LMFDB - Embedded newform 7098.2.a.o.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:	$1$
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -3.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +7.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +3.00000 q^{38} -1.00000 q^{40} -4.00000 q^{41} +1.00000 q^{42} +5.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} -7.00000 q^{51} -6.00000 q^{53} -1.00000 q^{54} +3.00000 q^{55} +1.00000 q^{56} -3.00000 q^{57} +1.00000 q^{58} +10.0000 q^{59} +1.00000 q^{60} -13.0000 q^{61} -8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +8.00000 q^{67} -7.00000 q^{68} -1.00000 q^{69} +1.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} -13.0000 q^{73} +1.00000 q^{74} -4.00000 q^{75} -3.00000 q^{76} -3.00000 q^{77} -12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} -7.00000 q^{85} -5.00000 q^{86} -1.00000 q^{87} -3.00000 q^{88} +12.0000 q^{89} -1.00000 q^{90} -1.00000 q^{92} +8.00000 q^{93} -3.00000 q^{95} -1.00000 q^{96} -6.00000 q^{97} -1.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	−1.00000	−0.235702
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	1.00000	0.223607
$21$	−1.00000	−0.218218
$22$	−3.00000	−0.639602
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	−1.00000	−0.204124
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	−1.00000	−0.182574
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	7.00000	1.20049
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−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
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	3.00000	0.452267
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	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
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	3.00000	0.404520
$56$	1.00000	0.133631
$57$	−3.00000	−0.397360
$58$	1.00000	0.131306
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	1.00000	0.129099
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	−8.00000	−1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−7.00000	−0.848875
$69$	−1.00000	−0.120386
$70$	1.00000	0.119523
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	1.00000	0.116248
$75$	−4.00000	−0.461880
$76$	−3.00000	−0.344124
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	−1.00000	−0.109109
$85$	−7.00000	−0.759257
$86$	−5.00000	−0.539164
$87$	−1.00000	−0.107211
$88$	−3.00000	−0.319801
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−1.00000	−0.104257
$93$	8.00000	0.829561
$94$	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
$99$	3.00000	0.301511
$100$	−4.00000	−0.400000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	7.00000	0.693103
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	6.00000	0.582772
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	1.00000	0.0962250
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	−3.00000	−0.286039
$111$	−1.00000	−0.0949158
$112$	−1.00000	−0.0944911
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	3.00000	0.280976
$115$	−1.00000	−0.0932505
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	−10.0000	−0.920575
$119$	7.00000	0.641689
$120$	−1.00000	−0.0912871
$121$	−2.00000	−0.181818
$122$	13.0000	1.17696
$123$	−4.00000	−0.360668
$124$	8.00000	0.718421
$125$	−9.00000	−0.804984
$126$	1.00000	0.0890871
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	5.00000	0.440225
$130$	0	0
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	3.00000	0.261116
$133$	3.00000	0.260133
$134$	−8.00000	−0.691095
$135$	1.00000	0.0860663
$136$	7.00000	0.600245
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	1.00000	0.0851257
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	6.00000	0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	−1.00000	−0.0830455
$146$	13.0000	1.07589
$147$	1.00000	0.0824786
$148$	−1.00000	−0.0821995
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	4.00000	0.326599
$151$	−11.0000	−0.895167	−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000	−0.895167	−0.447584	+	0.894242i	$0.647715\pi$
$152$	3.00000	0.243332
$153$	−7.00000	−0.565916
$154$	3.00000	0.241747
$155$	8.00000	0.642575
$156$	0	0
$157$	19.0000	1.51637	0.758183	−	0.652042i	$-0.226088\pi$
$157$	19.0000	1.51637	0.758183	+	0.652042i	$0.226088\pi$
$158$	12.0000	0.954669
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−4.00000	−0.312348
$165$	3.00000	0.233550
$166$	2.00000	0.155230
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	7.00000	0.536875
$171$	−3.00000	−0.229416
$172$	5.00000	0.381246
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	1.00000	0.0758098
$175$	4.00000	0.302372
$176$	3.00000	0.226134
$177$	10.0000	0.751646
$178$	−12.0000	−0.899438
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	1.00000	0.0745356
$181$	26.0000	1.93256	0.966282	−	0.257485i	$-0.0828937\pi$
$181$	26.0000	1.93256	0.966282	+	0.257485i	$0.0828937\pi$
$182$	0	0
$183$	−13.0000	−0.960988
$184$	1.00000	0.0737210
$185$	−1.00000	−0.0735215
$186$	−8.00000	−0.586588
$187$	−21.0000	−1.53567
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	3.00000	0.217643
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	1.00000	0.0721688
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	1.00000	0.0714286
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	−3.00000	−0.213201
$199$	−9.00000	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$200$	4.00000	0.282843
$201$	8.00000	0.564276
$202$	14.0000	0.985037
$203$	1.00000	0.0701862
$204$	−7.00000	−0.490098
$205$	−4.00000	−0.279372
$206$	−1.00000	−0.0696733
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−9.00000	−0.622543
$210$	1.00000	0.0690066
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	−6.00000	−0.412082
$213$	−6.00000	−0.411113
$214$	−2.00000	−0.136717
$215$	5.00000	0.340997
$216$	−1.00000	−0.0680414
$217$	−8.00000	−0.543075
$218$	7.00000	0.474100
$219$	−13.0000	−0.878459
$220$	3.00000	0.202260
$221$	0	0
$222$	1.00000	0.0671156
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	1.00000	0.0668153
$225$	−4.00000	−0.266667
$226$	10.0000	0.665190
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	−3.00000	−0.198680
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	1.00000	0.0659380
$231$	−3.00000	−0.197386
$232$	1.00000	0.0656532
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	10.0000	0.650945
$237$	−12.0000	−0.779484
$238$	−7.00000	−0.453743
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	1.00000	0.0645497
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	−13.0000	−0.832240
$245$	1.00000	0.0638877
$246$	4.00000	0.255031
$247$	0	0
$248$	−8.00000	−0.508001
$249$	−2.00000	−0.126745
$250$	9.00000	0.569210
$251$	−17.0000	−1.07303	−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000	−1.07303	−0.536515	+	0.843891i	$0.680260\pi$
$252$	−1.00000	−0.0629941
$253$	−3.00000	−0.188608
$254$	−4.00000	−0.250982
$255$	−7.00000	−0.438357
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−5.00000	−0.311286
$259$	1.00000	0.0621370
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	17.0000	1.05026
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−3.00000	−0.184637
$265$	−6.00000	−0.368577
$266$	−3.00000	−0.183942
$267$	12.0000	0.734388
$268$	8.00000	0.488678
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	−1.00000	−0.0608581
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−7.00000	−0.424437
$273$	0	0
$274$	17.0000	1.02701
$275$	−12.0000	−0.723627
$276$	−1.00000	−0.0601929
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	8.00000	0.479808
$279$	8.00000	0.478947
$280$	1.00000	0.0597614
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−6.00000	−0.356034
$285$	−3.00000	−0.177705
$286$	0	0
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	32.0000	1.88235
$290$	1.00000	0.0587220
$291$	−6.00000	−0.351726
$292$	−13.0000	−0.760767
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−1.00000	−0.0583212
$295$	10.0000	0.582223
$296$	1.00000	0.0581238
$297$	3.00000	0.174078
$298$	−22.0000	−1.27443
$299$	0	0
$300$	−4.00000	−0.230940
$301$	−5.00000	−0.288195
$302$	11.0000	0.632979
$303$	−14.0000	−0.804279
$304$	−3.00000	−0.172062
$305$	−13.0000	−0.744378
$306$	7.00000	0.400163
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	−3.00000	−0.170941
$309$	1.00000	0.0568880
$310$	−8.00000	−0.454369
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−19.0000	−1.07223
$315$	−1.00000	−0.0563436
$316$	−12.0000	−0.675053
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	6.00000	0.336463
$319$	−3.00000	−0.167968
$320$	1.00000	0.0559017
$321$	2.00000	0.111629
$322$	−1.00000	−0.0557278
$323$	21.0000	1.16847
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−7.00000	−0.387101
$328$	4.00000	0.220863
$329$	0	0
$330$	−3.00000	−0.165145
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	−2.00000	−0.109764
$333$	−1.00000	−0.0547997
$334$	−15.0000	−0.820763
$335$	8.00000	0.437087
$336$	−1.00000	−0.0545545
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	−7.00000	−0.379628
$341$	24.0000	1.29967
$342$	3.00000	0.162221
$343$	−1.00000	−0.0539949
$344$	−5.00000	−0.269582
$345$	−1.00000	−0.0538382
$346$	−6.00000	−0.322562
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	−1.00000	−0.0536056
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	−3.00000	−0.159901
$353$	16.0000	0.851594	0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000	0.851594	0.425797	+	0.904819i	$0.359994\pi$
$354$	−10.0000	−0.531494
$355$	−6.00000	−0.318447
$356$	12.0000	0.635999
$357$	7.00000	0.370479
$358$	−6.00000	−0.317110
$359$	−26.0000	−1.37223	−0.686114	−	0.727494i	$-0.740685\pi$
$359$	−26.0000	−1.37223	−0.686114	+	0.727494i	$0.740685\pi$
$360$	−1.00000	−0.0527046
$361$	−10.0000	−0.526316
$362$	−26.0000	−1.36653
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−13.0000	−0.680451
$366$	13.0000	0.679521
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−1.00000	−0.0521286
$369$	−4.00000	−0.208232
$370$	1.00000	0.0519875
$371$	6.00000	0.311504
$372$	8.00000	0.414781
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	21.0000	1.08588
$375$	−9.00000	−0.464758
$376$	0	0
$377$	0	0
$378$	1.00000	0.0514344
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	−3.00000	−0.153897
$381$	4.00000	0.204926
$382$	9.00000	0.460480
$383$	−17.0000	−0.868659	−0.434330	−	0.900754i	$-0.643015\pi$
$383$	−17.0000	−0.868659	−0.434330	+	0.900754i	$0.643015\pi$
$384$	−1.00000	−0.0510310
$385$	−3.00000	−0.152894
$386$	8.00000	0.407189
$387$	5.00000	0.254164
$388$	−6.00000	−0.304604
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	7.00000	0.354005
$392$	−1.00000	−0.0505076
$393$	−17.0000	−0.857537
$394$	14.0000	0.705310
$395$	−12.0000	−0.603786
$396$	3.00000	0.150756
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	9.00000	0.451129
$399$	3.00000	0.150188
$400$	−4.00000	−0.200000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−8.00000	−0.399004
$403$	0	0
$404$	−14.0000	−0.696526
$405$	1.00000	0.0496904
$406$	−1.00000	−0.0496292
$407$	−3.00000	−0.148704
$408$	7.00000	0.346552
$409$	29.0000	1.43396	0.716979	−	0.697095i	$-0.245524\pi$
$409$	29.0000	1.43396	0.716979	+	0.697095i	$0.245524\pi$
$410$	4.00000	0.197546
$411$	−17.0000	−0.838548
$412$	1.00000	0.0492665
$413$	−10.0000	−0.492068
$414$	1.00000	0.0491473
$415$	−2.00000	−0.0981761
$416$	0	0
$417$	−8.00000	−0.391762
$418$	9.00000	0.440204
$419$	−11.0000	−0.537385	−0.268693	−	0.963226i	$-0.586592\pi$
$419$	−11.0000	−0.537385	−0.268693	+	0.963226i	$0.586592\pi$
$420$	−1.00000	−0.0487950
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	−15.0000	−0.730189
$423$	0	0
$424$	6.00000	0.291386
$425$	28.0000	1.35820
$426$	6.00000	0.290701
$427$	13.0000	0.629114
$428$	2.00000	0.0966736
$429$	0	0
$430$	−5.00000	−0.241121
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	1.00000	0.0481125
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	8.00000	0.384012
$435$	−1.00000	−0.0479463
$436$	−7.00000	−0.335239
$437$	3.00000	0.143509
$438$	13.0000	0.621164
$439$	−27.0000	−1.28864	−0.644320	−	0.764756i	$-0.722859\pi$
$439$	−27.0000	−1.28864	−0.644320	+	0.764756i	$0.722859\pi$
$440$	−3.00000	−0.143019
$441$	1.00000	0.0476190
$442$	0	0
$443$	−34.0000	−1.61539	−0.807694	−	0.589601i	$-0.799285\pi$
$443$	−34.0000	−1.61539	−0.807694	+	0.589601i	$0.799285\pi$
$444$	−1.00000	−0.0474579
$445$	12.0000	0.568855
$446$	4.00000	0.189405
$447$	22.0000	1.04056
$448$	−1.00000	−0.0472456
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	4.00000	0.188562
$451$	−12.0000	−0.565058
$452$	−10.0000	−0.470360
$453$	−11.0000	−0.516825
$454$	20.0000	0.938647
$455$	0	0
$456$	3.00000	0.140488
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	−10.0000	−0.467269
$459$	−7.00000	−0.326732
$460$	−1.00000	−0.0466252
$461$	27.0000	1.25752	0.628758	−	0.777601i	$-0.283564\pi$
$461$	27.0000	1.25752	0.628758	+	0.777601i	$0.283564\pi$
$462$	3.00000	0.139573
$463$	−15.0000	−0.697109	−0.348555	−	0.937288i	$-0.613327\pi$
$463$	−15.0000	−0.697109	−0.348555	+	0.937288i	$0.613327\pi$
$464$	−1.00000	−0.0464238
$465$	8.00000	0.370991
$466$	18.0000	0.833834
$467$	−9.00000	−0.416470	−0.208235	−	0.978079i	$-0.566772\pi$
$467$	−9.00000	−0.416470	−0.208235	+	0.978079i	$0.566772\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	19.0000	0.875474
$472$	−10.0000	−0.460287
$473$	15.0000	0.689701
$474$	12.0000	0.551178
$475$	12.0000	0.550598
$476$	7.00000	0.320844
$477$	−6.00000	−0.274721
$478$	8.00000	0.365911
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−26.0000	−1.18427
$483$	1.00000	0.0455016
$484$	−2.00000	−0.0909091
$485$	−6.00000	−0.272446
$486$	−1.00000	−0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	13.0000	0.588482
$489$	4.00000	0.180886
$490$	−1.00000	−0.0451754
$491$	40.0000	1.80517	0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000	1.80517	0.902587	+	0.430507i	$0.141665\pi$
$492$	−4.00000	−0.180334
$493$	7.00000	0.315264
$494$	0	0
$495$	3.00000	0.134840
$496$	8.00000	0.359211
$497$	6.00000	0.269137
$498$	2.00000	0.0896221
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	−9.00000	−0.402492
$501$	15.0000	0.670151
$502$	17.0000	0.758747
$503$	−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
$505$	−14.0000	−0.622992
$506$	3.00000	0.133366
$507$	0	0
$508$	4.00000	0.177471
$509$	27.0000	1.19675	0.598377	−	0.801215i	$-0.295813\pi$
$509$	27.0000	1.19675	0.598377	+	0.801215i	$0.295813\pi$
$510$	7.00000	0.309965
$511$	13.0000	0.575086
$512$	−1.00000	−0.0441942
$513$	−3.00000	−0.132453
$514$	−6.00000	−0.264649
$515$	1.00000	0.0440653
$516$	5.00000	0.220113
$517$	0	0
$518$	−1.00000	−0.0439375
$519$	6.00000	0.263371
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	1.00000	0.0437688
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−17.0000	−0.742648
$525$	4.00000	0.174574
$526$	24.0000	1.04645
$527$	−56.0000	−2.43940
$528$	3.00000	0.130558
$529$	−22.0000	−0.956522
$530$	6.00000	0.260623
$531$	10.0000	0.433963
$532$	3.00000	0.130066
$533$	0	0
$534$	−12.0000	−0.519291
$535$	2.00000	0.0864675
$536$	−8.00000	−0.345547
$537$	6.00000	0.258919
$538$	4.00000	0.172452
$539$	3.00000	0.129219
$540$	1.00000	0.0430331
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	−20.0000	−0.859074
$543$	26.0000	1.11577
$544$	7.00000	0.300123
$545$	−7.00000	−0.299847
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−17.0000	−0.726204
$549$	−13.0000	−0.554826
$550$	12.0000	0.511682
$551$	3.00000	0.127804
$552$	1.00000	0.0425628
$553$	12.0000	0.510292
$554$	28.0000	1.18961
$555$	−1.00000	−0.0424476
$556$	−8.00000	−0.339276
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	−8.00000	−0.338667
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	−21.0000	−0.886621
$562$	10.0000	0.421825
$563$	−41.0000	−1.72794	−0.863972	−	0.503540i	$-0.832031\pi$
$563$	−41.0000	−1.72794	−0.863972	+	0.503540i	$0.832031\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	−14.0000	−0.588464
$567$	−1.00000	−0.0419961
$568$	6.00000	0.251754
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	3.00000	0.125656
$571$	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
$573$	−9.00000	−0.375980
$574$	−4.00000	−0.166957
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	−32.0000	−1.33102
$579$	−8.00000	−0.332469
$580$	−1.00000	−0.0415227
$581$	2.00000	0.0829740
$582$	6.00000	0.248708
$583$	−18.0000	−0.745484
$584$	13.0000	0.537944
$585$	0	0
$586$	14.0000	0.578335
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	1.00000	0.0412393
$589$	−24.0000	−0.988903
$590$	−10.0000	−0.411693
$591$	−14.0000	−0.575883
$592$	−1.00000	−0.0410997
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	−3.00000	−0.123091
$595$	7.00000	0.286972
$596$	22.0000	0.901155
$597$	−9.00000	−0.368345
$598$	0	0
$599$	−3.00000	−0.122577	−0.0612883	−	0.998120i	$-0.519521\pi$
$599$	−3.00000	−0.122577	−0.0612883	+	0.998120i	$0.519521\pi$
$600$	4.00000	0.163299
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	5.00000	0.203785
$603$	8.00000	0.325785
$604$	−11.0000	−0.447584
$605$	−2.00000	−0.0813116
$606$	14.0000	0.568711
$607$	−11.0000	−0.446476	−0.223238	−	0.974764i	$-0.571663\pi$
$607$	−11.0000	−0.446476	−0.223238	+	0.974764i	$0.571663\pi$
$608$	3.00000	0.121666
$609$	1.00000	0.0405220
$610$	13.0000	0.526355
$611$	0	0
$612$	−7.00000	−0.282958
$613$	−31.0000	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$614$	−16.0000	−0.645707
$615$	−4.00000	−0.161296
$616$	3.00000	0.120873
$617$	37.0000	1.48956	0.744782	−	0.667308i	$-0.232553\pi$
$617$	37.0000	1.48956	0.744782	+	0.667308i	$0.232553\pi$
$618$	−1.00000	−0.0402259
$619$	−23.0000	−0.924448	−0.462224	−	0.886763i	$-0.652948\pi$
$619$	−23.0000	−0.924448	−0.462224	+	0.886763i	$0.652948\pi$
$620$	8.00000	0.321288
$621$	−1.00000	−0.0401286
$622$	−4.00000	−0.160385
$623$	−12.0000	−0.480770
$624$	0	0
$625$	11.0000	0.440000
$626$	26.0000	1.03917
$627$	−9.00000	−0.359425
$628$	19.0000	0.758183
$629$	7.00000	0.279108
$630$	1.00000	0.0398410
$631$	−47.0000	−1.87104	−0.935520	−	0.353273i	$-0.885069\pi$
$631$	−47.0000	−1.87104	−0.935520	+	0.353273i	$0.885069\pi$
$632$	12.0000	0.477334
$633$	15.0000	0.596196
$634$	24.0000	0.953162
$635$	4.00000	0.158735
$636$	−6.00000	−0.237915
$637$	0	0
$638$	3.00000	0.118771
$639$	−6.00000	−0.237356
$640$	−1.00000	−0.0395285
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	−2.00000	−0.0789337
$643$	39.0000	1.53801	0.769005	−	0.639243i	$-0.220752\pi$
$643$	39.0000	1.53801	0.769005	+	0.639243i	$0.220752\pi$
$644$	1.00000	0.0394055
$645$	5.00000	0.196875
$646$	−21.0000	−0.826234
$647$	−22.0000	−0.864909	−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000	−0.864909	−0.432455	+	0.901656i	$0.642352\pi$
$648$	−1.00000	−0.0392837
$649$	30.0000	1.17760
$650$	0	0
$651$	−8.00000	−0.313545
$652$	4.00000	0.156652
$653$	−35.0000	−1.36966	−0.684828	−	0.728705i	$-0.740123\pi$
$653$	−35.0000	−1.36966	−0.684828	+	0.728705i	$0.740123\pi$
$654$	7.00000	0.273722
$655$	−17.0000	−0.664245
$656$	−4.00000	−0.156174
$657$	−13.0000	−0.507178
$658$	0	0
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	3.00000	0.116775
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	2.00000	0.0776151
$665$	3.00000	0.116335
$666$	1.00000	0.0387492
$667$	1.00000	0.0387202
$668$	15.0000	0.580367
$669$	−4.00000	−0.154649
$670$	−8.00000	−0.309067
$671$	−39.0000	−1.50558
$672$	1.00000	0.0385758
$673$	3.00000	0.115642	0.0578208	−	0.998327i	$-0.481585\pi$
$673$	3.00000	0.115642	0.0578208	+	0.998327i	$0.481585\pi$
$674$	−23.0000	−0.885927
$675$	−4.00000	−0.153960
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	10.0000	0.384048
$679$	6.00000	0.230259
$680$	7.00000	0.268438
$681$	−20.0000	−0.766402
$682$	−24.0000	−0.919007
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	−3.00000	−0.114708
$685$	−17.0000	−0.649537
$686$	1.00000	0.0381802
$687$	10.0000	0.381524
$688$	5.00000	0.190623
$689$	0	0
$690$	1.00000	0.0380693
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	6.00000	0.228086
$693$	−3.00000	−0.113961
$694$	24.0000	0.911028
$695$	−8.00000	−0.303457
$696$	1.00000	0.0379049
$697$	28.0000	1.06058
$698$	16.0000	0.605609
$699$	−18.0000	−0.680823
$700$	4.00000	0.151186
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	3.00000	0.113147
$704$	3.00000	0.113067
$705$	0	0
$706$	−16.0000	−0.602168
$707$	14.0000	0.526524
$708$	10.0000	0.375823
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	6.00000	0.225176
$711$	−12.0000	−0.450035
$712$	−12.0000	−0.449719
$713$	−8.00000	−0.299602
$714$	−7.00000	−0.261968
$715$	0	0
$716$	6.00000	0.224231
$717$	−8.00000	−0.298765
$718$	26.0000	0.970311
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	1.00000	0.0372678
$721$	−1.00000	−0.0372419
$722$	10.0000	0.372161
$723$	26.0000	0.966950
$724$	26.0000	0.966282
$725$	4.00000	0.148556
$726$	2.00000	0.0742270
$727$	−35.0000	−1.29808	−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000	−1.29808	−0.649039	+	0.760755i	$0.724829\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	13.0000	0.481152
$731$	−35.0000	−1.29452
$732$	−13.0000	−0.480494
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	−32.0000	−1.18114
$735$	1.00000	0.0368856
$736$	1.00000	0.0368605
$737$	24.0000	0.884051
$738$	4.00000	0.147242
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−6.00000	−0.220267
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	−8.00000	−0.293294
$745$	22.0000	0.806018
$746$	−6.00000	−0.219676
$747$	−2.00000	−0.0731762
$748$	−21.0000	−0.767836
$749$	−2.00000	−0.0730784
$750$	9.00000	0.328634
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	−17.0000	−0.619514
$754$	0	0
$755$	−11.0000	−0.400331
$756$	−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	$0.343354\pi$
$758$	4.00000	0.145287
$759$	−3.00000	−0.108893
$760$	3.00000	0.108821
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−4.00000	−0.144905
$763$	7.00000	0.253417
$764$	−9.00000	−0.325609
$765$	−7.00000	−0.253086
$766$	17.0000	0.614235
$767$	0	0
$768$	1.00000	0.0360844
$769$	−39.0000	−1.40638	−0.703188	−	0.711004i	$-0.748241\pi$
$769$	−39.0000	−1.40638	−0.703188	+	0.711004i	$0.748241\pi$
$770$	3.00000	0.108112
$771$	6.00000	0.216085
$772$	−8.00000	−0.287926
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	−5.00000	−0.179721
$775$	−32.0000	−1.14947
$776$	6.00000	0.215387
$777$	1.00000	0.0358748
$778$	−18.0000	−0.645331
$779$	12.0000	0.429945
$780$	0	0
$781$	−18.0000	−0.644091
$782$	−7.00000	−0.250319
$783$	−1.00000	−0.0357371
$784$	1.00000	0.0357143
$785$	19.0000	0.678139
$786$	17.0000	0.606370
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	−14.0000	−0.498729
$789$	−24.0000	−0.854423
$790$	12.0000	0.426941
$791$	10.0000	0.355559
$792$	−3.00000	−0.106600
$793$	0	0
$794$	−8.00000	−0.283909
$795$	−6.00000	−0.212798
$796$	−9.00000	−0.318997
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	−3.00000	−0.106199
$799$	0	0
$800$	4.00000	0.141421
$801$	12.0000	0.423999
$802$	6.00000	0.211867
$803$	−39.0000	−1.37628
$804$	8.00000	0.282138
$805$	1.00000	0.0352454
$806$	0	0
$807$	−4.00000	−0.140807
$808$	14.0000	0.492518
$809$	52.0000	1.82822	0.914111	−	0.405463i	$-0.132890\pi$
$809$	52.0000	1.82822	0.914111	+	0.405463i	$0.132890\pi$
$810$	−1.00000	−0.0351364
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	1.00000	0.0350931
$813$	20.0000	0.701431
$814$	3.00000	0.105150
$815$	4.00000	0.140114
$816$	−7.00000	−0.245049
$817$	−15.0000	−0.524784
$818$	−29.0000	−1.01396
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	17.0000	0.592943
$823$	−12.0000	−0.418294	−0.209147	−	0.977884i	$-0.567069\pi$
$823$	−12.0000	−0.418294	−0.209147	+	0.977884i	$0.567069\pi$
$824$	−1.00000	−0.0348367
$825$	−12.0000	−0.417786
$826$	10.0000	0.347945
$827$	−9.00000	−0.312961	−0.156480	−	0.987681i	$-0.550015\pi$
$827$	−9.00000	−0.312961	−0.156480	+	0.987681i	$0.550015\pi$
$828$	−1.00000	−0.0347524
$829$	15.0000	0.520972	0.260486	−	0.965478i	$-0.416117\pi$
$829$	15.0000	0.520972	0.260486	+	0.965478i	$0.416117\pi$
$830$	2.00000	0.0694210
$831$	−28.0000	−0.971309
$832$	0	0
$833$	−7.00000	−0.242536
$834$	8.00000	0.277017
$835$	15.0000	0.519096
$836$	−9.00000	−0.311272
$837$	8.00000	0.276520
$838$	11.0000	0.379989
$839$	−28.0000	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$840$	1.00000	0.0345033
$841$	−28.0000	−0.965517
$842$	−34.0000	−1.17172
$843$	−10.0000	−0.344418
$844$	15.0000	0.516321
$845$	0	0
$846$	0	0
$847$	2.00000	0.0687208
$848$	−6.00000	−0.206041
$849$	14.0000	0.480479
$850$	−28.0000	−0.960392
$851$	1.00000	0.0342796
$852$	−6.00000	−0.205557
$853$	32.0000	1.09566	0.547830	−	0.836590i	$-0.315454\pi$
$853$	32.0000	1.09566	0.547830	+	0.836590i	$0.315454\pi$
$854$	−13.0000	−0.444851
$855$	−3.00000	−0.102598
$856$	−2.00000	−0.0683586
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	5.00000	0.170499
$861$	4.00000	0.136320
$862$	−14.0000	−0.476842
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	−1.00000	−0.0340207
$865$	6.00000	0.204006
$866$	−8.00000	−0.271851
$867$	32.0000	1.08678
$868$	−8.00000	−0.271538
$869$	−36.0000	−1.22122
$870$	1.00000	0.0339032
$871$	0	0
$872$	7.00000	0.237050
$873$	−6.00000	−0.203069
$874$	−3.00000	−0.101477
$875$	9.00000	0.304256
$876$	−13.0000	−0.439229
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	27.0000	0.911206
$879$	−14.0000	−0.472208
$880$	3.00000	0.101130
$881$	−51.0000	−1.71823	−0.859117	−	0.511780i	$-0.828986\pi$
$881$	−51.0000	−1.71823	−0.859117	+	0.511780i	$0.828986\pi$
$882$	−1.00000	−0.0336718
$883$	−3.00000	−0.100958	−0.0504790	−	0.998725i	$-0.516075\pi$
$883$	−3.00000	−0.100958	−0.0504790	+	0.998725i	$0.516075\pi$
$884$	0	0
$885$	10.0000	0.336146
$886$	34.0000	1.14225
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	1.00000	0.0335578
$889$	−4.00000	−0.134156
$890$	−12.0000	−0.402241
$891$	3.00000	0.100504
$892$	−4.00000	−0.133930
$893$	0	0
$894$	−22.0000	−0.735790
$895$	6.00000	0.200558
$896$	1.00000	0.0334077
$897$	0	0
$898$	9.00000	0.300334
$899$	−8.00000	−0.266815
$900$	−4.00000	−0.133333
$901$	42.0000	1.39922
$902$	12.0000	0.399556
$903$	−5.00000	−0.166390
$904$	10.0000	0.332595
$905$	26.0000	0.864269
$906$	11.0000	0.365451
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−20.0000	−0.663723
$909$	−14.0000	−0.464351
$910$	0	0
$911$	57.0000	1.88849	0.944247	−	0.329238i	$-0.106792\pi$
$911$	57.0000	1.88849	0.944247	+	0.329238i	$0.106792\pi$
$912$	−3.00000	−0.0993399
$913$	−6.00000	−0.198571
$914$	28.0000	0.926158
$915$	−13.0000	−0.429767
$916$	10.0000	0.330409
$917$	17.0000	0.561389
$918$	7.00000	0.231034
$919$	−30.0000	−0.989609	−0.494804	−	0.869004i	$-0.664760\pi$
$919$	−30.0000	−0.989609	−0.494804	+	0.869004i	$0.664760\pi$
$920$	1.00000	0.0329690
$921$	16.0000	0.527218
$922$	−27.0000	−0.889198
$923$	0	0
$924$	−3.00000	−0.0986928
$925$	4.00000	0.131519
$926$	15.0000	0.492931
$927$	1.00000	0.0328443
$928$	1.00000	0.0328266
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	−8.00000	−0.262330
$931$	−3.00000	−0.0983210
$932$	−18.0000	−0.589610
$933$	4.00000	0.130954
$934$	9.00000	0.294489
$935$	−21.0000	−0.686773
$936$	0	0
$937$	4.00000	0.130674	0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000	0.130674	0.0653372	+	0.997863i	$0.479188\pi$
$938$	8.00000	0.261209
$939$	−26.0000	−0.848478
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−19.0000	−0.619053
$943$	4.00000	0.130258
$944$	10.0000	0.325472
$945$	−1.00000	−0.0325300
$946$	−15.0000	−0.487692
$947$	−25.0000	−0.812391	−0.406195	−	0.913786i	$-0.633145\pi$
$947$	−25.0000	−0.812391	−0.406195	+	0.913786i	$0.633145\pi$
$948$	−12.0000	−0.389742
$949$	0	0
$950$	−12.0000	−0.389331
$951$	−24.0000	−0.778253
$952$	−7.00000	−0.226871
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	6.00000	0.194257
$955$	−9.00000	−0.291233
$956$	−8.00000	−0.258738
$957$	−3.00000	−0.0969762
$958$	3.00000	0.0969256
$959$	17.0000	0.548959
$960$	1.00000	0.0322749
$961$	33.0000	1.06452
$962$	0	0
$963$	2.00000	0.0644491
$964$	26.0000	0.837404
$965$	−8.00000	−0.257529
$966$	−1.00000	−0.0321745
$967$	43.0000	1.38279	0.691393	−	0.722478i	$-0.256997\pi$
$967$	43.0000	1.38279	0.691393	+	0.722478i	$0.256997\pi$
$968$	2.00000	0.0642824
$969$	21.0000	0.674617
$970$	6.00000	0.192648
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	8.00000	0.256468
$974$	−24.0000	−0.769010
$975$	0	0
$976$	−13.0000	−0.416120
$977$	−15.0000	−0.479893	−0.239946	−	0.970786i	$-0.577130\pi$
$977$	−15.0000	−0.479893	−0.239946	+	0.970786i	$0.577130\pi$
$978$	−4.00000	−0.127906
$979$	36.0000	1.15056
$980$	1.00000	0.0319438
$981$	−7.00000	−0.223493
$982$	−40.0000	−1.27645
$983$	−51.0000	−1.62665	−0.813324	−	0.581811i	$-0.802344\pi$
$983$	−51.0000	−1.62665	−0.813324	+	0.581811i	$0.802344\pi$
$984$	4.00000	0.127515
$985$	−14.0000	−0.446077
$986$	−7.00000	−0.222925
$987$	0	0
$988$	0	0
$989$	−5.00000	−0.158991
$990$	−3.00000	−0.0953463
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	−8.00000	−0.254000
$993$	−8.00000	−0.253872
$994$	−6.00000	−0.190308
$995$	−9.00000	−0.285319
$996$	−2.00000	−0.0633724
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	14.0000	0.443162
$999$	−1.00000	−0.0316386

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.c.337.1	✓	2	13.8	odd	4
546.2.c.c.337.2	yes	2	13.5	odd	4
1638.2.c.e.883.1		2	39.5	even	4
1638.2.c.e.883.2		2	39.8	even	4
3822.2.c.b.883.1		2	91.34	even	4
3822.2.c.b.883.2		2	91.83	even	4
4368.2.h.h.337.1		2	52.31	even	4
4368.2.h.h.337.2		2	52.47	even	4
7098.2.a.o.1.1		1	1.1	even	1	trivial
7098.2.a.y.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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