LMFDB - Embedded newform 12.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [12,8,Mod(1,12)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(12, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("12.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	12.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:	$3.74862030581$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	12.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-27.0000 q^{3} -378.000 q^{5} -832.000 q^{7} +729.000 q^{9} +O(q^{10})$

$q-27.0000 q^{3} -378.000 q^{5} -832.000 q^{7} +729.000 q^{9} -2484.00 q^{11} +14870.0 q^{13} +10206.0 q^{15} -22302.0 q^{17} -16300.0 q^{19} +22464.0 q^{21} -115128. q^{23} +64759.0 q^{25} -19683.0 q^{27} +157086. q^{29} -16456.0 q^{31} +67068.0 q^{33} +314496. q^{35} -149266. q^{37} -401490. q^{39} -241110. q^{41} -443188. q^{43} -275562. q^{45} +922752. q^{47} -131319. q^{49} +602154. q^{51} -697626. q^{53} +938952. q^{55} +440100. q^{57} +870156. q^{59} +2.06706e6 q^{61} -606528. q^{63} -5.62086e6 q^{65} -1.68075e6 q^{67} +3.10846e6 q^{69} -1.07028e6 q^{71} -2.40333e6 q^{73} -1.74849e6 q^{75} +2.06669e6 q^{77} +2.30151e6 q^{79} +531441. q^{81} +4.70869e6 q^{83} +8.43016e6 q^{85} -4.24132e6 q^{87} +4.14369e6 q^{89} -1.23718e7 q^{91} +444312. q^{93} +6.16140e6 q^{95} -1.62297e6 q^{97} -1.81084e6 q^{99} +O(q^{100})$

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
$2$	0	0
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−378.000	−1.35237	−0.676187	−	0.736730i	$-0.736369\pi$
$5$	−378.000	−1.35237	−0.676187	+	0.736730i	$0.736369\pi$
$6$	0	0
$7$	−832.000	−0.916812	−0.458406	−	0.888743i	$-0.651579\pi$
$7$	−832.000	−0.916812	−0.458406	+	0.888743i	$0.651579\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−2484.00	−0.562700	−0.281350	−	0.959605i	$-0.590782\pi$
$11$	−2484.00	−0.562700	−0.281350	+	0.959605i	$0.590782\pi$
$12$	0	0
$13$	14870.0	1.87719	0.938597	−	0.345015i	$-0.112126\pi$
$13$	14870.0	1.87719	0.938597	+	0.345015i	$0.112126\pi$
$14$	0	0
$15$	10206.0	0.780793
$16$	0	0
$17$	−22302.0	−1.10096	−0.550481	−	0.834847i	$-0.685556\pi$
$17$	−22302.0	−1.10096	−0.550481	+	0.834847i	$0.685556\pi$
$18$	0	0
$19$	−16300.0	−0.545193	−0.272596	−	0.962128i	$-0.587882\pi$
$19$	−16300.0	−0.545193	−0.272596	+	0.962128i	$0.587882\pi$
$20$	0	0
$21$	22464.0	0.529322
$22$	0	0
$23$	−115128.	−1.97303	−0.986515	−	0.163673i	$-0.947666\pi$
$23$	−115128.	−1.97303	−0.986515	+	0.163673i	$0.947666\pi$
$24$	0	0
$25$	64759.0	0.828915
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	157086.	1.19604	0.598018	−	0.801482i	$-0.295955\pi$
$29$	157086.	1.19604	0.598018	+	0.801482i	$0.295955\pi$
$30$	0	0
$31$	−16456.0	−0.0992107	−0.0496053	−	0.998769i	$-0.515796\pi$
$31$	−16456.0	−0.0992107	−0.0496053	+	0.998769i	$0.515796\pi$
$32$	0	0
$33$	67068.0	0.324875
$34$	0	0
$35$	314496.	1.23987
$36$	0	0
$37$	−149266.	−0.484457	−0.242228	−	0.970219i	$-0.577878\pi$
$37$	−149266.	−0.484457	−0.242228	+	0.970219i	$0.577878\pi$
$38$	0	0
$39$	−401490.	−1.08380
$40$	0	0
$41$	−241110.	−0.546351	−0.273175	−	0.961964i	$-0.588074\pi$
$41$	−241110.	−0.546351	−0.273175	+	0.961964i	$0.588074\pi$
$42$	0	0
$43$	−443188.	−0.850058	−0.425029	−	0.905180i	$-0.639736\pi$
$43$	−443188.	−0.850058	−0.425029	+	0.905180i	$0.639736\pi$
$44$	0	0
$45$	−275562.	−0.450791
$46$	0	0
$47$	922752.	1.29641	0.648205	−	0.761466i	$-0.275520\pi$
$47$	922752.	1.29641	0.648205	+	0.761466i	$0.275520\pi$
$48$	0	0
$49$	−131319.	−0.159456
$50$	0	0
$51$	602154.	0.635641
$52$	0	0
$53$	−697626.	−0.643661	−0.321830	−	0.946797i	$-0.604298\pi$
$53$	−697626.	−0.643661	−0.321830	+	0.946797i	$0.604298\pi$
$54$	0	0
$55$	938952.	0.760981
$56$	0	0
$57$	440100.	0.314767
$58$	0	0
$59$	870156.	0.551588	0.275794	−	0.961217i	$-0.411059\pi$
$59$	870156.	0.551588	0.275794	+	0.961217i	$0.411059\pi$
$60$	0	0
$61$	2.06706e6	1.16600	0.583001	−	0.812472i	$-0.301878\pi$
$61$	2.06706e6	1.16600	0.583001	+	0.812472i	$0.301878\pi$
$62$	0	0
$63$	−606528.	−0.305604
$64$	0	0
$65$	−5.62086e6	−2.53867
$66$	0	0
$67$	−1.68075e6	−0.682717	−0.341359	−	0.939933i	$-0.610887\pi$
$67$	−1.68075e6	−0.682717	−0.341359	+	0.939933i	$0.610887\pi$
$68$	0	0
$69$	3.10846e6	1.13913
$70$	0	0
$71$	−1.07028e6	−0.354890	−0.177445	−	0.984131i	$-0.556783\pi$
$71$	−1.07028e6	−0.354890	−0.177445	+	0.984131i	$0.556783\pi$
$72$	0	0
$73$	−2.40333e6	−0.723076	−0.361538	−	0.932357i	$-0.617748\pi$
$73$	−2.40333e6	−0.723076	−0.361538	+	0.932357i	$0.617748\pi$
$74$	0	0
$75$	−1.74849e6	−0.478574
$76$	0	0
$77$	2.06669e6	0.515890
$78$	0	0
$79$	2.30151e6	0.525192	0.262596	−	0.964906i	$-0.415421\pi$
$79$	2.30151e6	0.525192	0.262596	+	0.964906i	$0.415421\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	4.70869e6	0.903914	0.451957	−	0.892040i	$-0.350726\pi$
$83$	4.70869e6	0.903914	0.451957	+	0.892040i	$0.350726\pi$
$84$	0	0
$85$	8.43016e6	1.48891
$86$	0	0
$87$	−4.24132e6	−0.690532
$88$	0	0
$89$	4.14369e6	0.623049	0.311525	−	0.950238i	$-0.399160\pi$
$89$	4.14369e6	0.623049	0.311525	+	0.950238i	$0.399160\pi$
$90$	0	0
$91$	−1.23718e7	−1.72103
$92$	0	0
$93$	444312.	0.0572793
$94$	0	0
$95$	6.16140e6	0.737304
$96$	0	0
$97$	−1.62297e6	−0.180555	−0.0902777	−	0.995917i	$-0.528775\pi$
$97$	−1.62297e6	−0.180555	−0.0902777	+	0.995917i	$0.528775\pi$
$98$	0	0
$99$	−1.81084e6	−0.187567
$100$	0	0
$101$	1.09388e6	0.105644	0.0528219	−	0.998604i	$-0.483178\pi$
$101$	1.09388e6	0.105644	0.0528219	+	0.998604i	$0.483178\pi$
$102$	0	0
$103$	−1.56332e7	−1.40967	−0.704837	−	0.709370i	$-0.748980\pi$
$103$	−1.56332e7	−1.40967	−0.704837	+	0.709370i	$0.748980\pi$
$104$	0	0
$105$	−8.49139e6	−0.715841
$106$	0	0
$107$	−2.37028e6	−0.187049	−0.0935246	−	0.995617i	$-0.529813\pi$
$107$	−2.37028e6	−0.187049	−0.0935246	+	0.995617i	$0.529813\pi$
$108$	0	0
$109$	2.12706e6	0.157321	0.0786606	−	0.996901i	$-0.474936\pi$
$109$	2.12706e6	0.157321	0.0786606	+	0.996901i	$0.474936\pi$
$110$	0	0
$111$	4.03018e6	0.279701
$112$	0	0
$113$	−2.17091e7	−1.41536	−0.707682	−	0.706531i	$-0.750259\pi$
$113$	−2.17091e7	−1.41536	−0.707682	+	0.706531i	$0.750259\pi$
$114$	0	0
$115$	4.35184e7	2.66827
$116$	0	0
$117$	1.08402e7	0.625731
$118$	0	0
$119$	1.85553e7	1.00938
$120$	0	0
$121$	−1.33169e7	−0.683368
$122$	0	0
$123$	6.50997e6	0.315436
$124$	0	0
$125$	5.05235e6	0.231371
$126$	0	0
$127$	−1.02998e7	−0.446187	−0.223094	−	0.974797i	$-0.571616\pi$
$127$	−1.02998e7	−0.446187	−0.223094	+	0.974797i	$0.571616\pi$
$128$	0	0
$129$	1.19661e7	0.490781
$130$	0	0
$131$	−3.98823e7	−1.55000	−0.774999	−	0.631962i	$-0.782250\pi$
$131$	−3.98823e7	−1.55000	−0.774999	+	0.631962i	$0.782250\pi$
$132$	0	0
$133$	1.35616e7	0.499839
$134$	0	0
$135$	7.44017e6	0.260264
$136$	0	0
$137$	−2.57412e7	−0.855277	−0.427639	−	0.903950i	$-0.640654\pi$
$137$	−2.57412e7	−0.855277	−0.427639	+	0.903950i	$0.640654\pi$
$138$	0	0
$139$	2.62409e7	0.828757	0.414379	−	0.910105i	$-0.363999\pi$
$139$	2.62409e7	0.828757	0.414379	+	0.910105i	$0.363999\pi$
$140$	0	0
$141$	−2.49143e7	−0.748483
$142$	0	0
$143$	−3.69371e7	−1.05630
$144$	0	0
$145$	−5.93785e7	−1.61749
$146$	0	0
$147$	3.54561e6	0.0920621
$148$	0	0
$149$	6.98130e7	1.72896	0.864479	−	0.502670i	$-0.167649\pi$
$149$	6.98130e7	1.72896	0.864479	+	0.502670i	$0.167649\pi$
$150$	0	0
$151$	2.71335e7	0.641338	0.320669	−	0.947191i	$-0.396092\pi$
$151$	2.71335e7	0.641338	0.320669	+	0.947191i	$0.396092\pi$
$152$	0	0
$153$	−1.62582e7	−0.366988
$154$	0	0
$155$	6.22037e6	0.134170
$156$	0	0
$157$	−1.09857e7	−0.226557	−0.113279	−	0.993563i	$-0.536135\pi$
$157$	−1.09857e7	−0.226557	−0.113279	+	0.993563i	$0.536135\pi$
$158$	0	0
$159$	1.88359e7	0.371618
$160$	0	0
$161$	9.57865e7	1.80890
$162$	0	0
$163$	−8.43924e7	−1.52632	−0.763162	−	0.646207i	$-0.776354\pi$
$163$	−8.43924e7	−1.52632	−0.763162	+	0.646207i	$0.776354\pi$
$164$	0	0
$165$	−2.53517e7	−0.439353
$166$	0	0
$167$	5.95467e7	0.989350	0.494675	−	0.869078i	$-0.335287\pi$
$167$	5.95467e7	0.989350	0.494675	+	0.869078i	$0.335287\pi$
$168$	0	0
$169$	1.58368e8	2.52386
$170$	0	0
$171$	−1.18827e7	−0.181731
$172$	0	0
$173$	−2.12157e7	−0.311527	−0.155763	−	0.987794i	$-0.549784\pi$
$173$	−2.12157e7	−0.311527	−0.155763	+	0.987794i	$0.549784\pi$
$174$	0	0
$175$	−5.38795e7	−0.759959
$176$	0	0
$177$	−2.34942e7	−0.318460
$178$	0	0
$179$	−9.50932e7	−1.23926	−0.619632	−	0.784892i	$-0.712718\pi$
$179$	−9.50932e7	−1.23926	−0.619632	+	0.784892i	$0.712718\pi$
$180$	0	0
$181$	−1.31732e8	−1.65126	−0.825629	−	0.564214i	$-0.809179\pi$
$181$	−1.31732e8	−1.65126	−0.825629	+	0.564214i	$0.809179\pi$
$182$	0	0
$183$	−5.58107e7	−0.673191
$184$	0	0
$185$	5.64225e7	0.655166
$186$	0	0
$187$	5.53982e7	0.619512
$188$	0	0
$189$	1.63763e7	0.176441
$190$	0	0
$191$	−5.80470e7	−0.602786	−0.301393	−	0.953500i	$-0.597452\pi$
$191$	−5.80470e7	−0.602786	−0.301393	+	0.953500i	$0.597452\pi$
$192$	0	0
$193$	−6.92087e7	−0.692963	−0.346482	−	0.938057i	$-0.612624\pi$
$193$	−6.92087e7	−0.692963	−0.346482	+	0.938057i	$0.612624\pi$
$194$	0	0
$195$	1.51763e8	1.46570
$196$	0	0
$197$	−5.33848e7	−0.497491	−0.248746	−	0.968569i	$-0.580018\pi$
$197$	−5.33848e7	−0.497491	−0.248746	+	0.968569i	$0.580018\pi$
$198$	0	0
$199$	7.19134e7	0.646880	0.323440	−	0.946249i	$-0.395161\pi$
$199$	7.19134e7	0.646880	0.323440	+	0.946249i	$0.395161\pi$
$200$	0	0
$201$	4.53802e7	0.394167
$202$	0	0
$203$	−1.30696e8	−1.09654
$204$	0	0
$205$	9.11396e7	0.738871
$206$	0	0
$207$	−8.39283e7	−0.657676
$208$	0	0
$209$	4.04892e7	0.306780
$210$	0	0
$211$	2.05463e8	1.50572	0.752861	−	0.658180i	$-0.228673\pi$
$211$	2.05463e8	1.50572	0.752861	+	0.658180i	$0.228673\pi$
$212$	0	0
$213$	2.88976e7	0.204896
$214$	0	0
$215$	1.67525e8	1.14960
$216$	0	0
$217$	1.36914e7	0.0909575
$218$	0	0
$219$	6.48900e7	0.417468
$220$	0	0
$221$	−3.31631e8	−2.06672
$222$	0	0
$223$	5.12508e7	0.309480	0.154740	−	0.987955i	$-0.450546\pi$
$223$	5.12508e7	0.309480	0.154740	+	0.987955i	$0.450546\pi$
$224$	0	0
$225$	4.72093e7	0.276305
$226$	0	0
$227$	1.17076e8	0.664319	0.332159	−	0.943223i	$-0.392223\pi$
$227$	1.17076e8	0.664319	0.332159	+	0.943223i	$0.392223\pi$
$228$	0	0
$229$	−1.95800e8	−1.07743	−0.538715	−	0.842488i	$-0.681090\pi$
$229$	−1.95800e8	−1.07743	−0.538715	+	0.842488i	$0.681090\pi$
$230$	0	0
$231$	−5.58006e7	−0.297849
$232$	0	0
$233$	−1.30949e8	−0.678199	−0.339099	−	0.940751i	$-0.610122\pi$
$233$	−1.30949e8	−0.678199	−0.339099	+	0.940751i	$0.610122\pi$
$234$	0	0
$235$	−3.48800e8	−1.75323
$236$	0	0
$237$	−6.21408e7	−0.303220
$238$	0	0
$239$	1.72546e8	0.817544	0.408772	−	0.912637i	$-0.365957\pi$
$239$	1.72546e8	0.817544	0.408772	+	0.912637i	$0.365957\pi$
$240$	0	0
$241$	1.05073e8	0.483538	0.241769	−	0.970334i	$-0.422272\pi$
$241$	1.05073e8	0.483538	0.241769	+	0.970334i	$0.422272\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	4.96386e7	0.215644
$246$	0	0
$247$	−2.42381e8	−1.02343
$248$	0	0
$249$	−1.27135e8	−0.521875
$250$	0	0
$251$	1.33754e8	0.533885	0.266943	−	0.963712i	$-0.413987\pi$
$251$	1.33754e8	0.533885	0.266943	+	0.963712i	$0.413987\pi$
$252$	0	0
$253$	2.85978e8	1.11022
$254$	0	0
$255$	−2.27614e8	−0.859624
$256$	0	0
$257$	2.97590e8	1.09359	0.546793	−	0.837268i	$-0.315849\pi$
$257$	2.97590e8	1.09359	0.546793	+	0.837268i	$0.315849\pi$
$258$	0	0
$259$	1.24189e8	0.444156
$260$	0	0
$261$	1.14516e8	0.398679
$262$	0	0
$263$	3.39541e8	1.15092	0.575462	−	0.817829i	$-0.304822\pi$
$263$	3.39541e8	1.15092	0.575462	+	0.817829i	$0.304822\pi$
$264$	0	0
$265$	2.63703e8	0.870470
$266$	0	0
$267$	−1.11880e8	−0.359718
$268$	0	0
$269$	−2.68088e7	−0.0839739	−0.0419869	−	0.999118i	$-0.513369\pi$
$269$	−2.68088e7	−0.0839739	−0.0419869	+	0.999118i	$0.513369\pi$
$270$	0	0
$271$	−3.49721e8	−1.06741	−0.533703	−	0.845672i	$-0.679200\pi$
$271$	−3.49721e8	−1.06741	−0.533703	+	0.845672i	$0.679200\pi$
$272$	0	0
$273$	3.34040e8	0.993639
$274$	0	0
$275$	−1.60861e8	−0.466431
$276$	0	0
$277$	−4.34818e8	−1.22922	−0.614608	−	0.788833i	$-0.710686\pi$
$277$	−4.34818e8	−1.22922	−0.614608	+	0.788833i	$0.710686\pi$
$278$	0	0
$279$	−1.19964e7	−0.0330702
$280$	0	0
$281$	−1.27669e8	−0.343252	−0.171626	−	0.985162i	$-0.554902\pi$
$281$	−1.27669e8	−0.343252	−0.171626	+	0.985162i	$0.554902\pi$
$282$	0	0
$283$	2.78115e8	0.729411	0.364705	−	0.931123i	$-0.381170\pi$
$283$	2.78115e8	0.729411	0.364705	+	0.931123i	$0.381170\pi$
$284$	0	0
$285$	−1.66358e8	−0.425683
$286$	0	0
$287$	2.00604e8	0.500901
$288$	0	0
$289$	8.70405e7	0.212119
$290$	0	0
$291$	4.38203e7	0.104244
$292$	0	0
$293$	−2.26385e7	−0.0525788	−0.0262894	−	0.999654i	$-0.508369\pi$
$293$	−2.26385e7	−0.0525788	−0.0262894	+	0.999654i	$0.508369\pi$
$294$	0	0
$295$	−3.28919e8	−0.745954
$296$	0	0
$297$	4.88926e7	0.108292
$298$	0	0
$299$	−1.71195e9	−3.70376
$300$	0	0
$301$	3.68732e8	0.779343
$302$	0	0
$303$	−2.95347e7	−0.0609935
$304$	0	0
$305$	−7.81349e8	−1.57687
$306$	0	0
$307$	8.19497e8	1.61645	0.808226	−	0.588872i	$-0.200428\pi$
$307$	8.19497e8	1.61645	0.808226	+	0.588872i	$0.200428\pi$
$308$	0	0
$309$	4.22097e8	0.813875
$310$	0	0
$311$	9.24456e8	1.74271	0.871355	−	0.490654i	$-0.163242\pi$
$311$	9.24456e8	1.74271	0.871355	+	0.490654i	$0.163242\pi$
$312$	0	0
$313$	−3.23653e8	−0.596588	−0.298294	−	0.954474i	$-0.596418\pi$
$313$	−3.23653e8	−0.596588	−0.298294	+	0.954474i	$0.596418\pi$
$314$	0	0
$315$	2.29268e8	0.413291
$316$	0	0
$317$	7.12783e8	1.25675	0.628376	−	0.777910i	$-0.283720\pi$
$317$	7.12783e8	1.25675	0.628376	+	0.777910i	$0.283720\pi$
$318$	0	0
$319$	−3.90202e8	−0.673010
$320$	0	0
$321$	6.39975e7	0.107993
$322$	0	0
$323$	3.63523e8	0.600237
$324$	0	0
$325$	9.62966e8	1.55603
$326$	0	0
$327$	−5.74307e7	−0.0908295
$328$	0	0
$329$	−7.67730e8	−1.18856
$330$	0	0
$331$	−1.65095e8	−0.250229	−0.125114	−	0.992142i	$-0.539930\pi$
$331$	−1.65095e8	−0.250229	−0.125114	+	0.992142i	$0.539930\pi$
$332$	0	0
$333$	−1.08815e8	−0.161486
$334$	0	0
$335$	6.35323e8	0.923289
$336$	0	0
$337$	−8.09761e8	−1.15253	−0.576265	−	0.817263i	$-0.695490\pi$
$337$	−8.09761e8	−1.15253	−0.576265	+	0.817263i	$0.695490\pi$
$338$	0	0
$339$	5.86147e8	0.817160
$340$	0	0
$341$	4.08767e7	0.0558259
$342$	0	0
$343$	7.94445e8	1.06300
$344$	0	0
$345$	−1.17500e9	−1.54053
$346$	0	0
$347$	−4.44349e8	−0.570915	−0.285458	−	0.958391i	$-0.592146\pi$
$347$	−4.44349e8	−0.570915	−0.285458	+	0.958391i	$0.592146\pi$
$348$	0	0
$349$	−9.48806e8	−1.19478	−0.597390	−	0.801951i	$-0.703796\pi$
$349$	−9.48806e8	−1.19478	−0.597390	+	0.801951i	$0.703796\pi$
$350$	0	0
$351$	−2.92686e8	−0.361266
$352$	0	0
$353$	4.38524e8	0.530618	0.265309	−	0.964163i	$-0.414526\pi$
$353$	4.38524e8	0.530618	0.265309	+	0.964163i	$0.414526\pi$
$354$	0	0
$355$	4.04566e8	0.479943
$356$	0	0
$357$	−5.00992e8	−0.582763
$358$	0	0
$359$	−5.00653e8	−0.571092	−0.285546	−	0.958365i	$-0.592175\pi$
$359$	−5.00653e8	−0.571092	−0.285546	+	0.958365i	$0.592175\pi$
$360$	0	0
$361$	−6.28182e8	−0.702765
$362$	0	0
$363$	3.59557e8	0.394543
$364$	0	0
$365$	9.08460e8	0.977870
$366$	0	0
$367$	−1.61229e9	−1.70259	−0.851297	−	0.524685i	$-0.824183\pi$
$367$	−1.61229e9	−1.70259	−0.851297	+	0.524685i	$0.824183\pi$
$368$	0	0
$369$	−1.75769e8	−0.182117
$370$	0	0
$371$	5.80425e8	0.590116
$372$	0	0
$373$	9.55028e7	0.0952873	0.0476437	−	0.998864i	$-0.484829\pi$
$373$	9.55028e7	0.0952873	0.0476437	+	0.998864i	$0.484829\pi$
$374$	0	0
$375$	−1.36413e8	−0.133582
$376$	0	0
$377$	2.33587e9	2.24519
$378$	0	0
$379$	3.51478e8	0.331635	0.165818	−	0.986156i	$-0.446974\pi$
$379$	3.51478e8	0.331635	0.165818	+	0.986156i	$0.446974\pi$
$380$	0	0
$381$	2.78095e8	0.257606
$382$	0	0
$383$	−1.21120e9	−1.10159	−0.550797	−	0.834639i	$-0.685676\pi$
$383$	−1.21120e9	−1.10159	−0.550797	+	0.834639i	$0.685676\pi$
$384$	0	0
$385$	−7.81208e8	−0.697677
$386$	0	0
$387$	−3.23084e8	−0.283353
$388$	0	0
$389$	−1.43013e7	−0.0123183	−0.00615917	−	0.999981i	$-0.501961\pi$
$389$	−1.43013e7	−0.0123183	−0.00615917	+	0.999981i	$0.501961\pi$
$390$	0	0
$391$	2.56758e9	2.17223
$392$	0	0
$393$	1.07682e9	0.894892
$394$	0	0
$395$	−8.69972e8	−0.710256
$396$	0	0
$397$	1.83153e9	1.46908	0.734541	−	0.678564i	$-0.237397\pi$
$397$	1.83153e9	1.46908	0.734541	+	0.678564i	$0.237397\pi$
$398$	0	0
$399$	−3.66163e8	−0.288582
$400$	0	0
$401$	−3.14244e8	−0.243367	−0.121684	−	0.992569i	$-0.538829\pi$
$401$	−3.14244e8	−0.243367	−0.121684	+	0.992569i	$0.538829\pi$
$402$	0	0
$403$	−2.44701e8	−0.186238
$404$	0	0
$405$	−2.00885e8	−0.150264
$406$	0	0
$407$	3.70777e8	0.272604
$408$	0	0
$409$	1.39490e8	0.100812	0.0504059	−	0.998729i	$-0.483948\pi$
$409$	1.39490e8	0.100812	0.0504059	+	0.998729i	$0.483948\pi$
$410$	0	0
$411$	6.95013e8	0.493795
$412$	0	0
$413$	−7.23970e8	−0.505703
$414$	0	0
$415$	−1.77989e9	−1.22243
$416$	0	0
$417$	−7.08505e8	−0.478483
$418$	0	0
$419$	−2.31748e9	−1.53910	−0.769549	−	0.638588i	$-0.779519\pi$
$419$	−2.31748e9	−1.53910	−0.769549	+	0.638588i	$0.779519\pi$
$420$	0	0
$421$	−1.05935e9	−0.691913	−0.345957	−	0.938250i	$-0.612446\pi$
$421$	−1.05935e9	−0.691913	−0.345957	+	0.938250i	$0.612446\pi$
$422$	0	0
$423$	6.72686e8	0.432137
$424$	0	0
$425$	−1.44426e9	−0.912605
$426$	0	0
$427$	−1.71980e9	−1.06900
$428$	0	0
$429$	9.97301e8	0.609854
$430$	0	0
$431$	−1.70587e9	−1.02630	−0.513150	−	0.858299i	$-0.671522\pi$
$431$	−1.70587e9	−1.02630	−0.513150	+	0.858299i	$0.671522\pi$
$432$	0	0
$433$	2.71320e9	1.60611	0.803054	−	0.595906i	$-0.203207\pi$
$433$	2.71320e9	1.60611	0.803054	+	0.595906i	$0.203207\pi$
$434$	0	0
$435$	1.60322e9	0.933858
$436$	0	0
$437$	1.87659e9	1.07568
$438$	0	0
$439$	1.66778e9	0.940832	0.470416	−	0.882445i	$-0.344104\pi$
$439$	1.66778e9	0.940832	0.470416	+	0.882445i	$0.344104\pi$
$440$	0	0
$441$	−9.57316e7	−0.0531521
$442$	0	0
$443$	−7.96843e8	−0.435472	−0.217736	−	0.976008i	$-0.569867\pi$
$443$	−7.96843e8	−0.435472	−0.217736	+	0.976008i	$0.569867\pi$
$444$	0	0
$445$	−1.56631e9	−0.842595
$446$	0	0
$447$	−1.88495e9	−0.998214
$448$	0	0
$449$	−9.88997e8	−0.515624	−0.257812	−	0.966195i	$-0.583001\pi$
$449$	−9.88997e8	−0.515624	−0.257812	+	0.966195i	$0.583001\pi$
$450$	0	0
$451$	5.98917e8	0.307432
$452$	0	0
$453$	−7.32605e8	−0.370277
$454$	0	0
$455$	4.67656e9	2.32748
$456$	0	0
$457$	7.56748e8	0.370890	0.185445	−	0.982655i	$-0.440627\pi$
$457$	7.56748e8	0.370890	0.185445	+	0.982655i	$0.440627\pi$
$458$	0	0
$459$	4.38970e8	0.211880
$460$	0	0
$461$	−2.54215e9	−1.20851	−0.604253	−	0.796793i	$-0.706528\pi$
$461$	−2.54215e9	−1.20851	−0.604253	+	0.796793i	$0.706528\pi$
$462$	0	0
$463$	1.35745e9	0.635611	0.317806	−	0.948156i	$-0.397054\pi$
$463$	1.35745e9	0.635611	0.317806	+	0.948156i	$0.397054\pi$
$464$	0	0
$465$	−1.67950e8	−0.0774630
$466$	0	0
$467$	3.65551e9	1.66088	0.830441	−	0.557107i	$-0.188089\pi$
$467$	3.65551e9	1.66088	0.830441	+	0.557107i	$0.188089\pi$
$468$	0	0
$469$	1.39838e9	0.625923
$470$	0	0
$471$	2.96613e8	0.130803
$472$	0	0
$473$	1.10088e9	0.478328
$474$	0	0
$475$	−1.05557e9	−0.451918
$476$	0	0
$477$	−5.08569e8	−0.214554
$478$	0	0
$479$	1.22597e9	0.509687	0.254844	−	0.966982i	$-0.417976\pi$
$479$	1.22597e9	0.509687	0.254844	+	0.966982i	$0.417976\pi$
$480$	0	0
$481$	−2.21959e9	−0.909419
$482$	0	0
$483$	−2.58624e9	−1.04437
$484$	0	0
$485$	6.13484e8	0.244178
$486$	0	0
$487$	−4.96327e9	−1.94723	−0.973613	−	0.228204i	$-0.926715\pi$
$487$	−4.96327e9	−1.94723	−0.973613	+	0.228204i	$0.926715\pi$
$488$	0	0
$489$	2.27859e9	0.881223
$490$	0	0
$491$	2.58972e8	0.0987344	0.0493672	−	0.998781i	$-0.484280\pi$
$491$	2.58972e8	0.0987344	0.0493672	+	0.998781i	$0.484280\pi$
$492$	0	0
$493$	−3.50333e9	−1.31679
$494$	0	0
$495$	6.84496e8	0.253660
$496$	0	0
$497$	8.90473e8	0.325367
$498$	0	0
$499$	3.60546e9	1.29900	0.649500	−	0.760362i	$-0.274978\pi$
$499$	3.60546e9	1.29900	0.649500	+	0.760362i	$0.274978\pi$
$500$	0	0
$501$	−1.60776e9	−0.571201
$502$	0	0
$503$	−2.25140e9	−0.788795	−0.394398	−	0.918940i	$-0.629047\pi$
$503$	−2.25140e9	−0.788795	−0.394398	+	0.918940i	$0.629047\pi$
$504$	0	0
$505$	−4.13486e8	−0.142870
$506$	0	0
$507$	−4.27595e9	−1.45715
$508$	0	0
$509$	−5.03059e9	−1.69086	−0.845428	−	0.534090i	$-0.820654\pi$
$509$	−5.03059e9	−1.69086	−0.845428	+	0.534090i	$0.820654\pi$
$510$	0	0
$511$	1.99957e9	0.662925
$512$	0	0
$513$	3.20833e8	0.104922
$514$	0	0
$515$	5.90936e9	1.90641
$516$	0	0
$517$	−2.29212e9	−0.729491
$518$	0	0
$519$	5.72823e8	0.179860
$520$	0	0
$521$	8.62267e8	0.267122	0.133561	−	0.991041i	$-0.457359\pi$
$521$	8.62267e8	0.267122	0.133561	+	0.991041i	$0.457359\pi$
$522$	0	0
$523$	−2.60059e9	−0.794907	−0.397453	−	0.917622i	$-0.630106\pi$
$523$	−2.60059e9	−0.794907	−0.397453	+	0.917622i	$0.630106\pi$
$524$	0	0
$525$	1.45475e9	0.438763
$526$	0	0
$527$	3.67002e8	0.109227
$528$	0	0
$529$	9.84963e9	2.89284
$530$	0	0
$531$	6.34344e8	0.183863
$532$	0	0
$533$	−3.58531e9	−1.02561
$534$	0	0
$535$	8.95964e8	0.252960
$536$	0	0
$537$	2.56752e9	0.715490
$538$	0	0
$539$	3.26196e8	0.0897260
$540$	0	0
$541$	−3.78391e8	−0.102742	−0.0513712	−	0.998680i	$-0.516359\pi$
$541$	−3.78391e8	−0.102742	−0.0513712	+	0.998680i	$0.516359\pi$
$542$	0	0
$543$	3.55675e9	0.953354
$544$	0	0
$545$	−8.04029e8	−0.212757
$546$	0	0
$547$	1.35980e9	0.355238	0.177619	−	0.984099i	$-0.443160\pi$
$547$	1.35980e9	0.355238	0.177619	+	0.984099i	$0.443160\pi$
$548$	0	0
$549$	1.50689e9	0.388667
$550$	0	0
$551$	−2.56050e9	−0.652070
$552$	0	0
$553$	−1.91486e9	−0.481503
$554$	0	0
$555$	−1.52341e9	−0.378261
$556$	0	0
$557$	3.33022e9	0.816544	0.408272	−	0.912860i	$-0.366131\pi$
$557$	3.33022e9	0.816544	0.408272	+	0.912860i	$0.366131\pi$
$558$	0	0
$559$	−6.59021e9	−1.59572
$560$	0	0
$561$	−1.49575e9	−0.357675
$562$	0	0
$563$	−6.37690e9	−1.50602	−0.753009	−	0.658010i	$-0.771398\pi$
$563$	−6.37690e9	−1.50602	−0.753009	+	0.658010i	$0.771398\pi$
$564$	0	0
$565$	8.20605e9	1.91410
$566$	0	0
$567$	−4.42159e8	−0.101868
$568$	0	0
$569$	2.87492e9	0.654235	0.327117	−	0.944984i	$-0.393923\pi$
$569$	2.87492e9	0.654235	0.327117	+	0.944984i	$0.393923\pi$
$570$	0	0
$571$	−2.26452e8	−0.0509037	−0.0254519	−	0.999676i	$-0.508102\pi$
$571$	−2.26452e8	−0.0509037	−0.0254519	+	0.999676i	$0.508102\pi$
$572$	0	0
$573$	1.56727e9	0.348018
$574$	0	0
$575$	−7.45557e9	−1.63547
$576$	0	0
$577$	6.03280e8	0.130738	0.0653692	−	0.997861i	$-0.479177\pi$
$577$	6.03280e8	0.130738	0.0653692	+	0.997861i	$0.479177\pi$
$578$	0	0
$579$	1.86863e9	0.400082
$580$	0	0
$581$	−3.91763e9	−0.828719
$582$	0	0
$583$	1.73290e9	0.362188
$584$	0	0
$585$	−4.09761e9	−0.846223
$586$	0	0
$587$	−6.79902e7	−0.0138744	−0.00693718	−	0.999976i	$-0.502208\pi$
$587$	−6.79902e7	−0.0138744	−0.00693718	+	0.999976i	$0.502208\pi$
$588$	0	0
$589$	2.68233e8	0.0540889
$590$	0	0
$591$	1.44139e9	0.287227
$592$	0	0
$593$	−5.80175e9	−1.14253	−0.571265	−	0.820766i	$-0.693547\pi$
$593$	−5.80175e9	−1.14253	−0.571265	+	0.820766i	$0.693547\pi$
$594$	0	0
$595$	−7.01389e9	−1.36505
$596$	0	0
$597$	−1.94166e9	−0.373477
$598$	0	0
$599$	5.05519e9	0.961046	0.480523	−	0.876982i	$-0.340447\pi$
$599$	5.05519e9	0.961046	0.480523	+	0.876982i	$0.340447\pi$
$600$	0	0
$601$	−5.97061e9	−1.12191	−0.560955	−	0.827846i	$-0.689566\pi$
$601$	−5.97061e9	−1.12191	−0.560955	+	0.827846i	$0.689566\pi$
$602$	0	0
$603$	−1.22527e9	−0.227572
$604$	0	0
$605$	5.03379e9	0.924169
$606$	0	0
$607$	4.03712e9	0.732675	0.366337	−	0.930482i	$-0.380612\pi$
$607$	4.03712e9	0.732675	0.366337	+	0.930482i	$0.380612\pi$
$608$	0	0
$609$	3.52878e9	0.633088
$610$	0	0
$611$	1.37213e10	2.43361
$612$	0	0
$613$	−8.33838e8	−0.146208	−0.0731038	−	0.997324i	$-0.523290\pi$
$613$	−8.33838e8	−0.146208	−0.0731038	+	0.997324i	$0.523290\pi$
$614$	0	0
$615$	−2.46077e9	−0.426587
$616$	0	0
$617$	8.21270e9	1.40763	0.703814	−	0.710384i	$-0.251479\pi$
$617$	8.21270e9	1.40763	0.703814	+	0.710384i	$0.251479\pi$
$618$	0	0
$619$	5.43752e9	0.921475	0.460737	−	0.887537i	$-0.347585\pi$
$619$	5.43752e9	0.921475	0.460737	+	0.887537i	$0.347585\pi$
$620$	0	0
$621$	2.26606e9	0.379710
$622$	0	0
$623$	−3.44755e9	−0.571219
$624$	0	0
$625$	−6.96908e9	−1.14181
$626$	0	0
$627$	−1.09321e9	−0.177120
$628$	0	0
$629$	3.32893e9	0.533369
$630$	0	0
$631$	6.20797e9	0.983664	0.491832	−	0.870690i	$-0.336327\pi$
$631$	6.20797e9	0.983664	0.491832	+	0.870690i	$0.336327\pi$
$632$	0	0
$633$	−5.54750e9	−0.869329
$634$	0	0
$635$	3.89334e9	0.603412
$636$	0	0
$637$	−1.95271e9	−0.299330
$638$	0	0
$639$	−7.80234e8	−0.118297
$640$	0	0
$641$	−3.31503e8	−0.0497147	−0.0248573	−	0.999691i	$-0.507913\pi$
$641$	−3.31503e8	−0.0497147	−0.0248573	+	0.999691i	$0.507913\pi$
$642$	0	0
$643$	−6.37327e8	−0.0945418	−0.0472709	−	0.998882i	$-0.515052\pi$
$643$	−6.37327e8	−0.0945418	−0.0472709	+	0.998882i	$0.515052\pi$
$644$	0	0
$645$	−4.52318e9	−0.663719
$646$	0	0
$647$	−7.36821e9	−1.06954	−0.534770	−	0.844998i	$-0.679602\pi$
$647$	−7.36821e9	−1.06954	−0.534770	+	0.844998i	$0.679602\pi$
$648$	0	0
$649$	−2.16147e9	−0.310379
$650$	0	0
$651$	−3.69668e8	−0.0525143
$652$	0	0
$653$	9.87503e9	1.38785	0.693924	−	0.720048i	$-0.255880\pi$
$653$	9.87503e9	1.38785	0.693924	+	0.720048i	$0.255880\pi$
$654$	0	0
$655$	1.50755e10	2.09618
$656$	0	0
$657$	−1.75203e9	−0.241025
$658$	0	0
$659$	−1.21767e10	−1.65741	−0.828705	−	0.559686i	$-0.810922\pi$
$659$	−1.21767e10	−1.65741	−0.828705	+	0.559686i	$0.810922\pi$
$660$	0	0
$661$	−1.11032e10	−1.49536	−0.747678	−	0.664062i	$-0.768831\pi$
$661$	−1.11032e10	−1.49536	−0.747678	+	0.664062i	$0.768831\pi$
$662$	0	0
$663$	8.95403e9	1.19322
$664$	0	0
$665$	−5.12628e9	−0.675969
$666$	0	0
$667$	−1.80850e10	−2.35982
$668$	0	0
$669$	−1.38377e9	−0.178679
$670$	0	0
$671$	−5.13458e9	−0.656109
$672$	0	0
$673$	−3.49663e9	−0.442177	−0.221089	−	0.975254i	$-0.570961\pi$
$673$	−3.49663e9	−0.442177	−0.221089	+	0.975254i	$0.570961\pi$
$674$	0	0
$675$	−1.27465e9	−0.159525
$676$	0	0
$677$	6.40744e9	0.793641	0.396820	−	0.917896i	$-0.370114\pi$
$677$	6.40744e9	0.793641	0.396820	+	0.917896i	$0.370114\pi$
$678$	0	0
$679$	1.35031e9	0.165535
$680$	0	0
$681$	−3.16105e9	−0.383545
$682$	0	0
$683$	1.48217e10	1.78002	0.890011	−	0.455938i	$-0.150696\pi$
$683$	1.48217e10	1.78002	0.890011	+	0.455938i	$0.150696\pi$
$684$	0	0
$685$	9.73018e9	1.15665
$686$	0	0
$687$	5.28661e9	0.622055
$688$	0	0
$689$	−1.03737e10	−1.20828
$690$	0	0
$691$	1.91226e9	0.220482	0.110241	−	0.993905i	$-0.464838\pi$
$691$	1.91226e9	0.220482	0.110241	+	0.993905i	$0.464838\pi$
$692$	0	0
$693$	1.50662e9	0.171963
$694$	0	0
$695$	−9.91907e9	−1.12079
$696$	0	0
$697$	5.37724e9	0.601512
$698$	0	0
$699$	3.53563e9	0.391558
$700$	0	0
$701$	−1.15650e10	−1.26804	−0.634020	−	0.773317i	$-0.718596\pi$
$701$	−1.15650e10	−1.26804	−0.634020	+	0.773317i	$0.718596\pi$
$702$	0	0
$703$	2.43304e9	0.264122
$704$	0	0
$705$	9.41761e9	1.01223
$706$	0	0
$707$	−9.10106e8	−0.0968555
$708$	0	0
$709$	5.17466e9	0.545281	0.272641	−	0.962116i	$-0.412103\pi$
$709$	5.17466e9	0.545281	0.272641	+	0.962116i	$0.412103\pi$
$710$	0	0
$711$	1.67780e9	0.175064
$712$	0	0
$713$	1.89455e9	0.195746
$714$	0	0
$715$	1.39622e10	1.42851
$716$	0	0
$717$	−4.65873e9	−0.472009
$718$	0	0
$719$	−9.97915e9	−1.00125	−0.500625	−	0.865664i	$-0.666896\pi$
$719$	−9.97915e9	−1.00125	−0.500625	+	0.865664i	$0.666896\pi$
$720$	0	0
$721$	1.30068e10	1.29241
$722$	0	0
$723$	−2.83697e9	−0.279171
$724$	0	0
$725$	1.01727e10	0.991413
$726$	0	0
$727$	−5.25105e9	−0.506846	−0.253423	−	0.967356i	$-0.581556\pi$
$727$	−5.25105e9	−0.506846	−0.253423	+	0.967356i	$0.581556\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	9.88398e9	0.935882
$732$	0	0
$733$	1.21211e10	1.13679	0.568393	−	0.822757i	$-0.307565\pi$
$733$	1.21211e10	1.13679	0.568393	+	0.822757i	$0.307565\pi$
$734$	0	0
$735$	−1.34024e9	−0.124502
$736$	0	0
$737$	4.17498e9	0.384165
$738$	0	0
$739$	−1.85860e10	−1.69406	−0.847032	−	0.531542i	$-0.821613\pi$
$739$	−1.85860e10	−1.69406	−0.847032	+	0.531542i	$0.821613\pi$
$740$	0	0
$741$	6.54429e9	0.590879
$742$	0	0
$743$	−1.45147e10	−1.29822	−0.649109	−	0.760696i	$-0.724858\pi$
$743$	−1.45147e10	−1.29822	−0.649109	+	0.760696i	$0.724858\pi$
$744$	0	0
$745$	−2.63893e10	−2.33820
$746$	0	0
$747$	3.43264e9	0.301305
$748$	0	0
$749$	1.97207e9	0.171489
$750$	0	0
$751$	−4.06991e9	−0.350626	−0.175313	−	0.984513i	$-0.556094\pi$
$751$	−4.06991e9	−0.350626	−0.175313	+	0.984513i	$0.556094\pi$
$752$	0	0
$753$	−3.61135e9	−0.308239
$754$	0	0
$755$	−1.02565e10	−0.867328
$756$	0	0
$757$	5.68835e9	0.476596	0.238298	−	0.971192i	$-0.423410\pi$
$757$	5.68835e9	0.476596	0.238298	+	0.971192i	$0.423410\pi$
$758$	0	0
$759$	−7.72140e9	−0.640988
$760$	0	0
$761$	−1.26392e10	−1.03962	−0.519809	−	0.854282i	$-0.673997\pi$
$761$	−1.26392e10	−1.03962	−0.519809	+	0.854282i	$0.673997\pi$
$762$	0	0
$763$	−1.76972e9	−0.144234
$764$	0	0
$765$	6.14558e9	0.496304
$766$	0	0
$767$	1.29392e10	1.03544
$768$	0	0
$769$	−1.40530e10	−1.11436	−0.557182	−	0.830391i	$-0.688117\pi$
$769$	−1.40530e10	−1.11436	−0.557182	+	0.830391i	$0.688117\pi$
$770$	0	0
$771$	−8.03494e9	−0.631382
$772$	0	0
$773$	1.74521e10	1.35900	0.679500	−	0.733675i	$-0.262197\pi$
$773$	1.74521e10	1.35900	0.679500	+	0.733675i	$0.262197\pi$
$774$	0	0
$775$	−1.06567e9	−0.0822372
$776$	0	0
$777$	−3.35311e9	−0.256433
$778$	0	0
$779$	3.93009e9	0.297867
$780$	0	0
$781$	2.65858e9	0.199696
$782$	0	0
$783$	−3.09192e9	−0.230177
$784$	0	0
$785$	4.15258e9	0.306390
$786$	0	0
$787$	1.57180e10	1.14944	0.574720	−	0.818350i	$-0.305111\pi$
$787$	1.57180e10	1.14944	0.574720	+	0.818350i	$0.305111\pi$
$788$	0	0
$789$	−9.16759e9	−0.664486
$790$	0	0
$791$	1.80620e10	1.29762
$792$	0	0
$793$	3.07372e10	2.18881
$794$	0	0
$795$	−7.11997e9	−0.502566
$796$	0	0
$797$	−1.80766e10	−1.26477	−0.632385	−	0.774654i	$-0.717924\pi$
$797$	−1.80766e10	−1.26477	−0.632385	+	0.774654i	$0.717924\pi$
$798$	0	0
$799$	−2.05792e10	−1.42730
$800$	0	0
$801$	3.02075e9	0.207683
$802$	0	0
$803$	5.96988e9	0.406875
$804$	0	0
$805$	−3.62073e10	−2.44630
$806$	0	0
$807$	7.23837e8	0.0484823
$808$	0	0
$809$	2.14745e10	1.42595	0.712973	−	0.701192i	$-0.247348\pi$
$809$	2.14745e10	1.42595	0.712973	+	0.701192i	$0.247348\pi$
$810$	0	0
$811$	2.05508e10	1.35287	0.676436	−	0.736502i	$-0.263524\pi$
$811$	2.05508e10	1.35287	0.676436	+	0.736502i	$0.263524\pi$
$812$	0	0
$813$	9.44248e9	0.616267
$814$	0	0
$815$	3.19003e10	2.06416
$816$	0	0
$817$	7.22396e9	0.463445
$818$	0	0
$819$	−9.01907e9	−0.573678
$820$	0	0
$821$	−5.11303e9	−0.322461	−0.161231	−	0.986917i	$-0.551546\pi$
$821$	−5.11303e9	−0.322461	−0.161231	+	0.986917i	$0.551546\pi$
$822$	0	0
$823$	−1.96455e9	−0.122847	−0.0614233	−	0.998112i	$-0.519564\pi$
$823$	−1.96455e9	−0.122847	−0.0614233	+	0.998112i	$0.519564\pi$
$824$	0	0
$825$	4.34326e9	0.269294
$826$	0	0
$827$	−8.45961e9	−0.520093	−0.260047	−	0.965596i	$-0.583738\pi$
$827$	−8.45961e9	−0.520093	−0.260047	+	0.965596i	$0.583738\pi$
$828$	0	0
$829$	3.79163e9	0.231145	0.115573	−	0.993299i	$-0.463130\pi$
$829$	3.79163e9	0.231145	0.115573	+	0.993299i	$0.463130\pi$
$830$	0	0
$831$	1.17401e10	0.709688
$832$	0	0
$833$	2.92868e9	0.175555
$834$	0	0
$835$	−2.25086e10	−1.33797
$836$	0	0
$837$	3.23903e8	0.0190931
$838$	0	0
$839$	−5.75205e9	−0.336245	−0.168122	−	0.985766i	$-0.553770\pi$
$839$	−5.75205e9	−0.336245	−0.168122	+	0.985766i	$0.553770\pi$
$840$	0	0
$841$	7.42614e9	0.430504
$842$	0	0
$843$	3.44706e9	0.198177
$844$	0	0
$845$	−5.98632e10	−3.41320
$846$	0	0
$847$	1.10797e10	0.626520
$848$	0	0
$849$	−7.50911e9	−0.421125
$850$	0	0
$851$	1.71847e10	0.955847
$852$	0	0
$853$	−1.54604e10	−0.852904	−0.426452	−	0.904510i	$-0.640237\pi$
$853$	−1.54604e10	−0.852904	−0.426452	+	0.904510i	$0.640237\pi$
$854$	0	0
$855$	4.49166e9	0.245768
$856$	0	0
$857$	−2.91908e10	−1.58421	−0.792105	−	0.610385i	$-0.791015\pi$
$857$	−2.91908e10	−1.58421	−0.792105	+	0.610385i	$0.791015\pi$
$858$	0	0
$859$	1.51850e10	0.817410	0.408705	−	0.912667i	$-0.365981\pi$
$859$	1.51850e10	0.817410	0.408705	+	0.912667i	$0.365981\pi$
$860$	0	0
$861$	−5.41630e9	−0.289195
$862$	0	0
$863$	1.19335e10	0.632021	0.316011	−	0.948756i	$-0.397656\pi$
$863$	1.19335e10	0.632021	0.316011	+	0.948756i	$0.397656\pi$
$864$	0	0
$865$	8.01953e9	0.421301
$866$	0	0
$867$	−2.35009e9	−0.122467
$868$	0	0
$869$	−5.71696e9	−0.295526
$870$	0	0
$871$	−2.49927e10	−1.28159
$872$	0	0
$873$	−1.18315e9	−0.0601851
$874$	0	0
$875$	−4.20355e9	−0.212123
$876$	0	0
$877$	2.25544e10	1.12910	0.564549	−	0.825399i	$-0.309050\pi$
$877$	2.25544e10	1.12910	0.564549	+	0.825399i	$0.309050\pi$
$878$	0	0
$879$	6.11239e8	0.0303564
$880$	0	0
$881$	−6.33664e9	−0.312208	−0.156104	−	0.987741i	$-0.549893\pi$
$881$	−6.33664e9	−0.312208	−0.156104	+	0.987741i	$0.549893\pi$
$882$	0	0
$883$	−9.72527e9	−0.475378	−0.237689	−	0.971341i	$-0.576390\pi$
$883$	−9.72527e9	−0.475378	−0.237689	+	0.971341i	$0.576390\pi$
$884$	0	0
$885$	8.88081e9	0.430677
$886$	0	0
$887$	−3.36353e10	−1.61831	−0.809156	−	0.587593i	$-0.800075\pi$
$887$	−3.36353e10	−1.61831	−0.809156	+	0.587593i	$0.800075\pi$
$888$	0	0
$889$	8.56946e9	0.409070
$890$	0	0
$891$	−1.32010e9	−0.0625223
$892$	0	0
$893$	−1.50409e10	−0.706793
$894$	0	0
$895$	3.59452e10	1.67595
$896$	0	0
$897$	4.62227e10	2.13837
$898$	0	0
$899$	−2.58501e9	−0.118660
$900$	0	0
$901$	1.55585e10	0.708647
$902$	0	0
$903$	−9.95578e9	−0.449954
$904$	0	0
$905$	4.97945e10	2.23312
$906$	0	0
$907$	−1.24681e10	−0.554851	−0.277425	−	0.960747i	$-0.589481\pi$
$907$	−1.24681e10	−0.554851	−0.277425	+	0.960747i	$0.589481\pi$
$908$	0	0
$909$	7.97437e8	0.0352146
$910$	0	0
$911$	1.89363e10	0.829814	0.414907	−	0.909864i	$-0.363814\pi$
$911$	1.89363e10	0.829814	0.414907	+	0.909864i	$0.363814\pi$
$912$	0	0
$913$	−1.16964e10	−0.508633
$914$	0	0
$915$	2.10964e10	0.910406
$916$	0	0
$917$	3.31821e10	1.42106
$918$	0	0
$919$	−1.24971e10	−0.531135	−0.265568	−	0.964092i	$-0.585559\pi$
$919$	−1.24971e10	−0.531135	−0.265568	+	0.964092i	$0.585559\pi$
$920$	0	0
$921$	−2.21264e10	−0.933259
$922$	0	0
$923$	−1.59151e10	−0.666197
$924$	0	0
$925$	−9.66632e9	−0.401573
$926$	0	0
$927$	−1.13966e10	−0.469891
$928$	0	0
$929$	−7.35427e9	−0.300943	−0.150472	−	0.988614i	$-0.548079\pi$
$929$	−7.35427e9	−0.300943	−0.150472	+	0.988614i	$0.548079\pi$
$930$	0	0
$931$	2.14050e9	0.0869343
$932$	0	0
$933$	−2.49603e10	−1.00615
$934$	0	0
$935$	−2.09405e10	−0.837812
$936$	0	0
$937$	1.36199e10	0.540861	0.270430	−	0.962740i	$-0.412834\pi$
$937$	1.36199e10	0.540861	0.270430	+	0.962740i	$0.412834\pi$
$938$	0	0
$939$	8.73863e9	0.344440
$940$	0	0
$941$	1.02933e10	0.402707	0.201353	−	0.979519i	$-0.435466\pi$
$941$	1.02933e10	0.402707	0.201353	+	0.979519i	$0.435466\pi$
$942$	0	0
$943$	2.77585e10	1.07797
$944$	0	0
$945$	−6.19022e9	−0.238614
$946$	0	0
$947$	−2.39142e10	−0.915022	−0.457511	−	0.889204i	$-0.651259\pi$
$947$	−2.39142e10	−0.915022	−0.457511	+	0.889204i	$0.651259\pi$
$948$	0	0
$949$	−3.57376e10	−1.35735
$950$	0	0
$951$	−1.92451e10	−0.725586
$952$	0	0
$953$	2.83252e10	1.06010	0.530051	−	0.847966i	$-0.322173\pi$
$953$	2.83252e10	1.06010	0.530051	+	0.847966i	$0.322173\pi$
$954$	0	0
$955$	2.19418e10	0.815191
$956$	0	0
$957$	1.05354e10	0.388563
$958$	0	0
$959$	2.14167e10	0.784128
$960$	0	0
$961$	−2.72418e10	−0.990157
$962$	0	0
$963$	−1.72793e9	−0.0623497
$964$	0	0
$965$	2.61609e10	0.937145
$966$	0	0
$967$	−3.64982e10	−1.29801	−0.649007	−	0.760783i	$-0.724815\pi$
$967$	−3.64982e10	−1.29801	−0.649007	+	0.760783i	$0.724815\pi$
$968$	0	0
$969$	−9.81511e9	−0.346547
$970$	0	0
$971$	−2.88087e10	−1.00985	−0.504924	−	0.863164i	$-0.668479\pi$
$971$	−2.88087e10	−1.00985	−0.504924	+	0.863164i	$0.668479\pi$
$972$	0	0
$973$	−2.18324e10	−0.759814
$974$	0	0
$975$	−2.60001e10	−0.898377
$976$	0	0
$977$	−4.93806e10	−1.69405	−0.847023	−	0.531556i	$-0.821607\pi$
$977$	−4.93806e10	−1.69405	−0.847023	+	0.531556i	$0.821607\pi$
$978$	0	0
$979$	−1.02929e10	−0.350590
$980$	0	0
$981$	1.55063e9	0.0524404
$982$	0	0
$983$	5.11399e10	1.71721	0.858603	−	0.512641i	$-0.171333\pi$
$983$	5.11399e10	1.71721	0.858603	+	0.512641i	$0.171333\pi$
$984$	0	0
$985$	2.01794e10	0.672795
$986$	0	0
$987$	2.07287e10	0.686218
$988$	0	0
$989$	5.10233e10	1.67719
$990$	0	0
$991$	−5.74784e10	−1.87606	−0.938030	−	0.346553i	$-0.887352\pi$
$991$	−5.74784e10	−1.87606	−0.938030	+	0.346553i	$0.887352\pi$
$992$	0	0
$993$	4.45758e9	0.144470
$994$	0	0
$995$	−2.71833e10	−0.874824
$996$	0	0
$997$	1.98127e10	0.633155	0.316577	−	0.948567i	$-0.397466\pi$
$997$	1.98127e10	0.633155	0.316577	+	0.948567i	$0.397466\pi$
$998$	0	0
$999$	2.93800e9	0.0932337

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	12.8.a.a.1.1	✓	1
3.2	odd	2		36.8.a.c.1.1		1
4.3	odd	2		48.8.a.e.1.1		1
5.2	odd	4		300.8.d.c.49.2		2
5.3	odd	4		300.8.d.c.49.1		2
5.4	even	2		300.8.a.g.1.1		1
7.2	even	3		588.8.i.h.361.1		2
7.3	odd	6		588.8.i.a.373.1		2
7.4	even	3		588.8.i.h.373.1		2
7.5	odd	6		588.8.i.a.361.1		2
7.6	odd	2		588.8.a.d.1.1		1
8.3	odd	2		192.8.a.g.1.1		1
8.5	even	2		192.8.a.o.1.1		1
9.2	odd	6		324.8.e.a.109.1		2
9.4	even	3		324.8.e.f.217.1		2
9.5	odd	6		324.8.e.a.217.1		2
9.7	even	3		324.8.e.f.109.1		2
12.11	even	2		144.8.a.j.1.1		1
24.5	odd	2		576.8.a.d.1.1		1
24.11	even	2		576.8.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.8.a.a.1.1	✓	1	1.1	even	1	trivial
36.8.a.c.1.1		1	3.2	odd	2
48.8.a.e.1.1		1	4.3	odd	2
144.8.a.j.1.1		1	12.11	even	2
192.8.a.g.1.1		1	8.3	odd	2
192.8.a.o.1.1		1	8.5	even	2
300.8.a.g.1.1		1	5.4	even	2
300.8.d.c.49.1		2	5.3	odd	4
300.8.d.c.49.2		2	5.2	odd	4
324.8.e.a.109.1		2	9.2	odd	6
324.8.e.a.217.1		2	9.5	odd	6
324.8.e.f.109.1		2	9.7	even	3
324.8.e.f.217.1		2	9.4	even	3
576.8.a.d.1.1		1	24.5	odd	2
576.8.a.e.1.1		1	24.11	even	2
588.8.a.d.1.1		1	7.6	odd	2
588.8.i.a.361.1		2	7.5	odd	6
588.8.i.a.373.1		2	7.3	odd	6
588.8.i.h.361.1		2	7.2	even	3
588.8.i.h.373.1		2	7.4	even	3

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:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	12.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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