LMFDB - Embedded newform 4998.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4998,2,Mod(1,4998)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4998, 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("4998.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4998.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:	$39.9092309302$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 714)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4998.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +3.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +3.00000 q^{19} +3.00000 q^{20} +2.00000 q^{22} -5.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +3.00000 q^{30} +1.00000 q^{32} +2.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +3.00000 q^{37} +3.00000 q^{38} +3.00000 q^{40} -1.00000 q^{43} +2.00000 q^{44} +3.00000 q^{45} -5.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} +4.00000 q^{50} -1.00000 q^{51} -6.00000 q^{53} +1.00000 q^{54} +6.00000 q^{55} +3.00000 q^{57} +6.00000 q^{58} -5.00000 q^{59} +3.00000 q^{60} -10.0000 q^{61} +1.00000 q^{64} +2.00000 q^{66} -7.00000 q^{67} -1.00000 q^{68} -5.00000 q^{69} +3.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +3.00000 q^{74} +4.00000 q^{75} +3.00000 q^{76} +16.0000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +12.0000 q^{83} -3.00000 q^{85} -1.00000 q^{86} +6.00000 q^{87} +2.00000 q^{88} -5.00000 q^{89} +3.00000 q^{90} -5.00000 q^{92} +6.00000 q^{94} +9.00000 q^{95} +1.00000 q^{96} +4.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	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	$0.265942\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.00000	0.235702
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	2.00000	0.426401
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	3.00000	0.547723
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	3.00000	0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	2.00000	0.301511
$45$	3.00000	0.447214
$46$	−5.00000	−0.737210
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	6.00000	0.809040
$56$	0	0
$57$	3.00000	0.397360
$58$	6.00000	0.787839
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	3.00000	0.387298
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−1.00000	−0.121268
$69$	−5.00000	−0.601929
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	3.00000	0.348743
$75$	4.00000	0.461880
$76$	3.00000	0.344124
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	−1.00000	−0.107833
$87$	6.00000	0.643268
$88$	2.00000	0.213201
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	−5.00000	−0.521286
$93$	0	0
$94$	6.00000	0.618853
$95$	9.00000	0.923381
$96$	1.00000	0.102062
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	4.00000	0.400000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	−1.00000	−0.0990148
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	1.00000	0.0962250
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	6.00000	0.572078
$111$	3.00000	0.284747
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	3.00000	0.280976
$115$	−15.0000	−1.39876
$116$	6.00000	0.557086
$117$	0	0
$118$	−5.00000	−0.460287
$119$	0	0
$120$	3.00000	0.273861
$121$	−7.00000	−0.636364
$122$	−10.0000	−0.905357
$123$	0	0
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	−7.00000	−0.604708
$135$	3.00000	0.258199
$136$	−1.00000	−0.0857493
$137$	11.0000	0.939793	0.469897	−	0.882721i	$-0.344291\pi$
$137$	11.0000	0.939793	0.469897	+	0.882721i	$0.344291\pi$
$138$	−5.00000	−0.425628
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	3.00000	0.251754
$143$	0	0
$144$	1.00000	0.0833333
$145$	18.0000	1.49482
$146$	4.00000	0.331042
$147$	0	0
$148$	3.00000	0.246598
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	4.00000	0.326599
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	3.00000	0.243332
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	16.0000	1.27289
$159$	−6.00000	−0.475831
$160$	3.00000	0.237171
$161$	0	0
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	6.00000	0.467099
$166$	12.0000	0.931381
$167$	25.0000	1.93456	0.967279	−	0.253715i	$-0.0816525\pi$
$167$	25.0000	1.93456	0.967279	+	0.253715i	$0.0816525\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−3.00000	−0.230089
$171$	3.00000	0.229416
$172$	−1.00000	−0.0762493
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	2.00000	0.150756
$177$	−5.00000	−0.375823
$178$	−5.00000	−0.374766
$179$	−7.00000	−0.523205	−0.261602	−	0.965176i	$-0.584251\pi$
$179$	−7.00000	−0.523205	−0.261602	+	0.965176i	$0.584251\pi$
$180$	3.00000	0.223607
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	−5.00000	−0.368605
$185$	9.00000	0.661693
$186$	0	0
$187$	−2.00000	−0.146254
$188$	6.00000	0.437595
$189$	0	0
$190$	9.00000	0.652929
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000	0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	0	0
$197$	−21.0000	−1.49619	−0.748094	−	0.663593i	$-0.769031\pi$
$197$	−21.0000	−1.49619	−0.748094	+	0.663593i	$0.769031\pi$
$198$	2.00000	0.142134
$199$	−21.0000	−1.48865	−0.744325	−	0.667817i	$-0.767229\pi$
$199$	−21.0000	−1.48865	−0.744325	+	0.667817i	$0.767229\pi$
$200$	4.00000	0.282843
$201$	−7.00000	−0.493742
$202$	12.0000	0.844317
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	14.0000	0.975426
$207$	−5.00000	−0.347524
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	−6.00000	−0.412082
$213$	3.00000	0.205557
$214$	−16.0000	−1.09374
$215$	−3.00000	−0.204598
$216$	1.00000	0.0680414
$217$	0	0
$218$	−11.0000	−0.745014
$219$	4.00000	0.270295
$220$	6.00000	0.404520
$221$	0	0
$222$	3.00000	0.201347
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	−14.0000	−0.931266
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	3.00000	0.198680
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	−15.0000	−0.989071
$231$	0	0
$232$	6.00000	0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	−5.00000	−0.325472
$237$	16.0000	1.03931
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	3.00000	0.193649
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000	0.760469
$250$	−3.00000	−0.189737
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−10.0000	−0.628695
$254$	−20.0000	−1.25491
$255$	−3.00000	−0.187867
$256$	1.00000	0.0625000
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	−1.00000	−0.0622573
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	10.0000	0.617802
$263$	−26.0000	−1.60323	−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000	−1.60323	−0.801614	+	0.597841i	$0.796025\pi$
$264$	2.00000	0.123091
$265$	−18.0000	−1.10573
$266$	0	0
$267$	−5.00000	−0.305995
$268$	−7.00000	−0.427593
$269$	19.0000	1.15845	0.579225	−	0.815168i	$-0.303355\pi$
$269$	19.0000	1.15845	0.579225	+	0.815168i	$0.303355\pi$
$270$	3.00000	0.182574
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	11.0000	0.664534
$275$	8.00000	0.482418
$276$	−5.00000	−0.300965
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−16.0000	−0.959616
$279$	0	0
$280$	0	0
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	6.00000	0.357295
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	3.00000	0.178017
$285$	9.00000	0.533114
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	18.0000	1.05700
$291$	4.00000	0.234484
$292$	4.00000	0.234082
$293$	−8.00000	−0.467365	−0.233682	−	0.972313i	$-0.575078\pi$
$293$	−8.00000	−0.467365	−0.233682	+	0.972313i	$0.575078\pi$
$294$	0	0
$295$	−15.0000	−0.873334
$296$	3.00000	0.174371
$297$	2.00000	0.116052
$298$	−10.0000	−0.579284
$299$	0	0
$300$	4.00000	0.230940
$301$	0	0
$302$	0	0
$303$	12.0000	0.689382
$304$	3.00000	0.172062
$305$	−30.0000	−1.71780
$306$	−1.00000	−0.0571662
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	16.0000	0.900070
$317$	−1.00000	−0.0561656	−0.0280828	−	0.999606i	$-0.508940\pi$
$317$	−1.00000	−0.0561656	−0.0280828	+	0.999606i	$0.508940\pi$
$318$	−6.00000	−0.336463
$319$	12.0000	0.671871
$320$	3.00000	0.167705
$321$	−16.0000	−0.893033
$322$	0	0
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	−11.0000	−0.608301
$328$	0	0
$329$	0	0
$330$	6.00000	0.330289
$331$	9.00000	0.494685	0.247342	−	0.968928i	$-0.420443\pi$
$331$	9.00000	0.494685	0.247342	+	0.968928i	$0.420443\pi$
$332$	12.0000	0.658586
$333$	3.00000	0.164399
$334$	25.0000	1.36794
$335$	−21.0000	−1.14735
$336$	0	0
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	−13.0000	−0.707107
$339$	−14.0000	−0.760376
$340$	−3.00000	−0.162698
$341$	0	0
$342$	3.00000	0.162221
$343$	0	0
$344$	−1.00000	−0.0539164
$345$	−15.0000	−0.807573
$346$	−9.00000	−0.483843
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	6.00000	0.321634
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	−5.00000	−0.265747
$355$	9.00000	0.477670
$356$	−5.00000	−0.264999
$357$	0	0
$358$	−7.00000	−0.369961
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	3.00000	0.158114
$361$	−10.0000	−0.526316
$362$	1.00000	0.0525588
$363$	−7.00000	−0.367405
$364$	0	0
$365$	12.0000	0.628109
$366$	−10.0000	−0.522708
$367$	−29.0000	−1.51379	−0.756894	−	0.653538i	$-0.773284\pi$
$367$	−29.0000	−1.51379	−0.756894	+	0.653538i	$0.773284\pi$
$368$	−5.00000	−0.260643
$369$	0	0
$370$	9.00000	0.467888
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−2.00000	−0.103418
$375$	−3.00000	−0.154919
$376$	6.00000	0.309426
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	9.00000	0.461690
$381$	−20.0000	−1.02463
$382$	12.0000	0.613973
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−1.00000	−0.0508329
$388$	4.00000	0.203069
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	10.0000	0.504433
$394$	−21.0000	−1.05796
$395$	48.0000	2.41514
$396$	2.00000	0.100504
$397$	−23.0000	−1.15434	−0.577168	−	0.816625i	$-0.695842\pi$
$397$	−23.0000	−1.15434	−0.577168	+	0.816625i	$0.695842\pi$
$398$	−21.0000	−1.05263
$399$	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	$0.547867\pi$
$402$	−7.00000	−0.349128
$403$	0	0
$404$	12.0000	0.597022
$405$	3.00000	0.149071
$406$	0	0
$407$	6.00000	0.297409
$408$	−1.00000	−0.0495074
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	11.0000	0.542590
$412$	14.0000	0.689730
$413$	0	0
$414$	−5.00000	−0.245737
$415$	36.0000	1.76717
$416$	0	0
$417$	−16.0000	−0.783523
$418$	6.00000	0.293470
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	−14.0000	−0.681509
$423$	6.00000	0.291730
$424$	−6.00000	−0.291386
$425$	−4.00000	−0.194029
$426$	3.00000	0.145350
$427$	0	0
$428$	−16.0000	−0.773389
$429$	0	0
$430$	−3.00000	−0.144673
$431$	−37.0000	−1.78223	−0.891114	−	0.453780i	$-0.850075\pi$
$431$	−37.0000	−1.78223	−0.891114	+	0.453780i	$0.850075\pi$
$432$	1.00000	0.0481125
$433$	−37.0000	−1.77811	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−37.0000	−1.77811	−0.889053	+	0.457804i	$0.848636\pi$
$434$	0	0
$435$	18.0000	0.863034
$436$	−11.0000	−0.526804
$437$	−15.0000	−0.717547
$438$	4.00000	0.191127
$439$	41.0000	1.95682	0.978412	−	0.206666i	$-0.0662612\pi$
$439$	41.0000	1.95682	0.978412	+	0.206666i	$0.0662612\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	0	0
$443$	35.0000	1.66290	0.831450	−	0.555599i	$-0.187511\pi$
$443$	35.0000	1.66290	0.831450	+	0.555599i	$0.187511\pi$
$444$	3.00000	0.142374
$445$	−15.0000	−0.711068
$446$	4.00000	0.189405
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	4.00000	0.188562
$451$	0	0
$452$	−14.0000	−0.658505
$453$	0	0
$454$	18.0000	0.844782
$455$	0	0
$456$	3.00000	0.140488
$457$	21.0000	0.982339	0.491169	−	0.871064i	$-0.336570\pi$
$457$	21.0000	0.982339	0.491169	+	0.871064i	$0.336570\pi$
$458$	12.0000	0.560723
$459$	−1.00000	−0.0466760
$460$	−15.0000	−0.699379
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	0	0
$469$	0	0
$470$	18.0000	0.830278
$471$	−2.00000	−0.0921551
$472$	−5.00000	−0.230144
$473$	−2.00000	−0.0919601
$474$	16.0000	0.734904
$475$	12.0000	0.550598
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−20.0000	−0.914779
$479$	−27.0000	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$479$	−27.0000	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$480$	3.00000	0.136931
$481$	0	0
$482$	8.00000	0.364390
$483$	0	0
$484$	−7.00000	−0.318182
$485$	12.0000	0.544892
$486$	1.00000	0.0453609
$487$	17.0000	0.770344	0.385172	−	0.922845i	$-0.374142\pi$
$487$	17.0000	0.770344	0.385172	+	0.922845i	$0.374142\pi$
$488$	−10.0000	−0.452679
$489$	20.0000	0.904431
$490$	0	0
$491$	−27.0000	−1.21849	−0.609246	−	0.792981i	$-0.708528\pi$
$491$	−27.0000	−1.21849	−0.609246	+	0.792981i	$0.708528\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	0	0
$498$	12.0000	0.537733
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	−3.00000	−0.134164
$501$	25.0000	1.11692
$502$	12.0000	0.535586
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	−10.0000	−0.444554
$507$	−13.0000	−0.577350
$508$	−20.0000	−0.887357
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	−3.00000	−0.132842
$511$	0	0
$512$	1.00000	0.0441942
$513$	3.00000	0.132453
$514$	13.0000	0.573405
$515$	42.0000	1.85074
$516$	−1.00000	−0.0440225
$517$	12.0000	0.527759
$518$	0	0
$519$	−9.00000	−0.395056
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	6.00000	0.262613
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	10.0000	0.436852
$525$	0	0
$526$	−26.0000	−1.13365
$527$	0	0
$528$	2.00000	0.0870388
$529$	2.00000	0.0869565
$530$	−18.0000	−0.781870
$531$	−5.00000	−0.216982
$532$	0	0
$533$	0	0
$534$	−5.00000	−0.216371
$535$	−48.0000	−2.07522
$536$	−7.00000	−0.302354
$537$	−7.00000	−0.302072
$538$	19.0000	0.819148
$539$	0	0
$540$	3.00000	0.129099
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	2.00000	0.0859074
$543$	1.00000	0.0429141
$544$	−1.00000	−0.0428746
$545$	−33.0000	−1.41356
$546$	0	0
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	11.0000	0.469897
$549$	−10.0000	−0.426790
$550$	8.00000	0.341121
$551$	18.0000	0.766826
$552$	−5.00000	−0.212814
$553$	0	0
$554$	22.0000	0.934690
$555$	9.00000	0.382029
$556$	−16.0000	−0.678551
$557$	4.00000	0.169485	0.0847427	−	0.996403i	$-0.472993\pi$
$557$	4.00000	0.169485	0.0847427	+	0.996403i	$0.472993\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	−15.0000	−0.632737
$563$	41.0000	1.72794	0.863972	−	0.503540i	$-0.167969\pi$
$563$	41.0000	1.72794	0.863972	+	0.503540i	$0.167969\pi$
$564$	6.00000	0.252646
$565$	−42.0000	−1.76695
$566$	8.00000	0.336265
$567$	0	0
$568$	3.00000	0.125877
$569$	−17.0000	−0.712677	−0.356339	−	0.934357i	$-0.615975\pi$
$569$	−17.0000	−0.712677	−0.356339	+	0.934357i	$0.615975\pi$
$570$	9.00000	0.376969
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	−20.0000	−0.834058
$576$	1.00000	0.0416667
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	1.00000	0.0415945
$579$	−14.0000	−0.581820
$580$	18.0000	0.747409
$581$	0	0
$582$	4.00000	0.165805
$583$	−12.0000	−0.496989
$584$	4.00000	0.165521
$585$	0	0
$586$	−8.00000	−0.330477
$587$	27.0000	1.11441	0.557205	−	0.830375i	$-0.311874\pi$
$587$	27.0000	1.11441	0.557205	+	0.830375i	$0.311874\pi$
$588$	0	0
$589$	0	0
$590$	−15.0000	−0.617540
$591$	−21.0000	−0.863825
$592$	3.00000	0.123299
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−21.0000	−0.859473
$598$	0	0
$599$	34.0000	1.38920	0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000	1.38920	0.694601	+	0.719395i	$0.255581\pi$
$600$	4.00000	0.163299
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−7.00000	−0.285062
$604$	0	0
$605$	−21.0000	−0.853771
$606$	12.0000	0.487467
$607$	47.0000	1.90767	0.953836	−	0.300329i	$-0.0970966\pi$
$607$	47.0000	1.90767	0.953836	+	0.300329i	$0.0970966\pi$
$608$	3.00000	0.121666
$609$	0	0
$610$	−30.0000	−1.21466
$611$	0	0
$612$	−1.00000	−0.0404226
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−19.0000	−0.766778
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	14.0000	0.563163
$619$	−42.0000	−1.68812	−0.844061	−	0.536247i	$-0.819842\pi$
$619$	−42.0000	−1.68812	−0.844061	+	0.536247i	$0.819842\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	−15.0000	−0.601445
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	2.00000	0.0799361
$627$	6.00000	0.239617
$628$	−2.00000	−0.0798087
$629$	−3.00000	−0.119618
$630$	0	0
$631$	30.0000	1.19428	0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000	1.19428	0.597141	+	0.802137i	$0.296303\pi$
$632$	16.0000	0.636446
$633$	−14.0000	−0.556450
$634$	−1.00000	−0.0397151
$635$	−60.0000	−2.38103
$636$	−6.00000	−0.237915
$637$	0	0
$638$	12.0000	0.475085
$639$	3.00000	0.118678
$640$	3.00000	0.118585
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	−16.0000	−0.631470
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	−3.00000	−0.118125
$646$	−3.00000	−0.118033
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	1.00000	0.0392837
$649$	−10.0000	−0.392534
$650$	0	0
$651$	0	0
$652$	20.0000	0.783260
$653$	45.0000	1.76099	0.880493	−	0.474059i	$-0.157212\pi$
$653$	45.0000	1.76099	0.880493	+	0.474059i	$0.157212\pi$
$654$	−11.0000	−0.430134
$655$	30.0000	1.17220
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	6.00000	0.233550
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	9.00000	0.349795
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	3.00000	0.116248
$667$	−30.0000	−1.16160
$668$	25.0000	0.967279
$669$	4.00000	0.154649
$670$	−21.0000	−0.811301
$671$	−20.0000	−0.772091
$672$	0	0
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	12.0000	0.462223
$675$	4.00000	0.153960
$676$	−13.0000	−0.500000
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−14.0000	−0.537667
$679$	0	0
$680$	−3.00000	−0.115045
$681$	18.0000	0.689761
$682$	0	0
$683$	−32.0000	−1.22445	−0.612223	−	0.790685i	$-0.709725\pi$
$683$	−32.0000	−1.22445	−0.612223	+	0.790685i	$0.709725\pi$
$684$	3.00000	0.114708
$685$	33.0000	1.26087
$686$	0	0
$687$	12.0000	0.457829
$688$	−1.00000	−0.0381246
$689$	0	0
$690$	−15.0000	−0.571040
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−9.00000	−0.342129
$693$	0	0
$694$	12.0000	0.455514
$695$	−48.0000	−1.82074
$696$	6.00000	0.227429
$697$	0	0
$698$	−16.0000	−0.605609
$699$	10.0000	0.378235
$700$	0	0
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	0	0
$703$	9.00000	0.339441
$704$	2.00000	0.0753778
$705$	18.0000	0.677919
$706$	9.00000	0.338719
$707$	0	0
$708$	−5.00000	−0.187912
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	9.00000	0.337764
$711$	16.0000	0.600047
$712$	−5.00000	−0.187383
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−7.00000	−0.261602
$717$	−20.0000	−0.746914
$718$	−6.00000	−0.223918
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−10.0000	−0.372161
$723$	8.00000	0.297523
$724$	1.00000	0.0371647
$725$	24.0000	0.891338
$726$	−7.00000	−0.259794
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	1.00000	0.0369863
$732$	−10.0000	−0.369611
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	−29.0000	−1.07041
$735$	0	0
$736$	−5.00000	−0.184302
$737$	−14.0000	−0.515697
$738$	0	0
$739$	−3.00000	−0.110357	−0.0551784	−	0.998477i	$-0.517573\pi$
$739$	−3.00000	−0.110357	−0.0551784	+	0.998477i	$0.517573\pi$
$740$	9.00000	0.330847
$741$	0	0
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−30.0000	−1.09911
$746$	26.0000	0.951928
$747$	12.0000	0.439057
$748$	−2.00000	−0.0731272
$749$	0	0
$750$	−3.00000	−0.109545
$751$	−43.0000	−1.56909	−0.784546	−	0.620070i	$-0.787104\pi$
$751$	−43.0000	−1.56909	−0.784546	+	0.620070i	$0.787104\pi$
$752$	6.00000	0.218797
$753$	12.0000	0.437304
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.00000	0.145382	0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000	0.145382	0.0726912	+	0.997354i	$0.476841\pi$
$758$	0	0
$759$	−10.0000	−0.362977
$760$	9.00000	0.326464
$761$	−47.0000	−1.70375	−0.851874	−	0.523746i	$-0.824534\pi$
$761$	−47.0000	−1.70375	−0.851874	+	0.523746i	$0.824534\pi$
$762$	−20.0000	−0.724524
$763$	0	0
$764$	12.0000	0.434145
$765$	−3.00000	−0.108465
$766$	−14.0000	−0.505841
$767$	0	0
$768$	1.00000	0.0360844
$769$	−19.0000	−0.685158	−0.342579	−	0.939489i	$-0.611300\pi$
$769$	−19.0000	−0.685158	−0.342579	+	0.939489i	$0.611300\pi$
$770$	0	0
$771$	13.0000	0.468184
$772$	−14.0000	−0.503871
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	4.00000	0.143592
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	6.00000	0.214697
$782$	5.00000	0.178800
$783$	6.00000	0.214423
$784$	0	0
$785$	−6.00000	−0.214149
$786$	10.0000	0.356688
$787$	30.0000	1.06938	0.534692	−	0.845047i	$-0.320428\pi$
$787$	30.0000	1.06938	0.534692	+	0.845047i	$0.320428\pi$
$788$	−21.0000	−0.748094
$789$	−26.0000	−0.925625
$790$	48.0000	1.70776
$791$	0	0
$792$	2.00000	0.0710669
$793$	0	0
$794$	−23.0000	−0.816239
$795$	−18.0000	−0.638394
$796$	−21.0000	−0.744325
$797$	−44.0000	−1.55856	−0.779280	−	0.626676i	$-0.784415\pi$
$797$	−44.0000	−1.55856	−0.779280	+	0.626676i	$0.784415\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	4.00000	0.141421
$801$	−5.00000	−0.176666
$802$	−6.00000	−0.211867
$803$	8.00000	0.282314
$804$	−7.00000	−0.246871
$805$	0	0
$806$	0	0
$807$	19.0000	0.668832
$808$	12.0000	0.422159
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	3.00000	0.105409
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	6.00000	0.210300
$815$	60.0000	2.10171
$816$	−1.00000	−0.0350070
$817$	−3.00000	−0.104957
$818$	14.0000	0.489499
$819$	0	0
$820$	0	0
$821$	−53.0000	−1.84971	−0.924856	−	0.380317i	$-0.875815\pi$
$821$	−53.0000	−1.84971	−0.924856	+	0.380317i	$0.875815\pi$
$822$	11.0000	0.383669
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	14.0000	0.487713
$825$	8.00000	0.278524
$826$	0	0
$827$	22.0000	0.765015	0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000	0.765015	0.382507	+	0.923952i	$0.375061\pi$
$828$	−5.00000	−0.173762
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	36.0000	1.24958
$831$	22.0000	0.763172
$832$	0	0
$833$	0	0
$834$	−16.0000	−0.554035
$835$	75.0000	2.59548
$836$	6.00000	0.207514
$837$	0	0
$838$	−24.0000	−0.829066
$839$	51.0000	1.76072	0.880358	−	0.474310i	$-0.157302\pi$
$839$	51.0000	1.76072	0.880358	+	0.474310i	$0.157302\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−8.00000	−0.275698
$843$	−15.0000	−0.516627
$844$	−14.0000	−0.481900
$845$	−39.0000	−1.34164
$846$	6.00000	0.206284
$847$	0	0
$848$	−6.00000	−0.206041
$849$	8.00000	0.274559
$850$	−4.00000	−0.137199
$851$	−15.0000	−0.514193
$852$	3.00000	0.102778
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	0	0
$855$	9.00000	0.307794
$856$	−16.0000	−0.546869
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	15.0000	0.511793	0.255897	−	0.966704i	$-0.417629\pi$
$859$	15.0000	0.511793	0.255897	+	0.966704i	$0.417629\pi$
$860$	−3.00000	−0.102299
$861$	0	0
$862$	−37.0000	−1.26023
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	1.00000	0.0340207
$865$	−27.0000	−0.918028
$866$	−37.0000	−1.25731
$867$	1.00000	0.0339618
$868$	0	0
$869$	32.0000	1.08553
$870$	18.0000	0.610257
$871$	0	0
$872$	−11.0000	−0.372507
$873$	4.00000	0.135379
$874$	−15.0000	−0.507383
$875$	0	0
$876$	4.00000	0.135147
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	41.0000	1.38368
$879$	−8.00000	−0.269833
$880$	6.00000	0.202260
$881$	40.0000	1.34763	0.673817	−	0.738898i	$-0.264654\pi$
$881$	40.0000	1.34763	0.673817	+	0.738898i	$0.264654\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	−15.0000	−0.504219
$886$	35.0000	1.17585
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	3.00000	0.100673
$889$	0	0
$890$	−15.0000	−0.502801
$891$	2.00000	0.0670025
$892$	4.00000	0.133930
$893$	18.0000	0.602347
$894$	−10.0000	−0.334450
$895$	−21.0000	−0.701953
$896$	0	0
$897$	0	0
$898$	−24.0000	−0.800890
$899$	0	0
$900$	4.00000	0.133333
$901$	6.00000	0.199889
$902$	0	0
$903$	0	0
$904$	−14.0000	−0.465633
$905$	3.00000	0.0997234
$906$	0	0
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	18.0000	0.597351
$909$	12.0000	0.398015
$910$	0	0
$911$	−33.0000	−1.09334	−0.546669	−	0.837349i	$-0.684105\pi$
$911$	−33.0000	−1.09334	−0.546669	+	0.837349i	$0.684105\pi$
$912$	3.00000	0.0993399
$913$	24.0000	0.794284
$914$	21.0000	0.694618
$915$	−30.0000	−0.991769
$916$	12.0000	0.396491
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−15.0000	−0.494535
$921$	−19.0000	−0.626071
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	12.0000	0.394558
$926$	20.0000	0.657241
$927$	14.0000	0.459820
$928$	6.00000	0.196960
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	10.0000	0.327561
$933$	−15.0000	−0.491078
$934$	21.0000	0.687141
$935$	−6.00000	−0.196221
$936$	0	0
$937$	39.0000	1.27407	0.637037	−	0.770833i	$-0.280160\pi$
$937$	39.0000	1.27407	0.637037	+	0.770833i	$0.280160\pi$
$938$	0	0
$939$	2.00000	0.0652675
$940$	18.0000	0.587095
$941$	−27.0000	−0.880175	−0.440087	−	0.897955i	$-0.645053\pi$
$941$	−27.0000	−0.880175	−0.440087	+	0.897955i	$0.645053\pi$
$942$	−2.00000	−0.0651635
$943$	0	0
$944$	−5.00000	−0.162736
$945$	0	0
$946$	−2.00000	−0.0650256
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	16.0000	0.519656
$949$	0	0
$950$	12.0000	0.389331
$951$	−1.00000	−0.0324272
$952$	0	0
$953$	−41.0000	−1.32812	−0.664060	−	0.747679i	$-0.731168\pi$
$953$	−41.0000	−1.32812	−0.664060	+	0.747679i	$0.731168\pi$
$954$	−6.00000	−0.194257
$955$	36.0000	1.16493
$956$	−20.0000	−0.646846
$957$	12.0000	0.387905
$958$	−27.0000	−0.872330
$959$	0	0
$960$	3.00000	0.0968246
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−16.0000	−0.515593
$964$	8.00000	0.257663
$965$	−42.0000	−1.35203
$966$	0	0
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	−7.00000	−0.224989
$969$	−3.00000	−0.0963739
$970$	12.0000	0.385297
$971$	1.00000	0.0320915	0.0160458	−	0.999871i	$-0.494892\pi$
$971$	1.00000	0.0320915	0.0160458	+	0.999871i	$0.494892\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	17.0000	0.544715
$975$	0	0
$976$	−10.0000	−0.320092
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	20.0000	0.639529
$979$	−10.0000	−0.319601
$980$	0	0
$981$	−11.0000	−0.351203
$982$	−27.0000	−0.861605
$983$	−19.0000	−0.606006	−0.303003	−	0.952990i	$-0.597989\pi$
$983$	−19.0000	−0.606006	−0.303003	+	0.952990i	$0.597989\pi$
$984$	0	0
$985$	−63.0000	−2.00735
$986$	−6.00000	−0.191079
$987$	0	0
$988$	0	0
$989$	5.00000	0.158991
$990$	6.00000	0.190693
$991$	−3.00000	−0.0952981	−0.0476491	−	0.998864i	$-0.515173\pi$
$991$	−3.00000	−0.0952981	−0.0476491	+	0.998864i	$0.515173\pi$
$992$	0	0
$993$	9.00000	0.285606
$994$	0	0
$995$	−63.0000	−1.99723
$996$	12.0000	0.380235
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	16.0000	0.506471
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4998.2.a.br.1.1		1
7.3	odd	6		714.2.i.f.205.1	✓	2
7.5	odd	6		714.2.i.f.613.1	yes	2
7.6	odd	2		4998.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.i.f.205.1	✓	2	7.3	odd	6
714.2.i.f.613.1	yes	2	7.5	odd	6
4998.2.a.y.1.1		1	7.6	odd	2
4998.2.a.br.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:	\( N \)	\(=\)	\( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4998.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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