LMFDB - Embedded newform 7098.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} +6.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{30} -1.00000 q^{32} -2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{40} -1.00000 q^{42} -4.00000 q^{43} -2.00000 q^{45} -6.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} +4.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} -6.00000 q^{59} -2.00000 q^{60} +12.0000 q^{61} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{67} +2.00000 q^{68} +6.00000 q^{69} +2.00000 q^{70} -1.00000 q^{72} -14.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{80} +1.00000 q^{81} +14.0000 q^{83} +1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} +4.00000 q^{89} +2.00000 q^{90} +6.00000 q^{92} -8.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−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	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	1.00000	0.218218
$22$	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
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	2.00000	0.365148
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	2.00000	0.316228
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−1.00000	−0.154303
$43$	−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
$45$	−2.00000	−0.298142
$46$	−6.00000	−0.884652
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	2.00000	0.280056
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−2.00000	−0.258199
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	2.00000	0.242536
$69$	6.00000	0.722315
$70$	2.00000	0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	0	0
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	1.00000	0.109109
$85$	−4.00000	−0.433861
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	−8.00000	−0.825137
$95$	8.00000	0.820783
$96$	−1.00000	−0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	−2.00000	−0.198030
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	−4.00000	−0.388514
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	1.00000	0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	4.00000	0.374634
$115$	−12.0000	−1.11901
$116$	0	0
$117$	0	0
$118$	6.00000	0.552345
$119$	2.00000	0.183340
$120$	2.00000	0.182574
$121$	−11.0000	−1.00000
$122$	−12.0000	−1.08643
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	−1.00000	−0.0890871
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	−2.00000	−0.172774
$135$	−2.00000	−0.172133
$136$	−2.00000	−0.171499
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−6.00000	−0.510754
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	14.0000	1.15865
$147$	1.00000	0.0824786
$148$	−2.00000	−0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	4.00000	0.324443
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	2.00000	0.158114
$161$	6.00000	0.472866
$162$	−1.00000	−0.0785674
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	0	0
$165$	0	0
$166$	−14.0000	−1.08661
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	4.00000	0.306786
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−6.00000	−0.450988
$178$	−4.00000	−0.299813
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−2.00000	−0.149071
$181$	−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
$183$	12.0000	0.887066
$184$	−6.00000	−0.442326
$185$	4.00000	0.294086
$186$	0	0
$187$	0	0
$188$	8.00000	0.583460
$189$	1.00000	0.0727393
$190$	−8.00000	−0.580381
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	1.00000	0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	1.00000	0.0707107
$201$	2.00000	0.141069
$202$	2.00000	0.140720
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	14.0000	0.975426
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	2.00000	0.138013
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	12.0000	0.820303
$215$	8.00000	0.545595
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−6.00000	−0.406371
$219$	−14.0000	−0.946032
$220$	0	0
$221$	0	0
$222$	2.00000	0.134231
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−1.00000	−0.0668153
$225$	−1.00000	−0.0666667
$226$	−14.0000	−0.931266
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	−4.00000	−0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	12.0000	0.791257
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	−6.00000	−0.390567
$237$	0	0
$238$	−2.00000	−0.129641
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	−2.00000	−0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	11.0000	0.707107
$243$	1.00000	0.0641500
$244$	12.0000	0.768221
$245$	−2.00000	−0.127775
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.0000	0.887214
$250$	−12.0000	−0.758947
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−12.0000	−0.752947
$255$	−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
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	−12.0000	−0.741362
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	−8.00000	−0.491436
$266$	4.00000	0.245256
$267$	4.00000	0.244796
$268$	2.00000	0.122169
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	2.00000	0.121716
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	2.00000	0.120824
$275$	0	0
$276$	6.00000	0.361158
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	2.00000	0.119523
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−8.00000	−0.476393
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	2.00000	0.117242
$292$	−14.0000	−0.819288
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−1.00000	−0.0583212
$295$	12.0000	0.698667
$296$	2.00000	0.116248
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−4.00000	−0.230556
$302$	0	0
$303$	−2.00000	−0.114897
$304$	−4.00000	−0.229416
$305$	−24.0000	−1.37424
$306$	−2.00000	−0.114332
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−8.00000	−0.451466
$315$	−2.00000	−0.112687
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	−4.00000	−0.224309
$319$	0	0
$320$	−2.00000	−0.111803
$321$	−12.0000	−0.669775
$322$	−6.00000	−0.334367
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	0	0
$326$	14.0000	0.775388
$327$	6.00000	0.331801
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	14.0000	0.768350
$333$	−2.00000	−0.109599
$334$	12.0000	0.656611
$335$	−4.00000	−0.218543
$336$	1.00000	0.0545545
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	14.0000	0.760376
$340$	−4.00000	−0.216930
$341$	0	0
$342$	4.00000	0.216295
$343$	1.00000	0.0539949
$344$	4.00000	0.215666
$345$	−12.0000	−0.646058
$346$	−6.00000	−0.322562
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	4.00000	0.212000
$357$	2.00000	0.105851
$358$	−20.0000	−1.05703
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	2.00000	0.105409
$361$	−3.00000	−0.157895
$362$	12.0000	0.630706
$363$	−11.0000	−0.577350
$364$	0	0
$365$	28.0000	1.46559
$366$	−12.0000	−0.627250
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	−4.00000	−0.207950
$371$	4.00000	0.207670
$372$	0	0
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	−8.00000	−0.412568
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	8.00000	0.410391
$381$	12.0000	0.614779
$382$	−2.00000	−0.102329
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	4.00000	0.203595
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	−1.00000	−0.0505076
$393$	12.0000	0.605320
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−10.0000	−0.501255
$399$	−4.00000	−0.200250
$400$	−1.00000	−0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	−2.00000	−0.0997509
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	0	0
$408$	−2.00000	−0.0990148
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	−14.0000	−0.689730
$413$	−6.00000	−0.295241
$414$	−6.00000	−0.294884
$415$	−28.0000	−1.37447
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	−2.00000	−0.0975900
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	28.0000	1.36302
$423$	8.00000	0.388973
$424$	−4.00000	−0.194257
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	12.0000	0.580721
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−8.00000	−0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	0	0
$436$	6.00000	0.287348
$437$	−24.0000	−1.14808
$438$	14.0000	0.668946
$439$	−30.0000	−1.43182	−0.715911	−	0.698192i	$-0.753988\pi$
$439$	−30.0000	−1.43182	−0.715911	+	0.698192i	$0.753988\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−2.00000	−0.0949158
$445$	−8.00000	−0.379236
$446$	16.0000	0.757622
$447$	6.00000	0.283790
$448$	1.00000	0.0472456
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	14.0000	0.658505
$453$	0	0
$454$	−22.0000	−1.03251
$455$	0	0
$456$	4.00000	0.187317
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	−14.0000	−0.654177
$459$	2.00000	0.0933520
$460$	−12.0000	−0.559503
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	0	0
$466$	−6.00000	−0.277945
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	16.0000	0.738025
$471$	8.00000	0.368621
$472$	6.00000	0.276172
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	2.00000	0.0916698
$477$	4.00000	0.183147
$478$	−16.0000	−0.731823
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	2.00000	0.0912871
$481$	0	0
$482$	−10.0000	−0.455488
$483$	6.00000	0.273009
$484$	−11.0000	−0.500000
$485$	−4.00000	−0.181631
$486$	−1.00000	−0.0453609
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	−12.0000	−0.543214
$489$	−14.0000	−0.633102
$490$	2.00000	0.0903508
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	−14.0000	−0.627355
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	12.0000	0.536656
$501$	−12.0000	−0.536120
$502$	−28.0000	−1.24970
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	−1.00000	−0.0445435
$505$	4.00000	0.177998
$506$	0	0
$507$	0	0
$508$	12.0000	0.532414
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	4.00000	0.177123
$511$	−14.0000	−0.619324
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	18.0000	0.793946
$515$	28.0000	1.23383
$516$	−4.00000	−0.176090
$517$	0	0
$518$	2.00000	0.0878750
$519$	6.00000	0.263371
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	12.0000	0.524222
$525$	−1.00000	−0.0436436
$526$	6.00000	0.261612
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	8.00000	0.347498
$531$	−6.00000	−0.260378
$532$	−4.00000	−0.173422
$533$	0	0
$534$	−4.00000	−0.173097
$535$	24.0000	1.03761
$536$	−2.00000	−0.0863868
$537$	20.0000	0.863064
$538$	−10.0000	−0.431131
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	−12.0000	−0.514969
$544$	−2.00000	−0.0857493
$545$	−12.0000	−0.514024
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−2.00000	−0.0854358
$549$	12.0000	0.512148
$550$	0	0
$551$	0	0
$552$	−6.00000	−0.255377
$553$	0	0
$554$	−22.0000	−0.934690
$555$	4.00000	0.169791
$556$	20.0000	0.848189
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−30.0000	−1.26547
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	8.00000	0.336861
$565$	−28.0000	−1.17797
$566$	−16.0000	−0.672530
$567$	1.00000	0.0419961
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	−8.00000	−0.335083
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	2.00000	0.0835512
$574$	0	0
$575$	−6.00000	−0.250217
$576$	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	$0.348589\pi$
$578$	13.0000	0.540729
$579$	−4.00000	−0.166234
$580$	0	0
$581$	14.0000	0.580818
$582$	−2.00000	−0.0829027
$583$	0	0
$584$	14.0000	0.579324
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	1.00000	0.0412393
$589$	0	0
$590$	−12.0000	−0.494032
$591$	−18.0000	−0.740421
$592$	−2.00000	−0.0821995
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	6.00000	0.245770
$597$	10.0000	0.409273
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	1.00000	0.0408248
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	4.00000	0.163028
$603$	2.00000	0.0814463
$604$	0	0
$605$	22.0000	0.894427
$606$	2.00000	0.0812444
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	24.0000	0.971732
$611$	0	0
$612$	2.00000	0.0808452
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	−28.0000	−1.12999
$615$	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
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	6.00000	0.240772
$622$	12.0000	0.481156
$623$	4.00000	0.160257
$624$	0	0
$625$	−19.0000	−0.760000
$626$	6.00000	0.239808
$627$	0	0
$628$	8.00000	0.319235
$629$	−4.00000	−0.159490
$630$	2.00000	0.0796819
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	−28.0000	−1.11290
$634$	−2.00000	−0.0794301
$635$	−24.0000	−0.952411
$636$	4.00000	0.158610
$637$	0	0
$638$	0	0
$639$	0	0
$640$	2.00000	0.0790569
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	12.0000	0.473602
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	6.00000	0.236433
$645$	8.00000	0.315000
$646$	8.00000	0.314756
$647$	−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
$650$	0	0
$651$	0	0
$652$	−14.0000	−0.548282
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	−6.00000	−0.234619
$655$	−24.0000	−0.937758
$656$	0	0
$657$	−14.0000	−0.546192
$658$	−8.00000	−0.311872
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	−14.0000	−0.543305
$665$	8.00000	0.310227
$666$	2.00000	0.0774984
$667$	0	0
$668$	−12.0000	−0.464294
$669$	−16.0000	−0.618596
$670$	4.00000	0.154533
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	−22.0000	−0.847408
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−14.0000	−0.537667
$679$	2.00000	0.0767530
$680$	4.00000	0.153393
$681$	22.0000	0.843042
$682$	0	0
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	−4.00000	−0.152944
$685$	4.00000	0.152832
$686$	−1.00000	−0.0381802
$687$	14.0000	0.534133
$688$	−4.00000	−0.152499
$689$	0	0
$690$	12.0000	0.456832
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	32.0000	1.21470
$695$	−40.0000	−1.51729
$696$	0	0
$697$	0	0
$698$	−14.0000	−0.529908
$699$	6.00000	0.226941
$700$	−1.00000	−0.0377964
$701$	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
$703$	8.00000	0.301726
$704$	0	0
$705$	−16.0000	−0.602595
$706$	−24.0000	−0.903252
$707$	−2.00000	−0.0752177
$708$	−6.00000	−0.225494
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	0	0
$712$	−4.00000	−0.149906
$713$	0	0
$714$	−2.00000	−0.0748481
$715$	0	0
$716$	20.0000	0.747435
$717$	16.0000	0.597531
$718$	−24.0000	−0.895672
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	−2.00000	−0.0745356
$721$	−14.0000	−0.521387
$722$	3.00000	0.111648
$723$	10.0000	0.371904
$724$	−12.0000	−0.445976
$725$	0	0
$726$	11.0000	0.408248
$727$	−38.0000	−1.40934	−0.704671	−	0.709534i	$-0.748905\pi$
$727$	−38.0000	−1.40934	−0.704671	+	0.709534i	$0.748905\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−28.0000	−1.03633
$731$	−8.00000	−0.295891
$732$	12.0000	0.443533
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	22.0000	0.812035
$735$	−2.00000	−0.0737711
$736$	−6.00000	−0.221163
$737$	0	0
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	−4.00000	−0.146845
$743$	−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
$745$	−12.0000	−0.439646
$746$	−34.0000	−1.24483
$747$	14.0000	0.512233
$748$	0	0
$749$	−12.0000	−0.438470
$750$	−12.0000	−0.438178
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	8.00000	0.291730
$753$	28.0000	1.02038
$754$	0	0
$755$	0	0
$756$	1.00000	0.0363696
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	−26.0000	−0.944363
$759$	0	0
$760$	−8.00000	−0.290191
$761$	40.0000	1.45000	0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000	1.45000	0.724999	+	0.688749i	$0.241840\pi$
$762$	−12.0000	−0.434714
$763$	6.00000	0.217215
$764$	2.00000	0.0723575
$765$	−4.00000	−0.144620
$766$	−24.0000	−0.867155
$767$	0	0
$768$	1.00000	0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−4.00000	−0.143963
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	−2.00000	−0.0717496
$778$	20.0000	0.717035
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−12.0000	−0.429119
$783$	0	0
$784$	1.00000	0.0357143
$785$	−16.0000	−0.571064
$786$	−12.0000	−0.428026
$787$	48.0000	1.71102	0.855508	−	0.517790i	$-0.173245\pi$
$787$	48.0000	1.71102	0.855508	+	0.517790i	$0.173245\pi$
$788$	−18.0000	−0.641223
$789$	−6.00000	−0.213606
$790$	0	0
$791$	14.0000	0.497783
$792$	0	0
$793$	0	0
$794$	−18.0000	−0.638796
$795$	−8.00000	−0.283731
$796$	10.0000	0.354441
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	4.00000	0.141598
$799$	16.0000	0.566039
$800$	1.00000	0.0353553
$801$	4.00000	0.141333
$802$	30.0000	1.05934
$803$	0	0
$804$	2.00000	0.0705346
$805$	−12.0000	−0.422944
$806$	0	0
$807$	10.0000	0.352017
$808$	2.00000	0.0703598
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	2.00000	0.0702728
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	28.0000	0.980797
$816$	2.00000	0.0700140
$817$	16.0000	0.559769
$818$	14.0000	0.489499
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	2.00000	0.0697580
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	6.00000	0.208767
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	6.00000	0.208514
$829$	40.0000	1.38926	0.694629	−	0.719368i	$-0.255569\pi$
$829$	40.0000	1.38926	0.694629	+	0.719368i	$0.255569\pi$
$830$	28.0000	0.971894
$831$	22.0000	0.763172
$832$	0	0
$833$	2.00000	0.0692959
$834$	−20.0000	−0.692543
$835$	24.0000	0.830554
$836$	0	0
$837$	0	0
$838$	−20.0000	−0.690889
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	2.00000	0.0690066
$841$	−29.0000	−1.00000
$842$	−30.0000	−1.03387
$843$	30.0000	1.03325
$844$	−28.0000	−0.963800
$845$	0	0
$846$	−8.00000	−0.275046
$847$	−11.0000	−0.377964
$848$	4.00000	0.137361
$849$	16.0000	0.549119
$850$	2.00000	0.0685994
$851$	−12.0000	−0.411355
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−12.0000	−0.410632
$855$	8.00000	0.273594
$856$	12.0000	0.410152
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	−1.00000	−0.0340207
$865$	−12.0000	−0.408012
$866$	−26.0000	−0.883516
$867$	−13.0000	−0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−6.00000	−0.203186
$873$	2.00000	0.0676897
$874$	24.0000	0.811812
$875$	12.0000	0.405674
$876$	−14.0000	−0.473016
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	30.0000	1.01245
$879$	6.00000	0.202375
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	−1.00000	−0.0336718
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	−4.00000	−0.134383
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	2.00000	0.0671156
$889$	12.0000	0.402467
$890$	8.00000	0.268161
$891$	0	0
$892$	−16.0000	−0.535720
$893$	−32.0000	−1.07084
$894$	−6.00000	−0.200670
$895$	−40.0000	−1.33705
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−14.0000	−0.467186
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	8.00000	0.266519
$902$	0	0
$903$	−4.00000	−0.133112
$904$	−14.0000	−0.465633
$905$	24.0000	0.797787
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	22.0000	0.730096
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	22.0000	0.728893	0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000	0.728893	0.364446	+	0.931224i	$0.381258\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	8.00000	0.264616
$915$	−24.0000	−0.793416
$916$	14.0000	0.462573
$917$	12.0000	0.396275
$918$	−2.00000	−0.0660098
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	12.0000	0.395628
$921$	28.0000	0.922631
$922$	10.0000	0.329332
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	−16.0000	−0.525793
$927$	−14.0000	−0.459820
$928$	0	0
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	6.00000	0.196537
$933$	−12.0000	−0.392862
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	−2.00000	−0.0653023
$939$	−6.00000	−0.195803
$940$	−16.0000	−0.521862
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	−8.00000	−0.260654
$943$	0	0
$944$	−6.00000	−0.195283
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	0	0
$949$	0	0
$950$	−4.00000	−0.129777
$951$	2.00000	0.0648544
$952$	−2.00000	−0.0648204
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	−4.00000	−0.129505
$955$	−4.00000	−0.129437
$956$	16.0000	0.517477
$957$	0	0
$958$	16.0000	0.516937
$959$	−2.00000	−0.0645834
$960$	−2.00000	−0.0645497
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−12.0000	−0.386695
$964$	10.0000	0.322078
$965$	8.00000	0.257529
$966$	−6.00000	−0.193047
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	11.0000	0.353553
$969$	−8.00000	−0.256997
$970$	4.00000	0.128432
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	1.00000	0.0320750
$973$	20.0000	0.641171
$974$	28.0000	0.897178
$975$	0	0
$976$	12.0000	0.384111
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	14.0000	0.447671
$979$	0	0
$980$	−2.00000	−0.0638877
$981$	6.00000	0.191565
$982$	−8.00000	−0.255290
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	0	0
$995$	−20.0000	−0.634043
$996$	14.0000	0.443607
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	14.0000	0.443162
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.k.1.1		1
13.5	odd	4		546.2.c.b.337.2	yes	2
13.8	odd	4		546.2.c.b.337.1	✓	2
13.12	even	2		7098.2.a.bc.1.1		1
39.5	even	4		1638.2.c.b.883.1		2
39.8	even	4		1638.2.c.b.883.2		2
52.31	even	4		4368.2.h.f.337.2		2
52.47	even	4		4368.2.h.f.337.1		2
91.34	even	4		3822.2.c.c.883.1		2
91.83	even	4		3822.2.c.c.883.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.b.337.1	✓	2	13.8	odd	4
546.2.c.b.337.2	yes	2	13.5	odd	4
1638.2.c.b.883.1		2	39.5	even	4
1638.2.c.b.883.2		2	39.8	even	4
3822.2.c.c.883.1		2	91.34	even	4
3822.2.c.c.883.2		2	91.83	even	4
4368.2.h.f.337.1		2	52.47	even	4
4368.2.h.f.337.2		2	52.31	even	4
7098.2.a.k.1.1		1	1.1	even	1	trivial
7098.2.a.bc.1.1		1	13.12	even	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:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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