LMFDB - Embedded newform 2100.2.s.b.1457.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1457,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 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("2100.1457");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.s (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.8
Character	$\chi$	$=$	2100.1457
Dual form			2100.2.s.b.1793.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.04182 - 1.38370i) q^{3} +(-0.707107 - 0.707107i) q^{7} +(-0.829228 - 2.88312i) q^{9} +O(q^{10})$	$q+(1.04182 - 1.38370i) q^{3} +(-0.707107 - 0.707107i) q^{7} +(-0.829228 - 2.88312i) q^{9} +1.21059i q^{11} +(1.31886 - 1.31886i) q^{13} +(4.79377 - 4.79377i) q^{17} -3.66781i q^{19} +(-1.71510 + 0.241744i) q^{21} +(6.15405 + 6.15405i) q^{23} +(-4.85327 - 1.85629i) q^{27} -6.65350 q^{29} +0.677592 q^{31} +(1.67509 + 1.26122i) q^{33} +(-5.16682 - 5.16682i) q^{37} +(-0.450887 - 3.19891i) q^{39} +4.65662i q^{41} +(2.35753 - 2.35753i) q^{43} +(1.91004 - 1.91004i) q^{47} +1.00000i q^{49} +(-1.63888 - 11.6274i) q^{51} +(-8.64475 - 8.64475i) q^{53} +(-5.07514 - 3.82120i) q^{57} +6.62929 q^{59} -7.07370 q^{61} +(-1.45232 + 2.62503i) q^{63} +(-4.94717 - 4.94717i) q^{67} +(14.9267 - 2.10393i) q^{69} -1.87014i q^{71} +(7.93894 - 7.93894i) q^{73} +(0.856017 - 0.856017i) q^{77} -10.6399i q^{79} +(-7.62476 + 4.78153i) q^{81} +(-9.69039 - 9.69039i) q^{83} +(-6.93174 + 9.20642i) q^{87} +2.84828 q^{89} -1.86515 q^{91} +(0.705928 - 0.937581i) q^{93} +(-3.90018 - 3.90018i) q^{97} +(3.49028 - 1.00386i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{3}+O(q^{10}) $ 24 * q + 4 * q^3	$ 24 q + 4 q^{3} - 24 q^{13} + 4 q^{21} - 8 q^{27} - 16 q^{31} + 20 q^{33} - 32 q^{37} + 8 q^{43} + 52 q^{51} + 28 q^{57} - 8 q^{63} + 24 q^{67} - 12 q^{81} + 20 q^{87} - 24 q^{91} - 20 q^{93} + 104 q^{97}+O(q^{100}) $ 24 * q + 4 * q^3 - 24 * q^13 + 4 * q^21 - 8 * q^27 - 16 * q^31 + 20 * q^33 - 32 * q^37 + 8 * q^43 + 52 * q^51 + 28 * q^57 - 8 * q^63 + 24 * q^67 - 12 * q^81 + 20 * q^87 - 24 * q^91 - 20 * q^93 + 104 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	1.04182	−	1.38370i	0.601494	−	0.798877i
$3$	1.04182	−	1.38370i	0.601494	−	0.798877i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.707107	−	0.707107i	−0.267261	−	0.267261i
$7$	−0.707107	−	0.707107i	−0.267261	−	0.267261i
$8$		0			0
$9$	−0.829228	−	2.88312i	−0.276409	−	0.961040i
$10$		0			0
$11$			1.21059i			0.365007i	0.983205	+	0.182504i	$0.0584201\pi$
$11$			1.21059i			0.365007i	−0.983205	+	0.182504i	$0.941580\pi$
$12$		0			0
$13$	1.31886	−	1.31886i	0.365785	−	0.365785i	−0.500152	−	0.865938i	$-0.666723\pi$
$13$	1.31886	−	1.31886i	0.365785	−	0.365785i	0.865938	+	0.500152i	$0.166723\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.79377	−	4.79377i	1.16266	−	1.16266i	0.178770	−	0.983891i	$-0.442788\pi$
$17$	4.79377	−	4.79377i	1.16266	−	1.16266i	0.983891	−	0.178770i	$-0.0572117\pi$
$18$		0			0
$19$		−	3.66781i		−	0.841454i	−0.907187	−	0.420727i	$-0.861775\pi$
$19$		−	3.66781i		−	0.841454i	0.907187	−	0.420727i	$-0.138225\pi$
$20$		0			0
$21$	−1.71510	+	0.241744i	−0.374265	+	0.0527528i
$22$		0			0
$23$	6.15405	+	6.15405i	1.28321	+	1.28321i	0.938832	+	0.344376i	$0.111909\pi$
$23$	6.15405	+	6.15405i	1.28321	+	1.28321i	0.344376	+	0.938832i	$0.388091\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.85327	−	1.85629i	−0.934011	−	0.357243i
$28$		0			0
$29$	−6.65350			−1.23552			−0.617762	−	0.786365i	$-0.711960\pi$
$29$	−6.65350			−1.23552			−0.617762	+	0.786365i	$0.711960\pi$
$30$		0			0
$31$	0.677592			0.121699			0.0608495	−	0.998147i	$-0.480619\pi$
$31$	0.677592			0.121699			0.0608495	+	0.998147i	$0.480619\pi$
$32$		0			0
$33$	1.67509	+	1.26122i	0.291596	+	0.219550i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.16682	−	5.16682i	−0.849419	−	0.849419i	0.140641	−	0.990061i	$-0.455084\pi$
$37$	−5.16682	−	5.16682i	−0.849419	−	0.849419i	−0.990061	+	0.140641i	$0.955084\pi$
$38$		0			0
$39$	−0.450887	−	3.19891i	−0.0721997	−	0.512235i
$40$		0			0
$41$			4.65662i			0.727242i	0.931547	+	0.363621i	$0.118460\pi$
$41$			4.65662i			0.727242i	−0.931547	+	0.363621i	$0.881540\pi$
$42$		0			0
$43$	2.35753	−	2.35753i	0.359520	−	0.359520i	−0.504116	−	0.863636i	$-0.668182\pi$
$43$	2.35753	−	2.35753i	0.359520	−	0.359520i	0.863636	+	0.504116i	$0.168182\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.91004	−	1.91004i	0.278608	−	0.278608i	−0.553945	−	0.832553i	$-0.686878\pi$
$47$	1.91004	−	1.91004i	0.278608	−	0.278608i	0.832553	+	0.553945i	$0.186878\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$	−1.63888	−	11.6274i	−0.229489	−	1.62816i
$52$		0			0
$53$	−8.64475	−	8.64475i	−1.18745	−	1.18745i	−0.977771	−	0.209678i	$-0.932759\pi$
$53$	−8.64475	−	8.64475i	−1.18745	−	1.18745i	−0.209678	−	0.977771i	$-0.567241\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−5.07514	−	3.82120i	−0.672219	−	0.506130i
$58$		0			0
$59$	6.62929			0.863061			0.431530	−	0.902098i	$-0.357974\pi$
$59$	6.62929			0.863061			0.431530	+	0.902098i	$0.357974\pi$
$60$		0			0
$61$	−7.07370			−0.905695			−0.452847	−	0.891588i	$-0.649592\pi$
$61$	−7.07370			−0.905695			−0.452847	+	0.891588i	$0.649592\pi$
$62$		0			0
$63$	−1.45232	+	2.62503i	−0.182975	+	0.330722i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.94717	−	4.94717i	−0.604394	−	0.604394i	0.337082	−	0.941475i	$-0.390560\pi$
$67$	−4.94717	−	4.94717i	−0.604394	−	0.604394i	−0.941475	+	0.337082i	$0.890560\pi$
$68$		0			0
$69$	14.9267	−	2.10393i	1.79697	−	0.253283i
$70$		0			0
$71$		−	1.87014i		−	0.221945i	−0.993823	−	0.110972i	$-0.964603\pi$
$71$		−	1.87014i		−	0.221945i	0.993823	−	0.110972i	$-0.0353965\pi$
$72$		0			0
$73$	7.93894	−	7.93894i	0.929183	−	0.929183i	−0.0684701	−	0.997653i	$-0.521812\pi$
$73$	7.93894	−	7.93894i	0.929183	−	0.929183i	0.997653	+	0.0684701i	$0.0218118\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.856017	−	0.856017i	0.0975522	−	0.0975522i
$78$		0			0
$79$		−	10.6399i		−	1.19708i	−0.801093	−	0.598539i	$-0.795748\pi$
$79$		−	10.6399i		−	1.19708i	0.801093	−	0.598539i	$-0.204252\pi$
$80$		0			0
$81$	−7.62476	+	4.78153i	−0.847196	+	0.531281i
$82$		0			0
$83$	−9.69039	−	9.69039i	−1.06366	−	1.06366i	−0.997831	−	0.0658277i	$-0.979031\pi$
$83$	−9.69039	−	9.69039i	−1.06366	−	1.06366i	−0.0658277	−	0.997831i	$-0.520969\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.93174	+	9.20642i	−0.743161	+	0.987032i
$88$		0			0
$89$	2.84828			0.301917			0.150958	−	0.988540i	$-0.451764\pi$
$89$	2.84828			0.301917			0.150958	+	0.988540i	$0.451764\pi$
$90$		0			0
$91$	−1.86515			−0.195520
$92$		0			0
$93$	0.705928	−	0.937581i	0.0732013	−	0.0972226i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.90018	−	3.90018i	−0.396003	−	0.396003i	0.480817	−	0.876821i	$-0.340340\pi$
$97$	−3.90018	−	3.90018i	−0.396003	−	0.396003i	−0.876821	+	0.480817i	$0.840340\pi$
$98$		0			0
$99$	3.49028	−	1.00386i	0.350786	−	0.100891i
$100$		0			0
$101$			7.69232i			0.765415i	0.923870	+	0.382707i	$0.125008\pi$
$101$			7.69232i			0.765415i	−0.923870	+	0.382707i	$0.874992\pi$
$102$		0			0
$103$	3.59549	−	3.59549i	0.354274	−	0.354274i	−0.507423	−	0.861697i	$-0.669402\pi$
$103$	3.59549	−	3.59549i	0.354274	−	0.354274i	0.861697	+	0.507423i	$0.169402\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.72919	+	1.72919i	−0.167167	+	0.167167i	−0.785733	−	0.618566i	$-0.787714\pi$
$107$	−1.72919	+	1.72919i	−0.167167	+	0.167167i	0.618566	+	0.785733i	$0.287714\pi$
$108$		0			0
$109$			19.6004i			1.87738i	0.344766	+	0.938688i	$0.387958\pi$
$109$			19.6004i			1.87738i	−0.344766	+	0.938688i	$0.612042\pi$
$110$		0			0
$111$	−12.5322	+	1.76642i	−1.18950	+	0.167661i
$112$		0			0
$113$	11.9813	+	11.9813i	1.12711	+	1.12711i	0.990645	+	0.136465i	$0.0435742\pi$
$113$	11.9813	+	11.9813i	1.12711	+	1.12711i	0.136465	+	0.990645i	$0.456426\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.89606	−	2.70879i	−0.452641	−	0.250428i
$118$		0			0
$119$	−6.77942			−0.621468
$120$		0			0
$121$	9.53447			0.866770
$122$		0			0
$123$	6.44334	+	4.85135i	0.580977	+	0.437432i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−2.34817	−	2.34817i	−0.208366	−	0.208366i	0.595207	−	0.803573i	$-0.297070\pi$
$127$	−2.34817	−	2.34817i	−0.208366	−	0.208366i	−0.803573	+	0.595207i	$0.797070\pi$
$128$		0			0
$129$	−0.805985	−	5.71822i	−0.0709630	−	0.503461i
$130$		0			0
$131$		−	11.9036i		−	1.04002i	−0.854160	−	0.520010i	$-0.825928\pi$
$131$		−	11.9036i		−	1.04002i	0.854160	−	0.520010i	$-0.174072\pi$
$132$		0			0
$133$	−2.59354	+	2.59354i	−0.224888	+	0.224888i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.88006	+	7.88006i	−0.673239	+	0.673239i	−0.958461	−	0.285222i	$-0.907933\pi$
$137$	−7.88006	+	7.88006i	−0.673239	+	0.673239i	0.285222	+	0.958461i	$0.407933\pi$
$138$		0			0
$139$		−	5.15820i		−	0.437513i	−0.975779	−	0.218756i	$-0.929800\pi$
$139$		−	5.15820i		−	0.437513i	0.975779	−	0.218756i	$-0.0702000\pi$
$140$		0			0
$141$	−0.652998	−	4.63283i	−0.0549924	−	0.390154i
$142$		0			0
$143$	1.59660	+	1.59660i	0.133514	+	0.133514i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.38370	+	1.04182i	0.114125	+	0.0859278i
$148$		0			0
$149$	−9.16856			−0.751118			−0.375559	−	0.926798i	$-0.622549\pi$
$149$	−9.16856			−0.751118			−0.375559	+	0.926798i	$0.622549\pi$
$150$		0			0
$151$	−4.55639			−0.370794			−0.185397	−	0.982664i	$-0.559357\pi$
$151$	−4.55639			−0.370794			−0.185397	+	0.982664i	$0.559357\pi$
$152$		0			0
$153$	−17.7961	−	9.84589i	−1.43873	−	0.795993i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	12.5673	+	12.5673i	1.00298	+	1.00298i	0.999996	+	0.00298395i	$0.000949822\pi$
$157$	12.5673	+	12.5673i	1.00298	+	1.00298i	0.00298395	+	0.999996i	$0.499050\pi$
$158$		0			0
$159$	−20.9680	+	2.95544i	−1.66287	+	0.234382i
$160$		0			0
$161$		−	8.70314i		−	0.685903i
$162$		0			0
$163$	17.8032	−	17.8032i	1.39446	−	1.39446i	0.579444	−	0.815012i	$-0.303270\pi$
$163$	17.8032	−	17.8032i	1.39446	−	1.39446i	0.815012	−	0.579444i	$-0.196730\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.53053	−	8.53053i	0.660112	−	0.660112i	−0.295294	−	0.955406i	$-0.595418\pi$
$167$	8.53053	−	8.53053i	0.660112	−	0.660112i	0.955406	+	0.295294i	$0.0954177\pi$
$168$		0			0
$169$			9.52123i			0.732402i
$170$		0			0
$171$	−10.5748	+	3.04145i	−0.808671	+	0.232586i
$172$		0			0
$173$	3.46780	+	3.46780i	0.263652	+	0.263652i	0.826536	−	0.562884i	$-0.190308\pi$
$173$	3.46780	+	3.46780i	0.263652	+	0.263652i	−0.562884	+	0.826536i	$0.690308\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	6.90652	−	9.17293i	0.519126	−	0.689479i
$178$		0			0
$179$	20.3729			1.52274			0.761370	−	0.648318i	$-0.224527\pi$
$179$	20.3729			1.52274			0.761370	+	0.648318i	$0.224527\pi$
$180$		0			0
$181$	3.23245			0.240266			0.120133	−	0.992758i	$-0.461668\pi$
$181$	3.23245			0.240266			0.120133	+	0.992758i	$0.461668\pi$
$182$		0			0
$183$	−7.36952	+	9.78785i	−0.544770	+	0.723539i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	5.80330	+	5.80330i	0.424379	+	0.424379i
$188$		0			0
$189$	2.11918	+	4.74437i	0.154148	+	0.345102i
$190$		0			0
$191$			11.7248i			0.848377i	0.905574	+	0.424189i	$0.139441\pi$
$191$			11.7248i			0.848377i	−0.905574	+	0.424189i	$0.860559\pi$
$192$		0			0
$193$	−9.23315	+	9.23315i	−0.664617	+	0.664617i	−0.956465	−	0.291848i	$-0.905730\pi$
$193$	−9.23315	+	9.23315i	−0.664617	+	0.664617i	0.291848	+	0.956465i	$0.405730\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.73126	−	6.73126i	0.479583	−	0.479583i	−0.425415	−	0.904998i	$-0.639872\pi$
$197$	6.73126	−	6.73126i	0.479583	−	0.479583i	0.904998	+	0.425415i	$0.139872\pi$
$198$		0			0
$199$			3.77719i			0.267758i	0.990998	+	0.133879i	$0.0427434\pi$
$199$			3.77719i			0.267758i	−0.990998	+	0.133879i	$0.957257\pi$
$200$		0			0
$201$	−11.9994	+	1.69133i	−0.846376	+	0.119297i
$202$		0			0
$203$	4.70474	+	4.70474i	0.330208	+	0.330208i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	12.6398	−	22.8460i	0.878523	−	1.58790i
$208$		0			0
$209$	4.44023			0.307137
$210$		0			0
$211$	−3.31292			−0.228071			−0.114035	−	0.993477i	$-0.536378\pi$
$211$	−3.31292			−0.228071			−0.114035	+	0.993477i	$0.536378\pi$
$212$		0			0
$213$	−2.58770	−	1.94835i	−0.177307	−	0.133499i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.479130	−	0.479130i	−0.0325254	−	0.0325254i
$218$		0			0
$219$	−2.71414	−	19.2560i	−0.183405	−	1.30120i
$220$		0			0
$221$		−	12.6446i		−	0.850568i
$222$		0			0
$223$	3.06323	−	3.06323i	0.205129	−	0.205129i	−0.597064	−	0.802193i	$-0.703666\pi$
$223$	3.06323	−	3.06323i	0.205129	−	0.205129i	0.802193	+	0.597064i	$0.203666\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.6081	+	13.6081i	−0.903204	+	0.903204i	−0.995712	−	0.0925080i	$-0.970512\pi$
$227$	−13.6081	+	13.6081i	−0.903204	+	0.903204i	0.0925080	+	0.995712i	$0.470512\pi$
$228$		0			0
$229$			17.3847i			1.14882i	0.818569	+	0.574408i	$0.194768\pi$
$229$			17.3847i			1.14882i	−0.818569	+	0.574408i	$0.805232\pi$
$230$		0			0
$231$	−0.292653	−	2.07628i	−0.0192551	−	0.136609i
$232$		0			0
$233$	2.37732	+	2.37732i	0.155743	+	0.155743i	0.780678	−	0.624934i	$-0.214874\pi$
$233$	2.37732	+	2.37732i	0.155743	+	0.155743i	−0.624934	+	0.780678i	$0.714874\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−14.7223	−	11.0848i	−0.956319	−	0.720036i
$238$		0			0
$239$	−15.2026			−0.983376			−0.491688	−	0.870771i	$-0.663620\pi$
$239$	−15.2026			−0.983376			−0.491688	+	0.870771i	$0.663620\pi$
$240$		0			0
$241$	11.5037			0.741021			0.370511	−	0.928828i	$-0.379183\pi$
$241$	11.5037			0.741021			0.370511	+	0.928828i	$0.379183\pi$
$242$		0			0
$243$	−1.32744	+	15.5318i	−0.0851555	+	0.996368i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.83733	−	4.83733i	−0.307792	−	0.307792i
$248$		0			0
$249$	−23.5042	+	3.31292i	−1.48952	+	0.209948i
$250$		0			0
$251$			17.0928i			1.07889i	0.842021	+	0.539445i	$0.181366\pi$
$251$			17.0928i			1.07889i	−0.842021	+	0.539445i	$0.818634\pi$
$252$		0			0
$253$	−7.45004	+	7.45004i	−0.468380	+	0.468380i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.85821	+	7.85821i	−0.490182	+	0.490182i	−0.908363	−	0.418182i	$-0.862668\pi$
$257$	−7.85821	+	7.85821i	−0.490182	+	0.490182i	0.418182	+	0.908363i	$0.362668\pi$
$258$		0			0
$259$			7.30698i			0.454034i
$260$		0			0
$261$	5.51727	+	19.1828i	0.341510	+	1.18739i
$262$		0			0
$263$	−10.6251	−	10.6251i	−0.655172	−	0.655172i	0.299062	−	0.954234i	$-0.403326\pi$
$263$	−10.6251	−	10.6251i	−0.655172	−	0.655172i	−0.954234	+	0.299062i	$0.903326\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.96739	−	3.94115i	0.181601	−	0.241194i
$268$		0			0
$269$	−18.5570			−1.13144			−0.565720	−	0.824598i	$-0.691402\pi$
$269$	−18.5570			−1.13144			−0.565720	+	0.824598i	$0.691402\pi$
$270$		0			0
$271$	19.7027			1.19685			0.598427	−	0.801177i	$-0.295792\pi$
$271$	19.7027			1.19685			0.598427	+	0.801177i	$0.295792\pi$
$272$		0			0
$273$	−1.94314	+	2.58080i	−0.117604	+	0.156197i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	16.3407	+	16.3407i	0.981815	+	0.981815i	0.999838	−	0.0180230i	$-0.00573721\pi$
$277$	16.3407	+	16.3407i	0.981815	+	0.981815i	−0.0180230	+	0.999838i	$0.505737\pi$
$278$		0			0
$279$	−0.561878	−	1.95358i	−0.0336387	−	0.116958i
$280$		0			0
$281$		−	10.7751i		−	0.642788i	−0.946945	−	0.321394i	$-0.895849\pi$
$281$		−	10.7751i		−	0.642788i	0.946945	−	0.321394i	$-0.104151\pi$
$282$		0			0
$283$	12.2145	−	12.2145i	0.726074	−	0.726074i	−0.243761	−	0.969835i	$-0.578381\pi$
$283$	12.2145	−	12.2145i	0.726074	−	0.726074i	0.969835	+	0.243761i	$0.0783812\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.29273	−	3.29273i	0.194364	−	0.194364i
$288$		0			0
$289$		−	28.9605i		−	1.70356i
$290$		0			0
$291$	−9.45994	+	1.33338i	−0.554552	+	0.0781642i
$292$		0			0
$293$	12.4296	+	12.4296i	0.726148	+	0.726148i	0.969850	−	0.243702i	$-0.0783620\pi$
$293$	12.4296	+	12.4296i	0.726148	+	0.726148i	−0.243702	+	0.969850i	$0.578362\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.24721	−	5.87532i	0.130396	−	0.340921i
$298$		0			0
$299$	16.2326			0.938757
$300$		0			0
$301$	−3.33405			−0.192171
$302$		0			0
$303$	10.6438	+	8.01401i	0.611472	+	0.460393i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	6.76706	+	6.76706i	0.386217	+	0.386217i	0.873336	−	0.487119i	$-0.161952\pi$
$307$	6.76706	+	6.76706i	0.386217	+	0.386217i	−0.487119	+	0.873336i	$0.661952\pi$
$308$		0			0
$309$	−1.22921	−	8.72091i	−0.0699276	−	0.496115i
$310$		0			0
$311$		−	24.8435i		−	1.40875i	−0.709830	−	0.704373i	$-0.751228\pi$
$311$		−	24.8435i		−	1.40875i	0.709830	−	0.704373i	$-0.248772\pi$
$312$		0			0
$313$	−2.53954	+	2.53954i	−0.143543	+	0.143543i	−0.775227	−	0.631683i	$-0.782364\pi$
$313$	−2.53954	+	2.53954i	−0.143543	+	0.143543i	0.631683	+	0.775227i	$0.282364\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.68000	+	4.68000i	−0.262855	+	0.262855i	−0.826213	−	0.563358i	$-0.809509\pi$
$317$	−4.68000	+	4.68000i	−0.262855	+	0.262855i	0.563358	+	0.826213i	$0.309509\pi$
$318$		0			0
$319$		−	8.05467i		−	0.450975i
$320$		0			0
$321$	0.591170	+	4.19417i	0.0329959	+	0.234096i
$322$		0			0
$323$	−17.5827	−	17.5827i	−0.978326	−	0.978326i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	27.1210	+	20.4201i	1.49979	+	1.12923i
$328$		0			0
$329$	−2.70120			−0.148922
$330$		0			0
$331$	32.4286			1.78244			0.891218	−	0.453576i	$-0.149852\pi$
$331$	32.4286			1.78244			0.891218	+	0.453576i	$0.149852\pi$
$332$		0			0
$333$	−10.6121	+	19.1810i	−0.581539	+	1.05111i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−17.6095	−	17.6095i	−0.959250	−	0.959250i	0.0399517	−	0.999202i	$-0.487280\pi$
$337$	−17.6095	−	17.6095i	−0.959250	−	0.959250i	−0.999202	+	0.0399517i	$0.987280\pi$
$338$		0			0
$339$	29.0609	−	4.09615i	1.57837	−	0.222472i
$340$		0			0
$341$			0.820287i			0.0444210i
$342$		0			0
$343$	0.707107	−	0.707107i	0.0381802	−	0.0381802i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.5184	+	12.5184i	−0.672023	+	0.672023i	−0.958182	−	0.286159i	$-0.907622\pi$
$347$	−12.5184	+	12.5184i	−0.672023	+	0.672023i	0.286159	+	0.958182i	$0.407622\pi$
$348$		0			0
$349$			17.7929i			0.952429i	0.879329	+	0.476215i	$0.157992\pi$
$349$			17.7929i			0.952429i	−0.879329	+	0.476215i	$0.842008\pi$
$350$		0			0
$351$	−8.84895	+	3.95259i	−0.472322	+	0.210973i
$352$		0			0
$353$	3.66929	+	3.66929i	0.195297	+	0.195297i	0.797980	−	0.602684i	$-0.205902\pi$
$353$	3.66929	+	3.66929i	0.195297	+	0.195297i	−0.602684	+	0.797980i	$0.705902\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−7.06292	+	9.38065i	−0.373810	+	0.496477i
$358$		0			0
$359$	29.5503			1.55960			0.779802	−	0.626026i	$-0.215320\pi$
$359$	29.5503			1.55960			0.779802	+	0.626026i	$0.215320\pi$
$360$		0			0
$361$	5.54713			0.291954
$362$		0			0
$363$	9.93319	−	13.1928i	0.521357	−	0.692443i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	21.5412	+	21.5412i	1.12444	+	1.12444i	0.991065	+	0.133377i	$0.0425823\pi$
$367$	21.5412	+	21.5412i	1.12444	+	1.12444i	0.133377	+	0.991065i	$0.457418\pi$
$368$		0			0
$369$	13.4256	−	3.86140i	0.698909	−	0.201016i
$370$		0			0
$371$			12.2255i			0.634718i
$372$		0			0
$373$	12.4363	−	12.4363i	0.643929	−	0.643929i	−0.307590	−	0.951519i	$-0.599523\pi$
$373$	12.4363	−	12.4363i	0.643929	−	0.643929i	0.951519	+	0.307590i	$0.0995225\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.77502	+	8.77502i	−0.451937	+	0.451937i
$378$		0			0
$379$		−	13.4965i		−	0.693270i	−0.938000	−	0.346635i	$-0.887324\pi$
$379$		−	13.4965i		−	0.693270i	0.938000	−	0.346635i	$-0.112676\pi$
$380$		0			0
$381$	−5.69551	+	0.802784i	−0.291790	+	0.0411279i
$382$		0			0
$383$	−6.78763	−	6.78763i	−0.346832	−	0.346832i	0.512096	−	0.858928i	$-0.328869\pi$
$383$	−6.78763	−	6.78763i	−0.346832	−	0.346832i	−0.858928	+	0.512096i	$0.828869\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.75197	−	4.84211i	−0.444887	−	0.246138i
$388$		0			0
$389$	31.3012			1.58703			0.793517	−	0.608548i	$-0.208248\pi$
$389$	31.3012			1.58703			0.793517	+	0.608548i	$0.208248\pi$
$390$		0			0
$391$	59.0022			2.98387
$392$		0			0
$393$	−16.4709	−	12.4014i	−0.830849	−	0.625567i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.83384	+	1.83384i	0.0920379	+	0.0920379i	0.751627	−	0.659589i	$-0.229270\pi$
$397$	1.83384	+	1.83384i	0.0920379	+	0.0920379i	−0.659589	+	0.751627i	$0.729270\pi$
$398$		0			0
$399$	0.886671	+	6.29066i	0.0443891	+	0.314927i
$400$		0			0
$401$		−	22.1431i		−	1.10577i	−0.833257	−	0.552886i	$-0.813526\pi$
$401$		−	22.1431i		−	1.10577i	0.833257	−	0.552886i	$-0.186474\pi$
$402$		0			0
$403$	0.893647	−	0.893647i	0.0445157	−	0.0445157i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.25490	−	6.25490i	0.310044	−	0.310044i
$408$		0			0
$409$			22.1801i			1.09673i	0.836238	+	0.548366i	$0.184750\pi$
$409$			22.1801i			1.09673i	−0.836238	+	0.548366i	$0.815250\pi$
$410$		0			0
$411$	2.69401	+	19.1132i	0.132886	+	0.942784i
$412$		0			0
$413$	−4.68762	−	4.68762i	−0.230663	−	0.230663i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−7.13738	−	5.37391i	−0.349519	−	0.263162i
$418$		0			0
$419$	−16.0297			−0.783102			−0.391551	−	0.920156i	$-0.628061\pi$
$419$	−16.0297			−0.783102			−0.391551	+	0.920156i	$0.628061\pi$
$420$		0			0
$421$	0.809448			0.0394501			0.0197250	−	0.999805i	$-0.493721\pi$
$421$	0.809448			0.0394501			0.0197250	+	0.999805i	$0.493721\pi$
$422$		0			0
$423$	−7.09073	−	3.92301i	−0.344763	−	0.190743i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	5.00186	+	5.00186i	0.242057	+	0.242057i
$428$		0			0
$429$	3.87257	−	0.545840i	0.186969	−	0.0263534i
$430$		0			0
$431$			27.5870i			1.32882i	0.747369	+	0.664409i	$0.231317\pi$
$431$			27.5870i			1.32882i	−0.747369	+	0.664409i	$0.768683\pi$
$432$		0			0
$433$	−10.3930	+	10.3930i	−0.499458	+	0.499458i	−0.911269	−	0.411811i	$-0.864896\pi$
$433$	−10.3930	+	10.3930i	−0.499458	+	0.499458i	0.411811	+	0.911269i	$0.364896\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	22.5719	−	22.5719i	1.07976	−	1.07976i
$438$		0			0
$439$			27.3915i			1.30733i	0.756786	+	0.653663i	$0.226769\pi$
$439$			27.3915i			1.30733i	−0.756786	+	0.653663i	$0.773231\pi$
$440$		0			0
$441$	2.88312	−	0.829228i	0.137291	−	0.0394870i
$442$		0			0
$443$	−14.5303	−	14.5303i	−0.690356	−	0.690356i	0.271954	−	0.962310i	$-0.412330\pi$
$443$	−14.5303	−	14.5303i	−0.690356	−	0.690356i	−0.962310	+	0.271954i	$0.912330\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−9.55198	+	12.6865i	−0.451793	+	0.600051i
$448$		0			0
$449$	27.2047			1.28387			0.641935	−	0.766759i	$-0.278132\pi$
$449$	27.2047			1.28387			0.641935	+	0.766759i	$0.278132\pi$
$450$		0			0
$451$	−5.63726			−0.265448
$452$		0			0
$453$	−4.74693	+	6.30466i	−0.223030	+	0.296219i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.74086	−	5.74086i	−0.268546	−	0.268546i	0.559968	−	0.828514i	$-0.310813\pi$
$457$	−5.74086	−	5.74086i	−0.268546	−	0.268546i	−0.828514	+	0.559968i	$0.810813\pi$
$458$		0			0
$459$	−32.1641	+	14.3668i	−1.50129	+	0.670586i
$460$		0			0
$461$		−	32.0803i		−	1.49413i	−0.664751	−	0.747065i	$-0.731462\pi$
$461$		−	32.0803i		−	1.49413i	0.664751	−	0.747065i	$-0.268538\pi$
$462$		0			0
$463$	−1.99168	+	1.99168i	−0.0925610	+	0.0925610i	−0.751871	−	0.659310i	$-0.770848\pi$
$463$	−1.99168	+	1.99168i	−0.0925610	+	0.0925610i	0.659310	+	0.751871i	$0.270848\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−14.9386	+	14.9386i	−0.691278	+	0.691278i	−0.962513	−	0.271236i	$-0.912568\pi$
$467$	−14.9386	+	14.9386i	−0.691278	+	0.691278i	0.271236	+	0.962513i	$0.412568\pi$
$468$		0			0
$469$			6.99636i			0.323062i
$470$		0			0
$471$	30.4822	−	4.29647i	1.40454	−	0.197971i
$472$		0			0
$473$	2.85400	+	2.85400i	0.131227	+	0.131227i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−17.7554	+	32.0923i	−0.812964	+	1.46941i
$478$		0			0
$479$	−21.1482			−0.966286			−0.483143	−	0.875541i	$-0.660505\pi$
$479$	−21.1482			−0.966286			−0.483143	+	0.875541i	$0.660505\pi$
$480$		0			0
$481$	−13.6286			−0.621410
$482$		0			0
$483$	−12.0425	−	9.06709i	−0.547952	−	0.412567i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	15.6516	+	15.6516i	0.709241	+	0.709241i	0.966376	−	0.257135i	$-0.0827785\pi$
$487$	15.6516	+	15.6516i	0.709241	+	0.709241i	−0.257135	+	0.966376i	$0.582778\pi$
$488$		0			0
$489$	−6.08651	−	43.1820i	−0.275242	−	1.95276i
$490$		0			0
$491$			17.3816i			0.784420i	0.919876	+	0.392210i	$0.128289\pi$
$491$			17.3816i			0.784420i	−0.919876	+	0.392210i	$0.871711\pi$
$492$		0			0
$493$	−31.8954	+	31.8954i	−1.43650	+	1.43650i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.32239	+	1.32239i	−0.0593172	+	0.0593172i
$498$		0			0
$499$		−	31.0765i		−	1.39118i	−0.718440	−	0.695588i	$-0.755144\pi$
$499$		−	31.0765i		−	1.39118i	0.718440	−	0.695588i	$-0.244856\pi$
$500$		0			0
$501$	−2.91639	−	20.6909i	−0.130295	−	0.924402i
$502$		0			0
$503$	25.5230	+	25.5230i	1.13802	+	1.13802i	0.988805	+	0.149211i	$0.0476733\pi$
$503$	25.5230	+	25.5230i	1.13802	+	1.13802i	0.149211	+	0.988805i	$0.452327\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	13.1745	+	9.91939i	0.585099	+	0.440536i
$508$		0			0
$509$	−28.0369			−1.24271			−0.621356	−	0.783528i	$-0.713418\pi$
$509$	−28.0369			−1.24271			−0.621356	+	0.783528i	$0.713418\pi$
$510$		0			0
$511$	−11.2274			−0.496669
$512$		0			0
$513$	−6.80853	+	17.8009i	−0.300604	+	0.785928i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	2.31228	+	2.31228i	0.101694	+	0.101694i
$518$		0			0
$519$	8.41121	−	1.18556i	0.369211	−	0.0520404i
$520$		0			0
$521$			35.5908i			1.55926i	0.626241	+	0.779630i	$0.284593\pi$
$521$			35.5908i			1.55926i	−0.626241	+	0.779630i	$0.715407\pi$
$522$		0			0
$523$	12.7863	−	12.7863i	0.559105	−	0.559105i	−0.369948	−	0.929053i	$-0.620624\pi$
$523$	12.7863	−	12.7863i	0.559105	−	0.559105i	0.929053	+	0.369948i	$0.120624\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.24822	−	3.24822i	0.141495	−	0.141495i
$528$		0			0
$529$			52.7446i			2.29324i
$530$		0			0
$531$	−5.49719	−	19.1131i	−0.238558	−	0.829436i
$532$		0			0
$533$	6.14142	+	6.14142i	0.266014	+	0.266014i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	21.2248	−	28.1899i	0.915919	−	1.21648i
$538$		0			0
$539$	−1.21059			−0.0521439
$540$		0			0
$541$	4.50612			0.193733			0.0968667	−	0.995297i	$-0.469118\pi$
$541$	4.50612			0.193733			0.0968667	+	0.995297i	$0.469118\pi$
$542$		0			0
$543$	3.36763	−	4.47273i	0.144519	−	0.191943i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	4.60733	+	4.60733i	0.196995	+	0.196995i	0.798711	−	0.601715i	$-0.205516\pi$
$547$	4.60733	+	4.60733i	0.196995	+	0.196995i	−0.601715	+	0.798711i	$0.705516\pi$
$548$		0			0
$549$	5.86571	+	20.3943i	0.250342	+	0.870409i
$550$		0			0
$551$			24.4038i			1.03964i
$552$		0			0
$553$	−7.52352	+	7.52352i	−0.319933	+	0.319933i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.41412	+	4.41412i	−0.187032	+	0.187032i	−0.794412	−	0.607380i	$-0.792221\pi$
$557$	−4.41412	+	4.41412i	−0.187032	+	0.187032i	0.607380	+	0.794412i	$0.292221\pi$
$558$		0			0
$559$		−	6.21849i		−	0.263014i
$560$		0			0
$561$	14.0760	−	1.98401i	0.594289	−	0.0837652i
$562$		0			0
$563$	20.0660	+	20.0660i	0.845683	+	0.845683i	0.989591	−	0.143908i	$-0.0459671\pi$
$563$	20.0660	+	20.0660i	0.845683	+	0.845683i	−0.143908	+	0.989591i	$0.545967\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	8.77257	+	2.01047i	0.368413	+	0.0844319i
$568$		0			0
$569$	8.35737			0.350359			0.175180	−	0.984537i	$-0.443949\pi$
$569$	8.35737			0.350359			0.175180	+	0.984537i	$0.443949\pi$
$570$		0			0
$571$	10.5864			0.443026			0.221513	−	0.975157i	$-0.428900\pi$
$571$	10.5864			0.443026			0.221513	+	0.975157i	$0.428900\pi$
$572$		0			0
$573$	16.2236	+	12.2151i	0.677749	+	0.510294i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	21.8642	+	21.8642i	0.910217	+	0.910217i	0.996289	−	0.0860719i	$-0.0274315\pi$
$577$	21.8642	+	21.8642i	0.910217	+	0.910217i	−0.0860719	+	0.996289i	$0.527431\pi$
$578$		0			0
$579$	3.15660	+	22.3952i	0.131184	+	0.930711i
$580$		0			0
$581$			13.7043i			0.568549i
$582$		0			0
$583$	10.4653	−	10.4653i	0.433427	−	0.433427i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	29.0468	−	29.0468i	1.19889	−	1.19889i	0.224392	−	0.974499i	$-0.427960\pi$
$587$	29.0468	−	29.0468i	1.19889	−	1.19889i	0.974499	−	0.224392i	$-0.0720397\pi$
$588$		0			0
$589$		−	2.48528i		−	0.102404i
$590$		0			0
$591$	−2.30126	−	16.3268i	−0.0946614	−	0.671594i
$592$		0			0
$593$	−23.3117	−	23.3117i	−0.957298	−	0.957298i	0.0418264	−	0.999125i	$-0.486682\pi$
$593$	−23.3117	−	23.3117i	−0.957298	−	0.957298i	−0.999125	+	0.0418264i	$0.986682\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	5.22649	+	3.93515i	0.213906	+	0.161055i
$598$		0			0
$599$	−41.2640			−1.68600			−0.843000	−	0.537914i	$-0.819213\pi$
$599$	−41.2640			−1.68600			−0.843000	+	0.537914i	$0.819213\pi$
$600$		0			0
$601$	−6.93221			−0.282771			−0.141385	−	0.989955i	$-0.545156\pi$
$601$	−6.93221			−0.282771			−0.141385	+	0.989955i	$0.545156\pi$
$602$		0			0
$603$	−10.1610	+	18.3656i	−0.413786	+	0.747906i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	24.1329	+	24.1329i	0.979523	+	0.979523i	0.999795	−	0.0202711i	$-0.00645293\pi$
$607$	24.1329	+	24.1329i	0.979523	+	0.979523i	−0.0202711	+	0.999795i	$0.506453\pi$
$608$		0			0
$609$	11.4114	−	1.60844i	0.462413	−	0.0651773i
$610$		0			0
$611$		−	5.03814i		−	0.203821i
$612$		0			0
$613$	−15.4289	+	15.4289i	−0.623167	+	0.623167i	−0.946340	−	0.323173i	$-0.895250\pi$
$613$	−15.4289	+	15.4289i	−0.623167	+	0.623167i	0.323173	+	0.946340i	$0.395250\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.98976	+	3.98976i	−0.160622	+	0.160622i	−0.782842	−	0.622220i	$-0.786231\pi$
$617$	−3.98976	+	3.98976i	−0.160622	+	0.160622i	0.622220	+	0.782842i	$0.286231\pi$
$618$		0			0
$619$			18.4604i			0.741985i	0.928636	+	0.370993i	$0.120982\pi$
$619$			18.4604i			0.741985i	−0.928636	+	0.370993i	$0.879018\pi$
$620$		0			0
$621$	−18.4435	−	41.2909i	−0.740113	−	1.65695i
$622$		0			0
$623$	−2.01404	−	2.01404i	−0.0806906	−	0.0806906i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	4.62591	−	6.14392i	0.184741	−	0.245365i
$628$		0			0
$629$	−49.5371			−1.97517
$630$		0			0
$631$	21.4925			0.855603			0.427802	−	0.903873i	$-0.359288\pi$
$631$	21.4925			0.855603			0.427802	+	0.903873i	$0.359288\pi$
$632$		0			0
$633$	−3.45146	+	4.58408i	−0.137183	+	0.182201i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.31886	+	1.31886i	0.0522550	+	0.0522550i
$638$		0			0
$639$	−5.39184	+	1.55077i	−0.213298	+	0.0613476i
$640$		0			0
$641$		−	14.3498i		−	0.566784i	−0.959004	−	0.283392i	$-0.908540\pi$
$641$		−	14.3498i		−	0.566784i	0.959004	−	0.283392i	$-0.0914598\pi$
$642$		0			0
$643$	−8.65934	+	8.65934i	−0.341491	+	0.341491i	−0.856928	−	0.515437i	$-0.827630\pi$
$643$	−8.65934	+	8.65934i	−0.341491	+	0.341491i	0.515437	+	0.856928i	$0.327630\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11.2705	+	11.2705i	−0.443088	+	0.443088i	−0.893048	−	0.449961i	$-0.851438\pi$
$647$	−11.2705	+	11.2705i	−0.443088	+	0.443088i	0.449961	+	0.893048i	$0.351438\pi$
$648$		0			0
$649$			8.02537i			0.315023i
$650$		0			0
$651$	−1.16214	+	0.163803i	−0.0455477	+	0.00641996i
$652$		0			0
$653$	−16.6089	−	16.6089i	−0.649958	−	0.649958i	0.303025	−	0.952983i	$-0.402003\pi$
$653$	−16.6089	−	16.6089i	−0.649958	−	0.649958i	−0.952983	+	0.303025i	$0.902003\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−29.4721	−	16.3057i	−1.14982	−	0.636147i
$658$		0			0
$659$	−4.99323			−0.194509			−0.0972543	−	0.995260i	$-0.531006\pi$
$659$	−4.99323			−0.194509			−0.0972543	+	0.995260i	$0.531006\pi$
$660$		0			0
$661$	−27.6925			−1.07711			−0.538556	−	0.842589i	$-0.681030\pi$
$661$	−27.6925			−1.07711			−0.538556	+	0.842589i	$0.681030\pi$
$662$		0			0
$663$	−17.4963	−	13.1734i	−0.679500	−	0.511612i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−40.9460	−	40.9460i	−1.58543	−	1.58543i
$668$		0			0
$669$	−1.04725	−	7.42991i	−0.0404889	−	0.287257i
$670$		0			0
$671$		−	8.56337i		−	0.330585i
$672$		0			0
$673$	30.8390	−	30.8390i	1.18876	−	1.18876i	0.211344	−	0.977412i	$-0.432216\pi$
$673$	30.8390	−	30.8390i	1.18876	−	1.18876i	0.977412	−	0.211344i	$-0.0677841\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.2980	−	15.2980i	0.587952	−	0.587952i	−0.349125	−	0.937076i	$-0.613521\pi$
$677$	15.2980	−	15.2980i	0.587952	−	0.587952i	0.937076	+	0.349125i	$0.113521\pi$
$678$		0			0
$679$			5.51569i			0.211673i
$680$		0			0
$681$	4.65231	+	33.0067i	0.178277	+	1.26482i
$682$		0			0
$683$	13.5834	+	13.5834i	0.519753	+	0.519753i	0.917497	−	0.397743i	$-0.130207\pi$
$683$	13.5834	+	13.5834i	0.519753	+	0.519753i	−0.397743	+	0.917497i	$0.630207\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	24.0552	+	18.1117i	0.917763	+	0.691006i
$688$		0			0
$689$	−22.8024			−0.868702
$690$		0			0
$691$	−9.91319			−0.377116			−0.188558	−	0.982062i	$-0.560381\pi$
$691$	−9.91319			−0.377116			−0.188558	+	0.982062i	$0.560381\pi$
$692$		0			0
$693$	−3.17783	−	1.75817i	−0.120716	−	0.0667873i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	22.3228	+	22.3228i	0.845535	+	0.845535i
$698$		0			0
$699$	5.76622	−	0.812751i	0.218099	−	0.0307411i
$700$		0			0
$701$			24.1977i			0.913936i	0.889483	+	0.456968i	$0.151065\pi$
$701$			24.1977i			0.913936i	−0.889483	+	0.456968i	$0.848935\pi$
$702$		0			0
$703$	−18.9509	+	18.9509i	−0.714748	+	0.714748i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.43929	−	5.43929i	0.204566	−	0.204566i
$708$		0			0
$709$		−	9.04474i		−	0.339683i	−0.985471	−	0.169841i	$-0.945675\pi$
$709$		−	9.04474i		−	0.339683i	0.985471	−	0.169841i	$-0.0543255\pi$
$710$		0			0
$711$	−30.6760	+	8.82288i	−1.15044	+	0.330884i
$712$		0			0
$713$	4.16993	+	4.16993i	0.156165	+	0.156165i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−15.8384	+	21.0358i	−0.591495	+	0.785596i
$718$		0			0
$719$	32.9027			1.22706			0.613531	−	0.789670i	$-0.289748\pi$
$719$	32.9027			1.22706			0.613531	+	0.789670i	$0.289748\pi$
$720$		0			0
$721$	−5.08479			−0.189367
$722$		0			0
$723$	11.9848	−	15.9177i	0.445720	−	0.591985i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	30.5266	+	30.5266i	1.13217	+	1.13217i	0.989816	+	0.142351i	$0.0454663\pi$
$727$	30.5266	+	30.5266i	1.13217	+	1.13217i	0.142351	+	0.989816i	$0.454534\pi$
$728$		0			0
$729$	20.1084	+	18.0181i	0.744755	+	0.667338i
$730$		0			0
$731$		−	22.6029i		−	0.835999i
$732$		0			0
$733$	−18.0111	+	18.0111i	−0.665257	+	0.665257i	−0.956614	−	0.291358i	$-0.905893\pi$
$733$	−18.0111	+	18.0111i	−0.665257	+	0.665257i	0.291358	+	0.956614i	$0.405893\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.98901	−	5.98901i	0.220608	−	0.220608i
$738$		0			0
$739$		−	23.3893i		−	0.860391i	−0.902736	−	0.430195i	$-0.858445\pi$
$739$		−	23.3893i		−	0.860391i	0.902736	−	0.430195i	$-0.141555\pi$
$740$		0			0
$741$	−11.7330	+	1.65377i	−0.431023	+	0.0607528i
$742$		0			0
$743$	10.7432	+	10.7432i	0.394129	+	0.394129i	0.876156	−	0.482027i	$-0.160099\pi$
$743$	10.7432	+	10.7432i	0.394129	+	0.394129i	−0.482027	+	0.876156i	$0.660099\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−19.9030	+	35.9741i	−0.728214	+	1.31622i
$748$		0			0
$749$	2.44544			0.0893545
$750$		0			0
$751$	2.66315			0.0971798			0.0485899	−	0.998819i	$-0.484527\pi$
$751$	2.66315			0.0971798			0.0485899	+	0.998819i	$0.484527\pi$
$752$		0			0
$753$	23.6513	+	17.8076i	0.861901	+	0.648947i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−4.00745	−	4.00745i	−0.145653	−	0.145653i	0.630520	−	0.776173i	$-0.282842\pi$
$757$	−4.00745	−	4.00745i	−0.145653	−	0.145653i	−0.776173	+	0.630520i	$0.782842\pi$
$758$		0			0
$759$	2.54700	+	18.0702i	0.0924501	+	0.655906i
$760$		0			0
$761$			25.2284i			0.914530i	0.889330	+	0.457265i	$0.151171\pi$
$761$			25.2284i			0.914530i	−0.889330	+	0.457265i	$0.848829\pi$
$762$		0			0
$763$	13.8596	−	13.8596i	0.501750	−	0.501750i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.74310	−	8.74310i	0.315695	−	0.315695i
$768$		0			0
$769$		−	37.7217i		−	1.36028i	−0.733082	−	0.680140i	$-0.761919\pi$
$769$		−	37.7217i		−	1.36028i	0.733082	−	0.680140i	$-0.238081\pi$
$770$		0			0
$771$	2.68654	+	19.0602i	0.0967534	+	0.686437i
$772$		0			0
$773$	−18.6191	−	18.6191i	−0.669682	−	0.669682i	0.287960	−	0.957642i	$-0.407023\pi$
$773$	−18.6191	−	18.6191i	−0.669682	−	0.669682i	−0.957642	+	0.287960i	$0.907023\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	10.1106	+	7.61255i	0.362717	+	0.273099i
$778$		0			0
$779$	17.0796			0.611941
$780$		0			0
$781$	2.26398			0.0810114
$782$		0			0
$783$	32.2912	+	12.3508i	1.15399	+	0.441382i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−38.9229	−	38.9229i	−1.38745	−	1.38745i	−0.830636	−	0.556816i	$-0.812023\pi$
$787$	−38.9229	−	38.9229i	−1.38745	−	1.38745i	−0.556816	−	0.830636i	$-0.687977\pi$
$788$		0			0
$789$	−25.7713	+	3.63248i	−0.917484	+	0.129320i
$790$		0			0
$791$		−	16.9442i		−	0.602466i
$792$		0			0
$793$	−9.32921	+	9.32921i	−0.331290	+	0.331290i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.0056	−	17.0056i	0.602368	−	0.602368i	−0.338573	−	0.940940i	$-0.609944\pi$
$797$	17.0056	−	17.0056i	0.602368	−	0.602368i	0.940940	+	0.338573i	$0.109944\pi$
$798$		0			0
$799$		−	18.3126i		−	0.647853i
$800$		0			0
$801$	−2.36187	−	8.21192i	−0.0834526	−	0.290154i
$802$		0			0
$803$	9.61082	+	9.61082i	0.339158	+	0.339158i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−19.3330	+	25.6772i	−0.680554	+	0.903881i
$808$		0			0
$809$	7.65158			0.269015			0.134508	−	0.990913i	$-0.457055\pi$
$809$	7.65158			0.269015			0.134508	+	0.990913i	$0.457055\pi$
$810$		0			0
$811$	13.4460			0.472152			0.236076	−	0.971735i	$-0.424139\pi$
$811$	13.4460			0.472152			0.236076	+	0.971735i	$0.424139\pi$
$812$		0			0
$813$	20.5267	−	27.2626i	0.719901	−	0.956140i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.64698	−	8.64698i	−0.302520	−	0.302520i
$818$		0			0
$819$	1.54663	+	5.37744i	0.0540437	+	0.187903i
$820$		0			0
$821$			43.2148i			1.50821i	0.656755	+	0.754104i	$0.271928\pi$
$821$			43.2148i			1.50821i	−0.656755	+	0.754104i	$0.728072\pi$
$822$		0			0
$823$	14.9428	−	14.9428i	0.520875	−	0.520875i	−0.396961	−	0.917836i	$-0.629935\pi$
$823$	14.9428	−	14.9428i	0.520875	−	0.520875i	0.917836	+	0.396961i	$0.129935\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.4905	−	16.4905i	0.573430	−	0.573430i	−0.359655	−	0.933085i	$-0.617106\pi$
$827$	16.4905	−	16.4905i	0.573430	−	0.573430i	0.933085	+	0.359655i	$0.117106\pi$
$828$		0			0
$829$			55.7459i			1.93614i	0.250689	+	0.968068i	$0.419343\pi$
$829$			55.7459i			1.93614i	−0.250689	+	0.968068i	$0.580657\pi$
$830$		0			0
$831$	39.6345	−	5.58649i	1.37491	−	0.193793i
$832$		0			0
$833$	4.79377	+	4.79377i	0.166094	+	0.166094i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.28853	−	1.25781i	−0.113668	−	0.0434762i
$838$		0			0
$839$	−1.77729			−0.0613588			−0.0306794	−	0.999529i	$-0.509767\pi$
$839$	−1.77729			−0.0613588			−0.0306794	+	0.999529i	$0.509767\pi$
$840$		0			0
$841$	15.2691			0.526520
$842$		0			0
$843$	−14.9095	−	11.2257i	−0.513509	−	0.386634i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−6.74189	−	6.74189i	−0.231654	−	0.231654i
$848$		0			0
$849$	−4.17584	−	29.6264i	−0.143315	−	1.01677i
$850$		0			0
$851$		−	63.5937i		−	2.17996i
$852$		0			0
$853$	−9.70145	+	9.70145i	−0.332171	+	0.332171i	−0.853411	−	0.521239i	$-0.825470\pi$
$853$	−9.70145	+	9.70145i	−0.332171	+	0.332171i	0.521239	+	0.853411i	$0.325470\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.21818	−	4.21818i	0.144090	−	0.144090i	−0.631382	−	0.775472i	$-0.717512\pi$
$857$	4.21818	−	4.21818i	0.144090	−	0.144090i	0.775472	+	0.631382i	$0.217512\pi$
$858$		0			0
$859$			2.26158i			0.0771642i	0.999255	+	0.0385821i	$0.0122841\pi$
$859$			2.26158i			0.0771642i	−0.999255	+	0.0385821i	$0.987716\pi$
$860$		0			0
$861$	−1.12571	−	7.98656i	−0.0383640	−	0.272181i
$862$		0			0
$863$	−19.7520	−	19.7520i	−0.672366	−	0.672366i	0.285895	−	0.958261i	$-0.407709\pi$
$863$	−19.7520	−	19.7520i	−0.672366	−	0.672366i	−0.958261	+	0.285895i	$0.907709\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−40.0725	−	30.1716i	−1.36093	−	1.02468i
$868$		0			0
$869$	12.8805			0.436942
$870$		0			0
$871$	−13.0492			−0.442157
$872$		0			0
$873$	−8.01055	+	14.4788i	−0.271116	+	0.490034i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.4841	−	23.4841i	−0.793003	−	0.793003i	0.188978	−	0.981981i	$-0.439482\pi$
$877$	−23.4841	−	23.4841i	−0.793003	−	0.793003i	−0.981981	+	0.188978i	$0.939482\pi$
$878$		0			0
$879$	30.1483	−	4.24941i	1.01688	−	0.143329i
$880$		0			0
$881$		−	15.3754i		−	0.518009i	−0.965876	−	0.259004i	$-0.916606\pi$
$881$		−	15.3754i		−	0.518009i	0.965876	−	0.259004i	$-0.0833944\pi$
$882$		0			0
$883$	26.4444	−	26.4444i	0.889925	−	0.889925i	−0.104591	−	0.994515i	$-0.533353\pi$
$883$	26.4444	−	26.4444i	0.889925	−	0.889925i	0.994515	+	0.104591i	$0.0333533\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−29.6576	+	29.6576i	−0.995806	+	0.995806i	−0.999991	−	0.00418537i	$-0.998668\pi$
$887$	−29.6576	+	29.6576i	−0.995806	+	0.995806i	0.00418537	+	0.999991i	$0.498668\pi$
$888$		0			0
$889$			3.32081i			0.111376i
$890$		0			0
$891$	−5.78847	−	9.23047i	−0.193921	−	0.309232i
$892$		0			0
$893$	−7.00567	−	7.00567i	−0.234436	−	0.234436i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	16.9115	−	22.4610i	0.564657	−	0.749951i
$898$		0			0
$899$	−4.50836			−0.150362
$900$		0			0
$901$	−82.8820			−2.76120
$902$		0			0
$903$	−3.47347	+	4.61331i	−0.115590	+	0.153521i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−8.87348	−	8.87348i	−0.294639	−	0.294639i	0.544271	−	0.838910i	$-0.316806\pi$
$907$	−8.87348	−	8.87348i	−0.294639	−	0.294639i	−0.838910	+	0.544271i	$0.816806\pi$
$908$		0			0
$909$	22.1779	−	6.37869i	0.735594	−	0.211568i
$910$		0			0
$911$		−	4.64169i		−	0.153786i	−0.997039	−	0.0768930i	$-0.975500\pi$
$911$		−	4.64169i		−	0.153786i	0.997039	−	0.0768930i	$-0.0245000\pi$
$912$		0			0
$913$	11.7311	−	11.7311i	0.388243	−	0.388243i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.41710	+	8.41710i	−0.277957	+	0.277957i
$918$		0			0
$919$		−	40.8491i		−	1.34749i	−0.738965	−	0.673743i	$-0.764685\pi$
$919$		−	40.8491i		−	1.34749i	0.738965	−	0.673743i	$-0.235315\pi$
$920$		0			0
$921$	16.4136	−	2.31350i	0.540847	−	0.0762325i
$922$		0			0
$923$	−2.46645	−	2.46645i	−0.0811841	−	0.0811841i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−13.3477	−	7.38475i	−0.438396	−	0.242547i
$928$		0			0
$929$	21.5171			0.705954			0.352977	−	0.935632i	$-0.385169\pi$
$929$	21.5171			0.705954			0.352977	+	0.935632i	$0.385169\pi$
$930$		0			0
$931$	3.66781			0.120208
$932$		0			0
$933$	−34.3759	−	25.8824i	−1.12541	−	0.847353i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−14.0454	−	14.0454i	−0.458842	−	0.458842i	0.439433	−	0.898275i	$-0.355179\pi$
$937$	−14.0454	−	14.0454i	−0.458842	−	0.458842i	−0.898275	+	0.439433i	$0.855179\pi$
$938$		0			0
$939$	0.868209	+	6.15968i	0.0283329	+	0.201014i
$940$		0			0
$941$			18.0981i			0.589983i	0.955500	+	0.294991i	$0.0953167\pi$
$941$			18.0981i			0.589983i	−0.955500	+	0.294991i	$0.904683\pi$
$942$		0			0
$943$	−28.6571	+	28.6571i	−0.933202	+	0.933202i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.91624	−	9.91624i	0.322235	−	0.322235i	−0.527389	−	0.849624i	$-0.676829\pi$
$947$	9.91624	−	9.91624i	0.322235	−	0.322235i	0.849624	+	0.527389i	$0.176829\pi$
$948$		0			0
$949$		−	20.9407i		−	0.679763i
$950$		0			0
$951$	1.59998	+	11.3514i	0.0518831	+	0.368095i
$952$		0			0
$953$	−38.3878	−	38.3878i	−1.24350	−	1.24350i	−0.958537	−	0.284967i	$-0.908017\pi$
$953$	−38.3878	−	38.3878i	−1.24350	−	1.24350i	−0.284967	−	0.958537i	$-0.591983\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−11.1452	−	8.39151i	−0.360274	−	0.271259i
$958$		0			0
$959$	11.1441			0.359861
$960$		0			0
$961$	−30.5409			−0.985189
$962$		0			0
$963$	6.41935	+	3.55157i	0.206861	+	0.114448i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−20.8316	−	20.8316i	−0.669898	−	0.669898i	0.287794	−	0.957692i	$-0.407078\pi$
$967$	−20.8316	−	20.8316i	−0.669898	−	0.669898i	−0.957692	+	0.287794i	$0.907078\pi$
$968$		0			0
$969$	−42.6470	+	6.01111i	−1.37002	+	0.193105i
$970$		0			0
$971$		−	49.8127i		−	1.59856i	−0.600956	−	0.799282i	$-0.705213\pi$
$971$		−	49.8127i		−	1.59856i	0.600956	−	0.799282i	$-0.294787\pi$
$972$		0			0
$973$	−3.64740	+	3.64740i	−0.116930	+	0.116930i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−22.7612	+	22.7612i	−0.728195	+	0.728195i	−0.970260	−	0.242065i	$-0.922175\pi$
$977$	−22.7612	+	22.7612i	−0.728195	+	0.728195i	0.242065	+	0.970260i	$0.422175\pi$
$978$		0			0
$979$			3.44810i			0.110202i
$980$		0			0
$981$	56.5103	−	16.2532i	1.80423	−	0.518924i
$982$		0			0
$983$	9.92066	+	9.92066i	0.316420	+	0.316420i	0.847390	−	0.530970i	$-0.178172\pi$
$983$	9.92066	+	9.92066i	0.316420	+	0.316420i	−0.530970	+	0.847390i	$0.678172\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−2.81416	+	3.73764i	−0.0895758	+	0.118970i
$988$		0			0
$989$	29.0167			0.922677
$990$		0			0
$991$	−0.717293			−0.0227856			−0.0113928	−	0.999935i	$-0.503627\pi$
$991$	−0.717293			−0.0227856			−0.0113928	+	0.999935i	$0.503627\pi$
$992$		0			0
$993$	33.7847	−	44.8713i	1.07212	−	1.42395i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	8.49889	+	8.49889i	0.269163	+	0.269163i	0.828763	−	0.559600i	$-0.189045\pi$
$997$	8.49889	+	8.49889i	0.269163	+	0.269163i	−0.559600	+	0.828763i	$0.689045\pi$
$998$		0			0
$999$	15.4848	+	34.6670i	0.489918	+	1.09682i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.s.b.1457.8		24
3.2	odd	2	inner	2100.2.s.b.1457.10		24
5.2	odd	4		420.2.s.a.113.3	✓	24
5.3	odd	4	inner	2100.2.s.b.1793.10		24
5.4	even	2		420.2.s.a.197.5	yes	24
15.2	even	4		420.2.s.a.113.5	yes	24
15.8	even	4	inner	2100.2.s.b.1793.8		24
15.14	odd	2		420.2.s.a.197.3	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.s.a.113.3	✓	24	5.2	odd	4
420.2.s.a.113.5	yes	24	15.2	even	4
420.2.s.a.197.3	yes	24	15.14	odd	2
420.2.s.a.197.5	yes	24	5.4	even	2
2100.2.s.b.1457.8		24	1.1	even	1	trivial
2100.2.s.b.1457.10		24	3.2	odd	2	inner
2100.2.s.b.1793.8		24	15.8	even	4	inner
2100.2.s.b.1793.10		24	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.04182 - 1.38370i) q^{3} +(-0.707107 - 0.707107i) q^{7} +(-0.829228 - 2.88312i) q^{9} +O(q^{10})\)	\(q+(1.04182 - 1.38370i) q^{3} +(-0.707107 - 0.707107i) q^{7} +(-0.829228 - 2.88312i) q^{9} +1.21059i q^{11} +(1.31886 - 1.31886i) q^{13} +(4.79377 - 4.79377i) q^{17} -3.66781i q^{19} +(-1.71510 + 0.241744i) q^{21} +(6.15405 + 6.15405i) q^{23} +(-4.85327 - 1.85629i) q^{27} -6.65350 q^{29} +0.677592 q^{31} +(1.67509 + 1.26122i) q^{33} +(-5.16682 - 5.16682i) q^{37} +(-0.450887 - 3.19891i) q^{39} +4.65662i q^{41} +(2.35753 - 2.35753i) q^{43} +(1.91004 - 1.91004i) q^{47} +1.00000i q^{49} +(-1.63888 - 11.6274i) q^{51} +(-8.64475 - 8.64475i) q^{53} +(-5.07514 - 3.82120i) q^{57} +6.62929 q^{59} -7.07370 q^{61} +(-1.45232 + 2.62503i) q^{63} +(-4.94717 - 4.94717i) q^{67} +(14.9267 - 2.10393i) q^{69} -1.87014i q^{71} +(7.93894 - 7.93894i) q^{73} +(0.856017 - 0.856017i) q^{77} -10.6399i q^{79} +(-7.62476 + 4.78153i) q^{81} +(-9.69039 - 9.69039i) q^{83} +(-6.93174 + 9.20642i) q^{87} +2.84828 q^{89} -1.86515 q^{91} +(0.705928 - 0.937581i) q^{93} +(-3.90018 - 3.90018i) q^{97} +(3.49028 - 1.00386i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{3}+O(q^{10}) \) 24 * q + 4 * q^3	\( 24 q + 4 q^{3} - 24 q^{13} + 4 q^{21} - 8 q^{27} - 16 q^{31} + 20 q^{33} - 32 q^{37} + 8 q^{43} + 52 q^{51} + 28 q^{57} - 8 q^{63} + 24 q^{67} - 12 q^{81} + 20 q^{87} - 24 q^{91} - 20 q^{93} + 104 q^{97}+O(q^{100}) \) 24 * q + 4 * q^3 - 24 * q^13 + 4 * q^21 - 8 * q^27 - 16 * q^31 + 20 * q^33 - 32 * q^37 + 8 * q^43 + 52 * q^51 + 28 * q^57 - 8 * q^63 + 24 * q^67 - 12 * q^81 + 20 * q^87 - 24 * q^91 - 20 * q^93 + 104 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more