LMFDB - Embedded newform 9702.2.a.di.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +2.82843 q^{10} -1.00000 q^{11} +4.24264 q^{13} +1.00000 q^{16} +5.65685 q^{17} +7.07107 q^{19} +2.82843 q^{20} -1.00000 q^{22} +3.00000 q^{25} +4.24264 q^{26} -8.00000 q^{29} +1.41421 q^{31} +1.00000 q^{32} +5.65685 q^{34} +6.00000 q^{37} +7.07107 q^{38} +2.82843 q^{40} +4.00000 q^{43} -1.00000 q^{44} -7.07107 q^{47} +3.00000 q^{50} +4.24264 q^{52} +6.00000 q^{53} -2.82843 q^{55} -8.00000 q^{58} +2.82843 q^{59} -12.7279 q^{61} +1.41421 q^{62} +1.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} +5.65685 q^{68} +6.00000 q^{71} -8.48528 q^{73} +6.00000 q^{74} +7.07107 q^{76} +12.0000 q^{79} +2.82843 q^{80} -7.07107 q^{83} +16.0000 q^{85} +4.00000 q^{86} -1.00000 q^{88} -15.5563 q^{89} -7.07107 q^{94} +20.0000 q^{95} -7.07107 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 2 q^{16} - 2 q^{22} + 6 q^{25} - 16 q^{29} + 2 q^{32} + 12 q^{37} + 8 q^{43} - 2 q^{44} + 6 q^{50} + 12 q^{53} - 16 q^{58} + 2 q^{64} + 24 q^{65} - 8 q^{67} + 12 q^{71} + 12 q^{74} + 24 q^{79} + 32 q^{85} + 8 q^{86} - 2 q^{88} + 40 q^{95}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 + 2 * q^16 - 2 * q^22 + 6 * q^25 - 16 * q^29 + 2 * q^32 + 12 * q^37 + 8 * q^43 - 2 * q^44 + 6 * q^50 + 12 * q^53 - 16 * q^58 + 2 * q^64 + 24 * q^65 - 8 * q^67 + 12 * q^71 + 12 * q^74 + 24 * q^79 + 32 * q^85 + 8 * q^86 - 2 * q^88 + 40 * q^95

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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	2.82843	0.894427
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	0	0
$19$	7.07107	1.62221	0.811107	−	0.584898i	$-0.198865\pi$
$19$	7.07107	1.62221	0.811107	+	0.584898i	$0.198865\pi$
$20$	2.82843	0.632456
$21$	0	0
$22$	−1.00000	−0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	4.24264	0.832050
$27$	0	0
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.65685	0.970143
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	7.07107	1.14708
$39$	0	0
$40$	2.82843	0.447214
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0	0
$47$	−7.07107	−1.03142	−0.515711	−	0.856763i	$-0.672472\pi$
$47$	−7.07107	−1.03142	−0.515711	+	0.856763i	$0.672472\pi$
$48$	0	0
$49$	0	0
$50$	3.00000	0.424264
$51$	0	0
$52$	4.24264	0.588348
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−2.82843	−0.381385
$56$	0	0
$57$	0	0
$58$	−8.00000	−1.05045
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	−12.7279	−1.62964	−0.814822	−	0.579712i	$-0.803165\pi$
$61$	−12.7279	−1.62964	−0.814822	+	0.579712i	$0.803165\pi$
$62$	1.41421	0.179605
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	5.65685	0.685994
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	7.07107	0.811107
$77$	0	0
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	2.82843	0.316228
$81$	0	0
$82$	0	0
$83$	−7.07107	−0.776151	−0.388075	−	0.921628i	$-0.626860\pi$
$83$	−7.07107	−0.776151	−0.388075	+	0.921628i	$0.626860\pi$
$84$	0	0
$85$	16.0000	1.73544
$86$	4.00000	0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−15.5563	−1.64897	−0.824485	−	0.565884i	$-0.808535\pi$
$89$	−15.5563	−1.64897	−0.824485	+	0.565884i	$0.808535\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−7.07107	−0.729325
$95$	20.0000	2.05196
$96$	0	0
$97$	−7.07107	−0.717958	−0.358979	−	0.933346i	$-0.616875\pi$
$97$	−7.07107	−0.717958	−0.358979	+	0.933346i	$0.616875\pi$
$98$	0	0
$99$	0	0
$100$	3.00000	0.300000
$101$	15.5563	1.54791	0.773957	−	0.633238i	$-0.218274\pi$
$101$	15.5563	1.54791	0.773957	+	0.633238i	$0.218274\pi$
$102$	0	0
$103$	−4.24264	−0.418040	−0.209020	−	0.977911i	$-0.567027\pi$
$103$	−4.24264	−0.418040	−0.209020	+	0.977911i	$0.567027\pi$
$104$	4.24264	0.416025
$105$	0	0
$106$	6.00000	0.582772
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−2.82843	−0.269680
$111$	0	0
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−8.00000	−0.742781
$117$	0	0
$118$	2.82843	0.260378
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−12.7279	−1.15233
$123$	0	0
$124$	1.41421	0.127000
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	12.0000	1.05247
$131$	15.5563	1.35916	0.679582	−	0.733599i	$-0.262161\pi$
$131$	15.5563	1.35916	0.679582	+	0.733599i	$0.262161\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	5.65685	0.485071
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	4.24264	0.359856	0.179928	−	0.983680i	$-0.442414\pi$
$139$	4.24264	0.359856	0.179928	+	0.983680i	$0.442414\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	−4.24264	−0.354787
$144$	0	0
$145$	−22.6274	−1.87910
$146$	−8.48528	−0.702247
$147$	0	0
$148$	6.00000	0.493197
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	7.07107	0.573539
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	8.48528	0.677199	0.338600	−	0.940931i	$-0.390047\pi$
$157$	8.48528	0.677199	0.338600	+	0.940931i	$0.390047\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	2.82843	0.223607
$161$	0	0
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	0	0
$166$	−7.07107	−0.548821
$167$	2.82843	0.218870	0.109435	−	0.993994i	$-0.465096\pi$
$167$	2.82843	0.218870	0.109435	+	0.993994i	$0.465096\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	16.0000	1.22714
$171$	0	0
$172$	4.00000	0.304997
$173$	9.89949	0.752645	0.376322	−	0.926489i	$-0.377189\pi$
$173$	9.89949	0.752645	0.376322	+	0.926489i	$0.377189\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−15.5563	−1.16600
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−16.9706	−1.26141	−0.630706	−	0.776022i	$-0.717235\pi$
$181$	−16.9706	−1.26141	−0.630706	+	0.776022i	$0.717235\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.9706	1.24770
$186$	0	0
$187$	−5.65685	−0.413670
$188$	−7.07107	−0.515711
$189$	0	0
$190$	20.0000	1.45095
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	−7.07107	−0.507673
$195$	0	0
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	4.24264	0.300753	0.150376	−	0.988629i	$-0.451951\pi$
$199$	4.24264	0.300753	0.150376	+	0.988629i	$0.451951\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	15.5563	1.09454
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−4.24264	−0.295599
$207$	0	0
$208$	4.24264	0.294174
$209$	−7.07107	−0.489116
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−6.00000	−0.410152
$215$	11.3137	0.771589
$216$	0	0
$217$	0	0
$218$	−16.0000	−1.08366
$219$	0	0
$220$	−2.82843	−0.190693
$221$	24.0000	1.61441
$222$	0	0
$223$	21.2132	1.42054	0.710271	−	0.703929i	$-0.248573\pi$
$223$	21.2132	1.42054	0.710271	+	0.703929i	$0.248573\pi$
$224$	0	0
$225$	0	0
$226$	−14.0000	−0.931266
$227$	18.3848	1.22024	0.610120	−	0.792309i	$-0.291121\pi$
$227$	18.3848	1.22024	0.610120	+	0.792309i	$0.291121\pi$
$228$	0	0
$229$	−16.9706	−1.12145	−0.560723	−	0.828003i	$-0.689477\pi$
$229$	−16.9706	−1.12145	−0.560723	+	0.828003i	$0.689477\pi$
$230$	0	0
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	−20.0000	−1.30466
$236$	2.82843	0.184115
$237$	0	0
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−14.1421	−0.910975	−0.455488	−	0.890242i	$-0.650535\pi$
$241$	−14.1421	−0.910975	−0.455488	+	0.890242i	$0.650535\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−12.7279	−0.814822
$245$	0	0
$246$	0	0
$247$	30.0000	1.90885
$248$	1.41421	0.0898027
$249$	0	0
$250$	−5.65685	−0.357771
$251$	−8.48528	−0.535586	−0.267793	−	0.963476i	$-0.586294\pi$
$251$	−8.48528	−0.535586	−0.267793	+	0.963476i	$0.586294\pi$
$252$	0	0
$253$	0	0
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.89949	0.617514	0.308757	−	0.951141i	$-0.400087\pi$
$257$	9.89949	0.617514	0.308757	+	0.951141i	$0.400087\pi$
$258$	0	0
$259$	0	0
$260$	12.0000	0.744208
$261$	0	0
$262$	15.5563	0.961074
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	16.9706	1.04249
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−8.48528	−0.517357	−0.258678	−	0.965964i	$-0.583287\pi$
$269$	−8.48528	−0.517357	−0.258678	+	0.965964i	$0.583287\pi$
$270$	0	0
$271$	−25.4558	−1.54633	−0.773166	−	0.634203i	$-0.781328\pi$
$271$	−25.4558	−1.54633	−0.773166	+	0.634203i	$0.781328\pi$
$272$	5.65685	0.342997
$273$	0	0
$274$	−4.00000	−0.241649
$275$	−3.00000	−0.180907
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	4.24264	0.254457
$279$	0	0
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	21.2132	1.26099	0.630497	−	0.776192i	$-0.282851\pi$
$283$	21.2132	1.26099	0.630497	+	0.776192i	$0.282851\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−4.24264	−0.250873
$287$	0	0
$288$	0	0
$289$	15.0000	0.882353
$290$	−22.6274	−1.32873
$291$	0	0
$292$	−8.48528	−0.496564
$293$	−7.07107	−0.413096	−0.206548	−	0.978436i	$-0.566223\pi$
$293$	−7.07107	−0.413096	−0.206548	+	0.978436i	$0.566223\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	6.00000	0.348743
$297$	0	0
$298$	−4.00000	−0.231714
$299$	0	0
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	0	0
$304$	7.07107	0.405554
$305$	−36.0000	−2.06135
$306$	0	0
$307$	7.07107	0.403567	0.201784	−	0.979430i	$-0.435326\pi$
$307$	7.07107	0.403567	0.201784	+	0.979430i	$0.435326\pi$
$308$	0	0
$309$	0	0
$310$	4.00000	0.227185
$311$	1.41421	0.0801927	0.0400963	−	0.999196i	$-0.487234\pi$
$311$	1.41421	0.0801927	0.0400963	+	0.999196i	$0.487234\pi$
$312$	0	0
$313$	−12.7279	−0.719425	−0.359712	−	0.933063i	$-0.617125\pi$
$313$	−12.7279	−0.719425	−0.359712	+	0.933063i	$0.617125\pi$
$314$	8.48528	0.478852
$315$	0	0
$316$	12.0000	0.675053
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	2.82843	0.158114
$321$	0	0
$322$	0	0
$323$	40.0000	2.22566
$324$	0	0
$325$	12.7279	0.706018
$326$	20.0000	1.10770
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−7.07107	−0.388075
$333$	0	0
$334$	2.82843	0.154765
$335$	−11.3137	−0.618134
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	16.0000	0.867722
$341$	−1.41421	−0.0765840
$342$	0	0
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	9.89949	0.532200
$347$	−34.0000	−1.82522	−0.912608	−	0.408836i	$-0.865935\pi$
$347$	−34.0000	−1.82522	−0.912608	+	0.408836i	$0.865935\pi$
$348$	0	0
$349$	26.8701	1.43832	0.719161	−	0.694844i	$-0.244527\pi$
$349$	26.8701	1.43832	0.719161	+	0.694844i	$0.244527\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−12.7279	−0.677439	−0.338719	−	0.940887i	$-0.609994\pi$
$353$	−12.7279	−0.677439	−0.338719	+	0.940887i	$0.609994\pi$
$354$	0	0
$355$	16.9706	0.900704
$356$	−15.5563	−0.824485
$357$	0	0
$358$	12.0000	0.634220
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	31.0000	1.63158
$362$	−16.9706	−0.891953
$363$	0	0
$364$	0	0
$365$	−24.0000	−1.25622
$366$	0	0
$367$	35.3553	1.84553	0.922767	−	0.385359i	$-0.125922\pi$
$367$	35.3553	1.84553	0.922767	+	0.385359i	$0.125922\pi$
$368$	0	0
$369$	0	0
$370$	16.9706	0.882258
$371$	0	0
$372$	0	0
$373$	8.00000	0.414224	0.207112	−	0.978317i	$-0.433593\pi$
$373$	8.00000	0.414224	0.207112	+	0.978317i	$0.433593\pi$
$374$	−5.65685	−0.292509
$375$	0	0
$376$	−7.07107	−0.364662
$377$	−33.9411	−1.74806
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	20.0000	1.02598
$381$	0	0
$382$	−8.00000	−0.409316
$383$	−24.0416	−1.22847	−0.614235	−	0.789123i	$-0.710535\pi$
$383$	−24.0416	−1.22847	−0.614235	+	0.789123i	$0.710535\pi$
$384$	0	0
$385$	0	0
$386$	22.0000	1.11977
$387$	0	0
$388$	−7.07107	−0.358979
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−10.0000	−0.503793
$395$	33.9411	1.70776
$396$	0	0
$397$	−14.1421	−0.709773	−0.354887	−	0.934909i	$-0.615481\pi$
$397$	−14.1421	−0.709773	−0.354887	+	0.934909i	$0.615481\pi$
$398$	4.24264	0.212664
$399$	0	0
$400$	3.00000	0.150000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	15.5563	0.773957
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	11.3137	0.559427	0.279713	−	0.960084i	$-0.409761\pi$
$409$	11.3137	0.559427	0.279713	+	0.960084i	$0.409761\pi$
$410$	0	0
$411$	0	0
$412$	−4.24264	−0.209020
$413$	0	0
$414$	0	0
$415$	−20.0000	−0.981761
$416$	4.24264	0.208013
$417$	0	0
$418$	−7.07107	−0.345857
$419$	11.3137	0.552711	0.276355	−	0.961056i	$-0.410873\pi$
$419$	11.3137	0.552711	0.276355	+	0.961056i	$0.410873\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	6.00000	0.291386
$425$	16.9706	0.823193
$426$	0	0
$427$	0	0
$428$	−6.00000	−0.290021
$429$	0	0
$430$	11.3137	0.545595
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−32.5269	−1.56314	−0.781572	−	0.623815i	$-0.785582\pi$
$433$	−32.5269	−1.56314	−0.781572	+	0.623815i	$0.785582\pi$
$434$	0	0
$435$	0	0
$436$	−16.0000	−0.766261
$437$	0	0
$438$	0	0
$439$	−19.7990	−0.944954	−0.472477	−	0.881343i	$-0.656640\pi$
$439$	−19.7990	−0.944954	−0.472477	+	0.881343i	$0.656640\pi$
$440$	−2.82843	−0.134840
$441$	0	0
$442$	24.0000	1.14156
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	−44.0000	−2.08580
$446$	21.2132	1.00447
$447$	0	0
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	0	0
$452$	−14.0000	−0.658505
$453$	0	0
$454$	18.3848	0.862840
$455$	0	0
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	−16.9706	−0.792982
$459$	0	0
$460$	0	0
$461$	−24.0416	−1.11973	−0.559865	−	0.828584i	$-0.689147\pi$
$461$	−24.0416	−1.11973	−0.559865	+	0.828584i	$0.689147\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−14.0000	−0.648537
$467$	−16.9706	−0.785304	−0.392652	−	0.919687i	$-0.628442\pi$
$467$	−16.9706	−0.785304	−0.392652	+	0.919687i	$0.628442\pi$
$468$	0	0
$469$	0	0
$470$	−20.0000	−0.922531
$471$	0	0
$472$	2.82843	0.130189
$473$	−4.00000	−0.183920
$474$	0	0
$475$	21.2132	0.973329
$476$	0	0
$477$	0	0
$478$	16.0000	0.731823
$479$	−28.2843	−1.29234	−0.646171	−	0.763193i	$-0.723631\pi$
$479$	−28.2843	−1.29234	−0.646171	+	0.763193i	$0.723631\pi$
$480$	0	0
$481$	25.4558	1.16069
$482$	−14.1421	−0.644157
$483$	0	0
$484$	1.00000	0.0454545
$485$	−20.0000	−0.908153
$486$	0	0
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	−12.7279	−0.576166
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−45.2548	−2.03818
$494$	30.0000	1.34976
$495$	0	0
$496$	1.41421	0.0635001
$497$	0	0
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−5.65685	−0.252982
$501$	0	0
$502$	−8.48528	−0.378717
$503$	19.7990	0.882793	0.441397	−	0.897312i	$-0.354483\pi$
$503$	19.7990	0.882793	0.441397	+	0.897312i	$0.354483\pi$
$504$	0	0
$505$	44.0000	1.95797
$506$	0	0
$507$	0	0
$508$	−4.00000	−0.177471
$509$	2.82843	0.125368	0.0626839	−	0.998033i	$-0.480034\pi$
$509$	2.82843	0.125368	0.0626839	+	0.998033i	$0.480034\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	9.89949	0.436648
$515$	−12.0000	−0.528783
$516$	0	0
$517$	7.07107	0.310985
$518$	0	0
$519$	0	0
$520$	12.0000	0.526235
$521$	1.41421	0.0619578	0.0309789	−	0.999520i	$-0.490138\pi$
$521$	1.41421	0.0619578	0.0309789	+	0.999520i	$0.490138\pi$
$522$	0	0
$523$	−32.5269	−1.42230	−0.711151	−	0.703039i	$-0.751826\pi$
$523$	−32.5269	−1.42230	−0.711151	+	0.703039i	$0.751826\pi$
$524$	15.5563	0.679582
$525$	0	0
$526$	−24.0000	−1.04645
$527$	8.00000	0.348485
$528$	0	0
$529$	−23.0000	−1.00000
$530$	16.9706	0.737154
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−16.9706	−0.733701
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−8.48528	−0.365826
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−25.4558	−1.09342
$543$	0	0
$544$	5.65685	0.242536
$545$	−45.2548	−1.93850
$546$	0	0
$547$	6.00000	0.256541	0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000	0.256541	0.128271	+	0.991739i	$0.459057\pi$
$548$	−4.00000	−0.170872
$549$	0	0
$550$	−3.00000	−0.127920
$551$	−56.5685	−2.40990
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	4.24264	0.179928
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	16.9706	0.717778
$560$	0	0
$561$	0	0
$562$	30.0000	1.26547
$563$	−9.89949	−0.417214	−0.208607	−	0.978000i	$-0.566893\pi$
$563$	−9.89949	−0.417214	−0.208607	+	0.978000i	$0.566893\pi$
$564$	0	0
$565$	−39.5980	−1.66590
$566$	21.2132	0.891657
$567$	0	0
$568$	6.00000	0.251754
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	30.0000	1.25546	0.627730	−	0.778431i	$-0.283984\pi$
$571$	30.0000	1.25546	0.627730	+	0.778431i	$0.283984\pi$
$572$	−4.24264	−0.177394
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.6985	1.23636	0.618182	−	0.786035i	$-0.287869\pi$
$577$	29.6985	1.23636	0.618182	+	0.786035i	$0.287869\pi$
$578$	15.0000	0.623918
$579$	0	0
$580$	−22.6274	−0.939552
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	−8.48528	−0.351123
$585$	0	0
$586$	−7.07107	−0.292103
$587$	−19.7990	−0.817192	−0.408596	−	0.912715i	$-0.633981\pi$
$587$	−19.7990	−0.817192	−0.408596	+	0.912715i	$0.633981\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	8.00000	0.329355
$591$	0	0
$592$	6.00000	0.246598
$593$	−22.6274	−0.929197	−0.464598	−	0.885522i	$-0.653801\pi$
$593$	−22.6274	−0.929197	−0.464598	+	0.885522i	$0.653801\pi$
$594$	0	0
$595$	0	0
$596$	−4.00000	−0.163846
$597$	0	0
$598$	0	0
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	0	0
$601$	16.9706	0.692244	0.346122	−	0.938190i	$-0.387498\pi$
$601$	16.9706	0.692244	0.346122	+	0.938190i	$0.387498\pi$
$602$	0	0
$603$	0	0
$604$	16.0000	0.651031
$605$	2.82843	0.114992
$606$	0	0
$607$	16.9706	0.688814	0.344407	−	0.938820i	$-0.388080\pi$
$607$	16.9706	0.688814	0.344407	+	0.938820i	$0.388080\pi$
$608$	7.07107	0.286770
$609$	0	0
$610$	−36.0000	−1.45760
$611$	−30.0000	−1.21367
$612$	0	0
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	7.07107	0.285365
$615$	0	0
$616$	0	0
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	39.5980	1.59158	0.795789	−	0.605575i	$-0.207057\pi$
$619$	39.5980	1.59158	0.795789	+	0.605575i	$0.207057\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	1.41421	0.0567048
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	−12.7279	−0.508710
$627$	0	0
$628$	8.48528	0.338600
$629$	33.9411	1.35332
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	12.0000	0.477334
$633$	0	0
$634$	18.0000	0.714871
$635$	−11.3137	−0.448971
$636$	0	0
$637$	0	0
$638$	8.00000	0.316723
$639$	0	0
$640$	2.82843	0.111803
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	8.48528	0.334627	0.167313	−	0.985904i	$-0.446491\pi$
$643$	8.48528	0.334627	0.167313	+	0.985904i	$0.446491\pi$
$644$	0	0
$645$	0	0
$646$	40.0000	1.57378
$647$	18.3848	0.722780	0.361390	−	0.932415i	$-0.382302\pi$
$647$	18.3848	0.722780	0.361390	+	0.932415i	$0.382302\pi$
$648$	0	0
$649$	−2.82843	−0.111025
$650$	12.7279	0.499230
$651$	0	0
$652$	20.0000	0.783260
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	0	0
$655$	44.0000	1.71922
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	−7.07107	−0.274411
$665$	0	0
$666$	0	0
$667$	0	0
$668$	2.82843	0.109435
$669$	0	0
$670$	−11.3137	−0.437087
$671$	12.7279	0.491356
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	5.00000	0.192308
$677$	−35.3553	−1.35882	−0.679408	−	0.733761i	$-0.737763\pi$
$677$	−35.3553	−1.35882	−0.679408	+	0.733761i	$0.737763\pi$
$678$	0	0
$679$	0	0
$680$	16.0000	0.613572
$681$	0	0
$682$	−1.41421	−0.0541530
$683$	−32.0000	−1.22445	−0.612223	−	0.790685i	$-0.709725\pi$
$683$	−32.0000	−1.22445	−0.612223	+	0.790685i	$0.709725\pi$
$684$	0	0
$685$	−11.3137	−0.432275
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	25.4558	0.969790
$690$	0	0
$691$	−42.4264	−1.61398	−0.806988	−	0.590567i	$-0.798904\pi$
$691$	−42.4264	−1.61398	−0.806988	+	0.590567i	$0.798904\pi$
$692$	9.89949	0.376322
$693$	0	0
$694$	−34.0000	−1.29062
$695$	12.0000	0.455186
$696$	0	0
$697$	0	0
$698$	26.8701	1.01705
$699$	0	0
$700$	0	0
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	0	0
$703$	42.4264	1.60014
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−12.7279	−0.479022
$707$	0	0
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	16.9706	0.636894
$711$	0	0
$712$	−15.5563	−0.582999
$713$	0	0
$714$	0	0
$715$	−12.0000	−0.448775
$716$	12.0000	0.448461
$717$	0	0
$718$	−36.0000	−1.34351
$719$	4.24264	0.158224	0.0791119	−	0.996866i	$-0.474792\pi$
$719$	4.24264	0.158224	0.0791119	+	0.996866i	$0.474792\pi$
$720$	0	0
$721$	0	0
$722$	31.0000	1.15370
$723$	0	0
$724$	−16.9706	−0.630706
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−7.07107	−0.262251	−0.131126	−	0.991366i	$-0.541859\pi$
$727$	−7.07107	−0.262251	−0.131126	+	0.991366i	$0.541859\pi$
$728$	0	0
$729$	0	0
$730$	−24.0000	−0.888280
$731$	22.6274	0.836905
$732$	0	0
$733$	−4.24264	−0.156706	−0.0783528	−	0.996926i	$-0.524966\pi$
$733$	−4.24264	−0.156706	−0.0783528	+	0.996926i	$0.524966\pi$
$734$	35.3553	1.30499
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	16.9706	0.623850
$741$	0	0
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−11.3137	−0.414502
$746$	8.00000	0.292901
$747$	0	0
$748$	−5.65685	−0.206835
$749$	0	0
$750$	0	0
$751$	46.0000	1.67856	0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000	1.67856	0.839282	+	0.543696i	$0.182976\pi$
$752$	−7.07107	−0.257855
$753$	0	0
$754$	−33.9411	−1.23606
$755$	45.2548	1.64699
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	32.0000	1.16229
$759$	0	0
$760$	20.0000	0.725476
$761$	31.1127	1.12783	0.563917	−	0.825831i	$-0.309294\pi$
$761$	31.1127	1.12783	0.563917	+	0.825831i	$0.309294\pi$
$762$	0	0
$763$	0	0
$764$	−8.00000	−0.289430
$765$	0	0
$766$	−24.0416	−0.868659
$767$	12.0000	0.433295
$768$	0	0
$769$	−16.9706	−0.611974	−0.305987	−	0.952036i	$-0.598986\pi$
$769$	−16.9706	−0.611974	−0.305987	+	0.952036i	$0.598986\pi$
$770$	0	0
$771$	0	0
$772$	22.0000	0.791797
$773$	−5.65685	−0.203463	−0.101731	−	0.994812i	$-0.532438\pi$
$773$	−5.65685	−0.203463	−0.101731	+	0.994812i	$0.532438\pi$
$774$	0	0
$775$	4.24264	0.152400
$776$	−7.07107	−0.253837
$777$	0	0
$778$	−6.00000	−0.215110
$779$	0	0
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	9.89949	0.352879	0.176439	−	0.984311i	$-0.443542\pi$
$787$	9.89949	0.352879	0.176439	+	0.984311i	$0.443542\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	33.9411	1.20757
$791$	0	0
$792$	0	0
$793$	−54.0000	−1.91760
$794$	−14.1421	−0.501886
$795$	0	0
$796$	4.24264	0.150376
$797$	−5.65685	−0.200376	−0.100188	−	0.994969i	$-0.531944\pi$
$797$	−5.65685	−0.200376	−0.100188	+	0.994969i	$0.531944\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	3.00000	0.106066
$801$	0	0
$802$	−12.0000	−0.423735
$803$	8.48528	0.299439
$804$	0	0
$805$	0	0
$806$	6.00000	0.211341
$807$	0	0
$808$	15.5563	0.547270
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	24.0416	0.844216	0.422108	−	0.906546i	$-0.361290\pi$
$811$	24.0416	0.844216	0.422108	+	0.906546i	$0.361290\pi$
$812$	0	0
$813$	0	0
$814$	−6.00000	−0.210300
$815$	56.5685	1.98151
$816$	0	0
$817$	28.2843	0.989541
$818$	11.3137	0.395575
$819$	0	0
$820$	0	0
$821$	20.0000	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$822$	0	0
$823$	18.0000	0.627441	0.313720	−	0.949515i	$-0.398425\pi$
$823$	18.0000	0.627441	0.313720	+	0.949515i	$0.398425\pi$
$824$	−4.24264	−0.147799
$825$	0	0
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	16.9706	0.589412	0.294706	−	0.955588i	$-0.404778\pi$
$829$	16.9706	0.589412	0.294706	+	0.955588i	$0.404778\pi$
$830$	−20.0000	−0.694210
$831$	0	0
$832$	4.24264	0.147087
$833$	0	0
$834$	0	0
$835$	8.00000	0.276851
$836$	−7.07107	−0.244558
$837$	0	0
$838$	11.3137	0.390826
$839$	−49.4975	−1.70884	−0.854421	−	0.519581i	$-0.826088\pi$
$839$	−49.4975	−1.70884	−0.854421	+	0.519581i	$0.826088\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	2.00000	0.0689246
$843$	0	0
$844$	−20.0000	−0.688428
$845$	14.1421	0.486504
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	16.9706	0.582086
$851$	0	0
$852$	0	0
$853$	−29.6985	−1.01686	−0.508428	−	0.861104i	$-0.669773\pi$
$853$	−29.6985	−1.01686	−0.508428	+	0.861104i	$0.669773\pi$
$854$	0	0
$855$	0	0
$856$	−6.00000	−0.205076
$857$	−19.7990	−0.676321	−0.338160	−	0.941089i	$-0.609805\pi$
$857$	−19.7990	−0.676321	−0.338160	+	0.941089i	$0.609805\pi$
$858$	0	0
$859$	−31.1127	−1.06155	−0.530776	−	0.847512i	$-0.678099\pi$
$859$	−31.1127	−1.06155	−0.530776	+	0.847512i	$0.678099\pi$
$860$	11.3137	0.385794
$861$	0	0
$862$	24.0000	0.817443
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	0	0
$865$	28.0000	0.952029
$866$	−32.5269	−1.10531
$867$	0	0
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−16.9706	−0.575026
$872$	−16.0000	−0.541828
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.0000	1.21563	0.607817	−	0.794077i	$-0.292045\pi$
$877$	36.0000	1.21563	0.607817	+	0.794077i	$0.292045\pi$
$878$	−19.7990	−0.668184
$879$	0	0
$880$	−2.82843	−0.0953463
$881$	43.8406	1.47703	0.738514	−	0.674238i	$-0.235528\pi$
$881$	43.8406	1.47703	0.738514	+	0.674238i	$0.235528\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	24.0000	0.806296
$887$	5.65685	0.189939	0.0949693	−	0.995480i	$-0.469725\pi$
$887$	5.65685	0.189939	0.0949693	+	0.995480i	$0.469725\pi$
$888$	0	0
$889$	0	0
$890$	−44.0000	−1.47488
$891$	0	0
$892$	21.2132	0.710271
$893$	−50.0000	−1.67319
$894$	0	0
$895$	33.9411	1.13453
$896$	0	0
$897$	0	0
$898$	22.0000	0.734150
$899$	−11.3137	−0.377333
$900$	0	0
$901$	33.9411	1.13074
$902$	0	0
$903$	0	0
$904$	−14.0000	−0.465633
$905$	−48.0000	−1.59557
$906$	0	0
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	18.3848	0.610120
$909$	0	0
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	0	0
$913$	7.07107	0.234018
$914$	−34.0000	−1.12462
$915$	0	0
$916$	−16.9706	−0.560723
$917$	0	0
$918$	0	0
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	0	0
$921$	0	0
$922$	−24.0416	−0.791769
$923$	25.4558	0.837889
$924$	0	0
$925$	18.0000	0.591836
$926$	32.0000	1.05159
$927$	0	0
$928$	−8.00000	−0.262613
$929$	−24.0416	−0.788780	−0.394390	−	0.918943i	$-0.629044\pi$
$929$	−24.0416	−0.788780	−0.394390	+	0.918943i	$0.629044\pi$
$930$	0	0
$931$	0	0
$932$	−14.0000	−0.458585
$933$	0	0
$934$	−16.9706	−0.555294
$935$	−16.0000	−0.523256
$936$	0	0
$937$	19.7990	0.646805	0.323402	−	0.946262i	$-0.395173\pi$
$937$	19.7990	0.646805	0.323402	+	0.946262i	$0.395173\pi$
$938$	0	0
$939$	0	0
$940$	−20.0000	−0.652328
$941$	−29.6985	−0.968143	−0.484071	−	0.875028i	$-0.660843\pi$
$941$	−29.6985	−0.968143	−0.484071	+	0.875028i	$0.660843\pi$
$942$	0	0
$943$	0	0
$944$	2.82843	0.0920575
$945$	0	0
$946$	−4.00000	−0.130051
$947$	40.0000	1.29983	0.649913	−	0.760009i	$-0.274805\pi$
$947$	40.0000	1.29983	0.649913	+	0.760009i	$0.274805\pi$
$948$	0	0
$949$	−36.0000	−1.16861
$950$	21.2132	0.688247
$951$	0	0
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	−22.6274	−0.732206
$956$	16.0000	0.517477
$957$	0	0
$958$	−28.2843	−0.913823
$959$	0	0
$960$	0	0
$961$	−29.0000	−0.935484
$962$	25.4558	0.820729
$963$	0	0
$964$	−14.1421	−0.455488
$965$	62.2254	2.00311
$966$	0	0
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−20.0000	−0.642161
$971$	22.6274	0.726148	0.363074	−	0.931760i	$-0.381727\pi$
$971$	22.6274	0.726148	0.363074	+	0.931760i	$0.381727\pi$
$972$	0	0
$973$	0	0
$974$	−38.0000	−1.21760
$975$	0	0
$976$	−12.7279	−0.407411
$977$	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
$979$	15.5563	0.497183
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	46.6690	1.48851	0.744256	−	0.667895i	$-0.232804\pi$
$983$	46.6690	1.48851	0.744256	+	0.667895i	$0.232804\pi$
$984$	0	0
$985$	−28.2843	−0.901212
$986$	−45.2548	−1.44121
$987$	0	0
$988$	30.0000	0.954427
$989$	0	0
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	1.41421	0.0449013
$993$	0	0
$994$	0	0
$995$	12.0000	0.380426
$996$	0	0
$997$	−24.0416	−0.761406	−0.380703	−	0.924697i	$-0.624318\pi$
$997$	−24.0416	−0.761406	−0.380703	+	0.924697i	$0.624318\pi$
$998$	28.0000	0.886325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.di.1.2	yes	2
3.2	odd	2		9702.2.a.cr.1.1	✓	2
7.6	odd	2	inner	9702.2.a.di.1.1	yes	2
21.20	even	2		9702.2.a.cr.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9702.2.a.cr.1.1	✓	2	3.2	odd	2
9702.2.a.cr.1.2	yes	2	21.20	even	2
9702.2.a.di.1.1	yes	2	7.6	odd	2	inner
9702.2.a.di.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} +1.00000 q^{8} +2.82843 q^{10} -1.00000 q^{11} +4.24264 q^{13} +1.00000 q^{16} +5.65685 q^{17} +7.07107 q^{19} +2.82843 q^{20} -1.00000 q^{22} +3.00000 q^{25} +4.24264 q^{26} -8.00000 q^{29} +1.41421 q^{31} +1.00000 q^{32} +5.65685 q^{34} +6.00000 q^{37} +7.07107 q^{38} +2.82843 q^{40} +4.00000 q^{43} -1.00000 q^{44} -7.07107 q^{47} +3.00000 q^{50} +4.24264 q^{52} +6.00000 q^{53} -2.82843 q^{55} -8.00000 q^{58} +2.82843 q^{59} -12.7279 q^{61} +1.41421 q^{62} +1.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} +5.65685 q^{68} +6.00000 q^{71} -8.48528 q^{73} +6.00000 q^{74} +7.07107 q^{76} +12.0000 q^{79} +2.82843 q^{80} -7.07107 q^{83} +16.0000 q^{85} +4.00000 q^{86} -1.00000 q^{88} -15.5563 q^{89} -7.07107 q^{94} +20.0000 q^{95} -7.07107 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} + 2 q^{16} - 2 q^{22} + 6 q^{25} - 16 q^{29} + 2 q^{32} + 12 q^{37} + 8 q^{43} - 2 q^{44} + 6 q^{50} + 12 q^{53} - 16 q^{58} + 2 q^{64} + 24 q^{65} - 8 q^{67} + 12 q^{71} + 12 q^{74} + 24 q^{79} + 32 q^{85} + 8 q^{86} - 2 q^{88} + 40 q^{95}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 + 2 * q^16 - 2 * q^22 + 6 * q^25 - 16 * q^29 + 2 * q^32 + 12 * q^37 + 8 * q^43 - 2 * q^44 + 6 * q^50 + 12 * q^53 - 16 * q^58 + 2 * q^64 + 24 * q^65 - 8 * q^67 + 12 * q^71 + 12 * q^74 + 24 * q^79 + 32 * q^85 + 8 * q^86 - 2 * q^88 + 40 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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