LMFDB - Embedded newform 5415.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5415,2,Mod(1,5415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5415, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5415 = 3 \cdot 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5415.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:	$43.2389926945$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} +2.00000 q^{18} -2.00000 q^{20} -2.00000 q^{21} +2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +5.00000 q^{29} -2.00000 q^{30} -9.00000 q^{31} -8.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +2.00000 q^{35} +2.00000 q^{36} +6.00000 q^{37} -2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} -10.0000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} -4.00000 q^{48} -3.00000 q^{49} +2.00000 q^{50} +2.00000 q^{51} -4.00000 q^{52} +2.00000 q^{53} +2.00000 q^{54} -1.00000 q^{55} +10.0000 q^{58} -7.00000 q^{59} -2.00000 q^{60} -7.00000 q^{61} -18.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} -8.00000 q^{67} +4.00000 q^{68} -4.00000 q^{69} +4.00000 q^{70} -3.00000 q^{71} -2.00000 q^{73} +12.0000 q^{74} +1.00000 q^{75} -2.00000 q^{77} -4.00000 q^{78} +11.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} +6.00000 q^{83} -4.00000 q^{84} -2.00000 q^{85} -20.0000 q^{86} +5.00000 q^{87} -15.0000 q^{89} -2.00000 q^{90} +4.00000 q^{91} -8.00000 q^{92} -9.00000 q^{93} -8.00000 q^{96} -8.00000 q^{97} -6.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	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	$0.589456\pi$
$14$	−4.00000	−1.06904
$15$	−1.00000	−0.258199
$16$	−4.00000	−1.00000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	2.00000	0.471405
$19$	0	0
$19$	0	0
$20$	−2.00000	−0.447214
$21$	−2.00000	−0.436436
$22$	2.00000	0.426401
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	−4.00000	−0.755929
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	−2.00000	−0.365148
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−8.00000	−1.41421
$33$	1.00000	0.174078
$34$	4.00000	0.685994
$35$	2.00000	0.338062
$36$	2.00000	0.333333
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−4.00000	−0.617213
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	2.00000	0.301511
$45$	−1.00000	−0.149071
$46$	−8.00000	−1.17954
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−4.00000	−0.577350
$49$	−3.00000	−0.428571
$50$	2.00000	0.282843
$51$	2.00000	0.280056
$52$	−4.00000	−0.554700
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	2.00000	0.272166
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	10.0000	1.31306
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	−2.00000	−0.258199
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	−18.0000	−2.28600
$63$	−2.00000	−0.251976
$64$	−8.00000	−1.00000
$65$	2.00000	0.248069
$66$	2.00000	0.246183
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	4.00000	0.485071
$69$	−4.00000	−0.481543
$70$	4.00000	0.478091
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	12.0000	1.39497
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.00000	−0.227921
$78$	−4.00000	−0.452911
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−4.00000	−0.436436
$85$	−2.00000	−0.216930
$86$	−20.0000	−2.15666
$87$	5.00000	0.536056
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	−2.00000	−0.210819
$91$	4.00000	0.419314
$92$	−8.00000	−0.834058
$93$	−9.00000	−0.933257
$94$	0	0
$95$	0	0
$96$	−8.00000	−0.816497
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−6.00000	−0.606092
$99$	1.00000	0.100504
$100$	2.00000	0.200000
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	4.00000	0.396059
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	4.00000	0.388514
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	2.00000	0.192450
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	−2.00000	−0.190693
$111$	6.00000	0.569495
$112$	8.00000	0.755929
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	10.0000	0.928477
$117$	−2.00000	−0.184900
$118$	−14.0000	−1.28880
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−14.0000	−1.26750
$123$	−6.00000	−0.541002
$124$	−18.0000	−1.61645
$125$	−1.00000	−0.0894427
$126$	−4.00000	−0.356348
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	4.00000	0.350823
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	−16.0000	−1.38219
$135$	−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	$0.582506\pi$
$138$	−8.00000	−0.681005
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−2.00000	−0.167248
$144$	−4.00000	−0.333333
$145$	−5.00000	−0.415227
$146$	−4.00000	−0.331042
$147$	−3.00000	−0.247436
$148$	12.0000	0.986394
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	2.00000	0.163299
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	−4.00000	−0.322329
$155$	9.00000	0.722897
$156$	−4.00000	−0.320256
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	22.0000	1.75023
$159$	2.00000	0.158610
$160$	8.00000	0.632456
$161$	8.00000	0.630488
$162$	2.00000	0.157135
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	−12.0000	−0.937043
$165$	−1.00000	−0.0778499
$166$	12.0000	0.931381
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−20.0000	−1.52499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	10.0000	0.758098
$175$	−2.00000	−0.151186
$176$	−4.00000	−0.301511
$177$	−7.00000	−0.526152
$178$	−30.0000	−2.24860
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	−2.00000	−0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	8.00000	0.592999
$183$	−7.00000	−0.517455
$184$	0	0
$185$	−6.00000	−0.441129
$186$	−18.0000	−1.31982
$187$	2.00000	0.146254
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	−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	$0.114715\pi$
$194$	−16.0000	−1.14873
$195$	2.00000	0.143223
$196$	−6.00000	−0.428571
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	2.00000	0.142134
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	−14.0000	−0.985037
$203$	−10.0000	−0.701862
$204$	4.00000	0.280056
$205$	6.00000	0.419058
$206$	16.0000	1.11477
$207$	−4.00000	−0.278019
$208$	8.00000	0.554700
$209$	0	0
$210$	4.00000	0.276026
$211$	27.0000	1.85876	0.929378	−	0.369129i	$-0.120344\pi$
$211$	27.0000	1.85876	0.929378	+	0.369129i	$0.120344\pi$
$212$	4.00000	0.274721
$213$	−3.00000	−0.205557
$214$	0	0
$215$	10.0000	0.681994
$216$	0	0
$217$	18.0000	1.22192
$218$	−38.0000	−2.57368
$219$	−2.00000	−0.135147
$220$	−2.00000	−0.134840
$221$	−4.00000	−0.269069
$222$	12.0000	0.805387
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	16.0000	1.06904
$225$	1.00000	0.0666667
$226$	20.0000	1.33038
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	8.00000	0.527504
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	−14.0000	−0.911322
$237$	11.0000	0.714527
$238$	−8.00000	−0.518563
$239$	27.0000	1.74648	0.873242	−	0.487286i	$-0.162013\pi$
$239$	27.0000	1.74648	0.873242	+	0.487286i	$0.162013\pi$
$240$	4.00000	0.258199
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	−20.0000	−1.28565
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	3.00000	0.191663
$246$	−12.0000	−0.765092
$247$	0	0
$248$	0	0
$249$	6.00000	0.380235
$250$	−2.00000	−0.126491
$251$	−17.0000	−1.07303	−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000	−1.07303	−0.536515	+	0.843891i	$0.680260\pi$
$252$	−4.00000	−0.251976
$253$	−4.00000	−0.251478
$254$	−8.00000	−0.501965
$255$	−2.00000	−0.125245
$256$	16.0000	1.00000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	−20.0000	−1.24515
$259$	−12.0000	−0.745644
$260$	4.00000	0.248069
$261$	5.00000	0.309492
$262$	32.0000	1.97697
$263$	20.0000	1.23325	0.616626	−	0.787256i	$-0.288499\pi$
$263$	20.0000	1.23325	0.616626	+	0.787256i	$0.288499\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−15.0000	−0.917985
$268$	−16.0000	−0.977356
$269$	−1.00000	−0.0609711	−0.0304855	−	0.999535i	$-0.509705\pi$
$269$	−1.00000	−0.0609711	−0.0304855	+	0.999535i	$0.509705\pi$
$270$	−2.00000	−0.121716
$271$	27.0000	1.64013	0.820067	−	0.572268i	$-0.193936\pi$
$271$	27.0000	1.64013	0.820067	+	0.572268i	$0.193936\pi$
$272$	−8.00000	−0.485071
$273$	4.00000	0.242091
$274$	−12.0000	−0.724947
$275$	1.00000	0.0603023
$276$	−8.00000	−0.481543
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	40.0000	2.39904
$279$	−9.00000	−0.538816
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−4.00000	−0.236525
$287$	12.0000	0.708338
$288$	−8.00000	−0.471405
$289$	−13.0000	−0.764706
$290$	−10.0000	−0.587220
$291$	−8.00000	−0.468968
$292$	−4.00000	−0.234082
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	−6.00000	−0.349927
$295$	7.00000	0.407556
$296$	0	0
$297$	1.00000	0.0580259
$298$	18.0000	1.04271
$299$	8.00000	0.462652
$300$	2.00000	0.115470
$301$	20.0000	1.15278
$302$	6.00000	0.345261
$303$	−7.00000	−0.402139
$304$	0	0
$305$	7.00000	0.400819
$306$	4.00000	0.228665
$307$	24.0000	1.36975	0.684876	−	0.728659i	$-0.259856\pi$
$307$	24.0000	1.36975	0.684876	+	0.728659i	$0.259856\pi$
$308$	−4.00000	−0.227921
$309$	8.00000	0.455104
$310$	18.0000	1.02233
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	24.0000	1.35440
$315$	2.00000	0.112687
$316$	22.0000	1.23760
$317$	−32.0000	−1.79730	−0.898650	−	0.438667i	$-0.855451\pi$
$317$	−32.0000	−1.79730	−0.898650	+	0.438667i	$0.855451\pi$
$318$	4.00000	0.224309
$319$	5.00000	0.279946
$320$	8.00000	0.447214
$321$	0	0
$322$	16.0000	0.891645
$323$	0	0
$324$	2.00000	0.111111
$325$	−2.00000	−0.110940
$326$	−36.0000	−1.99386
$327$	−19.0000	−1.05070
$328$	0	0
$329$	0	0
$330$	−2.00000	−0.110096
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	12.0000	0.658586
$333$	6.00000	0.328798
$334$	4.00000	0.218870
$335$	8.00000	0.437087
$336$	8.00000	0.436436
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	−18.0000	−0.979071
$339$	10.0000	0.543125
$340$	−4.00000	−0.216930
$341$	−9.00000	−0.487377
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	4.00000	0.215353
$346$	−12.0000	−0.645124
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	10.0000	0.536056
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	−4.00000	−0.213809
$351$	−2.00000	−0.106752
$352$	−8.00000	−0.426401
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−14.0000	−0.744092
$355$	3.00000	0.159223
$356$	−30.0000	−1.59000
$357$	−4.00000	−0.211702
$358$	6.00000	0.317110
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	0	0
$362$	−20.0000	−1.05118
$363$	−10.0000	−0.524864
$364$	8.00000	0.419314
$365$	2.00000	0.104685
$366$	−14.0000	−0.731792
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	16.0000	0.834058
$369$	−6.00000	−0.312348
$370$	−12.0000	−0.623850
$371$	−4.00000	−0.207670
$372$	−18.0000	−0.933257
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	4.00000	0.206835
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−10.0000	−0.515026
$378$	−4.00000	−0.205738
$379$	−31.0000	−1.59236	−0.796182	−	0.605058i	$-0.793150\pi$
$379$	−31.0000	−1.59236	−0.796182	+	0.605058i	$0.793150\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	30.0000	1.53493
$383$	−26.0000	−1.32854	−0.664269	−	0.747494i	$-0.731257\pi$
$383$	−26.0000	−1.32854	−0.664269	+	0.747494i	$0.731257\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	52.0000	2.64673
$387$	−10.0000	−0.508329
$388$	−16.0000	−0.812277
$389$	33.0000	1.67317	0.836583	−	0.547840i	$-0.184550\pi$
$389$	33.0000	1.67317	0.836583	+	0.547840i	$0.184550\pi$
$390$	4.00000	0.202548
$391$	−8.00000	−0.404577
$392$	0	0
$393$	16.0000	0.807093
$394$	20.0000	1.00759
$395$	−11.0000	−0.553470
$396$	2.00000	0.100504
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−6.00000	−0.300753
$399$	0	0
$400$	−4.00000	−0.200000
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	−16.0000	−0.798007
$403$	18.0000	0.896644
$404$	−14.0000	−0.696526
$405$	−1.00000	−0.0496904
$406$	−20.0000	−0.992583
$407$	6.00000	0.297409
$408$	0	0
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	12.0000	0.592638
$411$	−6.00000	−0.295958
$412$	16.0000	0.788263
$413$	14.0000	0.688895
$414$	−8.00000	−0.393179
$415$	−6.00000	−0.294528
$416$	16.0000	0.784465
$417$	20.0000	0.979404
$418$	0	0
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	4.00000	0.195180
$421$	7.00000	0.341159	0.170580	−	0.985344i	$-0.445436\pi$
$421$	7.00000	0.341159	0.170580	+	0.985344i	$0.445436\pi$
$422$	54.0000	2.62868
$423$	0	0
$424$	0	0
$425$	2.00000	0.0970143
$426$	−6.00000	−0.290701
$427$	14.0000	0.677507
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	20.0000	0.964486
$431$	−23.0000	−1.10787	−0.553936	−	0.832560i	$-0.686875\pi$
$431$	−23.0000	−1.10787	−0.553936	+	0.832560i	$0.686875\pi$
$432$	−4.00000	−0.192450
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	36.0000	1.72806
$435$	−5.00000	−0.239732
$436$	−38.0000	−1.81987
$437$	0	0
$438$	−4.00000	−0.191127
$439$	25.0000	1.19318	0.596592	−	0.802544i	$-0.296521\pi$
$439$	25.0000	1.19318	0.596592	+	0.802544i	$0.296521\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−8.00000	−0.380521
$443$	−8.00000	−0.380091	−0.190046	−	0.981775i	$-0.560864\pi$
$443$	−8.00000	−0.380091	−0.190046	+	0.981775i	$0.560864\pi$
$444$	12.0000	0.569495
$445$	15.0000	0.711068
$446$	24.0000	1.13643
$447$	9.00000	0.425685
$448$	16.0000	0.755929
$449$	39.0000	1.84052	0.920262	−	0.391303i	$-0.127976\pi$
$449$	39.0000	1.84052	0.920262	+	0.391303i	$0.127976\pi$
$450$	2.00000	0.0942809
$451$	−6.00000	−0.282529
$452$	20.0000	0.940721
$453$	3.00000	0.140952
$454$	−36.0000	−1.68956
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	−22.0000	−1.02799
$459$	2.00000	0.0933520
$460$	8.00000	0.373002
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	−4.00000	−0.186097
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	−20.0000	−0.928477
$465$	9.00000	0.417365
$466$	−48.0000	−2.22356
$467$	−38.0000	−1.75843	−0.879215	−	0.476425i	$-0.841932\pi$
$467$	−38.0000	−1.75843	−0.879215	+	0.476425i	$0.841932\pi$
$468$	−4.00000	−0.184900
$469$	16.0000	0.738811
$470$	0	0
$471$	12.0000	0.552931
$472$	0	0
$473$	−10.0000	−0.459800
$474$	22.0000	1.01049
$475$	0	0
$476$	−8.00000	−0.366679
$477$	2.00000	0.0915737
$478$	54.0000	2.46990
$479$	5.00000	0.228456	0.114228	−	0.993455i	$-0.463561\pi$
$479$	5.00000	0.228456	0.114228	+	0.993455i	$0.463561\pi$
$480$	8.00000	0.365148
$481$	−12.0000	−0.547153
$482$	−10.0000	−0.455488
$483$	8.00000	0.364013
$484$	−20.0000	−0.909091
$485$	8.00000	0.363261
$486$	2.00000	0.0907218
$487$	−14.0000	−0.634401	−0.317200	−	0.948359i	$-0.602743\pi$
$487$	−14.0000	−0.634401	−0.317200	+	0.948359i	$0.602743\pi$
$488$	0	0
$489$	−18.0000	−0.813988
$490$	6.00000	0.271052
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	−12.0000	−0.541002
$493$	10.0000	0.450377
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	36.0000	1.61645
$497$	6.00000	0.269137
$498$	12.0000	0.537733
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−2.00000	−0.0894427
$501$	2.00000	0.0893534
$502$	−34.0000	−1.51749
$503$	−44.0000	−1.96186	−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−44.0000	−1.96186	−0.980932	+	0.194354i	$0.937739\pi$
$504$	0	0
$505$	7.00000	0.311496
$506$	−8.00000	−0.355643
$507$	−9.00000	−0.399704
$508$	−8.00000	−0.354943
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	−4.00000	−0.177123
$511$	4.00000	0.176950
$512$	32.0000	1.41421
$513$	0	0
$514$	−44.0000	−1.94076
$515$	−8.00000	−0.352522
$516$	−20.0000	−0.880451
$517$	0	0
$518$	−24.0000	−1.05450
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−21.0000	−0.920027	−0.460013	−	0.887912i	$-0.652155\pi$
$521$	−21.0000	−0.920027	−0.460013	+	0.887912i	$0.652155\pi$
$522$	10.0000	0.437688
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	32.0000	1.39793
$525$	−2.00000	−0.0872872
$526$	40.0000	1.74408
$527$	−18.0000	−0.784092
$528$	−4.00000	−0.174078
$529$	−7.00000	−0.304348
$530$	−4.00000	−0.173749
$531$	−7.00000	−0.303774
$532$	0	0
$533$	12.0000	0.519778
$534$	−30.0000	−1.29823
$535$	0	0
$536$	0	0
$537$	3.00000	0.129460
$538$	−2.00000	−0.0862261
$539$	−3.00000	−0.129219
$540$	−2.00000	−0.0860663
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	54.0000	2.31950
$543$	−10.0000	−0.429141
$544$	−16.0000	−0.685994
$545$	19.0000	0.813871
$546$	8.00000	0.342368
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−12.0000	−0.512615
$549$	−7.00000	−0.298753
$550$	2.00000	0.0852803
$551$	0	0
$552$	0	0
$553$	−22.0000	−0.935535
$554$	−16.0000	−0.679775
$555$	−6.00000	−0.254686
$556$	40.0000	1.69638
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	−18.0000	−0.762001
$559$	20.0000	0.845910
$560$	−8.00000	−0.338062
$561$	2.00000	0.0844401
$562$	52.0000	2.19349
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	4.00000	0.168133
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	39.0000	1.63497	0.817483	−	0.575953i	$-0.195369\pi$
$569$	39.0000	1.63497	0.817483	+	0.575953i	$0.195369\pi$
$570$	0	0
$571$	−15.0000	−0.627730	−0.313865	−	0.949468i	$-0.601624\pi$
$571$	−15.0000	−0.627730	−0.313865	+	0.949468i	$0.601624\pi$
$572$	−4.00000	−0.167248
$573$	15.0000	0.626634
$574$	24.0000	1.00174
$575$	−4.00000	−0.166812
$576$	−8.00000	−0.333333
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	−26.0000	−1.08146
$579$	26.0000	1.08052
$580$	−10.0000	−0.415227
$581$	−12.0000	−0.497844
$582$	−16.0000	−0.663221
$583$	2.00000	0.0828315
$584$	0	0
$585$	2.00000	0.0826898
$586$	8.00000	0.330477
$587$	22.0000	0.908037	0.454019	−	0.890992i	$-0.349990\pi$
$587$	22.0000	0.908037	0.454019	+	0.890992i	$0.349990\pi$
$588$	−6.00000	−0.247436
$589$	0	0
$590$	14.0000	0.576371
$591$	10.0000	0.411345
$592$	−24.0000	−0.986394
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	2.00000	0.0820610
$595$	4.00000	0.163984
$596$	18.0000	0.737309
$597$	−3.00000	−0.122782
$598$	16.0000	0.654289
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	15.0000	0.611863	0.305931	−	0.952054i	$-0.401032\pi$
$601$	15.0000	0.611863	0.305931	+	0.952054i	$0.401032\pi$
$602$	40.0000	1.63028
$603$	−8.00000	−0.325785
$604$	6.00000	0.244137
$605$	10.0000	0.406558
$606$	−14.0000	−0.568711
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	0	0
$609$	−10.0000	−0.405220
$610$	14.0000	0.566843
$611$	0	0
$612$	4.00000	0.161690
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	48.0000	1.93712
$615$	6.00000	0.241943
$616$	0	0
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	16.0000	0.643614
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	18.0000	0.722897
$621$	−4.00000	−0.160514
$622$	48.0000	1.92462
$623$	30.0000	1.20192
$624$	8.00000	0.320256
$625$	1.00000	0.0400000
$626$	−40.0000	−1.59872
$627$	0	0
$628$	24.0000	0.957704
$629$	12.0000	0.478471
$630$	4.00000	0.159364
$631$	−3.00000	−0.119428	−0.0597141	−	0.998216i	$-0.519019\pi$
$631$	−3.00000	−0.119428	−0.0597141	+	0.998216i	$0.519019\pi$
$632$	0	0
$633$	27.0000	1.07315
$634$	−64.0000	−2.54176
$635$	4.00000	0.158735
$636$	4.00000	0.158610
$637$	6.00000	0.237729
$638$	10.0000	0.395904
$639$	−3.00000	−0.118678
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	16.0000	0.630488
$645$	10.0000	0.393750
$646$	0	0
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	−7.00000	−0.274774
$650$	−4.00000	−0.156893
$651$	18.0000	0.705476
$652$	−36.0000	−1.40987
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−38.0000	−1.48592
$655$	−16.0000	−0.625172
$656$	24.0000	0.937043
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	−2.00000	−0.0778499
$661$	−9.00000	−0.350059	−0.175030	−	0.984563i	$-0.556002\pi$
$661$	−9.00000	−0.350059	−0.175030	+	0.984563i	$0.556002\pi$
$662$	−16.0000	−0.621858
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	12.0000	0.464991
$667$	−20.0000	−0.774403
$668$	4.00000	0.154765
$669$	12.0000	0.463947
$670$	16.0000	0.618134
$671$	−7.00000	−0.270232
$672$	16.0000	0.617213
$673$	−36.0000	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$673$	−36.0000	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$674$	16.0000	0.616297
$675$	1.00000	0.0384900
$676$	−18.0000	−0.692308
$677$	−46.0000	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$678$	20.0000	0.768095
$679$	16.0000	0.614024
$680$	0	0
$681$	−18.0000	−0.689761
$682$	−18.0000	−0.689256
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	40.0000	1.52721
$687$	−11.0000	−0.419676
$688$	40.0000	1.52499
$689$	−4.00000	−0.152388
$690$	8.00000	0.304555
$691$	23.0000	0.874961	0.437481	−	0.899228i	$-0.355871\pi$
$691$	23.0000	0.874961	0.437481	+	0.899228i	$0.355871\pi$
$692$	−12.0000	−0.456172
$693$	−2.00000	−0.0759737
$694$	−44.0000	−1.67022
$695$	−20.0000	−0.758643
$696$	0	0
$697$	−12.0000	−0.454532
$698$	12.0000	0.454207
$699$	−24.0000	−0.907763
$700$	−4.00000	−0.151186
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	−4.00000	−0.150970
$703$	0	0
$704$	−8.00000	−0.301511
$705$	0	0
$706$	−28.0000	−1.05379
$707$	14.0000	0.526524
$708$	−14.0000	−0.526152
$709$	−27.0000	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$709$	−27.0000	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$710$	6.00000	0.225176
$711$	11.0000	0.412532
$712$	0	0
$713$	36.0000	1.34821
$714$	−8.00000	−0.299392
$715$	2.00000	0.0747958
$716$	6.00000	0.224231
$717$	27.0000	1.00833
$718$	−16.0000	−0.597115
$719$	−27.0000	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$720$	4.00000	0.149071
$721$	−16.0000	−0.595871
$722$	0	0
$723$	−5.00000	−0.185952
$724$	−20.0000	−0.743294
$725$	5.00000	0.185695
$726$	−20.0000	−0.742270
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	−20.0000	−0.739727
$732$	−14.0000	−0.517455
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	−40.0000	−1.47643
$735$	3.00000	0.110657
$736$	32.0000	1.17954
$737$	−8.00000	−0.294684
$738$	−12.0000	−0.441726
$739$	−53.0000	−1.94964	−0.974818	−	0.223001i	$-0.928415\pi$
$739$	−53.0000	−1.94964	−0.974818	+	0.223001i	$0.928415\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−8.00000	−0.293689
$743$	−22.0000	−0.807102	−0.403551	−	0.914957i	$-0.632224\pi$
$743$	−22.0000	−0.807102	−0.403551	+	0.914957i	$0.632224\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	64.0000	2.34321
$747$	6.00000	0.219529
$748$	4.00000	0.146254
$749$	0	0
$750$	−2.00000	−0.0730297
$751$	3.00000	0.109472	0.0547358	−	0.998501i	$-0.482568\pi$
$751$	3.00000	0.109472	0.0547358	+	0.998501i	$0.482568\pi$
$752$	0	0
$753$	−17.0000	−0.619514
$754$	−20.0000	−0.728357
$755$	−3.00000	−0.109181
$756$	−4.00000	−0.145479
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	−62.0000	−2.25194
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−8.00000	−0.289809
$763$	38.0000	1.37569
$764$	30.0000	1.08536
$765$	−2.00000	−0.0723102
$766$	−52.0000	−1.87884
$767$	14.0000	0.505511
$768$	16.0000	0.577350
$769$	−43.0000	−1.55062	−0.775310	−	0.631581i	$-0.782406\pi$
$769$	−43.0000	−1.55062	−0.775310	+	0.631581i	$0.782406\pi$
$770$	4.00000	0.144150
$771$	−22.0000	−0.792311
$772$	52.0000	1.87152
$773$	−20.0000	−0.719350	−0.359675	−	0.933078i	$-0.617112\pi$
$773$	−20.0000	−0.719350	−0.359675	+	0.933078i	$0.617112\pi$
$774$	−20.0000	−0.718885
$775$	−9.00000	−0.323290
$776$	0	0
$777$	−12.0000	−0.430498
$778$	66.0000	2.36621
$779$	0	0
$780$	4.00000	0.143223
$781$	−3.00000	−0.107348
$782$	−16.0000	−0.572159
$783$	5.00000	0.178685
$784$	12.0000	0.428571
$785$	−12.0000	−0.428298
$786$	32.0000	1.14140
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	20.0000	0.712470
$789$	20.0000	0.712019
$790$	−22.0000	−0.782725
$791$	−20.0000	−0.711118
$792$	0	0
$793$	14.0000	0.497155
$794$	−40.0000	−1.41955
$795$	−2.00000	−0.0709327
$796$	−6.00000	−0.212664
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	0	0
$799$	0	0
$800$	−8.00000	−0.282843
$801$	−15.0000	−0.529999
$802$	54.0000	1.90681
$803$	−2.00000	−0.0705785
$804$	−16.0000	−0.564276
$805$	−8.00000	−0.281963
$806$	36.0000	1.26805
$807$	−1.00000	−0.0352017
$808$	0	0
$809$	−1.00000	−0.0351581	−0.0175791	−	0.999845i	$-0.505596\pi$
$809$	−1.00000	−0.0351581	−0.0175791	+	0.999845i	$0.505596\pi$
$810$	−2.00000	−0.0702728
$811$	−49.0000	−1.72062	−0.860311	−	0.509769i	$-0.829731\pi$
$811$	−49.0000	−1.72062	−0.860311	+	0.509769i	$0.829731\pi$
$812$	−20.0000	−0.701862
$813$	27.0000	0.946931
$814$	12.0000	0.420600
$815$	18.0000	0.630512
$816$	−8.00000	−0.280056
$817$	0	0
$818$	58.0000	2.02792
$819$	4.00000	0.139771
$820$	12.0000	0.419058
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	−12.0000	−0.418548
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	28.0000	0.974245
$827$	−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
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−12.0000	−0.416526
$831$	−8.00000	−0.277517
$832$	16.0000	0.554700
$833$	−6.00000	−0.207888
$834$	40.0000	1.38509
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	−9.00000	−0.311086
$838$	−10.0000	−0.345444
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	14.0000	0.482472
$843$	26.0000	0.895488
$844$	54.0000	1.85876
$845$	9.00000	0.309609
$846$	0	0
$847$	20.0000	0.687208
$848$	−8.00000	−0.274721
$849$	2.00000	0.0686398
$850$	4.00000	0.137199
$851$	−24.0000	−0.822709
$852$	−6.00000	−0.205557
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	28.0000	0.958140
$855$	0	0
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	−4.00000	−0.136558
$859$	45.0000	1.53538	0.767690	−	0.640821i	$-0.221406\pi$
$859$	45.0000	1.53538	0.767690	+	0.640821i	$0.221406\pi$
$860$	20.0000	0.681994
$861$	12.0000	0.408959
$862$	−46.0000	−1.56677
$863$	38.0000	1.29354	0.646768	−	0.762687i	$-0.276120\pi$
$863$	38.0000	1.29354	0.646768	+	0.762687i	$0.276120\pi$
$864$	−8.00000	−0.272166
$865$	6.00000	0.204006
$866$	−12.0000	−0.407777
$867$	−13.0000	−0.441503
$868$	36.0000	1.22192
$869$	11.0000	0.373149
$870$	−10.0000	−0.339032
$871$	16.0000	0.542139
$872$	0	0
$873$	−8.00000	−0.270759
$874$	0	0
$875$	2.00000	0.0676123
$876$	−4.00000	−0.135147
$877$	−24.0000	−0.810422	−0.405211	−	0.914223i	$-0.632802\pi$
$877$	−24.0000	−0.810422	−0.405211	+	0.914223i	$0.632802\pi$
$878$	50.0000	1.68742
$879$	4.00000	0.134917
$880$	4.00000	0.134840
$881$	−1.00000	−0.0336909	−0.0168454	−	0.999858i	$-0.505362\pi$
$881$	−1.00000	−0.0336909	−0.0168454	+	0.999858i	$0.505362\pi$
$882$	−6.00000	−0.202031
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	−8.00000	−0.269069
$885$	7.00000	0.235302
$886$	−16.0000	−0.537531
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	30.0000	1.00560
$891$	1.00000	0.0335013
$892$	24.0000	0.803579
$893$	0	0
$894$	18.0000	0.602010
$895$	−3.00000	−0.100279
$896$	0	0
$897$	8.00000	0.267112
$898$	78.0000	2.60289
$899$	−45.0000	−1.50083
$900$	2.00000	0.0666667
$901$	4.00000	0.133259
$902$	−12.0000	−0.399556
$903$	20.0000	0.665558
$904$	0	0
$905$	10.0000	0.332411
$906$	6.00000	0.199337
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−36.0000	−1.19470
$909$	−7.00000	−0.232175
$910$	−8.00000	−0.265197
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	−16.0000	−0.529233
$915$	7.00000	0.231413
$916$	−22.0000	−0.726900
$917$	−32.0000	−1.05673
$918$	4.00000	0.132020
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	24.0000	0.790827
$922$	−6.00000	−0.197599
$923$	6.00000	0.197492
$924$	−4.00000	−0.131590
$925$	6.00000	0.197279
$926$	−20.0000	−0.657241
$927$	8.00000	0.262754
$928$	−40.0000	−1.31306
$929$	−11.0000	−0.360898	−0.180449	−	0.983584i	$-0.557755\pi$
$929$	−11.0000	−0.360898	−0.180449	+	0.983584i	$0.557755\pi$
$930$	18.0000	0.590243
$931$	0	0
$932$	−48.0000	−1.57229
$933$	24.0000	0.785725
$934$	−76.0000	−2.48680
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	32.0000	1.04484
$939$	−20.0000	−0.652675
$940$	0	0
$941$	−21.0000	−0.684580	−0.342290	−	0.939594i	$-0.611203\pi$
$941$	−21.0000	−0.684580	−0.342290	+	0.939594i	$0.611203\pi$
$942$	24.0000	0.781962
$943$	24.0000	0.781548
$944$	28.0000	0.911322
$945$	2.00000	0.0650600
$946$	−20.0000	−0.650256
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	22.0000	0.714527
$949$	4.00000	0.129845
$950$	0	0
$951$	−32.0000	−1.03767
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	4.00000	0.129505
$955$	−15.0000	−0.485389
$956$	54.0000	1.74648
$957$	5.00000	0.161627
$958$	10.0000	0.323085
$959$	12.0000	0.387500
$960$	8.00000	0.258199
$961$	50.0000	1.61290
$962$	−24.0000	−0.773791
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−26.0000	−0.836970
$966$	16.0000	0.514792
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	0	0
$969$	0	0
$970$	16.0000	0.513729
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	2.00000	0.0641500
$973$	−40.0000	−1.28234
$974$	−28.0000	−0.897178
$975$	−2.00000	−0.0640513
$976$	28.0000	0.896258
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	−36.0000	−1.15115
$979$	−15.0000	−0.479402
$980$	6.00000	0.191663
$981$	−19.0000	−0.606623
$982$	−66.0000	−2.10614
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	20.0000	0.636930
$987$	0	0
$988$	0	0
$989$	40.0000	1.27193
$990$	−2.00000	−0.0635642
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	72.0000	2.28600
$993$	−8.00000	−0.253872
$994$	12.0000	0.380617
$995$	3.00000	0.0951064
$996$	12.0000	0.380235
$997$	52.0000	1.64686	0.823428	−	0.567420i	$-0.192059\pi$
$997$	52.0000	1.64686	0.823428	+	0.567420i	$0.192059\pi$
$998$	40.0000	1.26618
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5415.2.a.l.1.1		1
19.8	odd	6		285.2.i.c.121.1	yes	2
19.12	odd	6		285.2.i.c.106.1	✓	2
19.18	odd	2		5415.2.a.b.1.1		1
57.8	even	6		855.2.k.a.406.1		2
57.50	even	6		855.2.k.a.676.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.i.c.106.1	✓	2	19.12	odd	6
285.2.i.c.121.1	yes	2	19.8	odd	6
855.2.k.a.406.1		2	57.8	even	6
855.2.k.a.676.1		2	57.50	even	6
5415.2.a.b.1.1		1	19.18	odd	2
5415.2.a.l.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5415.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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