LMFDB - Embedded newform 2100.2.x.c.1693.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1357,2100)]

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, 0, 1, 2]))

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

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

chi := DirichletCharacter("2100.1357");

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.x (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:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	16.0.478584585616890104119296.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 31x^{12} + 336x^{8} - 19375x^{4} + 390625 $ x^16 - 31x^12 + 336x^8 - 19375*x^4 + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1693.6
Root			$-1.04705 + 1.97578i$ of defining polynomial
Character	$\chi$	$=$	2100.1693
Dual form			2100.2.x.c.1357.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{3} +(0.884806 - 2.49342i) q^{7} -1.00000i q^{9} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{3} +(0.884806 - 2.49342i) q^{7} -1.00000i q^{9} +3.27492 q^{11} +(3.02282 - 3.02282i) q^{13} +(-1.41421 - 1.41421i) q^{17} -2.15068 q^{19} +(-1.13746 - 2.38876i) q^{21} +(-0.296014 - 0.296014i) q^{23} +(-0.707107 - 0.707107i) q^{27} +0.725083i q^{29} -1.31342i q^{31} +(2.31572 - 2.31572i) q^{33} +(1.56145 - 1.56145i) q^{37} -4.27492i q^{39} +5.25370i q^{41} +(0.632717 + 0.632717i) q^{43} +(-1.41421 - 1.41421i) q^{47} +(-5.43424 - 4.41238i) q^{49} -2.00000 q^{51} +(3.71492 + 3.71492i) q^{53} +(-1.52076 + 1.52076i) q^{57} -8.71780 q^{59} +1.31342i q^{61} +(-2.49342 - 0.884806i) q^{63} +(-3.08221 + 3.08221i) q^{67} -0.418627 q^{69} +9.27492 q^{71} +(7.07107 - 7.07107i) q^{73} +(2.89767 - 8.16573i) q^{77} -11.8248i q^{79} -1.00000 q^{81} +(11.7025 - 11.7025i) q^{83} +(0.512711 + 0.512711i) q^{87} -18.2728 q^{89} +(-4.86254 - 10.2118i) q^{91} +(-0.928731 - 0.928731i) q^{93} +(-7.26546 - 7.26546i) q^{97} -3.27492i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 8 q^{11} + 12 q^{21} - 32 q^{51} + 88 q^{71} - 16 q^{81} - 108 q^{91}+O(q^{100}) $ 16 * q - 8 * q^11 + 12 * q^21 - 32 * q^51 + 88 * q^71 - 16 * q^81 - 108 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.884806	−	2.49342i	0.334425	−	0.942422i
$7$	0.884806	−	2.49342i	0.334425	−	0.942422i
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	3.27492			0.987425			0.493712	−	0.869625i	$-0.335640\pi$
$11$	3.27492			0.987425			0.493712	+	0.869625i	$0.335640\pi$
$12$		0			0
$13$	3.02282	−	3.02282i	0.838380	−	0.838380i	−0.150265	−	0.988646i	$-0.548013\pi$
$13$	3.02282	−	3.02282i	0.838380	−	0.838380i	0.988646	+	0.150265i	$0.0480128\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.41421	−	1.41421i	−0.342997	−	0.342997i	0.514496	−	0.857493i	$-0.327979\pi$
$17$	−1.41421	−	1.41421i	−0.342997	−	0.342997i	−0.857493	+	0.514496i	$0.827979\pi$
$18$		0			0
$19$	−2.15068			−0.493399			−0.246700	−	0.969092i	$-0.579346\pi$
$19$	−2.15068			−0.493399			−0.246700	+	0.969092i	$0.579346\pi$
$20$		0			0
$21$	−1.13746	−	2.38876i	−0.248214	−	0.521271i
$22$		0			0
$23$	−0.296014	−	0.296014i	−0.0617231	−	0.0617231i	0.675571	−	0.737295i	$-0.263897\pi$
$23$	−0.296014	−	0.296014i	−0.0617231	−	0.0617231i	−0.737295	+	0.675571i	$0.763897\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−0.707107	−	0.707107i	−0.136083	−	0.136083i
$28$		0			0
$29$			0.725083i			0.134644i	0.997731	+	0.0673222i	$0.0214456\pi$
$29$			0.725083i			0.134644i	−0.997731	+	0.0673222i	$0.978554\pi$
$30$		0			0
$31$		−	1.31342i		−	0.235898i	−0.993020	−	0.117949i	$-0.962368\pi$
$31$		−	1.31342i		−	0.235898i	0.993020	−	0.117949i	$-0.0376319\pi$
$32$		0			0
$33$	2.31572	−	2.31572i	0.403114	−	0.403114i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.56145	−	1.56145i	0.256701	−	0.256701i	−0.567010	−	0.823711i	$-0.691900\pi$
$37$	1.56145	−	1.56145i	0.256701	−	0.256701i	0.823711	+	0.567010i	$0.191900\pi$
$38$		0			0
$39$		−	4.27492i		−	0.684535i
$40$		0			0
$41$			5.25370i			0.820490i	0.911975	+	0.410245i	$0.134557\pi$
$41$			5.25370i			0.820490i	−0.911975	+	0.410245i	$0.865443\pi$
$42$		0			0
$43$	0.632717	+	0.632717i	0.0964885	+	0.0964885i	0.753703	−	0.657215i	$-0.228266\pi$
$43$	0.632717	+	0.632717i	0.0964885	+	0.0964885i	−0.657215	+	0.753703i	$0.728266\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.41421	−	1.41421i	−0.206284	−	0.206284i	0.596402	−	0.802686i	$-0.296597\pi$
$47$	−1.41421	−	1.41421i	−0.206284	−	0.206284i	−0.802686	+	0.596402i	$0.796597\pi$
$48$		0			0
$49$	−5.43424	−	4.41238i	−0.776320	−	0.630339i
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$	3.71492	+	3.71492i	0.510284	+	0.510284i	0.914613	−	0.404329i	$-0.132495\pi$
$53$	3.71492	+	3.71492i	0.510284	+	0.510284i	−0.404329	+	0.914613i	$0.632495\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.52076	+	1.52076i	−0.201429	+	0.201429i
$58$		0			0
$59$	−8.71780			−1.13496			−0.567480	−	0.823387i	$-0.692082\pi$
$59$	−8.71780			−1.13496			−0.567480	+	0.823387i	$0.692082\pi$
$60$		0			0
$61$			1.31342i			0.168167i	0.996459	+	0.0840834i	$0.0267962\pi$
$61$			1.31342i			0.168167i	−0.996459	+	0.0840834i	$0.973204\pi$
$62$		0			0
$63$	−2.49342	−	0.884806i	−0.314141	−	0.111475i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.08221	+	3.08221i	−0.376552	+	0.376552i	−0.869856	−	0.493305i	$-0.835789\pi$
$67$	−3.08221	+	3.08221i	−0.376552	+	0.376552i	0.493305	+	0.869856i	$0.335789\pi$
$68$		0			0
$69$	−0.418627			−0.0503967
$70$		0			0
$71$	9.27492			1.10073			0.550365	−	0.834924i	$-0.314489\pi$
$71$	9.27492			1.10073			0.550365	+	0.834924i	$0.314489\pi$
$72$		0			0
$73$	7.07107	−	7.07107i	0.827606	−	0.827606i	−0.159579	−	0.987185i	$-0.551014\pi$
$73$	7.07107	−	7.07107i	0.827606	−	0.827606i	0.987185	+	0.159579i	$0.0510137\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.89767	−	8.16573i	0.330220	−	0.930571i
$78$		0			0
$79$		−	11.8248i		−	1.33039i	−0.746670	−	0.665194i	$-0.768349\pi$
$79$		−	11.8248i		−	1.33039i	0.746670	−	0.665194i	$-0.231651\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	11.7025	−	11.7025i	1.28452	−	1.28452i	0.346447	−	0.938070i	$-0.387388\pi$
$83$	11.7025	−	11.7025i	1.28452	−	1.28452i	0.938070	−	0.346447i	$-0.112612\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.512711	+	0.512711i	0.0549684	+	0.0549684i
$88$		0			0
$89$	−18.2728			−1.93692			−0.968459	−	0.249173i	$-0.919841\pi$
$89$	−18.2728			−1.93692			−0.968459	+	0.249173i	$0.919841\pi$
$90$		0			0
$91$	−4.86254	−	10.2118i	−0.509733	−	1.07048i
$92$		0			0
$93$	−0.928731	−	0.928731i	−0.0963049	−	0.0963049i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−7.26546	−	7.26546i	−0.737696	−	0.737696i	0.234436	−	0.972132i	$-0.424676\pi$
$97$	−7.26546	−	7.26546i	−0.737696	−	0.737696i	−0.972132	+	0.234436i	$0.924676\pi$
$98$		0			0
$99$		−	3.27492i		−	0.329142i
$100$		0			0
$101$			7.76546i			0.772692i	0.922354	+	0.386346i	$0.126263\pi$
$101$			7.76546i			0.772692i	−0.922354	+	0.386346i	$0.873737\pi$
$102$		0			0
$103$	4.63143	−	4.63143i	0.456349	−	0.456349i	−0.441106	−	0.897455i	$-0.645414\pi$
$103$	4.63143	−	4.63143i	0.456349	−	0.456349i	0.897455	+	0.441106i	$0.145414\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.30695	−	4.30695i	0.416369	−	0.416369i	−0.467581	−	0.883950i	$-0.654874\pi$
$107$	4.30695	−	4.30695i	0.416369	−	0.416369i	0.883950	+	0.467581i	$0.154874\pi$
$108$		0			0
$109$		−	1.00000i		−	0.0957826i	−0.998853	−	0.0478913i	$-0.984750\pi$
$109$		−	1.00000i		−	0.0957826i	0.998853	−	0.0478913i	$-0.0152501\pi$
$110$		0			0
$111$		−	2.20822i		−	0.209595i
$112$		0			0
$113$	11.3594	+	11.3594i	1.06860	+	1.06860i	0.997467	+	0.0711366i	$0.0226626\pi$
$113$	11.3594	+	11.3594i	1.06860	+	1.06860i	0.0711366	+	0.997467i	$0.477337\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.02282	−	3.02282i	−0.279460	−	0.279460i
$118$		0			0
$119$	−4.77753	+	2.27492i	−0.437955	+	0.208541i
$120$		0			0
$121$	−0.274917			−0.0249925
$122$		0			0
$123$	3.71492	+	3.71492i	0.334963	+	0.334963i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−9.50195	+	9.50195i	−0.843161	+	0.843161i	−0.989269	−	0.146107i	$-0.953325\pi$
$127$	−9.50195	+	9.50195i	−0.843161	+	0.843161i	0.146107	+	0.989269i	$0.453325\pi$
$128$		0			0
$129$	0.894797			0.0787825
$130$		0			0
$131$		−	19.9474i		−	1.74281i	−0.490566	−	0.871404i	$-0.663210\pi$
$131$		−	19.9474i		−	1.74281i	0.490566	−	0.871404i	$-0.336790\pi$
$132$		0			0
$133$	−1.90293	+	5.36253i	−0.165005	+	0.464990i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.3703	−	15.3703i	1.31318	−	1.31318i	0.394117	−	0.919060i	$-0.371051\pi$
$137$	15.3703	−	15.3703i	1.31318	−	1.31318i	0.919060	−	0.394117i	$-0.128949\pi$
$138$		0			0
$139$	−3.46410			−0.293821			−0.146911	−	0.989150i	$-0.546933\pi$
$139$	−3.46410			−0.293821			−0.146911	+	0.989150i	$0.546933\pi$
$140$		0			0
$141$	−2.00000			−0.168430
$142$		0			0
$143$	9.89949	−	9.89949i	0.827837	−	0.827837i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6.96261	+	0.722565i	−0.574266	+	0.0595962i
$148$		0			0
$149$			23.2749i			1.90676i	0.301779	+	0.953378i	$0.402419\pi$
$149$			23.2749i			1.90676i	−0.301779	+	0.953378i	$0.597581\pi$
$150$		0			0
$151$	−11.0000			−0.895167			−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000			−0.895167			−0.447584	+	0.894242i	$0.647715\pi$
$152$		0			0
$153$	−1.41421	+	1.41421i	−0.114332	+	0.114332i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−7.65426	−	7.65426i	−0.610876	−	0.610876i	0.332298	−	0.943174i	$-0.392176\pi$
$157$	−7.65426	−	7.65426i	−0.610876	−	0.610876i	−0.943174	+	0.332298i	$0.892176\pi$
$158$		0			0
$159$	5.25370			0.416645
$160$		0			0
$161$	−1.00000	+	0.476171i	−0.0788110	+	0.0375275i
$162$		0			0
$163$	15.7070	+	15.7070i	1.23027	+	1.23027i	0.963859	+	0.266412i	$0.0858381\pi$
$163$	15.7070	+	15.7070i	1.23027	+	1.23027i	0.266412	+	0.963859i	$0.414162\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.45986	+	7.45986i	0.577261	+	0.577261i	0.934148	−	0.356886i	$-0.116162\pi$
$167$	7.45986	+	7.45986i	0.577261	+	0.577261i	−0.356886	+	0.934148i	$0.616162\pi$
$168$		0			0
$169$		−	5.27492i		−	0.405763i
$170$		0			0
$171$			2.15068i			0.164466i
$172$		0			0
$173$	−13.5055	+	13.5055i	−1.02680	+	1.02680i	−0.0271738	+	0.999631i	$0.508651\pi$
$173$	−13.5055	+	13.5055i	−1.02680	+	1.02680i	−0.999631	+	0.0271738i	$0.991349\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−6.16441	+	6.16441i	−0.463346	+	0.463346i
$178$		0			0
$179$			7.45017i			0.556852i	0.960458	+	0.278426i	$0.0898126\pi$
$179$			7.45017i			0.556852i	−0.960458	+	0.278426i	$0.910187\pi$
$180$		0			0
$181$		−	3.10302i		−	0.230646i	−0.993328	−	0.115323i	$-0.963210\pi$
$181$		−	3.10302i		−	0.230646i	0.993328	−	0.115323i	$-0.0367902\pi$
$182$		0			0
$183$	0.928731	+	0.928731i	0.0686538	+	0.0686538i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−4.63143	−	4.63143i	−0.338684	−	0.338684i
$188$		0			0
$189$	−2.38876	+	1.13746i	−0.173757	+	0.0827379i
$190$		0			0
$191$	12.5498			0.908074			0.454037	−	0.890983i	$-0.349983\pi$
$191$	12.5498			0.908074			0.454037	+	0.890983i	$0.349983\pi$
$192$		0			0
$193$	−13.5536	−	13.5536i	−0.975608	−	0.975608i	0.0241020	−	0.999710i	$-0.492327\pi$
$193$	−13.5536	−	13.5536i	−0.975608	−	0.975608i	−0.999710	+	0.0241020i	$0.992327\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0328	−	12.0328i	0.857303	−	0.857303i	−0.133717	−	0.991020i	$-0.542691\pi$
$197$	12.0328	−	12.0328i	0.857303	−	0.857303i	0.991020	+	0.133717i	$0.0426913\pi$
$198$		0			0
$199$	15.1698			1.07536			0.537680	−	0.843149i	$-0.319301\pi$
$199$	15.1698			1.07536			0.537680	+	0.843149i	$0.319301\pi$
$200$		0			0
$201$			4.35890i			0.307453i
$202$		0			0
$203$	1.80793	+	0.641557i	0.126892	+	0.0450285i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.296014	+	0.296014i	−0.0205744	+	0.0205744i
$208$		0			0
$209$	−7.04329			−0.487195
$210$		0			0
$211$	1.72508			0.118760			0.0593798	−	0.998235i	$-0.481088\pi$
$211$	1.72508			0.118760			0.0593798	+	0.998235i	$0.481088\pi$
$212$		0			0
$213$	6.55836	−	6.55836i	0.449371	−	0.449371i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.27491	−	1.16213i	−0.222315	−	0.0788902i
$218$		0			0
$219$		−	10.0000i		−	0.675737i
$220$		0			0
$221$	−8.54983			−0.575124
$222$		0			0
$223$	−3.41161	+	3.41161i	−0.228459	+	0.228459i	−0.812049	−	0.583590i	$-0.801648\pi$
$223$	−3.41161	+	3.41161i	−0.228459	+	0.228459i	0.583590	+	0.812049i	$0.301648\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.8943	+	13.8943i	0.922197	+	0.922197i	0.997184	−	0.0749876i	$-0.0238917\pi$
$227$	13.8943	+	13.8943i	0.922197	+	0.922197i	−0.0749876	+	0.997184i	$0.523892\pi$
$228$		0			0
$229$	−12.5430			−0.828864			−0.414432	−	0.910080i	$-0.636020\pi$
$229$	−12.5430			−0.828864			−0.414432	+	0.910080i	$0.636020\pi$
$230$		0			0
$231$	−3.72508	−	7.82300i	−0.245092	−	0.514716i
$232$		0			0
$233$	−6.46043	−	6.46043i	−0.423237	−	0.423237i	0.463080	−	0.886317i	$-0.346744\pi$
$233$	−6.46043	−	6.46043i	−0.423237	−	0.423237i	−0.886317	+	0.463080i	$0.846744\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−8.36136	−	8.36136i	−0.543129	−	0.543129i
$238$		0			0
$239$			9.09967i			0.588609i	0.955712	+	0.294304i	$0.0950879\pi$
$239$			9.09967i			0.588609i	−0.955712	+	0.294304i	$0.904912\pi$
$240$		0			0
$241$			28.1890i			1.81581i	0.419174	+	0.907906i	$0.362320\pi$
$241$			28.1890i			1.81581i	−0.419174	+	0.907906i	$0.637680\pi$
$242$		0			0
$243$	−0.707107	+	0.707107i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.50112	+	6.50112i	−0.413656	+	0.413656i
$248$		0			0
$249$		−	16.5498i		−	1.04880i
$250$		0			0
$251$			6.09095i			0.384457i	0.981350	+	0.192229i	$0.0615715\pi$
$251$			6.09095i			0.384457i	−0.981350	+	0.192229i	$0.938428\pi$
$252$		0			0
$253$	−0.969421	−	0.969421i	−0.0609470	−	0.0609470i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.60601	+	3.60601i	0.224937	+	0.224937i	0.810574	−	0.585637i	$-0.199156\pi$
$257$	3.60601	+	3.60601i	0.224937	+	0.224937i	−0.585637	+	0.810574i	$0.699156\pi$
$258$		0			0
$259$	−2.51176	−	5.27492i	−0.156073	−	0.327767i
$260$		0			0
$261$	0.725083			0.0448815
$262$		0			0
$263$	−7.05246	−	7.05246i	−0.434873	−	0.434873i	0.455409	−	0.890282i	$-0.349493\pi$
$263$	−7.05246	−	7.05246i	−0.434873	−	0.434873i	−0.890282	+	0.455409i	$0.849493\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−12.9209	+	12.9209i	−0.790744	+	0.790744i
$268$		0			0
$269$	18.1578			1.10710			0.553549	−	0.832817i	$-0.313273\pi$
$269$	18.1578			1.10710			0.553549	+	0.832817i	$0.313273\pi$
$270$		0			0
$271$		−	22.5742i		−	1.37129i	−0.727938	−	0.685643i	$-0.759521\pi$
$271$		−	22.5742i		−	1.37129i	0.727938	−	0.685643i	$-0.240479\pi$
$272$		0			0
$273$	−10.6591	−	3.78247i	−0.645121	−	0.228926i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3.97025	−	3.97025i	0.238549	−	0.238549i	−0.577700	−	0.816249i	$-0.696050\pi$
$277$	3.97025	−	3.97025i	0.238549	−	0.238549i	0.816249	+	0.577700i	$0.196050\pi$
$278$		0			0
$279$	−1.31342			−0.0786326
$280$		0			0
$281$	12.7251			0.759115			0.379557	−	0.925168i	$-0.376076\pi$
$281$	12.7251			0.759115			0.379557	+	0.925168i	$0.376076\pi$
$282$		0			0
$283$	18.9680	−	18.9680i	1.12753	−	1.12753i	0.136951	−	0.990578i	$-0.456270\pi$
$283$	18.9680	−	18.9680i	1.12753	−	1.12753i	0.990578	−	0.136951i	$-0.0437301\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.0996	+	4.64850i	0.773248	+	0.274392i
$288$		0			0
$289$		−	13.0000i		−	0.764706i
$290$		0			0
$291$	−10.2749			−0.602326
$292$		0			0
$293$	−10.6771	+	10.6771i	−0.623762	+	0.623762i	−0.946491	−	0.322730i	$-0.895400\pi$
$293$	−10.6771	+	10.6771i	−0.623762	+	0.623762i	0.322730	+	0.946491i	$0.395400\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−2.31572	−	2.31572i	−0.134371	−	0.134371i
$298$		0			0
$299$	−1.78959			−0.103495
$300$		0			0
$301$	2.13746	−	1.01779i	0.123201	−	0.0586647i
$302$		0			0
$303$	5.49101	+	5.49101i	0.315450	+	0.315450i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	11.1193	+	11.1193i	0.634613	+	0.634613i	0.949221	−	0.314609i	$-0.101873\pi$
$307$	11.1193	+	11.1193i	0.634613	+	0.634613i	−0.314609	+	0.949221i	$0.601873\pi$
$308$		0			0
$309$		−	6.54983i		−	0.372607i
$310$		0			0
$311$			33.0816i			1.87589i	0.346790	+	0.937943i	$0.387271\pi$
$311$			33.0816i			1.87589i	−0.346790	+	0.937943i	$0.612729\pi$
$312$		0			0
$313$	−19.6046	+	19.6046i	−1.10812	+	1.10812i	−0.114719	+	0.993398i	$0.536597\pi$
$313$	−19.6046	+	19.6046i	−1.10812	+	1.10812i	−0.993398	+	0.114719i	$0.963403\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.64448	+	7.64448i	−0.429357	+	0.429357i	−0.888409	−	0.459052i	$-0.848189\pi$
$317$	−7.64448	+	7.64448i	−0.429357	+	0.429357i	0.459052	+	0.888409i	$0.348189\pi$
$318$		0			0
$319$			2.37459i			0.132951i
$320$		0			0
$321$		−	6.09095i		−	0.339964i
$322$		0			0
$323$	3.04152	+	3.04152i	0.169235	+	0.169235i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−0.707107	−	0.707107i	−0.0391031	−	0.0391031i
$328$		0			0
$329$	−4.77753	+	2.27492i	−0.263394	+	0.125420i
$330$		0			0
$331$	10.3746			0.570239			0.285119	−	0.958492i	$-0.407967\pi$
$331$	10.3746			0.570239			0.285119	+	0.958492i	$0.407967\pi$
$332$		0			0
$333$	−1.56145	−	1.56145i	−0.0855668	−	0.0855668i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−12.5842	+	12.5842i	−0.685502	+	0.685502i	−0.961235	−	0.275732i	$-0.911080\pi$
$337$	−12.5842	+	12.5842i	−0.685502	+	0.685502i	0.275732	+	0.961235i	$0.411080\pi$
$338$		0			0
$339$	16.0646			0.872511
$340$		0			0
$341$		−	4.30136i		−	0.232931i
$342$		0			0
$343$	−15.8101	+	9.64572i	−0.853667	+	0.520820i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.50195	−	9.50195i	0.510091	−	0.510091i	−0.404463	−	0.914554i	$-0.632542\pi$
$347$	9.50195	−	9.50195i	0.510091	−	0.510091i	0.914554	+	0.404463i	$0.132542\pi$
$348$		0			0
$349$	18.2728			0.978123			0.489062	−	0.872249i	$-0.337339\pi$
$349$	18.2728			0.978123			0.489062	+	0.872249i	$0.337339\pi$
$350$		0			0
$351$	−4.27492			−0.228178
$352$		0			0
$353$	5.26806	−	5.26806i	0.280391	−	0.280391i	−0.552874	−	0.833265i	$-0.686469\pi$
$353$	5.26806	−	5.26806i	0.280391	−	0.280391i	0.833265	+	0.552874i	$0.186469\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−1.76961	+	4.98683i	−0.0936578	+	0.263931i
$358$		0			0
$359$			26.3746i			1.39200i	0.718043	+	0.695999i	$0.245038\pi$
$359$			26.3746i			1.39200i	−0.718043	+	0.695999i	$0.754962\pi$
$360$		0			0
$361$	−14.3746			−0.756557
$362$		0			0
$363$	−0.194396	+	0.194396i	−0.0102031	+	0.0102031i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	18.9680	+	18.9680i	0.990120	+	0.990120i	0.999952	−	0.00983201i	$-0.00312968\pi$
$367$	18.9680	+	18.9680i	0.990120	+	0.990120i	−0.00983201	+	0.999952i	$0.503130\pi$
$368$		0			0
$369$	5.25370			0.273497
$370$		0			0
$371$	12.5498	−	5.97586i	0.651555	−	0.310251i
$372$		0			0
$373$	−12.9615	−	12.9615i	−0.671123	−	0.671123i	0.286852	−	0.957975i	$-0.407391\pi$
$373$	−12.9615	−	12.9615i	−0.671123	−	0.671123i	−0.957975	+	0.286852i	$0.907391\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.19180	+	2.19180i	0.112883	+	0.112883i
$378$		0			0
$379$			15.0000i			0.770498i	0.922813	+	0.385249i	$0.125884\pi$
$379$			15.0000i			0.770498i	−0.922813	+	0.385249i	$0.874116\pi$
$380$		0			0
$381$			13.4378i			0.688438i
$382$		0			0
$383$	−15.1676	+	15.1676i	−0.775026	+	0.775026i	−0.978980	−	0.203954i	$-0.934621\pi$
$383$	−15.1676	+	15.1676i	−0.775026	+	0.775026i	0.203954	+	0.978980i	$0.434621\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.632717	−	0.632717i	0.0321628	−	0.0321628i
$388$		0			0
$389$			18.3746i			0.931628i	0.884883	+	0.465814i	$0.154239\pi$
$389$			18.3746i			0.931628i	−0.884883	+	0.465814i	$0.845761\pi$
$390$		0			0
$391$			0.837253i			0.0423417i
$392$		0			0
$393$	−14.1049	−	14.1049i	−0.711499	−	0.711499i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−4.43704	−	4.43704i	−0.222688	−	0.222688i	0.586941	−	0.809630i	$-0.300332\pi$
$397$	−4.43704	−	4.43704i	−0.222688	−	0.222688i	−0.809630	+	0.586941i	$0.800332\pi$
$398$		0			0
$399$	2.44631	+	5.13746i	0.122469	+	0.257195i
$400$		0			0
$401$	−1.27492			−0.0636663			−0.0318332	−	0.999493i	$-0.510135\pi$
$401$	−1.27492			−0.0636663			−0.0318332	+	0.999493i	$0.510135\pi$
$402$		0			0
$403$	−3.97025	−	3.97025i	−0.197772	−	0.197772i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	5.11361	−	5.11361i	0.253472	−	0.253472i
$408$		0			0
$409$	31.7682			1.57084			0.785418	−	0.618966i	$-0.212448\pi$
$409$	31.7682			1.57084			0.785418	+	0.618966i	$0.212448\pi$
$410$		0			0
$411$		−	21.7370i		−	1.07220i
$412$		0			0
$413$	−7.71356	+	21.7371i	−0.379559	+	1.06961i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.44949	+	2.44949i	−0.119952	+	0.119952i
$418$		0			0
$419$	12.1819			0.595125			0.297562	−	0.954702i	$-0.403826\pi$
$419$	12.1819			0.595125			0.297562	+	0.954702i	$0.403826\pi$
$420$		0			0
$421$	0.725083			0.0353384			0.0176692	−	0.999844i	$-0.494375\pi$
$421$	0.725083			0.0353384			0.0176692	+	0.999844i	$0.494375\pi$
$422$		0			0
$423$	−1.41421	+	1.41421i	−0.0687614	+	0.0687614i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.27491	+	1.16213i	0.158484	+	0.0562392i
$428$		0			0
$429$		−	14.0000i		−	0.675926i
$430$		0			0
$431$	25.6495			1.23549			0.617747	−	0.786377i	$-0.288046\pi$
$431$	25.6495			1.23549			0.617747	+	0.786377i	$0.288046\pi$
$432$		0			0
$433$	−4.43704	+	4.43704i	−0.213230	+	0.213230i	−0.805638	−	0.592408i	$-0.798177\pi$
$433$	−4.43704	+	4.43704i	−0.213230	+	0.213230i	0.592408	+	0.805638i	$0.298177\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.636630	+	0.636630i	0.0304542	+	0.0304542i
$438$		0			0
$439$	33.4427			1.59613			0.798066	−	0.602570i	$-0.205857\pi$
$439$	33.4427			1.59613			0.798066	+	0.602570i	$0.205857\pi$
$440$		0			0
$441$	−4.41238	+	5.43424i	−0.210113	+	0.258773i
$442$		0			0
$443$	−19.0039	−	19.0039i	−0.902902	−	0.902902i	0.0927842	−	0.995686i	$-0.470423\pi$
$443$	−19.0039	−	19.0039i	−0.902902	−	0.902902i	−0.995686	+	0.0927842i	$0.970423\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	16.4579	+	16.4579i	0.778430	+	0.778430i
$448$		0			0
$449$			7.82475i			0.369273i	0.982807	+	0.184636i	$0.0591108\pi$
$449$			7.82475i			0.369273i	−0.982807	+	0.184636i	$0.940889\pi$
$450$		0			0
$451$			17.2054i			0.810172i
$452$		0			0
$453$	−7.77817	+	7.77817i	−0.365451	+	0.365451i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.86840	−	5.86840i	0.274512	−	0.274512i	−0.556401	−	0.830914i	$-0.687818\pi$
$457$	5.86840	−	5.86840i	0.274512	−	0.274512i	0.830914	+	0.556401i	$0.187818\pi$
$458$		0			0
$459$			2.00000i			0.0933520i
$460$		0			0
$461$			21.6219i			1.00703i	0.863986	+	0.503515i	$0.167960\pi$
$461$			21.6219i			1.00703i	−0.863986	+	0.503515i	$0.832040\pi$
$462$		0			0
$463$	−14.7783	−	14.7783i	−0.686807	−	0.686807i	0.274718	−	0.961525i	$-0.411415\pi$
$463$	−14.7783	−	14.7783i	−0.686807	−	0.686807i	−0.961525	+	0.274718i	$0.911415\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.388792	+	0.388792i	0.0179911	+	0.0179911i	0.716045	−	0.698054i	$-0.245950\pi$
$467$	0.388792	+	0.388792i	0.0179911	+	0.0179911i	−0.698054	+	0.716045i	$0.745950\pi$
$468$		0			0
$469$	4.95807	+	10.4124i	0.228942	+	0.480799i
$470$		0			0
$471$	−10.8248			−0.498778
$472$		0			0
$473$	2.07210	+	2.07210i	0.0952751	+	0.0952751i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	3.71492	−	3.71492i	0.170095	−	0.170095i
$478$		0			0
$479$	24.2487			1.10795			0.553976	−	0.832533i	$-0.313110\pi$
$479$	24.2487			1.10795			0.553976	+	0.832533i	$0.313110\pi$
$480$		0			0
$481$		−	9.43996i		−	0.430425i
$482$		0			0
$483$	−0.370403	+	1.04381i	−0.0168539	+	0.0474950i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−15.4110	+	15.4110i	−0.698341	+	0.698341i	−0.964052	−	0.265712i	$-0.914393\pi$
$487$	−15.4110	+	15.4110i	−0.698341	+	0.698341i	0.265712	+	0.964052i	$0.414393\pi$
$488$		0			0
$489$	22.2131			1.00451
$490$		0			0
$491$	28.9244			1.30534			0.652670	−	0.757642i	$-0.273649\pi$
$491$	28.9244			1.30534			0.652670	+	0.757642i	$0.273649\pi$
$492$		0			0
$493$	1.02542	−	1.02542i	0.0461827	−	0.0461827i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.20650	−	23.1262i	0.368112	−	1.03735i
$498$		0			0
$499$		−	9.17525i		−	0.410741i	−0.978684	−	0.205370i	$-0.934160\pi$
$499$		−	9.17525i		−	0.410741i	0.978684	−	0.205370i	$-0.0658399\pi$
$500$		0			0
$501$	10.5498			0.471332
$502$		0			0
$503$	−8.48528	+	8.48528i	−0.378340	+	0.378340i	−0.870503	−	0.492163i	$-0.836206\pi$
$503$	−8.48528	+	8.48528i	−0.378340	+	0.378340i	0.492163	+	0.870503i	$0.336206\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3.72993	−	3.72993i	−0.165652	−	0.165652i
$508$		0			0
$509$	−17.3205			−0.767718			−0.383859	−	0.923392i	$-0.625405\pi$
$509$	−17.3205			−0.767718			−0.383859	+	0.923392i	$0.625405\pi$
$510$		0			0
$511$	−11.3746	−	23.8876i	−0.503182	−	1.05673i
$512$		0			0
$513$	1.52076	+	1.52076i	0.0671431	+	0.0671431i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−4.63143	−	4.63143i	−0.203690	−	0.203690i
$518$		0			0
$519$			19.0997i			0.838382i
$520$		0			0
$521$			32.0142i			1.40257i	0.712883	+	0.701283i	$0.247389\pi$
$521$			32.0142i			1.40257i	−0.712883	+	0.701283i	$0.752611\pi$
$522$		0			0
$523$	−20.1343	+	20.1343i	−0.880413	+	0.880413i	−0.993576	−	0.113163i	$-0.963902\pi$
$523$	−20.1343	+	20.1343i	−0.880413	+	0.880413i	0.113163	+	0.993576i	$0.463902\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.85746	+	1.85746i	−0.0809123	+	0.0809123i
$528$		0			0
$529$		−	22.8248i		−	0.992381i
$530$		0			0
$531$			8.71780i			0.378320i
$532$		0			0
$533$	15.8810	+	15.8810i	0.687882	+	0.687882i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	5.26806	+	5.26806i	0.227334	+	0.227334i
$538$		0			0
$539$	−17.7967	−	14.4502i	−0.766557	−	0.622413i
$540$		0			0
$541$	−33.1993			−1.42735			−0.713676	−	0.700476i	$-0.752971\pi$
$541$	−33.1993			−1.42735			−0.713676	+	0.700476i	$0.752971\pi$
$542$		0			0
$543$	−2.19417	−	2.19417i	−0.0941607	−	0.0941607i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−5.19499	+	5.19499i	−0.222122	+	0.222122i	−0.809391	−	0.587270i	$-0.800203\pi$
$547$	−5.19499	+	5.19499i	−0.222122	+	0.222122i	0.587270	+	0.809391i	$0.300203\pi$
$548$		0			0
$549$	1.31342			0.0560556
$550$		0			0
$551$		−	1.55942i		−	0.0664335i
$552$		0			0
$553$	−29.4840	−	10.4626i	−1.25379	−	0.444915i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16.9318	+	16.9318i	−0.717423	+	0.717423i	−0.968077	−	0.250654i	$-0.919354\pi$
$557$	−16.9318	+	16.9318i	−0.717423	+	0.717423i	0.250654	+	0.968077i	$0.419354\pi$
$558$		0			0
$559$	3.82518			0.161788
$560$		0			0
$561$	−6.54983			−0.276534
$562$		0			0
$563$	19.4102	−	19.4102i	0.818042	−	0.818042i	−0.167782	−	0.985824i	$-0.553660\pi$
$563$	19.4102	−	19.4102i	0.818042	−	0.818042i	0.985824	+	0.167782i	$0.0536604\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.884806	+	2.49342i	−0.0371583	+	0.104714i
$568$		0			0
$569$			41.8248i			1.75338i	0.481051	+	0.876692i	$0.340255\pi$
$569$			41.8248i			1.75338i	−0.481051	+	0.876692i	$0.659745\pi$
$570$		0			0
$571$	18.0997			0.757448			0.378724	−	0.925510i	$-0.376363\pi$
$571$	18.0997			0.757448			0.378724	+	0.925510i	$0.376363\pi$
$572$		0			0
$573$	8.87407	−	8.87407i	0.370720	−	0.370720i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	8.67968	+	8.67968i	0.361340	+	0.361340i	0.864306	−	0.502966i	$-0.167758\pi$
$577$	8.67968	+	8.67968i	0.361340	+	0.361340i	−0.502966	+	0.864306i	$0.667758\pi$
$578$		0			0
$579$	−19.1676			−0.796580
$580$		0			0
$581$	−18.8248	−	39.5336i	−0.780982	−	1.64013i
$582$		0			0
$583$	12.1661	+	12.1661i	0.503867	+	0.503867i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.63143	+	4.63143i	0.191160	+	0.191160i	0.796197	−	0.605037i	$-0.206842\pi$
$587$	4.63143	+	4.63143i	0.191160	+	0.191160i	−0.605037	+	0.796197i	$0.706842\pi$
$588$		0			0
$589$			2.82475i			0.116392i
$590$		0			0
$591$		−	17.0170i		−	0.699985i
$592$		0			0
$593$	−14.1421	+	14.1421i	−0.580748	+	0.580748i	−0.935109	−	0.354361i	$-0.884698\pi$
$593$	−14.1421	+	14.1421i	−0.580748	+	0.580748i	0.354361	+	0.935109i	$0.384698\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	10.7267	−	10.7267i	0.439014	−	0.439014i
$598$		0			0
$599$			22.3746i			0.914201i	0.889415	+	0.457100i	$0.151112\pi$
$599$			22.3746i			0.914201i	−0.889415	+	0.457100i	$0.848888\pi$
$600$		0			0
$601$			24.0027i			0.979091i	0.871978	+	0.489546i	$0.162837\pi$
$601$			24.0027i			0.979091i	−0.871978	+	0.489546i	$0.837163\pi$
$602$		0			0
$603$	3.08221	+	3.08221i	0.125517	+	0.125517i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	5.40902	+	5.40902i	0.219545	+	0.219545i	0.808307	−	0.588762i	$-0.200384\pi$
$607$	5.40902	+	5.40902i	0.219545	+	0.219545i	−0.588762	+	0.808307i	$0.700384\pi$
$608$		0			0
$609$	1.73205	−	0.824752i	0.0701862	−	0.0334206i
$610$		0			0
$611$	−8.54983			−0.345889
$612$		0			0
$613$	29.7713	+	29.7713i	1.20245	+	1.20245i	0.973419	+	0.229031i	$0.0735558\pi$
$613$	29.7713	+	29.7713i	1.20245	+	1.20245i	0.229031	+	0.973419i	$0.426444\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.27637	+	5.27637i	−0.212419	+	0.212419i	−0.805294	−	0.592875i	$-0.797993\pi$
$617$	−5.27637	+	5.27637i	−0.212419	+	0.212419i	0.592875	+	0.805294i	$0.297993\pi$
$618$		0			0
$619$	8.12654			0.326633			0.163317	−	0.986574i	$-0.447781\pi$
$619$	8.12654			0.326633			0.163317	+	0.986574i	$0.447781\pi$
$620$		0			0
$621$			0.418627i			0.0167989i
$622$		0			0
$623$	−16.1679	+	45.5618i	−0.647754	+	1.82539i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−4.98036	+	4.98036i	−0.198896	+	0.198896i
$628$		0			0
$629$	−4.41644			−0.176095
$630$		0			0
$631$	−29.0000			−1.15447			−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000			−1.15447			−0.577236	+	0.816577i	$0.695869\pi$
$632$		0			0
$633$	1.21982	−	1.21982i	0.0484834	−	0.0484834i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−29.7646	+	3.08891i	−1.17932	+	0.122387i
$638$		0			0
$639$		−	9.27492i		−	0.366910i
$640$		0			0
$641$	−13.2749			−0.524328			−0.262164	−	0.965023i	$-0.584436\pi$
$641$	−13.2749			−0.524328			−0.262164	+	0.965023i	$0.584436\pi$
$642$		0			0
$643$	21.3542	−	21.3542i	0.842126	−	0.842126i	−0.147009	−	0.989135i	$-0.546965\pi$
$643$	21.3542	−	21.3542i	0.842126	−	0.842126i	0.989135	+	0.147009i	$0.0469647\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.7481	+	17.7481i	0.697752	+	0.697752i	0.963925	−	0.266174i	$-0.0857594\pi$
$647$	17.7481	+	17.7481i	0.697752	+	0.697752i	−0.266174	+	0.963925i	$0.585759\pi$
$648$		0			0
$649$	−28.5501			−1.12069
$650$		0			0
$651$	−3.13746	+	1.49397i	−0.122967	+	0.0585531i
$652$		0			0
$653$	−14.0235	−	14.0235i	−0.548783	−	0.548783i	0.377306	−	0.926089i	$-0.376851\pi$
$653$	−14.0235	−	14.0235i	−0.548783	−	0.548783i	−0.926089	+	0.377306i	$0.876851\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−7.07107	−	7.07107i	−0.275869	−	0.275869i
$658$		0			0
$659$		−	33.0997i		−	1.28938i	−0.764444	−	0.644690i	$-0.776986\pi$
$659$		−	33.0997i		−	1.28938i	0.764444	−	0.644690i	$-0.223014\pi$
$660$		0			0
$661$			33.8038i			1.31481i	0.753536	+	0.657407i	$0.228347\pi$
$661$			33.8038i			1.31481i	−0.753536	+	0.657407i	$0.771653\pi$
$662$		0			0
$663$	−6.04565	+	6.04565i	−0.234793	+	0.234793i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.214635	−	0.214635i	0.00831068	−	0.00831068i
$668$		0			0
$669$			4.82475i			0.186536i
$670$		0			0
$671$			4.30136i			0.166052i
$672$		0			0
$673$	−7.26709	−	7.26709i	−0.280126	−	0.280126i	0.553033	−	0.833159i	$-0.313470\pi$
$673$	−7.26709	−	7.26709i	−0.280126	−	0.280126i	−0.833159	+	0.553033i	$0.813470\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.26806	+	5.26806i	0.202468	+	0.202468i	0.801057	−	0.598589i	$-0.204272\pi$
$677$	5.26806	+	5.26806i	0.202468	+	0.202468i	−0.598589	+	0.801057i	$0.704272\pi$
$678$		0			0
$679$	−24.5443	+	11.6873i	−0.941925	+	0.448517i
$680$		0			0
$681$	19.6495			0.752971
$682$		0			0
$683$	−13.7275	−	13.7275i	−0.525269	−	0.525269i	0.393889	−	0.919158i	$-0.371129\pi$
$683$	−13.7275	−	13.7275i	−0.525269	−	0.525269i	−0.919158	+	0.393889i	$0.871129\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−8.86923	+	8.86923i	−0.338382	+	0.338382i
$688$		0			0
$689$	22.4591			0.855624
$690$		0			0
$691$		−	47.7753i		−	1.81746i	−0.417388	−	0.908728i	$-0.637055\pi$
$691$		−	47.7753i		−	1.81746i	0.417388	−	0.908728i	$-0.362945\pi$
$692$		0			0
$693$	−8.16573	−	2.89767i	−0.310190	−	0.110073i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	7.42985	−	7.42985i	0.281426	−	0.281426i
$698$		0			0
$699$	−9.13642			−0.345571
$700$		0			0
$701$	−45.2990			−1.71092			−0.855460	−	0.517869i	$-0.826725\pi$
$701$	−45.2990			−1.71092			−0.855460	+	0.517869i	$0.826725\pi$
$702$		0			0
$703$	−3.35817	+	3.35817i	−0.126656	+	0.126656i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19.3625	+	6.87092i	0.728202	+	0.258408i
$708$		0			0
$709$		−	4.27492i		−	0.160548i	−0.996773	−	0.0802739i	$-0.974420\pi$
$709$		−	4.27492i		−	0.160548i	0.996773	−	0.0802739i	$-0.0255795\pi$
$710$		0			0
$711$	−11.8248			−0.443463
$712$		0			0
$713$	−0.388792	+	0.388792i	−0.0145604	+	0.0145604i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	6.43444	+	6.43444i	0.240298	+	0.240298i
$718$		0			0
$719$	32.2443			1.20251			0.601256	−	0.799057i	$-0.294667\pi$
$719$	32.2443			1.20251			0.601256	+	0.799057i	$0.294667\pi$
$720$		0			0
$721$	−7.45017	−	15.6460i	−0.277459	−	0.582688i
$722$		0			0
$723$	19.9326	+	19.9326i	0.741302	+	0.741302i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	20.7710	+	20.7710i	0.770353	+	0.770353i	0.978168	−	0.207815i	$-0.0666353\pi$
$727$	20.7710	+	20.7710i	0.770353	+	0.770353i	−0.207815	+	0.978168i	$0.566635\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$		−	1.78959i		−	0.0661905i
$732$		0			0
$733$	−16.5818	+	16.5818i	−0.612462	+	0.612462i	−0.943587	−	0.331125i	$-0.892572\pi$
$733$	−16.5818	+	16.5818i	−0.612462	+	0.612462i	0.331125	+	0.943587i	$0.392572\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10.0940	+	10.0940i	−0.371816	+	0.371816i
$738$		0			0
$739$			13.6254i			0.501219i	0.968088	+	0.250609i	$0.0806310\pi$
$739$			13.6254i			0.501219i	−0.968088	+	0.250609i	$0.919369\pi$
$740$		0			0
$741$			9.19397i			0.337749i
$742$		0			0
$743$	4.30695	+	4.30695i	0.158007	+	0.158007i	0.781683	−	0.623676i	$-0.214362\pi$
$743$	4.30695	+	4.30695i	0.158007	+	0.158007i	−0.623676	+	0.781683i	$0.714362\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−11.7025	−	11.7025i	−0.428172	−	0.428172i
$748$		0			0
$749$	−6.92820	−	14.5498i	−0.253151	−	0.531639i
$750$		0			0
$751$	24.0000			0.875772			0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000			0.875772			0.437886	+	0.899030i	$0.355727\pi$
$752$		0			0
$753$	4.30695	+	4.30695i	0.156954	+	0.156954i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	9.16524	−	9.16524i	0.333116	−	0.333116i	−0.520652	−	0.853769i	$-0.674311\pi$
$757$	9.16524	−	9.16524i	0.333116	−	0.333116i	0.853769	+	0.520652i	$0.174311\pi$
$758$		0			0
$759$	−1.37097			−0.0497630
$760$		0			0
$761$			13.8564i			0.502294i	0.967949	+	0.251147i	$0.0808078\pi$
$761$			13.8564i			0.502294i	−0.967949	+	0.251147i	$0.919192\pi$
$762$		0			0
$763$	−2.49342	−	0.884806i	−0.0902677	−	0.0320321i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−26.3524	+	26.3524i	−0.951529	+	0.951529i
$768$		0			0
$769$	37.8591			1.36523			0.682617	−	0.730776i	$-0.260842\pi$
$769$	37.8591			1.36523			0.682617	+	0.730776i	$0.260842\pi$
$770$		0			0
$771$	5.09967			0.183660
$772$		0			0
$773$	16.9706	−	16.9706i	0.610389	−	0.610389i	−0.332659	−	0.943047i	$-0.607946\pi$
$773$	16.9706	−	16.9706i	0.610389	−	0.610389i	0.943047	+	0.332659i	$0.107946\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−5.50601	−	1.95385i	−0.197527	−	0.0700939i
$778$		0			0
$779$		−	11.2990i		−	0.404829i
$780$		0			0
$781$	30.3746			1.08689
$782$		0			0
$783$	0.512711	−	0.512711i	0.0183228	−	0.0183228i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−24.3770	−	24.3770i	−0.868945	−	0.868945i	0.123410	−	0.992356i	$-0.460617\pi$
$787$	−24.3770	−	24.3770i	−0.868945	−	0.868945i	−0.992356	+	0.123410i	$0.960617\pi$
$788$		0			0
$789$	−9.97368			−0.355072
$790$		0			0
$791$	38.3746	−	18.2728i	1.36444	−	0.649708i
$792$		0			0
$793$	3.97025	+	3.97025i	0.140988	+	0.140988i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.1167	−	13.1167i	−0.464618	−	0.464618i	0.435548	−	0.900166i	$-0.356555\pi$
$797$	−13.1167	−	13.1167i	−0.464618	−	0.464618i	−0.900166	+	0.435548i	$0.856555\pi$
$798$		0			0
$799$			4.00000i			0.141510i
$800$		0			0
$801$			18.2728i			0.645639i
$802$		0			0
$803$	23.1572	−	23.1572i	0.817198	−	0.817198i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	12.8395	−	12.8395i	0.451971	−	0.451971i
$808$		0			0
$809$			6.72508i			0.236441i	0.992987	+	0.118221i	$0.0377190\pi$
$809$			6.72508i			0.236441i	−0.992987	+	0.118221i	$0.962281\pi$
$810$		0			0
$811$			19.4712i			0.683726i	0.939750	+	0.341863i	$0.111058\pi$
$811$			19.4712i			0.683726i	−0.939750	+	0.341863i	$0.888942\pi$
$812$		0			0
$813$	−15.9624	−	15.9624i	−0.559825	−	0.559825i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.36077	−	1.36077i	−0.0476073	−	0.0476073i
$818$		0			0
$819$	−10.2118	+	4.86254i	−0.356828	+	0.169911i
$820$		0			0
$821$	−34.0000			−1.18661			−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000			−1.18661			−0.593304	+	0.804978i	$0.702177\pi$
$822$		0			0
$823$	6.20510	+	6.20510i	0.216296	+	0.216296i	0.806936	−	0.590639i	$-0.201124\pi$
$823$	6.20510	+	6.20510i	0.216296	+	0.216296i	−0.590639	+	0.806936i	$0.701124\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	37.2011	−	37.2011i	1.29361	−	1.29361i	0.361072	−	0.932538i	$-0.382411\pi$
$827$	37.2011	−	37.2011i	1.29361	−	1.29361i	0.932538	−	0.361072i	$-0.117589\pi$
$828$		0			0
$829$	−35.7084			−1.24021			−0.620103	−	0.784521i	$-0.712909\pi$
$829$	−35.7084			−1.24021			−0.620103	+	0.784521i	$0.712909\pi$
$830$		0			0
$831$		−	5.61478i		−	0.194775i
$832$		0			0
$833$	1.44513	+	13.9252i	0.0500708	+	0.482480i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.928731	+	0.928731i	−0.0321016	+	0.0321016i
$838$		0			0
$839$	19.2252			0.663727			0.331864	−	0.943327i	$-0.392323\pi$
$839$	19.2252			0.663727			0.331864	+	0.943327i	$0.392323\pi$
$840$		0			0
$841$	28.4743			0.981871
$842$		0			0
$843$	8.99799	−	8.99799i	0.309907	−	0.309907i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.243248	+	0.685483i	−0.00835811	+	0.0235535i
$848$		0			0
$849$		−	26.8248i		−	0.920623i
$850$		0			0
$851$	−0.924421			−0.0316887
$852$		0			0
$853$	24.2360	−	24.2360i	0.829826	−	0.829826i	−0.157667	−	0.987492i	$-0.550397\pi$
$853$	24.2360	−	24.2360i	0.829826	−	0.829826i	0.987492	+	0.157667i	$0.0503971\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.26286	−	9.26286i	−0.316413	−	0.316413i	0.530974	−	0.847388i	$-0.321826\pi$
$857$	−9.26286	−	9.26286i	−0.316413	−	0.316413i	−0.847388	+	0.530974i	$0.821826\pi$
$858$		0			0
$859$	−51.3544			−1.75219			−0.876096	−	0.482138i	$-0.839861\pi$
$859$	−51.3544			−1.75219			−0.876096	+	0.482138i	$0.839861\pi$
$860$		0			0
$861$	12.5498	−	5.97586i	0.427697	−	0.203657i
$862$		0			0
$863$	−29.7713	−	29.7713i	−1.01343	−	1.01343i	−0.999909	−	0.0135172i	$-0.995697\pi$
$863$	−29.7713	−	29.7713i	−1.01343	−	1.01343i	−0.0135172	−	0.999909i	$-0.504303\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−9.19239	−	9.19239i	−0.312190	−	0.312190i
$868$		0			0
$869$		−	38.7251i		−	1.31366i
$870$		0			0
$871$			18.6339i			0.631387i
$872$		0			0
$873$	−7.26546	+	7.26546i	−0.245899	+	0.245899i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.0555	−	23.0555i	0.778530	−	0.778530i	−0.201051	−	0.979581i	$-0.564436\pi$
$877$	23.0555	−	23.0555i	0.778530	−	0.778530i	0.979581	+	0.201051i	$0.0644358\pi$
$878$		0			0
$879$			15.0997i			0.509299i
$880$		0			0
$881$		−	45.1484i		−	1.52109i	−0.649286	−	0.760544i	$-0.724932\pi$
$881$		−	45.1484i		−	1.52109i	0.649286	−	0.760544i	$-0.275068\pi$
$882$		0			0
$883$	20.9834	+	20.9834i	0.706148	+	0.706148i	0.965723	−	0.259575i	$-0.0835824\pi$
$883$	20.9834	+	20.9834i	0.706148	+	0.706148i	−0.259575	+	0.965723i	$0.583582\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.26806	−	5.26806i	−0.176884	−	0.176884i	0.613112	−	0.789996i	$-0.289917\pi$
$887$	−5.26806	−	5.26806i	−0.176884	−	0.176884i	−0.789996	+	0.613112i	$0.789917\pi$
$888$		0			0
$889$	15.2849	+	32.0997i	0.512640	+	1.07659i
$890$		0			0
$891$	−3.27492			−0.109714
$892$		0			0
$893$	3.04152	+	3.04152i	0.101780	+	0.101780i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.26543	+	1.26543i	−0.0422516	+	0.0422516i
$898$		0			0
$899$	0.952341			0.0317624
$900$		0			0
$901$		−	10.5074i		−	0.350052i
$902$		0			0
$903$	0.791722	−	2.23110i	0.0263469	−	0.0742464i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	22.2082	−	22.2082i	0.737410	−	0.737410i	−0.234666	−	0.972076i	$-0.575400\pi$
$907$	22.2082	−	22.2082i	0.737410	−	0.737410i	0.972076	+	0.234666i	$0.0753996\pi$
$908$		0			0
$909$	7.76546			0.257564
$910$		0			0
$911$	−12.7251			−0.421601			−0.210800	−	0.977529i	$-0.567607\pi$
$911$	−12.7251			−0.421601			−0.210800	+	0.977529i	$0.567607\pi$
$912$		0			0
$913$	38.3247	−	38.3247i	1.26836	−	1.26836i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−49.7370	−	17.6495i	−1.64246	−	0.582839i
$918$		0			0
$919$			36.2990i			1.19739i	0.800976	+	0.598697i	$0.204315\pi$
$919$			36.2990i			1.19739i	−0.800976	+	0.598697i	$0.795685\pi$
$920$		0			0
$921$	15.7251			0.518159
$922$		0			0
$923$	28.0364	−	28.0364i	0.922830	−	0.922830i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−4.63143	−	4.63143i	−0.152116	−	0.152116i
$928$		0			0
$929$	−39.7796			−1.30513			−0.652564	−	0.757734i	$-0.726306\pi$
$929$	−39.7796			−1.30513			−0.652564	+	0.757734i	$0.726306\pi$
$930$		0			0
$931$	11.6873	+	9.48960i	0.383036	+	0.311009i
$932$		0			0
$933$	23.3922	+	23.3922i	0.765827	+	0.765827i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−24.6248	−	24.6248i	−0.804458	−	0.804458i	0.179331	−	0.983789i	$-0.442607\pi$
$937$	−24.6248	−	24.6248i	−0.804458	−	0.804458i	−0.983789	+	0.179331i	$0.942607\pi$
$938$		0			0
$939$			27.7251i			0.904774i
$940$		0			0
$941$			7.65037i			0.249395i	0.992195	+	0.124697i	$0.0397960\pi$
$941$			7.65037i			0.249395i	−0.992195	+	0.124697i	$0.960204\pi$
$942$		0			0
$943$	1.55517	−	1.55517i	0.0506432	−	0.0506432i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.71658	+	9.71658i	−0.315746	+	0.315746i	−0.847131	−	0.531384i	$-0.821672\pi$
$947$	−9.71658	+	9.71658i	−0.315746	+	0.315746i	0.531384	+	0.847131i	$0.321672\pi$
$948$		0			0
$949$		−	42.7492i		−	1.38770i
$950$		0			0
$951$			10.8109i			0.350568i
$952$		0			0
$953$	9.42057	+	9.42057i	0.305162	+	0.305162i	0.843029	−	0.537867i	$-0.180770\pi$
$953$	9.42057	+	9.42057i	0.305162	+	0.305162i	−0.537867	+	0.843029i	$0.680770\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	1.67909	+	1.67909i	0.0542771	+	0.0542771i
$958$		0			0
$959$	−24.7249	−	51.9244i	−0.798408	−	1.67673i
$960$		0			0
$961$	29.2749			0.944352
$962$		0			0
$963$	−4.30695	−	4.30695i	−0.138790	−	0.138790i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	24.5763	−	24.5763i	0.790320	−	0.790320i	−0.191226	−	0.981546i	$-0.561246\pi$
$967$	24.5763	−	24.5763i	0.790320	−	0.790320i	0.981546	+	0.191226i	$0.0612464\pi$
$968$		0			0
$969$	4.30136			0.138179
$970$		0			0
$971$		−	25.9232i		−	0.831916i	−0.909384	−	0.415958i	$-0.863446\pi$
$971$		−	25.9232i		−	0.831916i	0.909384	−	0.415958i	$-0.136554\pi$
$972$		0			0
$973$	−3.06506	+	8.63744i	−0.0982612	+	0.276904i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12.0328	−	12.0328i	0.384964	−	0.384964i	−0.487923	−	0.872887i	$-0.662245\pi$
$977$	12.0328	−	12.0328i	0.384964	−	0.384964i	0.872887	+	0.487923i	$0.162245\pi$
$978$		0			0
$979$	−59.8421			−1.91256
$980$		0			0
$981$	−1.00000			−0.0319275
$982$		0			0
$983$	30.6170	−	30.6170i	0.976531	−	0.976531i	−0.0231995	−	0.999731i	$-0.507385\pi$
$983$	30.6170	−	30.6170i	0.976531	−	0.976531i	0.999731	+	0.0231995i	$0.00738530\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.76961	+	4.98683i	−0.0563274	+	0.158733i
$988$		0			0
$989$		−	0.374586i		−	0.0119111i
$990$		0			0
$991$	−21.5498			−0.684553			−0.342277	−	0.939599i	$-0.611198\pi$
$991$	−21.5498			−0.684553			−0.342277	+	0.939599i	$0.611198\pi$
$992$		0			0
$993$	7.33594	−	7.33594i	0.232799	−	0.232799i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	32.0312	+	32.0312i	1.01444	+	1.01444i	0.999894	+	0.0145452i	$0.00463003\pi$
$997$	32.0312	+	32.0312i	1.01444	+	1.01444i	0.0145452	+	0.999894i	$0.495370\pi$
$998$		0			0
$999$	−2.20822			−0.0698650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.x.c.1693.6	yes	16
5.2	odd	4	inner	2100.2.x.c.1357.4	yes	16
5.3	odd	4	inner	2100.2.x.c.1357.5	yes	16
5.4	even	2	inner	2100.2.x.c.1693.3	yes	16
7.6	odd	2	inner	2100.2.x.c.1693.4	yes	16
35.13	even	4	inner	2100.2.x.c.1357.3	✓	16
35.27	even	4	inner	2100.2.x.c.1357.6	yes	16
35.34	odd	2	inner	2100.2.x.c.1693.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.x.c.1357.3	✓	16	35.13	even	4	inner
2100.2.x.c.1357.4	yes	16	5.2	odd	4	inner
2100.2.x.c.1357.5	yes	16	5.3	odd	4	inner
2100.2.x.c.1357.6	yes	16	35.27	even	4	inner
2100.2.x.c.1693.3	yes	16	5.4	even	2	inner
2100.2.x.c.1693.4	yes	16	7.6	odd	2	inner
2100.2.x.c.1693.5	yes	16	35.34	odd	2	inner
2100.2.x.c.1693.6	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{3} +(0.884806 - 2.49342i) q^{7} -1.00000i q^{9} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{3} +(0.884806 - 2.49342i) q^{7} -1.00000i q^{9} +3.27492 q^{11} +(3.02282 - 3.02282i) q^{13} +(-1.41421 - 1.41421i) q^{17} -2.15068 q^{19} +(-1.13746 - 2.38876i) q^{21} +(-0.296014 - 0.296014i) q^{23} +(-0.707107 - 0.707107i) q^{27} +0.725083i q^{29} -1.31342i q^{31} +(2.31572 - 2.31572i) q^{33} +(1.56145 - 1.56145i) q^{37} -4.27492i q^{39} +5.25370i q^{41} +(0.632717 + 0.632717i) q^{43} +(-1.41421 - 1.41421i) q^{47} +(-5.43424 - 4.41238i) q^{49} -2.00000 q^{51} +(3.71492 + 3.71492i) q^{53} +(-1.52076 + 1.52076i) q^{57} -8.71780 q^{59} +1.31342i q^{61} +(-2.49342 - 0.884806i) q^{63} +(-3.08221 + 3.08221i) q^{67} -0.418627 q^{69} +9.27492 q^{71} +(7.07107 - 7.07107i) q^{73} +(2.89767 - 8.16573i) q^{77} -11.8248i q^{79} -1.00000 q^{81} +(11.7025 - 11.7025i) q^{83} +(0.512711 + 0.512711i) q^{87} -18.2728 q^{89} +(-4.86254 - 10.2118i) q^{91} +(-0.928731 - 0.928731i) q^{93} +(-7.26546 - 7.26546i) q^{97} -3.27492i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 8 q^{11} + 12 q^{21} - 32 q^{51} + 88 q^{71} - 16 q^{81} - 108 q^{91}+O(q^{100}) \) 16 * q - 8 * q^11 + 12 * q^21 - 32 * q^51 + 88 * q^71 - 16 * q^81 - 108 * q^91

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