LMFDB - Embedded newform 1155.2.a.n.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)

$f(q)$

$=$

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

\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} -5.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} -2.00000 q^{22} +3.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -3.00000 q^{29} +2.00000 q^{30} -8.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{35} +2.00000 q^{36} -6.00000 q^{37} -10.0000 q^{38} +2.00000 q^{39} -4.00000 q^{41} +2.00000 q^{42} +7.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} +6.00000 q^{46} -4.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} +2.00000 q^{50} +3.00000 q^{51} -4.00000 q^{52} +9.00000 q^{53} -2.00000 q^{54} +1.00000 q^{55} +5.00000 q^{57} -6.00000 q^{58} -11.0000 q^{59} +2.00000 q^{60} -1.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} -2.00000 q^{67} -6.00000 q^{68} -3.00000 q^{69} +2.00000 q^{70} -8.00000 q^{71} +4.00000 q^{73} -12.0000 q^{74} -1.00000 q^{75} -10.0000 q^{76} +1.00000 q^{77} +4.00000 q^{78} +10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} +11.0000 q^{83} +2.00000 q^{84} +3.00000 q^{85} +14.0000 q^{86} +3.00000 q^{87} -7.00000 q^{89} -2.00000 q^{90} +2.00000 q^{91} +6.00000 q^{92} -8.00000 q^{94} +5.00000 q^{95} +8.00000 q^{96} +1.00000 q^{97} +2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.00000	−0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.00000	−0.577350
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	−4.00000	−1.00000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	2.00000	0.471405
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	−2.00000	−0.447214
$21$	1.00000	0.218218
$22$	−2.00000	−0.426401
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	2.00000	0.365148
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−8.00000	−1.41421
$33$	1.00000	0.174078
$34$	−6.00000	−1.02899
$35$	1.00000	0.169031
$36$	2.00000	0.333333
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−10.0000	−1.62221
$39$	2.00000	0.320256
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	2.00000	0.308607
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	−2.00000	−0.301511
$45$	−1.00000	−0.149071
$46$	6.00000	0.884652
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	4.00000	0.577350
$49$	1.00000	0.142857
$50$	2.00000	0.282843
$51$	3.00000	0.420084
$52$	−4.00000	−0.554700
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−2.00000	−0.272166
$55$	1.00000	0.134840
$56$	0	0
$57$	5.00000	0.662266
$58$	−6.00000	−0.787839
$59$	−11.0000	−1.43208	−0.716039	−	0.698060i	$-0.754047\pi$
$59$	−11.0000	−1.43208	−0.716039	+	0.698060i	$0.754047\pi$
$60$	2.00000	0.258199
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	−8.00000	−1.00000
$65$	2.00000	0.248069
$66$	2.00000	0.246183
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−6.00000	−0.727607
$69$	−3.00000	−0.361158
$70$	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	$0.657448\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−12.0000	−1.39497
$75$	−1.00000	−0.115470
$76$	−10.0000	−1.14708
$77$	1.00000	0.113961
$78$	4.00000	0.452911
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−8.00000	−0.883452
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	2.00000	0.218218
$85$	3.00000	0.325396
$86$	14.0000	1.50966
$87$	3.00000	0.321634
$88$	0	0
$89$	−7.00000	−0.741999	−0.370999	−	0.928633i	$-0.620985\pi$
$89$	−7.00000	−0.741999	−0.370999	+	0.928633i	$0.620985\pi$
$90$	−2.00000	−0.210819
$91$	2.00000	0.209657
$92$	6.00000	0.625543
$93$	0	0
$94$	−8.00000	−0.825137
$95$	5.00000	0.512989
$96$	8.00000	0.816497
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	2.00000	0.202031
$99$	−1.00000	−0.100504
$100$	2.00000	0.200000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000	0.594089
$103$	−3.00000	−0.295599	−0.147799	−	0.989017i	$-0.547219\pi$
$103$	−3.00000	−0.295599	−0.147799	+	0.989017i	$0.547219\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	18.0000	1.74831
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−2.00000	−0.192450
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	2.00000	0.190693
$111$	6.00000	0.569495
$112$	4.00000	0.377964
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	10.0000	0.936586
$115$	−3.00000	−0.279751
$116$	−6.00000	−0.557086
$117$	−2.00000	−0.184900
$118$	−22.0000	−2.02526
$119$	3.00000	0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	4.00000	0.360668
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−2.00000	−0.178174
$127$	19.0000	1.68598	0.842989	−	0.537931i	$-0.180794\pi$
$127$	19.0000	1.68598	0.842989	+	0.537931i	$0.180794\pi$
$128$	0	0
$129$	−7.00000	−0.616316
$130$	4.00000	0.350823
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	2.00000	0.174078
$133$	5.00000	0.433555
$134$	−4.00000	−0.345547
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−6.00000	−0.510754
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	2.00000	0.169031
$141$	4.00000	0.336861
$142$	−16.0000	−1.34269
$143$	2.00000	0.167248
$144$	−4.00000	−0.333333
$145$	3.00000	0.249136
$146$	8.00000	0.662085
$147$	−1.00000	−0.0824786
$148$	−12.0000	−0.986394
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	−2.00000	−0.163299
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	2.00000	0.161165
$155$	0	0
$156$	4.00000	0.320256
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	20.0000	1.59111
$159$	−9.00000	−0.713746
$160$	8.00000	0.632456
$161$	−3.00000	−0.236433
$162$	2.00000	0.157135
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	−8.00000	−0.624695
$165$	−1.00000	−0.0778499
$166$	22.0000	1.70753
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	−5.00000	−0.382360
$172$	14.0000	1.06749
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	6.00000	0.454859
$175$	−1.00000	−0.0755929
$176$	4.00000	0.301511
$177$	11.0000	0.826811
$178$	−14.0000	−1.04934
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−2.00000	−0.149071
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	4.00000	0.296500
$183$	1.00000	0.0739221
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	3.00000	0.219382
$188$	−8.00000	−0.583460
$189$	1.00000	0.0727393
$190$	10.0000	0.725476
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	8.00000	0.577350
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	2.00000	0.143592
$195$	−2.00000	−0.143223
$196$	2.00000	0.142857
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	−2.00000	−0.142134
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	−12.0000	−0.844317
$203$	3.00000	0.210559
$204$	6.00000	0.420084
$205$	4.00000	0.279372
$206$	−6.00000	−0.418040
$207$	3.00000	0.208514
$208$	8.00000	0.554700
$209$	5.00000	0.345857
$210$	−2.00000	−0.138013
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	18.0000	1.23625
$213$	8.00000	0.548151
$214$	−8.00000	−0.546869
$215$	−7.00000	−0.477396
$216$	0	0
$217$	0	0
$218$	−8.00000	−0.541828
$219$	−4.00000	−0.270295
$220$	2.00000	0.134840
$221$	6.00000	0.403604
$222$	12.0000	0.805387
$223$	11.0000	0.736614	0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000	0.736614	0.368307	+	0.929704i	$0.379937\pi$
$224$	8.00000	0.534522
$225$	1.00000	0.0666667
$226$	26.0000	1.72949
$227$	−21.0000	−1.39382	−0.696909	−	0.717159i	$-0.745442\pi$
$227$	−21.0000	−1.39382	−0.696909	+	0.717159i	$0.745442\pi$
$228$	10.0000	0.662266
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	−6.00000	−0.395628
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−4.00000	−0.261488
$235$	4.00000	0.260931
$236$	−22.0000	−1.43208
$237$	−10.0000	−0.649570
$238$	6.00000	0.388922
$239$	−21.0000	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$240$	−4.00000	−0.258199
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	−1.00000	−0.0638877
$246$	8.00000	0.510061
$247$	10.0000	0.636285
$248$	0	0
$249$	−11.0000	−0.697097
$250$	−2.00000	−0.126491
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	−2.00000	−0.125988
$253$	−3.00000	−0.188608
$254$	38.0000	2.38433
$255$	−3.00000	−0.187867
$256$	16.0000	1.00000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−14.0000	−0.871602
$259$	6.00000	0.372822
$260$	4.00000	0.248069
$261$	−3.00000	−0.185695
$262$	20.0000	1.23560
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	10.0000	0.613139
$267$	7.00000	0.428393
$268$	−4.00000	−0.244339
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	2.00000	0.121716
$271$	29.0000	1.76162	0.880812	−	0.473466i	$-0.156997\pi$
$271$	29.0000	1.76162	0.880812	+	0.473466i	$0.156997\pi$
$272$	12.0000	0.727607
$273$	−2.00000	−0.121046
$274$	−12.0000	−0.724947
$275$	−1.00000	−0.0603023
$276$	−6.00000	−0.361158
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	32.0000	1.91923
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	8.00000	0.476393
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−16.0000	−0.949425
$285$	−5.00000	−0.296174
$286$	4.00000	0.236525
$287$	4.00000	0.236113
$288$	−8.00000	−0.471405
$289$	−8.00000	−0.470588
$290$	6.00000	0.352332
$291$	−1.00000	−0.0586210
$292$	8.00000	0.468165
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	−2.00000	−0.116642
$295$	11.0000	0.640445
$296$	0	0
$297$	1.00000	0.0580259
$298$	−28.0000	−1.62200
$299$	−6.00000	−0.346989
$300$	−2.00000	−0.115470
$301$	−7.00000	−0.403473
$302$	36.0000	2.07157
$303$	6.00000	0.344691
$304$	20.0000	1.14708
$305$	1.00000	0.0572598
$306$	−6.00000	−0.342997
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	2.00000	0.113961
$309$	3.00000	0.170664
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	−6.00000	−0.338600
$315$	1.00000	0.0563436
$316$	20.0000	1.12509
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	−18.0000	−1.00939
$319$	3.00000	0.167968
$320$	8.00000	0.447214
$321$	4.00000	0.223258
$322$	−6.00000	−0.334367
$323$	15.0000	0.834622
$324$	2.00000	0.111111
$325$	−2.00000	−0.110940
$326$	−28.0000	−1.55078
$327$	4.00000	0.221201
$328$	0	0
$329$	4.00000	0.220527
$330$	−2.00000	−0.110096
$331$	−11.0000	−0.604615	−0.302307	−	0.953211i	$-0.597757\pi$
$331$	−11.0000	−0.604615	−0.302307	+	0.953211i	$0.597757\pi$
$332$	22.0000	1.20741
$333$	−6.00000	−0.328798
$334$	16.0000	0.875481
$335$	2.00000	0.109272
$336$	−4.00000	−0.218218
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	−18.0000	−0.979071
$339$	−13.0000	−0.706063
$340$	6.00000	0.325396
$341$	0	0
$342$	−10.0000	−0.540738
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.00000	0.161515
$346$	−12.0000	−0.645124
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	6.00000	0.321634
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	−2.00000	−0.106904
$351$	2.00000	0.106752
$352$	8.00000	0.426401
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	22.0000	1.16929
$355$	8.00000	0.424596
$356$	−14.0000	−0.741999
$357$	−3.00000	−0.158777
$358$	0	0
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−28.0000	−1.47165
$363$	−1.00000	−0.0524864
$364$	4.00000	0.209657
$365$	−4.00000	−0.209370
$366$	2.00000	0.104542
$367$	13.0000	0.678594	0.339297	−	0.940679i	$-0.389811\pi$
$367$	13.0000	0.678594	0.339297	+	0.940679i	$0.389811\pi$
$368$	−12.0000	−0.625543
$369$	−4.00000	−0.208232
$370$	12.0000	0.623850
$371$	−9.00000	−0.467257
$372$	0	0
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	6.00000	0.310253
$375$	1.00000	0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	2.00000	0.102869
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	10.0000	0.512989
$381$	−19.0000	−0.973399
$382$	−24.0000	−1.22795
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−36.0000	−1.83235
$387$	7.00000	0.355830
$388$	2.00000	0.101535
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	−4.00000	−0.202548
$391$	−9.00000	−0.455150
$392$	0	0
$393$	−10.0000	−0.504433
$394$	8.00000	0.403034
$395$	−10.0000	−0.503155
$396$	−2.00000	−0.100504
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	−8.00000	−0.401004
$399$	−5.00000	−0.250313
$400$	−4.00000	−0.200000
$401$	−28.0000	−1.39825	−0.699127	−	0.714998i	$-0.746428\pi$
$401$	−28.0000	−1.39825	−0.699127	+	0.714998i	$0.746428\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	−12.0000	−0.597022
$405$	−1.00000	−0.0496904
$406$	6.00000	0.297775
$407$	6.00000	0.297409
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	8.00000	0.395092
$411$	6.00000	0.295958
$412$	−6.00000	−0.295599
$413$	11.0000	0.541275
$414$	6.00000	0.294884
$415$	−11.0000	−0.539969
$416$	16.0000	0.784465
$417$	−16.0000	−0.783523
$418$	10.0000	0.489116
$419$	23.0000	1.12362	0.561812	−	0.827265i	$-0.310105\pi$
$419$	23.0000	1.12362	0.561812	+	0.827265i	$0.310105\pi$
$420$	−2.00000	−0.0975900
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	20.0000	0.973585
$423$	−4.00000	−0.194487
$424$	0	0
$425$	−3.00000	−0.145521
$426$	16.0000	0.775203
$427$	1.00000	0.0483934
$428$	−8.00000	−0.386695
$429$	−2.00000	−0.0965609
$430$	−14.0000	−0.675140
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	4.00000	0.192450
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	−8.00000	−0.383131
$437$	−15.0000	−0.717547
$438$	−8.00000	−0.382255
$439$	27.0000	1.28864	0.644320	−	0.764756i	$-0.277141\pi$
$439$	27.0000	1.28864	0.644320	+	0.764756i	$0.277141\pi$
$440$	0	0
$441$	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	$0.407984\pi$
$444$	12.0000	0.569495
$445$	7.00000	0.331832
$446$	22.0000	1.04173
$447$	14.0000	0.662177
$448$	8.00000	0.377964
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	2.00000	0.0942809
$451$	4.00000	0.188353
$452$	26.0000	1.22294
$453$	−18.0000	−0.845714
$454$	−42.0000	−1.97116
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	−32.0000	−1.49526
$459$	3.00000	0.140028
$460$	−6.00000	−0.279751
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	−2.00000	−0.0930484
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	12.0000	0.557086
$465$	0	0
$466$	36.0000	1.66767
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	−4.00000	−0.184900
$469$	2.00000	0.0923514
$470$	8.00000	0.369012
$471$	3.00000	0.138233
$472$	0	0
$473$	−7.00000	−0.321860
$474$	−20.0000	−0.918630
$475$	−5.00000	−0.229416
$476$	6.00000	0.275010
$477$	9.00000	0.412082
$478$	−42.0000	−1.92104
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	−8.00000	−0.365148
$481$	12.0000	0.547153
$482$	−4.00000	−0.182195
$483$	3.00000	0.136505
$484$	2.00000	0.0909091
$485$	−1.00000	−0.0454077
$486$	−2.00000	−0.0907218
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	14.0000	0.633102
$490$	−2.00000	−0.0903508
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	8.00000	0.360668
$493$	9.00000	0.405340
$494$	20.0000	0.899843
$495$	1.00000	0.0449467
$496$	0	0
$497$	8.00000	0.358849
$498$	−22.0000	−0.985844
$499$	23.0000	1.02962	0.514811	−	0.857304i	$-0.327862\pi$
$499$	23.0000	1.02962	0.514811	+	0.857304i	$0.327862\pi$
$500$	−2.00000	−0.0894427
$501$	−8.00000	−0.357414
$502$	−8.00000	−0.357057
$503$	17.0000	0.757993	0.378996	−	0.925398i	$-0.376269\pi$
$503$	17.0000	0.757993	0.378996	+	0.925398i	$0.376269\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−6.00000	−0.266733
$507$	9.00000	0.399704
$508$	38.0000	1.68598
$509$	19.0000	0.842160	0.421080	−	0.907023i	$-0.361651\pi$
$509$	19.0000	0.842160	0.421080	+	0.907023i	$0.361651\pi$
$510$	−6.00000	−0.265684
$511$	−4.00000	−0.176950
$512$	32.0000	1.41421
$513$	5.00000	0.220755
$514$	−12.0000	−0.529297
$515$	3.00000	0.132196
$516$	−14.0000	−0.616316
$517$	4.00000	0.175920
$518$	12.0000	0.527250
$519$	6.00000	0.263371
$520$	0	0
$521$	−25.0000	−1.09527	−0.547635	−	0.836717i	$-0.684472\pi$
$521$	−25.0000	−1.09527	−0.547635	+	0.836717i	$0.684472\pi$
$522$	−6.00000	−0.262613
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	20.0000	0.873704
$525$	1.00000	0.0436436
$526$	−48.0000	−2.09290
$527$	0	0
$528$	−4.00000	−0.174078
$529$	−14.0000	−0.608696
$530$	−18.0000	−0.781870
$531$	−11.0000	−0.477359
$532$	10.0000	0.433555
$533$	8.00000	0.346518
$534$	14.0000	0.605839
$535$	4.00000	0.172935
$536$	0	0
$537$	0	0
$538$	−30.0000	−1.29339
$539$	−1.00000	−0.0430730
$540$	2.00000	0.0860663
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	58.0000	2.49131
$543$	14.0000	0.600798
$544$	24.0000	1.02899
$545$	4.00000	0.171341
$546$	−4.00000	−0.171184
$547$	1.00000	0.0427569	0.0213785	−	0.999771i	$-0.493195\pi$
$547$	1.00000	0.0427569	0.0213785	+	0.999771i	$0.493195\pi$
$548$	−12.0000	−0.512615
$549$	−1.00000	−0.0426790
$550$	−2.00000	−0.0852803
$551$	15.0000	0.639021
$552$	0	0
$553$	−10.0000	−0.425243
$554$	44.0000	1.86938
$555$	−6.00000	−0.254686
$556$	32.0000	1.35710
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	−14.0000	−0.592137
$560$	−4.00000	−0.169031
$561$	−3.00000	−0.126660
$562$	−12.0000	−0.506189
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	8.00000	0.336861
$565$	−13.0000	−0.546914
$566$	−28.0000	−1.17693
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	−10.0000	−0.418854
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	4.00000	0.167248
$573$	12.0000	0.501307
$574$	8.00000	0.333914
$575$	3.00000	0.125109
$576$	−8.00000	−0.333333
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	−16.0000	−0.665512
$579$	18.0000	0.748054
$580$	6.00000	0.249136
$581$	−11.0000	−0.456357
$582$	−2.00000	−0.0829027
$583$	−9.00000	−0.372742
$584$	0	0
$585$	2.00000	0.0826898
$586$	18.0000	0.743573
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	−2.00000	−0.0824786
$589$	0	0
$590$	22.0000	0.905726
$591$	−4.00000	−0.164538
$592$	24.0000	0.986394
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	2.00000	0.0820610
$595$	−3.00000	−0.122988
$596$	−28.0000	−1.14692
$597$	4.00000	0.163709
$598$	−12.0000	−0.490716
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	−14.0000	−0.570597
$603$	−2.00000	−0.0814463
$604$	36.0000	1.46482
$605$	−1.00000	−0.0406558
$606$	12.0000	0.487467
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	40.0000	1.62221
$609$	−3.00000	−0.121566
$610$	2.00000	0.0809776
$611$	8.00000	0.323645
$612$	−6.00000	−0.242536
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	−52.0000	−2.09855
$615$	−4.00000	−0.161296
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	6.00000	0.241355
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	8.00000	0.320771
$623$	7.00000	0.280449
$624$	−8.00000	−0.320256
$625$	1.00000	0.0400000
$626$	38.0000	1.51879
$627$	−5.00000	−0.199681
$628$	−6.00000	−0.239426
$629$	18.0000	0.717707
$630$	2.00000	0.0796819
$631$	−35.0000	−1.39333	−0.696664	−	0.717398i	$-0.745333\pi$
$631$	−35.0000	−1.39333	−0.696664	+	0.717398i	$0.745333\pi$
$632$	0	0
$633$	−10.0000	−0.397464
$634$	−44.0000	−1.74746
$635$	−19.0000	−0.753992
$636$	−18.0000	−0.713746
$637$	−2.00000	−0.0792429
$638$	6.00000	0.237542
$639$	−8.00000	−0.316475
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	8.00000	0.315735
$643$	9.00000	0.354925	0.177463	−	0.984128i	$-0.443211\pi$
$643$	9.00000	0.354925	0.177463	+	0.984128i	$0.443211\pi$
$644$	−6.00000	−0.236433
$645$	7.00000	0.275625
$646$	30.0000	1.18033
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	11.0000	0.431788
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−28.0000	−1.09656
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	8.00000	0.312825
$655$	−10.0000	−0.390732
$656$	16.0000	0.624695
$657$	4.00000	0.156055
$658$	8.00000	0.311872
$659$	41.0000	1.59713	0.798567	−	0.601906i	$-0.205592\pi$
$659$	41.0000	1.59713	0.798567	+	0.601906i	$0.205592\pi$
$660$	−2.00000	−0.0778499
$661$	16.0000	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$661$	16.0000	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$662$	−22.0000	−0.855054
$663$	−6.00000	−0.233021
$664$	0	0
$665$	−5.00000	−0.193892
$666$	−12.0000	−0.464991
$667$	−9.00000	−0.348481
$668$	16.0000	0.619059
$669$	−11.0000	−0.425285
$670$	4.00000	0.154533
$671$	1.00000	0.0386046
$672$	−8.00000	−0.308607
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	62.0000	2.38815
$675$	−1.00000	−0.0384900
$676$	−18.0000	−0.692308
$677$	−15.0000	−0.576497	−0.288248	−	0.957556i	$-0.593073\pi$
$677$	−15.0000	−0.576497	−0.288248	+	0.957556i	$0.593073\pi$
$678$	−26.0000	−0.998524
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	21.0000	0.804722
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	−10.0000	−0.382360
$685$	6.00000	0.229248
$686$	−2.00000	−0.0763604
$687$	16.0000	0.610438
$688$	−28.0000	−1.06749
$689$	−18.0000	−0.685745
$690$	6.00000	0.228416
$691$	−50.0000	−1.90209	−0.951045	−	0.309053i	$-0.899988\pi$
$691$	−50.0000	−1.90209	−0.951045	+	0.309053i	$0.899988\pi$
$692$	−12.0000	−0.456172
$693$	1.00000	0.0379869
$694$	−72.0000	−2.73308
$695$	−16.0000	−0.606915
$696$	0	0
$697$	12.0000	0.454532
$698$	14.0000	0.529908
$699$	−18.0000	−0.680823
$700$	−2.00000	−0.0755929
$701$	−1.00000	−0.0377695	−0.0188847	−	0.999822i	$-0.506012\pi$
$701$	−1.00000	−0.0377695	−0.0188847	+	0.999822i	$0.506012\pi$
$702$	4.00000	0.150970
$703$	30.0000	1.13147
$704$	8.00000	0.301511
$705$	−4.00000	−0.150649
$706$	−60.0000	−2.25813
$707$	6.00000	0.225653
$708$	22.0000	0.826811
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	16.0000	0.600469
$711$	10.0000	0.375029
$712$	0	0
$713$	0	0
$714$	−6.00000	−0.224544
$715$	−2.00000	−0.0747958
$716$	0	0
$717$	21.0000	0.784259
$718$	−18.0000	−0.671754
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	4.00000	0.149071
$721$	3.00000	0.111726
$722$	12.0000	0.446594
$723$	2.00000	0.0743808
$724$	−28.0000	−1.04061
$725$	−3.00000	−0.111417
$726$	−2.00000	−0.0742270
$727$	−37.0000	−1.37225	−0.686127	−	0.727482i	$-0.740691\pi$
$727$	−37.0000	−1.37225	−0.686127	+	0.727482i	$0.740691\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	−21.0000	−0.776713
$732$	2.00000	0.0739221
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	26.0000	0.959678
$735$	1.00000	0.0368856
$736$	−24.0000	−0.884652
$737$	2.00000	0.0736709
$738$	−8.00000	−0.294484
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	12.0000	0.441129
$741$	−10.0000	−0.367359
$742$	−18.0000	−0.660801
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	0	0
$745$	14.0000	0.512920
$746$	−50.0000	−1.83063
$747$	11.0000	0.402469
$748$	6.00000	0.219382
$749$	4.00000	0.146157
$750$	2.00000	0.0730297
$751$	9.00000	0.328415	0.164207	−	0.986426i	$-0.447493\pi$
$751$	9.00000	0.328415	0.164207	+	0.986426i	$0.447493\pi$
$752$	16.0000	0.583460
$753$	4.00000	0.145768
$754$	12.0000	0.437014
$755$	−18.0000	−0.655087
$756$	2.00000	0.0727393
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	10.0000	0.363216
$759$	3.00000	0.108893
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	−38.0000	−1.37659
$763$	4.00000	0.144810
$764$	−24.0000	−0.868290
$765$	3.00000	0.108465
$766$	12.0000	0.433578
$767$	22.0000	0.794374
$768$	−16.0000	−0.577350
$769$	−45.0000	−1.62274	−0.811371	−	0.584532i	$-0.801278\pi$
$769$	−45.0000	−1.62274	−0.811371	+	0.584532i	$0.801278\pi$
$770$	−2.00000	−0.0720750
$771$	6.00000	0.216085
$772$	−36.0000	−1.29567
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	14.0000	0.503220
$775$	0	0
$776$	0	0
$777$	−6.00000	−0.215249
$778$	−68.0000	−2.43792
$779$	20.0000	0.716574
$780$	−4.00000	−0.143223
$781$	8.00000	0.286263
$782$	−18.0000	−0.643679
$783$	3.00000	0.107211
$784$	−4.00000	−0.142857
$785$	3.00000	0.107075
$786$	−20.0000	−0.713376
$787$	24.0000	0.855508	0.427754	−	0.903895i	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	8.00000	0.284988
$789$	24.0000	0.854423
$790$	−20.0000	−0.711568
$791$	−13.0000	−0.462227
$792$	0	0
$793$	2.00000	0.0710221
$794$	20.0000	0.709773
$795$	9.00000	0.319197
$796$	−8.00000	−0.283552
$797$	38.0000	1.34603	0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000	1.34603	0.673015	+	0.739629i	$0.264999\pi$
$798$	−10.0000	−0.353996
$799$	12.0000	0.424529
$800$	−8.00000	−0.282843
$801$	−7.00000	−0.247333
$802$	−56.0000	−1.97743
$803$	−4.00000	−0.141157
$804$	4.00000	0.141069
$805$	3.00000	0.105736
$806$	0	0
$807$	15.0000	0.528025
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−2.00000	−0.0702728
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	6.00000	0.210559
$813$	−29.0000	−1.01707
$814$	12.0000	0.420600
$815$	14.0000	0.490399
$816$	−12.0000	−0.420084
$817$	−35.0000	−1.22449
$818$	−4.00000	−0.139857
$819$	2.00000	0.0698857
$820$	8.00000	0.279372
$821$	−53.0000	−1.84971	−0.924856	−	0.380317i	$-0.875815\pi$
$821$	−53.0000	−1.84971	−0.924856	+	0.380317i	$0.875815\pi$
$822$	12.0000	0.418548
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	22.0000	0.765478
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	6.00000	0.208514
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	−22.0000	−0.763631
$831$	−22.0000	−0.763172
$832$	16.0000	0.554700
$833$	−3.00000	−0.103944
$834$	−32.0000	−1.10807
$835$	−8.00000	−0.276851
$836$	10.0000	0.345857
$837$	0	0
$838$	46.0000	1.58904
$839$	−15.0000	−0.517858	−0.258929	−	0.965896i	$-0.583369\pi$
$839$	−15.0000	−0.517858	−0.258929	+	0.965896i	$0.583369\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	2.00000	0.0689246
$843$	6.00000	0.206651
$844$	20.0000	0.688428
$845$	9.00000	0.309609
$846$	−8.00000	−0.275046
$847$	−1.00000	−0.0343604
$848$	−36.0000	−1.23625
$849$	14.0000	0.480479
$850$	−6.00000	−0.205798
$851$	−18.0000	−0.617032
$852$	16.0000	0.548151
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	2.00000	0.0684386
$855$	5.00000	0.170996
$856$	0	0
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	−4.00000	−0.136558
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	−14.0000	−0.477396
$861$	−4.00000	−0.136320
$862$	−48.0000	−1.63489
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	8.00000	0.272166
$865$	6.00000	0.204006
$866$	4.00000	0.135926
$867$	8.00000	0.271694
$868$	0	0
$869$	−10.0000	−0.339227
$870$	−6.00000	−0.203419
$871$	4.00000	0.135535
$872$	0	0
$873$	1.00000	0.0338449
$874$	−30.0000	−1.01477
$875$	1.00000	0.0338062
$876$	−8.00000	−0.270295
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	54.0000	1.82241
$879$	−9.00000	−0.303562
$880$	−4.00000	−0.134840
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	2.00000	0.0673435
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	12.0000	0.403604
$885$	−11.0000	−0.369761
$886$	24.0000	0.806296
$887$	−33.0000	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$887$	−33.0000	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$888$	0	0
$889$	−19.0000	−0.637240
$890$	14.0000	0.469281
$891$	−1.00000	−0.0335013
$892$	22.0000	0.736614
$893$	20.0000	0.669274
$894$	28.0000	0.936460
$895$	0	0
$896$	0	0
$897$	6.00000	0.200334
$898$	−16.0000	−0.533927
$899$	0	0
$900$	2.00000	0.0666667
$901$	−27.0000	−0.899500
$902$	8.00000	0.266371
$903$	7.00000	0.232945
$904$	0	0
$905$	14.0000	0.465376
$906$	−36.0000	−1.19602
$907$	58.0000	1.92586	0.962929	−	0.269754i	$-0.0869425\pi$
$907$	58.0000	1.92586	0.962929	+	0.269754i	$0.0869425\pi$
$908$	−42.0000	−1.39382
$909$	−6.00000	−0.199007
$910$	−4.00000	−0.132599
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	−20.0000	−0.662266
$913$	−11.0000	−0.364047
$914$	−74.0000	−2.44770
$915$	−1.00000	−0.0330590
$916$	−32.0000	−1.05731
$917$	−10.0000	−0.330229
$918$	6.00000	0.198030
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	0	0
$921$	26.0000	0.856729
$922$	−60.0000	−1.97599
$923$	16.0000	0.526646
$924$	−2.00000	−0.0657952
$925$	−6.00000	−0.197279
$926$	44.0000	1.44593
$927$	−3.00000	−0.0985329
$928$	24.0000	0.787839
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	−5.00000	−0.163868
$932$	36.0000	1.17922
$933$	−4.00000	−0.130954
$934$	40.0000	1.30884
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	4.00000	0.130605
$939$	−19.0000	−0.620042
$940$	8.00000	0.260931
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	6.00000	0.195491
$943$	−12.0000	−0.390774
$944$	44.0000	1.43208
$945$	−1.00000	−0.0325300
$946$	−14.0000	−0.455179
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	−20.0000	−0.649570
$949$	−8.00000	−0.259691
$950$	−10.0000	−0.324443
$951$	22.0000	0.713399
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	18.0000	0.582772
$955$	12.0000	0.388311
$956$	−42.0000	−1.35838
$957$	−3.00000	−0.0969762
$958$	−60.0000	−1.93851
$959$	6.00000	0.193750
$960$	−8.00000	−0.258199
$961$	−31.0000	−1.00000
$962$	24.0000	0.773791
$963$	−4.00000	−0.128898
$964$	−4.00000	−0.128831
$965$	18.0000	0.579441
$966$	6.00000	0.193047
$967$	−5.00000	−0.160789	−0.0803946	−	0.996763i	$-0.525618\pi$
$967$	−5.00000	−0.160789	−0.0803946	+	0.996763i	$0.525618\pi$
$968$	0	0
$969$	−15.0000	−0.481869
$970$	−2.00000	−0.0642161
$971$	29.0000	0.930654	0.465327	−	0.885139i	$-0.345937\pi$
$971$	29.0000	0.930654	0.465327	+	0.885139i	$0.345937\pi$
$972$	−2.00000	−0.0641500
$973$	−16.0000	−0.512936
$974$	−4.00000	−0.128168
$975$	2.00000	0.0640513
$976$	4.00000	0.128037
$977$	45.0000	1.43968	0.719839	−	0.694141i	$-0.244216\pi$
$977$	45.0000	1.43968	0.719839	+	0.694141i	$0.244216\pi$
$978$	28.0000	0.895341
$979$	7.00000	0.223721
$980$	−2.00000	−0.0638877
$981$	−4.00000	−0.127710
$982$	−6.00000	−0.191468
$983$	54.0000	1.72233	0.861166	−	0.508323i	$-0.169735\pi$
$983$	54.0000	1.72233	0.861166	+	0.508323i	$0.169735\pi$
$984$	0	0
$985$	−4.00000	−0.127451
$986$	18.0000	0.573237
$987$	−4.00000	−0.127321
$988$	20.0000	0.636285
$989$	21.0000	0.667761
$990$	2.00000	0.0635642
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	0	0
$993$	11.0000	0.349074
$994$	16.0000	0.507489
$995$	4.00000	0.126809
$996$	−22.0000	−0.697097
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	46.0000	1.45610
$999$	6.00000	0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.n.1.1	✓	1	1.1	even	1	trivial
3465.2.a.a.1.1		1	3.2	odd	2
5775.2.a.a.1.1		1	5.4	even	2
8085.2.a.z.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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