LMFDB - Embedded newform 75.10.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,10,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	75.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:	$38.6276877123$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+36.0000 q^{2} +81.0000 q^{3} +784.000 q^{4} +2916.00 q^{6} +4480.00 q^{7} +9792.00 q^{8} +6561.00 q^{9} +O(q^{10})$

\(q+36.0000 q^{2} +81.0000 q^{3} +784.000 q^{4} +2916.00 q^{6} +4480.00 q^{7} +9792.00 q^{8} +6561.00 q^{9} +1476.00 q^{11} +63504.0 q^{12} +151522. q^{13} +161280. q^{14} -48896.0 q^{16} -108162. q^{17} +236196. q^{18} +593084. q^{19} +362880. q^{21} +53136.0 q^{22} +969480. q^{23} +793152. q^{24} +5.45479e6 q^{26} +531441. q^{27} +3.51232e6 q^{28} -6.64252e6 q^{29} +7.07060e6 q^{31} -6.77376e6 q^{32} +119556. q^{33} -3.89383e6 q^{34} +5.14382e6 q^{36} +7.47241e6 q^{37} +2.13510e7 q^{38} +1.22733e7 q^{39} -4.35015e6 q^{41} +1.30637e7 q^{42} +4.35872e6 q^{43} +1.15718e6 q^{44} +3.49013e7 q^{46} -2.83092e7 q^{47} -3.96058e6 q^{48} -2.02832e7 q^{49} -8.76112e6 q^{51} +1.18793e8 q^{52} -1.61117e7 q^{53} +1.91319e7 q^{54} +4.38682e7 q^{56} +4.80398e7 q^{57} -2.39131e8 q^{58} -8.60760e7 q^{59} +3.22139e7 q^{61} +2.54542e8 q^{62} +2.93933e7 q^{63} -2.18821e8 q^{64} +4.30402e6 q^{66} -9.95315e7 q^{67} -8.47990e7 q^{68} +7.85279e7 q^{69} -4.41705e7 q^{71} +6.42453e7 q^{72} +2.35606e7 q^{73} +2.69007e8 q^{74} +4.64978e8 q^{76} +6.61248e6 q^{77} +4.41838e8 q^{78} -4.01755e8 q^{79} +4.30467e7 q^{81} -1.56605e8 q^{82} +7.44529e8 q^{83} +2.84498e8 q^{84} +1.56914e8 q^{86} -5.38044e8 q^{87} +1.44530e7 q^{88} +7.69871e8 q^{89} +6.78819e8 q^{91} +7.60072e8 q^{92} +5.72719e8 q^{93} -1.01913e9 q^{94} -5.48675e8 q^{96} -9.07131e8 q^{97} -7.30195e8 q^{98} +9.68404e6 q^{99} +O(q^{100})\)

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 $

$2$	36.0000	1.59099	0.795495	−	0.605960i	$-0.207211\pi$
$2$	36.0000	1.59099	0.795495	+	0.605960i	$0.207211\pi$
$3$	81.0000	0.577350
$3$	81.0000	0.577350
$4$	784.000	1.53125
$5$	0	0
$5$	0	0
$6$	2916.00	0.918559
$7$	4480.00	0.705240	0.352620	−	0.935767i	$-0.385291\pi$
$7$	4480.00	0.705240	0.352620	+	0.935767i	$0.385291\pi$
$8$	9792.00	0.845214
$9$	6561.00	0.333333
$10$	0	0
$11$	1476.00	0.0303962	0.0151981	−	0.999885i	$-0.495162\pi$
$11$	1476.00	0.0303962	0.0151981	+	0.999885i	$0.495162\pi$
$12$	63504.0	0.884068
$13$	151522.	1.47140	0.735700	−	0.677308i	$-0.236853\pi$
$13$	151522.	1.47140	0.735700	+	0.677308i	$0.236853\pi$
$14$	161280.	1.12203
$15$	0	0
$16$	−48896.0	−0.186523
$17$	−108162.	−0.314090	−0.157045	−	0.987591i	$-0.550197\pi$
$17$	−108162.	−0.314090	−0.157045	+	0.987591i	$0.550197\pi$
$18$	236196.	0.530330
$19$	593084.	1.04406	0.522029	−	0.852927i	$-0.325175\pi$
$19$	593084.	1.04406	0.522029	+	0.852927i	$0.325175\pi$
$20$	0	0
$21$	362880.	0.407170
$22$	53136.0	0.0483601
$23$	969480.	0.722376	0.361188	−	0.932493i	$-0.382371\pi$
$23$	969480.	0.722376	0.361188	+	0.932493i	$0.382371\pi$
$24$	793152.	0.487984
$25$	0	0
$26$	5.45479e6	2.34098
$27$	531441.	0.192450
$28$	3.51232e6	1.07990
$29$	−6.64252e6	−1.74398	−0.871991	−	0.489522i	$-0.837171\pi$
$29$	−6.64252e6	−1.74398	−0.871991	+	0.489522i	$0.837171\pi$
$30$	0	0
$31$	7.07060e6	1.37508	0.687541	−	0.726145i	$-0.258690\pi$
$31$	7.07060e6	1.37508	0.687541	+	0.726145i	$0.258690\pi$
$32$	−6.77376e6	−1.14197
$33$	119556.	0.0175493
$34$	−3.89383e6	−0.499715
$35$	0	0
$36$	5.14382e6	0.510417
$37$	7.47241e6	0.655470	0.327735	−	0.944770i	$-0.393715\pi$
$37$	7.47241e6	0.655470	0.327735	+	0.944770i	$0.393715\pi$
$38$	2.13510e7	1.66109
$39$	1.22733e7	0.849513
$40$	0	0
$41$	−4.35015e6	−0.240423	−0.120212	−	0.992748i	$-0.538357\pi$
$41$	−4.35015e6	−0.240423	−0.120212	+	0.992748i	$0.538357\pi$
$42$	1.30637e7	0.647804
$43$	4.35872e6	0.194424	0.0972121	−	0.995264i	$-0.469007\pi$
$43$	4.35872e6	0.194424	0.0972121	+	0.995264i	$0.469007\pi$
$44$	1.15718e6	0.0465442
$45$	0	0
$46$	3.49013e7	1.14929
$47$	−2.83092e7	−0.846229	−0.423115	−	0.906076i	$-0.639063\pi$
$47$	−2.83092e7	−0.846229	−0.423115	+	0.906076i	$0.639063\pi$
$48$	−3.96058e6	−0.107689
$49$	−2.02832e7	−0.502637
$50$	0	0
$51$	−8.76112e6	−0.181340
$52$	1.18793e8	2.25308
$53$	−1.61117e7	−0.280479	−0.140239	−	0.990118i	$-0.544787\pi$
$53$	−1.61117e7	−0.280479	−0.140239	+	0.990118i	$0.544787\pi$
$54$	1.91319e7	0.306186
$55$	0	0
$56$	4.38682e7	0.596078
$57$	4.80398e7	0.602788
$58$	−2.39131e8	−2.77466
$59$	−8.60760e7	−0.924800	−0.462400	−	0.886671i	$-0.653012\pi$
$59$	−8.60760e7	−0.924800	−0.462400	+	0.886671i	$0.653012\pi$
$60$	0	0
$61$	3.22139e7	0.297892	0.148946	−	0.988845i	$-0.452412\pi$
$61$	3.22139e7	0.297892	0.148946	+	0.988845i	$0.452412\pi$
$62$	2.54542e8	2.18774
$63$	2.93933e7	0.235080
$64$	−2.18821e8	−1.63034
$65$	0	0
$66$	4.30402e6	0.0279207
$67$	−9.95315e7	−0.603426	−0.301713	−	0.953399i	$-0.597558\pi$
$67$	−9.95315e7	−0.603426	−0.301713	+	0.953399i	$0.597558\pi$
$68$	−8.47990e7	−0.480951
$69$	7.85279e7	0.417064
$70$	0	0
$71$	−4.41705e7	−0.206286	−0.103143	−	0.994667i	$-0.532890\pi$
$71$	−4.41705e7	−0.206286	−0.103143	+	0.994667i	$0.532890\pi$
$72$	6.42453e7	0.281738
$73$	2.35606e7	0.0971033	0.0485517	−	0.998821i	$-0.484539\pi$
$73$	2.35606e7	0.0971033	0.0485517	+	0.998821i	$0.484539\pi$
$74$	2.69007e8	1.04285
$75$	0	0
$76$	4.64978e8	1.59872
$77$	6.61248e6	0.0214366
$78$	4.41838e8	1.35157
$79$	−4.01755e8	−1.16048	−0.580242	−	0.814444i	$-0.697042\pi$
$79$	−4.01755e8	−1.16048	−0.580242	+	0.814444i	$0.697042\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	−1.56605e8	−0.382511
$83$	7.44529e8	1.72199	0.860994	−	0.508615i	$-0.169842\pi$
$83$	7.44529e8	1.72199	0.860994	+	0.508615i	$0.169842\pi$
$84$	2.84498e8	0.623480
$85$	0	0
$86$	1.56914e8	0.309327
$87$	−5.38044e8	−1.00689
$88$	1.44530e7	0.0256913
$89$	7.69871e8	1.30066	0.650329	−	0.759653i	$-0.274631\pi$
$89$	7.69871e8	1.30066	0.650329	+	0.759653i	$0.274631\pi$
$90$	0	0
$91$	6.78819e8	1.03769
$92$	7.60072e8	1.10614
$93$	5.72719e8	0.793904
$94$	−1.01913e9	−1.34634
$95$	0	0
$96$	−5.48675e8	−0.659317
$97$	−9.07131e8	−1.04039	−0.520196	−	0.854047i	$-0.674141\pi$
$97$	−9.07131e8	−1.04039	−0.520196	+	0.854047i	$0.674141\pi$
$98$	−7.30195e8	−0.799690
$99$	9.68404e6	0.0101321
$100$	0	0
$101$	−4.21902e8	−0.403427	−0.201714	−	0.979445i	$-0.564651\pi$
$101$	−4.21902e8	−0.403427	−0.201714	+	0.979445i	$0.564651\pi$
$102$	−3.15400e8	−0.288510
$103$	−5.79043e8	−0.506924	−0.253462	−	0.967345i	$-0.581569\pi$
$103$	−5.79043e8	−0.506924	−0.253462	+	0.967345i	$0.581569\pi$
$104$	1.48370e9	1.24365
$105$	0	0
$106$	−5.80022e8	−0.446239
$107$	−1.39714e9	−1.03042	−0.515208	−	0.857065i	$-0.672285\pi$
$107$	−1.39714e9	−1.03042	−0.515208	+	0.857065i	$0.672285\pi$
$108$	4.16650e8	0.294689
$109$	−2.68530e9	−1.82211	−0.911054	−	0.412286i	$-0.864730\pi$
$109$	−2.68530e9	−1.82211	−0.911054	+	0.412286i	$0.864730\pi$
$110$	0	0
$111$	6.05265e8	0.378436
$112$	−2.19054e8	−0.131544
$113$	−4.54749e8	−0.262373	−0.131186	−	0.991358i	$-0.541879\pi$
$113$	−4.54749e8	−0.262373	−0.131186	+	0.991358i	$0.541879\pi$
$114$	1.72943e9	0.959029
$115$	0	0
$116$	−5.20774e9	−2.67047
$117$	9.94136e8	0.490467
$118$	−3.09873e9	−1.47135
$119$	−4.84566e8	−0.221509
$120$	0	0
$121$	−2.35577e9	−0.999076
$122$	1.15970e9	0.473944
$123$	−3.52362e8	−0.138809
$124$	5.54335e9	2.10559
$125$	0	0
$126$	1.05816e9	0.374010
$127$	8.38650e7	0.0286064	0.0143032	−	0.999898i	$-0.495447\pi$
$127$	8.38650e7	0.0286064	0.0143032	+	0.999898i	$0.495447\pi$
$128$	−4.40938e9	−1.45189
$129$	3.53056e8	0.112251
$130$	0	0
$131$	−3.73601e9	−1.10838	−0.554188	−	0.832391i	$-0.686971\pi$
$131$	−3.73601e9	−1.10838	−0.554188	+	0.832391i	$0.686971\pi$
$132$	9.37319e7	0.0268723
$133$	2.65702e9	0.736312
$134$	−3.58313e9	−0.960044
$135$	0	0
$136$	−1.05912e9	−0.265473
$137$	−6.43598e9	−1.56089	−0.780444	−	0.625225i	$-0.785007\pi$
$137$	−6.43598e9	−1.56089	−0.780444	+	0.625225i	$0.785007\pi$
$138$	2.82700e9	0.663545
$139$	1.81833e9	0.413148	0.206574	−	0.978431i	$-0.433769\pi$
$139$	1.81833e9	0.413148	0.206574	+	0.978431i	$0.433769\pi$
$140$	0	0
$141$	−2.29305e9	−0.488571
$142$	−1.59014e9	−0.328199
$143$	2.23646e8	0.0447250
$144$	−3.20807e8	−0.0621745
$145$	0	0
$146$	8.48183e8	0.154490
$147$	−1.64294e9	−0.290197
$148$	5.85837e9	1.00369
$149$	−8.30199e9	−1.37989	−0.689944	−	0.723863i	$-0.742365\pi$
$149$	−8.30199e9	−1.37989	−0.689944	+	0.723863i	$0.742365\pi$
$150$	0	0
$151$	3.84057e9	0.601173	0.300587	−	0.953755i	$-0.402818\pi$
$151$	3.84057e9	0.601173	0.300587	+	0.953755i	$0.402818\pi$
$152$	5.80748e9	0.882453
$153$	−7.09651e8	−0.104697
$154$	2.38049e8	0.0341054
$155$	0	0
$156$	9.62225e9	1.30082
$157$	2.17912e9	0.286242	0.143121	−	0.989705i	$-0.454286\pi$
$157$	2.17912e9	0.286242	0.143121	+	0.989705i	$0.454286\pi$
$158$	−1.44632e10	−1.84632
$159$	−1.30505e9	−0.161935
$160$	0	0
$161$	4.34327e9	0.509449
$162$	1.54968e9	0.176777
$163$	1.54147e10	1.71038	0.855188	−	0.518317i	$-0.173441\pi$
$163$	1.54147e10	1.71038	0.855188	+	0.518317i	$0.173441\pi$
$164$	−3.41052e9	−0.368148
$165$	0	0
$166$	2.68030e10	2.73967
$167$	5.65506e9	0.562617	0.281309	−	0.959617i	$-0.409232\pi$
$167$	5.65506e9	0.562617	0.281309	+	0.959617i	$0.409232\pi$
$168$	3.55332e9	0.344146
$169$	1.23544e10	1.16502
$170$	0	0
$171$	3.89122e9	0.348020
$172$	3.41723e9	0.297712
$173$	−7.69892e7	−0.00653465	−0.00326733	−	0.999995i	$-0.501040\pi$
$173$	−7.69892e7	−0.00653465	−0.00326733	+	0.999995i	$0.501040\pi$
$174$	−1.93696e10	−1.60195
$175$	0	0
$176$	−7.21705e7	−0.00566960
$177$	−6.97215e9	−0.533934
$178$	2.77154e10	2.06933
$179$	2.32247e10	1.69087	0.845436	−	0.534077i	$-0.179341\pi$
$179$	2.32247e10	1.69087	0.845436	+	0.534077i	$0.179341\pi$
$180$	0	0
$181$	−1.23532e10	−0.855513	−0.427756	−	0.903894i	$-0.640696\pi$
$181$	−1.23532e10	−0.855513	−0.427756	+	0.903894i	$0.640696\pi$
$182$	2.44375e10	1.65095
$183$	2.60933e9	0.171988
$184$	9.49315e9	0.610562
$185$	0	0
$186$	2.06179e10	1.26309
$187$	−1.59647e8	−0.00954715
$188$	−2.21945e10	−1.29579
$189$	2.38086e9	0.135723
$190$	0	0
$191$	−4.20433e9	−0.228584	−0.114292	−	0.993447i	$-0.536460\pi$
$191$	−4.20433e9	−0.228584	−0.114292	+	0.993447i	$0.536460\pi$
$192$	−1.77245e10	−0.941278
$193$	4.38611e9	0.227547	0.113774	−	0.993507i	$-0.463706\pi$
$193$	4.38611e9	0.227547	0.113774	+	0.993507i	$0.463706\pi$
$194$	−3.26567e10	−1.65525
$195$	0	0
$196$	−1.59020e10	−0.769663
$197$	3.36694e10	1.59271	0.796356	−	0.604828i	$-0.206758\pi$
$197$	3.36694e10	1.59271	0.796356	+	0.604828i	$0.206758\pi$
$198$	3.48625e8	0.0161200
$199$	1.02732e10	0.464374	0.232187	−	0.972671i	$-0.425412\pi$
$199$	1.02732e10	0.464374	0.232187	+	0.972671i	$0.425412\pi$
$200$	0	0
$201$	−8.06205e9	−0.348388
$202$	−1.51885e10	−0.641849
$203$	−2.97585e10	−1.22993
$204$	−6.86872e9	−0.277677
$205$	0	0
$206$	−2.08455e10	−0.806512
$207$	6.36076e9	0.240792
$208$	−7.40882e9	−0.274450
$209$	8.75392e8	0.0317354
$210$	0	0
$211$	7.96696e9	0.276708	0.138354	−	0.990383i	$-0.455819\pi$
$211$	7.96696e9	0.276708	0.138354	+	0.990383i	$0.455819\pi$
$212$	−1.26316e10	−0.429483
$213$	−3.57781e9	−0.119099
$214$	−5.02970e10	−1.63938
$215$	0	0
$216$	5.20387e9	0.162661
$217$	3.16763e10	0.969763
$218$	−9.66710e10	−2.89896
$219$	1.90841e9	0.0560626
$220$	0	0
$221$	−1.63889e10	−0.462152
$222$	2.17895e10	0.602088
$223$	−6.96581e9	−0.188625	−0.0943126	−	0.995543i	$-0.530065\pi$
$223$	−6.96581e9	−0.188625	−0.0943126	+	0.995543i	$0.530065\pi$
$224$	−3.03464e10	−0.805363
$225$	0	0
$226$	−1.63709e10	−0.417432
$227$	−3.35697e10	−0.839133	−0.419567	−	0.907725i	$-0.637818\pi$
$227$	−3.35697e10	−0.839133	−0.419567	+	0.907725i	$0.637818\pi$
$228$	3.76632e10	0.923019
$229$	2.93198e10	0.704534	0.352267	−	0.935900i	$-0.385411\pi$
$229$	2.93198e10	0.704534	0.352267	+	0.935900i	$0.385411\pi$
$230$	0	0
$231$	5.35611e8	0.0123764
$232$	−6.50436e10	−1.47404
$233$	8.20079e10	1.82286	0.911431	−	0.411453i	$-0.134978\pi$
$233$	8.20079e10	1.82286	0.911431	+	0.411453i	$0.134978\pi$
$234$	3.57889e10	0.780327
$235$	0	0
$236$	−6.74836e10	−1.41610
$237$	−3.25421e10	−0.670006
$238$	−1.74444e10	−0.352419
$239$	6.26609e10	1.24224	0.621121	−	0.783715i	$-0.286677\pi$
$239$	6.26609e10	1.24224	0.621121	+	0.783715i	$0.286677\pi$
$240$	0	0
$241$	7.75548e10	1.48092	0.740460	−	0.672100i	$-0.234608\pi$
$241$	7.75548e10	1.48092	0.740460	+	0.672100i	$0.234608\pi$
$242$	−8.48077e10	−1.58952
$243$	3.48678e9	0.0641500
$244$	2.52557e10	0.456148
$245$	0	0
$246$	−1.26850e10	−0.220843
$247$	8.98653e10	1.53623
$248$	6.92353e10	1.16224
$249$	6.03068e10	0.994190
$250$	0	0
$251$	−5.81901e10	−0.925374	−0.462687	−	0.886522i	$-0.653115\pi$
$251$	−5.81901e10	−0.925374	−0.462687	+	0.886522i	$0.653115\pi$
$252$	2.30443e10	0.359966
$253$	1.43095e9	0.0219575
$254$	3.01914e9	0.0455126
$255$	0	0
$256$	−4.67014e10	−0.679595
$257$	7.41485e9	0.106024	0.0530119	−	0.998594i	$-0.483118\pi$
$257$	7.41485e9	0.106024	0.0530119	+	0.998594i	$0.483118\pi$
$258$	1.27100e10	0.178590
$259$	3.34764e10	0.462264
$260$	0	0
$261$	−4.35816e10	−0.581327
$262$	−1.34496e11	−1.76342
$263$	1.05271e11	1.35677	0.678387	−	0.734705i	$-0.262679\pi$
$263$	1.05271e11	1.35677	0.678387	+	0.734705i	$0.262679\pi$
$264$	1.17069e9	0.0148329
$265$	0	0
$266$	9.56526e10	1.17147
$267$	6.23596e10	0.750935
$268$	−7.80327e10	−0.923995
$269$	4.67239e10	0.544069	0.272034	−	0.962288i	$-0.412304\pi$
$269$	4.67239e10	0.544069	0.272034	+	0.962288i	$0.412304\pi$
$270$	0	0
$271$	2.86868e10	0.323087	0.161544	−	0.986866i	$-0.448353\pi$
$271$	2.86868e10	0.323087	0.161544	+	0.986866i	$0.448353\pi$
$272$	5.28869e9	0.0585852
$273$	5.49843e10	0.599110
$274$	−2.31695e11	−2.48336
$275$	0	0
$276$	6.15659e10	0.638630
$277$	−8.50676e10	−0.868171	−0.434085	−	0.900872i	$-0.642928\pi$
$277$	−8.50676e10	−0.868171	−0.434085	+	0.900872i	$0.642928\pi$
$278$	6.54598e10	0.657314
$279$	4.63902e10	0.458361
$280$	0	0
$281$	−7.87257e8	−0.00753248	−0.00376624	−	0.999993i	$-0.501199\pi$
$281$	−7.87257e8	−0.00753248	−0.00376624	+	0.999993i	$0.501199\pi$
$282$	−8.25498e10	−0.777311
$283$	−2.48961e10	−0.230724	−0.115362	−	0.993324i	$-0.536803\pi$
$283$	−2.48961e10	−0.230724	−0.115362	+	0.993324i	$0.536803\pi$
$284$	−3.46297e10	−0.315875
$285$	0	0
$286$	8.05127e9	0.0711570
$287$	−1.94887e10	−0.169556
$288$	−4.44426e10	−0.380657
$289$	−1.06889e11	−0.901347
$290$	0	0
$291$	−7.34776e10	−0.600671
$292$	1.84715e10	0.148689
$293$	−1.57074e11	−1.24509	−0.622543	−	0.782586i	$-0.713900\pi$
$293$	−1.57074e11	−1.24509	−0.622543	+	0.782586i	$0.713900\pi$
$294$	−5.91458e10	−0.461701
$295$	0	0
$296$	7.31698e10	0.554012
$297$	7.84407e8	0.00584975
$298$	−2.98871e11	−2.19539
$299$	1.46898e11	1.06290
$300$	0	0
$301$	1.95270e10	0.137116
$302$	1.38261e11	0.956461
$303$	−3.41741e10	−0.232919
$304$	−2.89994e10	−0.194741
$305$	0	0
$306$	−2.55474e10	−0.166572
$307$	2.45737e11	1.57887	0.789437	−	0.613831i	$-0.210372\pi$
$307$	2.45737e11	1.57887	0.789437	+	0.613831i	$0.210372\pi$
$308$	5.18418e9	0.0328248
$309$	−4.69025e10	−0.292673
$310$	0	0
$311$	−1.61050e11	−0.976201	−0.488101	−	0.872787i	$-0.662310\pi$
$311$	−1.61050e11	−0.976201	−0.488101	+	0.872787i	$0.662310\pi$
$312$	1.20180e11	0.718020
$313$	2.44646e11	1.44075	0.720374	−	0.693586i	$-0.243970\pi$
$313$	2.44646e11	1.44075	0.720374	+	0.693586i	$0.243970\pi$
$314$	7.84484e10	0.455408
$315$	0	0
$316$	−3.14976e11	−1.77699
$317$	−1.12832e11	−0.627575	−0.313787	−	0.949493i	$-0.601598\pi$
$317$	−1.12832e11	−0.627575	−0.313787	+	0.949493i	$0.601598\pi$
$318$	−4.69817e10	−0.257636
$319$	−9.80436e9	−0.0530104
$320$	0	0
$321$	−1.13168e11	−0.594911
$322$	1.56358e11	0.810528
$323$	−6.41492e10	−0.327929
$324$	3.37486e10	0.170139
$325$	0	0
$326$	5.54930e11	2.72119
$327$	−2.17510e11	−1.05200
$328$	−4.25967e10	−0.203209
$329$	−1.26825e11	−0.596794
$330$	0	0
$331$	2.87348e11	1.31578	0.657889	−	0.753115i	$-0.271450\pi$
$331$	2.87348e11	1.31578	0.657889	+	0.753115i	$0.271450\pi$
$332$	5.83711e11	2.63679
$333$	4.90265e10	0.218490
$334$	2.03582e11	0.895119
$335$	0	0
$336$	−1.77434e10	−0.0759468
$337$	2.52635e10	0.106699	0.0533494	−	0.998576i	$-0.483010\pi$
$337$	2.52635e10	0.106699	0.0533494	+	0.998576i	$0.483010\pi$
$338$	4.44759e11	1.85353
$339$	−3.68346e10	−0.151481
$340$	0	0
$341$	1.04362e10	0.0417973
$342$	1.40084e11	0.553696
$343$	−2.71653e11	−1.05972
$344$	4.26805e10	0.164330
$345$	0	0
$346$	−2.77161e9	−0.0103966
$347$	−9.04803e10	−0.335020	−0.167510	−	0.985870i	$-0.553573\pi$
$347$	−9.04803e10	−0.335020	−0.167510	+	0.985870i	$0.553573\pi$
$348$	−4.21827e11	−1.54180
$349$	−1.53822e10	−0.0555016	−0.0277508	−	0.999615i	$-0.508834\pi$
$349$	−1.53822e10	−0.0555016	−0.0277508	+	0.999615i	$0.508834\pi$
$350$	0	0
$351$	8.05250e10	0.283171
$352$	−9.99807e9	−0.0347116
$353$	1.46875e11	0.503457	0.251728	−	0.967798i	$-0.419001\pi$
$353$	1.46875e11	0.503457	0.251728	+	0.967798i	$0.419001\pi$
$354$	−2.50998e11	−0.849483
$355$	0	0
$356$	6.03579e11	1.99163
$357$	−3.92498e10	−0.127888
$358$	8.36088e11	2.69016
$359$	−4.42246e11	−1.40520	−0.702602	−	0.711583i	$-0.747979\pi$
$359$	−4.42246e11	−1.40520	−0.702602	+	0.711583i	$0.747979\pi$
$360$	0	0
$361$	2.90609e10	0.0900590
$362$	−4.44716e11	−1.36111
$363$	−1.90817e11	−0.576817
$364$	5.32194e11	1.58896
$365$	0	0
$366$	9.39358e10	0.273632
$367$	−1.48110e11	−0.426175	−0.213088	−	0.977033i	$-0.568352\pi$
$367$	−1.48110e11	−0.426175	−0.213088	+	0.977033i	$0.568352\pi$
$368$	−4.74037e10	−0.134740
$369$	−2.85413e10	−0.0801412
$370$	0	0
$371$	−7.21805e10	−0.197805
$372$	4.49011e11	1.21567
$373$	−7.63489e10	−0.204227	−0.102114	−	0.994773i	$-0.532560\pi$
$373$	−7.63489e10	−0.204227	−0.102114	+	0.994773i	$0.532560\pi$
$374$	−5.74730e9	−0.0151894
$375$	0	0
$376$	−2.77204e11	−0.715244
$377$	−1.00649e12	−2.56609
$378$	8.57108e10	0.215935
$379$	−2.70192e11	−0.672660	−0.336330	−	0.941744i	$-0.609186\pi$
$379$	−2.70192e11	−0.672660	−0.336330	+	0.941744i	$0.609186\pi$
$380$	0	0
$381$	6.79306e9	0.0165159
$382$	−1.51356e11	−0.363676
$383$	6.61033e11	1.56974	0.784872	−	0.619658i	$-0.212729\pi$
$383$	6.61033e11	1.56974	0.784872	+	0.619658i	$0.212729\pi$
$384$	−3.57160e11	−0.838246
$385$	0	0
$386$	1.57900e11	0.362026
$387$	2.85975e10	0.0648081
$388$	−7.11191e11	−1.59310
$389$	−3.09861e11	−0.686109	−0.343054	−	0.939316i	$-0.611462\pi$
$389$	−3.09861e11	−0.686109	−0.343054	+	0.939316i	$0.611462\pi$
$390$	0	0
$391$	−1.04861e11	−0.226891
$392$	−1.98613e11	−0.424835
$393$	−3.02617e11	−0.639921
$394$	1.21210e12	2.53399
$395$	0	0
$396$	7.59228e9	0.0155147
$397$	6.50589e11	1.31447	0.657233	−	0.753688i	$-0.271727\pi$
$397$	6.50589e11	1.31447	0.657233	+	0.753688i	$0.271727\pi$
$398$	3.69836e11	0.738814
$399$	2.15218e11	0.425110
$400$	0	0
$401$	2.76701e10	0.0534393	0.0267196	−	0.999643i	$-0.491494\pi$
$401$	2.76701e10	0.0534393	0.0267196	+	0.999643i	$0.491494\pi$
$402$	−2.90234e11	−0.554282
$403$	1.07135e12	2.02330
$404$	−3.30771e11	−0.617748
$405$	0	0
$406$	−1.07131e12	−1.95680
$407$	1.10293e10	0.0199238
$408$	−8.57889e10	−0.153271
$409$	2.08505e11	0.368436	0.184218	−	0.982885i	$-0.441025\pi$
$409$	2.08505e11	0.368436	0.184218	+	0.982885i	$0.441025\pi$
$410$	0	0
$411$	−5.21314e11	−0.901180
$412$	−4.53969e11	−0.776228
$413$	−3.85620e11	−0.652206
$414$	2.28987e11	0.383098
$415$	0	0
$416$	−1.02637e12	−1.68029
$417$	1.47285e11	0.238531
$418$	3.15141e10	0.0504907
$419$	4.50465e11	0.714000	0.357000	−	0.934104i	$-0.383800\pi$
$419$	4.50465e11	0.714000	0.357000	+	0.934104i	$0.383800\pi$
$420$	0	0
$421$	8.60883e11	1.33559	0.667797	−	0.744343i	$-0.267237\pi$
$421$	8.60883e11	1.33559	0.667797	+	0.744343i	$0.267237\pi$
$422$	2.86810e11	0.440239
$423$	−1.85737e11	−0.282076
$424$	−1.57766e11	−0.237065
$425$	0	0
$426$	−1.28801e11	−0.189486
$427$	1.44318e11	0.210086
$428$	−1.09536e12	−1.57782
$429$	1.81154e10	0.0258220
$430$	0	0
$431$	−3.02405e10	−0.0422125	−0.0211063	−	0.999777i	$-0.506719\pi$
$431$	−3.02405e10	−0.0422125	−0.0211063	+	0.999777i	$0.506719\pi$
$432$	−2.59853e10	−0.0358965
$433$	−1.03636e12	−1.41682	−0.708410	−	0.705802i	$-0.750587\pi$
$433$	−1.03636e12	−1.41682	−0.708410	+	0.705802i	$0.750587\pi$
$434$	1.14035e12	1.54288
$435$	0	0
$436$	−2.10528e12	−2.79010
$437$	5.74983e11	0.754204
$438$	6.87028e10	0.0891951
$439$	−5.90670e11	−0.759022	−0.379511	−	0.925187i	$-0.623908\pi$
$439$	−5.90670e11	−0.759022	−0.379511	+	0.925187i	$0.623908\pi$
$440$	0	0
$441$	−1.33078e11	−0.167546
$442$	−5.90001e11	−0.735280
$443$	−1.27097e12	−1.56790	−0.783948	−	0.620827i	$-0.786797\pi$
$443$	−1.27097e12	−1.56790	−0.783948	+	0.620827i	$0.786797\pi$
$444$	4.74528e11	0.579480
$445$	0	0
$446$	−2.50769e11	−0.300101
$447$	−6.72461e11	−0.796679
$448$	−9.80316e11	−1.14978
$449$	−9.34644e11	−1.08527	−0.542635	−	0.839969i	$-0.682573\pi$
$449$	−9.34644e11	−1.08527	−0.542635	+	0.839969i	$0.682573\pi$
$450$	0	0
$451$	−6.42082e9	−0.00730796
$452$	−3.56523e11	−0.401758
$453$	3.11086e11	0.347087
$454$	−1.20851e12	−1.33505
$455$	0	0
$456$	4.70406e11	0.509484
$457$	4.52481e11	0.485263	0.242632	−	0.970119i	$-0.421989\pi$
$457$	4.52481e11	0.485263	0.242632	+	0.970119i	$0.421989\pi$
$458$	1.05551e12	1.12091
$459$	−5.74817e10	−0.0604467
$460$	0	0
$461$	−8.56467e11	−0.883195	−0.441597	−	0.897213i	$-0.645588\pi$
$461$	−8.56467e11	−0.883195	−0.441597	+	0.897213i	$0.645588\pi$
$462$	1.92820e10	0.0196908
$463$	−9.21380e11	−0.931803	−0.465902	−	0.884836i	$-0.654270\pi$
$463$	−9.21380e11	−0.931803	−0.465902	+	0.884836i	$0.654270\pi$
$464$	3.24793e11	0.325294
$465$	0	0
$466$	2.95228e12	2.90016
$467$	−8.65382e10	−0.0841941	−0.0420971	−	0.999114i	$-0.513404\pi$
$467$	−8.65382e10	−0.0841941	−0.0420971	+	0.999114i	$0.513404\pi$
$468$	7.79403e11	0.751027
$469$	−4.45901e11	−0.425560
$470$	0	0
$471$	1.76509e11	0.165262
$472$	−8.42856e11	−0.781654
$473$	6.43346e9	0.00590976
$474$	−1.17152e12	−1.06597
$475$	0	0
$476$	−3.79900e11	−0.339186
$477$	−1.05709e11	−0.0934930
$478$	2.25579e12	1.97640
$479$	7.63707e11	0.662852	0.331426	−	0.943481i	$-0.392470\pi$
$479$	7.63707e11	0.662852	0.331426	+	0.943481i	$0.392470\pi$
$480$	0	0
$481$	1.13223e12	0.964458
$482$	2.79197e12	2.35613
$483$	3.51805e11	0.294130
$484$	−1.84692e12	−1.52984
$485$	0	0
$486$	1.25524e11	0.102062
$487$	5.25531e11	0.423368	0.211684	−	0.977338i	$-0.432105\pi$
$487$	5.25531e11	0.423368	0.211684	+	0.977338i	$0.432105\pi$
$488$	3.15439e11	0.251783
$489$	1.24859e12	0.987487
$490$	0	0
$491$	2.37265e12	1.84233	0.921163	−	0.389177i	$-0.127241\pi$
$491$	2.37265e12	1.84233	0.921163	+	0.389177i	$0.127241\pi$
$492$	−2.76252e11	−0.212551
$493$	7.18468e11	0.547768
$494$	3.23515e12	2.44412
$495$	0	0
$496$	−3.45724e11	−0.256485
$497$	−1.97884e11	−0.145481
$498$	2.17105e12	1.58175
$499$	−1.33387e11	−0.0963080	−0.0481540	−	0.998840i	$-0.515334\pi$
$499$	−1.33387e11	−0.0963080	−0.0481540	+	0.998840i	$0.515334\pi$
$500$	0	0
$501$	4.58060e11	0.324827
$502$	−2.09484e12	−1.47226
$503$	6.58632e11	0.458762	0.229381	−	0.973337i	$-0.426330\pi$
$503$	6.58632e11	0.458762	0.229381	+	0.973337i	$0.426330\pi$
$504$	2.87819e11	0.198693
$505$	0	0
$506$	5.15143e10	0.0349342
$507$	1.00071e12	0.672623
$508$	6.57501e10	0.0438036
$509$	−1.01965e12	−0.673322	−0.336661	−	0.941626i	$-0.609298\pi$
$509$	−1.01965e12	−0.673322	−0.336661	+	0.941626i	$0.609298\pi$
$510$	0	0
$511$	1.05552e11	0.0684811
$512$	5.76350e11	0.370656
$513$	3.15189e11	0.200929
$514$	2.66935e11	0.168683
$515$	0	0
$516$	2.76796e11	0.171884
$517$	−4.17845e10	−0.0257221
$518$	1.20515e12	0.735457
$519$	−6.23613e9	−0.00377278
$520$	0	0
$521$	−5.57535e11	−0.331514	−0.165757	−	0.986167i	$-0.553007\pi$
$521$	−5.57535e11	−0.331514	−0.165757	+	0.986167i	$0.553007\pi$
$522$	−1.56894e12	−0.924886
$523$	−2.12050e12	−1.23931	−0.619657	−	0.784873i	$-0.712728\pi$
$523$	−2.12050e12	−1.23931	−0.619657	+	0.784873i	$0.712728\pi$
$524$	−2.92903e12	−1.69720
$525$	0	0
$526$	3.78975e12	2.15861
$527$	−7.64770e11	−0.431900
$528$	−5.84581e9	−0.00327335
$529$	−8.61261e11	−0.478172
$530$	0	0
$531$	−5.64744e11	−0.308267
$532$	2.08310e12	1.12748
$533$	−6.59143e11	−0.353759
$534$	2.24494e12	1.19473
$535$	0	0
$536$	−9.74612e11	−0.510024
$537$	1.88120e12	0.976225
$538$	1.68206e12	0.865608
$539$	−2.99380e10	−0.0152782
$540$	0	0
$541$	1.92746e12	0.967379	0.483690	−	0.875240i	$-0.339296\pi$
$541$	1.92746e12	0.967379	0.483690	+	0.875240i	$0.339296\pi$
$542$	1.03272e12	0.514029
$543$	−1.00061e12	−0.493931
$544$	7.32663e11	0.358682
$545$	0	0
$546$	1.97943e12	0.953179
$547$	2.32751e12	1.11160	0.555799	−	0.831317i	$-0.312412\pi$
$547$	2.32751e12	1.11160	0.555799	+	0.831317i	$0.312412\pi$
$548$	−5.04581e12	−2.39011
$549$	2.11356e11	0.0992974
$550$	0	0
$551$	−3.93957e12	−1.82082
$552$	7.68945e11	0.352508
$553$	−1.79986e12	−0.818419
$554$	−3.06243e12	−1.38125
$555$	0	0
$556$	1.42557e12	0.632633
$557$	−4.94739e11	−0.217785	−0.108892	−	0.994054i	$-0.534730\pi$
$557$	−4.94739e11	−0.217785	−0.108892	+	0.994054i	$0.534730\pi$
$558$	1.67005e12	0.729247
$559$	6.60441e11	0.286076
$560$	0	0
$561$	−1.29314e10	−0.00551205
$562$	−2.83413e10	−0.0119841
$563$	−1.14083e12	−0.478557	−0.239279	−	0.970951i	$-0.576911\pi$
$563$	−1.14083e12	−0.478557	−0.239279	+	0.970951i	$0.576911\pi$
$564$	−1.79775e12	−0.748124
$565$	0	0
$566$	−8.96260e11	−0.367079
$567$	1.92849e11	0.0783600
$568$	−4.32517e11	−0.174356
$569$	−1.64398e10	−0.00657495	−0.00328747	−	0.999995i	$-0.501046\pi$
$569$	−1.64398e10	−0.00657495	−0.00328747	+	0.999995i	$0.501046\pi$
$570$	0	0
$571$	−3.67652e12	−1.44735	−0.723676	−	0.690139i	$-0.757549\pi$
$571$	−3.67652e12	−1.44735	−0.723676	+	0.690139i	$0.757549\pi$
$572$	1.75339e11	0.0684851
$573$	−3.40551e11	−0.131973
$574$	−7.01592e11	−0.269762
$575$	0	0
$576$	−1.43568e12	−0.543447
$577$	2.29045e12	0.860260	0.430130	−	0.902767i	$-0.358468\pi$
$577$	2.29045e12	0.860260	0.430130	+	0.902767i	$0.358468\pi$
$578$	−3.84800e12	−1.43403
$579$	3.55275e11	0.131375
$580$	0	0
$581$	3.33549e12	1.21441
$582$	−2.64519e12	−0.955661
$583$	−2.37809e10	−0.00852549
$584$	2.30706e11	0.0820730
$585$	0	0
$586$	−5.65465e12	−1.98092
$587$	−4.68750e12	−1.62956	−0.814780	−	0.579771i	$-0.803142\pi$
$587$	−4.68750e12	−1.62956	−0.814780	+	0.579771i	$0.803142\pi$
$588$	−1.28806e12	−0.444365
$589$	4.19346e12	1.43567
$590$	0	0
$591$	2.72722e12	0.919553
$592$	−3.65371e11	−0.122261
$593$	2.33770e12	0.776323	0.388162	−	0.921591i	$-0.373110\pi$
$593$	2.33770e12	0.776323	0.388162	+	0.921591i	$0.373110\pi$
$594$	2.82386e10	0.00930690
$595$	0	0
$596$	−6.50876e12	−2.11295
$597$	8.32131e11	0.268106
$598$	5.28831e12	1.69107
$599$	−4.66995e12	−1.48215	−0.741075	−	0.671423i	$-0.765684\pi$
$599$	−4.66995e12	−1.48215	−0.741075	+	0.671423i	$0.765684\pi$
$600$	0	0
$601$	−3.96517e12	−1.23973	−0.619864	−	0.784709i	$-0.712812\pi$
$601$	−3.96517e12	−1.23973	−0.619864	+	0.784709i	$0.712812\pi$
$602$	7.02974e11	0.218150
$603$	−6.53026e11	−0.201142
$604$	3.01101e12	0.920546
$605$	0	0
$606$	−1.23027e12	−0.370572
$607$	−6.24743e12	−1.86790	−0.933948	−	0.357409i	$-0.883660\pi$
$607$	−6.24743e12	−1.86790	−0.933948	+	0.357409i	$0.883660\pi$
$608$	−4.01741e12	−1.19228
$609$	−2.41044e12	−0.710098
$610$	0	0
$611$	−4.28947e12	−1.24514
$612$	−5.56366e11	−0.160317
$613$	3.73193e12	1.06748	0.533742	−	0.845647i	$-0.320785\pi$
$613$	3.73193e12	1.06748	0.533742	+	0.845647i	$0.320785\pi$
$614$	8.84653e12	2.51197
$615$	0	0
$616$	6.47494e10	0.0181185
$617$	6.05181e12	1.68113	0.840567	−	0.541708i	$-0.182222\pi$
$617$	6.05181e12	1.68113	0.840567	+	0.541708i	$0.182222\pi$
$618$	−1.68849e12	−0.465640
$619$	4.69849e12	1.28632	0.643162	−	0.765730i	$-0.277622\pi$
$619$	4.69849e12	1.28632	0.643162	+	0.765730i	$0.277622\pi$
$620$	0	0
$621$	5.15221e11	0.139021
$622$	−5.79780e12	−1.55313
$623$	3.44902e12	0.917275
$624$	−6.00114e11	−0.158454
$625$	0	0
$626$	8.80724e12	2.29222
$627$	7.09068e10	0.0183225
$628$	1.70843e12	0.438308
$629$	−8.08231e11	−0.205877
$630$	0	0
$631$	2.16875e12	0.544600	0.272300	−	0.962212i	$-0.412216\pi$
$631$	2.16875e12	0.544600	0.272300	+	0.962212i	$0.412216\pi$
$632$	−3.93398e12	−0.980857
$633$	6.45323e11	0.159757
$634$	−4.06195e12	−0.998466
$635$	0	0
$636$	−1.02316e12	−0.247962
$637$	−3.07335e12	−0.739580
$638$	−3.52957e11	−0.0843391
$639$	−2.89803e11	−0.0687620
$640$	0	0
$641$	3.56446e12	0.833936	0.416968	−	0.908921i	$-0.363093\pi$
$641$	3.56446e12	0.833936	0.416968	+	0.908921i	$0.363093\pi$
$642$	−4.07406e12	−0.946497
$643$	5.37917e12	1.24098	0.620491	−	0.784213i	$-0.286933\pi$
$643$	5.37917e12	1.24098	0.620491	+	0.784213i	$0.286933\pi$
$644$	3.40512e12	0.780093
$645$	0	0
$646$	−2.30937e12	−0.521732
$647$	−6.01827e12	−1.35021	−0.675107	−	0.737720i	$-0.735903\pi$
$647$	−6.01827e12	−1.35021	−0.675107	+	0.737720i	$0.735903\pi$
$648$	4.21513e11	0.0939126
$649$	−1.27048e11	−0.0281104
$650$	0	0
$651$	2.56578e12	0.559893
$652$	1.20852e13	2.61901
$653$	3.68383e11	0.0792849	0.0396424	−	0.999214i	$-0.487378\pi$
$653$	3.68383e11	0.0792849	0.0396424	+	0.999214i	$0.487378\pi$
$654$	−7.83035e12	−1.67371
$655$	0	0
$656$	2.12705e11	0.0448446
$657$	1.54581e11	0.0323678
$658$	−4.56572e12	−0.949494
$659$	−5.24810e12	−1.08397	−0.541985	−	0.840388i	$-0.682327\pi$
$659$	−5.24810e12	−1.08397	−0.541985	+	0.840388i	$0.682327\pi$
$660$	0	0
$661$	1.55039e11	0.0315888	0.0157944	−	0.999875i	$-0.494972\pi$
$661$	1.55039e11	0.0315888	0.0157944	+	0.999875i	$0.494972\pi$
$662$	1.03445e13	2.09339
$663$	−1.32750e12	−0.266824
$664$	7.29043e12	1.45545
$665$	0	0
$666$	1.76495e12	0.347616
$667$	−6.43979e12	−1.25981
$668$	4.43357e12	0.861508
$669$	−5.64230e11	−0.108903
$670$	0	0
$671$	4.75477e10	0.00905479
$672$	−2.45806e12	−0.464977
$673$	−5.89588e11	−0.110785	−0.0553925	−	0.998465i	$-0.517641\pi$
$673$	−5.89588e11	−0.110785	−0.0553925	+	0.998465i	$0.517641\pi$
$674$	9.09487e11	0.169757
$675$	0	0
$676$	9.68586e12	1.78393
$677$	−3.32980e12	−0.609214	−0.304607	−	0.952478i	$-0.598525\pi$
$677$	−3.32980e12	−0.609214	−0.304607	+	0.952478i	$0.598525\pi$
$678$	−1.32605e12	−0.241005
$679$	−4.06395e12	−0.733726
$680$	0	0
$681$	−2.71914e12	−0.484474
$682$	3.75703e11	0.0664991
$683$	−2.28928e12	−0.402536	−0.201268	−	0.979536i	$-0.564506\pi$
$683$	−2.28928e12	−0.402536	−0.201268	+	0.979536i	$0.564506\pi$
$684$	3.05072e12	0.532905
$685$	0	0
$686$	−9.77951e12	−1.68600
$687$	2.37491e12	0.406763
$688$	−2.13124e11	−0.0362647
$689$	−2.44128e12	−0.412697
$690$	0	0
$691$	−9.46129e12	−1.57870	−0.789349	−	0.613945i	$-0.789582\pi$
$691$	−9.46129e12	−1.57870	−0.789349	+	0.613945i	$0.789582\pi$
$692$	−6.03596e10	−0.0100062
$693$	4.33845e10	0.00714554
$694$	−3.25729e12	−0.533014
$695$	0	0
$696$	−5.26853e12	−0.851036
$697$	4.70521e11	0.0755147
$698$	−5.53761e11	−0.0883024
$699$	6.64264e12	1.05243
$700$	0	0
$701$	8.22209e12	1.28603	0.643015	−	0.765854i	$-0.277683\pi$
$701$	8.22209e12	1.28603	0.643015	+	0.765854i	$0.277683\pi$
$702$	2.89890e12	0.450522
$703$	4.43177e12	0.684349
$704$	−3.22979e11	−0.0495562
$705$	0	0
$706$	5.28751e12	0.800995
$707$	−1.89012e12	−0.284513
$708$	−5.46617e12	−0.817586
$709$	7.61957e12	1.13246	0.566230	−	0.824247i	$-0.308401\pi$
$709$	7.61957e12	1.13246	0.566230	+	0.824247i	$0.308401\pi$
$710$	0	0
$711$	−2.63591e12	−0.386828
$712$	7.53858e12	1.09933
$713$	6.85481e12	0.993327
$714$	−1.41299e12	−0.203469
$715$	0	0
$716$	1.82081e13	2.58915
$717$	5.07554e12	0.717209
$718$	−1.59209e13	−2.23567
$719$	7.94823e12	1.10915	0.554575	−	0.832134i	$-0.312881\pi$
$719$	7.94823e12	1.10915	0.554575	+	0.832134i	$0.312881\pi$
$720$	0	0
$721$	−2.59411e12	−0.357503
$722$	1.04619e12	0.143283
$723$	6.28194e12	0.855010
$724$	−9.68492e12	−1.31000
$725$	0	0
$726$	−6.86942e12	−0.917710
$727$	5.75787e12	0.764464	0.382232	−	0.924066i	$-0.375156\pi$
$727$	5.75787e12	0.764464	0.382232	+	0.924066i	$0.375156\pi$
$728$	6.64699e12	0.877069
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−4.71447e11	−0.0610668
$732$	2.04571e12	0.263357
$733$	−1.07808e13	−1.37938	−0.689690	−	0.724105i	$-0.742253\pi$
$733$	−1.07808e13	−1.37938	−0.689690	+	0.724105i	$0.742253\pi$
$734$	−5.33198e12	−0.678041
$735$	0	0
$736$	−6.56702e12	−0.824933
$737$	−1.46908e11	−0.0183418
$738$	−1.02749e12	−0.127504
$739$	7.37756e12	0.909940	0.454970	−	0.890507i	$-0.349650\pi$
$739$	7.37756e12	0.909940	0.454970	+	0.890507i	$0.349650\pi$
$740$	0	0
$741$	7.27909e12	0.886942
$742$	−2.59850e12	−0.314706
$743$	1.02899e13	1.23869	0.619346	−	0.785118i	$-0.287398\pi$
$743$	1.02899e13	1.23869	0.619346	+	0.785118i	$0.287398\pi$
$744$	5.60806e12	0.671018
$745$	0	0
$746$	−2.74856e12	−0.324923
$747$	4.88485e12	0.573996
$748$	−1.25163e11	−0.0146191
$749$	−6.25918e12	−0.726690
$750$	0	0
$751$	−7.56555e12	−0.867882	−0.433941	−	0.900941i	$-0.642877\pi$
$751$	−7.56555e12	−0.867882	−0.433941	+	0.900941i	$0.642877\pi$
$752$	1.38421e12	0.157842
$753$	−4.71340e12	−0.534265
$754$	−3.62336e13	−4.08263
$755$	0	0
$756$	1.86659e12	0.207827
$757$	6.88713e12	0.762267	0.381133	−	0.924520i	$-0.375534\pi$
$757$	6.88713e12	0.762267	0.381133	+	0.924520i	$0.375534\pi$
$758$	−9.72690e12	−1.07020
$759$	1.15907e11	0.0126772
$760$	0	0
$761$	4.30641e12	0.465462	0.232731	−	0.972541i	$-0.425234\pi$
$761$	4.30641e12	0.465462	0.232731	+	0.972541i	$0.425234\pi$
$762$	2.44550e11	0.0262767
$763$	−1.20302e13	−1.28502
$764$	−3.29619e12	−0.350020
$765$	0	0
$766$	2.37972e13	2.49745
$767$	−1.30424e13	−1.36075
$768$	−3.78281e12	−0.392364
$769$	2.16296e12	0.223039	0.111519	−	0.993762i	$-0.464428\pi$
$769$	2.16296e12	0.223039	0.111519	+	0.993762i	$0.464428\pi$
$770$	0	0
$771$	6.00603e11	0.0612129
$772$	3.43871e12	0.348432
$773$	8.15447e12	0.821463	0.410731	−	0.911756i	$-0.365273\pi$
$773$	8.15447e12	0.821463	0.410731	+	0.911756i	$0.365273\pi$
$774$	1.02951e12	0.103109
$775$	0	0
$776$	−8.88263e12	−0.879354
$777$	2.71159e12	0.266888
$778$	−1.11550e13	−1.09159
$779$	−2.58000e12	−0.251016
$780$	0	0
$781$	−6.51956e10	−0.00627031
$782$	−3.77499e12	−0.360982
$783$	−3.53011e12	−0.335630
$784$	9.91768e11	0.0937535
$785$	0	0
$786$	−1.08942e13	−1.01811
$787$	2.25199e12	0.209257	0.104629	−	0.994511i	$-0.466635\pi$
$787$	2.25199e12	0.209257	0.104629	+	0.994511i	$0.466635\pi$
$788$	2.63968e13	2.43884
$789$	8.52694e12	0.783334
$790$	0	0
$791$	−2.03727e12	−0.185036
$792$	9.48261e10	0.00856376
$793$	4.88112e12	0.438319
$794$	2.34212e13	2.09130
$795$	0	0
$796$	8.05420e12	0.711072
$797$	1.16128e13	1.01947	0.509737	−	0.860331i	$-0.329743\pi$
$797$	1.16128e13	1.01947	0.509737	+	0.860331i	$0.329743\pi$
$798$	7.74786e12	0.676346
$799$	3.06198e12	0.265792
$800$	0	0
$801$	5.05112e12	0.433553
$802$	9.96122e11	0.0850214
$803$	3.47755e10	0.00295157
$804$	−6.32065e12	−0.533469
$805$	0	0
$806$	3.85687e13	3.21904
$807$	3.78464e12	0.314118
$808$	−4.13126e12	−0.340982
$809$	2.31768e12	0.190233	0.0951165	−	0.995466i	$-0.469678\pi$
$809$	2.31768e12	0.190233	0.0951165	+	0.995466i	$0.469678\pi$
$810$	0	0
$811$	−9.18815e12	−0.745821	−0.372910	−	0.927867i	$-0.621640\pi$
$811$	−9.18815e12	−0.745821	−0.372910	+	0.927867i	$0.621640\pi$
$812$	−2.33307e13	−1.88332
$813$	2.32363e12	0.186535
$814$	3.97054e11	0.0316986
$815$	0	0
$816$	4.28384e11	0.0338242
$817$	2.58508e12	0.202990
$818$	7.50620e12	0.586179
$819$	4.45373e12	0.345897
$820$	0	0
$821$	−1.92679e13	−1.48010	−0.740048	−	0.672554i	$-0.765197\pi$
$821$	−1.92679e13	−1.48010	−0.740048	+	0.672554i	$0.765197\pi$
$822$	−1.87673e13	−1.43377
$823$	1.04990e13	0.797713	0.398857	−	0.917013i	$-0.369407\pi$
$823$	1.04990e13	0.797713	0.398857	+	0.917013i	$0.369407\pi$
$824$	−5.66999e12	−0.428459
$825$	0	0
$826$	−1.38823e13	−1.03765
$827$	8.10053e12	0.602197	0.301098	−	0.953593i	$-0.402647\pi$
$827$	8.10053e12	0.602197	0.301098	+	0.953593i	$0.402647\pi$
$828$	4.98683e12	0.368713
$829$	−1.52452e13	−1.12108	−0.560542	−	0.828126i	$-0.689407\pi$
$829$	−1.52452e13	−1.12108	−0.560542	+	0.828126i	$0.689407\pi$
$830$	0	0
$831$	−6.89048e12	−0.501239
$832$	−3.31561e13	−2.39888
$833$	2.19387e12	0.157873
$834$	5.30224e12	0.379501
$835$	0	0
$836$	6.86307e11	0.0485949
$837$	3.75761e12	0.264635
$838$	1.62168e13	1.13597
$839$	2.72258e13	1.89694	0.948468	−	0.316874i	$-0.102633\pi$
$839$	2.72258e13	1.89694	0.948468	+	0.316874i	$0.102633\pi$
$840$	0	0
$841$	2.96160e13	2.04147
$842$	3.09918e13	2.12492
$843$	−6.37678e10	−0.00434888
$844$	6.24609e12	0.423709
$845$	0	0
$846$	−6.68653e12	−0.448781
$847$	−1.05538e13	−0.704588
$848$	7.87798e11	0.0523159
$849$	−2.01658e12	−0.133208
$850$	0	0
$851$	7.24435e12	0.473496
$852$	−2.80500e12	−0.182371
$853$	−5.38234e12	−0.348097	−0.174048	−	0.984737i	$-0.555685\pi$
$853$	−5.38234e12	−0.348097	−0.174048	+	0.984737i	$0.555685\pi$
$854$	5.19546e12	0.334244
$855$	0	0
$856$	−1.36808e13	−0.870922
$857$	−2.13114e12	−0.134958	−0.0674791	−	0.997721i	$-0.521496\pi$
$857$	−2.13114e12	−0.134958	−0.0674791	+	0.997721i	$0.521496\pi$
$858$	6.52153e11	0.0410825
$859$	1.07650e13	0.674598	0.337299	−	0.941398i	$-0.390487\pi$
$859$	1.07650e13	0.674598	0.337299	+	0.941398i	$0.390487\pi$
$860$	0	0
$861$	−1.57858e12	−0.0978933
$862$	−1.08866e12	−0.0671597
$863$	−2.36698e13	−1.45260	−0.726299	−	0.687379i	$-0.758761\pi$
$863$	−2.36698e13	−1.45260	−0.726299	+	0.687379i	$0.758761\pi$
$864$	−3.59985e12	−0.219772
$865$	0	0
$866$	−3.73089e13	−2.25415
$867$	−8.65800e12	−0.520393
$868$	2.48342e13	1.48495
$869$	−5.92990e11	−0.0352743
$870$	0	0
$871$	−1.50812e13	−0.887880
$872$	−2.62945e13	−1.54007
$873$	−5.95169e12	−0.346797
$874$	2.06994e13	1.19993
$875$	0	0
$876$	1.49619e12	0.0858459
$877$	9.91806e12	0.566146	0.283073	−	0.959098i	$-0.408646\pi$
$877$	9.91806e12	0.566146	0.283073	+	0.959098i	$0.408646\pi$
$878$	−2.12641e13	−1.20760
$879$	−1.27230e13	−0.718851
$880$	0	0
$881$	1.28992e13	0.721391	0.360696	−	0.932684i	$-0.382539\pi$
$881$	1.28992e13	0.721391	0.360696	+	0.932684i	$0.382539\pi$
$882$	−4.79081e12	−0.266563
$883$	−1.00956e13	−0.558866	−0.279433	−	0.960165i	$-0.590146\pi$
$883$	−1.00956e13	−0.558866	−0.279433	+	0.960165i	$0.590146\pi$
$884$	−1.28489e13	−0.707671
$885$	0	0
$886$	−4.57548e13	−2.49451
$887$	2.22403e12	0.120638	0.0603190	−	0.998179i	$-0.480788\pi$
$887$	2.22403e12	0.120638	0.0603190	+	0.998179i	$0.480788\pi$
$888$	5.92676e12	0.319859
$889$	3.75715e11	0.0201744
$890$	0	0
$891$	6.35370e10	0.00337736
$892$	−5.46119e12	−0.288832
$893$	−1.67898e13	−0.883513
$894$	−2.42086e13	−1.26751
$895$	0	0
$896$	−1.97540e13	−1.02393
$897$	1.18987e13	0.613668
$898$	−3.36472e13	−1.72665
$899$	−4.69666e13	−2.39812
$900$	0	0
$901$	1.74267e12	0.0880957
$902$	−2.31150e11	−0.0116269
$903$	1.58169e12	0.0791638
$904$	−4.45290e12	−0.221761
$905$	0	0
$906$	1.11991e13	0.552213
$907$	6.35542e12	0.311825	0.155913	−	0.987771i	$-0.450168\pi$
$907$	6.35542e12	0.311825	0.155913	+	0.987771i	$0.450168\pi$
$908$	−2.63186e13	−1.28492
$909$	−2.76810e12	−0.134476
$910$	0	0
$911$	−1.91432e13	−0.920837	−0.460418	−	0.887702i	$-0.652301\pi$
$911$	−1.91432e13	−0.920837	−0.460418	+	0.887702i	$0.652301\pi$
$912$	−2.34895e12	−0.112434
$913$	1.09892e12	0.0523419
$914$	1.62893e13	0.772049
$915$	0	0
$916$	2.29867e13	1.07882
$917$	−1.67373e13	−0.781671
$918$	−2.06934e12	−0.0961701
$919$	2.86053e13	1.32290	0.661449	−	0.749991i	$-0.269942\pi$
$919$	2.86053e13	1.32290	0.661449	+	0.749991i	$0.269942\pi$
$920$	0	0
$921$	1.99047e13	0.911564
$922$	−3.08328e13	−1.40515
$923$	−6.69280e12	−0.303529
$924$	4.19919e11	0.0189514
$925$	0	0
$926$	−3.31697e13	−1.48249
$927$	−3.79910e12	−0.168975
$928$	4.49948e13	1.99158
$929$	−1.18899e13	−0.523728	−0.261864	−	0.965105i	$-0.584337\pi$
$929$	−1.18899e13	−0.523728	−0.261864	+	0.965105i	$0.584337\pi$
$930$	0	0
$931$	−1.20296e13	−0.524782
$932$	6.42942e13	2.79126
$933$	−1.30451e13	−0.563610
$934$	−3.11537e12	−0.133952
$935$	0	0
$936$	9.73458e12	0.414549
$937$	3.45871e13	1.46584	0.732920	−	0.680315i	$-0.238157\pi$
$937$	3.45871e13	1.46584	0.732920	+	0.680315i	$0.238157\pi$
$938$	−1.60524e13	−0.677061
$939$	1.98163e13	0.831816
$940$	0	0
$941$	2.59742e13	1.07991	0.539956	−	0.841693i	$-0.318441\pi$
$941$	2.59742e13	1.07991	0.539956	+	0.841693i	$0.318441\pi$
$942$	6.35432e12	0.262930
$943$	−4.21738e12	−0.173676
$944$	4.20877e12	0.172497
$945$	0	0
$946$	2.31605e11	0.00940237
$947$	−2.43739e13	−0.984804	−0.492402	−	0.870368i	$-0.663881\pi$
$947$	−2.43739e13	−0.984804	−0.492402	+	0.870368i	$0.663881\pi$
$948$	−2.55130e13	−1.02595
$949$	3.56995e12	0.142878
$950$	0	0
$951$	−9.13939e12	−0.362331
$952$	−4.74487e12	−0.187222
$953$	2.34891e13	0.922461	0.461230	−	0.887280i	$-0.347408\pi$
$953$	2.34891e13	0.922461	0.461230	+	0.887280i	$0.347408\pi$
$954$	−3.80552e12	−0.148746
$955$	0	0
$956$	4.91262e13	1.90218
$957$	−7.94153e11	−0.0306056
$958$	2.74934e13	1.05459
$959$	−2.88332e13	−1.10080
$960$	0	0
$961$	2.35538e13	0.890851
$962$	4.07604e13	1.53444
$963$	−9.16663e12	−0.343472
$964$	6.08029e13	2.26766
$965$	0	0
$966$	1.26650e13	0.467958
$967$	−4.68639e12	−0.172353	−0.0861766	−	0.996280i	$-0.527465\pi$
$967$	−4.68639e12	−0.172353	−0.0861766	+	0.996280i	$0.527465\pi$
$968$	−2.30677e13	−0.844433
$969$	−5.19608e12	−0.189330
$970$	0	0
$971$	−4.63936e13	−1.67483	−0.837416	−	0.546566i	$-0.815935\pi$
$971$	−4.63936e13	−1.67483	−0.837416	+	0.546566i	$0.815935\pi$
$972$	2.73364e12	0.0982297
$973$	8.14611e12	0.291368
$974$	1.89191e13	0.673574
$975$	0	0
$976$	−1.57513e12	−0.0555639
$977$	−2.53890e13	−0.891496	−0.445748	−	0.895159i	$-0.647062\pi$
$977$	−2.53890e13	−0.891496	−0.445748	+	0.895159i	$0.647062\pi$
$978$	4.49494e13	1.57108
$979$	1.13633e12	0.0395350
$980$	0	0
$981$	−1.76183e13	−0.607370
$982$	8.54153e13	2.93112
$983$	8.90780e12	0.304284	0.152142	−	0.988359i	$-0.451383\pi$
$983$	8.90780e12	0.304284	0.152142	+	0.988359i	$0.451383\pi$
$984$	−3.45033e12	−0.117323
$985$	0	0
$986$	2.58649e13	0.871493
$987$	−1.02729e13	−0.344559
$988$	7.04544e13	2.35235
$989$	4.22569e12	0.140448
$990$	0	0
$991$	3.90840e13	1.28727	0.643633	−	0.765335i	$-0.277427\pi$
$991$	3.90840e13	1.28727	0.643633	+	0.765335i	$0.277427\pi$
$992$	−4.78945e13	−1.57030
$993$	2.32752e13	0.759665
$994$	−7.12382e12	−0.231459
$995$	0	0
$996$	4.72806e13	1.52235
$997$	1.94045e12	0.0621975	0.0310988	−	0.999516i	$-0.490099\pi$
$997$	1.94045e12	0.0621975	0.0310988	+	0.999516i	$0.490099\pi$
$998$	−4.80195e12	−0.153225
$999$	3.97115e12	0.126145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.10.a.d.1.1		1
3.2	odd	2		225.10.a.a.1.1		1
5.2	odd	4		75.10.b.a.49.2		2
5.3	odd	4		75.10.b.a.49.1		2
5.4	even	2		3.10.a.a.1.1	✓	1
15.2	even	4		225.10.b.a.199.1		2
15.8	even	4		225.10.b.a.199.2		2
15.14	odd	2		9.10.a.c.1.1		1
20.19	odd	2		48.10.a.e.1.1		1
35.34	odd	2		147.10.a.a.1.1		1
40.19	odd	2		192.10.a.f.1.1		1
40.29	even	2		192.10.a.m.1.1		1
45.4	even	6		81.10.c.e.55.1		2
45.14	odd	6		81.10.c.a.55.1		2
45.29	odd	6		81.10.c.a.28.1		2
45.34	even	6		81.10.c.e.28.1		2
55.54	odd	2		363.10.a.b.1.1		1
60.59	even	2		144.10.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.10.a.a.1.1	✓	1	5.4	even	2
9.10.a.c.1.1		1	15.14	odd	2
48.10.a.e.1.1		1	20.19	odd	2
75.10.a.d.1.1		1	1.1	even	1	trivial
75.10.b.a.49.1		2	5.3	odd	4
75.10.b.a.49.2		2	5.2	odd	4
81.10.c.a.28.1		2	45.29	odd	6
81.10.c.a.55.1		2	45.14	odd	6
81.10.c.e.28.1		2	45.34	even	6
81.10.c.e.55.1		2	45.4	even	6
144.10.a.l.1.1		1	60.59	even	2
147.10.a.a.1.1		1	35.34	odd	2
192.10.a.f.1.1		1	40.19	odd	2
192.10.a.m.1.1		1	40.29	even	2
225.10.a.a.1.1		1	3.2	odd	2
225.10.b.a.199.1		2	15.2	even	4
225.10.b.a.199.2		2	15.8	even	4
363.10.a.b.1.1		1	55.54	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more