LMFDB - Embedded newform 4100.2.d.c.1149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4100,2,Mod(1149,4100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4100.1149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4100 = 2^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4100.d (of order $2$, degree $1$, 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:	$32.7386648287$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 4x^{5} + 14x^{4} - 14x^{3} + 8x^{2} + 4x + 1 $ x^8 - 2x^7 + 2x^6 + 4x^5 + 14x^4 - 14x^3 + 8x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 164)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.2
Root			$-0.187509 + 0.187509i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.c.1149.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.92968i q^{3} +2.77840i q^{7} -5.58303 q^{9} +O(q^{10})$	$q-2.92968i q^{3} +2.77840i q^{7} -5.58303 q^{9} +2.02837 q^{11} +3.10932i q^{13} -7.91609i q^{17} -2.17964 q^{19} +8.13983 q^{21} +4.75004i q^{23} +7.56744i q^{27} -3.85936 q^{29} -10.6661 q^{31} -5.94246i q^{33} +6.58303i q^{37} +9.10932 q^{39} -1.00000 q^{41} -4.80677i q^{43} +4.03900i q^{47} -0.719526 q^{49} -23.1916 q^{51} +1.94327i q^{53} +6.38566i q^{57} -4.75004 q^{59} +2.89068 q^{61} -15.5119i q^{63} +5.97163i q^{67} +13.9161 q^{69} +14.4026 q^{71} +9.24916i q^{73} +5.63562i q^{77} +0.320282 q^{79} +5.42111 q^{81} +3.69745i q^{83} +11.3067i q^{87} -8.96868 q^{89} -8.63895 q^{91} +31.2484i q^{93} +17.8322i q^{97} -11.3244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} + 8 q^{11} - 12 q^{19} + 8 q^{29} - 16 q^{31} + 48 q^{39} - 8 q^{41} - 32 q^{49} - 8 q^{51} - 24 q^{59} + 48 q^{61} + 56 q^{69} - 4 q^{71} + 36 q^{79} + 56 q^{81} - 8 q^{89} + 72 q^{91} - 116 q^{99}+O(q^{100}) $ 8 * q - 24 * q^9 + 8 * q^11 - 12 * q^19 + 8 * q^29 - 16 * q^31 + 48 * q^39 - 8 * q^41 - 32 * q^49 - 8 * q^51 - 24 * q^59 + 48 * q^61 + 56 * q^69 - 4 * q^71 + 36 * q^79 + 56 * q^81 - 8 * q^89 + 72 * q^91 - 116 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times$.

$n$	$1477$	$2051$	$3901$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 2.92968i	− 1.69145i	−0.533618	−	0.845726i	$-0.679168\pi$
$3$	− 2.92968i	− 1.69145i	0.533618	−	0.845726i	$-0.320832\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.77840i	1.05014i	0.851060	+	0.525069i	$0.175960\pi$
$7$	2.77840i	1.05014i	−0.851060	+	0.525069i	$0.824040\pi$
$8$	0	0
$9$	−5.58303	−1.86101
$10$	0	0
$11$	2.02837	0.611575	0.305788	−	0.952100i	$-0.401080\pi$
$11$	2.02837	0.611575	0.305788	+	0.952100i	$0.401080\pi$
$12$	0	0
$13$	3.10932i	0.862371i	0.902263	+	0.431186i	$0.141905\pi$
$13$	3.10932i	0.862371i	−0.902263	+	0.431186i	$0.858095\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.91609i	− 1.91993i	−0.280112	−	0.959967i	$-0.590372\pi$
$17$	− 7.91609i	− 1.91993i	0.280112	−	0.959967i	$-0.409628\pi$
$18$	0	0
$19$	−2.17964	−0.500044	−0.250022	−	0.968240i	$-0.580438\pi$
$19$	−2.17964	−0.500044	−0.250022	+	0.968240i	$0.580438\pi$
$20$	0	0
$21$	8.13983	1.77626
$22$	0	0
$23$	4.75004i	0.990451i	0.868764	+	0.495226i	$0.164915\pi$
$23$	4.75004i	0.990451i	−0.868764	+	0.495226i	$0.835085\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	7.56744i	1.45636i
$28$	0	0
$29$	−3.85936	−0.716665	−0.358333	−	0.933594i	$-0.616655\pi$
$29$	−3.85936	−0.716665	−0.358333	+	0.933594i	$0.616655\pi$
$30$	0	0
$31$	−10.6661	−1.91569	−0.957847	−	0.287280i	$-0.907249\pi$
$31$	−10.6661	−1.91569	−0.957847	+	0.287280i	$0.907249\pi$
$32$	0	0
$33$	− 5.94246i	− 1.03445i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.58303i	1.08224i	0.840944	+	0.541122i	$0.182000\pi$
$37$	6.58303i	1.08224i	−0.840944	+	0.541122i	$0.818000\pi$
$38$	0	0
$39$	9.10932	1.45866
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	− 4.80677i	− 0.733025i	−0.930413	−	0.366513i	$-0.880552\pi$
$43$	− 4.80677i	− 0.733025i	0.930413	−	0.366513i	$-0.119448\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.03900i	0.589149i	0.955628	+	0.294575i	$0.0951779\pi$
$47$	4.03900i	0.589149i	−0.955628	+	0.294575i	$0.904822\pi$
$48$	0	0
$49$	−0.719526	−0.102789
$50$	0	0
$51$	−23.1916	−3.24748
$52$	0	0
$53$	1.94327i	0.266928i	0.991054	+	0.133464i	$0.0426101\pi$
$53$	1.94327i	0.266928i	−0.991054	+	0.133464i	$0.957390\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.38566i	0.845801i
$58$	0	0
$59$	−4.75004	−0.618402	−0.309201	−	0.950997i	$-0.600062\pi$
$59$	−4.75004	−0.618402	−0.309201	+	0.950997i	$0.600062\pi$
$60$	0	0
$61$	2.89068	0.370113	0.185057	−	0.982728i	$-0.440753\pi$
$61$	2.89068	0.370113	0.185057	+	0.982728i	$0.440753\pi$
$62$	0	0
$63$	− 15.5119i	− 1.95432i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.97163i	0.729551i	0.931095	+	0.364776i	$0.118854\pi$
$67$	5.97163i	0.729551i	−0.931095	+	0.364776i	$0.881146\pi$
$68$	0	0
$69$	13.9161	1.67530
$70$	0	0
$71$	14.4026	1.70927	0.854636	−	0.519228i	$-0.173780\pi$
$71$	14.4026	1.70927	0.854636	+	0.519228i	$0.173780\pi$
$72$	0	0
$73$	9.24916i	1.08253i	0.840851	+	0.541266i	$0.182055\pi$
$73$	9.24916i	1.08253i	−0.840851	+	0.541266i	$0.817945\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.63562i	0.642238i
$78$	0	0
$79$	0.320282	0.0360345	0.0180173	−	0.999838i	$-0.494265\pi$
$79$	0.320282	0.0360345	0.0180173	+	0.999838i	$0.494265\pi$
$80$	0	0
$81$	5.42111	0.602346
$82$	0	0
$83$	3.69745i	0.405847i	0.979195	+	0.202924i	$0.0650443\pi$
$83$	3.69745i	0.405847i	−0.979195	+	0.202924i	$0.934956\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.3067i	1.21220i
$88$	0	0
$89$	−8.96868	−0.950679	−0.475339	−	0.879803i	$-0.657675\pi$
$89$	−8.96868	−0.950679	−0.475339	+	0.879803i	$0.657675\pi$
$90$	0	0
$91$	−8.63895	−0.905608
$92$	0	0
$93$	31.2484i	3.24030i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.8322i	1.81058i	0.424790	+	0.905292i	$0.360348\pi$
$97$	17.8322i	1.81058i	−0.424790	+	0.905292i	$0.639652\pi$
$98$	0	0
$99$	−11.3244	−1.13815
$100$	0	0
$101$	−13.0254	−1.29608	−0.648039	−	0.761607i	$-0.724410\pi$
$101$	−13.0254	−1.29608	−0.648039	+	0.761607i	$0.724410\pi$
$102$	0	0
$103$	8.05673i	0.793853i	0.917850	+	0.396927i	$0.129923\pi$
$103$	8.05673i	0.793853i	−0.917850	+	0.396927i	$0.870077\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.41602i	− 0.426912i	−0.976953	−	0.213456i	$-0.931528\pi$
$107$	− 4.41602i	− 0.426912i	0.976953	−	0.213456i	$-0.0684721\pi$
$108$	0	0
$109$	−18.5255	−1.77442	−0.887210	−	0.461366i	$-0.847360\pi$
$109$	−18.5255	−1.77442	−0.887210	+	0.461366i	$0.847360\pi$
$110$	0	0
$111$	19.2862	1.83056
$112$	0	0
$113$	8.08310i	0.760394i	0.924905	+	0.380197i	$0.124144\pi$
$113$	8.08310i	0.760394i	−0.924905	+	0.380197i	$0.875856\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 17.3594i	− 1.60488i
$118$	0	0
$119$	21.9941	2.01620
$120$	0	0
$121$	−6.88573	−0.625976
$122$	0	0
$123$	2.92968i	0.264160i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.44748i	0.394650i	0.980338	+	0.197325i	$0.0632255\pi$
$127$	4.44748i	0.394650i	−0.980338	+	0.197325i	$0.936775\pi$
$128$	0	0
$129$	−14.0823	−1.23988
$130$	0	0
$131$	9.85936	0.861416	0.430708	−	0.902491i	$-0.358264\pi$
$131$	9.85936	0.861416	0.430708	+	0.902491i	$0.358264\pi$
$132$	0	0
$133$	− 6.05593i	− 0.525115i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.30255i	− 0.196720i	−0.995151	−	0.0983602i	$-0.968640\pi$
$137$	− 2.30255i	− 0.196720i	0.995151	−	0.0983602i	$-0.0313597\pi$
$138$	0	0
$139$	−5.16605	−0.438179	−0.219090	−	0.975705i	$-0.570309\pi$
$139$	−5.16605	−0.438179	−0.219090	+	0.975705i	$0.570309\pi$
$140$	0	0
$141$	11.8330	0.996517
$142$	0	0
$143$	6.30684i	0.527405i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.10798i	0.173863i
$148$	0	0
$149$	−3.85936	−0.316171	−0.158086	−	0.987425i	$-0.550532\pi$
$149$	−3.85936	−0.316171	−0.158086	+	0.987425i	$0.550532\pi$
$150$	0	0
$151$	−4.71103	−0.383379	−0.191689	−	0.981456i	$-0.561397\pi$
$151$	−4.71103	−0.383379	−0.191689	+	0.981456i	$0.561397\pi$
$152$	0	0
$153$	44.1958i	3.57302i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.30684i	0.662958i	0.943463	+	0.331479i	$0.107548\pi$
$157$	8.30684i	0.662958i	−0.943463	+	0.331479i	$0.892452\pi$
$158$	0	0
$159$	5.69316	0.451497
$160$	0	0
$161$	−13.1975	−1.04011
$162$	0	0
$163$	13.6348i	1.06796i	0.845497	+	0.533981i	$0.179304\pi$
$163$	13.6348i	1.06796i	−0.845497	+	0.533981i	$0.820696\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.97163i	0.771628i	0.922577	+	0.385814i	$0.126079\pi$
$167$	9.97163i	0.771628i	−0.922577	+	0.385814i	$0.873921\pi$
$168$	0	0
$169$	3.33211	0.256316
$170$	0	0
$171$	12.1690	0.930587
$172$	0	0
$173$	− 9.11361i	− 0.692895i	−0.938069	−	0.346448i	$-0.887388\pi$
$173$	− 9.11361i	− 0.692895i	0.938069	−	0.346448i	$-0.112612\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.9161i	1.04600i
$178$	0	0
$179$	12.9403	0.967205	0.483602	−	0.875288i	$-0.339328\pi$
$179$	12.9403	0.967205	0.483602	+	0.875288i	$0.339328\pi$
$180$	0	0
$181$	8.16191	0.606670	0.303335	−	0.952884i	$-0.401900\pi$
$181$	8.16191	0.606670	0.303335	+	0.952884i	$0.401900\pi$
$182$	0	0
$183$	− 8.46876i	− 0.626029i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 16.0567i	− 1.17418i
$188$	0	0
$189$	−21.0254	−1.52937
$190$	0	0
$191$	4.52829	0.327656	0.163828	−	0.986489i	$-0.447616\pi$
$191$	4.52829	0.327656	0.163828	+	0.986489i	$0.447616\pi$
$192$	0	0
$193$	14.1662i	1.01971i	0.860262	+	0.509853i	$0.170300\pi$
$193$	14.1662i	1.01971i	−0.860262	+	0.509853i	$0.829700\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.5042i	− 0.890888i	−0.895310	−	0.445444i	$-0.853046\pi$
$197$	− 12.5042i	− 0.890888i	0.895310	−	0.445444i	$-0.146954\pi$
$198$	0	0
$199$	14.5853	1.03393	0.516963	−	0.856008i	$-0.327062\pi$
$199$	14.5853	1.03393	0.516963	+	0.856008i	$0.327062\pi$
$200$	0	0
$201$	17.4950	1.23400
$202$	0	0
$203$	− 10.7229i	− 0.752597i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 26.5196i	− 1.84324i
$208$	0	0
$209$	−4.42111	−0.305815
$210$	0	0
$211$	−26.3860	−1.81649	−0.908245	−	0.418439i	$-0.862577\pi$
$211$	−26.3860	−1.81649	−0.908245	+	0.418439i	$0.862577\pi$
$212$	0	0
$213$	− 42.1950i	− 2.89115i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 29.6348i	− 2.01174i
$218$	0	0
$219$	27.0971	1.83105
$220$	0	0
$221$	24.6137	1.65570
$222$	0	0
$223$	1.89482i	0.126886i	0.997985	+	0.0634432i	$0.0202082\pi$
$223$	1.89482i	0.126886i	−0.997985	+	0.0634432i	$0.979792\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.9805i	− 1.65801i	−0.559238	−	0.829007i	$-0.688906\pi$
$227$	− 24.9805i	− 1.65801i	0.559238	−	0.829007i	$-0.311094\pi$
$228$	0	0
$229$	2.52549	0.166889	0.0834446	−	0.996512i	$-0.473408\pi$
$229$	2.52549	0.166889	0.0834446	+	0.996512i	$0.473408\pi$
$230$	0	0
$231$	16.5106	1.08632
$232$	0	0
$233$	− 9.24835i	− 0.605880i	−0.953010	−	0.302940i	$-0.902032\pi$
$233$	− 9.24835i	− 0.605880i	0.953010	−	0.302940i	$-0.0979681\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 0.938323i	− 0.0609507i
$238$	0	0
$239$	−2.34746	−0.151844	−0.0759222	−	0.997114i	$-0.524190\pi$
$239$	−2.34746	−0.151844	−0.0759222	+	0.997114i	$0.524190\pi$
$240$	0	0
$241$	2.89068	0.186205	0.0931024	−	0.995657i	$-0.470322\pi$
$241$	2.89068	0.186205	0.0931024	+	0.995657i	$0.470322\pi$
$242$	0	0
$243$	6.82021i	0.437516i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.77721i	− 0.431224i
$248$	0	0
$249$	10.8323	0.686471
$250$	0	0
$251$	24.9730	1.57628	0.788140	−	0.615496i	$-0.211044\pi$
$251$	24.9730	1.57628	0.788140	+	0.615496i	$0.211044\pi$
$252$	0	0
$253$	9.63481i	0.605736i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.7500i	0.670569i	0.942117	+	0.335284i	$0.108832\pi$
$257$	10.7500i	0.670569i	−0.942117	+	0.335284i	$0.891168\pi$
$258$	0	0
$259$	−18.2903	−1.13650
$260$	0	0
$261$	21.5469	1.33372
$262$	0	0
$263$	4.90250i	0.302301i	0.988511	+	0.151151i	$0.0482979\pi$
$263$	4.90250i	0.302301i	−0.988511	+	0.151151i	$0.951702\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	26.2754i	1.60803i
$268$	0	0
$269$	−3.32797	−0.202910	−0.101455	−	0.994840i	$-0.532350\pi$
$269$	−3.32797	−0.202910	−0.101455	+	0.994840i	$0.532350\pi$
$270$	0	0
$271$	4.58222	0.278350	0.139175	−	0.990268i	$-0.455555\pi$
$271$	4.58222	0.278350	0.139175	+	0.990268i	$0.455555\pi$
$272$	0	0
$273$	25.3094i	1.53179i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 24.2178i	− 1.45511i	−0.686050	−	0.727555i	$-0.740657\pi$
$277$	− 24.2178i	− 1.45511i	0.686050	−	0.727555i	$-0.259343\pi$
$278$	0	0
$279$	59.5493	3.56512
$280$	0	0
$281$	6.61369	0.394540	0.197270	−	0.980349i	$-0.436792\pi$
$281$	6.61369	0.394540	0.197270	+	0.980349i	$0.436792\pi$
$282$	0	0
$283$	− 5.16605i	− 0.307090i	−0.988142	−	0.153545i	$-0.950931\pi$
$283$	− 5.16605i	− 0.307090i	0.988142	−	0.153545i	$-0.0490690\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.77840i	− 0.164004i
$288$	0	0
$289$	−45.6645	−2.68615
$290$	0	0
$291$	52.2426	3.06252
$292$	0	0
$293$	10.2458i	0.598567i	0.954164	+	0.299284i	$0.0967477\pi$
$293$	10.2458i	0.598567i	−0.954164	+	0.299284i	$0.903252\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	15.3495i	0.890671i
$298$	0	0
$299$	−14.7694	−0.854137
$300$	0	0
$301$	13.3551	0.769778
$302$	0	0
$303$	38.1603i	2.19225i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 18.2754i	− 1.04303i	−0.853242	−	0.521515i	$-0.825367\pi$
$307$	− 18.2754i	− 1.04303i	0.853242	−	0.521515i	$-0.174633\pi$
$308$	0	0
$309$	23.6036	1.34276
$310$	0	0
$311$	4.50303	0.255343	0.127672	−	0.991816i	$-0.459250\pi$
$311$	4.50303	0.255343	0.127672	+	0.991816i	$0.459250\pi$
$312$	0	0
$313$	− 6.56095i	− 0.370847i	−0.982659	−	0.185423i	$-0.940634\pi$
$313$	− 6.56095i	− 0.370847i	0.982659	−	0.185423i	$-0.0593656\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.8363i	0.720960i	0.932767	+	0.360480i	$0.117387\pi$
$317$	12.8363i	0.720960i	−0.932767	+	0.360480i	$0.882613\pi$
$318$	0	0
$319$	−7.82820	−0.438295
$320$	0	0
$321$	−12.9375	−0.722102
$322$	0	0
$323$	17.2543i	0.960052i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	54.2738i	3.00135i
$328$	0	0
$329$	−11.2220	−0.618688
$330$	0	0
$331$	−8.99705	−0.494523	−0.247261	−	0.968949i	$-0.579531\pi$
$331$	−8.99705	−0.494523	−0.247261	+	0.968949i	$0.579531\pi$
$332$	0	0
$333$	− 36.7532i	− 2.01406i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 35.0502i	− 1.90930i	−0.297723	−	0.954652i	$-0.596227\pi$
$337$	− 35.0502i	− 1.90930i	0.297723	−	0.954652i	$-0.403773\pi$
$338$	0	0
$339$	23.6809	1.28617
$340$	0	0
$341$	−21.6348	−1.17159
$342$	0	0
$343$	17.4497i	0.942195i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.8982i	1.01451i	0.861796	+	0.507255i	$0.169340\pi$
$347$	18.8982i	1.01451i	−0.861796	+	0.507255i	$0.830660\pi$
$348$	0	0
$349$	23.1653	1.24001	0.620004	−	0.784599i	$-0.287131\pi$
$349$	23.1653	1.24001	0.620004	+	0.784599i	$0.287131\pi$
$350$	0	0
$351$	−23.5296	−1.25592
$352$	0	0
$353$	1.55171i	0.0825893i	0.999147	+	0.0412946i	$0.0131482\pi$
$353$	1.55171i	0.0825893i	−0.999147	+	0.0412946i	$0.986852\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 64.4357i	− 3.41030i
$358$	0	0
$359$	4.28128	0.225957	0.112979	−	0.993597i	$-0.463961\pi$
$359$	4.28128	0.225957	0.112979	+	0.993597i	$0.463961\pi$
$360$	0	0
$361$	−14.2492	−0.749956
$362$	0	0
$363$	20.1730i	1.05881i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 15.3067i	− 0.799003i	−0.916733	−	0.399501i	$-0.869183\pi$
$367$	− 15.3067i	− 0.799003i	0.916733	−	0.399501i	$-0.130817\pi$
$368$	0	0
$369$	5.58303	0.290641
$370$	0	0
$371$	−5.39918	−0.280312
$372$	0	0
$373$	5.39489i	0.279337i	0.990198	+	0.139668i	$0.0446037\pi$
$373$	5.39489i	0.279337i	−0.990198	+	0.139668i	$0.955396\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	−26.8009	−1.37667	−0.688334	−	0.725394i	$-0.741658\pi$
$379$	−26.8009	−1.37667	−0.688334	+	0.725394i	$0.741658\pi$
$380$	0	0
$381$	13.0297	0.667532
$382$	0	0
$383$	− 21.6675i	− 1.10716i	−0.832797	−	0.553578i	$-0.813262\pi$
$383$	− 21.6675i	− 1.10716i	0.832797	−	0.553578i	$-0.186738\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.8363i	1.36417i
$388$	0	0
$389$	−25.7187	−1.30399	−0.651995	−	0.758223i	$-0.726068\pi$
$389$	−25.7187	−1.30399	−0.651995	+	0.758223i	$0.726068\pi$
$390$	0	0
$391$	37.6017	1.90160
$392$	0	0
$393$	− 28.8848i	− 1.45704i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	28.1603i	1.41333i	0.707551	+	0.706663i	$0.249800\pi$
$397$	28.1603i	1.41333i	−0.707551	+	0.706663i	$0.750200\pi$
$398$	0	0
$399$	−17.7419	−0.888208
$400$	0	0
$401$	−10.7178	−0.535220	−0.267610	−	0.963527i	$-0.586234\pi$
$401$	−10.7178	−0.535220	−0.267610	+	0.963527i	$0.586234\pi$
$402$	0	0
$403$	− 33.1644i	− 1.65204i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13.3528i	0.661873i
$408$	0	0
$409$	−22.7492	−1.12488	−0.562439	−	0.826839i	$-0.690137\pi$
$409$	−22.7492	−1.12488	−0.562439	+	0.826839i	$0.690137\pi$
$410$	0	0
$411$	−6.74575	−0.332743
$412$	0	0
$413$	− 13.1975i	− 0.649408i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	15.1349i	0.741159i
$418$	0	0
$419$	7.02542	0.343214	0.171607	−	0.985165i	$-0.445104\pi$
$419$	7.02542	0.343214	0.171607	+	0.985165i	$0.445104\pi$
$420$	0	0
$421$	16.5992	0.808996	0.404498	−	0.914539i	$-0.367446\pi$
$421$	16.5992	0.808996	0.404498	+	0.914539i	$0.367446\pi$
$422$	0	0
$423$	− 22.5499i	− 1.09641i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.03147i	0.388670i
$428$	0	0
$429$	18.4770	0.892080
$430$	0	0
$431$	−10.7229	−0.516502	−0.258251	−	0.966078i	$-0.583146\pi$
$431$	−10.7229	−0.516502	−0.258251	+	0.966078i	$0.583146\pi$
$432$	0	0
$433$	− 31.1644i	− 1.49767i	−0.662758	−	0.748834i	$-0.730614\pi$
$433$	− 31.1644i	− 1.49767i	0.662758	−	0.748834i	$-0.269386\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 10.3534i	− 0.495270i
$438$	0	0
$439$	34.7981	1.66082	0.830411	−	0.557152i	$-0.188106\pi$
$439$	34.7981	1.66082	0.830411	+	0.557152i	$0.188106\pi$
$440$	0	0
$441$	4.01714	0.191292
$442$	0	0
$443$	25.8806i	1.22963i	0.788673	+	0.614813i	$0.210769\pi$
$443$	25.8806i	1.22963i	−0.788673	+	0.614813i	$0.789231\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	11.3067i	0.534788i
$448$	0	0
$449$	−27.2231	−1.28474	−0.642368	−	0.766396i	$-0.722048\pi$
$449$	−27.2231	−1.28474	−0.642368	+	0.766396i	$0.722048\pi$
$450$	0	0
$451$	−2.02837	−0.0955120
$452$	0	0
$453$	13.8018i	0.648466i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 6.19767i	− 0.289915i	−0.989438	−	0.144957i	$-0.953695\pi$
$457$	− 6.19767i	− 0.289915i	0.989438	−	0.144957i	$-0.0463045\pi$
$458$	0	0
$459$	59.9046	2.79611
$460$	0	0
$461$	−10.9170	−0.508458	−0.254229	−	0.967144i	$-0.581822\pi$
$461$	−10.9170	−0.508458	−0.254229	+	0.967144i	$0.581822\pi$
$462$	0	0
$463$	− 4.07047i	− 0.189171i	−0.995517	−	0.0945854i	$-0.969847\pi$
$463$	− 4.07047i	− 0.189171i	0.995517	−	0.0945854i	$-0.0301525\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.63657i	− 0.399653i	−0.979831	−	0.199826i	$-0.935962\pi$
$467$	− 8.63657i	− 0.399653i	0.979831	−	0.199826i	$-0.0640379\pi$
$468$	0	0
$469$	−16.5916	−0.766129
$470$	0	0
$471$	24.3364	1.12136
$472$	0	0
$473$	− 9.74989i	− 0.448300i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 10.8493i	− 0.496756i
$478$	0	0
$479$	23.5537	1.07620	0.538098	−	0.842882i	$-0.319143\pi$
$479$	23.5537	1.07620	0.538098	+	0.842882i	$0.319143\pi$
$480$	0	0
$481$	−20.4688	−0.933295
$482$	0	0
$483$	38.6645i	1.75930i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19.9102i	0.902217i	0.892469	+	0.451108i	$0.148971\pi$
$487$	19.9102i	0.902217i	−0.892469	+	0.451108i	$0.851029\pi$
$488$	0	0
$489$	39.9456	1.80640
$490$	0	0
$491$	29.7585	1.34298	0.671490	−	0.741013i	$-0.265654\pi$
$491$	29.7585	1.34298	0.671490	+	0.741013i	$0.265654\pi$
$492$	0	0
$493$	30.5511i	1.37595i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.0162i	1.79497i
$498$	0	0
$499$	−18.9297	−0.847409	−0.423704	−	0.905801i	$-0.639270\pi$
$499$	−18.9297	−0.847409	−0.423704	+	0.905801i	$0.639270\pi$
$500$	0	0
$501$	29.2137	1.30517
$502$	0	0
$503$	1.93382i	0.0862248i	0.999070	+	0.0431124i	$0.0137274\pi$
$503$	1.93382i	0.0862248i	−0.999070	+	0.0431124i	$0.986273\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.76202i	− 0.433546i
$508$	0	0
$509$	−34.6602	−1.53629	−0.768144	−	0.640277i	$-0.778819\pi$
$509$	−34.6602	−1.53629	−0.768144	+	0.640277i	$0.778819\pi$
$510$	0	0
$511$	−25.6979	−1.13681
$512$	0	0
$513$	− 16.4943i	− 0.728242i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.19258i	0.360309i
$518$	0	0
$519$	−26.7000	−1.17200
$520$	0	0
$521$	−6.33402	−0.277498	−0.138749	−	0.990328i	$-0.544308\pi$
$521$	−6.33402	−0.277498	−0.138749	+	0.990328i	$0.544308\pi$
$522$	0	0
$523$	29.9460i	1.30944i	0.755869	+	0.654722i	$0.227215\pi$
$523$	29.9460i	1.30944i	−0.755869	+	0.654722i	$0.772785\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	84.4341i	3.67801i
$528$	0	0
$529$	0.437142	0.0190062
$530$	0	0
$531$	26.5196	1.15085
$532$	0	0
$533$	− 3.10932i	− 0.134680i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 37.9110i	− 1.63598i
$538$	0	0
$539$	−1.45946	−0.0628635
$540$	0	0
$541$	0.834750	0.0358887	0.0179443	−	0.999839i	$-0.494288\pi$
$541$	0.834750	0.0358887	0.0179443	+	0.999839i	$0.494288\pi$
$542$	0	0
$543$	− 23.9118i	− 1.02615i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.6527i	− 0.883045i	−0.897250	−	0.441523i	$-0.854438\pi$
$547$	− 20.6527i	− 0.883045i	0.897250	−	0.441523i	$-0.145562\pi$
$548$	0	0
$549$	−16.1387	−0.688784
$550$	0	0
$551$	8.41203	0.358364
$552$	0	0
$553$	0.889872i	0.0378412i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8.72447i	− 0.369668i	−0.982770	−	0.184834i	$-0.940825\pi$
$557$	− 8.72447i	− 0.369668i	0.982770	−	0.184834i	$-0.0591747\pi$
$558$	0	0
$559$	14.9458	0.632140
$560$	0	0
$561$	−47.0411	−1.98608
$562$	0	0
$563$	− 9.80144i	− 0.413081i	−0.978438	−	0.206541i	$-0.933779\pi$
$563$	− 9.80144i	− 0.413081i	0.978438	−	0.206541i	$-0.0662206\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	15.0620i	0.632546i
$568$	0	0
$569$	−25.1570	−1.05464	−0.527318	−	0.849668i	$-0.676802\pi$
$569$	−25.1570	−1.05464	−0.527318	+	0.849668i	$0.676802\pi$
$570$	0	0
$571$	27.1928	1.13798	0.568992	−	0.822343i	$-0.307334\pi$
$571$	27.1928	1.13798	0.568992	+	0.822343i	$0.307334\pi$
$572$	0	0
$573$	− 13.2664i	− 0.554214i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 0.365186i	− 0.0152029i	−0.999971	−	0.00760144i	$-0.997580\pi$
$577$	− 0.365186i	− 0.0152029i	0.999971	−	0.00760144i	$-0.00241964\pi$
$578$	0	0
$579$	41.5025	1.72478
$580$	0	0
$581$	−10.2730	−0.426196
$582$	0	0
$583$	3.94166i	0.163247i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 37.7242i	− 1.55704i	−0.627617	−	0.778522i	$-0.715970\pi$
$587$	− 37.7242i	− 1.55704i	0.627617	−	0.778522i	$-0.284030\pi$
$588$	0	0
$589$	23.2484	0.957932
$590$	0	0
$591$	−36.6334	−1.50689
$592$	0	0
$593$	45.1333i	1.85340i	0.375800	+	0.926701i	$0.377368\pi$
$593$	45.1333i	1.85340i	−0.375800	+	0.926701i	$0.622632\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 42.7303i	− 1.74884i
$598$	0	0
$599$	−15.4982	−0.633238	−0.316619	−	0.948553i	$-0.602548\pi$
$599$	−15.4982	−0.633238	−0.316619	+	0.948553i	$0.602548\pi$
$600$	0	0
$601$	−5.18924	−0.211674	−0.105837	−	0.994384i	$-0.533752\pi$
$601$	−5.18924	−0.211674	−0.105837	+	0.994384i	$0.533752\pi$
$602$	0	0
$603$	− 33.3398i	− 1.35770i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16.8280i	− 0.683029i	−0.939876	−	0.341515i	$-0.889060\pi$
$607$	− 16.8280i	− 0.683029i	0.939876	−	0.341515i	$-0.110940\pi$
$608$	0	0
$609$	−31.4146	−1.27298
$610$	0	0
$611$	−12.5586	−0.508065
$612$	0	0
$613$	− 25.1042i	− 1.01395i	−0.861961	−	0.506975i	$-0.830764\pi$
$613$	− 25.1042i	− 1.01395i	0.861961	−	0.506975i	$-0.169236\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.3324i	0.577001i	0.957480	+	0.288501i	$0.0931567\pi$
$617$	14.3324i	0.577001i	−0.957480	+	0.288501i	$0.906843\pi$
$618$	0	0
$619$	−19.2712	−0.774576	−0.387288	−	0.921959i	$-0.626588\pi$
$619$	−19.2712	−0.774576	−0.387288	+	0.921959i	$0.626588\pi$
$620$	0	0
$621$	−35.9456	−1.44245
$622$	0	0
$623$	− 24.9186i	− 0.998344i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12.9524i	0.517271i
$628$	0	0
$629$	52.1119	2.07784
$630$	0	0
$631$	−30.1856	−1.20167	−0.600834	−	0.799374i	$-0.705165\pi$
$631$	−30.1856	−1.20167	−0.600834	+	0.799374i	$0.705165\pi$
$632$	0	0
$633$	77.3027i	3.07251i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.23724i	− 0.0886427i
$638$	0	0
$639$	−80.4100	−3.18097
$640$	0	0
$641$	−25.9984	−1.02687	−0.513437	−	0.858127i	$-0.671628\pi$
$641$	−25.9984	−1.02687	−0.513437	+	0.858127i	$0.671628\pi$
$642$	0	0
$643$	− 11.4402i	− 0.451159i	−0.974225	−	0.225580i	$-0.927572\pi$
$643$	− 11.4402i	− 0.451159i	0.974225	−	0.225580i	$-0.0724276\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.6915i	1.48181i	0.671611	+	0.740904i	$0.265603\pi$
$647$	37.6915i	1.48181i	−0.671611	+	0.740904i	$0.734397\pi$
$648$	0	0
$649$	−9.63481	−0.378200
$650$	0	0
$651$	−86.8205	−3.40277
$652$	0	0
$653$	43.6814i	1.70938i	0.519136	+	0.854692i	$0.326254\pi$
$653$	43.6814i	1.70938i	−0.519136	+	0.854692i	$0.673746\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 51.6383i	− 2.01460i
$658$	0	0
$659$	−25.0900	−0.977367	−0.488684	−	0.872461i	$-0.662523\pi$
$659$	−25.0900	−0.977367	−0.488684	+	0.872461i	$0.662523\pi$
$660$	0	0
$661$	−37.7392	−1.46788	−0.733942	−	0.679212i	$-0.762322\pi$
$661$	−37.7392	−1.46788	−0.733942	+	0.679212i	$0.762322\pi$
$662$	0	0
$663$	− 72.1102i	− 2.80053i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 18.3321i	− 0.709822i
$668$	0	0
$669$	5.55121	0.214622
$670$	0	0
$671$	5.86335	0.226352
$672$	0	0
$673$	− 3.63642i	− 0.140174i	−0.997541	−	0.0700869i	$-0.977672\pi$
$673$	− 3.63642i	− 0.140174i	0.997541	−	0.0700869i	$-0.0223277\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.1948i	1.12205i	0.827800	+	0.561024i	$0.189592\pi$
$677$	29.1948i	1.12205i	−0.827800	+	0.561024i	$0.810408\pi$
$678$	0	0
$679$	−49.5450	−1.90136
$680$	0	0
$681$	−73.1849	−2.80445
$682$	0	0
$683$	28.6527i	1.09636i	0.836359	+	0.548182i	$0.184680\pi$
$683$	28.6527i	1.09636i	−0.836359	+	0.548182i	$0.815320\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7.39888i	− 0.282285i
$688$	0	0
$689$	−6.04225	−0.230191
$690$	0	0
$691$	17.6880	0.672883	0.336442	−	0.941704i	$-0.390777\pi$
$691$	17.6880	0.672883	0.336442	+	0.941704i	$0.390777\pi$
$692$	0	0
$693$	− 31.4638i	− 1.19521i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.91609i	0.299843i
$698$	0	0
$699$	−27.0947	−1.02482
$700$	0	0
$701$	2.79339	0.105505	0.0527525	−	0.998608i	$-0.483201\pi$
$701$	2.79339	0.105505	0.0527525	+	0.998608i	$0.483201\pi$
$702$	0	0
$703$	− 14.3486i	− 0.541169i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 36.1899i	− 1.36106i
$708$	0	0
$709$	−20.4117	−0.766578	−0.383289	−	0.923628i	$-0.625209\pi$
$709$	−20.4117	−0.766578	−0.383289	+	0.923628i	$0.625209\pi$
$710$	0	0
$711$	−1.78814	−0.0670606
$712$	0	0
$713$	− 50.6645i	− 1.89740i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.87730i	0.256838i
$718$	0	0
$719$	−34.9848	−1.30471	−0.652356	−	0.757912i	$-0.726219\pi$
$719$	−34.9848	−1.30471	−0.652356	+	0.757912i	$0.726219\pi$
$720$	0	0
$721$	−22.3849	−0.833655
$722$	0	0
$723$	− 8.46876i	− 0.314957i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 24.8796i	− 0.922734i	−0.887209	−	0.461367i	$-0.847359\pi$
$727$	− 24.8796i	− 0.922734i	0.887209	−	0.461367i	$-0.152641\pi$
$728$	0	0
$729$	36.2444	1.34238
$730$	0	0
$731$	−38.0508	−1.40736
$732$	0	0
$733$	28.7737i	1.06278i	0.847127	+	0.531390i	$0.178330\pi$
$733$	28.7737i	1.06278i	−0.847127	+	0.531390i	$0.821670\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.1127i	0.446176i
$738$	0	0
$739$	30.1856	1.11039	0.555197	−	0.831719i	$-0.312643\pi$
$739$	30.1856	1.11039	0.555197	+	0.831719i	$0.312643\pi$
$740$	0	0
$741$	−19.8551	−0.729394
$742$	0	0
$743$	22.7127i	0.833247i	0.909079	+	0.416624i	$0.136787\pi$
$743$	22.7127i	0.833247i	−0.909079	+	0.416624i	$0.863213\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 20.6429i	− 0.755286i
$748$	0	0
$749$	12.2695	0.448317
$750$	0	0
$751$	−0.0200854	−0.000732927 0	−0.000366463	−	1.00000i	$-0.500117\pi$
$751$	−0.0200854	−0.000732927 0	−0.000366463		1.00000i	$0.500117\pi$
$752$	0	0
$753$	− 73.1628i	− 2.66620i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	44.9002i	1.63192i	0.578106	+	0.815962i	$0.303792\pi$
$757$	44.9002i	1.63192i	−0.578106	+	0.815962i	$0.696208\pi$
$758$	0	0
$759$	28.2269	1.02457
$760$	0	0
$761$	−19.0203	−0.689486	−0.344743	−	0.938697i	$-0.612034\pi$
$761$	−19.0203	−0.689486	−0.344743	+	0.938697i	$0.612034\pi$
$762$	0	0
$763$	− 51.4713i	− 1.86339i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 14.7694i	− 0.533292i
$768$	0	0
$769$	−42.1220	−1.51896	−0.759480	−	0.650531i	$-0.774546\pi$
$769$	−42.1220	−1.51896	−0.759480	+	0.650531i	$0.774546\pi$
$770$	0	0
$771$	31.4942	1.13424
$772$	0	0
$773$	− 10.7753i	− 0.387561i	−0.981045	−	0.193780i	$-0.937925\pi$
$773$	− 10.7753i	− 0.387561i	0.981045	−	0.193780i	$-0.0620749\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	53.5848i	1.92234i
$778$	0	0
$779$	2.17964	0.0780938
$780$	0	0
$781$	29.2137	1.04535
$782$	0	0
$783$	− 29.2055i	− 1.04372i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22.8592i	0.814843i	0.913240	+	0.407421i	$0.133572\pi$
$787$	22.8592i	0.814843i	−0.913240	+	0.407421i	$0.866428\pi$
$788$	0	0
$789$	14.3628	0.511328
$790$	0	0
$791$	−22.4581	−0.798519
$792$	0	0
$793$	8.98805i	0.319175i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 5.00444i	− 0.177266i	−0.996064	−	0.0886332i	$-0.971750\pi$
$797$	− 5.00444i	− 0.177266i	0.996064	−	0.0886332i	$-0.0282499\pi$
$798$	0	0
$799$	31.9731	1.13113
$800$	0	0
$801$	50.0724	1.76922
$802$	0	0
$803$	18.7607i	0.662050i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9.74989i	0.343212i
$808$	0	0
$809$	−7.52135	−0.264437	−0.132218	−	0.991221i	$-0.542210\pi$
$809$	−7.52135	−0.264437	−0.132218	+	0.991221i	$0.542210\pi$
$810$	0	0
$811$	29.4103	1.03273	0.516367	−	0.856367i	$-0.327284\pi$
$811$	29.4103	1.03273	0.516367	+	0.856367i	$0.327284\pi$
$812$	0	0
$813$	− 13.4244i	− 0.470816i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.4770i	0.366545i
$818$	0	0
$819$	48.2315	1.68535
$820$	0	0
$821$	38.0205	1.32692	0.663462	−	0.748210i	$-0.269087\pi$
$821$	38.0205	1.32692	0.663462	+	0.748210i	$0.269087\pi$
$822$	0	0
$823$	− 25.7849i	− 0.898805i	−0.893329	−	0.449403i	$-0.851637\pi$
$823$	− 25.7849i	− 0.898805i	0.893329	−	0.449403i	$-0.148363\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.00295i	0.243516i	0.992560	+	0.121758i	$0.0388533\pi$
$827$	7.00295i	0.243516i	−0.992560	+	0.121758i	$0.961147\pi$
$828$	0	0
$829$	−0.166709	−0.00579003	−0.00289501	−	0.999996i	$-0.500922\pi$
$829$	−0.166709	−0.00579003	−0.00289501	+	0.999996i	$0.500922\pi$
$830$	0	0
$831$	−70.9505	−2.46125
$832$	0	0
$833$	5.69584i	0.197349i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 80.7154i	− 2.78993i
$838$	0	0
$839$	13.1105	0.452625	0.226313	−	0.974055i	$-0.427333\pi$
$839$	13.1105	0.452625	0.226313	+	0.974055i	$0.427333\pi$
$840$	0	0
$841$	−14.1053	−0.486391
$842$	0	0
$843$	− 19.3760i	− 0.667345i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 19.1313i	− 0.657361i
$848$	0	0
$849$	−15.1349	−0.519428
$850$	0	0
$851$	−31.2696	−1.07191
$852$	0	0
$853$	0.445573i	0.0152561i	0.999971	+	0.00762806i	$0.00242811\pi$
$853$	0.445573i	0.0152561i	−0.999971	+	0.00762806i	$0.997572\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 20.5519i	− 0.702038i	−0.936368	−	0.351019i	$-0.885835\pi$
$857$	− 20.5519i	− 0.702038i	0.936368	−	0.351019i	$-0.114165\pi$
$858$	0	0
$859$	18.8891	0.644487	0.322243	−	0.946657i	$-0.395563\pi$
$859$	18.8891	0.644487	0.322243	+	0.946657i	$0.395563\pi$
$860$	0	0
$861$	−8.13983	−0.277405
$862$	0	0
$863$	47.9204i	1.63123i	0.578596	+	0.815614i	$0.303601\pi$
$863$	47.9204i	1.63123i	−0.578596	+	0.815614i	$0.696399\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	133.782i	4.54349i
$868$	0	0
$869$	0.649649	0.0220378
$870$	0	0
$871$	−18.5677	−0.629144
$872$	0	0
$873$	− 99.5576i	− 3.36951i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 50.2738i	− 1.69762i	−0.528694	−	0.848812i	$-0.677318\pi$
$877$	− 50.2738i	− 1.69762i	0.528694	−	0.848812i	$-0.322682\pi$
$878$	0	0
$879$	30.0170	1.01245
$880$	0	0
$881$	−38.6019	−1.30053	−0.650265	−	0.759707i	$-0.725342\pi$
$881$	−38.6019	−1.30053	−0.650265	+	0.759707i	$0.725342\pi$
$882$	0	0
$883$	− 3.07860i	− 0.103603i	−0.998657	−	0.0518016i	$-0.983504\pi$
$883$	− 3.07860i	− 0.103603i	0.998657	−	0.0518016i	$-0.0164963\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 31.0770i	− 1.04346i	−0.853110	−	0.521731i	$-0.825286\pi$
$887$	− 31.0770i	− 1.04346i	0.853110	−	0.521731i	$-0.174714\pi$
$888$	0	0
$889$	−12.3569	−0.414437
$890$	0	0
$891$	10.9960	0.368380
$892$	0	0
$893$	− 8.80358i	− 0.294601i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	43.2696i	1.44473i
$898$	0	0
$899$	41.1644	1.37291
$900$	0	0
$901$	15.3831	0.512485
$902$	0	0
$903$	− 39.1263i	− 1.30204i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	42.5534i	1.41296i	0.707731	+	0.706482i	$0.249719\pi$
$907$	42.5534i	1.41296i	−0.707731	+	0.706482i	$0.750281\pi$
$908$	0	0
$909$	72.7213	2.41201
$910$	0	0
$911$	−14.8694	−0.492645	−0.246323	−	0.969188i	$-0.579222\pi$
$911$	−14.8694	−0.492645	−0.246323	+	0.969188i	$0.579222\pi$
$912$	0	0
$913$	7.49977i	0.248206i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.3933i	0.904606i
$918$	0	0
$919$	−26.5539	−0.875931	−0.437965	−	0.898992i	$-0.644301\pi$
$919$	−26.5539	−0.875931	−0.437965	+	0.898992i	$0.644301\pi$
$920$	0	0
$921$	−53.5410	−1.76424
$922$	0	0
$923$	44.7823i	1.47403i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 44.9810i	− 1.47737i
$928$	0	0
$929$	−2.70749	−0.0888298	−0.0444149	−	0.999013i	$-0.514142\pi$
$929$	−2.70749	−0.0888298	−0.0444149	+	0.999013i	$0.514142\pi$
$930$	0	0
$931$	1.56831	0.0513993
$932$	0	0
$933$	− 13.1924i	− 0.431901i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 36.0067i	− 1.17629i	−0.808757	−	0.588143i	$-0.799859\pi$
$937$	− 36.0067i	− 1.17629i	0.808757	−	0.588143i	$-0.200141\pi$
$938$	0	0
$939$	−19.2215	−0.627269
$940$	0	0
$941$	11.6135	0.378591	0.189295	−	0.981920i	$-0.439380\pi$
$941$	11.6135	0.378591	0.189295	+	0.981920i	$0.439380\pi$
$942$	0	0
$943$	− 4.75004i	− 0.154683i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.9061i	− 0.679355i	−0.940542	−	0.339678i	$-0.889682\pi$
$947$	− 20.9061i	− 0.679355i	0.940542	−	0.339678i	$-0.110318\pi$
$948$	0	0
$949$	−28.7586	−0.933544
$950$	0	0
$951$	37.6063	1.21947
$952$	0	0
$953$	13.7510i	0.445438i	0.974883	+	0.222719i	$0.0714933\pi$
$953$	13.7510i	0.445438i	−0.974883	+	0.222719i	$0.928507\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	22.9341i	0.741355i
$958$	0	0
$959$	6.39742	0.206584
$960$	0	0
$961$	82.7663	2.66988
$962$	0	0
$963$	24.6547i	0.794488i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.07935i	0.227656i	0.993500	+	0.113828i	$0.0363114\pi$
$967$	7.07935i	0.227656i	−0.993500	+	0.113828i	$0.963689\pi$
$968$	0	0
$969$	50.5494	1.62388
$970$	0	0
$971$	−18.7382	−0.601338	−0.300669	−	0.953729i	$-0.597210\pi$
$971$	−18.7382	−0.601338	−0.300669	+	0.953729i	$0.597210\pi$
$972$	0	0
$973$	− 14.3534i	− 0.460148i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	60.6756i	1.94118i	0.240729	+	0.970592i	$0.422613\pi$
$977$	60.6756i	1.94118i	−0.240729	+	0.970592i	$0.577387\pi$
$978$	0	0
$979$	−18.1918	−0.581412
$980$	0	0
$981$	103.428	3.30221
$982$	0	0
$983$	− 47.3720i	− 1.51093i	−0.655188	−	0.755466i	$-0.727410\pi$
$983$	− 47.3720i	− 1.51093i	0.655188	−	0.755466i	$-0.272590\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	32.8768i	1.04648i
$988$	0	0
$989$	22.8323	0.726026
$990$	0	0
$991$	−19.2110	−0.610256	−0.305128	−	0.952311i	$-0.598699\pi$
$991$	−19.2110	−0.610256	−0.305128	+	0.952311i	$0.598699\pi$
$992$	0	0
$993$	26.3585i	0.836461i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 59.1077i	− 1.87196i	−0.352053	−	0.935980i	$-0.614516\pi$
$997$	− 59.1077i	− 1.87196i	0.352053	−	0.935980i	$-0.385484\pi$
$998$	0	0
$999$	−49.8167	−1.57613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.c.1149.2		8
5.2	odd	4		4100.2.a.c.1.1		4
5.3	odd	4		164.2.a.a.1.4	✓	4
5.4	even	2	inner	4100.2.d.c.1149.7		8
15.8	even	4		1476.2.a.g.1.4		4
20.3	even	4		656.2.a.i.1.1		4
35.13	even	4		8036.2.a.i.1.1		4
40.3	even	4		2624.2.a.y.1.4		4
40.13	odd	4		2624.2.a.v.1.1		4
60.23	odd	4		5904.2.a.bp.1.4		4
205.163	odd	4		6724.2.a.c.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.a.a.1.4	✓	4	5.3	odd	4
656.2.a.i.1.1		4	20.3	even	4
1476.2.a.g.1.4		4	15.8	even	4
2624.2.a.v.1.1		4	40.13	odd	4
2624.2.a.y.1.4		4	40.3	even	4
4100.2.a.c.1.1		4	5.2	odd	4
4100.2.d.c.1149.2		8	1.1	even	1	trivial
4100.2.d.c.1149.7		8	5.4	even	2	inner
5904.2.a.bp.1.4		4	60.23	odd	4
6724.2.a.c.1.1		4	205.163	odd	4
8036.2.a.i.1.1		4	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.92968i q^{3} +2.77840i q^{7} -5.58303 q^{9} +O(q^{10})\)	\(q-2.92968i q^{3} +2.77840i q^{7} -5.58303 q^{9} +2.02837 q^{11} +3.10932i q^{13} -7.91609i q^{17} -2.17964 q^{19} +8.13983 q^{21} +4.75004i q^{23} +7.56744i q^{27} -3.85936 q^{29} -10.6661 q^{31} -5.94246i q^{33} +6.58303i q^{37} +9.10932 q^{39} -1.00000 q^{41} -4.80677i q^{43} +4.03900i q^{47} -0.719526 q^{49} -23.1916 q^{51} +1.94327i q^{53} +6.38566i q^{57} -4.75004 q^{59} +2.89068 q^{61} -15.5119i q^{63} +5.97163i q^{67} +13.9161 q^{69} +14.4026 q^{71} +9.24916i q^{73} +5.63562i q^{77} +0.320282 q^{79} +5.42111 q^{81} +3.69745i q^{83} +11.3067i q^{87} -8.96868 q^{89} -8.63895 q^{91} +31.2484i q^{93} +17.8322i q^{97} -11.3244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} + 8 q^{11} - 12 q^{19} + 8 q^{29} - 16 q^{31} + 48 q^{39} - 8 q^{41} - 32 q^{49} - 8 q^{51} - 24 q^{59} + 48 q^{61} + 56 q^{69} - 4 q^{71} + 36 q^{79} + 56 q^{81} - 8 q^{89} + 72 q^{91} - 116 q^{99}+O(q^{100}) \) 8 * q - 24 * q^9 + 8 * q^11 - 12 * q^19 + 8 * q^29 - 16 * q^31 + 48 * q^39 - 8 * q^41 - 32 * q^49 - 8 * q^51 - 24 * q^59 + 48 * q^61 + 56 * q^69 - 4 * q^71 + 36 * q^79 + 56 * q^81 - 8 * q^89 + 72 * q^91 - 116 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more