LMFDB - Embedded newform 4998.2.a.bf.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +3.00000 q^{20} -2.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{29} -3.00000 q^{30} +9.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} -1.00000 q^{39} +3.00000 q^{40} -5.00000 q^{41} -2.00000 q^{44} +3.00000 q^{45} +4.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} +4.00000 q^{50} +1.00000 q^{51} +1.00000 q^{52} -4.00000 q^{53} -1.00000 q^{54} -6.00000 q^{55} +1.00000 q^{58} +13.0000 q^{59} -3.00000 q^{60} +9.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +2.00000 q^{66} -8.00000 q^{67} -1.00000 q^{68} -4.00000 q^{69} +12.0000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -4.00000 q^{74} -4.00000 q^{75} -1.00000 q^{78} +3.00000 q^{80} +1.00000 q^{81} -5.00000 q^{82} +7.00000 q^{83} -3.00000 q^{85} -1.00000 q^{87} -2.00000 q^{88} +6.00000 q^{89} +3.00000 q^{90} +4.00000 q^{92} -9.00000 q^{93} +9.00000 q^{94} -1.00000 q^{96} +12.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−2.00000	−0.426401
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	−3.00000	−0.547723
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	3.00000	0.474342
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−2.00000	−0.301511
$45$	3.00000	0.447214
$46$	4.00000	0.589768
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	4.00000	0.565685
$51$	1.00000	0.140028
$52$	1.00000	0.138675
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	−1.00000	−0.136083
$55$	−6.00000	−0.809040
$56$	0	0
$57$	0	0
$58$	1.00000	0.131306
$59$	13.0000	1.69246	0.846228	−	0.532821i	$-0.178868\pi$
$59$	13.0000	1.69246	0.846228	+	0.532821i	$0.178868\pi$
$60$	−3.00000	−0.387298
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	9.00000	1.14300
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	2.00000	0.246183
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−1.00000	−0.121268
$69$	−4.00000	−0.481543
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−4.00000	−0.464991
$75$	−4.00000	−0.461880
$76$	0	0
$77$	0	0
$78$	−1.00000	−0.113228
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	−5.00000	−0.552158
$83$	7.00000	0.768350	0.384175	−	0.923260i	$-0.374486\pi$
$83$	7.00000	0.768350	0.384175	+	0.923260i	$0.374486\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	−1.00000	−0.107211
$88$	−2.00000	−0.213201
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	4.00000	0.417029
$93$	−9.00000	−0.933257
$94$	9.00000	0.928279
$95$	0	0
$96$	−1.00000	−0.102062
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	4.00000	0.400000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	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	$0.257729\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−4.00000	−0.388514
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−1.00000	−0.0962250
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	−6.00000	−0.572078
$111$	4.00000	0.379663
$112$	0	0
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	0	0
$115$	12.0000	1.11901
$116$	1.00000	0.0928477
$117$	1.00000	0.0924500
$118$	13.0000	1.19675
$119$	0	0
$120$	−3.00000	−0.273861
$121$	−7.00000	−0.636364
$122$	0	0
$123$	5.00000	0.450835
$124$	9.00000	0.808224
$125$	−3.00000	−0.268328
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.00000	0.263117
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	−8.00000	−0.691095
$135$	−3.00000	−0.258199
$136$	−1.00000	−0.0857493
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−4.00000	−0.340503
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	12.0000	1.00702
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	3.00000	0.249136
$146$	10.0000	0.827606
$147$	0	0
$148$	−4.00000	−0.328798
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−4.00000	−0.326599
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	27.0000	2.16869
$156$	−1.00000	−0.0800641
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	3.00000	0.237171
$161$	0	0
$162$	1.00000	0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−5.00000	−0.390434
$165$	6.00000	0.467099
$166$	7.00000	0.543305
$167$	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
$170$	−3.00000	−0.230089
$171$	0	0
$172$	0	0
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	−2.00000	−0.150756
$177$	−13.0000	−0.977140
$178$	6.00000	0.449719
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	3.00000	0.223607
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	0	0
$184$	4.00000	0.294884
$185$	−12.0000	−0.882258
$186$	−9.00000	−0.659912
$187$	2.00000	0.146254
$188$	9.00000	0.656392
$189$	0	0
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	−1.00000	−0.0721688
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	12.0000	0.861550
$195$	−3.00000	−0.214834
$196$	0	0
$197$	11.0000	0.783718	0.391859	−	0.920025i	$-0.371832\pi$
$197$	11.0000	0.783718	0.391859	+	0.920025i	$0.371832\pi$
$198$	−2.00000	−0.142134
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	4.00000	0.282843
$201$	8.00000	0.564276
$202$	−10.0000	−0.703598
$203$	0	0
$204$	1.00000	0.0700140
$205$	−15.0000	−1.04765
$206$	14.0000	0.975426
$207$	4.00000	0.278019
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	−4.00000	−0.274721
$213$	−12.0000	−0.822226
$214$	−8.00000	−0.546869
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	4.00000	0.270914
$219$	−10.0000	−0.675737
$220$	−6.00000	−0.404520
$221$	−1.00000	−0.0672673
$222$	4.00000	0.268462
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	−3.00000	−0.199557
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	1.00000	0.0656532
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	1.00000	0.0653720
$235$	27.0000	1.76129
$236$	13.0000	0.846228
$237$	0	0
$238$	0	0
$239$	27.0000	1.74648	0.873242	−	0.487286i	$-0.162013\pi$
$239$	27.0000	1.74648	0.873242	+	0.487286i	$0.162013\pi$
$240$	−3.00000	−0.193649
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−7.00000	−0.449977
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	5.00000	0.318788
$247$	0	0
$248$	9.00000	0.571501
$249$	−7.00000	−0.443607
$250$	−3.00000	−0.189737
$251$	−31.0000	−1.95670	−0.978351	−	0.206951i	$-0.933646\pi$
$251$	−31.0000	−1.95670	−0.978351	+	0.206951i	$0.933646\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	8.00000	0.501965
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	3.00000	0.186052
$261$	1.00000	0.0618984
$262$	−10.0000	−0.617802
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	2.00000	0.123091
$265$	−12.0000	−0.737154
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−8.00000	−0.488678
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−3.00000	−0.182574
$271$	30.0000	1.82237	0.911185	−	0.411997i	$-0.135169\pi$
$271$	30.0000	1.82237	0.911185	+	0.411997i	$0.135169\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−8.00000	−0.482418
$276$	−4.00000	−0.240772
$277$	24.0000	1.44202	0.721010	−	0.692925i	$-0.243678\pi$
$277$	24.0000	1.44202	0.721010	+	0.692925i	$0.243678\pi$
$278$	8.00000	0.479808
$279$	9.00000	0.538816
$280$	0	0
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	−9.00000	−0.535942
$283$	15.0000	0.891657	0.445829	−	0.895118i	$-0.352909\pi$
$283$	15.0000	0.891657	0.445829	+	0.895118i	$0.352909\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	3.00000	0.176166
$291$	−12.0000	−0.703452
$292$	10.0000	0.585206
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	39.0000	2.27067
$296$	−4.00000	−0.232495
$297$	2.00000	0.116052
$298$	6.00000	0.347571
$299$	4.00000	0.231326
$300$	−4.00000	−0.230940
$301$	0	0
$302$	−20.0000	−1.15087
$303$	10.0000	0.574485
$304$	0	0
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	27.0000	1.53350
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	−1.00000	−0.0566139
$313$	12.0000	0.678280	0.339140	−	0.940736i	$-0.389864\pi$
$313$	12.0000	0.678280	0.339140	+	0.940736i	$0.389864\pi$
$314$	−7.00000	−0.395033
$315$	0	0
$316$	0	0
$317$	15.0000	0.842484	0.421242	−	0.906948i	$-0.361594\pi$
$317$	15.0000	0.842484	0.421242	+	0.906948i	$0.361594\pi$
$318$	4.00000	0.224309
$319$	−2.00000	−0.111979
$320$	3.00000	0.167705
$321$	8.00000	0.446516
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	−8.00000	−0.443079
$327$	−4.00000	−0.221201
$328$	−5.00000	−0.276079
$329$	0	0
$330$	6.00000	0.330289
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	7.00000	0.384175
$333$	−4.00000	−0.219199
$334$	18.0000	0.984916
$335$	−24.0000	−1.31126
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−12.0000	−0.652714
$339$	3.00000	0.162938
$340$	−3.00000	−0.162698
$341$	−18.0000	−0.974755
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−12.0000	−0.646058
$346$	−9.00000	−0.483843
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	−1.00000	−0.0536056
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−2.00000	−0.106600
$353$	32.0000	1.70319	0.851594	−	0.524202i	$-0.175636\pi$
$353$	32.0000	1.70319	0.851594	+	0.524202i	$0.175636\pi$
$354$	−13.0000	−0.690942
$355$	36.0000	1.91068
$356$	6.00000	0.317999
$357$	0	0
$358$	−3.00000	−0.158555
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	3.00000	0.158114
$361$	−19.0000	−1.00000
$362$	−12.0000	−0.630706
$363$	7.00000	0.367405
$364$	0	0
$365$	30.0000	1.57027
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	4.00000	0.208514
$369$	−5.00000	−0.260290
$370$	−12.0000	−0.623850
$371$	0	0
$372$	−9.00000	−0.466628
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	2.00000	0.103418
$375$	3.00000	0.154919
$376$	9.00000	0.464140
$377$	1.00000	0.0515026
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	−3.00000	−0.153493
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−2.00000	−0.101797
$387$	0	0
$388$	12.0000	0.609208
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	−3.00000	−0.151911
$391$	−4.00000	−0.202289
$392$	0	0
$393$	10.0000	0.504433
$394$	11.0000	0.554172
$395$	0	0
$396$	−2.00000	−0.100504
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	−25.0000	−1.25314
$399$	0	0
$400$	4.00000	0.200000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	8.00000	0.399004
$403$	9.00000	0.448322
$404$	−10.0000	−0.497519
$405$	3.00000	0.149071
$406$	0	0
$407$	8.00000	0.396545
$408$	1.00000	0.0495074
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−15.0000	−0.740797
$411$	12.0000	0.591916
$412$	14.0000	0.689730
$413$	0	0
$414$	4.00000	0.196589
$415$	21.0000	1.03085
$416$	1.00000	0.0490290
$417$	−8.00000	−0.391762
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	9.00000	0.438633	0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000	0.438633	0.219317	+	0.975654i	$0.429617\pi$
$422$	23.0000	1.11962
$423$	9.00000	0.437595
$424$	−4.00000	−0.194257
$425$	−4.00000	−0.194029
$426$	−12.0000	−0.581402
$427$	0	0
$428$	−8.00000	−0.386695
$429$	2.00000	0.0965609
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	−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
$435$	−3.00000	−0.143839
$436$	4.00000	0.191565
$437$	0	0
$438$	−10.0000	−0.477818
$439$	−5.00000	−0.238637	−0.119318	−	0.992856i	$-0.538071\pi$
$439$	−5.00000	−0.238637	−0.119318	+	0.992856i	$0.538071\pi$
$440$	−6.00000	−0.286039
$441$	0	0
$442$	−1.00000	−0.0475651
$443$	−25.0000	−1.18779	−0.593893	−	0.804544i	$-0.702410\pi$
$443$	−25.0000	−1.18779	−0.593893	+	0.804544i	$0.702410\pi$
$444$	4.00000	0.189832
$445$	18.0000	0.853282
$446$	24.0000	1.13643
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	4.00000	0.188562
$451$	10.0000	0.470882
$452$	−3.00000	−0.141108
$453$	20.0000	0.939682
$454$	−18.0000	−0.844782
$455$	0	0
$456$	0	0
$457$	−7.00000	−0.327446	−0.163723	−	0.986506i	$-0.552350\pi$
$457$	−7.00000	−0.327446	−0.163723	+	0.986506i	$0.552350\pi$
$458$	7.00000	0.327089
$459$	1.00000	0.0466760
$460$	12.0000	0.559503
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	1.00000	0.0464238
$465$	−27.0000	−1.25210
$466$	−9.00000	−0.416917
$467$	−29.0000	−1.34196	−0.670980	−	0.741475i	$-0.734126\pi$
$467$	−29.0000	−1.34196	−0.670980	+	0.741475i	$0.734126\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	27.0000	1.24542
$471$	7.00000	0.322543
$472$	13.0000	0.598374
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.00000	−0.183147
$478$	27.0000	1.23495
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	−3.00000	−0.136931
$481$	−4.00000	−0.182384
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−7.00000	−0.318182
$485$	36.0000	1.63468
$486$	−1.00000	−0.0453609
$487$	−31.0000	−1.40474	−0.702372	−	0.711810i	$-0.747876\pi$
$487$	−31.0000	−1.40474	−0.702372	+	0.711810i	$0.747876\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	5.00000	0.225417
$493$	−1.00000	−0.0450377
$494$	0	0
$495$	−6.00000	−0.269680
$496$	9.00000	0.404112
$497$	0	0
$498$	−7.00000	−0.313678
$499$	−39.0000	−1.74588	−0.872940	−	0.487828i	$-0.837789\pi$
$499$	−39.0000	−1.74588	−0.872940	+	0.487828i	$0.837789\pi$
$500$	−3.00000	−0.134164
$501$	−18.0000	−0.804181
$502$	−31.0000	−1.38360
$503$	−34.0000	−1.51599	−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000	−1.51599	−0.757993	+	0.652263i	$0.773820\pi$
$504$	0	0
$505$	−30.0000	−1.33498
$506$	−8.00000	−0.355643
$507$	12.0000	0.532939
$508$	8.00000	0.354943
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	3.00000	0.132842
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	42.0000	1.85074
$516$	0	0
$517$	−18.0000	−0.791639
$518$	0	0
$519$	9.00000	0.395056
$520$	3.00000	0.131559
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	1.00000	0.0437688
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	16.0000	0.697633
$527$	−9.00000	−0.392046
$528$	2.00000	0.0870388
$529$	−7.00000	−0.304348
$530$	−12.0000	−0.521247
$531$	13.0000	0.564152
$532$	0	0
$533$	−5.00000	−0.216574
$534$	−6.00000	−0.259645
$535$	−24.0000	−1.03761
$536$	−8.00000	−0.345547
$537$	3.00000	0.129460
$538$	9.00000	0.388018
$539$	0	0
$540$	−3.00000	−0.129099
$541$	16.0000	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$542$	30.0000	1.28861
$543$	12.0000	0.514969
$544$	−1.00000	−0.0428746
$545$	12.0000	0.514024
$546$	0	0
$547$	35.0000	1.49649	0.748246	−	0.663421i	$-0.230896\pi$
$547$	35.0000	1.49649	0.748246	+	0.663421i	$0.230896\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	−8.00000	−0.341121
$551$	0	0
$552$	−4.00000	−0.170251
$553$	0	0
$554$	24.0000	1.01966
$555$	12.0000	0.509372
$556$	8.00000	0.339276
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	9.00000	0.381000
$559$	0	0
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	−28.0000	−1.18111
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	−9.00000	−0.378968
$565$	−9.00000	−0.378633
$566$	15.0000	0.630497
$567$	0	0
$568$	12.0000	0.503509
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−2.00000	−0.0836242
$573$	3.00000	0.125327
$574$	0	0
$575$	16.0000	0.667246
$576$	1.00000	0.0416667
$577$	−31.0000	−1.29055	−0.645273	−	0.763952i	$-0.723257\pi$
$577$	−31.0000	−1.29055	−0.645273	+	0.763952i	$0.723257\pi$
$578$	1.00000	0.0415945
$579$	2.00000	0.0831172
$580$	3.00000	0.124568
$581$	0	0
$582$	−12.0000	−0.497416
$583$	8.00000	0.331326
$584$	10.0000	0.413803
$585$	3.00000	0.124035
$586$	−24.0000	−0.991431
$587$	−20.0000	−0.825488	−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000	−0.825488	−0.412744	+	0.910847i	$0.635430\pi$
$588$	0	0
$589$	0	0
$590$	39.0000	1.60560
$591$	−11.0000	−0.452480
$592$	−4.00000	−0.164399
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	6.00000	0.245770
$597$	25.0000	1.02318
$598$	4.00000	0.163572
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	−4.00000	−0.163299
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	−20.0000	−0.813788
$605$	−21.0000	−0.853771
$606$	10.0000	0.406222
$607$	−35.0000	−1.42061	−0.710303	−	0.703896i	$-0.751442\pi$
$607$	−35.0000	−1.42061	−0.710303	+	0.703896i	$0.751442\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.00000	0.364101
$612$	−1.00000	−0.0404226
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	4.00000	0.161427
$615$	15.0000	0.604858
$616$	0	0
$617$	−29.0000	−1.16750	−0.583748	−	0.811935i	$-0.698414\pi$
$617$	−29.0000	−1.16750	−0.583748	+	0.811935i	$0.698414\pi$
$618$	−14.0000	−0.563163
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	27.0000	1.08435
$621$	−4.00000	−0.160514
$622$	30.0000	1.20289
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	−29.0000	−1.16000
$626$	12.0000	0.479616
$627$	0	0
$628$	−7.00000	−0.279330
$629$	4.00000	0.159490
$630$	0	0
$631$	−18.0000	−0.716569	−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000	−0.716569	−0.358284	+	0.933613i	$0.616638\pi$
$632$	0	0
$633$	−23.0000	−0.914168
$634$	15.0000	0.595726
$635$	24.0000	0.952411
$636$	4.00000	0.158610
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	12.0000	0.474713
$640$	3.00000	0.118585
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	8.00000	0.315735
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	1.00000	0.0392837
$649$	−26.0000	−1.02059
$650$	4.00000	0.156893
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	−4.00000	−0.156412
$655$	−30.0000	−1.17220
$656$	−5.00000	−0.195217
$657$	10.0000	0.390137
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	6.00000	0.233550
$661$	−49.0000	−1.90588	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−49.0000	−1.90588	−0.952940	+	0.303160i	$0.901958\pi$
$662$	−26.0000	−1.01052
$663$	1.00000	0.0388368
$664$	7.00000	0.271653
$665$	0	0
$666$	−4.00000	−0.154997
$667$	4.00000	0.154881
$668$	18.0000	0.696441
$669$	−24.0000	−0.927894
$670$	−24.0000	−0.927201
$671$	0	0
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	2.00000	0.0770371
$675$	−4.00000	−0.153960
$676$	−12.0000	−0.461538
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	3.00000	0.115214
$679$	0	0
$680$	−3.00000	−0.115045
$681$	18.0000	0.689761
$682$	−18.0000	−0.689256
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−36.0000	−1.37549
$686$	0	0
$687$	−7.00000	−0.267067
$688$	0	0
$689$	−4.00000	−0.152388
$690$	−12.0000	−0.456832
$691$	−3.00000	−0.114125	−0.0570627	−	0.998371i	$-0.518173\pi$
$691$	−3.00000	−0.114125	−0.0570627	+	0.998371i	$0.518173\pi$
$692$	−9.00000	−0.342129
$693$	0	0
$694$	24.0000	0.911028
$695$	24.0000	0.910372
$696$	−1.00000	−0.0379049
$697$	5.00000	0.189389
$698$	11.0000	0.416356
$699$	9.00000	0.340411
$700$	0	0
$701$	44.0000	1.66186	0.830929	−	0.556379i	$-0.187810\pi$
$701$	44.0000	1.66186	0.830929	+	0.556379i	$0.187810\pi$
$702$	−1.00000	−0.0377426
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	−27.0000	−1.01688
$706$	32.0000	1.20434
$707$	0	0
$708$	−13.0000	−0.488570
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	36.0000	1.35106
$711$	0	0
$712$	6.00000	0.224860
$713$	36.0000	1.34821
$714$	0	0
$715$	−6.00000	−0.224387
$716$	−3.00000	−0.112115
$717$	−27.0000	−1.00833
$718$	25.0000	0.932992
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−19.0000	−0.707107
$723$	2.00000	0.0743808
$724$	−12.0000	−0.445976
$725$	4.00000	0.148556
$726$	7.00000	0.259794
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	30.0000	1.11035
$731$	0	0
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−24.0000	−0.885856
$735$	0	0
$736$	4.00000	0.147442
$737$	16.0000	0.589368
$738$	−5.00000	−0.184053
$739$	18.0000	0.662141	0.331070	−	0.943606i	$-0.392590\pi$
$739$	18.0000	0.662141	0.331070	+	0.943606i	$0.392590\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	−9.00000	−0.329956
$745$	18.0000	0.659469
$746$	−13.0000	−0.475964
$747$	7.00000	0.256117
$748$	2.00000	0.0731272
$749$	0	0
$750$	3.00000	0.109545
$751$	−15.0000	−0.547358	−0.273679	−	0.961821i	$-0.588241\pi$
$751$	−15.0000	−0.547358	−0.273679	+	0.961821i	$0.588241\pi$
$752$	9.00000	0.328196
$753$	31.0000	1.12970
$754$	1.00000	0.0364179
$755$	−60.0000	−2.18362
$756$	0	0
$757$	−21.0000	−0.763258	−0.381629	−	0.924316i	$-0.624637\pi$
$757$	−21.0000	−0.763258	−0.381629	+	0.924316i	$0.624637\pi$
$758$	−16.0000	−0.581146
$759$	8.00000	0.290382
$760$	0	0
$761$	−8.00000	−0.290000	−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000	−0.290000	−0.145000	+	0.989432i	$0.546318\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	−3.00000	−0.108536
$765$	−3.00000	−0.108465
$766$	−24.0000	−0.867155
$767$	13.0000	0.469403
$768$	−1.00000	−0.0360844
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	−2.00000	−0.0719816
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	36.0000	1.29316
$776$	12.0000	0.430775
$777$	0	0
$778$	6.00000	0.215110
$779$	0	0
$780$	−3.00000	−0.107417
$781$	−24.0000	−0.858788
$782$	−4.00000	−0.143040
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−21.0000	−0.749522
$786$	10.0000	0.356688
$787$	−17.0000	−0.605985	−0.302992	−	0.952993i	$-0.597986\pi$
$787$	−17.0000	−0.605985	−0.302992	+	0.952993i	$0.597986\pi$
$788$	11.0000	0.391859
$789$	−16.0000	−0.569615
$790$	0	0
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	0	0
$794$	−4.00000	−0.141955
$795$	12.0000	0.425596
$796$	−25.0000	−0.886102
$797$	52.0000	1.84193	0.920967	−	0.389640i	$-0.127401\pi$
$797$	52.0000	1.84193	0.920967	+	0.389640i	$0.127401\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	4.00000	0.141421
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	−20.0000	−0.705785
$804$	8.00000	0.282138
$805$	0	0
$806$	9.00000	0.317011
$807$	−9.00000	−0.316815
$808$	−10.0000	−0.351799
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	3.00000	0.105409
$811$	−17.0000	−0.596951	−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	0	0
$813$	−30.0000	−1.05215
$814$	8.00000	0.280400
$815$	−24.0000	−0.840683
$816$	1.00000	0.0350070
$817$	0	0
$818$	−22.0000	−0.769212
$819$	0	0
$820$	−15.0000	−0.523823
$821$	34.0000	1.18661	0.593304	−	0.804978i	$-0.297823\pi$
$821$	34.0000	1.18661	0.593304	+	0.804978i	$0.297823\pi$
$822$	12.0000	0.418548
$823$	15.0000	0.522867	0.261434	−	0.965221i	$-0.415805\pi$
$823$	15.0000	0.522867	0.261434	+	0.965221i	$0.415805\pi$
$824$	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	$0.375061\pi$
$828$	4.00000	0.139010
$829$	50.0000	1.73657	0.868286	−	0.496064i	$-0.165222\pi$
$829$	50.0000	1.73657	0.868286	+	0.496064i	$0.165222\pi$
$830$	21.0000	0.728921
$831$	−24.0000	−0.832551
$832$	1.00000	0.0346688
$833$	0	0
$834$	−8.00000	−0.277017
$835$	54.0000	1.86875
$836$	0	0
$837$	−9.00000	−0.311086
$838$	4.00000	0.138178
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	9.00000	0.310160
$843$	28.0000	0.964371
$844$	23.0000	0.791693
$845$	−36.0000	−1.23844
$846$	9.00000	0.309426
$847$	0	0
$848$	−4.00000	−0.137361
$849$	−15.0000	−0.514799
$850$	−4.00000	−0.137199
$851$	−16.0000	−0.548473
$852$	−12.0000	−0.411113
$853$	−4.00000	−0.136957	−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000	−0.136957	−0.0684787	+	0.997653i	$0.521815\pi$
$854$	0	0
$855$	0	0
$856$	−8.00000	−0.273434
$857$	−7.00000	−0.239115	−0.119558	−	0.992827i	$-0.538148\pi$
$857$	−7.00000	−0.239115	−0.119558	+	0.992827i	$0.538148\pi$
$858$	2.00000	0.0682789
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	−1.00000	−0.0340207
$865$	−27.0000	−0.918028
$866$	−37.0000	−1.25731
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	0	0
$870$	−3.00000	−0.101710
$871$	−8.00000	−0.271070
$872$	4.00000	0.135457
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	−10.0000	−0.337869
$877$	−58.0000	−1.95852	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−58.0000	−1.95852	−0.979260	+	0.202606i	$0.935059\pi$
$878$	−5.00000	−0.168742
$879$	24.0000	0.809500
$880$	−6.00000	−0.202260
$881$	−1.00000	−0.0336909	−0.0168454	−	0.999858i	$-0.505362\pi$
$881$	−1.00000	−0.0336909	−0.0168454	+	0.999858i	$0.505362\pi$
$882$	0	0
$883$	−56.0000	−1.88455	−0.942275	−	0.334840i	$-0.891318\pi$
$883$	−56.0000	−1.88455	−0.942275	+	0.334840i	$0.891318\pi$
$884$	−1.00000	−0.0336336
$885$	−39.0000	−1.31097
$886$	−25.0000	−0.839891
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	4.00000	0.134231
$889$	0	0
$890$	18.0000	0.603361
$891$	−2.00000	−0.0670025
$892$	24.0000	0.803579
$893$	0	0
$894$	−6.00000	−0.200670
$895$	−9.00000	−0.300837
$896$	0	0
$897$	−4.00000	−0.133556
$898$	−10.0000	−0.333704
$899$	9.00000	0.300167
$900$	4.00000	0.133333
$901$	4.00000	0.133259
$902$	10.0000	0.332964
$903$	0	0
$904$	−3.00000	−0.0997785
$905$	−36.0000	−1.19668
$906$	20.0000	0.664455
$907$	−5.00000	−0.166022	−0.0830111	−	0.996549i	$-0.526454\pi$
$907$	−5.00000	−0.166022	−0.0830111	+	0.996549i	$0.526454\pi$
$908$	−18.0000	−0.597351
$909$	−10.0000	−0.331679
$910$	0	0
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	0	0
$913$	−14.0000	−0.463332
$914$	−7.00000	−0.231539
$915$	0	0
$916$	7.00000	0.231287
$917$	0	0
$918$	1.00000	0.0330049
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	12.0000	0.395628
$921$	−4.00000	−0.131804
$922$	20.0000	0.658665
$923$	12.0000	0.394985
$924$	0	0
$925$	−16.0000	−0.526077
$926$	14.0000	0.460069
$927$	14.0000	0.459820
$928$	1.00000	0.0328266
$929$	−23.0000	−0.754606	−0.377303	−	0.926090i	$-0.623148\pi$
$929$	−23.0000	−0.754606	−0.377303	+	0.926090i	$0.623148\pi$
$930$	−27.0000	−0.885365
$931$	0	0
$932$	−9.00000	−0.294805
$933$	−30.0000	−0.982156
$934$	−29.0000	−0.948909
$935$	6.00000	0.196221
$936$	1.00000	0.0326860
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	27.0000	0.880643
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	7.00000	0.228072
$943$	−20.0000	−0.651290
$944$	13.0000	0.423114
$945$	0	0
$946$	0	0
$947$	10.0000	0.324956	0.162478	−	0.986712i	$-0.448051\pi$
$947$	10.0000	0.324956	0.162478	+	0.986712i	$0.448051\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	−15.0000	−0.486408
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	−4.00000	−0.129505
$955$	−9.00000	−0.291233
$956$	27.0000	0.873242
$957$	2.00000	0.0646508
$958$	−6.00000	−0.193851
$959$	0	0
$960$	−3.00000	−0.0968246
$961$	50.0000	1.61290
$962$	−4.00000	−0.128965
$963$	−8.00000	−0.257796
$964$	−2.00000	−0.0644157
$965$	−6.00000	−0.193147
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	36.0000	1.15589
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−31.0000	−0.993304
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−16.0000	−0.511885	−0.255943	−	0.966692i	$-0.582386\pi$
$977$	−16.0000	−0.511885	−0.255943	+	0.966692i	$0.582386\pi$
$978$	8.00000	0.255812
$979$	−12.0000	−0.383522
$980$	0	0
$981$	4.00000	0.127710
$982$	20.0000	0.638226
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	5.00000	0.159394
$985$	33.0000	1.05147
$986$	−1.00000	−0.0318465
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−6.00000	−0.190693
$991$	−51.0000	−1.62007	−0.810034	−	0.586383i	$-0.800552\pi$
$991$	−51.0000	−1.62007	−0.810034	+	0.586383i	$0.800552\pi$
$992$	9.00000	0.285750
$993$	26.0000	0.825085
$994$	0	0
$995$	−75.0000	−2.37766
$996$	−7.00000	−0.221803
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−39.0000	−1.23452
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4998.2.a.bf.1.1		1
7.2	even	3		714.2.i.c.613.1	yes	2
7.4	even	3		714.2.i.c.205.1	✓	2
7.6	odd	2		4998.2.a.bh.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.i.c.205.1	✓	2	7.4	even	3
714.2.i.c.613.1	yes	2	7.2	even	3
4998.2.a.bf.1.1		1	1.1	even	1	trivial
4998.2.a.bh.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4998.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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