LMFDB - Embedded newform 354.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,2,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("354.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	354.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:	$2.82670423155$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -4.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{20} -4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -1.00000 q^{27} +4.00000 q^{29} -4.00000 q^{30} -10.0000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +4.00000 q^{40} -2.00000 q^{41} -12.0000 q^{43} -4.00000 q^{44} +4.00000 q^{45} +4.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +11.0000 q^{50} +2.00000 q^{51} -1.00000 q^{54} -16.0000 q^{55} -4.00000 q^{57} +4.00000 q^{58} -1.00000 q^{59} -4.00000 q^{60} +4.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +4.00000 q^{66} -8.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +6.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} -4.00000 q^{74} -11.0000 q^{75} +4.00000 q^{76} +8.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} -8.00000 q^{85} -12.0000 q^{86} -4.00000 q^{87} -4.00000 q^{88} -18.0000 q^{89} +4.00000 q^{90} +4.00000 q^{92} +10.0000 q^{93} +4.00000 q^{94} +16.0000 q^{95} -1.00000 q^{96} +14.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	−4.00000	−0.852803
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	−4.00000	−0.730297
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	4.00000	0.632456
$41$	−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
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	−4.00000	−0.603023
$45$	4.00000	0.596285
$46$	4.00000	0.589768
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	11.0000	1.55563
$51$	2.00000	0.280056
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−1.00000	−0.136083
$55$	−16.0000	−2.15744
$56$	0	0
$57$	−4.00000	−0.529813
$58$	4.00000	0.525226
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	−4.00000	−0.516398
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−2.00000	−0.242536
$69$	−4.00000	−0.481543
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−4.00000	−0.464991
$75$	−11.0000	−1.27017
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	−12.0000	−1.29399
$87$	−4.00000	−0.428845
$88$	−4.00000	−0.426401
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	4.00000	0.421637
$91$	0	0
$92$	4.00000	0.417029
$93$	10.0000	1.03695
$94$	4.00000	0.412568
$95$	16.0000	1.64157
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−7.00000	−0.707107
$99$	−4.00000	−0.402015
$100$	11.0000	1.10000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	2.00000	0.198030
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−1.00000	−0.0962250
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	−16.0000	−1.52554
$111$	4.00000	0.379663
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−4.00000	−0.374634
$115$	16.0000	1.49201
$116$	4.00000	0.371391
$117$	0	0
$118$	−1.00000	−0.0920575
$119$	0	0
$120$	−4.00000	−0.365148
$121$	5.00000	0.454545
$122$	4.00000	0.362143
$123$	2.00000	0.180334
$124$	−10.0000	−0.898027
$125$	24.0000	2.14663
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000	0.0883883
$129$	12.0000	1.05654
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−8.00000	−0.691095
$135$	−4.00000	−0.344265
$136$	−2.00000	−0.171499
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−4.00000	−0.340503
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	6.00000	0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	16.0000	1.32873
$146$	−14.0000	−1.15865
$147$	7.00000	0.577350
$148$	−4.00000	−0.328798
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	−11.0000	−0.898146
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	4.00000	0.324443
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−40.0000	−3.21288
$156$	0	0
$157$	−24.0000	−1.91541	−0.957704	−	0.287754i	$-0.907091\pi$
$157$	−24.0000	−1.91541	−0.957704	+	0.287754i	$0.907091\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	4.00000	0.316228
$161$	0	0
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−2.00000	−0.156174
$165$	16.0000	1.24560
$166$	−4.00000	−0.310460
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−8.00000	−0.613572
$171$	4.00000	0.305888
$172$	−12.0000	−0.914991
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−4.00000	−0.301511
$177$	1.00000	0.0751646
$178$	−18.0000	−1.34916
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	4.00000	0.298142
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	4.00000	0.294884
$185$	−16.0000	−1.17634
$186$	10.0000	0.733236
$187$	8.00000	0.585018
$188$	4.00000	0.291730
$189$	0	0
$190$	16.0000	1.16076
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	−4.00000	−0.284268
$199$	28.0000	1.98487	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	28.0000	1.98487	0.992434	+	0.122782i	$0.0391815\pi$
$200$	11.0000	0.777817
$201$	8.00000	0.564276
$202$	18.0000	1.26648
$203$	0	0
$204$	2.00000	0.140028
$205$	−8.00000	−0.558744
$206$	14.0000	0.975426
$207$	4.00000	0.278019
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	0	0
$215$	−48.0000	−3.27357
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−8.00000	−0.541828
$219$	14.0000	0.946032
$220$	−16.0000	−1.07872
$221$	0	0
$222$	4.00000	0.268462
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	2.00000	0.133038
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−4.00000	−0.264906
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	16.0000	1.05501
$231$	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
$235$	16.0000	1.04372
$236$	−1.00000	−0.0650945
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	−4.00000	−0.258199
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	4.00000	0.256074
$245$	−28.0000	−1.78885
$246$	2.00000	0.127515
$247$	0	0
$248$	−10.0000	−0.635001
$249$	4.00000	0.253490
$250$	24.0000	1.51789
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	−20.0000	−1.25491
$255$	8.00000	0.500979
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	12.0000	0.747087
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	12.0000	0.741362
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	0	0
$267$	18.0000	1.10158
$268$	−8.00000	−0.488678
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	−4.00000	−0.243432
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	6.00000	0.362473
$275$	−44.0000	−2.65330
$276$	−4.00000	−0.240772
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−4.00000	−0.239904
$279$	−10.0000	−0.598684
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	−4.00000	−0.238197
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	6.00000	0.356034
$285$	−16.0000	−0.947758
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	16.0000	0.939552
$291$	−14.0000	−0.820695
$292$	−14.0000	−0.819288
$293$	28.0000	1.63578	0.817889	−	0.575376i	$-0.195144\pi$
$293$	28.0000	1.63578	0.817889	+	0.575376i	$0.195144\pi$
$294$	7.00000	0.408248
$295$	−4.00000	−0.232889
$296$	−4.00000	−0.232495
$297$	4.00000	0.232104
$298$	−2.00000	−0.115857
$299$	0	0
$300$	−11.0000	−0.635085
$301$	0	0
$302$	2.00000	0.115087
$303$	−18.0000	−1.03407
$304$	4.00000	0.229416
$305$	16.0000	0.916157
$306$	−2.00000	−0.114332
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	−40.0000	−2.27185
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	−24.0000	−1.35440
$315$	0	0
$316$	8.00000	0.450035
$317$	−20.0000	−1.12331	−0.561656	−	0.827371i	$-0.689836\pi$
$317$	−20.0000	−1.12331	−0.561656	+	0.827371i	$0.689836\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	4.00000	0.223607
$321$	0	0
$322$	0	0
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	8.00000	0.442401
$328$	−2.00000	−0.110432
$329$	0	0
$330$	16.0000	0.880771
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−4.00000	−0.219529
$333$	−4.00000	−0.219199
$334$	−6.00000	−0.328305
$335$	−32.0000	−1.74835
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−13.0000	−0.707107
$339$	−2.00000	−0.108625
$340$	−8.00000	−0.433861
$341$	40.0000	2.16612
$342$	4.00000	0.216295
$343$	0	0
$344$	−12.0000	−0.646997
$345$	−16.0000	−0.861411
$346$	10.0000	0.537603
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−4.00000	−0.214423
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	0	0
$352$	−4.00000	−0.213201
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	1.00000	0.0531494
$355$	24.0000	1.27379
$356$	−18.0000	−0.953998
$357$	0	0
$358$	20.0000	1.05703
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	4.00000	0.210819
$361$	−3.00000	−0.157895
$362$	14.0000	0.735824
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−56.0000	−2.93117
$366$	−4.00000	−0.209083
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	4.00000	0.208514
$369$	−2.00000	−0.104116
$370$	−16.0000	−0.831800
$371$	0	0
$372$	10.0000	0.518476
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	8.00000	0.413670
$375$	−24.0000	−1.23935
$376$	4.00000	0.206284
$377$	0	0
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	16.0000	0.820783
$381$	20.0000	1.02463
$382$	12.0000	0.613973
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−12.0000	−0.609994
$388$	14.0000	0.710742
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	−7.00000	−0.353553
$393$	−12.0000	−0.605320
$394$	12.0000	0.604551
$395$	32.0000	1.61009
$396$	−4.00000	−0.201008
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	28.0000	1.40351
$399$	0	0
$400$	11.0000	0.550000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	8.00000	0.399004
$403$	0	0
$404$	18.0000	0.895533
$405$	4.00000	0.198762
$406$	0	0
$407$	16.0000	0.793091
$408$	2.00000	0.0990148
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	−8.00000	−0.395092
$411$	−6.00000	−0.295958
$412$	14.0000	0.689730
$413$	0	0
$414$	4.00000	0.196589
$415$	−16.0000	−0.785409
$416$	0	0
$417$	4.00000	0.195881
$418$	−16.0000	−0.782586
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−32.0000	−1.55958	−0.779792	−	0.626038i	$-0.784675\pi$
$421$	−32.0000	−1.55958	−0.779792	+	0.626038i	$0.784675\pi$
$422$	16.0000	0.778868
$423$	4.00000	0.194487
$424$	0	0
$425$	−22.0000	−1.06716
$426$	−6.00000	−0.290701
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−48.0000	−2.31477
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−1.00000	−0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	−16.0000	−0.767141
$436$	−8.00000	−0.383131
$437$	16.0000	0.765384
$438$	14.0000	0.668946
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	−16.0000	−0.762770
$441$	−7.00000	−0.333333
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	4.00000	0.189832
$445$	−72.0000	−3.41313
$446$	0	0
$447$	2.00000	0.0945968
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	11.0000	0.518545
$451$	8.00000	0.376705
$452$	2.00000	0.0940721
$453$	−2.00000	−0.0939682
$454$	−4.00000	−0.187729
$455$	0	0
$456$	−4.00000	−0.187317
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	20.0000	0.934539
$459$	2.00000	0.0933520
$460$	16.0000	0.746004
$461$	36.0000	1.67669	0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000	1.67669	0.838344	+	0.545142i	$0.183524\pi$
$462$	0	0
$463$	30.0000	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$463$	30.0000	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$464$	4.00000	0.185695
$465$	40.0000	1.85496
$466$	6.00000	0.277945
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	0	0
$470$	16.0000	0.738025
$471$	24.0000	1.10586
$472$	−1.00000	−0.0460287
$473$	48.0000	2.20704
$474$	−8.00000	−0.367452
$475$	44.0000	2.01886
$476$	0	0
$477$	0	0
$478$	−22.0000	−1.00626
$479$	−14.0000	−0.639676	−0.319838	−	0.947472i	$-0.603629\pi$
$479$	−14.0000	−0.639676	−0.319838	+	0.947472i	$0.603629\pi$
$480$	−4.00000	−0.182574
$481$	0	0
$482$	14.0000	0.637683
$483$	0	0
$484$	5.00000	0.227273
$485$	56.0000	2.54283
$486$	−1.00000	−0.0453609
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	4.00000	0.181071
$489$	−4.00000	−0.180886
$490$	−28.0000	−1.26491
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	2.00000	0.0901670
$493$	−8.00000	−0.360302
$494$	0	0
$495$	−16.0000	−0.719147
$496$	−10.0000	−0.449013
$497$	0	0
$498$	4.00000	0.179244
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	24.0000	1.07331
$501$	6.00000	0.268060
$502$	28.0000	1.24970
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	72.0000	3.20396
$506$	−16.0000	−0.711287
$507$	13.0000	0.577350
$508$	−20.0000	−0.887357
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	8.00000	0.354246
$511$	0	0
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	18.0000	0.793946
$515$	56.0000	2.46765
$516$	12.0000	0.528271
$517$	−16.0000	−0.703679
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−2.00000	−0.0876216	−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000	−0.0876216	−0.0438108	+	0.999040i	$0.513950\pi$
$522$	4.00000	0.175075
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−18.0000	−0.784837
$527$	20.0000	0.871214
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−1.00000	−0.0433963
$532$	0	0
$533$	0	0
$534$	18.0000	0.778936
$535$	0	0
$536$	−8.00000	−0.345547
$537$	−20.0000	−0.863064
$538$	−10.0000	−0.431131
$539$	28.0000	1.20605
$540$	−4.00000	−0.172133
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	8.00000	0.343629
$543$	−14.0000	−0.600798
$544$	−2.00000	−0.0857493
$545$	−32.0000	−1.37073
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	6.00000	0.256307
$549$	4.00000	0.170716
$550$	−44.0000	−1.87617
$551$	16.0000	0.681623
$552$	−4.00000	−0.170251
$553$	0	0
$554$	18.0000	0.764747
$555$	16.0000	0.679162
$556$	−4.00000	−0.169638
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	−10.0000	−0.423334
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	2.00000	0.0843649
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−4.00000	−0.168430
$565$	8.00000	0.336563
$566$	−4.00000	−0.168133
$567$	0	0
$568$	6.00000	0.251754
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	−16.0000	−0.670166
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	0	0
$575$	44.0000	1.83493
$576$	1.00000	0.0416667
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	−13.0000	−0.540729
$579$	14.0000	0.581820
$580$	16.0000	0.664364
$581$	0	0
$582$	−14.0000	−0.580319
$583$	0	0
$584$	−14.0000	−0.579324
$585$	0	0
$586$	28.0000	1.15667
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	7.00000	0.288675
$589$	−40.0000	−1.64817
$590$	−4.00000	−0.164677
$591$	−12.0000	−0.493614
$592$	−4.00000	−0.164399
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	−28.0000	−1.14596
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	−11.0000	−0.449073
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	2.00000	0.0813788
$605$	20.0000	0.813116
$606$	−18.0000	−0.731200
$607$	−36.0000	−1.46119	−0.730597	−	0.682808i	$-0.760758\pi$
$607$	−36.0000	−1.46119	−0.730597	+	0.682808i	$0.760758\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	16.0000	0.647821
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	−28.0000	−1.13091	−0.565455	−	0.824779i	$-0.691299\pi$
$613$	−28.0000	−1.13091	−0.565455	+	0.824779i	$0.691299\pi$
$614$	−20.0000	−0.807134
$615$	8.00000	0.322591
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	−14.0000	−0.563163
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	−40.0000	−1.60644
$621$	−4.00000	−0.160514
$622$	10.0000	0.400963
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	30.0000	1.19904
$627$	16.0000	0.638978
$628$	−24.0000	−0.957704
$629$	8.00000	0.318981
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	8.00000	0.318223
$633$	−16.0000	−0.635943
$634$	−20.0000	−0.794301
$635$	−80.0000	−3.17470
$636$	0	0
$637$	0	0
$638$	−16.0000	−0.633446
$639$	6.00000	0.237356
$640$	4.00000	0.158114
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	48.0000	1.89000
$646$	−8.00000	−0.314756
$647$	−46.0000	−1.80845	−0.904223	−	0.427060i	$-0.859549\pi$
$647$	−46.0000	−1.80845	−0.904223	+	0.427060i	$0.859549\pi$
$648$	1.00000	0.0392837
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	8.00000	0.312825
$655$	48.0000	1.87552
$656$	−2.00000	−0.0780869
$657$	−14.0000	−0.546192
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	16.0000	0.622799
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−4.00000	−0.154997
$667$	16.0000	0.619522
$668$	−6.00000	−0.232147
$669$	0	0
$670$	−32.0000	−1.23627
$671$	−16.0000	−0.617673
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	−14.0000	−0.539260
$675$	−11.0000	−0.423390
$676$	−13.0000	−0.500000
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	−8.00000	−0.306786
$681$	4.00000	0.153280
$682$	40.0000	1.53168
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	4.00000	0.152944
$685$	24.0000	0.916993
$686$	0	0
$687$	−20.0000	−0.763048
$688$	−12.0000	−0.457496
$689$	0	0
$690$	−16.0000	−0.609110
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−16.0000	−0.606915
$696$	−4.00000	−0.151620
$697$	4.00000	0.151511
$698$	28.0000	1.05982
$699$	−6.00000	−0.226941
$700$	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
$703$	−16.0000	−0.603451
$704$	−4.00000	−0.150756
$705$	−16.0000	−0.602595
$706$	−14.0000	−0.526897
$707$	0	0
$708$	1.00000	0.0375823
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	24.0000	0.900704
$711$	8.00000	0.300023
$712$	−18.0000	−0.674579
$713$	−40.0000	−1.49801
$714$	0	0
$715$	0	0
$716$	20.0000	0.747435
$717$	22.0000	0.821605
$718$	6.00000	0.223918
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	4.00000	0.149071
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−14.0000	−0.520666
$724$	14.0000	0.520306
$725$	44.0000	1.63412
$726$	−5.00000	−0.185567
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−56.0000	−2.07265
$731$	24.0000	0.887672
$732$	−4.00000	−0.147844
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	10.0000	0.369107
$735$	28.0000	1.03280
$736$	4.00000	0.147442
$737$	32.0000	1.17874
$738$	−2.00000	−0.0736210
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−16.0000	−0.588172
$741$	0	0
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	10.0000	0.366618
$745$	−8.00000	−0.293097
$746$	−26.0000	−0.951928
$747$	−4.00000	−0.146352
$748$	8.00000	0.292509
$749$	0	0
$750$	−24.0000	−0.876356
$751$	−6.00000	−0.218943	−0.109472	−	0.993990i	$-0.534916\pi$
$751$	−6.00000	−0.218943	−0.109472	+	0.993990i	$0.534916\pi$
$752$	4.00000	0.145865
$753$	−28.0000	−1.02038
$754$	0	0
$755$	8.00000	0.291150
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−4.00000	−0.145287
$759$	16.0000	0.580763
$760$	16.0000	0.580381
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	20.0000	0.724524
$763$	0	0
$764$	12.0000	0.434145
$765$	−8.00000	−0.289241
$766$	−34.0000	−1.22847
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−14.0000	−0.503871
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−12.0000	−0.431331
$775$	−110.000	−3.95132
$776$	14.0000	0.502571
$777$	0	0
$778$	20.0000	0.717035
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−24.0000	−0.858788
$782$	−8.00000	−0.286079
$783$	−4.00000	−0.142948
$784$	−7.00000	−0.250000
$785$	−96.0000	−3.42639
$786$	−12.0000	−0.428026
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	12.0000	0.427482
$789$	18.0000	0.640817
$790$	32.0000	1.13851
$791$	0	0
$792$	−4.00000	−0.142134
$793$	0	0
$794$	−8.00000	−0.283909
$795$	0	0
$796$	28.0000	0.992434
$797$	38.0000	1.34603	0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000	1.34603	0.673015	+	0.739629i	$0.264999\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	11.0000	0.388909
$801$	−18.0000	−0.635999
$802$	−18.0000	−0.635602
$803$	56.0000	1.97620
$804$	8.00000	0.282138
$805$	0	0
$806$	0	0
$807$	10.0000	0.352017
$808$	18.0000	0.633238
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	4.00000	0.140546
$811$	48.0000	1.68551	0.842754	−	0.538299i	$-0.180933\pi$
$811$	48.0000	1.68551	0.842754	+	0.538299i	$0.180933\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	16.0000	0.560800
$815$	16.0000	0.560456
$816$	2.00000	0.0700140
$817$	−48.0000	−1.67931
$818$	−26.0000	−0.909069
$819$	0	0
$820$	−8.00000	−0.279372
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	−6.00000	−0.209274
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	14.0000	0.487713
$825$	44.0000	1.53188
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	4.00000	0.139010
$829$	22.0000	0.764092	0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000	0.764092	0.382046	+	0.924143i	$0.375220\pi$
$830$	−16.0000	−0.555368
$831$	−18.0000	−0.624413
$832$	0	0
$833$	14.0000	0.485071
$834$	4.00000	0.138509
$835$	−24.0000	−0.830554
$836$	−16.0000	−0.553372
$837$	10.0000	0.345651
$838$	12.0000	0.414533
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−32.0000	−1.10279
$843$	−2.00000	−0.0688837
$844$	16.0000	0.550743
$845$	−52.0000	−1.78885
$846$	4.00000	0.137523
$847$	0	0
$848$	0	0
$849$	4.00000	0.137280
$850$	−22.0000	−0.754594
$851$	−16.0000	−0.548473
$852$	−6.00000	−0.205557
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	0	0
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	−48.0000	−1.63679
$861$	0	0
$862$	−12.0000	−0.408722
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	−1.00000	−0.0340207
$865$	40.0000	1.36004
$866$	−2.00000	−0.0679628
$867$	13.0000	0.441503
$868$	0	0
$869$	−32.0000	−1.08553
$870$	−16.0000	−0.542451
$871$	0	0
$872$	−8.00000	−0.270914
$873$	14.0000	0.473828
$874$	16.0000	0.541208
$875$	0	0
$876$	14.0000	0.473016
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−4.00000	−0.134993
$879$	−28.0000	−0.944417
$880$	−16.0000	−0.539360
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	−7.00000	−0.235702
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	12.0000	0.403148
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	4.00000	0.134231
$889$	0	0
$890$	−72.0000	−2.41345
$891$	−4.00000	−0.134005
$892$	0	0
$893$	16.0000	0.535420
$894$	2.00000	0.0668900
$895$	80.0000	2.67411
$896$	0	0
$897$	0	0
$898$	−10.0000	−0.333704
$899$	−40.0000	−1.33407
$900$	11.0000	0.366667
$901$	0	0
$902$	8.00000	0.266371
$903$	0	0
$904$	2.00000	0.0665190
$905$	56.0000	1.86150
$906$	−2.00000	−0.0664455
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−4.00000	−0.132745
$909$	18.0000	0.597022
$910$	0	0
$911$	38.0000	1.25900	0.629498	−	0.777002i	$-0.283261\pi$
$911$	38.0000	1.25900	0.629498	+	0.777002i	$0.283261\pi$
$912$	−4.00000	−0.132453
$913$	16.0000	0.529523
$914$	−34.0000	−1.12462
$915$	−16.0000	−0.528944
$916$	20.0000	0.660819
$917$	0	0
$918$	2.00000	0.0660098
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	16.0000	0.527504
$921$	20.0000	0.659022
$922$	36.0000	1.18560
$923$	0	0
$924$	0	0
$925$	−44.0000	−1.44671
$926$	30.0000	0.985861
$927$	14.0000	0.459820
$928$	4.00000	0.131306
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	40.0000	1.31165
$931$	−28.0000	−0.917663
$932$	6.00000	0.196537
$933$	−10.0000	−0.327385
$934$	28.0000	0.916188
$935$	32.0000	1.04651
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−30.0000	−0.979013
$940$	16.0000	0.521862
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	24.0000	0.781962
$943$	−8.00000	−0.260516
$944$	−1.00000	−0.0325472
$945$	0	0
$946$	48.0000	1.56061
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	44.0000	1.42755
$951$	20.0000	0.648544
$952$	0	0
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	0	0
$955$	48.0000	1.55324
$956$	−22.0000	−0.711531
$957$	16.0000	0.517207
$958$	−14.0000	−0.452319
$959$	0	0
$960$	−4.00000	−0.129099
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	14.0000	0.450910
$965$	−56.0000	−1.80270
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	5.00000	0.160706
$969$	8.00000	0.256997
$970$	56.0000	1.79805
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	4.00000	0.128168
$975$	0	0
$976$	4.00000	0.128037
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−4.00000	−0.127906
$979$	72.0000	2.30113
$980$	−28.0000	−0.894427
$981$	−8.00000	−0.255420
$982$	−8.00000	−0.255290
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	2.00000	0.0637577
$985$	48.0000	1.52941
$986$	−8.00000	−0.254772
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	−16.0000	−0.508513
$991$	−30.0000	−0.952981	−0.476491	−	0.879180i	$-0.658091\pi$
$991$	−30.0000	−0.952981	−0.476491	+	0.879180i	$0.658091\pi$
$992$	−10.0000	−0.317500
$993$	−12.0000	−0.380808
$994$	0	0
$995$	112.000	3.55064
$996$	4.00000	0.126745
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	12.0000	0.379853
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.2.a.f.1.1	✓	1
3.2	odd	2		1062.2.a.a.1.1		1
4.3	odd	2		2832.2.a.g.1.1		1
5.4	even	2		8850.2.a.j.1.1		1
12.11	even	2		8496.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.2.a.f.1.1	✓	1	1.1	even	1	trivial
1062.2.a.a.1.1		1	3.2	odd	2
2832.2.a.g.1.1		1	4.3	odd	2
8496.2.a.b.1.1		1	12.11	even	2
8850.2.a.j.1.1		1	5.4	even	2

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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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