LMFDB - Embedded newform 9702.2.a.dc.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1386)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.16228 q^{5} +1.00000 q^{8} +3.16228 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -7.16228 q^{17} -5.16228 q^{19} +3.16228 q^{20} -1.00000 q^{22} -6.32456 q^{23} +5.00000 q^{25} -2.00000 q^{26} +4.00000 q^{29} -9.16228 q^{31} +1.00000 q^{32} -7.16228 q^{34} +8.32456 q^{37} -5.16228 q^{38} +3.16228 q^{40} +7.16228 q^{41} -6.32456 q^{43} -1.00000 q^{44} -6.32456 q^{46} -11.4868 q^{47} +5.00000 q^{50} -2.00000 q^{52} +4.32456 q^{53} -3.16228 q^{55} +4.00000 q^{58} +1.67544 q^{59} -10.0000 q^{61} -9.16228 q^{62} +1.00000 q^{64} -6.32456 q^{65} +4.32456 q^{67} -7.16228 q^{68} -8.00000 q^{71} +5.48683 q^{73} +8.32456 q^{74} -5.16228 q^{76} -4.32456 q^{79} +3.16228 q^{80} +7.16228 q^{82} +5.16228 q^{83} -22.6491 q^{85} -6.32456 q^{86} -1.00000 q^{88} -16.3246 q^{89} -6.32456 q^{92} -11.4868 q^{94} -16.3246 q^{95} +10.6491 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} - 4 q^{13} + 2 q^{16} - 8 q^{17} - 4 q^{19} - 2 q^{22} + 10 q^{25} - 4 q^{26} + 8 q^{29} - 12 q^{31} + 2 q^{32} - 8 q^{34} + 4 q^{37} - 4 q^{38} + 8 q^{41} - 2 q^{44}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 - 4 * q^13 + 2 * q^16 - 8 * q^17 - 4 * q^19 - 2 * q^22 + 10 * q^25 - 4 * q^26 + 8 * q^29 - 12 * q^31 + 2 * q^32 - 8 * q^34 + 4 * q^37 - 4 * q^38 + 8 * q^41 - 2 * q^44 - 4 * q^47 + 10 * q^50 - 4 * q^52 - 4 * q^53 + 8 * q^58 + 16 * q^59 - 20 * q^61 - 12 * q^62 + 2 * q^64 - 4 * q^67 - 8 * q^68 - 16 * q^71 - 8 * q^73 + 4 * q^74 - 4 * q^76 + 4 * q^79 + 8 * q^82 + 4 * q^83 - 20 * q^85 - 2 * q^88 - 20 * q^89 - 4 * q^94 - 20 * q^95 - 4 * q^97

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
$5$	3.16228	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$5$	3.16228	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	3.16228	1.00000
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.16228	−1.73711	−0.868554	−	0.495595i	$-0.834950\pi$
$17$	−7.16228	−1.73711	−0.868554	+	0.495595i	$0.834950\pi$
$18$	0	0
$19$	−5.16228	−1.18431	−0.592154	−	0.805825i	$-0.701722\pi$
$19$	−5.16228	−1.18431	−0.592154	+	0.805825i	$0.701722\pi$
$20$	3.16228	0.707107
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−6.32456	−1.31876	−0.659380	−	0.751809i	$-0.729181\pi$
$23$	−6.32456	−1.31876	−0.659380	+	0.751809i	$0.729181\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−9.16228	−1.64559	−0.822797	−	0.568336i	$-0.807588\pi$
$31$	−9.16228	−1.64559	−0.822797	+	0.568336i	$0.807588\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.16228	−1.22832
$35$	0	0
$36$	0	0
$37$	8.32456	1.36855	0.684274	−	0.729225i	$-0.260119\pi$
$37$	8.32456	1.36855	0.684274	+	0.729225i	$0.260119\pi$
$38$	−5.16228	−0.837432
$39$	0	0
$40$	3.16228	0.500000
$41$	7.16228	1.11856	0.559280	−	0.828979i	$-0.311078\pi$
$41$	7.16228	1.11856	0.559280	+	0.828979i	$0.311078\pi$
$42$	0	0
$43$	−6.32456	−0.964486	−0.482243	−	0.876038i	$-0.660178\pi$
$43$	−6.32456	−0.964486	−0.482243	+	0.876038i	$0.660178\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−6.32456	−0.932505
$47$	−11.4868	−1.67553	−0.837763	−	0.546033i	$-0.816137\pi$
$47$	−11.4868	−1.67553	−0.837763	+	0.546033i	$0.816137\pi$
$48$	0	0
$49$	0	0
$50$	5.00000	0.707107
$51$	0	0
$52$	−2.00000	−0.277350
$53$	4.32456	0.594023	0.297012	−	0.954874i	$-0.404010\pi$
$53$	4.32456	0.594023	0.297012	+	0.954874i	$0.404010\pi$
$54$	0	0
$55$	−3.16228	−0.426401
$56$	0	0
$57$	0	0
$58$	4.00000	0.525226
$59$	1.67544	0.218124	0.109062	−	0.994035i	$-0.465215\pi$
$59$	1.67544	0.218124	0.109062	+	0.994035i	$0.465215\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−9.16228	−1.16361
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.32456	−0.784465
$66$	0	0
$67$	4.32456	0.528329	0.264164	−	0.964478i	$-0.414904\pi$
$67$	4.32456	0.528329	0.264164	+	0.964478i	$0.414904\pi$
$68$	−7.16228	−0.868554
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	5.48683	0.642185	0.321093	−	0.947048i	$-0.395950\pi$
$73$	5.48683	0.642185	0.321093	+	0.947048i	$0.395950\pi$
$74$	8.32456	0.967710
$75$	0	0
$76$	−5.16228	−0.592154
$77$	0	0
$78$	0	0
$79$	−4.32456	−0.486550	−0.243275	−	0.969957i	$-0.578222\pi$
$79$	−4.32456	−0.486550	−0.243275	+	0.969957i	$0.578222\pi$
$80$	3.16228	0.353553
$81$	0	0
$82$	7.16228	0.790941
$83$	5.16228	0.566634	0.283317	−	0.959026i	$-0.408565\pi$
$83$	5.16228	0.566634	0.283317	+	0.959026i	$0.408565\pi$
$84$	0	0
$85$	−22.6491	−2.45664
$86$	−6.32456	−0.681994
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−16.3246	−1.73040	−0.865200	−	0.501427i	$-0.832808\pi$
$89$	−16.3246	−1.73040	−0.865200	+	0.501427i	$0.832808\pi$
$90$	0	0
$91$	0	0
$92$	−6.32456	−0.659380
$93$	0	0
$94$	−11.4868	−1.18478
$95$	−16.3246	−1.67486
$96$	0	0
$97$	10.6491	1.08125	0.540627	−	0.841263i	$-0.318187\pi$
$97$	10.6491	1.08125	0.540627	+	0.841263i	$0.318187\pi$
$98$	0	0
$99$	0	0
$100$	5.00000	0.500000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−13.8114	−1.36088	−0.680438	−	0.732805i	$-0.738211\pi$
$103$	−13.8114	−1.36088	−0.680438	+	0.732805i	$0.738211\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	4.32456	0.420038
$107$	−14.3246	−1.38481	−0.692404	−	0.721510i	$-0.743448\pi$
$107$	−14.3246	−1.38481	−0.692404	+	0.721510i	$0.743448\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	−3.16228	−0.301511
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−20.0000	−1.86501
$116$	4.00000	0.371391
$117$	0	0
$118$	1.67544	0.154237
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−9.16228	−0.822797
$125$	0	0
$126$	0	0
$127$	20.9737	1.86111	0.930556	−	0.366150i	$-0.119324\pi$
$127$	20.9737	1.86111	0.930556	+	0.366150i	$0.119324\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−6.32456	−0.554700
$131$	−5.16228	−0.451030	−0.225515	−	0.974240i	$-0.572407\pi$
$131$	−5.16228	−0.451030	−0.225515	+	0.974240i	$0.572407\pi$
$132$	0	0
$133$	0	0
$134$	4.32456	0.373585
$135$	0	0
$136$	−7.16228	−0.614160
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	7.48683	0.635025	0.317512	−	0.948254i	$-0.397152\pi$
$139$	7.48683	0.635025	0.317512	+	0.948254i	$0.397152\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	2.00000	0.167248
$144$	0	0
$145$	12.6491	1.05045
$146$	5.48683	0.454094
$147$	0	0
$148$	8.32456	0.684274
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	20.6491	1.68040	0.840200	−	0.542276i	$-0.182437\pi$
$151$	20.6491	1.68040	0.840200	+	0.542276i	$0.182437\pi$
$152$	−5.16228	−0.418716
$153$	0	0
$154$	0	0
$155$	−28.9737	−2.32722
$156$	0	0
$157$	5.48683	0.437897	0.218948	−	0.975736i	$-0.429737\pi$
$157$	5.48683	0.437897	0.218948	+	0.975736i	$0.429737\pi$
$158$	−4.32456	−0.344043
$159$	0	0
$160$	3.16228	0.250000
$161$	0	0
$162$	0	0
$163$	−16.6491	−1.30406	−0.652029	−	0.758194i	$-0.726082\pi$
$163$	−16.6491	−1.30406	−0.652029	+	0.758194i	$0.726082\pi$
$164$	7.16228	0.559280
$165$	0	0
$166$	5.16228	0.400670
$167$	6.32456	0.489409	0.244704	−	0.969598i	$-0.421309\pi$
$167$	6.32456	0.489409	0.244704	+	0.969598i	$0.421309\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−22.6491	−1.73711
$171$	0	0
$172$	−6.32456	−0.482243
$173$	12.3246	0.937019	0.468509	−	0.883459i	$-0.344791\pi$
$173$	12.3246	0.937019	0.468509	+	0.883459i	$0.344791\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−16.3246	−1.22358
$179$	−3.67544	−0.274716	−0.137358	−	0.990521i	$-0.543861\pi$
$179$	−3.67544	−0.274716	−0.137358	+	0.990521i	$0.543861\pi$
$180$	0	0
$181$	−17.4868	−1.29979	−0.649893	−	0.760026i	$-0.725186\pi$
$181$	−17.4868	−1.29979	−0.649893	+	0.760026i	$0.725186\pi$
$182$	0	0
$183$	0	0
$184$	−6.32456	−0.466252
$185$	26.3246	1.93542
$186$	0	0
$187$	7.16228	0.523758
$188$	−11.4868	−0.837763
$189$	0	0
$190$	−16.3246	−1.18431
$191$	−5.67544	−0.410661	−0.205330	−	0.978693i	$-0.565827\pi$
$191$	−5.67544	−0.410661	−0.205330	+	0.978693i	$0.565827\pi$
$192$	0	0
$193$	10.6491	0.766540	0.383270	−	0.923636i	$-0.374798\pi$
$193$	10.6491	0.766540	0.383270	+	0.923636i	$0.374798\pi$
$194$	10.6491	0.764562
$195$	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
$199$	−19.4868	−1.38138	−0.690692	−	0.723149i	$-0.742694\pi$
$199$	−19.4868	−1.38138	−0.690692	+	0.723149i	$0.742694\pi$
$200$	5.00000	0.353553
$201$	0	0
$202$	6.00000	0.422159
$203$	0	0
$204$	0	0
$205$	22.6491	1.58188
$206$	−13.8114	−0.962285
$207$	0	0
$208$	−2.00000	−0.138675
$209$	5.16228	0.357082
$210$	0	0
$211$	−18.3246	−1.26151	−0.630757	−	0.775980i	$-0.717256\pi$
$211$	−18.3246	−1.26151	−0.630757	+	0.775980i	$0.717256\pi$
$212$	4.32456	0.297012
$213$	0	0
$214$	−14.3246	−0.979206
$215$	−20.0000	−1.36399
$216$	0	0
$217$	0	0
$218$	−8.00000	−0.541828
$219$	0	0
$220$	−3.16228	−0.213201
$221$	14.3246	0.963574
$222$	0	0
$223$	11.4868	0.769215	0.384608	−	0.923080i	$-0.374337\pi$
$223$	11.4868	0.769215	0.384608	+	0.923080i	$0.374337\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	0.133038
$227$	−7.48683	−0.496919	−0.248459	−	0.968642i	$-0.579924\pi$
$227$	−7.48683	−0.496919	−0.248459	+	0.968642i	$0.579924\pi$
$228$	0	0
$229$	25.4868	1.68422	0.842109	−	0.539308i	$-0.181314\pi$
$229$	25.4868	1.68422	0.842109	+	0.539308i	$0.181314\pi$
$230$	−20.0000	−1.31876
$231$	0	0
$232$	4.00000	0.262613
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−36.3246	−2.36955
$236$	1.67544	0.109062
$237$	0	0
$238$	0	0
$239$	12.6491	0.818203	0.409101	−	0.912489i	$-0.365842\pi$
$239$	12.6491	0.818203	0.409101	+	0.912489i	$0.365842\pi$
$240$	0	0
$241$	−0.837722	−0.0539624	−0.0269812	−	0.999636i	$-0.508589\pi$
$241$	−0.837722	−0.0539624	−0.0269812	+	0.999636i	$0.508589\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	10.3246	0.656936
$248$	−9.16228	−0.581805
$249$	0	0
$250$	0	0
$251$	13.6754	0.863186	0.431593	−	0.902068i	$-0.357952\pi$
$251$	13.6754	0.863186	0.431593	+	0.902068i	$0.357952\pi$
$252$	0	0
$253$	6.32456	0.397621
$254$	20.9737	1.31600
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.67544	−0.478781	−0.239391	−	0.970923i	$-0.576948\pi$
$257$	−7.67544	−0.478781	−0.239391	+	0.970923i	$0.576948\pi$
$258$	0	0
$259$	0	0
$260$	−6.32456	−0.392232
$261$	0	0
$262$	−5.16228	−0.318927
$263$	24.9737	1.53994	0.769971	−	0.638079i	$-0.220271\pi$
$263$	24.9737	1.53994	0.769971	+	0.638079i	$0.220271\pi$
$264$	0	0
$265$	13.6754	0.840076
$266$	0	0
$267$	0	0
$268$	4.32456	0.264164
$269$	−11.1623	−0.680576	−0.340288	−	0.940321i	$-0.610525\pi$
$269$	−11.1623	−0.680576	−0.340288	+	0.940321i	$0.610525\pi$
$270$	0	0
$271$	6.32456	0.384189	0.192095	−	0.981376i	$-0.438472\pi$
$271$	6.32456	0.384189	0.192095	+	0.981376i	$0.438472\pi$
$272$	−7.16228	−0.434277
$273$	0	0
$274$	−18.0000	−1.08742
$275$	−5.00000	−0.301511
$276$	0	0
$277$	−24.0000	−1.44202	−0.721010	−	0.692925i	$-0.756322\pi$
$277$	−24.0000	−1.44202	−0.721010	+	0.692925i	$0.756322\pi$
$278$	7.48683	0.449030
$279$	0	0
$280$	0	0
$281$	22.6491	1.35113	0.675566	−	0.737299i	$-0.263899\pi$
$281$	22.6491	1.35113	0.675566	+	0.737299i	$0.263899\pi$
$282$	0	0
$283$	15.4868	0.920597	0.460298	−	0.887764i	$-0.347742\pi$
$283$	15.4868	0.920597	0.460298	+	0.887764i	$0.347742\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	0	0
$289$	34.2982	2.01754
$290$	12.6491	0.742781
$291$	0	0
$292$	5.48683	0.321093
$293$	−24.9737	−1.45898	−0.729489	−	0.683993i	$-0.760242\pi$
$293$	−24.9737	−1.45898	−0.729489	+	0.683993i	$0.760242\pi$
$294$	0	0
$295$	5.29822	0.308474
$296$	8.32456	0.483855
$297$	0	0
$298$	2.00000	0.115857
$299$	12.6491	0.731517
$300$	0	0
$301$	0	0
$302$	20.6491	1.18822
$303$	0	0
$304$	−5.16228	−0.296077
$305$	−31.6228	−1.81071
$306$	0	0
$307$	21.1623	1.20779	0.603897	−	0.797062i	$-0.293614\pi$
$307$	21.1623	1.20779	0.603897	+	0.797062i	$0.293614\pi$
$308$	0	0
$309$	0	0
$310$	−28.9737	−1.64559
$311$	29.8114	1.69045	0.845224	−	0.534412i	$-0.179467\pi$
$311$	29.8114	1.69045	0.845224	+	0.534412i	$0.179467\pi$
$312$	0	0
$313$	−20.3246	−1.14881	−0.574406	−	0.818571i	$-0.694767\pi$
$313$	−20.3246	−1.14881	−0.574406	+	0.818571i	$0.694767\pi$
$314$	5.48683	0.309640
$315$	0	0
$316$	−4.32456	−0.243275
$317$	12.9737	0.728674	0.364337	−	0.931267i	$-0.381296\pi$
$317$	12.9737	0.728674	0.364337	+	0.931267i	$0.381296\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	3.16228	0.176777
$321$	0	0
$322$	0	0
$323$	36.9737	2.05727
$324$	0	0
$325$	−10.0000	−0.554700
$326$	−16.6491	−0.922109
$327$	0	0
$328$	7.16228	0.395471
$329$	0	0
$330$	0	0
$331$	−8.97367	−0.493237	−0.246619	−	0.969113i	$-0.579320\pi$
$331$	−8.97367	−0.493237	−0.246619	+	0.969113i	$0.579320\pi$
$332$	5.16228	0.283317
$333$	0	0
$334$	6.32456	0.346064
$335$	13.6754	0.747169
$336$	0	0
$337$	−19.6754	−1.07179	−0.535895	−	0.844285i	$-0.680026\pi$
$337$	−19.6754	−1.07179	−0.535895	+	0.844285i	$0.680026\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	−22.6491	−1.22832
$341$	9.16228	0.496165
$342$	0	0
$343$	0	0
$344$	−6.32456	−0.340997
$345$	0	0
$346$	12.3246	0.662572
$347$	−14.3246	−0.768982	−0.384491	−	0.923129i	$-0.625623\pi$
$347$	−14.3246	−0.768982	−0.384491	+	0.923129i	$0.625623\pi$
$348$	0	0
$349$	−8.97367	−0.480349	−0.240175	−	0.970730i	$-0.577205\pi$
$349$	−8.97367	−0.480349	−0.240175	+	0.970730i	$0.577205\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−7.67544	−0.408523	−0.204261	−	0.978916i	$-0.565479\pi$
$353$	−7.67544	−0.408523	−0.204261	+	0.978916i	$0.565479\pi$
$354$	0	0
$355$	−25.2982	−1.34269
$356$	−16.3246	−0.865200
$357$	0	0
$358$	−3.67544	−0.194253
$359$	3.67544	0.193983	0.0969913	−	0.995285i	$-0.469078\pi$
$359$	3.67544	0.193983	0.0969913	+	0.995285i	$0.469078\pi$
$360$	0	0
$361$	7.64911	0.402585
$362$	−17.4868	−0.919088
$363$	0	0
$364$	0	0
$365$	17.3509	0.908187
$366$	0	0
$367$	1.16228	0.0606704	0.0303352	−	0.999540i	$-0.490343\pi$
$367$	1.16228	0.0606704	0.0303352	+	0.999540i	$0.490343\pi$
$368$	−6.32456	−0.329690
$369$	0	0
$370$	26.3246	1.36855
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	7.16228	0.370353
$375$	0	0
$376$	−11.4868	−0.592388
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−13.2982	−0.683084	−0.341542	−	0.939867i	$-0.610949\pi$
$379$	−13.2982	−0.683084	−0.341542	+	0.939867i	$0.610949\pi$
$380$	−16.3246	−0.837432
$381$	0	0
$382$	−5.67544	−0.290381
$383$	9.16228	0.468171	0.234085	−	0.972216i	$-0.424791\pi$
$383$	9.16228	0.468171	0.234085	+	0.972216i	$0.424791\pi$
$384$	0	0
$385$	0	0
$386$	10.6491	0.542025
$387$	0	0
$388$	10.6491	0.540627
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	45.2982	2.29083
$392$	0	0
$393$	0	0
$394$	24.0000	1.20910
$395$	−13.6754	−0.688086
$396$	0	0
$397$	−21.4868	−1.07839	−0.539197	−	0.842180i	$-0.681272\pi$
$397$	−21.4868	−1.07839	−0.539197	+	0.842180i	$0.681272\pi$
$398$	−19.4868	−0.976787
$399$	0	0
$400$	5.00000	0.250000
$401$	23.2982	1.16346	0.581729	−	0.813383i	$-0.302376\pi$
$401$	23.2982	1.16346	0.581729	+	0.813383i	$0.302376\pi$
$402$	0	0
$403$	18.3246	0.912811
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	−8.32456	−0.412633
$408$	0	0
$409$	−27.1623	−1.34309	−0.671544	−	0.740965i	$-0.734369\pi$
$409$	−27.1623	−1.34309	−0.671544	+	0.740965i	$0.734369\pi$
$410$	22.6491	1.11856
$411$	0	0
$412$	−13.8114	−0.680438
$413$	0	0
$414$	0	0
$415$	16.3246	0.801341
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	5.16228	0.252495
$419$	16.6491	0.813362	0.406681	−	0.913570i	$-0.366686\pi$
$419$	16.6491	0.813362	0.406681	+	0.913570i	$0.366686\pi$
$420$	0	0
$421$	−39.9473	−1.94691	−0.973457	−	0.228870i	$-0.926497\pi$
$421$	−39.9473	−1.94691	−0.973457	+	0.228870i	$0.926497\pi$
$422$	−18.3246	−0.892025
$423$	0	0
$424$	4.32456	0.210019
$425$	−35.8114	−1.73711
$426$	0	0
$427$	0	0
$428$	−14.3246	−0.692404
$429$	0	0
$430$	−20.0000	−0.964486
$431$	−8.97367	−0.432246	−0.216123	−	0.976366i	$-0.569341\pi$
$431$	−8.97367	−0.432246	−0.216123	+	0.976366i	$0.569341\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	0	0
$436$	−8.00000	−0.383131
$437$	32.6491	1.56182
$438$	0	0
$439$	6.32456	0.301855	0.150927	−	0.988545i	$-0.451774\pi$
$439$	6.32456	0.301855	0.150927	+	0.988545i	$0.451774\pi$
$440$	−3.16228	−0.150756
$441$	0	0
$442$	14.3246	0.681350
$443$	4.32456	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$443$	4.32456	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$444$	0	0
$445$	−51.6228	−2.44715
$446$	11.4868	0.543917
$447$	0	0
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−7.16228	−0.337258
$452$	2.00000	0.0940721
$453$	0	0
$454$	−7.48683	−0.351374
$455$	0	0
$456$	0	0
$457$	17.3509	0.811640	0.405820	−	0.913953i	$-0.366986\pi$
$457$	17.3509	0.811640	0.405820	+	0.913953i	$0.366986\pi$
$458$	25.4868	1.19092
$459$	0	0
$460$	−20.0000	−0.932505
$461$	23.2982	1.08511	0.542553	−	0.840021i	$-0.317458\pi$
$461$	23.2982	1.08511	0.542553	+	0.840021i	$0.317458\pi$
$462$	0	0
$463$	−28.6491	−1.33144	−0.665719	−	0.746203i	$-0.731875\pi$
$463$	−28.6491	−1.33144	−0.665719	+	0.746203i	$0.731875\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	6.00000	0.277945
$467$	17.2982	0.800466	0.400233	−	0.916413i	$-0.368929\pi$
$467$	17.2982	0.800466	0.400233	+	0.916413i	$0.368929\pi$
$468$	0	0
$469$	0	0
$470$	−36.3246	−1.67553
$471$	0	0
$472$	1.67544	0.0771186
$473$	6.32456	0.290803
$474$	0	0
$475$	−25.8114	−1.18431
$476$	0	0
$477$	0	0
$478$	12.6491	0.578557
$479$	21.2982	0.973141	0.486570	−	0.873641i	$-0.338248\pi$
$479$	21.2982	0.973141	0.486570	+	0.873641i	$0.338248\pi$
$480$	0	0
$481$	−16.6491	−0.759134
$482$	−0.837722	−0.0381572
$483$	0	0
$484$	1.00000	0.0454545
$485$	33.6754	1.52912
$486$	0	0
$487$	−14.3246	−0.649108	−0.324554	−	0.945867i	$-0.605214\pi$
$487$	−14.3246	−0.649108	−0.324554	+	0.945867i	$0.605214\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	0	0
$491$	8.64911	0.390329	0.195164	−	0.980771i	$-0.437476\pi$
$491$	8.64911	0.390329	0.195164	+	0.980771i	$0.437476\pi$
$492$	0	0
$493$	−28.6491	−1.29029
$494$	10.3246	0.464524
$495$	0	0
$496$	−9.16228	−0.411398
$497$	0	0
$498$	0	0
$499$	−16.9737	−0.759846	−0.379923	−	0.925018i	$-0.624049\pi$
$499$	−16.9737	−0.759846	−0.379923	+	0.925018i	$0.624049\pi$
$500$	0	0
$501$	0	0
$502$	13.6754	0.610365
$503$	5.02633	0.224113	0.112057	−	0.993702i	$-0.464256\pi$
$503$	5.02633	0.224113	0.112057	+	0.993702i	$0.464256\pi$
$504$	0	0
$505$	18.9737	0.844317
$506$	6.32456	0.281161
$507$	0	0
$508$	20.9737	0.930556
$509$	−39.8114	−1.76461	−0.882304	−	0.470679i	$-0.844009\pi$
$509$	−39.8114	−1.76461	−0.882304	+	0.470679i	$0.844009\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−7.67544	−0.338549
$515$	−43.6754	−1.92457
$516$	0	0
$517$	11.4868	0.505190
$518$	0	0
$519$	0	0
$520$	−6.32456	−0.277350
$521$	−20.3246	−0.890435	−0.445217	−	0.895422i	$-0.646874\pi$
$521$	−20.3246	−0.890435	−0.445217	+	0.895422i	$0.646874\pi$
$522$	0	0
$523$	4.13594	0.180852	0.0904261	−	0.995903i	$-0.471177\pi$
$523$	4.13594	0.180852	0.0904261	+	0.995903i	$0.471177\pi$
$524$	−5.16228	−0.225515
$525$	0	0
$526$	24.9737	1.08890
$527$	65.6228	2.85857
$528$	0	0
$529$	17.0000	0.739130
$530$	13.6754	0.594023
$531$	0	0
$532$	0	0
$533$	−14.3246	−0.620465
$534$	0	0
$535$	−45.2982	−1.95841
$536$	4.32456	0.186792
$537$	0	0
$538$	−11.1623	−0.481240
$539$	0	0
$540$	0	0
$541$	31.2982	1.34562	0.672808	−	0.739817i	$-0.265088\pi$
$541$	31.2982	1.34562	0.672808	+	0.739817i	$0.265088\pi$
$542$	6.32456	0.271663
$543$	0	0
$544$	−7.16228	−0.307080
$545$	−25.2982	−1.08366
$546$	0	0
$547$	−41.2982	−1.76578	−0.882892	−	0.469576i	$-0.844407\pi$
$547$	−41.2982	−1.76578	−0.882892	+	0.469576i	$0.844407\pi$
$548$	−18.0000	−0.768922
$549$	0	0
$550$	−5.00000	−0.213201
$551$	−20.6491	−0.879682
$552$	0	0
$553$	0	0
$554$	−24.0000	−1.01966
$555$	0	0
$556$	7.48683	0.317512
$557$	−33.2982	−1.41089	−0.705445	−	0.708764i	$-0.749253\pi$
$557$	−33.2982	−1.41089	−0.705445	+	0.708764i	$0.749253\pi$
$558$	0	0
$559$	12.6491	0.535000
$560$	0	0
$561$	0	0
$562$	22.6491	0.955395
$563$	−20.1359	−0.848629	−0.424314	−	0.905515i	$-0.639485\pi$
$563$	−20.1359	−0.848629	−0.424314	+	0.905515i	$0.639485\pi$
$564$	0	0
$565$	6.32456	0.266076
$566$	15.4868	0.650960
$567$	0	0
$568$	−8.00000	−0.335673
$569$	26.6491	1.11719	0.558594	−	0.829441i	$-0.311341\pi$
$569$	26.6491	1.11719	0.558594	+	0.829441i	$0.311341\pi$
$570$	0	0
$571$	12.6491	0.529349	0.264674	−	0.964338i	$-0.414736\pi$
$571$	12.6491	0.529349	0.264674	+	0.964338i	$0.414736\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	0	0
$575$	−31.6228	−1.31876
$576$	0	0
$577$	−15.6754	−0.652577	−0.326289	−	0.945270i	$-0.605798\pi$
$577$	−15.6754	−0.652577	−0.326289	+	0.945270i	$0.605798\pi$
$578$	34.2982	1.42662
$579$	0	0
$580$	12.6491	0.525226
$581$	0	0
$582$	0	0
$583$	−4.32456	−0.179105
$584$	5.48683	0.227047
$585$	0	0
$586$	−24.9737	−1.03165
$587$	−6.97367	−0.287834	−0.143917	−	0.989590i	$-0.545970\pi$
$587$	−6.97367	−0.287834	−0.143917	+	0.989590i	$0.545970\pi$
$588$	0	0
$589$	47.2982	1.94889
$590$	5.29822	0.218124
$591$	0	0
$592$	8.32456	0.342137
$593$	16.4605	0.675952	0.337976	−	0.941155i	$-0.390258\pi$
$593$	16.4605	0.675952	0.337976	+	0.941155i	$0.390258\pi$
$594$	0	0
$595$	0	0
$596$	2.00000	0.0819232
$597$	0	0
$598$	12.6491	0.517261
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−19.1623	−0.781646	−0.390823	−	0.920466i	$-0.627809\pi$
$601$	−19.1623	−0.781646	−0.390823	+	0.920466i	$0.627809\pi$
$602$	0	0
$603$	0	0
$604$	20.6491	0.840200
$605$	3.16228	0.128565
$606$	0	0
$607$	−4.64911	−0.188702	−0.0943508	−	0.995539i	$-0.530078\pi$
$607$	−4.64911	−0.188702	−0.0943508	+	0.995539i	$0.530078\pi$
$608$	−5.16228	−0.209358
$609$	0	0
$610$	−31.6228	−1.28037
$611$	22.9737	0.929415
$612$	0	0
$613$	−31.2982	−1.26412	−0.632062	−	0.774918i	$-0.717791\pi$
$613$	−31.2982	−1.26412	−0.632062	+	0.774918i	$0.717791\pi$
$614$	21.1623	0.854040
$615$	0	0
$616$	0	0
$617$	20.6491	0.831302	0.415651	−	0.909524i	$-0.363554\pi$
$617$	20.6491	0.831302	0.415651	+	0.909524i	$0.363554\pi$
$618$	0	0
$619$	−23.3509	−0.938551	−0.469276	−	0.883052i	$-0.655485\pi$
$619$	−23.3509	−0.938551	−0.469276	+	0.883052i	$0.655485\pi$
$620$	−28.9737	−1.16361
$621$	0	0
$622$	29.8114	1.19533
$623$	0	0
$624$	0	0
$625$	−25.0000	−1.00000
$626$	−20.3246	−0.812333
$627$	0	0
$628$	5.48683	0.218948
$629$	−59.6228	−2.37732
$630$	0	0
$631$	−27.6228	−1.09965	−0.549823	−	0.835281i	$-0.685305\pi$
$631$	−27.6228	−1.09965	−0.549823	+	0.835281i	$0.685305\pi$
$632$	−4.32456	−0.172022
$633$	0	0
$634$	12.9737	0.515250
$635$	66.3246	2.63201
$636$	0	0
$637$	0	0
$638$	−4.00000	−0.158362
$639$	0	0
$640$	3.16228	0.125000
$641$	37.9473	1.49883	0.749415	−	0.662101i	$-0.230335\pi$
$641$	37.9473	1.49883	0.749415	+	0.662101i	$0.230335\pi$
$642$	0	0
$643$	−34.9737	−1.37923	−0.689613	−	0.724178i	$-0.742220\pi$
$643$	−34.9737	−1.37923	−0.689613	+	0.724178i	$0.742220\pi$
$644$	0	0
$645$	0	0
$646$	36.9737	1.45471
$647$	−3.48683	−0.137082	−0.0685408	−	0.997648i	$-0.521834\pi$
$647$	−3.48683	−0.137082	−0.0685408	+	0.997648i	$0.521834\pi$
$648$	0	0
$649$	−1.67544	−0.0657670
$650$	−10.0000	−0.392232
$651$	0	0
$652$	−16.6491	−0.652029
$653$	−43.2982	−1.69439	−0.847195	−	0.531282i	$-0.821711\pi$
$653$	−43.2982	−1.69439	−0.847195	+	0.531282i	$0.821711\pi$
$654$	0	0
$655$	−16.3246	−0.637853
$656$	7.16228	0.279640
$657$	0	0
$658$	0	0
$659$	22.9737	0.894927	0.447463	−	0.894302i	$-0.352327\pi$
$659$	22.9737	0.894927	0.447463	+	0.894302i	$0.352327\pi$
$660$	0	0
$661$	22.1359	0.860988	0.430494	−	0.902593i	$-0.358339\pi$
$661$	22.1359	0.860988	0.430494	+	0.902593i	$0.358339\pi$
$662$	−8.97367	−0.348771
$663$	0	0
$664$	5.16228	0.200335
$665$	0	0
$666$	0	0
$667$	−25.2982	−0.979551
$668$	6.32456	0.244704
$669$	0	0
$670$	13.6754	0.528329
$671$	10.0000	0.386046
$672$	0	0
$673$	13.6228	0.525119	0.262560	−	0.964916i	$-0.415433\pi$
$673$	13.6228	0.525119	0.262560	+	0.964916i	$0.415433\pi$
$674$	−19.6754	−0.757870
$675$	0	0
$676$	−9.00000	−0.346154
$677$	8.32456	0.319939	0.159969	−	0.987122i	$-0.448860\pi$
$677$	8.32456	0.319939	0.159969	+	0.987122i	$0.448860\pi$
$678$	0	0
$679$	0	0
$680$	−22.6491	−0.868554
$681$	0	0
$682$	9.16228	0.350842
$683$	−24.9737	−0.955591	−0.477795	−	0.878471i	$-0.658564\pi$
$683$	−24.9737	−0.955591	−0.477795	+	0.878471i	$0.658564\pi$
$684$	0	0
$685$	−56.9210	−2.17484
$686$	0	0
$687$	0	0
$688$	−6.32456	−0.241121
$689$	−8.64911	−0.329505
$690$	0	0
$691$	6.97367	0.265291	0.132645	−	0.991164i	$-0.457653\pi$
$691$	6.97367	0.265291	0.132645	+	0.991164i	$0.457653\pi$
$692$	12.3246	0.468509
$693$	0	0
$694$	−14.3246	−0.543753
$695$	23.6754	0.898061
$696$	0	0
$697$	−51.2982	−1.94306
$698$	−8.97367	−0.339658
$699$	0	0
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−42.9737	−1.62078
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−7.67544	−0.288869
$707$	0	0
$708$	0	0
$709$	40.3246	1.51442	0.757210	−	0.653171i	$-0.226562\pi$
$709$	40.3246	1.51442	0.757210	+	0.653171i	$0.226562\pi$
$710$	−25.2982	−0.949425
$711$	0	0
$712$	−16.3246	−0.611789
$713$	57.9473	2.17014
$714$	0	0
$715$	6.32456	0.236525
$716$	−3.67544	−0.137358
$717$	0	0
$718$	3.67544	0.137166
$719$	−22.8377	−0.851703	−0.425852	−	0.904793i	$-0.640025\pi$
$719$	−22.8377	−0.851703	−0.425852	+	0.904793i	$0.640025\pi$
$720$	0	0
$721$	0	0
$722$	7.64911	0.284670
$723$	0	0
$724$	−17.4868	−0.649893
$725$	20.0000	0.742781
$726$	0	0
$727$	40.1359	1.48856	0.744280	−	0.667868i	$-0.232793\pi$
$727$	40.1359	1.48856	0.744280	+	0.667868i	$0.232793\pi$
$728$	0	0
$729$	0	0
$730$	17.3509	0.642185
$731$	45.2982	1.67542
$732$	0	0
$733$	−5.35089	−0.197640	−0.0988198	−	0.995105i	$-0.531507\pi$
$733$	−5.35089	−0.197640	−0.0988198	+	0.995105i	$0.531507\pi$
$734$	1.16228	0.0429005
$735$	0	0
$736$	−6.32456	−0.233126
$737$	−4.32456	−0.159297
$738$	0	0
$739$	3.35089	0.123264	0.0616322	−	0.998099i	$-0.480369\pi$
$739$	3.35089	0.123264	0.0616322	+	0.998099i	$0.480369\pi$
$740$	26.3246	0.967710
$741$	0	0
$742$	0	0
$743$	3.35089	0.122932	0.0614661	−	0.998109i	$-0.480422\pi$
$743$	3.35089	0.122932	0.0614661	+	0.998109i	$0.480422\pi$
$744$	0	0
$745$	6.32456	0.231714
$746$	−10.0000	−0.366126
$747$	0	0
$748$	7.16228	0.261879
$749$	0	0
$750$	0	0
$751$	5.67544	0.207100	0.103550	−	0.994624i	$-0.466980\pi$
$751$	5.67544	0.207100	0.103550	+	0.994624i	$0.466980\pi$
$752$	−11.4868	−0.418882
$753$	0	0
$754$	−8.00000	−0.291343
$755$	65.2982	2.37645
$756$	0	0
$757$	29.6228	1.07666	0.538329	−	0.842735i	$-0.319056\pi$
$757$	29.6228	1.07666	0.538329	+	0.842735i	$0.319056\pi$
$758$	−13.2982	−0.483013
$759$	0	0
$760$	−16.3246	−0.592154
$761$	9.48683	0.343897	0.171949	−	0.985106i	$-0.444994\pi$
$761$	9.48683	0.343897	0.171949	+	0.985106i	$0.444994\pi$
$762$	0	0
$763$	0	0
$764$	−5.67544	−0.205330
$765$	0	0
$766$	9.16228	0.331047
$767$	−3.35089	−0.120994
$768$	0	0
$769$	−25.4868	−0.919079	−0.459539	−	0.888157i	$-0.651985\pi$
$769$	−25.4868	−0.919079	−0.459539	+	0.888157i	$0.651985\pi$
$770$	0	0
$771$	0	0
$772$	10.6491	0.383270
$773$	5.86406	0.210915	0.105458	−	0.994424i	$-0.466369\pi$
$773$	5.86406	0.210915	0.105458	+	0.994424i	$0.466369\pi$
$774$	0	0
$775$	−45.8114	−1.64559
$776$	10.6491	0.382281
$777$	0	0
$778$	6.00000	0.215110
$779$	−36.9737	−1.32472
$780$	0	0
$781$	8.00000	0.286263
$782$	45.2982	1.61986
$783$	0	0
$784$	0	0
$785$	17.3509	0.619280
$786$	0	0
$787$	−6.46050	−0.230292	−0.115146	−	0.993349i	$-0.536734\pi$
$787$	−6.46050	−0.230292	−0.115146	+	0.993349i	$0.536734\pi$
$788$	24.0000	0.854965
$789$	0	0
$790$	−13.6754	−0.486550
$791$	0	0
$792$	0	0
$793$	20.0000	0.710221
$794$	−21.4868	−0.762539
$795$	0	0
$796$	−19.4868	−0.690692
$797$	−8.83772	−0.313048	−0.156524	−	0.987674i	$-0.550029\pi$
$797$	−8.83772	−0.313048	−0.156524	+	0.987674i	$0.550029\pi$
$798$	0	0
$799$	82.2719	2.91057
$800$	5.00000	0.176777
$801$	0	0
$802$	23.2982	0.822689
$803$	−5.48683	−0.193626
$804$	0	0
$805$	0	0
$806$	18.3246	0.645455
$807$	0	0
$808$	6.00000	0.211079
$809$	43.6754	1.53555	0.767773	−	0.640721i	$-0.221365\pi$
$809$	43.6754	1.53555	0.767773	+	0.640721i	$0.221365\pi$
$810$	0	0
$811$	7.48683	0.262898	0.131449	−	0.991323i	$-0.458037\pi$
$811$	7.48683	0.262898	0.131449	+	0.991323i	$0.458037\pi$
$812$	0	0
$813$	0	0
$814$	−8.32456	−0.291776
$815$	−52.6491	−1.84422
$816$	0	0
$817$	32.6491	1.14225
$818$	−27.1623	−0.949707
$819$	0	0
$820$	22.6491	0.790941
$821$	−52.5964	−1.83563	−0.917814	−	0.397010i	$-0.870048\pi$
$821$	−52.5964	−1.83563	−0.917814	+	0.397010i	$0.870048\pi$
$822$	0	0
$823$	−9.02633	−0.314638	−0.157319	−	0.987548i	$-0.550285\pi$
$823$	−9.02633	−0.314638	−0.157319	+	0.987548i	$0.550285\pi$
$824$	−13.8114	−0.481143
$825$	0	0
$826$	0	0
$827$	33.9473	1.18046	0.590232	−	0.807234i	$-0.299036\pi$
$827$	33.9473	1.18046	0.590232	+	0.807234i	$0.299036\pi$
$828$	0	0
$829$	12.4605	0.432771	0.216386	−	0.976308i	$-0.430573\pi$
$829$	12.4605	0.432771	0.216386	+	0.976308i	$0.430573\pi$
$830$	16.3246	0.566634
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	0	0
$835$	20.0000	0.692129
$836$	5.16228	0.178541
$837$	0	0
$838$	16.6491	0.575134
$839$	34.4605	1.18971	0.594854	−	0.803834i	$-0.297210\pi$
$839$	34.4605	1.18971	0.594854	+	0.803834i	$0.297210\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−39.9473	−1.37668
$843$	0	0
$844$	−18.3246	−0.630757
$845$	−28.4605	−0.979071
$846$	0	0
$847$	0	0
$848$	4.32456	0.148506
$849$	0	0
$850$	−35.8114	−1.22832
$851$	−52.6491	−1.80479
$852$	0	0
$853$	6.64911	0.227661	0.113831	−	0.993500i	$-0.463688\pi$
$853$	6.64911	0.227661	0.113831	+	0.993500i	$0.463688\pi$
$854$	0	0
$855$	0	0
$856$	−14.3246	−0.489603
$857$	−24.1886	−0.826267	−0.413134	−	0.910670i	$-0.635566\pi$
$857$	−24.1886	−0.826267	−0.413134	+	0.910670i	$0.635566\pi$
$858$	0	0
$859$	30.3246	1.03466	0.517330	−	0.855786i	$-0.326926\pi$
$859$	30.3246	1.03466	0.517330	+	0.855786i	$0.326926\pi$
$860$	−20.0000	−0.681994
$861$	0	0
$862$	−8.97367	−0.305644
$863$	17.6754	0.601679	0.300840	−	0.953675i	$-0.402733\pi$
$863$	17.6754	0.601679	0.300840	+	0.953675i	$0.402733\pi$
$864$	0	0
$865$	38.9737	1.32514
$866$	34.0000	1.15537
$867$	0	0
$868$	0	0
$869$	4.32456	0.146700
$870$	0	0
$871$	−8.64911	−0.293064
$872$	−8.00000	−0.270914
$873$	0	0
$874$	32.6491	1.10437
$875$	0	0
$876$	0	0
$877$	−23.3509	−0.788504	−0.394252	−	0.919002i	$-0.628996\pi$
$877$	−23.3509	−0.788504	−0.394252	+	0.919002i	$0.628996\pi$
$878$	6.32456	0.213443
$879$	0	0
$880$	−3.16228	−0.106600
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	−13.2982	−0.447521	−0.223760	−	0.974644i	$-0.571833\pi$
$883$	−13.2982	−0.447521	−0.223760	+	0.974644i	$0.571833\pi$
$884$	14.3246	0.481787
$885$	0	0
$886$	4.32456	0.145286
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	0	0
$890$	−51.6228	−1.73040
$891$	0	0
$892$	11.4868	0.384608
$893$	59.2982	1.98434
$894$	0	0
$895$	−11.6228	−0.388507
$896$	0	0
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−36.6491	−1.22232
$900$	0	0
$901$	−30.9737	−1.03188
$902$	−7.16228	−0.238478
$903$	0	0
$904$	2.00000	0.0665190
$905$	−55.2982	−1.83818
$906$	0	0
$907$	53.6228	1.78052	0.890258	−	0.455457i	$-0.150524\pi$
$907$	53.6228	1.78052	0.890258	+	0.455457i	$0.150524\pi$
$908$	−7.48683	−0.248459
$909$	0	0
$910$	0	0
$911$	−22.3246	−0.739646	−0.369823	−	0.929102i	$-0.620582\pi$
$911$	−22.3246	−0.739646	−0.369823	+	0.929102i	$0.620582\pi$
$912$	0	0
$913$	−5.16228	−0.170846
$914$	17.3509	0.573916
$915$	0	0
$916$	25.4868	0.842109
$917$	0	0
$918$	0	0
$919$	−17.2982	−0.570616	−0.285308	−	0.958436i	$-0.592096\pi$
$919$	−17.2982	−0.570616	−0.285308	+	0.958436i	$0.592096\pi$
$920$	−20.0000	−0.659380
$921$	0	0
$922$	23.2982	0.767286
$923$	16.0000	0.526646
$924$	0	0
$925$	41.6228	1.36855
$926$	−28.6491	−0.941468
$927$	0	0
$928$	4.00000	0.131306
$929$	−33.6228	−1.10313	−0.551564	−	0.834133i	$-0.685969\pi$
$929$	−33.6228	−1.10313	−0.551564	+	0.834133i	$0.685969\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	0	0
$934$	17.2982	0.566015
$935$	22.6491	0.740705
$936$	0	0
$937$	42.1359	1.37652	0.688261	−	0.725464i	$-0.258375\pi$
$937$	42.1359	1.37652	0.688261	+	0.725464i	$0.258375\pi$
$938$	0	0
$939$	0	0
$940$	−36.3246	−1.18478
$941$	27.6754	0.902194	0.451097	−	0.892475i	$-0.351033\pi$
$941$	27.6754	0.902194	0.451097	+	0.892475i	$0.351033\pi$
$942$	0	0
$943$	−45.2982	−1.47511
$944$	1.67544	0.0545311
$945$	0	0
$946$	6.32456	0.205629
$947$	2.70178	0.0877960	0.0438980	−	0.999036i	$-0.486022\pi$
$947$	2.70178	0.0877960	0.0438980	+	0.999036i	$0.486022\pi$
$948$	0	0
$949$	−10.9737	−0.356220
$950$	−25.8114	−0.837432
$951$	0	0
$952$	0	0
$953$	−4.97367	−0.161113	−0.0805564	−	0.996750i	$-0.525670\pi$
$953$	−4.97367	−0.161113	−0.0805564	+	0.996750i	$0.525670\pi$
$954$	0	0
$955$	−17.9473	−0.580762
$956$	12.6491	0.409101
$957$	0	0
$958$	21.2982	0.688114
$959$	0	0
$960$	0	0
$961$	52.9473	1.70798
$962$	−16.6491	−0.536789
$963$	0	0
$964$	−0.837722	−0.0269812
$965$	33.6754	1.08405
$966$	0	0
$967$	−57.2982	−1.84259	−0.921293	−	0.388868i	$-0.872866\pi$
$967$	−57.2982	−1.84259	−0.921293	+	0.388868i	$0.872866\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	33.6754	1.08125
$971$	7.35089	0.235901	0.117951	−	0.993019i	$-0.462368\pi$
$971$	7.35089	0.235901	0.117951	+	0.993019i	$0.462368\pi$
$972$	0	0
$973$	0	0
$974$	−14.3246	−0.458988
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−24.6491	−0.788595	−0.394297	−	0.918983i	$-0.629012\pi$
$977$	−24.6491	−0.788595	−0.394297	+	0.918983i	$0.629012\pi$
$978$	0	0
$979$	16.3246	0.521735
$980$	0	0
$981$	0	0
$982$	8.64911	0.276004
$983$	−48.1359	−1.53530	−0.767649	−	0.640870i	$-0.778574\pi$
$983$	−48.1359	−1.53530	−0.767649	+	0.640870i	$0.778574\pi$
$984$	0	0
$985$	75.8947	2.41821
$986$	−28.6491	−0.912374
$987$	0	0
$988$	10.3246	0.328468
$989$	40.0000	1.27193
$990$	0	0
$991$	−61.2982	−1.94720	−0.973601	−	0.228256i	$-0.926698\pi$
$991$	−61.2982	−1.94720	−0.973601	+	0.228256i	$0.926698\pi$
$992$	−9.16228	−0.290903
$993$	0	0
$994$	0	0
$995$	−61.6228	−1.95357
$996$	0	0
$997$	−48.9737	−1.55101	−0.775506	−	0.631340i	$-0.782505\pi$
$997$	−48.9737	−1.55101	−0.775506	+	0.631340i	$0.782505\pi$
$998$	−16.9737	−0.537292
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.dc.1.2		2
3.2	odd	2		9702.2.a.cn.1.1		2
7.6	odd	2		1386.2.a.o.1.1	yes	2
21.20	even	2		1386.2.a.n.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.a.n.1.2	✓	2	21.20	even	2
1386.2.a.o.1.1	yes	2	7.6	odd	2
9702.2.a.cn.1.1		2	3.2	odd	2
9702.2.a.dc.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.16228 q^{5} +1.00000 q^{8} +3.16228 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -7.16228 q^{17} -5.16228 q^{19} +3.16228 q^{20} -1.00000 q^{22} -6.32456 q^{23} +5.00000 q^{25} -2.00000 q^{26} +4.00000 q^{29} -9.16228 q^{31} +1.00000 q^{32} -7.16228 q^{34} +8.32456 q^{37} -5.16228 q^{38} +3.16228 q^{40} +7.16228 q^{41} -6.32456 q^{43} -1.00000 q^{44} -6.32456 q^{46} -11.4868 q^{47} +5.00000 q^{50} -2.00000 q^{52} +4.32456 q^{53} -3.16228 q^{55} +4.00000 q^{58} +1.67544 q^{59} -10.0000 q^{61} -9.16228 q^{62} +1.00000 q^{64} -6.32456 q^{65} +4.32456 q^{67} -7.16228 q^{68} -8.00000 q^{71} +5.48683 q^{73} +8.32456 q^{74} -5.16228 q^{76} -4.32456 q^{79} +3.16228 q^{80} +7.16228 q^{82} +5.16228 q^{83} -22.6491 q^{85} -6.32456 q^{86} -1.00000 q^{88} -16.3246 q^{89} -6.32456 q^{92} -11.4868 q^{94} -16.3246 q^{95} +10.6491 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{11} - 4 q^{13} + 2 q^{16} - 8 q^{17} - 4 q^{19} - 2 q^{22} + 10 q^{25} - 4 q^{26} + 8 q^{29} - 12 q^{31} + 2 q^{32} - 8 q^{34} + 4 q^{37} - 4 q^{38} + 8 q^{41} - 2 q^{44}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^11 - 4 * q^13 + 2 * q^16 - 8 * q^17 - 4 * q^19 - 2 * q^22 + 10 * q^25 - 4 * q^26 + 8 * q^29 - 12 * q^31 + 2 * q^32 - 8 * q^34 + 4 * q^37 - 4 * q^38 + 8 * q^41 - 2 * q^44 - 4 * q^47 + 10 * q^50 - 4 * q^52 - 4 * q^53 + 8 * q^58 + 16 * q^59 - 20 * q^61 - 12 * q^62 + 2 * q^64 - 4 * q^67 - 8 * q^68 - 16 * q^71 - 8 * q^73 + 4 * q^74 - 4 * q^76 + 4 * q^79 + 8 * q^82 + 4 * q^83 - 20 * q^85 - 2 * q^88 - 20 * q^89 - 4 * q^94 - 20 * q^95 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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