LMFDB - Embedded newform 3969.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{8} -1.00000 q^{10} +5.00000 q^{11} -5.00000 q^{13} -1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{19} +1.00000 q^{20} +5.00000 q^{22} +3.00000 q^{23} -4.00000 q^{25} -5.00000 q^{26} -1.00000 q^{29} +5.00000 q^{32} +3.00000 q^{34} +3.00000 q^{37} +1.00000 q^{38} +3.00000 q^{40} -5.00000 q^{41} -1.00000 q^{43} -5.00000 q^{44} +3.00000 q^{46} -4.00000 q^{50} +5.00000 q^{52} -9.00000 q^{53} -5.00000 q^{55} -1.00000 q^{58} -14.0000 q^{61} +7.00000 q^{64} +5.00000 q^{65} +4.00000 q^{67} -3.00000 q^{68} -12.0000 q^{71} +3.00000 q^{73} +3.00000 q^{74} -1.00000 q^{76} +8.00000 q^{79} +1.00000 q^{80} -5.00000 q^{82} -9.00000 q^{83} -3.00000 q^{85} -1.00000 q^{86} -15.0000 q^{88} -13.0000 q^{89} -3.00000 q^{92} -1.00000 q^{95} -9.00000 q^{97} +O(q^{100})$

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	0	0
$10$	−1.00000	−0.316228
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	5.00000	1.06600
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−5.00000	−0.980581
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	3.00000	0.474342
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	3.00000	0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	−4.00000	−0.565685
$51$	0	0
$52$	5.00000	0.693375
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−5.00000	−0.674200
$56$	0	0
$57$	0	0
$58$	−1.00000	−0.131306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	0	0
$64$	7.00000	0.875000
$65$	5.00000	0.620174
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−5.00000	−0.552158
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−15.0000	−1.59901
$89$	−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
$92$	−3.00000	−0.312772
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	0	0
$99$	0	0
$100$	4.00000	0.400000
$101$	−17.0000	−1.69156	−0.845782	−	0.533529i	$-0.820865\pi$
$101$	−17.0000	−1.69156	−0.845782	+	0.533529i	$0.820865\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	15.0000	1.47087
$105$	0	0
$106$	−9.00000	−0.874157
$107$	17.0000	1.64345	0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000	1.64345	0.821726	+	0.569883i	$0.193011\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	−5.00000	−0.476731
$111$	0	0
$112$	0	0
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	1.00000	0.0928477
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	−14.0000	−1.26750
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	5.00000	0.438529
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	−9.00000	−0.771744
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	9.00000	0.763370	0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000	0.763370	0.381685	+	0.924292i	$0.375344\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	−25.0000	−2.09061
$144$	0	0
$145$	1.00000	0.0830455
$146$	3.00000	0.248282
$147$	0	0
$148$	−3.00000	−0.246598
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	−5.00000	−0.395285
$161$	0	0
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	−9.00000	−0.698535
$167$	−19.0000	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	−3.00000	−0.230089
$171$	0	0
$172$	1.00000	0.0762493
$173$	−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
$176$	−5.00000	−0.376889
$177$	0	0
$178$	−13.0000	−0.974391
$179$	19.0000	1.42013	0.710063	−	0.704138i	$-0.248666\pi$
$179$	19.0000	1.42013	0.710063	+	0.704138i	$0.248666\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	0	0
$184$	−9.00000	−0.663489
$185$	−3.00000	−0.220564
$186$	0	0
$187$	15.0000	1.09691
$188$	0	0
$189$	0	0
$190$	−1.00000	−0.0725476
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−9.00000	−0.646162
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	12.0000	0.848528
$201$	0	0
$202$	−17.0000	−1.19612
$203$	0	0
$204$	0	0
$205$	5.00000	0.349215
$206$	−1.00000	−0.0696733
$207$	0	0
$208$	5.00000	0.346688
$209$	5.00000	0.345857
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	17.0000	1.16210
$215$	1.00000	0.0681994
$216$	0	0
$217$	0	0
$218$	−9.00000	−0.609557
$219$	0	0
$220$	5.00000	0.337100
$221$	−15.0000	−1.00901
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	0	0
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	3.00000	0.196960
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	11.0000	0.708572	0.354286	−	0.935137i	$-0.384724\pi$
$241$	11.0000	0.708572	0.354286	+	0.935137i	$0.384724\pi$
$242$	14.0000	0.899954
$243$	0	0
$244$	14.0000	0.896258
$245$	0	0
$246$	0	0
$247$	−5.00000	−0.318142
$248$	0	0
$249$	0	0
$250$	9.00000	0.569210
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	−12.0000	−0.752947
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−29.0000	−1.80897	−0.904485	−	0.426505i	$-0.859745\pi$
$257$	−29.0000	−1.80897	−0.904485	+	0.426505i	$0.859745\pi$
$258$	0	0
$259$	0	0
$260$	−5.00000	−0.310087
$261$	0	0
$262$	−1.00000	−0.0617802
$263$	5.00000	0.308313	0.154157	−	0.988046i	$-0.450734\pi$
$263$	5.00000	0.308313	0.154157	+	0.988046i	$0.450734\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	1.00000	0.0607457	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000	0.0607457	0.0303728	+	0.999539i	$0.490331\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−9.00000	−0.543710
$275$	−20.0000	−1.20605
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	9.00000	0.539784
$279$	0	0
$280$	0	0
$281$	−29.0000	−1.72999	−0.864997	−	0.501776i	$-0.832680\pi$
$281$	−29.0000	−1.72999	−0.864997	+	0.501776i	$0.832680\pi$
$282$	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
$285$	0	0
$286$	−25.0000	−1.47828
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	1.00000	0.0587220
$291$	0	0
$292$	−3.00000	−0.175562
$293$	−5.00000	−0.292103	−0.146052	−	0.989277i	$-0.546657\pi$
$293$	−5.00000	−0.292103	−0.146052	+	0.989277i	$0.546657\pi$
$294$	0	0
$295$	0	0
$296$	−9.00000	−0.523114
$297$	0	0
$298$	3.00000	0.173785
$299$	−15.0000	−0.867472
$300$	0	0
$301$	0	0
$302$	5.00000	0.287718
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	14.0000	0.801638
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−8.00000	−0.450035
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	−5.00000	−0.279946
$320$	−7.00000	−0.391312
$321$	0	0
$322$	0	0
$323$	3.00000	0.166924
$324$	0	0
$325$	20.0000	1.10940
$326$	−11.0000	−0.609234
$327$	0	0
$328$	15.0000	0.828236
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	9.00000	0.493939
$333$	0	0
$334$	−19.0000	−1.03963
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	3.00000	0.162698
$341$	0	0
$342$	0	0
$343$	0	0
$344$	3.00000	0.161749
$345$	0	0
$346$	−14.0000	−0.752645
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	0	0
$352$	25.0000	1.33250
$353$	11.0000	0.585471	0.292735	−	0.956193i	$-0.405434\pi$
$353$	11.0000	0.585471	0.292735	+	0.956193i	$0.405434\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	13.0000	0.688999
$357$	0	0
$358$	19.0000	1.00418
$359$	−11.0000	−0.580558	−0.290279	−	0.956942i	$-0.593748\pi$
$359$	−11.0000	−0.580558	−0.290279	+	0.956942i	$0.593748\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−14.0000	−0.735824
$363$	0	0
$364$	0	0
$365$	−3.00000	−0.157027
$366$	0	0
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	−3.00000	−0.155963
$371$	0	0
$372$	0	0
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	15.0000	0.775632
$375$	0	0
$376$	0	0
$377$	5.00000	0.257513
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	1.00000	0.0512989
$381$	0	0
$382$	8.00000	0.409316
$383$	27.0000	1.37964	0.689818	−	0.723983i	$-0.257691\pi$
$383$	27.0000	1.37964	0.689818	+	0.723983i	$0.257691\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	9.00000	0.456906
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	0	0
$393$	0	0
$394$	2.00000	0.100759
$395$	−8.00000	−0.402524
$396$	0	0
$397$	15.0000	0.752828	0.376414	−	0.926451i	$-0.377157\pi$
$397$	15.0000	0.752828	0.376414	+	0.926451i	$0.377157\pi$
$398$	3.00000	0.150376
$399$	0	0
$400$	4.00000	0.200000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	0	0
$404$	17.0000	0.845782
$405$	0	0
$406$	0	0
$407$	15.0000	0.743522
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	5.00000	0.246932
$411$	0	0
$412$	1.00000	0.0492665
$413$	0	0
$414$	0	0
$415$	9.00000	0.441793
$416$	−25.0000	−1.22573
$417$	0	0
$418$	5.00000	0.244558
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	13.0000	0.632830
$423$	0	0
$424$	27.0000	1.31124
$425$	−12.0000	−0.582086
$426$	0	0
$427$	0	0
$428$	−17.0000	−0.821726
$429$	0	0
$430$	1.00000	0.0482243
$431$	−9.00000	−0.433515	−0.216757	−	0.976226i	$-0.569548\pi$
$431$	−9.00000	−0.433515	−0.216757	+	0.976226i	$0.569548\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	9.00000	0.431022
$437$	3.00000	0.143509
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	15.0000	0.715097
$441$	0	0
$442$	−15.0000	−0.713477
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	13.0000	0.616259
$446$	19.0000	0.899676
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	−25.0000	−1.17720
$452$	1.00000	0.0470360
$453$	0	0
$454$	−3.00000	−0.140797
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−1.00000	−0.0467269
$459$	0	0
$460$	3.00000	0.139876
$461$	19.0000	0.884918	0.442459	−	0.896789i	$-0.354106\pi$
$461$	19.0000	0.884918	0.442459	+	0.896789i	$0.354106\pi$
$462$	0	0
$463$	13.0000	0.604161	0.302081	−	0.953282i	$-0.402319\pi$
$463$	13.0000	0.604161	0.302081	+	0.953282i	$0.402319\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	3.00000	0.138972
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.00000	−0.229900
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	−15.0000	−0.686084
$479$	25.0000	1.14228	0.571140	−	0.820853i	$-0.306501\pi$
$479$	25.0000	1.14228	0.571140	+	0.820853i	$0.306501\pi$
$480$	0	0
$481$	−15.0000	−0.683941
$482$	11.0000	0.501036
$483$	0	0
$484$	−14.0000	−0.636364
$485$	9.00000	0.408669
$486$	0	0
$487$	19.0000	0.860972	0.430486	−	0.902597i	$-0.358342\pi$
$487$	19.0000	0.860972	0.430486	+	0.902597i	$0.358342\pi$
$488$	42.0000	1.90125
$489$	0	0
$490$	0	0
$491$	13.0000	0.586682	0.293341	−	0.956008i	$-0.405233\pi$
$491$	13.0000	0.586682	0.293341	+	0.956008i	$0.405233\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	−5.00000	−0.224961
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	31.0000	1.38775	0.693875	−	0.720095i	$-0.255902\pi$
$499$	31.0000	1.38775	0.693875	+	0.720095i	$0.255902\pi$
$500$	−9.00000	−0.402492
$501$	0	0
$502$	−28.0000	−1.24970
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	17.0000	0.756490
$506$	15.0000	0.666831
$507$	0	0
$508$	12.0000	0.532414
$509$	−29.0000	−1.28540	−0.642701	−	0.766117i	$-0.722186\pi$
$509$	−29.0000	−1.28540	−0.642701	+	0.766117i	$0.722186\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−29.0000	−1.27914
$515$	1.00000	0.0440653
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−15.0000	−0.657794
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	1.00000	0.0437269	0.0218635	−	0.999761i	$-0.493040\pi$
$523$	1.00000	0.0437269	0.0218635	+	0.999761i	$0.493040\pi$
$524$	1.00000	0.0436852
$525$	0	0
$526$	5.00000	0.218010
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	9.00000	0.390935
$531$	0	0
$532$	0	0
$533$	25.0000	1.08287
$534$	0	0
$535$	−17.0000	−0.734974
$536$	−12.0000	−0.518321
$537$	0	0
$538$	3.00000	0.129339
$539$	0	0
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	1.00000	0.0429537
$543$	0	0
$544$	15.0000	0.643120
$545$	9.00000	0.385518
$546$	0	0
$547$	−29.0000	−1.23995	−0.619975	−	0.784621i	$-0.712857\pi$
$547$	−29.0000	−1.23995	−0.619975	+	0.784621i	$0.712857\pi$
$548$	9.00000	0.384461
$549$	0	0
$550$	−20.0000	−0.852803
$551$	−1.00000	−0.0426014
$552$	0	0
$553$	0	0
$554$	19.0000	0.807233
$555$	0	0
$556$	−9.00000	−0.381685
$557$	−37.0000	−1.56774	−0.783870	−	0.620925i	$-0.786757\pi$
$557$	−37.0000	−1.56774	−0.783870	+	0.620925i	$0.786757\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	0	0
$561$	0	0
$562$	−29.0000	−1.22329
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	1.00000	0.0420703
$566$	28.0000	1.17693
$567$	0	0
$568$	36.0000	1.51053
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	25.0000	1.04530
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	31.0000	1.29055	0.645273	−	0.763952i	$-0.276743\pi$
$577$	31.0000	1.29055	0.645273	+	0.763952i	$0.276743\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−1.00000	−0.0415227
$581$	0	0
$582$	0	0
$583$	−45.0000	−1.86371
$584$	−9.00000	−0.372423
$585$	0	0
$586$	−5.00000	−0.206548
$587$	−37.0000	−1.52715	−0.763577	−	0.645717i	$-0.776559\pi$
$587$	−37.0000	−1.52715	−0.763577	+	0.645717i	$0.776559\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−3.00000	−0.123299
$593$	15.0000	0.615976	0.307988	−	0.951390i	$-0.400344\pi$
$593$	15.0000	0.615976	0.307988	+	0.951390i	$0.400344\pi$
$594$	0	0
$595$	0	0
$596$	−3.00000	−0.122885
$597$	0	0
$598$	−15.0000	−0.613396
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−9.00000	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$602$	0	0
$603$	0	0
$604$	−5.00000	−0.203447
$605$	−14.0000	−0.569181
$606$	0	0
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	14.0000	0.566843
$611$	0	0
$612$	0	0
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	0	0
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	0	0
$619$	25.0000	1.00483	0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000	1.00483	0.502417	+	0.864625i	$0.332444\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	14.0000	0.559553
$627$	0	0
$628$	14.0000	0.558661
$629$	9.00000	0.358854
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−24.0000	−0.954669
$633$	0	0
$634$	−6.00000	−0.238290
$635$	12.0000	0.476205
$636$	0	0
$637$	0	0
$638$	−5.00000	−0.197952
$639$	0	0
$640$	3.00000	0.118585
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	0	0
$645$	0	0
$646$	3.00000	0.118033
$647$	31.0000	1.21874	0.609368	−	0.792888i	$-0.291423\pi$
$647$	31.0000	1.21874	0.609368	+	0.792888i	$0.291423\pi$
$648$	0	0
$649$	0	0
$650$	20.0000	0.784465
$651$	0	0
$652$	11.0000	0.430793
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	0	0
$655$	1.00000	0.0390732
$656$	5.00000	0.195217
$657$	0	0
$658$	0	0
$659$	27.0000	1.05177	0.525885	−	0.850555i	$-0.323734\pi$
$659$	27.0000	1.05177	0.525885	+	0.850555i	$0.323734\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	27.0000	1.04780
$665$	0	0
$666$	0	0
$667$	−3.00000	−0.116160
$668$	19.0000	0.735132
$669$	0	0
$670$	−4.00000	−0.154533
$671$	−70.0000	−2.70232
$672$	0	0
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	−29.0000	−1.11704
$675$	0	0
$676$	−12.0000	−0.461538
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	0	0
$680$	9.00000	0.345134
$681$	0	0
$682$	0	0
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	0	0
$685$	9.00000	0.343872
$686$	0	0
$687$	0	0
$688$	1.00000	0.0381246
$689$	45.0000	1.71436
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	4.00000	0.151838
$695$	−9.00000	−0.341389
$696$	0	0
$697$	−15.0000	−0.568166
$698$	19.0000	0.719161
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	3.00000	0.113147
$704$	35.0000	1.31911
$705$	0	0
$706$	11.0000	0.413990
$707$	0	0
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	39.0000	1.46159
$713$	0	0
$714$	0	0
$715$	25.0000	0.934947
$716$	−19.0000	−0.710063
$717$	0	0
$718$	−11.0000	−0.410516
$719$	−27.0000	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$720$	0	0
$721$	0	0
$722$	−18.0000	−0.669891
$723$	0	0
$724$	14.0000	0.520306
$725$	4.00000	0.148556
$726$	0	0
$727$	47.0000	1.74313	0.871567	−	0.490277i	$-0.163104\pi$
$727$	47.0000	1.74313	0.871567	+	0.490277i	$0.163104\pi$
$728$	0	0
$729$	0	0
$730$	−3.00000	−0.111035
$731$	−3.00000	−0.110959
$732$	0	0
$733$	27.0000	0.997268	0.498634	−	0.866813i	$-0.333835\pi$
$733$	27.0000	0.997268	0.498634	+	0.866813i	$0.333835\pi$
$734$	−3.00000	−0.110732
$735$	0	0
$736$	15.0000	0.552907
$737$	20.0000	0.736709
$738$	0	0
$739$	−9.00000	−0.331070	−0.165535	−	0.986204i	$-0.552935\pi$
$739$	−9.00000	−0.331070	−0.165535	+	0.986204i	$0.552935\pi$
$740$	3.00000	0.110282
$741$	0	0
$742$	0	0
$743$	−15.0000	−0.550297	−0.275148	−	0.961402i	$-0.588727\pi$
$743$	−15.0000	−0.550297	−0.275148	+	0.961402i	$0.588727\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	−25.0000	−0.915315
$747$	0	0
$748$	−15.0000	−0.548454
$749$	0	0
$750$	0	0
$751$	31.0000	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$752$	0	0
$753$	0	0
$754$	5.00000	0.182089
$755$	−5.00000	−0.181969
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−12.0000	−0.435860
$759$	0	0
$760$	3.00000	0.108821
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	0	0
$763$	0	0
$764$	−8.00000	−0.289430
$765$	0	0
$766$	27.0000	0.975550
$767$	0	0
$768$	0	0
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	0	0
$772$	10.0000	0.359908
$773$	31.0000	1.11499	0.557496	−	0.830179i	$-0.311762\pi$
$773$	31.0000	1.11499	0.557496	+	0.830179i	$0.311762\pi$
$774$	0	0
$775$	0	0
$776$	27.0000	0.969244
$777$	0	0
$778$	−9.00000	−0.322666
$779$	−5.00000	−0.179144
$780$	0	0
$781$	−60.0000	−2.14697
$782$	9.00000	0.321839
$783$	0	0
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	−8.00000	−0.284627
$791$	0	0
$792$	0	0
$793$	70.0000	2.48577
$794$	15.0000	0.532330
$795$	0	0
$796$	−3.00000	−0.106332
$797$	23.0000	0.814702	0.407351	−	0.913272i	$-0.366453\pi$
$797$	23.0000	0.814702	0.407351	+	0.913272i	$0.366453\pi$
$798$	0	0
$799$	0	0
$800$	−20.0000	−0.707107
$801$	0	0
$802$	3.00000	0.105934
$803$	15.0000	0.529339
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	51.0000	1.79417
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	0	0
$814$	15.0000	0.525750
$815$	11.0000	0.385313
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	14.0000	0.489499
$819$	0	0
$820$	−5.00000	−0.174608
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	3.00000	0.104510
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	9.00000	0.312395
$831$	0	0
$832$	−35.0000	−1.21341
$833$	0	0
$834$	0	0
$835$	19.0000	0.657522
$836$	−5.00000	−0.172929
$837$	0	0
$838$	9.00000	0.310900
$839$	−37.0000	−1.27738	−0.638691	−	0.769463i	$-0.720524\pi$
$839$	−37.0000	−1.27738	−0.638691	+	0.769463i	$0.720524\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−1.00000	−0.0344623
$843$	0	0
$844$	−13.0000	−0.447478
$845$	−12.0000	−0.412813
$846$	0	0
$847$	0	0
$848$	9.00000	0.309061
$849$	0	0
$850$	−12.0000	−0.411597
$851$	9.00000	0.308516
$852$	0	0
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	0	0
$855$	0	0
$856$	−51.0000	−1.74314
$857$	11.0000	0.375753	0.187876	−	0.982193i	$-0.439840\pi$
$857$	11.0000	0.375753	0.187876	+	0.982193i	$0.439840\pi$
$858$	0	0
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	−1.00000	−0.0340997
$861$	0	0
$862$	−9.00000	−0.306541
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	−14.0000	−0.475739
$867$	0	0
$868$	0	0
$869$	40.0000	1.35691
$870$	0	0
$871$	−20.0000	−0.677674
$872$	27.0000	0.914335
$873$	0	0
$874$	3.00000	0.101477
$875$	0	0
$876$	0	0
$877$	−53.0000	−1.78968	−0.894841	−	0.446384i	$-0.852711\pi$
$877$	−53.0000	−1.78968	−0.894841	+	0.446384i	$0.852711\pi$
$878$	0	0
$879$	0	0
$880$	5.00000	0.168550
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	15.0000	0.504505
$885$	0	0
$886$	36.0000	1.20944
$887$	−29.0000	−0.973725	−0.486862	−	0.873479i	$-0.661859\pi$
$887$	−29.0000	−0.973725	−0.486862	+	0.873479i	$0.661859\pi$
$888$	0	0
$889$	0	0
$890$	13.0000	0.435761
$891$	0	0
$892$	−19.0000	−0.636167
$893$	0	0
$894$	0	0
$895$	−19.0000	−0.635100
$896$	0	0
$897$	0	0
$898$	30.0000	1.00111
$899$	0	0
$900$	0	0
$901$	−27.0000	−0.899500
$902$	−25.0000	−0.832409
$903$	0	0
$904$	3.00000	0.0997785
$905$	14.0000	0.465376
$906$	0	0
$907$	5.00000	0.166022	0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000	0.166022	0.0830111	+	0.996549i	$0.473546\pi$
$908$	3.00000	0.0995585
$909$	0	0
$910$	0	0
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	0	0
$913$	−45.0000	−1.48928
$914$	22.0000	0.727695
$915$	0	0
$916$	1.00000	0.0330409
$917$	0	0
$918$	0	0
$919$	17.0000	0.560778	0.280389	−	0.959886i	$-0.409536\pi$
$919$	17.0000	0.560778	0.280389	+	0.959886i	$0.409536\pi$
$920$	9.00000	0.296721
$921$	0	0
$922$	19.0000	0.625732
$923$	60.0000	1.97492
$924$	0	0
$925$	−12.0000	−0.394558
$926$	13.0000	0.427207
$927$	0	0
$928$	−5.00000	−0.164133
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	0	0
$932$	−3.00000	−0.0982683
$933$	0	0
$934$	−27.0000	−0.883467
$935$	−15.0000	−0.490552
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	−15.0000	−0.488467
$944$	0	0
$945$	0	0
$946$	−5.00000	−0.162564
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	0	0
$949$	−15.0000	−0.486921
$950$	−4.00000	−0.129777
$951$	0	0
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	15.0000	0.485135
$957$	0	0
$958$	25.0000	0.807713
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−15.0000	−0.483619
$963$	0	0
$964$	−11.0000	−0.354286
$965$	10.0000	0.321911
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	−42.0000	−1.34993
$969$	0	0
$970$	9.00000	0.288973
$971$	57.0000	1.82922	0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000	1.82922	0.914609	+	0.404341i	$0.132499\pi$
$972$	0	0
$973$	0	0
$974$	19.0000	0.608799
$975$	0	0
$976$	14.0000	0.448129
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−65.0000	−2.07741
$980$	0	0
$981$	0	0
$982$	13.0000	0.414847
$983$	−3.00000	−0.0956851	−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000	−0.0956851	−0.0478426	+	0.998855i	$0.515235\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	−3.00000	−0.0955395
$987$	0	0
$988$	5.00000	0.159071
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	−37.0000	−1.17534	−0.587672	−	0.809099i	$-0.699955\pi$
$991$	−37.0000	−1.17534	−0.587672	+	0.809099i	$0.699955\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.00000	−0.0951064
$996$	0	0
$997$	−17.0000	−0.538395	−0.269198	−	0.963085i	$-0.586759\pi$
$997$	−17.0000	−0.538395	−0.269198	+	0.963085i	$0.586759\pi$
$998$	31.0000	0.981288
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.d.1.1		1
3.2	odd	2		3969.2.a.c.1.1		1
7.2	even	3		567.2.e.a.487.1		2
7.4	even	3		567.2.e.a.163.1		2
7.6	odd	2		3969.2.a.f.1.1		1
9.2	odd	6		1323.2.f.a.442.1		2
9.4	even	3		441.2.f.b.295.1		2
9.5	odd	6		1323.2.f.a.883.1		2
9.7	even	3		441.2.f.b.148.1		2
21.2	odd	6		567.2.e.b.487.1		2
21.11	odd	6		567.2.e.b.163.1		2
21.20	even	2		3969.2.a.a.1.1		1
63.2	odd	6		189.2.g.a.172.1		2
63.4	even	3		63.2.g.a.16.1	yes	2
63.5	even	6		1323.2.h.a.802.1		2
63.11	odd	6		189.2.h.a.37.1		2
63.13	odd	6		441.2.f.a.295.1		2
63.16	even	3		63.2.g.a.4.1	✓	2
63.20	even	6		1323.2.f.b.442.1		2
63.23	odd	6		189.2.h.a.46.1		2
63.25	even	3		63.2.h.a.58.1	yes	2
63.31	odd	6		441.2.g.a.79.1		2
63.32	odd	6		189.2.g.a.100.1		2
63.34	odd	6		441.2.f.a.148.1		2
63.38	even	6		1323.2.h.a.226.1		2
63.40	odd	6		441.2.h.a.214.1		2
63.41	even	6		1323.2.f.b.883.1		2
63.47	even	6		1323.2.g.a.361.1		2
63.52	odd	6		441.2.h.a.373.1		2
63.58	even	3		63.2.h.a.25.1	yes	2
63.59	even	6		1323.2.g.a.667.1		2
63.61	odd	6		441.2.g.a.67.1		2
252.11	even	6		3024.2.q.b.2305.1		2
252.23	even	6		3024.2.q.b.2881.1		2
252.67	odd	6		1008.2.t.d.961.1		2
252.79	odd	6		1008.2.t.d.193.1		2
252.95	even	6		3024.2.t.d.289.1		2
252.151	odd	6		1008.2.q.c.625.1		2
252.191	even	6		3024.2.t.d.1873.1		2
252.247	odd	6		1008.2.q.c.529.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.g.a.4.1	✓	2	63.16	even	3
63.2.g.a.16.1	yes	2	63.4	even	3
63.2.h.a.25.1	yes	2	63.58	even	3
63.2.h.a.58.1	yes	2	63.25	even	3
189.2.g.a.100.1		2	63.32	odd	6
189.2.g.a.172.1		2	63.2	odd	6
189.2.h.a.37.1		2	63.11	odd	6
189.2.h.a.46.1		2	63.23	odd	6
441.2.f.a.148.1		2	63.34	odd	6
441.2.f.a.295.1		2	63.13	odd	6
441.2.f.b.148.1		2	9.7	even	3
441.2.f.b.295.1		2	9.4	even	3
441.2.g.a.67.1		2	63.61	odd	6
441.2.g.a.79.1		2	63.31	odd	6
441.2.h.a.214.1		2	63.40	odd	6
441.2.h.a.373.1		2	63.52	odd	6
567.2.e.a.163.1		2	7.4	even	3
567.2.e.a.487.1		2	7.2	even	3
567.2.e.b.163.1		2	21.11	odd	6
567.2.e.b.487.1		2	21.2	odd	6
1008.2.q.c.529.1		2	252.247	odd	6
1008.2.q.c.625.1		2	252.151	odd	6
1008.2.t.d.193.1		2	252.79	odd	6
1008.2.t.d.961.1		2	252.67	odd	6
1323.2.f.a.442.1		2	9.2	odd	6
1323.2.f.a.883.1		2	9.5	odd	6
1323.2.f.b.442.1		2	63.20	even	6
1323.2.f.b.883.1		2	63.41	even	6
1323.2.g.a.361.1		2	63.47	even	6
1323.2.g.a.667.1		2	63.59	even	6
1323.2.h.a.226.1		2	63.38	even	6
1323.2.h.a.802.1		2	63.5	even	6
3024.2.q.b.2305.1		2	252.11	even	6
3024.2.q.b.2881.1		2	252.23	even	6
3024.2.t.d.289.1		2	252.95	even	6
3024.2.t.d.1873.1		2	252.191	even	6
3969.2.a.a.1.1		1	21.20	even	2
3969.2.a.c.1.1		1	3.2	odd	2
3969.2.a.d.1.1		1	1.1	even	1	trivial
3969.2.a.f.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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

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