LMFDB - Embedded newform 8085.2.a.k.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)

$f(q)$

$=$

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

$q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{12} +4.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} +5.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} -5.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.00000 q^{29} +6.00000 q^{31} +1.00000 q^{33} -2.00000 q^{36} -12.0000 q^{37} -4.00000 q^{39} +2.00000 q^{41} +13.0000 q^{43} +2.00000 q^{44} -1.00000 q^{45} +6.00000 q^{47} -4.00000 q^{48} -5.00000 q^{51} -8.00000 q^{52} +1.00000 q^{53} +1.00000 q^{55} +1.00000 q^{57} +11.0000 q^{59} -2.00000 q^{60} -5.00000 q^{61} -8.00000 q^{64} -4.00000 q^{65} -10.0000 q^{67} -10.0000 q^{68} +5.00000 q^{69} -6.00000 q^{71} +4.00000 q^{73} -1.00000 q^{75} +2.00000 q^{76} -8.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -5.00000 q^{83} -5.00000 q^{85} -3.00000 q^{87} -13.0000 q^{89} +10.0000 q^{92} -6.00000 q^{93} +1.00000 q^{95} +19.0000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	4.00000	1.00000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−12.0000	−1.97279	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−12.0000	−1.97279	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	13.0000	1.98248	0.991241	−	0.132068i	$-0.0421616\pi$
$43$	13.0000	1.98248	0.991241	+	0.132068i	$0.0421616\pi$
$44$	2.00000	0.301511
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−4.00000	−0.577350
$49$	0	0
$50$	0	0
$51$	−5.00000	−0.700140
$52$	−8.00000	−1.10940
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	−2.00000	−0.258199
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−10.0000	−1.21268
$69$	5.00000	0.601929
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.00000	−0.548821	−0.274411	−	0.961613i	$-0.588483\pi$
$83$	−5.00000	−0.548821	−0.274411	+	0.961613i	$0.588483\pi$
$84$	0	0
$85$	−5.00000	−0.542326
$86$	0	0
$87$	−3.00000	−0.321634
$88$	0	0
$89$	−13.0000	−1.37800	−0.688999	−	0.724763i	$-0.741949\pi$
$89$	−13.0000	−1.37800	−0.688999	+	0.724763i	$0.741949\pi$
$90$	0	0
$91$	0	0
$92$	10.0000	1.04257
$93$	−6.00000	−0.622171
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	19.0000	1.92916	0.964579	−	0.263795i	$-0.0849741\pi$
$97$	19.0000	1.92916	0.964579	+	0.263795i	$0.0849741\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−2.00000	−0.200000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	2.00000	0.192450
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	12.0000	1.13899
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	5.00000	0.466252
$116$	−6.00000	−0.557086
$117$	4.00000	0.369800
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.00000	−0.180334
$124$	−12.0000	−1.07763
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.0000	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$128$	0	0
$129$	−13.0000	−1.14459
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−4.00000	−0.334497
$144$	4.00000	0.333333
$145$	−3.00000	−0.249136
$146$	0	0
$147$	0	0
$148$	24.0000	1.97279
$149$	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
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	5.00000	0.404226
$154$	0	0
$155$	−6.00000	−0.481932
$156$	8.00000	0.640513
$157$	−21.0000	−1.67598	−0.837991	−	0.545684i	$-0.816270\pi$
$157$	−21.0000	−1.67598	−0.837991	+	0.545684i	$0.816270\pi$
$158$	0	0
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−4.00000	−0.312348
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−26.0000	−1.98248
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−11.0000	−0.826811
$178$	0	0
$179$	22.0000	1.64436	0.822179	−	0.569230i	$-0.192758\pi$
$179$	22.0000	1.64436	0.822179	+	0.569230i	$0.192758\pi$
$180$	2.00000	0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	−5.00000	−0.365636
$188$	−12.0000	−0.875190
$189$	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	$0.357051\pi$
$192$	8.00000	0.577350
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	0	0
$204$	10.0000	0.700140
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−5.00000	−0.347524
$208$	16.0000	1.10940
$209$	1.00000	0.0691714
$210$	0	0
$211$	26.0000	1.78991	0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000	1.78991	0.894957	+	0.446153i	$0.147206\pi$
$212$	−2.00000	−0.137361
$213$	6.00000	0.411113
$214$	0	0
$215$	−13.0000	−0.886593
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.00000	−0.270295
$220$	−2.00000	−0.134840
$221$	20.0000	1.34535
$222$	0	0
$223$	13.0000	0.870544	0.435272	−	0.900299i	$-0.356652\pi$
$223$	13.0000	0.870544	0.435272	+	0.900299i	$0.356652\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	−2.00000	−0.132453
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.0000	1.31024	0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000	1.31024	0.655122	+	0.755523i	$0.272617\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	−22.0000	−1.43208
$237$	8.00000	0.519656
$238$	0	0
$239$	25.0000	1.61712	0.808558	−	0.588417i	$-0.200249\pi$
$239$	25.0000	1.61712	0.808558	+	0.588417i	$0.200249\pi$
$240$	4.00000	0.258199
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	5.00000	0.316862
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	0	0
$255$	5.00000	0.313112
$256$	16.0000	1.00000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	0	0
$260$	8.00000	0.496139
$261$	3.00000	0.185695
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−1.00000	−0.0614295
$266$	0	0
$267$	13.0000	0.795587
$268$	20.0000	1.22169
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	20.0000	1.21268
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	−10.0000	−0.601929
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	12.0000	0.712069
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−19.0000	−1.11380
$292$	−8.00000	−0.468165
$293$	13.0000	0.759468	0.379734	−	0.925096i	$-0.376015\pi$
$293$	13.0000	0.759468	0.379734	+	0.925096i	$0.376015\pi$
$294$	0	0
$295$	−11.0000	−0.640445
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−20.0000	−1.15663
$300$	2.00000	0.115470
$301$	0	0
$302$	0	0
$303$	2.00000	0.114897
$304$	−4.00000	−0.229416
$305$	5.00000	0.286299
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	9.00000	0.511992
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	0	0
$315$	0	0
$316$	16.0000	0.900070
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	8.00000	0.447214
$321$	−18.0000	−1.00466
$322$	0	0
$323$	−5.00000	−0.278207
$324$	−2.00000	−0.111111
$325$	4.00000	0.221880
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.0000	1.37412	0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000	1.37412	0.687062	+	0.726599i	$0.258900\pi$
$332$	10.0000	0.548821
$333$	−12.0000	−0.657596
$334$	0	0
$335$	10.0000	0.546358
$336$	0	0
$337$	17.0000	0.926049	0.463025	−	0.886345i	$-0.346764\pi$
$337$	17.0000	0.926049	0.463025	+	0.886345i	$0.346764\pi$
$338$	0	0
$339$	−9.00000	−0.488813
$340$	10.0000	0.542326
$341$	−6.00000	−0.324918
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5.00000	−0.269191
$346$	0	0
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	6.00000	0.321634
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	26.0000	1.37800
$357$	0	0
$358$	0	0
$359$	1.00000	0.0527780	0.0263890	−	0.999652i	$-0.491599\pi$
$359$	1.00000	0.0527780	0.0263890	+	0.999652i	$0.491599\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	−20.0000	−1.04257
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	12.0000	0.622171
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	13.0000	0.667765	0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000	0.667765	0.333883	+	0.942615i	$0.391641\pi$
$380$	−2.00000	−0.102598
$381$	19.0000	0.973399
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.0000	0.660827
$388$	−38.0000	−1.92916
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−25.0000	−1.26430
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	8.00000	0.402524
$396$	2.00000	0.100504
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	4.00000	0.199007
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	18.0000	0.886796
$413$	0	0
$414$	0	0
$415$	5.00000	0.245440
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	33.0000	1.61216	0.806078	−	0.591810i	$-0.201586\pi$
$419$	33.0000	1.61216	0.806078	+	0.591810i	$0.201586\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	5.00000	0.242536
$426$	0	0
$427$	0	0
$428$	−36.0000	−1.74013
$429$	4.00000	0.193122
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	−4.00000	−0.192450
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	20.0000	0.957826
$437$	5.00000	0.239182
$438$	0	0
$439$	−21.0000	−1.00228	−0.501138	−	0.865368i	$-0.667085\pi$
$439$	−21.0000	−1.00228	−0.501138	+	0.865368i	$0.667085\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−24.0000	−1.13899
$445$	13.0000	0.616259
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	−18.0000	−0.846649
$453$	−6.00000	−0.281905
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.0000	−1.26301	−0.631503	−	0.775373i	$-0.717562\pi$
$457$	−27.0000	−1.26301	−0.631503	+	0.775373i	$0.717562\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	−10.0000	−0.466252
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	12.0000	0.557086
$465$	6.00000	0.278243
$466$	0	0
$467$	32.0000	1.48078	0.740392	−	0.672176i	$-0.234640\pi$
$467$	32.0000	1.48078	0.740392	+	0.672176i	$0.234640\pi$
$468$	−8.00000	−0.369800
$469$	0	0
$470$	0	0
$471$	21.0000	0.967629
$472$	0	0
$473$	−13.0000	−0.597741
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	1.00000	0.0457869
$478$	0	0
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	−48.0000	−2.18861
$482$	0	0
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−19.0000	−0.862746
$486$	0	0
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−37.0000	−1.66979	−0.834893	−	0.550412i	$-0.814471\pi$
$491$	−37.0000	−1.66979	−0.834893	+	0.550412i	$0.814471\pi$
$492$	4.00000	0.180334
$493$	15.0000	0.675566
$494$	0	0
$495$	1.00000	0.0449467
$496$	24.0000	1.07763
$497$	0	0
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	2.00000	0.0894427
$501$	12.0000	0.536120
$502$	0	0
$503$	21.0000	0.936344	0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000	0.936344	0.468172	+	0.883637i	$0.344913\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	−3.00000	−0.133235
$508$	38.0000	1.68598
$509$	−11.0000	−0.487566	−0.243783	−	0.969830i	$-0.578389\pi$
$509$	−11.0000	−0.487566	−0.243783	+	0.969830i	$0.578389\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	9.00000	0.396587
$516$	26.0000	1.14459
$517$	−6.00000	−0.263880
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	21.0000	0.920027	0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000	0.920027	0.460013	+	0.887912i	$0.347845\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	0	0
$527$	30.0000	1.30682
$528$	4.00000	0.174078
$529$	2.00000	0.0869565
$530$	0	0
$531$	11.0000	0.477359
$532$	0	0
$533$	8.00000	0.346518
$534$	0	0
$535$	−18.0000	−0.778208
$536$	0	0
$537$	−22.0000	−0.949370
$538$	0	0
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	11.0000	0.470326	0.235163	−	0.971956i	$-0.424438\pi$
$547$	11.0000	0.470326	0.235163	+	0.971956i	$0.424438\pi$
$548$	20.0000	0.854358
$549$	−5.00000	−0.213395
$550$	0	0
$551$	−3.00000	−0.127804
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−12.0000	−0.509372
$556$	8.00000	0.339276
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	52.0000	2.19937
$560$	0	0
$561$	5.00000	0.211100
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	12.0000	0.505291
$565$	−9.00000	−0.378633
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.0000	−1.04805	−0.524027	−	0.851701i	$-0.675571\pi$
$569$	−25.0000	−1.04805	−0.524027	+	0.851701i	$0.675571\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	8.00000	0.334497
$573$	−12.0000	−0.501307
$574$	0	0
$575$	−5.00000	−0.208514
$576$	−8.00000	−0.333333
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	6.00000	0.249136
$581$	0	0
$582$	0	0
$583$	−1.00000	−0.0414158
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	24.0000	0.987228
$592$	−48.0000	−1.97279
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−34.0000	−1.38920	−0.694601	−	0.719395i	$-0.744419\pi$
$599$	−34.0000	−1.38920	−0.694601	+	0.719395i	$0.744419\pi$
$600$	0	0
$601$	13.0000	0.530281	0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000	0.530281	0.265141	+	0.964210i	$0.414582\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	−12.0000	−0.488273
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	−10.0000	−0.404226
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	12.0000	0.481932
$621$	5.00000	0.200643
$622$	0	0
$623$	0	0
$624$	−16.0000	−0.640513
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.00000	−0.0399362
$628$	42.0000	1.67598
$629$	−60.0000	−2.39236
$630$	0	0
$631$	13.0000	0.517522	0.258761	−	0.965941i	$-0.416686\pi$
$631$	13.0000	0.517522	0.258761	+	0.965941i	$0.416686\pi$
$632$	0	0
$633$	−26.0000	−1.03341
$634$	0	0
$635$	19.0000	0.753992
$636$	2.00000	0.0793052
$637$	0	0
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	−13.0000	−0.512670	−0.256335	−	0.966588i	$-0.582515\pi$
$643$	−13.0000	−0.512670	−0.256335	+	0.966588i	$0.582515\pi$
$644$	0	0
$645$	13.0000	0.511875
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−11.0000	−0.431788
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	11.0000	0.430463	0.215232	−	0.976563i	$-0.430949\pi$
$653$	11.0000	0.430463	0.215232	+	0.976563i	$0.430949\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	8.00000	0.312348
$657$	4.00000	0.156055
$658$	0	0
$659$	−37.0000	−1.44132	−0.720658	−	0.693291i	$-0.756160\pi$
$659$	−37.0000	−1.44132	−0.720658	+	0.693291i	$0.756160\pi$
$660$	2.00000	0.0778499
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	−20.0000	−0.776736
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15.0000	−0.580802
$668$	24.0000	0.928588
$669$	−13.0000	−0.502609
$670$	0	0
$671$	5.00000	0.193023
$672$	0	0
$673$	11.0000	0.424019	0.212009	−	0.977268i	$-0.431999\pi$
$673$	11.0000	0.424019	0.212009	+	0.977268i	$0.431999\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	−6.00000	−0.230769
$677$	−31.0000	−1.19143	−0.595713	−	0.803197i	$-0.703131\pi$
$677$	−31.0000	−1.19143	−0.595713	+	0.803197i	$0.703131\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	2.00000	0.0764719
$685$	10.0000	0.382080
$686$	0	0
$687$	−14.0000	−0.534133
$688$	52.0000	1.98248
$689$	4.00000	0.152388
$690$	0	0
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	10.0000	0.378777
$698$	0	0
$699$	−20.0000	−0.756469
$700$	0	0
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	8.00000	0.301511
$705$	6.00000	0.225973
$706$	0	0
$707$	0	0
$708$	22.0000	0.826811
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−30.0000	−1.12351
$714$	0	0
$715$	4.00000	0.149592
$716$	−44.0000	−1.64436
$717$	−25.0000	−0.933642
$718$	0	0
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	−4.00000	−0.149071
$721$	0	0
$722$	0	0
$723$	2.00000	0.0743808
$724$	−4.00000	−0.148659
$725$	3.00000	0.111417
$726$	0	0
$727$	5.00000	0.185440	0.0927199	−	0.995692i	$-0.470444\pi$
$727$	5.00000	0.185440	0.0927199	+	0.995692i	$0.470444\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	65.0000	2.40411
$732$	−10.0000	−0.369611
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.0000	0.368355
$738$	0	0
$739$	−18.0000	−0.662141	−0.331070	−	0.943606i	$-0.607410\pi$
$739$	−18.0000	−0.662141	−0.331070	+	0.943606i	$0.607410\pi$
$740$	−24.0000	−0.882258
$741$	4.00000	0.146944
$742$	0	0
$743$	18.0000	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−5.00000	−0.182940
$748$	10.0000	0.365636
$749$	0	0
$750$	0	0
$751$	5.00000	0.182453	0.0912263	−	0.995830i	$-0.470921\pi$
$751$	5.00000	0.182453	0.0912263	+	0.995830i	$0.470921\pi$
$752$	24.0000	0.875190
$753$	−28.0000	−1.02038
$754$	0	0
$755$	−6.00000	−0.218362
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	−5.00000	−0.181489
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	0	0
$763$	0	0
$764$	−24.0000	−0.868290
$765$	−5.00000	−0.180775
$766$	0	0
$767$	44.0000	1.58875
$768$	−16.0000	−0.577350
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	28.0000	1.00774
$773$	34.0000	1.22290	0.611448	−	0.791285i	$-0.290588\pi$
$773$	34.0000	1.22290	0.611448	+	0.791285i	$0.290588\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	−8.00000	−0.286446
$781$	6.00000	0.214697
$782$	0	0
$783$	−3.00000	−0.107211
$784$	0	0
$785$	21.0000	0.749522
$786$	0	0
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	48.0000	1.70993
$789$	−4.00000	−0.142404
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	1.00000	0.0354663
$796$	−16.0000	−0.567105
$797$	26.0000	0.920967	0.460484	−	0.887668i	$-0.347676\pi$
$797$	26.0000	0.920967	0.460484	+	0.887668i	$0.347676\pi$
$798$	0	0
$799$	30.0000	1.06132
$800$	0	0
$801$	−13.0000	−0.459332
$802$	0	0
$803$	−4.00000	−0.141157
$804$	−20.0000	−0.705346
$805$	0	0
$806$	0	0
$807$	17.0000	0.598428
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	−5.00000	−0.175358
$814$	0	0
$815$	−4.00000	−0.140114
$816$	−20.0000	−0.700140
$817$	−13.0000	−0.454812
$818$	0	0
$819$	0	0
$820$	4.00000	0.139686
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	10.0000	0.347524
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	−32.0000	−1.10940
$833$	0	0
$834$	0	0
$835$	12.0000	0.415277
$836$	−2.00000	−0.0691714
$837$	−6.00000	−0.207390
$838$	0	0
$839$	39.0000	1.34643	0.673215	−	0.739447i	$-0.264913\pi$
$839$	39.0000	1.34643	0.673215	+	0.739447i	$0.264913\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	18.0000	0.619953
$844$	−52.0000	−1.78991
$845$	−3.00000	−0.103203
$846$	0	0
$847$	0	0
$848$	4.00000	0.137361
$849$	−28.0000	−0.960958
$850$	0	0
$851$	60.0000	2.05677
$852$	−12.0000	−0.411113
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−34.0000	−1.16007	−0.580033	−	0.814593i	$-0.696960\pi$
$859$	−34.0000	−1.16007	−0.580033	+	0.814593i	$0.696960\pi$
$860$	26.0000	0.886593
$861$	0	0
$862$	0	0
$863$	3.00000	0.102121	0.0510606	−	0.998696i	$-0.483740\pi$
$863$	3.00000	0.102121	0.0510606	+	0.998696i	$0.483740\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	19.0000	0.643053
$874$	0	0
$875$	0	0
$876$	8.00000	0.270295
$877$	11.0000	0.371444	0.185722	−	0.982602i	$-0.440538\pi$
$877$	11.0000	0.371444	0.185722	+	0.982602i	$0.440538\pi$
$878$	0	0
$879$	−13.0000	−0.438479
$880$	4.00000	0.134840
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	−40.0000	−1.34535
$885$	11.0000	0.369761
$886$	0	0
$887$	−33.0000	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$887$	−33.0000	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−26.0000	−0.870544
$893$	−6.00000	−0.200782
$894$	0	0
$895$	−22.0000	−0.735379
$896$	0	0
$897$	20.0000	0.667781
$898$	0	0
$899$	18.0000	0.600334
$900$	−2.00000	−0.0666667
$901$	5.00000	0.166574
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	−6.00000	−0.199117
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	56.0000	1.85536	0.927681	−	0.373373i	$-0.121799\pi$
$911$	56.0000	1.85536	0.927681	+	0.373373i	$0.121799\pi$
$912$	4.00000	0.132453
$913$	5.00000	0.165476
$914$	0	0
$915$	−5.00000	−0.165295
$916$	−28.0000	−0.925146
$917$	0	0
$918$	0	0
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	−12.0000	−0.394558
$926$	0	0
$927$	−9.00000	−0.295599
$928$	0	0
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	0	0
$932$	−40.0000	−1.31024
$933$	8.00000	0.261908
$934$	0	0
$935$	5.00000	0.163517
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−9.00000	−0.293704
$940$	12.0000	0.391397
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	−10.0000	−0.325645
$944$	44.0000	1.43208
$945$	0	0
$946$	0	0
$947$	47.0000	1.52729	0.763647	−	0.645634i	$-0.223407\pi$
$947$	47.0000	1.52729	0.763647	+	0.645634i	$0.223407\pi$
$948$	−16.0000	−0.519656
$949$	16.0000	0.519382
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	−50.0000	−1.61712
$957$	3.00000	0.0969762
$958$	0	0
$959$	0	0
$960$	−8.00000	−0.258199
$961$	5.00000	0.161290
$962$	0	0
$963$	18.0000	0.580042
$964$	4.00000	0.128831
$965$	14.0000	0.450676
$966$	0	0
$967$	1.00000	0.0321578	0.0160789	−	0.999871i	$-0.494882\pi$
$967$	1.00000	0.0321578	0.0160789	+	0.999871i	$0.494882\pi$
$968$	0	0
$969$	5.00000	0.160623
$970$	0	0
$971$	−57.0000	−1.82922	−0.914609	−	0.404341i	$-0.867501\pi$
$971$	−57.0000	−1.82922	−0.914609	+	0.404341i	$0.867501\pi$
$972$	2.00000	0.0641500
$973$	0	0
$974$	0	0
$975$	−4.00000	−0.128103
$976$	−20.0000	−0.640184
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	0	0
$979$	13.0000	0.415482
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	0	0
$988$	8.00000	0.254514
$989$	−65.0000	−2.06688
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	0	0
$993$	−25.0000	−0.793351
$994$	0	0
$995$	−8.00000	−0.253617
$996$	−10.0000	−0.316862
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.k.1.1		1
7.6	odd	2		1155.2.a.i.1.1	✓	1
21.20	even	2		3465.2.a.h.1.1		1
35.34	odd	2		5775.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.i.1.1	✓	1	7.6	odd	2
3465.2.a.h.1.1		1	21.20	even	2
5775.2.a.m.1.1		1	35.34	odd	2
8085.2.a.k.1.1		1	1.1	even	1	trivial

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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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