LMFDB - Embedded newform 5760.2.b.d.4031.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5760,2,Mod(4031,5760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5760, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5760.4031");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5760 = 2^{7} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5760.b (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:	$45.9938315643$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4031.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	5760.4031
Dual form			5760.2.b.d.4031.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -4.24264i q^{7} +4.24264i q^{11} -4.24264i q^{13} +5.65685i q^{17} +6.00000 q^{19} +1.00000 q^{25} +6.00000 q^{29} +4.24264i q^{35} -4.24264i q^{37} -1.41421i q^{41} -6.00000 q^{47} -11.0000 q^{49} -4.24264i q^{55} +12.7279i q^{59} +8.48528i q^{61} +4.24264i q^{65} +12.0000 q^{67} -2.00000 q^{73} +18.0000 q^{77} +16.9706i q^{79} -8.48528i q^{83} -5.65685i q^{85} -7.07107i q^{89} -18.0000 q^{91} -6.00000 q^{95} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 12 q^{19} + 2 q^{25} + 12 q^{29} - 12 q^{47} - 22 q^{49} + 24 q^{67} - 4 q^{73} + 36 q^{77} - 36 q^{91} - 12 q^{95} + 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 12 * q^19 + 2 * q^25 + 12 * q^29 - 12 * q^47 - 22 * q^49 + 24 * q^67 - 4 * q^73 + 36 * q^77 - 36 * q^91 - 12 * q^95 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5760\mathbb{Z}\right)^\times$.

$n$	$641$	$901$	$2431$	$3457$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	− 4.24264i	− 1.60357i	−0.597614	−	0.801784i	$-0.703885\pi$
$7$	− 4.24264i	− 1.60357i	0.597614	−	0.801784i	$-0.296115\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.24264i	1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$	4.24264i	1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$	0	0
$13$	− 4.24264i	− 1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$	− 4.24264i	− 1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.65685i	1.37199i	0.727607	+	0.685994i	$0.240633\pi$
$17$	5.65685i	1.37199i	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.24264i	0.717137i
$36$	0	0
$37$	− 4.24264i	− 0.697486i	−0.937218	−	0.348743i	$-0.886609\pi$
$37$	− 4.24264i	− 0.697486i	0.937218	−	0.348743i	$-0.113391\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.41421i	− 0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$	− 1.41421i	− 0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−11.0000	−1.57143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	− 4.24264i	− 0.572078i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.7279i	1.65703i	0.559964	+	0.828517i	$0.310815\pi$
$59$	12.7279i	1.65703i	−0.559964	+	0.828517i	$0.689185\pi$
$60$	0	0
$61$	8.48528i	1.08643i	0.839594	+	0.543214i	$0.182793\pi$
$61$	8.48528i	1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.24264i	0.526235i
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.0000	2.05129
$78$	0	0
$79$	16.9706i	1.90934i	0.297670	+	0.954669i	$0.403790\pi$
$79$	16.9706i	1.90934i	−0.297670	+	0.954669i	$0.596210\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.48528i	− 0.931381i	−0.884948	−	0.465690i	$-0.845806\pi$
$83$	− 8.48528i	− 0.931381i	0.884948	−	0.465690i	$-0.154194\pi$
$84$	0	0
$85$	− 5.65685i	− 0.613572i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 7.07107i	− 0.749532i	−0.927119	−	0.374766i	$-0.877723\pi$
$89$	− 7.07107i	− 0.749532i	0.927119	−	0.374766i	$-0.122277\pi$
$90$	0	0
$91$	−18.0000	−1.88691
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	12.7279i	1.25412i	0.778971	+	0.627060i	$0.215742\pi$
$103$	12.7279i	1.25412i	−0.778971	+	0.627060i	$0.784258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.48528i	− 0.820303i	−0.912017	−	0.410152i	$-0.865476\pi$
$107$	− 8.48528i	− 0.820303i	0.912017	−	0.410152i	$-0.134524\pi$
$108$	0	0
$109$	− 8.48528i	− 0.812743i	−0.913708	−	0.406371i	$-0.866794\pi$
$109$	− 8.48528i	− 0.812743i	0.913708	−	0.406371i	$-0.133206\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 14.1421i	− 1.33038i	−0.746674	−	0.665190i	$-0.768350\pi$
$113$	− 14.1421i	− 1.33038i	0.746674	−	0.665190i	$-0.231650\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.0000	2.20008
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	− 21.2132i	− 1.88237i	−0.337895	−	0.941184i	$-0.609715\pi$
$127$	− 21.2132i	− 1.88237i	0.337895	−	0.941184i	$-0.390285\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 12.7279i	− 1.11204i	−0.831168	−	0.556022i	$-0.812327\pi$
$131$	− 12.7279i	− 1.11204i	0.831168	−	0.556022i	$-0.187673\pi$
$132$	0	0
$133$	− 25.4558i	− 2.20730i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.82843i	0.241649i	0.992674	+	0.120824i	$0.0385538\pi$
$137$	2.82843i	0.241649i	−0.992674	+	0.120824i	$0.961446\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	18.0000	1.50524
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	8.48528i	0.690522i	0.938507	+	0.345261i	$0.112210\pi$
$151$	8.48528i	0.690522i	−0.938507	+	0.345261i	$0.887790\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.24264i	0.338600i	0.985565	+	0.169300i	$0.0541506\pi$
$157$	4.24264i	0.338600i	−0.985565	+	0.169300i	$0.945849\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	− 4.24264i	− 0.320713i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4.24264i	− 0.317110i	−0.987350	−	0.158555i	$-0.949317\pi$
$179$	− 4.24264i	− 0.317110i	0.987350	−	0.158555i	$-0.0506835\pi$
$180$	0	0
$181$	− 25.4558i	− 1.89212i	−0.323994	−	0.946059i	$-0.605026\pi$
$181$	− 25.4558i	− 1.89212i	0.323994	−	0.946059i	$-0.394974\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.24264i	0.311925i
$186$	0	0
$187$	−24.0000	−1.75505
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	8.48528i	0.601506i	0.953702	+	0.300753i	$0.0972379\pi$
$199$	8.48528i	0.601506i	−0.953702	+	0.300753i	$0.902762\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 25.4558i	− 1.78665i
$204$	0	0
$205$	1.41421i	0.0987730i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	25.4558i	1.76082i
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	− 4.24264i	− 0.284108i	−0.989859	−	0.142054i	$-0.954629\pi$
$223$	− 4.24264i	− 0.284108i	0.989859	−	0.142054i	$-0.0453707\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 16.9706i	− 1.12145i	−0.828003	−	0.560723i	$-0.810523\pi$
$229$	− 16.9706i	− 1.12145i	0.828003	−	0.560723i	$-0.189477\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.65685i	0.370593i	0.982683	+	0.185296i	$0.0593245\pi$
$233$	5.65685i	0.370593i	−0.982683	+	0.185296i	$0.940675\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	11.0000	0.702764
$246$	0	0
$247$	− 25.4558i	− 1.61972i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4.24264i	− 0.267793i	−0.990995	−	0.133897i	$-0.957251\pi$
$251$	− 4.24264i	− 0.267793i	0.990995	−	0.133897i	$-0.0427490\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.1421i	− 0.882162i	−0.897467	−	0.441081i	$-0.854595\pi$
$257$	− 14.1421i	− 0.882162i	0.897467	−	0.441081i	$-0.145405\pi$
$258$	0	0
$259$	−18.0000	−1.11847
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	− 8.48528i	− 0.515444i	−0.966219	−	0.257722i	$-0.917028\pi$
$271$	− 8.48528i	− 0.515444i	0.966219	−	0.257722i	$-0.0829719\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.24264i	0.255841i
$276$	0	0
$277$	− 12.7279i	− 0.764747i	−0.924008	−	0.382373i	$-0.875107\pi$
$277$	− 12.7279i	− 0.764747i	0.924008	−	0.382373i	$-0.124893\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 15.5563i	− 0.928014i	−0.885832	−	0.464007i	$-0.846411\pi$
$281$	− 15.5563i	− 0.928014i	0.885832	−	0.464007i	$-0.153589\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	− 12.7279i	− 0.741048i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.48528i	− 0.485866i
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	25.4558i	1.42525i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	33.9411i	1.88853i
$324$	0	0
$325$	− 4.24264i	− 0.235339i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	25.4558i	1.40343i
$330$	0	0
$331$	30.0000	1.64895	0.824475	−	0.565899i	$-0.191471\pi$
$331$	30.0000	1.64895	0.824475	+	0.565899i	$0.191471\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	16.9706i	0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.9706i	0.911028i	0.890229	+	0.455514i	$0.150544\pi$
$347$	16.9706i	0.911028i	−0.890229	+	0.455514i	$0.849456\pi$
$348$	0	0
$349$	25.4558i	1.36262i	0.731995	+	0.681310i	$0.238589\pi$
$349$	25.4558i	1.36262i	−0.731995	+	0.681310i	$0.761411\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 22.6274i	− 1.20434i	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	− 22.6274i	− 1.20434i	0.798369	−	0.602168i	$-0.205696\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	− 4.24264i	− 0.221464i	−0.993850	−	0.110732i	$-0.964680\pi$
$367$	− 4.24264i	− 0.221464i	0.993850	−	0.110732i	$-0.0353195\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.7279i	0.659027i	0.944151	+	0.329513i	$0.106885\pi$
$373$	12.7279i	0.659027i	−0.944151	+	0.329513i	$0.893115\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 25.4558i	− 1.31104i
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	0	0
$385$	−18.0000	−0.917365
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 16.9706i	− 0.853882i
$396$	0	0
$397$	− 29.6985i	− 1.49052i	−0.666772	−	0.745262i	$-0.732324\pi$
$397$	− 29.6985i	− 1.49052i	0.666772	−	0.745262i	$-0.267676\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.41421i	0.0706225i	0.999376	+	0.0353112i	$0.0112422\pi$
$401$	1.41421i	0.0706225i	−0.999376	+	0.0353112i	$0.988758\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	18.0000	0.892227
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	54.0000	2.65717
$414$	0	0
$415$	8.48528i	0.416526i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.2132i	1.03633i	0.855280	+	0.518166i	$0.173385\pi$
$419$	21.2132i	1.03633i	−0.855280	+	0.518166i	$0.826615\pi$
$420$	0	0
$421$	− 33.9411i	− 1.65419i	−0.562063	−	0.827095i	$-0.689992\pi$
$421$	− 33.9411i	− 1.65419i	0.562063	−	0.827095i	$-0.310008\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.65685i	0.274398i
$426$	0	0
$427$	36.0000	1.74216
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 25.4558i	− 1.21494i	−0.794342	−	0.607471i	$-0.792184\pi$
$439$	− 25.4558i	− 1.21494i	0.794342	−	0.607471i	$-0.207816\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	33.9411i	1.61259i	0.591513	+	0.806296i	$0.298531\pi$
$443$	33.9411i	1.61259i	−0.591513	+	0.806296i	$0.701469\pi$
$444$	0	0
$445$	7.07107i	0.335201i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.3848i	0.867631i	0.901002	+	0.433816i	$0.142833\pi$
$449$	18.3848i	0.867631i	−0.901002	+	0.433816i	$0.857167\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	0	0
$454$	0	0
$455$	18.0000	0.843853
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 12.7279i	− 0.591517i	−0.955263	−	0.295758i	$-0.904428\pi$
$463$	− 12.7279i	− 0.591517i	0.955263	−	0.295758i	$-0.0955723\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.4558i	1.17796i	0.808149	+	0.588978i	$0.200470\pi$
$467$	25.4558i	1.17796i	−0.808149	+	0.588978i	$0.799530\pi$
$468$	0	0
$469$	− 50.9117i	− 2.35088i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−18.0000	−0.820729
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	29.6985i	1.34577i	0.739749	+	0.672883i	$0.234944\pi$
$487$	29.6985i	1.34577i	−0.739749	+	0.672883i	$0.765056\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.7279i	0.574403i	0.957870	+	0.287202i	$0.0927249\pi$
$491$	12.7279i	0.574403i	−0.957870	+	0.287202i	$0.907275\pi$
$492$	0	0
$493$	33.9411i	1.52863i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	18.0000	0.805791	0.402895	−	0.915246i	$-0.368004\pi$
$499$	18.0000	0.805791	0.402895	+	0.915246i	$0.368004\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	8.48528i	0.375367i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 12.7279i	− 0.560859i
$516$	0	0
$517$	− 25.4558i	− 1.11955i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 18.3848i	− 0.805452i	−0.915321	−	0.402726i	$-0.868063\pi$
$521$	− 18.3848i	− 0.805452i	0.915321	−	0.402726i	$-0.131937\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.00000	−0.259889
$534$	0	0
$535$	8.48528i	0.366851i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 46.6690i	− 2.01018i
$540$	0	0
$541$	8.48528i	0.364811i	0.983223	+	0.182405i	$0.0583883\pi$
$541$	8.48528i	0.364811i	−0.983223	+	0.182405i	$0.941612\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.48528i	0.363470i
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	72.0000	3.06175
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	14.1421i	0.594964i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	41.0122i	1.71932i	0.510866	+	0.859660i	$0.329325\pi$
$569$	41.0122i	1.71932i	−0.510866	+	0.859660i	$0.670675\pi$
$570$	0	0
$571$	−18.0000	−0.753277	−0.376638	−	0.926360i	$-0.622920\pi$
$571$	−18.0000	−0.753277	−0.376638	+	0.926360i	$0.622920\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−36.0000	−1.49353
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.48528i	0.350225i	0.984548	+	0.175113i	$0.0560289\pi$
$587$	8.48528i	0.350225i	−0.984548	+	0.175113i	$0.943971\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 31.1127i	− 1.27765i	−0.769354	−	0.638823i	$-0.779422\pi$
$593$	− 31.1127i	− 1.27765i	0.769354	−	0.638823i	$-0.220578\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	29.6985i	1.20542i	0.797959	+	0.602712i	$0.205913\pi$
$607$	29.6985i	1.20542i	−0.797959	+	0.602712i	$0.794087\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	25.4558i	1.02983i
$612$	0	0
$613$	− 38.1838i	− 1.54223i	−0.636697	−	0.771114i	$-0.719700\pi$
$613$	− 38.1838i	− 1.54223i	0.636697	−	0.771114i	$-0.280300\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 11.3137i	− 0.455473i	−0.973723	−	0.227736i	$-0.926868\pi$
$617$	− 11.3137i	− 0.455473i	0.973723	−	0.227736i	$-0.0731324\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−30.0000	−1.20192
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24.0000	0.956943
$630$	0	0
$631$	16.9706i	0.675587i	0.941220	+	0.337794i	$0.109681\pi$
$631$	16.9706i	0.675587i	−0.941220	+	0.337794i	$0.890319\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	21.2132i	0.841820i
$636$	0	0
$637$	46.6690i	1.84909i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	35.3553i	1.39645i	0.715877	+	0.698226i	$0.246027\pi$
$641$	35.3553i	1.39645i	−0.715877	+	0.698226i	$0.753973\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	−54.0000	−2.11969
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	12.7279i	0.497321i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 46.6690i	− 1.81797i	−0.416831	−	0.908984i	$-0.636859\pi$
$659$	− 46.6690i	− 1.81797i	0.416831	−	0.908984i	$-0.363141\pi$
$660$	0	0
$661$	8.48528i	0.330039i	0.986290	+	0.165020i	$0.0527687\pi$
$661$	8.48528i	0.330039i	−0.986290	+	0.165020i	$0.947231\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	25.4558i	0.987135i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−36.0000	−1.38976
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	− 42.4264i	− 1.62818i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 8.48528i	− 0.324680i	−0.986735	−	0.162340i	$-0.948096\pi$
$683$	− 8.48528i	− 0.324680i	0.986735	−	0.162340i	$-0.0519042\pi$
$684$	0	0
$685$	− 2.82843i	− 0.108069i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	6.00000	0.228251	0.114125	−	0.993466i	$-0.463593\pi$
$691$	6.00000	0.228251	0.114125	+	0.993466i	$0.463593\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.0000	−0.455186
$696$	0	0
$697$	8.00000	0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	− 25.4558i	− 0.960085i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 76.3675i	− 2.87210i
$708$	0	0
$709$	33.9411i	1.27469i	0.770580	+	0.637343i	$0.219966\pi$
$709$	33.9411i	1.27469i	−0.770580	+	0.637343i	$0.780034\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−18.0000	−0.673162
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	54.0000	2.01107
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	21.2132i	0.786754i	0.919377	+	0.393377i	$0.128693\pi$
$727$	21.2132i	0.786754i	−0.919377	+	0.393377i	$0.871307\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	38.1838i	1.41035i	0.709034	+	0.705175i	$0.249131\pi$
$733$	38.1838i	1.41035i	−0.709034	+	0.705175i	$0.750869\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	50.9117i	1.87536i
$738$	0	0
$739$	−18.0000	−0.662141	−0.331070	−	0.943606i	$-0.607410\pi$
$739$	−18.0000	−0.662141	−0.331070	+	0.943606i	$0.607410\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−36.0000	−1.31541
$750$	0	0
$751$	− 33.9411i	− 1.23853i	−0.785182	−	0.619265i	$-0.787431\pi$
$751$	− 33.9411i	− 1.23853i	0.785182	−	0.619265i	$-0.212569\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 8.48528i	− 0.308811i
$756$	0	0
$757$	− 12.7279i	− 0.462604i	−0.972882	−	0.231302i	$-0.925701\pi$
$757$	− 12.7279i	− 0.462604i	0.972882	−	0.231302i	$-0.0742986\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 26.8701i	− 0.974039i	−0.873391	−	0.487019i	$-0.838084\pi$
$761$	− 26.8701i	− 0.974039i	0.873391	−	0.487019i	$-0.161916\pi$
$762$	0	0
$763$	−36.0000	−1.30329
$764$	0	0
$765$	0	0
$766$	0	0
$767$	54.0000	1.94983
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 8.48528i	− 0.304017i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 4.24264i	− 0.151426i
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−60.0000	−2.13335
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	− 33.9411i	− 1.20075i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 8.48528i	− 0.299439i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.5269i	1.14359i	0.820398	+	0.571793i	$0.193752\pi$
$809$	32.5269i	1.14359i	−0.820398	+	0.571793i	$0.806248\pi$
$810$	0	0
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	24.0000	0.840683
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	− 21.2132i	− 0.739446i	−0.929142	−	0.369723i	$-0.879453\pi$
$823$	− 21.2132i	− 0.739446i	0.929142	−	0.369723i	$-0.120547\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 16.9706i	− 0.590124i	−0.955478	−	0.295062i	$-0.904660\pi$
$827$	− 16.9706i	− 0.590124i	0.955478	−	0.295062i	$-0.0953404\pi$
$828$	0	0
$829$	25.4558i	0.884118i	0.896986	+	0.442059i	$0.145752\pi$
$829$	25.4558i	0.884118i	−0.896986	+	0.442059i	$0.854248\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 62.2254i	− 2.15598i
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.00000	0.172005
$846$	0	0
$847$	29.6985i	1.02045i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	12.7279i	0.435796i	0.975972	+	0.217898i	$0.0699200\pi$
$853$	12.7279i	0.435796i	−0.975972	+	0.217898i	$0.930080\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.2843i	0.966172i	0.875573	+	0.483086i	$0.160484\pi$
$857$	28.2843i	0.966172i	−0.875573	+	0.483086i	$0.839516\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−72.0000	−2.44243
$870$	0	0
$871$	− 50.9117i	− 1.72508i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.24264i	0.143427i
$876$	0	0
$877$	46.6690i	1.57590i	0.615738	+	0.787951i	$0.288858\pi$
$877$	46.6690i	1.57590i	−0.615738	+	0.787951i	$0.711142\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 43.8406i	− 1.47703i	−0.674238	−	0.738514i	$-0.735528\pi$
$881$	− 43.8406i	− 1.47703i	0.674238	−	0.738514i	$-0.264472\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	−90.0000	−3.01850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−36.0000	−1.20469
$894$	0	0
$895$	4.24264i	0.141816i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	25.4558i	0.846181i
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	60.0000	1.98789	0.993944	−	0.109885i	$-0.0350482\pi$
$911$	60.0000	1.98789	0.993944	+	0.109885i	$0.0350482\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−54.0000	−1.78324
$918$	0	0
$919$	− 16.9706i	− 0.559807i	−0.960028	−	0.279904i	$-0.909697\pi$
$919$	− 16.9706i	− 0.559807i	0.960028	−	0.279904i	$-0.0903025\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 4.24264i	− 0.139497i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 15.5563i	− 0.510387i	−0.966890	−	0.255194i	$-0.917861\pi$
$929$	− 15.5563i	− 0.510387i	0.966890	−	0.255194i	$-0.0821392\pi$
$930$	0	0
$931$	−66.0000	−2.16306
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	8.48528i	0.275444i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 31.1127i	− 1.00784i	−0.863751	−	0.503920i	$-0.831891\pi$
$953$	− 31.1127i	− 1.00784i	0.863751	−	0.503920i	$-0.168109\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−22.0000	−0.708205
$966$	0	0
$967$	38.1838i	1.22791i	0.789342	+	0.613954i	$0.210422\pi$
$967$	38.1838i	1.22791i	−0.789342	+	0.613954i	$0.789578\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.24264i	0.136153i	0.997680	+	0.0680764i	$0.0216862\pi$
$971$	4.24264i	0.136153i	−0.997680	+	0.0680764i	$0.978314\pi$
$972$	0	0
$973$	− 50.9117i	− 1.63215i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 11.3137i	− 0.361958i	−0.983487	−	0.180979i	$-0.942073\pi$
$977$	− 11.3137i	− 0.361958i	0.983487	−	0.180979i	$-0.0579265\pi$
$978$	0	0
$979$	30.0000	0.958804
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−54.0000	−1.72233	−0.861166	−	0.508323i	$-0.830265\pi$
$983$	−54.0000	−1.72233	−0.861166	+	0.508323i	$0.830265\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	25.4558i	0.808632i	0.914619	+	0.404316i	$0.132490\pi$
$991$	25.4558i	0.808632i	−0.914619	+	0.404316i	$0.867510\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 8.48528i	− 0.269002i
$996$	0	0
$997$	4.24264i	0.134366i	0.997741	+	0.0671829i	$0.0214011\pi$
$997$	4.24264i	0.134366i	−0.997741	+	0.0671829i	$0.978599\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5760.2.b.d.4031.1	yes	2
3.2	odd	2		5760.2.b.h.4031.1	yes	2
4.3	odd	2		5760.2.b.a.4031.2	yes	2
8.3	odd	2		5760.2.b.h.4031.2	yes	2
8.5	even	2		5760.2.b.e.4031.1	yes	2
12.11	even	2		5760.2.b.e.4031.2	yes	2
24.5	odd	2		5760.2.b.a.4031.1	✓	2
24.11	even	2	inner	5760.2.b.d.4031.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5760.2.b.a.4031.1	✓	2	24.5	odd	2
5760.2.b.a.4031.2	yes	2	4.3	odd	2
5760.2.b.d.4031.1	yes	2	1.1	even	1	trivial
5760.2.b.d.4031.2	yes	2	24.11	even	2	inner
5760.2.b.e.4031.1	yes	2	8.5	even	2
5760.2.b.e.4031.2	yes	2	12.11	even	2
5760.2.b.h.4031.1	yes	2	3.2	odd	2
5760.2.b.h.4031.2	yes	2	8.3	odd	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:	\( N \)	\(=\)	\( 5760 = 2^{7} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5760.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -4.24264i q^{7} +4.24264i q^{11} -4.24264i q^{13} +5.65685i q^{17} +6.00000 q^{19} +1.00000 q^{25} +6.00000 q^{29} +4.24264i q^{35} -4.24264i q^{37} -1.41421i q^{41} -6.00000 q^{47} -11.0000 q^{49} -4.24264i q^{55} +12.7279i q^{59} +8.48528i q^{61} +4.24264i q^{65} +12.0000 q^{67} -2.00000 q^{73} +18.0000 q^{77} +16.9706i q^{79} -8.48528i q^{83} -5.65685i q^{85} -7.07107i q^{89} -18.0000 q^{91} -6.00000 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 12 q^{19} + 2 q^{25} + 12 q^{29} - 12 q^{47} - 22 q^{49} + 24 q^{67} - 4 q^{73} + 36 q^{77} - 36 q^{91} - 12 q^{95} + 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 12 * q^19 + 2 * q^25 + 12 * q^29 - 12 * q^47 - 22 * q^49 + 24 * q^67 - 4 * q^73 + 36 * q^77 - 36 * q^91 - 12 * q^95 + 20 * q^97

Introduction

L-functions

Modular forms

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