LMFDB - Embedded newform 1045.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
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	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{12} +6.00000 q^{13} +2.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} -1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} +4.00000 q^{21} +1.00000 q^{22} -6.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} +4.00000 q^{27} +2.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} +4.00000 q^{31} -5.00000 q^{32} +2.00000 q^{33} +2.00000 q^{35} -1.00000 q^{36} +1.00000 q^{38} -12.0000 q^{39} -3.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} -6.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} -4.00000 q^{47} +2.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} +12.0000 q^{53} -4.00000 q^{54} +1.00000 q^{55} -6.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} -2.00000 q^{60} -14.0000 q^{61} -4.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} -6.00000 q^{65} -2.00000 q^{66} +6.00000 q^{67} -2.00000 q^{70} -8.00000 q^{71} +3.00000 q^{72} +16.0000 q^{73} -2.00000 q^{75} +1.00000 q^{76} +2.00000 q^{77} +12.0000 q^{78} -8.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -4.00000 q^{84} +6.00000 q^{86} -12.0000 q^{87} -3.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} -12.0000 q^{91} -8.00000 q^{93} +4.00000 q^{94} +1.00000 q^{95} +10.0000 q^{96} +4.00000 q^{97} +3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	−1.00000	−0.500000
$5$	−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	$0.623376\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.00000	0.577350
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	2.00000	0.534522
$15$	2.00000	0.516398
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.00000	0.223607
$21$	4.00000	0.872872
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−6.00000	−1.22474
$25$	1.00000	0.200000
$26$	−6.00000	−1.17670
$27$	4.00000	0.769800
$28$	2.00000	0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−2.00000	−0.365148
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−5.00000	−0.883883
$33$	2.00000	0.348155
$34$	0	0
$35$	2.00000	0.338062
$36$	−1.00000	−0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	1.00000	0.162221
$39$	−12.0000	−1.92154
$40$	−3.00000	−0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	−4.00000	−0.617213
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	1.00000	0.150756
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	2.00000	0.288675
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−6.00000	−0.832050
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	−4.00000	−0.544331
$55$	1.00000	0.134840
$56$	−6.00000	−0.801784
$57$	2.00000	0.264906
$58$	−6.00000	−0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	−2.00000	−0.258199
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	−4.00000	−0.508001
$63$	−2.00000	−0.251976
$64$	7.00000	0.875000
$65$	−6.00000	−0.744208
$66$	−2.00000	−0.246183
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	0	0
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	3.00000	0.353553
$73$	16.0000	1.87266	0.936329	−	0.351123i	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	1.00000	0.114708
$77$	2.00000	0.227921
$78$	12.0000	1.35873
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	6.00000	0.662589
$83$	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
$85$	0	0
$86$	6.00000	0.646997
$87$	−12.0000	−1.28654
$88$	−3.00000	−0.319801
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	1.00000	0.105409
$91$	−12.0000	−1.25794
$92$	0	0
$93$	−8.00000	−0.829561
$94$	4.00000	0.412568
$95$	1.00000	0.102598
$96$	10.0000	1.02062
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	3.00000	0.303046
$99$	−1.00000	−0.100504
$100$	−1.00000	−0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	18.0000	1.76505
$105$	−4.00000	−0.390360
$106$	−12.0000	−1.16554
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	−4.00000	−0.384900
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	2.00000	0.188982
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−6.00000	−0.557086
$117$	6.00000	0.554700
$118$	8.00000	0.736460
$119$	0	0
$120$	6.00000	0.547723
$121$	1.00000	0.0909091
$122$	14.0000	1.26750
$123$	12.0000	1.08200
$124$	−4.00000	−0.359211
$125$	−1.00000	−0.0894427
$126$	2.00000	0.178174
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	3.00000	0.265165
$129$	12.0000	1.05654
$130$	6.00000	0.526235
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−2.00000	−0.174078
$133$	2.00000	0.173422
$134$	−6.00000	−0.518321
$135$	−4.00000	−0.344265
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	8.00000	0.671345
$143$	−6.00000	−0.501745
$144$	−1.00000	−0.0833333
$145$	−6.00000	−0.498273
$146$	−16.0000	−1.32417
$147$	6.00000	0.494872
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	2.00000	0.163299
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	−2.00000	−0.161165
$155$	−4.00000	−0.321288
$156$	12.0000	0.960769
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	8.00000	0.636446
$159$	−24.0000	−1.90332
$160$	5.00000	0.395285
$161$	0	0
$162$	11.0000	0.864242
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	6.00000	0.468521
$165$	−2.00000	−0.155700
$166$	−6.00000	−0.465690
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	12.0000	0.925820
$169$	23.0000	1.76923
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	6.00000	0.457496
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	12.0000	0.909718
$175$	−2.00000	−0.151186
$176$	1.00000	0.0753778
$177$	16.0000	1.20263
$178$	2.00000	0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	1.00000	0.0745356
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	12.0000	0.889499
$183$	28.0000	2.06982
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	4.00000	0.291730
$189$	−8.00000	−0.581914
$190$	−1.00000	−0.0725476
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−14.0000	−1.01036
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−4.00000	−0.287183
$195$	12.0000	0.859338
$196$	3.00000	0.214286
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	1.00000	0.0710669
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	3.00000	0.212132
$201$	−12.0000	−0.846415
$202$	18.0000	1.26648
$203$	−12.0000	−0.842235
$204$	0	0
$205$	6.00000	0.419058
$206$	10.0000	0.696733
$207$	0	0
$208$	−6.00000	−0.416025
$209$	1.00000	0.0691714
$210$	4.00000	0.276026
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	−12.0000	−0.824163
$213$	16.0000	1.09630
$214$	20.0000	1.36717
$215$	6.00000	0.409197
$216$	12.0000	0.816497
$217$	−8.00000	−0.543075
$218$	2.00000	0.135457
$219$	−32.0000	−2.16236
$220$	−1.00000	−0.0674200
$221$	0	0
$222$	0	0
$223$	−22.0000	−1.47323	−0.736614	−	0.676313i	$-0.763577\pi$
$223$	−22.0000	−1.47323	−0.736614	+	0.676313i	$0.763577\pi$
$224$	10.0000	0.668153
$225$	1.00000	0.0666667
$226$	−4.00000	−0.266076
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	−2.00000	−0.132453
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	18.0000	1.18176
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	−6.00000	−0.392232
$235$	4.00000	0.260931
$236$	8.00000	0.520756
$237$	16.0000	1.03931
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−2.00000	−0.129099
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−1.00000	−0.0642824
$243$	10.0000	0.641500
$244$	14.0000	0.896258
$245$	3.00000	0.191663
$246$	−12.0000	−0.765092
$247$	−6.00000	−0.381771
$248$	12.0000	0.762001
$249$	−12.0000	−0.760469
$250$	1.00000	0.0632456
$251$	−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
$253$	0	0
$254$	−4.00000	−0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	6.00000	0.372104
$261$	6.00000	0.371391
$262$	−4.00000	−0.247121
$263$	10.0000	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$264$	6.00000	0.369274
$265$	−12.0000	−0.737154
$266$	−2.00000	−0.122628
$267$	4.00000	0.244796
$268$	−6.00000	−0.366508
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	4.00000	0.243432
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	24.0000	1.45255
$274$	−2.00000	−0.120824
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−12.0000	−0.719712
$279$	4.00000	0.239474
$280$	6.00000	0.358569
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−8.00000	−0.476393
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	8.00000	0.474713
$285$	−2.00000	−0.118470
$286$	6.00000	0.354787
$287$	12.0000	0.708338
$288$	−5.00000	−0.294628
$289$	−17.0000	−1.00000
$290$	6.00000	0.352332
$291$	−8.00000	−0.468968
$292$	−16.0000	−0.936329
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−6.00000	−0.349927
$295$	8.00000	0.465778
$296$	0	0
$297$	−4.00000	−0.232104
$298$	−6.00000	−0.347571
$299$	0	0
$300$	2.00000	0.115470
$301$	12.0000	0.691669
$302$	8.00000	0.460348
$303$	36.0000	2.06815
$304$	1.00000	0.0573539
$305$	14.0000	0.801638
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	−2.00000	−0.113961
$309$	20.0000	1.13776
$310$	4.00000	0.227185
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−36.0000	−2.03810
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	10.0000	0.564333
$315$	2.00000	0.112687
$316$	8.00000	0.450035
$317$	−20.0000	−1.12331	−0.561656	−	0.827371i	$-0.689836\pi$
$317$	−20.0000	−1.12331	−0.561656	+	0.827371i	$0.689836\pi$
$318$	24.0000	1.34585
$319$	−6.00000	−0.335936
$320$	−7.00000	−0.391312
$321$	40.0000	2.23258
$322$	0	0
$323$	0	0
$324$	11.0000	0.611111
$325$	6.00000	0.332820
$326$	−4.00000	−0.221540
$327$	4.00000	0.221201
$328$	−18.0000	−0.993884
$329$	8.00000	0.441054
$330$	2.00000	0.110096
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	8.00000	0.437741
$335$	−6.00000	−0.327815
$336$	−4.00000	−0.218218
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−23.0000	−1.25104
$339$	−8.00000	−0.434500
$340$	0	0
$341$	−4.00000	−0.216612
$342$	1.00000	0.0540738
$343$	20.0000	1.07990
$344$	−18.0000	−0.970495
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	12.0000	0.643268
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	2.00000	0.106904
$351$	24.0000	1.28103
$352$	5.00000	0.266501
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	−16.0000	−0.850390
$355$	8.00000	0.424596
$356$	2.00000	0.106000
$357$	0	0
$358$	12.0000	0.634220
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	−3.00000	−0.158114
$361$	1.00000	0.0526316
$362$	−14.0000	−0.735824
$363$	−2.00000	−0.104973
$364$	12.0000	0.628971
$365$	−16.0000	−0.837478
$366$	−28.0000	−1.46358
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	8.00000	0.414781
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	2.00000	0.103280
$376$	−12.0000	−0.618853
$377$	36.0000	1.85409
$378$	8.00000	0.411476
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−1.00000	−0.0512989
$381$	−8.00000	−0.409852
$382$	8.00000	0.409316
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	−6.00000	−0.306186
$385$	−2.00000	−0.101929
$386$	6.00000	0.305392
$387$	−6.00000	−0.304997
$388$	−4.00000	−0.203069
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	−12.0000	−0.607644
$391$	0	0
$392$	−9.00000	−0.454569
$393$	−8.00000	−0.403547
$394$	24.0000	1.20910
$395$	8.00000	0.402524
$396$	1.00000	0.0502519
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−16.0000	−0.802008
$399$	−4.00000	−0.200250
$400$	−1.00000	−0.0500000
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	12.0000	0.598506
$403$	24.0000	1.19553
$404$	18.0000	0.895533
$405$	11.0000	0.546594
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−6.00000	−0.296319
$411$	−4.00000	−0.197305
$412$	10.0000	0.492665
$413$	16.0000	0.787309
$414$	0	0
$415$	−6.00000	−0.294528
$416$	−30.0000	−1.47087
$417$	−24.0000	−1.17529
$418$	−1.00000	−0.0489116
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	4.00000	0.195180
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	−28.0000	−1.36302
$423$	−4.00000	−0.194487
$424$	36.0000	1.74831
$425$	0	0
$426$	−16.0000	−0.775203
$427$	28.0000	1.35501
$428$	20.0000	0.966736
$429$	12.0000	0.579365
$430$	−6.00000	−0.289346
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−4.00000	−0.192450
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	8.00000	0.384012
$435$	12.0000	0.575356
$436$	2.00000	0.0957826
$437$	0	0
$438$	32.0000	1.52902
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	3.00000	0.143019
$441$	−3.00000	−0.142857
$442$	0	0
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	22.0000	1.04173
$447$	−12.0000	−0.567581
$448$	−14.0000	−0.661438
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	−1.00000	−0.0471405
$451$	6.00000	0.282529
$452$	−4.00000	−0.188144
$453$	16.0000	0.751746
$454$	8.00000	0.375459
$455$	12.0000	0.562569
$456$	6.00000	0.280976
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	4.00000	0.186097
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−6.00000	−0.278543
$465$	8.00000	0.370991
$466$	24.0000	1.11178
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	−6.00000	−0.277350
$469$	−12.0000	−0.554109
$470$	−4.00000	−0.184506
$471$	20.0000	0.921551
$472$	−24.0000	−1.10469
$473$	6.00000	0.275880
$474$	−16.0000	−0.734904
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	12.0000	0.549442
$478$	24.0000	1.09773
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	−10.0000	−0.456435
$481$	0	0
$482$	−14.0000	−0.637683
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	−4.00000	−0.181631
$486$	−10.0000	−0.453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−42.0000	−1.90125
$489$	−8.00000	−0.361773
$490$	−3.00000	−0.135526
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	−12.0000	−0.541002
$493$	0	0
$494$	6.00000	0.269953
$495$	1.00000	0.0449467
$496$	−4.00000	−0.179605
$497$	16.0000	0.717698
$498$	12.0000	0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	1.00000	0.0447214
$501$	16.0000	0.714827
$502$	4.00000	0.178529
$503$	−10.0000	−0.445878	−0.222939	−	0.974832i	$-0.571565\pi$
$503$	−10.0000	−0.445878	−0.222939	+	0.974832i	$0.571565\pi$
$504$	−6.00000	−0.267261
$505$	18.0000	0.800989
$506$	0	0
$507$	−46.0000	−2.04293
$508$	−4.00000	−0.177471
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−32.0000	−1.41560
$512$	11.0000	0.486136
$513$	−4.00000	−0.176604
$514$	−20.0000	−0.882162
$515$	10.0000	0.440653
$516$	−12.0000	−0.528271
$517$	4.00000	0.175920
$518$	0	0
$519$	−12.0000	−0.526742
$520$	−18.0000	−0.789352
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−6.00000	−0.262613
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	−4.00000	−0.174741
$525$	4.00000	0.174574
$526$	−10.0000	−0.436021
$527$	0	0
$528$	−2.00000	−0.0870388
$529$	−23.0000	−1.00000
$530$	12.0000	0.521247
$531$	−8.00000	−0.347170
$532$	−2.00000	−0.0867110
$533$	−36.0000	−1.55933
$534$	−4.00000	−0.173097
$535$	20.0000	0.864675
$536$	18.0000	0.777482
$537$	24.0000	1.03568
$538$	10.0000	0.431131
$539$	3.00000	0.129219
$540$	4.00000	0.172133
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−20.0000	−0.859074
$543$	−28.0000	−1.20160
$544$	0	0
$545$	2.00000	0.0856706
$546$	−24.0000	−1.02711
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−2.00000	−0.0854358
$549$	−14.0000	−0.597505
$550$	1.00000	0.0426401
$551$	−6.00000	−0.255609
$552$	0	0
$553$	16.0000	0.680389
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−12.0000	−0.508913
$557$	28.0000	1.18640	0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000	1.18640	0.593199	+	0.805056i	$0.297865\pi$
$558$	−4.00000	−0.169334
$559$	−36.0000	−1.52264
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	−8.00000	−0.336861
$565$	−4.00000	−0.168281
$566$	−6.00000	−0.252199
$567$	22.0000	0.923913
$568$	−24.0000	−1.00702
$569$	46.0000	1.92842	0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000	1.92842	0.964210	+	0.265139i	$0.0854179\pi$
$570$	2.00000	0.0837708
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	6.00000	0.250873
$573$	16.0000	0.668410
$574$	−12.0000	−0.500870
$575$	0	0
$576$	7.00000	0.291667
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	17.0000	0.707107
$579$	12.0000	0.498703
$580$	6.00000	0.249136
$581$	−12.0000	−0.497844
$582$	8.00000	0.331611
$583$	−12.0000	−0.496989
$584$	48.0000	1.98625
$585$	−6.00000	−0.248069
$586$	14.0000	0.578335
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	−6.00000	−0.247436
$589$	−4.00000	−0.164817
$590$	−8.00000	−0.329355
$591$	48.0000	1.97446
$592$	0	0
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−32.0000	−1.30967
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	−6.00000	−0.244949
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	−12.0000	−0.489083
$603$	6.00000	0.244339
$604$	8.00000	0.325515
$605$	−1.00000	−0.0406558
$606$	−36.0000	−1.46240
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	5.00000	0.202777
$609$	24.0000	0.972529
$610$	−14.0000	−0.566843
$611$	−24.0000	−0.970936
$612$	0	0
$613$	−36.0000	−1.45403	−0.727013	−	0.686624i	$-0.759092\pi$
$613$	−36.0000	−1.45403	−0.727013	+	0.686624i	$0.759092\pi$
$614$	12.0000	0.484281
$615$	−12.0000	−0.483887
$616$	6.00000	0.241747
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	−20.0000	−0.804518
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	0	0
$623$	4.00000	0.160257
$624$	12.0000	0.480384
$625$	1.00000	0.0400000
$626$	−26.0000	−1.03917
$627$	−2.00000	−0.0798723
$628$	10.0000	0.399043
$629$	0	0
$630$	−2.00000	−0.0796819
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	−24.0000	−0.954669
$633$	−56.0000	−2.22580
$634$	20.0000	0.794301
$635$	−4.00000	−0.158735
$636$	24.0000	0.951662
$637$	−18.0000	−0.713186
$638$	6.00000	0.237542
$639$	−8.00000	−0.316475
$640$	−3.00000	−0.118585
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	−40.0000	−1.57867
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	−33.0000	−1.29636
$649$	8.00000	0.314027
$650$	−6.00000	−0.235339
$651$	16.0000	0.627089
$652$	−4.00000	−0.156652
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	−4.00000	−0.156412
$655$	−4.00000	−0.156293
$656$	6.00000	0.234261
$657$	16.0000	0.624219
$658$	−8.00000	−0.311872
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	2.00000	0.0778499
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	18.0000	0.698535
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	0	0
$668$	8.00000	0.309529
$669$	44.0000	1.70114
$670$	6.00000	0.231800
$671$	14.0000	0.540464
$672$	−20.0000	−0.771517
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	−10.0000	−0.385186
$675$	4.00000	0.153960
$676$	−23.0000	−0.884615
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	8.00000	0.307238
$679$	−8.00000	−0.307012
$680$	0	0
$681$	16.0000	0.613121
$682$	4.00000	0.153168
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	1.00000	0.0382360
$685$	−2.00000	−0.0764161
$686$	−20.0000	−0.763604
$687$	12.0000	0.457829
$688$	6.00000	0.228748
$689$	72.0000	2.74298
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−6.00000	−0.228086
$693$	2.00000	0.0759737
$694$	26.0000	0.986947
$695$	−12.0000	−0.455186
$696$	−36.0000	−1.36458
$697$	0	0
$698$	30.0000	1.13552
$699$	48.0000	1.81553
$700$	2.00000	0.0755929
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	−24.0000	−0.905822
$703$	0	0
$704$	−7.00000	−0.263822
$705$	−8.00000	−0.301297
$706$	18.0000	0.677439
$707$	36.0000	1.35392
$708$	−16.0000	−0.601317
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	−8.00000	−0.300235
$711$	−8.00000	−0.300023
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	6.00000	0.224387
$716$	12.0000	0.448461
$717$	48.0000	1.79259
$718$	−12.0000	−0.447836
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	1.00000	0.0372678
$721$	20.0000	0.744839
$722$	−1.00000	−0.0372161
$723$	−28.0000	−1.04133
$724$	−14.0000	−0.520306
$725$	6.00000	0.222834
$726$	2.00000	0.0742270
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−36.0000	−1.33425
$729$	13.0000	0.481481
$730$	16.0000	0.592187
$731$	0	0
$732$	−28.0000	−1.03491
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	28.0000	1.03350
$735$	−6.00000	−0.221313
$736$	0	0
$737$	−6.00000	−0.221013
$738$	6.00000	0.220863
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	24.0000	0.881068
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	−24.0000	−0.879883
$745$	−6.00000	−0.219823
$746$	−14.0000	−0.512576
$747$	6.00000	0.219529
$748$	0	0
$749$	40.0000	1.46157
$750$	−2.00000	−0.0730297
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	4.00000	0.145865
$753$	8.00000	0.291536
$754$	−36.0000	−1.31104
$755$	8.00000	0.291150
$756$	8.00000	0.290957
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	3.00000	0.108821
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	8.00000	0.289809
$763$	4.00000	0.144810
$764$	8.00000	0.289430
$765$	0	0
$766$	30.0000	1.08394
$767$	−48.0000	−1.73318
$768$	34.0000	1.22687
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	2.00000	0.0720750
$771$	−40.0000	−1.44056
$772$	6.00000	0.215945
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	6.00000	0.215666
$775$	4.00000	0.143684
$776$	12.0000	0.430775
$777$	0	0
$778$	26.0000	0.932145
$779$	6.00000	0.214972
$780$	−12.0000	−0.429669
$781$	8.00000	0.286263
$782$	0	0
$783$	24.0000	0.857690
$784$	3.00000	0.107143
$785$	10.0000	0.356915
$786$	8.00000	0.285351
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	24.0000	0.854965
$789$	−20.0000	−0.712019
$790$	−8.00000	−0.284627
$791$	−8.00000	−0.284447
$792$	−3.00000	−0.106600
$793$	−84.0000	−2.98293
$794$	18.0000	0.638796
$795$	24.0000	0.851192
$796$	−16.0000	−0.567105
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	−5.00000	−0.176777
$801$	−2.00000	−0.0706665
$802$	22.0000	0.776847
$803$	−16.0000	−0.564628
$804$	12.0000	0.423207
$805$	0	0
$806$	−24.0000	−0.845364
$807$	20.0000	0.704033
$808$	−54.0000	−1.89971
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	−11.0000	−0.386501
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	12.0000	0.421117
$813$	−40.0000	−1.40286
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	6.00000	0.209913
$818$	22.0000	0.769212
$819$	−12.0000	−0.419314
$820$	−6.00000	−0.209529
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	4.00000	0.139516
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−30.0000	−1.04510
$825$	2.00000	0.0696311
$826$	−16.0000	−0.556711
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	6.00000	0.208263
$831$	−16.0000	−0.555034
$832$	42.0000	1.45609
$833$	0	0
$834$	24.0000	0.831052
$835$	8.00000	0.276851
$836$	−1.00000	−0.0345857
$837$	16.0000	0.553041
$838$	20.0000	0.690889
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	−12.0000	−0.414039
$841$	7.00000	0.241379
$842$	14.0000	0.482472
$843$	−12.0000	−0.413302
$844$	−28.0000	−0.963800
$845$	−23.0000	−0.791224
$846$	4.00000	0.137523
$847$	−2.00000	−0.0687208
$848$	−12.0000	−0.412082
$849$	−12.0000	−0.411839
$850$	0	0
$851$	0	0
$852$	−16.0000	−0.548151
$853$	−24.0000	−0.821744	−0.410872	−	0.911693i	$-0.634776\pi$
$853$	−24.0000	−0.821744	−0.410872	+	0.911693i	$0.634776\pi$
$854$	−28.0000	−0.958140
$855$	1.00000	0.0341993
$856$	−60.0000	−2.05076
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	−12.0000	−0.409673
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−6.00000	−0.204598
$861$	−24.0000	−0.817918
$862$	8.00000	0.272481
$863$	10.0000	0.340404	0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000	0.340404	0.170202	+	0.985409i	$0.445558\pi$
$864$	−20.0000	−0.680414
$865$	−6.00000	−0.204006
$866$	4.00000	0.135926
$867$	34.0000	1.15470
$868$	8.00000	0.271538
$869$	8.00000	0.271381
$870$	−12.0000	−0.406838
$871$	36.0000	1.21981
$872$	−6.00000	−0.203186
$873$	4.00000	0.135379
$874$	0	0
$875$	2.00000	0.0676123
$876$	32.0000	1.08118
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	16.0000	0.539974
$879$	28.0000	0.944417
$880$	−1.00000	−0.0337100
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	3.00000	0.101015
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	−8.00000	−0.268765
$887$	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
$890$	−2.00000	−0.0670402
$891$	11.0000	0.368514
$892$	22.0000	0.736614
$893$	4.00000	0.133855
$894$	12.0000	0.401340
$895$	12.0000	0.401116
$896$	−6.00000	−0.200446
$897$	0	0
$898$	14.0000	0.467186
$899$	24.0000	0.800445
$900$	−1.00000	−0.0333333
$901$	0	0
$902$	−6.00000	−0.199778
$903$	−24.0000	−0.798670
$904$	12.0000	0.399114
$905$	−14.0000	−0.465376
$906$	−16.0000	−0.531564
$907$	50.0000	1.66022	0.830111	−	0.557598i	$-0.188277\pi$
$907$	50.0000	1.66022	0.830111	+	0.557598i	$0.188277\pi$
$908$	8.00000	0.265489
$909$	−18.0000	−0.597022
$910$	−12.0000	−0.397796
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	−2.00000	−0.0662266
$913$	−6.00000	−0.198571
$914$	−32.0000	−1.05847
$915$	−28.0000	−0.925651
$916$	6.00000	0.198246
$917$	−8.00000	−0.264183
$918$	0	0
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	24.0000	0.790827
$922$	−30.0000	−0.987997
$923$	−48.0000	−1.57994
$924$	4.00000	0.131590
$925$	0	0
$926$	−8.00000	−0.262896
$927$	−10.0000	−0.328443
$928$	−30.0000	−0.984798
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	−8.00000	−0.262330
$931$	3.00000	0.0983210
$932$	24.0000	0.786146
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	18.0000	0.588348
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	12.0000	0.391814
$939$	−52.0000	−1.69696
$940$	−4.00000	−0.130466
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−20.0000	−0.651635
$943$	0	0
$944$	8.00000	0.260378
$945$	8.00000	0.260240
$946$	−6.00000	−0.195077
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−16.0000	−0.519656
$949$	96.0000	3.11629
$950$	1.00000	0.0324443
$951$	40.0000	1.29709
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	−12.0000	−0.388514
$955$	8.00000	0.258874
$956$	24.0000	0.776215
$957$	12.0000	0.387905
$958$	−28.0000	−0.904639
$959$	−4.00000	−0.129167
$960$	14.0000	0.451848
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−20.0000	−0.644491
$964$	−14.0000	−0.450910
$965$	6.00000	0.193147
$966$	0	0
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	3.00000	0.0964237
$969$	0	0
$970$	4.00000	0.128432
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	−10.0000	−0.320750
$973$	−24.0000	−0.769405
$974$	−2.00000	−0.0640841
$975$	−12.0000	−0.384308
$976$	14.0000	0.448129
$977$	32.0000	1.02377	0.511885	−	0.859054i	$-0.328947\pi$
$977$	32.0000	1.02377	0.511885	+	0.859054i	$0.328947\pi$
$978$	8.00000	0.255812
$979$	2.00000	0.0639203
$980$	−3.00000	−0.0958315
$981$	−2.00000	−0.0638551
$982$	16.0000	0.510581
$983$	−38.0000	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$983$	−38.0000	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$984$	36.0000	1.14764
$985$	24.0000	0.764704
$986$	0	0
$987$	−16.0000	−0.509286
$988$	6.00000	0.190885
$989$	0	0
$990$	−1.00000	−0.0317821
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−20.0000	−0.635001
$993$	56.0000	1.77711
$994$	−16.0000	−0.507489
$995$	−16.0000	−0.507234
$996$	12.0000	0.380235
$997$	12.0000	0.380044	0.190022	−	0.981780i	$-0.439144\pi$
$997$	12.0000	0.380044	0.190022	+	0.981780i	$0.439144\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.a.1.1	✓	1
3.2	odd	2		9405.2.a.j.1.1		1
5.4	even	2		5225.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.a.1.1	✓	1	1.1	even	1	trivial
5225.2.a.c.1.1		1	5.4	even	2
9405.2.a.j.1.1		1	3.2	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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