LMFDB - Embedded newform 9702.2.a.eb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.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:	$77.4708600410$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.14336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 14 $ x^4 - 8*x^2 + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3234)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.32685$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +3.29066 q^{10} +1.00000 q^{11} +3.23948 q^{13} +1.00000 q^{16} +0.462230 q^{17} -2.77725 q^{19} +3.29066 q^{20} +1.00000 q^{22} +2.90080 q^{23} +5.82843 q^{25} +3.23948 q^{26} -2.82843 q^{29} +0.704871 q^{31} +1.00000 q^{32} +0.462230 q^{34} +0.996838 q^{37} -2.77725 q^{38} +3.29066 q^{40} -0.462230 q^{41} +6.75605 q^{43} +1.00000 q^{44} +2.90080 q^{46} +3.94882 q^{47} +5.82843 q^{50} +3.23948 q^{52} -12.3105 q^{53} +3.29066 q^{55} -2.82843 q^{58} +10.8284 q^{59} +2.06791 q^{61} +0.704871 q^{62} +1.00000 q^{64} +10.6600 q^{65} +15.3374 q^{67} +0.462230 q^{68} -12.1358 q^{71} -0.359873 q^{73} +0.996838 q^{74} -2.77725 q^{76} +0.168411 q^{79} +3.29066 q^{80} -0.462230 q^{82} +7.53330 q^{83} +1.52104 q^{85} +6.75605 q^{86} +1.00000 q^{88} -11.3753 q^{89} +2.90080 q^{92} +3.94882 q^{94} -9.13897 q^{95} -3.89317 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{11} + 4 q^{16} + 4 q^{22} + 8 q^{23} + 12 q^{25} - 16 q^{31} + 4 q^{32} + 8 q^{37} + 8 q^{43} + 4 q^{44} + 8 q^{46} + 16 q^{47} + 12 q^{50} - 8 q^{53} + 32 q^{59} - 16 q^{61} - 16 q^{62} + 4 q^{64} + 16 q^{65} + 16 q^{67} + 8 q^{74} + 16 q^{79} + 32 q^{85} + 8 q^{86} + 4 q^{88} + 16 q^{89} + 8 q^{92} + 16 q^{94} + 16 q^{95} + 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^11 + 4 * q^16 + 4 * q^22 + 8 * q^23 + 12 * q^25 - 16 * q^31 + 4 * q^32 + 8 * q^37 + 8 * q^43 + 4 * q^44 + 8 * q^46 + 16 * q^47 + 12 * q^50 - 8 * q^53 + 32 * q^59 - 16 * q^61 - 16 * q^62 + 4 * q^64 + 16 * q^65 + 16 * q^67 + 8 * q^74 + 16 * q^79 + 32 * q^85 + 8 * q^86 + 4 * q^88 + 16 * q^89 + 8 * q^92 + 16 * q^94 + 16 * q^95 + 16 * q^97

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
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.29066	1.47163	0.735813	−	0.677184i	$-0.236800\pi$
$5$	3.29066	1.47163	0.735813	+	0.677184i	$0.236800\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	3.29066	1.04060
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.23948	0.898470	0.449235	−	0.893414i	$-0.351697\pi$
$13$	3.23948	0.898470	0.449235	+	0.893414i	$0.351697\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0.462230	0.112107	0.0560537	−	0.998428i	$-0.482148\pi$
$17$	0.462230	0.112107	0.0560537	+	0.998428i	$0.482148\pi$
$18$	0	0
$19$	−2.77725	−0.637145	−0.318572	−	0.947899i	$-0.603203\pi$
$19$	−2.77725	−0.637145	−0.318572	+	0.947899i	$0.603203\pi$
$20$	3.29066	0.735813
$21$	0	0
$22$	1.00000	0.213201
$23$	2.90080	0.604860	0.302430	−	0.953172i	$-0.402202\pi$
$23$	2.90080	0.604860	0.302430	+	0.953172i	$0.402202\pi$
$24$	0	0
$25$	5.82843	1.16569
$26$	3.23948	0.635314
$27$	0	0
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	0	0
$31$	0.704871	0.126599	0.0632993	−	0.997995i	$-0.479838\pi$
$31$	0.704871	0.126599	0.0632993	+	0.997995i	$0.479838\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.462230	0.0792719
$35$	0	0
$36$	0	0
$37$	0.996838	0.163879	0.0819396	−	0.996637i	$-0.473889\pi$
$37$	0.996838	0.163879	0.0819396	+	0.996637i	$0.473889\pi$
$38$	−2.77725	−0.450529
$39$	0	0
$40$	3.29066	0.520299
$41$	−0.462230	−0.0721883	−0.0360941	−	0.999348i	$-0.511492\pi$
$41$	−0.462230	−0.0721883	−0.0360941	+	0.999348i	$0.511492\pi$
$42$	0	0
$43$	6.75605	1.03029	0.515144	−	0.857104i	$-0.327738\pi$
$43$	6.75605	1.03029	0.515144	+	0.857104i	$0.327738\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	2.90080	0.427700
$47$	3.94882	0.575995	0.287997	−	0.957631i	$-0.407011\pi$
$47$	3.94882	0.575995	0.287997	+	0.957631i	$0.407011\pi$
$48$	0	0
$49$	0	0
$50$	5.82843	0.824264
$51$	0	0
$52$	3.23948	0.449235
$53$	−12.3105	−1.69098	−0.845492	−	0.533988i	$-0.820693\pi$
$53$	−12.3105	−1.69098	−0.845492	+	0.533988i	$0.820693\pi$
$54$	0	0
$55$	3.29066	0.443712
$56$	0	0
$57$	0	0
$58$	−2.82843	−0.371391
$59$	10.8284	1.40974	0.704871	−	0.709336i	$-0.251005\pi$
$59$	10.8284	1.40974	0.704871	+	0.709336i	$0.251005\pi$
$60$	0	0
$61$	2.06791	0.264768	0.132384	−	0.991198i	$-0.457737\pi$
$61$	2.06791	0.264768	0.132384	+	0.991198i	$0.457737\pi$
$62$	0.704871	0.0895187
$63$	0	0
$64$	1.00000	0.125000
$65$	10.6600	1.32221
$66$	0	0
$67$	15.3374	1.87376	0.936879	−	0.349655i	$-0.113701\pi$
$67$	15.3374	1.87376	0.936879	+	0.349655i	$0.113701\pi$
$68$	0.462230	0.0560537
$69$	0	0
$70$	0	0
$71$	−12.1358	−1.44026	−0.720128	−	0.693841i	$-0.755917\pi$
$71$	−12.1358	−1.44026	−0.720128	+	0.693841i	$0.755917\pi$
$72$	0	0
$73$	−0.359873	−0.0421200	−0.0210600	−	0.999778i	$-0.506704\pi$
$73$	−0.359873	−0.0421200	−0.0210600	+	0.999778i	$0.506704\pi$
$74$	0.996838	0.115880
$75$	0	0
$76$	−2.77725	−0.318572
$77$	0	0
$78$	0	0
$79$	0.168411	0.0189477	0.00947387	−	0.999955i	$-0.496984\pi$
$79$	0.168411	0.0189477	0.00947387	+	0.999955i	$0.496984\pi$
$80$	3.29066	0.367907
$81$	0	0
$82$	−0.462230	−0.0510448
$83$	7.53330	0.826887	0.413443	−	0.910530i	$-0.364326\pi$
$83$	7.53330	0.826887	0.413443	+	0.910530i	$0.364326\pi$
$84$	0	0
$85$	1.52104	0.164980
$86$	6.75605	0.728524
$87$	0	0
$88$	1.00000	0.106600
$89$	−11.3753	−1.20578	−0.602889	−	0.797825i	$-0.705984\pi$
$89$	−11.3753	−1.20578	−0.602889	+	0.797825i	$0.705984\pi$
$90$	0	0
$91$	0	0
$92$	2.90080	0.302430
$93$	0	0
$94$	3.94882	0.407290
$95$	−9.13897	−0.937639
$96$	0	0
$97$	−3.89317	−0.395292	−0.197646	−	0.980273i	$-0.563330\pi$
$97$	−3.89317	−0.395292	−0.197646	+	0.980273i	$0.563330\pi$
$98$	0	0
$99$	0	0
$100$	5.82843	0.582843
$101$	10.9687	1.09143	0.545714	−	0.837972i	$-0.316259\pi$
$101$	10.9687	1.09143	0.545714	+	0.837972i	$0.316259\pi$
$102$	0	0
$103$	−12.8072	−1.26193	−0.630967	−	0.775810i	$-0.717342\pi$
$103$	−12.8072	−1.26193	−0.630967	+	0.775810i	$0.717342\pi$
$104$	3.23948	0.317657
$105$	0	0
$106$	−12.3105	−1.19571
$107$	11.9611	1.15632	0.578161	−	0.815923i	$-0.303771\pi$
$107$	11.9611	1.15632	0.578161	+	0.815923i	$0.303771\pi$
$108$	0	0
$109$	12.9642	1.24175	0.620874	−	0.783910i	$-0.286778\pi$
$109$	12.9642	1.24175	0.620874	+	0.783910i	$0.286778\pi$
$110$	3.29066	0.313752
$111$	0	0
$112$	0	0
$113$	13.4097	1.26148	0.630741	−	0.775993i	$-0.282751\pi$
$113$	13.4097	1.26148	0.630741	+	0.775993i	$0.282751\pi$
$114$	0	0
$115$	9.54555	0.890128
$116$	−2.82843	−0.262613
$117$	0	0
$118$	10.8284	0.996838
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.06791	0.187219
$123$	0	0
$124$	0.704871	0.0632993
$125$	2.72607	0.243827
$126$	0	0
$127$	−22.3016	−1.97895	−0.989474	−	0.144713i	$-0.953774\pi$
$127$	−22.3016	−1.97895	−0.989474	+	0.144713i	$0.953774\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	10.6600	0.934945
$131$	−3.53330	−0.308706	−0.154353	−	0.988016i	$-0.549329\pi$
$131$	−3.53330	−0.308706	−0.154353	+	0.988016i	$0.549329\pi$
$132$	0	0
$133$	0	0
$134$	15.3374	1.32495
$135$	0	0
$136$	0.462230	0.0396359
$137$	10.1358	0.865961	0.432980	−	0.901403i	$-0.357462\pi$
$137$	10.1358	0.865961	0.432980	+	0.901403i	$0.357462\pi$
$138$	0	0
$139$	−6.63249	−0.562561	−0.281280	−	0.959626i	$-0.590759\pi$
$139$	−6.63249	−0.562561	−0.281280	+	0.959626i	$0.590759\pi$
$140$	0	0
$141$	0	0
$142$	−12.1358	−1.01841
$143$	3.23948	0.270899
$144$	0	0
$145$	−9.30739	−0.772936
$146$	−0.359873	−0.0297833
$147$	0	0
$148$	0.996838	0.0819396
$149$	20.8132	1.70508	0.852540	−	0.522662i	$-0.175061\pi$
$149$	20.8132	1.70508	0.852540	+	0.522662i	$0.175061\pi$
$150$	0	0
$151$	0.144755	0.0117800	0.00588999	−	0.999983i	$-0.498125\pi$
$151$	0.144755	0.0117800	0.00588999	+	0.999983i	$0.498125\pi$
$152$	−2.77725	−0.225265
$153$	0	0
$154$	0	0
$155$	2.31949	0.186306
$156$	0	0
$157$	−6.94119	−0.553967	−0.276984	−	0.960875i	$-0.589335\pi$
$157$	−6.94119	−0.553967	−0.276984	+	0.960875i	$0.589335\pi$
$158$	0.168411	0.0133981
$159$	0	0
$160$	3.29066	0.260149
$161$	0	0
$162$	0	0
$163$	2.34315	0.183529	0.0917647	−	0.995781i	$-0.470749\pi$
$163$	2.34315	0.183529	0.0917647	+	0.995781i	$0.470749\pi$
$164$	−0.462230	−0.0360941
$165$	0	0
$166$	7.53330	0.584697
$167$	−17.4432	−1.34980	−0.674898	−	0.737911i	$-0.735812\pi$
$167$	−17.4432	−1.34980	−0.674898	+	0.737911i	$0.735812\pi$
$168$	0	0
$169$	−2.50578	−0.192752
$170$	1.52104	0.116659
$171$	0	0
$172$	6.75605	0.515144
$173$	−15.7484	−1.19733	−0.598665	−	0.801000i	$-0.704302\pi$
$173$	−15.7484	−1.19733	−0.598665	+	0.801000i	$0.704302\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−11.3753	−0.852614
$179$	17.3310	1.29538	0.647691	−	0.761903i	$-0.275735\pi$
$179$	17.3310	1.29538	0.647691	+	0.761903i	$0.275735\pi$
$180$	0	0
$181$	−23.9207	−1.77801	−0.889006	−	0.457896i	$-0.848603\pi$
$181$	−23.9207	−1.77801	−0.889006	+	0.457896i	$0.848603\pi$
$182$	0	0
$183$	0	0
$184$	2.90080	0.213850
$185$	3.28025	0.241169
$186$	0	0
$187$	0.462230	0.0338016
$188$	3.94882	0.287997
$189$	0	0
$190$	−9.13897	−0.663011
$191$	7.10552	0.514137	0.257069	−	0.966393i	$-0.417243\pi$
$191$	7.10552	0.514137	0.257069	+	0.966393i	$0.417243\pi$
$192$	0	0
$193$	−5.50578	−0.396314	−0.198157	−	0.980170i	$-0.563496\pi$
$193$	−5.50578	−0.396314	−0.198157	+	0.980170i	$0.563496\pi$
$194$	−3.89317	−0.279513
$195$	0	0
$196$	0	0
$197$	1.86419	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$197$	1.86419	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$198$	0	0
$199$	−8.95198	−0.634589	−0.317295	−	0.948327i	$-0.602774\pi$
$199$	−8.95198	−0.634589	−0.317295	+	0.948327i	$0.602774\pi$
$200$	5.82843	0.412132
$201$	0	0
$202$	10.9687	0.771756
$203$	0	0
$204$	0	0
$205$	−1.52104	−0.106234
$206$	−12.8072	−0.892322
$207$	0	0
$208$	3.23948	0.224617
$209$	−2.77725	−0.192106
$210$	0	0
$211$	16.0634	1.10585	0.552926	−	0.833230i	$-0.313511\pi$
$211$	16.0634	1.10585	0.552926	+	0.833230i	$0.313511\pi$
$212$	−12.3105	−0.845492
$213$	0	0
$214$	11.9611	0.817642
$215$	22.2318	1.51620
$216$	0	0
$217$	0	0
$218$	12.9642	0.878049
$219$	0	0
$220$	3.29066	0.221856
$221$	1.49739	0.100725
$222$	0	0
$223$	15.3196	1.02588	0.512940	−	0.858425i	$-0.328557\pi$
$223$	15.3196	1.02588	0.512940	+	0.858425i	$0.328557\pi$
$224$	0	0
$225$	0	0
$226$	13.4097	0.892003
$227$	11.3885	0.755884	0.377942	−	0.925829i	$-0.376632\pi$
$227$	11.3885	0.755884	0.377942	+	0.925829i	$0.376632\pi$
$228$	0	0
$229$	0.462230	0.0305450	0.0152725	−	0.999883i	$-0.495138\pi$
$229$	0.462230	0.0305450	0.0152725	+	0.999883i	$0.495138\pi$
$230$	9.54555	0.629415
$231$	0	0
$232$	−2.82843	−0.185695
$233$	11.6569	0.763666	0.381833	−	0.924231i	$-0.375293\pi$
$233$	11.6569	0.763666	0.381833	+	0.924231i	$0.375293\pi$
$234$	0	0
$235$	12.9942	0.847649
$236$	10.8284	0.704871
$237$	0	0
$238$	0	0
$239$	2.69261	0.174171	0.0870854	−	0.996201i	$-0.472245\pi$
$239$	2.69261	0.174171	0.0870854	+	0.996201i	$0.472245\pi$
$240$	0	0
$241$	10.7491	0.692412	0.346206	−	0.938159i	$-0.387470\pi$
$241$	10.7491	0.692412	0.346206	+	0.938159i	$0.387470\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	2.06791	0.132384
$245$	0	0
$246$	0	0
$247$	−8.99684	−0.572455
$248$	0.704871	0.0447594
$249$	0	0
$250$	2.72607	0.172412
$251$	−1.75921	−0.111040	−0.0555202	−	0.998458i	$-0.517682\pi$
$251$	−1.75921	−0.111040	−0.0555202	+	0.998458i	$0.517682\pi$
$252$	0	0
$253$	2.90080	0.182372
$254$	−22.3016	−1.39933
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.5442	−1.65578	−0.827892	−	0.560887i	$-0.810460\pi$
$257$	−26.5442	−1.65578	−0.827892	+	0.560887i	$0.810460\pi$
$258$	0	0
$259$	0	0
$260$	10.6600	0.661106
$261$	0	0
$262$	−3.53330	−0.218288
$263$	−3.68683	−0.227340	−0.113670	−	0.993519i	$-0.536261\pi$
$263$	−3.68683	−0.227340	−0.113670	+	0.993519i	$0.536261\pi$
$264$	0	0
$265$	−40.5098	−2.48850
$266$	0	0
$267$	0	0
$268$	15.3374	0.936879
$269$	0.471173	0.0287279	0.0143640	−	0.999897i	$-0.495428\pi$
$269$	0.471173	0.0287279	0.0143640	+	0.999897i	$0.495428\pi$
$270$	0	0
$271$	−29.7927	−1.80978	−0.904888	−	0.425650i	$-0.860045\pi$
$271$	−29.7927	−1.80978	−0.904888	+	0.425650i	$0.860045\pi$
$272$	0.462230	0.0280268
$273$	0	0
$274$	10.1358	0.612327
$275$	5.82843	0.351467
$276$	0	0
$277$	−32.3650	−1.94463	−0.972313	−	0.233681i	$-0.924923\pi$
$277$	−32.3650	−1.94463	−0.972313	+	0.233681i	$0.924923\pi$
$278$	−6.63249	−0.397791
$279$	0	0
$280$	0	0
$281$	30.6211	1.82670	0.913351	−	0.407174i	$-0.133486\pi$
$281$	30.6211	1.82670	0.913351	+	0.407174i	$0.133486\pi$
$282$	0	0
$283$	18.3381	1.09009	0.545043	−	0.838408i	$-0.316514\pi$
$283$	18.3381	1.09009	0.545043	+	0.838408i	$0.316514\pi$
$284$	−12.1358	−0.720128
$285$	0	0
$286$	3.23948	0.191554
$287$	0	0
$288$	0	0
$289$	−16.7863	−0.987432
$290$	−9.30739	−0.546548
$291$	0	0
$292$	−0.359873	−0.0210600
$293$	−6.09156	−0.355873	−0.177936	−	0.984042i	$-0.556942\pi$
$293$	−6.09156	−0.355873	−0.177936	+	0.984042i	$0.556942\pi$
$294$	0	0
$295$	35.6326	2.07461
$296$	0.996838	0.0579400
$297$	0	0
$298$	20.8132	1.20567
$299$	9.39710	0.543448
$300$	0	0
$301$	0	0
$302$	0.144755	0.00832971
$303$	0	0
$304$	−2.77725	−0.159286
$305$	6.80477	0.389640
$306$	0	0
$307$	9.32511	0.532212	0.266106	−	0.963944i	$-0.414263\pi$
$307$	9.32511	0.532212	0.266106	+	0.963944i	$0.414263\pi$
$308$	0	0
$309$	0	0
$310$	2.31949	0.131738
$311$	25.0641	1.42126	0.710628	−	0.703567i	$-0.248411\pi$
$311$	25.0641	1.42126	0.710628	+	0.703567i	$0.248411\pi$
$312$	0	0
$313$	−19.1734	−1.08375	−0.541873	−	0.840460i	$-0.682285\pi$
$313$	−19.1734	−1.08375	−0.541873	+	0.840460i	$0.682285\pi$
$314$	−6.94119	−0.391714
$315$	0	0
$316$	0.168411	0.00947387
$317$	−28.1658	−1.58195	−0.790974	−	0.611849i	$-0.790426\pi$
$317$	−28.1658	−1.58195	−0.790974	+	0.611849i	$0.790426\pi$
$318$	0	0
$319$	−2.82843	−0.158362
$320$	3.29066	0.183953
$321$	0	0
$322$	0	0
$323$	−1.28373	−0.0714286
$324$	0	0
$325$	18.8811	1.04733
$326$	2.34315	0.129775
$327$	0	0
$328$	−0.462230	−0.0255224
$329$	0	0
$330$	0	0
$331$	5.13897	0.282464	0.141232	−	0.989977i	$-0.454894\pi$
$331$	5.13897	0.282464	0.141232	+	0.989977i	$0.454894\pi$
$332$	7.53330	0.413443
$333$	0	0
$334$	−17.4432	−0.954449
$335$	50.4700	2.75747
$336$	0	0
$337$	19.8163	1.07946	0.539732	−	0.841837i	$-0.318526\pi$
$337$	19.8163	1.07946	0.539732	+	0.841837i	$0.318526\pi$
$338$	−2.50578	−0.136296
$339$	0	0
$340$	1.52104	0.0824901
$341$	0.704871	0.0381709
$342$	0	0
$343$	0	0
$344$	6.75605	0.364262
$345$	0	0
$346$	−15.7484	−0.846640
$347$	−4.10583	−0.220413	−0.110206	−	0.993909i	$-0.535151\pi$
$347$	−4.10583	−0.220413	−0.110206	+	0.993909i	$0.535151\pi$
$348$	0	0
$349$	−20.9897	−1.12356	−0.561778	−	0.827288i	$-0.689882\pi$
$349$	−20.9897	−1.12356	−0.561778	+	0.827288i	$0.689882\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	−10.7453	−0.571912	−0.285956	−	0.958243i	$-0.592311\pi$
$353$	−10.7453	−0.571912	−0.285956	+	0.958243i	$0.592311\pi$
$354$	0	0
$355$	−39.9348	−2.11952
$356$	−11.3753	−0.602889
$357$	0	0
$358$	17.3310	0.915974
$359$	26.5911	1.40343	0.701713	−	0.712460i	$-0.252419\pi$
$359$	26.5911	1.40343	0.701713	+	0.712460i	$0.252419\pi$
$360$	0	0
$361$	−11.2869	−0.594047
$362$	−23.9207	−1.25724
$363$	0	0
$364$	0	0
$365$	−1.18422	−0.0619849
$366$	0	0
$367$	−20.5601	−1.07323	−0.536615	−	0.843827i	$-0.680297\pi$
$367$	−20.5601	−1.07323	−0.536615	+	0.843827i	$0.680297\pi$
$368$	2.90080	0.151215
$369$	0	0
$370$	3.28025	0.170532
$371$	0	0
$372$	0	0
$373$	−29.4495	−1.52484	−0.762419	−	0.647083i	$-0.775989\pi$
$373$	−29.4495	−1.52484	−0.762419	+	0.647083i	$0.775989\pi$
$374$	0.462230	0.0239014
$375$	0	0
$376$	3.94882	0.203645
$377$	−9.16263	−0.471899
$378$	0	0
$379$	1.86157	0.0956224	0.0478112	−	0.998856i	$-0.484775\pi$
$379$	1.86157	0.0956224	0.0478112	+	0.998856i	$0.484775\pi$
$380$	−9.13897	−0.468819
$381$	0	0
$382$	7.10552	0.363550
$383$	9.55063	0.488014	0.244007	−	0.969773i	$-0.421538\pi$
$383$	9.55063	0.488014	0.244007	+	0.969773i	$0.421538\pi$
$384$	0	0
$385$	0	0
$386$	−5.50578	−0.280237
$387$	0	0
$388$	−3.89317	−0.197646
$389$	3.01788	0.153012	0.0765062	−	0.997069i	$-0.475624\pi$
$389$	3.01788	0.153012	0.0765062	+	0.997069i	$0.475624\pi$
$390$	0	0
$391$	1.34084	0.0678092
$392$	0	0
$393$	0	0
$394$	1.86419	0.0939164
$395$	0.554183	0.0278840
$396$	0	0
$397$	34.8425	1.74870	0.874348	−	0.485299i	$-0.161289\pi$
$397$	34.8425	1.74870	0.874348	+	0.485299i	$0.161289\pi$
$398$	−8.95198	−0.448722
$399$	0	0
$400$	5.82843	0.291421
$401$	−9.44952	−0.471887	−0.235943	−	0.971767i	$-0.575818\pi$
$401$	−9.44952	−0.471887	−0.235943	+	0.971767i	$0.575818\pi$
$402$	0	0
$403$	2.28342	0.113745
$404$	10.9687	0.545714
$405$	0	0
$406$	0	0
$407$	0.996838	0.0494114
$408$	0	0
$409$	−29.0372	−1.43580	−0.717899	−	0.696147i	$-0.754896\pi$
$409$	−29.0372	−1.43580	−0.717899	+	0.696147i	$0.754896\pi$
$410$	−1.52104	−0.0751189
$411$	0	0
$412$	−12.8072	−0.630967
$413$	0	0
$414$	0	0
$415$	24.7895	1.21687
$416$	3.23948	0.158829
$417$	0	0
$418$	−2.77725	−0.135840
$419$	18.2318	0.890684	0.445342	−	0.895361i	$-0.353082\pi$
$419$	18.2318	0.890684	0.445342	+	0.895361i	$0.353082\pi$
$420$	0	0
$421$	1.79529	0.0874969	0.0437484	−	0.999043i	$-0.486070\pi$
$421$	1.79529	0.0874969	0.0437484	+	0.999043i	$0.486070\pi$
$422$	16.0634	0.781956
$423$	0	0
$424$	−12.3105	−0.597853
$425$	2.69408	0.130682
$426$	0	0
$427$	0	0
$428$	11.9611	0.578161
$429$	0	0
$430$	22.2318	1.07211
$431$	12.9942	0.625910	0.312955	−	0.949768i	$-0.398681\pi$
$431$	12.9942	0.625910	0.312955	+	0.949768i	$0.398681\pi$
$432$	0	0
$433$	0.387396	0.0186170	0.00930852	−	0.999957i	$-0.497037\pi$
$433$	0.387396	0.0186170	0.00930852	+	0.999957i	$0.497037\pi$
$434$	0	0
$435$	0	0
$436$	12.9642	0.620874
$437$	−8.05626	−0.385383
$438$	0	0
$439$	6.82843	0.325903	0.162952	−	0.986634i	$-0.447899\pi$
$439$	6.82843	0.325903	0.162952	+	0.986634i	$0.447899\pi$
$440$	3.29066	0.156876
$441$	0	0
$442$	1.49739	0.0712234
$443$	−18.5911	−0.883290	−0.441645	−	0.897190i	$-0.645605\pi$
$443$	−18.5911	−0.883290	−0.441645	+	0.897190i	$0.645605\pi$
$444$	0	0
$445$	−37.4322	−1.77446
$446$	15.3196	0.725406
$447$	0	0
$448$	0	0
$449$	−6.10868	−0.288286	−0.144143	−	0.989557i	$-0.546043\pi$
$449$	−6.10868	−0.288286	−0.144143	+	0.989557i	$0.546043\pi$
$450$	0	0
$451$	−0.462230	−0.0217656
$452$	13.4097	0.630741
$453$	0	0
$454$	11.3885	0.534491
$455$	0	0
$456$	0	0
$457$	29.9348	1.40029	0.700145	−	0.714000i	$-0.253118\pi$
$457$	29.9348	1.40029	0.700145	+	0.714000i	$0.253118\pi$
$458$	0.462230	0.0215986
$459$	0	0
$460$	9.54555	0.445064
$461$	5.65238	0.263258	0.131629	−	0.991299i	$-0.457979\pi$
$461$	5.65238	0.263258	0.131629	+	0.991299i	$0.457979\pi$
$462$	0	0
$463$	7.79529	0.362278	0.181139	−	0.983458i	$-0.442022\pi$
$463$	7.79529	0.362278	0.181139	+	0.983458i	$0.442022\pi$
$464$	−2.82843	−0.131306
$465$	0	0
$466$	11.6569	0.539993
$467$	−34.9464	−1.61712	−0.808562	−	0.588411i	$-0.799754\pi$
$467$	−34.9464	−1.61712	−0.808562	+	0.588411i	$0.799754\pi$
$468$	0	0
$469$	0	0
$470$	12.9942	0.599379
$471$	0	0
$472$	10.8284	0.498419
$473$	6.75605	0.310643
$474$	0	0
$475$	−16.1870	−0.742710
$476$	0	0
$477$	0	0
$478$	2.69261	0.123157
$479$	38.5035	1.75927	0.879634	−	0.475651i	$-0.157787\pi$
$479$	38.5035	1.75927	0.879634	+	0.475651i	$0.157787\pi$
$480$	0	0
$481$	3.22924	0.147241
$482$	10.7491	0.489609
$483$	0	0
$484$	1.00000	0.0454545
$485$	−12.8111	−0.581722
$486$	0	0
$487$	−18.7137	−0.847997	−0.423998	−	0.905663i	$-0.639374\pi$
$487$	−18.7137	−0.847997	−0.423998	+	0.905663i	$0.639374\pi$
$488$	2.06791	0.0936097
$489$	0	0
$490$	0	0
$491$	31.3137	1.41317	0.706584	−	0.707629i	$-0.250235\pi$
$491$	31.3137	1.41317	0.706584	+	0.707629i	$0.250235\pi$
$492$	0	0
$493$	−1.30739	−0.0588817
$494$	−8.99684	−0.404787
$495$	0	0
$496$	0.704871	0.0316496
$497$	0	0
$498$	0	0
$499$	−31.4169	−1.40641	−0.703207	−	0.710985i	$-0.748249\pi$
$499$	−31.4169	−1.40641	−0.703207	+	0.710985i	$0.748249\pi$
$500$	2.72607	0.121914
$501$	0	0
$502$	−1.75921	−0.0785174
$503$	−43.2921	−1.93030	−0.965150	−	0.261697i	$-0.915718\pi$
$503$	−43.2921	−1.93030	−0.965150	+	0.261697i	$0.915718\pi$
$504$	0	0
$505$	36.0943	1.60617
$506$	2.90080	0.128956
$507$	0	0
$508$	−22.3016	−0.989474
$509$	−38.1562	−1.69125	−0.845623	−	0.533781i	$-0.820771\pi$
$509$	−38.1562	−1.69125	−0.845623	+	0.533781i	$0.820771\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−26.5442	−1.17082
$515$	−42.1442	−1.85710
$516$	0	0
$517$	3.94882	0.173669
$518$	0	0
$519$	0	0
$520$	10.6600	0.467473
$521$	−3.79366	−0.166203	−0.0831017	−	0.996541i	$-0.526483\pi$
$521$	−3.79366	−0.166203	−0.0831017	+	0.996541i	$0.526483\pi$
$522$	0	0
$523$	8.27670	0.361915	0.180957	−	0.983491i	$-0.442080\pi$
$523$	8.27670	0.361915	0.180957	+	0.983491i	$0.442080\pi$
$524$	−3.53330	−0.154353
$525$	0	0
$526$	−3.68683	−0.160754
$527$	0.325813	0.0141926
$528$	0	0
$529$	−14.5853	−0.634145
$530$	−40.5098	−1.75963
$531$	0	0
$532$	0	0
$533$	−1.49739	−0.0648590
$534$	0	0
$535$	39.3598	1.70167
$536$	15.3374	0.662473
$537$	0	0
$538$	0.471173	0.0203137
$539$	0	0
$540$	0	0
$541$	16.4916	0.709029	0.354515	−	0.935050i	$-0.384646\pi$
$541$	16.4916	0.709029	0.354515	+	0.935050i	$0.384646\pi$
$542$	−29.7927	−1.27970
$543$	0	0
$544$	0.462230	0.0198180
$545$	42.6609	1.82739
$546$	0	0
$547$	4.00894	0.171410	0.0857050	−	0.996321i	$-0.472686\pi$
$547$	4.00894	0.171410	0.0857050	+	0.996321i	$0.472686\pi$
$548$	10.1358	0.432980
$549$	0	0
$550$	5.82843	0.248525
$551$	7.85525	0.334645
$552$	0	0
$553$	0	0
$554$	−32.3650	−1.37506
$555$	0	0
$556$	−6.63249	−0.281280
$557$	−19.7800	−0.838106	−0.419053	−	0.907962i	$-0.637638\pi$
$557$	−19.7800	−0.838106	−0.419053	+	0.907962i	$0.637638\pi$
$558$	0	0
$559$	21.8861	0.925683
$560$	0	0
$561$	0	0
$562$	30.6211	1.29167
$563$	39.9199	1.68242	0.841212	−	0.540705i	$-0.181843\pi$
$563$	39.9199	1.68242	0.841212	+	0.540705i	$0.181843\pi$
$564$	0	0
$565$	44.1269	1.85643
$566$	18.3381	0.770807
$567$	0	0
$568$	−12.1358	−0.509207
$569$	−35.7238	−1.49762	−0.748809	−	0.662786i	$-0.769374\pi$
$569$	−35.7238	−1.49762	−0.748809	+	0.662786i	$0.769374\pi$
$570$	0	0
$571$	1.44320	0.0603959	0.0301980	−	0.999544i	$-0.490386\pi$
$571$	1.44320	0.0603959	0.0301980	+	0.999544i	$0.490386\pi$
$572$	3.23948	0.135449
$573$	0	0
$574$	0	0
$575$	16.9071	0.705076
$576$	0	0
$577$	−7.64606	−0.318310	−0.159155	−	0.987254i	$-0.550877\pi$
$577$	−7.64606	−0.318310	−0.159155	+	0.987254i	$0.550877\pi$
$578$	−16.7863	−0.698220
$579$	0	0
$580$	−9.30739	−0.386468
$581$	0	0
$582$	0	0
$583$	−12.3105	−0.509851
$584$	−0.359873	−0.0148917
$585$	0	0
$586$	−6.09156	−0.251640
$587$	19.3382	0.798174	0.399087	−	0.916913i	$-0.369327\pi$
$587$	19.3382	0.798174	0.399087	+	0.916913i	$0.369327\pi$
$588$	0	0
$589$	−1.95760	−0.0806616
$590$	35.6326	1.46697
$591$	0	0
$592$	0.996838	0.0409698
$593$	−30.1191	−1.23684	−0.618421	−	0.785847i	$-0.712227\pi$
$593$	−30.1191	−1.23684	−0.618421	+	0.785847i	$0.712227\pi$
$594$	0	0
$595$	0	0
$596$	20.8132	0.852540
$597$	0	0
$598$	9.39710	0.384276
$599$	30.9080	1.26287	0.631433	−	0.775430i	$-0.282467\pi$
$599$	30.9080	1.26287	0.631433	+	0.775430i	$0.282467\pi$
$600$	0	0
$601$	−24.3970	−0.995176	−0.497588	−	0.867414i	$-0.665781\pi$
$601$	−24.3970	−0.995176	−0.497588	+	0.867414i	$0.665781\pi$
$602$	0	0
$603$	0	0
$604$	0.144755	0.00588999
$605$	3.29066	0.133784
$606$	0	0
$607$	−39.9348	−1.62090	−0.810452	−	0.585805i	$-0.800778\pi$
$607$	−39.9348	−1.62090	−0.810452	+	0.585805i	$0.800778\pi$
$608$	−2.77725	−0.112632
$609$	0	0
$610$	6.80477	0.275517
$611$	12.7921	0.517514
$612$	0	0
$613$	23.4034	0.945255	0.472628	−	0.881262i	$-0.343306\pi$
$613$	23.4034	0.945255	0.472628	+	0.881262i	$0.343306\pi$
$614$	9.32511	0.376331
$615$	0	0
$616$	0	0
$617$	35.7439	1.43900	0.719499	−	0.694494i	$-0.244372\pi$
$617$	35.7439	1.43900	0.719499	+	0.694494i	$0.244372\pi$
$618$	0	0
$619$	12.2382	0.491894	0.245947	−	0.969283i	$-0.420901\pi$
$619$	12.2382	0.491894	0.245947	+	0.969283i	$0.420901\pi$
$620$	2.31949	0.0931529
$621$	0	0
$622$	25.0641	1.00498
$623$	0	0
$624$	0	0
$625$	−20.1716	−0.806863
$626$	−19.1734	−0.766324
$627$	0	0
$628$	−6.94119	−0.276984
$629$	0.460769	0.0183721
$630$	0	0
$631$	−13.8100	−0.549767	−0.274884	−	0.961477i	$-0.588639\pi$
$631$	−13.8100	−0.549767	−0.274884	+	0.961477i	$0.588639\pi$
$632$	0.168411	0.00669904
$633$	0	0
$634$	−28.1658	−1.11861
$635$	−73.3869	−2.91227
$636$	0	0
$637$	0	0
$638$	−2.82843	−0.111979
$639$	0	0
$640$	3.29066	0.130075
$641$	47.6188	1.88083	0.940415	−	0.340030i	$-0.110437\pi$
$641$	47.6188	1.88083	0.940415	+	0.340030i	$0.110437\pi$
$642$	0	0
$643$	−23.2921	−0.918552	−0.459276	−	0.888294i	$-0.651891\pi$
$643$	−23.2921	−0.918552	−0.459276	+	0.888294i	$0.651891\pi$
$644$	0	0
$645$	0	0
$646$	−1.28373	−0.0505076
$647$	41.0463	1.61369	0.806847	−	0.590760i	$-0.201172\pi$
$647$	41.0463	1.61369	0.806847	+	0.590760i	$0.201172\pi$
$648$	0	0
$649$	10.8284	0.425053
$650$	18.8811	0.740576
$651$	0	0
$652$	2.34315	0.0917647
$653$	33.9348	1.32797	0.663986	−	0.747745i	$-0.268864\pi$
$653$	33.9348	1.32797	0.663986	+	0.747745i	$0.268864\pi$
$654$	0	0
$655$	−11.6269	−0.454300
$656$	−0.462230	−0.0180471
$657$	0	0
$658$	0	0
$659$	−30.1721	−1.17534	−0.587669	−	0.809101i	$-0.699954\pi$
$659$	−30.1721	−1.17534	−0.587669	+	0.809101i	$0.699954\pi$
$660$	0	0
$661$	7.82728	0.304446	0.152223	−	0.988346i	$-0.451357\pi$
$661$	7.82728	0.304446	0.152223	+	0.988346i	$0.451357\pi$
$662$	5.13897	0.199732
$663$	0	0
$664$	7.53330	0.292349
$665$	0	0
$666$	0	0
$667$	−8.20471	−0.317688
$668$	−17.4432	−0.674898
$669$	0	0
$670$	50.4700	1.94983
$671$	2.06791	0.0798306
$672$	0	0
$673$	22.1721	0.854672	0.427336	−	0.904093i	$-0.359452\pi$
$673$	22.1721	0.854672	0.427336	+	0.904093i	$0.359452\pi$
$674$	19.8163	0.763296
$675$	0	0
$676$	−2.50578	−0.0963760
$677$	0.494668	0.0190116	0.00950581	−	0.999955i	$-0.496974\pi$
$677$	0.494668	0.0190116	0.00950581	+	0.999955i	$0.496974\pi$
$678$	0	0
$679$	0	0
$680$	1.52104	0.0583293
$681$	0	0
$682$	0.704871	0.0269909
$683$	2.51156	0.0961021	0.0480510	−	0.998845i	$-0.484699\pi$
$683$	2.51156	0.0961021	0.0480510	+	0.998845i	$0.484699\pi$
$684$	0	0
$685$	33.3535	1.27437
$686$	0	0
$687$	0	0
$688$	6.75605	0.257572
$689$	−39.8798	−1.51930
$690$	0	0
$691$	23.2587	0.884801	0.442401	−	0.896818i	$-0.354127\pi$
$691$	23.2587	0.884801	0.442401	+	0.896818i	$0.354127\pi$
$692$	−15.7484	−0.598665
$693$	0	0
$694$	−4.10583	−0.155855
$695$	−21.8253	−0.827880
$696$	0	0
$697$	−0.213657	−0.00809283
$698$	−20.9897	−0.794474
$699$	0	0
$700$	0	0
$701$	−20.3253	−0.767674	−0.383837	−	0.923401i	$-0.625398\pi$
$701$	−20.3253	−0.767674	−0.383837	+	0.923401i	$0.625398\pi$
$702$	0	0
$703$	−2.76847	−0.104415
$704$	1.00000	0.0376889
$705$	0	0
$706$	−10.7453	−0.404403
$707$	0	0
$708$	0	0
$709$	24.7869	0.930891	0.465446	−	0.885077i	$-0.345894\pi$
$709$	24.7869	0.930891	0.465446	+	0.885077i	$0.345894\pi$
$710$	−39.9348	−1.49873
$711$	0	0
$712$	−11.3753	−0.426307
$713$	2.04469	0.0765744
$714$	0	0
$715$	10.6600	0.398662
$716$	17.3310	0.647691
$717$	0	0
$718$	26.5911	0.992372
$719$	−38.5212	−1.43660	−0.718299	−	0.695734i	$-0.755079\pi$
$719$	−38.5212	−1.43660	−0.718299	+	0.695734i	$0.755079\pi$
$720$	0	0
$721$	0	0
$722$	−11.2869	−0.420055
$723$	0	0
$724$	−23.9207	−0.889006
$725$	−16.4853	−0.612248
$726$	0	0
$727$	−28.2657	−1.04832	−0.524158	−	0.851621i	$-0.675620\pi$
$727$	−28.2657	−1.04832	−0.524158	+	0.851621i	$0.675620\pi$
$728$	0	0
$729$	0	0
$730$	−1.18422	−0.0438299
$731$	3.12285	0.115503
$732$	0	0
$733$	4.28418	0.158240	0.0791199	−	0.996865i	$-0.474789\pi$
$733$	4.28418	0.158240	0.0791199	+	0.996865i	$0.474789\pi$
$734$	−20.5601	−0.758888
$735$	0	0
$736$	2.90080	0.106925
$737$	15.3374	0.564959
$738$	0	0
$739$	20.1958	0.742913	0.371457	−	0.928450i	$-0.378858\pi$
$739$	20.1958	0.742913	0.371457	+	0.928450i	$0.378858\pi$
$740$	3.28025	0.120585
$741$	0	0
$742$	0	0
$743$	32.8258	1.20426	0.602131	−	0.798397i	$-0.294318\pi$
$743$	32.8258	1.20426	0.602131	+	0.798397i	$0.294318\pi$
$744$	0	0
$745$	68.4890	2.50924
$746$	−29.4495	−1.07822
$747$	0	0
$748$	0.462230	0.0169008
$749$	0	0
$750$	0	0
$751$	−18.1468	−0.662187	−0.331093	−	0.943598i	$-0.607418\pi$
$751$	−18.1468	−0.662187	−0.331093	+	0.943598i	$0.607418\pi$
$752$	3.94882	0.143999
$753$	0	0
$754$	−9.16263	−0.333683
$755$	0.476339	0.0173357
$756$	0	0
$757$	−10.9905	−0.399457	−0.199729	−	0.979851i	$-0.564006\pi$
$757$	−10.9905	−0.399457	−0.199729	+	0.979851i	$0.564006\pi$
$758$	1.86157	0.0676152
$759$	0	0
$760$	−9.13897	−0.331505
$761$	−18.4596	−0.669160	−0.334580	−	0.942367i	$-0.608595\pi$
$761$	−18.4596	−0.669160	−0.334580	+	0.942367i	$0.608595\pi$
$762$	0	0
$763$	0	0
$764$	7.10552	0.257069
$765$	0	0
$766$	9.55063	0.345078
$767$	35.0785	1.26661
$768$	0	0
$769$	47.8567	1.72576	0.862878	−	0.505411i	$-0.168659\pi$
$769$	47.8567	1.72576	0.862878	+	0.505411i	$0.168659\pi$
$770$	0	0
$771$	0	0
$772$	−5.50578	−0.198157
$773$	−45.4201	−1.63365	−0.816825	−	0.576886i	$-0.804268\pi$
$773$	−45.4201	−1.63365	−0.816825	+	0.576886i	$0.804268\pi$
$774$	0	0
$775$	4.10829	0.147574
$776$	−3.89317	−0.139757
$777$	0	0
$778$	3.01788	0.108196
$779$	1.28373	0.0459944
$780$	0	0
$781$	−12.1358	−0.434254
$782$	1.34084	0.0479483
$783$	0	0
$784$	0	0
$785$	−22.8411	−0.815233
$786$	0	0
$787$	13.5298	0.482286	0.241143	−	0.970490i	$-0.422478\pi$
$787$	13.5298	0.482286	0.241143	+	0.970490i	$0.422478\pi$
$788$	1.86419	0.0664089
$789$	0	0
$790$	0.554183	0.0197170
$791$	0	0
$792$	0	0
$793$	6.69894	0.237886
$794$	34.8425	1.23652
$795$	0	0
$796$	−8.95198	−0.317295
$797$	−13.8031	−0.488930	−0.244465	−	0.969658i	$-0.578612\pi$
$797$	−13.8031	−0.488930	−0.244465	+	0.969658i	$0.578612\pi$
$798$	0	0
$799$	1.82527	0.0645732
$800$	5.82843	0.206066
$801$	0	0
$802$	−9.44952	−0.333674
$803$	−0.359873	−0.0126997
$804$	0	0
$805$	0	0
$806$	2.28342	0.0804299
$807$	0	0
$808$	10.9687	0.385878
$809$	−38.0816	−1.33888	−0.669439	−	0.742867i	$-0.733466\pi$
$809$	−38.0816	−1.33888	−0.669439	+	0.742867i	$0.733466\pi$
$810$	0	0
$811$	42.0359	1.47608	0.738040	−	0.674757i	$-0.235751\pi$
$811$	42.0359	1.47608	0.738040	+	0.674757i	$0.235751\pi$
$812$	0	0
$813$	0	0
$814$	0.996838	0.0349392
$815$	7.71049	0.270087
$816$	0	0
$817$	−18.7632	−0.656442
$818$	−29.0372	−1.01526
$819$	0	0
$820$	−1.52104	−0.0531171
$821$	−49.0974	−1.71351	−0.856756	−	0.515722i	$-0.827524\pi$
$821$	−49.0974	−1.71351	−0.856756	+	0.515722i	$0.827524\pi$
$822$	0	0
$823$	−24.7073	−0.861243	−0.430622	−	0.902533i	$-0.641706\pi$
$823$	−24.7073	−0.861243	−0.430622	+	0.902533i	$0.641706\pi$
$824$	−12.8072	−0.446161
$825$	0	0
$826$	0	0
$827$	−19.7774	−0.687728	−0.343864	−	0.939020i	$-0.611736\pi$
$827$	−19.7774	−0.687728	−0.343864	+	0.939020i	$0.611736\pi$
$828$	0	0
$829$	−25.4789	−0.884919	−0.442459	−	0.896789i	$-0.645894\pi$
$829$	−25.4789	−0.884919	−0.442459	+	0.896789i	$0.645894\pi$
$830$	24.7895	0.860456
$831$	0	0
$832$	3.23948	0.112309
$833$	0	0
$834$	0	0
$835$	−57.3996	−1.98639
$836$	−2.77725	−0.0960531
$837$	0	0
$838$	18.2318	0.629809
$839$	11.9064	0.411055	0.205528	−	0.978651i	$-0.434109\pi$
$839$	11.9064	0.411055	0.205528	+	0.978651i	$0.434109\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	1.79529	0.0618696
$843$	0	0
$844$	16.0634	0.552926
$845$	−8.24565	−0.283659
$846$	0	0
$847$	0	0
$848$	−12.3105	−0.422746
$849$	0	0
$850$	2.69408	0.0924061
$851$	2.89163	0.0991239
$852$	0	0
$853$	41.2968	1.41398	0.706988	−	0.707225i	$-0.250053\pi$
$853$	41.2968	1.41398	0.706988	+	0.707225i	$0.250053\pi$
$854$	0	0
$855$	0	0
$856$	11.9611	0.408821
$857$	−1.80506	−0.0616598	−0.0308299	−	0.999525i	$-0.509815\pi$
$857$	−1.80506	−0.0616598	−0.0308299	+	0.999525i	$0.509815\pi$
$858$	0	0
$859$	−49.9974	−1.70589	−0.852944	−	0.522002i	$-0.825185\pi$
$859$	−49.9974	−1.70589	−0.852944	+	0.522002i	$0.825185\pi$
$860$	22.2318	0.758100
$861$	0	0
$862$	12.9942	0.442585
$863$	−22.5640	−0.768087	−0.384043	−	0.923315i	$-0.625469\pi$
$863$	−22.5640	−0.768087	−0.384043	+	0.923315i	$0.625469\pi$
$864$	0	0
$865$	−51.8226	−1.76202
$866$	0.387396	0.0131642
$867$	0	0
$868$	0	0
$869$	0.168411	0.00571296
$870$	0	0
$871$	49.6851	1.68351
$872$	12.9642	0.439025
$873$	0	0
$874$	−8.05626	−0.272507
$875$	0	0
$876$	0	0
$877$	9.98211	0.337072	0.168536	−	0.985695i	$-0.446096\pi$
$877$	9.98211	0.337072	0.168536	+	0.985695i	$0.446096\pi$
$878$	6.82843	0.230448
$879$	0	0
$880$	3.29066	0.110928
$881$	−37.6108	−1.26714	−0.633571	−	0.773685i	$-0.718411\pi$
$881$	−37.6108	−1.26714	−0.633571	+	0.773685i	$0.718411\pi$
$882$	0	0
$883$	−24.2843	−0.817231	−0.408615	−	0.912707i	$-0.633988\pi$
$883$	−24.2843	−0.817231	−0.408615	+	0.912707i	$0.633988\pi$
$884$	1.49739	0.0503625
$885$	0	0
$886$	−18.5911	−0.624581
$887$	−7.66318	−0.257304	−0.128652	−	0.991690i	$-0.541065\pi$
$887$	−7.66318	−0.257304	−0.128652	+	0.991690i	$0.541065\pi$
$888$	0	0
$889$	0	0
$890$	−37.4322	−1.25473
$891$	0	0
$892$	15.3196	0.512940
$893$	−10.9669	−0.366992
$894$	0	0
$895$	57.0305	1.90632
$896$	0	0
$897$	0	0
$898$	−6.10868	−0.203849
$899$	−1.99368	−0.0664928
$900$	0	0
$901$	−5.69031	−0.189572
$902$	−0.462230	−0.0153906
$903$	0	0
$904$	13.4097	0.446001
$905$	−78.7148	−2.61657
$906$	0	0
$907$	23.9458	0.795108	0.397554	−	0.917579i	$-0.369859\pi$
$907$	23.9458	0.795108	0.397554	+	0.917579i	$0.369859\pi$
$908$	11.3885	0.377942
$909$	0	0
$910$	0	0
$911$	2.61129	0.0865161	0.0432580	−	0.999064i	$-0.486226\pi$
$911$	2.61129	0.0865161	0.0432580	+	0.999064i	$0.486226\pi$
$912$	0	0
$913$	7.53330	0.249316
$914$	29.9348	0.990155
$915$	0	0
$916$	0.462230	0.0152725
$917$	0	0
$918$	0	0
$919$	43.9222	1.44886	0.724429	−	0.689349i	$-0.242103\pi$
$919$	43.9222	1.44886	0.724429	+	0.689349i	$0.242103\pi$
$920$	9.54555	0.314708
$921$	0	0
$922$	5.65238	0.186151
$923$	−39.3137	−1.29403
$924$	0	0
$925$	5.81000	0.191032
$926$	7.79529	0.256169
$927$	0	0
$928$	−2.82843	−0.0928477
$929$	40.3369	1.32341	0.661706	−	0.749764i	$-0.269833\pi$
$929$	40.3369	1.32341	0.661706	+	0.749764i	$0.269833\pi$
$930$	0	0
$931$	0	0
$932$	11.6569	0.381833
$933$	0	0
$934$	−34.9464	−1.14348
$935$	1.52104	0.0497434
$936$	0	0
$937$	10.6378	0.347522	0.173761	−	0.984788i	$-0.444408\pi$
$937$	10.6378	0.347522	0.173761	+	0.984788i	$0.444408\pi$
$938$	0	0
$939$	0	0
$940$	12.9942	0.423825
$941$	5.45258	0.177749	0.0888746	−	0.996043i	$-0.471673\pi$
$941$	5.45258	0.177749	0.0888746	+	0.996043i	$0.471673\pi$
$942$	0	0
$943$	−1.34084	−0.0436638
$944$	10.8284	0.352435
$945$	0	0
$946$	6.75605	0.219658
$947$	11.9152	0.387192	0.193596	−	0.981081i	$-0.437985\pi$
$947$	11.9152	0.387192	0.193596	+	0.981081i	$0.437985\pi$
$948$	0	0
$949$	−1.16580	−0.0378435
$950$	−16.1870	−0.525175
$951$	0	0
$952$	0	0
$953$	−19.5447	−0.633115	−0.316557	−	0.948573i	$-0.602527\pi$
$953$	−19.5447	−0.633115	−0.316557	+	0.948573i	$0.602527\pi$
$954$	0	0
$955$	23.3818	0.756618
$956$	2.69261	0.0870854
$957$	0	0
$958$	38.5035	1.24399
$959$	0	0
$960$	0	0
$961$	−30.5032	−0.983973
$962$	3.22924	0.104115
$963$	0	0
$964$	10.7491	0.346206
$965$	−18.1176	−0.583227
$966$	0	0
$967$	−55.0311	−1.76968	−0.884841	−	0.465893i	$-0.845733\pi$
$967$	−55.0311	−1.76968	−0.884841	+	0.465893i	$0.845733\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−12.8111	−0.411339
$971$	7.36505	0.236356	0.118178	−	0.992992i	$-0.462295\pi$
$971$	7.36505	0.236356	0.118178	+	0.992992i	$0.462295\pi$
$972$	0	0
$973$	0	0
$974$	−18.7137	−0.599624
$975$	0	0
$976$	2.06791	0.0661921
$977$	32.9192	1.05318	0.526590	−	0.850120i	$-0.323470\pi$
$977$	32.9192	1.05318	0.526590	+	0.850120i	$0.323470\pi$
$978$	0	0
$979$	−11.3753	−0.363556
$980$	0	0
$981$	0	0
$982$	31.3137	0.999261
$983$	−45.3957	−1.44790	−0.723949	−	0.689853i	$-0.757675\pi$
$983$	−45.3957	−1.44790	−0.723949	+	0.689853i	$0.757675\pi$
$984$	0	0
$985$	6.13440	0.195458
$986$	−1.30739	−0.0416356
$987$	0	0
$988$	−8.99684	−0.286228
$989$	19.5980	0.623180
$990$	0	0
$991$	−11.5657	−0.367398	−0.183699	−	0.982983i	$-0.558807\pi$
$991$	−11.5657	−0.367398	−0.183699	+	0.982983i	$0.558807\pi$
$992$	0.704871	0.0223797
$993$	0	0
$994$	0	0
$995$	−29.4579	−0.933879
$996$	0	0
$997$	2.58294	0.0818025	0.0409012	−	0.999163i	$-0.486977\pi$
$997$	2.58294	0.0818025	0.0409012	+	0.999163i	$0.486977\pi$
$998$	−31.4169	−0.994485
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.eb.1.4		4
3.2	odd	2		3234.2.a.bj.1.1	✓	4
7.6	odd	2		9702.2.a.ec.1.1		4
21.20	even	2		3234.2.a.bk.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bj.1.1	✓	4	3.2	odd	2
3234.2.a.bk.1.4	yes	4	21.20	even	2
9702.2.a.eb.1.4		4	1.1	even	1	trivial
9702.2.a.ec.1.1		4	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:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +3.29066 q^{10} +1.00000 q^{11} +3.23948 q^{13} +1.00000 q^{16} +0.462230 q^{17} -2.77725 q^{19} +3.29066 q^{20} +1.00000 q^{22} +2.90080 q^{23} +5.82843 q^{25} +3.23948 q^{26} -2.82843 q^{29} +0.704871 q^{31} +1.00000 q^{32} +0.462230 q^{34} +0.996838 q^{37} -2.77725 q^{38} +3.29066 q^{40} -0.462230 q^{41} +6.75605 q^{43} +1.00000 q^{44} +2.90080 q^{46} +3.94882 q^{47} +5.82843 q^{50} +3.23948 q^{52} -12.3105 q^{53} +3.29066 q^{55} -2.82843 q^{58} +10.8284 q^{59} +2.06791 q^{61} +0.704871 q^{62} +1.00000 q^{64} +10.6600 q^{65} +15.3374 q^{67} +0.462230 q^{68} -12.1358 q^{71} -0.359873 q^{73} +0.996838 q^{74} -2.77725 q^{76} +0.168411 q^{79} +3.29066 q^{80} -0.462230 q^{82} +7.53330 q^{83} +1.52104 q^{85} +6.75605 q^{86} +1.00000 q^{88} -11.3753 q^{89} +2.90080 q^{92} +3.94882 q^{94} -9.13897 q^{95} -3.89317 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{11} + 4 q^{16} + 4 q^{22} + 8 q^{23} + 12 q^{25} - 16 q^{31} + 4 q^{32} + 8 q^{37} + 8 q^{43} + 4 q^{44} + 8 q^{46} + 16 q^{47} + 12 q^{50} - 8 q^{53} + 32 q^{59} - 16 q^{61} - 16 q^{62} + 4 q^{64} + 16 q^{65} + 16 q^{67} + 8 q^{74} + 16 q^{79} + 32 q^{85} + 8 q^{86} + 4 q^{88} + 16 q^{89} + 8 q^{92} + 16 q^{94} + 16 q^{95} + 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^11 + 4 * q^16 + 4 * q^22 + 8 * q^23 + 12 * q^25 - 16 * q^31 + 4 * q^32 + 8 * q^37 + 8 * q^43 + 4 * q^44 + 8 * q^46 + 16 * q^47 + 12 * q^50 - 8 * q^53 + 32 * q^59 - 16 * q^61 - 16 * q^62 + 4 * q^64 + 16 * q^65 + 16 * q^67 + 8 * q^74 + 16 * q^79 + 32 * q^85 + 8 * q^86 + 4 * q^88 + 16 * q^89 + 8 * q^92 + 16 * q^94 + 16 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more