LMFDB - Embedded newform 9702.2.a.dp.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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +1.41421 q^{10} +1.00000 q^{11} +1.41421 q^{13} +1.00000 q^{16} -1.41421 q^{17} -2.82843 q^{19} +1.41421 q^{20} +1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{25} +1.41421 q^{26} -6.00000 q^{29} -2.82843 q^{31} +1.00000 q^{32} -1.41421 q^{34} -8.00000 q^{37} -2.82843 q^{38} +1.41421 q^{40} -7.07107 q^{41} -8.00000 q^{43} +1.00000 q^{44} -4.00000 q^{46} -8.48528 q^{47} -3.00000 q^{50} +1.41421 q^{52} +1.41421 q^{55} -6.00000 q^{58} -7.07107 q^{61} -2.82843 q^{62} +1.00000 q^{64} +2.00000 q^{65} -8.00000 q^{67} -1.41421 q^{68} -8.00000 q^{71} +1.41421 q^{73} -8.00000 q^{74} -2.82843 q^{76} +4.00000 q^{79} +1.41421 q^{80} -7.07107 q^{82} -2.82843 q^{83} -2.00000 q^{85} -8.00000 q^{86} +1.00000 q^{88} +9.89949 q^{89} -4.00000 q^{92} -8.48528 q^{94} -4.00000 q^{95} +15.5563 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{11} + 2 q^{16} + 2 q^{22} - 8 q^{23} - 6 q^{25} - 12 q^{29} + 2 q^{32} - 16 q^{37} - 16 q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{50} - 12 q^{58} + 2 q^{64} + 4 q^{65} - 16 q^{67} - 16 q^{71} - 16 q^{74} + 8 q^{79} - 4 q^{85} - 16 q^{86} + 2 q^{88} - 8 q^{92} - 8 q^{95}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^11 + 2 * q^16 + 2 * q^22 - 8 * q^23 - 6 * q^25 - 12 * q^29 + 2 * q^32 - 16 * q^37 - 16 * q^43 + 2 * q^44 - 8 * q^46 - 6 * q^50 - 12 * q^58 + 2 * q^64 + 4 * q^65 - 16 * q^67 - 16 * q^71 - 16 * q^74 + 8 * q^79 - 4 * q^85 - 16 * q^86 + 2 * q^88 - 8 * q^92 - 8 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	1.41421	0.447214
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.41421	−0.342997	−0.171499	−	0.985184i	$-0.554861\pi$
$17$	−1.41421	−0.342997	−0.171499	+	0.985184i	$0.554861\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	1.41421	0.316228
$21$	0	0
$22$	1.00000	0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	1.41421	0.277350
$27$	0	0
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.41421	−0.242536
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−2.82843	−0.458831
$39$	0	0
$40$	1.41421	0.223607
$41$	−7.07107	−1.10432	−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107	−1.10432	−0.552158	+	0.833740i	$0.686195\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	0	0
$49$	0	0
$50$	−3.00000	−0.424264
$51$	0	0
$52$	1.41421	0.196116
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	0	0
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−7.07107	−0.905357	−0.452679	−	0.891674i	$-0.649532\pi$
$61$	−7.07107	−0.905357	−0.452679	+	0.891674i	$0.649532\pi$
$62$	−2.82843	−0.359211
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−1.41421	−0.171499
$69$	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
$73$	1.41421	0.165521	0.0827606	−	0.996569i	$-0.473626\pi$
$73$	1.41421	0.165521	0.0827606	+	0.996569i	$0.473626\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	−2.82843	−0.324443
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	1.41421	0.158114
$81$	0	0
$82$	−7.07107	−0.780869
$83$	−2.82843	−0.310460	−0.155230	−	0.987878i	$-0.549612\pi$
$83$	−2.82843	−0.310460	−0.155230	+	0.987878i	$0.549612\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−8.00000	−0.862662
$87$	0	0
$88$	1.00000	0.106600
$89$	9.89949	1.04934	0.524672	−	0.851304i	$-0.324188\pi$
$89$	9.89949	1.04934	0.524672	+	0.851304i	$0.324188\pi$
$90$	0	0
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−8.48528	−0.875190
$95$	−4.00000	−0.410391
$96$	0	0
$97$	15.5563	1.57951	0.789754	−	0.613424i	$-0.210208\pi$
$97$	15.5563	1.57951	0.789754	+	0.613424i	$0.210208\pi$
$98$	0	0
$99$	0	0
$100$	−3.00000	−0.300000
$101$	15.5563	1.54791	0.773957	−	0.633238i	$-0.218274\pi$
$101$	15.5563	1.54791	0.773957	+	0.633238i	$0.218274\pi$
$102$	0	0
$103$	8.48528	0.836080	0.418040	−	0.908429i	$-0.362717\pi$
$103$	8.48528	0.836080	0.418040	+	0.908429i	$0.362717\pi$
$104$	1.41421	0.138675
$105$	0	0
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	1.41421	0.134840
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	−6.00000	−0.557086
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.07107	−0.640184
$123$	0	0
$124$	−2.82843	−0.254000
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	2.82843	0.247121	0.123560	−	0.992337i	$-0.460569\pi$
$131$	2.82843	0.247121	0.123560	+	0.992337i	$0.460569\pi$
$132$	0	0
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−1.41421	−0.121268
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	2.82843	0.239904	0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843	0.239904	0.119952	+	0.992780i	$0.461726\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	1.41421	0.118262
$144$	0	0
$145$	−8.48528	−0.704664
$146$	1.41421	0.117041
$147$	0	0
$148$	−8.00000	−0.657596
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−2.82843	−0.229416
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−1.41421	−0.112867	−0.0564333	−	0.998406i	$-0.517973\pi$
$157$	−1.41421	−0.112867	−0.0564333	+	0.998406i	$0.517973\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	1.41421	0.111803
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−7.07107	−0.552158
$165$	0	0
$166$	−2.82843	−0.219529
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	−2.00000	−0.153393
$171$	0	0
$172$	−8.00000	−0.609994
$173$	1.41421	0.107521	0.0537603	−	0.998554i	$-0.482879\pi$
$173$	1.41421	0.107521	0.0537603	+	0.998554i	$0.482879\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	9.89949	0.741999
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	12.7279	0.946059	0.473029	−	0.881047i	$-0.343160\pi$
$181$	12.7279	0.946059	0.473029	+	0.881047i	$0.343160\pi$
$182$	0	0
$183$	0	0
$184$	−4.00000	−0.294884
$185$	−11.3137	−0.831800
$186$	0	0
$187$	−1.41421	−0.103418
$188$	−8.48528	−0.618853
$189$	0	0
$190$	−4.00000	−0.290191
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	15.5563	1.11688
$195$	0	0
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	25.4558	1.80452	0.902258	−	0.431196i	$-0.141908\pi$
$199$	25.4558	1.80452	0.902258	+	0.431196i	$0.141908\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	15.5563	1.09454
$203$	0	0
$204$	0	0
$205$	−10.0000	−0.698430
$206$	8.48528	0.591198
$207$	0	0
$208$	1.41421	0.0980581
$209$	−2.82843	−0.195646
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0	0
$214$	8.00000	0.546869
$215$	−11.3137	−0.771589
$216$	0	0
$217$	0	0
$218$	4.00000	0.270914
$219$	0	0
$220$	1.41421	0.0953463
$221$	−2.00000	−0.134535
$222$	0	0
$223$	2.82843	0.189405	0.0947027	−	0.995506i	$-0.469810\pi$
$223$	2.82843	0.189405	0.0947027	+	0.995506i	$0.469810\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.1421	0.938647	0.469323	−	0.883026i	$-0.344498\pi$
$227$	14.1421	0.938647	0.469323	+	0.883026i	$0.344498\pi$
$228$	0	0
$229$	−21.2132	−1.40181	−0.700904	−	0.713256i	$-0.747220\pi$
$229$	−21.2132	−1.40181	−0.700904	+	0.713256i	$0.747220\pi$
$230$	−5.65685	−0.373002
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	9.89949	0.637683	0.318841	−	0.947808i	$-0.396706\pi$
$241$	9.89949	0.637683	0.318841	+	0.947808i	$0.396706\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−7.07107	−0.452679
$245$	0	0
$246$	0	0
$247$	−4.00000	−0.254514
$248$	−2.82843	−0.179605
$249$	0	0
$250$	−11.3137	−0.715542
$251$	−11.3137	−0.714115	−0.357057	−	0.934082i	$-0.616220\pi$
$251$	−11.3137	−0.714115	−0.357057	+	0.934082i	$0.616220\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.07107	0.441081	0.220541	−	0.975378i	$-0.429218\pi$
$257$	7.07107	0.441081	0.220541	+	0.975378i	$0.429218\pi$
$258$	0	0
$259$	0	0
$260$	2.00000	0.124035
$261$	0	0
$262$	2.82843	0.174741
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−8.00000	−0.488678
$269$	9.89949	0.603583	0.301791	−	0.953374i	$-0.402415\pi$
$269$	9.89949	0.603583	0.301791	+	0.953374i	$0.402415\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	−1.41421	−0.0857493
$273$	0	0
$274$	−10.0000	−0.604122
$275$	−3.00000	−0.180907
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	2.82843	0.169638
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	1.41421	0.0836242
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	−8.48528	−0.498273
$291$	0	0
$292$	1.41421	0.0827606
$293$	21.2132	1.23929	0.619644	−	0.784883i	$-0.287277\pi$
$293$	21.2132	1.23929	0.619644	+	0.784883i	$0.287277\pi$
$294$	0	0
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	4.00000	0.231714
$299$	−5.65685	−0.327144
$300$	0	0
$301$	0	0
$302$	16.0000	0.920697
$303$	0	0
$304$	−2.82843	−0.162221
$305$	−10.0000	−0.572598
$306$	0	0
$307$	2.82843	0.161427	0.0807134	−	0.996737i	$-0.474280\pi$
$307$	2.82843	0.161427	0.0807134	+	0.996737i	$0.474280\pi$
$308$	0	0
$309$	0	0
$310$	−4.00000	−0.227185
$311$	19.7990	1.12270	0.561349	−	0.827579i	$-0.310283\pi$
$311$	19.7990	1.12270	0.561349	+	0.827579i	$0.310283\pi$
$312$	0	0
$313$	7.07107	0.399680	0.199840	−	0.979829i	$-0.435958\pi$
$313$	7.07107	0.399680	0.199840	+	0.979829i	$0.435958\pi$
$314$	−1.41421	−0.0798087
$315$	0	0
$316$	4.00000	0.225018
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	1.41421	0.0790569
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−4.24264	−0.235339
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−7.07107	−0.390434
$329$	0	0
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−2.82843	−0.155230
$333$	0	0
$334$	−11.3137	−0.619059
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	−11.0000	−0.598321
$339$	0	0
$340$	−2.00000	−0.108465
$341$	−2.82843	−0.153168
$342$	0	0
$343$	0	0
$344$	−8.00000	−0.431331
$345$	0	0
$346$	1.41421	0.0760286
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	18.3848	0.984115	0.492057	−	0.870563i	$-0.336245\pi$
$349$	18.3848	0.984115	0.492057	+	0.870563i	$0.336245\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	26.8701	1.43015	0.715074	−	0.699048i	$-0.246393\pi$
$353$	26.8701	1.43015	0.715074	+	0.699048i	$0.246393\pi$
$354$	0	0
$355$	−11.3137	−0.600469
$356$	9.89949	0.524672
$357$	0	0
$358$	16.0000	0.845626
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	12.7279	0.668965
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	−8.48528	−0.442928	−0.221464	−	0.975169i	$-0.571084\pi$
$367$	−8.48528	−0.442928	−0.221464	+	0.975169i	$0.571084\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	−11.3137	−0.588172
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−1.41421	−0.0731272
$375$	0	0
$376$	−8.48528	−0.437595
$377$	−8.48528	−0.437014
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	12.0000	0.613973
$383$	14.1421	0.722629	0.361315	−	0.932444i	$-0.382328\pi$
$383$	14.1421	0.722629	0.361315	+	0.932444i	$0.382328\pi$
$384$	0	0
$385$	0	0
$386$	−18.0000	−0.916176
$387$	0	0
$388$	15.5563	0.789754
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	0	0
$394$	12.0000	0.604551
$395$	5.65685	0.284627
$396$	0	0
$397$	−9.89949	−0.496841	−0.248421	−	0.968652i	$-0.579912\pi$
$397$	−9.89949	−0.496841	−0.248421	+	0.968652i	$0.579912\pi$
$398$	25.4558	1.27599
$399$	0	0
$400$	−3.00000	−0.150000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	15.5563	0.773957
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	24.0416	1.18878	0.594391	−	0.804176i	$-0.297393\pi$
$409$	24.0416	1.18878	0.594391	+	0.804176i	$0.297393\pi$
$410$	−10.0000	−0.493865
$411$	0	0
$412$	8.48528	0.418040
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	1.41421	0.0693375
$417$	0	0
$418$	−2.82843	−0.138343
$419$	−16.9706	−0.829066	−0.414533	−	0.910034i	$-0.636055\pi$
$419$	−16.9706	−0.829066	−0.414533	+	0.910034i	$0.636055\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	0	0
$425$	4.24264	0.205798
$426$	0	0
$427$	0	0
$428$	8.00000	0.386695
$429$	0	0
$430$	−11.3137	−0.545595
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	0	0
$433$	4.24264	0.203888	0.101944	−	0.994790i	$-0.467494\pi$
$433$	4.24264	0.203888	0.101944	+	0.994790i	$0.467494\pi$
$434$	0	0
$435$	0	0
$436$	4.00000	0.191565
$437$	11.3137	0.541208
$438$	0	0
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	1.41421	0.0674200
$441$	0	0
$442$	−2.00000	−0.0951303
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	14.0000	0.663664
$446$	2.82843	0.133930
$447$	0	0
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	−7.07107	−0.332964
$452$	0	0
$453$	0	0
$454$	14.1421	0.663723
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−21.2132	−0.991228
$459$	0	0
$460$	−5.65685	−0.263752
$461$	−26.8701	−1.25146	−0.625732	−	0.780038i	$-0.715200\pi$
$461$	−26.8701	−1.25146	−0.625732	+	0.780038i	$0.715200\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−16.9706	−0.785304	−0.392652	−	0.919687i	$-0.628442\pi$
$467$	−16.9706	−0.785304	−0.392652	+	0.919687i	$0.628442\pi$
$468$	0	0
$469$	0	0
$470$	−12.0000	−0.553519
$471$	0	0
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	8.48528	0.389331
$476$	0	0
$477$	0	0
$478$	−8.00000	−0.365911
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−11.3137	−0.515861
$482$	9.89949	0.450910
$483$	0	0
$484$	1.00000	0.0454545
$485$	22.0000	0.998969
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	−7.07107	−0.320092
$489$	0	0
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	8.48528	0.382158
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−2.82843	−0.127000
$497$	0	0
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	−11.3137	−0.505964
$501$	0	0
$502$	−11.3137	−0.504956
$503$	−5.65685	−0.252227	−0.126113	−	0.992016i	$-0.540250\pi$
$503$	−5.65685	−0.252227	−0.126113	+	0.992016i	$0.540250\pi$
$504$	0	0
$505$	22.0000	0.978987
$506$	−4.00000	−0.177822
$507$	0	0
$508$	−12.0000	−0.532414
$509$	−12.7279	−0.564155	−0.282078	−	0.959392i	$-0.591024\pi$
$509$	−12.7279	−0.564155	−0.282078	+	0.959392i	$0.591024\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	7.07107	0.311891
$515$	12.0000	0.528783
$516$	0	0
$517$	−8.48528	−0.373182
$518$	0	0
$519$	0	0
$520$	2.00000	0.0877058
$521$	−7.07107	−0.309789	−0.154895	−	0.987931i	$-0.549504\pi$
$521$	−7.07107	−0.309789	−0.154895	+	0.987931i	$0.549504\pi$
$522$	0	0
$523$	−42.4264	−1.85518	−0.927589	−	0.373603i	$-0.878122\pi$
$523$	−42.4264	−1.85518	−0.927589	+	0.373603i	$0.878122\pi$
$524$	2.82843	0.123560
$525$	0	0
$526$	−12.0000	−0.523225
$527$	4.00000	0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10.0000	−0.433148
$534$	0	0
$535$	11.3137	0.489134
$536$	−8.00000	−0.345547
$537$	0	0
$538$	9.89949	0.426798
$539$	0	0
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−16.9706	−0.728948
$543$	0	0
$544$	−1.41421	−0.0606339
$545$	5.65685	0.242313
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−10.0000	−0.427179
$549$	0	0
$550$	−3.00000	−0.127920
$551$	16.9706	0.722970
$552$	0	0
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	2.82843	0.119952
$557$	36.0000	1.52537	0.762684	−	0.646771i	$-0.223881\pi$
$557$	36.0000	1.52537	0.762684	+	0.646771i	$0.223881\pi$
$558$	0	0
$559$	−11.3137	−0.478519
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	2.82843	0.119204	0.0596020	−	0.998222i	$-0.481017\pi$
$563$	2.82843	0.119204	0.0596020	+	0.998222i	$0.481017\pi$
$564$	0	0
$565$	0	0
$566$	8.48528	0.356663
$567$	0	0
$568$	−8.00000	−0.335673
$569$	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
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	1.41421	0.0591312
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	38.1838	1.58961	0.794805	−	0.606864i	$-0.207573\pi$
$577$	38.1838	1.58961	0.794805	+	0.606864i	$0.207573\pi$
$578$	−15.0000	−0.623918
$579$	0	0
$580$	−8.48528	−0.352332
$581$	0	0
$582$	0	0
$583$	0	0
$584$	1.41421	0.0585206
$585$	0	0
$586$	21.2132	0.876309
$587$	45.2548	1.86787	0.933933	−	0.357447i	$-0.116353\pi$
$587$	45.2548	1.86787	0.933933	+	0.357447i	$0.116353\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	0	0
$592$	−8.00000	−0.328798
$593$	−18.3848	−0.754972	−0.377486	−	0.926015i	$-0.623211\pi$
$593$	−18.3848	−0.754972	−0.377486	+	0.926015i	$0.623211\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	0	0
$598$	−5.65685	−0.231326
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−7.07107	−0.288435	−0.144217	−	0.989546i	$-0.546066\pi$
$601$	−7.07107	−0.288435	−0.144217	+	0.989546i	$0.546066\pi$
$602$	0	0
$603$	0	0
$604$	16.0000	0.651031
$605$	1.41421	0.0574960
$606$	0	0
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	−2.82843	−0.114708
$609$	0	0
$610$	−10.0000	−0.404888
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	2.82843	0.114146
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	16.9706	0.682105	0.341052	−	0.940044i	$-0.389217\pi$
$619$	16.9706	0.682105	0.341052	+	0.940044i	$0.389217\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	19.7990	0.793867
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	7.07107	0.282617
$627$	0	0
$628$	−1.41421	−0.0564333
$629$	11.3137	0.451107
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	8.00000	0.317721
$635$	−16.9706	−0.673456
$636$	0	0
$637$	0	0
$638$	−6.00000	−0.237542
$639$	0	0
$640$	1.41421	0.0559017
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	4.00000	0.157378
$647$	36.7696	1.44556	0.722780	−	0.691078i	$-0.242864\pi$
$647$	36.7696	1.44556	0.722780	+	0.691078i	$0.242864\pi$
$648$	0	0
$649$	0	0
$650$	−4.24264	−0.166410
$651$	0	0
$652$	−4.00000	−0.156652
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	−7.07107	−0.276079
$657$	0	0
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	−29.6985	−1.15514	−0.577569	−	0.816342i	$-0.695998\pi$
$661$	−29.6985	−1.15514	−0.577569	+	0.816342i	$0.695998\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	−2.82843	−0.109764
$665$	0	0
$666$	0	0
$667$	24.0000	0.929284
$668$	−11.3137	−0.437741
$669$	0	0
$670$	−11.3137	−0.437087
$671$	−7.07107	−0.272976
$672$	0	0
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	−11.0000	−0.423077
$677$	−15.5563	−0.597879	−0.298940	−	0.954272i	$-0.596633\pi$
$677$	−15.5563	−0.597879	−0.298940	+	0.954272i	$0.596633\pi$
$678$	0	0
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	0	0
$682$	−2.82843	−0.108306
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	−14.1421	−0.540343
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	0	0
$690$	0	0
$691$	−22.6274	−0.860788	−0.430394	−	0.902641i	$-0.641625\pi$
$691$	−22.6274	−0.860788	−0.430394	+	0.902641i	$0.641625\pi$
$692$	1.41421	0.0537603
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	10.0000	0.378777
$698$	18.3848	0.695874
$699$	0	0
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	22.6274	0.853409
$704$	1.00000	0.0376889
$705$	0	0
$706$	26.8701	1.01127
$707$	0	0
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	−11.3137	−0.424596
$711$	0	0
$712$	9.89949	0.370999
$713$	11.3137	0.423702
$714$	0	0
$715$	2.00000	0.0747958
$716$	16.0000	0.597948
$717$	0	0
$718$	4.00000	0.149279
$719$	−36.7696	−1.37127	−0.685636	−	0.727944i	$-0.740476\pi$
$719$	−36.7696	−1.37127	−0.685636	+	0.727944i	$0.740476\pi$
$720$	0	0
$721$	0	0
$722$	−11.0000	−0.409378
$723$	0	0
$724$	12.7279	0.473029
$725$	18.0000	0.668503
$726$	0	0
$727$	2.82843	0.104901	0.0524503	−	0.998624i	$-0.483297\pi$
$727$	2.82843	0.104901	0.0524503	+	0.998624i	$0.483297\pi$
$728$	0	0
$729$	0	0
$730$	2.00000	0.0740233
$731$	11.3137	0.418453
$732$	0	0
$733$	−29.6985	−1.09694	−0.548469	−	0.836171i	$-0.684789\pi$
$733$	−29.6985	−1.09694	−0.548469	+	0.836171i	$0.684789\pi$
$734$	−8.48528	−0.313197
$735$	0	0
$736$	−4.00000	−0.147442
$737$	−8.00000	−0.294684
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	−11.3137	−0.415900
$741$	0	0
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	5.65685	0.207251
$746$	10.0000	0.366126
$747$	0	0
$748$	−1.41421	−0.0517088
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−8.48528	−0.309426
$753$	0	0
$754$	−8.48528	−0.309016
$755$	22.6274	0.823496
$756$	0	0
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	−12.0000	−0.435860
$759$	0	0
$760$	−4.00000	−0.145095
$761$	12.7279	0.461387	0.230693	−	0.973026i	$-0.425901\pi$
$761$	12.7279	0.461387	0.230693	+	0.973026i	$0.425901\pi$
$762$	0	0
$763$	0	0
$764$	12.0000	0.434145
$765$	0	0
$766$	14.1421	0.510976
$767$	0	0
$768$	0	0
$769$	29.6985	1.07095	0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985	1.07095	0.535477	+	0.844550i	$0.320132\pi$
$770$	0	0
$771$	0	0
$772$	−18.0000	−0.647834
$773$	−18.3848	−0.661254	−0.330627	−	0.943761i	$-0.607260\pi$
$773$	−18.3848	−0.661254	−0.330627	+	0.943761i	$0.607260\pi$
$774$	0	0
$775$	8.48528	0.304800
$776$	15.5563	0.558440
$777$	0	0
$778$	10.0000	0.358517
$779$	20.0000	0.716574
$780$	0	0
$781$	−8.00000	−0.286263
$782$	5.65685	0.202289
$783$	0	0
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−42.4264	−1.51234	−0.756169	−	0.654376i	$-0.772931\pi$
$787$	−42.4264	−1.51234	−0.756169	+	0.654376i	$0.772931\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	5.65685	0.201262
$791$	0	0
$792$	0	0
$793$	−10.0000	−0.355110
$794$	−9.89949	−0.351320
$795$	0	0
$796$	25.4558	0.902258
$797$	−26.8701	−0.951786	−0.475893	−	0.879503i	$-0.657875\pi$
$797$	−26.8701	−0.951786	−0.475893	+	0.879503i	$0.657875\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	−3.00000	−0.106066
$801$	0	0
$802$	18.0000	0.635602
$803$	1.41421	0.0499065
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	15.5563	0.547270
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	8.48528	0.297959	0.148979	−	0.988840i	$-0.452401\pi$
$811$	8.48528	0.297959	0.148979	+	0.988840i	$0.452401\pi$
$812$	0	0
$813$	0	0
$814$	−8.00000	−0.280400
$815$	−5.65685	−0.198151
$816$	0	0
$817$	22.6274	0.791633
$818$	24.0416	0.840596
$819$	0	0
$820$	−10.0000	−0.349215
$821$	20.0000	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$822$	0	0
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	8.48528	0.295599
$825$	0	0
$826$	0	0
$827$	−52.0000	−1.80822	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000	−1.80822	−0.904109	+	0.427303i	$0.859464\pi$
$828$	0	0
$829$	9.89949	0.343824	0.171912	−	0.985112i	$-0.445006\pi$
$829$	9.89949	0.343824	0.171912	+	0.985112i	$0.445006\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	1.41421	0.0490290
$833$	0	0
$834$	0	0
$835$	−16.0000	−0.553703
$836$	−2.82843	−0.0978232
$837$	0	0
$838$	−16.9706	−0.586238
$839$	53.7401	1.85531	0.927657	−	0.373432i	$-0.121819\pi$
$839$	53.7401	1.85531	0.927657	+	0.373432i	$0.121819\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−22.0000	−0.758170
$843$	0	0
$844$	−8.00000	−0.275371
$845$	−15.5563	−0.535155
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	4.24264	0.145521
$851$	32.0000	1.09695
$852$	0	0
$853$	41.0122	1.40423	0.702115	−	0.712063i	$-0.252239\pi$
$853$	41.0122	1.40423	0.702115	+	0.712063i	$0.252239\pi$
$854$	0	0
$855$	0	0
$856$	8.00000	0.273434
$857$	12.7279	0.434778	0.217389	−	0.976085i	$-0.430246\pi$
$857$	12.7279	0.434778	0.217389	+	0.976085i	$0.430246\pi$
$858$	0	0
$859$	16.9706	0.579028	0.289514	−	0.957174i	$-0.406506\pi$
$859$	16.9706	0.579028	0.289514	+	0.957174i	$0.406506\pi$
$860$	−11.3137	−0.385794
$861$	0	0
$862$	−28.0000	−0.953684
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	4.24264	0.144171
$867$	0	0
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−11.3137	−0.383350
$872$	4.00000	0.135457
$873$	0	0
$874$	11.3137	0.382692
$875$	0	0
$876$	0	0
$877$	−52.0000	−1.75592	−0.877958	−	0.478738i	$-0.841094\pi$
$877$	−52.0000	−1.75592	−0.877958	+	0.478738i	$0.841094\pi$
$878$	−33.9411	−1.14546
$879$	0	0
$880$	1.41421	0.0476731
$881$	−55.1543	−1.85820	−0.929098	−	0.369833i	$-0.879415\pi$
$881$	−55.1543	−1.85820	−0.929098	+	0.369833i	$0.879415\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−45.2548	−1.51951	−0.759754	−	0.650210i	$-0.774681\pi$
$887$	−45.2548	−1.51951	−0.759754	+	0.650210i	$0.774681\pi$
$888$	0	0
$889$	0	0
$890$	14.0000	0.469281
$891$	0	0
$892$	2.82843	0.0947027
$893$	24.0000	0.803129
$894$	0	0
$895$	22.6274	0.756351
$896$	0	0
$897$	0	0
$898$	24.0000	0.800890
$899$	16.9706	0.566000
$900$	0	0
$901$	0	0
$902$	−7.07107	−0.235441
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	14.1421	0.469323
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	−2.82843	−0.0936073
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−21.2132	−0.700904
$917$	0	0
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	−5.65685	−0.186501
$921$	0	0
$922$	−26.8701	−0.884918
$923$	−11.3137	−0.372395
$924$	0	0
$925$	24.0000	0.789115
$926$	−32.0000	−1.05159
$927$	0	0
$928$	−6.00000	−0.196960
$929$	43.8406	1.43836	0.719182	−	0.694822i	$-0.244517\pi$
$929$	43.8406	1.43836	0.719182	+	0.694822i	$0.244517\pi$
$930$	0	0
$931$	0	0
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−16.9706	−0.555294
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	−46.6690	−1.52461	−0.762306	−	0.647217i	$-0.775933\pi$
$937$	−46.6690	−1.52461	−0.762306	+	0.647217i	$0.775933\pi$
$938$	0	0
$939$	0	0
$940$	−12.0000	−0.391397
$941$	−15.5563	−0.507122	−0.253561	−	0.967319i	$-0.581602\pi$
$941$	−15.5563	−0.507122	−0.253561	+	0.967319i	$0.581602\pi$
$942$	0	0
$943$	28.2843	0.921063
$944$	0	0
$945$	0	0
$946$	−8.00000	−0.260102
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	8.48528	0.275299
$951$	0	0
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	16.9706	0.549155
$956$	−8.00000	−0.258738
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	−11.3137	−0.364769
$963$	0	0
$964$	9.89949	0.318841
$965$	−25.4558	−0.819453
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	22.0000	0.706377
$971$	−28.2843	−0.907685	−0.453843	−	0.891082i	$-0.649947\pi$
$971$	−28.2843	−0.907685	−0.453843	+	0.891082i	$0.649947\pi$
$972$	0	0
$973$	0	0
$974$	28.0000	0.897178
$975$	0	0
$976$	−7.07107	−0.226339
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	9.89949	0.316389
$980$	0	0
$981$	0	0
$982$	20.0000	0.638226
$983$	2.82843	0.0902128	0.0451064	−	0.998982i	$-0.485637\pi$
$983$	2.82843	0.0902128	0.0451064	+	0.998982i	$0.485637\pi$
$984$	0	0
$985$	16.9706	0.540727
$986$	8.48528	0.270226
$987$	0	0
$988$	−4.00000	−0.127257
$989$	32.0000	1.01754
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−2.82843	−0.0898027
$993$	0	0
$994$	0	0
$995$	36.0000	1.14128
$996$	0	0
$997$	29.6985	0.940560	0.470280	−	0.882517i	$-0.344153\pi$
$997$	29.6985	0.940560	0.470280	+	0.882517i	$0.344153\pi$
$998$	−24.0000	−0.759707
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +1.41421 q^{10} +1.00000 q^{11} +1.41421 q^{13} +1.00000 q^{16} -1.41421 q^{17} -2.82843 q^{19} +1.41421 q^{20} +1.00000 q^{22} -4.00000 q^{23} -3.00000 q^{25} +1.41421 q^{26} -6.00000 q^{29} -2.82843 q^{31} +1.00000 q^{32} -1.41421 q^{34} -8.00000 q^{37} -2.82843 q^{38} +1.41421 q^{40} -7.07107 q^{41} -8.00000 q^{43} +1.00000 q^{44} -4.00000 q^{46} -8.48528 q^{47} -3.00000 q^{50} +1.41421 q^{52} +1.41421 q^{55} -6.00000 q^{58} -7.07107 q^{61} -2.82843 q^{62} +1.00000 q^{64} +2.00000 q^{65} -8.00000 q^{67} -1.41421 q^{68} -8.00000 q^{71} +1.41421 q^{73} -8.00000 q^{74} -2.82843 q^{76} +4.00000 q^{79} +1.41421 q^{80} -7.07107 q^{82} -2.82843 q^{83} -2.00000 q^{85} -8.00000 q^{86} +1.00000 q^{88} +9.89949 q^{89} -4.00000 q^{92} -8.48528 q^{94} -4.00000 q^{95} +15.5563 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{11} + 2 q^{16} + 2 q^{22} - 8 q^{23} - 6 q^{25} - 12 q^{29} + 2 q^{32} - 16 q^{37} - 16 q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{50} - 12 q^{58} + 2 q^{64} + 4 q^{65} - 16 q^{67} - 16 q^{71} - 16 q^{74} + 8 q^{79} - 4 q^{85} - 16 q^{86} + 2 q^{88} - 8 q^{92} - 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^11 + 2 * q^16 + 2 * q^22 - 8 * q^23 - 6 * q^25 - 12 * q^29 + 2 * q^32 - 16 * q^37 - 16 * q^43 + 2 * q^44 - 8 * q^46 - 6 * q^50 - 12 * q^58 + 2 * q^64 + 4 * q^65 - 16 * q^67 - 16 * q^71 - 16 * q^74 + 8 * q^79 - 4 * q^85 - 16 * q^86 + 2 * q^88 - 8 * q^92 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more