LMFDB - Embedded newform 9702.2.a.ce.1.1

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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{8} +3.00000 q^{10} +1.00000 q^{11} -6.00000 q^{13} +1.00000 q^{16} -5.00000 q^{17} -6.00000 q^{19} +3.00000 q^{20} +1.00000 q^{22} -5.00000 q^{23} +4.00000 q^{25} -6.00000 q^{26} +6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -5.00000 q^{34} -2.00000 q^{37} -6.00000 q^{38} +3.00000 q^{40} +5.00000 q^{41} -10.0000 q^{43} +1.00000 q^{44} -5.00000 q^{46} +9.00000 q^{47} +4.00000 q^{50} -6.00000 q^{52} -2.00000 q^{53} +3.00000 q^{55} +6.00000 q^{58} -12.0000 q^{59} +5.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -18.0000 q^{65} +5.00000 q^{67} -5.00000 q^{68} -4.00000 q^{71} -12.0000 q^{73} -2.00000 q^{74} -6.00000 q^{76} -1.00000 q^{79} +3.00000 q^{80} +5.00000 q^{82} +1.00000 q^{83} -15.0000 q^{85} -10.0000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -5.00000 q^{92} +9.00000 q^{94} -18.0000 q^{95} -9.00000 q^{97} +O(q^{100})$

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
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	3.00000	0.948683
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	1.00000	0.213201
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−6.00000	−1.17670
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.00000	−0.857493
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	3.00000	0.474342
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−5.00000	−0.737210
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	0	0
$50$	4.00000	0.565685
$51$	0	0
$52$	−6.00000	−0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	0	0
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−18.0000	−2.23263
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	5.00000	0.552158
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	0	0
$85$	−15.0000	−1.62698
$86$	−10.0000	−1.07833
$87$	0	0
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	−5.00000	−0.521286
$93$	0	0
$94$	9.00000	0.928279
$95$	−18.0000	−1.84676
$96$	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
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	3.00000	0.286039
$111$	0	0
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−15.0000	−1.39876
$116$	6.00000	0.557086
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.00000	0.452679
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−18.0000	−1.57870
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0	0
$134$	5.00000	0.431934
$135$	0	0
$136$	−5.00000	−0.428746
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	0	0
$142$	−4.00000	−0.335673
$143$	−6.00000	−0.501745
$144$	0	0
$145$	18.0000	1.49482
$146$	−12.0000	−0.993127
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	0	0
$155$	−12.0000	−0.963863
$156$	0	0
$157$	24.0000	1.91541	0.957704	−	0.287754i	$-0.0929087\pi$
$157$	24.0000	1.91541	0.957704	+	0.287754i	$0.0929087\pi$
$158$	−1.00000	−0.0795557
$159$	0	0
$160$	3.00000	0.237171
$161$	0	0
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	1.00000	0.0776151
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−15.0000	−1.15045
$171$	0	0
$172$	−10.0000	−0.762493
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	6.00000	0.449719
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	0	0
$184$	−5.00000	−0.368605
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−5.00000	−0.365636
$188$	9.00000	0.656392
$189$	0	0
$190$	−18.0000	−1.30586
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−9.00000	−0.646162
$195$	0	0
$196$	0	0
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−8.00000	−0.562878
$203$	0	0
$204$	0	0
$205$	15.0000	1.04765
$206$	2.00000	0.139347
$207$	0	0
$208$	−6.00000	−0.416025
$209$	−6.00000	−0.415029
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	13.0000	0.888662
$215$	−30.0000	−2.04598
$216$	0	0
$217$	0	0
$218$	−11.0000	−0.745014
$219$	0	0
$220$	3.00000	0.202260
$221$	30.0000	2.01802
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	−10.0000	−0.665190
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	−15.0000	−0.989071
$231$	0	0
$232$	6.00000	0.393919
$233$	29.0000	1.89985	0.949927	−	0.312473i	$-0.101157\pi$
$233$	29.0000	1.89985	0.949927	+	0.312473i	$0.101157\pi$
$234$	0	0
$235$	27.0000	1.76129
$236$	−12.0000	−0.781133
$237$	0	0
$238$	0	0
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	5.00000	0.320092
$245$	0	0
$246$	0	0
$247$	36.0000	2.29063
$248$	−4.00000	−0.254000
$249$	0	0
$250$	−3.00000	−0.189737
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	5.00000	0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	0	0
$260$	−18.0000	−1.11631
$261$	0	0
$262$	−4.00000	−0.247121
$263$	−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
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	5.00000	0.305424
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	2.00000	0.120824
$275$	4.00000	0.241209
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−22.0000	−1.31947
$279$	0	0
$280$	0	0
$281$	−11.0000	−0.656205	−0.328102	−	0.944642i	$-0.606409\pi$
$281$	−11.0000	−0.656205	−0.328102	+	0.944642i	$0.606409\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	18.0000	1.05700
$291$	0	0
$292$	−12.0000	−0.702247
$293$	8.00000	0.467365	0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000	0.467365	0.233682	+	0.972313i	$0.424922\pi$
$294$	0	0
$295$	−36.0000	−2.09600
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−18.0000	−1.04271
$299$	30.0000	1.73494
$300$	0	0
$301$	0	0
$302$	9.00000	0.517892
$303$	0	0
$304$	−6.00000	−0.344124
$305$	15.0000	0.858898
$306$	0	0
$307$	−18.0000	−1.02731	−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000	−1.02731	−0.513657	+	0.857996i	$0.671710\pi$
$308$	0	0
$309$	0	0
$310$	−12.0000	−0.681554
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	24.0000	1.35440
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	3.00000	0.167705
$321$	0	0
$322$	0	0
$323$	30.0000	1.66924
$324$	0	0
$325$	−24.0000	−1.33128
$326$	−1.00000	−0.0553849
$327$	0	0
$328$	5.00000	0.276079
$329$	0	0
$330$	0	0
$331$	35.0000	1.92377	0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000	1.92377	0.961887	+	0.273447i	$0.0881639\pi$
$332$	1.00000	0.0548821
$333$	0	0
$334$	−24.0000	−1.31322
$335$	15.0000	0.819538
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	23.0000	1.25104
$339$	0	0
$340$	−15.0000	−0.813489
$341$	−4.00000	−0.216612
$342$	0	0
$343$	0	0
$344$	−10.0000	−0.539164
$345$	0	0
$346$	−12.0000	−0.645124
$347$	−7.00000	−0.375780	−0.187890	−	0.982190i	$-0.560165\pi$
$347$	−7.00000	−0.375780	−0.187890	+	0.982190i	$0.560165\pi$
$348$	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
$352$	1.00000	0.0533002
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	6.00000	0.317999
$357$	0	0
$358$	8.00000	0.422813
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	22.0000	1.15629
$363$	0	0
$364$	0	0
$365$	−36.0000	−1.88433
$366$	0	0
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	−5.00000	−0.260643
$369$	0	0
$370$	−6.00000	−0.311925
$371$	0	0
$372$	0	0
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	−5.00000	−0.258544
$375$	0	0
$376$	9.00000	0.464140
$377$	−36.0000	−1.85409
$378$	0	0
$379$	13.0000	0.667765	0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000	0.667765	0.333883	+	0.942615i	$0.391641\pi$
$380$	−18.0000	−0.923381
$381$	0	0
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	0	0
$386$	6.00000	0.305392
$387$	0	0
$388$	−9.00000	−0.456906
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	25.0000	1.26430
$392$	0	0
$393$	0	0
$394$	24.0000	1.20910
$395$	−3.00000	−0.150946
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	−10.0000	−0.501255
$399$	0	0
$400$	4.00000	0.200000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	−8.00000	−0.398015
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	15.0000	0.740797
$411$	0	0
$412$	2.00000	0.0985329
$413$	0	0
$414$	0	0
$415$	3.00000	0.147264
$416$	−6.00000	−0.294174
$417$	0	0
$418$	−6.00000	−0.293470
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	14.0000	0.681509
$423$	0	0
$424$	−2.00000	−0.0971286
$425$	−20.0000	−0.970143
$426$	0	0
$427$	0	0
$428$	13.0000	0.628379
$429$	0	0
$430$	−30.0000	−1.44673
$431$	−22.0000	−1.05970	−0.529851	−	0.848091i	$-0.677752\pi$
$431$	−22.0000	−1.05970	−0.529851	+	0.848091i	$0.677752\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	0	0
$436$	−11.0000	−0.526804
$437$	30.0000	1.43509
$438$	0	0
$439$	5.00000	0.238637	0.119318	−	0.992856i	$-0.461929\pi$
$439$	5.00000	0.238637	0.119318	+	0.992856i	$0.461929\pi$
$440$	3.00000	0.143019
$441$	0	0
$442$	30.0000	1.42695
$443$	−2.00000	−0.0950229	−0.0475114	−	0.998871i	$-0.515129\pi$
$443$	−2.00000	−0.0950229	−0.0475114	+	0.998871i	$0.515129\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	−16.0000	−0.757622
$447$	0	0
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	5.00000	0.235441
$452$	−10.0000	−0.470360
$453$	0	0
$454$	3.00000	0.140797
$455$	0	0
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	−26.0000	−1.21490
$459$	0	0
$460$	−15.0000	−0.699379
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	28.0000	1.30127	0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000	1.30127	0.650635	+	0.759390i	$0.274503\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	29.0000	1.34340
$467$	16.0000	0.740392	0.370196	−	0.928954i	$-0.379291\pi$
$467$	16.0000	0.740392	0.370196	+	0.928954i	$0.379291\pi$
$468$	0	0
$469$	0	0
$470$	27.0000	1.24542
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−10.0000	−0.459800
$474$	0	0
$475$	−24.0000	−1.10120
$476$	0	0
$477$	0	0
$478$	−26.0000	−1.18921
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	26.0000	1.18427
$483$	0	0
$484$	1.00000	0.0454545
$485$	−27.0000	−1.22601
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	5.00000	0.226339
$489$	0	0
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	−30.0000	−1.35113
$494$	36.0000	1.61972
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	12.0000	0.535586
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	−5.00000	−0.222277
$507$	0	0
$508$	5.00000	0.221839
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−14.0000	−0.617514
$515$	6.00000	0.264392
$516$	0	0
$517$	9.00000	0.395820
$518$	0	0
$519$	0	0
$520$	−18.0000	−0.789352
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−24.0000	−1.04645
$527$	20.0000	0.871214
$528$	0	0
$529$	2.00000	0.0869565
$530$	−6.00000	−0.260623
$531$	0	0
$532$	0	0
$533$	−30.0000	−1.29944
$534$	0	0
$535$	39.0000	1.68612
$536$	5.00000	0.215967
$537$	0	0
$538$	−9.00000	−0.388018
$539$	0	0
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	24.0000	1.03089
$543$	0	0
$544$	−5.00000	−0.214373
$545$	−33.0000	−1.41356
$546$	0	0
$547$	−42.0000	−1.79579	−0.897895	−	0.440209i	$-0.854904\pi$
$547$	−42.0000	−1.79579	−0.897895	+	0.440209i	$0.854904\pi$
$548$	2.00000	0.0854358
$549$	0	0
$550$	4.00000	0.170561
$551$	−36.0000	−1.53365
$552$	0	0
$553$	0	0
$554$	18.0000	0.764747
$555$	0	0
$556$	−22.0000	−0.933008
$557$	32.0000	1.35588	0.677942	−	0.735116i	$-0.262872\pi$
$557$	32.0000	1.35588	0.677942	+	0.735116i	$0.262872\pi$
$558$	0	0
$559$	60.0000	2.53773
$560$	0	0
$561$	0	0
$562$	−11.0000	−0.464007
$563$	−16.0000	−0.674320	−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000	−0.674320	−0.337160	+	0.941447i	$0.609466\pi$
$564$	0	0
$565$	−30.0000	−1.26211
$566$	6.00000	0.252199
$567$	0	0
$568$	−4.00000	−0.167836
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	0	0
$575$	−20.0000	−0.834058
$576$	0	0
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	18.0000	0.747409
$581$	0	0
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	−12.0000	−0.496564
$585$	0	0
$586$	8.00000	0.330477
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	−36.0000	−1.48210
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	0	0
$598$	30.0000	1.22679
$599$	3.00000	0.122577	0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000	0.122577	0.0612883	+	0.998120i	$0.480479\pi$
$600$	0	0
$601$	44.0000	1.79480	0.897399	−	0.441221i	$-0.145454\pi$
$601$	44.0000	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$602$	0	0
$603$	0	0
$604$	9.00000	0.366205
$605$	3.00000	0.121967
$606$	0	0
$607$	−33.0000	−1.33943	−0.669714	−	0.742619i	$-0.733583\pi$
$607$	−33.0000	−1.33943	−0.669714	+	0.742619i	$0.733583\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	15.0000	0.607332
$611$	−54.0000	−2.18461
$612$	0	0
$613$	−41.0000	−1.65597	−0.827987	−	0.560747i	$-0.810514\pi$
$613$	−41.0000	−1.65597	−0.827987	+	0.560747i	$0.810514\pi$
$614$	−18.0000	−0.726421
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	29.0000	1.16561	0.582804	−	0.812613i	$-0.301955\pi$
$619$	29.0000	1.16561	0.582804	+	0.812613i	$0.301955\pi$
$620$	−12.0000	−0.481932
$621$	0	0
$622$	−21.0000	−0.842023
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	6.00000	0.239808
$627$	0	0
$628$	24.0000	0.957704
$629$	10.0000	0.398726
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	−1.00000	−0.0397779
$633$	0	0
$634$	−9.00000	−0.357436
$635$	15.0000	0.595257
$636$	0	0
$637$	0	0
$638$	6.00000	0.237542
$639$	0	0
$640$	3.00000	0.118585
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	0	0
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	0	0
$646$	30.0000	1.18033
$647$	17.0000	0.668339	0.334169	−	0.942513i	$-0.391544\pi$
$647$	17.0000	0.668339	0.334169	+	0.942513i	$0.391544\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	−24.0000	−0.941357
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	5.00000	0.195217
$657$	0	0
$658$	0	0
$659$	−31.0000	−1.20759	−0.603794	−	0.797140i	$-0.706345\pi$
$659$	−31.0000	−1.20759	−0.603794	+	0.797140i	$0.706345\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	35.0000	1.36031
$663$	0	0
$664$	1.00000	0.0388075
$665$	0	0
$666$	0	0
$667$	−30.0000	−1.16160
$668$	−24.0000	−0.928588
$669$	0	0
$670$	15.0000	0.579501
$671$	5.00000	0.193023
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	23.0000	0.884615
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	0	0
$680$	−15.0000	−0.575224
$681$	0	0
$682$	−4.00000	−0.153168
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	0	0
$688$	−10.0000	−0.381246
$689$	12.0000	0.457164
$690$	0	0
$691$	27.0000	1.02713	0.513564	−	0.858051i	$-0.328325\pi$
$691$	27.0000	1.02713	0.513564	+	0.858051i	$0.328325\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	−7.00000	−0.265716
$695$	−66.0000	−2.50352
$696$	0	0
$697$	−25.0000	−0.946943
$698$	19.0000	0.719161
$699$	0	0
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	1.00000	0.0376889
$705$	0	0
$706$	24.0000	0.903252
$707$	0	0
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	−12.0000	−0.450352
$711$	0	0
$712$	6.00000	0.224860
$713$	20.0000	0.749006
$714$	0	0
$715$	−18.0000	−0.673162
$716$	8.00000	0.298974
$717$	0	0
$718$	10.0000	0.373197
$719$	37.0000	1.37987	0.689934	−	0.723873i	$-0.257640\pi$
$719$	37.0000	1.37987	0.689934	+	0.723873i	$0.257640\pi$
$720$	0	0
$721$	0	0
$722$	17.0000	0.632674
$723$	0	0
$724$	22.0000	0.817624
$725$	24.0000	0.891338
$726$	0	0
$727$	−22.0000	−0.815935	−0.407967	−	0.912996i	$-0.633762\pi$
$727$	−22.0000	−0.815935	−0.407967	+	0.912996i	$0.633762\pi$
$728$	0	0
$729$	0	0
$730$	−36.0000	−1.33242
$731$	50.0000	1.84932
$732$	0	0
$733$	15.0000	0.554038	0.277019	−	0.960864i	$-0.410654\pi$
$733$	15.0000	0.554038	0.277019	+	0.960864i	$0.410654\pi$
$734$	14.0000	0.516749
$735$	0	0
$736$	−5.00000	−0.184302
$737$	5.00000	0.184177
$738$	0	0
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	−6.00000	−0.220564
$741$	0	0
$742$	0	0
$743$	−22.0000	−0.807102	−0.403551	−	0.914957i	$-0.632224\pi$
$743$	−22.0000	−0.807102	−0.403551	+	0.914957i	$0.632224\pi$
$744$	0	0
$745$	−54.0000	−1.97841
$746$	−7.00000	−0.256288
$747$	0	0
$748$	−5.00000	−0.182818
$749$	0	0
$750$	0	0
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	9.00000	0.328196
$753$	0	0
$754$	−36.0000	−1.31104
$755$	27.0000	0.982631
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	13.0000	0.472181
$759$	0	0
$760$	−18.0000	−0.652929
$761$	47.0000	1.70375	0.851874	−	0.523746i	$-0.175466\pi$
$761$	47.0000	1.70375	0.851874	+	0.523746i	$0.175466\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	8.00000	0.289052
$767$	72.0000	2.59977
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	0	0
$772$	6.00000	0.215945
$773$	11.0000	0.395643	0.197821	−	0.980238i	$-0.436613\pi$
$773$	11.0000	0.395643	0.197821	+	0.980238i	$0.436613\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	−9.00000	−0.323081
$777$	0	0
$778$	21.0000	0.752886
$779$	−30.0000	−1.07486
$780$	0	0
$781$	−4.00000	−0.143131
$782$	25.0000	0.893998
$783$	0	0
$784$	0	0
$785$	72.0000	2.56979
$786$	0	0
$787$	50.0000	1.78231	0.891154	−	0.453701i	$-0.149897\pi$
$787$	50.0000	1.78231	0.891154	+	0.453701i	$0.149897\pi$
$788$	24.0000	0.854965
$789$	0	0
$790$	−3.00000	−0.106735
$791$	0	0
$792$	0	0
$793$	−30.0000	−1.06533
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−53.0000	−1.87736	−0.938678	−	0.344795i	$-0.887949\pi$
$797$	−53.0000	−1.87736	−0.938678	+	0.344795i	$0.887949\pi$
$798$	0	0
$799$	−45.0000	−1.59199
$800$	4.00000	0.141421
$801$	0	0
$802$	6.00000	0.211867
$803$	−12.0000	−0.423471
$804$	0	0
$805$	0	0
$806$	24.0000	0.845364
$807$	0	0
$808$	−8.00000	−0.281439
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	0	0
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	0	0
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	−3.00000	−0.105085
$816$	0	0
$817$	60.0000	2.09913
$818$	−10.0000	−0.349642
$819$	0	0
$820$	15.0000	0.523823
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	0	0
$823$	10.0000	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$823$	10.0000	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$824$	2.00000	0.0696733
$825$	0	0
$826$	0	0
$827$	15.0000	0.521601	0.260801	−	0.965393i	$-0.416014\pi$
$827$	15.0000	0.521601	0.260801	+	0.965393i	$0.416014\pi$
$828$	0	0
$829$	54.0000	1.87550	0.937749	−	0.347314i	$-0.112906\pi$
$829$	54.0000	1.87550	0.937749	+	0.347314i	$0.112906\pi$
$830$	3.00000	0.104132
$831$	0	0
$832$	−6.00000	−0.208013
$833$	0	0
$834$	0	0
$835$	−72.0000	−2.49166
$836$	−6.00000	−0.207514
$837$	0	0
$838$	−20.0000	−0.690889
$839$	−9.00000	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$839$	−9.00000	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	8.00000	0.275698
$843$	0	0
$844$	14.0000	0.481900
$845$	69.0000	2.37367
$846$	0	0
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	−20.0000	−0.685994
$851$	10.0000	0.342796
$852$	0	0
$853$	−15.0000	−0.513590	−0.256795	−	0.966466i	$-0.582667\pi$
$853$	−15.0000	−0.513590	−0.256795	+	0.966466i	$0.582667\pi$
$854$	0	0
$855$	0	0
$856$	13.0000	0.444331
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	45.0000	1.53538	0.767690	−	0.640821i	$-0.221406\pi$
$859$	45.0000	1.53538	0.767690	+	0.640821i	$0.221406\pi$
$860$	−30.0000	−1.02299
$861$	0	0
$862$	−22.0000	−0.749323
$863$	39.0000	1.32758	0.663788	−	0.747921i	$-0.268948\pi$
$863$	39.0000	1.32758	0.663788	+	0.747921i	$0.268948\pi$
$864$	0	0
$865$	−36.0000	−1.22404
$866$	−11.0000	−0.373795
$867$	0	0
$868$	0	0
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	−30.0000	−1.01651
$872$	−11.0000	−0.372507
$873$	0	0
$874$	30.0000	1.01477
$875$	0	0
$876$	0	0
$877$	−3.00000	−0.101303	−0.0506514	−	0.998716i	$-0.516130\pi$
$877$	−3.00000	−0.101303	−0.0506514	+	0.998716i	$0.516130\pi$
$878$	5.00000	0.168742
$879$	0	0
$880$	3.00000	0.101130
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	−13.0000	−0.437485	−0.218742	−	0.975783i	$-0.570195\pi$
$883$	−13.0000	−0.437485	−0.218742	+	0.975783i	$0.570195\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	−2.00000	−0.0671913
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	0	0
$890$	18.0000	0.603361
$891$	0	0
$892$	−16.0000	−0.535720
$893$	−54.0000	−1.80704
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	−24.0000	−0.800445
$900$	0	0
$901$	10.0000	0.333148
$902$	5.00000	0.166482
$903$	0	0
$904$	−10.0000	−0.332595
$905$	66.0000	2.19391
$906$	0	0
$907$	23.0000	0.763702	0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000	0.763702	0.381851	+	0.924224i	$0.375287\pi$
$908$	3.00000	0.0995585
$909$	0	0
$910$	0	0
$911$	−51.0000	−1.68971	−0.844853	−	0.534999i	$-0.820312\pi$
$911$	−51.0000	−1.68971	−0.844853	+	0.534999i	$0.820312\pi$
$912$	0	0
$913$	1.00000	0.0330952
$914$	−26.0000	−0.860004
$915$	0	0
$916$	−26.0000	−0.859064
$917$	0	0
$918$	0	0
$919$	−17.0000	−0.560778	−0.280389	−	0.959886i	$-0.590464\pi$
$919$	−17.0000	−0.560778	−0.280389	+	0.959886i	$0.590464\pi$
$920$	−15.0000	−0.494535
$921$	0	0
$922$	−10.0000	−0.329332
$923$	24.0000	0.789970
$924$	0	0
$925$	−8.00000	−0.263038
$926$	28.0000	0.920137
$927$	0	0
$928$	6.00000	0.196960
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	0	0
$932$	29.0000	0.949927
$933$	0	0
$934$	16.0000	0.523536
$935$	−15.0000	−0.490552
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	0	0
$939$	0	0
$940$	27.0000	0.880643
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	−25.0000	−0.814112
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−10.0000	−0.325128
$947$	30.0000	0.974869	0.487435	−	0.873160i	$-0.337933\pi$
$947$	30.0000	0.974869	0.487435	+	0.873160i	$0.337933\pi$
$948$	0	0
$949$	72.0000	2.33722
$950$	−24.0000	−0.778663
$951$	0	0
$952$	0	0
$953$	21.0000	0.680257	0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000	0.680257	0.340128	+	0.940379i	$0.389529\pi$
$954$	0	0
$955$	0	0
$956$	−26.0000	−0.840900
$957$	0	0
$958$	12.0000	0.387702
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	12.0000	0.386896
$963$	0	0
$964$	26.0000	0.837404
$965$	18.0000	0.579441
$966$	0	0
$967$	−21.0000	−0.675314	−0.337657	−	0.941269i	$-0.609634\pi$
$967$	−21.0000	−0.675314	−0.337657	+	0.941269i	$0.609634\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−27.0000	−0.866918
$971$	−34.0000	−1.09111	−0.545556	−	0.838074i	$-0.683681\pi$
$971$	−34.0000	−1.09111	−0.545556	+	0.838074i	$0.683681\pi$
$972$	0	0
$973$	0	0
$974$	−4.00000	−0.128168
$975$	0	0
$976$	5.00000	0.160046
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	0	0
$982$	3.00000	0.0957338
$983$	−59.0000	−1.88181	−0.940904	−	0.338674i	$-0.890022\pi$
$983$	−59.0000	−1.88181	−0.940904	+	0.338674i	$0.890022\pi$
$984$	0	0
$985$	72.0000	2.29411
$986$	−30.0000	−0.955395
$987$	0	0
$988$	36.0000	1.14531
$989$	50.0000	1.58991
$990$	0	0
$991$	22.0000	0.698853	0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000	0.698853	0.349427	+	0.936964i	$0.386376\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	0	0
$995$	−30.0000	−0.951064
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	12.0000	0.379853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.ce.1.1		1
3.2	odd	2		3234.2.a.a.1.1		1
7.3	odd	6		1386.2.k.j.793.1		2
7.5	odd	6		1386.2.k.j.991.1		2
7.6	odd	2		9702.2.a.be.1.1		1
21.5	even	6		462.2.i.a.67.1	✓	2
21.17	even	6		462.2.i.a.331.1	yes	2
21.20	even	2		3234.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.a.67.1	✓	2	21.5	even	6
462.2.i.a.331.1	yes	2	21.17	even	6
1386.2.k.j.793.1		2	7.3	odd	6
1386.2.k.j.991.1		2	7.5	odd	6
3234.2.a.a.1.1		1	3.2	odd	2
3234.2.a.o.1.1		1	21.20	even	2
9702.2.a.be.1.1		1	7.6	odd	2
9702.2.a.ce.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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Representations

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