LMFDB - Embedded newform 2520.2.k.a.1889.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,2,Mod(1889,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, 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("2520.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2520.k (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:	$20.1223013094$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.2
Character	$\chi$	$=$	2520.1889
Dual form			2520.2.k.a.1889.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.22844 + 0.184526i) q^{5} +(0.741091 + 2.53984i) q^{7} +O(q^{10})$	$q+(-2.22844 + 0.184526i) q^{5} +(0.741091 + 2.53984i) q^{7} -2.92735i q^{11} +0.587717 q^{13} +4.81256i q^{17} +1.64486i q^{19} -2.14663 q^{23} +(4.93190 - 0.822409i) q^{25} -5.05303i q^{29} +5.69350i q^{31} +(-2.12014 - 5.52313i) q^{35} -1.78663i q^{37} -7.74332 q^{41} -4.04741i q^{43} -0.204218i q^{47} +(-5.90157 + 3.76450i) q^{49} +6.67331 q^{53} +(0.540171 + 6.52342i) q^{55} -10.6144 q^{59} +4.35046i q^{61} +(-1.30969 + 0.108449i) q^{65} +11.7543i q^{67} +3.07869i q^{71} -10.7506 q^{73} +(7.43499 - 2.16943i) q^{77} -9.71043 q^{79} -8.45973i q^{83} +(-0.888042 - 10.7245i) q^{85} -6.74979 q^{89} +(0.435552 + 1.49271i) q^{91} +(-0.303518 - 3.66547i) q^{95} -15.1141 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{23} - 16 q^{25} + 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} - 48 q^{95}+O(q^{100}) $ 24 * q - 8 * q^23 - 16 * q^25 + 4 * q^35 - 12 * q^49 + 24 * q^53 + 8 * q^65 + 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 - 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$1$	$-1$

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
$5$	−2.22844	+	0.184526i	−0.996589	+	0.0825224i
$5$	−2.22844	+	0.184526i	−0.996589	+	0.0825224i
$6$		0			0
$7$	0.741091	+	2.53984i	0.280106	+	0.959969i
$7$	0.741091	+	2.53984i	0.280106	+	0.959969i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.92735i		−	0.882628i	−0.897353	−	0.441314i	$-0.854513\pi$
$11$		−	2.92735i		−	0.882628i	0.897353	−	0.441314i	$-0.145487\pi$
$12$		0			0
$13$	0.587717			0.163003			0.0815017	−	0.996673i	$-0.474028\pi$
$13$	0.587717			0.163003			0.0815017	+	0.996673i	$0.474028\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.81256i			1.16722i	0.812035	+	0.583609i	$0.198360\pi$
$17$			4.81256i			1.16722i	−0.812035	+	0.583609i	$0.801640\pi$
$18$		0			0
$19$			1.64486i			0.377356i	0.982039	+	0.188678i	$0.0604202\pi$
$19$			1.64486i			0.377356i	−0.982039	+	0.188678i	$0.939580\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.14663			−0.447604			−0.223802	−	0.974635i	$-0.571847\pi$
$23$	−2.14663			−0.447604			−0.223802	+	0.974635i	$0.571847\pi$
$24$		0			0
$25$	4.93190	−	0.822409i	0.986380	−	0.164482i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.05303i		−	0.938325i	−0.883112	−	0.469162i	$-0.844556\pi$
$29$		−	5.05303i		−	0.938325i	0.883112	−	0.469162i	$-0.155444\pi$
$30$		0			0
$31$			5.69350i			1.02258i	0.859407	+	0.511292i	$0.170833\pi$
$31$			5.69350i			1.02258i	−0.859407	+	0.511292i	$0.829167\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.12014	−	5.52313i	−0.358370	−	0.933580i
$36$		0			0
$37$		−	1.78663i		−	0.293721i	−0.989157	−	0.146860i	$-0.953083\pi$
$37$		−	1.78663i		−	0.293721i	0.989157	−	0.146860i	$-0.0469168\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.74332			−1.20930			−0.604652	−	0.796490i	$-0.706688\pi$
$41$	−7.74332			−1.20930			−0.604652	+	0.796490i	$0.706688\pi$
$42$		0			0
$43$		−	4.04741i		−	0.617225i	−0.951188	−	0.308612i	$-0.900135\pi$
$43$		−	4.04741i		−	0.617225i	0.951188	−	0.308612i	$-0.0998646\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	0.204218i		−	0.0297883i	−0.999889	−	0.0148942i	$-0.995259\pi$
$47$		−	0.204218i		−	0.0297883i	0.999889	−	0.0148942i	$-0.00474113\pi$
$48$		0			0
$49$	−5.90157	+	3.76450i	−0.843081	+	0.537786i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.67331			0.916650			0.458325	−	0.888785i	$-0.348450\pi$
$53$	6.67331			0.916650			0.458325	+	0.888785i	$0.348450\pi$
$54$		0			0
$55$	0.540171	+	6.52342i	0.0728366	+	0.879617i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.6144			−1.38187			−0.690936	−	0.722916i	$-0.742801\pi$
$59$	−10.6144			−1.38187			−0.690936	+	0.722916i	$0.742801\pi$
$60$		0			0
$61$			4.35046i			0.557019i	0.960433	+	0.278510i	$0.0898404\pi$
$61$			4.35046i			0.557019i	−0.960433	+	0.278510i	$0.910160\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.30969	+	0.108449i	−0.162447	+	0.0134514i
$66$		0			0
$67$			11.7543i			1.43602i	0.696035	+	0.718008i	$0.254946\pi$
$67$			11.7543i			1.43602i	−0.696035	+	0.718008i	$0.745054\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.07869i			0.365373i	0.983171	+	0.182687i	$0.0584794\pi$
$71$			3.07869i			0.365373i	−0.983171	+	0.182687i	$0.941521\pi$
$72$		0			0
$73$	−10.7506			−1.25826			−0.629129	−	0.777301i	$-0.716588\pi$
$73$	−10.7506			−1.25826			−0.629129	+	0.777301i	$0.716588\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.43499	−	2.16943i	0.847296	−	0.247229i
$78$		0			0
$79$	−9.71043			−1.09251			−0.546254	−	0.837619i	$-0.683947\pi$
$79$	−9.71043			−1.09251			−0.546254	+	0.837619i	$0.683947\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	8.45973i		−	0.928576i	−0.885684	−	0.464288i	$-0.846310\pi$
$83$		−	8.45973i		−	0.928576i	0.885684	−	0.464288i	$-0.153690\pi$
$84$		0			0
$85$	−0.888042	−	10.7245i	−0.0963217	−	1.16324i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.74979			−0.715476			−0.357738	−	0.933822i	$-0.616452\pi$
$89$	−6.74979			−0.715476			−0.357738	+	0.933822i	$0.616452\pi$
$90$		0			0
$91$	0.435552	+	1.49271i	0.0456582	+	0.156478i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.303518	−	3.66547i	−0.0311403	−	0.376069i
$96$		0			0
$97$	−15.1141			−1.53460			−0.767302	−	0.641286i	$-0.778401\pi$
$97$	−15.1141			−1.53460			−0.767302	+	0.641286i	$0.778401\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.60763			−0.557980			−0.278990	−	0.960294i	$-0.590000\pi$
$101$	−5.60763			−0.557980			−0.278990	+	0.960294i	$0.590000\pi$
$102$		0			0
$103$	17.2019			1.69496			0.847479	−	0.530829i	$-0.178119\pi$
$103$	17.2019			1.69496			0.847479	+	0.530829i	$0.178119\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.99944			0.676661			0.338331	−	0.941027i	$-0.390138\pi$
$107$	6.99944			0.676661			0.338331	+	0.941027i	$0.390138\pi$
$108$		0			0
$109$	−11.6498			−1.11585			−0.557923	−	0.829893i	$-0.688401\pi$
$109$	−11.6498			−1.11585			−0.557923	+	0.829893i	$0.688401\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−15.6102			−1.46848			−0.734240	−	0.678890i	$-0.762462\pi$
$113$	−15.6102			−1.46848			−0.734240	+	0.678890i	$0.762462\pi$
$114$		0			0
$115$	4.78364	−	0.396109i	0.446077	−	0.0369373i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.2231	+	3.56655i	−1.12049	+	0.326945i
$120$		0			0
$121$	2.43065			0.220968
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.8387	+	2.74275i	−0.969442	+	0.245319i
$126$		0			0
$127$		−	7.01766i		−	0.622716i	−0.950293	−	0.311358i	$-0.899216\pi$
$127$		−	7.01766i		−	0.622716i	0.950293	−	0.311358i	$-0.100784\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14.4195			−1.25984			−0.629918	−	0.776662i	$-0.716911\pi$
$131$	−14.4195			−1.25984			−0.629918	+	0.776662i	$0.716911\pi$
$132$		0			0
$133$	−4.17767	+	1.21899i	−0.362250	+	0.105700i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.56879			−0.304902			−0.152451	−	0.988311i	$-0.548717\pi$
$137$	−3.56879			−0.304902			−0.152451	+	0.988311i	$0.548717\pi$
$138$		0			0
$139$		−	4.51151i		−	0.382661i	−0.981526	−	0.191331i	$-0.938720\pi$
$139$		−	4.51151i		−	0.382661i	0.981526	−	0.191331i	$-0.0612803\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	1.72045i		−	0.143871i
$144$		0			0
$145$	0.932414	+	11.2604i	0.0774328	+	0.935124i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			8.89649i			0.728829i	0.931237	+	0.364414i	$0.118731\pi$
$149$			8.89649i			0.728829i	−0.931237	+	0.364414i	$0.881269\pi$
$150$		0			0
$151$	−5.00618			−0.407397			−0.203698	−	0.979034i	$-0.565296\pi$
$151$	−5.00618			−0.407397			−0.203698	+	0.979034i	$0.565296\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.05060	−	12.6876i	−0.0843860	−	1.01910i
$156$		0			0
$157$	1.10963			0.0885585			0.0442793	−	0.999019i	$-0.485901\pi$
$157$	1.10963			0.0885585			0.0442793	+	0.999019i	$0.485901\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.59085	−	5.45210i	−0.125376	−	0.429686i
$162$		0			0
$163$		−	6.66847i		−	0.522315i	−0.965296	−	0.261158i	$-0.915896\pi$
$163$		−	6.66847i		−	0.522315i	0.965296	−	0.261158i	$-0.0841042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.68932i		−	0.672399i	−0.941791	−	0.336200i	$-0.890858\pi$
$167$		−	8.68932i		−	0.672399i	0.941791	−	0.336200i	$-0.109142\pi$
$168$		0			0
$169$	−12.6546			−0.973430
$170$		0			0
$171$		0			0
$172$		0			0
$173$			20.5040i			1.55889i	0.626469	+	0.779446i	$0.284499\pi$
$173$			20.5040i			1.55889i	−0.626469	+	0.779446i	$0.715501\pi$
$174$		0			0
$175$	5.74377	+	11.9168i	0.434188	+	0.900822i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			10.0604i			0.751953i	0.926629	+	0.375976i	$0.122693\pi$
$179$			10.0604i			0.751953i	−0.926629	+	0.375976i	$0.877307\pi$
$180$		0			0
$181$			18.3215i			1.36183i	0.732364	+	0.680913i	$0.238417\pi$
$181$			18.3215i			1.36183i	−0.732364	+	0.680913i	$0.761583\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.329680	+	3.98141i	0.0242385	+	0.292719i
$186$		0			0
$187$	14.0880			1.03022
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	22.7971i		−	1.64954i	−0.565468	−	0.824771i	$-0.691304\pi$
$191$		−	22.7971i		−	1.64954i	0.565468	−	0.824771i	$-0.308696\pi$
$192$		0			0
$193$			8.32623i			0.599335i	0.954044	+	0.299668i	$0.0968758\pi$
$193$			8.32623i			0.599335i	−0.954044	+	0.299668i	$0.903124\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−5.87054			−0.418259			−0.209129	−	0.977888i	$-0.567063\pi$
$197$	−5.87054			−0.418259			−0.209129	+	0.977888i	$0.567063\pi$
$198$		0			0
$199$			19.0568i			1.35090i	0.737406	+	0.675449i	$0.236050\pi$
$199$			19.0568i			1.35090i	−0.737406	+	0.675449i	$0.763950\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.8339	−	3.74476i	0.900763	−	0.262830i
$204$		0			0
$205$	17.2555	−	1.42884i	1.20518	−	0.0997947i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.81506			0.333065
$210$		0			0
$211$	9.66325			0.665245			0.332623	−	0.943060i	$-0.392066\pi$
$211$	9.66325			0.665245			0.332623	+	0.943060i	$0.392066\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.746852	+	9.01943i	0.0509349	+	0.615120i
$216$		0			0
$217$	−14.4606	+	4.21940i	−0.981648	+	0.286432i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			2.82843i			0.190261i
$222$		0			0
$223$	1.77099			0.118594			0.0592971	−	0.998240i	$-0.481114\pi$
$223$	1.77099			0.118594			0.0592971	+	0.998240i	$0.481114\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	5.47889i		−	0.363647i	−0.983331	−	0.181823i	$-0.941800\pi$
$227$		−	5.47889i		−	0.363647i	0.983331	−	0.181823i	$-0.0581999\pi$
$228$		0			0
$229$		−	0.763511i		−	0.0504542i	−0.999682	−	0.0252271i	$-0.991969\pi$
$229$		−	0.763511i		−	0.0504542i	0.999682	−	0.0252271i	$-0.00803089\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.6430			−1.15583			−0.577917	−	0.816096i	$-0.696134\pi$
$233$	−17.6430			−1.15583			−0.577917	+	0.816096i	$0.696134\pi$
$234$		0			0
$235$	0.0376835	+	0.455089i	0.00245820	+	0.0296867i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			24.8204i			1.60550i	0.596319	+	0.802748i	$0.296629\pi$
$239$			24.8204i			1.60550i	−0.596319	+	0.802748i	$0.703371\pi$
$240$		0			0
$241$		−	11.5311i		−	0.742786i	−0.928476	−	0.371393i	$-0.878880\pi$
$241$		−	11.5311i		−	0.742786i	0.928476	−	0.371393i	$-0.121120\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.4567	−	9.47797i	0.795826	−	0.605525i
$246$		0			0
$247$			0.966710i			0.0615103i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3.77185			0.238077			0.119039	−	0.992890i	$-0.462019\pi$
$251$	3.77185			0.238077			0.119039	+	0.992890i	$0.462019\pi$
$252$		0			0
$253$			6.28393i			0.395067i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			24.5916i			1.53398i	0.641657	+	0.766991i	$0.278247\pi$
$257$			24.5916i			1.53398i	−0.641657	+	0.766991i	$0.721753\pi$
$258$		0			0
$259$	4.53776	−	1.32406i	0.281963	−	0.0822729i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	31.1200			1.91894			0.959472	−	0.281804i	$-0.0909328\pi$
$263$	31.1200			1.91894			0.959472	+	0.281804i	$0.0909328\pi$
$264$		0			0
$265$	−14.8711	+	1.23140i	−0.913523	+	0.0756441i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.1486			−1.10654			−0.553271	−	0.833001i	$-0.686621\pi$
$269$	−18.1486			−1.10654			−0.553271	+	0.833001i	$0.686621\pi$
$270$		0			0
$271$			2.62612i			0.159525i	0.996814	+	0.0797627i	$0.0254162\pi$
$271$			2.62612i			0.159525i	−0.996814	+	0.0797627i	$0.974584\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.40748	−	14.4374i	−0.145176	−	0.870607i
$276$		0			0
$277$		−	4.07759i		−	0.244999i	−0.992469	−	0.122499i	$-0.960909\pi$
$277$		−	4.07759i		−	0.244999i	0.992469	−	0.122499i	$-0.0390909\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	20.8037i		−	1.24104i	−0.784189	−	0.620522i	$-0.786921\pi$
$281$		−	20.8037i		−	1.24104i	0.784189	−	0.620522i	$-0.213079\pi$
$282$		0			0
$283$	17.9291			1.06577			0.532887	−	0.846186i	$-0.321107\pi$
$283$	17.9291			1.06577			0.532887	+	0.846186i	$0.321107\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.73851	−	19.6668i	−0.338733	−	1.16089i
$288$		0			0
$289$	−6.16077			−0.362398
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	23.8651i		−	1.39421i	−0.716967	−	0.697107i	$-0.754470\pi$
$293$		−	23.8651i		−	1.39421i	0.716967	−	0.697107i	$-0.245530\pi$
$294$		0			0
$295$	23.6535	−	1.95862i	1.37716	−	0.114035i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.26161			−0.0729609
$300$		0			0
$301$	10.2798	−	2.99950i	0.592517	−	0.172888i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.802771	−	9.69474i	−0.0459665	−	0.555119i
$306$		0			0
$307$	5.35174			0.305440			0.152720	−	0.988270i	$-0.451197\pi$
$307$	5.35174			0.305440			0.152720	+	0.988270i	$0.451197\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.6995			1.06035			0.530174	−	0.847889i	$-0.322126\pi$
$311$	18.6995			1.06035			0.530174	+	0.847889i	$0.322126\pi$
$312$		0			0
$313$	−20.5373			−1.16084			−0.580419	−	0.814318i	$-0.697111\pi$
$313$	−20.5373			−1.16084			−0.580419	+	0.814318i	$0.697111\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.82051			0.102250			0.0511249	−	0.998692i	$-0.483719\pi$
$317$	1.82051			0.102250			0.0511249	+	0.998692i	$0.483719\pi$
$318$		0			0
$319$	−14.7920			−0.828192
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.91598			−0.440457
$324$		0			0
$325$	2.89856	−	0.483344i	0.160783	−	0.0268111i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.518682	−	0.151344i	0.0285959	−	0.00834388i
$330$		0			0
$331$	7.05958			0.388029			0.194015	−	0.980999i	$-0.437849\pi$
$331$	7.05958			0.388029			0.194015	+	0.980999i	$0.437849\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.16897	−	26.1938i	−0.118503	−	1.43112i
$336$		0			0
$337$		−	25.3657i		−	1.38176i	−0.722969	−	0.690880i	$-0.757223\pi$
$337$		−	25.3657i		−	1.38176i	0.722969	−	0.690880i	$-0.242777\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.6669			0.902561
$342$		0			0
$343$	−13.9348	−	12.1992i	−0.752410	−	0.658695i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.03712			−0.485138			−0.242569	−	0.970134i	$-0.577990\pi$
$347$	−9.03712			−0.485138			−0.242569	+	0.970134i	$0.577990\pi$
$348$		0			0
$349$		−	30.2929i		−	1.62154i	−0.585362	−	0.810772i	$-0.699048\pi$
$349$		−	30.2929i		−	1.62154i	0.585362	−	0.810772i	$-0.300952\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			19.5945i			1.04291i	0.853279	+	0.521454i	$0.174610\pi$
$353$			19.5945i			1.04291i	−0.853279	+	0.521454i	$0.825390\pi$
$354$		0			0
$355$	−0.568097	−	6.86068i	−0.0301515	−	0.364127i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			15.7638i			0.831981i	0.909369	+	0.415990i	$0.136565\pi$
$359$			15.7638i			0.831981i	−0.909369	+	0.415990i	$0.863435\pi$
$360$		0			0
$361$	16.2944			0.857603
$362$		0			0
$363$		0			0
$364$		0			0
$365$	23.9570	−	1.98375i	1.25397	−	0.103834i
$366$		0			0
$367$	20.6990			1.08048			0.540239	−	0.841512i	$-0.318334\pi$
$367$	20.6990			1.08048			0.540239	+	0.841512i	$0.318334\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.94553	+	16.9491i	0.256759	+	0.879955i
$372$		0			0
$373$			2.06973i			0.107167i	0.998563	+	0.0535834i	$0.0170643\pi$
$373$			2.06973i			0.107167i	−0.998563	+	0.0535834i	$0.982936\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	2.96975i		−	0.152950i
$378$		0			0
$379$	−29.0841			−1.49395			−0.746975	−	0.664852i	$-0.768495\pi$
$379$	−29.0841			−1.49395			−0.746975	+	0.664852i	$0.768495\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			13.7663i			0.703424i	0.936108	+	0.351712i	$0.114400\pi$
$383$			13.7663i			0.703424i	−0.936108	+	0.351712i	$0.885600\pi$
$384$		0			0
$385$	−16.1681	+	6.20639i	−0.824004	+	0.316307i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			22.5433i			1.14299i	0.820605	+	0.571496i	$0.193637\pi$
$389$			22.5433i			1.14299i	−0.820605	+	0.571496i	$0.806363\pi$
$390$		0			0
$391$		−	10.3308i		−	0.522451i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	21.6391	−	1.79182i	1.08878	−	0.0901565i
$396$		0			0
$397$	27.3655			1.37344			0.686718	−	0.726924i	$-0.259051\pi$
$397$	27.3655			1.37344			0.686718	+	0.726924i	$0.259051\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			21.4163i			1.06948i	0.845018	+	0.534739i	$0.179590\pi$
$401$			21.4163i			1.06948i	−0.845018	+	0.534739i	$0.820410\pi$
$402$		0			0
$403$			3.34617i			0.166685i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.23009			−0.259246
$408$		0			0
$409$		−	34.8966i		−	1.72553i	−0.505607	−	0.862764i	$-0.668731\pi$
$409$		−	34.8966i		−	1.72553i	0.505607	−	0.862764i	$-0.331269\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.86620	−	26.9588i	−0.387070	−	1.32655i
$414$		0			0
$415$	1.56104	+	18.8520i	0.0766283	+	0.925409i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.4695			0.902295			0.451148	−	0.892449i	$-0.351015\pi$
$419$	18.4695			0.902295			0.451148	+	0.892449i	$0.351015\pi$
$420$		0			0
$421$	4.78348			0.233133			0.116566	−	0.993183i	$-0.462811\pi$
$421$	4.78348			0.233133			0.116566	+	0.993183i	$0.462811\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.95790	+	23.7351i	0.191986	+	1.15132i
$426$		0			0
$427$	−11.0495	+	3.22408i	−0.534721	+	0.156024i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	16.7979i		−	0.809125i	−0.914510	−	0.404563i	$-0.867424\pi$
$431$		−	16.7979i		−	0.809125i	0.914510	−	0.404563i	$-0.132576\pi$
$432$		0			0
$433$	−24.7365			−1.18876			−0.594379	−	0.804185i	$-0.702602\pi$
$433$	−24.7365			−1.18876			−0.594379	+	0.804185i	$0.702602\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	3.53090i		−	0.168906i
$438$		0			0
$439$		−	33.4470i		−	1.59634i	−0.602433	−	0.798169i	$-0.705802\pi$
$439$		−	33.4470i		−	1.59634i	0.602433	−	0.798169i	$-0.294198\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.6434			0.980797			0.490399	−	0.871498i	$-0.336851\pi$
$443$	20.6434			0.980797			0.490399	+	0.871498i	$0.336851\pi$
$444$		0			0
$445$	15.0415	−	1.24551i	0.713036	−	0.0590428i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			19.4928i			0.919922i	0.887939	+	0.459961i	$0.152137\pi$
$449$			19.4928i			0.919922i	−0.887939	+	0.459961i	$0.847863\pi$
$450$		0			0
$451$			22.6674i			1.06737i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.24604	−	3.24604i	−0.0584155	−	0.152177i
$456$		0			0
$457$		−	6.77440i		−	0.316893i	−0.987368	−	0.158447i	$-0.949351\pi$
$457$		−	6.77440i		−	0.316893i	0.987368	−	0.158447i	$-0.0506485\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.3561			0.482334			0.241167	−	0.970484i	$-0.422470\pi$
$461$	10.3561			0.482334			0.241167	+	0.970484i	$0.422470\pi$
$462$		0			0
$463$			24.7918i			1.15217i	0.817389	+	0.576087i	$0.195421\pi$
$463$			24.7918i			1.15217i	−0.817389	+	0.576087i	$0.804579\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			32.1955i			1.48983i	0.667160	+	0.744914i	$0.267510\pi$
$467$			32.1955i			1.48983i	−0.667160	+	0.744914i	$0.732490\pi$
$468$		0			0
$469$	−29.8540	+	8.71100i	−1.37853	+	0.402237i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.8482			−0.544780
$474$		0			0
$475$	1.35275	+	8.11227i	0.0620682	+	0.372216i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.5203			0.891905			0.445952	−	0.895057i	$-0.352865\pi$
$479$	19.5203			0.891905			0.445952	+	0.895057i	$0.352865\pi$
$480$		0			0
$481$		−	1.05004i		−	0.0478775i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	33.6809	−	2.78894i	1.52937	−	0.126639i
$486$		0			0
$487$		−	16.6327i		−	0.753701i	−0.926274	−	0.376850i	$-0.877007\pi$
$487$		−	16.6327i		−	0.753701i	0.926274	−	0.376850i	$-0.122993\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			13.9384i			0.629030i	0.949253	+	0.314515i	$0.101842\pi$
$491$			13.9384i			0.629030i	−0.949253	+	0.314515i	$0.898158\pi$
$492$		0			0
$493$	24.3180			1.09523
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.81938	+	2.28159i	−0.350747	+	0.102343i
$498$		0			0
$499$	−37.1816			−1.66448			−0.832239	−	0.554417i	$-0.812941\pi$
$499$	−37.1816			−1.66448			−0.832239	+	0.554417i	$0.812941\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			3.70436i			0.165169i	0.996584	+	0.0825847i	$0.0263175\pi$
$503$			3.70436i			0.165169i	−0.996584	+	0.0825847i	$0.973683\pi$
$504$		0			0
$505$	12.4963	−	1.03475i	0.556076	−	0.0460458i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	21.7679			0.964845			0.482422	−	0.875939i	$-0.339757\pi$
$509$	21.7679			0.964845			0.482422	+	0.875939i	$0.339757\pi$
$510$		0			0
$511$	−7.96714	−	27.3047i	−0.352445	−	1.20789i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−38.3335	+	3.17420i	−1.68918	+	0.139872i
$516$		0			0
$517$	−0.597818			−0.0262920
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.49214			0.284426			0.142213	−	0.989836i	$-0.454578\pi$
$521$	6.49214			0.284426			0.142213	+	0.989836i	$0.454578\pi$
$522$		0			0
$523$	36.4915			1.59566			0.797832	−	0.602880i	$-0.205980\pi$
$523$	36.4915			1.59566			0.797832	+	0.602880i	$0.205980\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−27.4003			−1.19358
$528$		0			0
$529$	−18.3920			−0.799651
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.55089			−0.197121
$534$		0			0
$535$	−15.5978	+	1.29158i	−0.674353	+	0.0558397i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.0200	+	17.2759i	0.474665	+	0.744127i
$540$		0			0
$541$	−7.46934			−0.321132			−0.160566	−	0.987025i	$-0.551332\pi$
$541$	−7.46934			−0.321132			−0.160566	+	0.987025i	$0.551332\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	25.9608	−	2.14968i	1.11204	−	0.0920822i
$546$		0			0
$547$			19.7180i			0.843081i	0.906809	+	0.421541i	$0.138511\pi$
$547$			19.7180i			0.843081i	−0.906809	+	0.421541i	$0.861489\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.31151			0.354082
$552$		0			0
$553$	−7.19631	−	24.6629i	−0.306018	−	1.04877i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.10509			−0.258681			−0.129340	−	0.991600i	$-0.541286\pi$
$557$	−6.10509			−0.258681			−0.129340	+	0.991600i	$0.541286\pi$
$558$		0			0
$559$		−	2.37874i		−	0.100610i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			16.2824i			0.686223i	0.939295	+	0.343111i	$0.111481\pi$
$563$			16.2824i			0.686223i	−0.939295	+	0.343111i	$0.888519\pi$
$564$		0			0
$565$	34.7863	−	2.88048i	1.46347	−	0.121183i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			21.6471i			0.907495i	0.891130	+	0.453748i	$0.149913\pi$
$569$			21.6471i			0.907495i	−0.891130	+	0.453748i	$0.850087\pi$
$570$		0			0
$571$	34.4628			1.44222			0.721111	−	0.692819i	$-0.243632\pi$
$571$	34.4628			1.44222			0.721111	+	0.692819i	$0.243632\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.5870	+	1.76541i	−0.441507	+	0.0736227i
$576$		0			0
$577$	−5.04500			−0.210026			−0.105013	−	0.994471i	$-0.533488\pi$
$577$	−5.04500			−0.210026			−0.105013	+	0.994471i	$0.533488\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	21.4864	−	6.26943i	0.891404	−	0.260100i
$582$		0			0
$583$		−	19.5351i		−	0.809061i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	38.1567i		−	1.57490i	−0.616381	−	0.787448i	$-0.711402\pi$
$587$		−	38.1567i		−	1.57490i	0.616381	−	0.787448i	$-0.288598\pi$
$588$		0			0
$589$	−9.36499			−0.385878
$590$		0			0
$591$		0			0
$592$		0			0
$593$			38.3198i			1.57361i	0.617205	+	0.786803i	$0.288265\pi$
$593$			38.3198i			1.57361i	−0.617205	+	0.786803i	$0.711735\pi$
$594$		0			0
$595$	26.5804	−	10.2033i	1.08969	−	0.418295i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	4.30380i		−	0.175848i	−0.996127	−	0.0879242i	$-0.971977\pi$
$599$		−	4.30380i		−	0.175848i	0.996127	−	0.0879242i	$-0.0280233\pi$
$600$		0			0
$601$		−	40.6313i		−	1.65739i	−0.559703	−	0.828693i	$-0.689085\pi$
$601$		−	40.6313i		−	1.65739i	0.559703	−	0.828693i	$-0.310915\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.41655	+	0.448517i	−0.220214	+	0.0182348i
$606$		0			0
$607$	17.3728			0.705139			0.352570	−	0.935786i	$-0.385308\pi$
$607$	17.3728			0.705139			0.352570	+	0.935786i	$0.385308\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	0.120023i		−	0.00485560i
$612$		0			0
$613$			25.9635i			1.04866i	0.851516	+	0.524329i	$0.175684\pi$
$613$			25.9635i			1.04866i	−0.851516	+	0.524329i	$0.824316\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	44.0449			1.77318			0.886590	−	0.462556i	$-0.153068\pi$
$617$	44.0449			1.77318			0.886590	+	0.462556i	$0.153068\pi$
$618$		0			0
$619$		−	16.5230i		−	0.664116i	−0.943259	−	0.332058i	$-0.892257\pi$
$619$		−	16.5230i		−	0.664116i	0.943259	−	0.332058i	$-0.107743\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.00221	−	17.1434i	−0.200409	−	0.686835i
$624$		0			0
$625$	23.6473	−	8.11208i	0.945891	−	0.324483i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.59828			0.342836
$630$		0			0
$631$	10.2090			0.406416			0.203208	−	0.979136i	$-0.434863\pi$
$631$	10.2090			0.406416			0.203208	+	0.979136i	$0.434863\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.29494	+	15.6384i	0.0513880	+	0.620592i
$636$		0			0
$637$	−3.46845	+	2.21246i	−0.137425	+	0.0876610i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			2.21940i			0.0876610i	0.999039	+	0.0438305i	$0.0139562\pi$
$641$			2.21940i			0.0876610i	−0.999039	+	0.0438305i	$0.986044\pi$
$642$		0			0
$643$	−47.8784			−1.88814			−0.944070	−	0.329744i	$-0.893038\pi$
$643$	−47.8784			−1.88814			−0.944070	+	0.329744i	$0.893038\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			12.9865i			0.510552i	0.966868	+	0.255276i	$0.0821664\pi$
$647$			12.9865i			0.510552i	−0.966868	+	0.255276i	$0.917834\pi$
$648$		0			0
$649$			31.0719i			1.21968i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−46.2552			−1.81011			−0.905053	−	0.425298i	$-0.860169\pi$
$653$	−46.2552			−1.81011			−0.905053	+	0.425298i	$0.860169\pi$
$654$		0			0
$655$	32.1330	−	2.66076i	1.25554	−	0.103965i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	9.76283i		−	0.380306i	−0.981754	−	0.190153i	$-0.939102\pi$
$659$		−	9.76283i		−	0.380306i	0.981754	−	0.190153i	$-0.0608984\pi$
$660$		0			0
$661$		−	22.7190i		−	0.883666i	−0.897097	−	0.441833i	$-0.854328\pi$
$661$		−	22.7190i		−	0.883666i	0.897097	−	0.441833i	$-0.145672\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	9.08476	−	3.48733i	0.352292	−	0.135233i
$666$		0			0
$667$			10.8470i			0.419997i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12.7353			0.491641
$672$		0			0
$673$			35.7915i			1.37966i	0.723971	+	0.689830i	$0.242315\pi$
$673$			35.7915i			1.37966i	−0.723971	+	0.689830i	$0.757685\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			5.66939i			0.217892i	0.994048	+	0.108946i	$0.0347476\pi$
$677$			5.66939i			0.217892i	−0.994048	+	0.108946i	$0.965252\pi$
$678$		0			0
$679$	−11.2009	−	38.3874i	−0.429852	−	1.47317i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	29.0285			1.11074			0.555372	−	0.831602i	$-0.312576\pi$
$683$	29.0285			1.11074			0.555372	+	0.831602i	$0.312576\pi$
$684$		0			0
$685$	7.95284	−	0.658533i	0.303862	−	0.0251613i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.92202			0.149417
$690$		0			0
$691$		−	31.7831i		−	1.20909i	−0.796572	−	0.604544i	$-0.793355\pi$
$691$		−	31.7831i		−	1.20909i	0.796572	−	0.604544i	$-0.206645\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.832490	+	10.0536i	0.0315781	+	0.381356i
$696$		0			0
$697$		−	37.2652i		−	1.41152i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	21.9446i		−	0.828837i	−0.910086	−	0.414418i	$-0.863985\pi$
$701$		−	21.9446i		−	0.828837i	0.910086	−	0.414418i	$-0.136015\pi$
$702$		0			0
$703$	2.93875			0.110837
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.15576	−	14.2425i	−0.156293	−	0.535643i
$708$		0			0
$709$	−1.71523			−0.0644167			−0.0322083	−	0.999481i	$-0.510254\pi$
$709$	−1.71523			−0.0644167			−0.0322083	+	0.999481i	$0.510254\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	12.2219i		−	0.457712i
$714$		0			0
$715$	0.317468	+	3.83393i	0.0118726	+	0.143381i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.0682			−0.599244			−0.299622	−	0.954058i	$-0.596861\pi$
$719$	−16.0682			−0.599244			−0.299622	+	0.954058i	$0.596861\pi$
$720$		0			0
$721$	12.7482	+	43.6902i	0.474768	+	1.62711i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.15566	−	24.9211i	−0.154337	−	0.925545i
$726$		0			0
$727$	−42.0750			−1.56048			−0.780238	−	0.625482i	$-0.784902\pi$
$727$	−42.0750			−1.56048			−0.780238	+	0.625482i	$0.784902\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	19.4784			0.720436
$732$		0			0
$733$	41.1089			1.51839			0.759197	−	0.650861i	$-0.225592\pi$
$733$	41.1089			1.51839			0.759197	+	0.650861i	$0.225592\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	34.4089			1.26747
$738$		0			0
$739$	−24.4504			−0.899423			−0.449712	−	0.893174i	$-0.648473\pi$
$739$	−24.4504			−0.899423			−0.449712	+	0.893174i	$0.648473\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	38.4935			1.41219			0.706094	−	0.708118i	$-0.250456\pi$
$743$	38.4935			1.41219			0.706094	+	0.708118i	$0.250456\pi$
$744$		0			0
$745$	−1.64163	−	19.8253i	−0.0601447	−	0.726343i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.18722	+	17.7774i	0.189537	+	0.649574i
$750$		0			0
$751$	39.8627			1.45461			0.727305	−	0.686314i	$-0.240772\pi$
$751$	39.8627			1.45461			0.727305	+	0.686314i	$0.240772\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.1560	−	0.923768i	0.406007	−	0.0336194i
$756$		0			0
$757$			33.5071i			1.21784i	0.793234	+	0.608918i	$0.208396\pi$
$757$			33.5071i			1.21784i	−0.793234	+	0.608918i	$0.791604\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9.38102			0.340062			0.170031	−	0.985439i	$-0.445613\pi$
$761$	9.38102			0.340062			0.170031	+	0.985439i	$0.445613\pi$
$762$		0			0
$763$	−8.63354	−	29.5885i	−0.312555	−	1.07118i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.23824			−0.225250
$768$		0			0
$769$			0.233161i			0.00840802i	0.999991	+	0.00420401i	$0.00133818\pi$
$769$			0.233161i			0.00840802i	−0.999991	+	0.00420401i	$0.998662\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	32.2756i		−	1.16087i	−0.814305	−	0.580437i	$-0.802882\pi$
$773$		−	32.2756i		−	1.16087i	0.814305	−	0.580437i	$-0.197118\pi$
$774$		0			0
$775$	4.68239	+	28.0798i	0.168196	+	1.00866i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	12.7367i		−	0.456338i
$780$		0			0
$781$	9.01239			0.322489
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.47276	+	0.204756i	−0.0882564	+	0.00730806i
$786$		0			0
$787$	−44.2399			−1.57698			−0.788490	−	0.615047i	$-0.789137\pi$
$787$	−44.2399			−1.57698			−0.788490	+	0.615047i	$0.789137\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11.5686	−	39.6473i	−0.411330	−	1.40970i
$792$		0			0
$793$			2.55684i			0.0907960i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	54.7878i		−	1.94068i	−0.241737	−	0.970342i	$-0.577717\pi$
$797$		−	54.7878i		−	1.94068i	0.241737	−	0.970342i	$-0.422283\pi$
$798$		0			0
$799$	0.982814			0.0347695
$800$		0			0
$801$		0			0
$802$		0			0
$803$			31.4706i			1.11057i
$804$		0			0
$805$	4.55116	+	11.8561i	0.160407	+	0.417874i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			16.2930i			0.572830i	0.958106	+	0.286415i	$0.0924637\pi$
$809$			16.2930i			0.572830i	−0.958106	+	0.286415i	$0.907536\pi$
$810$		0			0
$811$			50.1487i			1.76096i	0.474085	+	0.880479i	$0.342779\pi$
$811$			50.1487i			1.76096i	−0.474085	+	0.880479i	$0.657221\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.23050	+	14.8603i	0.0431027	+	0.520534i
$816$		0			0
$817$	6.65742			0.232913
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	3.55630i		−	0.124116i	−0.998073	−	0.0620579i	$-0.980234\pi$
$821$		−	3.55630i		−	0.124116i	0.998073	−	0.0620579i	$-0.0197663\pi$
$822$		0			0
$823$		−	21.3288i		−	0.743474i	−0.928338	−	0.371737i	$-0.878762\pi$
$823$		−	21.3288i		−	0.743474i	0.928338	−	0.371737i	$-0.121238\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.90839			0.275002			0.137501	−	0.990502i	$-0.456093\pi$
$827$	7.90839			0.275002			0.137501	+	0.990502i	$0.456093\pi$
$828$		0			0
$829$		−	5.93152i		−	0.206010i	−0.994681	−	0.103005i	$-0.967154\pi$
$829$		−	5.93152i		−	0.206010i	0.994681	−	0.103005i	$-0.0328458\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.1169	−	28.4017i	−0.627714	−	0.984060i
$834$		0			0
$835$	1.60340	+	19.3636i	0.0554880	+	0.670106i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.0159			−0.829122			−0.414561	−	0.910021i	$-0.636065\pi$
$839$	−24.0159			−0.829122			−0.414561	+	0.910021i	$0.636065\pi$
$840$		0			0
$841$	3.46686			0.119547
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.2000	−	2.33510i	0.970110	−	0.0803298i
$846$		0			0
$847$	1.80133	+	6.17345i	0.0618944	+	0.212122i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			3.83524i			0.131470i
$852$		0			0
$853$	21.8071			0.746660			0.373330	−	0.927699i	$-0.378216\pi$
$853$	21.8071			0.746660			0.373330	+	0.927699i	$0.378216\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			42.7205i			1.45931i	0.683818	+	0.729653i	$0.260318\pi$
$857$			42.7205i			1.45931i	−0.683818	+	0.729653i	$0.739682\pi$
$858$		0			0
$859$		−	37.7172i		−	1.28689i	−0.765491	−	0.643447i	$-0.777504\pi$
$859$		−	37.7172i		−	1.28689i	0.765491	−	0.643447i	$-0.222496\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.6223			0.599872			0.299936	−	0.953959i	$-0.403035\pi$
$863$	17.6223			0.599872			0.299936	+	0.953959i	$0.403035\pi$
$864$		0			0
$865$	−3.78352	−	45.6920i	−0.128644	−	1.55358i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			28.4258i			0.964279i
$870$		0			0
$871$			6.90820i			0.234075i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−14.9986	−	25.4959i	−0.507046	−	0.861919i
$876$		0			0
$877$			55.8083i			1.88451i	0.334893	+	0.942256i	$0.391300\pi$
$877$			55.8083i			1.88451i	−0.334893	+	0.942256i	$0.608700\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.7179			0.428477			0.214239	−	0.976781i	$-0.431273\pi$
$881$	12.7179			0.428477			0.214239	+	0.976781i	$0.431273\pi$
$882$		0			0
$883$		−	24.9236i		−	0.838745i	−0.907814	−	0.419372i	$-0.862250\pi$
$883$		−	24.9236i		−	0.838745i	0.907814	−	0.419372i	$-0.137750\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			53.7929i			1.80619i	0.429443	+	0.903094i	$0.358710\pi$
$887$			53.7929i			1.80619i	−0.429443	+	0.903094i	$0.641290\pi$
$888$		0			0
$889$	17.8237	−	5.20072i	0.597788	−	0.174427i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.335910			0.0112408
$894$		0			0
$895$	−1.85641	−	22.4191i	−0.0620529	−	0.749388i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	28.7695			0.959515
$900$		0			0
$901$			32.1157i			1.06993i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.38079	−	40.8284i	−0.112381	−	1.35718i
$906$		0			0
$907$		−	16.2281i		−	0.538844i	−0.963022	−	0.269422i	$-0.913167\pi$
$907$		−	16.2281i		−	0.538844i	0.963022	−	0.269422i	$-0.0868327\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			33.7286i			1.11748i	0.829344	+	0.558739i	$0.188715\pi$
$911$			33.7286i			1.11748i	−0.829344	+	0.558739i	$0.811285\pi$
$912$		0			0
$913$	−24.7645			−0.819587
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.6861	−	36.6232i	−0.352887	−	1.20940i
$918$		0			0
$919$	−12.8627			−0.424302			−0.212151	−	0.977237i	$-0.568047\pi$
$919$	−12.8627			−0.424302			−0.212151	+	0.977237i	$0.568047\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1.80940i			0.0595571i
$924$		0			0
$925$	−1.46934	−	8.81150i	−0.0483117	−	0.289720i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.49658			−0.213146			−0.106573	−	0.994305i	$-0.533988\pi$
$929$	−6.49658			−0.213146			−0.106573	+	0.994305i	$0.533988\pi$
$930$		0			0
$931$	−6.19207	−	9.70723i	−0.202937	−	0.318142i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−31.3944	+	2.59961i	−1.02671	+	0.0850162i
$936$		0			0
$937$	−18.9393			−0.618721			−0.309360	−	0.950945i	$-0.600115\pi$
$937$	−18.9393			−0.618721			−0.309360	+	0.950945i	$0.600115\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−44.1924			−1.44063			−0.720316	−	0.693647i	$-0.756003\pi$
$941$	−44.1924			−1.44063			−0.720316	+	0.693647i	$0.756003\pi$
$942$		0			0
$943$	16.6221			0.541289
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.74015			0.316512			0.158256	−	0.987398i	$-0.449413\pi$
$947$	9.74015			0.316512			0.158256	+	0.987398i	$0.449413\pi$
$948$		0			0
$949$	−6.31829			−0.205100
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−30.4050			−0.984914			−0.492457	−	0.870337i	$-0.663901\pi$
$953$	−30.4050			−0.984914			−0.492457	+	0.870337i	$0.663901\pi$
$954$		0			0
$955$	4.20665	+	50.8020i	0.136124	+	1.64391i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.64480	−	9.06415i	−0.0854049	−	0.292697i
$960$		0			0
$961$	−1.41598			−0.0456769
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.53640	−	18.5545i	−0.0494586	−	0.597291i
$966$		0			0
$967$			11.0870i			0.356534i	0.983982	+	0.178267i	$0.0570491\pi$
$967$			11.0870i			0.356534i	−0.983982	+	0.178267i	$0.942951\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−25.8732			−0.830310			−0.415155	−	0.909751i	$-0.636273\pi$
$971$	−25.8732			−0.830310			−0.415155	+	0.909751i	$0.636273\pi$
$972$		0			0
$973$	11.4585	−	3.34344i	0.367343	−	0.107186i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.0105			0.768165			0.384082	−	0.923299i	$-0.374518\pi$
$977$	24.0105			0.768165			0.384082	+	0.923299i	$0.374518\pi$
$978$		0			0
$979$			19.7590i			0.631499i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			29.9238i			0.954420i	0.878789	+	0.477210i	$0.158352\pi$
$983$			29.9238i			0.954420i	−0.878789	+	0.477210i	$0.841648\pi$
$984$		0			0
$985$	13.0822	−	1.08327i	0.416832	−	0.0345157i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			8.68831i			0.276272i
$990$		0			0
$991$	−51.9628			−1.65065			−0.825326	−	0.564657i	$-0.809009\pi$
$991$	−51.9628			−1.65065			−0.825326	+	0.564657i	$0.809009\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.51646	−	42.4669i	−0.111479	−	1.34629i
$996$		0			0
$997$	60.3535			1.91141			0.955707	−	0.294320i	$-0.0950932\pi$
$997$	60.3535			1.91141			0.955707	+	0.294320i	$0.0950932\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.k.a.1889.2	yes	24
3.2	odd	2		2520.2.k.b.1889.23	yes	24
4.3	odd	2		5040.2.k.i.1889.2		24
5.4	even	2		2520.2.k.b.1889.1	yes	24
7.6	odd	2	inner	2520.2.k.a.1889.23	yes	24
12.11	even	2		5040.2.k.h.1889.23		24
15.14	odd	2	inner	2520.2.k.a.1889.24	yes	24
20.19	odd	2		5040.2.k.h.1889.1		24
21.20	even	2		2520.2.k.b.1889.2	yes	24
28.27	even	2		5040.2.k.i.1889.23		24
35.34	odd	2		2520.2.k.b.1889.24	yes	24
60.59	even	2		5040.2.k.i.1889.24		24
84.83	odd	2		5040.2.k.h.1889.2		24
105.104	even	2	inner	2520.2.k.a.1889.1	✓	24
140.139	even	2		5040.2.k.h.1889.24		24
420.419	odd	2		5040.2.k.i.1889.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2520.2.k.a.1889.1	✓	24	105.104	even	2	inner
2520.2.k.a.1889.2	yes	24	1.1	even	1	trivial
2520.2.k.a.1889.23	yes	24	7.6	odd	2	inner
2520.2.k.a.1889.24	yes	24	15.14	odd	2	inner
2520.2.k.b.1889.1	yes	24	5.4	even	2
2520.2.k.b.1889.2	yes	24	21.20	even	2
2520.2.k.b.1889.23	yes	24	3.2	odd	2
2520.2.k.b.1889.24	yes	24	35.34	odd	2
5040.2.k.h.1889.1		24	20.19	odd	2
5040.2.k.h.1889.2		24	84.83	odd	2
5040.2.k.h.1889.23		24	12.11	even	2
5040.2.k.h.1889.24		24	140.139	even	2
5040.2.k.i.1889.1		24	420.419	odd	2
5040.2.k.i.1889.2		24	4.3	odd	2
5040.2.k.i.1889.23		24	28.27	even	2
5040.2.k.i.1889.24		24	60.59	even	2

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:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2520.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.22844 + 0.184526i) q^{5} +(0.741091 + 2.53984i) q^{7} +O(q^{10})\)	\(q+(-2.22844 + 0.184526i) q^{5} +(0.741091 + 2.53984i) q^{7} -2.92735i q^{11} +0.587717 q^{13} +4.81256i q^{17} +1.64486i q^{19} -2.14663 q^{23} +(4.93190 - 0.822409i) q^{25} -5.05303i q^{29} +5.69350i q^{31} +(-2.12014 - 5.52313i) q^{35} -1.78663i q^{37} -7.74332 q^{41} -4.04741i q^{43} -0.204218i q^{47} +(-5.90157 + 3.76450i) q^{49} +6.67331 q^{53} +(0.540171 + 6.52342i) q^{55} -10.6144 q^{59} +4.35046i q^{61} +(-1.30969 + 0.108449i) q^{65} +11.7543i q^{67} +3.07869i q^{71} -10.7506 q^{73} +(7.43499 - 2.16943i) q^{77} -9.71043 q^{79} -8.45973i q^{83} +(-0.888042 - 10.7245i) q^{85} -6.74979 q^{89} +(0.435552 + 1.49271i) q^{91} +(-0.303518 - 3.66547i) q^{95} -15.1141 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{23} - 16 q^{25} + 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} - 48 q^{95}+O(q^{100}) \) 24 * q - 8 * q^23 - 16 * q^25 + 4 * q^35 - 12 * q^49 + 24 * q^53 + 8 * q^65 + 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 - 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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