LMFDB - Embedded newform 1155.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, 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("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1155.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:	$9.22272143346$
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	$\chi$	$=$	1155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{22} +8.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{30} +8.00000 q^{31} -5.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -1.00000 q^{35} -1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +3.00000 q^{40} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -3.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} -12.0000 q^{67} +6.00000 q^{68} -8.00000 q^{69} +1.00000 q^{70} -8.00000 q^{71} +3.00000 q^{72} -6.00000 q^{73} -6.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{77} -2.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -1.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} -6.00000 q^{87} +3.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -8.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} +5.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	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
$15$	−1.00000	−0.258199
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	−1.00000	−0.213201
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	1.00000	0.182574
$31$	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
$34$	6.00000	1.02899
$35$	−1.00000	−0.169031
$36$	−1.00000	−0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	3.00000	0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−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	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	1.00000	0.149071
$46$	−8.00000	−1.17954
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	6.00000	0.840168
$52$	2.00000	0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	−3.00000	−0.400892
$57$	4.00000	0.529813
$58$	−6.00000	−0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	1.00000	0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−8.00000	−1.01600
$63$	−1.00000	−0.125988
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	1.00000	0.123091
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	6.00000	0.727607
$69$	−8.00000	−0.963087
$70$	1.00000	0.119523
$71$	−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
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−6.00000	−0.697486
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	−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	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	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
$85$	−6.00000	−0.650791
$86$	4.00000	0.431331
$87$	−6.00000	−0.643268
$88$	3.00000	0.319801
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	−1.00000	−0.105409
$91$	2.00000	0.209657
$92$	−8.00000	−0.834058
$93$	−8.00000	−0.829561
$94$	8.00000	0.825137
$95$	−4.00000	−0.410391
$96$	5.00000	0.510310
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−1.00000	−0.101015
$99$	1.00000	0.100504
$100$	−1.00000	−0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−6.00000	−0.594089
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−6.00000	−0.588348
$105$	1.00000	0.0975900
$106$	10.0000	0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−1.00000	−0.0953463
$111$	−6.00000	−0.569495
$112$	1.00000	0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−4.00000	−0.374634
$115$	8.00000	0.746004
$116$	−6.00000	−0.557086
$117$	−2.00000	−0.184900
$118$	12.0000	1.10469
$119$	6.00000	0.550019
$120$	−3.00000	−0.273861
$121$	1.00000	0.0909091
$122$	10.0000	0.905357
$123$	6.00000	0.541002
$124$	−8.00000	−0.718421
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	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	$0.675644\pi$
$132$	1.00000	0.0870388
$133$	4.00000	0.346844
$134$	12.0000	1.03664
$135$	−1.00000	−0.0860663
$136$	−18.0000	−1.54349
$137$	−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
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	1.00000	0.0845154
$141$	8.00000	0.673722
$142$	8.00000	0.671345
$143$	−2.00000	−0.167248
$144$	−1.00000	−0.0833333
$145$	6.00000	0.498273
$146$	6.00000	0.496564
$147$	−1.00000	−0.0824786
$148$	−6.00000	−0.493197
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	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	$0.274337\pi$
$152$	−12.0000	−0.973329
$153$	−6.00000	−0.485071
$154$	1.00000	0.0805823
$155$	8.00000	0.642575
$156$	−2.00000	−0.160128
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−8.00000	−0.636446
$159$	10.0000	0.793052
$160$	−5.00000	−0.395285
$161$	−8.00000	−0.630488
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	6.00000	0.468521
$165$	−1.00000	−0.0778499
$166$	4.00000	0.310460
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	3.00000	0.231455
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	−4.00000	−0.305888
$172$	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	$0.426753\pi$
$174$	6.00000	0.454859
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	12.0000	0.901975
$178$	−10.0000	−0.749532
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−1.00000	−0.0745356
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	−2.00000	−0.148250
$183$	10.0000	0.739221
$184$	24.0000	1.76930
$185$	6.00000	0.441129
$186$	8.00000	0.586588
$187$	−6.00000	−0.438763
$188$	8.00000	0.583460
$189$	1.00000	0.0727393
$190$	4.00000	0.290191
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−7.00000	−0.505181
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	14.0000	1.00514
$195$	2.00000	0.143223
$196$	−1.00000	−0.0714286
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	−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
$201$	12.0000	0.846415
$202$	−6.00000	−0.422159
$203$	−6.00000	−0.421117
$204$	−6.00000	−0.420084
$205$	−6.00000	−0.419058
$206$	−16.0000	−1.11477
$207$	8.00000	0.556038
$208$	2.00000	0.138675
$209$	−4.00000	−0.276686
$210$	−1.00000	−0.0690066
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	10.0000	0.686803
$213$	8.00000	0.548151
$214$	12.0000	0.820303
$215$	−4.00000	−0.272798
$216$	−3.00000	−0.204124
$217$	−8.00000	−0.543075
$218$	2.00000	0.135457
$219$	6.00000	0.405442
$220$	−1.00000	−0.0674200
$221$	12.0000	0.807207
$222$	6.00000	0.402694
$223$	−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
$226$	14.0000	0.931266
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−4.00000	−0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−8.00000	−0.527504
$231$	1.00000	0.0657952
$232$	18.0000	1.18176
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	2.00000	0.130744
$235$	−8.00000	−0.521862
$236$	12.0000	0.781133
$237$	−8.00000	−0.519656
$238$	−6.00000	−0.388922
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	1.00000	0.0645497
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	1.00000	0.0638877
$246$	−6.00000	−0.382546
$247$	8.00000	0.509028
$248$	24.0000	1.52400
$249$	4.00000	0.253490
$250$	−1.00000	−0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	1.00000	0.0629941
$253$	8.00000	0.502956
$254$	0	0
$255$	6.00000	0.375735
$256$	−17.0000	−1.06250
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	−4.00000	−0.249029
$259$	−6.00000	−0.372822
$260$	2.00000	0.124035
$261$	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	$0.335790\pi$
$264$	−3.00000	−0.184637
$265$	−10.0000	−0.614295
$266$	−4.00000	−0.245256
$267$	−10.0000	−0.611990
$268$	12.0000	0.733017
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	1.00000	0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	6.00000	0.363803
$273$	−2.00000	−0.121046
$274$	22.0000	1.32907
$275$	1.00000	0.0603023
$276$	8.00000	0.481543
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−4.00000	−0.239904
$279$	8.00000	0.478947
$280$	−3.00000	−0.179284
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−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
$285$	4.00000	0.236940
$286$	2.00000	0.118262
$287$	6.00000	0.354169
$288$	−5.00000	−0.294628
$289$	19.0000	1.11765
$290$	−6.00000	−0.352332
$291$	14.0000	0.820695
$292$	6.00000	0.351123
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	1.00000	0.0583212
$295$	−12.0000	−0.698667
$296$	18.0000	1.04623
$297$	−1.00000	−0.0580259
$298$	18.0000	1.04271
$299$	−16.0000	−0.925304
$300$	1.00000	0.0577350
$301$	4.00000	0.230556
$302$	−16.0000	−0.920697
$303$	−6.00000	−0.344691
$304$	4.00000	0.229416
$305$	−10.0000	−0.572598
$306$	6.00000	0.342997
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	1.00000	0.0569803
$309$	−16.0000	−0.910208
$310$	−8.00000	−0.454369
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	6.00000	0.339683
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	2.00000	0.112867
$315$	−1.00000	−0.0563436
$316$	−8.00000	−0.450035
$317$	−34.0000	−1.90963	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−34.0000	−1.90963	−0.954815	+	0.297200i	$0.903947\pi$
$318$	−10.0000	−0.560772
$319$	6.00000	0.335936
$320$	7.00000	0.391312
$321$	12.0000	0.669775
$322$	8.00000	0.445823
$323$	24.0000	1.33540
$324$	−1.00000	−0.0555556
$325$	−2.00000	−0.110940
$326$	−20.0000	−1.10770
$327$	2.00000	0.110600
$328$	−18.0000	−0.993884
$329$	8.00000	0.441054
$330$	1.00000	0.0550482
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	4.00000	0.219529
$333$	6.00000	0.328798
$334$	16.0000	0.875481
$335$	−12.0000	−0.655630
$336$	−1.00000	−0.0545545
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	9.00000	0.489535
$339$	14.0000	0.760376
$340$	6.00000	0.325396
$341$	8.00000	0.433224
$342$	4.00000	0.216295
$343$	−1.00000	−0.0539949
$344$	−12.0000	−0.646997
$345$	−8.00000	−0.430706
$346$	−6.00000	−0.322562
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	6.00000	0.321634
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	1.00000	0.0534522
$351$	2.00000	0.106752
$352$	−5.00000	−0.266501
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	−12.0000	−0.637793
$355$	−8.00000	−0.424596
$356$	−10.0000	−0.529999
$357$	−6.00000	−0.317554
$358$	−4.00000	−0.211407
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	3.00000	0.158114
$361$	−3.00000	−0.157895
$362$	−6.00000	−0.315353
$363$	−1.00000	−0.0524864
$364$	−2.00000	−0.104828
$365$	−6.00000	−0.314054
$366$	−10.0000	−0.522708
$367$	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
$370$	−6.00000	−0.311925
$371$	10.0000	0.519174
$372$	8.00000	0.414781
$373$	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
$375$	−1.00000	−0.0516398
$376$	−24.0000	−1.23771
$377$	−12.0000	−0.618031
$378$	−1.00000	−0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−16.0000	−0.818631
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−3.00000	−0.153093
$385$	−1.00000	−0.0509647
$386$	22.0000	1.11977
$387$	−4.00000	−0.203331
$388$	14.0000	0.710742
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	−2.00000	−0.101274
$391$	−48.0000	−2.42746
$392$	3.00000	0.151523
$393$	12.0000	0.605320
$394$	−22.0000	−1.10834
$395$	8.00000	0.402524
$396$	−1.00000	−0.0502519
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−16.0000	−0.802008
$399$	−4.00000	−0.200250
$400$	−1.00000	−0.0500000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	−12.0000	−0.598506
$403$	−16.0000	−0.797017
$404$	−6.00000	−0.298511
$405$	1.00000	0.0496904
$406$	6.00000	0.297775
$407$	6.00000	0.297409
$408$	18.0000	0.891133
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	6.00000	0.296319
$411$	22.0000	1.08518
$412$	−16.0000	−0.788263
$413$	12.0000	0.590481
$414$	−8.00000	−0.393179
$415$	−4.00000	−0.196352
$416$	10.0000	0.490290
$417$	−4.00000	−0.195881
$418$	4.00000	0.195646
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	−1.00000	−0.0487950
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	4.00000	0.194717
$423$	−8.00000	−0.388973
$424$	−30.0000	−1.45693
$425$	−6.00000	−0.291043
$426$	−8.00000	−0.387601
$427$	10.0000	0.483934
$428$	12.0000	0.580042
$429$	2.00000	0.0965609
$430$	4.00000	0.192897
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	8.00000	0.384012
$435$	−6.00000	−0.287678
$436$	2.00000	0.0957826
$437$	−32.0000	−1.53077
$438$	−6.00000	−0.286691
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	3.00000	0.143019
$441$	1.00000	0.0476190
$442$	−12.0000	−0.570782
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	6.00000	0.284747
$445$	10.0000	0.474045
$446$	8.00000	0.378811
$447$	18.0000	0.851371
$448$	−7.00000	−0.330719
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	−1.00000	−0.0471405
$451$	−6.00000	−0.282529
$452$	14.0000	0.658505
$453$	−16.0000	−0.751746
$454$	−12.0000	−0.563188
$455$	2.00000	0.0937614
$456$	12.0000	0.561951
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	10.0000	0.467269
$459$	6.00000	0.280056
$460$	−8.00000	−0.373002
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	−1.00000	−0.0465242
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−6.00000	−0.278543
$465$	−8.00000	−0.370991
$466$	22.0000	1.01913
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	2.00000	0.0924500
$469$	12.0000	0.554109
$470$	8.00000	0.369012
$471$	2.00000	0.0921551
$472$	−36.0000	−1.65703
$473$	−4.00000	−0.183920
$474$	8.00000	0.367452
$475$	−4.00000	−0.183533
$476$	−6.00000	−0.275010
$477$	−10.0000	−0.457869
$478$	16.0000	0.731823
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	5.00000	0.228218
$481$	−12.0000	−0.547153
$482$	22.0000	1.00207
$483$	8.00000	0.364013
$484$	−1.00000	−0.0454545
$485$	−14.0000	−0.635707
$486$	1.00000	0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−30.0000	−1.35804
$489$	−20.0000	−0.904431
$490$	−1.00000	−0.0451754
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−6.00000	−0.270501
$493$	−36.0000	−1.62136
$494$	−8.00000	−0.359937
$495$	1.00000	0.0449467
$496$	−8.00000	−0.359211
$497$	8.00000	0.358849
$498$	−4.00000	−0.179244
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	−1.00000	−0.0447214
$501$	16.0000	0.714827
$502$	−20.0000	−0.892644
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	−3.00000	−0.133631
$505$	6.00000	0.266996
$506$	−8.00000	−0.355643
$507$	9.00000	0.399704
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−6.00000	−0.265684
$511$	6.00000	0.265424
$512$	11.0000	0.486136
$513$	4.00000	0.176604
$514$	30.0000	1.32324
$515$	16.0000	0.705044
$516$	−4.00000	−0.176090
$517$	−8.00000	−0.351840
$518$	6.00000	0.263625
$519$	−6.00000	−0.263371
$520$	−6.00000	−0.263117
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−6.00000	−0.262613
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	12.0000	0.524222
$525$	1.00000	0.0436436
$526$	−16.0000	−0.697633
$527$	−48.0000	−2.09091
$528$	1.00000	0.0435194
$529$	41.0000	1.78261
$530$	10.0000	0.434372
$531$	−12.0000	−0.520756
$532$	−4.00000	−0.173422
$533$	12.0000	0.519778
$534$	10.0000	0.432742
$535$	−12.0000	−0.518805
$536$	−36.0000	−1.55496
$537$	−4.00000	−0.172613
$538$	−14.0000	−0.603583
$539$	1.00000	0.0430730
$540$	1.00000	0.0430331
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−8.00000	−0.343629
$543$	−6.00000	−0.257485
$544$	30.0000	1.28624
$545$	−2.00000	−0.0856706
$546$	2.00000	0.0855921
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	22.0000	0.939793
$549$	−10.0000	−0.426790
$550$	−1.00000	−0.0426401
$551$	−24.0000	−1.02243
$552$	−24.0000	−1.02151
$553$	−8.00000	−0.340195
$554$	18.0000	0.764747
$555$	−6.00000	−0.254686
$556$	−4.00000	−0.169638
$557$	−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
$559$	8.00000	0.338364
$560$	1.00000	0.0422577
$561$	6.00000	0.253320
$562$	−18.0000	−0.759284
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	−8.00000	−0.336861
$565$	−14.0000	−0.588984
$566$	4.00000	0.168133
$567$	−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	$0.376853\pi$
$570$	−4.00000	−0.167542
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	2.00000	0.0836242
$573$	−16.0000	−0.668410
$574$	−6.00000	−0.250435
$575$	8.00000	0.333623
$576$	7.00000	0.291667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	−19.0000	−0.790296
$579$	22.0000	0.914289
$580$	−6.00000	−0.249136
$581$	4.00000	0.165948
$582$	−14.0000	−0.580319
$583$	−10.0000	−0.414158
$584$	−18.0000	−0.744845
$585$	−2.00000	−0.0826898
$586$	18.0000	0.743573
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	1.00000	0.0412393
$589$	−32.0000	−1.31854
$590$	12.0000	0.494032
$591$	−22.0000	−0.904959
$592$	−6.00000	−0.246598
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	1.00000	0.0410305
$595$	6.00000	0.245976
$596$	18.0000	0.737309
$597$	−16.0000	−0.654836
$598$	16.0000	0.654289
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−3.00000	−0.122474
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−4.00000	−0.163028
$603$	−12.0000	−0.488678
$604$	−16.0000	−0.651031
$605$	1.00000	0.0406558
$606$	6.00000	0.243733
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	20.0000	0.811107
$609$	6.00000	0.243132
$610$	10.0000	0.404888
$611$	16.0000	0.647291
$612$	6.00000	0.242536
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−4.00000	−0.161427
$615$	6.00000	0.241943
$616$	−3.00000	−0.120873
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	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	$0.690240\pi$
$620$	−8.00000	−0.321288
$621$	−8.00000	−0.321029
$622$	16.0000	0.641542
$623$	−10.0000	−0.400642
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−10.0000	−0.399680
$627$	4.00000	0.159745
$628$	2.00000	0.0798087
$629$	−36.0000	−1.43541
$630$	1.00000	0.0398410
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	24.0000	0.954669
$633$	4.00000	0.158986
$634$	34.0000	1.35031
$635$	0	0
$636$	−10.0000	−0.396526
$637$	−2.00000	−0.0792429
$638$	−6.00000	−0.237542
$639$	−8.00000	−0.316475
$640$	3.00000	0.118585
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	−12.0000	−0.473602
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	8.00000	0.315244
$645$	4.00000	0.157500
$646$	−24.0000	−0.944267
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	3.00000	0.117851
$649$	−12.0000	−0.471041
$650$	2.00000	0.0784465
$651$	8.00000	0.313545
$652$	−20.0000	−0.783260
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−2.00000	−0.0782062
$655$	−12.0000	−0.468879
$656$	6.00000	0.234261
$657$	−6.00000	−0.234082
$658$	−8.00000	−0.311872
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	1.00000	0.0389249
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	−28.0000	−1.08825
$663$	−12.0000	−0.466041
$664$	−12.0000	−0.465690
$665$	4.00000	0.155113
$666$	−6.00000	−0.232495
$667$	48.0000	1.85857
$668$	16.0000	0.619059
$669$	8.00000	0.309298
$670$	12.0000	0.463600
$671$	−10.0000	−0.386046
$672$	−5.00000	−0.192879
$673$	−38.0000	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$674$	−26.0000	−1.00148
$675$	−1.00000	−0.0384900
$676$	9.00000	0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−14.0000	−0.537667
$679$	14.0000	0.537271
$680$	−18.0000	−0.690268
$681$	−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	$0.741841\pi$
$684$	4.00000	0.152944
$685$	−22.0000	−0.840577
$686$	1.00000	0.0381802
$687$	10.0000	0.381524
$688$	4.00000	0.152499
$689$	20.0000	0.761939
$690$	8.00000	0.304555
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−6.00000	−0.228086
$693$	−1.00000	−0.0379869
$694$	−20.0000	−0.759190
$695$	4.00000	0.151729
$696$	−18.0000	−0.682288
$697$	36.0000	1.36360
$698$	10.0000	0.378506
$699$	22.0000	0.832116
$700$	1.00000	0.0377964
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	−2.00000	−0.0754851
$703$	−24.0000	−0.905177
$704$	7.00000	0.263822
$705$	8.00000	0.301297
$706$	−34.0000	−1.27961
$707$	−6.00000	−0.225653
$708$	−12.0000	−0.450988
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	8.00000	0.300235
$711$	8.00000	0.300023
$712$	30.0000	1.12430
$713$	64.0000	2.39682
$714$	6.00000	0.224544
$715$	−2.00000	−0.0747958
$716$	−4.00000	−0.149487
$717$	16.0000	0.597531
$718$	−8.00000	−0.298557
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−1.00000	−0.0372678
$721$	−16.0000	−0.595871
$722$	3.00000	0.111648
$723$	22.0000	0.818189
$724$	−6.00000	−0.222988
$725$	6.00000	0.222834
$726$	1.00000	0.0371135
$727$	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
$730$	6.00000	0.222070
$731$	24.0000	0.887672
$732$	−10.0000	−0.369611
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−8.00000	−0.295285
$735$	−1.00000	−0.0368856
$736$	−40.0000	−1.47442
$737$	−12.0000	−0.442026
$738$	6.00000	0.220863
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−6.00000	−0.220564
$741$	−8.00000	−0.293887
$742$	−10.0000	−0.367112
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	−24.0000	−0.879883
$745$	−18.0000	−0.659469
$746$	−14.0000	−0.512576
$747$	−4.00000	−0.146352
$748$	6.00000	0.219382
$749$	12.0000	0.438470
$750$	1.00000	0.0365148
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	8.00000	0.291730
$753$	−20.0000	−0.728841
$754$	12.0000	0.437014
$755$	16.0000	0.582300
$756$	−1.00000	−0.0363696
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	20.0000	0.726433
$759$	−8.00000	−0.290382
$760$	−12.0000	−0.435286
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	−16.0000	−0.578860
$765$	−6.00000	−0.216930
$766$	−24.0000	−0.867155
$767$	24.0000	0.866590
$768$	17.0000	0.613435
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	1.00000	0.0360375
$771$	30.0000	1.08042
$772$	22.0000	0.791797
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	4.00000	0.143777
$775$	8.00000	0.287368
$776$	−42.0000	−1.50771
$777$	6.00000	0.215249
$778$	10.0000	0.358517
$779$	24.0000	0.859889
$780$	−2.00000	−0.0716115
$781$	−8.00000	−0.286263
$782$	48.0000	1.71648
$783$	−6.00000	−0.214423
$784$	−1.00000	−0.0357143
$785$	−2.00000	−0.0713831
$786$	−12.0000	−0.428026
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−22.0000	−0.783718
$789$	−16.0000	−0.569615
$790$	−8.00000	−0.284627
$791$	14.0000	0.497783
$792$	3.00000	0.106600
$793$	20.0000	0.710221
$794$	18.0000	0.638796
$795$	10.0000	0.354663
$796$	−16.0000	−0.567105
$797$	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
$799$	48.0000	1.69812
$800$	−5.00000	−0.176777
$801$	10.0000	0.353333
$802$	−2.00000	−0.0706225
$803$	−6.00000	−0.211735
$804$	−12.0000	−0.423207
$805$	−8.00000	−0.281963
$806$	16.0000	0.563576
$807$	−14.0000	−0.492823
$808$	18.0000	0.633238
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−1.00000	−0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	6.00000	0.210559
$813$	−8.00000	−0.280572
$814$	−6.00000	−0.210300
$815$	20.0000	0.700569
$816$	−6.00000	−0.210042
$817$	16.0000	0.559769
$818$	14.0000	0.489499
$819$	2.00000	0.0698857
$820$	6.00000	0.209529
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−22.0000	−0.767338
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	48.0000	1.67216
$825$	−1.00000	−0.0348155
$826$	−12.0000	−0.417533
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−8.00000	−0.278019
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	4.00000	0.138842
$831$	18.0000	0.624413
$832$	−14.0000	−0.485363
$833$	−6.00000	−0.207888
$834$	4.00000	0.138509
$835$	−16.0000	−0.553703
$836$	4.00000	0.138343
$837$	−8.00000	−0.276520
$838$	36.0000	1.24360
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	3.00000	0.103510
$841$	7.00000	0.241379
$842$	−6.00000	−0.206774
$843$	−18.0000	−0.619953
$844$	4.00000	0.137686
$845$	−9.00000	−0.309609
$846$	8.00000	0.275046
$847$	−1.00000	−0.0343604
$848$	10.0000	0.343401
$849$	4.00000	0.137280
$850$	6.00000	0.205798
$851$	48.0000	1.64542
$852$	−8.00000	−0.274075
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	−10.0000	−0.342193
$855$	−4.00000	−0.136797
$856$	−36.0000	−1.23045
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	−2.00000	−0.0682789
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	4.00000	0.136399
$861$	−6.00000	−0.204479
$862$	16.0000	0.544962
$863$	−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
$865$	6.00000	0.204006
$866$	−2.00000	−0.0679628
$867$	−19.0000	−0.645274
$868$	8.00000	0.271538
$869$	8.00000	0.271381
$870$	6.00000	0.203419
$871$	24.0000	0.813209
$872$	−6.00000	−0.203186
$873$	−14.0000	−0.473828
$874$	32.0000	1.08242
$875$	−1.00000	−0.0338062
$876$	−6.00000	−0.202721
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−32.0000	−1.07995
$879$	18.0000	0.607125
$880$	−1.00000	−0.0337100
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	−1.00000	−0.0336718
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−12.0000	−0.403604
$885$	12.0000	0.403376
$886$	−12.0000	−0.403148
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	−18.0000	−0.604040
$889$	0	0
$890$	−10.0000	−0.335201
$891$	1.00000	0.0335013
$892$	8.00000	0.267860
$893$	32.0000	1.07084
$894$	−18.0000	−0.602010
$895$	4.00000	0.133705
$896$	−3.00000	−0.100223
$897$	16.0000	0.534224
$898$	−2.00000	−0.0667409
$899$	48.0000	1.60089
$900$	−1.00000	−0.0333333
$901$	60.0000	1.99889
$902$	6.00000	0.199778
$903$	−4.00000	−0.133112
$904$	−42.0000	−1.39690
$905$	6.00000	0.199447
$906$	16.0000	0.531564
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−12.0000	−0.398234
$909$	6.00000	0.199007
$910$	−2.00000	−0.0662994
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	−4.00000	−0.132453
$913$	−4.00000	−0.132381
$914$	−18.0000	−0.595387
$915$	10.0000	0.330590
$916$	10.0000	0.330409
$917$	12.0000	0.396275
$918$	−6.00000	−0.198030
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	24.0000	0.791257
$921$	−4.00000	−0.131804
$922$	−14.0000	−0.461065
$923$	16.0000	0.526646
$924$	−1.00000	−0.0328976
$925$	6.00000	0.197279
$926$	−16.0000	−0.525793
$927$	16.0000	0.525509
$928$	−30.0000	−0.984798
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	8.00000	0.262330
$931$	−4.00000	−0.131095
$932$	22.0000	0.720634
$933$	16.0000	0.523816
$934$	−28.0000	−0.916188
$935$	−6.00000	−0.196221
$936$	−6.00000	−0.196116
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	−12.0000	−0.391814
$939$	−10.0000	−0.326338
$940$	8.00000	0.260931
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	−2.00000	−0.0651635
$943$	−48.0000	−1.56310
$944$	12.0000	0.390567
$945$	1.00000	0.0325300
$946$	4.00000	0.130051
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	8.00000	0.259828
$949$	12.0000	0.389536
$950$	4.00000	0.129777
$951$	34.0000	1.10253
$952$	18.0000	0.583383
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	10.0000	0.323762
$955$	16.0000	0.517748
$956$	16.0000	0.517477
$957$	−6.00000	−0.193952
$958$	16.0000	0.516937
$959$	22.0000	0.710417
$960$	−7.00000	−0.225924
$961$	33.0000	1.06452
$962$	12.0000	0.386896
$963$	−12.0000	−0.386695
$964$	22.0000	0.708572
$965$	−22.0000	−0.708205
$966$	−8.00000	−0.257396
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	3.00000	0.0964237
$969$	−24.0000	−0.770991
$970$	14.0000	0.449513
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	1.00000	0.0320750
$973$	−4.00000	−0.128234
$974$	−24.0000	−0.769010
$975$	2.00000	0.0640513
$976$	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	$0.407031\pi$
$978$	20.0000	0.639529
$979$	10.0000	0.319601
$980$	−1.00000	−0.0319438
$981$	−2.00000	−0.0638551
$982$	20.0000	0.638226
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	18.0000	0.573819
$985$	22.0000	0.700978
$986$	36.0000	1.14647
$987$	−8.00000	−0.254643
$988$	−8.00000	−0.254514
$989$	−32.0000	−1.01754
$990$	−1.00000	−0.0317821
$991$	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
$993$	−28.0000	−0.888553
$994$	−8.00000	−0.253745
$995$	16.0000	0.507234
$996$	−4.00000	−0.126745
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	12.0000	0.379853
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.e.1.1	✓	1
3.2	odd	2		3465.2.a.l.1.1		1
5.4	even	2		5775.2.a.v.1.1		1
7.6	odd	2		8085.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.e.1.1	✓	1	1.1	even	1	trivial
3465.2.a.l.1.1		1	3.2	odd	2
5775.2.a.v.1.1		1	5.4	even	2
8085.2.a.g.1.1		1	7.6	odd	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:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1155.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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