LMFDB - Embedded newform 6300.2.d.f.3401.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(3401,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6300.3401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.d (of order $2$, degree $1$, 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:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.101415451701035401216.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 18x^{12} + 145x^{8} - 72x^{4} + 16 $ x^16 - 18x^12 + 145x^8 - 72*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3401.8
Root			$0.137538 + 0.752908i$ of defining polynomial
Character	$\chi$	$=$	6300.3401
Dual form			6300.2.d.f.3401.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.16372 + 2.37608i) q^{7} +O(q^{10})$	$q+(-1.16372 + 2.37608i) q^{7} +3.74166i q^{11} -0.841723i q^{13} -3.36028 q^{17} +4.55066i q^{19} +7.64575i q^{23} -1.41421i q^{29} +0.979531i q^{31} -2.32744 q^{37} +10.3460 q^{41} +10.8127 q^{43} -7.91094 q^{47} +(-4.29150 - 5.53019i) q^{49} -4.35425i q^{53} +1.38527 q^{59} -13.1402 q^{67} +3.74166i q^{71} -8.66259i q^{73} +(-8.89047 - 4.35425i) q^{77} -14.5830 q^{79} -3.14944 q^{83} -3.91044 q^{89} +(2.00000 + 0.979531i) q^{91} -14.8000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{49} - 64 q^{79} + 32 q^{91}+O(q^{100}) $ 16 * q + 16 * q^49 - 64 * q^79 + 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.16372	+	2.37608i	−0.439846	+	0.898073i
$7$	−1.16372	+	2.37608i	−0.439846	+	0.898073i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.74166i			1.12815i	0.825723	+	0.564076i	$0.190768\pi$
$11$			3.74166i			1.12815i	−0.825723	+	0.564076i	$0.809232\pi$
$12$		0			0
$13$		−	0.841723i		−	0.233452i	−0.993164	−	0.116726i	$-0.962760\pi$
$13$		−	0.841723i		−	0.233452i	0.993164	−	0.116726i	$-0.0372399\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.36028			−0.814988			−0.407494	−	0.913208i	$-0.633597\pi$
$17$	−3.36028			−0.814988			−0.407494	+	0.913208i	$0.633597\pi$
$18$		0			0
$19$			4.55066i			1.04399i	0.852948	+	0.521996i	$0.174813\pi$
$19$			4.55066i			1.04399i	−0.852948	+	0.521996i	$0.825187\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.64575i			1.59425i	0.603815	+	0.797125i	$0.293647\pi$
$23$			7.64575i			1.59425i	−0.603815	+	0.797125i	$0.706353\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	1.41421i		−	0.262613i	−0.991342	−	0.131306i	$-0.958083\pi$
$29$		−	1.41421i		−	0.262613i	0.991342	−	0.131306i	$-0.0419172\pi$
$30$		0			0
$31$			0.979531i			0.175929i	0.996124	+	0.0879645i	$0.0280362\pi$
$31$			0.979531i			0.175929i	−0.996124	+	0.0879645i	$0.971964\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.32744			−0.382629			−0.191315	−	0.981529i	$-0.561275\pi$
$37$	−2.32744			−0.382629			−0.191315	+	0.981529i	$0.561275\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.3460			1.61578			0.807890	−	0.589333i	$-0.200610\pi$
$41$	10.3460			1.61578			0.807890	+	0.589333i	$0.200610\pi$
$42$		0			0
$43$	10.8127			1.64893			0.824463	−	0.565916i	$-0.191478\pi$
$43$	10.8127			1.64893			0.824463	+	0.565916i	$0.191478\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.91094			−1.15393			−0.576965	−	0.816769i	$-0.695763\pi$
$47$	−7.91094			−1.15393			−0.576965	+	0.816769i	$0.695763\pi$
$48$		0			0
$49$	−4.29150	−	5.53019i	−0.613072	−	0.790027i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4.35425i		−	0.598102i	−0.954237	−	0.299051i	$-0.903330\pi$
$53$		−	4.35425i		−	0.598102i	0.954237	−	0.299051i	$-0.0966701\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.38527			0.180346			0.0901732	−	0.995926i	$-0.471258\pi$
$59$	1.38527			0.180346			0.0901732	+	0.995926i	$0.471258\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−13.1402			−1.60533			−0.802664	−	0.596432i	$-0.796585\pi$
$67$	−13.1402			−1.60533			−0.802664	+	0.596432i	$0.796585\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.74166i			0.444053i	0.975041	+	0.222027i	$0.0712672\pi$
$71$			3.74166i			0.444053i	−0.975041	+	0.222027i	$0.928733\pi$
$72$		0			0
$73$		−	8.66259i		−	1.01388i	−0.861981	−	0.506940i	$-0.830777\pi$
$73$		−	8.66259i		−	1.01388i	0.861981	−	0.506940i	$-0.169223\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.89047	−	4.35425i	−1.01316	−	0.496213i
$78$		0			0
$79$	−14.5830			−1.64072			−0.820358	−	0.571850i	$-0.806226\pi$
$79$	−14.5830			−1.64072			−0.820358	+	0.571850i	$0.806226\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.14944			−0.345696			−0.172848	−	0.984949i	$-0.555297\pi$
$83$	−3.14944			−0.345696			−0.172848	+	0.984949i	$0.555297\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.91044			−0.414505			−0.207253	−	0.978287i	$-0.566452\pi$
$89$	−3.91044			−0.414505			−0.207253	+	0.978287i	$0.566452\pi$
$90$		0			0
$91$	2.00000	+	0.979531i	0.209657	+	0.102683i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	14.8000i		−	1.50271i	−0.659896	−	0.751357i	$-0.729400\pi$
$97$		−	14.8000i		−	1.50271i	0.659896	−	0.751357i	$-0.270600\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.3460			−1.02947			−0.514735	−	0.857350i	$-0.672110\pi$
$101$	−10.3460			−1.02947			−0.514735	+	0.857350i	$0.672110\pi$
$102$		0			0
$103$		−	3.36689i		−	0.331750i	−0.986147	−	0.165875i	$-0.946955\pi$
$103$		−	3.36689i		−	0.331750i	0.986147	−	0.165875i	$-0.0530448\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.06275i		−	0.102740i	−0.998680	−	0.0513698i	$-0.983641\pi$
$107$		−	1.06275i		−	0.102740i	0.998680	−	0.0513698i	$-0.0163587\pi$
$108$		0			0
$109$	13.8745			1.32894			0.664468	−	0.747316i	$-0.268658\pi$
$109$	13.8745			1.32894			0.664468	+	0.747316i	$0.268658\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.9373i			1.02889i	0.857523	+	0.514445i	$0.172002\pi$
$113$			10.9373i			1.02889i	−0.857523	+	0.514445i	$0.827998\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.91044	−	7.98430i	0.358469	−	0.731919i
$120$		0			0
$121$	−3.00000			−0.272727
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−4.65489			−0.413054			−0.206527	−	0.978441i	$-0.566216\pi$
$127$	−4.65489			−0.413054			−0.206527	+	0.978441i	$0.566216\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.43560			0.562281			0.281141	−	0.959667i	$-0.409287\pi$
$131$	6.43560			0.562281			0.281141	+	0.959667i	$0.409287\pi$
$132$		0			0
$133$	−10.8127	−	5.29570i	−0.937582	−	0.459196i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			14.2288i			1.21564i	0.794073	+	0.607822i	$0.207957\pi$
$137$			14.2288i			1.21564i	−0.794073	+	0.607822i	$0.792043\pi$
$138$		0			0
$139$			21.1412i			1.79318i	0.442866	+	0.896588i	$0.353962\pi$
$139$			21.1412i			1.79318i	−0.442866	+	0.896588i	$0.646038\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.14944			0.263369
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			2.41618i			0.197941i	0.995090	+	0.0989706i	$0.0315550\pi$
$149$			2.41618i			0.197941i	−0.995090	+	0.0989706i	$0.968445\pi$
$150$		0			0
$151$	5.29150			0.430616			0.215308	−	0.976546i	$-0.430924\pi$
$151$	5.29150			0.430616			0.215308	+	0.976546i	$0.430924\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	18.4651i		−	1.47367i	−0.676070	−	0.736837i	$-0.736318\pi$
$157$		−	18.4651i		−	1.47367i	0.676070	−	0.736837i	$-0.263682\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−18.1669	−	8.89753i	−1.43175	−	0.701223i
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.4821			−0.888509			−0.444255	−	0.895901i	$-0.646531\pi$
$167$	−11.4821			−0.888509			−0.444255	+	0.895901i	$0.646531\pi$
$168$		0			0
$169$	12.2915			0.945500
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−20.3725			−1.54890			−0.774448	−	0.632638i	$-0.781972\pi$
$173$	−20.3725			−1.54890			−0.774448	+	0.632638i	$0.781972\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			14.0534i			1.05040i	0.850979	+	0.525200i	$0.176010\pi$
$179$			14.0534i			1.05040i	−0.850979	+	0.525200i	$0.823990\pi$
$180$		0			0
$181$		−	18.5496i		−	1.37878i	−0.724389	−	0.689392i	$-0.757878\pi$
$181$		−	18.5496i		−	1.37878i	0.724389	−	0.689392i	$-0.242122\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	12.5730i		−	0.919431i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	5.56812i		−	0.402895i	−0.979499	−	0.201447i	$-0.935435\pi$
$191$		−	5.56812i		−	0.402895i	0.979499	−	0.201447i	$-0.0645645\pi$
$192$		0			0
$193$	−6.98233			−0.502599			−0.251300	−	0.967909i	$-0.580858\pi$
$193$	−6.98233			−0.502599			−0.251300	+	0.967909i	$0.580858\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			20.8118i			1.48278i	0.671076	+	0.741388i	$0.265832\pi$
$197$			20.8118i			1.48278i	−0.671076	+	0.741388i	$0.734168\pi$
$198$		0			0
$199$		−	26.6714i		−	1.89069i	−0.326076	−	0.945343i	$-0.605727\pi$
$199$		−	26.6714i		−	1.89069i	0.326076	−	0.945343i	$-0.394273\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.36028	+	1.64575i	0.235846	+	0.115509i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−17.0270			−1.17778
$210$		0			0
$211$	5.29150			0.364282			0.182141	−	0.983272i	$-0.441697\pi$
$211$	5.29150			0.364282			0.182141	+	0.983272i	$0.441697\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.32744	−	1.13990i	−0.157997	−	0.0773816i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			2.82843i			0.190261i
$222$		0			0
$223$			18.7105i			1.25294i	0.779444	+	0.626472i	$0.215502\pi$
$223$			18.7105i			1.25294i	−0.779444	+	0.626472i	$0.784498\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.9304			−1.38920			−0.694599	−	0.719397i	$-0.744418\pi$
$227$	−20.9304			−1.38920			−0.694599	+	0.719397i	$0.744418\pi$
$228$		0			0
$229$		−	12.6724i		−	0.837419i	−0.908120	−	0.418709i	$-0.862483\pi$
$229$		−	12.6724i		−	0.837419i	0.908120	−	0.418709i	$-0.137517\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	16.3542i		−	1.07140i	−0.844408	−	0.535701i	$-0.820047\pi$
$233$		−	16.3542i		−	1.07140i	0.844408	−	0.535701i	$-0.179953\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	7.57205i		−	0.489795i	−0.969549	−	0.244898i	$-0.921246\pi$
$239$		−	7.57205i		−	0.489795i	0.969549	−	0.244898i	$-0.0787544\pi$
$240$		0			0
$241$			14.6315i			0.942498i	0.882000	+	0.471249i	$0.156197\pi$
$241$			14.6315i			0.942498i	−0.882000	+	0.471249i	$0.843803\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.83039			0.243722
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.20614			0.581086			0.290543	−	0.956862i	$-0.406164\pi$
$251$	9.20614			0.581086			0.290543	+	0.956862i	$0.406164\pi$
$252$		0			0
$253$	−28.6078			−1.79856
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−23.5220			−1.46726			−0.733630	−	0.679549i	$-0.762176\pi$
$257$	−23.5220			−1.46726			−0.733630	+	0.679549i	$0.762176\pi$
$258$		0			0
$259$	2.70850	−	5.53019i	0.168298	−	0.343629i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.35425i			0.268494i	0.990948	+	0.134247i	$0.0428616\pi$
$263$			4.35425i			0.268494i	−0.990948	+	0.134247i	$0.957138\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.3964			0.938734			0.469367	−	0.883003i	$-0.344482\pi$
$269$	15.3964			0.938734			0.469367	+	0.883003i	$0.344482\pi$
$270$		0			0
$271$		−	12.0399i		−	0.731373i	−0.930738	−	0.365686i	$-0.880834\pi$
$271$		−	12.0399i		−	0.731373i	0.930738	−	0.365686i	$-0.119166\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	25.4558			1.52949			0.764747	−	0.644331i	$-0.222864\pi$
$277$	25.4558			1.52949			0.764747	+	0.644331i	$0.222864\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	24.8661i		−	1.48339i	−0.670738	−	0.741694i	$-0.734023\pi$
$281$		−	24.8661i		−	1.48339i	0.670738	−	0.741694i	$-0.265977\pi$
$282$		0			0
$283$		−	15.3436i		−	0.912080i	−0.889959	−	0.456040i	$-0.849267\pi$
$283$		−	15.3436i		−	0.912080i	0.889959	−	0.456040i	$-0.150733\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.0399	+	24.5830i	−0.710694	+	1.45109i
$288$		0			0
$289$	−5.70850			−0.335794
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−29.8209			−1.74215			−0.871077	−	0.491147i	$-0.836578\pi$
$293$	−29.8209			−1.74215			−0.871077	+	0.491147i	$0.836578\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.43560			0.372181
$300$		0			0
$301$	−12.5830	+	25.6919i	−0.725272	+	1.48086i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	26.5313i		−	1.51422i	−0.653286	−	0.757111i	$-0.726610\pi$
$307$		−	26.5313i		−	1.51422i	0.653286	−	0.757111i	$-0.273390\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−17.0270			−0.965513			−0.482756	−	0.875755i	$-0.660364\pi$
$311$	−17.0270			−0.965513			−0.482756	+	0.875755i	$0.660364\pi$
$312$		0			0
$313$			5.59388i			0.316185i	0.987424	+	0.158092i	$0.0505344\pi$
$313$			5.59388i			0.316185i	−0.987424	+	0.158092i	$0.949466\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			32.8118i			1.84289i	0.388506	+	0.921446i	$0.372991\pi$
$317$			32.8118i			1.84289i	−0.388506	+	0.921446i	$0.627009\pi$
$318$		0			0
$319$	5.29150			0.296267
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	15.2915i		−	0.850842i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.20614	−	18.7970i	0.507551	−	1.03631i
$330$		0			0
$331$	−19.8745			−1.09240			−0.546201	−	0.837654i	$-0.683926\pi$
$331$	−19.8745			−1.09240			−0.546201	+	0.837654i	$0.683926\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.50295			0.0818709			0.0409354	−	0.999162i	$-0.486966\pi$
$337$	1.50295			0.0818709			0.0409354	+	0.999162i	$0.486966\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.66507			−0.198475
$342$		0			0
$343$	18.1343	−	3.76135i	0.979159	−	0.203094i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	13.0627i		−	0.701245i	−0.936517	−	0.350622i	$-0.885970\pi$
$347$		−	13.0627i		−	0.701245i	0.936517	−	0.350622i	$-0.114030\pi$
$348$		0			0
$349$			24.0798i			1.28896i	0.764620	+	0.644482i	$0.222927\pi$
$349$			24.0798i			1.28896i	−0.764620	+	0.644482i	$0.777073\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−20.3725			−1.08432			−0.542161	−	0.840275i	$-0.682394\pi$
$353$	−20.3725			−1.08432			−0.542161	+	0.840275i	$0.682394\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			21.7142i			1.14603i	0.819545	+	0.573015i	$0.194227\pi$
$359$			21.7142i			1.14603i	−0.819545	+	0.573015i	$0.805773\pi$
$360$		0			0
$361$	−1.70850			−0.0899209
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$			15.3436i			0.800927i	0.916313	+	0.400464i	$0.131151\pi$
$367$			15.3436i			0.800927i	−0.916313	+	0.400464i	$0.868849\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.3460	+	5.06713i	0.537140	+	0.263073i
$372$		0			0
$373$	−13.9647			−0.723063			−0.361531	−	0.932360i	$-0.617746\pi$
$373$	−13.9647			−0.723063			−0.361531	+	0.932360i	$0.617746\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.19038			−0.0613075
$378$		0			0
$379$	13.2915			0.682739			0.341369	−	0.939929i	$-0.389109\pi$
$379$	13.2915			0.682739			0.341369	+	0.939929i	$0.389109\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.4821			0.586706			0.293353	−	0.956004i	$-0.405229\pi$
$383$	11.4821			0.586706			0.293353	+	0.956004i	$0.405229\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			24.8661i			1.26076i	0.776286	+	0.630381i	$0.217101\pi$
$389$			24.8661i			1.26076i	−0.776286	+	0.630381i	$0.782899\pi$
$390$		0			0
$391$		−	25.6919i		−	1.29929i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	26.2860i		−	1.31925i	−0.751593	−	0.659627i	$-0.770714\pi$
$397$		−	26.2860i		−	1.31925i	0.751593	−	0.659627i	$-0.229286\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			32.3494i			1.61545i	0.589557	+	0.807727i	$0.299302\pi$
$401$			32.3494i			1.61545i	−0.589557	+	0.807727i	$0.700698\pi$
$402$		0			0
$403$	0.824494			0.0410710
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.70850i		−	0.431664i
$408$		0			0
$409$			3.57113i			0.176581i	0.996095	+	0.0882904i	$0.0281404\pi$
$409$			3.57113i			0.176581i	−0.996095	+	0.0882904i	$0.971860\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.61206	+	3.29150i	−0.0793245	+	0.161964i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28.5129			−1.39295			−0.696474	−	0.717582i	$-0.745249\pi$
$419$	−28.5129			−1.39295			−0.696474	+	0.717582i	$0.745249\pi$
$420$		0			0
$421$	−7.87451			−0.383780			−0.191890	−	0.981416i	$-0.561462\pi$
$421$	−7.87451			−0.383780			−0.191890	+	0.981416i	$0.561462\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	14.0534i		−	0.676928i	−0.940979	−	0.338464i	$-0.890093\pi$
$431$		−	14.0534i		−	0.676928i	0.940979	−	0.338464i	$-0.109907\pi$
$432$		0			0
$433$		−	4.50679i		−	0.216583i	−0.994119	−	0.108291i	$-0.965462\pi$
$433$		−	4.50679i		−	0.216583i	0.994119	−	0.108291i	$-0.0345379\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−34.7932			−1.66438
$438$		0			0
$439$			35.7727i			1.70734i	0.520815	+	0.853670i	$0.325628\pi$
$439$			35.7727i			1.70734i	−0.520815	+	0.853670i	$0.674372\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			4.35425i			0.206877i	0.994636	+	0.103438i	$0.0329844\pi$
$443$			4.35425i			0.206877i	−0.994636	+	0.103438i	$0.967016\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	13.7299i		−	0.647954i	−0.946065	−	0.323977i	$-0.894980\pi$
$449$		−	13.7299i		−	0.647954i	0.946065	−	0.323977i	$-0.105020\pi$
$450$		0			0
$451$			38.7113i			1.82285i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−32.4382			−1.51739			−0.758697	−	0.651444i	$-0.774164\pi$
$457$	−32.4382			−1.51739			−0.758697	+	0.651444i	$0.774164\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.5018			0.675418			0.337709	−	0.941251i	$-0.390348\pi$
$461$	14.5018			0.675418			0.337709	+	0.941251i	$0.390348\pi$
$462$		0			0
$463$	−16.9706			−0.788689			−0.394344	−	0.918963i	$-0.629028\pi$
$463$	−16.9706			−0.788689			−0.394344	+	0.918963i	$0.629028\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14.2098			0.657552			0.328776	−	0.944408i	$-0.393364\pi$
$467$	14.2098			0.657552			0.328776	+	0.944408i	$0.393364\pi$
$468$		0			0
$469$	15.2915	−	31.2221i	0.706096	−	1.44170i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			40.4575i			1.86024i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.0270			0.777984			0.388992	−	0.921241i	$-0.372823\pi$
$479$	17.0270			0.777984			0.388992	+	0.921241i	$0.372823\pi$
$480$		0			0
$481$			1.95906i			0.0893256i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−15.4676			−0.700904			−0.350452	−	0.936581i	$-0.613972\pi$
$487$	−15.4676			−0.700904			−0.350452	+	0.936581i	$0.613972\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			26.1916i			1.18201i	0.806668	+	0.591005i	$0.201269\pi$
$491$			26.1916i			1.18201i	−0.806668	+	0.591005i	$0.798731\pi$
$492$		0			0
$493$			4.75216i			0.214026i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.89047	−	4.35425i	−0.398792	−	0.195315i
$498$		0			0
$499$	−29.2915			−1.31127			−0.655634	−	0.755079i	$-0.727599\pi$
$499$	−29.2915			−1.31127			−0.655634	+	0.755079i	$0.727599\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.33981			0.193503			0.0967514	−	0.995309i	$-0.469155\pi$
$503$	4.33981			0.193503			0.0967514	+	0.995309i	$0.469155\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.57551			−0.335778			−0.167889	−	0.985806i	$-0.553695\pi$
$509$	−7.57551			−0.335778			−0.167889	+	0.985806i	$0.553695\pi$
$510$		0			0
$511$	20.5830	+	10.0808i	0.910539	+	0.445951i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	29.6000i		−	1.30181i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	31.0381			1.35980			0.679902	−	0.733303i	$-0.262022\pi$
$521$	31.0381			1.35980			0.679902	+	0.733303i	$0.262022\pi$
$522$		0			0
$523$		−	3.36689i		−	0.147224i	−0.997287	−	0.0736119i	$-0.976547\pi$
$523$		−	3.36689i		−	0.147224i	0.997287	−	0.0736119i	$-0.0234526\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3.29150i		−	0.143380i
$528$		0			0
$529$	−35.4575			−1.54163
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	8.70850i		−	0.377207i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	20.6921	−	16.0573i	0.891271	−	0.691638i
$540$		0			0
$541$	11.8745			0.510525			0.255262	−	0.966872i	$-0.417838\pi$
$541$	11.8745			0.510525			0.255262	+	0.966872i	$0.417838\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−25.4558			−1.08841			−0.544207	−	0.838951i	$-0.683169\pi$
$547$	−25.4558			−1.08841			−0.544207	+	0.838951i	$0.683169\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.43560			0.274166
$552$		0			0
$553$	16.9706	−	34.6504i	0.721662	−	1.47348i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			17.5203i			0.742357i	0.928561	+	0.371179i	$0.121046\pi$
$557$			17.5203i			0.742357i	−0.928561	+	0.371179i	$0.878954\pi$
$558$		0			0
$559$		−	9.10132i		−	0.384945i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	44.6632			1.88233			0.941165	−	0.337947i	$-0.109733\pi$
$563$	44.6632			1.88233			0.941165	+	0.337947i	$0.109733\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			37.1818i			1.55874i	0.626563	+	0.779371i	$0.284461\pi$
$569$			37.1818i			1.55874i	−0.626563	+	0.779371i	$0.715539\pi$
$570$		0			0
$571$	1.41699			0.0592994			0.0296497	−	0.999560i	$-0.490561\pi$
$571$	1.41699			0.0592994			0.0296497	+	0.999560i	$0.490561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			20.1485i			0.838794i	0.907803	+	0.419397i	$0.137759\pi$
$577$			20.1485i			0.838794i	−0.907803	+	0.419397i	$0.862241\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.66507	−	7.48331i	0.152053	−	0.310460i
$582$		0			0
$583$	16.2921			0.674750
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.6387			0.439106			0.219553	−	0.975601i	$-0.429540\pi$
$587$	10.6387			0.439106			0.219553	+	0.975601i	$0.429540\pi$
$588$		0			0
$589$	−4.45751			−0.183669
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15.1894			−0.623752			−0.311876	−	0.950123i	$-0.600957\pi$
$593$	−15.1894			−0.623752			−0.311876	+	0.950123i	$0.600957\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	26.1916i		−	1.07016i	−0.844801	−	0.535080i	$-0.820281\pi$
$599$		−	26.1916i		−	1.07016i	0.844801	−	0.535080i	$-0.179719\pi$
$600$		0			0
$601$		−	38.3643i		−	1.56491i	−0.622705	−	0.782457i	$-0.713966\pi$
$601$		−	38.3643i		−	1.56491i	0.622705	−	0.782457i	$-0.286034\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	13.6601i		−	0.554447i	−0.960805	−	0.277223i	$-0.910586\pi$
$607$		−	13.6601i		−	0.554447i	0.960805	−	0.277223i	$-0.0894142\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			6.65882i			0.269387i
$612$		0			0
$613$	12.3157			0.497425			0.248713	−	0.968577i	$-0.419992\pi$
$613$	12.3157			0.497425			0.248713	+	0.968577i	$0.419992\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	29.5203i		−	1.18844i	−0.804302	−	0.594220i	$-0.797461\pi$
$617$		−	29.5203i		−	1.18844i	0.804302	−	0.594220i	$-0.202539\pi$
$618$		0			0
$619$		−	23.1003i		−	0.928479i	−0.885710	−	0.464240i	$-0.846328\pi$
$619$		−	23.1003i		−	0.928479i	0.885710	−	0.464240i	$-0.153672\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.55066	−	9.29150i	0.182318	−	0.372256i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.82087			0.311839
$630$		0			0
$631$	21.1660			0.842606			0.421303	−	0.906920i	$-0.361573\pi$
$631$	21.1660			0.842606			0.421303	+	0.906920i	$0.361573\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−4.65489	+	3.61226i	−0.184433	+	0.143123i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			27.6946i			1.09387i	0.837175	+	0.546935i	$0.184205\pi$
$641$			27.6946i			1.09387i	−0.837175	+	0.546935i	$0.815795\pi$
$642$		0			0
$643$			5.83925i			0.230277i	0.993349	+	0.115139i	$0.0367312\pi$
$643$			5.83925i			0.230277i	−0.993349	+	0.115139i	$0.963269\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.29888			−0.247634			−0.123817	−	0.992305i	$-0.539514\pi$
$647$	−6.29888			−0.247634			−0.123817	+	0.992305i	$0.539514\pi$
$648$		0			0
$649$			5.18319i			0.203458i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			36.1033i			1.41283i	0.707798	+	0.706415i	$0.249689\pi$
$653$			36.1033i			1.41283i	−0.707798	+	0.706415i	$0.750311\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	38.5073i		−	1.50003i	−0.661421	−	0.750015i	$-0.730046\pi$
$659$		−	38.5073i		−	1.50003i	0.661421	−	0.750015i	$-0.269954\pi$
$660$		0			0
$661$		−	11.0604i		−	0.430199i	−0.976592	−	0.215099i	$-0.930992\pi$
$661$		−	11.0604i		−	0.430199i	0.976592	−	0.215099i	$-0.0690076\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10.8127			0.418670
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	45.5783			1.75692			0.878458	−	0.477820i	$-0.158573\pi$
$673$	45.5783			1.75692			0.878458	+	0.477820i	$0.158573\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.27841			−0.279732			−0.139866	−	0.990170i	$-0.544667\pi$
$677$	−7.27841			−0.279732			−0.139866	+	0.990170i	$0.544667\pi$
$678$		0			0
$679$	35.1660	+	17.2231i	1.34955	+	0.660962i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	17.5203i		−	0.670394i	−0.942148	−	0.335197i	$-0.891197\pi$
$683$		−	17.5203i		−	0.670394i	0.942148	−	0.335197i	$-0.108803\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.66507			−0.139628
$690$		0			0
$691$		−	10.0808i		−	0.383494i	−0.981444	−	0.191747i	$-0.938585\pi$
$691$		−	10.0808i		−	0.383494i	0.981444	−	0.191747i	$-0.0614152\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−34.7656			−1.31684
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	28.6965i		−	1.08385i	−0.840426	−	0.541926i	$-0.817695\pi$
$701$		−	28.6965i		−	1.08385i	0.840426	−	0.541926i	$-0.182305\pi$
$702$		0			0
$703$		−	10.5914i		−	0.399462i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.0399	−	24.5830i	0.452807	−	0.924539i
$708$		0			0
$709$	−6.83399			−0.256656			−0.128328	−	0.991732i	$-0.540961\pi$
$709$	−6.83399			−0.256656			−0.128328	+	0.991732i	$0.540961\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.48925			−0.280475
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−48.3105			−1.80168			−0.900839	−	0.434154i	$-0.857047\pi$
$719$	−48.3105			−1.80168			−0.900839	+	0.434154i	$0.857047\pi$
$720$		0			0
$721$	8.00000	+	3.91813i	0.297936	+	0.145919i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			34.6504i			1.28511i	0.766239	+	0.642556i	$0.222126\pi$
$727$			34.6504i			1.28511i	−0.766239	+	0.642556i	$0.777874\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36.3338			−1.34385
$732$		0			0
$733$		−	32.1252i		−	1.18657i	−0.804992	−	0.593286i	$-0.797830\pi$
$733$		−	32.1252i		−	1.18657i	0.804992	−	0.593286i	$-0.202170\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	49.1660i		−	1.81105i
$738$		0			0
$739$	−33.1660			−1.22003			−0.610016	−	0.792389i	$-0.708837\pi$
$739$	−33.1660			−1.22003			−0.610016	+	0.792389i	$0.708837\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			13.0627i			0.479226i	0.970869	+	0.239613i	$0.0770205\pi$
$743$			13.0627i			0.479226i	−0.970869	+	0.239613i	$0.922979\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.52517	+	1.23674i	0.0922677	+	0.0451895i
$750$		0			0
$751$	−18.7085			−0.682683			−0.341341	−	0.939939i	$-0.610881\pi$
$751$	−18.7085			−0.682683			−0.341341	+	0.939939i	$0.610881\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	6.15784			0.223810			0.111905	−	0.993719i	$-0.464305\pi$
$757$	6.15784			0.223810			0.111905	+	0.993719i	$0.464305\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.45150			−0.342617			−0.171308	−	0.985217i	$-0.554799\pi$
$761$	−9.45150			−0.342617			−0.171308	+	0.985217i	$0.554799\pi$
$762$		0			0
$763$	−16.1461	+	32.9669i	−0.584527	+	1.19348i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	1.16601i		−	0.0421022i
$768$		0			0
$769$			38.3643i			1.38345i	0.722159	+	0.691727i	$0.243150\pi$
$769$			38.3643i			1.38345i	−0.722159	+	0.691727i	$0.756850\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−44.4524			−1.59884			−0.799420	−	0.600772i	$-0.794860\pi$
$773$	−44.4524			−1.59884			−0.799420	+	0.600772i	$0.794860\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			47.0813i			1.68686i
$780$		0			0
$781$	−14.0000			−0.500959
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			29.6000i			1.05513i	0.849516	+	0.527564i	$0.176894\pi$
$787$			29.6000i			1.05513i	−0.849516	+	0.527564i	$0.823106\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−25.9878	−	12.7279i	−0.924019	−	0.452553i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−16.0327			−0.567908			−0.283954	−	0.958838i	$-0.591646\pi$
$797$	−16.0327			−0.567908			−0.283954	+	0.958838i	$0.591646\pi$
$798$		0			0
$799$	26.5830			0.940439
$800$		0			0
$801$		0			0
$802$		0			0
$803$	32.4125			1.14381
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	24.6887i		−	0.868007i	−0.900911	−	0.434003i	$-0.857101\pi$
$809$		−	24.6887i		−	0.868007i	0.900911	−	0.434003i	$-0.142899\pi$
$810$		0			0
$811$		−	12.0399i		−	0.422778i	−0.977402	−	0.211389i	$-0.932201\pi$
$811$		−	12.0399i		−	0.422778i	0.977402	−	0.211389i	$-0.0677988\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$			49.2050i			1.72147i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			9.89949i			0.345495i	0.984966	+	0.172747i	$0.0552644\pi$
$821$			9.89949i			0.345495i	−0.984966	+	0.172747i	$0.944736\pi$
$822$		0			0
$823$	12.3157			0.429297			0.214649	−	0.976691i	$-0.431139\pi$
$823$	12.3157			0.429297			0.214649	+	0.976691i	$0.431139\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			37.0627i			1.28880i	0.764689	+	0.644399i	$0.222892\pi$
$827$			37.0627i			1.28880i	−0.764689	+	0.644399i	$0.777108\pi$
$828$		0			0
$829$			27.6510i			0.960357i	0.877171	+	0.480179i	$0.159428\pi$
$829$			27.6510i			0.960357i	−0.877171	+	0.480179i	$0.840572\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	14.4207	+	18.5830i	0.499646	+	0.643863i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.82087			−0.270006			−0.135003	−	0.990845i	$-0.543104\pi$
$839$	−7.82087			−0.270006			−0.135003	+	0.990845i	$0.543104\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	3.49117	−	7.12824i	0.119958	−	0.244929i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	17.7951i		−	0.610007i
$852$		0			0
$853$			13.4148i			0.459312i	0.973272	+	0.229656i	$0.0737602\pi$
$853$			13.4148i			0.459312i	−0.973272	+	0.229656i	$0.926240\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.2907			0.829753			0.414877	−	0.909878i	$-0.363825\pi$
$857$	24.2907			0.829753			0.414877	+	0.909878i	$0.363825\pi$
$858$		0			0
$859$		−	31.8546i		−	1.08687i	−0.839453	−	0.543433i	$-0.817124\pi$
$859$		−	31.8546i		−	1.08687i	0.839453	−	0.543433i	$-0.182876\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	30.6863i		−	1.04457i	−0.852770	−	0.522286i	$-0.825079\pi$
$863$		−	30.6863i		−	1.04457i	0.852770	−	0.522286i	$-0.174921\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	54.5646i		−	1.85098i
$870$		0			0
$871$			11.0604i			0.374767i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	16.9706			0.573055			0.286528	−	0.958072i	$-0.407499\pi$
$877$	16.9706			0.573055			0.286528	+	0.958072i	$0.407499\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27.3730			−0.922221			−0.461111	−	0.887343i	$-0.652549\pi$
$881$	−27.3730			−0.922221			−0.461111	+	0.887343i	$0.652549\pi$
$882$		0			0
$883$	23.9529			0.806079			0.403040	−	0.915183i	$-0.367954\pi$
$883$	23.9529			0.806079			0.403040	+	0.915183i	$0.367954\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	32.4125			1.08830			0.544152	−	0.838987i	$-0.316852\pi$
$887$	32.4125			1.08830			0.544152	+	0.838987i	$0.316852\pi$
$888$		0			0
$889$	5.41699	−	11.0604i	0.181680	−	0.370953i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	36.0000i		−	1.20469i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.38527			0.0462012
$900$		0			0
$901$			14.6315i			0.487446i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	37.9176			1.25903			0.629516	−	0.776988i	$-0.283253\pi$
$907$	37.9176			1.25903			0.629516	+	0.776988i	$0.283253\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	32.6729i		−	1.08250i	−0.840861	−	0.541252i	$-0.817951\pi$
$911$		−	32.6729i		−	1.08250i	0.840861	−	0.541252i	$-0.182049\pi$
$912$		0			0
$913$		−	11.7841i		−	0.389997i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.48925	+	15.2915i	−0.247317	+	0.504970i
$918$		0			0
$919$	−16.1255			−0.531931			−0.265965	−	0.963983i	$-0.585691\pi$
$919$	−16.1255			−0.531931			−0.265965	+	0.963983i	$0.585691\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.14944			0.103665
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	48.0651			1.57697			0.788483	−	0.615057i	$-0.210867\pi$
$929$	48.0651			1.57697			0.788483	+	0.615057i	$0.210867\pi$
$930$		0			0
$931$	25.1660	−	19.5292i	0.824783	−	0.640043i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			27.9694i			0.913721i	0.889538	+	0.456860i	$0.151026\pi$
$937$			27.9694i			0.913721i	−0.889538	+	0.456860i	$0.848974\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.7313			0.382430			0.191215	−	0.981548i	$-0.438757\pi$
$941$	11.7313			0.382430			0.191215	+	0.981548i	$0.438757\pi$
$942$		0			0
$943$			79.1032i			2.57596i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			13.0627i			0.424482i	0.977217	+	0.212241i	$0.0680762\pi$
$947$			13.0627i			0.424482i	−0.977217	+	0.212241i	$0.931924\pi$
$948$		0			0
$949$	−7.29150			−0.236692
$950$		0			0
$951$		0			0
$952$		0			0
$953$			44.8118i			1.45160i	0.687908	+	0.725798i	$0.258529\pi$
$953$			44.8118i			1.45160i	−0.687908	+	0.725798i	$0.741471\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−33.8086	−	16.5583i	−1.09174	−	0.534696i
$960$		0			0
$961$	30.0405			0.969049
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	39.4205			1.26768			0.633839	−	0.773465i	$-0.281478\pi$
$967$	39.4205			1.26768			0.633839	+	0.773465i	$0.281478\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	34.0540			1.09285			0.546423	−	0.837510i	$-0.315989\pi$
$971$	34.0540			1.09285			0.546423	+	0.837510i	$0.315989\pi$
$972$		0			0
$973$	−50.2332	−	24.6025i	−1.61040	−	0.788720i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	18.6863i		−	0.597827i	−0.954280	−	0.298913i	$-0.903376\pi$
$977$		−	18.6863i		−	0.597827i	0.954280	−	0.298913i	$-0.0966242\pi$
$978$		0			0
$979$		−	14.6315i		−	0.467625i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	41.0921			1.31063			0.655317	−	0.755354i	$-0.272535\pi$
$983$	41.0921			1.31063			0.655317	+	0.755354i	$0.272535\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			82.6714i			2.62880i
$990$		0			0
$991$	−38.4575			−1.22164			−0.610822	−	0.791768i	$-0.709161\pi$
$991$	−38.4575			−1.22164			−0.610822	+	0.791768i	$0.709161\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	29.0565i		−	0.920228i	−0.887860	−	0.460114i	$-0.847808\pi$
$997$		−	29.0565i		−	0.920228i	0.887860	−	0.460114i	$-0.152192\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.d.f.3401.8		16
3.2	odd	2	inner	6300.2.d.f.3401.7		16
5.2	odd	4		1260.2.f.b.629.2	yes	8
5.3	odd	4		1260.2.f.a.629.1	✓	8
5.4	even	2	inner	6300.2.d.f.3401.10		16
7.6	odd	2	inner	6300.2.d.f.3401.6		16
15.2	even	4		1260.2.f.a.629.7	yes	8
15.8	even	4		1260.2.f.b.629.8	yes	8
15.14	odd	2	inner	6300.2.d.f.3401.9		16
20.3	even	4		5040.2.k.f.1889.1		8
20.7	even	4		5040.2.k.e.1889.2		8
21.20	even	2	inner	6300.2.d.f.3401.5		16
35.13	even	4		1260.2.f.a.629.8	yes	8
35.27	even	4		1260.2.f.b.629.7	yes	8
35.34	odd	2	inner	6300.2.d.f.3401.12		16
60.23	odd	4		5040.2.k.e.1889.8		8
60.47	odd	4		5040.2.k.f.1889.7		8
105.62	odd	4		1260.2.f.a.629.2	yes	8
105.83	odd	4		1260.2.f.b.629.1	yes	8
105.104	even	2	inner	6300.2.d.f.3401.11		16
140.27	odd	4		5040.2.k.e.1889.7		8
140.83	odd	4		5040.2.k.f.1889.8		8
420.83	even	4		5040.2.k.e.1889.1		8
420.167	even	4		5040.2.k.f.1889.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.f.a.629.1	✓	8	5.3	odd	4
1260.2.f.a.629.2	yes	8	105.62	odd	4
1260.2.f.a.629.7	yes	8	15.2	even	4
1260.2.f.a.629.8	yes	8	35.13	even	4
1260.2.f.b.629.1	yes	8	105.83	odd	4
1260.2.f.b.629.2	yes	8	5.2	odd	4
1260.2.f.b.629.7	yes	8	35.27	even	4
1260.2.f.b.629.8	yes	8	15.8	even	4
5040.2.k.e.1889.1		8	420.83	even	4
5040.2.k.e.1889.2		8	20.7	even	4
5040.2.k.e.1889.7		8	140.27	odd	4
5040.2.k.e.1889.8		8	60.23	odd	4
5040.2.k.f.1889.1		8	20.3	even	4
5040.2.k.f.1889.2		8	420.167	even	4
5040.2.k.f.1889.7		8	60.47	odd	4
5040.2.k.f.1889.8		8	140.83	odd	4
6300.2.d.f.3401.5		16	21.20	even	2	inner
6300.2.d.f.3401.6		16	7.6	odd	2	inner
6300.2.d.f.3401.7		16	3.2	odd	2	inner
6300.2.d.f.3401.8		16	1.1	even	1	trivial
6300.2.d.f.3401.9		16	15.14	odd	2	inner
6300.2.d.f.3401.10		16	5.4	even	2	inner
6300.2.d.f.3401.11		16	105.104	even	2	inner
6300.2.d.f.3401.12		16	35.34	odd	2	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:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.16372 + 2.37608i) q^{7} +O(q^{10})\)	\(q+(-1.16372 + 2.37608i) q^{7} +3.74166i q^{11} -0.841723i q^{13} -3.36028 q^{17} +4.55066i q^{19} +7.64575i q^{23} -1.41421i q^{29} +0.979531i q^{31} -2.32744 q^{37} +10.3460 q^{41} +10.8127 q^{43} -7.91094 q^{47} +(-4.29150 - 5.53019i) q^{49} -4.35425i q^{53} +1.38527 q^{59} -13.1402 q^{67} +3.74166i q^{71} -8.66259i q^{73} +(-8.89047 - 4.35425i) q^{77} -14.5830 q^{79} -3.14944 q^{83} -3.91044 q^{89} +(2.00000 + 0.979531i) q^{91} -14.8000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{49} - 64 q^{79} + 32 q^{91}+O(q^{100}) \) 16 * q + 16 * q^49 - 64 * q^79 + 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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