LMFDB - Embedded newform 9702.2.a.cc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$77.4708600410$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
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			1.1
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{8} +2.00000 q^{10} +1.00000 q^{11} +4.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} +6.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} +2.00000 q^{23} -1.00000 q^{25} +4.00000 q^{26} +2.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} -2.00000 q^{37} +6.00000 q^{38} +2.00000 q^{40} -2.00000 q^{41} -2.00000 q^{43} +1.00000 q^{44} +2.00000 q^{46} +2.00000 q^{47} -1.00000 q^{50} +4.00000 q^{52} -6.00000 q^{53} +2.00000 q^{55} +2.00000 q^{58} -4.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +8.00000 q^{65} +2.00000 q^{68} -10.0000 q^{71} +2.00000 q^{73} -2.00000 q^{74} +6.00000 q^{76} -12.0000 q^{79} +2.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} +4.00000 q^{85} -2.00000 q^{86} +1.00000 q^{88} +12.0000 q^{89} +2.00000 q^{92} +2.00000 q^{94} +12.0000 q^{95} +2.00000 q^{97} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	1.00000	0.213201
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	4.00000	0.784465
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	2.00000	0.316228
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	2.00000	0.294884
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	0	0
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.00000	0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	2.00000	0.262613
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	6.00000	0.688247
$77$	0	0
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−2.00000	−0.215666
$87$	0	0
$88$	1.00000	0.106600
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	2.00000	0.208514
$93$	0	0
$94$	2.00000	0.206284
$95$	12.0000	1.23117
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	2.00000	0.185695
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.00000	−0.362143
$123$	0	0
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	8.00000	0.701646
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	2.00000	0.171499
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	−10.0000	−0.839181
$143$	4.00000	0.334497
$144$	0	0
$145$	4.00000	0.332182
$146$	2.00000	0.165521
$147$	0	0
$148$	−2.00000	−0.164399
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	0	0
$155$	16.0000	1.28515
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	2.00000	0.158114
$161$	0	0
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	4.00000	0.306786
$171$	0	0
$172$	−2.00000	−0.152499
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	12.0000	0.899438
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	0	0
$184$	2.00000	0.147442
$185$	−4.00000	−0.294086
$186$	0	0
$187$	2.00000	0.146254
$188$	2.00000	0.145865
$189$	0	0
$190$	12.0000	0.870572
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	0	0
$208$	4.00000	0.277350
$209$	6.00000	0.415029
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	4.00000	0.273434
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	−8.00000	−0.541828
$219$	0	0
$220$	2.00000	0.134840
$221$	8.00000	0.538138
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	4.00000	0.266076
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	2.00000	0.131306
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−28.0000	−1.81117	−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000	−1.81117	−0.905585	+	0.424165i	$0.860568\pi$
$240$	0	0
$241$	30.0000	1.93247	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000	1.93247	0.966235	+	0.257663i	$0.0829523\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	24.0000	1.52708
$248$	8.00000	0.508001
$249$	0	0
$250$	−12.0000	−0.758947
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	0	0
$260$	8.00000	0.496139
$261$	0	0
$262$	−12.0000	−0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	−2.00000	−0.119952
$279$	0	0
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	4.00000	0.234888
$291$	0	0
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	22.0000	1.27443
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	−12.0000	−0.690522
$303$	0	0
$304$	6.00000	0.344124
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	0	0
$310$	16.0000	0.908739
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	2.00000	0.111803
$321$	0	0
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	−4.00000	−0.221880
$326$	−8.00000	−0.443079
$327$	0	0
$328$	−2.00000	−0.110432
$329$	0	0
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	−8.00000	−0.437741
$335$	0	0
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	4.00000	0.216930
$341$	8.00000	0.433224
$342$	0	0
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	22.0000	1.18273
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	−36.0000	−1.91609	−0.958043	−	0.286623i	$-0.907467\pi$
$353$	−36.0000	−1.91609	−0.958043	+	0.286623i	$0.907467\pi$
$354$	0	0
$355$	−20.0000	−1.06149
$356$	12.0000	0.635999
$357$	0	0
$358$	20.0000	1.05703
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−14.0000	−0.735824
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	2.00000	0.104257
$369$	0	0
$370$	−4.00000	−0.207950
$371$	0	0
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	2.00000	0.103142
$377$	8.00000	0.412021
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	12.0000	0.615587
$381$	0	0
$382$	10.0000	0.511645
$383$	2.00000	0.102195	0.0510976	−	0.998694i	$-0.483728\pi$
$383$	2.00000	0.102195	0.0510976	+	0.998694i	$0.483728\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	2.00000	0.101535
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	0	0
$394$	−18.0000	−0.906827
$395$	−24.0000	−1.20757
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−24.0000	−1.17811
$416$	4.00000	0.196116
$417$	0	0
$418$	6.00000	0.293470
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	10.0000	0.486792
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	4.00000	0.193347
$429$	0	0
$430$	−4.00000	−0.192897
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	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
$437$	12.0000	0.574038
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	8.00000	0.380521
$443$	−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
$445$	24.0000	1.13771
$446$	−16.0000	−0.757622
$447$	0	0
$448$	0	0
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	4.00000	0.188144
$453$	0	0
$454$	−20.0000	−0.938647
$455$	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	$0.672044\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	4.00000	0.186501
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−18.0000	−0.833834
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	0	0
$470$	4.00000	0.184506
$471$	0	0
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	0	0
$478$	−28.0000	−1.28069
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	30.0000	1.36646
$483$	0	0
$484$	1.00000	0.0454545
$485$	4.00000	0.181631
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−4.00000	−0.181071
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	24.0000	1.07981
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	12.0000	0.535586
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	2.00000	0.0889108
$507$	0	0
$508$	20.0000	0.887357
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	12.0000	0.529297
$515$	0	0
$516$	0	0
$517$	2.00000	0.0879599
$518$	0	0
$519$	0	0
$520$	8.00000	0.350823
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	0	0
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	16.0000	0.696971
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−12.0000	−0.521247
$531$	0	0
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	0	0
$538$	6.00000	0.258678
$539$	0	0
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	20.0000	0.859074
$543$	0	0
$544$	2.00000	0.0857493
$545$	−16.0000	−0.685365
$546$	0	0
$547$	18.0000	0.769624	0.384812	−	0.922995i	$-0.374266\pi$
$547$	18.0000	0.769624	0.384812	+	0.922995i	$0.374266\pi$
$548$	0	0
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	12.0000	0.509831
$555$	0	0
$556$	−2.00000	−0.0848189
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	−2.00000	−0.0843649
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	8.00000	0.336563
$566$	6.00000	0.252199
$567$	0	0
$568$	−10.0000	−0.419591
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	−2.00000	−0.0834058
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	4.00000	0.166091
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	2.00000	0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	−48.0000	−1.98117	−0.990586	−	0.136892i	$-0.956289\pi$
$587$	−48.0000	−1.98117	−0.990586	+	0.136892i	$0.956289\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	0	0
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	22.0000	0.901155
$597$	0	0
$598$	8.00000	0.327144
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	0	0
$604$	−12.0000	−0.488273
$605$	2.00000	0.0813116
$606$	0	0
$607$	4.00000	0.162355	0.0811775	−	0.996700i	$-0.474132\pi$
$607$	4.00000	0.162355	0.0811775	+	0.996700i	$0.474132\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	−8.00000	−0.323911
$611$	8.00000	0.323645
$612$	0	0
$613$	−8.00000	−0.323117	−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000	−0.323117	−0.161558	+	0.986863i	$0.551652\pi$
$614$	−26.0000	−1.04927
$615$	0	0
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	16.0000	0.642575
$621$	0	0
$622$	2.00000	0.0801927
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	2.00000	0.0798087
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	−12.0000	−0.477334
$633$	0	0
$634$	−30.0000	−1.19145
$635$	40.0000	1.58735
$636$	0	0
$637$	0	0
$638$	2.00000	0.0791808
$639$	0	0
$640$	2.00000	0.0790569
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	0	0
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	16.0000	0.621858
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	4.00000	0.154881
$668$	−8.00000	−0.309529
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	30.0000	1.15556
$675$	0	0
$676$	3.00000	0.115385
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	0	0
$679$	0	0
$680$	4.00000	0.153393
$681$	0	0
$682$	8.00000	0.306336
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−2.00000	−0.0762493
$689$	−24.0000	−0.914327
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	22.0000	0.836315
$693$	0	0
$694$	4.00000	0.151838
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−4.00000	−0.151511
$698$	−8.00000	−0.302804
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	1.00000	0.0376889
$705$	0	0
$706$	−36.0000	−1.35488
$707$	0	0
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	−20.0000	−0.750587
$711$	0	0
$712$	12.0000	0.449719
$713$	16.0000	0.599205
$714$	0	0
$715$	8.00000	0.299183
$716$	20.0000	0.747435
$717$	0	0
$718$	−32.0000	−1.19423
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	0	0
$721$	0	0
$722$	17.0000	0.632674
$723$	0	0
$724$	−14.0000	−0.520306
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	0	0
$730$	4.00000	0.148047
$731$	−4.00000	−0.147945
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	2.00000	0.0737210
$737$	0	0
$738$	0	0
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	44.0000	1.61204
$746$	−4.00000	−0.146450
$747$	0	0
$748$	2.00000	0.0731272
$749$	0	0
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	2.00000	0.0729325
$753$	0	0
$754$	8.00000	0.291343
$755$	−24.0000	−0.873449
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	12.0000	0.435286
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	0	0
$764$	10.0000	0.361787
$765$	0	0
$766$	2.00000	0.0722629
$767$	0	0
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	14.0000	0.503871
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	2.00000	0.0717958
$777$	0	0
$778$	−18.0000	−0.645331
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−10.0000	−0.357828
$782$	4.00000	0.143040
$783$	0	0
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	−24.0000	−0.853882
$791$	0	0
$792$	0	0
$793$	−16.0000	−0.568177
$794$	18.0000	0.638796
$795$	0	0
$796$	8.00000	0.283552
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	0	0
$803$	2.00000	0.0705785
$804$	0	0
$805$	0	0
$806$	32.0000	1.12715
$807$	0	0
$808$	−6.00000	−0.211079
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	0	0
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	−16.0000	−0.560456
$816$	0	0
$817$	−12.0000	−0.419827
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−24.0000	−0.833052
$831$	0	0
$832$	4.00000	0.138675
$833$	0	0
$834$	0	0
$835$	−16.0000	−0.553703
$836$	6.00000	0.207514
$837$	0	0
$838$	4.00000	0.138178
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−26.0000	−0.896019
$843$	0	0
$844$	10.0000	0.344214
$845$	6.00000	0.206406
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	−4.00000	−0.137118
$852$	0	0
$853$	52.0000	1.78045	0.890223	−	0.455525i	$-0.150548\pi$
$853$	52.0000	1.78045	0.890223	+	0.455525i	$0.150548\pi$
$854$	0	0
$855$	0	0
$856$	4.00000	0.136717
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	32.0000	1.08992
$863$	−10.0000	−0.340404	−0.170202	−	0.985409i	$-0.554442\pi$
$863$	−10.0000	−0.340404	−0.170202	+	0.985409i	$0.554442\pi$
$864$	0	0
$865$	44.0000	1.49604
$866$	34.0000	1.15537
$867$	0	0
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	0	0
$872$	−8.00000	−0.270914
$873$	0	0
$874$	12.0000	0.405906
$875$	0	0
$876$	0	0
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	2.00000	0.0674200
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−32.0000	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$883$	−32.0000	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−20.0000	−0.671534	−0.335767	−	0.941945i	$-0.608996\pi$
$887$	−20.0000	−0.671534	−0.335767	+	0.941945i	$0.608996\pi$
$888$	0	0
$889$	0	0
$890$	24.0000	0.804482
$891$	0	0
$892$	−16.0000	−0.535720
$893$	12.0000	0.401565
$894$	0	0
$895$	40.0000	1.33705
$896$	0	0
$897$	0	0
$898$	−36.0000	−1.20134
$899$	16.0000	0.533630
$900$	0	0
$901$	−12.0000	−0.399778
$902$	−2.00000	−0.0665927
$903$	0	0
$904$	4.00000	0.133038
$905$	−28.0000	−0.930751
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	−22.0000	−0.727695
$915$	0	0
$916$	10.0000	0.330409
$917$	0	0
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	4.00000	0.131876
$921$	0	0
$922$	−18.0000	−0.592798
$923$	−40.0000	−1.31662
$924$	0	0
$925$	2.00000	0.0657596
$926$	16.0000	0.525793
$927$	0	0
$928$	2.00000	0.0656532
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	0	0
$932$	−18.0000	−0.589610
$933$	0	0
$934$	12.0000	0.392652
$935$	4.00000	0.130814
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	0	0
$940$	4.00000	0.130466
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	−4.00000	−0.130258
$944$	0	0
$945$	0	0
$946$	−2.00000	−0.0650256
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	−6.00000	−0.194666
$951$	0	0
$952$	0	0
$953$	−38.0000	−1.23094	−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000	−1.23094	−0.615470	+	0.788160i	$0.711034\pi$
$954$	0	0
$955$	20.0000	0.647185
$956$	−28.0000	−0.905585
$957$	0	0
$958$	20.0000	0.646171
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	−8.00000	−0.257930
$963$	0	0
$964$	30.0000	0.966235
$965$	28.0000	0.901352
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	4.00000	0.128432
$971$	−16.0000	−0.513464	−0.256732	−	0.966483i	$-0.582646\pi$
$971$	−16.0000	−0.513464	−0.256732	+	0.966483i	$0.582646\pi$
$972$	0	0
$973$	0	0
$974$	32.0000	1.02535
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	12.0000	0.382935
$983$	42.0000	1.33959	0.669796	−	0.742545i	$-0.266382\pi$
$983$	42.0000	1.33959	0.669796	+	0.742545i	$0.266382\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	4.00000	0.127386
$987$	0	0
$988$	24.0000	0.763542
$989$	−4.00000	−0.127193
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−48.0000	−1.52018	−0.760088	−	0.649821i	$-0.774844\pi$
$997$	−48.0000	−1.52018	−0.760088	+	0.649821i	$0.774844\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.cc.1.1		1
3.2	odd	2		9702.2.a.h.1.1		1
7.6	odd	2		1386.2.a.h.1.1	yes	1
21.20	even	2		1386.2.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.a.d.1.1	✓	1	21.20	even	2
1386.2.a.h.1.1	yes	1	7.6	odd	2
9702.2.a.h.1.1		1	3.2	odd	2
9702.2.a.cc.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more