4.d.d.337.5		8
52.7	even	12		676.4.e.h.529.4		16
52.11	even	12		676.4.a.g.1.6		8
52.15	even	12		676.4.a.g.1.5		8
52.19	even	12		676.4.e.h.529.3		16
52.23	odd	6		676.4.d.d.337.6		8
52.31	even	4		676.4.e.h.653.3		16
52.35	odd	6		676.4.h.e.485.2		8
52.43	odd	6		52.4.h.a.17.2	✓	8
52.47	even	4		676.4.e.h.653.4		16
52.51	odd	2		676.4.h.e.361.2		8
156.95	even	6		468.4.t.g.433.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.4.h.a.17.2	✓	8	52.43	odd	6
52.4.h.a.49.2	yes	8	4.3	odd	2
208.4.w.c.17.3		8	13.4	even	6	inner
208.4.w.c.49.3		8	1.1	even	1	trivial
468.4.t.g.361.4		8	12.11	even	2
468.4.t.g.433.1		8	156.95	even	6
676.4.a.g.1.5		8	52.15	even	12
676.4.a.g.1.6		8	52.11	even	12
676.4.d.d.337.5		8	52.3	odd	6
676.4.d.d.337.6		8	52.23	odd	6
676.4.e.h.529.3		16	52.19	even	12
676.4.e.h.529.4		16	52.7	even	12
676.4.e.h.653.3		16	52.31	even	4
676.4.e.h.653.4		16	52.47	even	4
676.4.h.e.361.2		8	52.51	odd	2
676.4.h.e.485.2		8	52.35	odd	6

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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.643469 + 1.11452i) q^{3} -15.8968i q^{5} +(1.02552 + 0.592083i) q^{7} +(12.6719 - 21.9484i) q^{9} +O(q^{10})\)	\(q+(0.643469 + 1.11452i) q^{3} -15.8968i q^{5} +(1.02552 + 0.592083i) q^{7} +(12.6719 - 21.9484i) q^{9} +(-33.6987 + 19.4560i) q^{11} +(-45.4488 + 11.4634i) q^{13} +(17.7173 - 10.2291i) q^{15} +(-18.4601 + 31.9739i) q^{17} +(-37.1887 - 21.4709i) q^{19} +1.52395i q^{21} +(-101.967 - 176.612i) q^{23} -127.707 q^{25} +67.3632 q^{27} +(-29.4958 - 51.0882i) q^{29} -77.3896i q^{31} +(-43.3681 - 25.0386i) q^{33} +(9.41221 - 16.3024i) q^{35} +(223.689 - 129.147i) q^{37} +(-42.0210 - 43.2773i) q^{39} +(146.884 - 84.8034i) q^{41} +(-183.168 + 317.255i) q^{43} +(-348.908 - 201.442i) q^{45} -249.834i q^{47} +(-170.799 - 295.832i) q^{49} -47.5141 q^{51} +157.459 q^{53} +(309.287 + 535.701i) q^{55} -55.2634i q^{57} +(-582.168 - 336.115i) q^{59} +(-290.125 + 502.512i) q^{61} +(25.9905 - 15.0056i) q^{63} +(182.231 + 722.489i) q^{65} +(-156.637 + 90.4343i) q^{67} +(131.225 - 227.289i) q^{69} +(449.996 + 259.805i) q^{71} +982.663i q^{73} +(-82.1757 - 142.332i) q^{75} -46.0782 q^{77} +1265.85 q^{79} +(-298.795 - 517.528i) q^{81} -1026.42i q^{83} +(508.282 + 293.457i) q^{85} +(37.9592 - 65.7473i) q^{87} +(1286.09 - 742.526i) q^{89} +(-53.3958 - 15.1535i) q^{91} +(86.2523 - 49.7978i) q^{93} +(-341.318 + 591.180i) q^{95} +(-303.313 - 175.118i) q^{97} +986.176i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{7} - 70 q^{9}+O(q^{10}) \) 8 * q + 36 * q^7 - 70 * q^9	\( 8 q + 36 q^{7} - 70 q^{9} - 72 q^{11} + 62 q^{13} - 96 q^{15} + 88 q^{17} + 144 q^{19} + 20 q^{23} - 84 q^{25} + 432 q^{27} - 484 q^{29} + 1038 q^{33} - 40 q^{35} + 996 q^{37} + 236 q^{39} + 156 q^{41} - 504 q^{43} - 1530 q^{45} + 922 q^{49} + 1808 q^{51} - 1164 q^{53} + 1128 q^{55} - 600 q^{59} - 1224 q^{61} - 6480 q^{63} + 670 q^{65} - 960 q^{67} + 1738 q^{69} + 2964 q^{71} - 1448 q^{75} - 3972 q^{77} + 3968 q^{79} - 4132 q^{81} + 3870 q^{85} + 1660 q^{87} + 5430 q^{89} + 1720 q^{91} + 3324 q^{93} - 2400 q^{95} - 3042 q^{97}+O(q^{100}) \) 8 * q + 36 * q^7 - 70 * q^9 - 72 * q^11 + 62 * q^13 - 96 * q^15 + 88 * q^17 + 144 * q^19 + 20 * q^23 - 84 * q^25 + 432 * q^27 - 484 * q^29 + 1038 * q^33 - 40 * q^35 + 996 * q^37 + 236 * q^39 + 156 * q^41 - 504 * q^43 - 1530 * q^45 + 922 * q^49 + 1808 * q^51 - 1164 * q^53 + 1128 * q^55 - 600 * q^59 - 1224 * q^61 - 6480 * q^63 + 670 * q^65 - 960 * q^67 + 1738 * q^69 + 2964 * q^71 - 1448 * q^75 - 3972 * q^77 + 3968 * q^79 - 4132 * q^81 + 3870 * q^85 + 1660 * q^87 + 5430 * q^89 + 1720 * q^91 + 3324 * q^93 - 2400 * q^95 - 3042 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more