LMFDB - Embedded newform 1098.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1098,2,Mod(1,1098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1098, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1098 = 2 \cdot 3^{2} \cdot 61$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1098.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:	$8.76757414194$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 366)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1098.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -3.00000 q^{7} -1.00000 q^{8} -3.00000 q^{10} +1.00000 q^{11} -5.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -8.00000 q^{19} +3.00000 q^{20} -1.00000 q^{22} -5.00000 q^{23} +4.00000 q^{25} +5.00000 q^{26} -3.00000 q^{28} -4.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} -9.00000 q^{35} +4.00000 q^{37} +8.00000 q^{38} -3.00000 q^{40} -3.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +5.00000 q^{46} -2.00000 q^{47} +2.00000 q^{49} -4.00000 q^{50} -5.00000 q^{52} +3.00000 q^{55} +3.00000 q^{56} +7.00000 q^{59} +1.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -15.0000 q^{65} -13.0000 q^{67} -2.00000 q^{68} +9.00000 q^{70} +16.0000 q^{71} +9.00000 q^{73} -4.00000 q^{74} -8.00000 q^{76} -3.00000 q^{77} -1.00000 q^{79} +3.00000 q^{80} +3.00000 q^{82} -14.0000 q^{83} -6.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} +4.00000 q^{89} +15.0000 q^{91} -5.00000 q^{92} +2.00000 q^{94} -24.0000 q^{95} +14.0000 q^{97} -2.00000 q^{98} +O(q^{100})

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
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	3.00000	0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	5.00000	0.980581
$27$	0	0
$28$	−3.00000	−0.566947
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	−9.00000	−1.52128
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	5.00000	0.737210
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−5.00000	−0.693375
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	3.00000	0.400892
$57$	0	0
$58$	0	0
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−15.0000	−1.86052
$66$	0	0
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	9.00000	1.07571
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−8.00000	−0.917663
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	3.00000	0.331295
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	15.0000	1.57243
$92$	−5.00000	−0.521286
$93$	0	0
$94$	2.00000	0.206284
$95$	−24.0000	−2.46235
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	4.00000	0.400000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	0	0
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	−3.00000	−0.286039
$111$	0	0
$112$	−3.00000	−0.283473
$113$	−11.0000	−1.03479	−0.517396	−	0.855746i	$-0.673099\pi$
$113$	−11.0000	−1.03479	−0.517396	+	0.855746i	$0.673099\pi$
$114$	0	0
$115$	−15.0000	−1.39876
$116$	0	0
$117$	0	0
$118$	−7.00000	−0.644402
$119$	6.00000	0.550019
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	−4.00000	−0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−22.0000	−1.95218	−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−22.0000	−1.95218	−0.976092	+	0.217357i	$0.930256\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	15.0000	1.31559
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	24.0000	2.08106
$134$	13.0000	1.12303
$135$	0	0
$136$	2.00000	0.171499
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	−9.00000	−0.760639
$141$	0	0
$142$	−16.0000	−1.34269
$143$	−5.00000	−0.418121
$144$	0	0
$145$	0	0
$146$	−9.00000	−0.744845
$147$	0	0
$148$	4.00000	0.328798
$149$	21.0000	1.72039	0.860194	−	0.509968i	$-0.170343\pi$
$149$	21.0000	1.72039	0.860194	+	0.509968i	$0.170343\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	8.00000	0.648886
$153$	0	0
$154$	3.00000	0.241747
$155$	−12.0000	−0.963863
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	1.00000	0.0795557
$159$	0	0
$160$	−3.00000	−0.237171
$161$	15.0000	1.18217
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	14.0000	1.08661
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	6.00000	0.460179
$171$	0	0
$172$	4.00000	0.304997
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	1.00000	0.0753778
$177$	0	0
$178$	−4.00000	−0.299813
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	−15.0000	−1.11187
$183$	0	0
$184$	5.00000	0.368605
$185$	12.0000	0.882258
$186$	0	0
$187$	−2.00000	−0.146254
$188$	−2.00000	−0.145865
$189$	0	0
$190$	24.0000	1.74114
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	2.00000	0.142857
$197$	−21.0000	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$197$	−21.0000	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$198$	0	0
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−8.00000	−0.562878
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	14.0000	0.975426
$207$	0	0
$208$	−5.00000	−0.346688
$209$	−8.00000	−0.553372
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	−2.00000	−0.136717
$215$	12.0000	0.818393
$216$	0	0
$217$	12.0000	0.814613
$218$	15.0000	1.01593
$219$	0	0
$220$	3.00000	0.202260
$221$	10.0000	0.672673
$222$	0	0
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	3.00000	0.200446
$225$	0	0
$226$	11.0000	0.731709
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	0	0
$229$	−17.0000	−1.12339	−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000	−1.12339	−0.561696	+	0.827344i	$0.689851\pi$
$230$	15.0000	0.989071
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	7.00000	0.455661
$237$	0	0
$238$	−6.00000	−0.388922
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	1.00000	0.0640184
$245$	6.00000	0.383326
$246$	0	0
$247$	40.0000	2.54514
$248$	4.00000	0.254000
$249$	0	0
$250$	3.00000	0.189737
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	22.0000	1.38040
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	−15.0000	−0.930261
$261$	0	0
$262$	10.0000	0.617802
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	−24.0000	−1.47153
$267$	0	0
$268$	−13.0000	−0.794101
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	15.0000	0.906183
$275$	4.00000	0.241209
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	5.00000	0.299880
$279$	0	0
$280$	9.00000	0.537853
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	5.00000	0.295656
$287$	9.00000	0.531253
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	9.00000	0.526685
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	21.0000	1.22267
$296$	−4.00000	−0.232495
$297$	0	0
$298$	−21.0000	−1.21650
$299$	25.0000	1.44579
$300$	0	0
$301$	−12.0000	−0.691669
$302$	9.00000	0.517892
$303$	0	0
$304$	−8.00000	−0.458831
$305$	3.00000	0.171780
$306$	0	0
$307$	19.0000	1.08439	0.542194	−	0.840254i	$-0.317594\pi$
$307$	19.0000	1.08439	0.542194	+	0.840254i	$0.317594\pi$
$308$	−3.00000	−0.170941
$309$	0	0
$310$	12.0000	0.681554
$311$	−35.0000	−1.98467	−0.992334	−	0.123585i	$-0.960561\pi$
$311$	−35.0000	−1.98467	−0.992334	+	0.123585i	$0.960561\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	0	0
$320$	3.00000	0.167705
$321$	0	0
$322$	−15.0000	−0.835917
$323$	16.0000	0.890264
$324$	0	0
$325$	−20.0000	−1.10940
$326$	4.00000	0.221540
$327$	0	0
$328$	3.00000	0.165647
$329$	6.00000	0.330791
$330$	0	0
$331$	9.00000	0.494685	0.247342	−	0.968928i	$-0.420443\pi$
$331$	9.00000	0.494685	0.247342	+	0.968928i	$0.420443\pi$
$332$	−14.0000	−0.768350
$333$	0	0
$334$	−18.0000	−0.984916
$335$	−39.0000	−2.13080
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	−6.00000	−0.325396
$341$	−4.00000	−0.216612
$342$	0	0
$343$	15.0000	0.809924
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−22.0000	−1.18273
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	12.0000	0.641427
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−25.0000	−1.33062	−0.665308	−	0.746569i	$-0.731700\pi$
$353$	−25.0000	−1.33062	−0.665308	+	0.746569i	$0.731700\pi$
$354$	0	0
$355$	48.0000	2.54758
$356$	4.00000	0.212000
$357$	0	0
$358$	0	0
$359$	−36.0000	−1.90001	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	−36.0000	−1.90001	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−26.0000	−1.36653
$363$	0	0
$364$	15.0000	0.786214
$365$	27.0000	1.41324
$366$	0	0
$367$	36.0000	1.87918	0.939592	−	0.342296i	$-0.111204\pi$
$367$	36.0000	1.87918	0.939592	+	0.342296i	$0.111204\pi$
$368$	−5.00000	−0.260643
$369$	0	0
$370$	−12.0000	−0.623850
$371$	0	0
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	2.00000	0.103142
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	−24.0000	−1.23117
$381$	0	0
$382$	−15.0000	−0.767467
$383$	−7.00000	−0.357683	−0.178842	−	0.983878i	$-0.557235\pi$
$383$	−7.00000	−0.357683	−0.178842	+	0.983878i	$0.557235\pi$
$384$	0	0
$385$	−9.00000	−0.458682
$386$	18.0000	0.916176
$387$	0	0
$388$	14.0000	0.710742
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	10.0000	0.505722
$392$	−2.00000	−0.101015
$393$	0	0
$394$	21.0000	1.05796
$395$	−3.00000	−0.150946
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−18.0000	−0.902258
$399$	0	0
$400$	4.00000	0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	8.00000	0.398015
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	9.00000	0.444478
$411$	0	0
$412$	−14.0000	−0.689730
$413$	−21.0000	−1.03334
$414$	0	0
$415$	−42.0000	−2.06170
$416$	5.00000	0.245145
$417$	0	0
$418$	8.00000	0.391293
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	−3.00000	−0.145180
$428$	2.00000	0.0966736
$429$	0	0
$430$	−12.0000	−0.578691
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−12.0000	−0.576018
$435$	0	0
$436$	−15.0000	−0.718370
$437$	40.0000	1.91346
$438$	0	0
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	−10.0000	−0.475651
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	21.0000	0.994379
$447$	0	0
$448$	−3.00000	−0.141737
$449$	17.0000	0.802280	0.401140	−	0.916017i	$-0.368614\pi$
$449$	17.0000	0.802280	0.401140	+	0.916017i	$0.368614\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	−11.0000	−0.517396
$453$	0	0
$454$	−15.0000	−0.703985
$455$	45.0000	2.10963
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	17.0000	0.794358
$459$	0	0
$460$	−15.0000	−0.699379
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.0000	1.15686	0.578431	−	0.815731i	$-0.303665\pi$
$467$	25.0000	1.15686	0.578431	+	0.815731i	$0.303665\pi$
$468$	0	0
$469$	39.0000	1.80085
$470$	6.00000	0.276759
$471$	0	0
$472$	−7.00000	−0.322201
$473$	4.00000	0.183920
$474$	0	0
$475$	−32.0000	−1.46826
$476$	6.00000	0.275010
$477$	0	0
$478$	−12.0000	−0.548867
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	−25.0000	−1.13872
$483$	0	0
$484$	−10.0000	−0.454545
$485$	42.0000	1.90712
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−1.00000	−0.0452679
$489$	0	0
$490$	−6.00000	−0.271052
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	0	0
$494$	−40.0000	−1.79969
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−48.0000	−2.15309
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	8.00000	0.357057
$503$	10.0000	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$503$	10.0000	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	5.00000	0.222277
$507$	0	0
$508$	−22.0000	−0.976092
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	−27.0000	−1.19441
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	−42.0000	−1.85074
$516$	0	0
$517$	−2.00000	−0.0879599
$518$	12.0000	0.527250
$519$	0	0
$520$	15.0000	0.657794
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	5.00000	0.218635	0.109317	−	0.994007i	$-0.465134\pi$
$523$	5.00000	0.218635	0.109317	+	0.994007i	$0.465134\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	24.0000	1.04645
$527$	8.00000	0.348485
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	24.0000	1.04053
$533$	15.0000	0.649722
$534$	0	0
$535$	6.00000	0.259403
$536$	13.0000	0.561514
$537$	0	0
$538$	−18.0000	−0.776035
$539$	2.00000	0.0861461
$540$	0	0
$541$	−40.0000	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$541$	−40.0000	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	2.00000	0.0857493
$545$	−45.0000	−1.92759
$546$	0	0
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	−15.0000	−0.640768
$549$	0	0
$550$	−4.00000	−0.170561
$551$	0	0
$552$	0	0
$553$	3.00000	0.127573
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−5.00000	−0.212047
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	−9.00000	−0.380319
$561$	0	0
$562$	24.0000	1.01238
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	−33.0000	−1.38832
$566$	14.0000	0.588464
$567$	0	0
$568$	−16.0000	−0.671345
$569$	17.0000	0.712677	0.356339	−	0.934357i	$-0.384025\pi$
$569$	17.0000	0.712677	0.356339	+	0.934357i	$0.384025\pi$
$570$	0	0
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	−5.00000	−0.209061
$573$	0	0
$574$	−9.00000	−0.375653
$575$	−20.0000	−0.834058
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	0	0
$581$	42.0000	1.74245
$582$	0	0
$583$	0	0
$584$	−9.00000	−0.372423
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	32.0000	1.31854
$590$	−21.0000	−0.864556
$591$	0	0
$592$	4.00000	0.164399
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	18.0000	0.737928
$596$	21.0000	0.860194
$597$	0	0
$598$	−25.0000	−1.02233
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	0	0
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	12.0000	0.489083
$603$	0	0
$604$	−9.00000	−0.366205
$605$	−30.0000	−1.21967
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	−3.00000	−0.121466
$611$	10.0000	0.404557
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	−19.0000	−0.766778
$615$	0	0
$616$	3.00000	0.120873
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	−12.0000	−0.481932
$621$	0	0
$622$	35.0000	1.40337
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−29.0000	−1.16000
$626$	8.00000	0.319744
$627$	0	0
$628$	6.00000	0.239426
$629$	−8.00000	−0.318981
$630$	0	0
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	1.00000	0.0397779
$633$	0	0
$634$	10.0000	0.397151
$635$	−66.0000	−2.61913
$636$	0	0
$637$	−10.0000	−0.396214
$638$	0	0
$639$	0	0
$640$	−3.00000	−0.118585
$641$	44.0000	1.73790	0.868948	−	0.494904i	$-0.164797\pi$
$641$	44.0000	1.73790	0.868948	+	0.494904i	$0.164797\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	15.0000	0.591083
$645$	0	0
$646$	−16.0000	−0.629512
$647$	3.00000	0.117942	0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000	0.117942	0.0589711	+	0.998260i	$0.481218\pi$
$648$	0	0
$649$	7.00000	0.274774
$650$	20.0000	0.784465
$651$	0	0
$652$	−4.00000	−0.156652
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	0	0
$655$	−30.0000	−1.17220
$656$	−3.00000	−0.117130
$657$	0	0
$658$	−6.00000	−0.233904
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	−9.00000	−0.349795
$663$	0	0
$664$	14.0000	0.543305
$665$	72.0000	2.79204
$666$	0	0
$667$	0	0
$668$	18.0000	0.696441
$669$	0	0
$670$	39.0000	1.50670
$671$	1.00000	0.0386046
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	16.0000	0.616297
$675$	0	0
$676$	12.0000	0.461538
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	−42.0000	−1.61181
$680$	6.00000	0.230089
$681$	0	0
$682$	4.00000	0.153168
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	−45.0000	−1.71936
$686$	−15.0000	−0.572703
$687$	0	0
$688$	4.00000	0.152499
$689$	0	0
$690$	0	0
$691$	−24.0000	−0.913003	−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000	−0.913003	−0.456502	+	0.889723i	$0.650898\pi$
$692$	22.0000	0.836315
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−15.0000	−0.568982
$696$	0	0
$697$	6.00000	0.227266
$698$	22.0000	0.832712
$699$	0	0
$700$	−12.0000	−0.453557
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	1.00000	0.0376889
$705$	0	0
$706$	25.0000	0.940887
$707$	−24.0000	−0.902613
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	−48.0000	−1.80141
$711$	0	0
$712$	−4.00000	−0.149906
$713$	20.0000	0.749006
$714$	0	0
$715$	−15.0000	−0.560968
$716$	0	0
$717$	0	0
$718$	36.0000	1.34351
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	−45.0000	−1.67473
$723$	0	0
$724$	26.0000	0.966282
$725$	0	0
$726$	0	0
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	−15.0000	−0.555937
$729$	0	0
$730$	−27.0000	−0.999315
$731$	−8.00000	−0.295891
$732$	0	0
$733$	−1.00000	−0.0369358	−0.0184679	−	0.999829i	$-0.505879\pi$
$733$	−1.00000	−0.0369358	−0.0184679	+	0.999829i	$0.505879\pi$
$734$	−36.0000	−1.32878
$735$	0	0
$736$	5.00000	0.184302
$737$	−13.0000	−0.478861
$738$	0	0
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	0	0
$743$	31.0000	1.13728	0.568640	−	0.822587i	$-0.307470\pi$
$743$	31.0000	1.13728	0.568640	+	0.822587i	$0.307470\pi$
$744$	0	0
$745$	63.0000	2.30814
$746$	−4.00000	−0.146450
$747$	0	0
$748$	−2.00000	−0.0731272
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	−2.00000	−0.0729325
$753$	0	0
$754$	0	0
$755$	−27.0000	−0.982631
$756$	0	0
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	0	0
$759$	0	0
$760$	24.0000	0.870572
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	45.0000	1.62911
$764$	15.0000	0.542681
$765$	0	0
$766$	7.00000	0.252920
$767$	−35.0000	−1.26378
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	9.00000	0.324337
$771$	0	0
$772$	−18.0000	−0.647834
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	−14.0000	−0.502571
$777$	0	0
$778$	18.0000	0.645331
$779$	24.0000	0.859889
$780$	0	0
$781$	16.0000	0.572525
$782$	−10.0000	−0.357599
$783$	0	0
$784$	2.00000	0.0714286
$785$	18.0000	0.642448
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	−21.0000	−0.748094
$789$	0	0
$790$	3.00000	0.106735
$791$	33.0000	1.17334
$792$	0	0
$793$	−5.00000	−0.177555
$794$	2.00000	0.0709773
$795$	0	0
$796$	18.0000	0.637993
$797$	−29.0000	−1.02723	−0.513616	−	0.858020i	$-0.671695\pi$
$797$	−29.0000	−1.02723	−0.513616	+	0.858020i	$0.671695\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	−4.00000	−0.141421
$801$	0	0
$802$	6.00000	0.211867
$803$	9.00000	0.317603
$804$	0	0
$805$	45.0000	1.58604
$806$	−20.0000	−0.704470
$807$	0	0
$808$	−8.00000	−0.281439
$809$	−23.0000	−0.808637	−0.404318	−	0.914618i	$-0.632491\pi$
$809$	−23.0000	−0.808637	−0.404318	+	0.914618i	$0.632491\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	0	0
$814$	−4.00000	−0.140200
$815$	−12.0000	−0.420342
$816$	0	0
$817$	−32.0000	−1.11954
$818$	20.0000	0.699284
$819$	0	0
$820$	−9.00000	−0.314294
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	21.0000	0.730683
$827$	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
$829$	−7.00000	−0.243120	−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000	−0.243120	−0.121560	+	0.992584i	$0.538790\pi$
$830$	42.0000	1.45784
$831$	0	0
$832$	−5.00000	−0.173344
$833$	−4.00000	−0.138592
$834$	0	0
$835$	54.0000	1.86875
$836$	−8.00000	−0.276686
$837$	0	0
$838$	4.00000	0.138178
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−2.00000	−0.0689246
$843$	0	0
$844$	20.0000	0.688428
$845$	36.0000	1.23844
$846$	0	0
$847$	30.0000	1.03081
$848$	0	0
$849$	0	0
$850$	8.00000	0.274398
$851$	−20.0000	−0.685591
$852$	0	0
$853$	39.0000	1.33533	0.667667	−	0.744460i	$-0.267293\pi$
$853$	39.0000	1.33533	0.667667	+	0.744460i	$0.267293\pi$
$854$	3.00000	0.102658
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	53.0000	1.81045	0.905223	−	0.424937i	$-0.139704\pi$
$857$	53.0000	1.81045	0.905223	+	0.424937i	$0.139704\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	12.0000	0.409197
$861$	0	0
$862$	−2.00000	−0.0681203
$863$	56.0000	1.90626	0.953131	−	0.302558i	$-0.0978405\pi$
$863$	56.0000	1.90626	0.953131	+	0.302558i	$0.0978405\pi$
$864$	0	0
$865$	66.0000	2.24407
$866$	2.00000	0.0679628
$867$	0	0
$868$	12.0000	0.407307
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	65.0000	2.20244
$872$	15.0000	0.507964
$873$	0	0
$874$	−40.0000	−1.35302
$875$	9.00000	0.304256
$876$	0	0
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	12.0000	0.404980
$879$	0	0
$880$	3.00000	0.101130
$881$	35.0000	1.17918	0.589590	−	0.807703i	$-0.299289\pi$
$881$	35.0000	1.17918	0.589590	+	0.807703i	$0.299289\pi$
$882$	0	0
$883$	15.0000	0.504790	0.252395	−	0.967624i	$-0.418782\pi$
$883$	15.0000	0.504790	0.252395	+	0.967624i	$0.418782\pi$
$884$	10.0000	0.336336
$885$	0	0
$886$	−16.0000	−0.537531
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	66.0000	2.21357
$890$	−12.0000	−0.402241
$891$	0	0
$892$	−21.0000	−0.703132
$893$	16.0000	0.535420
$894$	0	0
$895$	0	0
$896$	3.00000	0.100223
$897$	0	0
$898$	−17.0000	−0.567297
$899$	0	0
$900$	0	0
$901$	0	0
$902$	3.00000	0.0998891
$903$	0	0
$904$	11.0000	0.365855
$905$	78.0000	2.59281
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	15.0000	0.497792
$909$	0	0
$910$	−45.0000	−1.49174
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−14.0000	−0.463332
$914$	−8.00000	−0.264616
$915$	0	0
$916$	−17.0000	−0.561696
$917$	30.0000	0.990687
$918$	0	0
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	15.0000	0.494535
$921$	0	0
$922$	−6.00000	−0.197599
$923$	−80.0000	−2.63323
$924$	0	0
$925$	16.0000	0.526077
$926$	−8.00000	−0.262896
$927$	0	0
$928$	0	0
$929$	−43.0000	−1.41078	−0.705392	−	0.708817i	$-0.749229\pi$
$929$	−43.0000	−1.41078	−0.705392	+	0.708817i	$0.749229\pi$
$930$	0	0
$931$	−16.0000	−0.524379
$932$	0	0
$933$	0	0
$934$	−25.0000	−0.818025
$935$	−6.00000	−0.196221
$936$	0	0
$937$	−41.0000	−1.33941	−0.669706	−	0.742627i	$-0.733580\pi$
$937$	−41.0000	−1.33941	−0.669706	+	0.742627i	$0.733580\pi$
$938$	−39.0000	−1.27340
$939$	0	0
$940$	−6.00000	−0.195698
$941$	−40.0000	−1.30396	−0.651981	−	0.758235i	$-0.726062\pi$
$941$	−40.0000	−1.30396	−0.651981	+	0.758235i	$0.726062\pi$
$942$	0	0
$943$	15.0000	0.488467
$944$	7.00000	0.227831
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−35.0000	−1.13735	−0.568674	−	0.822563i	$-0.692543\pi$
$947$	−35.0000	−1.13735	−0.568674	+	0.822563i	$0.692543\pi$
$948$	0	0
$949$	−45.0000	−1.46076
$950$	32.0000	1.03822
$951$	0	0
$952$	−6.00000	−0.194461
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	0	0
$955$	45.0000	1.45617
$956$	12.0000	0.388108
$957$	0	0
$958$	−36.0000	−1.16311
$959$	45.0000	1.45313
$960$	0	0
$961$	−15.0000	−0.483871
$962$	20.0000	0.644826
$963$	0	0
$964$	25.0000	0.805196
$965$	−54.0000	−1.73832
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	−42.0000	−1.34854
$971$	14.0000	0.449281	0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000	0.449281	0.224641	+	0.974442i	$0.427879\pi$
$972$	0	0
$973$	15.0000	0.480878
$974$	16.0000	0.512673
$975$	0	0
$976$	1.00000	0.0320092
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	6.00000	0.191663
$981$	0	0
$982$	28.0000	0.893516
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	−63.0000	−2.00735
$986$	0	0
$987$	0	0
$988$	40.0000	1.27257
$989$	−20.0000	−0.635963
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	48.0000	1.52247
$995$	54.0000	1.71192
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	5.00000	0.158272
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1098.2.a.e.1.1		1
3.2	odd	2		366.2.a.d.1.1	✓	1
4.3	odd	2		8784.2.a.z.1.1		1
12.11	even	2		2928.2.a.h.1.1		1
15.14	odd	2		9150.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
366.2.a.d.1.1	✓	1	3.2	odd	2
1098.2.a.e.1.1		1	1.1	even	1	trivial
2928.2.a.h.1.1		1	12.11	even	2
8784.2.a.z.1.1		1	4.3	odd	2
9150.2.a.n.1.1		1	15.14	odd	2

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

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Newspace parameters

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Embedding invariants

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Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1098.2.a.e.1.1

Level	$1098$
Weight	$2$
Character	1098.1
Self dual	yes
Analytic conductor	$8.768$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$