LMFDB - Embedded newform 4641.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4641,2,Mod(1,4641)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4641, 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("4641.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4641 = 3 \cdot 7 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4641.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:	$37.0585715781$
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	$\chi$	$=$	4641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} -8.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} -4.00000 q^{33} -1.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -1.00000 q^{39} +6.00000 q^{40} +6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +8.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} -3.00000 q^{56} -4.00000 q^{57} +6.00000 q^{58} +2.00000 q^{60} -14.0000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} +8.00000 q^{67} -1.00000 q^{68} +8.00000 q^{69} +2.00000 q^{70} -8.00000 q^{71} +3.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} +1.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -1.00000 q^{84} +2.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} +12.0000 q^{88} -14.0000 q^{89} -2.00000 q^{90} -1.00000 q^{91} +8.00000 q^{92} +8.00000 q^{93} +12.0000 q^{94} +8.00000 q^{95} +5.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−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
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.00000	0.267261
$15$	−2.00000	−0.516398
$16$	−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	$0.348268\pi$
$20$	−2.00000	−0.447214
$21$	1.00000	0.218218
$22$	−4.00000	−0.852803
$23$	−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
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	2.00000	0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−5.00000	−0.883883
$33$	−4.00000	−0.696311
$34$	−1.00000	−0.171499
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−4.00000	−0.648886
$39$	−1.00000	−0.160128
$40$	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	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	2.00000	0.298142
$46$	8.00000	1.17954
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−1.00000	−0.140028
$52$	−1.00000	−0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	8.00000	1.07872
$56$	−3.00000	−0.400892
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.00000	0.258199
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	2.00000	0.248069
$66$	4.00000	0.492366
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−1.00000	−0.121268
$69$	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	$0.462659\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	1.00000	0.113228
$79$	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
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−1.00000	−0.109109
$85$	2.00000	0.216930
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	12.0000	1.27920
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	−2.00000	−0.210819
$91$	−1.00000	−0.104828
$92$	8.00000	0.834058
$93$	8.00000	0.829561
$94$	12.0000	1.23771
$95$	8.00000	0.820783
$96$	5.00000	0.510310
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−1.00000	−0.101015
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	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
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	3.00000	0.294174
$105$	2.00000	0.195180
$106$	−6.00000	−0.582772
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	−8.00000	−0.762770
$111$	−2.00000	−0.189832
$112$	1.00000	0.0944911
$113$	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
$115$	−16.0000	−1.49201
$116$	6.00000	0.557086
$117$	1.00000	0.0924500
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	−6.00000	−0.547723
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	−6.00000	−0.541002
$124$	8.00000	0.718421
$125$	−12.0000	−1.07331
$126$	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	$0.384500\pi$
$128$	3.00000	0.265165
$129$	4.00000	0.352180
$130$	−2.00000	−0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	4.00000	0.348155
$133$	−4.00000	−0.346844
$134$	−8.00000	−0.691095
$135$	−2.00000	−0.172133
$136$	3.00000	0.257248
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	−8.00000	−0.681005
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	2.00000	0.169031
$141$	12.0000	1.01058
$142$	8.00000	0.671345
$143$	4.00000	0.334497
$144$	−1.00000	−0.0833333
$145$	−12.0000	−0.996546
$146$	−2.00000	−0.165521
$147$	−1.00000	−0.0824786
$148$	−2.00000	−0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	12.0000	0.973329
$153$	1.00000	0.0808452
$154$	4.00000	0.322329
$155$	−16.0000	−1.28515
$156$	1.00000	0.0800641
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	−8.00000	−0.636446
$159$	−6.00000	−0.475831
$160$	−10.0000	−0.790569
$161$	8.00000	0.630488
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−6.00000	−0.468521
$165$	−8.00000	−0.622799
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	3.00000	0.231455
$169$	1.00000	0.0769231
$170$	−2.00000	−0.153393
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−6.00000	−0.454859
$175$	1.00000	0.0755929
$176$	−4.00000	−0.301511
$177$	0	0
$178$	14.0000	1.04934
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	−2.00000	−0.149071
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	1.00000	0.0741249
$183$	14.0000	1.03491
$184$	−24.0000	−1.76930
$185$	4.00000	0.294086
$186$	−8.00000	−0.586588
$187$	4.00000	0.292509
$188$	12.0000	0.875190
$189$	1.00000	0.0727393
$190$	−8.00000	−0.580381
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−7.00000	−0.505181
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−10.0000	−0.717958
$195$	−2.00000	−0.143223
$196$	−1.00000	−0.0714286
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	−4.00000	−0.284268
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	−3.00000	−0.212132
$201$	−8.00000	−0.564276
$202$	−6.00000	−0.422159
$203$	6.00000	0.421117
$204$	1.00000	0.0700140
$205$	12.0000	0.838116
$206$	−4.00000	−0.278693
$207$	−8.00000	−0.556038
$208$	−1.00000	−0.0693375
$209$	16.0000	1.10674
$210$	−2.00000	−0.138013
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−6.00000	−0.412082
$213$	8.00000	0.548151
$214$	4.00000	0.273434
$215$	−8.00000	−0.545595
$216$	−3.00000	−0.204124
$217$	8.00000	0.543075
$218$	6.00000	0.406371
$219$	−2.00000	−0.135147
$220$	−8.00000	−0.539360
$221$	1.00000	0.0672673
$222$	2.00000	0.134231
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	5.00000	0.334077
$225$	−1.00000	−0.0666667
$226$	−6.00000	−0.399114
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	4.00000	0.264906
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	16.0000	1.05501
$231$	4.00000	0.263181
$232$	−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
$235$	−24.0000	−1.56559
$236$	0	0
$237$	−8.00000	−0.519656
$238$	1.00000	0.0648204
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	2.00000	0.129099
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	14.0000	0.896258
$245$	2.00000	0.127775
$246$	6.00000	0.382546
$247$	4.00000	0.254514
$248$	−24.0000	−1.52400
$249$	0	0
$250$	12.0000	0.758947
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	1.00000	0.0629941
$253$	−32.0000	−2.01182
$254$	−8.00000	−0.501965
$255$	−2.00000	−0.125245
$256$	−17.0000	−1.06250
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	−4.00000	−0.249029
$259$	−2.00000	−0.124274
$260$	−2.00000	−0.124035
$261$	−6.00000	−0.371391
$262$	12.0000	0.741362
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−12.0000	−0.738549
$265$	12.0000	0.737154
$266$	4.00000	0.245256
$267$	14.0000	0.856786
$268$	−8.00000	−0.488678
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	2.00000	0.121716
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−1.00000	−0.0606339
$273$	1.00000	0.0605228
$274$	22.0000	1.32907
$275$	−4.00000	−0.241209
$276$	−8.00000	−0.481543
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	20.0000	1.19952
$279$	−8.00000	−0.478947
$280$	−6.00000	−0.358569
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	−12.0000	−0.714590
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	8.00000	0.474713
$285$	−8.00000	−0.473879
$286$	−4.00000	−0.236525
$287$	−6.00000	−0.354169
$288$	−5.00000	−0.294628
$289$	1.00000	0.0588235
$290$	12.0000	0.704664
$291$	−10.0000	−0.586210
$292$	−2.00000	−0.117041
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	6.00000	0.348743
$297$	−4.00000	−0.232104
$298$	−6.00000	−0.347571
$299$	−8.00000	−0.462652
$300$	−1.00000	−0.0577350
$301$	4.00000	0.230556
$302$	20.0000	1.15087
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	−28.0000	−1.60328
$306$	−1.00000	−0.0571662
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	4.00000	0.227921
$309$	−4.00000	−0.227552
$310$	16.0000	0.908739
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−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	$0.591202\pi$
$314$	−22.0000	−1.24153
$315$	−2.00000	−0.112687
$316$	−8.00000	−0.450035
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	6.00000	0.336463
$319$	−24.0000	−1.34374
$320$	14.0000	0.782624
$321$	4.00000	0.223258
$322$	−8.00000	−0.445823
$323$	4.00000	0.222566
$324$	−1.00000	−0.0555556
$325$	−1.00000	−0.0554700
$326$	12.0000	0.664619
$327$	6.00000	0.331801
$328$	18.0000	0.993884
$329$	12.0000	0.661581
$330$	8.00000	0.440386
$331$	−24.0000	−1.31916	−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000	−1.31916	−0.659580	+	0.751635i	$0.729266\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	−24.0000	−1.31322
$335$	16.0000	0.874173
$336$	−1.00000	−0.0545545
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−1.00000	−0.0543928
$339$	−6.00000	−0.325875
$340$	−2.00000	−0.108465
$341$	−32.0000	−1.73290
$342$	−4.00000	−0.216295
$343$	−1.00000	−0.0539949
$344$	−12.0000	−0.646997
$345$	16.0000	0.861411
$346$	18.0000	0.967686
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	−6.00000	−0.321634
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	−1.00000	−0.0534522
$351$	−1.00000	−0.0533761
$352$	−20.0000	−1.06600
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	14.0000	0.741999
$357$	1.00000	0.0529256
$358$	−24.0000	−1.26844
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	6.00000	0.316228
$361$	−3.00000	−0.157895
$362$	−10.0000	−0.525588
$363$	−5.00000	−0.262432
$364$	1.00000	0.0524142
$365$	4.00000	0.209370
$366$	−14.0000	−0.731792
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	8.00000	0.417029
$369$	6.00000	0.312348
$370$	−4.00000	−0.207950
$371$	−6.00000	−0.311504
$372$	−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	$0.583357\pi$
$374$	−4.00000	−0.206835
$375$	12.0000	0.619677
$376$	−36.0000	−1.85656
$377$	−6.00000	−0.309016
$378$	−1.00000	−0.0514344
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	−8.00000	−0.410391
$381$	−8.00000	−0.409852
$382$	12.0000	0.613973
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	−3.00000	−0.153093
$385$	−8.00000	−0.407718
$386$	−6.00000	−0.305392
$387$	−4.00000	−0.203331
$388$	−10.0000	−0.507673
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	2.00000	0.101274
$391$	−8.00000	−0.404577
$392$	3.00000	0.151523
$393$	12.0000	0.605320
$394$	10.0000	0.503793
$395$	16.0000	0.805047
$396$	−4.00000	−0.201008
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	24.0000	1.20301
$399$	4.00000	0.200250
$400$	1.00000	0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	8.00000	0.399004
$403$	−8.00000	−0.398508
$404$	−6.00000	−0.298511
$405$	2.00000	0.0993808
$406$	−6.00000	−0.297775
$407$	8.00000	0.396545
$408$	−3.00000	−0.148522
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−12.0000	−0.592638
$411$	22.0000	1.08518
$412$	−4.00000	−0.197066
$413$	0	0
$414$	8.00000	0.393179
$415$	0	0
$416$	−5.00000	−0.245145
$417$	20.0000	0.979404
$418$	−16.0000	−0.782586
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	−2.00000	−0.0975900
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	12.0000	0.584151
$423$	−12.0000	−0.583460
$424$	18.0000	0.874157
$425$	−1.00000	−0.0485071
$426$	−8.00000	−0.387601
$427$	14.0000	0.677507
$428$	4.00000	0.193347
$429$	−4.00000	−0.193122
$430$	8.00000	0.385794
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000	0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	−8.00000	−0.384012
$435$	12.0000	0.575356
$436$	6.00000	0.287348
$437$	−32.0000	−1.53077
$438$	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	$0.561144\pi$
$440$	24.0000	1.14416
$441$	1.00000	0.0476190
$442$	−1.00000	−0.0475651
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	2.00000	0.0949158
$445$	−28.0000	−1.32733
$446$	8.00000	0.378811
$447$	−6.00000	−0.283790
$448$	−7.00000	−0.330719
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	1.00000	0.0471405
$451$	24.0000	1.13012
$452$	−6.00000	−0.282216
$453$	20.0000	0.939682
$454$	−4.00000	−0.187729
$455$	−2.00000	−0.0937614
$456$	−12.0000	−0.561951
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−6.00000	−0.280362
$459$	−1.00000	−0.0466760
$460$	16.0000	0.746004
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	−4.00000	−0.186097
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	6.00000	0.278543
$465$	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	$0.410445\pi$
$468$	−1.00000	−0.0462250
$469$	−8.00000	−0.369406
$470$	24.0000	1.10704
$471$	−22.0000	−1.01371
$472$	0	0
$473$	−16.0000	−0.735681
$474$	8.00000	0.367452
$475$	−4.00000	−0.183533
$476$	1.00000	0.0458349
$477$	6.00000	0.274721
$478$	−8.00000	−0.365911
$479$	−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
$481$	2.00000	0.0911922
$482$	22.0000	1.00207
$483$	−8.00000	−0.364013
$484$	−5.00000	−0.227273
$485$	20.0000	0.908153
$486$	1.00000	0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−42.0000	−1.90125
$489$	12.0000	0.542659
$490$	−2.00000	−0.0903508
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	6.00000	0.270501
$493$	−6.00000	−0.270226
$494$	−4.00000	−0.179969
$495$	8.00000	0.359573
$496$	8.00000	0.359211
$497$	8.00000	0.358849
$498$	0	0
$499$	−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
$501$	−24.0000	−1.07224
$502$	−12.0000	−0.535586
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	−3.00000	−0.133631
$505$	12.0000	0.533993
$506$	32.0000	1.42257
$507$	−1.00000	−0.0444116
$508$	−8.00000	−0.354943
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	2.00000	0.0885615
$511$	−2.00000	−0.0884748
$512$	11.0000	0.486136
$513$	−4.00000	−0.176604
$514$	−18.0000	−0.793946
$515$	8.00000	0.352522
$516$	−4.00000	−0.176090
$517$	−48.0000	−2.11104
$518$	2.00000	0.0878750
$519$	18.0000	0.790112
$520$	6.00000	0.263117
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	6.00000	0.262613
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	12.0000	0.524222
$525$	−1.00000	−0.0436436
$526$	−4.00000	−0.174408
$527$	−8.00000	−0.348485
$528$	4.00000	0.174078
$529$	41.0000	1.78261
$530$	−12.0000	−0.521247
$531$	0	0
$532$	4.00000	0.173422
$533$	6.00000	0.259889
$534$	−14.0000	−0.605839
$535$	−8.00000	−0.345870
$536$	24.0000	1.03664
$537$	−24.0000	−1.03568
$538$	10.0000	0.431131
$539$	4.00000	0.172292
$540$	2.00000	0.0860663
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	−5.00000	−0.214373
$545$	−12.0000	−0.514024
$546$	−1.00000	−0.0427960
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	22.0000	0.939793
$549$	−14.0000	−0.597505
$550$	4.00000	0.170561
$551$	−24.0000	−1.02243
$552$	24.0000	1.02151
$553$	−8.00000	−0.340195
$554$	10.0000	0.424859
$555$	−4.00000	−0.169791
$556$	20.0000	0.848189
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	8.00000	0.338667
$559$	−4.00000	−0.169182
$560$	2.00000	0.0845154
$561$	−4.00000	−0.168880
$562$	−26.0000	−1.09674
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	−12.0000	−0.505291
$565$	12.0000	0.504844
$566$	20.0000	0.840663
$567$	−1.00000	−0.0419961
$568$	−24.0000	−1.00702
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	8.00000	0.335083
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−4.00000	−0.167248
$573$	12.0000	0.501307
$574$	6.00000	0.250435
$575$	8.00000	0.333623
$576$	7.00000	0.291667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	−1.00000	−0.0415945
$579$	−6.00000	−0.249351
$580$	12.0000	0.498273
$581$	0	0
$582$	10.0000	0.414513
$583$	24.0000	0.993978
$584$	6.00000	0.248282
$585$	2.00000	0.0826898
$586$	2.00000	0.0826192
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	1.00000	0.0412393
$589$	−32.0000	−1.31854
$590$	0	0
$591$	10.0000	0.411345
$592$	−2.00000	−0.0821995
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	4.00000	0.164122
$595$	−2.00000	−0.0819920
$596$	−6.00000	−0.245770
$597$	24.0000	0.982255
$598$	8.00000	0.327144
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	3.00000	0.122474
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	−4.00000	−0.163028
$603$	8.00000	0.325785
$604$	20.0000	0.813788
$605$	10.0000	0.406558
$606$	6.00000	0.243733
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	−20.0000	−0.811107
$609$	−6.00000	−0.243132
$610$	28.0000	1.13369
$611$	−12.0000	−0.485468
$612$	−1.00000	−0.0404226
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−12.0000	−0.484281
$615$	−12.0000	−0.483887
$616$	−12.0000	−0.483494
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	4.00000	0.160904
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	16.0000	0.642575
$621$	8.00000	0.321029
$622$	−8.00000	−0.320771
$623$	14.0000	0.560898
$624$	1.00000	0.0400320
$625$	−19.0000	−0.760000
$626$	10.0000	0.399680
$627$	−16.0000	−0.638978
$628$	−22.0000	−0.877896
$629$	2.00000	0.0797452
$630$	2.00000	0.0796819
$631$	36.0000	1.43314	0.716569	−	0.697517i	$-0.245712\pi$
$631$	36.0000	1.43314	0.716569	+	0.697517i	$0.245712\pi$
$632$	24.0000	0.954669
$633$	12.0000	0.476957
$634$	2.00000	0.0794301
$635$	16.0000	0.634941
$636$	6.00000	0.237915
$637$	1.00000	0.0396214
$638$	24.0000	0.950169
$639$	−8.00000	−0.316475
$640$	6.00000	0.237171
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	−4.00000	−0.157867
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	−8.00000	−0.315244
$645$	8.00000	0.315000
$646$	−4.00000	−0.157378
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	3.00000	0.117851
$649$	0	0
$650$	1.00000	0.0392232
$651$	−8.00000	−0.313545
$652$	12.0000	0.469956
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−6.00000	−0.234619
$655$	−24.0000	−0.937758
$656$	−6.00000	−0.234261
$657$	2.00000	0.0780274
$658$	−12.0000	−0.467809
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	8.00000	0.311400
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	24.0000	0.932786
$663$	−1.00000	−0.0388368
$664$	0	0
$665$	−8.00000	−0.310227
$666$	−2.00000	−0.0774984
$667$	48.0000	1.85857
$668$	−24.0000	−0.928588
$669$	8.00000	0.309298
$670$	−16.0000	−0.618134
$671$	−56.0000	−2.16186
$672$	−5.00000	−0.192879
$673$	42.0000	1.61898	0.809491	−	0.587133i	$-0.199743\pi$
$673$	42.0000	1.61898	0.809491	+	0.587133i	$0.199743\pi$
$674$	−18.0000	−0.693334
$675$	1.00000	0.0384900
$676$	−1.00000	−0.0384615
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	6.00000	0.230429
$679$	−10.0000	−0.383765
$680$	6.00000	0.230089
$681$	−4.00000	−0.153280
$682$	32.0000	1.22534
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−4.00000	−0.152944
$685$	−44.0000	−1.68115
$686$	1.00000	0.0381802
$687$	−6.00000	−0.228914
$688$	4.00000	0.152499
$689$	6.00000	0.228582
$690$	−16.0000	−0.609110
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	18.0000	0.684257
$693$	−4.00000	−0.151947
$694$	−4.00000	−0.151838
$695$	−40.0000	−1.51729
$696$	18.0000	0.682288
$697$	6.00000	0.227266
$698$	34.0000	1.28692
$699$	−14.0000	−0.529529
$700$	−1.00000	−0.0377964
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	1.00000	0.0377426
$703$	8.00000	0.301726
$704$	28.0000	1.05529
$705$	24.0000	0.903892
$706$	−18.0000	−0.677439
$707$	−6.00000	−0.225653
$708$	0	0
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	16.0000	0.600469
$711$	8.00000	0.300023
$712$	−42.0000	−1.57402
$713$	64.0000	2.39682
$714$	−1.00000	−0.0374241
$715$	8.00000	0.299183
$716$	−24.0000	−0.896922
$717$	−8.00000	−0.298765
$718$	32.0000	1.19423
$719$	32.0000	1.19340	0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	−2.00000	−0.0745356
$721$	−4.00000	−0.148968
$722$	3.00000	0.111648
$723$	22.0000	0.818189
$724$	−10.0000	−0.371647
$725$	6.00000	0.222834
$726$	5.00000	0.185567
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	−3.00000	−0.111187
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	−4.00000	−0.147945
$732$	−14.0000	−0.517455
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−8.00000	−0.295285
$735$	−2.00000	−0.0737711
$736$	40.0000	1.47442
$737$	32.0000	1.17874
$738$	−6.00000	−0.220863
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	−4.00000	−0.147043
$741$	−4.00000	−0.146944
$742$	6.00000	0.220267
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	24.0000	0.879883
$745$	12.0000	0.439646
$746$	10.0000	0.366126
$747$	0	0
$748$	−4.00000	−0.146254
$749$	4.00000	0.146157
$750$	−12.0000	−0.438178
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	12.0000	0.437595
$753$	−12.0000	−0.437304
$754$	6.00000	0.218507
$755$	−40.0000	−1.45575
$756$	−1.00000	−0.0363696
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	12.0000	0.435860
$759$	32.0000	1.16153
$760$	24.0000	0.870572
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	8.00000	0.289809
$763$	6.00000	0.217215
$764$	12.0000	0.434145
$765$	2.00000	0.0723102
$766$	20.0000	0.722629
$767$	0	0
$768$	17.0000	0.613435
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	8.00000	0.288300
$771$	−18.0000	−0.648254
$772$	−6.00000	−0.215945
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	4.00000	0.143777
$775$	8.00000	0.287368
$776$	30.0000	1.07694
$777$	2.00000	0.0717496
$778$	18.0000	0.645331
$779$	24.0000	0.859889
$780$	2.00000	0.0716115
$781$	−32.0000	−1.14505
$782$	8.00000	0.286079
$783$	6.00000	0.214423
$784$	−1.00000	−0.0357143
$785$	44.0000	1.57043
$786$	−12.0000	−0.428026
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	10.0000	0.356235
$789$	−4.00000	−0.142404
$790$	−16.0000	−0.569254
$791$	−6.00000	−0.213335
$792$	12.0000	0.426401
$793$	−14.0000	−0.497155
$794$	2.00000	0.0709773
$795$	−12.0000	−0.425596
$796$	24.0000	0.850657
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	−4.00000	−0.141598
$799$	−12.0000	−0.424529
$800$	5.00000	0.176777
$801$	−14.0000	−0.494666
$802$	−10.0000	−0.353112
$803$	8.00000	0.282314
$804$	8.00000	0.282138
$805$	16.0000	0.563926
$806$	8.00000	0.281788
$807$	10.0000	0.352017
$808$	18.0000	0.633238
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	−2.00000	−0.0702728
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	−8.00000	−0.280400
$815$	−24.0000	−0.840683
$816$	1.00000	0.0350070
$817$	−16.0000	−0.559769
$818$	−10.0000	−0.349642
$819$	−1.00000	−0.0349428
$820$	−12.0000	−0.419058
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−22.0000	−0.767338
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	12.0000	0.418040
$825$	4.00000	0.139262
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	8.00000	0.278019
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	7.00000	0.242681
$833$	1.00000	0.0346479
$834$	−20.0000	−0.692543
$835$	48.0000	1.66111
$836$	−16.0000	−0.553372
$837$	8.00000	0.276520
$838$	−28.0000	−0.967244
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	6.00000	0.207020
$841$	7.00000	0.241379
$842$	−6.00000	−0.206774
$843$	−26.0000	−0.895488
$844$	12.0000	0.413057
$845$	2.00000	0.0688021
$846$	12.0000	0.412568
$847$	−5.00000	−0.171802
$848$	−6.00000	−0.206041
$849$	20.0000	0.686398
$850$	1.00000	0.0342997
$851$	−16.0000	−0.548473
$852$	−8.00000	−0.274075
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	−14.0000	−0.479070
$855$	8.00000	0.273594
$856$	−12.0000	−0.410152
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	4.00000	0.136558
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	8.00000	0.272798
$861$	6.00000	0.204479
$862$	−24.0000	−0.817443
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	5.00000	0.170103
$865$	−36.0000	−1.22404
$866$	14.0000	0.475739
$867$	−1.00000	−0.0339618
$868$	−8.00000	−0.271538
$869$	32.0000	1.08553
$870$	−12.0000	−0.406838
$871$	8.00000	0.271070
$872$	−18.0000	−0.609557
$873$	10.0000	0.338449
$874$	32.0000	1.08242
$875$	12.0000	0.405674
$876$	2.00000	0.0675737
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	8.00000	0.269987
$879$	2.00000	0.0674583
$880$	−8.00000	−0.269680
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	−1.00000	−0.0336718
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	−1.00000	−0.0336336
$885$	0	0
$886$	−16.0000	−0.537531
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−6.00000	−0.201347
$889$	−8.00000	−0.268311
$890$	28.0000	0.938562
$891$	4.00000	0.134005
$892$	8.00000	0.267860
$893$	−48.0000	−1.60626
$894$	6.00000	0.200670
$895$	48.0000	1.60446
$896$	−3.00000	−0.100223
$897$	8.00000	0.267112
$898$	30.0000	1.00111
$899$	48.0000	1.60089
$900$	1.00000	0.0333333
$901$	6.00000	0.199889
$902$	−24.0000	−0.799113
$903$	−4.00000	−0.133112
$904$	18.0000	0.598671
$905$	20.0000	0.664822
$906$	−20.0000	−0.664455
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	−4.00000	−0.132745
$909$	6.00000	0.199007
$910$	2.00000	0.0662994
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	22.0000	0.727695
$915$	28.0000	0.925651
$916$	−6.00000	−0.198246
$917$	12.0000	0.396275
$918$	1.00000	0.0330049
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	−48.0000	−1.58251
$921$	−12.0000	−0.395413
$922$	−30.0000	−0.987997
$923$	−8.00000	−0.263323
$924$	−4.00000	−0.131590
$925$	−2.00000	−0.0657596
$926$	20.0000	0.657241
$927$	4.00000	0.131377
$928$	30.0000	0.984798
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	−16.0000	−0.524661
$931$	4.00000	0.131095
$932$	−14.0000	−0.458585
$933$	−8.00000	−0.261908
$934$	−12.0000	−0.392652
$935$	8.00000	0.261628
$936$	3.00000	0.0980581
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	8.00000	0.261209
$939$	10.0000	0.326338
$940$	24.0000	0.782794
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	22.0000	0.716799
$943$	−48.0000	−1.56310
$944$	0	0
$945$	2.00000	0.0650600
$946$	16.0000	0.520205
$947$	60.0000	1.94974	0.974869	−	0.222779i	$-0.0715128\pi$
$947$	60.0000	1.94974	0.974869	+	0.222779i	$0.0715128\pi$
$948$	8.00000	0.259828
$949$	2.00000	0.0649227
$950$	4.00000	0.129777
$951$	2.00000	0.0648544
$952$	−3.00000	−0.0972306
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	−6.00000	−0.194257
$955$	−24.0000	−0.776622
$956$	−8.00000	−0.258738
$957$	24.0000	0.775810
$958$	24.0000	0.775405
$959$	22.0000	0.710417
$960$	−14.0000	−0.451848
$961$	33.0000	1.06452
$962$	−2.00000	−0.0644826
$963$	−4.00000	−0.128898
$964$	22.0000	0.708572
$965$	12.0000	0.386294
$966$	8.00000	0.257396
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	15.0000	0.482118
$969$	−4.00000	−0.128499
$970$	−20.0000	−0.642161
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	1.00000	0.0320750
$973$	20.0000	0.641171
$974$	16.0000	0.512673
$975$	1.00000	0.0320256
$976$	14.0000	0.448129
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−12.0000	−0.383718
$979$	−56.0000	−1.78977
$980$	−2.00000	−0.0638877
$981$	−6.00000	−0.191565
$982$	8.00000	0.255290
$983$	−40.0000	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$983$	−40.0000	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$984$	−18.0000	−0.573819
$985$	−20.0000	−0.637253
$986$	6.00000	0.191079
$987$	−12.0000	−0.381964
$988$	−4.00000	−0.127257
$989$	32.0000	1.01754
$990$	−8.00000	−0.254257
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	40.0000	1.27000
$993$	24.0000	0.761617
$994$	−8.00000	−0.253745
$995$	−48.0000	−1.52170
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	4.00000	0.126618
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4641.2.a.c.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4641.2.a.c.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:	\( N \)	\(=\)	\( 4641 = 3 \cdot 7 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4641.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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