LMFDB - Embedded newform 4998.2.a.x.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 102)
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)

$f(q)$

$=$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} +1.00000 q^{12} +6.00000 q^{13} +4.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +4.00000 q^{20} +6.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -6.00000 q^{26} +1.00000 q^{27} -4.00000 q^{29} -4.00000 q^{30} +6.00000 q^{31} -1.00000 q^{32} -1.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +4.00000 q^{38} +6.00000 q^{39} -4.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{45} -6.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} -11.0000 q^{50} +1.00000 q^{51} +6.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -4.00000 q^{57} +4.00000 q^{58} -12.0000 q^{59} +4.00000 q^{60} +4.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} +24.0000 q^{65} -12.0000 q^{67} +1.00000 q^{68} +6.00000 q^{69} -6.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +4.00000 q^{74} +11.0000 q^{75} -4.00000 q^{76} -6.00000 q^{78} +10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000 q^{83} +4.00000 q^{85} +4.00000 q^{86} -4.00000 q^{87} +2.00000 q^{89} -4.00000 q^{90} +6.00000 q^{92} +6.00000 q^{93} +4.00000 q^{94} -16.0000 q^{95} -1.00000 q^{96} -6.00000 q^{97} +O(q^{100})\)

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−4.00000	−1.26491
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−1.00000	−0.204124
$25$	11.0000	2.20000
$26$	−6.00000	−1.17670
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−4.00000	−0.730297
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−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	$0.606644\pi$
$38$	4.00000	0.648886
$39$	6.00000	0.960769
$40$	−4.00000	−0.632456
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	−6.00000	−0.884652
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−11.0000	−1.55563
$51$	1.00000	0.140028
$52$	6.00000	0.832050
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	4.00000	0.525226
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	4.00000	0.516398
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	24.0000	2.97683
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	1.00000	0.121268
$69$	6.00000	0.722315
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	4.00000	0.464991
$75$	11.0000	1.27017
$76$	−4.00000	−0.458831
$77$	0	0
$78$	−6.00000	−0.679366
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	4.00000	0.447214
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	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
$85$	4.00000	0.433861
$86$	4.00000	0.431331
$87$	−4.00000	−0.428845
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	−4.00000	−0.421637
$91$	0	0
$92$	6.00000	0.625543
$93$	6.00000	0.622171
$94$	4.00000	0.412568
$95$	−16.0000	−1.64157
$96$	−1.00000	−0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	0	0
$100$	11.0000	1.10000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	−1.00000	−0.0990148
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	4.00000	0.374634
$115$	24.0000	2.23801
$116$	−4.00000	−0.371391
$117$	6.00000	0.554700
$118$	12.0000	1.10469
$119$	0	0
$120$	−4.00000	−0.365148
$121$	−11.0000	−1.00000
$122$	−4.00000	−0.362143
$123$	10.0000	0.901670
$124$	6.00000	0.538816
$125$	24.0000	2.14663
$126$	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
$129$	−4.00000	−0.352180
$130$	−24.0000	−2.10494
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	0	0
$134$	12.0000	1.03664
$135$	4.00000	0.344265
$136$	−1.00000	−0.0857493
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−6.00000	−0.510754
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	6.00000	0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	−16.0000	−1.32873
$146$	2.00000	0.165521
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−11.0000	−0.898146
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	4.00000	0.324443
$153$	1.00000	0.0808452
$154$	0	0
$155$	24.0000	1.92773
$156$	6.00000	0.480384
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−10.0000	−0.795557
$159$	−2.00000	−0.158610
$160$	−4.00000	−0.316228
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−12.0000	−0.931381
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−4.00000	−0.306786
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	0	0
$177$	−12.0000	−0.901975
$178$	−2.00000	−0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	4.00000	0.298142
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	−6.00000	−0.442326
$185$	−16.0000	−1.17634
$186$	−6.00000	−0.439941
$187$	0	0
$188$	−4.00000	−0.291730
$189$	0	0
$190$	16.0000	1.16076
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	1.00000	0.0721688
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	6.00000	0.430775
$195$	24.0000	1.71868
$196$	0	0
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	−11.0000	−0.777817
$201$	−12.0000	−0.846415
$202$	14.0000	0.985037
$203$	0	0
$204$	1.00000	0.0700140
$205$	40.0000	2.79372
$206$	4.00000	0.278693
$207$	6.00000	0.417029
$208$	6.00000	0.416025
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−2.00000	−0.137361
$213$	−6.00000	−0.411113
$214$	0	0
$215$	−16.0000	−1.09119
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−16.0000	−1.08366
$219$	−2.00000	−0.135147
$220$	0	0
$221$	6.00000	0.403604
$222$	4.00000	0.268462
$223$	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
$225$	11.0000	0.733333
$226$	−2.00000	−0.133038
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−4.00000	−0.264906
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−24.0000	−1.58251
$231$	0	0
$232$	4.00000	0.262613
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−6.00000	−0.392232
$235$	−16.0000	−1.04372
$236$	−12.0000	−0.781133
$237$	10.0000	0.649570
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	4.00000	0.258199
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	11.0000	0.707107
$243$	1.00000	0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−10.0000	−0.637577
$247$	−24.0000	−1.52708
$248$	−6.00000	−0.381000
$249$	12.0000	0.760469
$250$	−24.0000	−1.51789
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	24.0000	1.48842
$261$	−4.00000	−0.247594
$262$	−16.0000	−0.988483
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	0	0
$267$	2.00000	0.122398
$268$	−12.0000	−0.733017
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	−4.00000	−0.243432
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	6.00000	0.361158
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	8.00000	0.479808
$279$	6.00000	0.359211
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	4.00000	0.238197
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	−6.00000	−0.356034
$285$	−16.0000	−0.947758
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	16.0000	0.939552
$291$	−6.00000	−0.351726
$292$	−2.00000	−0.117041
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	−48.0000	−2.79467
$296$	4.00000	0.232495
$297$	0	0
$298$	6.00000	0.347571
$299$	36.0000	2.08193
$300$	11.0000	0.635085
$301$	0	0
$302$	24.0000	1.38104
$303$	−14.0000	−0.804279
$304$	−4.00000	−0.229416
$305$	16.0000	0.916157
$306$	−1.00000	−0.0571662
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	−24.0000	−1.36311
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	−6.00000	−0.339683
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	10.0000	0.562544
$317$	−16.0000	−0.898650	−0.449325	−	0.893368i	$-0.648335\pi$
$317$	−16.0000	−0.898650	−0.449325	+	0.893368i	$0.648335\pi$
$318$	2.00000	0.112154
$319$	0	0
$320$	4.00000	0.223607
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	1.00000	0.0555556
$325$	66.0000	3.66102
$326$	−12.0000	−0.664619
$327$	16.0000	0.884802
$328$	−10.0000	−0.552158
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	12.0000	0.658586
$333$	−4.00000	−0.219199
$334$	−2.00000	−0.109435
$335$	−48.0000	−2.62252
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−23.0000	−1.25104
$339$	2.00000	0.108625
$340$	4.00000	0.216930
$341$	0	0
$342$	4.00000	0.216295
$343$	0	0
$344$	4.00000	0.215666
$345$	24.0000	1.29212
$346$	−4.00000	−0.215041
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−4.00000	−0.214423
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	12.0000	0.637793
$355$	−24.0000	−1.27379
$356$	2.00000	0.106000
$357$	0	0
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−4.00000	−0.210819
$361$	−3.00000	−0.157895
$362$	−20.0000	−1.05118
$363$	−11.0000	−0.577350
$364$	0	0
$365$	−8.00000	−0.418739
$366$	−4.00000	−0.209083
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	6.00000	0.312772
$369$	10.0000	0.520579
$370$	16.0000	0.831800
$371$	0	0
$372$	6.00000	0.311086
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	4.00000	0.206284
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−16.0000	−0.820783
$381$	8.00000	0.409852
$382$	4.00000	0.204658
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	−4.00000	−0.203331
$388$	−6.00000	−0.304604
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	−24.0000	−1.21529
$391$	6.00000	0.303433
$392$	0	0
$393$	16.0000	0.807093
$394$	−8.00000	−0.403034
$395$	40.0000	2.01262
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	11.0000	0.550000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	12.0000	0.598506
$403$	36.0000	1.79329
$404$	−14.0000	−0.696526
$405$	4.00000	0.198762
$406$	0	0
$407$	0	0
$408$	−1.00000	−0.0495074
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−40.0000	−1.97546
$411$	−6.00000	−0.295958
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−6.00000	−0.294884
$415$	48.0000	2.35623
$416$	−6.00000	−0.294174
$417$	−8.00000	−0.391762
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	−8.00000	−0.389434
$423$	−4.00000	−0.194487
$424$	2.00000	0.0971286
$425$	11.0000	0.533578
$426$	6.00000	0.290701
$427$	0	0
$428$	0	0
$429$	0	0
$430$	16.0000	0.771589
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	1.00000	0.0481125
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	−16.0000	−0.767141
$436$	16.0000	0.766261
$437$	−24.0000	−1.14808
$438$	2.00000	0.0955637
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	0	0
$442$	−6.00000	−0.285391
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−4.00000	−0.189832
$445$	8.00000	0.379236
$446$	−4.00000	−0.189405
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	−11.0000	−0.518545
$451$	0	0
$452$	2.00000	0.0940721
$453$	−24.0000	−1.12762
$454$	4.00000	0.187729
$455$	0	0
$456$	4.00000	0.187317
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−2.00000	−0.0934539
$459$	1.00000	0.0466760
$460$	24.0000	1.11901
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−4.00000	−0.185695
$465$	24.0000	1.11297
$466$	6.00000	0.277945
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	16.0000	0.738025
$471$	−6.00000	−0.276465
$472$	12.0000	0.552345
$473$	0	0
$474$	−10.0000	−0.459315
$475$	−44.0000	−2.01886
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	20.0000	0.914779
$479$	−10.0000	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$480$	−4.00000	−0.182574
$481$	−24.0000	−1.09431
$482$	−18.0000	−0.819878
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−24.0000	−1.08978
$486$	−1.00000	−0.0453609
$487$	−38.0000	−1.72194	−0.860972	−	0.508652i	$-0.830144\pi$
$487$	−38.0000	−1.72194	−0.860972	+	0.508652i	$0.830144\pi$
$488$	−4.00000	−0.181071
$489$	12.0000	0.542659
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	10.0000	0.450835
$493$	−4.00000	−0.180151
$494$	24.0000	1.07981
$495$	0	0
$496$	6.00000	0.269408
$497$	0	0
$498$	−12.0000	−0.537733
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	24.0000	1.07331
$501$	2.00000	0.0893534
$502$	12.0000	0.535586
$503$	26.0000	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$503$	26.0000	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$504$	0	0
$505$	−56.0000	−2.49197
$506$	0	0
$507$	23.0000	1.02147
$508$	8.00000	0.354943
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	−4.00000	−0.177123
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	18.0000	0.793946
$515$	−16.0000	−0.705044
$516$	−4.00000	−0.176090
$517$	0	0
$518$	0	0
$519$	4.00000	0.175581
$520$	−24.0000	−1.05247
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	4.00000	0.175075
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	12.0000	0.523225
$527$	6.00000	0.261364
$528$	0	0
$529$	13.0000	0.565217
$530$	8.00000	0.347498
$531$	−12.0000	−0.520756
$532$	0	0
$533$	60.0000	2.59889
$534$	−2.00000	−0.0865485
$535$	0	0
$536$	12.0000	0.518321
$537$	−12.0000	−0.517838
$538$	−12.0000	−0.517357
$539$	0	0
$540$	4.00000	0.172133
$541$	−12.0000	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$541$	−12.0000	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$542$	−16.0000	−0.687259
$543$	20.0000	0.858282
$544$	−1.00000	−0.0428746
$545$	64.0000	2.74146
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−6.00000	−0.256307
$549$	4.00000	0.170716
$550$	0	0
$551$	16.0000	0.681623
$552$	−6.00000	−0.255377
$553$	0	0
$554$	8.00000	0.339887
$555$	−16.0000	−0.679162
$556$	−8.00000	−0.339276
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	−6.00000	−0.254000
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	−4.00000	−0.168430
$565$	8.00000	0.336563
$566$	32.0000	1.34506
$567$	0	0
$568$	6.00000	0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	16.0000	0.670166
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	0	0
$575$	66.0000	2.75239
$576$	1.00000	0.0416667
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	−1.00000	−0.0415945
$579$	6.00000	0.249351
$580$	−16.0000	−0.664364
$581$	0	0
$582$	6.00000	0.248708
$583$	0	0
$584$	2.00000	0.0827606
$585$	24.0000	0.992278
$586$	2.00000	0.0826192
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	48.0000	1.97613
$591$	8.00000	0.329076
$592$	−4.00000	−0.164399
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−14.0000	−0.572982
$598$	−36.0000	−1.47215
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	−11.0000	−0.449073
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	−24.0000	−0.976546
$605$	−44.0000	−1.78885
$606$	14.0000	0.568711
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−16.0000	−0.647821
$611$	−24.0000	−0.970936
$612$	1.00000	0.0404226
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	−12.0000	−0.484281
$615$	40.0000	1.61296
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	4.00000	0.160904
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	24.0000	0.963863
$621$	6.00000	0.240772
$622$	30.0000	1.20289
$623$	0	0
$624$	6.00000	0.240192
$625$	41.0000	1.64000
$626$	−26.0000	−1.03917
$627$	0	0
$628$	−6.00000	−0.239426
$629$	−4.00000	−0.159490
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−10.0000	−0.397779
$633$	8.00000	0.317971
$634$	16.0000	0.635441
$635$	32.0000	1.26988
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	0	0
$639$	−6.00000	−0.237356
$640$	−4.00000	−0.158114
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	4.00000	0.157378
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−66.0000	−2.58873
$651$	0	0
$652$	12.0000	0.469956
$653$	20.0000	0.782660	0.391330	−	0.920250i	$-0.372015\pi$
$653$	20.0000	0.782660	0.391330	+	0.920250i	$0.372015\pi$
$654$	−16.0000	−0.625650
$655$	64.0000	2.50069
$656$	10.0000	0.390434
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−20.0000	−0.777322
$663$	6.00000	0.233021
$664$	−12.0000	−0.465690
$665$	0	0
$666$	4.00000	0.154997
$667$	−24.0000	−0.929284
$668$	2.00000	0.0773823
$669$	4.00000	0.154649
$670$	48.0000	1.85440
$671$	0	0
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	6.00000	0.231111
$675$	11.0000	0.423390
$676$	23.0000	0.884615
$677$	−8.00000	−0.307465	−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000	−0.307465	−0.153732	+	0.988113i	$0.549129\pi$
$678$	−2.00000	−0.0768095
$679$	0	0
$680$	−4.00000	−0.153393
$681$	−4.00000	−0.153280
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−4.00000	−0.152944
$685$	−24.0000	−0.916993
$686$	0	0
$687$	2.00000	0.0763048
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	−24.0000	−0.913664
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	4.00000	0.151838
$695$	−32.0000	−1.21383
$696$	4.00000	0.151620
$697$	10.0000	0.378777
$698$	−30.0000	−1.13552
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	−6.00000	−0.226455
$703$	16.0000	0.603451
$704$	0	0
$705$	−16.0000	−0.602595
$706$	14.0000	0.526897
$707$	0	0
$708$	−12.0000	−0.450988
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	24.0000	0.900704
$711$	10.0000	0.375029
$712$	−2.00000	−0.0749532
$713$	36.0000	1.34821
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−20.0000	−0.746914
$718$	0	0
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	4.00000	0.149071
$721$	0	0
$722$	3.00000	0.111648
$723$	18.0000	0.669427
$724$	20.0000	0.743294
$725$	−44.0000	−1.63412
$726$	11.0000	0.408248
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	8.00000	0.296093
$731$	−4.00000	−0.147945
$732$	4.00000	0.147844
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	−6.00000	−0.221163
$737$	0	0
$738$	−10.0000	−0.368105
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	−16.0000	−0.588172
$741$	−24.0000	−0.881662
$742$	0	0
$743$	38.0000	1.39408	0.697042	−	0.717030i	$-0.254499\pi$
$743$	38.0000	1.39408	0.697042	+	0.717030i	$0.254499\pi$
$744$	−6.00000	−0.219971
$745$	−24.0000	−0.879292
$746$	14.0000	0.512576
$747$	12.0000	0.439057
$748$	0	0
$749$	0	0
$750$	−24.0000	−0.876356
$751$	−34.0000	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$751$	−34.0000	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$752$	−4.00000	−0.145865
$753$	−12.0000	−0.437304
$754$	24.0000	0.874028
$755$	−96.0000	−3.49380
$756$	0	0
$757$	14.0000	0.508839	0.254419	−	0.967094i	$-0.418116\pi$
$757$	14.0000	0.508839	0.254419	+	0.967094i	$0.418116\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	16.0000	0.580381
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	−4.00000	−0.144715
$765$	4.00000	0.144620
$766$	28.0000	1.01168
$767$	−72.0000	−2.59977
$768$	1.00000	0.0360844
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	6.00000	0.215945
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	4.00000	0.143777
$775$	66.0000	2.37079
$776$	6.00000	0.215387
$777$	0	0
$778$	14.0000	0.501924
$779$	−40.0000	−1.43315
$780$	24.0000	0.859338
$781$	0	0
$782$	−6.00000	−0.214560
$783$	−4.00000	−0.142948
$784$	0	0
$785$	−24.0000	−0.856597
$786$	−16.0000	−0.570701
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	8.00000	0.284988
$789$	−12.0000	−0.427211
$790$	−40.0000	−1.42314
$791$	0	0
$792$	0	0
$793$	24.0000	0.852265
$794$	20.0000	0.709773
$795$	−8.00000	−0.283731
$796$	−14.0000	−0.496217
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	−11.0000	−0.388909
$801$	2.00000	0.0706665
$802$	−30.0000	−1.05934
$803$	0	0
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−36.0000	−1.26805
$807$	12.0000	0.422420
$808$	14.0000	0.492518
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	−4.00000	−0.140546
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	48.0000	1.68137
$816$	1.00000	0.0350070
$817$	16.0000	0.559769
$818$	−26.0000	−0.909069
$819$	0	0
$820$	40.0000	1.39686
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	6.00000	0.209274
$823$	18.0000	0.627441	0.313720	−	0.949515i	$-0.398425\pi$
$823$	18.0000	0.627441	0.313720	+	0.949515i	$0.398425\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	16.0000	0.556375	0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000	0.556375	0.278187	+	0.960527i	$0.410266\pi$
$828$	6.00000	0.208514
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	−48.0000	−1.66610
$831$	−8.00000	−0.277517
$832$	6.00000	0.208013
$833$	0	0
$834$	8.00000	0.277017
$835$	8.00000	0.276851
$836$	0	0
$837$	6.00000	0.207390
$838$	−12.0000	−0.414533
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−34.0000	−1.17172
$843$	−18.0000	−0.619953
$844$	8.00000	0.275371
$845$	92.0000	3.16490
$846$	4.00000	0.137523
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	−32.0000	−1.09824
$850$	−11.0000	−0.377297
$851$	−24.0000	−0.822709
$852$	−6.00000	−0.205557
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	0	0
$855$	−16.0000	−0.547188
$856$	0	0
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−16.0000	−0.545595
$861$	0	0
$862$	−14.0000	−0.476842
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	−1.00000	−0.0340207
$865$	16.0000	0.544016
$866$	−18.0000	−0.611665
$867$	1.00000	0.0339618
$868$	0	0
$869$	0	0
$870$	16.0000	0.542451
$871$	−72.0000	−2.43963
$872$	−16.0000	−0.541828
$873$	−6.00000	−0.203069
$874$	24.0000	0.811812
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	−24.0000	−0.810422	−0.405211	−	0.914223i	$-0.632802\pi$
$877$	−24.0000	−0.810422	−0.405211	+	0.914223i	$0.632802\pi$
$878$	−10.0000	−0.337484
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	−50.0000	−1.68454	−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000	−1.68454	−0.842271	+	0.539054i	$0.818782\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	6.00000	0.201802
$885$	−48.0000	−1.61350
$886$	−12.0000	−0.403148
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	4.00000	0.134231
$889$	0	0
$890$	−8.00000	−0.268161
$891$	0	0
$892$	4.00000	0.133930
$893$	16.0000	0.535420
$894$	6.00000	0.200670
$895$	−48.0000	−1.60446
$896$	0	0
$897$	36.0000	1.20201
$898$	26.0000	0.867631
$899$	−24.0000	−0.800445
$900$	11.0000	0.366667
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	80.0000	2.65929
$906$	24.0000	0.797347
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	−4.00000	−0.132745
$909$	−14.0000	−0.464351
$910$	0	0
$911$	26.0000	0.861418	0.430709	−	0.902491i	$-0.358263\pi$
$911$	26.0000	0.861418	0.430709	+	0.902491i	$0.358263\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	−22.0000	−0.727695
$915$	16.0000	0.528944
$916$	2.00000	0.0660819
$917$	0	0
$918$	−1.00000	−0.0330049
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−24.0000	−0.791257
$921$	12.0000	0.395413
$922$	−10.0000	−0.329332
$923$	−36.0000	−1.18495
$924$	0	0
$925$	−44.0000	−1.44671
$926$	4.00000	0.131448
$927$	−4.00000	−0.131377
$928$	4.00000	0.131306
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	−24.0000	−0.786991
$931$	0	0
$932$	−6.00000	−0.196537
$933$	−30.0000	−0.982156
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−6.00000	−0.196116
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	−16.0000	−0.521862
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	6.00000	0.195491
$943$	60.0000	1.95387
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	10.0000	0.324785
$949$	−12.0000	−0.389536
$950$	44.0000	1.42755
$951$	−16.0000	−0.518836
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	2.00000	0.0647524
$955$	−16.0000	−0.517748
$956$	−20.0000	−0.646846
$957$	0	0
$958$	10.0000	0.323085
$959$	0	0
$960$	4.00000	0.129099
$961$	5.00000	0.161290
$962$	24.0000	0.773791
$963$	0	0
$964$	18.0000	0.579741
$965$	24.0000	0.772587
$966$	0	0
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	11.0000	0.353553
$969$	−4.00000	−0.128499
$970$	24.0000	0.770594
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	38.0000	1.21760
$975$	66.0000	2.11369
$976$	4.00000	0.128037
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	−12.0000	−0.383718
$979$	0	0
$980$	0	0
$981$	16.0000	0.510841
$982$	20.0000	0.638226
$983$	38.0000	1.21201	0.606006	−	0.795460i	$-0.292771\pi$
$983$	38.0000	1.21201	0.606006	+	0.795460i	$0.292771\pi$
$984$	−10.0000	−0.318788
$985$	32.0000	1.01960
$986$	4.00000	0.127386
$987$	0	0
$988$	−24.0000	−0.763542
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	−6.00000	−0.190500
$993$	20.0000	0.634681
$994$	0	0
$995$	−56.0000	−1.77532
$996$	12.0000	0.380235
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	−32.0000	−1.01294
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4998.2.a.x.1.1		1
7.6	odd	2		102.2.a.a.1.1	✓	1
21.20	even	2		306.2.a.d.1.1		1
28.27	even	2		816.2.a.h.1.1		1
35.13	even	4		2550.2.d.q.2449.2		2
35.27	even	4		2550.2.d.q.2449.1		2
35.34	odd	2		2550.2.a.be.1.1		1
56.13	odd	2		3264.2.a.bf.1.1		1
56.27	even	2		3264.2.a.p.1.1		1
84.83	odd	2		2448.2.a.t.1.1		1
105.104	even	2		7650.2.a.z.1.1		1
119.13	odd	4		1734.2.b.d.577.1		2
119.55	odd	4		1734.2.b.d.577.2		2
119.76	odd	8		1734.2.f.g.829.2		4
119.83	odd	8		1734.2.f.g.1483.2		4
119.104	odd	8		1734.2.f.g.1483.1		4
119.111	odd	8		1734.2.f.g.829.1		4
119.118	odd	2		1734.2.a.h.1.1		1
168.83	odd	2		9792.2.a.b.1.1		1
168.125	even	2		9792.2.a.a.1.1		1
357.356	even	2		5202.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.a.a.1.1	✓	1	7.6	odd	2
306.2.a.d.1.1		1	21.20	even	2
816.2.a.h.1.1		1	28.27	even	2
1734.2.a.h.1.1		1	119.118	odd	2
1734.2.b.d.577.1		2	119.13	odd	4
1734.2.b.d.577.2		2	119.55	odd	4
1734.2.f.g.829.1		4	119.111	odd	8
1734.2.f.g.829.2		4	119.76	odd	8
1734.2.f.g.1483.1		4	119.104	odd	8
1734.2.f.g.1483.2		4	119.83	odd	8
2448.2.a.t.1.1		1	84.83	odd	2
2550.2.a.be.1.1		1	35.34	odd	2
2550.2.d.q.2449.1		2	35.27	even	4
2550.2.d.q.2449.2		2	35.13	even	4
3264.2.a.p.1.1		1	56.27	even	2
3264.2.a.bf.1.1		1	56.13	odd	2
4998.2.a.x.1.1		1	1.1	even	1	trivial
5202.2.a.g.1.1		1	357.356	even	2
7650.2.a.z.1.1		1	105.104	even	2
9792.2.a.a.1.1		1	168.125	even	2
9792.2.a.b.1.1		1	168.83	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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