LMFDB - Embedded newform 8085.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8085,2,Mod(1,8085)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8085, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8085.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8085.a (trivial)

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:	yes
Analytic conductor:	$64.5590500342$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8085.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{9} +1.00000 q^{11} +2.00000 q^{12} +3.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} +6.00000 q^{19} +2.00000 q^{20} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} -1.00000 q^{33} -2.00000 q^{36} +4.00000 q^{37} -3.00000 q^{39} +1.00000 q^{41} +1.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +9.00000 q^{47} -4.00000 q^{48} -6.00000 q^{52} +3.00000 q^{53} -1.00000 q^{55} -6.00000 q^{57} +4.00000 q^{59} -2.00000 q^{60} -10.0000 q^{61} -8.00000 q^{64} -3.00000 q^{65} -6.00000 q^{67} +1.00000 q^{69} -4.00000 q^{71} -6.00000 q^{73} -1.00000 q^{75} -12.0000 q^{76} -16.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +6.00000 q^{83} -5.00000 q^{87} +2.00000 q^{92} -6.00000 q^{95} +2.00000 q^{97} +1.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.00000	0.577350
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−2.00000	−0.301511
$45$	−1.00000	−0.149071
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−4.00000	−0.577350
$49$	0	0
$50$	0	0
$51$	0	0
$52$	−6.00000	−0.832050
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−2.00000	−0.258199
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−3.00000	−0.372104
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	−12.0000	−1.37649
$77$	0	0
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	2.00000	0.208514
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−2.00000	−0.200000
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	2.00000	0.192450
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	−10.0000	−0.928477
$117$	3.00000	0.277350
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	3.00000	0.250873
$144$	4.00000	0.333333
$145$	−5.00000	−0.415227
$146$	0	0
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	6.00000	0.480384
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−2.00000	−0.156174
$165$	1.00000	0.0778499
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	6.00000	0.458831
$172$	−2.00000	−0.152499
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	2.00000	0.149071
$181$	13.0000	0.966282	0.483141	−	0.875542i	$-0.339496\pi$
$181$	13.0000	0.966282	0.483141	+	0.875542i	$0.339496\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	−18.0000	−1.31278
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	8.00000	0.577350
$193$	15.0000	1.07972	0.539862	−	0.841754i	$-0.318476\pi$
$193$	15.0000	1.07972	0.539862	+	0.841754i	$0.318476\pi$
$194$	0	0
$195$	3.00000	0.214834
$196$	0	0
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.00000	−0.0698430
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	12.0000	0.832050
$209$	6.00000	0.415029
$210$	0	0
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	−6.00000	−0.412082
$213$	4.00000	0.274075
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.00000	0.405442
$220$	2.00000	0.134840
$221$	0	0
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	12.0000	0.794719
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	−8.00000	−0.520756
$237$	16.0000	1.03931
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	4.00000	0.258199
$241$	16.0000	1.03065	0.515325	−	0.856995i	$-0.327671\pi$
$241$	16.0000	1.03065	0.515325	+	0.856995i	$0.327671\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	20.0000	1.28037
$245$	0	0
$246$	0	0
$247$	18.0000	1.14531
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	0	0
$260$	6.00000	0.372104
$261$	5.00000	0.309492
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	0	0
$267$	0	0
$268$	12.0000	0.733017
$269$	32.0000	1.95107	0.975537	−	0.219834i	$-0.0705517\pi$
$269$	32.0000	1.95107	0.975537	+	0.219834i	$0.0705517\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	−2.00000	−0.120386
$277$	−25.0000	−1.50210	−0.751052	−	0.660243i	$-0.770453\pi$
$277$	−25.0000	−1.50210	−0.751052	+	0.660243i	$0.770453\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	29.0000	1.72999	0.864997	−	0.501776i	$-0.167320\pi$
$281$	29.0000	1.72999	0.864997	+	0.501776i	$0.167320\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	8.00000	0.474713
$285$	6.00000	0.355409
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−2.00000	−0.117242
$292$	12.0000	0.702247
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−3.00000	−0.173494
$300$	2.00000	0.115470
$301$	0	0
$302$	0	0
$303$	−7.00000	−0.402139
$304$	24.0000	1.37649
$305$	10.0000	0.572598
$306$	0	0
$307$	11.0000	0.627803	0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000	0.627803	0.313902	+	0.949456i	$0.398364\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	32.0000	1.80014
$317$	−7.00000	−0.393159	−0.196580	−	0.980488i	$-0.562983\pi$
$317$	−7.00000	−0.393159	−0.196580	+	0.980488i	$0.562983\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	8.00000	0.447214
$321$	6.00000	0.334887
$322$	0	0
$323$	0	0
$324$	−2.00000	−0.111111
$325$	3.00000	0.166410
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	−12.0000	−0.658586
$333$	4.00000	0.219199
$334$	0	0
$335$	6.00000	0.327815
$336$	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
$339$	−14.0000	−0.760376
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	10.0000	0.536056
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	0	0
$353$	35.0000	1.86286	0.931431	−	0.363918i	$-0.118561\pi$
$353$	35.0000	1.86286	0.931431	+	0.363918i	$0.118561\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	−4.00000	−0.208514
$369$	1.00000	0.0520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	15.0000	0.772539
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	12.0000	0.615587
$381$	−7.00000	−0.358621
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.00000	0.0508329
$388$	−4.00000	−0.203069
$389$	−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
$393$	−13.0000	−0.655763
$394$	0	0
$395$	16.0000	0.805047
$396$	−2.00000	−0.100504
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	20.0000	0.998752	0.499376	−	0.866385i	$-0.333563\pi$
$401$	20.0000	0.998752	0.499376	+	0.866385i	$0.333563\pi$
$402$	0	0
$403$	0	0
$404$	−14.0000	−0.696526
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	17.0000	0.838548
$412$	−32.0000	−1.57653
$413$	0	0
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	14.0000	0.685583
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	9.00000	0.437595
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	−3.00000	−0.144841
$430$	0	0
$431$	−39.0000	−1.87856	−0.939282	−	0.343146i	$-0.888507\pi$
$431$	−39.0000	−1.87856	−0.939282	+	0.343146i	$0.888507\pi$
$432$	−4.00000	−0.192450
$433$	40.0000	1.92228	0.961139	−	0.276066i	$-0.0890309\pi$
$433$	40.0000	1.92228	0.961139	+	0.276066i	$0.0890309\pi$
$434$	0	0
$435$	5.00000	0.239732
$436$	−8.00000	−0.383131
$437$	−6.00000	−0.287019
$438$	0	0
$439$	34.0000	1.62273	0.811366	−	0.584539i	$-0.198725\pi$
$439$	34.0000	1.62273	0.811366	+	0.584539i	$0.198725\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	0	0
$447$	15.0000	0.709476
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	1.00000	0.0470882
$452$	−28.0000	−1.31701
$453$	4.00000	0.187936
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	0	0
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	20.0000	0.928477
$465$	0	0
$466$	0	0
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	0	0
$471$	16.0000	0.737241
$472$	0	0
$473$	1.00000	0.0459800
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	2.00000	0.0901670
$493$	0	0
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	0	0
$498$	0	0
$499$	21.0000	0.940089	0.470045	−	0.882643i	$-0.344238\pi$
$499$	21.0000	0.940089	0.470045	+	0.882643i	$0.344238\pi$
$500$	2.00000	0.0894427
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−22.0000	−0.980932	−0.490466	−	0.871460i	$-0.663173\pi$
$503$	−22.0000	−0.980932	−0.490466	+	0.871460i	$0.663173\pi$
$504$	0	0
$505$	−7.00000	−0.311496
$506$	0	0
$507$	4.00000	0.177646
$508$	−14.0000	−0.621150
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−6.00000	−0.264906
$514$	0	0
$515$	−16.0000	−0.705044
$516$	2.00000	0.0880451
$517$	9.00000	0.395820
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	7.00000	0.306089	0.153044	−	0.988219i	$-0.451092\pi$
$523$	7.00000	0.306089	0.153044	+	0.988219i	$0.451092\pi$
$524$	−26.0000	−1.13582
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−4.00000	−0.174078
$529$	−22.0000	−0.956522
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	3.00000	0.129944
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	18.0000	0.776757
$538$	0	0
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	0	0
$543$	−13.0000	−0.557883
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	7.00000	0.299298	0.149649	−	0.988739i	$-0.452186\pi$
$547$	7.00000	0.299298	0.149649	+	0.988739i	$0.452186\pi$
$548$	34.0000	1.45241
$549$	−10.0000	−0.426790
$550$	0	0
$551$	30.0000	1.27804
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.00000	0.169791
$556$	28.0000	1.18746
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	3.00000	0.126886
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	18.0000	0.757937
$565$	−14.0000	−0.588984
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.0000	0.544988	0.272494	−	0.962157i	$-0.412151\pi$
$569$	13.0000	0.544988	0.272494	+	0.962157i	$0.412151\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	−6.00000	−0.250873
$573$	4.00000	0.167102
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	−8.00000	−0.333333
$577$	−36.0000	−1.49870	−0.749350	−	0.662174i	$-0.769634\pi$
$577$	−36.0000	−1.49870	−0.749350	+	0.662174i	$0.769634\pi$
$578$	0	0
$579$	−15.0000	−0.623379
$580$	10.0000	0.415227
$581$	0	0
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	−3.00000	−0.124035
$586$	0	0
$587$	−33.0000	−1.36206	−0.681028	−	0.732257i	$-0.738467\pi$
$587$	−33.0000	−1.36206	−0.681028	+	0.732257i	$0.738467\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−24.0000	−0.987228
$592$	16.0000	0.657596
$593$	32.0000	1.31408	0.657041	−	0.753855i	$-0.271808\pi$
$593$	32.0000	1.31408	0.657041	+	0.753855i	$0.271808\pi$
$594$	0	0
$595$	0	0
$596$	30.0000	1.22885
$597$	−15.0000	−0.613909
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	0	0
$603$	−6.00000	−0.244339
$604$	8.00000	0.325515
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	7.00000	0.284121	0.142061	−	0.989858i	$-0.454627\pi$
$607$	7.00000	0.284121	0.142061	+	0.989858i	$0.454627\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.0000	1.09230
$612$	0	0
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	0	0
$615$	1.00000	0.0403239
$616$	0	0
$617$	−43.0000	−1.73111	−0.865557	−	0.500810i	$-0.833036\pi$
$617$	−43.0000	−1.73111	−0.865557	+	0.500810i	$0.833036\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	0	0
$624$	−12.0000	−0.480384
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.00000	−0.239617
$628$	32.0000	1.27694
$629$	0	0
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	−7.00000	−0.277787
$636$	6.00000	0.237915
$637$	0	0
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	1.00000	0.0393750
$646$	0	0
$647$	5.00000	0.196570	0.0982851	−	0.995158i	$-0.468664\pi$
$647$	5.00000	0.196570	0.0982851	+	0.995158i	$0.468664\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	24.0000	0.939913
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	0	0
$655$	−13.0000	−0.507952
$656$	4.00000	0.156174
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	−2.00000	−0.0778499
$661$	−9.00000	−0.350059	−0.175030	−	0.984563i	$-0.556002\pi$
$661$	−9.00000	−0.350059	−0.175030	+	0.984563i	$0.556002\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.00000	−0.193601
$668$	−24.0000	−0.928588
$669$	−10.0000	−0.386622
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	8.00000	0.307692
$677$	50.0000	1.92166	0.960828	−	0.277145i	$-0.0893883\pi$
$677$	50.0000	1.92166	0.960828	+	0.277145i	$0.0893883\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−12.0000	−0.458831
$685$	17.0000	0.649537
$686$	0	0
$687$	5.00000	0.190762
$688$	4.00000	0.152499
$689$	9.00000	0.342873
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	28.0000	1.06440
$693$	0	0
$694$	0	0
$695$	14.0000	0.531050
$696$	0	0
$697$	0	0
$698$	0	0
$699$	16.0000	0.605176
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	−8.00000	−0.301511
$705$	9.00000	0.338960
$706$	0	0
$707$	0	0
$708$	8.00000	0.300658
$709$	−21.0000	−0.788672	−0.394336	−	0.918966i	$-0.629025\pi$
$709$	−21.0000	−0.788672	−0.394336	+	0.918966i	$0.629025\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−3.00000	−0.112194
$716$	36.0000	1.34538
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	−4.00000	−0.149071
$721$	0	0
$722$	0	0
$723$	−16.0000	−0.595046
$724$	−26.0000	−0.966282
$725$	5.00000	0.185695
$726$	0	0
$727$	−22.0000	−0.815935	−0.407967	−	0.912996i	$-0.633762\pi$
$727$	−22.0000	−0.815935	−0.407967	+	0.912996i	$0.633762\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−20.0000	−0.739221
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.00000	−0.221013
$738$	0	0
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	8.00000	0.294086
$741$	−18.0000	−0.661247
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	36.0000	1.31278
$753$	18.0000	0.655956
$754$	0	0
$755$	4.00000	0.145575
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	0	0
$759$	1.00000	0.0362977
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	0	0
$767$	12.0000	0.433295
$768$	−16.0000	−0.577350
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	−30.0000	−1.07972
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.00000	0.214972
$780$	−6.00000	−0.214834
$781$	−4.00000	−0.143131
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	16.0000	0.571064
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−48.0000	−1.70993
$789$	−16.0000	−0.569615
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−30.0000	−1.06533
$794$	0	0
$795$	3.00000	0.106399
$796$	−30.0000	−1.06332
$797$	−45.0000	−1.59398	−0.796991	−	0.603991i	$-0.793576\pi$
$797$	−45.0000	−1.59398	−0.796991	+	0.603991i	$0.793576\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.00000	−0.211735
$804$	−12.0000	−0.423207
$805$	0	0
$806$	0	0
$807$	−32.0000	−1.12645
$808$	0	0
$809$	7.00000	0.246107	0.123053	−	0.992400i	$-0.460731\pi$
$809$	7.00000	0.246107	0.123053	+	0.992400i	$0.460731\pi$
$810$	0	0
$811$	30.0000	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$812$	0	0
$813$	10.0000	0.350715
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	6.00000	0.209913
$818$	0	0
$819$	0	0
$820$	2.00000	0.0698430
$821$	26.0000	0.907406	0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000	0.907406	0.453703	+	0.891153i	$0.350103\pi$
$822$	0	0
$823$	36.0000	1.25488	0.627441	−	0.778664i	$-0.284103\pi$
$823$	36.0000	1.25488	0.627441	+	0.778664i	$0.284103\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	54.0000	1.87776	0.938882	−	0.344239i	$-0.111863\pi$
$827$	54.0000	1.87776	0.938882	+	0.344239i	$0.111863\pi$
$828$	2.00000	0.0695048
$829$	−13.0000	−0.451509	−0.225754	−	0.974184i	$-0.572485\pi$
$829$	−13.0000	−0.451509	−0.225754	+	0.974184i	$0.572485\pi$
$830$	0	0
$831$	25.0000	0.867240
$832$	−24.0000	−0.832050
$833$	0	0
$834$	0	0
$835$	−12.0000	−0.415277
$836$	−12.0000	−0.415029
$837$	0	0
$838$	0	0
$839$	22.0000	0.759524	0.379762	−	0.925084i	$-0.376006\pi$
$839$	22.0000	0.759524	0.379762	+	0.925084i	$0.376006\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−29.0000	−0.998813
$844$	−48.0000	−1.65223
$845$	4.00000	0.137604
$846$	0	0
$847$	0	0
$848$	12.0000	0.412082
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−4.00000	−0.137118
$852$	−8.00000	−0.274075
$853$	23.0000	0.787505	0.393753	−	0.919216i	$-0.371177\pi$
$853$	23.0000	0.787505	0.393753	+	0.919216i	$0.371177\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	0	0
$863$	9.00000	0.306364	0.153182	−	0.988198i	$-0.451048\pi$
$863$	9.00000	0.306364	0.153182	+	0.988198i	$0.451048\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	−18.0000	−0.609907
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	0	0
$876$	−12.0000	−0.405442
$877$	39.0000	1.31694	0.658468	−	0.752609i	$-0.271205\pi$
$877$	39.0000	1.31694	0.658468	+	0.752609i	$0.271205\pi$
$878$	0	0
$879$	2.00000	0.0674583
$880$	−4.00000	−0.134840
$881$	−48.0000	−1.61716	−0.808581	−	0.588386i	$-0.799764\pi$
$881$	−48.0000	−1.61716	−0.808581	+	0.588386i	$0.799764\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	−20.0000	−0.669650
$893$	54.0000	1.80704
$894$	0	0
$895$	18.0000	0.601674
$896$	0	0
$897$	3.00000	0.100167
$898$	0	0
$899$	0	0
$900$	−2.00000	−0.0666667
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.0000	−0.432135
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	−12.0000	−0.398234
$909$	7.00000	0.232175
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−24.0000	−0.794719
$913$	6.00000	0.198571
$914$	0	0
$915$	−10.0000	−0.330590
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−11.0000	−0.362462
$922$	0	0
$923$	−12.0000	−0.394985
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	0	0
$932$	32.0000	1.04819
$933$	−28.0000	−0.916679
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.00000	0.0326686	0.0163343	−	0.999867i	$-0.494800\pi$
$937$	1.00000	0.0326686	0.0163343	+	0.999867i	$0.494800\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	18.0000	0.587095
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	0	0
$943$	−1.00000	−0.0325645
$944$	16.0000	0.520756
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−32.0000	−1.03931
$949$	−18.0000	−0.584305
$950$	0	0
$951$	7.00000	0.226991
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	−5.00000	−0.161627
$958$	0	0
$959$	0	0
$960$	−8.00000	−0.258199
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−6.00000	−0.193347
$964$	−32.0000	−1.03065
$965$	−15.0000	−0.482867
$966$	0	0
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	2.00000	0.0641500
$973$	0	0
$974$	0	0
$975$	−3.00000	−0.0960769
$976$	−40.0000	−1.28037
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4.00000	0.127710
$982$	0	0
$983$	−45.0000	−1.43528	−0.717639	−	0.696416i	$-0.754777\pi$
$983$	−45.0000	−1.43528	−0.717639	+	0.696416i	$0.754777\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	0	0
$988$	−36.0000	−1.14531
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	43.0000	1.36594	0.682970	−	0.730446i	$-0.260688\pi$
$991$	43.0000	1.36594	0.682970	+	0.730446i	$0.260688\pi$
$992$	0	0
$993$	7.00000	0.222138
$994$	0	0
$995$	−15.0000	−0.475532
$996$	12.0000	0.380235
$997$	3.00000	0.0950110	0.0475055	−	0.998871i	$-0.484873\pi$
$997$	3.00000	0.0950110	0.0475055	+	0.998871i	$0.484873\pi$
$998$	0	0
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.l.1.1		1
7.2	even	3		1155.2.q.d.991.1	yes	2
7.4	even	3		1155.2.q.d.331.1	✓	2
7.6	odd	2		8085.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.q.d.331.1	✓	2	7.4	even	3
1155.2.q.d.991.1	yes	2	7.2	even	3
8085.2.a.l.1.1		1	1.1	even	1	trivial
8085.2.a.p.1.1		1	7.6	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:	\( N \)	\(=\)	\( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8085.a (trivial)

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L-functions

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