LMFDB - Embedded newform 75.16.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,16,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, 16, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 16 $
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:	$107.020128825$
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)

$f(q)$

$=$

$q+72.0000 q^{2} -2187.00 q^{3} -27584.0 q^{4} -157464. q^{6} +2.14900e6 q^{7} -4.34534e6 q^{8} +4.78297e6 q^{9} +O(q^{10})$

\(q+72.0000 q^{2} -2187.00 q^{3} -27584.0 q^{4} -157464. q^{6} +2.14900e6 q^{7} -4.34534e6 q^{8} +4.78297e6 q^{9} +3.71693e7 q^{11} +6.03262e7 q^{12} +2.79974e8 q^{13} +1.54728e8 q^{14} +5.91008e8 q^{16} -2.49291e9 q^{17} +3.44374e8 q^{18} -4.66978e9 q^{19} -4.69986e9 q^{21} +2.67619e9 q^{22} +1.84679e10 q^{23} +9.50327e9 q^{24} +2.01581e10 q^{26} -1.04604e10 q^{27} -5.92780e10 q^{28} -1.15953e11 q^{29} -5.61870e10 q^{31} +1.84941e11 q^{32} -8.12893e10 q^{33} -1.79490e11 q^{34} -1.31933e11 q^{36} -6.14765e11 q^{37} -3.36224e11 q^{38} -6.12304e11 q^{39} +5.49860e11 q^{41} -3.38390e11 q^{42} +9.82884e11 q^{43} -1.02528e12 q^{44} +1.32969e12 q^{46} -2.07614e12 q^{47} -1.29253e12 q^{48} -1.29361e11 q^{49} +5.45200e12 q^{51} -7.72281e12 q^{52} +1.20484e13 q^{53} -7.53145e11 q^{54} -9.33814e12 q^{56} +1.02128e13 q^{57} -8.34865e12 q^{58} +2.30879e13 q^{59} -8.50581e12 q^{61} -4.04547e12 q^{62} +1.02786e13 q^{63} -6.05040e12 q^{64} -5.85283e12 q^{66} +1.23310e13 q^{67} +6.87645e13 q^{68} -4.03894e13 q^{69} +5.89892e13 q^{71} -2.07836e13 q^{72} +5.60983e12 q^{73} -4.42631e13 q^{74} +1.28811e14 q^{76} +7.98769e13 q^{77} -4.40859e13 q^{78} +1.59919e14 q^{79} +2.28768e13 q^{81} +3.95899e13 q^{82} -5.76759e13 q^{83} +1.29641e14 q^{84} +7.07677e13 q^{86} +2.53590e14 q^{87} -1.61513e14 q^{88} -3.62288e14 q^{89} +6.01665e14 q^{91} -5.09419e14 q^{92} +1.22881e14 q^{93} -1.49482e14 q^{94} -4.04466e14 q^{96} +5.39787e14 q^{97} -9.31396e12 q^{98} +1.77780e14 q^{99} +O(q^{100})\)

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 $

$2$	72.0000	0.397748	0.198874	−	0.980025i	$-0.436272\pi$
$2$	72.0000	0.397748	0.198874	+	0.980025i	$0.436272\pi$
$3$	−2187.00	−0.577350
$3$	−2187.00	−0.577350
$4$	−27584.0	−0.841797
$5$	0	0
$5$	0	0
$6$	−157464.	−0.229640
$7$	2.14900e6	0.986282	0.493141	−	0.869949i	$-0.335849\pi$
$7$	2.14900e6	0.986282	0.493141	+	0.869949i	$0.335849\pi$
$8$	−4.34534e6	−0.732570
$9$	4.78297e6	0.333333
$10$	0	0
$11$	3.71693e7	0.575095	0.287547	−	0.957766i	$-0.407160\pi$
$11$	3.71693e7	0.575095	0.287547	+	0.957766i	$0.407160\pi$
$12$	6.03262e7	0.486012
$13$	2.79974e8	1.23749	0.618747	−	0.785590i	$-0.287641\pi$
$13$	2.79974e8	1.23749	0.618747	+	0.785590i	$0.287641\pi$
$14$	1.54728e8	0.392291
$15$	0	0
$16$	5.91008e8	0.550419
$17$	−2.49291e9	−1.47347	−0.736733	−	0.676184i	$-0.763633\pi$
$17$	−2.49291e9	−1.47347	−0.736733	+	0.676184i	$0.763633\pi$
$18$	3.44374e8	0.132583
$19$	−4.66978e9	−1.19852	−0.599259	−	0.800555i	$-0.704538\pi$
$19$	−4.66978e9	−1.19852	−0.599259	+	0.800555i	$0.704538\pi$
$20$	0	0
$21$	−4.69986e9	−0.569430
$22$	2.67619e9	0.228743
$23$	1.84679e10	1.13099	0.565496	−	0.824751i	$-0.308685\pi$
$23$	1.84679e10	1.13099	0.565496	+	0.824751i	$0.308685\pi$
$24$	9.50327e9	0.422950
$25$	0	0
$26$	2.01581e10	0.492210
$27$	−1.04604e10	−0.192450
$28$	−5.92780e10	−0.830249
$29$	−1.15953e11	−1.24824	−0.624121	−	0.781328i	$-0.714543\pi$
$29$	−1.15953e11	−1.24824	−0.624121	+	0.781328i	$0.714543\pi$
$30$	0	0
$31$	−5.61870e10	−0.366795	−0.183397	−	0.983039i	$-0.558710\pi$
$31$	−5.61870e10	−0.366795	−0.183397	+	0.983039i	$0.558710\pi$
$32$	1.84941e11	0.951498
$33$	−8.12893e10	−0.332031
$34$	−1.79490e11	−0.586068
$35$	0	0
$36$	−1.31933e11	−0.280599
$37$	−6.14765e11	−1.06462	−0.532312	−	0.846548i	$-0.678677\pi$
$37$	−6.14765e11	−1.06462	−0.532312	+	0.846548i	$0.678677\pi$
$38$	−3.36224e11	−0.476708
$39$	−6.12304e11	−0.714467
$40$	0	0
$41$	5.49860e11	0.440933	0.220467	−	0.975394i	$-0.429242\pi$
$41$	5.49860e11	0.440933	0.220467	+	0.975394i	$0.429242\pi$
$42$	−3.38390e11	−0.226489
$43$	9.82884e11	0.551428	0.275714	−	0.961240i	$-0.411086\pi$
$43$	9.82884e11	0.551428	0.275714	+	0.961240i	$0.411086\pi$
$44$	−1.02528e12	−0.484113
$45$	0	0
$46$	1.32969e12	0.449849
$47$	−2.07614e12	−0.597756	−0.298878	−	0.954291i	$-0.596612\pi$
$47$	−2.07614e12	−0.597756	−0.298878	+	0.954291i	$0.596612\pi$
$48$	−1.29253e12	−0.317784
$49$	−1.29361e11	−0.0272478
$50$	0	0
$51$	5.45200e12	0.850706
$52$	−7.72281e12	−1.04172
$53$	1.20484e13	1.40883	0.704417	−	0.709787i	$-0.251209\pi$
$53$	1.20484e13	1.40883	0.704417	+	0.709787i	$0.251209\pi$
$54$	−7.53145e11	−0.0765466
$55$	0	0
$56$	−9.33814e12	−0.722521
$57$	1.02128e13	0.691965
$58$	−8.34865e12	−0.496485
$59$	2.30879e13	1.20780	0.603899	−	0.797061i	$-0.293613\pi$
$59$	2.30879e13	1.20780	0.603899	+	0.797061i	$0.293613\pi$
$60$	0	0
$61$	−8.50581e12	−0.346531	−0.173265	−	0.984875i	$-0.555432\pi$
$61$	−8.50581e12	−0.346531	−0.173265	+	0.984875i	$0.555432\pi$
$62$	−4.04547e12	−0.145892
$63$	1.02786e13	0.328761
$64$	−6.05040e12	−0.171963
$65$	0	0
$66$	−5.85283e12	−0.132065
$67$	1.23310e13	0.248564	0.124282	−	0.992247i	$-0.460337\pi$
$67$	1.23310e13	0.248564	0.124282	+	0.992247i	$0.460337\pi$
$68$	6.87645e13	1.24036
$69$	−4.03894e13	−0.652979
$70$	0	0
$71$	5.89892e13	0.769724	0.384862	−	0.922974i	$-0.374249\pi$
$71$	5.89892e13	0.769724	0.384862	+	0.922974i	$0.374249\pi$
$72$	−2.07836e13	−0.244190
$73$	5.60983e12	0.0594331	0.0297165	−	0.999558i	$-0.490540\pi$
$73$	5.60983e12	0.0594331	0.0297165	+	0.999558i	$0.490540\pi$
$74$	−4.42631e13	−0.423452
$75$	0	0
$76$	1.28811e14	1.00891
$77$	7.98769e13	0.567206
$78$	−4.40859e13	−0.284178
$79$	1.59919e14	0.936906	0.468453	−	0.883488i	$-0.344812\pi$
$79$	1.59919e14	0.936906	0.468453	+	0.883488i	$0.344812\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	3.95899e13	0.175380
$83$	−5.76759e13	−0.233297	−0.116648	−	0.993173i	$-0.537215\pi$
$83$	−5.76759e13	−0.233297	−0.116648	+	0.993173i	$0.537215\pi$
$84$	1.29641e14	0.479345
$85$	0	0
$86$	7.07677e13	0.219329
$87$	2.53590e14	0.720673
$88$	−1.61513e14	−0.421297
$89$	−3.62288e14	−0.868217	−0.434109	−	0.900861i	$-0.642937\pi$
$89$	−3.62288e14	−0.868217	−0.434109	+	0.900861i	$0.642937\pi$
$90$	0	0
$91$	6.01665e14	1.22052
$92$	−5.09419e14	−0.952066
$93$	1.22881e14	0.211769
$94$	−1.49482e14	−0.237756
$95$	0	0
$96$	−4.04466e14	−0.549348
$97$	5.39787e14	0.678319	0.339160	−	0.940729i	$-0.389857\pi$
$97$	5.39787e14	0.678319	0.339160	+	0.940729i	$0.389857\pi$
$98$	−9.31396e12	−0.0108377
$99$	1.77780e14	0.191698
$100$	0	0
$101$	−4.22221e14	−0.391859	−0.195929	−	0.980618i	$-0.562772\pi$
$101$	−4.22221e14	−0.391859	−0.195929	+	0.980618i	$0.562772\pi$
$102$	3.92544e14	0.338366
$103$	1.45689e14	0.116720	0.0583602	−	0.998296i	$-0.481413\pi$
$103$	1.45689e14	0.116720	0.0583602	+	0.998296i	$0.481413\pi$
$104$	−1.21658e15	−0.906551
$105$	0	0
$106$	8.67483e14	0.560360
$107$	4.08696e14	0.246049	0.123024	−	0.992404i	$-0.460741\pi$
$107$	4.08696e14	0.246049	0.123024	+	0.992404i	$0.460741\pi$
$108$	2.88538e14	0.162004
$109$	1.10134e15	0.577060	0.288530	−	0.957471i	$-0.406833\pi$
$109$	1.10134e15	0.577060	0.288530	+	0.957471i	$0.406833\pi$
$110$	0	0
$111$	1.34449e15	0.614661
$112$	1.27008e15	0.542868
$113$	2.13023e15	0.851802	0.425901	−	0.904770i	$-0.359957\pi$
$113$	2.13023e15	0.851802	0.425901	+	0.904770i	$0.359957\pi$
$114$	7.35323e14	0.275227
$115$	0	0
$116$	3.19846e15	1.05077
$117$	1.33911e15	0.412498
$118$	1.66233e15	0.480399
$119$	−5.35727e15	−1.45325
$120$	0	0
$121$	−2.79569e15	−0.669266
$122$	−6.12418e14	−0.137832
$123$	−1.20254e15	−0.254573
$124$	1.54986e15	0.308767
$125$	0	0
$126$	7.40059e14	0.130764
$127$	6.25962e15	1.04236	0.521182	−	0.853445i	$-0.325491\pi$
$127$	6.25962e15	1.04236	0.521182	+	0.853445i	$0.325491\pi$
$128$	−6.49577e15	−1.01990
$129$	−2.14957e15	−0.318367
$130$	0	0
$131$	1.01059e16	1.33365	0.666825	−	0.745214i	$-0.267653\pi$
$131$	1.01059e16	1.33365	0.666825	+	0.745214i	$0.267653\pi$
$132$	2.24228e15	0.279503
$133$	−1.00354e16	−1.18208
$134$	8.87833e14	0.0988656
$135$	0	0
$136$	1.08326e16	1.07942
$137$	9.93739e15	0.937277	0.468638	−	0.883390i	$-0.344745\pi$
$137$	9.93739e15	0.937277	0.468638	+	0.883390i	$0.344745\pi$
$138$	−2.90803e15	−0.259721
$139$	−1.95078e16	−1.65043	−0.825216	−	0.564817i	$-0.808947\pi$
$139$	−1.95078e16	−1.65043	−0.825216	+	0.564817i	$0.808947\pi$
$140$	0	0
$141$	4.54053e15	0.345114
$142$	4.24722e15	0.306156
$143$	1.04065e16	0.711676
$144$	2.82677e15	0.183473
$145$	0	0
$146$	4.03908e14	0.0236394
$147$	2.82911e14	0.0157315
$148$	1.69577e16	0.896197
$149$	1.58935e16	0.798589	0.399295	−	0.916823i	$-0.369255\pi$
$149$	1.58935e16	0.798589	0.399295	+	0.916823i	$0.369255\pi$
$150$	0	0
$151$	3.10597e16	1.41211	0.706057	−	0.708155i	$-0.250472\pi$
$151$	3.10597e16	1.41211	0.706057	+	0.708155i	$0.250472\pi$
$152$	2.02918e16	0.877999
$153$	−1.19235e16	−0.491155
$154$	5.75113e15	0.225605
$155$	0	0
$156$	1.68898e16	0.601436
$157$	5.41372e16	1.83759	0.918794	−	0.394738i	$-0.129165\pi$
$157$	5.41372e16	1.83759	0.918794	+	0.394738i	$0.129165\pi$
$158$	1.15141e16	0.372652
$159$	−2.63498e16	−0.813390
$160$	0	0
$161$	3.96876e16	1.11548
$162$	1.64713e15	0.0441942
$163$	5.54612e16	1.42096	0.710481	−	0.703717i	$-0.248478\pi$
$163$	5.54612e16	1.42096	0.710481	+	0.703717i	$0.248478\pi$
$164$	−1.51673e16	−0.371176
$165$	0	0
$166$	−4.15266e15	−0.0927932
$167$	5.83552e15	0.124654	0.0623270	−	0.998056i	$-0.480148\pi$
$167$	5.83552e15	0.124654	0.0623270	+	0.998056i	$0.480148\pi$
$168$	2.04225e16	0.417148
$169$	2.71997e16	0.531390
$170$	0	0
$171$	−2.23354e16	−0.399506
$172$	−2.71119e16	−0.464191
$173$	−4.40958e16	−0.722855	−0.361428	−	0.932400i	$-0.617711\pi$
$173$	−4.40958e16	−0.722855	−0.361428	+	0.932400i	$0.617711\pi$
$174$	1.82585e16	0.286646
$175$	0	0
$176$	2.19674e16	0.316543
$177$	−5.04932e16	−0.697323
$178$	−2.60847e16	−0.345331
$179$	−5.10594e16	−0.648153	−0.324077	−	0.946031i	$-0.605054\pi$
$179$	−5.10594e16	−0.648153	−0.324077	+	0.946031i	$0.605054\pi$
$180$	0	0
$181$	3.16143e16	0.369228	0.184614	−	0.982811i	$-0.440897\pi$
$181$	3.16143e16	0.369228	0.184614	+	0.982811i	$0.440897\pi$
$182$	4.33199e16	0.485458
$183$	1.86022e16	0.200070
$184$	−8.02495e16	−0.828531
$185$	0	0
$186$	8.84743e15	0.0842307
$187$	−9.26599e16	−0.847383
$188$	5.72684e16	0.503189
$189$	−2.24793e16	−0.189810
$190$	0	0
$191$	9.44291e16	0.736810	0.368405	−	0.929665i	$-0.379904\pi$
$191$	9.44291e16	0.736810	0.368405	+	0.929665i	$0.379904\pi$
$192$	1.32322e16	0.0992828
$193$	−4.17744e16	−0.301460	−0.150730	−	0.988575i	$-0.548162\pi$
$193$	−4.17744e16	−0.301460	−0.150730	+	0.988575i	$0.548162\pi$
$194$	3.88646e16	0.269800
$195$	0	0
$196$	3.56828e15	0.0229371
$197$	1.48383e17	0.918094	0.459047	−	0.888412i	$-0.348191\pi$
$197$	1.48383e17	0.918094	0.459047	+	0.888412i	$0.348191\pi$
$198$	1.28001e16	0.0762475
$199$	1.97555e17	1.13316	0.566579	−	0.824007i	$-0.308267\pi$
$199$	1.97555e17	1.13316	0.566579	+	0.824007i	$0.308267\pi$
$200$	0	0
$201$	−2.69679e16	−0.143508
$202$	−3.03999e16	−0.155861
$203$	−2.49184e17	−1.23112
$204$	−1.50388e17	−0.716122
$205$	0	0
$206$	1.04896e16	0.0464253
$207$	8.83316e16	0.376997
$208$	1.65467e17	0.681140
$209$	−1.73573e17	−0.689262
$210$	0	0
$211$	−4.48570e17	−1.65849	−0.829244	−	0.558887i	$-0.811228\pi$
$211$	−4.48570e17	−1.65849	−0.829244	+	0.558887i	$0.811228\pi$
$212$	−3.32342e17	−1.18595
$213$	−1.29009e17	−0.444400
$214$	2.94261e16	0.0978654
$215$	0	0
$216$	4.54538e16	0.140983
$217$	−1.20746e17	−0.361763
$218$	7.92962e16	0.229524
$219$	−1.22687e16	−0.0343137
$220$	0	0
$221$	−6.97951e17	−1.82341
$222$	9.68033e16	0.244480
$223$	−1.59977e17	−0.390634	−0.195317	−	0.980740i	$-0.562574\pi$
$223$	−1.59977e17	−0.390634	−0.195317	+	0.980740i	$0.562574\pi$
$224$	3.97438e17	0.938445
$225$	0	0
$226$	1.53377e17	0.338802
$227$	−1.45803e17	−0.311582	−0.155791	−	0.987790i	$-0.549793\pi$
$227$	−1.45803e17	−0.311582	−0.155791	+	0.987790i	$0.549793\pi$
$228$	−2.81710e17	−0.582494
$229$	6.81194e17	1.36303	0.681514	−	0.731805i	$-0.261322\pi$
$229$	6.81194e17	1.36303	0.681514	+	0.731805i	$0.261322\pi$
$230$	0	0
$231$	−1.74691e17	−0.327476
$232$	5.03858e17	0.914425
$233$	3.89035e17	0.683628	0.341814	−	0.939768i	$-0.388959\pi$
$233$	3.89035e17	0.683628	0.341814	+	0.939768i	$0.388959\pi$
$234$	9.64158e16	0.164070
$235$	0	0
$236$	−6.36857e17	−1.01672
$237$	−3.49742e17	−0.540923
$238$	−3.85723e17	−0.578028
$239$	6.76227e16	0.0981993	0.0490996	−	0.998794i	$-0.484365\pi$
$239$	6.76227e16	0.0981993	0.0490996	+	0.998794i	$0.484365\pi$
$240$	0	0
$241$	−9.00528e17	−1.22848	−0.614242	−	0.789118i	$-0.710538\pi$
$241$	−9.00528e17	−1.22848	−0.614242	+	0.789118i	$0.710538\pi$
$242$	−2.01290e17	−0.266199
$243$	−5.00315e16	−0.0641500
$244$	2.34624e17	0.291709
$245$	0	0
$246$	−8.65831e16	−0.101256
$247$	−1.30742e18	−1.48316
$248$	2.44152e17	0.268703
$249$	1.26137e17	0.134694
$250$	0	0
$251$	1.58929e18	1.59827	0.799134	−	0.601153i	$-0.205292\pi$
$251$	1.58929e18	1.59827	0.799134	+	0.601153i	$0.205292\pi$
$252$	−2.83525e17	−0.276750
$253$	6.86440e17	0.650428
$254$	4.50692e17	0.414598
$255$	0	0
$256$	−2.69436e17	−0.233698
$257$	−2.29738e18	−1.93524	−0.967618	−	0.252418i	$-0.918774\pi$
$257$	−2.29738e18	−1.93524	−0.967618	+	0.252418i	$0.918774\pi$
$258$	−1.54769e17	−0.126630
$259$	−1.32113e18	−1.05002
$260$	0	0
$261$	−5.54602e17	−0.416081
$262$	7.27628e17	0.530456
$263$	5.05657e17	0.358252	0.179126	−	0.983826i	$-0.442673\pi$
$263$	5.05657e17	0.358252	0.179126	+	0.983826i	$0.442673\pi$
$264$	3.53230e17	0.243236
$265$	0	0
$266$	−7.22546e17	−0.470168
$267$	7.92323e17	0.501265
$268$	−3.40139e17	−0.209240
$269$	−2.44731e18	−1.46402	−0.732009	−	0.681295i	$-0.761417\pi$
$269$	−2.44731e18	−1.46402	−0.732009	+	0.681295i	$0.761417\pi$
$270$	0	0
$271$	−1.53807e18	−0.870373	−0.435187	−	0.900340i	$-0.643318\pi$
$271$	−1.53807e18	−0.870373	−0.435187	+	0.900340i	$0.643318\pi$
$272$	−1.47333e18	−0.811024
$273$	−1.31584e18	−0.704666
$274$	7.15492e17	0.372800
$275$	0	0
$276$	1.11410e18	0.549675
$277$	7.89959e17	0.379321	0.189660	−	0.981850i	$-0.439261\pi$
$277$	7.89959e17	0.379321	0.189660	+	0.981850i	$0.439261\pi$
$278$	−1.40456e18	−0.656456
$279$	−2.68741e17	−0.122265
$280$	0	0
$281$	−6.15303e17	−0.265333	−0.132667	−	0.991161i	$-0.542354\pi$
$281$	−6.15303e17	−0.265333	−0.132667	+	0.991161i	$0.542354\pi$
$282$	3.26918e17	0.137268
$283$	1.26279e18	0.516337	0.258169	−	0.966100i	$-0.416881\pi$
$283$	1.26279e18	0.516337	0.258169	+	0.966100i	$0.416881\pi$
$284$	−1.62716e18	−0.647951
$285$	0	0
$286$	7.49265e17	0.283067
$287$	1.18165e18	0.434885
$288$	8.84566e17	0.317166
$289$	3.35219e18	1.17110
$290$	0	0
$291$	−1.18051e18	−0.391628
$292$	−1.54742e17	−0.0500306
$293$	3.49864e18	1.10253	0.551267	−	0.834329i	$-0.314145\pi$
$293$	3.49864e18	1.10253	0.551267	+	0.834329i	$0.314145\pi$
$294$	2.03696e16	0.00625717
$295$	0	0
$296$	2.67137e18	0.779912
$297$	−3.88804e17	−0.110677
$298$	1.14433e18	0.317637
$299$	5.17055e18	1.39960
$300$	0	0
$301$	2.11222e18	0.543864
$302$	2.23630e18	0.561665
$303$	9.23398e17	0.226240
$304$	−2.75988e18	−0.659687
$305$	0	0
$306$	−8.58494e17	−0.195356
$307$	−1.28861e18	−0.286142	−0.143071	−	0.989712i	$-0.545698\pi$
$307$	−1.28861e18	−0.286142	−0.143071	+	0.989712i	$0.545698\pi$
$308$	−2.20332e18	−0.477472
$309$	−3.18621e17	−0.0673886
$310$	0	0
$311$	8.43030e18	1.69879	0.849396	−	0.527756i	$-0.176967\pi$
$311$	8.43030e18	1.69879	0.849396	+	0.527756i	$0.176967\pi$
$312$	2.66067e18	0.523397
$313$	−7.88359e18	−1.51405	−0.757027	−	0.653383i	$-0.773349\pi$
$313$	−7.88359e18	−1.51405	−0.757027	+	0.653383i	$0.773349\pi$
$314$	3.89788e18	0.730896
$315$	0	0
$316$	−4.41120e18	−0.788685
$317$	8.71539e18	1.52175	0.760873	−	0.648900i	$-0.224771\pi$
$317$	8.71539e18	1.52175	0.760873	+	0.648900i	$0.224771\pi$
$318$	−1.89719e18	−0.323524
$319$	−4.30991e18	−0.717857
$320$	0	0
$321$	−8.93817e17	−0.142056
$322$	2.85751e18	0.443678
$323$	1.16414e19	1.76598
$324$	−6.31033e17	−0.0935330
$325$	0	0
$326$	3.99321e18	0.565184
$327$	−2.40862e18	−0.333166
$328$	−2.38933e18	−0.323015
$329$	−4.46163e18	−0.589556
$330$	0	0
$331$	1.26995e19	1.60353	0.801763	−	0.597643i	$-0.203896\pi$
$331$	1.26995e19	1.60353	0.801763	+	0.597643i	$0.203896\pi$
$332$	1.59093e18	0.196388
$333$	−2.94040e18	−0.354875
$334$	4.20157e17	0.0495808
$335$	0	0
$336$	−2.77766e18	−0.313425
$337$	6.64250e18	0.733005	0.366503	−	0.930417i	$-0.380555\pi$
$337$	6.64250e18	0.733005	0.366503	+	0.930417i	$0.380555\pi$
$338$	1.95838e18	0.211359
$339$	−4.65882e18	−0.491788
$340$	0	0
$341$	−2.08843e18	−0.210942
$342$	−1.60815e18	−0.158903
$343$	−1.04805e19	−1.01316
$344$	−4.27097e18	−0.403960
$345$	0	0
$346$	−3.17490e18	−0.287514
$347$	−1.87835e18	−0.166458	−0.0832289	−	0.996530i	$-0.526523\pi$
$347$	−1.87835e18	−0.166458	−0.0832289	+	0.996530i	$0.526523\pi$
$348$	−6.99503e18	−0.606660
$349$	−1.48545e19	−1.26086	−0.630432	−	0.776245i	$-0.717122\pi$
$349$	−1.48545e19	−1.26086	−0.630432	+	0.776245i	$0.717122\pi$
$350$	0	0
$351$	−2.92863e18	−0.238156
$352$	6.87412e18	0.547202
$353$	8.23380e18	0.641638	0.320819	−	0.947140i	$-0.396042\pi$
$353$	8.23380e18	0.641638	0.320819	+	0.947140i	$0.396042\pi$
$354$	−3.63551e18	−0.277358
$355$	0	0
$356$	9.99334e18	0.730863
$357$	1.17163e19	0.839036
$358$	−3.67627e18	−0.257801
$359$	1.87417e19	1.28706	0.643532	−	0.765419i	$-0.277468\pi$
$359$	1.87417e19	1.28706	0.643532	+	0.765419i	$0.277468\pi$
$360$	0	0
$361$	6.62574e18	0.436446
$362$	2.27623e18	0.146859
$363$	6.11417e18	0.386401
$364$	−1.65963e19	−1.02743
$365$	0	0
$366$	1.33936e18	0.0795772
$367$	−8.63117e18	−0.502428	−0.251214	−	0.967932i	$-0.580830\pi$
$367$	−8.63117e18	−0.502428	−0.251214	+	0.967932i	$0.580830\pi$
$368$	1.09147e19	0.622519
$369$	2.62996e18	0.146978
$370$	0	0
$371$	2.58920e19	1.38951
$372$	−3.38955e18	−0.178267
$373$	9.95975e18	0.513372	0.256686	−	0.966495i	$-0.417369\pi$
$373$	9.95975e18	0.513372	0.256686	+	0.966495i	$0.417369\pi$
$374$	−6.67151e18	−0.337044
$375$	0	0
$376$	9.02156e18	0.437898
$377$	−3.24640e19	−1.54469
$378$	−1.61851e18	−0.0754965
$379$	1.00949e19	0.461645	0.230823	−	0.972996i	$-0.425858\pi$
$379$	1.00949e19	0.461645	0.230823	+	0.972996i	$0.425858\pi$
$380$	0	0
$381$	−1.36898e19	−0.601810
$382$	6.79890e18	0.293065
$383$	2.71764e19	1.14869	0.574343	−	0.818614i	$-0.305257\pi$
$383$	2.71764e19	1.14869	0.574343	+	0.818614i	$0.305257\pi$
$384$	1.42062e19	0.588837
$385$	0	0
$386$	−3.00775e18	−0.119905
$387$	4.70111e18	0.183809
$388$	−1.48895e19	−0.571007
$389$	−9.05285e18	−0.340536	−0.170268	−	0.985398i	$-0.554463\pi$
$389$	−9.05285e18	−0.340536	−0.170268	+	0.985398i	$0.554463\pi$
$390$	0	0
$391$	−4.60389e19	−1.66648
$392$	5.62116e17	0.0199609
$393$	−2.21017e19	−0.769984
$394$	1.06836e19	0.365170
$395$	0	0
$396$	−4.90387e18	−0.161371
$397$	1.76495e19	0.569908	0.284954	−	0.958541i	$-0.408022\pi$
$397$	1.76495e19	0.569908	0.284954	+	0.958541i	$0.408022\pi$
$398$	1.42240e19	0.450711
$399$	2.19473e19	0.682472
$400$	0	0
$401$	2.81135e19	0.842040	0.421020	−	0.907051i	$-0.361672\pi$
$401$	2.81135e19	0.842040	0.421020	+	0.907051i	$0.361672\pi$
$402$	−1.94169e18	−0.0570801
$403$	−1.57309e19	−0.453906
$404$	1.16465e19	0.329866
$405$	0	0
$406$	−1.79412e19	−0.489674
$407$	−2.28504e19	−0.612260
$408$	−2.36908e19	−0.623202
$409$	6.42955e19	1.66056	0.830282	−	0.557343i	$-0.188179\pi$
$409$	6.42955e19	1.66056	0.830282	+	0.557343i	$0.188179\pi$
$410$	0	0
$411$	−2.17331e19	−0.541137
$412$	−4.01868e18	−0.0982549
$413$	4.96159e19	1.19123
$414$	6.35987e18	0.149950
$415$	0	0
$416$	5.17787e19	1.17747
$417$	4.26636e19	0.952878
$418$	−1.24972e19	−0.274152
$419$	−2.34474e19	−0.505230	−0.252615	−	0.967567i	$-0.581291\pi$
$419$	−2.34474e19	−0.505230	−0.252615	+	0.967567i	$0.581291\pi$
$420$	0	0
$421$	−7.28227e19	−1.51409	−0.757044	−	0.653364i	$-0.773357\pi$
$421$	−7.28227e19	−1.51409	−0.757044	+	0.653364i	$0.773357\pi$
$422$	−3.22971e19	−0.659659
$423$	−9.93013e18	−0.199252
$424$	−5.23543e19	−1.03207
$425$	0	0
$426$	−9.28867e18	−0.176759
$427$	−1.82790e19	−0.341777
$428$	−1.12735e19	−0.207123
$429$	−2.27589e19	−0.410886
$430$	0	0
$431$	−4.71831e19	−0.822635	−0.411317	−	0.911492i	$-0.634931\pi$
$431$	−4.71831e19	−0.822635	−0.411317	+	0.911492i	$0.634931\pi$
$432$	−6.18215e18	−0.105928
$433$	−3.79099e19	−0.638400	−0.319200	−	0.947687i	$-0.603414\pi$
$433$	−3.79099e19	−0.638400	−0.319200	+	0.947687i	$0.603414\pi$
$434$	−8.69371e18	−0.143890
$435$	0	0
$436$	−3.03793e19	−0.485768
$437$	−8.62412e19	−1.35551
$438$	−8.83346e17	−0.0136482
$439$	−2.26832e19	−0.344525	−0.172262	−	0.985051i	$-0.555108\pi$
$439$	−2.26832e19	−0.344525	−0.172262	+	0.985051i	$0.555108\pi$
$440$	0	0
$441$	−6.18727e17	−0.00908259
$442$	−5.02525e19	−0.725255
$443$	9.94888e19	1.41171	0.705856	−	0.708355i	$-0.250562\pi$
$443$	9.94888e19	1.41171	0.705856	+	0.708355i	$0.250562\pi$
$444$	−3.70864e19	−0.517420
$445$	0	0
$446$	−1.15183e19	−0.155374
$447$	−3.47592e19	−0.461066
$448$	−1.30023e19	−0.169604
$449$	1.30159e20	1.66965	0.834825	−	0.550515i	$-0.185568\pi$
$449$	1.30159e20	1.66965	0.834825	+	0.550515i	$0.185568\pi$
$450$	0	0
$451$	2.04379e19	0.253579
$452$	−5.87604e19	−0.717045
$453$	−6.79275e19	−0.815285
$454$	−1.04978e19	−0.123931
$455$	0	0
$456$	−4.43782e19	−0.506913
$457$	1.01910e20	1.14510	0.572550	−	0.819870i	$-0.305954\pi$
$457$	1.01910e20	1.14510	0.572550	+	0.819870i	$0.305954\pi$
$458$	4.90460e19	0.542141
$459$	2.60767e19	0.283569
$460$	0	0
$461$	−1.08142e20	−1.13825	−0.569123	−	0.822253i	$-0.692717\pi$
$461$	−1.08142e20	−1.13825	−0.569123	+	0.822253i	$0.692717\pi$
$462$	−1.25777e19	−0.130253
$463$	−1.69113e20	−1.72314	−0.861569	−	0.507641i	$-0.830518\pi$
$463$	−1.69113e20	−1.72314	−0.861569	+	0.507641i	$0.830518\pi$
$464$	−6.85294e19	−0.687056
$465$	0	0
$466$	2.80105e19	0.271911
$467$	4.58418e19	0.437910	0.218955	−	0.975735i	$-0.429735\pi$
$467$	4.58418e19	0.437910	0.218955	+	0.975735i	$0.429735\pi$
$468$	−3.69380e19	−0.347239
$469$	2.64993e19	0.245154
$470$	0	0
$471$	−1.18398e20	−1.06093
$472$	−1.00325e20	−0.884797
$473$	3.65331e19	0.317124
$474$	−2.51814e19	−0.215151
$475$	0	0
$476$	1.47775e20	1.22334
$477$	5.76270e19	0.469611
$478$	4.86884e18	0.0390585
$479$	1.31513e20	1.03861	0.519303	−	0.854590i	$-0.326192\pi$
$479$	1.31513e20	1.03861	0.519303	+	0.854590i	$0.326192\pi$
$480$	0	0
$481$	−1.72118e20	−1.31747
$482$	−6.48380e19	−0.488626
$483$	−8.67968e19	−0.644021
$484$	7.71163e19	0.563386
$485$	0	0
$486$	−3.60227e18	−0.0255155
$487$	−6.13671e19	−0.428024	−0.214012	−	0.976831i	$-0.568653\pi$
$487$	−6.13671e19	−0.428024	−0.214012	+	0.976831i	$0.568653\pi$
$488$	3.69607e19	0.253858
$489$	−1.21294e20	−0.820392
$490$	0	0
$491$	2.42057e20	1.58784	0.793918	−	0.608024i	$-0.208038\pi$
$491$	2.42057e20	1.58784	0.793918	+	0.608024i	$0.208038\pi$
$492$	3.31710e19	0.214299
$493$	2.89062e20	1.83924
$494$	−9.41342e19	−0.589923
$495$	0	0
$496$	−3.32070e19	−0.201891
$497$	1.26768e20	0.759165
$498$	9.08188e18	0.0535742
$499$	1.46049e20	0.848680	0.424340	−	0.905503i	$-0.360506\pi$
$499$	1.46049e20	0.848680	0.424340	+	0.905503i	$0.360506\pi$
$500$	0	0
$501$	−1.27623e19	−0.0719690
$502$	1.14429e20	0.635707
$503$	1.68041e20	0.919720	0.459860	−	0.887991i	$-0.347900\pi$
$503$	1.68041e20	0.919720	0.459860	+	0.887991i	$0.347900\pi$
$504$	−4.46641e19	−0.240840
$505$	0	0
$506$	4.94237e19	0.258706
$507$	−5.94857e19	−0.306798
$508$	−1.72665e20	−0.877459
$509$	7.37709e19	0.369404	0.184702	−	0.982795i	$-0.440868\pi$
$509$	7.37709e19	0.369404	0.184702	+	0.982795i	$0.440868\pi$
$510$	0	0
$511$	1.20555e19	0.0586178
$512$	1.93454e20	0.926943
$513$	4.88476e19	0.230655
$514$	−1.65411e20	−0.769736
$515$	0	0
$516$	5.92937e19	0.268001
$517$	−7.71689e19	−0.343766
$518$	−9.51213e19	−0.417643
$519$	9.64375e19	0.417341
$520$	0	0
$521$	3.22023e20	1.35396	0.676978	−	0.736003i	$-0.263289\pi$
$521$	3.22023e20	1.35396	0.676978	+	0.736003i	$0.263289\pi$
$522$	−3.99313e19	−0.165495
$523$	1.19064e19	0.0486429	0.0243214	−	0.999704i	$-0.492257\pi$
$523$	1.19064e19	0.0486429	0.0243214	+	0.999704i	$0.492257\pi$
$524$	−2.78762e20	−1.12266
$525$	0	0
$526$	3.64073e19	0.142494
$527$	1.40069e20	0.540460
$528$	−4.80426e19	−0.182756
$529$	7.44293e19	0.279143
$530$	0	0
$531$	1.10429e20	0.402599
$532$	2.76815e20	0.995069
$533$	1.53947e20	0.545652
$534$	5.70473e19	0.199377
$535$	0	0
$536$	−5.35825e19	−0.182090
$537$	1.11667e20	0.374211
$538$	−1.76206e20	−0.582309
$539$	−4.80824e18	−0.0156701
$540$	0	0
$541$	1.59033e20	0.504089	0.252045	−	0.967716i	$-0.418897\pi$
$541$	1.59033e20	0.504089	0.252045	+	0.967716i	$0.418897\pi$
$542$	−1.10741e20	−0.346189
$543$	−6.91405e19	−0.213174
$544$	−4.61041e20	−1.40200
$545$	0	0
$546$	−9.47405e19	−0.280279
$547$	4.25405e20	1.24136	0.620680	−	0.784064i	$-0.286857\pi$
$547$	4.25405e20	1.24136	0.620680	+	0.784064i	$0.286857\pi$
$548$	−2.74113e20	−0.788997
$549$	−4.06830e19	−0.115510
$550$	0	0
$551$	5.41477e20	1.49604
$552$	1.75506e20	0.478353
$553$	3.43665e20	0.924054
$554$	5.68771e19	0.150874
$555$	0	0
$556$	5.38104e20	1.38933
$557$	−2.71579e20	−0.691803	−0.345901	−	0.938271i	$-0.612427\pi$
$557$	−2.71579e20	−0.691803	−0.345901	+	0.938271i	$0.612427\pi$
$558$	−1.93493e19	−0.0486306
$559$	2.75182e20	0.682389
$560$	0	0
$561$	2.02647e20	0.489237
$562$	−4.43018e19	−0.105536
$563$	2.68438e20	0.631002	0.315501	−	0.948925i	$-0.397827\pi$
$563$	2.68438e20	0.631002	0.315501	+	0.948925i	$0.397827\pi$
$564$	−1.25246e20	−0.290516
$565$	0	0
$566$	9.09210e19	0.205372
$567$	4.91622e19	0.109587
$568$	−2.56328e20	−0.563877
$569$	−1.34893e20	−0.292853	−0.146426	−	0.989222i	$-0.546777\pi$
$569$	−1.34893e20	−0.292853	−0.146426	+	0.989222i	$0.546777\pi$
$570$	0	0
$571$	−6.81153e20	−1.44037	−0.720185	−	0.693782i	$-0.755943\pi$
$571$	−6.81153e20	−1.44037	−0.720185	+	0.693782i	$0.755943\pi$
$572$	−2.87052e20	−0.599087
$573$	−2.06517e20	−0.425398
$574$	8.50787e19	0.172974
$575$	0	0
$576$	−2.89389e19	−0.0573209
$577$	2.42927e20	0.474961	0.237480	−	0.971392i	$-0.423678\pi$
$577$	2.42927e20	0.474961	0.237480	+	0.971392i	$0.423678\pi$
$578$	2.41358e20	0.465803
$579$	9.13605e19	0.174048
$580$	0	0
$581$	−1.23945e20	−0.230096
$582$	−8.49970e19	−0.155769
$583$	4.47830e20	0.810213
$584$	−2.43766e19	−0.0435389
$585$	0	0
$586$	2.51902e20	0.438530
$587$	1.06241e20	0.182602	0.0913012	−	0.995823i	$-0.470897\pi$
$587$	1.06241e20	0.182602	0.0913012	+	0.995823i	$0.470897\pi$
$588$	−7.80383e18	−0.0132427
$589$	2.62381e20	0.439610
$590$	0	0
$591$	−3.24513e20	−0.530062
$592$	−3.63331e20	−0.585989
$593$	7.70415e20	1.22692	0.613458	−	0.789727i	$-0.289778\pi$
$593$	7.70415e20	1.22692	0.613458	+	0.789727i	$0.289778\pi$
$594$	−2.79939e19	−0.0440215
$595$	0	0
$596$	−4.38407e20	−0.672250
$597$	−4.32053e20	−0.654229
$598$	3.72279e20	0.556686
$599$	−7.59796e19	−0.112201	−0.0561004	−	0.998425i	$-0.517867\pi$
$599$	−7.59796e19	−0.112201	−0.0561004	+	0.998425i	$0.517867\pi$
$600$	0	0
$601$	7.78003e20	1.12053	0.560264	−	0.828314i	$-0.310700\pi$
$601$	7.78003e20	1.12053	0.560264	+	0.828314i	$0.310700\pi$
$602$	1.52080e20	0.216321
$603$	5.89788e19	0.0828546
$604$	−8.56750e20	−1.18871
$605$	0	0
$606$	6.64846e19	0.0899863
$607$	−3.21867e20	−0.430290	−0.215145	−	0.976582i	$-0.569022\pi$
$607$	−3.21867e20	−0.430290	−0.215145	+	0.976582i	$0.569022\pi$
$608$	−8.63633e20	−1.14039
$609$	5.44965e20	0.710787
$610$	0	0
$611$	−5.81267e20	−0.739719
$612$	3.28898e20	0.413453
$613$	−6.68867e20	−0.830589	−0.415294	−	0.909687i	$-0.636321\pi$
$613$	−6.68867e20	−0.830589	−0.415294	+	0.909687i	$0.636321\pi$
$614$	−9.27796e19	−0.113812
$615$	0	0
$616$	−3.47092e20	−0.415518
$617$	8.47904e20	1.00279	0.501393	−	0.865220i	$-0.332821\pi$
$617$	8.47904e20	1.00279	0.501393	+	0.865220i	$0.332821\pi$
$618$	−2.29407e19	−0.0268036
$619$	−3.51027e20	−0.405192	−0.202596	−	0.979262i	$-0.564938\pi$
$619$	−3.51027e20	−0.405192	−0.202596	+	0.979262i	$0.564938\pi$
$620$	0	0
$621$	−1.93181e20	−0.217660
$622$	6.06982e20	0.675690
$623$	−7.78556e20	−0.856307
$624$	−3.61876e20	−0.393256
$625$	0	0
$626$	−5.67619e20	−0.602212
$627$	3.79603e20	0.397945
$628$	−1.49332e21	−1.54688
$629$	1.53256e21	1.56869
$630$	0	0
$631$	−1.30718e21	−1.30652	−0.653260	−	0.757134i	$-0.726599\pi$
$631$	−1.30718e21	−1.30652	−0.653260	+	0.757134i	$0.726599\pi$
$632$	−6.94902e20	−0.686349
$633$	9.81023e20	0.957528
$634$	6.27508e20	0.605271
$635$	0	0
$636$	7.26833e20	0.684709
$637$	−3.62176e19	−0.0337190
$638$	−3.10314e20	−0.285526
$639$	2.82143e20	0.256575
$640$	0	0
$641$	−1.77975e21	−1.58097	−0.790486	−	0.612480i	$-0.790172\pi$
$641$	−1.77975e21	−1.58097	−0.790486	+	0.612480i	$0.790172\pi$
$642$	−6.43548e19	−0.0565026
$643$	5.18269e19	0.0449752	0.0224876	−	0.999747i	$-0.492841\pi$
$643$	5.18269e19	0.0449752	0.0224876	+	0.999747i	$0.492841\pi$
$644$	−1.09474e21	−0.939005
$645$	0	0
$646$	8.38178e20	0.702413
$647$	−1.80268e21	−1.49327	−0.746634	−	0.665235i	$-0.768331\pi$
$647$	−1.80268e21	−1.49327	−0.746634	+	0.665235i	$0.768331\pi$
$648$	−9.94075e19	−0.0813967
$649$	8.58162e20	0.694599
$650$	0	0
$651$	2.64071e20	0.208864
$652$	−1.52984e21	−1.19616
$653$	−6.01117e20	−0.464633	−0.232317	−	0.972640i	$-0.574631\pi$
$653$	−6.01117e20	−0.464633	−0.232317	+	0.972640i	$0.574631\pi$
$654$	−1.73421e20	−0.132516
$655$	0	0
$656$	3.24971e20	0.242698
$657$	2.68316e19	0.0198110
$658$	−3.21238e20	−0.234494
$659$	−1.34492e21	−0.970635	−0.485318	−	0.874338i	$-0.661296\pi$
$659$	−1.34492e21	−0.970635	−0.485318	+	0.874338i	$0.661296\pi$
$660$	0	0
$661$	3.00301e20	0.211858	0.105929	−	0.994374i	$-0.466218\pi$
$661$	3.00301e20	0.211858	0.105929	+	0.994374i	$0.466218\pi$
$662$	9.14362e20	0.637798
$663$	1.52642e21	1.05274
$664$	2.50622e20	0.170906
$665$	0	0
$666$	−2.11709e20	−0.141151
$667$	−2.14142e21	−1.41175
$668$	−1.60967e20	−0.104933
$669$	3.49869e20	0.225533
$670$	0	0
$671$	−3.16155e20	−0.199288
$672$	−8.69196e20	−0.541812
$673$	2.48252e21	1.53031	0.765156	−	0.643845i	$-0.222662\pi$
$673$	2.48252e21	1.53031	0.765156	+	0.643845i	$0.222662\pi$
$674$	4.78260e20	0.291551
$675$	0	0
$676$	−7.50276e20	−0.447323
$677$	1.84000e21	1.08493	0.542467	−	0.840077i	$-0.317491\pi$
$677$	1.84000e21	1.08493	0.542467	+	0.840077i	$0.317491\pi$
$678$	−3.35435e20	−0.195608
$679$	1.16000e21	0.669014
$680$	0	0
$681$	3.18871e20	0.179892
$682$	−1.50367e20	−0.0839016
$683$	−2.19144e21	−1.20941	−0.604705	−	0.796450i	$-0.706709\pi$
$683$	−2.19144e21	−1.20941	−0.604705	+	0.796450i	$0.706709\pi$
$684$	6.16100e20	0.336303
$685$	0	0
$686$	−7.54596e20	−0.402980
$687$	−1.48977e21	−0.786945
$688$	5.80892e20	0.303517
$689$	3.37324e21	1.74342
$690$	0	0
$691$	−5.25387e20	−0.265701	−0.132851	−	0.991136i	$-0.542413\pi$
$691$	−5.25387e20	−0.265701	−0.132851	+	0.991136i	$0.542413\pi$
$692$	1.21634e21	0.608497
$693$	3.82049e20	0.189069
$694$	−1.35241e20	−0.0662082
$695$	0	0
$696$	−1.10194e21	−0.527943
$697$	−1.37075e21	−0.649701
$698$	−1.06953e21	−0.501505
$699$	−8.50820e20	−0.394693
$700$	0	0
$701$	−3.78227e21	−1.71739	−0.858693	−	0.512491i	$-0.828723\pi$
$701$	−3.78227e21	−1.71739	−0.858693	+	0.512491i	$0.828723\pi$
$702$	−2.10861e20	−0.0947259
$703$	2.87082e21	1.27597
$704$	−2.24889e20	−0.0988949
$705$	0	0
$706$	5.92834e20	0.255210
$707$	−9.07353e20	−0.386483
$708$	1.39281e21	0.587004
$709$	−4.24647e21	−1.77085	−0.885424	−	0.464784i	$-0.846132\pi$
$709$	−4.24647e21	−1.77085	−0.885424	+	0.464784i	$0.846132\pi$
$710$	0	0
$711$	7.64886e20	0.312302
$712$	1.57426e21	0.636030
$713$	−1.03766e21	−0.414842
$714$	8.43577e20	0.333725
$715$	0	0
$716$	1.40842e21	0.545613
$717$	−1.47891e20	−0.0566954
$718$	1.34940e21	0.511927
$719$	−3.17360e20	−0.119148	−0.0595738	−	0.998224i	$-0.518974\pi$
$719$	−3.17360e20	−0.119148	−0.0595738	+	0.998224i	$0.518974\pi$
$720$	0	0
$721$	3.13085e20	0.115119
$722$	4.77053e20	0.173595
$723$	1.96946e21	0.709266
$724$	−8.72049e20	−0.310815
$725$	0	0
$726$	4.40221e20	0.153690
$727$	−1.25112e21	−0.432307	−0.216154	−	0.976359i	$-0.569351\pi$
$727$	−1.25112e21	−0.432307	−0.216154	+	0.976359i	$0.569351\pi$
$728$	−2.61444e21	−0.894115
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	−2.45025e21	−0.812511
$732$	−5.13123e20	−0.168418
$733$	−4.26993e21	−1.38721	−0.693603	−	0.720357i	$-0.743978\pi$
$733$	−4.26993e21	−1.38721	−0.693603	+	0.720357i	$0.743978\pi$
$734$	−6.21444e20	−0.199840
$735$	0	0
$736$	3.41547e21	1.07614
$737$	4.58335e20	0.142948
$738$	1.89357e20	0.0584601
$739$	1.73302e21	0.529627	0.264813	−	0.964300i	$-0.414690\pi$
$739$	1.73302e21	0.529627	0.264813	+	0.964300i	$0.414690\pi$
$740$	0	0
$741$	2.85933e21	0.856302
$742$	1.86422e21	0.552673
$743$	2.63739e21	0.774032	0.387016	−	0.922073i	$-0.373506\pi$
$743$	2.63739e21	0.774032	0.387016	+	0.922073i	$0.373506\pi$
$744$	−5.33960e20	−0.155136
$745$	0	0
$746$	7.17102e20	0.204192
$747$	−2.75862e20	−0.0777656
$748$	2.55593e21	0.713324
$749$	8.78287e20	0.242674
$750$	0	0
$751$	2.27078e21	0.615001	0.307500	−	0.951548i	$-0.400507\pi$
$751$	2.27078e21	0.615001	0.307500	+	0.951548i	$0.400507\pi$
$752$	−1.22702e21	−0.329016
$753$	−3.47577e21	−0.922761
$754$	−2.33741e21	−0.614397
$755$	0	0
$756$	6.20069e20	0.159782
$757$	1.72959e21	0.441291	0.220645	−	0.975354i	$-0.429184\pi$
$757$	1.72959e21	0.441291	0.220645	+	0.975354i	$0.429184\pi$
$758$	7.26834e20	0.183618
$759$	−1.50125e21	−0.375525
$760$	0	0
$761$	−2.06739e20	−0.0507035	−0.0253518	−	0.999679i	$-0.508071\pi$
$761$	−2.06739e20	−0.0507035	−0.0253518	+	0.999679i	$0.508071\pi$
$762$	−9.85664e20	−0.239368
$763$	2.36677e21	0.569144
$764$	−2.60473e21	−0.620245
$765$	0	0
$766$	1.95670e21	0.456887
$767$	6.46402e21	1.49464
$768$	5.89256e20	0.134926
$769$	3.01490e21	0.683636	0.341818	−	0.939766i	$-0.388957\pi$
$769$	3.01490e21	0.683636	0.341818	+	0.939766i	$0.388957\pi$
$770$	0	0
$771$	5.02437e21	1.11731
$772$	1.15230e21	0.253768
$773$	−8.99357e21	−1.96149	−0.980745	−	0.195293i	$-0.937434\pi$
$773$	−8.99357e21	−1.96149	−0.980745	+	0.195293i	$0.937434\pi$
$774$	3.38480e20	0.0731098
$775$	0	0
$776$	−2.34556e21	−0.496916
$777$	2.88931e21	0.606229
$778$	−6.51805e20	−0.135447
$779$	−2.56773e21	−0.528467
$780$	0	0
$781$	2.19259e21	0.442664
$782$	−3.31480e21	−0.662838
$783$	1.21291e21	0.240224
$784$	−7.64531e19	−0.0149977
$785$	0	0
$786$	−1.59132e21	−0.306259
$787$	5.92807e21	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$787$	5.92807e21	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$788$	−4.09299e21	−0.772849
$789$	−1.10587e21	−0.206837
$790$	0	0
$791$	4.57787e21	0.840117
$792$	−7.72514e20	−0.140432
$793$	−2.38141e21	−0.428830
$794$	1.27077e21	0.226680
$795$	0	0
$796$	−5.44936e21	−0.953889
$797$	3.83837e21	0.665593	0.332797	−	0.942999i	$-0.392008\pi$
$797$	3.83837e21	0.665593	0.332797	+	0.942999i	$0.392008\pi$
$798$	1.58021e21	0.271452
$799$	5.17565e21	0.880773
$800$	0	0
$801$	−1.73281e21	−0.289406
$802$	2.02417e21	0.334919
$803$	2.08513e20	0.0341796
$804$	7.43883e20	0.120805
$805$	0	0
$806$	−1.13263e21	−0.180540
$807$	5.35226e21	0.845251
$808$	1.83470e21	0.287064
$809$	−6.17948e21	−0.957940	−0.478970	−	0.877831i	$-0.658990\pi$
$809$	−6.17948e21	−0.957940	−0.478970	+	0.877831i	$0.658990\pi$
$810$	0	0
$811$	1.87862e21	0.285879	0.142939	−	0.989731i	$-0.454345\pi$
$811$	1.87862e21	0.285879	0.142939	+	0.989731i	$0.454345\pi$
$812$	6.87349e21	1.03635
$813$	3.36375e21	0.502510
$814$	−1.64523e21	−0.243525
$815$	0	0
$816$	3.22217e21	0.468245
$817$	−4.58986e21	−0.660897
$818$	4.62927e21	0.660485
$819$	2.87774e21	0.406839
$820$	0	0
$821$	3.93227e21	0.545845	0.272923	−	0.962036i	$-0.412010\pi$
$821$	3.93227e21	0.545845	0.272923	+	0.962036i	$0.412010\pi$
$822$	−1.56478e21	−0.215236
$823$	7.92927e21	1.08077	0.540386	−	0.841417i	$-0.318278\pi$
$823$	7.92927e21	1.08077	0.540386	+	0.841417i	$0.318278\pi$
$824$	−6.33068e20	−0.0855059
$825$	0	0
$826$	3.57235e21	0.473809
$827$	1.97038e21	0.258976	0.129488	−	0.991581i	$-0.458667\pi$
$827$	1.97038e21	0.258976	0.129488	+	0.991581i	$0.458667\pi$
$828$	−2.43654e21	−0.317355
$829$	2.43930e21	0.314851	0.157426	−	0.987531i	$-0.449681\pi$
$829$	2.43930e21	0.314851	0.157426	+	0.987531i	$0.449681\pi$
$830$	0	0
$831$	−1.72764e21	−0.219001
$832$	−1.69396e21	−0.212803
$833$	3.22484e20	0.0401487
$834$	3.07178e21	0.379005
$835$	0	0
$836$	4.78783e21	0.580218
$837$	5.87736e20	0.0705897
$838$	−1.68821e21	−0.200954
$839$	−1.35318e22	−1.59639	−0.798196	−	0.602398i	$-0.794212\pi$
$839$	−1.35318e22	−1.59639	−0.798196	+	0.602398i	$0.794212\pi$
$840$	0	0
$841$	4.81601e21	0.558107
$842$	−5.24323e21	−0.602225
$843$	1.34567e21	0.153190
$844$	1.23734e22	1.39611
$845$	0	0
$846$	−7.14970e20	−0.0792519
$847$	−6.00794e21	−0.660085
$848$	7.12068e21	0.775448
$849$	−2.76173e21	−0.298107
$850$	0	0
$851$	−1.13534e22	−1.20408
$852$	3.55859e21	0.374095
$853$	7.30038e21	0.760725	0.380363	−	0.924837i	$-0.375799\pi$
$853$	7.30038e21	0.760725	0.380363	+	0.924837i	$0.375799\pi$
$854$	−1.31609e21	−0.135941
$855$	0	0
$856$	−1.77592e21	−0.180248
$857$	−1.16475e22	−1.17187	−0.585934	−	0.810359i	$-0.699272\pi$
$857$	−1.16475e22	−1.17187	−0.585934	+	0.810359i	$0.699272\pi$
$858$	−1.63864e21	−0.163429
$859$	−1.05473e22	−1.04278	−0.521388	−	0.853320i	$-0.674586\pi$
$859$	−1.05473e22	−1.04278	−0.521388	+	0.853320i	$0.674586\pi$
$860$	0	0
$861$	−2.58427e21	−0.251081
$862$	−3.39718e21	−0.327201
$863$	−1.25514e22	−1.19843	−0.599215	−	0.800588i	$-0.704520\pi$
$863$	−1.25514e22	−1.19843	−0.599215	+	0.800588i	$0.704520\pi$
$864$	−1.93455e21	−0.183116
$865$	0	0
$866$	−2.72951e21	−0.253922
$867$	−7.33124e21	−0.676136
$868$	3.33066e21	0.304531
$869$	5.94407e21	0.538810
$870$	0	0
$871$	3.45237e21	0.307596
$872$	−4.78569e21	−0.422737
$873$	2.58178e21	0.226106
$874$	−6.20937e21	−0.539153
$875$	0	0
$876$	3.38420e20	0.0288852
$877$	1.57294e22	1.33112	0.665558	−	0.746346i	$-0.268193\pi$
$877$	1.57294e22	1.33112	0.665558	+	0.746346i	$0.268193\pi$
$878$	−1.63319e21	−0.137034
$879$	−7.65152e21	−0.636548
$880$	0	0
$881$	1.80412e22	1.47553	0.737764	−	0.675059i	$-0.235882\pi$
$881$	1.80412e22	1.47553	0.737764	+	0.675059i	$0.235882\pi$
$882$	−4.45484e19	−0.00361258
$883$	−2.41866e22	−1.94478	−0.972388	−	0.233371i	$-0.925024\pi$
$883$	−2.41866e22	−1.94478	−0.972388	+	0.233371i	$0.925024\pi$
$884$	1.92523e22	1.53494
$885$	0	0
$886$	7.16320e21	0.561505
$887$	−7.05392e21	−0.548282	−0.274141	−	0.961690i	$-0.588393\pi$
$887$	−7.05392e21	−0.548282	−0.274141	+	0.961690i	$0.588393\pi$
$888$	−5.84228e21	−0.450282
$889$	1.34519e22	1.02807
$890$	0	0
$891$	8.50315e20	0.0638994
$892$	4.41279e21	0.328834
$893$	9.69514e21	0.716421
$894$	−2.50266e21	−0.183388
$895$	0	0
$896$	−1.39594e22	−1.00590
$897$	−1.13080e22	−0.808057
$898$	9.37143e21	0.664100
$899$	6.51508e21	0.457849
$900$	0	0
$901$	−3.00356e22	−2.07587
$902$	1.47153e21	0.100860
$903$	−4.61942e21	−0.314000
$904$	−9.25660e21	−0.624005
$905$	0	0
$906$	−4.89078e21	−0.324277
$907$	−2.48498e22	−1.63406	−0.817030	−	0.576596i	$-0.804381\pi$
$907$	−2.48498e22	−1.63406	−0.817030	+	0.576596i	$0.804381\pi$
$908$	4.02183e21	0.262289
$909$	−2.01947e21	−0.130620
$910$	0	0
$911$	−2.83848e21	−0.180592	−0.0902958	−	0.995915i	$-0.528781\pi$
$911$	−2.83848e21	−0.180592	−0.0902958	+	0.995915i	$0.528781\pi$
$912$	6.03585e21	0.380870
$913$	−2.14377e21	−0.134168
$914$	7.33749e21	0.455461
$915$	0	0
$916$	−1.87901e22	−1.14739
$917$	2.17177e22	1.31536
$918$	1.87753e21	0.112789
$919$	−1.87762e22	−1.11877	−0.559387	−	0.828907i	$-0.688963\pi$
$919$	−1.87762e22	−1.11877	−0.559387	+	0.828907i	$0.688963\pi$
$920$	0	0
$921$	2.81818e21	0.165204
$922$	−7.78620e21	−0.452734
$923$	1.65155e22	0.952528
$924$	4.81867e21	0.275669
$925$	0	0
$926$	−1.21761e22	−0.685374
$927$	6.96825e20	0.0389068
$928$	−2.14445e22	−1.18770
$929$	−2.61729e21	−0.143792	−0.0718958	−	0.997412i	$-0.522905\pi$
$929$	−2.61729e21	−0.143792	−0.0718958	+	0.997412i	$0.522905\pi$
$930$	0	0
$931$	6.04085e20	0.0326570
$932$	−1.07311e22	−0.575476
$933$	−1.84371e22	−0.980798
$934$	3.30061e21	0.174178
$935$	0	0
$936$	−5.81889e21	−0.302184
$937$	−1.24366e22	−0.640701	−0.320350	−	0.947299i	$-0.603801\pi$
$937$	−1.24366e22	−0.640701	−0.320350	+	0.947299i	$0.603801\pi$
$938$	1.90795e21	0.0975094
$939$	1.72414e22	0.874140
$940$	0	0
$941$	1.71266e22	0.854574	0.427287	−	0.904116i	$-0.359469\pi$
$941$	1.71266e22	0.854574	0.427287	+	0.904116i	$0.359469\pi$
$942$	−8.52465e21	−0.421983
$943$	1.01548e22	0.498692
$944$	1.36451e22	0.664795
$945$	0	0
$946$	2.63039e21	0.126135
$947$	−1.26035e21	−0.0599607	−0.0299804	−	0.999550i	$-0.509544\pi$
$947$	−1.26035e21	−0.0599607	−0.0299804	+	0.999550i	$0.509544\pi$
$948$	9.64729e21	0.455347
$949$	1.57061e21	0.0735480
$950$	0	0
$951$	−1.90606e22	−0.878581
$952$	2.32792e22	1.06461
$953$	2.19887e22	0.997706	0.498853	−	0.866687i	$-0.333755\pi$
$953$	2.19887e22	0.997706	0.498853	+	0.866687i	$0.333755\pi$
$954$	4.14915e21	0.186787
$955$	0	0
$956$	−1.86531e21	−0.0826639
$957$	9.42577e21	0.414455
$958$	9.46890e21	0.413103
$959$	2.13554e22	0.924419
$960$	0	0
$961$	−2.03083e22	−0.865461
$962$	−1.23925e22	−0.524019
$963$	1.95478e21	0.0820163
$964$	2.48402e22	1.03413
$965$	0	0
$966$	−6.24937e21	−0.256158
$967$	−1.21130e21	−0.0492666	−0.0246333	−	0.999697i	$-0.507842\pi$
$967$	−1.21130e21	−0.0492666	−0.0246333	+	0.999697i	$0.507842\pi$
$968$	1.21482e22	0.490284
$969$	−2.54597e22	−1.01959
$970$	0	0
$971$	1.26825e22	0.500103	0.250051	−	0.968233i	$-0.419552\pi$
$971$	1.26825e22	0.500103	0.250051	+	0.968233i	$0.419552\pi$
$972$	1.38007e21	0.0540013
$973$	−4.19223e22	−1.62779
$974$	−4.41843e21	−0.170245
$975$	0	0
$976$	−5.02700e21	−0.190737
$977$	3.29367e21	0.124014	0.0620070	−	0.998076i	$-0.480250\pi$
$977$	3.29367e21	0.124014	0.0620070	+	0.998076i	$0.480250\pi$
$978$	−8.73314e21	−0.326309
$979$	−1.34660e22	−0.499307
$980$	0	0
$981$	5.26766e21	0.192353
$982$	1.74281e22	0.631558
$983$	−3.74166e22	−1.34559	−0.672796	−	0.739828i	$-0.734907\pi$
$983$	−3.74166e22	−1.34559	−0.672796	+	0.739828i	$0.734907\pi$
$984$	5.22546e21	0.186493
$985$	0	0
$986$	2.08125e22	0.731554
$987$	9.75759e21	0.340380
$988$	3.60638e22	1.24852
$989$	1.81518e22	0.623661
$990$	0	0
$991$	−4.40305e22	−1.49005	−0.745026	−	0.667035i	$-0.767563\pi$
$991$	−4.40305e22	−1.49005	−0.745026	+	0.667035i	$0.767563\pi$
$992$	−1.03913e22	−0.349005
$993$	−2.77738e22	−0.925796
$994$	9.12728e21	0.301956
$995$	0	0
$996$	−3.47937e21	−0.113385
$997$	−2.62067e22	−0.847614	−0.423807	−	0.905753i	$-0.639306\pi$
$997$	−2.62067e22	−0.847614	−0.423807	+	0.905753i	$0.639306\pi$
$998$	1.05155e22	0.337560
$999$	6.43066e21	0.204887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.16.a.a.1.1		1
5.2	odd	4		75.16.b.b.49.2		2
5.3	odd	4		75.16.b.b.49.1		2
5.4	even	2		3.16.a.b.1.1	✓	1
15.14	odd	2		9.16.a.c.1.1		1
20.19	odd	2		48.16.a.a.1.1		1
35.34	odd	2		147.16.a.b.1.1		1
60.59	even	2		144.16.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.16.a.b.1.1	✓	1	5.4	even	2
9.16.a.c.1.1		1	15.14	odd	2
48.16.a.a.1.1		1	20.19	odd	2
75.16.a.a.1.1		1	1.1	even	1	trivial
75.16.b.b.49.1		2	5.3	odd	4
75.16.b.b.49.2		2	5.2	odd	4
144.16.a.l.1.1		1	60.59	even	2
147.16.a.b.1.1		1	35.34	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:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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