LMFDB - Embedded newform 1155.2.a.a.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)

$f(q)$

$=$

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

\(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} +1.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} +2.00000 q^{22} -7.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +1.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} +8.00000 q^{32} +1.00000 q^{33} +6.00000 q^{34} +1.00000 q^{35} +2.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} -8.00000 q^{41} +2.00000 q^{42} +9.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} +14.0000 q^{46} -12.0000 q^{47} +4.00000 q^{48} +1.00000 q^{49} -2.00000 q^{50} +3.00000 q^{51} -4.00000 q^{52} -5.00000 q^{53} +2.00000 q^{54} -1.00000 q^{55} -1.00000 q^{57} -2.00000 q^{58} +3.00000 q^{59} -2.00000 q^{60} +13.0000 q^{61} -16.0000 q^{62} +1.00000 q^{63} -8.00000 q^{64} -2.00000 q^{65} -2.00000 q^{66} -14.0000 q^{67} -6.00000 q^{68} +7.00000 q^{69} -2.00000 q^{70} +8.00000 q^{71} -8.00000 q^{73} +4.00000 q^{74} -1.00000 q^{75} +2.00000 q^{76} -1.00000 q^{77} -4.00000 q^{78} -14.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +16.0000 q^{82} +11.0000 q^{83} -2.00000 q^{84} -3.00000 q^{85} -18.0000 q^{86} -1.00000 q^{87} -9.00000 q^{89} -2.00000 q^{90} -2.00000 q^{91} -14.0000 q^{92} -8.00000 q^{93} +24.0000 q^{94} +1.00000 q^{95} -8.00000 q^{96} +5.00000 q^{97} -2.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−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
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	2.00000	0.447214
$21$	−1.00000	−0.218218
$22$	2.00000	0.426401
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	8.00000	1.41421
$33$	1.00000	0.174078
$34$	6.00000	1.02899
$35$	1.00000	0.169031
$36$	2.00000	0.333333
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−2.00000	−0.324443
$39$	2.00000	0.320256
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	2.00000	0.308607
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	−2.00000	−0.301511
$45$	1.00000	0.149071
$46$	14.0000	2.06419
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	4.00000	0.577350
$49$	1.00000	0.142857
$50$	−2.00000	−0.282843
$51$	3.00000	0.420084
$52$	−4.00000	−0.554700
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	2.00000	0.272166
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−2.00000	−0.262613
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	−2.00000	−0.258199
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	−16.0000	−2.03200
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	−2.00000	−0.248069
$66$	−2.00000	−0.246183
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	−6.00000	−0.727607
$69$	7.00000	0.842701
$70$	−2.00000	−0.239046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	4.00000	0.464991
$75$	−1.00000	−0.115470
$76$	2.00000	0.229416
$77$	−1.00000	−0.113961
$78$	−4.00000	−0.452911
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	16.0000	1.76690
$83$	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
$86$	−18.0000	−1.94099
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	−2.00000	−0.210819
$91$	−2.00000	−0.209657
$92$	−14.0000	−1.45960
$93$	−8.00000	−0.829561
$94$	24.0000	2.47541
$95$	1.00000	0.102598
$96$	−8.00000	−0.816497
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	−2.00000	−0.202031
$99$	−1.00000	−0.100504
$100$	2.00000	0.200000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	−6.00000	−0.594089
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	10.0000	0.971286
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−2.00000	−0.192450
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	2.00000	0.190693
$111$	2.00000	0.189832
$112$	−4.00000	−0.377964
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	2.00000	0.187317
$115$	−7.00000	−0.652753
$116$	2.00000	0.185695
$117$	−2.00000	−0.184900
$118$	−6.00000	−0.552345
$119$	−3.00000	−0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	−26.0000	−2.35393
$123$	8.00000	0.721336
$124$	16.0000	1.43684
$125$	1.00000	0.0894427
$126$	−2.00000	−0.178174
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	0	0
$129$	−9.00000	−0.792406
$130$	4.00000	0.350823
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	2.00000	0.174078
$133$	1.00000	0.0867110
$134$	28.0000	2.41883
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	−14.0000	−1.19176
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	2.00000	0.169031
$141$	12.0000	1.01058
$142$	−16.0000	−1.34269
$143$	2.00000	0.167248
$144$	−4.00000	−0.333333
$145$	1.00000	0.0830455
$146$	16.0000	1.32417
$147$	−1.00000	−0.0824786
$148$	−4.00000	−0.328798
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	2.00000	0.163299
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	2.00000	0.161165
$155$	8.00000	0.642575
$156$	4.00000	0.320256
$157$	−23.0000	−1.83560	−0.917800	−	0.397043i	$-0.870036\pi$
$157$	−23.0000	−1.83560	−0.917800	+	0.397043i	$0.870036\pi$
$158$	28.0000	2.22756
$159$	5.00000	0.396526
$160$	8.00000	0.632456
$161$	−7.00000	−0.551677
$162$	−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	$0.684718\pi$
$164$	−16.0000	−1.24939
$165$	1.00000	0.0778499
$166$	−22.0000	−1.70753
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	1.00000	0.0764719
$172$	18.0000	1.37249
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	2.00000	0.151620
$175$	1.00000	0.0755929
$176$	4.00000	0.301511
$177$	−3.00000	−0.225494
$178$	18.0000	1.34916
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	2.00000	0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	4.00000	0.296500
$183$	−13.0000	−0.960988
$184$	0	0
$185$	−2.00000	−0.147043
$186$	16.0000	1.17318
$187$	3.00000	0.219382
$188$	−24.0000	−1.75038
$189$	−1.00000	−0.0727393
$190$	−2.00000	−0.145095
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	8.00000	0.577350
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	−10.0000	−0.717958
$195$	2.00000	0.143223
$196$	2.00000	0.142857
$197$	−16.0000	−1.13995	−0.569976	−	0.821661i	$-0.693048\pi$
$197$	−16.0000	−1.13995	−0.569976	+	0.821661i	$0.693048\pi$
$198$	2.00000	0.142134
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	28.0000	1.97007
$203$	1.00000	0.0701862
$204$	6.00000	0.420084
$205$	−8.00000	−0.558744
$206$	14.0000	0.975426
$207$	−7.00000	−0.486534
$208$	8.00000	0.554700
$209$	−1.00000	−0.0691714
$210$	2.00000	0.138013
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−10.0000	−0.686803
$213$	−8.00000	−0.548151
$214$	16.0000	1.09374
$215$	9.00000	0.613795
$216$	0	0
$217$	8.00000	0.543075
$218$	32.0000	2.16731
$219$	8.00000	0.540590
$220$	−2.00000	−0.134840
$221$	6.00000	0.403604
$222$	−4.00000	−0.268462
$223$	15.0000	1.00447	0.502237	−	0.864730i	$-0.332510\pi$
$223$	15.0000	1.00447	0.502237	+	0.864730i	$0.332510\pi$
$224$	8.00000	0.534522
$225$	1.00000	0.0666667
$226$	18.0000	1.19734
$227$	11.0000	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$227$	11.0000	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$228$	−2.00000	−0.132453
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	14.0000	0.923133
$231$	1.00000	0.0657952
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	4.00000	0.261488
$235$	−12.0000	−0.782794
$236$	6.00000	0.390567
$237$	14.0000	0.909398
$238$	6.00000	0.388922
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	4.00000	0.258199
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−2.00000	−0.128565
$243$	−1.00000	−0.0641500
$244$	26.0000	1.66448
$245$	1.00000	0.0638877
$246$	−16.0000	−1.02012
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−11.0000	−0.697097
$250$	−2.00000	−0.126491
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	2.00000	0.125988
$253$	7.00000	0.440086
$254$	−10.0000	−0.627456
$255$	3.00000	0.187867
$256$	16.0000	1.00000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	18.0000	1.12063
$259$	−2.00000	−0.124274
$260$	−4.00000	−0.248069
$261$	1.00000	0.0618984
$262$	−4.00000	−0.247121
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	−5.00000	−0.307148
$266$	−2.00000	−0.122628
$267$	9.00000	0.550791
$268$	−28.0000	−1.71037
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	2.00000	0.121716
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	12.0000	0.727607
$273$	2.00000	0.121046
$274$	20.0000	1.20824
$275$	−1.00000	−0.0603023
$276$	14.0000	0.842701
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−16.0000	−0.959616
$279$	8.00000	0.478947
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−24.0000	−1.42918
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	16.0000	0.949425
$285$	−1.00000	−0.0592349
$286$	−4.00000	−0.236525
$287$	−8.00000	−0.472225
$288$	8.00000	0.471405
$289$	−8.00000	−0.470588
$290$	−2.00000	−0.117444
$291$	−5.00000	−0.293105
$292$	−16.0000	−0.936329
$293$	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
$295$	3.00000	0.174667
$296$	0	0
$297$	1.00000	0.0580259
$298$	12.0000	0.695141
$299$	14.0000	0.809641
$300$	−2.00000	−0.115470
$301$	9.00000	0.518751
$302$	−28.0000	−1.61122
$303$	14.0000	0.804279
$304$	−4.00000	−0.229416
$305$	13.0000	0.744378
$306$	6.00000	0.342997
$307$	30.0000	1.71219	0.856095	−	0.516818i	$-0.172884\pi$
$307$	30.0000	1.71219	0.856095	+	0.516818i	$0.172884\pi$
$308$	−2.00000	−0.113961
$309$	7.00000	0.398216
$310$	−16.0000	−0.908739
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	15.0000	0.847850	0.423925	−	0.905697i	$-0.360652\pi$
$313$	15.0000	0.847850	0.423925	+	0.905697i	$0.360652\pi$
$314$	46.0000	2.59593
$315$	1.00000	0.0563436
$316$	−28.0000	−1.57512
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−10.0000	−0.560772
$319$	−1.00000	−0.0559893
$320$	−8.00000	−0.447214
$321$	8.00000	0.446516
$322$	14.0000	0.780189
$323$	−3.00000	−0.166924
$324$	2.00000	0.111111
$325$	−2.00000	−0.110940
$326$	28.0000	1.55078
$327$	16.0000	0.884802
$328$	0	0
$329$	−12.0000	−0.661581
$330$	−2.00000	−0.110096
$331$	−27.0000	−1.48405	−0.742027	−	0.670370i	$-0.766135\pi$
$331$	−27.0000	−1.48405	−0.742027	+	0.670370i	$0.766135\pi$
$332$	22.0000	1.20741
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−14.0000	−0.764902
$336$	4.00000	0.218218
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	18.0000	0.979071
$339$	9.00000	0.488813
$340$	−6.00000	−0.325396
$341$	−8.00000	−0.433224
$342$	−2.00000	−0.108148
$343$	1.00000	0.0539949
$344$	0	0
$345$	7.00000	0.376867
$346$	−4.00000	−0.215041
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	−2.00000	−0.107211
$349$	−27.0000	−1.44528	−0.722638	−	0.691226i	$-0.757071\pi$
$349$	−27.0000	−1.44528	−0.722638	+	0.691226i	$0.757071\pi$
$350$	−2.00000	−0.106904
$351$	2.00000	0.106752
$352$	−8.00000	−0.426401
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	6.00000	0.318896
$355$	8.00000	0.424596
$356$	−18.0000	−0.953998
$357$	3.00000	0.158777
$358$	8.00000	0.422813
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−4.00000	−0.210235
$363$	−1.00000	−0.0524864
$364$	−4.00000	−0.209657
$365$	−8.00000	−0.418739
$366$	26.0000	1.35904
$367$	25.0000	1.30499	0.652495	−	0.757793i	$-0.273722\pi$
$367$	25.0000	1.30499	0.652495	+	0.757793i	$0.273722\pi$
$368$	28.0000	1.45960
$369$	−8.00000	−0.416463
$370$	4.00000	0.207950
$371$	−5.00000	−0.259587
$372$	−16.0000	−0.829561
$373$	−15.0000	−0.776671	−0.388335	−	0.921518i	$-0.626950\pi$
$373$	−15.0000	−0.776671	−0.388335	+	0.921518i	$0.626950\pi$
$374$	−6.00000	−0.310253
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−2.00000	−0.103005
$378$	2.00000	0.102869
$379$	13.0000	0.667765	0.333883	−	0.942615i	$-0.391641\pi$
$379$	13.0000	0.667765	0.333883	+	0.942615i	$0.391641\pi$
$380$	2.00000	0.102598
$381$	−5.00000	−0.256158
$382$	−8.00000	−0.409316
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−52.0000	−2.64673
$387$	9.00000	0.457496
$388$	10.0000	0.507673
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	−4.00000	−0.202548
$391$	21.0000	1.06202
$392$	0	0
$393$	−2.00000	−0.100887
$394$	32.0000	1.61214
$395$	−14.0000	−0.704416
$396$	−2.00000	−0.100504
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	40.0000	2.00502
$399$	−1.00000	−0.0500626
$400$	−4.00000	−0.200000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	−28.0000	−1.39651
$403$	−16.0000	−0.797017
$404$	−28.0000	−1.39305
$405$	1.00000	0.0496904
$406$	−2.00000	−0.0992583
$407$	2.00000	0.0991363
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	16.0000	0.790184
$411$	10.0000	0.493264
$412$	−14.0000	−0.689730
$413$	3.00000	0.147620
$414$	14.0000	0.688062
$415$	11.0000	0.539969
$416$	−16.0000	−0.784465
$417$	−8.00000	−0.391762
$418$	2.00000	0.0978232
$419$	1.00000	0.0488532	0.0244266	−	0.999702i	$-0.492224\pi$
$419$	1.00000	0.0488532	0.0244266	+	0.999702i	$0.492224\pi$
$420$	−2.00000	−0.0975900
$421$	9.00000	0.438633	0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000	0.438633	0.219317	+	0.975654i	$0.429617\pi$
$422$	44.0000	2.14189
$423$	−12.0000	−0.583460
$424$	0	0
$425$	−3.00000	−0.145521
$426$	16.0000	0.775203
$427$	13.0000	0.629114
$428$	−16.0000	−0.773389
$429$	−2.00000	−0.0965609
$430$	−18.0000	−0.868037
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	4.00000	0.192450
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	−16.0000	−0.768025
$435$	−1.00000	−0.0479463
$436$	−32.0000	−1.53252
$437$	−7.00000	−0.334855
$438$	−16.0000	−0.764510
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−12.0000	−0.570782
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	4.00000	0.189832
$445$	−9.00000	−0.426641
$446$	−30.0000	−1.42054
$447$	6.00000	0.283790
$448$	−8.00000	−0.377964
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	−2.00000	−0.0942809
$451$	8.00000	0.376705
$452$	−18.0000	−0.846649
$453$	−14.0000	−0.657777
$454$	−22.0000	−1.03251
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−3.00000	−0.140334	−0.0701670	−	0.997535i	$-0.522353\pi$
$457$	−3.00000	−0.140334	−0.0701670	+	0.997535i	$0.522353\pi$
$458$	8.00000	0.373815
$459$	3.00000	0.140028
$460$	−14.0000	−0.652753
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	−2.00000	−0.0930484
$463$	−2.00000	−0.0929479	−0.0464739	−	0.998920i	$-0.514798\pi$
$463$	−2.00000	−0.0929479	−0.0464739	+	0.998920i	$0.514798\pi$
$464$	−4.00000	−0.185695
$465$	−8.00000	−0.370991
$466$	−20.0000	−0.926482
$467$	32.0000	1.48078	0.740392	−	0.672176i	$-0.234640\pi$
$467$	32.0000	1.48078	0.740392	+	0.672176i	$0.234640\pi$
$468$	−4.00000	−0.184900
$469$	−14.0000	−0.646460
$470$	24.0000	1.10704
$471$	23.0000	1.05978
$472$	0	0
$473$	−9.00000	−0.413820
$474$	−28.0000	−1.28608
$475$	1.00000	0.0458831
$476$	−6.00000	−0.275010
$477$	−5.00000	−0.228934
$478$	−30.0000	−1.37217
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	−8.00000	−0.365148
$481$	4.00000	0.182384
$482$	−4.00000	−0.182195
$483$	7.00000	0.318511
$484$	2.00000	0.0909091
$485$	5.00000	0.227038
$486$	2.00000	0.0907218
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	0	0
$489$	14.0000	0.633102
$490$	−2.00000	−0.0903508
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	16.0000	0.721336
$493$	−3.00000	−0.135113
$494$	4.00000	0.179969
$495$	−1.00000	−0.0449467
$496$	−32.0000	−1.43684
$497$	8.00000	0.358849
$498$	22.0000	0.985844
$499$	−41.0000	−1.83541	−0.917706	−	0.397260i	$-0.869961\pi$
$499$	−41.0000	−1.83541	−0.917706	+	0.397260i	$0.869961\pi$
$500$	2.00000	0.0894427
$501$	0	0
$502$	24.0000	1.07117
$503$	25.0000	1.11469	0.557347	−	0.830279i	$-0.311819\pi$
$503$	25.0000	1.11469	0.557347	+	0.830279i	$0.311819\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	−14.0000	−0.622376
$507$	9.00000	0.399704
$508$	10.0000	0.443678
$509$	−3.00000	−0.132973	−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000	−0.132973	−0.0664863	+	0.997787i	$0.521179\pi$
$510$	−6.00000	−0.265684
$511$	−8.00000	−0.353899
$512$	−32.0000	−1.41421
$513$	−1.00000	−0.0441511
$514$	28.0000	1.23503
$515$	−7.00000	−0.308457
$516$	−18.0000	−0.792406
$517$	12.0000	0.527759
$518$	4.00000	0.175750
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	1.00000	0.0438108	0.0219054	−	0.999760i	$-0.493027\pi$
$521$	1.00000	0.0438108	0.0219054	+	0.999760i	$0.493027\pi$
$522$	−2.00000	−0.0875376
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	4.00000	0.174741
$525$	−1.00000	−0.0436436
$526$	−24.0000	−1.04645
$527$	−24.0000	−1.04546
$528$	−4.00000	−0.174078
$529$	26.0000	1.13043
$530$	10.0000	0.434372
$531$	3.00000	0.130189
$532$	2.00000	0.0867110
$533$	16.0000	0.693037
$534$	−18.0000	−0.778936
$535$	−8.00000	−0.345870
$536$	0	0
$537$	4.00000	0.172613
$538$	34.0000	1.46584
$539$	−1.00000	−0.0430730
$540$	−2.00000	−0.0860663
$541$	−12.0000	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$541$	−12.0000	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$542$	−14.0000	−0.601351
$543$	−2.00000	−0.0858282
$544$	−24.0000	−1.02899
$545$	−16.0000	−0.685365
$546$	−4.00000	−0.171184
$547$	−25.0000	−1.06892	−0.534461	−	0.845193i	$-0.679486\pi$
$547$	−25.0000	−1.06892	−0.534461	+	0.845193i	$0.679486\pi$
$548$	−20.0000	−0.854358
$549$	13.0000	0.554826
$550$	2.00000	0.0852803
$551$	1.00000	0.0426014
$552$	0	0
$553$	−14.0000	−0.595341
$554$	44.0000	1.86938
$555$	2.00000	0.0848953
$556$	16.0000	0.678551
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	−16.0000	−0.677334
$559$	−18.0000	−0.761319
$560$	−4.00000	−0.169031
$561$	−3.00000	−0.126660
$562$	−36.0000	−1.51857
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	24.0000	1.01058
$565$	−9.00000	−0.378633
$566$	−28.0000	−1.17693
$567$	1.00000	0.0419961
$568$	0	0
$569$	37.0000	1.55112	0.775560	−	0.631273i	$-0.217467\pi$
$569$	37.0000	1.55112	0.775560	+	0.631273i	$0.217467\pi$
$570$	2.00000	0.0837708
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	4.00000	0.167248
$573$	−4.00000	−0.167102
$574$	16.0000	0.667827
$575$	−7.00000	−0.291920
$576$	−8.00000	−0.333333
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	16.0000	0.665512
$579$	−26.0000	−1.08052
$580$	2.00000	0.0830455
$581$	11.0000	0.456357
$582$	10.0000	0.414513
$583$	5.00000	0.207079
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	−18.0000	−0.743573
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	−2.00000	−0.0824786
$589$	8.00000	0.329634
$590$	−6.00000	−0.247016
$591$	16.0000	0.658152
$592$	8.00000	0.328798
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	−2.00000	−0.0820610
$595$	−3.00000	−0.122988
$596$	−12.0000	−0.491539
$597$	20.0000	0.818546
$598$	−28.0000	−1.14501
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	0	0
$601$	−9.00000	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$602$	−18.0000	−0.733625
$603$	−14.0000	−0.570124
$604$	28.0000	1.13930
$605$	1.00000	0.0406558
$606$	−28.0000	−1.13742
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	8.00000	0.324443
$609$	−1.00000	−0.0405220
$610$	−26.0000	−1.05271
$611$	24.0000	0.970936
$612$	−6.00000	−0.242536
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−60.0000	−2.42140
$615$	8.00000	0.322591
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	−14.0000	−0.563163
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	16.0000	0.642575
$621$	7.00000	0.280900
$622$	8.00000	0.320771
$623$	−9.00000	−0.360577
$624$	−8.00000	−0.320256
$625$	1.00000	0.0400000
$626$	−30.0000	−1.19904
$627$	1.00000	0.0399362
$628$	−46.0000	−1.83560
$629$	6.00000	0.239236
$630$	−2.00000	−0.0796819
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	0	0
$633$	22.0000	0.874421
$634$	−60.0000	−2.38290
$635$	5.00000	0.198419
$636$	10.0000	0.396526
$637$	−2.00000	−0.0792429
$638$	2.00000	0.0791808
$639$	8.00000	0.316475
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	−16.0000	−0.631470
$643$	−3.00000	−0.118308	−0.0591542	−	0.998249i	$-0.518840\pi$
$643$	−3.00000	−0.118308	−0.0591542	+	0.998249i	$0.518840\pi$
$644$	−14.0000	−0.551677
$645$	−9.00000	−0.354375
$646$	6.00000	0.236067
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	4.00000	0.156893
$651$	−8.00000	−0.313545
$652$	−28.0000	−1.09656
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	−32.0000	−1.25130
$655$	2.00000	0.0781465
$656$	32.0000	1.24939
$657$	−8.00000	−0.312110
$658$	24.0000	0.935617
$659$	−3.00000	−0.116863	−0.0584317	−	0.998291i	$-0.518610\pi$
$659$	−3.00000	−0.116863	−0.0584317	+	0.998291i	$0.518610\pi$
$660$	2.00000	0.0778499
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	54.0000	2.09877
$663$	−6.00000	−0.233021
$664$	0	0
$665$	1.00000	0.0387783
$666$	4.00000	0.154997
$667$	−7.00000	−0.271041
$668$	0	0
$669$	−15.0000	−0.579934
$670$	28.0000	1.08173
$671$	−13.0000	−0.501859
$672$	−8.00000	−0.308607
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	46.0000	1.77185
$675$	−1.00000	−0.0384900
$676$	−18.0000	−0.692308
$677$	9.00000	0.345898	0.172949	−	0.984931i	$-0.444670\pi$
$677$	9.00000	0.345898	0.172949	+	0.984931i	$0.444670\pi$
$678$	−18.0000	−0.691286
$679$	5.00000	0.191882
$680$	0	0
$681$	−11.0000	−0.421521
$682$	16.0000	0.612672
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	2.00000	0.0764719
$685$	−10.0000	−0.382080
$686$	−2.00000	−0.0763604
$687$	4.00000	0.152610
$688$	−36.0000	−1.37249
$689$	10.0000	0.380970
$690$	−14.0000	−0.532971
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	4.00000	0.152057
$693$	−1.00000	−0.0379869
$694$	56.0000	2.12573
$695$	8.00000	0.303457
$696$	0	0
$697$	24.0000	0.909065
$698$	54.0000	2.04393
$699$	−10.0000	−0.378235
$700$	2.00000	0.0755929
$701$	43.0000	1.62409	0.812044	−	0.583597i	$-0.198355\pi$
$701$	43.0000	1.62409	0.812044	+	0.583597i	$0.198355\pi$
$702$	−4.00000	−0.150970
$703$	−2.00000	−0.0754314
$704$	8.00000	0.301511
$705$	12.0000	0.451946
$706$	−60.0000	−2.25813
$707$	−14.0000	−0.526524
$708$	−6.00000	−0.225494
$709$	21.0000	0.788672	0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000	0.788672	0.394336	+	0.918966i	$0.370975\pi$
$710$	−16.0000	−0.600469
$711$	−14.0000	−0.525041
$712$	0	0
$713$	−56.0000	−2.09722
$714$	−6.00000	−0.224544
$715$	2.00000	0.0747958
$716$	−8.00000	−0.298974
$717$	−15.0000	−0.560185
$718$	−6.00000	−0.223918
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	−4.00000	−0.149071
$721$	−7.00000	−0.260694
$722$	36.0000	1.33978
$723$	−2.00000	−0.0743808
$724$	4.00000	0.148659
$725$	1.00000	0.0371391
$726$	2.00000	0.0742270
$727$	31.0000	1.14973	0.574863	−	0.818250i	$-0.305055\pi$
$727$	31.0000	1.14973	0.574863	+	0.818250i	$0.305055\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	16.0000	0.592187
$731$	−27.0000	−0.998631
$732$	−26.0000	−0.960988
$733$	38.0000	1.40356	0.701781	−	0.712393i	$-0.252388\pi$
$733$	38.0000	1.40356	0.701781	+	0.712393i	$0.252388\pi$
$734$	−50.0000	−1.84553
$735$	−1.00000	−0.0368856
$736$	−56.0000	−2.06419
$737$	14.0000	0.515697
$738$	16.0000	0.588968
$739$	−6.00000	−0.220714	−0.110357	−	0.993892i	$-0.535199\pi$
$739$	−6.00000	−0.220714	−0.110357	+	0.993892i	$0.535199\pi$
$740$	−4.00000	−0.147043
$741$	2.00000	0.0734718
$742$	10.0000	0.367112
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	30.0000	1.09838
$747$	11.0000	0.402469
$748$	6.00000	0.219382
$749$	−8.00000	−0.292314
$750$	2.00000	0.0730297
$751$	17.0000	0.620339	0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000	0.620339	0.310169	+	0.950681i	$0.399614\pi$
$752$	48.0000	1.75038
$753$	12.0000	0.437304
$754$	4.00000	0.145671
$755$	14.0000	0.509512
$756$	−2.00000	−0.0727393
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	−26.0000	−0.944363
$759$	−7.00000	−0.254084
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	10.0000	0.362262
$763$	−16.0000	−0.579239
$764$	8.00000	0.289430
$765$	−3.00000	−0.108465
$766$	28.0000	1.01168
$767$	−6.00000	−0.216647
$768$	−16.0000	−0.577350
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	2.00000	0.0720750
$771$	14.0000	0.504198
$772$	52.0000	1.87152
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−18.0000	−0.646997
$775$	8.00000	0.287368
$776$	0	0
$777$	2.00000	0.0717496
$778$	4.00000	0.143407
$779$	−8.00000	−0.286630
$780$	4.00000	0.143223
$781$	−8.00000	−0.286263
$782$	−42.0000	−1.50192
$783$	−1.00000	−0.0357371
$784$	−4.00000	−0.142857
$785$	−23.0000	−0.820905
$786$	4.00000	0.142675
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−32.0000	−1.13995
$789$	−12.0000	−0.427211
$790$	28.0000	0.996195
$791$	−9.00000	−0.320003
$792$	0	0
$793$	−26.0000	−0.923287
$794$	60.0000	2.12932
$795$	5.00000	0.177332
$796$	−40.0000	−1.41776
$797$	38.0000	1.34603	0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000	1.34603	0.673015	+	0.739629i	$0.264999\pi$
$798$	2.00000	0.0707992
$799$	36.0000	1.27359
$800$	8.00000	0.282843
$801$	−9.00000	−0.317999
$802$	24.0000	0.847469
$803$	8.00000	0.282314
$804$	28.0000	0.987484
$805$	−7.00000	−0.246718
$806$	32.0000	1.12715
$807$	17.0000	0.598428
$808$	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
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	2.00000	0.0701862
$813$	−7.00000	−0.245501
$814$	−4.00000	−0.140200
$815$	−14.0000	−0.490399
$816$	−12.0000	−0.420084
$817$	9.00000	0.314870
$818$	−52.0000	−1.81814
$819$	−2.00000	−0.0698857
$820$	−16.0000	−0.558744
$821$	7.00000	0.244302	0.122151	−	0.992512i	$-0.461021\pi$
$821$	7.00000	0.244302	0.122151	+	0.992512i	$0.461021\pi$
$822$	−20.0000	−0.697580
$823$	−50.0000	−1.74289	−0.871445	−	0.490493i	$-0.836817\pi$
$823$	−50.0000	−1.74289	−0.871445	+	0.490493i	$0.836817\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	−6.00000	−0.208767
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	−14.0000	−0.486534
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−22.0000	−0.763631
$831$	22.0000	0.763172
$832$	16.0000	0.554700
$833$	−3.00000	−0.103944
$834$	16.0000	0.554035
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	−8.00000	−0.276520
$838$	−2.00000	−0.0690889
$839$	−9.00000	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$839$	−9.00000	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−18.0000	−0.620321
$843$	−18.0000	−0.619953
$844$	−44.0000	−1.51454
$845$	−9.00000	−0.309609
$846$	24.0000	0.825137
$847$	1.00000	0.0343604
$848$	20.0000	0.686803
$849$	−14.0000	−0.480479
$850$	6.00000	0.205798
$851$	14.0000	0.479914
$852$	−16.0000	−0.548151
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	−26.0000	−0.889702
$855$	1.00000	0.0341993
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	4.00000	0.136558
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	18.0000	0.613795
$861$	8.00000	0.272639
$862$	−32.0000	−1.08992
$863$	1.00000	0.0340404	0.0170202	−	0.999855i	$-0.494582\pi$
$863$	1.00000	0.0340404	0.0170202	+	0.999855i	$0.494582\pi$
$864$	−8.00000	−0.272166
$865$	2.00000	0.0680020
$866$	76.0000	2.58259
$867$	8.00000	0.271694
$868$	16.0000	0.543075
$869$	14.0000	0.474917
$870$	2.00000	0.0678064
$871$	28.0000	0.948744
$872$	0	0
$873$	5.00000	0.169224
$874$	14.0000	0.473557
$875$	1.00000	0.0338062
$876$	16.0000	0.540590
$877$	3.00000	0.101303	0.0506514	−	0.998716i	$-0.483870\pi$
$877$	3.00000	0.101303	0.0506514	+	0.998716i	$0.483870\pi$
$878$	−2.00000	−0.0674967
$879$	−9.00000	−0.303562
$880$	4.00000	0.134840
$881$	47.0000	1.58347	0.791735	−	0.610865i	$-0.209178\pi$
$881$	47.0000	1.58347	0.791735	+	0.610865i	$0.209178\pi$
$882$	−2.00000	−0.0673435
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	12.0000	0.403604
$885$	−3.00000	−0.100844
$886$	−8.00000	−0.268765
$887$	55.0000	1.84672	0.923360	−	0.383936i	$-0.125432\pi$
$887$	55.0000	1.84672	0.923360	+	0.383936i	$0.125432\pi$
$888$	0	0
$889$	5.00000	0.167695
$890$	18.0000	0.603361
$891$	−1.00000	−0.0335013
$892$	30.0000	1.00447
$893$	−12.0000	−0.401565
$894$	−12.0000	−0.401340
$895$	−4.00000	−0.133705
$896$	0	0
$897$	−14.0000	−0.467446
$898$	24.0000	0.800890
$899$	8.00000	0.266815
$900$	2.00000	0.0666667
$901$	15.0000	0.499722
$902$	−16.0000	−0.532742
$903$	−9.00000	−0.299501
$904$	0	0
$905$	2.00000	0.0664822
$906$	28.0000	0.930238
$907$	−6.00000	−0.199227	−0.0996134	−	0.995026i	$-0.531761\pi$
$907$	−6.00000	−0.199227	−0.0996134	+	0.995026i	$0.531761\pi$
$908$	22.0000	0.730096
$909$	−14.0000	−0.464351
$910$	4.00000	0.132599
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	4.00000	0.132453
$913$	−11.0000	−0.364047
$914$	6.00000	0.198462
$915$	−13.0000	−0.429767
$916$	−8.00000	−0.264327
$917$	2.00000	0.0660458
$918$	−6.00000	−0.198030
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	0	0
$921$	−30.0000	−0.988534
$922$	−12.0000	−0.395199
$923$	−16.0000	−0.526646
$924$	2.00000	0.0657952
$925$	−2.00000	−0.0657596
$926$	4.00000	0.131448
$927$	−7.00000	−0.229910
$928$	8.00000	0.262613
$929$	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
$931$	1.00000	0.0327737
$932$	20.0000	0.655122
$933$	4.00000	0.130954
$934$	−64.0000	−2.09414
$935$	3.00000	0.0981105
$936$	0	0
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	28.0000	0.914232
$939$	−15.0000	−0.489506
$940$	−24.0000	−0.782794
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	−46.0000	−1.49876
$943$	56.0000	1.82361
$944$	−12.0000	−0.390567
$945$	−1.00000	−0.0325300
$946$	18.0000	0.585230
$947$	57.0000	1.85225	0.926126	−	0.377215i	$-0.123118\pi$
$947$	57.0000	1.85225	0.926126	+	0.377215i	$0.123118\pi$
$948$	28.0000	0.909398
$949$	16.0000	0.519382
$950$	−2.00000	−0.0648886
$951$	−30.0000	−0.972817
$952$	0	0
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	10.0000	0.323762
$955$	4.00000	0.129437
$956$	30.0000	0.970269
$957$	1.00000	0.0323254
$958$	−20.0000	−0.646171
$959$	−10.0000	−0.322917
$960$	8.00000	0.258199
$961$	33.0000	1.06452
$962$	−8.00000	−0.257930
$963$	−8.00000	−0.257796
$964$	4.00000	0.128831
$965$	26.0000	0.836970
$966$	−14.0000	−0.450443
$967$	5.00000	0.160789	0.0803946	−	0.996763i	$-0.474382\pi$
$967$	5.00000	0.160789	0.0803946	+	0.996763i	$0.474382\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	−10.0000	−0.321081
$971$	43.0000	1.37994	0.689968	−	0.723840i	$-0.257625\pi$
$971$	43.0000	1.37994	0.689968	+	0.723840i	$0.257625\pi$
$972$	−2.00000	−0.0641500
$973$	8.00000	0.256468
$974$	−68.0000	−2.17886
$975$	2.00000	0.0640513
$976$	−52.0000	−1.66448
$977$	−41.0000	−1.31171	−0.655853	−	0.754889i	$-0.727691\pi$
$977$	−41.0000	−1.31171	−0.655853	+	0.754889i	$0.727691\pi$
$978$	−28.0000	−0.895341
$979$	9.00000	0.287641
$980$	2.00000	0.0638877
$981$	−16.0000	−0.510841
$982$	30.0000	0.957338
$983$	22.0000	0.701691	0.350846	−	0.936433i	$-0.385894\pi$
$983$	22.0000	0.701691	0.350846	+	0.936433i	$0.385894\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	6.00000	0.191079
$987$	12.0000	0.381964
$988$	−4.00000	−0.127257
$989$	−63.0000	−2.00328
$990$	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	$0.231732\pi$
$992$	64.0000	2.03200
$993$	27.0000	0.856819
$994$	−16.0000	−0.507489
$995$	−20.0000	−0.634043
$996$	−22.0000	−0.697097
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	82.0000	2.59566
$999$	2.00000	0.0632772

Twists

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

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