LMFDB - Embedded newform 354.2.a.b.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)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} -2.00000 q^{40} +2.00000 q^{41} +4.00000 q^{44} +2.00000 q^{45} -8.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -6.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} -4.00000 q^{57} -2.00000 q^{58} +1.00000 q^{59} -2.00000 q^{60} +10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +4.00000 q^{66} -8.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} -14.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -6.00000 q^{78} -16.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.00000 q^{83} +4.00000 q^{85} -2.00000 q^{87} -4.00000 q^{88} +6.00000 q^{89} -2.00000 q^{90} +8.00000 q^{92} -8.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	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
$10$	−2.00000	−0.632456
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	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	$0.422021\pi$
$18$	−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	$0.348268\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	6.00000	1.17670
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	2.00000	0.365148
$31$	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
$33$	−4.00000	−0.696311
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.00000	−0.648886
$39$	6.00000	0.960769
$40$	−2.00000	−0.316228
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	4.00000	0.603023
$45$	2.00000	0.298142
$46$	−8.00000	−1.17954
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	−6.00000	−0.832050
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	8.00000	1.07872
$56$	0	0
$57$	−4.00000	−0.529813
$58$	−2.00000	−0.262613
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	−2.00000	−0.258199
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	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
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−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	$0.805631\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	−6.00000	−0.679366
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	−2.00000	−0.214423
$88$	−4.00000	−0.426401
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	8.00000	0.834058
$93$	−8.00000	−0.829561
$94$	−8.00000	−0.825137
$95$	8.00000	0.820783
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	7.00000	0.707107
$99$	4.00000	0.402015
$100$	−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	$0.596490\pi$
$102$	2.00000	0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−8.00000	−0.762770
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	4.00000	0.374634
$115$	16.0000	1.49201
$116$	2.00000	0.185695
$117$	−6.00000	−0.554700
$118$	−1.00000	−0.0920575
$119$	0	0
$120$	2.00000	0.182574
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	−2.00000	−0.180334
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	12.0000	1.05247
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	8.00000	0.691095
$135$	−2.00000	−0.172133
$136$	−2.00000	−0.171499
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	8.00000	0.681005
$139$	−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
$141$	−8.00000	−0.673722
$142$	12.0000	1.00702
$143$	−24.0000	−2.00698
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	14.0000	1.15865
$147$	7.00000	0.577350
$148$	2.00000	0.164399
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	−1.00000	−0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−4.00000	−0.324443
$153$	2.00000	0.161690
$154$	0	0
$155$	16.0000	1.28515
$156$	6.00000	0.480384
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	16.0000	1.27289
$159$	6.00000	0.475831
$160$	−2.00000	−0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	2.00000	0.156174
$165$	−8.00000	−0.622799
$166$	−4.00000	−0.310460
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−4.00000	−0.306786
$171$	4.00000	0.305888
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	4.00000	0.301511
$177$	−1.00000	−0.0751646
$178$	−6.00000	−0.449719
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	2.00000	0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	−8.00000	−0.589768
$185$	4.00000	0.294086
$186$	8.00000	0.586588
$187$	8.00000	0.585018
$188$	8.00000	0.583460
$189$	0	0
$190$	−8.00000	−0.580381
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−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	$0.668091\pi$
$194$	−2.00000	−0.143592
$195$	12.0000	0.859338
$196$	−7.00000	−0.500000
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−4.00000	−0.284268
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	8.00000	0.564276
$202$	6.00000	0.422159
$203$	0	0
$204$	−2.00000	−0.140028
$205$	4.00000	0.279372
$206$	−8.00000	−0.557386
$207$	8.00000	0.556038
$208$	−6.00000	−0.416025
$209$	16.0000	1.10674
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−6.00000	−0.412082
$213$	12.0000	0.822226
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	14.0000	0.946032
$220$	8.00000	0.539360
$221$	−12.0000	−0.807207
$222$	2.00000	0.134231
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	2.00000	0.133038
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	−4.00000	−0.264906
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−16.0000	−1.05501
$231$	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	$0.700718\pi$
$234$	6.00000	0.392232
$235$	16.0000	1.04372
$236$	1.00000	0.0650945
$237$	16.0000	1.03931
$238$	0	0
$239$	28.0000	1.81117	0.905585	−	0.424165i	$-0.139432\pi$
$239$	28.0000	1.81117	0.905585	+	0.424165i	$0.139432\pi$
$240$	−2.00000	−0.129099
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	−14.0000	−0.894427
$246$	2.00000	0.127515
$247$	−24.0000	−1.52708
$248$	−8.00000	−0.508001
$249$	−4.00000	−0.253490
$250$	12.0000	0.758947
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	32.0000	2.01182
$254$	8.00000	0.501965
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	0	0
$260$	−12.0000	−0.744208
$261$	2.00000	0.123797
$262$	12.0000	0.741362
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	4.00000	0.246183
$265$	−12.0000	−0.737154
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−8.00000	−0.488678
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	2.00000	0.121716
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−18.0000	−1.08742
$275$	−4.00000	−0.241209
$276$	−8.00000	−0.481543
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	4.00000	0.239904
$279$	8.00000	0.478947
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	8.00000	0.476393
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−12.0000	−0.712069
$285$	−8.00000	−0.473879
$286$	24.0000	1.41915
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	−2.00000	−0.117242
$292$	−14.0000	−0.819288
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	−7.00000	−0.408248
$295$	2.00000	0.116445
$296$	−2.00000	−0.116248
$297$	−4.00000	−0.232104
$298$	22.0000	1.27443
$299$	−48.0000	−2.77591
$300$	1.00000	0.0577350
$301$	0	0
$302$	−8.00000	−0.460348
$303$	6.00000	0.344691
$304$	4.00000	0.229416
$305$	20.0000	1.14520
$306$	−2.00000	−0.114332
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	−16.0000	−0.908739
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	−6.00000	−0.339683
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	−18.0000	−1.01580
$315$	0	0
$316$	−16.0000	−0.900070
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−6.00000	−0.336463
$319$	8.00000	0.447914
$320$	2.00000	0.111803
$321$	12.0000	0.669775
$322$	0	0
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	6.00000	0.332820
$326$	20.0000	1.10770
$327$	−10.0000	−0.553001
$328$	−2.00000	−0.110432
$329$	0	0
$330$	8.00000	0.440386
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	4.00000	0.219529
$333$	2.00000	0.109599
$334$	−12.0000	−0.656611
$335$	−16.0000	−0.874173
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−23.0000	−1.25104
$339$	2.00000	0.108625
$340$	4.00000	0.216930
$341$	32.0000	1.73290
$342$	−4.00000	−0.216295
$343$	0	0
$344$	0	0
$345$	−16.0000	−0.861411
$346$	−2.00000	−0.107521
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	−2.00000	−0.107211
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	−4.00000	−0.213201
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	1.00000	0.0531494
$355$	−24.0000	−1.27379
$356$	6.00000	0.317999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	10.0000	0.525588
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−28.0000	−1.46559
$366$	10.0000	0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	8.00000	0.417029
$369$	2.00000	0.104116
$370$	−4.00000	−0.207950
$371$	0	0
$372$	−8.00000	−0.414781
$373$	−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
$375$	12.0000	0.619677
$376$	−8.00000	−0.412568
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	8.00000	0.410391
$381$	8.00000	0.409852
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	2.00000	0.101535
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	−12.0000	−0.607644
$391$	16.0000	0.809155
$392$	7.00000	0.353553
$393$	12.0000	0.605320
$394$	6.00000	0.302276
$395$	−32.0000	−1.61009
$396$	4.00000	0.201008
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	−8.00000	−0.399004
$403$	−48.0000	−2.39105
$404$	−6.00000	−0.298511
$405$	2.00000	0.0993808
$406$	0	0
$407$	8.00000	0.396545
$408$	2.00000	0.0990148
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−4.00000	−0.197546
$411$	−18.0000	−0.887875
$412$	8.00000	0.394132
$413$	0	0
$414$	−8.00000	−0.393179
$415$	8.00000	0.392705
$416$	6.00000	0.294174
$417$	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
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	8.00000	0.389434
$423$	8.00000	0.388973
$424$	6.00000	0.291386
$425$	−2.00000	−0.0970143
$426$	−12.0000	−0.581402
$427$	0	0
$428$	−12.0000	−0.580042
$429$	24.0000	1.15873
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−1.00000	−0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	10.0000	0.478913
$437$	32.0000	1.53077
$438$	−14.0000	−0.668946
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−8.00000	−0.381385
$441$	−7.00000	−0.333333
$442$	12.0000	0.570782
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	−2.00000	−0.0949158
$445$	12.0000	0.568855
$446$	0	0
$447$	22.0000	1.04056
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	1.00000	0.0471405
$451$	8.00000	0.376705
$452$	−2.00000	−0.0940721
$453$	−8.00000	−0.375873
$454$	−4.00000	−0.187729
$455$	0	0
$456$	4.00000	0.187317
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	22.0000	1.02799
$459$	−2.00000	−0.0933520
$460$	16.0000	0.746004
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	2.00000	0.0928477
$465$	−16.0000	−0.741982
$466$	18.0000	0.833834
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	−16.0000	−0.738025
$471$	−18.0000	−0.829396
$472$	−1.00000	−0.0460287
$473$	0	0
$474$	−16.0000	−0.734904
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−28.0000	−1.28069
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	2.00000	0.0912871
$481$	−12.0000	−0.547153
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	5.00000	0.227273
$485$	4.00000	0.181631
$486$	1.00000	0.0453609
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	−10.0000	−0.452679
$489$	20.0000	0.904431
$490$	14.0000	0.632456
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−2.00000	−0.0901670
$493$	4.00000	0.180151
$494$	24.0000	1.07981
$495$	8.00000	0.359573
$496$	8.00000	0.359211
$497$	0	0
$498$	4.00000	0.179244
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	−12.0000	−0.536656
$501$	−12.0000	−0.536120
$502$	4.00000	0.178529
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	−32.0000	−1.42257
$507$	−23.0000	−1.02147
$508$	−8.00000	−0.354943
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	6.00000	0.264649
$515$	16.0000	0.705044
$516$	0	0
$517$	32.0000	1.40736
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	12.0000	0.526235
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	−2.00000	−0.0875376
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−12.0000	−0.523225
$527$	16.0000	0.696971
$528$	−4.00000	−0.174078
$529$	41.0000	1.78261
$530$	12.0000	0.521247
$531$	1.00000	0.0433963
$532$	0	0
$533$	−12.0000	−0.519778
$534$	6.00000	0.259645
$535$	−24.0000	−1.03761
$536$	8.00000	0.345547
$537$	−4.00000	−0.172613
$538$	14.0000	0.603583
$539$	−28.0000	−1.20605
$540$	−2.00000	−0.0860663
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	16.0000	0.687259
$543$	10.0000	0.429141
$544$	−2.00000	−0.0857493
$545$	20.0000	0.856706
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	18.0000	0.768922
$549$	10.0000	0.426790
$550$	4.00000	0.170561
$551$	8.00000	0.340811
$552$	8.00000	0.340503
$553$	0	0
$554$	18.0000	0.764747
$555$	−4.00000	−0.169791
$556$	−4.00000	−0.169638
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	−8.00000	−0.338667
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	−10.0000	−0.421825
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−8.00000	−0.336861
$565$	−4.00000	−0.168281
$566$	16.0000	0.672530
$567$	0	0
$568$	12.0000	0.503509
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	8.00000	0.335083
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	−24.0000	−1.00349
$573$	0	0
$574$	0	0
$575$	−8.00000	−0.333623
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	13.0000	0.540729
$579$	14.0000	0.581820
$580$	4.00000	0.166091
$581$	0	0
$582$	2.00000	0.0829027
$583$	−24.0000	−0.993978
$584$	14.0000	0.579324
$585$	−12.0000	−0.496139
$586$	22.0000	0.908812
$587$	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
$589$	32.0000	1.31854
$590$	−2.00000	−0.0823387
$591$	6.00000	0.246807
$592$	2.00000	0.0821995
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−22.0000	−0.901155
$597$	−16.0000	−0.654836
$598$	48.0000	1.96287
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−1.00000	−0.0408248
$601$	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
$604$	8.00000	0.325515
$605$	10.0000	0.406558
$606$	−6.00000	−0.243733
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−20.0000	−0.809776
$611$	−48.0000	−1.94187
$612$	2.00000	0.0808452
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	−28.0000	−1.12999
$615$	−4.00000	−0.161296
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	8.00000	0.321807
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	16.0000	0.642575
$621$	−8.00000	−0.321029
$622$	−20.0000	−0.801927
$623$	0	0
$624$	6.00000	0.240192
$625$	−19.0000	−0.760000
$626$	−18.0000	−0.719425
$627$	−16.0000	−0.638978
$628$	18.0000	0.718278
$629$	4.00000	0.159490
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	16.0000	0.636446
$633$	8.00000	0.317971
$634$	−2.00000	−0.0794301
$635$	−16.0000	−0.634941
$636$	6.00000	0.237915
$637$	42.0000	1.66410
$638$	−8.00000	−0.316723
$639$	−12.0000	−0.474713
$640$	−2.00000	−0.0790569
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−12.0000	−0.473602
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	0	0
$646$	−8.00000	−0.314756
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	−1.00000	−0.0392837
$649$	4.00000	0.157014
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−20.0000	−0.783260
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	10.0000	0.391031
$655$	−24.0000	−0.937758
$656$	2.00000	0.0780869
$657$	−14.0000	−0.546192
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	−8.00000	−0.311400
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	12.0000	0.466393
$663$	12.0000	0.466041
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	16.0000	0.619522
$668$	12.0000	0.464294
$669$	0	0
$670$	16.0000	0.618134
$671$	40.0000	1.54418
$672$	0	0
$673$	42.0000	1.61898	0.809491	−	0.587133i	$-0.199743\pi$
$673$	42.0000	1.61898	0.809491	+	0.587133i	$0.199743\pi$
$674$	−10.0000	−0.385186
$675$	1.00000	0.0384900
$676$	23.0000	0.884615
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	−4.00000	−0.153393
$681$	−4.00000	−0.153280
$682$	−32.0000	−1.22534
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	4.00000	0.152944
$685$	36.0000	1.37549
$686$	0	0
$687$	22.0000	0.839352
$688$	0	0
$689$	36.0000	1.37149
$690$	16.0000	0.609110
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	20.0000	0.759190
$695$	−8.00000	−0.303457
$696$	2.00000	0.0758098
$697$	4.00000	0.151511
$698$	−10.0000	−0.378506
$699$	18.0000	0.680823
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	−6.00000	−0.226455
$703$	8.00000	0.301726
$704$	4.00000	0.150756
$705$	−16.0000	−0.602595
$706$	34.0000	1.27961
$707$	0	0
$708$	−1.00000	−0.0375823
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	24.0000	0.900704
$711$	−16.0000	−0.600047
$712$	−6.00000	−0.224860
$713$	64.0000	2.39682
$714$	0	0
$715$	−48.0000	−1.79510
$716$	4.00000	0.149487
$717$	−28.0000	−1.04568
$718$	−36.0000	−1.34351
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	3.00000	0.111648
$723$	−2.00000	−0.0743808
$724$	−10.0000	−0.371647
$725$	−2.00000	−0.0742781
$726$	5.00000	0.185567
$727$	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
$729$	1.00000	0.0370370
$730$	28.0000	1.03633
$731$	0	0
$732$	−10.0000	−0.369611
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	8.00000	0.295285
$735$	14.0000	0.516398
$736$	−8.00000	−0.294884
$737$	−32.0000	−1.17874
$738$	−2.00000	−0.0736210
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	4.00000	0.147043
$741$	24.0000	0.881662
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	8.00000	0.293294
$745$	−44.0000	−1.61204
$746$	26.0000	0.951928
$747$	4.00000	0.146352
$748$	8.00000	0.292509
$749$	0	0
$750$	−12.0000	−0.438178
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	8.00000	0.291730
$753$	4.00000	0.145768
$754$	12.0000	0.437014
$755$	16.0000	0.582300
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	28.0000	1.01701
$759$	−32.0000	−1.16153
$760$	−8.00000	−0.290191
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	0	0
$765$	4.00000	0.144620
$766$	−4.00000	−0.144526
$767$	−6.00000	−0.216647
$768$	−1.00000	−0.0360844
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−14.0000	−0.503871
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−34.0000	−1.21896
$779$	8.00000	0.286630
$780$	12.0000	0.429669
$781$	−48.0000	−1.71758
$782$	−16.0000	−0.572159
$783$	−2.00000	−0.0714742
$784$	−7.00000	−0.250000
$785$	36.0000	1.28490
$786$	−12.0000	−0.428026
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−6.00000	−0.213741
$789$	−12.0000	−0.427211
$790$	32.0000	1.13851
$791$	0	0
$792$	−4.00000	−0.142134
$793$	−60.0000	−2.13066
$794$	−34.0000	−1.20661
$795$	12.0000	0.425596
$796$	16.0000	0.567105
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	1.00000	0.0353553
$801$	6.00000	0.212000
$802$	−30.0000	−1.05934
$803$	−56.0000	−1.97620
$804$	8.00000	0.282138
$805$	0	0
$806$	48.0000	1.69073
$807$	14.0000	0.492823
$808$	6.00000	0.211079
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−2.00000	−0.0702728
$811$	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
$813$	16.0000	0.561144
$814$	−8.00000	−0.280400
$815$	−40.0000	−1.40114
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	−10.0000	−0.349642
$819$	0	0
$820$	4.00000	0.139686
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	18.0000	0.627822
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	−8.00000	−0.278693
$825$	4.00000	0.139262
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	8.00000	0.278019
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	−8.00000	−0.277684
$831$	18.0000	0.624413
$832$	−6.00000	−0.208013
$833$	−14.0000	−0.485071
$834$	−4.00000	−0.138509
$835$	24.0000	0.830554
$836$	16.0000	0.553372
$837$	−8.00000	−0.276520
$838$	−12.0000	−0.414533
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−34.0000	−1.17172
$843$	−10.0000	−0.344418
$844$	−8.00000	−0.275371
$845$	46.0000	1.58245
$846$	−8.00000	−0.275046
$847$	0	0
$848$	−6.00000	−0.206041
$849$	16.0000	0.549119
$850$	2.00000	0.0685994
$851$	16.0000	0.548473
$852$	12.0000	0.411113
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	12.0000	0.410152
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	−24.0000	−0.819346
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	1.00000	0.0340207
$865$	4.00000	0.136004
$866$	14.0000	0.475739
$867$	13.0000	0.441503
$868$	0	0
$869$	−64.0000	−2.17105
$870$	4.00000	0.135613
$871$	48.0000	1.62642
$872$	−10.0000	−0.338643
$873$	2.00000	0.0676897
$874$	−32.0000	−1.08242
$875$	0	0
$876$	14.0000	0.473016
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	−8.00000	−0.269987
$879$	22.0000	0.742042
$880$	8.00000	0.269680
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	7.00000	0.235702
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−12.0000	−0.403604
$885$	−2.00000	−0.0672293
$886$	−36.0000	−1.20944
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	−12.0000	−0.402241
$891$	4.00000	0.134005
$892$	0	0
$893$	32.0000	1.07084
$894$	−22.0000	−0.735790
$895$	8.00000	0.267411
$896$	0	0
$897$	48.0000	1.60267
$898$	14.0000	0.467186
$899$	16.0000	0.533630
$900$	−1.00000	−0.0333333
$901$	−12.0000	−0.399778
$902$	−8.00000	−0.266371
$903$	0	0
$904$	2.00000	0.0665190
$905$	−20.0000	−0.664822
$906$	8.00000	0.265782
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	4.00000	0.132745
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−44.0000	−1.45779	−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000	−1.45779	−0.728893	+	0.684628i	$0.759965\pi$
$912$	−4.00000	−0.132453
$913$	16.0000	0.529523
$914$	22.0000	0.727695
$915$	−20.0000	−0.661180
$916$	−22.0000	−0.726900
$917$	0	0
$918$	2.00000	0.0660098
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−16.0000	−0.527504
$921$	−28.0000	−0.922631
$922$	6.00000	0.197599
$923$	72.0000	2.36991
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	24.0000	0.788689
$927$	8.00000	0.262754
$928$	−2.00000	−0.0656532
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	16.0000	0.524661
$931$	−28.0000	−0.917663
$932$	−18.0000	−0.589610
$933$	−20.0000	−0.654771
$934$	−20.0000	−0.654420
$935$	16.0000	0.523256
$936$	6.00000	0.196116
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−18.0000	−0.587408
$940$	16.0000	0.521862
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	18.0000	0.586472
$943$	16.0000	0.521032
$944$	1.00000	0.0325472
$945$	0	0
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	16.0000	0.519656
$949$	84.0000	2.72676
$950$	4.00000	0.129777
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	28.0000	0.905585
$957$	−8.00000	−0.258603
$958$	4.00000	0.129234
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	33.0000	1.06452
$962$	12.0000	0.386896
$963$	−12.0000	−0.386695
$964$	2.00000	0.0644157
$965$	−28.0000	−0.901352
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	−5.00000	−0.160706
$969$	−8.00000	−0.256997
$970$	−4.00000	−0.128432
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−40.0000	−1.28168
$975$	−6.00000	−0.192154
$976$	10.0000	0.320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−20.0000	−0.639529
$979$	24.0000	0.767043
$980$	−14.0000	−0.447214
$981$	10.0000	0.319275
$982$	−20.0000	−0.638226
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	2.00000	0.0637577
$985$	−12.0000	−0.382352
$986$	−4.00000	−0.127386
$987$	0	0
$988$	−24.0000	−0.763542
$989$	0	0
$990$	−8.00000	−0.254257
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	−8.00000	−0.254000
$993$	12.0000	0.380808
$994$	0	0
$995$	32.0000	1.01447
$996$	−4.00000	−0.126745
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	12.0000	0.379853
$999$	−2.00000	−0.0632772

Twists

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

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

Related objects

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