LMFDB - Embedded newform 7098.2.a.a.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:	$2$
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -5.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -3.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} -9.00000 q^{29} -3.00000 q^{30} -1.00000 q^{32} +4.00000 q^{33} +5.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} -7.00000 q^{37} +4.00000 q^{38} +3.00000 q^{40} -7.00000 q^{41} +1.00000 q^{42} +8.00000 q^{43} -4.00000 q^{44} -3.00000 q^{45} +4.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.00000 q^{50} +5.00000 q^{51} +7.00000 q^{53} +1.00000 q^{54} +12.0000 q^{55} -1.00000 q^{56} +4.00000 q^{57} +9.00000 q^{58} +8.00000 q^{59} +3.00000 q^{60} +7.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -12.0000 q^{67} -5.00000 q^{68} +4.00000 q^{69} +3.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} +1.00000 q^{73} +7.00000 q^{74} -4.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} -16.0000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +7.00000 q^{82} +8.00000 q^{83} -1.00000 q^{84} +15.0000 q^{85} -8.00000 q^{86} +9.00000 q^{87} +4.00000 q^{88} +6.00000 q^{89} +3.00000 q^{90} -4.00000 q^{92} +12.0000 q^{94} +12.0000 q^{95} +1.00000 q^{96} -18.0000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−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
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	−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	$0.651732\pi$
$20$	−3.00000	−0.670820
$21$	−1.00000	−0.218218
$22$	4.00000	0.852803
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	−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
$33$	4.00000	0.696311
$34$	5.00000	0.857493
$35$	−3.00000	−0.507093
$36$	1.00000	0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	3.00000	0.474342
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	1.00000	0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−4.00000	−0.603023
$45$	−3.00000	−0.447214
$46$	4.00000	0.589768
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−4.00000	−0.565685
$51$	5.00000	0.700140
$52$	0	0
$53$	7.00000	0.961524	0.480762	−	0.876851i	$-0.340360\pi$
$53$	7.00000	0.961524	0.480762	+	0.876851i	$0.340360\pi$
$54$	1.00000	0.136083
$55$	12.0000	1.61808
$56$	−1.00000	−0.133631
$57$	4.00000	0.529813
$58$	9.00000	1.18176
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	3.00000	0.387298
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−5.00000	−0.606339
$69$	4.00000	0.481543
$70$	3.00000	0.358569
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	7.00000	0.813733
$75$	−4.00000	−0.461880
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	7.00000	0.773021
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−1.00000	−0.109109
$85$	15.0000	1.62698
$86$	−8.00000	−0.862662
$87$	9.00000	0.964901
$88$	4.00000	0.426401
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	3.00000	0.316228
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	12.0000	1.23771
$95$	12.0000	1.23117
$96$	1.00000	0.102062
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	−1.00000	−0.101015
$99$	−4.00000	−0.402015
$100$	4.00000	0.400000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	−5.00000	−0.495074
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	−7.00000	−0.679900
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−1.00000	−0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	−12.0000	−1.14416
$111$	7.00000	0.664411
$112$	1.00000	0.0944911
$113$	−5.00000	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$114$	−4.00000	−0.374634
$115$	12.0000	1.11901
$116$	−9.00000	−0.835629
$117$	0	0
$118$	−8.00000	−0.736460
$119$	−5.00000	−0.458349
$120$	−3.00000	−0.273861
$121$	5.00000	0.454545
$122$	−7.00000	−0.633750
$123$	7.00000	0.631169
$124$	0	0
$125$	3.00000	0.268328
$126$	−1.00000	−0.0890871
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	4.00000	0.348155
$133$	−4.00000	−0.346844
$134$	12.0000	1.03664
$135$	3.00000	0.258199
$136$	5.00000	0.428746
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	−4.00000	−0.340503
$139$	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
$141$	12.0000	1.01058
$142$	12.0000	1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	27.0000	2.24223
$146$	−1.00000	−0.0827606
$147$	−1.00000	−0.0824786
$148$	−7.00000	−0.575396
$149$	−7.00000	−0.573462	−0.286731	−	0.958011i	$-0.592569\pi$
$149$	−7.00000	−0.573462	−0.286731	+	0.958011i	$0.592569\pi$
$150$	4.00000	0.326599
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	4.00000	0.324443
$153$	−5.00000	−0.404226
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	16.0000	1.27289
$159$	−7.00000	−0.555136
$160$	3.00000	0.237171
$161$	−4.00000	−0.315244
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−7.00000	−0.546608
$165$	−12.0000	−0.934199
$166$	−8.00000	−0.620920
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	−15.0000	−1.15045
$171$	−4.00000	−0.305888
$172$	8.00000	0.609994
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−9.00000	−0.682288
$175$	4.00000	0.302372
$176$	−4.00000	−0.301511
$177$	−8.00000	−0.601317
$178$	−6.00000	−0.449719
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−3.00000	−0.223607
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	−7.00000	−0.517455
$184$	4.00000	0.294884
$185$	21.0000	1.54395
$186$	0	0
$187$	20.0000	1.46254
$188$	−12.0000	−0.875190
$189$	−1.00000	−0.0727393
$190$	−12.0000	−0.870572
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−1.00000	−0.0721688
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	1.00000	0.0714286
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	4.00000	0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−4.00000	−0.282843
$201$	12.0000	0.846415
$202$	−3.00000	−0.211079
$203$	−9.00000	−0.631676
$204$	5.00000	0.350070
$205$	21.0000	1.46670
$206$	4.00000	0.278693
$207$	−4.00000	−0.278019
$208$	0	0
$209$	16.0000	1.10674
$210$	−3.00000	−0.207020
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	7.00000	0.480762
$213$	12.0000	0.822226
$214$	0	0
$215$	−24.0000	−1.63679
$216$	1.00000	0.0680414
$217$	0	0
$218$	14.0000	0.948200
$219$	−1.00000	−0.0675737
$220$	12.0000	0.809040
$221$	0	0
$222$	−7.00000	−0.469809
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	−1.00000	−0.0668153
$225$	4.00000	0.266667
$226$	5.00000	0.332595
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	4.00000	0.264906
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−12.0000	−0.791257
$231$	4.00000	0.263181
$232$	9.00000	0.590879
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	36.0000	2.34838
$236$	8.00000	0.520756
$237$	16.0000	1.03931
$238$	5.00000	0.324102
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	3.00000	0.193649
$241$	9.00000	0.579741	0.289870	−	0.957066i	$-0.406388\pi$
$241$	9.00000	0.579741	0.289870	+	0.957066i	$0.406388\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	7.00000	0.448129
$245$	−3.00000	−0.191663
$246$	−7.00000	−0.446304
$247$	0	0
$248$	0	0
$249$	−8.00000	−0.506979
$250$	−3.00000	−0.189737
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	1.00000	0.0629941
$253$	16.0000	1.00591
$254$	8.00000	0.501965
$255$	−15.0000	−0.939336
$256$	1.00000	0.0625000
$257$	7.00000	0.436648	0.218324	−	0.975876i	$-0.429941\pi$
$257$	7.00000	0.436648	0.218324	+	0.975876i	$0.429941\pi$
$258$	8.00000	0.498058
$259$	−7.00000	−0.434959
$260$	0	0
$261$	−9.00000	−0.557086
$262$	12.0000	0.741362
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−4.00000	−0.246183
$265$	−21.0000	−1.29002
$266$	4.00000	0.245256
$267$	−6.00000	−0.367194
$268$	−12.0000	−0.733017
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	−3.00000	−0.182574
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	3.00000	0.181237
$275$	−16.0000	−0.964836
$276$	4.00000	0.240772
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	3.00000	0.179284
$281$	−15.0000	−0.894825	−0.447412	−	0.894328i	$-0.647654\pi$
$281$	−15.0000	−0.894825	−0.447412	+	0.894328i	$0.647654\pi$
$282$	−12.0000	−0.714590
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	−12.0000	−0.710819
$286$	0	0
$287$	−7.00000	−0.413197
$288$	−1.00000	−0.0589256
$289$	8.00000	0.470588
$290$	−27.0000	−1.58549
$291$	18.0000	1.05518
$292$	1.00000	0.0585206
$293$	33.0000	1.92788	0.963940	−	0.266119i	$-0.0857413\pi$
$293$	33.0000	1.92788	0.963940	+	0.266119i	$0.0857413\pi$
$294$	1.00000	0.0583212
$295$	−24.0000	−1.39733
$296$	7.00000	0.406867
$297$	4.00000	0.232104
$298$	7.00000	0.405499
$299$	0	0
$300$	−4.00000	−0.230940
$301$	8.00000	0.461112
$302$	−8.00000	−0.460348
$303$	−3.00000	−0.172345
$304$	−4.00000	−0.229416
$305$	−21.0000	−1.20246
$306$	5.00000	0.285831
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	−4.00000	−0.227921
$309$	4.00000	0.227552
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	13.0000	0.733632
$315$	−3.00000	−0.169031
$316$	−16.0000	−0.900070
$317$	−35.0000	−1.96580	−0.982898	−	0.184151i	$-0.941046\pi$
$317$	−35.0000	−1.96580	−0.982898	+	0.184151i	$0.941046\pi$
$318$	7.00000	0.392541
$319$	36.0000	2.01561
$320$	−3.00000	−0.167705
$321$	0	0
$322$	4.00000	0.222911
$323$	20.0000	1.11283
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	14.0000	0.774202
$328$	7.00000	0.386510
$329$	−12.0000	−0.661581
$330$	12.0000	0.660578
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	8.00000	0.439057
$333$	−7.00000	−0.383598
$334$	−8.00000	−0.437741
$335$	36.0000	1.96689
$336$	−1.00000	−0.0545545
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	0	0
$339$	5.00000	0.271563
$340$	15.0000	0.813489
$341$	0	0
$342$	4.00000	0.216295
$343$	1.00000	0.0539949
$344$	−8.00000	−0.431331
$345$	−12.0000	−0.646058
$346$	−14.0000	−0.752645
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	9.00000	0.482451
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	4.00000	0.213201
$353$	25.0000	1.33062	0.665308	−	0.746569i	$-0.268300\pi$
$353$	25.0000	1.33062	0.665308	+	0.746569i	$0.268300\pi$
$354$	8.00000	0.425195
$355$	36.0000	1.91068
$356$	6.00000	0.317999
$357$	5.00000	0.264628
$358$	−20.0000	−1.05703
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	3.00000	0.158114
$361$	−3.00000	−0.157895
$362$	5.00000	0.262794
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−3.00000	−0.157027
$366$	7.00000	0.365896
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−4.00000	−0.208514
$369$	−7.00000	−0.364405
$370$	−21.0000	−1.09174
$371$	7.00000	0.363422
$372$	0	0
$373$	15.0000	0.776671	0.388335	−	0.921518i	$-0.373050\pi$
$373$	15.0000	0.776671	0.388335	+	0.921518i	$0.373050\pi$
$374$	−20.0000	−1.03418
$375$	−3.00000	−0.154919
$376$	12.0000	0.618853
$377$	0	0
$378$	1.00000	0.0514344
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	12.0000	0.615587
$381$	8.00000	0.409852
$382$	24.0000	1.22795
$383$	28.0000	1.43073	0.715367	−	0.698749i	$-0.246260\pi$
$383$	28.0000	1.43073	0.715367	+	0.698749i	$0.246260\pi$
$384$	1.00000	0.0510310
$385$	12.0000	0.611577
$386$	19.0000	0.967075
$387$	8.00000	0.406663
$388$	−18.0000	−0.913812
$389$	7.00000	0.354914	0.177457	−	0.984129i	$-0.443213\pi$
$389$	7.00000	0.354914	0.177457	+	0.984129i	$0.443213\pi$
$390$	0	0
$391$	20.0000	1.01144
$392$	−1.00000	−0.0505076
$393$	12.0000	0.605320
$394$	22.0000	1.10834
$395$	48.0000	2.41514
$396$	−4.00000	−0.201008
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	4.00000	0.200000
$401$	17.0000	0.848939	0.424470	−	0.905442i	$-0.360461\pi$
$401$	17.0000	0.848939	0.424470	+	0.905442i	$0.360461\pi$
$402$	−12.0000	−0.598506
$403$	0	0
$404$	3.00000	0.149256
$405$	−3.00000	−0.149071
$406$	9.00000	0.446663
$407$	28.0000	1.38791
$408$	−5.00000	−0.247537
$409$	13.0000	0.642809	0.321404	−	0.946942i	$-0.395845\pi$
$409$	13.0000	0.642809	0.321404	+	0.946942i	$0.395845\pi$
$410$	−21.0000	−1.03712
$411$	3.00000	0.147979
$412$	−4.00000	−0.197066
$413$	8.00000	0.393654
$414$	4.00000	0.196589
$415$	−24.0000	−1.17811
$416$	0	0
$417$	−4.00000	−0.195881
$418$	−16.0000	−0.782586
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	3.00000	0.146385
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	8.00000	0.389434
$423$	−12.0000	−0.583460
$424$	−7.00000	−0.339950
$425$	−20.0000	−0.970143
$426$	−12.0000	−0.581402
$427$	7.00000	0.338754
$428$	0	0
$429$	0	0
$430$	24.0000	1.15738
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	−1.00000	−0.0481125
$433$	31.0000	1.48976	0.744882	−	0.667196i	$-0.232506\pi$
$433$	31.0000	1.48976	0.744882	+	0.667196i	$0.232506\pi$
$434$	0	0
$435$	−27.0000	−1.29455
$436$	−14.0000	−0.670478
$437$	16.0000	0.765384
$438$	1.00000	0.0477818
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	−12.0000	−0.572078
$441$	1.00000	0.0476190
$442$	0	0
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	7.00000	0.332205
$445$	−18.0000	−0.853282
$446$	−28.0000	−1.32584
$447$	7.00000	0.331089
$448$	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	$0.392833\pi$
$450$	−4.00000	−0.188562
$451$	28.0000	1.31847
$452$	−5.00000	−0.235180
$453$	−8.00000	−0.375873
$454$	24.0000	1.12638
$455$	0	0
$456$	−4.00000	−0.187317
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	6.00000	0.280362
$459$	5.00000	0.233380
$460$	12.0000	0.559503
$461$	−19.0000	−0.884918	−0.442459	−	0.896789i	$-0.645894\pi$
$461$	−19.0000	−0.884918	−0.442459	+	0.896789i	$0.645894\pi$
$462$	−4.00000	−0.186097
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	6.00000	0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	−36.0000	−1.66056
$471$	13.0000	0.599008
$472$	−8.00000	−0.368230
$473$	−32.0000	−1.47136
$474$	−16.0000	−0.734904
$475$	−16.0000	−0.734130
$476$	−5.00000	−0.229175
$477$	7.00000	0.320508
$478$	−4.00000	−0.182956
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	−3.00000	−0.136931
$481$	0	0
$482$	−9.00000	−0.409939
$483$	4.00000	0.182006
$484$	5.00000	0.227273
$485$	54.0000	2.45201
$486$	1.00000	0.0453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−7.00000	−0.316875
$489$	4.00000	0.180886
$490$	3.00000	0.135526
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	7.00000	0.315584
$493$	45.0000	2.02670
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	−12.0000	−0.538274
$498$	8.00000	0.358489
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	3.00000	0.134164
$501$	−8.00000	−0.357414
$502$	12.0000	0.535586
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	−1.00000	−0.0445435
$505$	−9.00000	−0.400495
$506$	−16.0000	−0.711287
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	15.0000	0.664211
$511$	1.00000	0.0442374
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−7.00000	−0.308757
$515$	12.0000	0.528783
$516$	−8.00000	−0.352180
$517$	48.0000	2.11104
$518$	7.00000	0.307562
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−17.0000	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$522$	9.00000	0.393919
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−12.0000	−0.524222
$525$	−4.00000	−0.174574
$526$	−8.00000	−0.348817
$527$	0	0
$528$	4.00000	0.174078
$529$	−7.00000	−0.304348
$530$	21.0000	0.912182
$531$	8.00000	0.347170
$532$	−4.00000	−0.173422
$533$	0	0
$534$	6.00000	0.259645
$535$	0	0
$536$	12.0000	0.518321
$537$	−20.0000	−0.863064
$538$	−14.0000	−0.603583
$539$	−4.00000	−0.172292
$540$	3.00000	0.129099
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	24.0000	1.03089
$543$	5.00000	0.214571
$544$	5.00000	0.214373
$545$	42.0000	1.79908
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	−3.00000	−0.128154
$549$	7.00000	0.298753
$550$	16.0000	0.682242
$551$	36.0000	1.53365
$552$	−4.00000	−0.170251
$553$	−16.0000	−0.680389
$554$	−19.0000	−0.807233
$555$	−21.0000	−0.891400
$556$	4.00000	0.169638
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	0	0
$559$	0	0
$560$	−3.00000	−0.126773
$561$	−20.0000	−0.844401
$562$	15.0000	0.632737
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	12.0000	0.505291
$565$	15.0000	0.631055
$566$	4.00000	0.168133
$567$	1.00000	0.0419961
$568$	12.0000	0.503509
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	12.0000	0.502625
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	7.00000	0.292174
$575$	−16.0000	−0.667246
$576$	1.00000	0.0416667
$577$	5.00000	0.208153	0.104076	−	0.994569i	$-0.466811\pi$
$577$	5.00000	0.208153	0.104076	+	0.994569i	$0.466811\pi$
$578$	−8.00000	−0.332756
$579$	19.0000	0.789613
$580$	27.0000	1.12111
$581$	8.00000	0.331896
$582$	−18.0000	−0.746124
$583$	−28.0000	−1.15964
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	−33.0000	−1.36322
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	−1.00000	−0.0412393
$589$	0	0
$590$	24.0000	0.988064
$591$	22.0000	0.904959
$592$	−7.00000	−0.287698
$593$	33.0000	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$593$	33.0000	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$594$	−4.00000	−0.164122
$595$	15.0000	0.614940
$596$	−7.00000	−0.286731
$597$	0	0
$598$	0	0
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	4.00000	0.163299
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	−8.00000	−0.326056
$603$	−12.0000	−0.488678
$604$	8.00000	0.325515
$605$	−15.0000	−0.609837
$606$	3.00000	0.121867
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	4.00000	0.162221
$609$	9.00000	0.364698
$610$	21.0000	0.850265
$611$	0	0
$612$	−5.00000	−0.202113
$613$	1.00000	0.0403896	0.0201948	−	0.999796i	$-0.493571\pi$
$613$	1.00000	0.0403896	0.0201948	+	0.999796i	$0.493571\pi$
$614$	4.00000	0.161427
$615$	−21.0000	−0.846802
$616$	4.00000	0.161165
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	−4.00000	−0.160904
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−8.00000	−0.320771
$623$	6.00000	0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	6.00000	0.239808
$627$	−16.0000	−0.638978
$628$	−13.0000	−0.518756
$629$	35.0000	1.39554
$630$	3.00000	0.119523
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	16.0000	0.636446
$633$	8.00000	0.317971
$634$	35.0000	1.39003
$635$	24.0000	0.952411
$636$	−7.00000	−0.277568
$637$	0	0
$638$	−36.0000	−1.42525
$639$	−12.0000	−0.474713
$640$	3.00000	0.118585
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	−12.0000	−0.473234	−0.236617	−	0.971603i	$-0.576039\pi$
$643$	−12.0000	−0.473234	−0.236617	+	0.971603i	$0.576039\pi$
$644$	−4.00000	−0.157622
$645$	24.0000	0.944999
$646$	−20.0000	−0.786889
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	−32.0000	−1.25611
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−14.0000	−0.547443
$655$	36.0000	1.40664
$656$	−7.00000	−0.273304
$657$	1.00000	0.0390137
$658$	12.0000	0.467809
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	−12.0000	−0.467099
$661$	1.00000	0.0388955	0.0194477	−	0.999811i	$-0.493809\pi$
$661$	1.00000	0.0388955	0.0194477	+	0.999811i	$0.493809\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−8.00000	−0.310460
$665$	12.0000	0.465340
$666$	7.00000	0.271244
$667$	36.0000	1.39393
$668$	8.00000	0.309529
$669$	−28.0000	−1.08254
$670$	−36.0000	−1.39080
$671$	−28.0000	−1.08093
$672$	1.00000	0.0385758
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	9.00000	0.346667
$675$	−4.00000	−0.153960
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−5.00000	−0.192024
$679$	−18.0000	−0.690777
$680$	−15.0000	−0.575224
$681$	24.0000	0.919682
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	−4.00000	−0.152944
$685$	9.00000	0.343872
$686$	−1.00000	−0.0381802
$687$	6.00000	0.228914
$688$	8.00000	0.304997
$689$	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	$0.224789\pi$
$692$	14.0000	0.532200
$693$	−4.00000	−0.151947
$694$	−20.0000	−0.759190
$695$	−12.0000	−0.455186
$696$	−9.00000	−0.341144
$697$	35.0000	1.32572
$698$	14.0000	0.529908
$699$	6.00000	0.226941
$700$	4.00000	0.151186
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	28.0000	1.05604
$704$	−4.00000	−0.150756
$705$	−36.0000	−1.35584
$706$	−25.0000	−0.940887
$707$	3.00000	0.112827
$708$	−8.00000	−0.300658
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	−36.0000	−1.35106
$711$	−16.0000	−0.600047
$712$	−6.00000	−0.224860
$713$	0	0
$714$	−5.00000	−0.187120
$715$	0	0
$716$	20.0000	0.747435
$717$	−4.00000	−0.149383
$718$	−12.0000	−0.447836
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	−3.00000	−0.111803
$721$	−4.00000	−0.148968
$722$	3.00000	0.111648
$723$	−9.00000	−0.334714
$724$	−5.00000	−0.185824
$725$	−36.0000	−1.33701
$726$	5.00000	0.185567
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	3.00000	0.111035
$731$	−40.0000	−1.47945
$732$	−7.00000	−0.258727
$733$	−23.0000	−0.849524	−0.424762	−	0.905305i	$-0.639642\pi$
$733$	−23.0000	−0.849524	−0.424762	+	0.905305i	$0.639642\pi$
$734$	4.00000	0.147643
$735$	3.00000	0.110657
$736$	4.00000	0.147442
$737$	48.0000	1.76810
$738$	7.00000	0.257674
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	21.0000	0.771975
$741$	0	0
$742$	−7.00000	−0.256978
$743$	−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
$745$	21.0000	0.769380
$746$	−15.0000	−0.549189
$747$	8.00000	0.292705
$748$	20.0000	0.731272
$749$	0	0
$750$	3.00000	0.109545
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	−12.0000	−0.437595
$753$	12.0000	0.437304
$754$	0	0
$755$	−24.0000	−0.873449
$756$	−1.00000	−0.0363696
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−28.0000	−1.01701
$759$	−16.0000	−0.580763
$760$	−12.0000	−0.435286
$761$	−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
$763$	−14.0000	−0.506834
$764$	−24.0000	−0.868290
$765$	15.0000	0.542326
$766$	−28.0000	−1.01168
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	−12.0000	−0.432450
$771$	−7.00000	−0.252099
$772$	−19.0000	−0.683825
$773$	−38.0000	−1.36677	−0.683383	−	0.730061i	$-0.739492\pi$
$773$	−38.0000	−1.36677	−0.683383	+	0.730061i	$0.739492\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	18.0000	0.646162
$777$	7.00000	0.251124
$778$	−7.00000	−0.250962
$779$	28.0000	1.00320
$780$	0	0
$781$	48.0000	1.71758
$782$	−20.0000	−0.715199
$783$	9.00000	0.321634
$784$	1.00000	0.0357143
$785$	39.0000	1.39197
$786$	−12.0000	−0.428026
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	−22.0000	−0.783718
$789$	−8.00000	−0.284808
$790$	−48.0000	−1.70776
$791$	−5.00000	−0.177780
$792$	4.00000	0.142134
$793$	0	0
$794$	−2.00000	−0.0709773
$795$	21.0000	0.744793
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	−4.00000	−0.141598
$799$	60.0000	2.12265
$800$	−4.00000	−0.141421
$801$	6.00000	0.212000
$802$	−17.0000	−0.600291
$803$	−4.00000	−0.141157
$804$	12.0000	0.423207
$805$	12.0000	0.422944
$806$	0	0
$807$	−14.0000	−0.492823
$808$	−3.00000	−0.105540
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	3.00000	0.105409
$811$	−24.0000	−0.842754	−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000	−0.842754	−0.421377	+	0.906886i	$0.638453\pi$
$812$	−9.00000	−0.315838
$813$	24.0000	0.841717
$814$	−28.0000	−0.981399
$815$	12.0000	0.420342
$816$	5.00000	0.175035
$817$	−32.0000	−1.11954
$818$	−13.0000	−0.454534
$819$	0	0
$820$	21.0000	0.733352
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	−3.00000	−0.104637
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	4.00000	0.139347
$825$	16.0000	0.557048
$826$	−8.00000	−0.278356
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	−4.00000	−0.139010
$829$	31.0000	1.07667	0.538337	−	0.842729i	$-0.319053\pi$
$829$	31.0000	1.07667	0.538337	+	0.842729i	$0.319053\pi$
$830$	24.0000	0.833052
$831$	−19.0000	−0.659103
$832$	0	0
$833$	−5.00000	−0.173240
$834$	4.00000	0.138509
$835$	−24.0000	−0.830554
$836$	16.0000	0.553372
$837$	0	0
$838$	16.0000	0.552711
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	−3.00000	−0.103510
$841$	52.0000	1.79310
$842$	−1.00000	−0.0344623
$843$	15.0000	0.516627
$844$	−8.00000	−0.275371
$845$	0	0
$846$	12.0000	0.412568
$847$	5.00000	0.171802
$848$	7.00000	0.240381
$849$	4.00000	0.137280
$850$	20.0000	0.685994
$851$	28.0000	0.959828
$852$	12.0000	0.411113
$853$	49.0000	1.67773	0.838864	−	0.544341i	$-0.183220\pi$
$853$	49.0000	1.67773	0.838864	+	0.544341i	$0.183220\pi$
$854$	−7.00000	−0.239535
$855$	12.0000	0.410391
$856$	0	0
$857$	31.0000	1.05894	0.529470	−	0.848329i	$-0.322391\pi$
$857$	31.0000	1.05894	0.529470	+	0.848329i	$0.322391\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−24.0000	−0.818393
$861$	7.00000	0.238559
$862$	−4.00000	−0.136241
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	1.00000	0.0340207
$865$	−42.0000	−1.42804
$866$	−31.0000	−1.05342
$867$	−8.00000	−0.271694
$868$	0	0
$869$	64.0000	2.17105
$870$	27.0000	0.915386
$871$	0	0
$872$	14.0000	0.474100
$873$	−18.0000	−0.609208
$874$	−16.0000	−0.541208
$875$	3.00000	0.101419
$876$	−1.00000	−0.0337869
$877$	−3.00000	−0.101303	−0.0506514	−	0.998716i	$-0.516130\pi$
$877$	−3.00000	−0.101303	−0.0506514	+	0.998716i	$0.516130\pi$
$878$	16.0000	0.539974
$879$	−33.0000	−1.11306
$880$	12.0000	0.404520
$881$	−41.0000	−1.38133	−0.690663	−	0.723177i	$-0.742681\pi$
$881$	−41.0000	−1.38133	−0.690663	+	0.723177i	$0.742681\pi$
$882$	−1.00000	−0.0336718
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	16.0000	0.537531
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−7.00000	−0.234905
$889$	−8.00000	−0.268311
$890$	18.0000	0.603361
$891$	−4.00000	−0.134005
$892$	28.0000	0.937509
$893$	48.0000	1.60626
$894$	−7.00000	−0.234115
$895$	−60.0000	−2.00558
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−14.0000	−0.467186
$899$	0	0
$900$	4.00000	0.133333
$901$	−35.0000	−1.16602
$902$	−28.0000	−0.932298
$903$	−8.00000	−0.266223
$904$	5.00000	0.166298
$905$	15.0000	0.498617
$906$	8.00000	0.265782
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−24.0000	−0.796468
$909$	3.00000	0.0995037
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	4.00000	0.132453
$913$	−32.0000	−1.05905
$914$	−1.00000	−0.0330771
$915$	21.0000	0.694239
$916$	−6.00000	−0.198246
$917$	−12.0000	−0.396275
$918$	−5.00000	−0.165025
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−12.0000	−0.395628
$921$	4.00000	0.131804
$922$	19.0000	0.625732
$923$	0	0
$924$	4.00000	0.131590
$925$	−28.0000	−0.920634
$926$	32.0000	1.05159
$927$	−4.00000	−0.131377
$928$	9.00000	0.295439
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	−6.00000	−0.196537
$933$	−8.00000	−0.261908
$934$	12.0000	0.392652
$935$	−60.0000	−1.96221
$936$	0	0
$937$	−49.0000	−1.60076	−0.800380	−	0.599493i	$-0.795369\pi$
$937$	−49.0000	−1.60076	−0.800380	+	0.599493i	$0.795369\pi$
$938$	12.0000	0.391814
$939$	6.00000	0.195803
$940$	36.0000	1.17419
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	−13.0000	−0.423563
$943$	28.0000	0.911805
$944$	8.00000	0.260378
$945$	3.00000	0.0975900
$946$	32.0000	1.04041
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	16.0000	0.519656
$949$	0	0
$950$	16.0000	0.519109
$951$	35.0000	1.13495
$952$	5.00000	0.162051
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−7.00000	−0.226633
$955$	72.0000	2.32987
$956$	4.00000	0.129369
$957$	−36.0000	−1.16371
$958$	28.0000	0.904639
$959$	−3.00000	−0.0968751
$960$	3.00000	0.0968246
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	9.00000	0.289870
$965$	57.0000	1.83489
$966$	−4.00000	−0.128698
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−5.00000	−0.160706
$969$	−20.0000	−0.642493
$970$	−54.0000	−1.73384
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	−1.00000	−0.0320750
$973$	4.00000	0.128234
$974$	4.00000	0.128168
$975$	0	0
$976$	7.00000	0.224065
$977$	−47.0000	−1.50366	−0.751832	−	0.659355i	$-0.770829\pi$
$977$	−47.0000	−1.50366	−0.751832	+	0.659355i	$0.770829\pi$
$978$	−4.00000	−0.127906
$979$	−24.0000	−0.767043
$980$	−3.00000	−0.0958315
$981$	−14.0000	−0.446986
$982$	−20.0000	−0.638226
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	−7.00000	−0.223152
$985$	66.0000	2.10293
$986$	−45.0000	−1.43309
$987$	12.0000	0.381964
$988$	0	0
$989$	−32.0000	−1.01754
$990$	−12.0000	−0.381385
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	12.0000	0.380617
$995$	0	0
$996$	−8.00000	−0.253490
$997$	39.0000	1.23514	0.617571	−	0.786515i	$-0.288117\pi$
$997$	39.0000	1.23514	0.617571	+	0.786515i	$0.288117\pi$
$998$	4.00000	0.126618
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.a.1.1		1
13.4	even	6		546.2.l.b.211.1	✓	2
13.10	even	6		546.2.l.b.295.1	yes	2
13.12	even	2		7098.2.a.v.1.1		1
39.17	odd	6		1638.2.r.o.757.1		2
39.23	odd	6		1638.2.r.o.1387.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.l.b.211.1	✓	2	13.4	even	6
546.2.l.b.295.1	yes	2	13.10	even	6
1638.2.r.o.757.1		2	39.17	odd	6
1638.2.r.o.1387.1		2	39.23	odd	6
7098.2.a.a.1.1		1	1.1	even	1	trivial
7098.2.a.v.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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