LMFDB - Embedded newform 3136.2.e.d.1567.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(1567,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3136.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.e (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:	$25.0410860739$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.4971563078713344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 16x^{8} + 8x^{7} + 8x^{6} + 32x^{5} + 240x^{4} + 120x^{3} + 32x^{2} + 16x + 4 $ x^12 - 4x^9 - 16x^8 + 8x^7 + 8x^6 + 32x^5 + 240x^4 + 120x^3 + 32x^2 + 16*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 448)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.5
Root			$-1.73080 + 0.463767i$ of defining polynomial
Character	$\chi$	$=$	3136.1567
Dual form			3136.2.e.d.1567.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.210756i q^{3} +1.53407 q^{5} +2.95558 q^{9} +O(q^{10})$	$q-0.210756i q^{3} +1.53407 q^{5} +2.95558 q^{9} -4.75418 q^{11} -4.95558 q^{13} -0.323314i q^{15} +2.65709i q^{17} -4.32331i q^{19} -0.632268i q^{23} -2.64663 q^{25} -1.25517i q^{27} +7.85324i q^{29} -1.29008 q^{31} +1.00197i q^{33} +1.73205i q^{37} +1.04442i q^{39} +7.85324i q^{41} -6.92820 q^{43} +4.53407 q^{45} -8.94836 q^{47} +0.559997 q^{51} +1.00197i q^{53} -7.29324 q^{55} -0.911164 q^{57} +9.70041i q^{59} -12.9112 q^{61} -7.60221 q^{65} +5.67921 q^{67} -0.133254 q^{69} +6.00000i q^{71} -11.2404i q^{73} +0.557793i q^{75} +5.27890i q^{79} +8.60221 q^{81} -15.0681i q^{83} +4.07616i q^{85} +1.65512 q^{87} +5.19615i q^{89} +0.271891i q^{93} -6.63227i q^{95} -6.96929i q^{97} -14.0514 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{5} - 8 q^{9}+O(q^{10}) $ 12 * q - 12 * q^5 - 8 * q^9	$ 12 q - 12 q^{5} - 8 q^{9} - 16 q^{13} + 16 q^{25} + 24 q^{45} + 76 q^{57} - 68 q^{61} - 132 q^{69} + 12 q^{81}+O(q^{100}) $ 12 * q - 12 * q^5 - 8 * q^9 - 16 * q^13 + 16 * q^25 + 24 * q^45 + 76 * q^57 - 68 * q^61 - 132 * q^69 + 12 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	− 0.210756i	− 0.121680i	−0.998148	−	0.0608400i	$-0.980622\pi$
$3$	− 0.210756i	− 0.121680i	0.998148	−	0.0608400i	$-0.0193779\pi$
$4$	0	0
$5$	1.53407	0.686057	0.343029	−	0.939325i	$-0.388547\pi$
$5$	1.53407	0.686057	0.343029	+	0.939325i	$0.388547\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.95558	0.985194
$10$	0	0
$11$	−4.75418	−1.43344	−0.716719	−	0.697362i	$-0.754357\pi$
$11$	−4.75418	−1.43344	−0.716719	+	0.697362i	$0.754357\pi$
$12$	0	0
$13$	−4.95558	−1.37443	−0.687216	−	0.726454i	$-0.741167\pi$
$13$	−4.95558	−1.37443	−0.687216	+	0.726454i	$0.741167\pi$
$14$	0	0
$15$	− 0.323314i	− 0.0834794i
$16$	0	0
$17$	2.65709i	0.644438i	0.946665	+	0.322219i	$0.104429\pi$
$17$	2.65709i	0.644438i	−0.946665	+	0.322219i	$0.895571\pi$
$18$	0	0
$19$	− 4.32331i	− 0.991836i	−0.868369	−	0.495918i	$-0.834832\pi$
$19$	− 4.32331i	− 0.991836i	0.868369	−	0.495918i	$-0.165168\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 0.632268i	− 0.131837i	−0.997825	−	0.0659185i	$-0.979002\pi$
$23$	− 0.632268i	− 0.131837i	0.997825	−	0.0659185i	$-0.0209977\pi$
$24$	0	0
$25$	−2.64663	−0.529326
$26$	0	0
$27$	− 1.25517i	− 0.241558i
$28$	0	0
$29$	7.85324i	1.45831i	0.684349	+	0.729155i	$0.260087\pi$
$29$	7.85324i	1.45831i	−0.684349	+	0.729155i	$0.739913\pi$
$30$	0	0
$31$	−1.29008	−0.231705	−0.115852	−	0.993266i	$-0.536960\pi$
$31$	−1.29008	−0.231705	−0.115852	+	0.993266i	$0.536960\pi$
$32$	0	0
$33$	1.00197i	0.174421i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.73205i	0.284747i	0.989813	+	0.142374i	$0.0454735\pi$
$37$	1.73205i	0.284747i	−0.989813	+	0.142374i	$0.954527\pi$
$38$	0	0
$39$	1.04442i	0.167241i
$40$	0	0
$41$	7.85324i	1.22647i	0.789901	+	0.613235i	$0.210132\pi$
$41$	7.85324i	1.22647i	−0.789901	+	0.613235i	$0.789868\pi$
$42$	0	0
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	0	0
$45$	4.53407	0.675899
$46$	0	0
$47$	−8.94836	−1.30525	−0.652626	−	0.757680i	$-0.726333\pi$
$47$	−8.94836	−1.30525	−0.652626	+	0.757680i	$0.726333\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.559997	0.0784152
$52$	0	0
$53$	1.00197i	0.137631i	0.997629	+	0.0688157i	$0.0219220\pi$
$53$	1.00197i	0.137631i	−0.997629	+	0.0688157i	$0.978078\pi$
$54$	0	0
$55$	−7.29324	−0.983421
$56$	0	0
$57$	−0.911164	−0.120687
$58$	0	0
$59$	9.70041i	1.26289i	0.775422	+	0.631443i	$0.217537\pi$
$59$	9.70041i	1.26289i	−0.775422	+	0.631443i	$0.782463\pi$
$60$	0	0
$61$	−12.9112	−1.65311	−0.826553	−	0.562860i	$-0.809701\pi$
$61$	−12.9112	−1.65311	−0.826553	+	0.562860i	$0.809701\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.60221	−0.942938
$66$	0	0
$67$	5.67921	0.693827	0.346913	−	0.937897i	$-0.387230\pi$
$67$	5.67921	0.693827	0.346913	+	0.937897i	$0.387230\pi$
$68$	0	0
$69$	−0.133254	−0.0160419
$70$	0	0
$71$	6.00000i	0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$	6.00000i	0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$	0	0
$73$	− 11.2404i	− 1.31559i	−0.753197	−	0.657795i	$-0.771489\pi$
$73$	− 11.2404i	− 1.31559i	0.753197	−	0.657795i	$-0.228511\pi$
$74$	0	0
$75$	0.557793i	0.0644083i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.27890i	0.593922i	0.954890	+	0.296961i	$0.0959732\pi$
$79$	5.27890i	0.593922i	−0.954890	+	0.296961i	$0.904027\pi$
$80$	0	0
$81$	8.60221	0.955801
$82$	0	0
$83$	− 15.0681i	− 1.65394i	−0.562244	−	0.826972i	$-0.690062\pi$
$83$	− 15.0681i	− 1.65394i	0.562244	−	0.826972i	$-0.309938\pi$
$84$	0	0
$85$	4.07616i	0.442121i
$86$	0	0
$87$	1.65512	0.177447
$88$	0	0
$89$	5.19615i	0.550791i	0.961331	+	0.275396i	$0.0888088\pi$
$89$	5.19615i	0.550791i	−0.961331	+	0.275396i	$0.911191\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.271891i	0.0281938i
$94$	0	0
$95$	− 6.63227i	− 0.680456i
$96$	0	0
$97$	− 6.96929i	− 0.707624i	−0.935316	−	0.353812i	$-0.884885\pi$
$97$	− 6.96929i	− 0.707624i	0.935316	−	0.353812i	$-0.115115\pi$
$98$	0	0
$99$	−14.0514	−1.41222
$100$	0	0
$101$	4.46593	0.444377	0.222188	−	0.975004i	$-0.428680\pi$
$101$	4.46593	0.444377	0.222188	+	0.975004i	$0.428680\pi$
$102$	0	0
$103$	5.67921	0.559590	0.279795	−	0.960060i	$-0.409734\pi$
$103$	5.67921	0.559590	0.279795	+	0.960060i	$0.409734\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.0684	−0.973344	−0.486672	−	0.873585i	$-0.661789\pi$
$107$	−10.0684	−0.973344	−0.486672	+	0.873585i	$0.661789\pi$
$108$	0	0
$109$	10.3154i	0.988033i	0.869452	+	0.494017i	$0.164472\pi$
$109$	10.3154i	0.988033i	−0.869452	+	0.494017i	$0.835528\pi$
$110$	0	0
$111$	0.365040	0.0346481
$112$	0	0
$113$	14.8667	1.39855	0.699273	−	0.714855i	$-0.253507\pi$
$113$	14.8667	1.39855	0.699273	+	0.714855i	$0.253507\pi$
$114$	0	0
$115$	− 0.969943i	− 0.0904476i
$116$	0	0
$117$	−14.6466	−1.35408
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.6022	1.05475
$122$	0	0
$123$	1.65512	0.149237
$124$	0	0
$125$	−11.7305	−1.04920
$126$	0	0
$127$	17.2044i	1.52665i	0.646017	+	0.763323i	$0.276433\pi$
$127$	17.2044i	1.52665i	−0.646017	+	0.763323i	$0.723567\pi$
$128$	0	0
$129$	1.46016i	0.128560i
$130$	0	0
$131$	− 12.7685i	− 1.11559i	−0.829978	−	0.557797i	$-0.811647\pi$
$131$	− 12.7685i	− 1.11559i	0.829978	−	0.557797i	$-0.188353\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 1.92552i	− 0.165723i
$136$	0	0
$137$	4.60221	0.393193	0.196597	−	0.980484i	$-0.437011\pi$
$137$	4.60221	0.393193	0.196597	+	0.980484i	$0.437011\pi$
$138$	0	0
$139$	− 15.9112i	− 1.34957i	−0.738016	−	0.674784i	$-0.764237\pi$
$139$	− 15.9112i	− 1.34957i	0.738016	−	0.674784i	$-0.235763\pi$
$140$	0	0
$141$	1.88592i	0.158823i
$142$	0	0
$143$	23.5597	1.97016
$144$	0	0
$145$	12.0474i	1.00048i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 4.66103i	− 0.381846i	−0.981605	−	0.190923i	$-0.938852\pi$
$149$	− 4.66103i	− 0.381846i	0.981605	−	0.190923i	$-0.0611481\pi$
$150$	0	0
$151$	18.2345i	1.48390i	0.670455	+	0.741950i	$0.266099\pi$
$151$	18.2345i	1.48390i	−0.670455	+	0.741950i	$0.733901\pi$
$152$	0	0
$153$	7.85324i	0.634897i
$154$	0	0
$155$	−1.97907	−0.158963
$156$	0	0
$157$	−0.911164	−0.0727188	−0.0363594	−	0.999339i	$-0.511576\pi$
$157$	−0.911164	−0.0727188	−0.0363594	+	0.999339i	$0.511576\pi$
$158$	0	0
$159$	0.211171	0.0167470
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.21511	0.173501	0.0867505	−	0.996230i	$-0.472352\pi$
$163$	2.21511	0.173501	0.0867505	+	0.996230i	$0.472352\pi$
$164$	0	0
$165$	1.53709i	0.119663i
$166$	0	0
$167$	−18.2455	−1.41188	−0.705941	−	0.708270i	$-0.749476\pi$
$167$	−18.2455	−1.41188	−0.705941	+	0.708270i	$0.749476\pi$
$168$	0	0
$169$	11.5578	0.889061
$170$	0	0
$171$	− 12.7779i	− 0.977151i
$172$	0	0
$173$	10.4008	0.790759	0.395380	−	0.918518i	$-0.370613\pi$
$173$	10.4008	0.790759	0.395380	+	0.918518i	$0.370613\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.04442	0.153668
$178$	0	0
$179$	−13.1425	−0.982320	−0.491160	−	0.871069i	$-0.663427\pi$
$179$	−13.1425	−0.982320	−0.491160	+	0.871069i	$0.663427\pi$
$180$	0	0
$181$	−23.8510	−1.77283	−0.886417	−	0.462887i	$-0.846814\pi$
$181$	−23.8510	−1.77283	−0.886417	+	0.462887i	$0.846814\pi$
$182$	0	0
$183$	2.72110i	0.201150i
$184$	0	0
$185$	2.65709i	0.195353i
$186$	0	0
$187$	− 12.6323i	− 0.923763i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 12.9699i	− 0.938472i	−0.883073	−	0.469236i	$-0.844529\pi$
$191$	− 12.9699i	− 0.938472i	0.883073	−	0.469236i	$-0.155471\pi$
$192$	0	0
$193$	0.0444180	0.00319728	0.00159864	−	0.999999i	$-0.499491\pi$
$193$	0.0444180	0.00319728	0.00159864	+	0.999999i	$0.499491\pi$
$194$	0	0
$195$	1.60221i	0.114737i
$196$	0	0
$197$	26.8700i	1.91440i	0.289419	+	0.957202i	$0.406538\pi$
$197$	26.8700i	1.91440i	−0.289419	+	0.957202i	$0.593462\pi$
$198$	0	0
$199$	−19.3407	−1.37102	−0.685512	−	0.728062i	$-0.740421\pi$
$199$	−19.3407	−1.37102	−0.685512	+	0.728062i	$0.740421\pi$
$200$	0	0
$201$	− 1.19693i	− 0.0844248i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	12.0474i	0.841428i
$206$	0	0
$207$	− 1.86872i	− 0.129885i
$208$	0	0
$209$	20.5538i	1.42174i
$210$	0	0
$211$	−12.5825	−0.866218	−0.433109	−	0.901342i	$-0.642583\pi$
$211$	−12.5825	−0.866218	−0.433109	+	0.901342i	$0.642583\pi$
$212$	0	0
$213$	1.26454	0.0866445
$214$	0	0
$215$	−10.6284	−0.724847
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.36898	−0.160081
$220$	0	0
$221$	− 13.1674i	− 0.885736i
$222$	0	0
$223$	11.5123	0.770921	0.385460	−	0.922724i	$-0.374043\pi$
$223$	11.5123	0.770921	0.385460	+	0.922724i	$0.374043\pi$
$224$	0	0
$225$	−7.82233	−0.521489
$226$	0	0
$227$	22.0381i	1.46272i	0.681992	+	0.731359i	$0.261114\pi$
$227$	22.0381i	1.46272i	−0.681992	+	0.731359i	$0.738886\pi$
$228$	0	0
$229$	−4.04442	−0.267263	−0.133631	−	0.991031i	$-0.542664\pi$
$229$	−4.04442	−0.267263	−0.133631	+	0.991031i	$0.542664\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.8667	−1.17049	−0.585245	−	0.810857i	$-0.699002\pi$
$233$	−17.8667	−1.17049	−0.585245	+	0.810857i	$0.699002\pi$
$234$	0	0
$235$	−13.7274	−0.895478
$236$	0	0
$237$	1.11256	0.0722684
$238$	0	0
$239$	− 24.0712i	− 1.55703i	−0.627623	−	0.778517i	$-0.715972\pi$
$239$	− 24.0712i	− 1.55703i	0.627623	−	0.778517i	$-0.284028\pi$
$240$	0	0
$241$	− 8.50639i	− 0.547944i	−0.961738	−	0.273972i	$-0.911662\pi$
$241$	− 8.50639i	− 0.547944i	0.961738	−	0.273972i	$-0.0883376\pi$
$242$	0	0
$243$	− 5.57849i	− 0.357860i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	21.4245i	1.36321i
$248$	0	0
$249$	−3.17570	−0.201252
$250$	0	0
$251$	5.93489i	0.374607i	0.982302	+	0.187303i	$0.0599747\pi$
$251$	5.93489i	0.374607i	−0.982302	+	0.187303i	$0.940025\pi$
$252$	0	0
$253$	3.00591i	0.188980i
$254$	0	0
$255$	0.859074	0.0537973
$256$	0	0
$257$	− 18.3636i	− 1.14549i	−0.819734	−	0.572744i	$-0.805879\pi$
$257$	− 18.3636i	− 1.14549i	0.819734	−	0.572744i	$-0.194121\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	23.2109i	1.43672i
$262$	0	0
$263$	18.6323i	1.14891i	0.818534	+	0.574457i	$0.194787\pi$
$263$	18.6323i	1.14891i	−0.818534	+	0.574457i	$0.805213\pi$
$264$	0	0
$265$	1.53709i	0.0944229i
$266$	0	0
$267$	1.09512	0.0670202
$268$	0	0
$269$	−0.0681404	−0.00415459	−0.00207730	−	0.999998i	$-0.500661\pi$
$269$	−0.0681404	−0.00415459	−0.00207730	+	0.999998i	$0.500661\pi$
$270$	0	0
$271$	−18.8055	−1.14236	−0.571178	−	0.820826i	$-0.693513\pi$
$271$	−18.8055	−1.14236	−0.571178	+	0.820826i	$0.693513\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.5825	0.758756
$276$	0	0
$277$	2.85204i	0.171363i	0.996323	+	0.0856814i	$0.0273067\pi$
$277$	2.85204i	0.171363i	−0.996323	+	0.0856814i	$0.972693\pi$
$278$	0	0
$279$	−3.81293	−0.228274
$280$	0	0
$281$	−0.337675	−0.0201440	−0.0100720	−	0.999949i	$-0.503206\pi$
$281$	−0.337675	−0.0201440	−0.0100720	+	0.999949i	$0.503206\pi$
$282$	0	0
$283$	6.14564i	0.365321i	0.983176	+	0.182660i	$0.0584708\pi$
$283$	6.14564i	0.365321i	−0.983176	+	0.182660i	$0.941529\pi$
$284$	0	0
$285$	−1.39779	−0.0827979
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.93989	0.584699
$290$	0	0
$291$	−1.46882	−0.0861037
$292$	0	0
$293$	0.136281	0.00796161	0.00398080	−	0.999992i	$-0.498733\pi$
$293$	0.136281	0.00796161	0.00398080	+	0.999992i	$0.498733\pi$
$294$	0	0
$295$	14.8811i	0.866412i
$296$	0	0
$297$	5.96732i	0.346259i
$298$	0	0
$299$	3.13325i	0.181201i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 0.941221i	− 0.0540717i
$304$	0	0
$305$	−19.8066	−1.13412
$306$	0	0
$307$	3.35337i	0.191387i	0.995411	+	0.0956935i	$0.0305069\pi$
$307$	3.35337i	0.191387i	−0.995411	+	0.0956935i	$0.969493\pi$
$308$	0	0
$309$	− 1.19693i	− 0.0680909i
$310$	0	0
$311$	−16.8016	−0.952731	−0.476366	−	0.879247i	$-0.658046\pi$
$311$	−16.8016	−0.952731	−0.476366	+	0.879247i	$0.658046\pi$
$312$	0	0
$313$	7.93017i	0.448240i	0.974562	+	0.224120i	$0.0719507\pi$
$313$	7.93017i	0.448240i	−0.974562	+	0.224120i	$0.928049\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 25.0968i	− 1.40958i	−0.709417	−	0.704789i	$-0.751042\pi$
$317$	− 25.0968i	− 1.40958i	0.709417	−	0.704789i	$-0.248958\pi$
$318$	0	0
$319$	− 37.3357i	− 2.09040i
$320$	0	0
$321$	2.12196i	0.118437i
$322$	0	0
$323$	11.4874	0.639177
$324$	0	0
$325$	13.1156	0.727522
$326$	0	0
$327$	2.17403	0.120224
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6.79921	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$331$	6.79921	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$332$	0	0
$333$	5.11922i	0.280531i
$334$	0	0
$335$	8.71231	0.476005
$336$	0	0
$337$	−1.69105	−0.0921172	−0.0460586	−	0.998939i	$-0.514666\pi$
$337$	−1.69105	−0.0921172	−0.0460586	+	0.998939i	$0.514666\pi$
$338$	0	0
$339$	− 3.13325i	− 0.170175i
$340$	0	0
$341$	6.13325	0.332134
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.204421	−0.0110057
$346$	0	0
$347$	24.1197	1.29481	0.647407	−	0.762144i	$-0.275853\pi$
$347$	24.1197	1.29481	0.647407	+	0.762144i	$0.275853\pi$
$348$	0	0
$349$	17.5134	0.937469	0.468735	−	0.883339i	$-0.344710\pi$
$349$	17.5134	0.937469	0.468735	+	0.883339i	$0.344710\pi$
$350$	0	0
$351$	6.22012i	0.332005i
$352$	0	0
$353$	− 9.92546i	− 0.528279i	−0.964485	−	0.264139i	$-0.914912\pi$
$353$	− 9.92546i	− 0.528279i	0.964485	−	0.264139i	$-0.0850879\pi$
$354$	0	0
$355$	9.20442i	0.488520i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.49901i	0.501339i	0.968073	+	0.250669i	$0.0806507\pi$
$359$	9.49901i	0.501339i	−0.968073	+	0.250669i	$0.919349\pi$
$360$	0	0
$361$	0.308953	0.0162607
$362$	0	0
$363$	− 2.44523i	− 0.128342i
$364$	0	0
$365$	− 17.2436i	− 0.902570i
$366$	0	0
$367$	35.9722	1.87773	0.938866	−	0.344282i	$-0.111878\pi$
$367$	35.9722	1.87773	0.938866	+	0.344282i	$0.111878\pi$
$368$	0	0
$369$	23.2109i	1.20831i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 16.4007i	− 0.849197i	−0.905382	−	0.424598i	$-0.860415\pi$
$373$	− 16.4007i	− 0.849197i	0.905382	−	0.424598i	$-0.139585\pi$
$374$	0	0
$375$	2.47226i	0.127667i
$376$	0	0
$377$	− 38.9174i	− 2.00435i
$378$	0	0
$379$	−6.23921	−0.320487	−0.160243	−	0.987078i	$-0.551228\pi$
$379$	−6.23921	−0.320487	−0.160243	+	0.987078i	$0.551228\pi$
$380$	0	0
$381$	3.62593	0.185762
$382$	0	0
$383$	29.9690	1.53135	0.765673	−	0.643231i	$-0.222406\pi$
$383$	29.9690	1.53135	0.765673	+	0.643231i	$0.222406\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−20.4769	−1.04090
$388$	0	0
$389$	4.66103i	0.236324i	0.992994	+	0.118162i	$0.0377001\pi$
$389$	4.66103i	0.236324i	−0.992994	+	0.118162i	$0.962300\pi$
$390$	0	0
$391$	1.67999	0.0849608
$392$	0	0
$393$	−2.69105	−0.135745
$394$	0	0
$395$	8.09820i	0.407465i
$396$	0	0
$397$	−17.1600	−0.861236	−0.430618	−	0.902534i	$-0.641704\pi$
$397$	−17.1600	−0.861236	−0.430618	+	0.902534i	$0.641704\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.3357	1.11539	0.557696	−	0.830045i	$-0.311686\pi$
$401$	22.3357	1.11539	0.557696	+	0.830045i	$0.311686\pi$
$402$	0	0
$403$	6.39308	0.318462
$404$	0	0
$405$	13.1964	0.655734
$406$	0	0
$407$	− 8.23448i	− 0.408168i
$408$	0	0
$409$	33.6801i	1.66538i	0.553743	+	0.832688i	$0.313199\pi$
$409$	33.6801i	1.66538i	−0.553743	+	0.832688i	$0.686801\pi$
$410$	0	0
$411$	− 0.969943i	− 0.0478438i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 23.1156i	− 1.13470i
$416$	0	0
$417$	−3.35337	−0.164215
$418$	0	0
$419$	− 10.3327i	− 0.504784i	−0.967625	−	0.252392i	$-0.918783\pi$
$419$	− 10.3327i	− 0.504784i	0.967625	−	0.252392i	$-0.0812173\pi$
$420$	0	0
$421$	− 18.9756i	− 0.924815i	−0.886667	−	0.462408i	$-0.846986\pi$
$421$	− 18.9756i	− 0.924815i	0.886667	−	0.462408i	$-0.153014\pi$
$422$	0	0
$423$	−26.4476	−1.28593
$424$	0	0
$425$	− 7.03232i	− 0.341118i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 4.96535i	− 0.239729i
$430$	0	0
$431$	− 17.0301i	− 0.820309i	−0.912016	−	0.410154i	$-0.865475\pi$
$431$	− 17.0301i	− 0.820309i	0.912016	−	0.410154i	$-0.134525\pi$
$432$	0	0
$433$	− 3.26914i	− 0.157105i	−0.996910	−	0.0785525i	$-0.974970\pi$
$433$	− 3.26914i	− 0.157105i	0.996910	−	0.0785525i	$-0.0250298\pi$
$434$	0	0
$435$	2.53906	0.121739
$436$	0	0
$437$	−2.73349	−0.130761
$438$	0	0
$439$	18.2704	0.872000	0.436000	−	0.899947i	$-0.356395\pi$
$439$	18.2704	0.872000	0.436000	+	0.899947i	$0.356395\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.1504	0.814841	0.407421	−	0.913241i	$-0.366428\pi$
$443$	17.1504	0.814841	0.407421	+	0.913241i	$0.366428\pi$
$444$	0	0
$445$	7.97126i	0.377874i
$446$	0	0
$447$	−0.982339	−0.0464631
$448$	0	0
$449$	−22.1313	−1.04444	−0.522220	−	0.852811i	$-0.674896\pi$
$449$	−22.1313	−1.04444	−0.522220	+	0.852811i	$0.674896\pi$
$450$	0	0
$451$	− 37.3357i	− 1.75807i
$452$	0	0
$453$	3.84302	0.180561
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.6466	−0.919031	−0.459515	−	0.888170i	$-0.651977\pi$
$457$	−19.6466	−0.919031	−0.459515	+	0.888170i	$0.651977\pi$
$458$	0	0
$459$	3.33511	0.155669
$460$	0	0
$461$	−4.67035	−0.217520	−0.108760	−	0.994068i	$-0.534688\pi$
$461$	−4.67035	−0.217520	−0.108760	+	0.994068i	$0.534688\pi$
$462$	0	0
$463$	− 19.1757i	− 0.891170i	−0.895240	−	0.445585i	$-0.852996\pi$
$463$	− 19.1757i	− 0.891170i	0.895240	−	0.445585i	$-0.147004\pi$
$464$	0	0
$465$	0.417100i	0.0193426i
$466$	0	0
$467$	36.9048i	1.70775i	0.520477	+	0.853876i	$0.325754\pi$
$467$	36.9048i	1.70775i	−0.520477	+	0.853876i	$0.674246\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.192033i	0.00884842i
$472$	0	0
$473$	32.9379	1.51449
$474$	0	0
$475$	11.4422i	0.525004i
$476$	0	0
$477$	2.96141i	0.135594i
$478$	0	0
$479$	3.98301	0.181988	0.0909942	−	0.995851i	$-0.470996\pi$
$479$	3.98301	0.181988	0.0909942	+	0.995851i	$0.470996\pi$
$480$	0	0
$481$	− 8.58332i	− 0.391366i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 10.6914i	− 0.485471i
$486$	0	0
$487$	8.10320i	0.367191i	0.983002	+	0.183595i	$0.0587736\pi$
$487$	8.10320i	0.367191i	−0.983002	+	0.183595i	$0.941226\pi$
$488$	0	0
$489$	− 0.466848i	− 0.0211116i
$490$	0	0
$491$	−0.771168	−0.0348023	−0.0174012	−	0.999849i	$-0.505539\pi$
$491$	−0.771168	−0.0348023	−0.0174012	+	0.999849i	$0.505539\pi$
$492$	0	0
$493$	−20.8667	−0.939791
$494$	0	0
$495$	−21.5558	−0.968860
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−34.3582	−1.53808	−0.769041	−	0.639199i	$-0.779266\pi$
$499$	−34.3582	−1.53808	−0.769041	+	0.639199i	$0.779266\pi$
$500$	0	0
$501$	3.84536i	0.171798i
$502$	0	0
$503$	−28.2890	−1.26135	−0.630673	−	0.776049i	$-0.717221\pi$
$503$	−28.2890	−1.26135	−0.630673	+	0.776049i	$0.717221\pi$
$504$	0	0
$505$	6.85105	0.304868
$506$	0	0
$507$	− 2.43587i	− 0.108181i
$508$	0	0
$509$	28.4659	1.26173	0.630865	−	0.775893i	$-0.282700\pi$
$509$	28.4659	1.26173	0.630865	+	0.775893i	$0.282700\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.42651	−0.239586
$514$	0	0
$515$	8.71231	0.383910
$516$	0	0
$517$	42.5421	1.87100
$518$	0	0
$519$	− 2.19203i	− 0.0962196i
$520$	0	0
$521$	− 20.9026i	− 0.915761i	−0.889014	−	0.457880i	$-0.848609\pi$
$521$	− 20.9026i	− 0.915761i	0.889014	−	0.457880i	$-0.151391\pi$
$522$	0	0
$523$	10.2058i	0.446267i	0.974788	+	0.223133i	$0.0716285\pi$
$523$	10.2058i	0.446267i	−0.974788	+	0.223133i	$0.928371\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.42785i	− 0.149319i
$528$	0	0
$529$	22.6002	0.982619
$530$	0	0
$531$	28.6704i	1.24419i
$532$	0	0
$533$	− 38.9174i	− 1.68570i
$534$	0	0
$535$	−15.4456	−0.667770
$536$	0	0
$537$	2.76987i	0.119529i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	38.8404i	1.66988i	0.550340	+	0.834940i	$0.314498\pi$
$541$	38.8404i	1.66988i	−0.550340	+	0.834940i	$0.685502\pi$
$542$	0	0
$543$	5.02675i	0.215718i
$544$	0	0
$545$	15.8245i	0.677847i
$546$	0	0
$547$	40.5725	1.73475	0.867377	−	0.497651i	$-0.165804\pi$
$547$	40.5725	1.73475	0.867377	+	0.497651i	$0.165804\pi$
$548$	0	0
$549$	−38.1600	−1.62863
$550$	0	0
$551$	33.9520	1.44640
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.559997	0.0237705
$556$	0	0
$557$	− 11.9294i	− 0.505465i	−0.967536	−	0.252732i	$-0.918671\pi$
$557$	− 11.9294i	− 0.505465i	0.967536	−	0.252732i	$-0.0813292\pi$
$558$	0	0
$559$	34.3333	1.45214
$560$	0	0
$561$	−2.66232	−0.112403
$562$	0	0
$563$	− 1.17134i	− 0.0493660i	−0.999695	−	0.0246830i	$-0.992142\pi$
$563$	− 1.17134i	− 0.0493660i	0.999695	−	0.0246830i	$-0.00785764\pi$
$564$	0	0
$565$	22.8066	0.959482
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.46896	−0.313115	−0.156557	−	0.987669i	$-0.550040\pi$
$569$	−7.46896	−0.313115	−0.156557	+	0.987669i	$0.550040\pi$
$570$	0	0
$571$	−23.3810	−0.978463	−0.489232	−	0.872154i	$-0.662723\pi$
$571$	−23.3810	−0.978463	−0.489232	+	0.872154i	$0.662723\pi$
$572$	0	0
$573$	−2.73349	−0.114193
$574$	0	0
$575$	1.67338i	0.0697847i
$576$	0	0
$577$	1.73205i	0.0721062i	0.999350	+	0.0360531i	$0.0114785\pi$
$577$	1.73205i	0.0721062i	−0.999350	+	0.0360531i	$0.988521\pi$
$578$	0	0
$579$	− 0.00936136i	0 0.000389045i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 4.76355i	− 0.197286i
$584$	0	0
$585$	−22.4690	−0.928977
$586$	0	0
$587$	16.1964i	0.668497i	0.942485	+	0.334248i	$0.108482\pi$
$587$	16.1964i	0.668497i	−0.942485	+	0.334248i	$0.891518\pi$
$588$	0	0
$589$	5.57741i	0.229813i
$590$	0	0
$591$	5.66300	0.232945
$592$	0	0
$593$	− 41.5745i	− 1.70726i	−0.520881	−	0.853629i	$-0.674397\pi$
$593$	− 41.5745i	− 1.70726i	0.520881	−	0.853629i	$-0.325603\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.07616i	0.166826i
$598$	0	0
$599$	41.7766i	1.70694i	0.521138	+	0.853472i	$0.325508\pi$
$599$	41.7766i	1.70694i	−0.521138	+	0.853472i	$0.674492\pi$
$600$	0	0
$601$	0.0410883i	0.00167603i	1.00000		0.000838014i	$0.000266748\pi$
$601$	0.0410883i	0.00167603i	−1.00000		0.000838014i	$0.999733\pi$
$602$	0	0
$603$	16.7854	0.683554
$604$	0	0
$605$	17.7986	0.723616
$606$	0	0
$607$	−30.3092	−1.23021	−0.615106	−	0.788445i	$-0.710887\pi$
$607$	−30.3092	−1.23021	−0.615106	+	0.788445i	$0.710887\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	44.3443	1.79398
$612$	0	0
$613$	− 36.4963i	− 1.47407i	−0.675853	−	0.737036i	$-0.736225\pi$
$613$	− 36.4963i	− 1.47407i	0.675853	−	0.737036i	$-0.263775\pi$
$614$	0	0
$615$	2.53906	0.102385
$616$	0	0
$617$	−31.6022	−1.27226	−0.636129	−	0.771583i	$-0.719465\pi$
$617$	−31.6022	−1.27226	−0.636129	+	0.771583i	$0.719465\pi$
$618$	0	0
$619$	− 13.3390i	− 0.536140i	−0.963399	−	0.268070i	$-0.913614\pi$
$619$	− 13.3390i	− 0.536140i	0.963399	−	0.268070i	$-0.0863859\pi$
$620$	0	0
$621$	−0.793606	−0.0318463
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.76221	−0.190489
$626$	0	0
$627$	4.33184	0.172997
$628$	0	0
$629$	−4.60221	−0.183502
$630$	0	0
$631$	− 11.9112i	− 0.474176i	−0.971488	−	0.237088i	$-0.923807\pi$
$631$	− 11.9112i	− 0.474176i	0.971488	−	0.237088i	$-0.0761930\pi$
$632$	0	0
$633$	2.65185i	0.105401i
$634$	0	0
$635$	26.3928i	1.04737i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	17.7335i	0.701526i
$640$	0	0
$641$	−24.2044	−0.956017	−0.478009	−	0.878355i	$-0.658641\pi$
$641$	−24.2044	−0.956017	−0.478009	+	0.878355i	$0.658641\pi$
$642$	0	0
$643$	23.1757i	0.913960i	0.889477	+	0.456980i	$0.151069\pi$
$643$	23.1757i	0.913960i	−0.889477	+	0.456980i	$0.848931\pi$
$644$	0	0
$645$	2.23999i	0.0881994i
$646$	0	0
$647$	−1.09512	−0.0430536	−0.0215268	−	0.999768i	$-0.506853\pi$
$647$	−1.09512	−0.0430536	−0.0215268	+	0.999768i	$0.506853\pi$
$648$	0	0
$649$	− 46.1175i	− 1.81027i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 21.7866i	− 0.852575i	−0.904588	−	0.426287i	$-0.859821\pi$
$653$	− 21.7866i	− 0.852575i	0.904588	−	0.426287i	$-0.140179\pi$
$654$	0	0
$655$	− 19.5878i	− 0.765361i
$656$	0	0
$657$	− 33.2219i	− 1.29611i
$658$	0	0
$659$	32.2969	1.25811	0.629054	−	0.777361i	$-0.283442\pi$
$659$	32.2969	1.25811	0.629054	+	0.777361i	$0.283442\pi$
$660$	0	0
$661$	−20.2933	−0.789316	−0.394658	−	0.918828i	$-0.629137\pi$
$661$	−20.2933	−0.789316	−0.394658	+	0.918828i	$0.629137\pi$
$662$	0	0
$663$	−2.77511	−0.107776
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.96535	0.192259
$668$	0	0
$669$	− 2.42628i	− 0.0938056i
$670$	0	0
$671$	61.3820	2.36962
$672$	0	0
$673$	6.13128	0.236344	0.118172	−	0.992993i	$-0.462297\pi$
$673$	6.13128	0.236344	0.118172	+	0.992993i	$0.462297\pi$
$674$	0	0
$675$	3.32198i	0.127863i
$676$	0	0
$677$	21.2014	0.814836	0.407418	−	0.913242i	$-0.366429\pi$
$677$	21.2014	0.814836	0.407418	+	0.913242i	$0.366429\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.64466	0.177984
$682$	0	0
$683$	21.7669	0.832889	0.416445	−	0.909161i	$-0.363276\pi$
$683$	21.7669	0.832889	0.416445	+	0.909161i	$0.363276\pi$
$684$	0	0
$685$	7.06011	0.269753
$686$	0	0
$687$	0.852385i	0.0325205i
$688$	0	0
$689$	− 4.96535i	− 0.189165i
$690$	0	0
$691$	− 17.2188i	− 0.655033i	−0.944845	−	0.327517i	$-0.893788\pi$
$691$	− 17.2188i	− 0.655033i	0.944845	−	0.327517i	$-0.106212\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 24.4088i	− 0.925880i
$696$	0	0
$697$	−20.8667	−0.790384
$698$	0	0
$699$	3.76552i	0.142425i
$700$	0	0
$701$	− 29.1730i	− 1.10185i	−0.834555	−	0.550924i	$-0.814275\pi$
$701$	− 29.1730i	− 1.10185i	0.834555	−	0.550924i	$-0.185725\pi$
$702$	0	0
$703$	7.48820	0.282423
$704$	0	0
$705$	2.89313i	0.108962i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 8.50639i	− 0.319464i	−0.987160	−	0.159732i	$-0.948937\pi$
$709$	− 8.50639i	− 0.319464i	0.987160	−	0.159732i	$-0.0510630\pi$
$710$	0	0
$711$	15.6022i	0.585129i
$712$	0	0
$713$	0.815674i	0.0305472i
$714$	0	0
$715$	36.1423	1.35164
$716$	0	0
$717$	−5.07314	−0.189460
$718$	0	0
$719$	14.3753	0.536109	0.268054	−	0.963404i	$-0.413619\pi$
$719$	14.3753	0.536109	0.268054	+	0.963404i	$0.413619\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.79277	−0.0666739
$724$	0	0
$725$	− 20.7846i	− 0.771921i
$726$	0	0
$727$	10.7325	0.398045	0.199023	−	0.979995i	$-0.436223\pi$
$727$	10.7325	0.398045	0.199023	+	0.979995i	$0.436223\pi$
$728$	0	0
$729$	24.6309	0.912257
$730$	0	0
$731$	− 18.4088i	− 0.680876i
$732$	0	0
$733$	14.1757	0.523591	0.261796	−	0.965123i	$-0.415685\pi$
$733$	14.1757	0.523591	0.261796	+	0.965123i	$0.415685\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27.0000	−0.994558
$738$	0	0
$739$	−11.1062	−0.408547	−0.204274	−	0.978914i	$-0.565483\pi$
$739$	−11.1062	−0.408547	−0.204274	+	0.978914i	$0.565483\pi$
$740$	0	0
$741$	4.51535	0.165875
$742$	0	0
$743$	3.47093i	0.127336i	0.997971	+	0.0636680i	$0.0202799\pi$
$743$	3.47093i	0.127336i	−0.997971	+	0.0636680i	$0.979720\pi$
$744$	0	0
$745$	− 7.15035i	− 0.261968i
$746$	0	0
$747$	− 44.5351i	− 1.62945i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.2345i	0.373461i	0.982411	+	0.186731i	$0.0597892\pi$
$751$	10.2345i	0.373461i	−0.982411	+	0.186731i	$0.940211\pi$
$752$	0	0
$753$	1.25081	0.0455821
$754$	0	0
$755$	27.9730i	1.01804i
$756$	0	0
$757$	− 24.6300i	− 0.895191i	−0.894236	−	0.447596i	$-0.852280\pi$
$757$	− 24.6300i	− 0.895191i	0.894236	−	0.447596i	$-0.147720\pi$
$758$	0	0
$759$	0.633514	0.0229951
$760$	0	0
$761$	28.0582i	1.01711i	0.861030	+	0.508555i	$0.169820\pi$
$761$	28.0582i	1.01711i	−0.861030	+	0.508555i	$0.830180\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	12.0474i	0.435575i
$766$	0	0
$767$	− 48.0712i	− 1.73575i
$768$	0	0
$769$	− 23.9496i	− 0.863646i	−0.901958	−	0.431823i	$-0.857871\pi$
$769$	− 23.9496i	− 0.863646i	0.901958	−	0.431823i	$-0.142129\pi$
$770$	0	0
$771$	−3.87023	−0.139383
$772$	0	0
$773$	31.9429	1.14891	0.574453	−	0.818537i	$-0.305215\pi$
$773$	31.9429	1.14891	0.574453	+	0.818537i	$0.305215\pi$
$774$	0	0
$775$	3.41435	0.122647
$776$	0	0
$777$	0	0
$778$	0	0
$779$	33.9520	1.21646
$780$	0	0
$781$	− 28.5251i	− 1.02071i
$782$	0	0
$783$	9.85718	0.352267
$784$	0	0
$785$	−1.39779	−0.0498892
$786$	0	0
$787$	22.9837i	0.819279i	0.912247	+	0.409640i	$0.134346\pi$
$787$	22.9837i	0.819279i	−0.912247	+	0.409640i	$0.865654\pi$
$788$	0	0
$789$	3.92686	0.139800
$790$	0	0
$791$	0	0
$792$	0	0
$793$	63.9823	2.27208
$794$	0	0
$795$	0.323952	0.0114894
$796$	0	0
$797$	33.7285	1.19472	0.597362	−	0.801972i	$-0.296215\pi$
$797$	33.7285	1.19472	0.597362	+	0.801972i	$0.296215\pi$
$798$	0	0
$799$	− 23.7766i	− 0.841155i
$800$	0	0
$801$	15.3577i	0.542636i
$802$	0	0
$803$	53.4389i	1.88582i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.0143610i	0 0.000505531i
$808$	0	0
$809$	−1.73546	−0.0610157	−0.0305078	−	0.999535i	$-0.509712\pi$
$809$	−1.73546	−0.0610157	−0.0305078	+	0.999535i	$0.509712\pi$
$810$	0	0
$811$	41.5845i	1.46023i	0.683324	+	0.730115i	$0.260534\pi$
$811$	41.5845i	1.46023i	−0.683324	+	0.730115i	$0.739466\pi$
$812$	0	0
$813$	3.96338i	0.139002i
$814$	0	0
$815$	3.39814	0.119032
$816$	0	0
$817$	29.9528i	1.04792i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 9.92546i	− 0.346401i	−0.984887	−	0.173200i	$-0.944589\pi$
$821$	− 9.92546i	− 0.346401i	0.984887	−	0.173200i	$-0.0554108\pi$
$822$	0	0
$823$	− 7.21878i	− 0.251631i	−0.992054	−	0.125815i	$-0.959845\pi$
$823$	− 7.21878i	− 0.251631i	0.992054	−	0.125815i	$-0.0401547\pi$
$824$	0	0
$825$	− 2.65185i	− 0.0923254i
$826$	0	0
$827$	5.07813	0.176584	0.0882919	−	0.996095i	$-0.471859\pi$
$827$	5.07813	0.176584	0.0882919	+	0.996095i	$0.471859\pi$
$828$	0	0
$829$	−28.0444	−0.974023	−0.487011	−	0.873396i	$-0.661913\pi$
$829$	−28.0444	−0.974023	−0.487011	+	0.873396i	$0.661913\pi$
$830$	0	0
$831$	0.601085	0.0208514
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−27.9899	−0.968632
$836$	0	0
$837$	1.61927i	0.0559702i
$838$	0	0
$839$	−36.4911	−1.25981	−0.629906	−	0.776671i	$-0.716907\pi$
$839$	−36.4911	−1.25981	−0.629906	+	0.776671i	$0.716907\pi$
$840$	0	0
$841$	−32.6734	−1.12667
$842$	0	0
$843$	0.0711671i	0.00245112i
$844$	0	0
$845$	17.7305	0.609947
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.29523	0.0444522
$850$	0	0
$851$	1.09512	0.0375402
$852$	0	0
$853$	14.4265	0.493954	0.246977	−	0.969021i	$-0.420563\pi$
$853$	14.4265	0.493954	0.246977	+	0.969021i	$0.420563\pi$
$854$	0	0
$855$	− 19.6022i	− 0.670381i
$856$	0	0
$857$	− 8.32009i	− 0.284209i	−0.989852	−	0.142104i	$-0.954613\pi$
$857$	− 8.32009i	− 0.284209i	0.989852	−	0.142104i	$-0.0453869\pi$
$858$	0	0
$859$	10.9987i	0.375270i	0.982239	+	0.187635i	$0.0600821\pi$
$859$	10.9987i	0.375270i	−0.982239	+	0.187635i	$0.939918\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 24.6323i	− 0.838492i	−0.907873	−	0.419246i	$-0.862294\pi$
$863$	− 24.6323i	− 0.838492i	0.907873	−	0.419246i	$-0.137706\pi$
$864$	0	0
$865$	15.9556	0.542506
$866$	0	0
$867$	− 2.09489i	− 0.0711462i
$868$	0	0
$869$	− 25.0968i	− 0.851351i
$870$	0	0
$871$	−28.1438	−0.953617
$872$	0	0
$873$	− 20.5983i	− 0.697147i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 30.9959i	− 1.04666i	−0.852131	−	0.523328i	$-0.824690\pi$
$877$	− 30.9959i	− 1.04666i	0.852131	−	0.523328i	$-0.175310\pi$
$878$	0	0
$879$	− 0.0287220i	0 0.000968768i
$880$	0	0
$881$	− 18.1328i	− 0.610908i	−0.952207	−	0.305454i	$-0.901192\pi$
$881$	− 18.1328i	− 0.610908i	0.952207	−	0.305454i	$-0.0988083\pi$
$882$	0	0
$883$	−22.8210	−0.767987	−0.383994	−	0.923336i	$-0.625452\pi$
$883$	−22.8210	−0.767987	−0.383994	+	0.923336i	$0.625452\pi$
$884$	0	0
$885$	3.13628	0.105425
$886$	0	0
$887$	48.1018	1.61510	0.807550	−	0.589799i	$-0.200793\pi$
$887$	48.1018	1.61510	0.807550	+	0.589799i	$0.200793\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−40.8964	−1.37008
$892$	0	0
$893$	38.6866i	1.29460i
$894$	0	0
$895$	−20.1616	−0.673927
$896$	0	0
$897$	0.660352	0.0220485
$898$	0	0
$899$	− 10.1313i	− 0.337897i
$900$	0	0
$901$	−2.66232	−0.0886949
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−36.5892	−1.21627
$906$	0	0
$907$	52.9028	1.75661	0.878304	−	0.478103i	$-0.158675\pi$
$907$	52.9028	1.75661	0.878304	+	0.478103i	$0.158675\pi$
$908$	0	0
$909$	13.1994	0.437797
$910$	0	0
$911$	− 10.0601i	− 0.333306i	−0.986016	−	0.166653i	$-0.946704\pi$
$911$	− 10.0601i	− 0.333306i	0.986016	−	0.166653i	$-0.0532960\pi$
$912$	0	0
$913$	71.6366i	2.37083i
$914$	0	0
$915$	4.17436i	0.138000i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	12.2058i	0.402631i	0.979526	+	0.201315i	$0.0645216\pi$
$919$	12.2058i	0.402631i	−0.979526	+	0.201315i	$0.935478\pi$
$920$	0	0
$921$	0.706743	0.0232880
$922$	0	0
$923$	− 29.7335i	− 0.978690i
$924$	0	0
$925$	− 4.58410i	− 0.150724i
$926$	0	0
$927$	16.7854	0.551304
$928$	0	0
$929$	− 8.08404i	− 0.265229i	−0.991168	−	0.132614i	$-0.957663\pi$
$929$	− 8.08404i	− 0.265229i	0.991168	−	0.132614i	$-0.0423372\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.54104i	0.115928i
$934$	0	0
$935$	− 19.3788i	− 0.633754i
$936$	0	0
$937$	28.1849i	0.920761i	0.887722	+	0.460380i	$0.152287\pi$
$937$	28.1849i	0.920761i	−0.887722	+	0.460380i	$0.847713\pi$
$938$	0	0
$939$	1.67133	0.0545418
$940$	0	0
$941$	41.2124	1.34349	0.671744	−	0.740784i	$-0.265546\pi$
$941$	41.2124	1.34349	0.671744	+	0.740784i	$0.265546\pi$
$942$	0	0
$943$	4.96535	0.161694
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.2018	1.01392	0.506961	−	0.861969i	$-0.330769\pi$
$947$	31.2018	1.01392	0.506961	+	0.861969i	$0.330769\pi$
$948$	0	0
$949$	55.7028i	1.80819i
$950$	0	0
$951$	−5.28930	−0.171517
$952$	0	0
$953$	−42.7465	−1.38470	−0.692348	−	0.721564i	$-0.743424\pi$
$953$	−42.7465	−1.38470	−0.692348	+	0.721564i	$0.743424\pi$
$954$	0	0
$955$	− 19.8968i	− 0.643845i
$956$	0	0
$957$	−7.86872	−0.254360
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.3357	−0.946313
$962$	0	0
$963$	−29.7578	−0.958933
$964$	0	0
$965$	0.0681404	0.00219352
$966$	0	0
$967$	− 34.9379i	− 1.12353i	−0.827298	−	0.561764i	$-0.810123\pi$
$967$	− 34.9379i	− 1.12353i	0.827298	−	0.561764i	$-0.189877\pi$
$968$	0	0
$969$	− 2.42104i	− 0.0777751i
$970$	0	0
$971$	59.3738i	1.90540i	0.303919	+	0.952698i	$0.401705\pi$
$971$	59.3738i	1.90540i	−0.303919	+	0.952698i	$0.598295\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 2.76419i	− 0.0885248i
$976$	0	0
$977$	−2.66232	−0.0851753	−0.0425877	−	0.999093i	$-0.513560\pi$
$977$	−2.66232	−0.0851753	−0.0425877	+	0.999093i	$0.513560\pi$
$978$	0	0
$979$	− 24.7034i	− 0.789525i
$980$	0	0
$981$	30.4879i	0.973404i
$982$	0	0
$983$	−21.7669	−0.694258	−0.347129	−	0.937817i	$-0.612843\pi$
$983$	−21.7669	−0.694258	−0.347129	+	0.937817i	$0.612843\pi$
$984$	0	0
$985$	41.2204i	1.31339i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.38048i	0.139291i
$990$	0	0
$991$	28.9236i	0.918787i	0.888233	+	0.459393i	$0.151933\pi$
$991$	28.9236i	0.918787i	−0.888233	+	0.459393i	$0.848067\pi$
$992$	0	0
$993$	− 1.43297i	− 0.0454740i
$994$	0	0
$995$	−29.6699	−0.940600
$996$	0	0
$997$	−6.64466	−0.210438	−0.105219	−	0.994449i	$-0.533554\pi$
$997$	−6.64466	−0.210438	−0.105219	+	0.994449i	$0.533554\pi$
$998$	0	0
$999$	2.17403	0.0687831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.e.d.1567.5		12
4.3	odd	2	inner	3136.2.e.d.1567.8		12
7.4	even	3		448.2.q.c.159.3	yes	12
7.5	odd	6		448.2.q.b.31.3	✓	12
7.6	odd	2		3136.2.e.e.1567.7		12
8.3	odd	2		3136.2.e.e.1567.5		12
8.5	even	2		3136.2.e.e.1567.8		12
28.11	odd	6		448.2.q.c.159.4	yes	12
28.19	even	6		448.2.q.b.31.4	yes	12
28.27	even	2		3136.2.e.e.1567.6		12
56.5	odd	6		448.2.q.c.31.4	yes	12
56.11	odd	6		448.2.q.b.159.3	yes	12
56.13	odd	2	inner	3136.2.e.d.1567.6		12
56.19	even	6		448.2.q.c.31.3	yes	12
56.27	even	2	inner	3136.2.e.d.1567.7		12
56.53	even	6		448.2.q.b.159.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.q.b.31.3	✓	12	7.5	odd	6
448.2.q.b.31.4	yes	12	28.19	even	6
448.2.q.b.159.3	yes	12	56.11	odd	6
448.2.q.b.159.4	yes	12	56.53	even	6
448.2.q.c.31.3	yes	12	56.19	even	6
448.2.q.c.31.4	yes	12	56.5	odd	6
448.2.q.c.159.3	yes	12	7.4	even	3
448.2.q.c.159.4	yes	12	28.11	odd	6
3136.2.e.d.1567.5		12	1.1	even	1	trivial
3136.2.e.d.1567.6		12	56.13	odd	2	inner
3136.2.e.d.1567.7		12	56.27	even	2	inner
3136.2.e.d.1567.8		12	4.3	odd	2	inner
3136.2.e.e.1567.5		12	8.3	odd	2
3136.2.e.e.1567.6		12	28.27	even	2
3136.2.e.e.1567.7		12	7.6	odd	2
3136.2.e.e.1567.8		12	8.5	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:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.210756i q^{3} +1.53407 q^{5} +2.95558 q^{9} +O(q^{10})\)	\(q-0.210756i q^{3} +1.53407 q^{5} +2.95558 q^{9} -4.75418 q^{11} -4.95558 q^{13} -0.323314i q^{15} +2.65709i q^{17} -4.32331i q^{19} -0.632268i q^{23} -2.64663 q^{25} -1.25517i q^{27} +7.85324i q^{29} -1.29008 q^{31} +1.00197i q^{33} +1.73205i q^{37} +1.04442i q^{39} +7.85324i q^{41} -6.92820 q^{43} +4.53407 q^{45} -8.94836 q^{47} +0.559997 q^{51} +1.00197i q^{53} -7.29324 q^{55} -0.911164 q^{57} +9.70041i q^{59} -12.9112 q^{61} -7.60221 q^{65} +5.67921 q^{67} -0.133254 q^{69} +6.00000i q^{71} -11.2404i q^{73} +0.557793i q^{75} +5.27890i q^{79} +8.60221 q^{81} -15.0681i q^{83} +4.07616i q^{85} +1.65512 q^{87} +5.19615i q^{89} +0.271891i q^{93} -6.63227i q^{95} -6.96929i q^{97} -14.0514 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5} - 8 q^{9}+O(q^{10}) \) 12 * q - 12 * q^5 - 8 * q^9	\( 12 q - 12 q^{5} - 8 q^{9} - 16 q^{13} + 16 q^{25} + 24 q^{45} + 76 q^{57} - 68 q^{61} - 132 q^{69} + 12 q^{81}+O(q^{100}) \) 12 * q - 12 * q^5 - 8 * q^9 - 16 * q^13 + 16 * q^25 + 24 * q^45 + 76 * q^57 - 68 * q^61 - 132 * q^69 + 12 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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