LMFDB - Embedded newform 5040.2.k.f.1889.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(1889,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5040.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5040.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$40.2446026187$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.10070523904.11
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{4} + 81 $ x^8 - 10*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.7
Root			$-0.420861 + 1.68014i$ of defining polynomial
Character	$\chi$	$=$	5040.1889
Dual form			5040.2.k.f.1889.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.95522 - 1.08495i) q^{5} +(2.37608 + 1.16372i) q^{7} +O(q^{10})$	$q+(1.95522 - 1.08495i) q^{5} +(2.37608 + 1.16372i) q^{7} +3.74166i q^{11} -0.841723 q^{13} +3.36028i q^{17} +4.55066i q^{19} +7.64575 q^{23} +(2.64575 - 4.24264i) q^{25} -1.41421i q^{29} -0.979531i q^{31} +(5.90834 - 0.302606i) q^{35} -2.32744i q^{37} -10.3460 q^{41} +10.8127i q^{43} -7.91094i q^{47} +(4.29150 + 5.53019i) q^{49} +4.35425 q^{53} +(4.05952 + 7.31575i) q^{55} -1.38527 q^{59} +(-1.64575 + 0.913230i) q^{65} +13.1402i q^{67} +3.74166i q^{71} -8.66259 q^{73} +(-4.35425 + 8.89047i) q^{77} -14.5830 q^{79} +3.14944i q^{83} +(3.64575 + 6.57008i) q^{85} -3.91044 q^{89} +(-2.00000 - 0.979531i) q^{91} +(4.93725 + 8.89753i) q^{95} +14.8000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 40 q^{23} + 16 q^{35} - 8 q^{49} + 56 q^{53} + 8 q^{65} - 56 q^{77} - 32 q^{79} + 8 q^{85} - 16 q^{91} - 24 q^{95}+O(q^{100}) $ 8 * q + 40 * q^23 + 16 * q^35 - 8 * q^49 + 56 * q^53 + 8 * q^65 - 56 * q^77 - 32 * q^79 + 8 * q^85 - 16 * q^91 - 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times$.

$n$	$2017$	$2801$	$3151$	$3601$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$-1$	$1$

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
$5$	1.95522	−	1.08495i	0.874400	−	0.485206i
$5$	1.95522	−	1.08495i	0.874400	−	0.485206i
$6$		0			0
$7$	2.37608	+	1.16372i	0.898073	+	0.439846i
$7$	2.37608	+	1.16372i	0.898073	+	0.439846i
$8$		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
$13$	−0.841723			−0.233452			−0.116726	−	0.993164i	$-0.537240\pi$
$13$	−0.841723			−0.233452			−0.116726	+	0.993164i	$0.537240\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.36028i			0.814988i	0.913208	+	0.407494i	$0.133597\pi$
$17$			3.36028i			0.814988i	−0.913208	+	0.407494i	$0.866403\pi$
$18$		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
$23$	7.64575			1.59425			0.797125	−	0.603815i	$-0.206353\pi$
$23$	7.64575			1.59425			0.797125	+	0.603815i	$0.206353\pi$
$24$		0			0
$25$	2.64575	−	4.24264i	0.529150	−	0.848528i
$26$		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	$-0.971964\pi$
$31$		−	0.979531i		−	0.175929i	0.996124	−	0.0879645i	$-0.0280362\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.90834	−	0.302606i	0.998691	−	0.0511497i
$36$		0			0
$37$		−	2.32744i		−	0.382629i	−0.981529	−	0.191315i	$-0.938725\pi$
$37$		−	2.32744i		−	0.382629i	0.981529	−	0.191315i	$-0.0612751\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.3460			−1.61578			−0.807890	−	0.589333i	$-0.799390\pi$
$41$	−10.3460			−1.61578			−0.807890	+	0.589333i	$0.799390\pi$
$42$		0			0
$43$			10.8127i			1.64893i	0.565916	+	0.824463i	$0.308522\pi$
$43$			10.8127i			1.64893i	−0.565916	+	0.824463i	$0.691478\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.91094i		−	1.15393i	−0.816769	−	0.576965i	$-0.804237\pi$
$47$		−	7.91094i		−	1.15393i	0.816769	−	0.576965i	$-0.195763\pi$
$48$		0			0
$49$	4.29150	+	5.53019i	0.613072	+	0.790027i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.35425			0.598102			0.299051	−	0.954237i	$-0.403330\pi$
$53$	4.35425			0.598102			0.299051	+	0.954237i	$0.403330\pi$
$54$		0			0
$55$	4.05952	+	7.31575i	0.547386	+	0.986456i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.38527			−0.180346			−0.0901732	−	0.995926i	$-0.528742\pi$
$59$	−1.38527			−0.180346			−0.0901732	+	0.995926i	$0.528742\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.64575	+	0.913230i	−0.204130	+	0.113272i
$66$		0			0
$67$			13.1402i			1.60533i	0.596432	+	0.802664i	$0.296585\pi$
$67$			13.1402i			1.60533i	−0.596432	+	0.802664i	$0.703415\pi$
$68$		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
$73$	−8.66259			−1.01388			−0.506940	−	0.861981i	$-0.669223\pi$
$73$	−8.66259			−1.01388			−0.506940	+	0.861981i	$0.669223\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.35425	+	8.89047i	−0.496213	+	1.01316i
$78$		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
$83$			3.14944i			0.345696i	0.984949	+	0.172848i	$0.0552969\pi$
$83$			3.14944i			0.345696i	−0.984949	+	0.172848i	$0.944703\pi$
$84$		0			0
$85$	3.64575	+	6.57008i	0.395437	+	0.712626i
$86$		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
$95$	4.93725	+	8.89753i	0.506552	+	0.912867i
$96$		0			0
$97$	14.8000			1.50271			0.751357	−	0.659896i	$-0.229400\pi$
$97$	14.8000			1.50271			0.751357	+	0.659896i	$0.229400\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.3460			1.02947			0.514735	−	0.857350i	$-0.327890\pi$
$101$	10.3460			1.02947			0.514735	+	0.857350i	$0.327890\pi$
$102$		0			0
$103$	3.36689			0.331750			0.165875	−	0.986147i	$-0.446955\pi$
$103$	3.36689			0.331750			0.165875	+	0.986147i	$0.446955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.06275			0.102740			0.0513698	−	0.998680i	$-0.483641\pi$
$107$	1.06275			0.102740			0.0513698	+	0.998680i	$0.483641\pi$
$108$		0			0
$109$	−13.8745			−1.32894			−0.664468	−	0.747316i	$-0.731342\pi$
$109$	−13.8745			−1.32894			−0.664468	+	0.747316i	$0.731342\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.9373			−1.02889			−0.514445	−	0.857523i	$-0.672002\pi$
$113$	−10.9373			−1.02889			−0.514445	+	0.857523i	$0.672002\pi$
$114$		0			0
$115$	14.9491	−	8.29529i	1.39401	−	0.773539i
$116$		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
$125$	0.569951	−	11.1658i	0.0509780	−	0.998700i
$126$		0			0
$127$			4.65489i			0.413054i	0.978441	+	0.206527i	$0.0662162\pi$
$127$			4.65489i			0.413054i	−0.978441	+	0.206527i	$0.933784\pi$
$128$		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
$133$	−5.29570	+	10.8127i	−0.459196	+	0.937582i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.2288			1.21564			0.607822	−	0.794073i	$-0.292043\pi$
$137$	14.2288			1.21564			0.607822	+	0.794073i	$0.292043\pi$
$138$		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
$143$		−	3.14944i		−	0.263369i
$144$		0			0
$145$	−1.53436	−	2.76510i	−0.127421	−	0.229629i
$146$		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	$-0.569076\pi$
$151$	−5.29150			−0.430616			−0.215308	+	0.976546i	$0.569076\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.06275	−	1.91520i	−0.0853618	−	0.153832i
$156$		0			0
$157$	18.4651			1.47367			0.736837	−	0.676070i	$-0.236318\pi$
$157$	18.4651			1.47367			0.736837	+	0.676070i	$0.236318\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	18.1669	+	8.89753i	1.43175	+	0.701223i
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	11.4821i		−	0.888509i	−0.895901	−	0.444255i	$-0.853469\pi$
$167$		−	11.4821i		−	0.888509i	0.895901	−	0.444255i	$-0.146531\pi$
$168$		0			0
$169$	−12.2915			−0.945500
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	20.3725i		−	1.54890i	−0.632638	−	0.774448i	$-0.718028\pi$
$173$		−	20.3725i		−	1.54890i	0.632638	−	0.774448i	$-0.281972\pi$
$174$		0			0
$175$	11.2238	−	7.00193i	0.848437	−	0.529296i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	14.0534i		−	1.05040i	−0.850979	−	0.525200i	$-0.823990\pi$
$179$		−	14.0534i		−	1.05040i	0.850979	−	0.525200i	$-0.176010\pi$
$180$		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
$185$	−2.52517	−	4.55066i	−0.185654	−	0.334571i
$186$		0			0
$187$	−12.5730			−0.919431
$188$		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
$193$			6.98233i			0.502599i	0.967909	+	0.251300i	$0.0808579\pi$
$193$			6.98233i			0.502599i	−0.967909	+	0.251300i	$0.919142\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	20.8118			1.48278			0.741388	−	0.671076i	$-0.234168\pi$
$197$	20.8118			1.48278			0.741388	+	0.671076i	$0.234168\pi$
$198$		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
$203$	1.64575	−	3.36028i	0.115509	−	0.235846i
$204$		0			0
$205$	−20.2288	+	11.2250i	−1.41284	+	0.783986i
$206$		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	$-0.558303\pi$
$211$	−5.29150			−0.364282			−0.182141	+	0.983272i	$0.558303\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	11.7313	+	21.1412i	0.800068	+	1.44182i
$216$		0			0
$217$	1.13990	−	2.32744i	0.0773816	−	0.157997i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.82843i		−	0.190261i
$222$		0			0
$223$	−18.7105			−1.25294			−0.626472	−	0.779444i	$-0.715502\pi$
$223$	−18.7105			−1.25294			−0.626472	+	0.779444i	$0.715502\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	20.9304i		−	1.38920i	−0.719397	−	0.694599i	$-0.755582\pi$
$227$		−	20.9304i		−	1.38920i	0.719397	−	0.694599i	$-0.244418\pi$
$228$		0			0
$229$			12.6724i			0.837419i	0.908120	+	0.418709i	$0.137517\pi$
$229$			12.6724i			0.837419i	−0.908120	+	0.418709i	$0.862483\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	16.3542			1.07140			0.535701	−	0.844408i	$-0.320047\pi$
$233$	16.3542			1.07140			0.535701	+	0.844408i	$0.320047\pi$
$234$		0			0
$235$	−8.58301	−	15.4676i	−0.559894	−	1.00900i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			7.57205i			0.489795i	0.969549	+	0.244898i	$0.0787544\pi$
$239$			7.57205i			0.489795i	−0.969549	+	0.244898i	$0.921246\pi$
$240$		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
$245$	14.3908	+	6.15664i	0.919396	+	0.393334i
$246$		0			0
$247$		−	3.83039i		−	0.243722i
$248$		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
$253$			28.6078i			1.79856i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			23.5220i			1.46726i	0.679549	+	0.733630i	$0.262176\pi$
$257$			23.5220i			1.46726i	−0.679549	+	0.733630i	$0.737824\pi$
$258$		0			0
$259$	2.70850	−	5.53019i	0.168298	−	0.343629i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.35425			0.268494			0.134247	−	0.990948i	$-0.457138\pi$
$263$	4.35425			0.268494			0.134247	+	0.990948i	$0.457138\pi$
$264$		0			0
$265$	8.51350	−	4.72416i	0.522980	−	0.290203i
$266$		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	$0.119166\pi$
$271$			12.0399i			0.731373i	−0.930738	+	0.365686i	$0.880834\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.8745	+	9.89949i	0.957269	+	0.596962i
$276$		0			0
$277$			25.4558i			1.52949i	0.644331	+	0.764747i	$0.277136\pi$
$277$			25.4558i			1.52949i	−0.644331	+	0.764747i	$0.722864\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			24.8661i			1.48339i	0.670738	+	0.741694i	$0.265977\pi$
$281$			24.8661i			1.48339i	−0.670738	+	0.741694i	$0.734023\pi$
$282$		0			0
$283$	15.3436			0.912080			0.456040	−	0.889959i	$-0.349267\pi$
$283$	15.3436			0.912080			0.456040	+	0.889959i	$0.349267\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.5830	−	12.0399i	−1.45109	−	0.710694i
$288$		0			0
$289$	5.70850			0.335794
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	29.8209i		−	1.74215i	−0.491147	−	0.871077i	$-0.663422\pi$
$293$		−	29.8209i		−	1.74215i	0.491147	−	0.871077i	$-0.336578\pi$
$294$		0			0
$295$	−2.70850	+	1.50295i	−0.157695	+	0.0875051i
$296$		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
$307$	−26.5313			−1.51422			−0.757111	−	0.653286i	$-0.773390\pi$
$307$	−26.5313			−1.51422			−0.757111	+	0.653286i	$0.773390\pi$
$308$		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
$313$	5.59388			0.316185			0.158092	−	0.987424i	$-0.449466\pi$
$313$	5.59388			0.316185			0.158092	+	0.987424i	$0.449466\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	32.8118			1.84289			0.921446	−	0.388506i	$-0.127009\pi$
$317$	32.8118			1.84289			0.921446	+	0.388506i	$0.127009\pi$
$318$		0			0
$319$	5.29150			0.296267
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−15.2915			−0.850842
$324$		0			0
$325$	−2.22699	+	3.57113i	−0.123531	+	0.198091i
$326$		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	$-0.316074\pi$
$331$	19.8745			1.09240			0.546201	+	0.837654i	$0.316074\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	14.2565	+	25.6919i	0.778914	+	1.40370i
$336$		0			0
$337$			1.50295i			0.0818709i	0.999162	+	0.0409354i	$0.0130338\pi$
$337$			1.50295i			0.0818709i	−0.999162	+	0.0409354i	$0.986966\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.66507			0.198475
$342$		0			0
$343$	3.76135	+	18.1343i	0.203094	+	0.979159i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.0627			0.701245			0.350622	−	0.936517i	$-0.385970\pi$
$347$	13.0627			0.701245			0.350622	+	0.936517i	$0.385970\pi$
$348$		0			0
$349$		−	24.0798i		−	1.28896i	−0.764620	−	0.644482i	$-0.777073\pi$
$349$		−	24.0798i		−	1.28896i	0.764620	−	0.644482i	$-0.222927\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	20.3725i		−	1.08432i	−0.840275	−	0.542161i	$-0.817606\pi$
$353$		−	20.3725i		−	1.08432i	0.840275	−	0.542161i	$-0.182394\pi$
$354$		0			0
$355$	4.05952	+	7.31575i	0.215457	+	0.388280i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	21.7142i		−	1.14603i	−0.819545	−	0.573015i	$-0.805773\pi$
$359$		−	21.7142i		−	1.14603i	0.819545	−	0.573015i	$-0.194227\pi$
$360$		0			0
$361$	−1.70850			−0.0899209
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.9373	+	9.39851i	−0.886536	+	0.491941i
$366$		0			0
$367$	15.3436			0.800927			0.400464	−	0.916313i	$-0.368849\pi$
$367$	15.3436			0.800927			0.400464	+	0.916313i	$0.368849\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.3460	+	5.06713i	0.537140	+	0.263073i
$372$		0			0
$373$			13.9647i			0.723063i	0.932360	+	0.361531i	$0.117746\pi$
$373$			13.9647i			0.723063i	−0.932360	+	0.361531i	$0.882254\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			1.19038i			0.0613075i
$378$		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
$383$		−	11.4821i		−	0.586706i	−0.956004	−	0.293353i	$-0.905229\pi$
$383$		−	11.4821i		−	0.586706i	0.956004	−	0.293353i	$-0.0947712\pi$
$384$		0			0
$385$	1.13225	+	22.1070i	0.0577047	+	1.12668i
$386$		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
$395$	−28.5129	+	15.8219i	−1.43464	+	0.796085i
$396$		0			0
$397$	26.2860			1.31925			0.659627	−	0.751593i	$-0.270714\pi$
$397$	26.2860			1.31925			0.659627	+	0.751593i	$0.270714\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	32.3494i		−	1.61545i	−0.589557	−	0.807727i	$-0.700698\pi$
$401$		−	32.3494i		−	1.61545i	0.589557	−	0.807727i	$-0.299302\pi$
$402$		0			0
$403$			0.824494i			0.0410710i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.70850			0.431664
$408$		0			0
$409$		−	3.57113i		−	0.176581i	−0.996095	−	0.0882904i	$-0.971860\pi$
$409$		−	3.57113i		−	0.176581i	0.996095	−	0.0882904i	$-0.0281404\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.29150	−	1.61206i	−0.161964	−	0.0793245i
$414$		0			0
$415$	3.41699	+	6.15784i	0.167734	+	0.302276i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	28.5129			1.39295			0.696474	−	0.717582i	$-0.254751\pi$
$419$	28.5129			1.39295			0.696474	+	0.717582i	$0.254751\pi$
$420$		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
$425$	14.2565	+	8.89047i	0.691541	+	0.431251i
$426$		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
$433$	−4.50679			−0.216583			−0.108291	−	0.994119i	$-0.534538\pi$
$433$	−4.50679			−0.216583			−0.108291	+	0.994119i	$0.534538\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			34.7932i			1.66438i
$438$		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
$443$	4.35425			0.206877			0.103438	−	0.994636i	$-0.467016\pi$
$443$	4.35425			0.206877			0.103438	+	0.994636i	$0.467016\pi$
$444$		0			0
$445$	−7.64575	+	4.24264i	−0.362443	+	0.201120i
$446$		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
$455$	−4.97318	+	0.254710i	−0.233146	+	0.0119410i
$456$		0			0
$457$		−	32.4382i		−	1.51739i	−0.651444	−	0.758697i	$-0.725836\pi$
$457$		−	32.4382i		−	1.51739i	0.651444	−	0.758697i	$-0.274164\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.5018			−0.675418			−0.337709	−	0.941251i	$-0.609652\pi$
$461$	−14.5018			−0.675418			−0.337709	+	0.941251i	$0.609652\pi$
$462$		0			0
$463$		−	16.9706i		−	0.788689i	−0.918963	−	0.394344i	$-0.870972\pi$
$463$		−	16.9706i		−	0.788689i	0.918963	−	0.394344i	$-0.129028\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			14.2098i			0.657552i	0.944408	+	0.328776i	$0.106636\pi$
$467$			14.2098i			0.657552i	−0.944408	+	0.328776i	$0.893364\pi$
$468$		0			0
$469$	−15.2915	+	31.2221i	−0.706096	+	1.44170i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−40.4575			−1.86024
$474$		0			0
$475$	19.3068	+	12.0399i	0.885857	+	0.552429i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−17.0270			−0.777984			−0.388992	−	0.921241i	$-0.627177\pi$
$479$	−17.0270			−0.777984			−0.388992	+	0.921241i	$0.627177\pi$
$480$		0			0
$481$			1.95906i			0.0893256i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	28.9373	−	16.0573i	1.31397	−	0.729126i
$486$		0			0
$487$			15.4676i			0.700904i	0.936581	+	0.350452i	$0.113972\pi$
$487$			15.4676i			0.700904i	−0.936581	+	0.350452i	$0.886028\pi$
$488$		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
$493$	4.75216			0.214026
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.35425	+	8.89047i	−0.195315	+	0.398792i
$498$		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
$503$		−	4.33981i		−	0.193503i	−0.995309	−	0.0967514i	$-0.969155\pi$
$503$		−	4.33981i		−	0.193503i	0.995309	−	0.0967514i	$-0.0308452\pi$
$504$		0			0
$505$	20.2288	−	11.2250i	0.900168	−	0.499505i
$506$		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
$515$	6.58301	−	3.65292i	0.290082	−	0.160967i
$516$		0			0
$517$	29.6000			1.30181
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−31.0381			−1.35980			−0.679902	−	0.733303i	$-0.737978\pi$
$521$	−31.0381			−1.35980			−0.679902	+	0.733303i	$0.737978\pi$
$522$		0			0
$523$	3.36689			0.147224			0.0736119	−	0.997287i	$-0.476547\pi$
$523$	3.36689			0.147224			0.0736119	+	0.997287i	$0.476547\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.29150			0.143380
$528$		0			0
$529$	35.4575			1.54163
$530$		0			0
$531$		0			0
$532$		0			0
$533$	8.70850			0.377207
$534$		0			0
$535$	2.07790	−	1.15303i	0.0898354	−	0.0498498i
$536$		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
$545$	−27.1277	+	15.0532i	−1.16202	+	0.644808i
$546$		0			0
$547$			25.4558i			1.08841i	0.838951	+	0.544207i	$0.183169\pi$
$547$			25.4558i			1.08841i	−0.838951	+	0.544207i	$0.816831\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.43560			0.274166
$552$		0			0
$553$	−34.6504	−	16.9706i	−1.47348	−	0.721662i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	17.5203			0.742357			0.371179	−	0.928561i	$-0.378954\pi$
$557$	17.5203			0.742357			0.371179	+	0.928561i	$0.378954\pi$
$558$		0			0
$559$		−	9.10132i		−	0.384945i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	44.6632i		−	1.88233i	−0.337947	−	0.941165i	$-0.609733\pi$
$563$		−	44.6632i		−	1.88233i	0.337947	−	0.941165i	$-0.390267\pi$
$564$		0			0
$565$	−21.3847	+	11.8664i	−0.899662	+	0.499224i
$566$		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	$-0.509439\pi$
$571$	−1.41699			−0.0592994			−0.0296497	+	0.999560i	$0.509439\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	20.2288	−	32.4382i	0.843597	−	1.35277i
$576$		0			0
$577$	−20.1485			−0.838794			−0.419397	−	0.907803i	$-0.637759\pi$
$577$	−20.1485			−0.838794			−0.419397	+	0.907803i	$0.637759\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.66507	+	7.48331i	−0.152053	+	0.310460i
$582$		0			0
$583$			16.2921i			0.674750i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			10.6387i			0.439106i	0.975601	+	0.219553i	$0.0704599\pi$
$587$			10.6387i			0.439106i	−0.975601	+	0.219553i	$0.929540\pi$
$588$		0			0
$589$	4.45751			0.183669
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	15.1894i		−	0.623752i	−0.950123	−	0.311876i	$-0.899043\pi$
$593$		−	15.1894i		−	0.623752i	0.950123	−	0.311876i	$-0.100957\pi$
$594$		0			0
$595$	1.01684	+	19.8537i	0.0416864	+	0.813922i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			26.1916i			1.07016i	0.844801	+	0.535080i	$0.179719\pi$
$599$			26.1916i			1.07016i	−0.844801	+	0.535080i	$0.820281\pi$
$600$		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
$605$	−5.86565	+	3.25486i	−0.238473	+	0.132329i
$606$		0			0
$607$	−13.6601			−0.554447			−0.277223	−	0.960805i	$-0.589414\pi$
$607$	−13.6601			−0.554447			−0.277223	+	0.960805i	$0.589414\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			6.65882i			0.269387i
$612$		0			0
$613$		−	12.3157i		−	0.497425i	−0.968577	−	0.248713i	$-0.919992\pi$
$613$		−	12.3157i		−	0.497425i	0.968577	−	0.248713i	$-0.0800075\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−29.5203			−1.18844			−0.594220	−	0.804302i	$-0.702539\pi$
$617$	−29.5203			−1.18844			−0.594220	+	0.804302i	$0.702539\pi$
$618$		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
$623$	−9.29150	−	4.55066i	−0.372256	−	0.182318i
$624$		0			0
$625$	−11.0000	−	22.4499i	−0.440000	−	0.897998i
$626$		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	$-0.638427\pi$
$631$	−21.1660			−0.842606			−0.421303	+	0.906920i	$0.638427\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.05034	+	9.10132i	0.200417	+	0.361175i
$636$		0			0
$637$	−3.61226	−	4.65489i	−0.143123	−	0.184433i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	27.6946i		−	1.09387i	−0.837175	−	0.546935i	$-0.815795\pi$
$641$		−	27.6946i		−	1.09387i	0.837175	−	0.546935i	$-0.184205\pi$
$642$		0			0
$643$	−5.83925			−0.230277			−0.115139	−	0.993349i	$-0.536731\pi$
$643$	−5.83925			−0.230277			−0.115139	+	0.993349i	$0.536731\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	6.29888i		−	0.247634i	−0.992305	−	0.123817i	$-0.960486\pi$
$647$		−	6.29888i		−	0.247634i	0.992305	−	0.123817i	$-0.0395136\pi$
$648$		0			0
$649$		−	5.18319i		−	0.203458i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−36.1033			−1.41283			−0.706415	−	0.707798i	$-0.749689\pi$
$653$	−36.1033			−1.41283			−0.706415	+	0.707798i	$0.749689\pi$
$654$		0			0
$655$	12.5830	−	6.98233i	0.491659	−	0.272822i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			38.5073i			1.50003i	0.661421	+	0.750015i	$0.269954\pi$
$659$			38.5073i			1.50003i	−0.661421	+	0.750015i	$0.730046\pi$
$660$		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
$665$	1.37706	+	26.8868i	0.0534000	+	1.04263i
$666$		0			0
$667$		−	10.8127i		−	0.418670i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	45.5783i		−	1.75692i	−0.477820	−	0.878458i	$-0.658573\pi$
$673$		−	45.5783i		−	1.75692i	0.477820	−	0.878458i	$-0.341427\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			7.27841i			0.279732i	0.990170	+	0.139866i	$0.0446672\pi$
$677$			7.27841i			0.279732i	−0.990170	+	0.139866i	$0.955333\pi$
$678$		0			0
$679$	35.1660	+	17.2231i	1.34955	+	0.660962i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−17.5203			−0.670394			−0.335197	−	0.942148i	$-0.608803\pi$
$683$	−17.5203			−0.670394			−0.335197	+	0.942148i	$0.608803\pi$
$684$		0			0
$685$	27.8203	−	15.4375i	1.06296	−	0.589838i
$686$		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	$0.0614152\pi$
$691$			10.0808i			0.383494i	−0.981444	+	0.191747i	$0.938585\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	22.9373	+	41.3357i	0.870060	+	1.56795i
$696$		0			0
$697$		−	34.7656i		−	1.31684i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			28.6965i			1.08385i	0.840426	+	0.541926i	$0.182305\pi$
$701$			28.6965i			1.08385i	−0.840426	+	0.541926i	$0.817695\pi$
$702$		0			0
$703$	10.5914			0.399462
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.5830	+	12.0399i	0.924539	+	0.452807i
$708$		0			0
$709$	6.83399			0.256656			0.128328	−	0.991732i	$-0.459039\pi$
$709$	6.83399			0.256656			0.128328	+	0.991732i	$0.459039\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7.48925i		−	0.280475i
$714$		0			0
$715$	−3.41699	−	6.15784i	−0.127788	−	0.230290i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	48.3105			1.80168			0.900839	−	0.434154i	$-0.142953\pi$
$719$	48.3105			1.80168			0.900839	+	0.434154i	$0.142953\pi$
$720$		0			0
$721$	8.00000	+	3.91813i	0.297936	+	0.145919i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.00000	−	3.74166i	−0.222834	−	0.138962i
$726$		0			0
$727$	34.6504			1.28511			0.642556	−	0.766239i	$-0.277874\pi$
$727$	34.6504			1.28511			0.642556	+	0.766239i	$0.277874\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36.3338			−1.34385
$732$		0			0
$733$	−32.1252			−1.18657			−0.593286	−	0.804992i	$-0.702170\pi$
$733$	−32.1252			−1.18657			−0.593286	+	0.804992i	$0.702170\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−49.1660			−1.81105
$738$		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
$743$	13.0627			0.479226			0.239613	−	0.970869i	$-0.422979\pi$
$743$	13.0627			0.479226			0.239613	+	0.970869i	$0.422979\pi$
$744$		0			0
$745$	2.62144	+	4.72416i	0.0960423	+	0.173080i
$746$		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	$-0.389119\pi$
$751$	18.7085			0.682683			0.341341	+	0.939939i	$0.389119\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.3460	+	5.74103i	−0.376531	+	0.208938i
$756$		0			0
$757$			6.15784i			0.223810i	0.993719	+	0.111905i	$0.0356953\pi$
$757$			6.15784i			0.223810i	−0.993719	+	0.111905i	$0.964305\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9.45150			0.342617			0.171308	−	0.985217i	$-0.445201\pi$
$761$	9.45150			0.342617			0.171308	+	0.985217i	$0.445201\pi$
$762$		0			0
$763$	−32.9669	−	16.1461i	−1.19348	−	0.584527i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.16601			0.0421022
$768$		0			0
$769$		−	38.3643i		−	1.38345i	−0.722159	−	0.691727i	$-0.756850\pi$
$769$		−	38.3643i		−	1.38345i	0.722159	−	0.691727i	$-0.243150\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	44.4524i		−	1.59884i	−0.600772	−	0.799420i	$-0.705140\pi$
$773$		−	44.4524i		−	1.59884i	0.600772	−	0.799420i	$-0.294860\pi$
$774$		0			0
$775$	−4.15580	−	2.59160i	−0.149281	−	0.0930929i
$776$		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
$785$	36.1033	−	20.0338i	1.28858	−	0.715036i
$786$		0			0
$787$	29.6000			1.05513			0.527564	−	0.849516i	$-0.323106\pi$
$787$	29.6000			1.05513			0.527564	+	0.849516i	$0.323106\pi$
$788$		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
$797$			16.0327i			0.567908i	0.958838	+	0.283954i	$0.0916464\pi$
$797$			16.0327i			0.567908i	−0.958838	+	0.283954i	$0.908354\pi$
$798$		0			0
$799$	26.5830			0.940439
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	32.4125i		−	1.14381i
$804$		0			0
$805$	45.1737	−	2.31365i	1.59216	−	0.0815454i
$806$		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	$0.0677988\pi$
$811$			12.0399i			0.422778i	−0.977402	+	0.211389i	$0.932201\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−49.2050			−1.72147
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	9.89949i		−	0.345495i	−0.984966	−	0.172747i	$-0.944736\pi$
$821$		−	9.89949i		−	0.345495i	0.984966	−	0.172747i	$-0.0552644\pi$
$822$		0			0
$823$			12.3157i			0.429297i	0.976691	+	0.214649i	$0.0688607\pi$
$823$			12.3157i			0.429297i	−0.976691	+	0.214649i	$0.931139\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−37.0627			−1.28880			−0.644399	−	0.764689i	$-0.722892\pi$
$827$	−37.0627			−1.28880			−0.644399	+	0.764689i	$0.722892\pi$
$828$		0			0
$829$		−	27.6510i		−	0.960357i	−0.877171	−	0.480179i	$-0.840572\pi$
$829$		−	27.6510i		−	0.960357i	0.877171	−	0.480179i	$-0.159428\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.5830	+	14.4207i	−0.643863	+	0.499646i
$834$		0			0
$835$	−12.4575	−	22.4499i	−0.431110	−	0.776912i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.82087			0.270006			0.135003	−	0.990845i	$-0.456896\pi$
$839$	7.82087			0.270006			0.135003	+	0.990845i	$0.456896\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−24.0326	+	13.3357i	−0.826745	+	0.458762i
$846$		0			0
$847$	−7.12824	−	3.49117i	−0.244929	−	0.119958i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	17.7951i		−	0.610007i
$852$		0			0
$853$	13.4148			0.459312			0.229656	−	0.973272i	$-0.426240\pi$
$853$	13.4148			0.459312			0.229656	+	0.973272i	$0.426240\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	24.2907i		−	0.829753i	−0.909878	−	0.414877i	$-0.863825\pi$
$857$		−	24.2907i		−	0.829753i	0.909878	−	0.414877i	$-0.136175\pi$
$858$		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
$863$	−30.6863			−1.04457			−0.522286	−	0.852770i	$-0.674921\pi$
$863$	−30.6863			−1.04457			−0.522286	+	0.852770i	$0.674921\pi$
$864$		0			0
$865$	−22.1033	−	39.8328i	−0.751534	−	1.35435i
$866$		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
$875$	14.3481	−	25.8676i	0.485056	−	0.874483i
$876$		0			0
$877$			16.9706i			0.573055i	0.958072	+	0.286528i	$0.0925010\pi$
$877$			16.9706i			0.573055i	−0.958072	+	0.286528i	$0.907499\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27.3730			0.922221			0.461111	−	0.887343i	$-0.347451\pi$
$881$	27.3730			0.922221			0.461111	+	0.887343i	$0.347451\pi$
$882$		0			0
$883$			23.9529i			0.806079i	0.915183	+	0.403040i	$0.132046\pi$
$883$			23.9529i			0.806079i	−0.915183	+	0.403040i	$0.867954\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			32.4125i			1.08830i	0.838987	+	0.544152i	$0.183148\pi$
$887$			32.4125i			1.08830i	−0.838987	+	0.544152i	$0.816852\pi$
$888$		0			0
$889$	−5.41699	+	11.0604i	−0.181680	+	0.370953i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	36.0000			1.20469
$894$		0			0
$895$	−15.2473	−	27.4775i	−0.509661	−	0.918470i
$896$		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
$905$	−20.1255	−	36.2686i	−0.668994	−	1.20561i
$906$		0			0
$907$		−	37.9176i		−	1.25903i	−0.776988	−	0.629516i	$-0.783253\pi$
$907$		−	37.9176i		−	1.25903i	0.776988	−	0.629516i	$-0.216747\pi$
$908$		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
$913$	−11.7841			−0.389997
$914$		0			0
$915$		0			0
$916$		0			0
$917$	15.2915	+	7.48925i	0.504970	+	0.247317i
$918$		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
$923$		−	3.14944i		−	0.103665i
$924$		0			0
$925$	−9.87451	−	6.15784i	−0.324672	−	0.202468i
$926$		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
$935$	−24.5830	+	13.6412i	−0.803950	+	0.446113i
$936$		0			0
$937$	−27.9694			−0.913721			−0.456860	−	0.889538i	$-0.651026\pi$
$937$	−27.9694			−0.913721			−0.456860	+	0.889538i	$0.651026\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−11.7313			−0.382430			−0.191215	−	0.981548i	$-0.561243\pi$
$941$	−11.7313			−0.382430			−0.191215	+	0.981548i	$0.561243\pi$
$942$		0			0
$943$	−79.1032			−2.57596
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.0627			−0.424482			−0.212241	−	0.977217i	$-0.568076\pi$
$947$	−13.0627			−0.424482			−0.212241	+	0.977217i	$0.568076\pi$
$948$		0			0
$949$	7.29150			0.236692
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−44.8118			−1.45160			−0.725798	−	0.687908i	$-0.758529\pi$
$953$	−44.8118			−1.45160			−0.725798	+	0.687908i	$0.758529\pi$
$954$		0			0
$955$	−6.04115	−	10.8869i	−0.195487	−	0.352291i
$956$		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
$965$	7.57551	+	13.6520i	0.243864	+	0.439473i
$966$		0			0
$967$		−	39.4205i		−	1.26768i	−0.773465	−	0.633839i	$-0.781478\pi$
$967$		−	39.4205i		−	1.26768i	0.773465	−	0.633839i	$-0.218522\pi$
$968$		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
$973$	−24.6025	+	50.2332i	−0.788720	+	1.61040i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.6863			−0.597827			−0.298913	−	0.954280i	$-0.596624\pi$
$977$	−18.6863			−0.597827			−0.298913	+	0.954280i	$0.596624\pi$
$978$		0			0
$979$		−	14.6315i		−	0.467625i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	41.0921i		−	1.31063i	−0.755354	−	0.655317i	$-0.772535\pi$
$983$		−	41.0921i		−	1.31063i	0.755354	−	0.655317i	$-0.227465\pi$
$984$		0			0
$985$	40.6915	−	22.5798i	1.29654	−	0.719452i
$986$		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	$-0.290839\pi$
$991$	38.4575			1.22164			0.610822	+	0.791768i	$0.290839\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−28.9373	−	52.1484i	−0.917373	−	1.65322i
$996$		0			0
$997$	29.0565			0.920228			0.460114	−	0.887860i	$-0.347808\pi$
$997$	29.0565			0.920228			0.460114	+	0.887860i	$0.347808\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5040.2.k.f.1889.7		8
3.2	odd	2		5040.2.k.e.1889.2		8
4.3	odd	2		1260.2.f.a.629.7	yes	8
5.4	even	2		5040.2.k.e.1889.8		8
7.6	odd	2	inner	5040.2.k.f.1889.2		8
12.11	even	2		1260.2.f.b.629.2	yes	8
15.14	odd	2	inner	5040.2.k.f.1889.1		8
20.3	even	4		6300.2.d.f.3401.7		16
20.7	even	4		6300.2.d.f.3401.9		16
20.19	odd	2		1260.2.f.b.629.8	yes	8
21.20	even	2		5040.2.k.e.1889.7		8
28.27	even	2		1260.2.f.a.629.2	yes	8
35.34	odd	2		5040.2.k.e.1889.1		8
60.23	odd	4		6300.2.d.f.3401.8		16
60.47	odd	4		6300.2.d.f.3401.10		16
60.59	even	2		1260.2.f.a.629.1	✓	8
84.83	odd	2		1260.2.f.b.629.7	yes	8
105.104	even	2	inner	5040.2.k.f.1889.8		8
140.27	odd	4		6300.2.d.f.3401.11		16
140.83	odd	4		6300.2.d.f.3401.5		16
140.139	even	2		1260.2.f.b.629.1	yes	8
420.83	even	4		6300.2.d.f.3401.6		16
420.167	even	4		6300.2.d.f.3401.12		16
420.419	odd	2		1260.2.f.a.629.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.f.a.629.1	✓	8	60.59	even	2
1260.2.f.a.629.2	yes	8	28.27	even	2
1260.2.f.a.629.7	yes	8	4.3	odd	2
1260.2.f.a.629.8	yes	8	420.419	odd	2
1260.2.f.b.629.1	yes	8	140.139	even	2
1260.2.f.b.629.2	yes	8	12.11	even	2
1260.2.f.b.629.7	yes	8	84.83	odd	2
1260.2.f.b.629.8	yes	8	20.19	odd	2
5040.2.k.e.1889.1		8	35.34	odd	2
5040.2.k.e.1889.2		8	3.2	odd	2
5040.2.k.e.1889.7		8	21.20	even	2
5040.2.k.e.1889.8		8	5.4	even	2
5040.2.k.f.1889.1		8	15.14	odd	2	inner
5040.2.k.f.1889.2		8	7.6	odd	2	inner
5040.2.k.f.1889.7		8	1.1	even	1	trivial
5040.2.k.f.1889.8		8	105.104	even	2	inner
6300.2.d.f.3401.5		16	140.83	odd	4
6300.2.d.f.3401.6		16	420.83	even	4
6300.2.d.f.3401.7		16	20.3	even	4
6300.2.d.f.3401.8		16	60.23	odd	4
6300.2.d.f.3401.9		16	20.7	even	4
6300.2.d.f.3401.10		16	60.47	odd	4
6300.2.d.f.3401.11		16	140.27	odd	4
6300.2.d.f.3401.12		16	420.167	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5040.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.95522 - 1.08495i) q^{5} +(2.37608 + 1.16372i) q^{7} +O(q^{10})\)	\(q+(1.95522 - 1.08495i) q^{5} +(2.37608 + 1.16372i) q^{7} +3.74166i q^{11} -0.841723 q^{13} +3.36028i q^{17} +4.55066i q^{19} +7.64575 q^{23} +(2.64575 - 4.24264i) q^{25} -1.41421i q^{29} -0.979531i q^{31} +(5.90834 - 0.302606i) q^{35} -2.32744i q^{37} -10.3460 q^{41} +10.8127i q^{43} -7.91094i q^{47} +(4.29150 + 5.53019i) q^{49} +4.35425 q^{53} +(4.05952 + 7.31575i) q^{55} -1.38527 q^{59} +(-1.64575 + 0.913230i) q^{65} +13.1402i q^{67} +3.74166i q^{71} -8.66259 q^{73} +(-4.35425 + 8.89047i) q^{77} -14.5830 q^{79} +3.14944i q^{83} +(3.64575 + 6.57008i) q^{85} -3.91044 q^{89} +(-2.00000 - 0.979531i) q^{91} +(4.93725 + 8.89753i) q^{95} +14.8000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 40 q^{23} + 16 q^{35} - 8 q^{49} + 56 q^{53} + 8 q^{65} - 56 q^{77} - 32 q^{79} + 8 q^{85} - 16 q^{91} - 24 q^{95}+O(q^{100}) \) 8 * q + 40 * q^23 + 16 * q^35 - 8 * q^49 + 56 * q^53 + 8 * q^65 - 56 * q^77 - 32 * q^79 + 8 * q^85 - 16 * q^91 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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