LMFDB - Embedded newform 6699.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6699,2,Mod(1,6699)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6699, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6699.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6699 = 3 \cdot 7 \cdot 11 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6699.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:	$53.4917843141$
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	$\chi$	$=$	6699.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{20} +1.00000 q^{21} +1.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -1.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} +5.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} -10.0000 q^{37} +2.00000 q^{39} -6.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} +2.00000 q^{45} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +2.00000 q^{52} +14.0000 q^{53} -1.00000 q^{54} +2.00000 q^{55} +3.00000 q^{56} -1.00000 q^{58} +2.00000 q^{60} +6.00000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} +2.00000 q^{68} +8.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} +1.00000 q^{75} -1.00000 q^{77} +2.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} +1.00000 q^{87} -3.00000 q^{88} +6.00000 q^{89} +2.00000 q^{90} +2.00000 q^{91} +8.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} -5.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$5$	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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−2.00000	−0.447214
$21$	1.00000	0.218218
$22$	1.00000	0.213201
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	3.00000	0.612372
$25$	−1.00000	−0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−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	$0.244865\pi$
$32$	5.00000	0.883883
$33$	−1.00000	−0.174078
$34$	−2.00000	−0.342997
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	−6.00000	−0.948683
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	1.00000	0.154303
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−1.00000	−0.150756
$45$	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
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	2.00000	0.280056
$52$	2.00000	0.277350
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	−1.00000	−0.136083
$55$	2.00000	0.269680
$56$	3.00000	0.400892
$57$	0	0
$58$	−1.00000	−0.131306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	−1.00000	−0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	2.00000	0.242536
$69$	8.00000	0.963087
$70$	−2.00000	−0.239046
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−3.00000	−0.353553
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−10.0000	−1.16248
$75$	1.00000	0.115470
$76$	0	0
$77$	−1.00000	−0.113961
$78$	2.00000	0.226455
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	−1.00000	−0.109109
$85$	−4.00000	−0.433861
$86$	4.00000	0.431331
$87$	1.00000	0.107211
$88$	−3.00000	−0.319801
$89$	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
$91$	2.00000	0.209657
$92$	8.00000	0.834058
$93$	−8.00000	−0.829561
$94$	8.00000	0.825137
$95$	0	0
$96$	−5.00000	−0.510310
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	1.00000	0.101015
$99$	1.00000	0.100504
$100$	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
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	6.00000	0.588348
$105$	2.00000	0.195180
$106$	14.0000	1.35980
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	1.00000	0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	2.00000	0.190693
$111$	10.0000	0.949158
$112$	1.00000	0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	1.00000	0.0928477
$117$	−2.00000	−0.184900
$118$	0	0
$119$	2.00000	0.183340
$120$	6.00000	0.547723
$121$	1.00000	0.0909091
$122$	6.00000	0.543214
$123$	−6.00000	−0.541002
$124$	−8.00000	−0.718421
$125$	−12.0000	−1.07331
$126$	−1.00000	−0.0890871
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−3.00000	−0.265165
$129$	−4.00000	−0.352180
$130$	−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	$0.324356\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−4.00000	−0.345547
$135$	−2.00000	−0.172133
$136$	6.00000	0.514496
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	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	$0.554260\pi$
$140$	2.00000	0.169031
$141$	−8.00000	−0.673722
$142$	−8.00000	−0.671345
$143$	−2.00000	−0.167248
$144$	−1.00000	−0.0833333
$145$	−2.00000	−0.166091
$146$	2.00000	0.165521
$147$	−1.00000	−0.0824786
$148$	10.0000	0.821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	1.00000	0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	−1.00000	−0.0805823
$155$	16.0000	1.28515
$156$	−2.00000	−0.160128
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	8.00000	0.636446
$159$	−14.0000	−1.11027
$160$	10.0000	0.790569
$161$	8.00000	0.630488
$162$	1.00000	0.0785674
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	−6.00000	−0.468521
$165$	−2.00000	−0.155700
$166$	−4.00000	−0.310460
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	−3.00000	−0.231455
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−4.00000	−0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	1.00000	0.0758098
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	6.00000	0.449719
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−2.00000	−0.149071
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	2.00000	0.148250
$183$	−6.00000	−0.443533
$184$	24.0000	1.76930
$185$	−20.0000	−1.47043
$186$	−8.00000	−0.586588
$187$	−2.00000	−0.146254
$188$	−8.00000	−0.583460
$189$	1.00000	0.0727393
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−7.00000	−0.505181
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−2.00000	−0.143592
$195$	4.00000	0.286446
$196$	−1.00000	−0.0714286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	1.00000	0.0710669
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	3.00000	0.212132
$201$	4.00000	0.282138
$202$	−6.00000	−0.422159
$203$	1.00000	0.0701862
$204$	−2.00000	−0.140028
$205$	12.0000	0.838116
$206$	−16.0000	−1.11477
$207$	−8.00000	−0.556038
$208$	2.00000	0.138675
$209$	0	0
$210$	2.00000	0.138013
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	−14.0000	−0.961524
$213$	8.00000	0.548151
$214$	−8.00000	−0.546869
$215$	8.00000	0.545595
$216$	3.00000	0.204124
$217$	−8.00000	−0.543075
$218$	14.0000	0.948200
$219$	−2.00000	−0.135147
$220$	−2.00000	−0.134840
$221$	4.00000	0.269069
$222$	10.0000	0.671156
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−5.00000	−0.334077
$225$	−1.00000	−0.0666667
$226$	14.0000	0.931266
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−16.0000	−1.05501
$231$	1.00000	0.0657952
$232$	3.00000	0.196960
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	−2.00000	−0.130744
$235$	16.0000	1.04372
$236$	0	0
$237$	−8.00000	−0.519656
$238$	2.00000	0.129641
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	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	$0.479482\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−6.00000	−0.384111
$245$	2.00000	0.127775
$246$	−6.00000	−0.382546
$247$	0	0
$248$	−24.0000	−1.52400
$249$	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	$0.540291\pi$
$252$	1.00000	0.0629941
$253$	−8.00000	−0.502956
$254$	−16.0000	−1.00393
$255$	4.00000	0.250490
$256$	−17.0000	−1.06250
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	−4.00000	−0.249029
$259$	10.0000	0.621370
$260$	4.00000	0.248069
$261$	−1.00000	−0.0618984
$262$	12.0000	0.741362
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	3.00000	0.184637
$265$	28.0000	1.72003
$266$	0	0
$267$	−6.00000	−0.367194
$268$	4.00000	0.244339
$269$	−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
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	2.00000	0.121268
$273$	−2.00000	−0.121046
$274$	−2.00000	−0.120824
$275$	−1.00000	−0.0603023
$276$	−8.00000	−0.481543
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	−4.00000	−0.239904
$279$	8.00000	0.478947
$280$	6.00000	0.358569
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	−8.00000	−0.476393
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−6.00000	−0.354169
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	2.00000	0.117242
$292$	−2.00000	−0.117041
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	−1.00000	−0.0583212
$295$	0	0
$296$	30.0000	1.74371
$297$	−1.00000	−0.0580259
$298$	−6.00000	−0.347571
$299$	16.0000	0.925304
$300$	−1.00000	−0.0577350
$301$	−4.00000	−0.230556
$302$	16.0000	0.920697
$303$	6.00000	0.344691
$304$	0	0
$305$	12.0000	0.687118
$306$	−2.00000	−0.114332
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	1.00000	0.0569803
$309$	16.0000	0.910208
$310$	16.0000	0.908739
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−6.00000	−0.339683
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	10.0000	0.564333
$315$	−2.00000	−0.112687
$316$	−8.00000	−0.450035
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	−14.0000	−0.785081
$319$	−1.00000	−0.0559893
$320$	14.0000	0.782624
$321$	8.00000	0.446516
$322$	8.00000	0.445823
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	2.00000	0.110940
$326$	16.0000	0.886158
$327$	−14.0000	−0.774202
$328$	−18.0000	−0.993884
$329$	−8.00000	−0.441054
$330$	−2.00000	−0.110096
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	4.00000	0.219529
$333$	−10.0000	−0.547997
$334$	16.0000	0.875481
$335$	−8.00000	−0.437087
$336$	−1.00000	−0.0545545
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−9.00000	−0.489535
$339$	−14.0000	−0.760376
$340$	4.00000	0.216930
$341$	8.00000	0.433224
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−12.0000	−0.646997
$345$	16.0000	0.861411
$346$	6.00000	0.322562
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	−1.00000	−0.0536056
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	1.00000	0.0534522
$351$	2.00000	0.106752
$352$	5.00000	0.266501
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	−6.00000	−0.317999
$357$	−2.00000	−0.105851
$358$	−4.00000	−0.211407
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−6.00000	−0.316228
$361$	−19.0000	−1.00000
$362$	14.0000	0.735824
$363$	−1.00000	−0.0524864
$364$	−2.00000	−0.104828
$365$	4.00000	0.209370
$366$	−6.00000	−0.313625
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	8.00000	0.417029
$369$	6.00000	0.312348
$370$	−20.0000	−1.03975
$371$	−14.0000	−0.726844
$372$	8.00000	0.414781
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−2.00000	−0.103418
$375$	12.0000	0.619677
$376$	−24.0000	−1.23771
$377$	2.00000	0.103005
$378$	1.00000	0.0514344
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	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
$385$	−2.00000	−0.101929
$386$	14.0000	0.712581
$387$	4.00000	0.203331
$388$	2.00000	0.101535
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	4.00000	0.202548
$391$	16.0000	0.809155
$392$	−3.00000	−0.151523
$393$	−12.0000	−0.605320
$394$	18.0000	0.906827
$395$	16.0000	0.805047
$396$	−1.00000	−0.0502519
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	4.00000	0.199502
$403$	−16.0000	−0.797017
$404$	6.00000	0.298511
$405$	2.00000	0.0993808
$406$	1.00000	0.0496292
$407$	−10.0000	−0.495682
$408$	−6.00000	−0.297044
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	12.0000	0.592638
$411$	2.00000	0.0986527
$412$	16.0000	0.788263
$413$	0	0
$414$	−8.00000	−0.393179
$415$	−8.00000	−0.392705
$416$	−10.0000	−0.490290
$417$	4.00000	0.195881
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	−2.00000	−0.0975900
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	28.0000	1.36302
$423$	8.00000	0.388973
$424$	−42.0000	−2.03970
$425$	2.00000	0.0970143
$426$	8.00000	0.387601
$427$	−6.00000	−0.290360
$428$	8.00000	0.386695
$429$	2.00000	0.0965609
$430$	8.00000	0.385794
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	1.00000	0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−8.00000	−0.384012
$435$	2.00000	0.0958927
$436$	−14.0000	−0.670478
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−6.00000	−0.286039
$441$	1.00000	0.0476190
$442$	4.00000	0.190261
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−10.0000	−0.474579
$445$	12.0000	0.568855
$446$	16.0000	0.757622
$447$	6.00000	0.283790
$448$	−7.00000	−0.330719
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	−1.00000	−0.0471405
$451$	6.00000	0.282529
$452$	−14.0000	−0.658505
$453$	−16.0000	−0.751746
$454$	−12.0000	−0.563188
$455$	4.00000	0.187523
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	10.0000	0.467269
$459$	2.00000	0.0933520
$460$	16.0000	0.746004
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	1.00000	0.0465242
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	1.00000	0.0464238
$465$	−16.0000	−0.741982
$466$	14.0000	0.648537
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	2.00000	0.0924500
$469$	4.00000	0.184703
$470$	16.0000	0.738025
$471$	−10.0000	−0.460776
$472$	0	0
$473$	4.00000	0.183920
$474$	−8.00000	−0.367452
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	14.0000	0.641016
$478$	−4.00000	−0.182956
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−10.0000	−0.456435
$481$	20.0000	0.911922
$482$	2.00000	0.0910975
$483$	−8.00000	−0.364013
$484$	−1.00000	−0.0454545
$485$	−4.00000	−0.181631
$486$	−1.00000	−0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−18.0000	−0.814822
$489$	−16.0000	−0.723545
$490$	2.00000	0.0903508
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	6.00000	0.270501
$493$	2.00000	0.0900755
$494$	0	0
$495$	2.00000	0.0898933
$496$	−8.00000	−0.359211
$497$	8.00000	0.358849
$498$	4.00000	0.179244
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	12.0000	0.536656
$501$	−16.0000	−0.714827
$502$	−4.00000	−0.178529
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	3.00000	0.133631
$505$	−12.0000	−0.533993
$506$	−8.00000	−0.355643
$507$	9.00000	0.399704
$508$	16.0000	0.709885
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	4.00000	0.177123
$511$	−2.00000	−0.0884748
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	−32.0000	−1.41009
$516$	4.00000	0.176090
$517$	8.00000	0.351840
$518$	10.0000	0.439375
$519$	−6.00000	−0.263371
$520$	12.0000	0.526235
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−1.00000	−0.0437688
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	−12.0000	−0.524222
$525$	−1.00000	−0.0436436
$526$	−24.0000	−1.04645
$527$	−16.0000	−0.696971
$528$	1.00000	0.0435194
$529$	41.0000	1.78261
$530$	28.0000	1.21624
$531$	0	0
$532$	0	0
$533$	−12.0000	−0.519778
$534$	−6.00000	−0.259645
$535$	−16.0000	−0.691740
$536$	12.0000	0.518321
$537$	4.00000	0.172613
$538$	−14.0000	−0.603583
$539$	1.00000	0.0430730
$540$	2.00000	0.0860663
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	20.0000	0.859074
$543$	−14.0000	−0.600798
$544$	−10.0000	−0.428746
$545$	28.0000	1.19939
$546$	−2.00000	−0.0855921
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	2.00000	0.0854358
$549$	6.00000	0.256074
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	−24.0000	−1.02151
$553$	−8.00000	−0.340195
$554$	−26.0000	−1.10463
$555$	20.0000	0.848953
$556$	4.00000	0.169638
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	8.00000	0.338667
$559$	−8.00000	−0.338364
$560$	2.00000	0.0845154
$561$	2.00000	0.0844401
$562$	−2.00000	−0.0843649
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	8.00000	0.336861
$565$	28.0000	1.17797
$566$	−4.00000	−0.168133
$567$	−1.00000	−0.0419961
$568$	24.0000	1.00702
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	−6.00000	−0.250435
$575$	8.00000	0.333623
$576$	7.00000	0.291667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−13.0000	−0.540729
$579$	−14.0000	−0.581820
$580$	2.00000	0.0830455
$581$	4.00000	0.165948
$582$	2.00000	0.0829027
$583$	14.0000	0.579821
$584$	−6.00000	−0.248282
$585$	−4.00000	−0.165380
$586$	18.0000	0.743573
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	1.00000	0.0412393
$589$	0	0
$590$	0	0
$591$	−18.0000	−0.740421
$592$	10.0000	0.410997
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	−1.00000	−0.0410305
$595$	4.00000	0.163984
$596$	6.00000	0.245770
$597$	−16.0000	−0.654836
$598$	16.0000	0.654289
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	−3.00000	−0.122474
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	−4.00000	−0.163028
$603$	−4.00000	−0.162893
$604$	−16.0000	−0.651031
$605$	2.00000	0.0813116
$606$	6.00000	0.243733
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	−1.00000	−0.0405220
$610$	12.0000	0.485866
$611$	−16.0000	−0.647291
$612$	2.00000	0.0808452
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	8.00000	0.322854
$615$	−12.0000	−0.483887
$616$	3.00000	0.120873
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	16.0000	0.643614
$619$	−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
$622$	0	0
$623$	−6.00000	−0.240385
$624$	−2.00000	−0.0800641
$625$	−19.0000	−0.760000
$626$	34.0000	1.35891
$627$	0	0
$628$	−10.0000	−0.399043
$629$	20.0000	0.797452
$630$	−2.00000	−0.0796819
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	−24.0000	−0.954669
$633$	−28.0000	−1.11290
$634$	26.0000	1.03259
$635$	−32.0000	−1.26988
$636$	14.0000	0.555136
$637$	−2.00000	−0.0792429
$638$	−1.00000	−0.0395904
$639$	−8.00000	−0.316475
$640$	−6.00000	−0.237171
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	8.00000	0.315735
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−8.00000	−0.315244
$645$	−8.00000	−0.315000
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	2.00000	0.0784465
$651$	8.00000	0.313545
$652$	−16.0000	−0.626608
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	−14.0000	−0.547443
$655$	24.0000	0.937758
$656$	−6.00000	−0.234261
$657$	2.00000	0.0780274
$658$	−8.00000	−0.311872
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	2.00000	0.0778499
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	32.0000	1.24372
$663$	−4.00000	−0.155347
$664$	12.0000	0.465690
$665$	0	0
$666$	−10.0000	−0.387492
$667$	8.00000	0.309761
$668$	−16.0000	−0.619059
$669$	−16.0000	−0.618596
$670$	−8.00000	−0.309067
$671$	6.00000	0.231627
$672$	5.00000	0.192879
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	14.0000	0.539260
$675$	1.00000	0.0384900
$676$	9.00000	0.346154
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−14.0000	−0.537667
$679$	2.00000	0.0767530
$680$	12.0000	0.460179
$681$	12.0000	0.459841
$682$	8.00000	0.306336
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	−1.00000	−0.0381802
$687$	−10.0000	−0.381524
$688$	−4.00000	−0.152499
$689$	−28.0000	−1.06672
$690$	16.0000	0.609110
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	−6.00000	−0.228086
$693$	−1.00000	−0.0379869
$694$	−8.00000	−0.303676
$695$	−8.00000	−0.303457
$696$	−3.00000	−0.113715
$697$	−12.0000	−0.454532
$698$	14.0000	0.529908
$699$	−14.0000	−0.529529
$700$	−1.00000	−0.0377964
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	7.00000	0.263822
$705$	−16.0000	−0.602595
$706$	6.00000	0.225813
$707$	6.00000	0.225653
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	−16.0000	−0.600469
$711$	8.00000	0.300023
$712$	−18.0000	−0.674579
$713$	−64.0000	−2.39682
$714$	−2.00000	−0.0748481
$715$	−4.00000	−0.149592
$716$	4.00000	0.149487
$717$	4.00000	0.149383
$718$	24.0000	0.895672
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	−2.00000	−0.0745356
$721$	16.0000	0.595871
$722$	−19.0000	−0.707107
$723$	−2.00000	−0.0743808
$724$	−14.0000	−0.520306
$725$	1.00000	0.0371391
$726$	−1.00000	−0.0371135
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	−6.00000	−0.222375
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	−8.00000	−0.295891
$732$	6.00000	0.221766
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−16.0000	−0.590571
$735$	−2.00000	−0.0737711
$736$	−40.0000	−1.47442
$737$	−4.00000	−0.147342
$738$	6.00000	0.220863
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	20.0000	0.735215
$741$	0	0
$742$	−14.0000	−0.513956
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	24.0000	0.879883
$745$	−12.0000	−0.439646
$746$	−10.0000	−0.366126
$747$	−4.00000	−0.146352
$748$	2.00000	0.0731272
$749$	8.00000	0.292314
$750$	12.0000	0.438178
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	−8.00000	−0.291730
$753$	4.00000	0.145768
$754$	2.00000	0.0728357
$755$	32.0000	1.16460
$756$	−1.00000	−0.0363696
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	32.0000	1.16229
$759$	8.00000	0.290382
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	16.0000	0.579619
$763$	−14.0000	−0.506834
$764$	0	0
$765$	−4.00000	−0.144620
$766$	12.0000	0.433578
$767$	0	0
$768$	17.0000	0.613435
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	−2.00000	−0.0720750
$771$	2.00000	0.0720282
$772$	−14.0000	−0.503871
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	4.00000	0.143777
$775$	−8.00000	−0.287368
$776$	6.00000	0.215387
$777$	−10.0000	−0.358748
$778$	−14.0000	−0.501924
$779$	0	0
$780$	−4.00000	−0.143223
$781$	−8.00000	−0.286263
$782$	16.0000	0.572159
$783$	1.00000	0.0357371
$784$	−1.00000	−0.0357143
$785$	20.0000	0.713831
$786$	−12.0000	−0.428026
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	−18.0000	−0.641223
$789$	24.0000	0.854423
$790$	16.0000	0.569254
$791$	−14.0000	−0.497783
$792$	−3.00000	−0.106600
$793$	−12.0000	−0.426132
$794$	−2.00000	−0.0709773
$795$	−28.0000	−0.993058
$796$	−16.0000	−0.567105
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	−5.00000	−0.176777
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	2.00000	0.0705785
$804$	−4.00000	−0.141069
$805$	16.0000	0.563926
$806$	−16.0000	−0.563576
$807$	14.0000	0.492823
$808$	18.0000	0.633238
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	2.00000	0.0702728
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	−1.00000	−0.0350931
$813$	−20.0000	−0.701431
$814$	−10.0000	−0.350500
$815$	32.0000	1.12091
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	−30.0000	−1.04893
$819$	2.00000	0.0698857
$820$	−12.0000	−0.419058
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	2.00000	0.0697580
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	48.0000	1.67216
$825$	1.00000	0.0348155
$826$	0	0
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	8.00000	0.278019
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	−8.00000	−0.277684
$831$	26.0000	0.901930
$832$	−14.0000	−0.485363
$833$	−2.00000	−0.0692959
$834$	4.00000	0.138509
$835$	32.0000	1.10741
$836$	0	0
$837$	−8.00000	−0.276520
$838$	24.0000	0.829066
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	−6.00000	−0.207020
$841$	1.00000	0.0344828
$842$	−18.0000	−0.620321
$843$	2.00000	0.0688837
$844$	−28.0000	−0.963800
$845$	−18.0000	−0.619219
$846$	8.00000	0.275046
$847$	−1.00000	−0.0343604
$848$	−14.0000	−0.480762
$849$	4.00000	0.137280
$850$	2.00000	0.0685994
$851$	80.0000	2.74236
$852$	−8.00000	−0.274075
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	24.0000	0.820303
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	2.00000	0.0682789
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	−8.00000	−0.272798
$861$	6.00000	0.204479
$862$	−12.0000	−0.408722
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−5.00000	−0.170103
$865$	12.0000	0.408012
$866$	−26.0000	−0.883516
$867$	13.0000	0.441503
$868$	8.00000	0.271538
$869$	8.00000	0.271381
$870$	2.00000	0.0678064
$871$	8.00000	0.271070
$872$	−42.0000	−1.42230
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	12.0000	0.405674
$876$	2.00000	0.0675737
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	−8.00000	−0.269987
$879$	−18.0000	−0.607125
$880$	−2.00000	−0.0674200
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	1.00000	0.0336718
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	4.00000	0.134383
$887$	−40.0000	−1.34307	−0.671534	−	0.740973i	$-0.734364\pi$
$887$	−40.0000	−1.34307	−0.671534	+	0.740973i	$0.734364\pi$
$888$	−30.0000	−1.00673
$889$	16.0000	0.536623
$890$	12.0000	0.402241
$891$	1.00000	0.0335013
$892$	−16.0000	−0.535720
$893$	0	0
$894$	6.00000	0.200670
$895$	−8.00000	−0.267411
$896$	3.00000	0.100223
$897$	−16.0000	−0.534224
$898$	−2.00000	−0.0667409
$899$	−8.00000	−0.266815
$900$	1.00000	0.0333333
$901$	−28.0000	−0.932815
$902$	6.00000	0.199778
$903$	4.00000	0.133112
$904$	−42.0000	−1.39690
$905$	28.0000	0.930751
$906$	−16.0000	−0.531564
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	12.0000	0.398234
$909$	−6.00000	−0.199007
$910$	4.00000	0.132599
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	2.00000	0.0661541
$915$	−12.0000	−0.396708
$916$	−10.0000	−0.330409
$917$	−12.0000	−0.396275
$918$	2.00000	0.0660098
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	48.0000	1.58251
$921$	−8.00000	−0.263609
$922$	26.0000	0.856264
$923$	16.0000	0.526646
$924$	−1.00000	−0.0328976
$925$	10.0000	0.328798
$926$	32.0000	1.05159
$927$	−16.0000	−0.525509
$928$	−5.00000	−0.164133
$929$	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
$931$	0	0
$932$	−14.0000	−0.458585
$933$	0	0
$934$	12.0000	0.392652
$935$	−4.00000	−0.130814
$936$	6.00000	0.196116
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	4.00000	0.130605
$939$	−34.0000	−1.10955
$940$	−16.0000	−0.521862
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	−10.0000	−0.325818
$943$	−48.0000	−1.56310
$944$	0	0
$945$	2.00000	0.0650600
$946$	4.00000	0.130051
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	8.00000	0.259828
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−26.0000	−0.843108
$952$	−6.00000	−0.194461
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	14.0000	0.453267
$955$	0	0
$956$	4.00000	0.129369
$957$	1.00000	0.0323254
$958$	−24.0000	−0.775405
$959$	2.00000	0.0645834
$960$	−14.0000	−0.451848
$961$	33.0000	1.06452
$962$	20.0000	0.644826
$963$	−8.00000	−0.257796
$964$	−2.00000	−0.0644157
$965$	28.0000	0.901352
$966$	−8.00000	−0.257396
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−3.00000	−0.0964237
$969$	0	0
$970$	−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
$973$	4.00000	0.128234
$974$	−32.0000	−1.02535
$975$	−2.00000	−0.0640513
$976$	−6.00000	−0.192055
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	−16.0000	−0.511624
$979$	6.00000	0.191761
$980$	−2.00000	−0.0638877
$981$	14.0000	0.446986
$982$	−36.0000	−1.14881
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	18.0000	0.573819
$985$	36.0000	1.14706
$986$	2.00000	0.0636930
$987$	8.00000	0.254643
$988$	0	0
$989$	−32.0000	−1.01754
$990$	2.00000	0.0635642
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	40.0000	1.27000
$993$	−32.0000	−1.01549
$994$	8.00000	0.253745
$995$	32.0000	1.01447
$996$	−4.00000	−0.126745
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	−4.00000	−0.126618
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6699.2.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6699.2.a.d.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6699 = 3 \cdot 7 \cdot 11 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6699.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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