LMFDB - Embedded newform 663.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [663,2,Mod(1,663)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(663, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("663.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 663 = 3 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	663.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:	$5.29408165401$
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	$\chi$	$=$	663.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} +4.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +4.00000 q^{20} -2.00000 q^{21} +6.00000 q^{22} +3.00000 q^{24} +11.0000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} +4.00000 q^{30} +10.0000 q^{31} +5.00000 q^{32} -6.00000 q^{33} +1.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} +12.0000 q^{40} -2.00000 q^{42} +12.0000 q^{43} -6.00000 q^{44} -4.00000 q^{45} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +11.0000 q^{50} -1.00000 q^{51} +1.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} -24.0000 q^{55} -6.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} -4.00000 q^{60} -10.0000 q^{61} +10.0000 q^{62} +2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{65} -6.00000 q^{66} +12.0000 q^{67} -1.00000 q^{68} -8.00000 q^{70} +2.00000 q^{71} -3.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} -11.0000 q^{75} -4.00000 q^{76} +12.0000 q^{77} +1.00000 q^{78} -4.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{85} +12.0000 q^{86} +6.00000 q^{87} -18.0000 q^{88} -6.00000 q^{89} -4.00000 q^{90} -2.00000 q^{91} -10.0000 q^{93} +8.00000 q^{94} -16.0000 q^{95} -5.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−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
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.00000	0.534522
$15$	4.00000	1.03280
$16$	−1.00000	−0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	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
$21$	−2.00000	−0.436436
$22$	6.00000	1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	11.0000	2.20000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	4.00000	0.730297
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	5.00000	0.883883
$33$	−6.00000	−1.04447
$34$	1.00000	0.171499
$35$	−8.00000	−1.35225
$36$	−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
$39$	1.00000	0.160128
$40$	12.0000	1.89737
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−2.00000	−0.308607
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	−6.00000	−0.904534
$45$	−4.00000	−0.596285
$46$	0	0
$47$	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
$49$	−3.00000	−0.428571
$50$	11.0000	1.55563
$51$	−1.00000	−0.140028
$52$	1.00000	0.138675
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−1.00000	−0.136083
$55$	−24.0000	−3.23616
$56$	−6.00000	−0.801784
$57$	−4.00000	−0.529813
$58$	−6.00000	−0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	−4.00000	−0.516398
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	10.0000	1.27000
$63$	2.00000	0.251976
$64$	7.00000	0.875000
$65$	4.00000	0.496139
$66$	−6.00000	−0.738549
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	−8.00000	−0.956183
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−3.00000	−0.353553
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−4.00000	−0.464991
$75$	−11.0000	−1.27017
$76$	−4.00000	−0.458831
$77$	12.0000	1.36753
$78$	1.00000	0.113228
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	2.00000	0.218218
$85$	−4.00000	−0.433861
$86$	12.0000	1.29399
$87$	6.00000	0.643268
$88$	−18.0000	−1.91881
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−4.00000	−0.421637
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−10.0000	−1.03695
$94$	8.00000	0.825137
$95$	−16.0000	−1.64157
$96$	−5.00000	−0.510310
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−3.00000	−0.303046
$99$	6.00000	0.603023
$100$	−11.0000	−1.10000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−1.00000	−0.0990148
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	3.00000	0.294174
$105$	8.00000	0.780720
$106$	2.00000	0.194257
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	1.00000	0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−24.0000	−2.28831
$111$	4.00000	0.379663
$112$	−2.00000	−0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	−1.00000	−0.0924500
$118$	−8.00000	−0.736460
$119$	2.00000	0.183340
$120$	−12.0000	−1.09545
$121$	25.0000	2.27273
$122$	−10.0000	−0.905357
$123$	0	0
$124$	−10.0000	−0.898027
$125$	−24.0000	−2.14663
$126$	2.00000	0.178174
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−3.00000	−0.265165
$129$	−12.0000	−1.05654
$130$	4.00000	0.350823
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	6.00000	0.522233
$133$	8.00000	0.693688
$134$	12.0000	1.03664
$135$	4.00000	0.344265
$136$	−3.00000	−0.257248
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	8.00000	0.676123
$141$	−8.00000	−0.673722
$142$	2.00000	0.167836
$143$	−6.00000	−0.501745
$144$	−1.00000	−0.0833333
$145$	24.0000	1.99309
$146$	4.00000	0.331042
$147$	3.00000	0.247436
$148$	4.00000	0.328798
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−11.0000	−0.898146
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	−12.0000	−0.973329
$153$	1.00000	0.0808452
$154$	12.0000	0.966988
$155$	−40.0000	−3.21288
$156$	−1.00000	−0.0800641
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−4.00000	−0.318223
$159$	−2.00000	−0.158610
$160$	−20.0000	−1.58114
$161$	0	0
$162$	1.00000	0.0785674
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	24.0000	1.86840
$166$	−12.0000	−0.931381
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	6.00000	0.462910
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	4.00000	0.305888
$172$	−12.0000	−0.914991
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	6.00000	0.454859
$175$	22.0000	1.66304
$176$	−6.00000	−0.452267
$177$	8.00000	0.601317
$178$	−6.00000	−0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	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	$0.621207\pi$
$182$	−2.00000	−0.148250
$183$	10.0000	0.739221
$184$	0	0
$185$	16.0000	1.17634
$186$	−10.0000	−0.733236
$187$	6.00000	0.438763
$188$	−8.00000	−0.583460
$189$	−2.00000	−0.145479
$190$	−16.0000	−1.16076
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−7.00000	−0.505181
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	8.00000	0.574367
$195$	−4.00000	−0.286446
$196$	3.00000	0.214286
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	6.00000	0.426401
$199$	−28.0000	−1.98487	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−28.0000	−1.98487	−0.992434	+	0.122782i	$0.960818\pi$
$200$	−33.0000	−2.33345
$201$	−12.0000	−0.846415
$202$	6.00000	0.422159
$203$	−12.0000	−0.842235
$204$	1.00000	0.0700140
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	1.00000	0.0693375
$209$	24.0000	1.66011
$210$	8.00000	0.552052
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−2.00000	−0.137361
$213$	−2.00000	−0.137038
$214$	16.0000	1.09374
$215$	−48.0000	−3.27357
$216$	3.00000	0.204124
$217$	20.0000	1.35769
$218$	−16.0000	−1.08366
$219$	−4.00000	−0.270295
$220$	24.0000	1.61808
$221$	−1.00000	−0.0672673
$222$	4.00000	0.268462
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	10.0000	0.668153
$225$	11.0000	0.733333
$226$	6.00000	0.399114
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	4.00000	0.264906
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	18.0000	1.18176
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	−1.00000	−0.0653720
$235$	−32.0000	−2.08745
$236$	8.00000	0.520756
$237$	4.00000	0.259828
$238$	2.00000	0.129641
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	−4.00000	−0.258199
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	25.0000	1.60706
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	12.0000	0.766652
$246$	0	0
$247$	−4.00000	−0.254514
$248$	−30.0000	−1.90500
$249$	12.0000	0.760469
$250$	−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	$0.717435\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	16.0000	1.00393
$255$	4.00000	0.250490
$256$	−17.0000	−1.06250
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	−12.0000	−0.747087
$259$	−8.00000	−0.497096
$260$	−4.00000	−0.248069
$261$	−6.00000	−0.371391
$262$	−12.0000	−0.741362
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	18.0000	1.10782
$265$	−8.00000	−0.491436
$266$	8.00000	0.490511
$267$	6.00000	0.367194
$268$	−12.0000	−0.733017
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	4.00000	0.243432
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−1.00000	−0.0606339
$273$	2.00000	0.121046
$274$	−6.00000	−0.362473
$275$	66.0000	3.97995
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	16.0000	0.959616
$279$	10.0000	0.598684
$280$	24.0000	1.43427
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	−8.00000	−0.476393
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−2.00000	−0.118678
$285$	16.0000	0.947758
$286$	−6.00000	−0.354787
$287$	0	0
$288$	5.00000	0.294628
$289$	1.00000	0.0588235
$290$	24.0000	1.40933
$291$	−8.00000	−0.468968
$292$	−4.00000	−0.234082
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	3.00000	0.174964
$295$	32.0000	1.86311
$296$	12.0000	0.697486
$297$	−6.00000	−0.348155
$298$	18.0000	1.04271
$299$	0	0
$300$	11.0000	0.635085
$301$	24.0000	1.38334
$302$	20.0000	1.15087
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	40.0000	2.29039
$306$	1.00000	0.0571662
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−12.0000	−0.683763
$309$	−8.00000	−0.455104
$310$	−40.0000	−2.27185
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	−3.00000	−0.169842
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	2.00000	0.112867
$315$	−8.00000	−0.450749
$316$	4.00000	0.225018
$317$	−4.00000	−0.224662	−0.112331	−	0.993671i	$-0.535832\pi$
$317$	−4.00000	−0.224662	−0.112331	+	0.993671i	$0.535832\pi$
$318$	−2.00000	−0.112154
$319$	−36.0000	−2.01561
$320$	−28.0000	−1.56525
$321$	−16.0000	−0.893033
$322$	0	0
$323$	4.00000	0.222566
$324$	−1.00000	−0.0555556
$325$	−11.0000	−0.610170
$326$	6.00000	0.332309
$327$	16.0000	0.884802
$328$	0	0
$329$	16.0000	0.882109
$330$	24.0000	1.32116
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	12.0000	0.658586
$333$	−4.00000	−0.219199
$334$	−14.0000	−0.766046
$335$	−48.0000	−2.62252
$336$	2.00000	0.109109
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	1.00000	0.0543928
$339$	−6.00000	−0.325875
$340$	4.00000	0.216930
$341$	60.0000	3.24918
$342$	4.00000	0.216295
$343$	−20.0000	−1.07990
$344$	−36.0000	−1.94099
$345$	0	0
$346$	2.00000	0.107521
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	−6.00000	−0.321634
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	22.0000	1.17595
$351$	1.00000	0.0533761
$352$	30.0000	1.59901
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	8.00000	0.425195
$355$	−8.00000	−0.424596
$356$	6.00000	0.317999
$357$	−2.00000	−0.105851
$358$	12.0000	0.634220
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	12.0000	0.632456
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	−25.0000	−1.31216
$364$	2.00000	0.104828
$365$	−16.0000	−0.837478
$366$	10.0000	0.522708
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	16.0000	0.831800
$371$	4.00000	0.207670
$372$	10.0000	0.518476
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	6.00000	0.310253
$375$	24.0000	1.23935
$376$	−24.0000	−1.23771
$377$	6.00000	0.309016
$378$	−2.00000	−0.102869
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	16.0000	0.820783
$381$	−16.0000	−0.819705
$382$	8.00000	0.409316
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	3.00000	0.153093
$385$	−48.0000	−2.44631
$386$	−4.00000	−0.203595
$387$	12.0000	0.609994
$388$	−8.00000	−0.406138
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	−4.00000	−0.202548
$391$	0	0
$392$	9.00000	0.454569
$393$	12.0000	0.605320
$394$	−12.0000	−0.604551
$395$	16.0000	0.805047
$396$	−6.00000	−0.301511
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−28.0000	−1.40351
$399$	−8.00000	−0.400501
$400$	−11.0000	−0.550000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	−12.0000	−0.598506
$403$	−10.0000	−0.498135
$404$	−6.00000	−0.298511
$405$	−4.00000	−0.198762
$406$	−12.0000	−0.595550
$407$	−24.0000	−1.18964
$408$	3.00000	0.148522
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	−8.00000	−0.394132
$413$	−16.0000	−0.787309
$414$	0	0
$415$	48.0000	2.35623
$416$	−5.00000	−0.245145
$417$	−16.0000	−0.783523
$418$	24.0000	1.17388
$419$	8.00000	0.390826	0.195413	−	0.980721i	$-0.437395\pi$
$419$	8.00000	0.390826	0.195413	+	0.980721i	$0.437395\pi$
$420$	−8.00000	−0.390360
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−8.00000	−0.389434
$423$	8.00000	0.388973
$424$	−6.00000	−0.291386
$425$	11.0000	0.533578
$426$	−2.00000	−0.0969003
$427$	−20.0000	−0.967868
$428$	−16.0000	−0.773389
$429$	6.00000	0.289683
$430$	−48.0000	−2.31477
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	1.00000	0.0481125
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	20.0000	0.960031
$435$	−24.0000	−1.15071
$436$	16.0000	0.766261
$437$	0	0
$438$	−4.00000	−0.191127
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	72.0000	3.43247
$441$	−3.00000	−0.142857
$442$	−1.00000	−0.0475651
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−4.00000	−0.189832
$445$	24.0000	1.13771
$446$	−12.0000	−0.568216
$447$	−18.0000	−0.851371
$448$	14.0000	0.661438
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	11.0000	0.518545
$451$	0	0
$452$	−6.00000	−0.282216
$453$	−20.0000	−0.939682
$454$	6.00000	0.281594
$455$	8.00000	0.375046
$456$	12.0000	0.561951
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	6.00000	0.280362
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	−12.0000	−0.558291
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	6.00000	0.278543
$465$	40.0000	1.85496
$466$	14.0000	0.648537
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	1.00000	0.0462250
$469$	24.0000	1.10822
$470$	−32.0000	−1.47605
$471$	−2.00000	−0.0921551
$472$	24.0000	1.10469
$473$	72.0000	3.31056
$474$	4.00000	0.183726
$475$	44.0000	2.01886
$476$	−2.00000	−0.0916698
$477$	2.00000	0.0915737
$478$	−12.0000	−0.548867
$479$	34.0000	1.55350	0.776750	−	0.629809i	$-0.216867\pi$
$479$	34.0000	1.55350	0.776750	+	0.629809i	$0.216867\pi$
$480$	20.0000	0.912871
$481$	4.00000	0.182384
$482$	−8.00000	−0.364390
$483$	0	0
$484$	−25.0000	−1.13636
$485$	−32.0000	−1.45305
$486$	−1.00000	−0.0453609
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	30.0000	1.35804
$489$	−6.00000	−0.271329
$490$	12.0000	0.542105
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	−4.00000	−0.179969
$495$	−24.0000	−1.07872
$496$	−10.0000	−0.449013
$497$	4.00000	0.179425
$498$	12.0000	0.537733
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	24.0000	1.07331
$501$	14.0000	0.625474
$502$	−20.0000	−0.892644
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	−6.00000	−0.267261
$505$	−24.0000	−1.06799
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	−16.0000	−0.709885
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	4.00000	0.177123
$511$	8.00000	0.353899
$512$	−11.0000	−0.486136
$513$	−4.00000	−0.176604
$514$	14.0000	0.617514
$515$	−32.0000	−1.41009
$516$	12.0000	0.528271
$517$	48.0000	2.11104
$518$	−8.00000	−0.351500
$519$	−2.00000	−0.0877903
$520$	−12.0000	−0.526235
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−6.00000	−0.262613
$523$	−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
$525$	−22.0000	−0.960159
$526$	16.0000	0.697633
$527$	10.0000	0.435607
$528$	6.00000	0.261116
$529$	−23.0000	−1.00000
$530$	−8.00000	−0.347498
$531$	−8.00000	−0.347170
$532$	−8.00000	−0.346844
$533$	0	0
$534$	6.00000	0.259645
$535$	−64.0000	−2.76696
$536$	−36.0000	−1.55496
$537$	−12.0000	−0.517838
$538$	−30.0000	−1.29339
$539$	−18.0000	−0.775315
$540$	−4.00000	−0.172133
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	−8.00000	−0.343629
$543$	10.0000	0.429141
$544$	5.00000	0.214373
$545$	64.0000	2.74146
$546$	2.00000	0.0855921
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	6.00000	0.256307
$549$	−10.0000	−0.426790
$550$	66.0000	2.81425
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−8.00000	−0.340195
$554$	2.00000	0.0849719
$555$	−16.0000	−0.679162
$556$	−16.0000	−0.678551
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	10.0000	0.423334
$559$	−12.0000	−0.507546
$560$	8.00000	0.338062
$561$	−6.00000	−0.253320
$562$	2.00000	0.0843649
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	8.00000	0.336861
$565$	−24.0000	−1.00969
$566$	12.0000	0.504398
$567$	2.00000	0.0839921
$568$	−6.00000	−0.251754
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	16.0000	0.670166
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	6.00000	0.250873
$573$	−8.00000	−0.334205
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	1.00000	0.0415945
$579$	4.00000	0.166234
$580$	−24.0000	−0.996546
$581$	−24.0000	−0.995688
$582$	−8.00000	−0.331611
$583$	12.0000	0.496989
$584$	−12.0000	−0.496564
$585$	4.00000	0.165380
$586$	2.00000	0.0826192
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−3.00000	−0.123718
$589$	40.0000	1.64817
$590$	32.0000	1.31742
$591$	12.0000	0.493614
$592$	4.00000	0.164399
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	−6.00000	−0.246183
$595$	−8.00000	−0.327968
$596$	−18.0000	−0.737309
$597$	28.0000	1.14596
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	33.0000	1.34722
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	24.0000	0.978167
$603$	12.0000	0.488678
$604$	−20.0000	−0.813788
$605$	−100.000	−4.06558
$606$	−6.00000	−0.243733
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	20.0000	0.811107
$609$	12.0000	0.486265
$610$	40.0000	1.61955
$611$	−8.00000	−0.323645
$612$	−1.00000	−0.0404226
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	−36.0000	−1.45048
$617$	−32.0000	−1.28827	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	−32.0000	−1.28827	−0.644136	+	0.764911i	$0.722783\pi$
$618$	−8.00000	−0.321807
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	40.0000	1.60644
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−12.0000	−0.480770
$624$	−1.00000	−0.0400320
$625$	41.0000	1.64000
$626$	−10.0000	−0.399680
$627$	−24.0000	−0.958468
$628$	−2.00000	−0.0798087
$629$	−4.00000	−0.159490
$630$	−8.00000	−0.318728
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	12.0000	0.477334
$633$	8.00000	0.317971
$634$	−4.00000	−0.158860
$635$	−64.0000	−2.53976
$636$	2.00000	0.0793052
$637$	3.00000	0.118864
$638$	−36.0000	−1.42525
$639$	2.00000	0.0791188
$640$	12.0000	0.474342
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	−16.0000	−0.631470
$643$	−46.0000	−1.81406	−0.907031	−	0.421063i	$-0.861657\pi$
$643$	−46.0000	−1.81406	−0.907031	+	0.421063i	$0.861657\pi$
$644$	0	0
$645$	48.0000	1.89000
$646$	4.00000	0.157378
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	−3.00000	−0.117851
$649$	−48.0000	−1.88416
$650$	−11.0000	−0.431455
$651$	−20.0000	−0.783862
$652$	−6.00000	−0.234978
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	16.0000	0.625650
$655$	48.0000	1.87552
$656$	0	0
$657$	4.00000	0.156055
$658$	16.0000	0.623745
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	−24.0000	−0.934199
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−8.00000	−0.310929
$663$	1.00000	0.0388368
$664$	36.0000	1.39707
$665$	−32.0000	−1.24091
$666$	−4.00000	−0.154997
$667$	0	0
$668$	14.0000	0.541676
$669$	12.0000	0.463947
$670$	−48.0000	−1.85440
$671$	−60.0000	−2.31627
$672$	−10.0000	−0.385758
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	2.00000	0.0770371
$675$	−11.0000	−0.423390
$676$	−1.00000	−0.0384615
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	−6.00000	−0.230429
$679$	16.0000	0.614024
$680$	12.0000	0.460179
$681$	−6.00000	−0.229920
$682$	60.0000	2.29752
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	−4.00000	−0.152944
$685$	24.0000	0.916993
$686$	−20.0000	−0.763604
$687$	−6.00000	−0.228914
$688$	−12.0000	−0.457496
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−6.00000	−0.228251	−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000	−0.228251	−0.114125	+	0.993466i	$0.536407\pi$
$692$	−2.00000	−0.0760286
$693$	12.0000	0.455842
$694$	−32.0000	−1.21470
$695$	−64.0000	−2.42766
$696$	−18.0000	−0.682288
$697$	0	0
$698$	10.0000	0.378506
$699$	−14.0000	−0.529529
$700$	−22.0000	−0.831522
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	1.00000	0.0377426
$703$	−16.0000	−0.603451
$704$	42.0000	1.58293
$705$	32.0000	1.20519
$706$	−26.0000	−0.978523
$707$	12.0000	0.451306
$708$	−8.00000	−0.300658
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	−8.00000	−0.300235
$711$	−4.00000	−0.150012
$712$	18.0000	0.674579
$713$	0	0
$714$	−2.00000	−0.0748481
$715$	24.0000	0.897549
$716$	−12.0000	−0.448461
$717$	12.0000	0.448148
$718$	−4.00000	−0.149279
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	4.00000	0.149071
$721$	16.0000	0.595871
$722$	−3.00000	−0.111648
$723$	8.00000	0.297523
$724$	10.0000	0.371647
$725$	−66.0000	−2.45118
$726$	−25.0000	−0.927837
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	−16.0000	−0.592187
$731$	12.0000	0.443836
$732$	−10.0000	−0.369611
$733$	−38.0000	−1.40356	−0.701781	−	0.712393i	$-0.747612\pi$
$733$	−38.0000	−1.40356	−0.701781	+	0.712393i	$0.747612\pi$
$734$	0	0
$735$	−12.0000	−0.442627
$736$	0	0
$737$	72.0000	2.65215
$738$	0	0
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	−16.0000	−0.588172
$741$	4.00000	0.146944
$742$	4.00000	0.146845
$743$	30.0000	1.10059	0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000	1.10059	0.550297	+	0.834969i	$0.314515\pi$
$744$	30.0000	1.09985
$745$	−72.0000	−2.63788
$746$	14.0000	0.512576
$747$	−12.0000	−0.439057
$748$	−6.00000	−0.219382
$749$	32.0000	1.16925
$750$	24.0000	0.876356
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−8.00000	−0.291730
$753$	20.0000	0.728841
$754$	6.00000	0.218507
$755$	−80.0000	−2.91150
$756$	2.00000	0.0727393
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	34.0000	1.23494
$759$	0	0
$760$	48.0000	1.74114
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	−16.0000	−0.579619
$763$	−32.0000	−1.15848
$764$	−8.00000	−0.289430
$765$	−4.00000	−0.144620
$766$	12.0000	0.433578
$767$	8.00000	0.288863
$768$	17.0000	0.613435
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	−48.0000	−1.72980
$771$	−14.0000	−0.504198
$772$	4.00000	0.143963
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	12.0000	0.431331
$775$	110.000	3.95132
$776$	−24.0000	−0.861550
$777$	8.00000	0.286998
$778$	10.0000	0.358517
$779$	0	0
$780$	4.00000	0.143223
$781$	12.0000	0.429394
$782$	0	0
$783$	6.00000	0.214423
$784$	3.00000	0.107143
$785$	−8.00000	−0.285532
$786$	12.0000	0.428026
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	12.0000	0.427482
$789$	−16.0000	−0.569615
$790$	16.0000	0.569254
$791$	12.0000	0.426671
$792$	−18.0000	−0.639602
$793$	10.0000	0.355110
$794$	−20.0000	−0.709773
$795$	8.00000	0.283731
$796$	28.0000	0.992434
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	−8.00000	−0.283197
$799$	8.00000	0.283020
$800$	55.0000	1.94454
$801$	−6.00000	−0.212000
$802$	12.0000	0.423735
$803$	24.0000	0.846942
$804$	12.0000	0.423207
$805$	0	0
$806$	−10.0000	−0.352235
$807$	30.0000	1.05605
$808$	−18.0000	−0.633238
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	−4.00000	−0.140546
$811$	−18.0000	−0.632065	−0.316033	−	0.948748i	$-0.602351\pi$
$811$	−18.0000	−0.632065	−0.316033	+	0.948748i	$0.602351\pi$
$812$	12.0000	0.421117
$813$	8.00000	0.280572
$814$	−24.0000	−0.841200
$815$	−24.0000	−0.840683
$816$	1.00000	0.0350070
$817$	48.0000	1.67931
$818$	−14.0000	−0.489499
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	6.00000	0.209274
$823$	36.0000	1.25488	0.627441	−	0.778664i	$-0.284103\pi$
$823$	36.0000	1.25488	0.627441	+	0.778664i	$0.284103\pi$
$824$	−24.0000	−0.836080
$825$	−66.0000	−2.29783
$826$	−16.0000	−0.556711
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	0	0
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	48.0000	1.66610
$831$	−2.00000	−0.0693792
$832$	−7.00000	−0.242681
$833$	−3.00000	−0.103944
$834$	−16.0000	−0.554035
$835$	56.0000	1.93796
$836$	−24.0000	−0.830057
$837$	−10.0000	−0.345651
$838$	8.00000	0.276355
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	−24.0000	−0.828079
$841$	7.00000	0.241379
$842$	2.00000	0.0689246
$843$	−2.00000	−0.0688837
$844$	8.00000	0.275371
$845$	−4.00000	−0.137604
$846$	8.00000	0.275046
$847$	50.0000	1.71802
$848$	−2.00000	−0.0686803
$849$	−12.0000	−0.411839
$850$	11.0000	0.377297
$851$	0	0
$852$	2.00000	0.0685189
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	−20.0000	−0.684386
$855$	−16.0000	−0.547188
$856$	−48.0000	−1.64061
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	6.00000	0.204837
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	48.0000	1.63679
$861$	0	0
$862$	18.0000	0.613082
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	−5.00000	−0.170103
$865$	−8.00000	−0.272008
$866$	18.0000	0.611665
$867$	−1.00000	−0.0339618
$868$	−20.0000	−0.678844
$869$	−24.0000	−0.814144
$870$	−24.0000	−0.813676
$871$	−12.0000	−0.406604
$872$	48.0000	1.62549
$873$	8.00000	0.270759
$874$	0	0
$875$	−48.0000	−1.62270
$876$	4.00000	0.135147
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	32.0000	1.07995
$879$	−2.00000	−0.0674583
$880$	24.0000	0.809040
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	−3.00000	−0.101015
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	1.00000	0.0336336
$885$	−32.0000	−1.07567
$886$	−4.00000	−0.134383
$887$	40.0000	1.34307	0.671534	−	0.740973i	$-0.265636\pi$
$887$	40.0000	1.34307	0.671534	+	0.740973i	$0.265636\pi$
$888$	−12.0000	−0.402694
$889$	32.0000	1.07325
$890$	24.0000	0.804482
$891$	6.00000	0.201008
$892$	12.0000	0.401790
$893$	32.0000	1.07084
$894$	−18.0000	−0.602010
$895$	−48.0000	−1.60446
$896$	−6.00000	−0.200446
$897$	0	0
$898$	−20.0000	−0.667409
$899$	−60.0000	−2.00111
$900$	−11.0000	−0.366667
$901$	2.00000	0.0666297
$902$	0	0
$903$	−24.0000	−0.798670
$904$	−18.0000	−0.598671
$905$	40.0000	1.32964
$906$	−20.0000	−0.664455
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−6.00000	−0.199117
$909$	6.00000	0.199007
$910$	8.00000	0.265197
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	4.00000	0.132453
$913$	−72.0000	−2.38285
$914$	10.0000	0.330771
$915$	−40.0000	−1.32236
$916$	−6.00000	−0.198246
$917$	−24.0000	−0.792550
$918$	−1.00000	−0.0330049
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	14.0000	0.461065
$923$	−2.00000	−0.0658308
$924$	12.0000	0.394771
$925$	−44.0000	−1.44671
$926$	−28.0000	−0.920137
$927$	8.00000	0.262754
$928$	−30.0000	−0.984798
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	40.0000	1.31165
$931$	−12.0000	−0.393284
$932$	−14.0000	−0.458585
$933$	12.0000	0.392862
$934$	28.0000	0.916188
$935$	−24.0000	−0.784884
$936$	3.00000	0.0980581
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	24.0000	0.783628
$939$	10.0000	0.326338
$940$	32.0000	1.04372
$941$	−44.0000	−1.43436	−0.717180	−	0.696888i	$-0.754567\pi$
$941$	−44.0000	−1.43436	−0.717180	+	0.696888i	$0.754567\pi$
$942$	−2.00000	−0.0651635
$943$	0	0
$944$	8.00000	0.260378
$945$	8.00000	0.260240
$946$	72.0000	2.34092
$947$	2.00000	0.0649913	0.0324956	−	0.999472i	$-0.489654\pi$
$947$	2.00000	0.0649913	0.0324956	+	0.999472i	$0.489654\pi$
$948$	−4.00000	−0.129914
$949$	−4.00000	−0.129845
$950$	44.0000	1.42755
$951$	4.00000	0.129709
$952$	−6.00000	−0.194461
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	2.00000	0.0647524
$955$	−32.0000	−1.03550
$956$	12.0000	0.388108
$957$	36.0000	1.16371
$958$	34.0000	1.09849
$959$	−12.0000	−0.387500
$960$	28.0000	0.903696
$961$	69.0000	2.22581
$962$	4.00000	0.128965
$963$	16.0000	0.515593
$964$	8.00000	0.257663
$965$	16.0000	0.515058
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−75.0000	−2.41059
$969$	−4.00000	−0.128499
$970$	−32.0000	−1.02746
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	1.00000	0.0320750
$973$	32.0000	1.02587
$974$	−2.00000	−0.0640841
$975$	11.0000	0.352282
$976$	10.0000	0.320092
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−6.00000	−0.191859
$979$	−36.0000	−1.15056
$980$	−12.0000	−0.383326
$981$	−16.0000	−0.510841
$982$	−36.0000	−1.14881
$983$	46.0000	1.46717	0.733586	−	0.679597i	$-0.237845\pi$
$983$	46.0000	1.46717	0.733586	+	0.679597i	$0.237845\pi$
$984$	0	0
$985$	48.0000	1.52941
$986$	−6.00000	−0.191079
$987$	−16.0000	−0.509286
$988$	4.00000	0.127257
$989$	0	0
$990$	−24.0000	−0.762770
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	50.0000	1.58750
$993$	8.00000	0.253872
$994$	4.00000	0.126872
$995$	112.000	3.55064
$996$	−12.0000	−0.380235
$997$	−34.0000	−1.07679	−0.538395	−	0.842692i	$-0.680969\pi$
$997$	−34.0000	−1.07679	−0.538395	+	0.842692i	$0.680969\pi$
$998$	−10.0000	−0.316544
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	663.2.a.c.1.1	✓	1
3.2	odd	2		1989.2.a.b.1.1		1
13.12	even	2		8619.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
663.2.a.c.1.1	✓	1	1.1	even	1	trivial
1989.2.a.b.1.1		1	3.2	odd	2
8619.2.a.e.1.1		1	13.12	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:	\( N \)	\(=\)	\( 663 = 3 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	663.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more