LMFDB - Embedded newform 354.2.a.d.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:	$1$
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)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -4.00000 q^{20} +1.00000 q^{21} -3.00000 q^{22} +2.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +4.00000 q^{30} +1.00000 q^{32} +3.00000 q^{33} -7.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +7.00000 q^{37} -4.00000 q^{38} +1.00000 q^{39} -4.00000 q^{40} +3.00000 q^{41} +1.00000 q^{42} +5.00000 q^{43} -3.00000 q^{44} -4.00000 q^{45} +2.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +11.0000 q^{50} +7.00000 q^{51} -1.00000 q^{52} -8.00000 q^{53} -1.00000 q^{54} +12.0000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +1.00000 q^{59} +4.00000 q^{60} -14.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +3.00000 q^{66} -4.00000 q^{67} -7.00000 q^{68} -2.00000 q^{69} +4.00000 q^{70} -15.0000 q^{71} +1.00000 q^{72} -4.00000 q^{73} +7.00000 q^{74} -11.0000 q^{75} -4.00000 q^{76} +3.00000 q^{77} +1.00000 q^{78} +5.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +3.00000 q^{82} +1.00000 q^{83} +1.00000 q^{84} +28.0000 q^{85} +5.00000 q^{86} +2.00000 q^{87} -3.00000 q^{88} +4.00000 q^{89} -4.00000 q^{90} +1.00000 q^{91} +2.00000 q^{92} +12.0000 q^{94} +16.0000 q^{95} -1.00000 q^{96} -4.00000 q^{97} -6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−1.00000	−0.267261
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−4.00000	−0.894427
$21$	1.00000	0.218218
$22$	−3.00000	−0.639602
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	4.00000	0.730297
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	3.00000	0.522233
$34$	−7.00000	−1.20049
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	−4.00000	−0.648886
$39$	1.00000	0.160128
$40$	−4.00000	−0.632456
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	1.00000	0.154303
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	−3.00000	−0.452267
$45$	−4.00000	−0.596285
$46$	2.00000	0.294884
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	11.0000	1.55563
$51$	7.00000	0.980196
$52$	−1.00000	−0.138675
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	−1.00000	−0.136083
$55$	12.0000	1.61808
$56$	−1.00000	−0.133631
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	4.00000	0.516398
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	3.00000	0.369274
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−7.00000	−0.848875
$69$	−2.00000	−0.240772
$70$	4.00000	0.478091
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	1.00000	0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	7.00000	0.813733
$75$	−11.0000	−1.27017
$76$	−4.00000	−0.458831
$77$	3.00000	0.341882
$78$	1.00000	0.113228
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	3.00000	0.331295
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	1.00000	0.109109
$85$	28.0000	3.03703
$86$	5.00000	0.539164
$87$	2.00000	0.214423
$88$	−3.00000	−0.319801
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	−4.00000	−0.421637
$91$	1.00000	0.104828
$92$	2.00000	0.208514
$93$	0	0
$94$	12.0000	1.23771
$95$	16.0000	1.64157
$96$	−1.00000	−0.102062
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	−6.00000	−0.606092
$99$	−3.00000	−0.301511
$100$	11.0000	1.10000
$101$	−17.0000	−1.69156	−0.845782	−	0.533529i	$-0.820865\pi$
$101$	−17.0000	−1.69156	−0.845782	+	0.533529i	$0.820865\pi$
$102$	7.00000	0.693103
$103$	20.0000	1.97066	0.985329	−	0.170664i	$-0.0545913\pi$
$103$	20.0000	1.97066	0.985329	+	0.170664i	$0.0545913\pi$
$104$	−1.00000	−0.0980581
$105$	−4.00000	−0.390360
$106$	−8.00000	−0.777029
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	−1.00000	−0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	12.0000	1.14416
$111$	−7.00000	−0.664411
$112$	−1.00000	−0.0944911
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	4.00000	0.374634
$115$	−8.00000	−0.746004
$116$	−2.00000	−0.185695
$117$	−1.00000	−0.0924500
$118$	1.00000	0.0920575
$119$	7.00000	0.641689
$120$	4.00000	0.365148
$121$	−2.00000	−0.181818
$122$	−14.0000	−1.26750
$123$	−3.00000	−0.270501
$124$	0	0
$125$	−24.0000	−2.14663
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−5.00000	−0.440225
$130$	4.00000	0.350823
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	3.00000	0.261116
$133$	4.00000	0.346844
$134$	−4.00000	−0.345547
$135$	4.00000	0.344265
$136$	−7.00000	−0.600245
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	−2.00000	−0.170251
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	4.00000	0.338062
$141$	−12.0000	−1.01058
$142$	−15.0000	−1.25877
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	8.00000	0.664364
$146$	−4.00000	−0.331042
$147$	6.00000	0.494872
$148$	7.00000	0.575396
$149$	19.0000	1.55654	0.778270	−	0.627929i	$-0.216097\pi$
$149$	19.0000	1.55654	0.778270	+	0.627929i	$0.216097\pi$
$150$	−11.0000	−0.898146
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−4.00000	−0.324443
$153$	−7.00000	−0.565916
$154$	3.00000	0.241747
$155$	0	0
$156$	1.00000	0.0800641
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	5.00000	0.397779
$159$	8.00000	0.634441
$160$	−4.00000	−0.316228
$161$	−2.00000	−0.157622
$162$	1.00000	0.0785674
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	3.00000	0.234261
$165$	−12.0000	−0.934199
$166$	1.00000	0.0776151
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	1.00000	0.0771517
$169$	−12.0000	−0.923077
$170$	28.0000	2.14750
$171$	−4.00000	−0.305888
$172$	5.00000	0.381246
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	2.00000	0.151620
$175$	−11.0000	−0.831522
$176$	−3.00000	−0.226134
$177$	−1.00000	−0.0751646
$178$	4.00000	0.299813
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	−4.00000	−0.298142
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	1.00000	0.0741249
$183$	14.0000	1.03491
$184$	2.00000	0.147442
$185$	−28.0000	−2.05860
$186$	0	0
$187$	21.0000	1.53567
$188$	12.0000	0.875190
$189$	1.00000	0.0727393
$190$	16.0000	1.16076
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	−1.00000	−0.0721688
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	−4.00000	−0.287183
$195$	−4.00000	−0.286446
$196$	−6.00000	−0.428571
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	−3.00000	−0.213201
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	11.0000	0.777817
$201$	4.00000	0.282138
$202$	−17.0000	−1.19612
$203$	2.00000	0.140372
$204$	7.00000	0.490098
$205$	−12.0000	−0.838116
$206$	20.0000	1.39347
$207$	2.00000	0.139010
$208$	−1.00000	−0.0693375
$209$	12.0000	0.830057
$210$	−4.00000	−0.276026
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−8.00000	−0.549442
$213$	15.0000	1.02778
$214$	−10.0000	−0.683586
$215$	−20.0000	−1.36399
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	2.00000	0.135457
$219$	4.00000	0.270295
$220$	12.0000	0.809040
$221$	7.00000	0.470871
$222$	−7.00000	−0.469809
$223$	7.00000	0.468755	0.234377	−	0.972146i	$-0.424695\pi$
$223$	7.00000	0.468755	0.234377	+	0.972146i	$0.424695\pi$
$224$	−1.00000	−0.0668153
$225$	11.0000	0.733333
$226$	−18.0000	−1.19734
$227$	17.0000	1.12833	0.564165	−	0.825662i	$-0.309198\pi$
$227$	17.0000	1.12833	0.564165	+	0.825662i	$0.309198\pi$
$228$	4.00000	0.264906
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	−8.00000	−0.527504
$231$	−3.00000	−0.197386
$232$	−2.00000	−0.131306
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	−1.00000	−0.0653720
$235$	−48.0000	−3.13117
$236$	1.00000	0.0650945
$237$	−5.00000	−0.324785
$238$	7.00000	0.453743
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	4.00000	0.258199
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	−2.00000	−0.128565
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	24.0000	1.53330
$246$	−3.00000	−0.191273
$247$	4.00000	0.254514
$248$	0	0
$249$	−1.00000	−0.0633724
$250$	−24.0000	−1.51789
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	−1.00000	−0.0629941
$253$	−6.00000	−0.377217
$254$	8.00000	0.501965
$255$	−28.0000	−1.75343
$256$	1.00000	0.0625000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	−5.00000	−0.311286
$259$	−7.00000	−0.434959
$260$	4.00000	0.248069
$261$	−2.00000	−0.123797
$262$	12.0000	0.741362
$263$	11.0000	0.678289	0.339145	−	0.940734i	$-0.389862\pi$
$263$	11.0000	0.678289	0.339145	+	0.940734i	$0.389862\pi$
$264$	3.00000	0.184637
$265$	32.0000	1.96574
$266$	4.00000	0.245256
$267$	−4.00000	−0.244796
$268$	−4.00000	−0.244339
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	4.00000	0.243432
$271$	1.00000	0.0607457	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000	0.0607457	0.0303728	+	0.999539i	$0.490331\pi$
$272$	−7.00000	−0.424437
$273$	−1.00000	−0.0605228
$274$	−13.0000	−0.785359
$275$	−33.0000	−1.98997
$276$	−2.00000	−0.120386
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−8.00000	−0.479808
$279$	0	0
$280$	4.00000	0.239046
$281$	−7.00000	−0.417585	−0.208792	−	0.977960i	$-0.566953\pi$
$281$	−7.00000	−0.417585	−0.208792	+	0.977960i	$0.566953\pi$
$282$	−12.0000	−0.714590
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	−15.0000	−0.890086
$285$	−16.0000	−0.947758
$286$	3.00000	0.177394
$287$	−3.00000	−0.177084
$288$	1.00000	0.0589256
$289$	32.0000	1.88235
$290$	8.00000	0.469776
$291$	4.00000	0.234484
$292$	−4.00000	−0.234082
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	6.00000	0.349927
$295$	−4.00000	−0.232889
$296$	7.00000	0.406867
$297$	3.00000	0.174078
$298$	19.0000	1.10064
$299$	−2.00000	−0.115663
$300$	−11.0000	−0.635085
$301$	−5.00000	−0.288195
$302$	−2.00000	−0.115087
$303$	17.0000	0.976624
$304$	−4.00000	−0.229416
$305$	56.0000	3.20655
$306$	−7.00000	−0.400163
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	3.00000	0.170941
$309$	−20.0000	−1.13776
$310$	0	0
$311$	−33.0000	−1.87126	−0.935629	−	0.352985i	$-0.885167\pi$
$311$	−33.0000	−1.87126	−0.935629	+	0.352985i	$0.885167\pi$
$312$	1.00000	0.0566139
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−2.00000	−0.112867
$315$	4.00000	0.225374
$316$	5.00000	0.281272
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	8.00000	0.448618
$319$	6.00000	0.335936
$320$	−4.00000	−0.223607
$321$	10.0000	0.558146
$322$	−2.00000	−0.111456
$323$	28.0000	1.55796
$324$	1.00000	0.0555556
$325$	−11.0000	−0.610170
$326$	−22.0000	−1.21847
$327$	−2.00000	−0.110600
$328$	3.00000	0.165647
$329$	−12.0000	−0.661581
$330$	−12.0000	−0.660578
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	1.00000	0.0548821
$333$	7.00000	0.383598
$334$	24.0000	1.31322
$335$	16.0000	0.874173
$336$	1.00000	0.0545545
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	−12.0000	−0.652714
$339$	18.0000	0.977626
$340$	28.0000	1.51851
$341$	0	0
$342$	−4.00000	−0.216295
$343$	13.0000	0.701934
$344$	5.00000	0.269582
$345$	8.00000	0.430706
$346$	9.00000	0.483843
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	2.00000	0.107211
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	−11.0000	−0.587975
$351$	1.00000	0.0533761
$352$	−3.00000	−0.159901
$353$	36.0000	1.91609	0.958043	−	0.286623i	$-0.0925328\pi$
$353$	36.0000	1.91609	0.958043	+	0.286623i	$0.0925328\pi$
$354$	−1.00000	−0.0531494
$355$	60.0000	3.18447
$356$	4.00000	0.212000
$357$	−7.00000	−0.370479
$358$	−5.00000	−0.264258
$359$	11.0000	0.580558	0.290279	−	0.956942i	$-0.406252\pi$
$359$	11.0000	0.580558	0.290279	+	0.956942i	$0.406252\pi$
$360$	−4.00000	−0.210819
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	2.00000	0.104973
$364$	1.00000	0.0524142
$365$	16.0000	0.837478
$366$	14.0000	0.731792
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	2.00000	0.104257
$369$	3.00000	0.156174
$370$	−28.0000	−1.45565
$371$	8.00000	0.415339
$372$	0	0
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	21.0000	1.08588
$375$	24.0000	1.23935
$376$	12.0000	0.618853
$377$	2.00000	0.103005
$378$	1.00000	0.0514344
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	16.0000	0.820783
$381$	−8.00000	−0.409852
$382$	4.00000	0.204658
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	−1.00000	−0.0510310
$385$	−12.0000	−0.611577
$386$	5.00000	0.254493
$387$	5.00000	0.254164
$388$	−4.00000	−0.203069
$389$	38.0000	1.92668	0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000	1.92668	0.963338	+	0.268290i	$0.0864585\pi$
$390$	−4.00000	−0.202548
$391$	−14.0000	−0.708010
$392$	−6.00000	−0.303046
$393$	−12.0000	−0.605320
$394$	−16.0000	−0.806068
$395$	−20.0000	−1.00631
$396$	−3.00000	−0.150756
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−20.0000	−1.00251
$399$	−4.00000	−0.200250
$400$	11.0000	0.550000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	−17.0000	−0.845782
$405$	−4.00000	−0.198762
$406$	2.00000	0.0992583
$407$	−21.0000	−1.04093
$408$	7.00000	0.346552
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−12.0000	−0.592638
$411$	13.0000	0.641243
$412$	20.0000	0.985329
$413$	−1.00000	−0.0492068
$414$	2.00000	0.0982946
$415$	−4.00000	−0.196352
$416$	−1.00000	−0.0490290
$417$	8.00000	0.391762
$418$	12.0000	0.586939
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	−4.00000	−0.195180
$421$	−25.0000	−1.21843	−0.609213	−	0.793007i	$-0.708514\pi$
$421$	−25.0000	−1.21843	−0.609213	+	0.793007i	$0.708514\pi$
$422$	−5.00000	−0.243396
$423$	12.0000	0.583460
$424$	−8.00000	−0.388514
$425$	−77.0000	−3.73505
$426$	15.0000	0.726752
$427$	14.0000	0.677507
$428$	−10.0000	−0.483368
$429$	−3.00000	−0.144841
$430$	−20.0000	−0.964486
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	−1.00000	−0.0481125
$433$	1.00000	0.0480569	0.0240285	−	0.999711i	$-0.492351\pi$
$433$	1.00000	0.0480569	0.0240285	+	0.999711i	$0.492351\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	2.00000	0.0957826
$437$	−8.00000	−0.382692
$438$	4.00000	0.191127
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	12.0000	0.572078
$441$	−6.00000	−0.285714
$442$	7.00000	0.332956
$443$	−1.00000	−0.0475114	−0.0237557	−	0.999718i	$-0.507562\pi$
$443$	−1.00000	−0.0475114	−0.0237557	+	0.999718i	$0.507562\pi$
$444$	−7.00000	−0.332205
$445$	−16.0000	−0.758473
$446$	7.00000	0.331460
$447$	−19.0000	−0.898669
$448$	−1.00000	−0.0472456
$449$	−7.00000	−0.330350	−0.165175	−	0.986264i	$-0.552819\pi$
$449$	−7.00000	−0.330350	−0.165175	+	0.986264i	$0.552819\pi$
$450$	11.0000	0.518545
$451$	−9.00000	−0.423793
$452$	−18.0000	−0.846649
$453$	2.00000	0.0939682
$454$	17.0000	0.797850
$455$	−4.00000	−0.187523
$456$	4.00000	0.187317
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	1.00000	0.0467269
$459$	7.00000	0.326732
$460$	−8.00000	−0.373002
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	−3.00000	−0.139573
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	12.0000	0.555889
$467$	35.0000	1.61961	0.809803	−	0.586701i	$-0.199574\pi$
$467$	35.0000	1.61961	0.809803	+	0.586701i	$0.199574\pi$
$468$	−1.00000	−0.0462250
$469$	4.00000	0.184703
$470$	−48.0000	−2.21407
$471$	2.00000	0.0921551
$472$	1.00000	0.0460287
$473$	−15.0000	−0.689701
$474$	−5.00000	−0.229658
$475$	−44.0000	−2.01886
$476$	7.00000	0.320844
$477$	−8.00000	−0.366295
$478$	24.0000	1.09773
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	4.00000	0.182574
$481$	−7.00000	−0.319173
$482$	−17.0000	−0.774329
$483$	2.00000	0.0910032
$484$	−2.00000	−0.0909091
$485$	16.0000	0.726523
$486$	−1.00000	−0.0453609
$487$	−23.0000	−1.04223	−0.521115	−	0.853487i	$-0.674484\pi$
$487$	−23.0000	−1.04223	−0.521115	+	0.853487i	$0.674484\pi$
$488$	−14.0000	−0.633750
$489$	22.0000	0.994874
$490$	24.0000	1.08421
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	−3.00000	−0.135250
$493$	14.0000	0.630528
$494$	4.00000	0.179969
$495$	12.0000	0.539360
$496$	0	0
$497$	15.0000	0.672842
$498$	−1.00000	−0.0448111
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−24.0000	−1.07331
$501$	−24.0000	−1.07224
$502$	−20.0000	−0.892644
$503$	26.0000	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$503$	26.0000	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$504$	−1.00000	−0.0445435
$505$	68.0000	3.02596
$506$	−6.00000	−0.266733
$507$	12.0000	0.532939
$508$	8.00000	0.354943
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	−28.0000	−1.23986
$511$	4.00000	0.176950
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	−21.0000	−0.926270
$515$	−80.0000	−3.52522
$516$	−5.00000	−0.220113
$517$	−36.0000	−1.58328
$518$	−7.00000	−0.307562
$519$	−9.00000	−0.395056
$520$	4.00000	0.175412
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	−2.00000	−0.0875376
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	12.0000	0.524222
$525$	11.0000	0.480079
$526$	11.0000	0.479623
$527$	0	0
$528$	3.00000	0.130558
$529$	−19.0000	−0.826087
$530$	32.0000	1.38999
$531$	1.00000	0.0433963
$532$	4.00000	0.173422
$533$	−3.00000	−0.129944
$534$	−4.00000	−0.173097
$535$	40.0000	1.72935
$536$	−4.00000	−0.172774
$537$	5.00000	0.215766
$538$	21.0000	0.905374
$539$	18.0000	0.775315
$540$	4.00000	0.172133
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	1.00000	0.0429537
$543$	10.0000	0.429141
$544$	−7.00000	−0.300123
$545$	−8.00000	−0.342682
$546$	−1.00000	−0.0427960
$547$	10.0000	0.427569	0.213785	−	0.976881i	$-0.431421\pi$
$547$	10.0000	0.427569	0.213785	+	0.976881i	$0.431421\pi$
$548$	−13.0000	−0.555332
$549$	−14.0000	−0.597505
$550$	−33.0000	−1.40712
$551$	8.00000	0.340811
$552$	−2.00000	−0.0851257
$553$	−5.00000	−0.212622
$554$	22.0000	0.934690
$555$	28.0000	1.18853
$556$	−8.00000	−0.339276
$557$	−36.0000	−1.52537	−0.762684	−	0.646771i	$-0.776119\pi$
$557$	−36.0000	−1.52537	−0.762684	+	0.646771i	$0.776119\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	4.00000	0.169031
$561$	−21.0000	−0.886621
$562$	−7.00000	−0.295277
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−12.0000	−0.505291
$565$	72.0000	3.02906
$566$	1.00000	0.0420331
$567$	−1.00000	−0.0419961
$568$	−15.0000	−0.629386
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−16.0000	−0.670166
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	3.00000	0.125436
$573$	−4.00000	−0.167102
$574$	−3.00000	−0.125218
$575$	22.0000	0.917463
$576$	1.00000	0.0416667
$577$	1.00000	0.0416305	0.0208153	−	0.999783i	$-0.493374\pi$
$577$	1.00000	0.0416305	0.0208153	+	0.999783i	$0.493374\pi$
$578$	32.0000	1.33102
$579$	−5.00000	−0.207793
$580$	8.00000	0.332182
$581$	−1.00000	−0.0414870
$582$	4.00000	0.165805
$583$	24.0000	0.993978
$584$	−4.00000	−0.165521
$585$	4.00000	0.165380
$586$	16.0000	0.660954
$587$	−21.0000	−0.866763	−0.433381	−	0.901211i	$-0.642680\pi$
$587$	−21.0000	−0.866763	−0.433381	+	0.901211i	$0.642680\pi$
$588$	6.00000	0.247436
$589$	0	0
$590$	−4.00000	−0.164677
$591$	16.0000	0.658152
$592$	7.00000	0.287698
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	3.00000	0.123091
$595$	−28.0000	−1.14789
$596$	19.0000	0.778270
$597$	20.0000	0.818546
$598$	−2.00000	−0.0817861
$599$	−25.0000	−1.02147	−0.510736	−	0.859738i	$-0.670627\pi$
$599$	−25.0000	−1.02147	−0.510736	+	0.859738i	$0.670627\pi$
$600$	−11.0000	−0.449073
$601$	−40.0000	−1.63163	−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000	−1.63163	−0.815817	+	0.578310i	$0.803712\pi$
$602$	−5.00000	−0.203785
$603$	−4.00000	−0.162893
$604$	−2.00000	−0.0813788
$605$	8.00000	0.325246
$606$	17.0000	0.690578
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	−4.00000	−0.162221
$609$	−2.00000	−0.0810441
$610$	56.0000	2.26737
$611$	−12.0000	−0.485468
$612$	−7.00000	−0.282958
$613$	−5.00000	−0.201948	−0.100974	−	0.994889i	$-0.532196\pi$
$613$	−5.00000	−0.201948	−0.100974	+	0.994889i	$0.532196\pi$
$614$	−26.0000	−1.04927
$615$	12.0000	0.483887
$616$	3.00000	0.120873
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	−20.0000	−0.804518
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	−33.0000	−1.32318
$623$	−4.00000	−0.160257
$624$	1.00000	0.0400320
$625$	41.0000	1.64000
$626$	−14.0000	−0.559553
$627$	−12.0000	−0.479234
$628$	−2.00000	−0.0798087
$629$	−49.0000	−1.95376
$630$	4.00000	0.159364
$631$	21.0000	0.835997	0.417998	−	0.908448i	$-0.362732\pi$
$631$	21.0000	0.835997	0.417998	+	0.908448i	$0.362732\pi$
$632$	5.00000	0.198889
$633$	5.00000	0.198732
$634$	−12.0000	−0.476581
$635$	−32.0000	−1.26988
$636$	8.00000	0.317221
$637$	6.00000	0.237729
$638$	6.00000	0.237542
$639$	−15.0000	−0.593391
$640$	−4.00000	−0.158114
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	10.0000	0.394669
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−2.00000	−0.0788110
$645$	20.0000	0.787499
$646$	28.0000	1.10165
$647$	−40.0000	−1.57256	−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000	−1.57256	−0.786281	+	0.617869i	$0.787996\pi$
$648$	1.00000	0.0392837
$649$	−3.00000	−0.117760
$650$	−11.0000	−0.431455
$651$	0	0
$652$	−22.0000	−0.861586
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	−2.00000	−0.0782062
$655$	−48.0000	−1.87552
$656$	3.00000	0.117130
$657$	−4.00000	−0.156055
$658$	−12.0000	−0.467809
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	−12.0000	−0.467099
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	10.0000	0.388661
$663$	−7.00000	−0.271857
$664$	1.00000	0.0388075
$665$	−16.0000	−0.620453
$666$	7.00000	0.271244
$667$	−4.00000	−0.154881
$668$	24.0000	0.928588
$669$	−7.00000	−0.270636
$670$	16.0000	0.618134
$671$	42.0000	1.62139
$672$	1.00000	0.0385758
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	34.0000	1.30963
$675$	−11.0000	−0.423390
$676$	−12.0000	−0.461538
$677$	−20.0000	−0.768662	−0.384331	−	0.923195i	$-0.625568\pi$
$677$	−20.0000	−0.768662	−0.384331	+	0.923195i	$0.625568\pi$
$678$	18.0000	0.691286
$679$	4.00000	0.153506
$680$	28.0000	1.07375
$681$	−17.0000	−0.651441
$682$	0	0
$683$	−1.00000	−0.0382639	−0.0191320	−	0.999817i	$-0.506090\pi$
$683$	−1.00000	−0.0382639	−0.0191320	+	0.999817i	$0.506090\pi$
$684$	−4.00000	−0.152944
$685$	52.0000	1.98682
$686$	13.0000	0.496342
$687$	−1.00000	−0.0381524
$688$	5.00000	0.190623
$689$	8.00000	0.304776
$690$	8.00000	0.304555
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	9.00000	0.342129
$693$	3.00000	0.113961
$694$	28.0000	1.06287
$695$	32.0000	1.21383
$696$	2.00000	0.0758098
$697$	−21.0000	−0.795432
$698$	7.00000	0.264954
$699$	−12.0000	−0.453882
$700$	−11.0000	−0.415761
$701$	−17.0000	−0.642081	−0.321041	−	0.947065i	$-0.604033\pi$
$701$	−17.0000	−0.642081	−0.321041	+	0.947065i	$0.604033\pi$
$702$	1.00000	0.0377426
$703$	−28.0000	−1.05604
$704$	−3.00000	−0.113067
$705$	48.0000	1.80778
$706$	36.0000	1.35488
$707$	17.0000	0.639351
$708$	−1.00000	−0.0375823
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	60.0000	2.25176
$711$	5.00000	0.187515
$712$	4.00000	0.149906
$713$	0	0
$714$	−7.00000	−0.261968
$715$	−12.0000	−0.448775
$716$	−5.00000	−0.186859
$717$	−24.0000	−0.896296
$718$	11.0000	0.410516
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	−4.00000	−0.149071
$721$	−20.0000	−0.744839
$722$	−3.00000	−0.111648
$723$	17.0000	0.632237
$724$	−10.0000	−0.371647
$725$	−22.0000	−0.817059
$726$	2.00000	0.0742270
$727$	−5.00000	−0.185440	−0.0927199	−	0.995692i	$-0.529556\pi$
$727$	−5.00000	−0.185440	−0.0927199	+	0.995692i	$0.529556\pi$
$728$	1.00000	0.0370625
$729$	1.00000	0.0370370
$730$	16.0000	0.592187
$731$	−35.0000	−1.29452
$732$	14.0000	0.517455
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−4.00000	−0.147643
$735$	−24.0000	−0.885253
$736$	2.00000	0.0737210
$737$	12.0000	0.442026
$738$	3.00000	0.110432
$739$	3.00000	0.110357	0.0551784	−	0.998477i	$-0.482427\pi$
$739$	3.00000	0.110357	0.0551784	+	0.998477i	$0.482427\pi$
$740$	−28.0000	−1.02930
$741$	−4.00000	−0.146944
$742$	8.00000	0.293689
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	−76.0000	−2.78442
$746$	−32.0000	−1.17160
$747$	1.00000	0.0365881
$748$	21.0000	0.767836
$749$	10.0000	0.365392
$750$	24.0000	0.876356
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	12.0000	0.437595
$753$	20.0000	0.728841
$754$	2.00000	0.0728357
$755$	8.00000	0.291150
$756$	1.00000	0.0363696
$757$	32.0000	1.16306	0.581530	−	0.813525i	$-0.302454\pi$
$757$	32.0000	1.16306	0.581530	+	0.813525i	$0.302454\pi$
$758$	2.00000	0.0726433
$759$	6.00000	0.217786
$760$	16.0000	0.580381
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	−8.00000	−0.289809
$763$	−2.00000	−0.0724049
$764$	4.00000	0.144715
$765$	28.0000	1.01234
$766$	−21.0000	−0.758761
$767$	−1.00000	−0.0361079
$768$	−1.00000	−0.0360844
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	−12.0000	−0.432450
$771$	21.0000	0.756297
$772$	5.00000	0.179954
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	5.00000	0.179721
$775$	0	0
$776$	−4.00000	−0.143592
$777$	7.00000	0.251124
$778$	38.0000	1.36237
$779$	−12.0000	−0.429945
$780$	−4.00000	−0.143223
$781$	45.0000	1.61023
$782$	−14.0000	−0.500639
$783$	2.00000	0.0714742
$784$	−6.00000	−0.214286
$785$	8.00000	0.285532
$786$	−12.0000	−0.428026
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	−16.0000	−0.569976
$789$	−11.0000	−0.391610
$790$	−20.0000	−0.711568
$791$	18.0000	0.640006
$792$	−3.00000	−0.106600
$793$	14.0000	0.497155
$794$	22.0000	0.780751
$795$	−32.0000	−1.13492
$796$	−20.0000	−0.708881
$797$	27.0000	0.956389	0.478195	−	0.878254i	$-0.341291\pi$
$797$	27.0000	0.956389	0.478195	+	0.878254i	$0.341291\pi$
$798$	−4.00000	−0.141598
$799$	−84.0000	−2.97171
$800$	11.0000	0.388909
$801$	4.00000	0.141333
$802$	24.0000	0.847469
$803$	12.0000	0.423471
$804$	4.00000	0.141069
$805$	8.00000	0.281963
$806$	0	0
$807$	−21.0000	−0.739235
$808$	−17.0000	−0.598058
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	−4.00000	−0.140546
$811$	37.0000	1.29925	0.649623	−	0.760257i	$-0.274927\pi$
$811$	37.0000	1.29925	0.649623	+	0.760257i	$0.274927\pi$
$812$	2.00000	0.0701862
$813$	−1.00000	−0.0350715
$814$	−21.0000	−0.736050
$815$	88.0000	3.08251
$816$	7.00000	0.245049
$817$	−20.0000	−0.699711
$818$	−10.0000	−0.349642
$819$	1.00000	0.0349428
$820$	−12.0000	−0.419058
$821$	13.0000	0.453703	0.226852	−	0.973929i	$-0.427157\pi$
$821$	13.0000	0.453703	0.226852	+	0.973929i	$0.427157\pi$
$822$	13.0000	0.453427
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	20.0000	0.696733
$825$	33.0000	1.14891
$826$	−1.00000	−0.0347945
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	2.00000	0.0695048
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	−4.00000	−0.138842
$831$	−22.0000	−0.763172
$832$	−1.00000	−0.0346688
$833$	42.0000	1.45521
$834$	8.00000	0.277017
$835$	−96.0000	−3.32222
$836$	12.0000	0.415029
$837$	0	0
$838$	−5.00000	−0.172722
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	−4.00000	−0.138013
$841$	−25.0000	−0.862069
$842$	−25.0000	−0.861557
$843$	7.00000	0.241093
$844$	−5.00000	−0.172107
$845$	48.0000	1.65125
$846$	12.0000	0.412568
$847$	2.00000	0.0687208
$848$	−8.00000	−0.274721
$849$	−1.00000	−0.0343199
$850$	−77.0000	−2.64108
$851$	14.0000	0.479914
$852$	15.0000	0.513892
$853$	−24.0000	−0.821744	−0.410872	−	0.911693i	$-0.634776\pi$
$853$	−24.0000	−0.821744	−0.410872	+	0.911693i	$0.634776\pi$
$854$	14.0000	0.479070
$855$	16.0000	0.547188
$856$	−10.0000	−0.341793
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	−3.00000	−0.102418
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−20.0000	−0.681994
$861$	3.00000	0.102240
$862$	−34.0000	−1.15804
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	−1.00000	−0.0340207
$865$	−36.0000	−1.22404
$866$	1.00000	0.0339814
$867$	−32.0000	−1.08678
$868$	0	0
$869$	−15.0000	−0.508840
$870$	−8.00000	−0.271225
$871$	4.00000	0.135535
$872$	2.00000	0.0677285
$873$	−4.00000	−0.135379
$874$	−8.00000	−0.270604
$875$	24.0000	0.811348
$876$	4.00000	0.135147
$877$	28.0000	0.945493	0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000	0.945493	0.472746	+	0.881199i	$0.343263\pi$
$878$	−19.0000	−0.641219
$879$	−16.0000	−0.539667
$880$	12.0000	0.404520
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	−6.00000	−0.202031
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	7.00000	0.235435
$885$	4.00000	0.134459
$886$	−1.00000	−0.0335957
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	−7.00000	−0.234905
$889$	−8.00000	−0.268311
$890$	−16.0000	−0.536321
$891$	−3.00000	−0.100504
$892$	7.00000	0.234377
$893$	−48.0000	−1.60626
$894$	−19.0000	−0.635455
$895$	20.0000	0.668526
$896$	−1.00000	−0.0334077
$897$	2.00000	0.0667781
$898$	−7.00000	−0.233593
$899$	0	0
$900$	11.0000	0.366667
$901$	56.0000	1.86563
$902$	−9.00000	−0.299667
$903$	5.00000	0.166390
$904$	−18.0000	−0.598671
$905$	40.0000	1.32964
$906$	2.00000	0.0664455
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	17.0000	0.564165
$909$	−17.0000	−0.563854
$910$	−4.00000	−0.132599
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	4.00000	0.132453
$913$	−3.00000	−0.0992855
$914$	−12.0000	−0.396925
$915$	−56.0000	−1.85130
$916$	1.00000	0.0330409
$917$	−12.0000	−0.396275
$918$	7.00000	0.231034
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	−8.00000	−0.263752
$921$	26.0000	0.856729
$922$	20.0000	0.658665
$923$	15.0000	0.493731
$924$	−3.00000	−0.0986928
$925$	77.0000	2.53174
$926$	4.00000	0.131448
$927$	20.0000	0.656886
$928$	−2.00000	−0.0656532
$929$	40.0000	1.31236	0.656179	−	0.754606i	$-0.272172\pi$
$929$	40.0000	1.31236	0.656179	+	0.754606i	$0.272172\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	12.0000	0.393073
$933$	33.0000	1.08037
$934$	35.0000	1.14523
$935$	−84.0000	−2.74709
$936$	−1.00000	−0.0326860
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	4.00000	0.130605
$939$	14.0000	0.456873
$940$	−48.0000	−1.56559
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	2.00000	0.0651635
$943$	6.00000	0.195387
$944$	1.00000	0.0325472
$945$	−4.00000	−0.130120
$946$	−15.0000	−0.487692
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	−5.00000	−0.162392
$949$	4.00000	0.129845
$950$	−44.0000	−1.42755
$951$	12.0000	0.389127
$952$	7.00000	0.226871
$953$	21.0000	0.680257	0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000	0.680257	0.340128	+	0.940379i	$0.389529\pi$
$954$	−8.00000	−0.259010
$955$	−16.0000	−0.517748
$956$	24.0000	0.776215
$957$	−6.00000	−0.193952
$958$	4.00000	0.129234
$959$	13.0000	0.419792
$960$	4.00000	0.129099
$961$	−31.0000	−1.00000
$962$	−7.00000	−0.225689
$963$	−10.0000	−0.322245
$964$	−17.0000	−0.547533
$965$	−20.0000	−0.643823
$966$	2.00000	0.0643489
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	−2.00000	−0.0642824
$969$	−28.0000	−0.899490
$970$	16.0000	0.513729
$971$	−26.0000	−0.834380	−0.417190	−	0.908819i	$-0.636985\pi$
$971$	−26.0000	−0.834380	−0.417190	+	0.908819i	$0.636985\pi$
$972$	−1.00000	−0.0320750
$973$	8.00000	0.256468
$974$	−23.0000	−0.736968
$975$	11.0000	0.352282
$976$	−14.0000	−0.448129
$977$	−4.00000	−0.127971	−0.0639857	−	0.997951i	$-0.520381\pi$
$977$	−4.00000	−0.127971	−0.0639857	+	0.997951i	$0.520381\pi$
$978$	22.0000	0.703482
$979$	−12.0000	−0.383522
$980$	24.0000	0.766652
$981$	2.00000	0.0638551
$982$	22.0000	0.702048
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	−3.00000	−0.0956365
$985$	64.0000	2.03921
$986$	14.0000	0.445851
$987$	12.0000	0.381964
$988$	4.00000	0.127257
$989$	10.0000	0.317982
$990$	12.0000	0.381385
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	−10.0000	−0.317340
$994$	15.0000	0.475771
$995$	80.0000	2.53617
$996$	−1.00000	−0.0316862
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	−4.00000	−0.126618
$999$	−7.00000	−0.221470

Twists

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

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