LMFDB - Embedded newform 9675.2.a.cw.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9675,2,Mod(1,9675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9675, 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("9675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9675 = 3^{2} \cdot 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9675.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:	$77.2552639556$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 13x^{8} + 58x^{6} - 103x^{4} + 65x^{2} - 13 $ x^10 - 13x^8 + 58x^6 - 103x^4 + 65x^2 - 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.756861$ of defining polynomial
Character	$\chi$	$=$	9675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.756861 q^{2} -1.42716 q^{4} -2.95974 q^{7} -2.59388 q^{8} +O(q^{10})$	$q+0.756861 q^{2} -1.42716 q^{4} -2.95974 q^{7} -2.59388 q^{8} -2.59388 q^{11} +5.38690 q^{13} -2.24011 q^{14} +0.891112 q^{16} -3.60010 q^{17} -2.06863 q^{19} -1.96321 q^{22} +5.48906 q^{23} +4.07714 q^{26} +4.22403 q^{28} +4.23088 q^{29} +3.82871 q^{31} +5.86222 q^{32} -2.72477 q^{34} -6.04651 q^{37} -1.56567 q^{38} +2.36882 q^{41} -1.00000 q^{43} +3.70189 q^{44} +4.15446 q^{46} +2.31687 q^{47} +1.76008 q^{49} -7.68798 q^{52} +0.622504 q^{53} +7.67723 q^{56} +3.20219 q^{58} +14.3268 q^{59} -3.56537 q^{61} +2.89780 q^{62} +2.65466 q^{64} +8.28573 q^{67} +5.13792 q^{68} -10.4796 q^{71} -3.93760 q^{73} -4.57637 q^{74} +2.95227 q^{76} +7.67723 q^{77} +0.684277 q^{79} +1.79286 q^{82} -16.1992 q^{83} -0.756861 q^{86} +6.72824 q^{88} +3.93448 q^{89} -15.9439 q^{91} -7.83378 q^{92} +1.75355 q^{94} +6.41993 q^{97} +1.33214 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{4}+O(q^{10}) $ 10 * q + 6 * q^4	$ 10 q + 6 q^{4} + 4 q^{13} - 10 q^{16} - 10 q^{19} + 2 q^{22} - 16 q^{31} - 8 q^{34} - 10 q^{37} - 10 q^{43} - 28 q^{46} - 26 q^{49} - 36 q^{52} + 16 q^{58} - 32 q^{61} - 28 q^{64} + 14 q^{67} - 4 q^{73} - 24 q^{76} - 18 q^{79} - 20 q^{82} + 56 q^{88} - 44 q^{91} - 12 q^{94} - 4 q^{97}+O(q^{100}) $ 10 * q + 6 * q^4 + 4 * q^13 - 10 * q^16 - 10 * q^19 + 2 * q^22 - 16 * q^31 - 8 * q^34 - 10 * q^37 - 10 * q^43 - 28 * q^46 - 26 * q^49 - 36 * q^52 + 16 * q^58 - 32 * q^61 - 28 * q^64 + 14 * q^67 - 4 * q^73 - 24 * q^76 - 18 * q^79 - 20 * q^82 + 56 * q^88 - 44 * q^91 - 12 * q^94 - 4 * 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 $

$2$	0.756861	0.535182	0.267591	−	0.963533i	$-0.413772\pi$
$2$	0.756861	0.535182	0.267591	+	0.963533i	$0.413772\pi$
$3$	0	0
$3$	0	0
$4$	−1.42716	−0.713581
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.95974	−1.11868	−0.559339	−	0.828939i	$-0.688945\pi$
$7$	−2.95974	−1.11868	−0.559339	+	0.828939i	$0.688945\pi$
$8$	−2.59388	−0.917077
$9$	0	0
$10$	0	0
$11$	−2.59388	−0.782086	−0.391043	−	0.920372i	$-0.627886\pi$
$11$	−2.59388	−0.782086	−0.391043	+	0.920372i	$0.627886\pi$
$12$	0	0
$13$	5.38690	1.49406	0.747029	−	0.664791i	$-0.231479\pi$
$13$	5.38690	1.49406	0.747029	+	0.664791i	$0.231479\pi$
$14$	−2.24011	−0.598696
$15$	0	0
$16$	0.891112	0.222778
$17$	−3.60010	−0.873151	−0.436576	−	0.899668i	$-0.643809\pi$
$17$	−3.60010	−0.873151	−0.436576	+	0.899668i	$0.643809\pi$
$18$	0	0
$19$	−2.06863	−0.474576	−0.237288	−	0.971439i	$-0.576259\pi$
$19$	−2.06863	−0.474576	−0.237288	+	0.971439i	$0.576259\pi$
$20$	0	0
$21$	0	0
$22$	−1.96321	−0.418558
$23$	5.48906	1.14455	0.572274	−	0.820062i	$-0.306061\pi$
$23$	5.48906	1.14455	0.572274	+	0.820062i	$0.306061\pi$
$24$	0	0
$25$	0	0
$26$	4.07714	0.799593
$27$	0	0
$28$	4.22403	0.798267
$29$	4.23088	0.785655	0.392827	−	0.919612i	$-0.371497\pi$
$29$	4.23088	0.785655	0.392827	+	0.919612i	$0.371497\pi$
$30$	0	0
$31$	3.82871	0.687657	0.343828	−	0.939032i	$-0.388276\pi$
$31$	3.82871	0.687657	0.343828	+	0.939032i	$0.388276\pi$
$32$	5.86222	1.03630
$33$	0	0
$34$	−2.72477	−0.467295
$35$	0	0
$36$	0	0
$37$	−6.04651	−0.994041	−0.497020	−	0.867739i	$-0.665573\pi$
$37$	−6.04651	−0.994041	−0.497020	+	0.867739i	$0.665573\pi$
$38$	−1.56567	−0.253985
$39$	0	0
$40$	0	0
$41$	2.36882	0.369947	0.184973	−	0.982744i	$-0.440780\pi$
$41$	2.36882	0.369947	0.184973	+	0.982744i	$0.440780\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	3.70189	0.558081
$45$	0	0
$46$	4.15446	0.612541
$47$	2.31687	0.337950	0.168975	−	0.985620i	$-0.445954\pi$
$47$	2.31687	0.337950	0.168975	+	0.985620i	$0.445954\pi$
$48$	0	0
$49$	1.76008	0.251440
$50$	0	0
$51$	0	0
$52$	−7.68798	−1.06613
$53$	0.622504	0.0855075	0.0427538	−	0.999086i	$-0.486387\pi$
$53$	0.622504	0.0855075	0.0427538	+	0.999086i	$0.486387\pi$
$54$	0	0
$55$	0	0
$56$	7.67723	1.02591
$57$	0	0
$58$	3.20219	0.420468
$59$	14.3268	1.86519	0.932594	−	0.360927i	$-0.117540\pi$
$59$	14.3268	1.86519	0.932594	+	0.360927i	$0.117540\pi$
$60$	0	0
$61$	−3.56537	−0.456498	−0.228249	−	0.973603i	$-0.573300\pi$
$61$	−3.56537	−0.456498	−0.228249	+	0.973603i	$0.573300\pi$
$62$	2.89780	0.368021
$63$	0	0
$64$	2.65466	0.331832
$65$	0	0
$66$	0	0
$67$	8.28573	1.01226	0.506131	−	0.862456i	$-0.331075\pi$
$67$	8.28573	1.01226	0.506131	+	0.862456i	$0.331075\pi$
$68$	5.13792	0.623064
$69$	0	0
$70$	0	0
$71$	−10.4796	−1.24370	−0.621850	−	0.783136i	$-0.713619\pi$
$71$	−10.4796	−1.24370	−0.621850	+	0.783136i	$0.713619\pi$
$72$	0	0
$73$	−3.93760	−0.460861	−0.230431	−	0.973089i	$-0.574013\pi$
$73$	−3.93760	−0.460861	−0.230431	+	0.973089i	$0.574013\pi$
$74$	−4.57637	−0.531992
$75$	0	0
$76$	2.95227	0.338649
$77$	7.67723	0.874902
$78$	0	0
$79$	0.684277	0.0769872	0.0384936	−	0.999259i	$-0.487744\pi$
$79$	0.684277	0.0769872	0.0384936	+	0.999259i	$0.487744\pi$
$80$	0	0
$81$	0	0
$82$	1.79286	0.197989
$83$	−16.1992	−1.77809	−0.889045	−	0.457820i	$-0.848630\pi$
$83$	−16.1992	−1.77809	−0.889045	+	0.457820i	$0.848630\pi$
$84$	0	0
$85$	0	0
$86$	−0.756861	−0.0816144
$87$	0	0
$88$	6.72824	0.717233
$89$	3.93448	0.417054	0.208527	−	0.978017i	$-0.433133\pi$
$89$	3.93448	0.417054	0.208527	+	0.978017i	$0.433133\pi$
$90$	0	0
$91$	−15.9439	−1.67137
$92$	−7.83378	−0.816728
$93$	0	0
$94$	1.75355	0.180865
$95$	0	0
$96$	0	0
$97$	6.41993	0.651845	0.325922	−	0.945397i	$-0.394325\pi$
$97$	6.41993	0.651845	0.325922	+	0.945397i	$0.394325\pi$
$98$	1.33214	0.134566
$99$	0	0
$100$	0	0
$101$	−11.7461	−1.16878	−0.584389	−	0.811474i	$-0.698666\pi$
$101$	−11.7461	−1.16878	−0.584389	+	0.811474i	$0.698666\pi$
$102$	0	0
$103$	18.7872	1.85116	0.925581	−	0.378550i	$-0.123577\pi$
$103$	18.7872	1.85116	0.925581	+	0.378550i	$0.123577\pi$
$104$	−13.9730	−1.37017
$105$	0	0
$106$	0.471149	0.0457620
$107$	12.9122	1.24827	0.624136	−	0.781315i	$-0.285451\pi$
$107$	12.9122	1.24827	0.624136	+	0.781315i	$0.285451\pi$
$108$	0	0
$109$	−10.1042	−0.967805	−0.483902	−	0.875122i	$-0.660781\pi$
$109$	−10.1042	−0.967805	−0.483902	+	0.875122i	$0.660781\pi$
$110$	0	0
$111$	0	0
$112$	−2.63746	−0.249217
$113$	1.73523	0.163236	0.0816182	−	0.996664i	$-0.473991\pi$
$113$	1.73523	0.163236	0.0816182	+	0.996664i	$0.473991\pi$
$114$	0	0
$115$	0	0
$116$	−6.03815	−0.560628
$117$	0	0
$118$	10.8434	0.998214
$119$	10.6554	0.976775
$120$	0	0
$121$	−4.27176	−0.388342
$122$	−2.69849	−0.244309
$123$	0	0
$124$	−5.46419	−0.490699
$125$	0	0
$126$	0	0
$127$	−11.3096	−1.00356	−0.501780	−	0.864995i	$-0.667321\pi$
$127$	−11.3096	−1.00356	−0.501780	+	0.864995i	$0.667321\pi$
$128$	−9.71523	−0.858713
$129$	0	0
$130$	0	0
$131$	−4.33005	−0.378318	−0.189159	−	0.981946i	$-0.560576\pi$
$131$	−4.33005	−0.378318	−0.189159	+	0.981946i	$0.560576\pi$
$132$	0	0
$133$	6.12262	0.530898
$134$	6.27114	0.541744
$135$	0	0
$136$	9.33823	0.800747
$137$	−16.5370	−1.41285	−0.706424	−	0.707789i	$-0.749693\pi$
$137$	−16.5370	−1.41285	−0.706424	+	0.707789i	$0.749693\pi$
$138$	0	0
$139$	13.8673	1.17621	0.588107	−	0.808783i	$-0.299874\pi$
$139$	13.8673	1.17621	0.588107	+	0.808783i	$0.299874\pi$
$140$	0	0
$141$	0	0
$142$	−7.93161	−0.665606
$143$	−13.9730	−1.16848
$144$	0	0
$145$	0	0
$146$	−2.98022	−0.246644
$147$	0	0
$148$	8.62935	0.709328
$149$	4.22879	0.346436	0.173218	−	0.984884i	$-0.444584\pi$
$149$	4.22879	0.346436	0.173218	+	0.984884i	$0.444584\pi$
$150$	0	0
$151$	−8.30014	−0.675455	−0.337728	−	0.941244i	$-0.609658\pi$
$151$	−8.30014	−0.675455	−0.337728	+	0.941244i	$0.609658\pi$
$152$	5.36579	0.435223
$153$	0	0
$154$	5.81060	0.468231
$155$	0	0
$156$	0	0
$157$	2.83272	0.226075	0.113038	−	0.993591i	$-0.463942\pi$
$157$	2.83272	0.226075	0.113038	+	0.993591i	$0.463942\pi$
$158$	0.517903	0.0412021
$159$	0	0
$160$	0	0
$161$	−16.2462	−1.28038
$162$	0	0
$163$	−15.5058	−1.21450	−0.607252	−	0.794509i	$-0.707728\pi$
$163$	−15.5058	−1.21450	−0.607252	+	0.794509i	$0.707728\pi$
$164$	−3.38068	−0.263987
$165$	0	0
$166$	−12.2605	−0.951601
$167$	−15.0069	−1.16127	−0.580634	−	0.814165i	$-0.697195\pi$
$167$	−15.0069	−1.16127	−0.580634	+	0.814165i	$0.697195\pi$
$168$	0	0
$169$	16.0187	1.23221
$170$	0	0
$171$	0	0
$172$	1.42716	0.108820
$173$	6.52009	0.495713	0.247856	−	0.968797i	$-0.420274\pi$
$173$	6.52009	0.495713	0.247856	+	0.968797i	$0.420274\pi$
$174$	0	0
$175$	0	0
$176$	−2.31144	−0.174232
$177$	0	0
$178$	2.97786	0.223200
$179$	−11.9190	−0.890871	−0.445435	−	0.895314i	$-0.646951\pi$
$179$	−11.9190	−0.890871	−0.445435	+	0.895314i	$0.646951\pi$
$180$	0	0
$181$	−20.5120	−1.52465	−0.762324	−	0.647195i	$-0.775942\pi$
$181$	−20.5120	−1.52465	−0.762324	+	0.647195i	$0.775942\pi$
$182$	−12.0673	−0.894486
$183$	0	0
$184$	−14.2380	−1.04964
$185$	0	0
$186$	0	0
$187$	9.33823	0.682879
$188$	−3.30655	−0.241155
$189$	0	0
$190$	0	0
$191$	17.5584	1.27048	0.635242	−	0.772313i	$-0.280900\pi$
$191$	17.5584	1.27048	0.635242	+	0.772313i	$0.280900\pi$
$192$	0	0
$193$	1.85621	0.133613	0.0668064	−	0.997766i	$-0.478719\pi$
$193$	1.85621	0.133613	0.0668064	+	0.997766i	$0.478719\pi$
$194$	4.85899	0.348855
$195$	0	0
$196$	−2.51192	−0.179423
$197$	13.3153	0.948677	0.474339	−	0.880342i	$-0.342687\pi$
$197$	13.3153	0.948677	0.474339	+	0.880342i	$0.342687\pi$
$198$	0	0
$199$	0.861495	0.0610698	0.0305349	−	0.999534i	$-0.490279\pi$
$199$	0.861495	0.0610698	0.0305349	+	0.999534i	$0.490279\pi$
$200$	0	0
$201$	0	0
$202$	−8.89014	−0.625508
$203$	−12.5223	−0.878894
$204$	0	0
$205$	0	0
$206$	14.2193	0.990708
$207$	0	0
$208$	4.80034	0.332843
$209$	5.36579	0.371160
$210$	0	0
$211$	−19.5118	−1.34325	−0.671623	−	0.740893i	$-0.734402\pi$
$211$	−19.5118	−1.34325	−0.671623	+	0.740893i	$0.734402\pi$
$212$	−0.888414	−0.0610165
$213$	0	0
$214$	9.77277	0.668052
$215$	0	0
$216$	0	0
$217$	−11.3320	−0.769266
$218$	−7.64746	−0.517951
$219$	0	0
$220$	0	0
$221$	−19.3934	−1.30454
$222$	0	0
$223$	7.15043	0.478828	0.239414	−	0.970918i	$-0.423045\pi$
$223$	7.15043	0.478828	0.239414	+	0.970918i	$0.423045\pi$
$224$	−17.3507	−1.15929
$225$	0	0
$226$	1.31333	0.0873611
$227$	−23.7199	−1.57434	−0.787171	−	0.616735i	$-0.788455\pi$
$227$	−23.7199	−1.57434	−0.787171	+	0.616735i	$0.788455\pi$
$228$	0	0
$229$	−14.2088	−0.938947	−0.469474	−	0.882947i	$-0.655556\pi$
$229$	−14.2088	−0.938947	−0.469474	+	0.882947i	$0.655556\pi$
$230$	0	0
$231$	0	0
$232$	−10.9744	−0.720506
$233$	8.56489	0.561104	0.280552	−	0.959839i	$-0.409482\pi$
$233$	8.56489	0.561104	0.280552	+	0.959839i	$0.409482\pi$
$234$	0	0
$235$	0	0
$236$	−20.4466	−1.33096
$237$	0	0
$238$	8.06463	0.522752
$239$	−22.7547	−1.47188	−0.735939	−	0.677048i	$-0.763259\pi$
$239$	−22.7547	−1.47188	−0.735939	+	0.677048i	$0.763259\pi$
$240$	0	0
$241$	−1.63554	−0.105354	−0.0526772	−	0.998612i	$-0.516775\pi$
$241$	−1.63554	−0.105354	−0.0526772	+	0.998612i	$0.516775\pi$
$242$	−3.23313	−0.207833
$243$	0	0
$244$	5.08835	0.325748
$245$	0	0
$246$	0	0
$247$	−11.1435	−0.709045
$248$	−9.93124	−0.630634
$249$	0	0
$250$	0	0
$251$	11.6253	0.733780	0.366890	−	0.930264i	$-0.380423\pi$
$251$	11.6253	0.733780	0.366890	+	0.930264i	$0.380423\pi$
$252$	0	0
$253$	−14.2380	−0.895135
$254$	−8.55976	−0.537087
$255$	0	0
$256$	−12.6624	−0.791400
$257$	−23.8196	−1.48582	−0.742912	−	0.669389i	$-0.766556\pi$
$257$	−23.8196	−1.48582	−0.742912	+	0.669389i	$0.766556\pi$
$258$	0	0
$259$	17.8961	1.11201
$260$	0	0
$261$	0	0
$262$	−3.27725	−0.202469
$263$	−4.12962	−0.254643	−0.127321	−	0.991862i	$-0.540638\pi$
$263$	−4.12962	−0.254643	−0.127321	+	0.991862i	$0.540638\pi$
$264$	0	0
$265$	0	0
$266$	4.63397	0.284127
$267$	0	0
$268$	−11.8251	−0.722331
$269$	−13.0865	−0.797898	−0.398949	−	0.916973i	$-0.630625\pi$
$269$	−13.0865	−0.797898	−0.398949	+	0.916973i	$0.630625\pi$
$270$	0	0
$271$	−0.757317	−0.0460038	−0.0230019	−	0.999735i	$-0.507322\pi$
$271$	−0.757317	−0.0460038	−0.0230019	+	0.999735i	$0.507322\pi$
$272$	−3.20809	−0.194519
$273$	0	0
$274$	−12.5162	−0.756130
$275$	0	0
$276$	0	0
$277$	−9.99505	−0.600544	−0.300272	−	0.953854i	$-0.597078\pi$
$277$	−9.99505	−0.600544	−0.300272	+	0.953854i	$0.597078\pi$
$278$	10.4957	0.629488
$279$	0	0
$280$	0	0
$281$	22.1017	1.31848	0.659239	−	0.751934i	$-0.270879\pi$
$281$	22.1017	1.31848	0.659239	+	0.751934i	$0.270879\pi$
$282$	0	0
$283$	3.40185	0.202219	0.101110	−	0.994875i	$-0.467761\pi$
$283$	3.40185	0.202219	0.101110	+	0.994875i	$0.467761\pi$
$284$	14.9561	0.887481
$285$	0	0
$286$	−10.5756	−0.625350
$287$	−7.01109	−0.413851
$288$	0	0
$289$	−4.03931	−0.237607
$290$	0	0
$291$	0	0
$292$	5.61959	0.328862
$293$	24.8930	1.45426	0.727132	−	0.686497i	$-0.240853\pi$
$293$	24.8930	1.45426	0.727132	+	0.686497i	$0.240853\pi$
$294$	0	0
$295$	0	0
$296$	15.6840	0.911612
$297$	0	0
$298$	3.20061	0.185406
$299$	29.5691	1.71002
$300$	0	0
$301$	2.95974	0.170597
$302$	−6.28205	−0.361491
$303$	0	0
$304$	−1.84338	−0.105725
$305$	0	0
$306$	0	0
$307$	2.76102	0.157580	0.0787899	−	0.996891i	$-0.474894\pi$
$307$	2.76102	0.157580	0.0787899	+	0.996891i	$0.474894\pi$
$308$	−10.9567	−0.624313
$309$	0	0
$310$	0	0
$311$	10.8195	0.613517	0.306758	−	0.951787i	$-0.400756\pi$
$311$	10.8195	0.613517	0.306758	+	0.951787i	$0.400756\pi$
$312$	0	0
$313$	−3.50976	−0.198383	−0.0991917	−	0.995068i	$-0.531626\pi$
$313$	−3.50976	−0.198383	−0.0991917	+	0.995068i	$0.531626\pi$
$314$	2.14397	0.120991
$315$	0	0
$316$	−0.976574	−0.0549366
$317$	25.0142	1.40494	0.702468	−	0.711716i	$-0.252082\pi$
$317$	25.0142	1.40494	0.702468	+	0.711716i	$0.252082\pi$
$318$	0	0
$319$	−10.9744	−0.614449
$320$	0	0
$321$	0	0
$322$	−12.2961	−0.685237
$323$	7.44727	0.414377
$324$	0	0
$325$	0	0
$326$	−11.7357	−0.649980
$327$	0	0
$328$	−6.14443	−0.339270
$329$	−6.85734	−0.378058
$330$	0	0
$331$	−35.7420	−1.96456	−0.982280	−	0.187420i	$-0.939988\pi$
$331$	−35.7420	−1.96456	−0.982280	+	0.187420i	$0.939988\pi$
$332$	23.1188	1.26881
$333$	0	0
$334$	−11.3581	−0.621489
$335$	0	0
$336$	0	0
$337$	−12.6768	−0.690550	−0.345275	−	0.938502i	$-0.612214\pi$
$337$	−12.6768	−0.690550	−0.345275	+	0.938502i	$0.612214\pi$
$338$	12.1240	0.659457
$339$	0	0
$340$	0	0
$341$	−9.93124	−0.537807
$342$	0	0
$343$	15.5088	0.837397
$344$	2.59388	0.139853
$345$	0	0
$346$	4.93480	0.265296
$347$	−1.03365	−0.0554891	−0.0277445	−	0.999615i	$-0.508832\pi$
$347$	−1.03365	−0.0554891	−0.0277445	+	0.999615i	$0.508832\pi$
$348$	0	0
$349$	−28.2699	−1.51325	−0.756626	−	0.653848i	$-0.773153\pi$
$349$	−28.2699	−1.51325	−0.756626	+	0.653848i	$0.773153\pi$
$350$	0	0
$351$	0	0
$352$	−15.2059	−0.810478
$353$	−25.0424	−1.33287	−0.666436	−	0.745562i	$-0.732181\pi$
$353$	−25.0424	−1.33287	−0.666436	+	0.745562i	$0.732181\pi$
$354$	0	0
$355$	0	0
$356$	−5.61514	−0.297602
$357$	0	0
$358$	−9.02105	−0.476778
$359$	3.48588	0.183978	0.0919889	−	0.995760i	$-0.470678\pi$
$359$	3.48588	0.183978	0.0919889	+	0.995760i	$0.470678\pi$
$360$	0	0
$361$	−14.7208	−0.774777
$362$	−15.5248	−0.815964
$363$	0	0
$364$	22.7545	1.19266
$365$	0	0
$366$	0	0
$367$	3.36108	0.175447	0.0877235	−	0.996145i	$-0.472041\pi$
$367$	3.36108	0.175447	0.0877235	+	0.996145i	$0.472041\pi$
$368$	4.89137	0.254980
$369$	0	0
$370$	0	0
$371$	−1.84245	−0.0956554
$372$	0	0
$373$	9.39630	0.486522	0.243261	−	0.969961i	$-0.421783\pi$
$373$	9.39630	0.486522	0.243261	+	0.969961i	$0.421783\pi$
$374$	7.06774	0.365464
$375$	0	0
$376$	−6.00970	−0.309926
$377$	22.7913	1.17381
$378$	0	0
$379$	−9.38613	−0.482133	−0.241067	−	0.970509i	$-0.577497\pi$
$379$	−9.38613	−0.482133	−0.241067	+	0.970509i	$0.577497\pi$
$380$	0	0
$381$	0	0
$382$	13.2893	0.679940
$383$	28.6009	1.46144	0.730719	−	0.682678i	$-0.239185\pi$
$383$	28.6009	1.46144	0.730719	+	0.682678i	$0.239185\pi$
$384$	0	0
$385$	0	0
$386$	1.40489	0.0715071
$387$	0	0
$388$	−9.16227	−0.465144
$389$	27.5838	1.39855	0.699277	−	0.714851i	$-0.253505\pi$
$389$	27.5838	1.39855	0.699277	+	0.714851i	$0.253505\pi$
$390$	0	0
$391$	−19.7612	−0.999364
$392$	−4.56545	−0.230590
$393$	0	0
$394$	10.0778	0.507715
$395$	0	0
$396$	0	0
$397$	−26.9067	−1.35041	−0.675203	−	0.737632i	$-0.735944\pi$
$397$	−26.9067	−1.35041	−0.675203	+	0.737632i	$0.735944\pi$
$398$	0.652032	0.0326834
$399$	0	0
$400$	0	0
$401$	5.73276	0.286280	0.143140	−	0.989702i	$-0.454280\pi$
$401$	5.73276	0.286280	0.143140	+	0.989702i	$0.454280\pi$
$402$	0	0
$403$	20.6249	1.02740
$404$	16.7635	0.834017
$405$	0	0
$406$	−9.47765	−0.470368
$407$	15.6840	0.777425
$408$	0	0
$409$	−16.5408	−0.817892	−0.408946	−	0.912558i	$-0.634104\pi$
$409$	−16.5408	−0.817892	−0.408946	+	0.912558i	$0.634104\pi$
$410$	0	0
$411$	0	0
$412$	−26.8124	−1.32095
$413$	−42.4036	−2.08654
$414$	0	0
$415$	0	0
$416$	31.5792	1.54830
$417$	0	0
$418$	4.06116	0.198638
$419$	20.8202	1.01713	0.508566	−	0.861023i	$-0.330176\pi$
$419$	20.8202	1.01713	0.508566	+	0.861023i	$0.330176\pi$
$420$	0	0
$421$	−14.8914	−0.725760	−0.362880	−	0.931836i	$-0.618207\pi$
$421$	−14.8914	−0.725760	−0.362880	+	0.931836i	$0.618207\pi$
$422$	−14.7677	−0.718880
$423$	0	0
$424$	−1.61470	−0.0784170
$425$	0	0
$426$	0	0
$427$	10.5526	0.510675
$428$	−18.4278	−0.890743
$429$	0	0
$430$	0	0
$431$	2.19026	0.105501	0.0527506	−	0.998608i	$-0.483201\pi$
$431$	2.19026	0.105501	0.0527506	+	0.998608i	$0.483201\pi$
$432$	0	0
$433$	−32.8565	−1.57898	−0.789492	−	0.613761i	$-0.789656\pi$
$433$	−32.8565	−1.57898	−0.789492	+	0.613761i	$0.789656\pi$
$434$	−8.57675	−0.411697
$435$	0	0
$436$	14.4203	0.690607
$437$	−11.3548	−0.543176
$438$	0	0
$439$	1.90828	0.0910774	0.0455387	−	0.998963i	$-0.485500\pi$
$439$	1.90828	0.0910774	0.0455387	+	0.998963i	$0.485500\pi$
$440$	0	0
$441$	0	0
$442$	−14.6781	−0.698165
$443$	−19.8154	−0.941460	−0.470730	−	0.882277i	$-0.656009\pi$
$443$	−19.8154	−0.941460	−0.470730	+	0.882277i	$0.656009\pi$
$444$	0	0
$445$	0	0
$446$	5.41188	0.256260
$447$	0	0
$448$	−7.85711	−0.371214
$449$	−14.2137	−0.670787	−0.335393	−	0.942078i	$-0.608869\pi$
$449$	−14.2137	−0.670787	−0.335393	+	0.942078i	$0.608869\pi$
$450$	0	0
$451$	−6.14443	−0.289330
$452$	−2.47645	−0.116482
$453$	0	0
$454$	−17.9526	−0.842559
$455$	0	0
$456$	0	0
$457$	3.45408	0.161575	0.0807875	−	0.996731i	$-0.474256\pi$
$457$	3.45408	0.161575	0.0807875	+	0.996731i	$0.474256\pi$
$458$	−10.7541	−0.502507
$459$	0	0
$460$	0	0
$461$	−3.95989	−0.184431	−0.0922153	−	0.995739i	$-0.529395\pi$
$461$	−3.95989	−0.184431	−0.0922153	+	0.995739i	$0.529395\pi$
$462$	0	0
$463$	14.2769	0.663502	0.331751	−	0.943367i	$-0.392361\pi$
$463$	14.2769	0.663502	0.331751	+	0.943367i	$0.392361\pi$
$464$	3.77019	0.175027
$465$	0	0
$466$	6.48243	0.300293
$467$	4.77767	0.221084	0.110542	−	0.993871i	$-0.464741\pi$
$467$	4.77767	0.221084	0.110542	+	0.993871i	$0.464741\pi$
$468$	0	0
$469$	−24.5236	−1.13240
$470$	0	0
$471$	0	0
$472$	−37.1620	−1.71052
$473$	2.59388	0.119267
$474$	0	0
$475$	0	0
$476$	−15.2069	−0.697008
$477$	0	0
$478$	−17.2221	−0.787722
$479$	2.13055	0.0973473	0.0486737	−	0.998815i	$-0.484501\pi$
$479$	2.13055	0.0973473	0.0486737	+	0.998815i	$0.484501\pi$
$480$	0	0
$481$	−32.5720	−1.48515
$482$	−1.23788	−0.0563837
$483$	0	0
$484$	6.09649	0.277113
$485$	0	0
$486$	0	0
$487$	15.0686	0.682823	0.341412	−	0.939914i	$-0.389095\pi$
$487$	15.0686	0.682823	0.341412	+	0.939914i	$0.389095\pi$
$488$	9.24815	0.418644
$489$	0	0
$490$	0	0
$491$	−16.0940	−0.726314	−0.363157	−	0.931728i	$-0.618301\pi$
$491$	−16.0940	−0.726314	−0.363157	+	0.931728i	$0.618301\pi$
$492$	0	0
$493$	−15.2316	−0.685995
$494$	−8.43409	−0.379468
$495$	0	0
$496$	3.41181	0.153195
$497$	31.0169	1.39130
$498$	0	0
$499$	−33.4914	−1.49928	−0.749642	−	0.661844i	$-0.769774\pi$
$499$	−33.4914	−1.49928	−0.749642	+	0.661844i	$0.769774\pi$
$500$	0	0
$501$	0	0
$502$	8.79871	0.392706
$503$	−12.6230	−0.562834	−0.281417	−	0.959586i	$-0.590804\pi$
$503$	−12.6230	−0.562834	−0.281417	+	0.959586i	$0.590804\pi$
$504$	0	0
$505$	0	0
$506$	−10.7762	−0.479060
$507$	0	0
$508$	16.1406	0.716122
$509$	−14.4922	−0.642354	−0.321177	−	0.947019i	$-0.604078\pi$
$509$	−14.4922	−0.642354	−0.321177	+	0.947019i	$0.604078\pi$
$510$	0	0
$511$	11.6543	0.515555
$512$	9.84678	0.435170
$513$	0	0
$514$	−18.0281	−0.795185
$515$	0	0
$516$	0	0
$517$	−6.00970	−0.264306
$518$	13.5449	0.595128
$519$	0	0
$520$	0	0
$521$	−33.0318	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$521$	−33.0318	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$522$	0	0
$523$	21.5802	0.943637	0.471818	−	0.881696i	$-0.343598\pi$
$523$	21.5802	0.943637	0.471818	+	0.881696i	$0.343598\pi$
$524$	6.17968	0.269961
$525$	0	0
$526$	−3.12555	−0.136280
$527$	−13.7837	−0.600429
$528$	0	0
$529$	7.12981	0.309992
$530$	0	0
$531$	0	0
$532$	−8.73796	−0.378839
$533$	12.7606	0.552722
$534$	0	0
$535$	0	0
$536$	−21.4922	−0.928323
$537$	0	0
$538$	−9.90467	−0.427021
$539$	−4.56545	−0.196648
$540$	0	0
$541$	27.4352	1.17953	0.589766	−	0.807574i	$-0.299220\pi$
$541$	27.4352	1.17953	0.589766	+	0.807574i	$0.299220\pi$
$542$	−0.573184	−0.0246204
$543$	0	0
$544$	−21.1045	−0.904850
$545$	0	0
$546$	0	0
$547$	24.3331	1.04041	0.520205	−	0.854042i	$-0.325856\pi$
$547$	24.3331	1.04041	0.520205	+	0.854042i	$0.325856\pi$
$548$	23.6009	1.00818
$549$	0	0
$550$	0	0
$551$	−8.75213	−0.372853
$552$	0	0
$553$	−2.02528	−0.0861239
$554$	−7.56487	−0.321400
$555$	0	0
$556$	−19.7909	−0.839323
$557$	17.5494	0.743590	0.371795	−	0.928315i	$-0.378742\pi$
$557$	17.5494	0.743590	0.371795	+	0.928315i	$0.378742\pi$
$558$	0	0
$559$	−5.38690	−0.227842
$560$	0	0
$561$	0	0
$562$	16.7279	0.705625
$563$	−31.0746	−1.30964	−0.654818	−	0.755786i	$-0.727255\pi$
$563$	−31.0746	−1.30964	−0.654818	+	0.755786i	$0.727255\pi$
$564$	0	0
$565$	0	0
$566$	2.57473	0.108224
$567$	0	0
$568$	27.1829	1.14057
$569$	−28.7842	−1.20670	−0.603349	−	0.797477i	$-0.706167\pi$
$569$	−28.7842	−1.20670	−0.603349	+	0.797477i	$0.706167\pi$
$570$	0	0
$571$	−38.1691	−1.59733	−0.798664	−	0.601777i	$-0.794459\pi$
$571$	−38.1691	−1.59733	−0.798664	+	0.601777i	$0.794459\pi$
$572$	19.9417	0.833806
$573$	0	0
$574$	−5.30642	−0.221486
$575$	0	0
$576$	0	0
$577$	−11.9329	−0.496772	−0.248386	−	0.968661i	$-0.579900\pi$
$577$	−11.9329	−0.496772	−0.248386	+	0.968661i	$0.579900\pi$
$578$	−3.05720	−0.127163
$579$	0	0
$580$	0	0
$581$	47.9454	1.98911
$582$	0	0
$583$	−1.61470	−0.0668742
$584$	10.2137	0.422645
$585$	0	0
$586$	18.8405	0.778296
$587$	−26.5275	−1.09491	−0.547454	−	0.836836i	$-0.684403\pi$
$587$	−26.5275	−1.09491	−0.547454	+	0.836836i	$0.684403\pi$
$588$	0	0
$589$	−7.92019	−0.326346
$590$	0	0
$591$	0	0
$592$	−5.38812	−0.221450
$593$	46.8656	1.92454	0.962269	−	0.272101i	$-0.0877184\pi$
$593$	46.8656	1.92454	0.962269	+	0.272101i	$0.0877184\pi$
$594$	0	0
$595$	0	0
$596$	−6.03516	−0.247210
$597$	0	0
$598$	22.3797	0.915173
$599$	−7.24103	−0.295860	−0.147930	−	0.988998i	$-0.547261\pi$
$599$	−7.24103	−0.295860	−0.147930	+	0.988998i	$0.547261\pi$
$600$	0	0
$601$	−28.6845	−1.17007	−0.585033	−	0.811009i	$-0.698919\pi$
$601$	−28.6845	−1.17007	−0.585033	+	0.811009i	$0.698919\pi$
$602$	2.24011	0.0913002
$603$	0	0
$604$	11.8456	0.481992
$605$	0	0
$606$	0	0
$607$	23.7062	0.962206	0.481103	−	0.876664i	$-0.340236\pi$
$607$	23.7062	0.962206	0.481103	+	0.876664i	$0.340236\pi$
$608$	−12.1268	−0.491805
$609$	0	0
$610$	0	0
$611$	12.4808	0.504918
$612$	0	0
$613$	25.7090	1.03838	0.519188	−	0.854660i	$-0.326234\pi$
$613$	25.7090	1.03838	0.519188	+	0.854660i	$0.326234\pi$
$614$	2.08971	0.0843338
$615$	0	0
$616$	−19.9139	−0.802352
$617$	29.3396	1.18117	0.590584	−	0.806976i	$-0.298897\pi$
$617$	29.3396	1.18117	0.590584	+	0.806976i	$0.298897\pi$
$618$	0	0
$619$	−15.6205	−0.627841	−0.313920	−	0.949449i	$-0.601642\pi$
$619$	−15.6205	−0.627841	−0.313920	+	0.949449i	$0.601642\pi$
$620$	0	0
$621$	0	0
$622$	8.18885	0.328343
$623$	−11.6451	−0.466549
$624$	0	0
$625$	0	0
$626$	−2.65640	−0.106171
$627$	0	0
$628$	−4.04274	−0.161323
$629$	21.7680	0.867948
$630$	0	0
$631$	21.0553	0.838200	0.419100	−	0.907940i	$-0.362346\pi$
$631$	21.0553	0.838200	0.419100	+	0.907940i	$0.362346\pi$
$632$	−1.77494	−0.0706032
$633$	0	0
$634$	18.9322	0.751896
$635$	0	0
$636$	0	0
$637$	9.48139	0.375666
$638$	−8.30611	−0.328842
$639$	0	0
$640$	0	0
$641$	−29.1190	−1.15013	−0.575065	−	0.818107i	$-0.695023\pi$
$641$	−29.1190	−1.15013	−0.575065	+	0.818107i	$0.695023\pi$
$642$	0	0
$643$	37.7779	1.48982	0.744908	−	0.667168i	$-0.232494\pi$
$643$	37.7779	1.48982	0.744908	+	0.667168i	$0.232494\pi$
$644$	23.1860	0.913655
$645$	0	0
$646$	5.63655	0.221767
$647$	−25.7453	−1.01215	−0.506076	−	0.862489i	$-0.668904\pi$
$647$	−25.7453	−1.01215	−0.506076	+	0.862489i	$0.668904\pi$
$648$	0	0
$649$	−37.1620	−1.45874
$650$	0	0
$651$	0	0
$652$	22.1292	0.866647
$653$	−22.1454	−0.866615	−0.433307	−	0.901246i	$-0.642654\pi$
$653$	−22.1454	−0.866615	−0.433307	+	0.901246i	$0.642654\pi$
$654$	0	0
$655$	0	0
$656$	2.11088	0.0824160
$657$	0	0
$658$	−5.19006	−0.202329
$659$	12.5936	0.490579	0.245289	−	0.969450i	$-0.421117\pi$
$659$	12.5936	0.490579	0.245289	+	0.969450i	$0.421117\pi$
$660$	0	0
$661$	−32.5669	−1.26671	−0.633353	−	0.773863i	$-0.718322\pi$
$661$	−32.5669	−1.26671	−0.633353	+	0.773863i	$0.718322\pi$
$662$	−27.0518	−1.05140
$663$	0	0
$664$	42.0188	1.63064
$665$	0	0
$666$	0	0
$667$	23.2236	0.899220
$668$	21.4172	0.828658
$669$	0	0
$670$	0	0
$671$	9.24815	0.357021
$672$	0	0
$673$	4.53605	0.174852	0.0874259	−	0.996171i	$-0.472136\pi$
$673$	4.53605	0.174852	0.0874259	+	0.996171i	$0.472136\pi$
$674$	−9.59458	−0.369569
$675$	0	0
$676$	−22.8613	−0.879282
$677$	7.73448	0.297260	0.148630	−	0.988893i	$-0.452514\pi$
$677$	7.73448	0.297260	0.148630	+	0.988893i	$0.452514\pi$
$678$	0	0
$679$	−19.0013	−0.729204
$680$	0	0
$681$	0	0
$682$	−7.51657	−0.287824
$683$	0.702986	0.0268990	0.0134495	−	0.999910i	$-0.495719\pi$
$683$	0.702986	0.0268990	0.0134495	+	0.999910i	$0.495719\pi$
$684$	0	0
$685$	0	0
$686$	11.7380	0.448160
$687$	0	0
$688$	−0.891112	−0.0339733
$689$	3.35337	0.127753
$690$	0	0
$691$	22.5031	0.856057	0.428028	−	0.903765i	$-0.359208\pi$
$691$	22.5031	0.856057	0.428028	+	0.903765i	$0.359208\pi$
$692$	−9.30522	−0.353731
$693$	0	0
$694$	−0.782327	−0.0296967
$695$	0	0
$696$	0	0
$697$	−8.52796	−0.323020
$698$	−21.3964	−0.809864
$699$	0	0
$700$	0	0
$701$	4.32501	0.163354	0.0816768	−	0.996659i	$-0.473972\pi$
$701$	4.32501	0.163354	0.0816768	+	0.996659i	$0.473972\pi$
$702$	0	0
$703$	12.5080	0.471748
$704$	−6.88588	−0.259521
$705$	0	0
$706$	−18.9536	−0.713328
$707$	34.7654	1.30749
$708$	0	0
$709$	3.38768	0.127227	0.0636135	−	0.997975i	$-0.479738\pi$
$709$	3.38768	0.127227	0.0636135	+	0.997975i	$0.479738\pi$
$710$	0	0
$711$	0	0
$712$	−10.2056	−0.382471
$713$	21.0160	0.787057
$714$	0	0
$715$	0	0
$716$	17.0104	0.635708
$717$	0	0
$718$	2.63833	0.0984616
$719$	−4.22113	−0.157421	−0.0787107	−	0.996897i	$-0.525080\pi$
$719$	−4.22113	−0.157421	−0.0787107	+	0.996897i	$0.525080\pi$
$720$	0	0
$721$	−55.6054	−2.07085
$722$	−11.1416	−0.414646
$723$	0	0
$724$	29.2740	1.08796
$725$	0	0
$726$	0	0
$727$	17.9156	0.664454	0.332227	−	0.943199i	$-0.392200\pi$
$727$	17.9156	0.664454	0.332227	+	0.943199i	$0.392200\pi$
$728$	41.3565	1.53277
$729$	0	0
$730$	0	0
$731$	3.60010	0.133154
$732$	0	0
$733$	7.60192	0.280783	0.140392	−	0.990096i	$-0.455164\pi$
$733$	7.60192	0.280783	0.140392	+	0.990096i	$0.455164\pi$
$734$	2.54387	0.0938960
$735$	0	0
$736$	32.1781	1.18610
$737$	−21.4922	−0.791676
$738$	0	0
$739$	−48.7666	−1.79391	−0.896954	−	0.442123i	$-0.854225\pi$
$739$	−48.7666	−1.79391	−0.896954	+	0.442123i	$0.854225\pi$
$740$	0	0
$741$	0	0
$742$	−1.39448	−0.0511930
$743$	36.3340	1.33296	0.666482	−	0.745521i	$-0.267799\pi$
$743$	36.3340	1.33296	0.666482	+	0.745521i	$0.267799\pi$
$744$	0	0
$745$	0	0
$746$	7.11169	0.260378
$747$	0	0
$748$	−13.3272	−0.487289
$749$	−38.2169	−1.39641
$750$	0	0
$751$	6.11198	0.223029	0.111515	−	0.993763i	$-0.464430\pi$
$751$	6.11198	0.223029	0.111515	+	0.993763i	$0.464430\pi$
$752$	2.06459	0.0752879
$753$	0	0
$754$	17.2499	0.628204
$755$	0	0
$756$	0	0
$757$	−5.06183	−0.183975	−0.0919877	−	0.995760i	$-0.529322\pi$
$757$	−5.06183	−0.183975	−0.0919877	+	0.995760i	$0.529322\pi$
$758$	−7.10399	−0.258029
$759$	0	0
$760$	0	0
$761$	36.4837	1.32253	0.661266	−	0.750152i	$-0.270020\pi$
$761$	36.4837	1.32253	0.661266	+	0.750152i	$0.270020\pi$
$762$	0	0
$763$	29.9058	1.08266
$764$	−25.0587	−0.906593
$765$	0	0
$766$	21.6469	0.782135
$767$	77.1770	2.78670
$768$	0	0
$769$	32.4486	1.17013	0.585064	−	0.810987i	$-0.301069\pi$
$769$	32.4486	1.17013	0.585064	+	0.810987i	$0.301069\pi$
$770$	0	0
$771$	0	0
$772$	−2.64911	−0.0953435
$773$	48.8900	1.75845	0.879225	−	0.476407i	$-0.158061\pi$
$773$	48.8900	1.75845	0.879225	+	0.476407i	$0.158061\pi$
$774$	0	0
$775$	0	0
$776$	−16.6525	−0.597792
$777$	0	0
$778$	20.8771	0.748480
$779$	−4.90020	−0.175568
$780$	0	0
$781$	27.1829	0.972681
$782$	−14.9564	−0.534841
$783$	0	0
$784$	1.56843	0.0560153
$785$	0	0
$786$	0	0
$787$	51.2232	1.82591	0.912956	−	0.408059i	$-0.133794\pi$
$787$	51.2232	1.82591	0.912956	+	0.408059i	$0.133794\pi$
$788$	−19.0031	−0.676958
$789$	0	0
$790$	0	0
$791$	−5.13583	−0.182609
$792$	0	0
$793$	−19.2063	−0.682035
$794$	−20.3646	−0.722712
$795$	0	0
$796$	−1.22949	−0.0435782
$797$	−13.4684	−0.477073	−0.238537	−	0.971133i	$-0.576668\pi$
$797$	−13.4684	−0.477073	−0.238537	+	0.971133i	$0.576668\pi$
$798$	0	0
$799$	−8.34096	−0.295082
$800$	0	0
$801$	0	0
$802$	4.33890	0.153212
$803$	10.2137	0.360433
$804$	0	0
$805$	0	0
$806$	15.6102	0.549845
$807$	0	0
$808$	30.4680	1.07186
$809$	40.7472	1.43260	0.716298	−	0.697794i	$-0.245835\pi$
$809$	40.7472	1.43260	0.716298	+	0.697794i	$0.245835\pi$
$810$	0	0
$811$	46.1734	1.62137	0.810684	−	0.585484i	$-0.199095\pi$
$811$	46.1734	1.62137	0.810684	+	0.585484i	$0.199095\pi$
$812$	17.8714	0.627162
$813$	0	0
$814$	11.8706	0.416063
$815$	0	0
$816$	0	0
$817$	2.06863	0.0723722
$818$	−12.5191	−0.437721
$819$	0	0
$820$	0	0
$821$	−37.8754	−1.32186	−0.660930	−	0.750448i	$-0.729838\pi$
$821$	−37.8754	−1.32186	−0.660930	+	0.750448i	$0.729838\pi$
$822$	0	0
$823$	−18.2341	−0.635602	−0.317801	−	0.948157i	$-0.602944\pi$
$823$	−18.2341	−0.635602	−0.317801	+	0.948157i	$0.602944\pi$
$824$	−48.7319	−1.69766
$825$	0	0
$826$	−32.0936	−1.11668
$827$	46.1628	1.60524	0.802620	−	0.596491i	$-0.203439\pi$
$827$	46.1628	1.60524	0.802620	+	0.596491i	$0.203439\pi$
$828$	0	0
$829$	−42.5952	−1.47939	−0.739696	−	0.672941i	$-0.765031\pi$
$829$	−42.5952	−1.47939	−0.739696	+	0.672941i	$0.765031\pi$
$830$	0	0
$831$	0	0
$832$	14.3004	0.495777
$833$	−6.33646	−0.219545
$834$	0	0
$835$	0	0
$836$	−7.65785	−0.264852
$837$	0	0
$838$	15.7580	0.544351
$839$	−13.6295	−0.470544	−0.235272	−	0.971930i	$-0.575598\pi$
$839$	−13.6295	−0.470544	−0.235272	+	0.971930i	$0.575598\pi$
$840$	0	0
$841$	−11.0997	−0.382747
$842$	−11.2707	−0.388413
$843$	0	0
$844$	27.8464	0.958514
$845$	0	0
$846$	0	0
$847$	12.6433	0.434429
$848$	0.554721	0.0190492
$849$	0	0
$850$	0	0
$851$	−33.1897	−1.13773
$852$	0	0
$853$	20.6294	0.706336	0.353168	−	0.935560i	$-0.385104\pi$
$853$	20.6294	0.706336	0.353168	+	0.935560i	$0.385104\pi$
$854$	7.98683	0.273304
$855$	0	0
$856$	−33.4928	−1.14476
$857$	−34.5503	−1.18021	−0.590107	−	0.807325i	$-0.700915\pi$
$857$	−34.5503	−1.18021	−0.590107	+	0.807325i	$0.700915\pi$
$858$	0	0
$859$	−12.6853	−0.432817	−0.216408	−	0.976303i	$-0.569434\pi$
$859$	−12.6853	−0.432817	−0.216408	+	0.976303i	$0.569434\pi$
$860$	0	0
$861$	0	0
$862$	1.65772	0.0564623
$863$	41.6898	1.41914	0.709569	−	0.704636i	$-0.248889\pi$
$863$	41.6898	1.41914	0.709569	+	0.704636i	$0.248889\pi$
$864$	0	0
$865$	0	0
$866$	−24.8678	−0.845043
$867$	0	0
$868$	16.1726	0.548934
$869$	−1.77494	−0.0602106
$870$	0	0
$871$	44.6344	1.51238
$872$	26.2091	0.887551
$873$	0	0
$874$	−8.59404	−0.290698
$875$	0	0
$876$	0	0
$877$	−39.9055	−1.34751	−0.673756	−	0.738954i	$-0.735320\pi$
$877$	−39.9055	−1.34751	−0.673756	+	0.738954i	$0.735320\pi$
$878$	1.44430	0.0487429
$879$	0	0
$880$	0	0
$881$	−41.5382	−1.39946	−0.699728	−	0.714409i	$-0.746696\pi$
$881$	−41.5382	−1.39946	−0.699728	+	0.714409i	$0.746696\pi$
$882$	0	0
$883$	−16.1580	−0.543759	−0.271879	−	0.962331i	$-0.587645\pi$
$883$	−16.1580	−0.543759	−0.271879	+	0.962331i	$0.587645\pi$
$884$	27.6775	0.930894
$885$	0	0
$886$	−14.9975	−0.503852
$887$	13.7994	0.463337	0.231668	−	0.972795i	$-0.425582\pi$
$887$	13.7994	0.463337	0.231668	+	0.972795i	$0.425582\pi$
$888$	0	0
$889$	33.4734	1.12266
$890$	0	0
$891$	0	0
$892$	−10.2048	−0.341682
$893$	−4.79275	−0.160383
$894$	0	0
$895$	0	0
$896$	28.7546	0.960623
$897$	0	0
$898$	−10.7578	−0.358993
$899$	16.1988	0.540261
$900$	0	0
$901$	−2.24107	−0.0746610
$902$	−4.65048	−0.154844
$903$	0	0
$904$	−4.50098	−0.149700
$905$	0	0
$906$	0	0
$907$	−49.4231	−1.64107	−0.820534	−	0.571598i	$-0.806324\pi$
$907$	−49.4231	−1.64107	−0.820534	+	0.571598i	$0.806324\pi$
$908$	33.8521	1.12342
$909$	0	0
$910$	0	0
$911$	−41.1921	−1.36475	−0.682377	−	0.731001i	$-0.739054\pi$
$911$	−41.1921	−1.36475	−0.682377	+	0.731001i	$0.739054\pi$
$912$	0	0
$913$	42.0188	1.39062
$914$	2.61426	0.0864720
$915$	0	0
$916$	20.2783	0.670014
$917$	12.8158	0.423216
$918$	0	0
$919$	−17.5638	−0.579377	−0.289689	−	0.957121i	$-0.593552\pi$
$919$	−17.5638	−0.579377	−0.289689	+	0.957121i	$0.593552\pi$
$920$	0	0
$921$	0	0
$922$	−2.99709	−0.0987039
$923$	−56.4526	−1.85816
$924$	0	0
$925$	0	0
$926$	10.8056	0.355094
$927$	0	0
$928$	24.8023	0.814177
$929$	−20.0021	−0.656247	−0.328123	−	0.944635i	$-0.606416\pi$
$929$	−20.0021	−0.656247	−0.328123	+	0.944635i	$0.606416\pi$
$930$	0	0
$931$	−3.64096	−0.119328
$932$	−12.2235	−0.400393
$933$	0	0
$934$	3.61603	0.118320
$935$	0	0
$936$	0	0
$937$	−34.9230	−1.14088	−0.570442	−	0.821338i	$-0.693228\pi$
$937$	−34.9230	−1.14088	−0.570442	+	0.821338i	$0.693228\pi$
$938$	−18.5610	−0.606037
$939$	0	0
$940$	0	0
$941$	44.6384	1.45517	0.727585	−	0.686018i	$-0.240643\pi$
$941$	44.6384	1.45517	0.727585	+	0.686018i	$0.240643\pi$
$942$	0	0
$943$	13.0026	0.423422
$944$	12.7668	0.415523
$945$	0	0
$946$	1.96321	0.0638295
$947$	−44.4665	−1.44497	−0.722483	−	0.691389i	$-0.756999\pi$
$947$	−44.4665	−1.44497	−0.722483	+	0.691389i	$0.756999\pi$
$948$	0	0
$949$	−21.2115	−0.688553
$950$	0	0
$951$	0	0
$952$	−27.6388	−0.895778
$953$	34.0454	1.10284	0.551420	−	0.834228i	$-0.314086\pi$
$953$	34.0454	1.10284	0.551420	+	0.834228i	$0.314086\pi$
$954$	0	0
$955$	0	0
$956$	32.4746	1.05030
$957$	0	0
$958$	1.61253	0.0520985
$959$	48.9452	1.58052
$960$	0	0
$961$	−16.3410	−0.527128
$962$	−24.6525	−0.794827
$963$	0	0
$964$	2.33418	0.0751789
$965$	0	0
$966$	0	0
$967$	−19.0252	−0.611808	−0.305904	−	0.952062i	$-0.598959\pi$
$967$	−19.0252	−0.611808	−0.305904	+	0.952062i	$0.598959\pi$
$968$	11.0805	0.356139
$969$	0	0
$970$	0	0
$971$	49.4345	1.58643	0.793215	−	0.608942i	$-0.208406\pi$
$971$	49.4345	1.58643	0.793215	+	0.608942i	$0.208406\pi$
$972$	0	0
$973$	−41.0438	−1.31580
$974$	11.4048	0.365434
$975$	0	0
$976$	−3.17714	−0.101698
$977$	−35.1408	−1.12426	−0.562128	−	0.827051i	$-0.690017\pi$
$977$	−35.1408	−1.12426	−0.562128	+	0.827051i	$0.690017\pi$
$978$	0	0
$979$	−10.2056	−0.326172
$980$	0	0
$981$	0	0
$982$	−12.1809	−0.388710
$983$	37.6520	1.20091	0.600456	−	0.799658i	$-0.294986\pi$
$983$	37.6520	1.20091	0.600456	+	0.799658i	$0.294986\pi$
$984$	0	0
$985$	0	0
$986$	−11.5282	−0.367132
$987$	0	0
$988$	15.9036	0.505961
$989$	−5.48906	−0.174542
$990$	0	0
$991$	2.99533	0.0951497	0.0475748	−	0.998868i	$-0.484851\pi$
$991$	2.99533	0.0951497	0.0475748	+	0.998868i	$0.484851\pi$
$992$	22.4447	0.712621
$993$	0	0
$994$	23.4755	0.744598
$995$	0	0
$996$	0	0
$997$	14.9444	0.473293	0.236646	−	0.971596i	$-0.423952\pi$
$997$	14.9444	0.473293	0.236646	+	0.971596i	$0.423952\pi$
$998$	−25.3484	−0.802389
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9675.2.a.cw.1.7	yes	10
3.2	odd	2	inner	9675.2.a.cw.1.4	yes	10
5.4	even	2		9675.2.a.cv.1.4	✓	10
15.14	odd	2		9675.2.a.cv.1.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9675.2.a.cv.1.4	✓	10	5.4	even	2
9675.2.a.cv.1.7	yes	10	15.14	odd	2
9675.2.a.cw.1.4	yes	10	3.2	odd	2	inner
9675.2.a.cw.1.7	yes	10	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:	\( N \)	\(=\)	\( 9675 = 3^{2} \cdot 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9675.a (trivial)

\(f(q)\)	\(=\)	\(q+0.756861 q^{2} -1.42716 q^{4} -2.95974 q^{7} -2.59388 q^{8} +O(q^{10})\)	\(q+0.756861 q^{2} -1.42716 q^{4} -2.95974 q^{7} -2.59388 q^{8} -2.59388 q^{11} +5.38690 q^{13} -2.24011 q^{14} +0.891112 q^{16} -3.60010 q^{17} -2.06863 q^{19} -1.96321 q^{22} +5.48906 q^{23} +4.07714 q^{26} +4.22403 q^{28} +4.23088 q^{29} +3.82871 q^{31} +5.86222 q^{32} -2.72477 q^{34} -6.04651 q^{37} -1.56567 q^{38} +2.36882 q^{41} -1.00000 q^{43} +3.70189 q^{44} +4.15446 q^{46} +2.31687 q^{47} +1.76008 q^{49} -7.68798 q^{52} +0.622504 q^{53} +7.67723 q^{56} +3.20219 q^{58} +14.3268 q^{59} -3.56537 q^{61} +2.89780 q^{62} +2.65466 q^{64} +8.28573 q^{67} +5.13792 q^{68} -10.4796 q^{71} -3.93760 q^{73} -4.57637 q^{74} +2.95227 q^{76} +7.67723 q^{77} +0.684277 q^{79} +1.79286 q^{82} -16.1992 q^{83} -0.756861 q^{86} +6.72824 q^{88} +3.93448 q^{89} -15.9439 q^{91} -7.83378 q^{92} +1.75355 q^{94} +6.41993 q^{97} +1.33214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{4}+O(q^{10}) \) 10 * q + 6 * q^4	\( 10 q + 6 q^{4} + 4 q^{13} - 10 q^{16} - 10 q^{19} + 2 q^{22} - 16 q^{31} - 8 q^{34} - 10 q^{37} - 10 q^{43} - 28 q^{46} - 26 q^{49} - 36 q^{52} + 16 q^{58} - 32 q^{61} - 28 q^{64} + 14 q^{67} - 4 q^{73} - 24 q^{76} - 18 q^{79} - 20 q^{82} + 56 q^{88} - 44 q^{91} - 12 q^{94} - 4 q^{97}+O(q^{100}) \) 10 * q + 6 * q^4 + 4 * q^13 - 10 * q^16 - 10 * q^19 + 2 * q^22 - 16 * q^31 - 8 * q^34 - 10 * q^37 - 10 * q^43 - 28 * q^46 - 26 * q^49 - 36 * q^52 + 16 * q^58 - 32 * q^61 - 28 * q^64 + 14 * q^67 - 4 * q^73 - 24 * q^76 - 18 * q^79 - 20 * q^82 + 56 * q^88 - 44 * q^91 - 12 * q^94 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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