LMFDB - Embedded newform 6975.2.a.cb.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,12,0,0,-8,0,0,0,0,0,0,0,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.361944768.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 40x^{2} - 27 $ x^6 - 12x^4 + 40x^2 - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 279)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.53758$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53758 q^{2} +4.43931 q^{4} -4.38955 q^{7} +6.18995 q^{8} +0.827370 q^{11} +4.87862 q^{13} -11.1388 q^{14} +6.82886 q^{16} -4.24779 q^{17} -0.389553 q^{19} +2.09952 q^{22} +7.30474 q^{23} +12.3799 q^{26} -19.4866 q^{28} +5.07516 q^{29} +1.00000 q^{31} +4.94889 q^{32} -10.7791 q^{34} +9.65773 q^{37} -0.988522 q^{38} +3.23868 q^{41} -2.87862 q^{43} +3.67295 q^{44} +18.5364 q^{46} +2.22958 q^{47} +12.2682 q^{49} +21.6577 q^{52} +4.24779 q^{53} -27.1711 q^{56} +12.8786 q^{58} -8.31384 q^{59} +6.77911 q^{61} +2.53758 q^{62} -1.09952 q^{64} +4.00000 q^{67} -18.8573 q^{68} -1.83648 q^{71} -4.09952 q^{73} +24.5073 q^{74} -1.72935 q^{76} -3.63178 q^{77} +4.97814 q^{79} +8.21842 q^{82} +16.6277 q^{83} -7.30474 q^{86} +5.12138 q^{88} +7.92074 q^{89} -21.4150 q^{91} +32.4280 q^{92} +5.65773 q^{94} -8.29004 q^{97} +31.1315 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4} - 8 q^{7} + 8 q^{16} + 16 q^{19} + 20 q^{22} - 18 q^{28} + 6 q^{31} - 28 q^{34} - 8 q^{37} + 12 q^{43} + 16 q^{46} + 26 q^{49} + 64 q^{52} + 48 q^{58} + 4 q^{61} - 14 q^{64} + 24 q^{67} - 32 q^{73}+ \cdots - 24 q^{97}+O(q^{100}) $ 6 * q + 12 * q^4 - 8 * q^7 + 8 * q^16 + 16 * q^19 + 20 * q^22 - 18 * q^28 + 6 * q^31 - 28 * q^34 - 8 * q^37 + 12 * q^43 + 16 * q^46 + 26 * q^49 + 64 * q^52 + 48 * q^58 + 4 * q^61 - 14 * q^64 + 24 * q^67 - 32 * q^73 + 30 * q^76 + 8 * q^79 - 2 * q^82 + 60 * q^88 - 4 * q^91 - 32 * q^94 - 24 * q^97

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 $

$2$	2.53758	1.79434	0.897170	−	0.441685i	$-0.145619\pi$
$2$	2.53758	1.79434	0.897170	+	0.441685i	$0.145619\pi$
$3$	0	0
$3$	0	0
$4$	4.43931	2.21966
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.38955	−1.65910	−0.829548	−	0.558436i	$-0.811402\pi$
$7$	−4.38955	−1.65910	−0.829548	+	0.558436i	$0.811402\pi$
$8$	6.18995	2.18848
$9$	0	0
$10$	0	0
$11$	0.827370	0.249461	0.124731	−	0.992191i	$-0.460193\pi$
$11$	0.827370	0.249461	0.124731	+	0.992191i	$0.460193\pi$
$12$	0	0
$13$	4.87862	1.35309	0.676543	−	0.736403i	$-0.263477\pi$
$13$	4.87862	1.35309	0.676543	+	0.736403i	$0.263477\pi$
$14$	−11.1388	−2.97698
$15$	0	0
$16$	6.82886	1.70722
$17$	−4.24779	−1.03024	−0.515120	−	0.857118i	$-0.672253\pi$
$17$	−4.24779	−1.03024	−0.515120	+	0.857118i	$0.672253\pi$
$18$	0	0
$19$	−0.389553	−0.0893696	−0.0446848	−	0.999001i	$-0.514228\pi$
$19$	−0.389553	−0.0893696	−0.0446848	+	0.999001i	$0.514228\pi$
$20$	0	0
$21$	0	0
$22$	2.09952	0.447619
$23$	7.30474	1.52314	0.761571	−	0.648081i	$-0.224428\pi$
$23$	7.30474	1.52314	0.761571	+	0.648081i	$0.224428\pi$
$24$	0	0
$25$	0	0
$26$	12.3799	2.42790
$27$	0	0
$28$	−19.4866	−3.68262
$29$	5.07516	0.942434	0.471217	−	0.882017i	$-0.343815\pi$
$29$	5.07516	0.942434	0.471217	+	0.882017i	$0.343815\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	4.94889	0.874849
$33$	0	0
$34$	−10.7791	−1.84860
$35$	0	0
$36$	0	0
$37$	9.65773	1.58772	0.793860	−	0.608100i	$-0.208068\pi$
$37$	9.65773	1.58772	0.793860	+	0.608100i	$0.208068\pi$
$38$	−0.988522	−0.160359
$39$	0	0
$40$	0	0
$41$	3.23868	0.505797	0.252899	−	0.967493i	$-0.418616\pi$
$41$	3.23868	0.505797	0.252899	+	0.967493i	$0.418616\pi$
$42$	0	0
$43$	−2.87862	−0.438986	−0.219493	−	0.975614i	$-0.570440\pi$
$43$	−2.87862	−0.438986	−0.219493	+	0.975614i	$0.570440\pi$
$44$	3.67295	0.553718
$45$	0	0
$46$	18.5364	2.73304
$47$	2.22958	0.325217	0.162609	−	0.986691i	$-0.448009\pi$
$47$	2.22958	0.325217	0.162609	+	0.986691i	$0.448009\pi$
$48$	0	0
$49$	12.2682	1.75260
$50$	0	0
$51$	0	0
$52$	21.6577	3.00339
$53$	4.24779	0.583479	0.291739	−	0.956498i	$-0.405766\pi$
$53$	4.24779	0.583479	0.291739	+	0.956498i	$0.405766\pi$
$54$	0	0
$55$	0	0
$56$	−27.1711	−3.63089
$57$	0	0
$58$	12.8786	1.69105
$59$	−8.31384	−1.08237	−0.541185	−	0.840903i	$-0.682024\pi$
$59$	−8.31384	−1.08237	−0.541185	+	0.840903i	$0.682024\pi$
$60$	0	0
$61$	6.77911	0.867976	0.433988	−	0.900919i	$-0.357106\pi$
$61$	6.77911	0.867976	0.433988	+	0.900919i	$0.357106\pi$
$62$	2.53758	0.322273
$63$	0	0
$64$	−1.09952	−0.137440
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−18.8573	−2.28678
$69$	0	0
$70$	0	0
$71$	−1.83648	−0.217950	−0.108975	−	0.994045i	$-0.534757\pi$
$71$	−1.83648	−0.217950	−0.108975	+	0.994045i	$0.534757\pi$
$72$	0	0
$73$	−4.09952	−0.479812	−0.239906	−	0.970796i	$-0.577117\pi$
$73$	−4.09952	−0.479812	−0.239906	+	0.970796i	$0.577117\pi$
$74$	24.5073	2.84891
$75$	0	0
$76$	−1.72935	−0.198370
$77$	−3.63178	−0.413880
$78$	0	0
$79$	4.97814	0.560085	0.280042	−	0.959988i	$-0.409652\pi$
$79$	4.97814	0.560085	0.280042	+	0.959988i	$0.409652\pi$
$80$	0	0
$81$	0	0
$82$	8.21842	0.907572
$83$	16.6277	1.82513	0.912563	−	0.408936i	$-0.134100\pi$
$83$	16.6277	1.82513	0.912563	+	0.408936i	$0.134100\pi$
$84$	0	0
$85$	0	0
$86$	−7.30474	−0.787690
$87$	0	0
$88$	5.12138	0.545941
$89$	7.92074	0.839597	0.419799	−	0.907617i	$-0.362101\pi$
$89$	7.92074	0.839597	0.419799	+	0.907617i	$0.362101\pi$
$90$	0	0
$91$	−21.4150	−2.24490
$92$	32.4280	3.38085
$93$	0	0
$94$	5.65773	0.583550
$95$	0	0
$96$	0	0
$97$	−8.29004	−0.841726	−0.420863	−	0.907124i	$-0.638273\pi$
$97$	−8.29004	−0.841726	−0.420863	+	0.907124i	$0.638273\pi$
$98$	31.1315	3.14475
$99$	0	0
$100$	0	0
$101$	9.71605	0.966783	0.483392	−	0.875404i	$-0.339405\pi$
$101$	9.71605	0.966783	0.483392	+	0.875404i	$0.339405\pi$
$102$	0	0
$103$	−4.38955	−0.432516	−0.216258	−	0.976336i	$-0.569385\pi$
$103$	−4.38955	−0.432516	−0.216258	+	0.976336i	$0.569385\pi$
$104$	30.1984	2.96120
$105$	0	0
$106$	10.7791	1.04696
$107$	−6.65910	−0.643760	−0.321880	−	0.946781i	$-0.604315\pi$
$107$	−6.65910	−0.643760	−0.321880	+	0.946781i	$0.604315\pi$
$108$	0	0
$109$	17.2682	1.65399	0.826996	−	0.562208i	$-0.190048\pi$
$109$	17.2682	1.65399	0.826996	+	0.562208i	$0.190048\pi$
$110$	0	0
$111$	0	0
$112$	−29.9757	−2.83243
$113$	11.7343	1.10387	0.551933	−	0.833888i	$-0.313890\pi$
$113$	11.7343	1.10387	0.551933	+	0.833888i	$0.313890\pi$
$114$	0	0
$115$	0	0
$116$	22.5302	2.09188
$117$	0	0
$118$	−21.0970	−1.94214
$119$	18.6459	1.70927
$120$	0	0
$121$	−10.3155	−0.937769
$122$	17.2025	1.55744
$123$	0	0
$124$	4.43931	0.398662
$125$	0	0
$126$	0	0
$127$	10.8786	0.965322	0.482661	−	0.875807i	$-0.339670\pi$
$127$	10.8786	0.965322	0.482661	+	0.875807i	$0.339670\pi$
$128$	−12.6879	−1.12146
$129$	0	0
$130$	0	0
$131$	−20.8755	−1.82390	−0.911949	−	0.410303i	$-0.865423\pi$
$131$	−20.8755	−1.82390	−0.911949	+	0.410303i	$0.865423\pi$
$132$	0	0
$133$	1.70996	0.148273
$134$	10.1503	0.876854
$135$	0	0
$136$	−26.2936	−2.25466
$137$	−8.74811	−0.747402	−0.373701	−	0.927549i	$-0.621911\pi$
$137$	−8.74811	−0.747402	−0.373701	+	0.927549i	$0.621911\pi$
$138$	0	0
$139$	−6.67959	−0.566555	−0.283278	−	0.959038i	$-0.591422\pi$
$139$	−6.67959	−0.566555	−0.283278	+	0.959038i	$0.591422\pi$
$140$	0	0
$141$	0	0
$142$	−4.66021	−0.391076
$143$	4.03643	0.337543
$144$	0	0
$145$	0	0
$146$	−10.4029	−0.860946
$147$	0	0
$148$	42.8737	3.52419
$149$	1.65474	0.135562	0.0677808	−	0.997700i	$-0.478408\pi$
$149$	1.65474	0.135562	0.0677808	+	0.997700i	$0.478408\pi$
$150$	0	0
$151$	−20.4368	−1.66313	−0.831563	−	0.555430i	$-0.812554\pi$
$151$	−20.4368	−1.66313	−0.831563	+	0.555430i	$0.812554\pi$
$152$	−2.41131	−0.195583
$153$	0	0
$154$	−9.21594	−0.742642
$155$	0	0
$156$	0	0
$157$	15.0254	1.19916	0.599580	−	0.800315i	$-0.295334\pi$
$157$	15.0254	1.19916	0.599580	+	0.800315i	$0.295334\pi$
$158$	12.6324	1.00498
$159$	0	0
$160$	0	0
$161$	−32.0645	−2.52704
$162$	0	0
$163$	−6.14680	−0.481455	−0.240727	−	0.970593i	$-0.577386\pi$
$163$	−6.14680	−0.481455	−0.240727	+	0.970593i	$0.577386\pi$
$164$	14.3775	1.12270
$165$	0	0
$166$	42.1941	3.27490
$167$	−4.50032	−0.348245	−0.174123	−	0.984724i	$-0.555709\pi$
$167$	−4.50032	−0.348245	−0.174123	+	0.984724i	$0.555709\pi$
$168$	0	0
$169$	10.8010	0.830844
$170$	0	0
$171$	0	0
$172$	−12.7791	−0.974398
$173$	−8.49558	−0.645907	−0.322953	−	0.946415i	$-0.604676\pi$
$173$	−8.49558	−0.645907	−0.322953	+	0.946415i	$0.604676\pi$
$174$	0	0
$175$	0	0
$176$	5.65000	0.425885
$177$	0	0
$178$	20.0995	1.50652
$179$	1.65474	0.123681	0.0618405	−	0.998086i	$-0.480303\pi$
$179$	1.65474	0.123681	0.0618405	+	0.998086i	$0.480303\pi$
$180$	0	0
$181$	−14.7791	−1.09852	−0.549261	−	0.835651i	$-0.685091\pi$
$181$	−14.7791	−1.09852	−0.549261	+	0.835651i	$0.685091\pi$
$182$	−54.3422	−4.02811
$183$	0	0
$184$	45.2159	3.33336
$185$	0	0
$186$	0	0
$187$	−3.51449	−0.257005
$188$	9.89779	0.721870
$189$	0	0
$190$	0	0
$191$	12.7730	0.924222	0.462111	−	0.886822i	$-0.347092\pi$
$191$	12.7730	0.924222	0.462111	+	0.886822i	$0.347092\pi$
$192$	0	0
$193$	6.04728	0.435293	0.217646	−	0.976028i	$-0.430162\pi$
$193$	6.04728	0.435293	0.217646	+	0.976028i	$0.430162\pi$
$194$	−21.0366	−1.51034
$195$	0	0
$196$	54.4623	3.89016
$197$	21.7028	1.54626	0.773132	−	0.634245i	$-0.218689\pi$
$197$	21.7028	1.54626	0.773132	+	0.634245i	$0.218689\pi$
$198$	0	0
$199$	−10.8786	−0.771165	−0.385583	−	0.922673i	$-0.626000\pi$
$199$	−10.8786	−0.771165	−0.385583	+	0.922673i	$0.626000\pi$
$200$	0	0
$201$	0	0
$202$	24.6553	1.73474
$203$	−22.2777	−1.56359
$204$	0	0
$205$	0	0
$206$	−11.1388	−0.776080
$207$	0	0
$208$	33.3155	2.31001
$209$	−0.322305	−0.0222943
$210$	0	0
$211$	−0.389553	−0.0268179	−0.0134090	−	0.999910i	$-0.504268\pi$
$211$	−0.389553	−0.0268179	−0.0134090	+	0.999910i	$0.504268\pi$
$212$	18.8573	1.29512
$213$	0	0
$214$	−16.8980	−1.15512
$215$	0	0
$216$	0	0
$217$	−4.38955	−0.297982
$218$	43.8194	2.96782
$219$	0	0
$220$	0	0
$221$	−20.7234	−1.39400
$222$	0	0
$223$	15.6577	1.04852	0.524260	−	0.851559i	$-0.324342\pi$
$223$	15.6577	1.04852	0.524260	+	0.851559i	$0.324342\pi$
$224$	−21.7234	−1.45146
$225$	0	0
$226$	29.7766	1.98071
$227$	−0.574837	−0.0381533	−0.0190766	−	0.999818i	$-0.506073\pi$
$227$	−0.574837	−0.0381533	−0.0190766	+	0.999818i	$0.506073\pi$
$228$	0	0
$229$	1.65773	0.109546	0.0547729	−	0.998499i	$-0.482557\pi$
$229$	1.65773	0.109546	0.0547729	+	0.998499i	$0.482557\pi$
$230$	0	0
$231$	0	0
$232$	31.4150	2.06249
$233$	−15.0437	−0.985548	−0.492774	−	0.870157i	$-0.664017\pi$
$233$	−15.0437	−0.985548	−0.492774	+	0.870157i	$0.664017\pi$
$234$	0	0
$235$	0	0
$236$	−36.9077	−2.40249
$237$	0	0
$238$	47.3155	3.06701
$239$	−6.47737	−0.418986	−0.209493	−	0.977810i	$-0.567181\pi$
$239$	−6.47737	−0.418986	−0.209493	+	0.977810i	$0.567181\pi$
$240$	0	0
$241$	17.8568	1.15026	0.575128	−	0.818064i	$-0.304952\pi$
$241$	17.8568	1.15026	0.575128	+	0.818064i	$0.304952\pi$
$242$	−26.1763	−1.68268
$243$	0	0
$244$	30.0946	1.92661
$245$	0	0
$246$	0	0
$247$	−1.90048	−0.120925
$248$	6.18995	0.393062
$249$	0	0
$250$	0	0
$251$	3.63178	0.229236	0.114618	−	0.993410i	$-0.463436\pi$
$251$	3.63178	0.229236	0.114618	+	0.993410i	$0.463436\pi$
$252$	0	0
$253$	6.04372	0.379965
$254$	27.6054	1.73212
$255$	0	0
$256$	−29.9975	−1.87485
$257$	−13.3890	−0.835183	−0.417592	−	0.908635i	$-0.637126\pi$
$257$	−13.3890	−0.835183	−0.417592	+	0.908635i	$0.637126\pi$
$258$	0	0
$259$	−42.3931	−2.63418
$260$	0	0
$261$	0	0
$262$	−52.9732	−3.27269
$263$	24.7598	1.52675	0.763377	−	0.645953i	$-0.223540\pi$
$263$	24.7598	1.52675	0.763377	+	0.645953i	$0.223540\pi$
$264$	0	0
$265$	0	0
$266$	4.33917	0.266052
$267$	0	0
$268$	17.7572	1.08470
$269$	3.05695	0.186385	0.0931927	−	0.995648i	$-0.470293\pi$
$269$	3.05695	0.186385	0.0931927	+	0.995648i	$0.470293\pi$
$270$	0	0
$271$	2.53635	0.154072	0.0770362	−	0.997028i	$-0.475454\pi$
$271$	2.53635	0.154072	0.0770362	+	0.997028i	$0.475454\pi$
$272$	−29.0076	−1.75884
$273$	0	0
$274$	−22.1990	−1.34109
$275$	0	0
$276$	0	0
$277$	7.55821	0.454129	0.227064	−	0.973880i	$-0.427087\pi$
$277$	7.55821	0.454129	0.227064	+	0.973880i	$0.427087\pi$
$278$	−16.9500	−1.01659
$279$	0	0
$280$	0	0
$281$	−25.5575	−1.52463	−0.762317	−	0.647203i	$-0.775938\pi$
$281$	−25.5575	−1.52463	−0.762317	+	0.647203i	$0.775938\pi$
$282$	0	0
$283$	−5.55821	−0.330401	−0.165201	−	0.986260i	$-0.552827\pi$
$283$	−5.55821	−0.330401	−0.165201	+	0.986260i	$0.552827\pi$
$284$	−8.15269	−0.483773
$285$	0	0
$286$	10.2428	0.605667
$287$	−14.2164	−0.839166
$288$	0	0
$289$	1.04372	0.0613952
$290$	0	0
$291$	0	0
$292$	−18.1990	−1.06502
$293$	−26.4145	−1.54315	−0.771577	−	0.636136i	$-0.780532\pi$
$293$	−26.4145	−1.54315	−0.771577	+	0.636136i	$0.780532\pi$
$294$	0	0
$295$	0	0
$296$	59.7808	3.47469
$297$	0	0
$298$	4.19903	0.243244
$299$	35.6371	2.06094
$300$	0	0
$301$	12.6359	0.728319
$302$	−51.8601	−2.98421
$303$	0	0
$304$	−2.66021	−0.152573
$305$	0	0
$306$	0	0
$307$	−9.16866	−0.523283	−0.261642	−	0.965165i	$-0.584264\pi$
$307$	−9.16866	−0.523283	−0.261642	+	0.965165i	$0.584264\pi$
$308$	−16.1226	−0.918672
$309$	0	0
$310$	0	0
$311$	−0.181737	−0.0103053	−0.00515267	−	0.999987i	$-0.501640\pi$
$311$	−0.181737	−0.0103053	−0.00515267	+	0.999987i	$0.501640\pi$
$312$	0	0
$313$	−12.7354	−0.719847	−0.359923	−	0.932982i	$-0.617197\pi$
$313$	−12.7354	−0.719847	−0.359923	+	0.932982i	$0.617197\pi$
$314$	38.1282	2.15170
$315$	0	0
$316$	22.0995	1.24319
$317$	−17.0620	−0.958295	−0.479147	−	0.877734i	$-0.659054\pi$
$317$	−17.0620	−0.958295	−0.479147	+	0.877734i	$0.659054\pi$
$318$	0	0
$319$	4.19903	0.235101
$320$	0	0
$321$	0	0
$322$	−81.3663	−4.53437
$323$	1.65474	0.0920722
$324$	0	0
$325$	0	0
$326$	−15.5980	−0.863893
$327$	0	0
$328$	20.0473	1.10693
$329$	−9.78685	−0.539566
$330$	0	0
$331$	22.3373	1.22777	0.613885	−	0.789395i	$-0.289606\pi$
$331$	22.3373	1.22777	0.613885	+	0.789395i	$0.289606\pi$
$332$	73.8155	4.05115
$333$	0	0
$334$	−11.4199	−0.624871
$335$	0	0
$336$	0	0
$337$	−23.5582	−1.28330	−0.641649	−	0.766999i	$-0.721749\pi$
$337$	−23.5582	−1.28330	−0.641649	+	0.766999i	$0.721749\pi$
$338$	27.4083	1.49082
$339$	0	0
$340$	0	0
$341$	0.827370	0.0448046
$342$	0	0
$343$	−23.1249	−1.24863
$344$	−17.8185	−0.960711
$345$	0	0
$346$	−21.5582	−1.15898
$347$	1.19084	0.0639278	0.0319639	−	0.999489i	$-0.489824\pi$
$347$	1.19084	0.0639278	0.0319639	+	0.999489i	$0.489824\pi$
$348$	0	0
$349$	11.5582	0.618697	0.309348	−	0.950949i	$-0.399889\pi$
$349$	11.5582	0.618697	0.309348	+	0.950949i	$0.399889\pi$
$350$	0	0
$351$	0	0
$352$	4.09457	0.218241
$353$	−7.09337	−0.377542	−0.188771	−	0.982021i	$-0.560450\pi$
$353$	−7.09337	−0.377542	−0.188771	+	0.982021i	$0.560450\pi$
$354$	0	0
$355$	0	0
$356$	35.1626	1.86362
$357$	0	0
$358$	4.19903	0.221926
$359$	−1.83648	−0.0969255	−0.0484628	−	0.998825i	$-0.515432\pi$
$359$	−1.83648	−0.0969255	−0.0484628	+	0.998825i	$0.515432\pi$
$360$	0	0
$361$	−18.8482	−0.992013
$362$	−37.5032	−1.97112
$363$	0	0
$364$	−95.0678	−4.98290
$365$	0	0
$366$	0	0
$367$	−16.2936	−0.850519	−0.425259	−	0.905071i	$-0.639817\pi$
$367$	−16.2936	−0.850519	−0.425259	+	0.905071i	$0.639817\pi$
$368$	49.8831	2.60033
$369$	0	0
$370$	0	0
$371$	−18.6459	−0.968047
$372$	0	0
$373$	20.4891	1.06088	0.530442	−	0.847721i	$-0.322026\pi$
$373$	20.4891	1.06088	0.530442	+	0.847721i	$0.322026\pi$
$374$	−8.91831	−0.461155
$375$	0	0
$376$	13.8010	0.711730
$377$	24.7598	1.27519
$378$	0	0
$379$	15.8010	0.811641	0.405821	−	0.913953i	$-0.366986\pi$
$379$	15.8010	0.811641	0.405821	+	0.913953i	$0.366986\pi$
$380$	0	0
$381$	0	0
$382$	32.4125	1.65837
$383$	−27.9277	−1.42704	−0.713519	−	0.700636i	$-0.752900\pi$
$383$	−27.9277	−1.42704	−0.713519	+	0.700636i	$0.752900\pi$
$384$	0	0
$385$	0	0
$386$	15.3455	0.781063
$387$	0	0
$388$	−36.8021	−1.86834
$389$	1.40221	0.0710947	0.0355474	−	0.999368i	$-0.488683\pi$
$389$	1.40221	0.0710947	0.0355474	+	0.999368i	$0.488683\pi$
$390$	0	0
$391$	−31.0290	−1.56920
$392$	75.9394	3.83552
$393$	0	0
$394$	55.0727	2.77452
$395$	0	0
$396$	0	0
$397$	17.4672	0.876654	0.438327	−	0.898816i	$-0.355571\pi$
$397$	17.4672	0.876654	0.438327	+	0.898816i	$0.355571\pi$
$398$	−27.6054	−1.38373
$399$	0	0
$400$	0	0
$401$	34.2941	1.71257	0.856283	−	0.516507i	$-0.172768\pi$
$401$	34.2941	1.71257	0.856283	+	0.516507i	$0.172768\pi$
$402$	0	0
$403$	4.87862	0.243022
$404$	43.1326	2.14593
$405$	0	0
$406$	−56.5314	−2.80561
$407$	7.99051	0.396075
$408$	0	0
$409$	−16.1941	−0.800746	−0.400373	−	0.916352i	$-0.631119\pi$
$409$	−16.1941	−0.800746	−0.400373	+	0.916352i	$0.631119\pi$
$410$	0	0
$411$	0	0
$412$	−19.4866	−0.960036
$413$	36.4941	1.79576
$414$	0	0
$415$	0	0
$416$	24.1438	1.18375
$417$	0	0
$418$	−0.817873	−0.0400035
$419$	−39.5510	−1.93219	−0.966096	−	0.258181i	$-0.916877\pi$
$419$	−39.5510	−1.93219	−0.966096	+	0.258181i	$0.916877\pi$
$420$	0	0
$421$	1.75368	0.0854693	0.0427346	−	0.999086i	$-0.486393\pi$
$421$	1.75368	0.0854693	0.0427346	+	0.999086i	$0.486393\pi$
$422$	−0.988522	−0.0481205
$423$	0	0
$424$	26.2936	1.27693
$425$	0	0
$426$	0	0
$427$	−29.7572	−1.44005
$428$	−29.5618	−1.42893
$429$	0	0
$430$	0	0
$431$	9.57548	0.461235	0.230617	−	0.973045i	$-0.425925\pi$
$431$	9.57548	0.461235	0.230617	+	0.973045i	$0.425925\pi$
$432$	0	0
$433$	25.5145	1.22615	0.613074	−	0.790025i	$-0.289933\pi$
$433$	25.5145	1.22615	0.613074	+	0.790025i	$0.289933\pi$
$434$	−11.1388	−0.534682
$435$	0	0
$436$	76.6588	3.67129
$437$	−2.84558	−0.136123
$438$	0	0
$439$	4.38955	0.209502	0.104751	−	0.994498i	$-0.466595\pi$
$439$	4.38955	0.209502	0.104751	+	0.994498i	$0.466595\pi$
$440$	0	0
$441$	0	0
$442$	−52.5872	−2.50132
$443$	−2.62268	−0.124607	−0.0623036	−	0.998057i	$-0.519845\pi$
$443$	−2.62268	−0.124607	−0.0623036	+	0.998057i	$0.519845\pi$
$444$	0	0
$445$	0	0
$446$	39.7327	1.88140
$447$	0	0
$448$	4.82639	0.228025
$449$	−20.9166	−0.987118	−0.493559	−	0.869712i	$-0.664304\pi$
$449$	−20.9166	−0.987118	−0.493559	+	0.869712i	$0.664304\pi$
$450$	0	0
$451$	2.67959	0.126177
$452$	52.0921	2.45020
$453$	0	0
$454$	−1.45870	−0.0684600
$455$	0	0
$456$	0	0
$457$	2.43684	0.113990	0.0569952	−	0.998374i	$-0.481848\pi$
$457$	2.43684	0.113990	0.0569952	+	0.998374i	$0.481848\pi$
$458$	4.20662	0.196562
$459$	0	0
$460$	0	0
$461$	−12.3799	−0.576589	−0.288295	−	0.957542i	$-0.593088\pi$
$461$	−12.3799	−0.576589	−0.288295	+	0.957542i	$0.593088\pi$
$462$	0	0
$463$	−34.3373	−1.59579	−0.797895	−	0.602796i	$-0.794053\pi$
$463$	−34.3373	−1.59579	−0.797895	+	0.602796i	$0.794053\pi$
$464$	34.6576	1.60894
$465$	0	0
$466$	−38.1747	−1.76841
$467$	39.5510	1.83020	0.915101	−	0.403225i	$-0.132111\pi$
$467$	39.5510	1.83020	0.915101	+	0.403225i	$0.132111\pi$
$468$	0	0
$469$	−17.5582	−0.810763
$470$	0	0
$471$	0	0
$472$	−51.4623	−2.36874
$473$	−2.38169	−0.109510
$474$	0	0
$475$	0	0
$476$	82.7750	3.79398
$477$	0	0
$478$	−16.4368	−0.751803
$479$	16.8094	0.768042	0.384021	−	0.923324i	$-0.374539\pi$
$479$	16.8094	0.768042	0.384021	+	0.923324i	$0.374539\pi$
$480$	0	0
$481$	47.1164	2.14832
$482$	45.3130	2.06395
$483$	0	0
$484$	−45.7935	−2.08152
$485$	0	0
$486$	0	0
$487$	13.9005	0.629891	0.314946	−	0.949110i	$-0.398014\pi$
$487$	13.9005	0.629891	0.314946	+	0.949110i	$0.398014\pi$
$488$	41.9623	1.89954
$489$	0	0
$490$	0	0
$491$	−38.9054	−1.75577	−0.877887	−	0.478867i	$-0.841048\pi$
$491$	−38.9054	−1.75577	−0.877887	+	0.478867i	$0.841048\pi$
$492$	0	0
$493$	−21.5582	−0.970933
$494$	−4.82263	−0.216980
$495$	0	0
$496$	6.82886	0.306625
$497$	8.06131	0.361599
$498$	0	0
$499$	4.97814	0.222852	0.111426	−	0.993773i	$-0.464458\pi$
$499$	4.97814	0.222852	0.111426	+	0.993773i	$0.464458\pi$
$500$	0	0
$501$	0	0
$502$	9.21594	0.411328
$503$	7.08183	0.315763	0.157882	−	0.987458i	$-0.449534\pi$
$503$	7.08183	0.315763	0.157882	+	0.987458i	$0.449534\pi$
$504$	0	0
$505$	0	0
$506$	15.3364	0.681787
$507$	0	0
$508$	48.2936	2.14268
$509$	−36.6758	−1.62563	−0.812813	−	0.582525i	$-0.802065\pi$
$509$	−36.6758	−1.62563	−0.812813	+	0.582525i	$0.802065\pi$
$510$	0	0
$511$	17.9950	0.796054
$512$	−50.7453	−2.24265
$513$	0	0
$514$	−33.9757	−1.49860
$515$	0	0
$516$	0	0
$517$	1.84468	0.0811291
$518$	−107.576	−4.72661
$519$	0	0
$520$	0	0
$521$	−17.9190	−0.785044	−0.392522	−	0.919743i	$-0.628397\pi$
$521$	−17.9190	−0.785044	−0.392522	+	0.919743i	$0.628397\pi$
$522$	0	0
$523$	−25.6140	−1.12002	−0.560011	−	0.828485i	$-0.689203\pi$
$523$	−25.6140	−1.12002	−0.560011	+	0.828485i	$0.689203\pi$
$524$	−92.6727	−4.04843
$525$	0	0
$526$	62.8300	2.73952
$527$	−4.24779	−0.185037
$528$	0	0
$529$	30.3592	1.31996
$530$	0	0
$531$	0	0
$532$	7.59106	0.329114
$533$	15.8003	0.684388
$534$	0	0
$535$	0	0
$536$	24.7598	1.06946
$537$	0	0
$538$	7.75725	0.334439
$539$	10.1503	0.437205
$540$	0	0
$541$	−15.0254	−0.645993	−0.322997	−	0.946400i	$-0.604690\pi$
$541$	−15.0254	−0.645993	−0.322997	+	0.946400i	$0.604690\pi$
$542$	6.43620	0.276458
$543$	0	0
$544$	−21.0219	−0.901305
$545$	0	0
$546$	0	0
$547$	32.1031	1.37263	0.686314	−	0.727305i	$-0.259227\pi$
$547$	32.1031	1.37263	0.686314	+	0.727305i	$0.259227\pi$
$548$	−38.8356	−1.65897
$549$	0	0
$550$	0	0
$551$	−1.97704	−0.0842249
$552$	0	0
$553$	−21.8518	−0.929234
$554$	19.1796	0.814862
$555$	0	0
$556$	−29.6528	−1.25756
$557$	−39.6218	−1.67883	−0.839415	−	0.543491i	$-0.817102\pi$
$557$	−39.6218	−1.67883	−0.839415	+	0.543491i	$0.817102\pi$
$558$	0	0
$559$	−14.0437	−0.593986
$560$	0	0
$561$	0	0
$562$	−64.8543	−2.73571
$563$	−2.62268	−0.110533	−0.0552663	−	0.998472i	$-0.517601\pi$
$563$	−2.62268	−0.110533	−0.0552663	+	0.998472i	$0.517601\pi$
$564$	0	0
$565$	0	0
$566$	−14.1044	−0.592852
$567$	0	0
$568$	−11.3677	−0.476978
$569$	20.0069	0.838734	0.419367	−	0.907817i	$-0.362252\pi$
$569$	20.0069	0.838734	0.419367	+	0.907817i	$0.362252\pi$
$570$	0	0
$571$	27.4587	1.14911	0.574555	−	0.818466i	$-0.305175\pi$
$571$	27.4587	1.14911	0.574555	+	0.818466i	$0.305175\pi$
$572$	17.9190	0.749229
$573$	0	0
$574$	−36.0752	−1.50575
$575$	0	0
$576$	0	0
$577$	−13.3155	−0.554330	−0.277165	−	0.960822i	$-0.589395\pi$
$577$	−13.3155	−0.554330	−0.277165	+	0.960822i	$0.589395\pi$
$578$	2.64852	0.110164
$579$	0	0
$580$	0	0
$581$	−72.9881	−3.02806
$582$	0	0
$583$	3.51449	0.145555
$584$	−25.3758	−1.05006
$585$	0	0
$586$	−67.0290	−2.76894
$587$	−10.1503	−0.418949	−0.209474	−	0.977814i	$-0.567175\pi$
$587$	−10.1503	−0.418949	−0.209474	+	0.977814i	$0.567175\pi$
$588$	0	0
$589$	−0.389553	−0.0160513
$590$	0	0
$591$	0	0
$592$	65.9513	2.71058
$593$	−44.6262	−1.83258	−0.916288	−	0.400519i	$-0.868830\pi$
$593$	−44.6262	−1.83258	−0.916288	+	0.400519i	$0.868830\pi$
$594$	0	0
$595$	0	0
$596$	7.34591	0.300900
$597$	0	0
$598$	90.4319	3.69803
$599$	12.7730	0.521891	0.260945	−	0.965354i	$-0.415966\pi$
$599$	12.7730	0.521891	0.260945	+	0.965354i	$0.415966\pi$
$600$	0	0
$601$	25.6577	1.04660	0.523300	−	0.852148i	$-0.324701\pi$
$601$	25.6577	1.04660	0.523300	+	0.852148i	$0.324701\pi$
$602$	32.0645	1.30685
$603$	0	0
$604$	−90.7255	−3.69157
$605$	0	0
$606$	0	0
$607$	25.5582	1.03738	0.518688	−	0.854964i	$-0.326421\pi$
$607$	25.5582	1.03738	0.518688	+	0.854964i	$0.326421\pi$
$608$	−1.92786	−0.0781849
$609$	0	0
$610$	0	0
$611$	10.8773	0.440047
$612$	0	0
$613$	29.1164	1.17600	0.588001	−	0.808860i	$-0.299915\pi$
$613$	29.1164	1.17600	0.588001	+	0.808860i	$0.299915\pi$
$614$	−23.2662	−0.938948
$615$	0	0
$616$	−22.4806	−0.905767
$617$	−41.2459	−1.66050	−0.830248	−	0.557394i	$-0.811801\pi$
$617$	−41.2459	−1.66050	−0.830248	+	0.557394i	$0.811801\pi$
$618$	0	0
$619$	30.7354	1.23536	0.617680	−	0.786430i	$-0.288073\pi$
$619$	30.7354	1.23536	0.617680	+	0.786430i	$0.288073\pi$
$620$	0	0
$621$	0	0
$622$	−0.461171	−0.0184913
$623$	−34.7685	−1.39297
$624$	0	0
$625$	0	0
$626$	−32.3171	−1.29165
$627$	0	0
$628$	66.7025	2.66172
$629$	−41.0240	−1.63573
$630$	0	0
$631$	−26.1383	−1.04055	−0.520274	−	0.853999i	$-0.674170\pi$
$631$	−26.1383	−1.04055	−0.520274	+	0.853999i	$0.674170\pi$
$632$	30.8144	1.22573
$633$	0	0
$634$	−43.2961	−1.71951
$635$	0	0
$636$	0	0
$637$	59.8518	2.37142
$638$	10.6554	0.421851
$639$	0	0
$640$	0	0
$641$	−33.9718	−1.34181	−0.670903	−	0.741545i	$-0.734093\pi$
$641$	−33.9718	−1.34181	−0.670903	+	0.741545i	$0.734093\pi$
$642$	0	0
$643$	2.73539	0.107873	0.0539366	−	0.998544i	$-0.482823\pi$
$643$	2.73539	0.107873	0.0539366	+	0.998544i	$0.482823\pi$
$644$	−142.344	−5.60916
$645$	0	0
$646$	4.19903	0.165209
$647$	−42.2148	−1.65964	−0.829818	−	0.558033i	$-0.811556\pi$
$647$	−42.2148	−1.65964	−0.829818	+	0.558033i	$0.811556\pi$
$648$	0	0
$649$	−6.87862	−0.270010
$650$	0	0
$651$	0	0
$652$	−27.2876	−1.06866
$653$	29.9459	1.17187	0.585937	−	0.810357i	$-0.300727\pi$
$653$	29.9459	1.17187	0.585937	+	0.810357i	$0.300727\pi$
$654$	0	0
$655$	0	0
$656$	22.1165	0.863505
$657$	0	0
$658$	−24.8349	−0.968165
$659$	9.60511	0.374162	0.187081	−	0.982344i	$-0.440097\pi$
$659$	9.60511	0.374162	0.187081	+	0.982344i	$0.440097\pi$
$660$	0	0
$661$	7.02542	0.273257	0.136629	−	0.990622i	$-0.456373\pi$
$661$	7.02542	0.273257	0.136629	+	0.990622i	$0.456373\pi$
$662$	56.6827	2.20304
$663$	0	0
$664$	102.925	3.99425
$665$	0	0
$666$	0	0
$667$	37.0727	1.43546
$668$	−19.9783	−0.772985
$669$	0	0
$670$	0	0
$671$	5.60883	0.216526
$672$	0	0
$673$	−29.1722	−1.12451	−0.562253	−	0.826965i	$-0.690065\pi$
$673$	−29.1722	−1.12451	−0.562253	+	0.826965i	$0.690065\pi$
$674$	−59.7808	−2.30267
$675$	0	0
$676$	47.9489	1.84419
$677$	16.3752	0.629348	0.314674	−	0.949200i	$-0.398105\pi$
$677$	16.3752	0.629348	0.314674	+	0.949200i	$0.398105\pi$
$678$	0	0
$679$	36.3896	1.39650
$680$	0	0
$681$	0	0
$682$	2.09952	0.0803947
$683$	−24.5781	−0.940453	−0.470227	−	0.882546i	$-0.655828\pi$
$683$	−24.5781	−0.940453	−0.470227	+	0.882546i	$0.655828\pi$
$684$	0	0
$685$	0	0
$686$	−58.6814	−2.24047
$687$	0	0
$688$	−19.6577	−0.749444
$689$	20.7234	0.789497
$690$	0	0
$691$	−23.7050	−0.901781	−0.450891	−	0.892579i	$-0.648894\pi$
$691$	−23.7050	−0.901781	−0.450891	+	0.892579i	$0.648894\pi$
$692$	−37.7145	−1.43369
$693$	0	0
$694$	3.02186	0.114708
$695$	0	0
$696$	0	0
$697$	−13.7572	−0.521093
$698$	29.3299	1.11015
$699$	0	0
$700$	0	0
$701$	17.4254	0.658149	0.329075	−	0.944304i	$-0.393263\pi$
$701$	17.4254	0.658149	0.329075	+	0.944304i	$0.393263\pi$
$702$	0	0
$703$	−3.76220	−0.141894
$704$	−0.909707	−0.0342859
$705$	0	0
$706$	−18.0000	−0.677439
$707$	−42.6491	−1.60399
$708$	0	0
$709$	32.5364	1.22193	0.610964	−	0.791658i	$-0.290782\pi$
$709$	32.5364	1.22193	0.610964	+	0.791658i	$0.290782\pi$
$710$	0	0
$711$	0	0
$712$	49.0290	1.83744
$713$	7.30474	0.273565
$714$	0	0
$715$	0	0
$716$	7.34591	0.274529
$717$	0	0
$718$	−4.66021	−0.173917
$719$	25.5460	0.952705	0.476352	−	0.879254i	$-0.341959\pi$
$719$	25.5460	0.952705	0.476352	+	0.879254i	$0.341959\pi$
$720$	0	0
$721$	19.2682	0.717584
$722$	−47.8289	−1.78001
$723$	0	0
$724$	−65.6091	−2.43834
$725$	0	0
$726$	0	0
$727$	−52.8822	−1.96129	−0.980646	−	0.195790i	$-0.937273\pi$
$727$	−52.8822	−1.96129	−0.980646	+	0.195790i	$0.937273\pi$
$728$	−132.558	−4.91291
$729$	0	0
$730$	0	0
$731$	12.2278	0.452261
$732$	0	0
$733$	42.0473	1.55305	0.776526	−	0.630085i	$-0.216980\pi$
$733$	42.0473	1.55305	0.776526	+	0.630085i	$0.216980\pi$
$734$	−41.3463	−1.52612
$735$	0	0
$736$	36.1504	1.33252
$737$	3.30948	0.121906
$738$	0	0
$739$	0.436836	0.0160693	0.00803463	−	0.999968i	$-0.497442\pi$
$739$	0.436836	0.0160693	0.00803463	+	0.999968i	$0.497442\pi$
$740$	0	0
$741$	0	0
$742$	−47.3155	−1.73701
$743$	35.6551	1.30806	0.654030	−	0.756468i	$-0.273077\pi$
$743$	35.6551	1.30806	0.654030	+	0.756468i	$0.273077\pi$
$744$	0	0
$745$	0	0
$746$	51.9927	1.90359
$747$	0	0
$748$	−15.6019	−0.570463
$749$	29.2305	1.06806
$750$	0	0
$751$	−4.48412	−0.163628	−0.0818139	−	0.996648i	$-0.526071\pi$
$751$	−4.48412	−0.163628	−0.0818139	+	0.996648i	$0.526071\pi$
$752$	15.2255	0.555216
$753$	0	0
$754$	62.8300	2.28813
$755$	0	0
$756$	0	0
$757$	−11.2159	−0.407650	−0.203825	−	0.979007i	$-0.565337\pi$
$757$	−11.2159	−0.407650	−0.203825	+	0.979007i	$0.565337\pi$
$758$	40.0962	1.45636
$759$	0	0
$760$	0	0
$761$	15.6482	0.567247	0.283624	−	0.958936i	$-0.408463\pi$
$761$	15.6482	0.567247	0.283624	+	0.958936i	$0.408463\pi$
$762$	0	0
$763$	−75.7996	−2.74413
$764$	56.7033	2.05145
$765$	0	0
$766$	−70.8687	−2.56059
$767$	−40.5601	−1.46454
$768$	0	0
$769$	35.2245	1.27023	0.635113	−	0.772419i	$-0.280953\pi$
$769$	35.2245	1.27023	0.635113	+	0.772419i	$0.280953\pi$
$770$	0	0
$771$	0	0
$772$	26.8458	0.966200
$773$	28.6029	1.02878	0.514388	−	0.857557i	$-0.328019\pi$
$773$	28.6029	1.02878	0.514388	+	0.857557i	$0.328019\pi$
$774$	0	0
$775$	0	0
$776$	−51.3149	−1.84210
$777$	0	0
$778$	3.55821	0.127568
$779$	−1.26164	−0.0452029
$780$	0	0
$781$	−1.51945	−0.0543700
$782$	−78.7385	−2.81568
$783$	0	0
$784$	83.7777	2.99206
$785$	0	0
$786$	0	0
$787$	16.5801	0.591016	0.295508	−	0.955340i	$-0.404511\pi$
$787$	16.5801	0.591016	0.295508	+	0.955340i	$0.404511\pi$
$788$	96.3457	3.43217
$789$	0	0
$790$	0	0
$791$	−51.5082	−1.83142
$792$	0	0
$793$	33.0727	1.17445
$794$	44.3244	1.57302
$795$	0	0
$796$	−48.2936	−1.71172
$797$	−31.8943	−1.12976	−0.564878	−	0.825175i	$-0.691077\pi$
$797$	−31.8943	−1.12976	−0.564878	+	0.825175i	$0.691077\pi$
$798$	0	0
$799$	−9.47077	−0.335052
$800$	0	0
$801$	0	0
$802$	87.0240	3.07293
$803$	−3.39182	−0.119695
$804$	0	0
$805$	0	0
$806$	12.3799	0.436063
$807$	0	0
$808$	60.1418	2.11578
$809$	−2.55188	−0.0897194	−0.0448597	−	0.998993i	$-0.514284\pi$
$809$	−2.55188	−0.0897194	−0.0448597	+	0.998993i	$0.514284\pi$
$810$	0	0
$811$	48.6746	1.70920	0.854599	−	0.519289i	$-0.173803\pi$
$811$	48.6746	1.70920	0.854599	+	0.519289i	$0.173803\pi$
$812$	−98.8976	−3.47062
$813$	0	0
$814$	20.2766	0.710693
$815$	0	0
$816$	0	0
$817$	1.12138	0.0392320
$818$	−41.0938	−1.43681
$819$	0	0
$820$	0	0
$821$	32.1755	1.12293	0.561466	−	0.827500i	$-0.310238\pi$
$821$	32.1755	1.12293	0.561466	+	0.827500i	$0.310238\pi$
$822$	0	0
$823$	−54.6309	−1.90431	−0.952157	−	0.305609i	$-0.901140\pi$
$823$	−54.6309	−1.90431	−0.952157	+	0.305609i	$0.901140\pi$
$824$	−27.1711	−0.946550
$825$	0	0
$826$	92.6066	3.22220
$827$	−50.2054	−1.74581	−0.872906	−	0.487889i	$-0.837767\pi$
$827$	−50.2054	−1.74581	−0.872906	+	0.487889i	$0.837767\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	−5.36413	−0.185968
$833$	−52.1126	−1.80560
$834$	0	0
$835$	0	0
$836$	−1.43081	−0.0494856
$837$	0	0
$838$	−100.364	−3.46701
$839$	−28.2214	−0.974310	−0.487155	−	0.873315i	$-0.661965\pi$
$839$	−28.2214	−0.974310	−0.487155	+	0.873315i	$0.661965\pi$
$840$	0	0
$841$	−3.24275	−0.111819
$842$	4.45011	0.153361
$843$	0	0
$844$	−1.72935	−0.0595266
$845$	0	0
$846$	0	0
$847$	45.2803	1.55585
$848$	29.0076	0.996124
$849$	0	0
$850$	0	0
$851$	70.5472	2.41833
$852$	0	0
$853$	−2.68454	−0.0919169	−0.0459585	−	0.998943i	$-0.514634\pi$
$853$	−2.68454	−0.0919169	−0.0459585	+	0.998943i	$0.514634\pi$
$854$	−75.5114	−2.58395
$855$	0	0
$856$	−41.2195	−1.40885
$857$	−4.54149	−0.155134	−0.0775672	−	0.996987i	$-0.524715\pi$
$857$	−4.54149	−0.155134	−0.0775672	+	0.996987i	$0.524715\pi$
$858$	0	0
$859$	−28.7304	−0.980270	−0.490135	−	0.871647i	$-0.663052\pi$
$859$	−28.7304	−0.980270	−0.490135	+	0.871647i	$0.663052\pi$
$860$	0	0
$861$	0	0
$862$	24.2986	0.827612
$863$	−47.7233	−1.62452	−0.812259	−	0.583297i	$-0.801763\pi$
$863$	−47.7233	−1.62452	−0.812259	+	0.583297i	$0.801763\pi$
$864$	0	0
$865$	0	0
$866$	64.7451	2.20013
$867$	0	0
$868$	−19.4866	−0.661418
$869$	4.11876	0.139719
$870$	0	0
$871$	19.5145	0.661223
$872$	106.889	3.61972
$873$	0	0
$874$	−7.22089	−0.244250
$875$	0	0
$876$	0	0
$877$	−1.06914	−0.0361024	−0.0180512	−	0.999837i	$-0.505746\pi$
$877$	−1.06914	−0.0361024	−0.0180512	+	0.999837i	$0.505746\pi$
$878$	11.1388	0.375918
$879$	0	0
$880$	0	0
$881$	−34.1937	−1.15201	−0.576007	−	0.817445i	$-0.695390\pi$
$881$	−34.1937	−1.15201	−0.576007	+	0.817445i	$0.695390\pi$
$882$	0	0
$883$	−15.8010	−0.531745	−0.265872	−	0.964008i	$-0.585660\pi$
$883$	−15.8010	−0.531745	−0.265872	+	0.964008i	$0.585660\pi$
$884$	−91.9975	−3.09421
$885$	0	0
$886$	−6.65525	−0.223588
$887$	−46.3918	−1.55769	−0.778843	−	0.627219i	$-0.784193\pi$
$887$	−46.3918	−1.55769	−0.778843	+	0.627219i	$0.784193\pi$
$888$	0	0
$889$	−47.7523	−1.60156
$890$	0	0
$891$	0	0
$892$	69.5095	2.32735
$893$	−0.868539	−0.0290645
$894$	0	0
$895$	0	0
$896$	55.6942	1.86061
$897$	0	0
$898$	−53.0777	−1.77122
$899$	5.07516	0.169266
$900$	0	0
$901$	−18.0437	−0.601123
$902$	6.79967	0.226404
$903$	0	0
$904$	72.6345	2.41579
$905$	0	0
$906$	0	0
$907$	44.6832	1.48368	0.741840	−	0.670577i	$-0.233953\pi$
$907$	44.6832	1.48368	0.741840	+	0.670577i	$0.233953\pi$
$908$	−2.55188	−0.0846872
$909$	0	0
$910$	0	0
$911$	−56.0381	−1.85663	−0.928313	−	0.371800i	$-0.878741\pi$
$911$	−56.0381	−1.85663	−0.928313	+	0.371800i	$0.878741\pi$
$912$	0	0
$913$	13.7572	0.455298
$914$	6.18367	0.204537
$915$	0	0
$916$	7.35918	0.243154
$917$	91.6340	3.02602
$918$	0	0
$919$	35.5145	1.17152	0.585758	−	0.810486i	$-0.300797\pi$
$919$	35.5145	1.17152	0.585758	+	0.810486i	$0.300797\pi$
$920$	0	0
$921$	0	0
$922$	−31.4150	−1.03460
$923$	−8.95948	−0.294905
$924$	0	0
$925$	0	0
$926$	−87.1337	−2.86339
$927$	0	0
$928$	25.1164	0.824487
$929$	15.0839	0.494886	0.247443	−	0.968902i	$-0.420410\pi$
$929$	15.0839	0.494886	0.247443	+	0.968902i	$0.420410\pi$
$930$	0	0
$931$	−4.77911	−0.156629
$932$	−66.7839	−2.18758
$933$	0	0
$934$	100.364	3.28400
$935$	0	0
$936$	0	0
$937$	19.5582	0.638939	0.319469	−	0.947597i	$-0.396495\pi$
$937$	19.5582	0.638939	0.319469	+	0.947597i	$0.396495\pi$
$938$	−44.5554	−1.45478
$939$	0	0
$940$	0	0
$941$	−19.2207	−0.626578	−0.313289	−	0.949658i	$-0.601431\pi$
$941$	−19.2207	−0.626578	−0.313289	+	0.949658i	$0.601431\pi$
$942$	0	0
$943$	23.6577	0.770402
$944$	−56.7741	−1.84784
$945$	0	0
$946$	−6.04372	−0.196498
$947$	34.5055	1.12128	0.560639	−	0.828061i	$-0.310556\pi$
$947$	34.5055	1.12128	0.560639	+	0.828061i	$0.310556\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	0	0
$952$	115.417	3.74069
$953$	−26.1208	−0.846137	−0.423068	−	0.906098i	$-0.639047\pi$
$953$	−26.1208	−0.846137	−0.423068	+	0.906098i	$0.639047\pi$
$954$	0	0
$955$	0	0
$956$	−28.7551	−0.930005
$957$	0	0
$958$	42.6553	1.37813
$959$	38.4003	1.24001
$960$	0	0
$961$	1.00000	0.0322581
$962$	119.562	3.85482
$963$	0	0
$964$	79.2717	2.55317
$965$	0	0
$966$	0	0
$967$	−41.2159	−1.32542	−0.662708	−	0.748878i	$-0.730593\pi$
$967$	−41.2159	−1.32542	−0.662708	+	0.748878i	$0.730593\pi$
$968$	−63.8522	−2.05229
$969$	0	0
$970$	0	0
$971$	5.03399	0.161548	0.0807742	−	0.996732i	$-0.474261\pi$
$971$	5.03399	0.161548	0.0807742	+	0.996732i	$0.474261\pi$
$972$	0	0
$973$	29.3204	0.939969
$974$	35.2736	1.13024
$975$	0	0
$976$	46.2936	1.48182
$977$	−4.89342	−0.156554	−0.0782772	−	0.996932i	$-0.524942\pi$
$977$	−4.89342	−0.156554	−0.0782772	+	0.996932i	$0.524942\pi$
$978$	0	0
$979$	6.55338	0.209447
$980$	0	0
$981$	0	0
$982$	−98.7255	−3.15046
$983$	43.7280	1.39471	0.697353	−	0.716728i	$-0.254361\pi$
$983$	43.7280	1.39471	0.697353	+	0.716728i	$0.254361\pi$
$984$	0	0
$985$	0	0
$986$	−54.7057	−1.74218
$987$	0	0
$988$	−8.43684	−0.268412
$989$	−21.0276	−0.668638
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	4.94889	0.157128
$993$	0	0
$994$	20.4562	0.648832
$995$	0	0
$996$	0	0
$997$	54.0473	1.71169	0.855847	−	0.517229i	$-0.173036\pi$
$997$	54.0473	1.71169	0.855847	+	0.517229i	$0.173036\pi$
$998$	12.6324	0.399873
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.cb.1.6		6
3.2	odd	2	inner	6975.2.a.cb.1.1		6
5.4	even	2		279.2.a.d.1.1	✓	6
15.14	odd	2		279.2.a.d.1.6	yes	6
20.19	odd	2		4464.2.a.bt.1.5		6
60.59	even	2		4464.2.a.bt.1.2		6
155.154	odd	2		8649.2.a.bb.1.1		6
465.464	even	2		8649.2.a.bb.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
279.2.a.d.1.1	✓	6	5.4	even	2
279.2.a.d.1.6	yes	6	15.14	odd	2
4464.2.a.bt.1.2		6	60.59	even	2
4464.2.a.bt.1.5		6	20.19	odd	2
6975.2.a.cb.1.1		6	3.2	odd	2	inner
6975.2.a.cb.1.6		6	1.1	even	1	trivial
8649.2.a.bb.1.1		6	155.154	odd	2
8649.2.a.bb.1.6		6	465.464	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53758 q^{2} +4.43931 q^{4} -4.38955 q^{7} +6.18995 q^{8} +0.827370 q^{11} +4.87862 q^{13} -11.1388 q^{14} +6.82886 q^{16} -4.24779 q^{17} -0.389553 q^{19} +2.09952 q^{22} +7.30474 q^{23} +12.3799 q^{26} -19.4866 q^{28} +5.07516 q^{29} +1.00000 q^{31} +4.94889 q^{32} -10.7791 q^{34} +9.65773 q^{37} -0.988522 q^{38} +3.23868 q^{41} -2.87862 q^{43} +3.67295 q^{44} +18.5364 q^{46} +2.22958 q^{47} +12.2682 q^{49} +21.6577 q^{52} +4.24779 q^{53} -27.1711 q^{56} +12.8786 q^{58} -8.31384 q^{59} +6.77911 q^{61} +2.53758 q^{62} -1.09952 q^{64} +4.00000 q^{67} -18.8573 q^{68} -1.83648 q^{71} -4.09952 q^{73} +24.5073 q^{74} -1.72935 q^{76} -3.63178 q^{77} +4.97814 q^{79} +8.21842 q^{82} +16.6277 q^{83} -7.30474 q^{86} +5.12138 q^{88} +7.92074 q^{89} -21.4150 q^{91} +32.4280 q^{92} +5.65773 q^{94} -8.29004 q^{97} +31.1315 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4} - 8 q^{7} + 8 q^{16} + 16 q^{19} + 20 q^{22} - 18 q^{28} + 6 q^{31} - 28 q^{34} - 8 q^{37} + 12 q^{43} + 16 q^{46} + 26 q^{49} + 64 q^{52} + 48 q^{58} + 4 q^{61} - 14 q^{64} + 24 q^{67} - 32 q^{73}+ \cdots - 24 q^{97}+O(q^{100}) \) 6 * q + 12 * q^4 - 8 * q^7 + 8 * q^16 + 16 * q^19 + 20 * q^22 - 18 * q^28 + 6 * q^31 - 28 * q^34 - 8 * q^37 + 12 * q^43 + 16 * q^46 + 26 * q^49 + 64 * q^52 + 48 * q^58 + 4 * q^61 - 14 * q^64 + 24 * q^67 - 32 * q^73 + 30 * q^76 + 8 * q^79 - 2 * q^82 + 60 * q^88 - 4 * q^91 - 32 * q^94 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

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