LMFDB - Embedded newform 7098.2.a.b.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)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} +7.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -3.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} -7.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} +3.00000 q^{29} -3.00000 q^{30} -1.00000 q^{32} +1.00000 q^{33} -7.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} +5.00000 q^{37} +1.00000 q^{38} +3.00000 q^{40} -4.00000 q^{41} +1.00000 q^{42} +11.0000 q^{43} -1.00000 q^{44} -3.00000 q^{45} +7.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} -7.00000 q^{51} -14.0000 q^{53} +1.00000 q^{54} +3.00000 q^{55} -1.00000 q^{56} +1.00000 q^{57} -3.00000 q^{58} -4.00000 q^{59} +3.00000 q^{60} +1.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} +6.00000 q^{67} +7.00000 q^{68} +7.00000 q^{69} +3.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} -5.00000 q^{73} -5.00000 q^{74} -4.00000 q^{75} -1.00000 q^{76} -1.00000 q^{77} -10.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} +14.0000 q^{83} -1.00000 q^{84} -21.0000 q^{85} -11.0000 q^{86} -3.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} +3.00000 q^{90} -7.00000 q^{92} +3.00000 q^{95} +1.00000 q^{96} -6.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−3.00000	−0.670820
$21$	−1.00000	−0.218218
$22$	1.00000	0.213201
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−3.00000	−0.547723
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	−7.00000	−1.20049
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	3.00000	0.474342
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	1.00000	0.154303
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	−1.00000	−0.150756
$45$	−3.00000	−0.447214
$46$	7.00000	1.03209
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−4.00000	−0.565685
$51$	−7.00000	−0.980196
$52$	0	0
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	1.00000	0.136083
$55$	3.00000	0.404520
$56$	−1.00000	−0.133631
$57$	1.00000	0.132453
$58$	−3.00000	−0.393919
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	3.00000	0.387298
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	7.00000	0.848875
$69$	7.00000	0.842701
$70$	3.00000	0.358569
$71$	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
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	−5.00000	−0.581238
$75$	−4.00000	−0.461880
$76$	−1.00000	−0.114708
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	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
$85$	−21.0000	−2.27777
$86$	−11.0000	−1.18616
$87$	−3.00000	−0.321634
$88$	1.00000	0.106600
$89$	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
$92$	−7.00000	−0.729800
$93$	0	0
$94$	0	0
$95$	3.00000	0.307794
$96$	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	$0.598524\pi$
$98$	−1.00000	−0.101015
$99$	−1.00000	−0.100504
$100$	4.00000	0.400000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	7.00000	0.693103
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	14.0000	1.35980
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	−1.00000	−0.0962250
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	−3.00000	−0.286039
$111$	−5.00000	−0.474579
$112$	1.00000	0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−1.00000	−0.0936586
$115$	21.0000	1.95826
$116$	3.00000	0.278543
$117$	0	0
$118$	4.00000	0.368230
$119$	7.00000	0.641689
$120$	−3.00000	−0.273861
$121$	−10.0000	−0.909091
$122$	−1.00000	−0.0905357
$123$	4.00000	0.360668
$124$	0	0
$125$	3.00000	0.268328
$126$	−1.00000	−0.0890871
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	−1.00000	−0.0883883
$129$	−11.0000	−0.968496
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	1.00000	0.0870388
$133$	−1.00000	−0.0867110
$134$	−6.00000	−0.518321
$135$	3.00000	0.258199
$136$	−7.00000	−0.600245
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	−7.00000	−0.595880
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−3.00000	−0.253546
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	−9.00000	−0.747409
$146$	5.00000	0.413803
$147$	−1.00000	−0.0824786
$148$	5.00000	0.410997
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	4.00000	0.326599
$151$	−7.00000	−0.569652	−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000	−0.569652	−0.284826	+	0.958579i	$0.591936\pi$
$152$	1.00000	0.0811107
$153$	7.00000	0.565916
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	10.0000	0.795557
$159$	14.0000	1.11027
$160$	3.00000	0.237171
$161$	−7.00000	−0.551677
$162$	−1.00000	−0.0785674
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−4.00000	−0.312348
$165$	−3.00000	−0.233550
$166$	−14.0000	−1.08661
$167$	−7.00000	−0.541676	−0.270838	−	0.962625i	$-0.587301\pi$
$167$	−7.00000	−0.541676	−0.270838	+	0.962625i	$0.587301\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	21.0000	1.61063
$171$	−1.00000	−0.0764719
$172$	11.0000	0.838742
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	3.00000	0.227429
$175$	4.00000	0.302372
$176$	−1.00000	−0.0753778
$177$	4.00000	0.300658
$178$	−6.00000	−0.449719
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	−3.00000	−0.223607
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	7.00000	0.516047
$185$	−15.0000	−1.10282
$186$	0	0
$187$	−7.00000	−0.511891
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	−3.00000	−0.217643
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	−1.00000	−0.0721688
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	1.00000	0.0714286
$197$	20.0000	1.42494	0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000	1.42494	0.712470	+	0.701702i	$0.247576\pi$
$198$	1.00000	0.0710669
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	−4.00000	−0.282843
$201$	−6.00000	−0.423207
$202$	12.0000	0.844317
$203$	3.00000	0.210559
$204$	−7.00000	−0.490098
$205$	12.0000	0.838116
$206$	7.00000	0.487713
$207$	−7.00000	−0.486534
$208$	0	0
$209$	1.00000	0.0691714
$210$	−3.00000	−0.207020
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	−14.0000	−0.961524
$213$	−12.0000	−0.822226
$214$	18.0000	1.23045
$215$	−33.0000	−2.25058
$216$	1.00000	0.0680414
$217$	0	0
$218$	−7.00000	−0.474100
$219$	5.00000	0.337869
$220$	3.00000	0.202260
$221$	0	0
$222$	5.00000	0.335578
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	−1.00000	−0.0668153
$225$	4.00000	0.266667
$226$	14.0000	0.931266
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	1.00000	0.0662266
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	−21.0000	−1.38470
$231$	1.00000	0.0657952
$232$	−3.00000	−0.196960
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	10.0000	0.649570
$238$	−7.00000	−0.453743
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	3.00000	0.193649
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	10.0000	0.642824
$243$	−1.00000	−0.0641500
$244$	1.00000	0.0640184
$245$	−3.00000	−0.191663
$246$	−4.00000	−0.255031
$247$	0	0
$248$	0	0
$249$	−14.0000	−0.887214
$250$	−3.00000	−0.189737
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	1.00000	0.0629941
$253$	7.00000	0.440086
$254$	−10.0000	−0.627456
$255$	21.0000	1.31507
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	11.0000	0.684830
$259$	5.00000	0.310685
$260$	0	0
$261$	3.00000	0.185695
$262$	−15.0000	−0.926703
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	−1.00000	−0.0615457
$265$	42.0000	2.58004
$266$	1.00000	0.0613139
$267$	−6.00000	−0.367194
$268$	6.00000	0.366508
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	−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	$0.135169\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	15.0000	0.906183
$275$	−4.00000	−0.241209
$276$	7.00000	0.421350
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	3.00000	0.179284
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	12.0000	0.712069
$285$	−3.00000	−0.177705
$286$	0	0
$287$	−4.00000	−0.236113
$288$	−1.00000	−0.0589256
$289$	32.0000	1.88235
$290$	9.00000	0.528498
$291$	6.00000	0.351726
$292$	−5.00000	−0.292603
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	1.00000	0.0583212
$295$	12.0000	0.698667
$296$	−5.00000	−0.290619
$297$	1.00000	0.0580259
$298$	16.0000	0.926855
$299$	0	0
$300$	−4.00000	−0.230940
$301$	11.0000	0.634029
$302$	7.00000	0.402805
$303$	12.0000	0.689382
$304$	−1.00000	−0.0573539
$305$	−3.00000	−0.171780
$306$	−7.00000	−0.400163
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	−1.00000	−0.0569803
$309$	7.00000	0.398216
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	−5.00000	−0.282166
$315$	−3.00000	−0.169031
$316$	−10.0000	−0.562544
$317$	28.0000	1.57264	0.786318	−	0.617822i	$-0.211985\pi$
$317$	28.0000	1.57264	0.786318	+	0.617822i	$0.211985\pi$
$318$	−14.0000	−0.785081
$319$	−3.00000	−0.167968
$320$	−3.00000	−0.167705
$321$	18.0000	1.00466
$322$	7.00000	0.390095
$323$	−7.00000	−0.389490
$324$	1.00000	0.0555556
$325$	0	0
$326$	10.0000	0.553849
$327$	−7.00000	−0.387101
$328$	4.00000	0.220863
$329$	0	0
$330$	3.00000	0.165145
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	14.0000	0.768350
$333$	5.00000	0.273998
$334$	7.00000	0.383023
$335$	−18.0000	−0.983445
$336$	−1.00000	−0.0545545
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	0	0
$339$	14.0000	0.760376
$340$	−21.0000	−1.13888
$341$	0	0
$342$	1.00000	0.0540738
$343$	1.00000	0.0539949
$344$	−11.0000	−0.593080
$345$	−21.0000	−1.13060
$346$	−2.00000	−0.107521
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	−3.00000	−0.160817
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	1.00000	0.0533002
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	−4.00000	−0.212598
$355$	−36.0000	−1.91068
$356$	6.00000	0.317999
$357$	−7.00000	−0.370479
$358$	−8.00000	−0.422813
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	3.00000	0.158114
$361$	−18.0000	−0.947368
$362$	−22.0000	−1.15629
$363$	10.0000	0.524864
$364$	0	0
$365$	15.0000	0.785136
$366$	1.00000	0.0522708
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−7.00000	−0.364900
$369$	−4.00000	−0.208232
$370$	15.0000	0.779813
$371$	−14.0000	−0.726844
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	7.00000	0.361961
$375$	−3.00000	−0.154919
$376$	0	0
$377$	0	0
$378$	1.00000	0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	3.00000	0.153897
$381$	−10.0000	−0.512316
$382$	15.0000	0.767467
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	1.00000	0.0510310
$385$	3.00000	0.152894
$386$	−8.00000	−0.407189
$387$	11.0000	0.559161
$388$	−6.00000	−0.304604
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	−49.0000	−2.47804
$392$	−1.00000	−0.0505076
$393$	−15.0000	−0.756650
$394$	−20.0000	−1.00759
$395$	30.0000	1.50946
$396$	−1.00000	−0.0502519
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−3.00000	−0.150376
$399$	1.00000	0.0500626
$400$	4.00000	0.200000
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	6.00000	0.299253
$403$	0	0
$404$	−12.0000	−0.597022
$405$	−3.00000	−0.149071
$406$	−3.00000	−0.148888
$407$	−5.00000	−0.247841
$408$	7.00000	0.346552
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	−12.0000	−0.592638
$411$	15.0000	0.739895
$412$	−7.00000	−0.344865
$413$	−4.00000	−0.196827
$414$	7.00000	0.344031
$415$	−42.0000	−2.06170
$416$	0	0
$417$	−4.00000	−0.195881
$418$	−1.00000	−0.0489116
$419$	29.0000	1.41674	0.708371	−	0.705840i	$-0.249430\pi$
$419$	29.0000	1.41674	0.708371	+	0.705840i	$0.249430\pi$
$420$	3.00000	0.146385
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−25.0000	−1.21698
$423$	0	0
$424$	14.0000	0.679900
$425$	28.0000	1.35820
$426$	12.0000	0.581402
$427$	1.00000	0.0483934
$428$	−18.0000	−0.870063
$429$	0	0
$430$	33.0000	1.59140
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	−1.00000	−0.0481125
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	7.00000	0.335239
$437$	7.00000	0.334855
$438$	−5.00000	−0.238909
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	−3.00000	−0.143019
$441$	1.00000	0.0476190
$442$	0	0
$443$	26.0000	1.23530	0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000	1.23530	0.617649	+	0.786454i	$0.288085\pi$
$444$	−5.00000	−0.237289
$445$	−18.0000	−0.853282
$446$	14.0000	0.662919
$447$	16.0000	0.756774
$448$	1.00000	0.0472456
$449$	−19.0000	−0.896665	−0.448333	−	0.893867i	$-0.647982\pi$
$449$	−19.0000	−0.896665	−0.448333	+	0.893867i	$0.647982\pi$
$450$	−4.00000	−0.188562
$451$	4.00000	0.188353
$452$	−14.0000	−0.658505
$453$	7.00000	0.328889
$454$	−18.0000	−0.844782
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	40.0000	1.87112	0.935561	−	0.353166i	$-0.114895\pi$
$457$	40.0000	1.87112	0.935561	+	0.353166i	$0.114895\pi$
$458$	18.0000	0.841085
$459$	−7.00000	−0.326732
$460$	21.0000	0.979130
$461$	23.0000	1.07122	0.535608	−	0.844466i	$-0.320082\pi$
$461$	23.0000	1.07122	0.535608	+	0.844466i	$0.320082\pi$
$462$	−1.00000	−0.0465242
$463$	−23.0000	−1.06890	−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000	−1.06890	−0.534450	+	0.845200i	$0.679481\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	12.0000	0.555889
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	0	0
$471$	−5.00000	−0.230388
$472$	4.00000	0.184115
$473$	−11.0000	−0.505781
$474$	−10.0000	−0.459315
$475$	−4.00000	−0.183533
$476$	7.00000	0.320844
$477$	−14.0000	−0.641016
$478$	20.0000	0.914779
$479$	11.0000	0.502603	0.251301	−	0.967909i	$-0.419141\pi$
$479$	11.0000	0.502603	0.251301	+	0.967909i	$0.419141\pi$
$480$	−3.00000	−0.136931
$481$	0	0
$482$	−18.0000	−0.819878
$483$	7.00000	0.318511
$484$	−10.0000	−0.454545
$485$	18.0000	0.817338
$486$	1.00000	0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−1.00000	−0.0452679
$489$	10.0000	0.452216
$490$	3.00000	0.135526
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	4.00000	0.180334
$493$	21.0000	0.945792
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	12.0000	0.538274
$498$	14.0000	0.627355
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	3.00000	0.134164
$501$	7.00000	0.312737
$502$	−3.00000	−0.133897
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	−1.00000	−0.0445435
$505$	36.0000	1.60198
$506$	−7.00000	−0.311188
$507$	0	0
$508$	10.0000	0.443678
$509$	39.0000	1.72864	0.864322	−	0.502938i	$-0.167748\pi$
$509$	39.0000	1.72864	0.864322	+	0.502938i	$0.167748\pi$
$510$	−21.0000	−0.929896
$511$	−5.00000	−0.221187
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	−10.0000	−0.441081
$515$	21.0000	0.925371
$516$	−11.0000	−0.484248
$517$	0	0
$518$	−5.00000	−0.219687
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−11.0000	−0.481919	−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000	−0.481919	−0.240959	+	0.970535i	$0.577462\pi$
$522$	−3.00000	−0.131306
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	15.0000	0.655278
$525$	−4.00000	−0.174574
$526$	4.00000	0.174408
$527$	0	0
$528$	1.00000	0.0435194
$529$	26.0000	1.13043
$530$	−42.0000	−1.82436
$531$	−4.00000	−0.173585
$532$	−1.00000	−0.0433555
$533$	0	0
$534$	6.00000	0.259645
$535$	54.0000	2.33462
$536$	−6.00000	−0.259161
$537$	−8.00000	−0.345225
$538$	−2.00000	−0.0862261
$539$	−1.00000	−0.0430730
$540$	3.00000	0.129099
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	−30.0000	−1.28861
$543$	−22.0000	−0.944110
$544$	−7.00000	−0.300123
$545$	−21.0000	−0.899541
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−15.0000	−0.640768
$549$	1.00000	0.0426790
$550$	4.00000	0.170561
$551$	−3.00000	−0.127804
$552$	−7.00000	−0.297940
$553$	−10.0000	−0.425243
$554$	−10.0000	−0.424859
$555$	15.0000	0.636715
$556$	4.00000	0.169638
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	0	0
$560$	−3.00000	−0.126773
$561$	7.00000	0.295540
$562$	6.00000	0.253095
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	42.0000	1.76695
$566$	16.0000	0.672530
$567$	1.00000	0.0419961
$568$	−12.0000	−0.503509
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	3.00000	0.125656
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	15.0000	0.626634
$574$	4.00000	0.166957
$575$	−28.0000	−1.16768
$576$	1.00000	0.0416667
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	−32.0000	−1.33102
$579$	−8.00000	−0.332469
$580$	−9.00000	−0.373705
$581$	14.0000	0.580818
$582$	−6.00000	−0.248708
$583$	14.0000	0.579821
$584$	5.00000	0.206901
$585$	0	0
$586$	30.0000	1.23929
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	−1.00000	−0.0412393
$589$	0	0
$590$	−12.0000	−0.494032
$591$	−20.0000	−0.822690
$592$	5.00000	0.205499
$593$	42.0000	1.72473	0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000	1.72473	0.862367	+	0.506284i	$0.168981\pi$
$594$	−1.00000	−0.0410305
$595$	−21.0000	−0.860916
$596$	−16.0000	−0.655386
$597$	−3.00000	−0.122782
$598$	0	0
$599$	−1.00000	−0.0408589	−0.0204294	−	0.999791i	$-0.506503\pi$
$599$	−1.00000	−0.0408589	−0.0204294	+	0.999791i	$0.506503\pi$
$600$	4.00000	0.163299
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	−11.0000	−0.448327
$603$	6.00000	0.244339
$604$	−7.00000	−0.284826
$605$	30.0000	1.21967
$606$	−12.0000	−0.487467
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	1.00000	0.0405554
$609$	−3.00000	−0.121566
$610$	3.00000	0.121466
$611$	0	0
$612$	7.00000	0.282958
$613$	7.00000	0.282727	0.141364	−	0.989958i	$-0.454851\pi$
$613$	7.00000	0.282727	0.141364	+	0.989958i	$0.454851\pi$
$614$	−32.0000	−1.29141
$615$	−12.0000	−0.483887
$616$	1.00000	0.0402911
$617$	27.0000	1.08698	0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000	1.08698	0.543490	+	0.839416i	$0.317103\pi$
$618$	−7.00000	−0.281581
$619$	−1.00000	−0.0401934	−0.0200967	−	0.999798i	$-0.506397\pi$
$619$	−1.00000	−0.0401934	−0.0200967	+	0.999798i	$0.506397\pi$
$620$	0	0
$621$	7.00000	0.280900
$622$	−2.00000	−0.0801927
$623$	6.00000	0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−1.00000	−0.0399362
$628$	5.00000	0.199522
$629$	35.0000	1.39554
$630$	3.00000	0.119523
$631$	37.0000	1.47295	0.736473	−	0.676467i	$-0.236490\pi$
$631$	37.0000	1.47295	0.736473	+	0.676467i	$0.236490\pi$
$632$	10.0000	0.397779
$633$	−25.0000	−0.993661
$634$	−28.0000	−1.11202
$635$	−30.0000	−1.19051
$636$	14.0000	0.555136
$637$	0	0
$638$	3.00000	0.118771
$639$	12.0000	0.474713
$640$	3.00000	0.118585
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	−18.0000	−0.710403
$643$	9.00000	0.354925	0.177463	−	0.984128i	$-0.443211\pi$
$643$	9.00000	0.354925	0.177463	+	0.984128i	$0.443211\pi$
$644$	−7.00000	−0.275839
$645$	33.0000	1.29937
$646$	7.00000	0.275411
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	−1.00000	−0.0392837
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	−10.0000	−0.391630
$653$	−43.0000	−1.68272	−0.841360	−	0.540475i	$-0.818245\pi$
$653$	−43.0000	−1.68272	−0.841360	+	0.540475i	$0.818245\pi$
$654$	7.00000	0.273722
$655$	−45.0000	−1.75830
$656$	−4.00000	−0.156174
$657$	−5.00000	−0.195069
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	−3.00000	−0.116775
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−14.0000	−0.544125
$663$	0	0
$664$	−14.0000	−0.543305
$665$	3.00000	0.116335
$666$	−5.00000	−0.193746
$667$	−21.0000	−0.813123
$668$	−7.00000	−0.270838
$669$	14.0000	0.541271
$670$	18.0000	0.695401
$671$	−1.00000	−0.0386046
$672$	1.00000	0.0385758
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	−3.00000	−0.115556
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	−14.0000	−0.537667
$679$	−6.00000	−0.230259
$680$	21.0000	0.805313
$681$	−18.0000	−0.689761
$682$	0	0
$683$	13.0000	0.497431	0.248716	−	0.968577i	$-0.419992\pi$
$683$	13.0000	0.497431	0.248716	+	0.968577i	$0.419992\pi$
$684$	−1.00000	−0.0382360
$685$	45.0000	1.71936
$686$	−1.00000	−0.0381802
$687$	18.0000	0.686743
$688$	11.0000	0.419371
$689$	0	0
$690$	21.0000	0.799456
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	2.00000	0.0760286
$693$	−1.00000	−0.0379869
$694$	−32.0000	−1.21470
$695$	−12.0000	−0.455186
$696$	3.00000	0.113715
$697$	−28.0000	−1.06058
$698$	2.00000	0.0757011
$699$	12.0000	0.453882
$700$	4.00000	0.151186
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	−5.00000	−0.188579
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−10.0000	−0.376355
$707$	−12.0000	−0.451306
$708$	4.00000	0.150329
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	36.0000	1.35106
$711$	−10.0000	−0.375029
$712$	−6.00000	−0.224860
$713$	0	0
$714$	7.00000	0.261968
$715$	0	0
$716$	8.00000	0.298974
$717$	20.0000	0.746914
$718$	12.0000	0.447836
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	−3.00000	−0.111803
$721$	−7.00000	−0.260694
$722$	18.0000	0.669891
$723$	−18.0000	−0.669427
$724$	22.0000	0.817624
$725$	12.0000	0.445669
$726$	−10.0000	−0.371135
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−15.0000	−0.555175
$731$	77.0000	2.84795
$732$	−1.00000	−0.0369611
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	−32.0000	−1.18114
$735$	3.00000	0.110657
$736$	7.00000	0.258023
$737$	−6.00000	−0.221013
$738$	4.00000	0.147242
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	−15.0000	−0.551411
$741$	0	0
$742$	14.0000	0.513956
$743$	26.0000	0.953847	0.476924	−	0.878945i	$-0.341752\pi$
$743$	26.0000	0.953847	0.476924	+	0.878945i	$0.341752\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	−12.0000	−0.439351
$747$	14.0000	0.512233
$748$	−7.00000	−0.255945
$749$	−18.0000	−0.657706
$750$	3.00000	0.109545
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	−3.00000	−0.109326
$754$	0	0
$755$	21.0000	0.764268
$756$	−1.00000	−0.0363696
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	20.0000	0.726433
$759$	−7.00000	−0.254084
$760$	−3.00000	−0.108821
$761$	32.0000	1.16000	0.580000	−	0.814617i	$-0.303053\pi$
$761$	32.0000	1.16000	0.580000	+	0.814617i	$0.303053\pi$
$762$	10.0000	0.362262
$763$	7.00000	0.253417
$764$	−15.0000	−0.542681
$765$	−21.0000	−0.759257
$766$	−1.00000	−0.0361315
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	−3.00000	−0.108112
$771$	−10.0000	−0.360141
$772$	8.00000	0.287926
$773$	7.00000	0.251773	0.125886	−	0.992045i	$-0.459823\pi$
$773$	7.00000	0.251773	0.125886	+	0.992045i	$0.459823\pi$
$774$	−11.0000	−0.395387
$775$	0	0
$776$	6.00000	0.215387
$777$	−5.00000	−0.179374
$778$	−34.0000	−1.21896
$779$	4.00000	0.143315
$780$	0	0
$781$	−12.0000	−0.429394
$782$	49.0000	1.75224
$783$	−3.00000	−0.107211
$784$	1.00000	0.0357143
$785$	−15.0000	−0.535373
$786$	15.0000	0.535032
$787$	31.0000	1.10503	0.552515	−	0.833503i	$-0.313668\pi$
$787$	31.0000	1.10503	0.552515	+	0.833503i	$0.313668\pi$
$788$	20.0000	0.712470
$789$	4.00000	0.142404
$790$	−30.0000	−1.06735
$791$	−14.0000	−0.497783
$792$	1.00000	0.0355335
$793$	0	0
$794$	−14.0000	−0.496841
$795$	−42.0000	−1.48959
$796$	3.00000	0.106332
$797$	36.0000	1.27519	0.637593	−	0.770374i	$-0.279930\pi$
$797$	36.0000	1.27519	0.637593	+	0.770374i	$0.279930\pi$
$798$	−1.00000	−0.0353996
$799$	0	0
$800$	−4.00000	−0.141421
$801$	6.00000	0.212000
$802$	−2.00000	−0.0706225
$803$	5.00000	0.176446
$804$	−6.00000	−0.211604
$805$	21.0000	0.740153
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	12.0000	0.422159
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	3.00000	0.105409
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	3.00000	0.105279
$813$	−30.0000	−1.05215
$814$	5.00000	0.175250
$815$	30.0000	1.05085
$816$	−7.00000	−0.245049
$817$	−11.0000	−0.384841
$818$	23.0000	0.804176
$819$	0	0
$820$	12.0000	0.419058
$821$	32.0000	1.11681	0.558404	−	0.829569i	$-0.311414\pi$
$821$	32.0000	1.11681	0.558404	+	0.829569i	$0.311414\pi$
$822$	−15.0000	−0.523185
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	7.00000	0.243857
$825$	4.00000	0.139262
$826$	4.00000	0.139178
$827$	−9.00000	−0.312961	−0.156480	−	0.987681i	$-0.550015\pi$
$827$	−9.00000	−0.312961	−0.156480	+	0.987681i	$0.550015\pi$
$828$	−7.00000	−0.243267
$829$	49.0000	1.70184	0.850920	−	0.525295i	$-0.176045\pi$
$829$	49.0000	1.70184	0.850920	+	0.525295i	$0.176045\pi$
$830$	42.0000	1.45784
$831$	−10.0000	−0.346896
$832$	0	0
$833$	7.00000	0.242536
$834$	4.00000	0.138509
$835$	21.0000	0.726735
$836$	1.00000	0.0345857
$837$	0	0
$838$	−29.0000	−1.00179
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	−3.00000	−0.103510
$841$	−20.0000	−0.689655
$842$	26.0000	0.896019
$843$	6.00000	0.206651
$844$	25.0000	0.860535
$845$	0	0
$846$	0	0
$847$	−10.0000	−0.343604
$848$	−14.0000	−0.480762
$849$	16.0000	0.549119
$850$	−28.0000	−0.960392
$851$	−35.0000	−1.19978
$852$	−12.0000	−0.411113
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	−1.00000	−0.0342193
$855$	3.00000	0.102598
$856$	18.0000	0.615227
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−33.0000	−1.12529
$861$	4.00000	0.136320
$862$	−10.0000	−0.340601
$863$	42.0000	1.42970	0.714848	−	0.699280i	$-0.246496\pi$
$863$	42.0000	1.42970	0.714848	+	0.699280i	$0.246496\pi$
$864$	1.00000	0.0340207
$865$	−6.00000	−0.204006
$866$	8.00000	0.271851
$867$	−32.0000	−1.08678
$868$	0	0
$869$	10.0000	0.339227
$870$	−9.00000	−0.305129
$871$	0	0
$872$	−7.00000	−0.237050
$873$	−6.00000	−0.203069
$874$	−7.00000	−0.236779
$875$	3.00000	0.101419
$876$	5.00000	0.168934
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	19.0000	0.641219
$879$	30.0000	1.01187
$880$	3.00000	0.101130
$881$	−17.0000	−0.572745	−0.286372	−	0.958118i	$-0.592449\pi$
$881$	−17.0000	−0.572745	−0.286372	+	0.958118i	$0.592449\pi$
$882$	−1.00000	−0.0336718
$883$	35.0000	1.17784	0.588922	−	0.808190i	$-0.299553\pi$
$883$	35.0000	1.17784	0.588922	+	0.808190i	$0.299553\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	−26.0000	−0.873487
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	5.00000	0.167789
$889$	10.0000	0.335389
$890$	18.0000	0.603361
$891$	−1.00000	−0.0335013
$892$	−14.0000	−0.468755
$893$	0	0
$894$	−16.0000	−0.535120
$895$	−24.0000	−0.802232
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	19.0000	0.634038
$899$	0	0
$900$	4.00000	0.133333
$901$	−98.0000	−3.26485
$902$	−4.00000	−0.133185
$903$	−11.0000	−0.366057
$904$	14.0000	0.465633
$905$	−66.0000	−2.19391
$906$	−7.00000	−0.232559
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	18.0000	0.597351
$909$	−12.0000	−0.398015
$910$	0	0
$911$	15.0000	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$911$	15.0000	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$912$	1.00000	0.0331133
$913$	−14.0000	−0.463332
$914$	−40.0000	−1.32308
$915$	3.00000	0.0991769
$916$	−18.0000	−0.594737
$917$	15.0000	0.495344
$918$	7.00000	0.231034
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	−21.0000	−0.692349
$921$	−32.0000	−1.05444
$922$	−23.0000	−0.757465
$923$	0	0
$924$	1.00000	0.0328976
$925$	20.0000	0.657596
$926$	23.0000	0.755827
$927$	−7.00000	−0.229910
$928$	−3.00000	−0.0984798
$929$	−48.0000	−1.57483	−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000	−1.57483	−0.787414	+	0.616424i	$0.788581\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−12.0000	−0.393073
$933$	−2.00000	−0.0654771
$934$	−3.00000	−0.0981630
$935$	21.0000	0.686773
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	−6.00000	−0.195907
$939$	0	0
$940$	0	0
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	5.00000	0.162909
$943$	28.0000	0.911805
$944$	−4.00000	−0.130189
$945$	3.00000	0.0975900
$946$	11.0000	0.357641
$947$	11.0000	0.357452	0.178726	−	0.983899i	$-0.442802\pi$
$947$	11.0000	0.357452	0.178726	+	0.983899i	$0.442802\pi$
$948$	10.0000	0.324785
$949$	0	0
$950$	4.00000	0.129777
$951$	−28.0000	−0.907962
$952$	−7.00000	−0.226871
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	14.0000	0.453267
$955$	45.0000	1.45617
$956$	−20.0000	−0.646846
$957$	3.00000	0.0969762
$958$	−11.0000	−0.355394
$959$	−15.0000	−0.484375
$960$	3.00000	0.0968246
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−18.0000	−0.580042
$964$	18.0000	0.579741
$965$	−24.0000	−0.772587
$966$	−7.00000	−0.225221
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	10.0000	0.321412
$969$	7.00000	0.224872
$970$	−18.0000	−0.577945
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	−1.00000	−0.0320750
$973$	4.00000	0.128234
$974$	−32.0000	−1.02535
$975$	0	0
$976$	1.00000	0.0320092
$977$	7.00000	0.223950	0.111975	−	0.993711i	$-0.464282\pi$
$977$	7.00000	0.223950	0.111975	+	0.993711i	$0.464282\pi$
$978$	−10.0000	−0.319765
$979$	−6.00000	−0.191761
$980$	−3.00000	−0.0958315
$981$	7.00000	0.223493
$982$	−2.00000	−0.0638226
$983$	19.0000	0.606006	0.303003	−	0.952990i	$-0.402011\pi$
$983$	19.0000	0.606006	0.303003	+	0.952990i	$0.402011\pi$
$984$	−4.00000	−0.127515
$985$	−60.0000	−1.91176
$986$	−21.0000	−0.668776
$987$	0	0
$988$	0	0
$989$	−77.0000	−2.44846
$990$	−3.00000	−0.0953463
$991$	34.0000	1.08005	0.540023	−	0.841650i	$-0.318416\pi$
$991$	34.0000	1.08005	0.540023	+	0.841650i	$0.318416\pi$
$992$	0	0
$993$	−14.0000	−0.444277
$994$	−12.0000	−0.380617
$995$	−9.00000	−0.285319
$996$	−14.0000	−0.443607
$997$	54.0000	1.71020	0.855099	−	0.518465i	$-0.173497\pi$
$997$	54.0000	1.71020	0.855099	+	0.518465i	$0.173497\pi$
$998$	16.0000	0.506471
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.b.1.1		1
13.12	even	2		546.2.a.e.1.1	✓	1
39.38	odd	2		1638.2.a.a.1.1		1
52.51	odd	2		4368.2.a.z.1.1		1
91.90	odd	2		3822.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.a.e.1.1	✓	1	13.12	even	2
1638.2.a.a.1.1		1	39.38	odd	2
3822.2.a.bc.1.1		1	91.90	odd	2
4368.2.a.z.1.1		1	52.51	odd	2
7098.2.a.b.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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