LMFDB - Embedded newform 75.12.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,12,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, 12, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
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:	$57.6257385420$
Analytic rank:	$1$
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)

$f(q)$

$=$

$q-78.0000 q^{2} +243.000 q^{3} +4036.00 q^{4} -18954.0 q^{6} +27760.0 q^{7} -155064. q^{8} +59049.0 q^{9} +O(q^{10})$

\(q-78.0000 q^{2} +243.000 q^{3} +4036.00 q^{4} -18954.0 q^{6} +27760.0 q^{7} -155064. q^{8} +59049.0 q^{9} +637836. q^{11} +980748. q^{12} -766214. q^{13} -2.16528e6 q^{14} +3.82926e6 q^{16} -3.08435e6 q^{17} -4.60582e6 q^{18} -1.95114e7 q^{19} +6.74568e6 q^{21} -4.97512e7 q^{22} -1.53124e7 q^{23} -3.76806e7 q^{24} +5.97647e7 q^{26} +1.43489e7 q^{27} +1.12039e8 q^{28} +1.07513e7 q^{29} -5.09374e7 q^{31} +1.88885e7 q^{32} +1.54994e8 q^{33} +2.40580e8 q^{34} +2.38322e8 q^{36} -6.64741e8 q^{37} +1.52189e9 q^{38} -1.86190e8 q^{39} +8.98833e8 q^{41} -5.26163e8 q^{42} +9.57947e8 q^{43} +2.57431e9 q^{44} +1.19436e9 q^{46} +1.55574e9 q^{47} +9.30511e8 q^{48} -1.20671e9 q^{49} -7.49498e8 q^{51} -3.09244e9 q^{52} -3.79242e9 q^{53} -1.11921e9 q^{54} -4.30458e9 q^{56} -4.74127e9 q^{57} -8.38598e8 q^{58} +5.55307e8 q^{59} +4.95042e9 q^{61} +3.97312e9 q^{62} +1.63920e9 q^{63} -9.31563e9 q^{64} -1.20895e10 q^{66} -5.29240e9 q^{67} -1.24485e10 q^{68} -3.72090e9 q^{69} -1.48311e10 q^{71} -9.15637e9 q^{72} -1.39710e10 q^{73} +5.18498e10 q^{74} -7.87480e10 q^{76} +1.77063e10 q^{77} +1.45228e10 q^{78} +3.72054e9 q^{79} +3.48678e9 q^{81} -7.01090e10 q^{82} -8.76845e9 q^{83} +2.72256e10 q^{84} -7.47199e10 q^{86} +2.61256e9 q^{87} -9.89054e10 q^{88} -2.54728e10 q^{89} -2.12701e10 q^{91} -6.18007e10 q^{92} -1.23778e10 q^{93} -1.21348e11 q^{94} +4.58990e9 q^{96} +3.90925e10 q^{97} +9.41233e10 q^{98} +3.76636e10 q^{99} +O(q^{100})\)

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 $

$2$	−78.0000	−1.72357	−0.861786	−	0.507271i	$-0.830654\pi$
$2$	−78.0000	−1.72357	−0.861786	+	0.507271i	$0.830654\pi$
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	4036.00	1.97070
$5$	0	0
$5$	0	0
$6$	−18954.0	−0.995105
$7$	27760.0	0.624281	0.312141	−	0.950036i	$-0.398954\pi$
$7$	27760.0	0.624281	0.312141	+	0.950036i	$0.398954\pi$
$8$	−155064.	−1.67308
$9$	59049.0	0.333333
$10$	0	0
$11$	637836.	1.19412	0.597062	−	0.802195i	$-0.296335\pi$
$11$	637836.	1.19412	0.597062	+	0.802195i	$0.296335\pi$
$12$	980748.	1.13779
$13$	−766214.	−0.572350	−0.286175	−	0.958177i	$-0.592384\pi$
$13$	−766214.	−0.572350	−0.286175	+	0.958177i	$0.592384\pi$
$14$	−2.16528e6	−1.07599
$15$	0	0
$16$	3.82926e6	0.912968
$17$	−3.08435e6	−0.526860	−0.263430	−	0.964679i	$-0.584854\pi$
$17$	−3.08435e6	−0.526860	−0.263430	+	0.964679i	$0.584854\pi$
$18$	−4.60582e6	−0.574524
$19$	−1.95114e7	−1.80777	−0.903886	−	0.427773i	$-0.859298\pi$
$19$	−1.95114e7	−1.80777	−0.903886	+	0.427773i	$0.859298\pi$
$20$	0	0
$21$	6.74568e6	0.360429
$22$	−4.97512e7	−2.05816
$23$	−1.53124e7	−0.496066	−0.248033	−	0.968752i	$-0.579784\pi$
$23$	−1.53124e7	−0.496066	−0.248033	+	0.968752i	$0.579784\pi$
$24$	−3.76806e7	−0.965952
$25$	0	0
$26$	5.97647e7	0.986487
$27$	1.43489e7	0.192450
$28$	1.12039e8	1.23027
$29$	1.07513e7	0.0973353	0.0486677	−	0.998815i	$-0.484502\pi$
$29$	1.07513e7	0.0973353	0.0486677	+	0.998815i	$0.484502\pi$
$30$	0	0
$31$	−5.09374e7	−0.319556	−0.159778	−	0.987153i	$-0.551078\pi$
$31$	−5.09374e7	−0.319556	−0.159778	+	0.987153i	$0.551078\pi$
$32$	1.88885e7	0.0995112
$33$	1.54994e8	0.689428
$34$	2.40580e8	0.908081
$35$	0	0
$36$	2.38322e8	0.656901
$37$	−6.64741e8	−1.57595	−0.787976	−	0.615706i	$-0.788871\pi$
$37$	−6.64741e8	−1.57595	−0.787976	+	0.615706i	$0.788871\pi$
$38$	1.52189e9	3.11583
$39$	−1.86190e8	−0.330446
$40$	0	0
$41$	8.98833e8	1.21162	0.605812	−	0.795608i	$-0.292848\pi$
$41$	8.98833e8	1.21162	0.605812	+	0.795608i	$0.292848\pi$
$42$	−5.26163e8	−0.621225
$43$	9.57947e8	0.993722	0.496861	−	0.867830i	$-0.334486\pi$
$43$	9.57947e8	0.993722	0.496861	+	0.867830i	$0.334486\pi$
$44$	2.57431e9	2.35326
$45$	0	0
$46$	1.19436e9	0.855005
$47$	1.55574e9	0.989462	0.494731	−	0.869046i	$-0.335267\pi$
$47$	1.55574e9	0.989462	0.494731	+	0.869046i	$0.335267\pi$
$48$	9.30511e8	0.527102
$49$	−1.20671e9	−0.610273
$50$	0	0
$51$	−7.49498e8	−0.304183
$52$	−3.09244e9	−1.12793
$53$	−3.79242e9	−1.24566	−0.622829	−	0.782358i	$-0.714017\pi$
$53$	−3.79242e9	−1.24566	−0.622829	+	0.782358i	$0.714017\pi$
$54$	−1.11921e9	−0.331702
$55$	0	0
$56$	−4.30458e9	−1.04447
$57$	−4.74127e9	−1.04372
$58$	−8.38598e8	−0.167765
$59$	5.55307e8	0.101122	0.0505612	−	0.998721i	$-0.483899\pi$
$59$	5.55307e8	0.101122	0.0505612	+	0.998721i	$0.483899\pi$
$60$	0	0
$61$	4.95042e9	0.750461	0.375230	−	0.926932i	$-0.377564\pi$
$61$	4.95042e9	0.750461	0.375230	+	0.926932i	$0.377564\pi$
$62$	3.97312e9	0.550779
$63$	1.63920e9	0.208094
$64$	−9.31563e9	−1.08448
$65$	0	0
$66$	−1.20895e10	−1.18828
$67$	−5.29240e9	−0.478896	−0.239448	−	0.970909i	$-0.576966\pi$
$67$	−5.29240e9	−0.478896	−0.239448	+	0.970909i	$0.576966\pi$
$68$	−1.24485e10	−1.03828
$69$	−3.72090e9	−0.286404
$70$	0	0
$71$	−1.48311e10	−0.975556	−0.487778	−	0.872968i	$-0.662192\pi$
$71$	−1.48311e10	−0.975556	−0.487778	+	0.872968i	$0.662192\pi$
$72$	−9.15637e9	−0.557692
$73$	−1.39710e10	−0.788773	−0.394386	−	0.918945i	$-0.629043\pi$
$73$	−1.39710e10	−0.788773	−0.394386	+	0.918945i	$0.629043\pi$
$74$	5.18498e10	2.71627
$75$	0	0
$76$	−7.87480e10	−3.56258
$77$	1.77063e10	0.745469
$78$	1.45228e10	0.569548
$79$	3.72054e9	0.136037	0.0680185	−	0.997684i	$-0.478332\pi$
$79$	3.72054e9	0.136037	0.0680185	+	0.997684i	$0.478332\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	−7.01090e10	−2.08832
$83$	−8.76845e9	−0.244339	−0.122170	−	0.992509i	$-0.538985\pi$
$83$	−8.76845e9	−0.244339	−0.122170	+	0.992509i	$0.538985\pi$
$84$	2.72256e10	0.710298
$85$	0	0
$86$	−7.47199e10	−1.71275
$87$	2.61256e9	0.0561966
$88$	−9.89054e10	−1.99786
$89$	−2.54728e10	−0.483539	−0.241769	−	0.970334i	$-0.577728\pi$
$89$	−2.54728e10	−0.483539	−0.241769	+	0.970334i	$0.577728\pi$
$90$	0	0
$91$	−2.12701e10	−0.357307
$92$	−6.18007e10	−0.977598
$93$	−1.23778e10	−0.184496
$94$	−1.21348e11	−1.70541
$95$	0	0
$96$	4.58990e9	0.0574528
$97$	3.90925e10	0.462220	0.231110	−	0.972928i	$-0.425764\pi$
$97$	3.90925e10	0.462220	0.231110	+	0.972928i	$0.425764\pi$
$98$	9.41233e10	1.05185
$99$	3.76636e10	0.398041
$100$	0	0
$101$	9.31078e9	0.0881492	0.0440746	−	0.999028i	$-0.485966\pi$
$101$	9.31078e9	0.0881492	0.0440746	+	0.999028i	$0.485966\pi$
$102$	5.84608e10	0.524281
$103$	−4.85751e10	−0.412865	−0.206433	−	0.978461i	$-0.566185\pi$
$103$	−4.85751e10	−0.412865	−0.206433	+	0.978461i	$0.566185\pi$
$104$	1.18812e11	0.957586
$105$	0	0
$106$	2.95809e11	2.14698
$107$	−2.25596e11	−1.55496	−0.777482	−	0.628906i	$-0.783503\pi$
$107$	−2.25596e11	−1.55496	−0.777482	+	0.628906i	$0.783503\pi$
$108$	5.79122e10	0.379262
$109$	−6.94512e10	−0.432348	−0.216174	−	0.976355i	$-0.569358\pi$
$109$	−6.94512e10	−0.432348	−0.216174	+	0.976355i	$0.569358\pi$
$110$	0	0
$111$	−1.61532e11	−0.909876
$112$	1.06300e11	0.569949
$113$	3.59665e11	1.83640	0.918198	−	0.396122i	$-0.129644\pi$
$113$	3.59665e11	1.83640	0.918198	+	0.396122i	$0.129644\pi$
$114$	3.69819e11	1.79892
$115$	0	0
$116$	4.33921e10	0.191819
$117$	−4.52442e10	−0.190783
$118$	−4.33139e10	−0.174292
$119$	−8.56217e10	−0.328909
$120$	0	0
$121$	1.21523e11	0.425931
$122$	−3.86133e11	−1.29347
$123$	2.18417e11	0.699532
$124$	−2.05583e11	−0.629751
$125$	0	0
$126$	−1.27858e11	−0.358665
$127$	−2.50273e11	−0.672192	−0.336096	−	0.941828i	$-0.609107\pi$
$127$	−2.50273e11	−0.672192	−0.336096	+	0.941828i	$0.609107\pi$
$128$	6.87936e11	1.76967
$129$	2.32781e11	0.573726
$130$	0	0
$131$	−9.78918e10	−0.221694	−0.110847	−	0.993837i	$-0.535356\pi$
$131$	−9.78918e10	−0.221694	−0.110847	+	0.993837i	$0.535356\pi$
$132$	6.25556e11	1.35866
$133$	−5.41637e11	−1.12856
$134$	4.12807e11	0.825412
$135$	0	0
$136$	4.78272e11	0.881477
$137$	1.55015e11	0.274416	0.137208	−	0.990542i	$-0.456187\pi$
$137$	1.55015e11	0.274416	0.137208	+	0.990542i	$0.456187\pi$
$138$	2.90230e11	0.493637
$139$	−1.12627e12	−1.84102	−0.920512	−	0.390715i	$-0.872228\pi$
$139$	−1.12627e12	−1.84102	−0.920512	+	0.390715i	$0.872228\pi$
$140$	0	0
$141$	3.78045e11	0.571266
$142$	1.15682e12	1.68144
$143$	−4.88719e11	−0.683457
$144$	2.26114e11	0.304323
$145$	0	0
$146$	1.08974e12	1.35951
$147$	−2.93230e11	−0.352341
$148$	−2.68289e12	−3.10573
$149$	−1.38458e12	−1.54452	−0.772261	−	0.635306i	$-0.780874\pi$
$149$	−1.38458e12	−1.54452	−0.772261	+	0.635306i	$0.780874\pi$
$150$	0	0
$151$	−6.98601e11	−0.724196	−0.362098	−	0.932140i	$-0.617939\pi$
$151$	−6.98601e11	−0.724196	−0.362098	+	0.932140i	$0.617939\pi$
$152$	3.02552e12	3.02454
$153$	−1.82128e11	−0.175620
$154$	−1.38109e12	−1.28487
$155$	0	0
$156$	−7.51463e11	−0.651212
$157$	−2.13127e12	−1.78316	−0.891580	−	0.452863i	$-0.850403\pi$
$157$	−2.13127e12	−1.78316	−0.891580	+	0.452863i	$0.850403\pi$
$158$	−2.90202e11	−0.234470
$159$	−9.21557e11	−0.719181
$160$	0	0
$161$	−4.25071e11	−0.309684
$162$	−2.71969e11	−0.191508
$163$	1.63564e11	0.111342	0.0556708	−	0.998449i	$-0.482270\pi$
$163$	1.63564e11	0.111342	0.0556708	+	0.998449i	$0.482270\pi$
$164$	3.62769e12	2.38775
$165$	0	0
$166$	6.83939e11	0.421137
$167$	8.80943e9	0.00524816	0.00262408	−	0.999997i	$-0.499165\pi$
$167$	8.80943e9	0.00524816	0.00262408	+	0.999997i	$0.499165\pi$
$168$	−1.04601e12	−0.603025
$169$	−1.20508e12	−0.672416
$170$	0	0
$171$	−1.15213e12	−0.602591
$172$	3.86627e12	1.95833
$173$	7.30852e11	0.358571	0.179286	−	0.983797i	$-0.442621\pi$
$173$	7.30852e11	0.358571	0.179286	+	0.983797i	$0.442621\pi$
$174$	−2.03779e11	−0.0968589
$175$	0	0
$176$	2.44244e12	1.09020
$177$	1.34940e11	0.0583830
$178$	1.98688e12	0.833414
$179$	3.92371e12	1.59590	0.797950	−	0.602724i	$-0.205918\pi$
$179$	3.92371e12	1.59590	0.797950	+	0.602724i	$0.205918\pi$
$180$	0	0
$181$	2.27931e12	0.872110	0.436055	−	0.899920i	$-0.356375\pi$
$181$	2.27931e12	0.872110	0.436055	+	0.899920i	$0.356375\pi$
$182$	1.65907e12	0.615845
$183$	1.20295e12	0.433279
$184$	2.37440e12	0.829956
$185$	0	0
$186$	9.65467e11	0.317992
$187$	−1.96731e12	−0.629136
$188$	6.27897e12	1.94994
$189$	3.98326e11	0.120143
$190$	0	0
$191$	3.38709e12	0.964147	0.482073	−	0.876131i	$-0.339884\pi$
$191$	3.38709e12	0.964147	0.482073	+	0.876131i	$0.339884\pi$
$192$	−2.26370e12	−0.626126
$193$	4.92921e12	1.32499	0.662494	−	0.749067i	$-0.269498\pi$
$193$	4.92921e12	1.32499	0.662494	+	0.749067i	$0.269498\pi$
$194$	−3.04921e12	−0.796670
$195$	0	0
$196$	−4.87028e12	−1.20267
$197$	−6.35785e12	−1.52667	−0.763337	−	0.646001i	$-0.776440\pi$
$197$	−6.35785e12	−1.52667	−0.763337	+	0.646001i	$0.776440\pi$
$198$	−2.93776e12	−0.686053
$199$	−3.78554e12	−0.859875	−0.429938	−	0.902859i	$-0.641464\pi$
$199$	−3.78554e12	−0.859875	−0.429938	+	0.902859i	$0.641464\pi$
$200$	0	0
$201$	−1.28605e12	−0.276491
$202$	−7.26241e11	−0.151932
$203$	2.98455e11	0.0607646
$204$	−3.02497e12	−0.599454
$205$	0	0
$206$	3.78885e12	0.711604
$207$	−9.04180e11	−0.165355
$208$	−2.93404e12	−0.522537
$209$	−1.24451e13	−2.15870
$210$	0	0
$211$	1.79494e11	0.0295458	0.0147729	−	0.999891i	$-0.495297\pi$
$211$	1.79494e11	0.0295458	0.0147729	+	0.999891i	$0.495297\pi$
$212$	−1.53062e13	−2.45482
$213$	−3.60395e12	−0.563237
$214$	1.75965e13	2.68009
$215$	0	0
$216$	−2.22500e12	−0.321984
$217$	−1.41402e12	−0.199493
$218$	5.41719e12	0.745184
$219$	−3.39495e12	−0.455398
$220$	0	0
$221$	2.36328e12	0.301548
$222$	1.25995e13	1.56824
$223$	−2.22568e12	−0.270263	−0.135132	−	0.990828i	$-0.543146\pi$
$223$	−2.22568e12	−0.270263	−0.135132	+	0.990828i	$0.543146\pi$
$224$	5.24344e11	0.0621230
$225$	0	0
$226$	−2.80538e13	−3.16516
$227$	4.15848e11	0.0457923	0.0228961	−	0.999738i	$-0.492711\pi$
$227$	4.15848e11	0.0457923	0.0228961	+	0.999738i	$0.492711\pi$
$228$	−1.91358e13	−2.05686
$229$	1.81248e13	1.90186	0.950928	−	0.309414i	$-0.100133\pi$
$229$	1.81248e13	1.90186	0.950928	+	0.309414i	$0.100133\pi$
$230$	0	0
$231$	4.30264e12	0.430397
$232$	−1.66713e12	−0.162850
$233$	1.87641e10	0.00179007	0.000895033	−	1.00000i	$-0.499715\pi$
$233$	1.87641e10	0.00179007	0.000895033		1.00000i	$0.499715\pi$
$234$	3.52905e12	0.328829
$235$	0	0
$236$	2.24122e12	0.199282
$237$	9.04092e11	0.0785410
$238$	6.67849e12	0.566898
$239$	−1.76252e13	−1.46200	−0.730999	−	0.682379i	$-0.760946\pi$
$239$	−1.76252e13	−1.46200	−0.730999	+	0.682379i	$0.760946\pi$
$240$	0	0
$241$	−8.90117e11	−0.0705267	−0.0352633	−	0.999378i	$-0.511227\pi$
$241$	−8.90117e11	−0.0705267	−0.0352633	+	0.999378i	$0.511227\pi$
$242$	−9.47880e12	−0.734123
$243$	8.47289e11	0.0641500
$244$	1.99799e13	1.47894
$245$	0	0
$246$	−1.70365e13	−1.20569
$247$	1.49499e13	1.03468
$248$	7.89856e12	0.534643
$249$	−2.13073e12	−0.141069
$250$	0	0
$251$	−2.42280e13	−1.53502	−0.767508	−	0.641040i	$-0.778503\pi$
$251$	−2.42280e13	−1.53502	−0.767508	+	0.641040i	$0.778503\pi$
$252$	6.61581e12	0.410091
$253$	−9.76677e12	−0.592364
$254$	1.95213e13	1.15857
$255$	0	0
$256$	−3.45806e13	−1.96568
$257$	−7.80492e12	−0.434246	−0.217123	−	0.976144i	$-0.569667\pi$
$257$	−7.80492e12	−0.434246	−0.217123	+	0.976144i	$0.569667\pi$
$258$	−1.81569e13	−0.988858
$259$	−1.84532e13	−0.983837
$260$	0	0
$261$	6.34851e11	0.0324451
$262$	7.63556e12	0.382106
$263$	−1.65956e13	−0.813272	−0.406636	−	0.913590i	$-0.633298\pi$
$263$	−1.65956e13	−0.813272	−0.406636	+	0.913590i	$0.633298\pi$
$264$	−2.40340e13	−1.15347
$265$	0	0
$266$	4.22477e13	1.94515
$267$	−6.18988e12	−0.279171
$268$	−2.13601e13	−0.943762
$269$	2.85236e13	1.23471	0.617357	−	0.786683i	$-0.288203\pi$
$269$	2.85236e13	1.23471	0.617357	+	0.786683i	$0.288203\pi$
$270$	0	0
$271$	2.33800e13	0.971658	0.485829	−	0.874054i	$-0.338518\pi$
$271$	2.33800e13	0.971658	0.485829	+	0.874054i	$0.338518\pi$
$272$	−1.18108e13	−0.481006
$273$	−5.16863e12	−0.206291
$274$	−1.20911e13	−0.472976
$275$	0	0
$276$	−1.50176e13	−0.564416
$277$	3.03641e13	1.11872	0.559361	−	0.828924i	$-0.311047\pi$
$277$	3.03641e13	1.11872	0.559361	+	0.828924i	$0.311047\pi$
$278$	8.78487e13	3.17314
$279$	−3.00780e12	−0.106519
$280$	0	0
$281$	1.59749e13	0.543942	0.271971	−	0.962306i	$-0.412325\pi$
$281$	1.59749e13	0.543942	0.271971	+	0.962306i	$0.412325\pi$
$282$	−2.94875e13	−0.984619
$283$	−2.87045e13	−0.939993	−0.469997	−	0.882668i	$-0.655745\pi$
$283$	−2.87045e13	−0.939993	−0.469997	+	0.882668i	$0.655745\pi$
$284$	−5.98583e13	−1.92253
$285$	0	0
$286$	3.81201e13	1.17799
$287$	2.49516e13	0.756394
$288$	1.11535e12	0.0331704
$289$	−2.47587e13	−0.722419
$290$	0	0
$291$	9.49948e12	0.266863
$292$	−5.63870e13	−1.55444
$293$	−4.81754e13	−1.30333	−0.651663	−	0.758508i	$-0.725929\pi$
$293$	−4.81754e13	−1.30333	−0.651663	+	0.758508i	$0.725929\pi$
$294$	2.28720e13	0.607286
$295$	0	0
$296$	1.03077e14	2.63669
$297$	9.15225e12	0.229809
$298$	1.07997e14	2.66209
$299$	1.17325e13	0.283923
$300$	0	0
$301$	2.65926e13	0.620362
$302$	5.44909e13	1.24820
$303$	2.26252e12	0.0508930
$304$	−7.47143e13	−1.65044
$305$	0	0
$306$	1.42060e13	0.302694
$307$	−2.57350e13	−0.538597	−0.269298	−	0.963057i	$-0.586792\pi$
$307$	−2.57350e13	−0.538597	−0.269298	+	0.963057i	$0.586792\pi$
$308$	7.14627e13	1.46910
$309$	−1.18037e13	−0.238368
$310$	0	0
$311$	3.46043e13	0.674446	0.337223	−	0.941425i	$-0.390512\pi$
$311$	3.46043e13	0.674446	0.337223	+	0.941425i	$0.390512\pi$
$312$	2.88714e13	0.552862
$313$	−1.26066e13	−0.237194	−0.118597	−	0.992942i	$-0.537840\pi$
$313$	−1.26066e13	−0.237194	−0.118597	+	0.992942i	$0.537840\pi$
$314$	1.66239e14	3.07341
$315$	0	0
$316$	1.50161e13	0.268089
$317$	8.18243e13	1.43568	0.717838	−	0.696210i	$-0.245132\pi$
$317$	8.18243e13	1.43568	0.717838	+	0.696210i	$0.245132\pi$
$318$	7.18815e13	1.23956
$319$	6.85754e12	0.116230
$320$	0	0
$321$	−5.48198e13	−0.897758
$322$	3.31555e13	0.533764
$323$	6.01801e13	0.952443
$324$	1.40727e13	0.218967
$325$	0	0
$326$	−1.27580e13	−0.191905
$327$	−1.68766e13	−0.249616
$328$	−1.39377e14	−2.02714
$329$	4.31874e13	0.617703
$330$	0	0
$331$	−3.40115e13	−0.470513	−0.235256	−	0.971933i	$-0.575593\pi$
$331$	−3.40115e13	−0.470513	−0.235256	+	0.971933i	$0.575593\pi$
$332$	−3.53895e13	−0.481520
$333$	−3.92523e13	−0.525317
$334$	−6.87135e11	−0.00904558
$335$	0	0
$336$	2.58310e13	0.329060
$337$	5.99439e13	0.751244	0.375622	−	0.926773i	$-0.377429\pi$
$337$	5.99439e13	0.751244	0.375622	+	0.926773i	$0.377429\pi$
$338$	9.39960e13	1.15896
$339$	8.73985e13	1.06024
$340$	0	0
$341$	−3.24897e13	−0.381590
$342$	8.98661e13	1.03861
$343$	−8.83888e13	−1.00526
$344$	−1.48543e14	−1.66257
$345$	0	0
$346$	−5.70064e13	−0.618024
$347$	−9.78685e13	−1.04431	−0.522157	−	0.852850i	$-0.674872\pi$
$347$	−9.78685e13	−1.04431	−0.522157	+	0.852850i	$0.674872\pi$
$348$	1.05443e13	0.110747
$349$	−1.42790e14	−1.47624	−0.738120	−	0.674670i	$-0.764286\pi$
$349$	−1.42790e14	−1.47624	−0.738120	+	0.674670i	$0.764286\pi$
$350$	0	0
$351$	−1.09943e13	−0.110149
$352$	1.20478e13	0.118829
$353$	−1.44246e14	−1.40069	−0.700346	−	0.713804i	$-0.746971\pi$
$353$	−1.44246e14	−1.40069	−0.700346	+	0.713804i	$0.746971\pi$
$354$	−1.05253e13	−0.100627
$355$	0	0
$356$	−1.02808e14	−0.952911
$357$	−2.08061e13	−0.189896
$358$	−3.06050e14	−2.75065
$359$	−1.24349e14	−1.10059	−0.550293	−	0.834972i	$-0.685484\pi$
$359$	−1.24349e14	−1.10059	−0.550293	+	0.834972i	$0.685484\pi$
$360$	0	0
$361$	2.64205e14	2.26804
$362$	−1.77786e14	−1.50314
$363$	2.95301e13	0.245911
$364$	−8.58461e13	−0.704147
$365$	0	0
$366$	−9.38303e13	−0.746788
$367$	1.60110e14	1.25532	0.627659	−	0.778488i	$-0.284013\pi$
$367$	1.60110e14	1.25532	0.627659	+	0.778488i	$0.284013\pi$
$368$	−5.86351e13	−0.452892
$369$	5.30752e13	0.403875
$370$	0	0
$371$	−1.05277e14	−0.777641
$372$	−4.99568e13	−0.363587
$373$	3.47258e13	0.249031	0.124516	−	0.992218i	$-0.460262\pi$
$373$	3.47258e13	0.249031	0.124516	+	0.992218i	$0.460262\pi$
$374$	1.53450e14	1.08436
$375$	0	0
$376$	−2.41239e14	−1.65545
$377$	−8.23777e12	−0.0557099
$378$	−3.10694e13	−0.207075
$379$	−1.46500e14	−0.962327	−0.481163	−	0.876631i	$-0.659786\pi$
$379$	−1.46500e14	−0.962327	−0.481163	+	0.876631i	$0.659786\pi$
$380$	0	0
$381$	−6.08164e13	−0.388090
$382$	−2.64193e14	−1.66178
$383$	−6.43419e13	−0.398933	−0.199467	−	0.979905i	$-0.563921\pi$
$383$	−6.43419e13	−0.398933	−0.199467	+	0.979905i	$0.563921\pi$
$384$	1.67168e14	1.02172
$385$	0	0
$386$	−3.84478e14	−2.28371
$387$	5.65658e13	0.331241
$388$	1.57777e14	0.910899
$389$	−3.98900e13	−0.227061	−0.113530	−	0.993535i	$-0.536216\pi$
$389$	−3.98900e13	−0.227061	−0.113530	+	0.993535i	$0.536216\pi$
$390$	0	0
$391$	4.72287e13	0.261357
$392$	1.87117e14	1.02103
$393$	−2.37877e13	−0.127995
$394$	4.95912e14	2.63133
$395$	0	0
$396$	1.52010e14	0.784421
$397$	−1.06552e14	−0.542268	−0.271134	−	0.962542i	$-0.587399\pi$
$397$	−1.06552e14	−0.542268	−0.271134	+	0.962542i	$0.587399\pi$
$398$	2.95272e14	1.48206
$399$	−1.31618e14	−0.651573
$400$	0	0
$401$	3.41445e13	0.164447	0.0822236	−	0.996614i	$-0.473798\pi$
$401$	3.41445e13	0.164447	0.0822236	+	0.996614i	$0.473798\pi$
$402$	1.00312e14	0.476552
$403$	3.90289e13	0.182898
$404$	3.75783e13	0.173716
$405$	0	0
$406$	−2.32795e13	−0.104732
$407$	−4.23996e14	−1.88188
$408$	1.16220e14	0.508921
$409$	5.33349e13	0.230427	0.115213	−	0.993341i	$-0.463245\pi$
$409$	5.33349e13	0.230427	0.115213	+	0.993341i	$0.463245\pi$
$410$	0	0
$411$	3.76686e13	0.158434
$412$	−1.96049e14	−0.813635
$413$	1.54153e13	0.0631288
$414$	7.05260e13	0.285002
$415$	0	0
$416$	−1.44726e13	−0.0569553
$417$	−2.73682e14	−1.06292
$418$	9.70716e14	3.72068
$419$	−1.01288e14	−0.383159	−0.191580	−	0.981477i	$-0.561361\pi$
$419$	−1.01288e14	−0.383159	−0.191580	+	0.981477i	$0.561361\pi$
$420$	0	0
$421$	−1.57928e14	−0.581981	−0.290991	−	0.956726i	$-0.593985\pi$
$421$	−1.57928e14	−0.581981	−0.290991	+	0.956726i	$0.593985\pi$
$422$	−1.40005e13	−0.0509244
$423$	9.18650e13	0.329821
$424$	5.88067e14	2.08408
$425$	0	0
$426$	2.81108e14	0.970780
$427$	1.37424e14	0.468499
$428$	−9.10504e14	−3.06437
$429$	−1.18759e14	−0.394594
$430$	0	0
$431$	5.13171e14	1.66202	0.831012	−	0.556254i	$-0.187762\pi$
$431$	5.13171e14	1.66202	0.831012	+	0.556254i	$0.187762\pi$
$432$	5.49458e13	0.175701
$433$	7.49248e13	0.236560	0.118280	−	0.992980i	$-0.462262\pi$
$433$	7.49248e13	0.236560	0.118280	+	0.992980i	$0.462262\pi$
$434$	1.10294e14	0.343841
$435$	0	0
$436$	−2.80305e14	−0.852030
$437$	2.98766e14	0.896773
$438$	2.64806e14	0.784912
$439$	−3.68335e14	−1.07817	−0.539086	−	0.842250i	$-0.681230\pi$
$439$	−3.68335e14	−1.07817	−0.539086	+	0.842250i	$0.681230\pi$
$440$	0	0
$441$	−7.12550e13	−0.203424
$442$	−1.84335e14	−0.519740
$443$	1.11248e14	0.309793	0.154896	−	0.987931i	$-0.450496\pi$
$443$	1.11248e14	0.309793	0.154896	+	0.987931i	$0.450496\pi$
$444$	−6.51943e14	−1.79310
$445$	0	0
$446$	1.73603e14	0.465818
$447$	−3.36453e14	−0.891730
$448$	−2.58602e14	−0.677022
$449$	−8.83314e13	−0.228434	−0.114217	−	0.993456i	$-0.536436\pi$
$449$	−8.83314e13	−0.228434	−0.114217	+	0.993456i	$0.536436\pi$
$450$	0	0
$451$	5.73308e14	1.44683
$452$	1.45161e15	3.61899
$453$	−1.69760e14	−0.418115
$454$	−3.24361e13	−0.0789263
$455$	0	0
$456$	7.35200e14	1.74622
$457$	9.42094e12	0.0221083	0.0110541	−	0.999939i	$-0.496481\pi$
$457$	9.42094e12	0.0221083	0.0110541	+	0.999939i	$0.496481\pi$
$458$	−1.41373e15	−3.27799
$459$	−4.42571e13	−0.101394
$460$	0	0
$461$	−6.97134e14	−1.55941	−0.779706	−	0.626146i	$-0.784631\pi$
$461$	−6.97134e14	−1.55941	−0.779706	+	0.626146i	$0.784631\pi$
$462$	−3.35606e14	−0.741820
$463$	1.87941e14	0.410513	0.205256	−	0.978708i	$-0.434197\pi$
$463$	1.87941e14	0.410513	0.205256	+	0.978708i	$0.434197\pi$
$464$	4.11694e13	0.0888640
$465$	0	0
$466$	−1.46360e12	−0.00308531
$467$	−3.20007e14	−0.666678	−0.333339	−	0.942807i	$-0.608175\pi$
$467$	−3.20007e14	−0.666678	−0.333339	+	0.942807i	$0.608175\pi$
$468$	−1.82605e14	−0.375977
$469$	−1.46917e14	−0.298966
$470$	0	0
$471$	−5.17899e14	−1.02951
$472$	−8.61081e13	−0.169185
$473$	6.11013e14	1.18663
$474$	−7.05192e13	−0.135371
$475$	0	0
$476$	−3.45569e14	−0.648181
$477$	−2.23938e14	−0.415219
$478$	1.37477e15	2.51986
$479$	−1.50382e14	−0.272491	−0.136245	−	0.990675i	$-0.543504\pi$
$479$	−1.50382e14	−0.272491	−0.136245	+	0.990675i	$0.543504\pi$
$480$	0	0
$481$	5.09334e14	0.901996
$482$	6.94291e13	0.121558
$483$	−1.03292e14	−0.178796
$484$	4.90467e14	0.839384
$485$	0	0
$486$	−6.60885e13	−0.110567
$487$	−1.76546e14	−0.292045	−0.146022	−	0.989281i	$-0.546647\pi$
$487$	−1.76546e14	−0.292045	−0.146022	+	0.989281i	$0.546647\pi$
$488$	−7.67632e14	−1.25558
$489$	3.97462e13	0.0642831
$490$	0	0
$491$	−8.60958e14	−1.36155	−0.680775	−	0.732492i	$-0.738357\pi$
$491$	−8.60958e14	−1.36155	−0.680775	+	0.732492i	$0.738357\pi$
$492$	8.81529e14	1.37857
$493$	−3.31607e13	−0.0512821
$494$	−1.16609e15	−1.78334
$495$	0	0
$496$	−1.95053e14	−0.291745
$497$	−4.11711e14	−0.609021
$498$	1.66197e14	0.243143
$499$	6.01209e14	0.869907	0.434953	−	0.900453i	$-0.356765\pi$
$499$	6.01209e14	0.869907	0.434953	+	0.900453i	$0.356765\pi$
$500$	0	0
$501$	2.14069e12	0.00303003
$502$	1.88979e15	2.64571
$503$	−1.09203e15	−1.51221	−0.756103	−	0.654453i	$-0.772899\pi$
$503$	−1.09203e15	−1.51221	−0.756103	+	0.654453i	$0.772899\pi$
$504$	−2.54181e14	−0.348157
$505$	0	0
$506$	7.61808e14	1.02098
$507$	−2.92834e14	−0.388219
$508$	−1.01010e15	−1.32469
$509$	8.76371e14	1.13695	0.568474	−	0.822702i	$-0.307534\pi$
$509$	8.76371e14	1.13695	0.568474	+	0.822702i	$0.307534\pi$
$510$	0	0
$511$	−3.87835e14	−0.492416
$512$	1.28839e15	1.61832
$513$	−2.79967e14	−0.347906
$514$	6.08784e14	0.748455
$515$	0	0
$516$	9.39505e14	1.13064
$517$	9.92308e14	1.18154
$518$	1.43935e15	1.69571
$519$	1.77597e14	0.207021
$520$	0	0
$521$	−3.71989e14	−0.424544	−0.212272	−	0.977211i	$-0.568086\pi$
$521$	−3.71989e14	−0.424544	−0.212272	+	0.977211i	$0.568086\pi$
$522$	−4.95184e13	−0.0559215
$523$	9.73507e14	1.08788	0.543939	−	0.839125i	$-0.316932\pi$
$523$	9.73507e14	1.08788	0.543939	+	0.839125i	$0.316932\pi$
$524$	−3.95091e14	−0.436893
$525$	0	0
$526$	1.29445e15	1.40173
$527$	1.57109e14	0.168361
$528$	5.93514e14	0.629425
$529$	−7.18341e14	−0.753919
$530$	0	0
$531$	3.27903e13	0.0337074
$532$	−2.18605e15	−2.22405
$533$	−6.88699e14	−0.693473
$534$	4.82811e14	0.481172
$535$	0	0
$536$	8.20661e14	0.801230
$537$	9.53463e14	0.921393
$538$	−2.22484e15	−2.12812
$539$	−7.69683e14	−0.728741
$540$	0	0
$541$	1.74606e15	1.61985	0.809925	−	0.586533i	$-0.199508\pi$
$541$	1.74606e15	1.61985	0.809925	+	0.586533i	$0.199508\pi$
$542$	−1.82364e15	−1.67472
$543$	5.53872e14	0.503513
$544$	−5.82588e13	−0.0524285
$545$	0	0
$546$	4.03153e14	0.355558
$547$	−1.74624e14	−0.152467	−0.0762333	−	0.997090i	$-0.524289\pi$
$547$	−1.74624e14	−0.152467	−0.0762333	+	0.997090i	$0.524289\pi$
$548$	6.25639e14	0.540793
$549$	2.92317e14	0.250154
$550$	0	0
$551$	−2.09772e14	−0.175960
$552$	5.76978e14	0.479175
$553$	1.03282e14	0.0849254
$554$	−2.36840e15	−1.92820
$555$	0	0
$556$	−4.54561e15	−3.62811
$557$	−1.58365e15	−1.25157	−0.625785	−	0.779995i	$-0.715221\pi$
$557$	−1.58365e15	−1.25157	−0.625785	+	0.779995i	$0.715221\pi$
$558$	2.34609e14	0.183593
$559$	−7.33993e14	−0.568757
$560$	0	0
$561$	−4.78057e14	−0.363232
$562$	−1.24604e15	−0.937523
$563$	9.75798e14	0.727049	0.363525	−	0.931585i	$-0.381573\pi$
$563$	9.75798e14	0.727049	0.363525	+	0.931585i	$0.381573\pi$
$564$	1.52579e15	1.12580
$565$	0	0
$566$	2.23895e15	1.62015
$567$	9.67931e13	0.0693646
$568$	2.29977e15	1.63218
$569$	1.92427e15	1.35254	0.676269	−	0.736655i	$-0.263596\pi$
$569$	1.92427e15	1.35254	0.676269	+	0.736655i	$0.263596\pi$
$570$	0	0
$571$	1.61132e15	1.11092	0.555461	−	0.831543i	$-0.312542\pi$
$571$	1.61132e15	1.11092	0.555461	+	0.831543i	$0.312542\pi$
$572$	−1.97247e15	−1.34689
$573$	8.23063e14	0.556650
$574$	−1.94623e15	−1.30370
$575$	0	0
$576$	−5.50079e14	−0.361494
$577$	2.53250e15	1.64848	0.824239	−	0.566242i	$-0.191603\pi$
$577$	2.53250e15	1.64848	0.824239	+	0.566242i	$0.191603\pi$
$578$	1.93118e15	1.24514
$579$	1.19780e15	0.764982
$580$	0	0
$581$	−2.43412e14	−0.152536
$582$	−7.40959e14	−0.459958
$583$	−2.41894e15	−1.48747
$584$	2.16640e15	1.31968
$585$	0	0
$586$	3.75768e15	2.24638
$587$	3.11508e15	1.84484	0.922421	−	0.386186i	$-0.126208\pi$
$587$	3.11508e15	1.84484	0.922421	+	0.386186i	$0.126208\pi$
$588$	−1.18348e15	−0.694360
$589$	9.93860e14	0.577685
$590$	0	0
$591$	−1.54496e15	−0.881425
$592$	−2.54547e15	−1.43879
$593$	2.20723e15	1.23608	0.618040	−	0.786146i	$-0.287927\pi$
$593$	2.20723e15	1.23608	0.618040	+	0.786146i	$0.287927\pi$
$594$	−7.13875e14	−0.396093
$595$	0	0
$596$	−5.58817e15	−3.04379
$597$	−9.19885e14	−0.496449
$598$	−9.15138e14	−0.489362
$599$	3.41159e15	1.80763	0.903813	−	0.427927i	$-0.140756\pi$
$599$	3.41159e15	1.80763	0.903813	+	0.427927i	$0.140756\pi$
$600$	0	0
$601$	−1.57545e15	−0.819588	−0.409794	−	0.912178i	$-0.634399\pi$
$601$	−1.57545e15	−0.819588	−0.409794	+	0.912178i	$0.634399\pi$
$602$	−2.07422e15	−1.06924
$603$	−3.12511e14	−0.159632
$604$	−2.81956e15	−1.42718
$605$	0	0
$606$	−1.76477e14	−0.0877178
$607$	5.05592e14	0.249036	0.124518	−	0.992217i	$-0.460262\pi$
$607$	5.05592e14	0.249036	0.124518	+	0.992217i	$0.460262\pi$
$608$	−3.68541e14	−0.179894
$609$	7.25246e13	0.0350825
$610$	0	0
$611$	−1.19203e15	−0.566319
$612$	−7.35069e14	−0.346095
$613$	1.98375e15	0.925665	0.462832	−	0.886446i	$-0.346833\pi$
$613$	1.98375e15	0.925665	0.462832	+	0.886446i	$0.346833\pi$
$614$	2.00733e15	0.928310
$615$	0	0
$616$	−2.74561e15	−1.24723
$617$	1.93099e15	0.869382	0.434691	−	0.900580i	$-0.356858\pi$
$617$	1.93099e15	0.869382	0.434691	+	0.900580i	$0.356858\pi$
$618$	9.20692e14	0.410845
$619$	9.47689e14	0.419148	0.209574	−	0.977793i	$-0.432792\pi$
$619$	9.47689e14	0.419148	0.209574	+	0.977793i	$0.432792\pi$
$620$	0	0
$621$	−2.19716e14	−0.0954679
$622$	−2.69913e15	−1.16246
$623$	−7.07124e14	−0.301864
$624$	−7.12971e14	−0.301687
$625$	0	0
$626$	9.83314e14	0.408821
$627$	−3.02415e15	−1.24633
$628$	−8.60181e15	−3.51408
$629$	2.05030e15	0.830306
$630$	0	0
$631$	1.00593e15	0.400318	0.200159	−	0.979763i	$-0.435854\pi$
$631$	1.00593e15	0.400318	0.200159	+	0.979763i	$0.435854\pi$
$632$	−5.76922e14	−0.227601
$633$	4.36170e13	0.0170583
$634$	−6.38230e15	−2.47449
$635$	0	0
$636$	−3.71941e15	−1.41729
$637$	9.24597e14	0.349290
$638$	−5.34888e14	−0.200332
$639$	−8.75761e14	−0.325185
$640$	0	0
$641$	1.13144e15	0.412966	0.206483	−	0.978450i	$-0.433798\pi$
$641$	1.13144e15	0.412966	0.206483	+	0.978450i	$0.433798\pi$
$642$	4.27594e15	1.54735
$643$	8.68306e14	0.311539	0.155769	−	0.987793i	$-0.450214\pi$
$643$	8.68306e14	0.311539	0.155769	+	0.987793i	$0.450214\pi$
$644$	−1.71559e15	−0.610296
$645$	0	0
$646$	−4.69405e15	−1.64160
$647$	1.37636e15	0.477265	0.238633	−	0.971110i	$-0.423301\pi$
$647$	1.37636e15	0.477265	0.238633	+	0.971110i	$0.423301\pi$
$648$	−5.40675e14	−0.185897
$649$	3.54195e14	0.120753
$650$	0	0
$651$	−3.43607e14	−0.115177
$652$	6.60146e14	0.219421
$653$	−2.55276e15	−0.841370	−0.420685	−	0.907207i	$-0.638210\pi$
$653$	−2.55276e15	−0.841370	−0.420685	+	0.907207i	$0.638210\pi$
$654$	1.31638e15	0.430232
$655$	0	0
$656$	3.44187e15	1.10617
$657$	−8.24974e14	−0.262924
$658$	−3.36862e15	−1.06466
$659$	9.26792e14	0.290478	0.145239	−	0.989397i	$-0.453605\pi$
$659$	9.26792e14	0.290478	0.145239	+	0.989397i	$0.453605\pi$
$660$	0	0
$661$	−1.90332e15	−0.586683	−0.293342	−	0.956008i	$-0.594767\pi$
$661$	−1.90332e15	−0.586683	−0.293342	+	0.956008i	$0.594767\pi$
$662$	2.65289e15	0.810963
$663$	5.74276e14	0.174099
$664$	1.35967e15	0.408799
$665$	0	0
$666$	3.06168e15	0.905422
$667$	−1.64627e14	−0.0482847
$668$	3.55548e13	0.0103426
$669$	−5.40841e14	−0.156037
$670$	0	0
$671$	3.15756e15	0.896143
$672$	1.27416e14	0.0358667
$673$	−4.92990e15	−1.37643	−0.688217	−	0.725505i	$-0.741606\pi$
$673$	−4.92990e15	−1.37643	−0.688217	+	0.725505i	$0.741606\pi$
$674$	−4.67563e15	−1.29482
$675$	0	0
$676$	−4.86369e15	−1.32513
$677$	−4.30293e14	−0.116286	−0.0581429	−	0.998308i	$-0.518518\pi$
$677$	−4.30293e14	−0.116286	−0.0581429	+	0.998308i	$0.518518\pi$
$678$	−6.81708e15	−1.82741
$679$	1.08521e15	0.288555
$680$	0	0
$681$	1.01051e14	0.0264382
$682$	2.53420e15	0.657698
$683$	3.81244e15	0.981498	0.490749	−	0.871301i	$-0.336723\pi$
$683$	3.81244e15	0.981498	0.490749	+	0.871301i	$0.336723\pi$
$684$	−4.64999e15	−1.18753
$685$	0	0
$686$	6.89433e15	1.73264
$687$	4.40432e15	1.09804
$688$	3.66823e15	0.907236
$689$	2.90580e15	0.712952
$690$	0	0
$691$	4.03729e15	0.974901	0.487450	−	0.873151i	$-0.337927\pi$
$691$	4.03729e15	0.974901	0.487450	+	0.873151i	$0.337927\pi$
$692$	2.94972e15	0.706638
$693$	1.04554e15	0.248490
$694$	7.63374e15	1.79995
$695$	0	0
$696$	−4.05113e14	−0.0940212
$697$	−2.77232e15	−0.638356
$698$	1.11376e16	2.54441
$699$	4.55967e12	0.00103350
$700$	0	0
$701$	−4.71267e15	−1.05152	−0.525761	−	0.850632i	$-0.676219\pi$
$701$	−4.71267e15	−1.05152	−0.525761	+	0.850632i	$0.676219\pi$
$702$	8.57558e14	0.189849
$703$	1.29700e16	2.84896
$704$	−5.94185e15	−1.29501
$705$	0	0
$706$	1.12512e16	2.41419
$707$	2.58467e14	0.0550299
$708$	5.44616e14	0.115056
$709$	8.66706e14	0.181684	0.0908422	−	0.995865i	$-0.471044\pi$
$709$	8.66706e14	0.181684	0.0908422	+	0.995865i	$0.471044\pi$
$710$	0	0
$711$	2.19694e14	0.0453457
$712$	3.94991e15	0.808997
$713$	7.79972e14	0.158521
$714$	1.62287e15	0.327299
$715$	0	0
$716$	1.58361e16	3.14505
$717$	−4.28293e15	−0.844085
$718$	9.69924e15	1.89694
$719$	2.62666e13	0.00509795	0.00254897	−	0.999997i	$-0.499189\pi$
$719$	2.62666e13	0.00509795	0.00254897	+	0.999997i	$0.499189\pi$
$720$	0	0
$721$	−1.34844e15	−0.257744
$722$	−2.06080e16	−3.90913
$723$	−2.16298e14	−0.0407186
$724$	9.19929e15	1.71867
$725$	0	0
$726$	−2.30335e15	−0.423846
$727$	5.68018e15	1.03734	0.518672	−	0.854973i	$-0.326427\pi$
$727$	5.68018e15	1.03734	0.518672	+	0.854973i	$0.326427\pi$
$728$	3.29823e15	0.597803
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−2.95465e15	−0.523552
$732$	4.85512e15	0.853864
$733$	1.62312e15	0.283320	0.141660	−	0.989915i	$-0.454756\pi$
$733$	1.62312e15	0.283320	0.141660	+	0.989915i	$0.454756\pi$
$734$	−1.24885e16	−2.16363
$735$	0	0
$736$	−2.89227e14	−0.0493641
$737$	−3.37568e15	−0.571861
$738$	−4.13987e15	−0.696108
$739$	3.64794e15	0.608840	0.304420	−	0.952538i	$-0.401537\pi$
$739$	3.64794e15	0.608840	0.304420	+	0.952538i	$0.401537\pi$
$740$	0	0
$741$	3.63283e15	0.597372
$742$	8.21164e15	1.34032
$743$	−6.76045e15	−1.09531	−0.547655	−	0.836704i	$-0.684479\pi$
$743$	−6.76045e15	−1.09531	−0.547655	+	0.836704i	$0.684479\pi$
$744$	1.91935e15	0.308676
$745$	0	0
$746$	−2.70861e15	−0.429224
$747$	−5.17768e14	−0.0814465
$748$	−7.94007e15	−1.23984
$749$	−6.26254e15	−0.970734
$750$	0	0
$751$	2.64833e13	0.00404532	0.00202266	−	0.999998i	$-0.499356\pi$
$751$	2.64833e13	0.00404532	0.00202266	+	0.999998i	$0.499356\pi$
$752$	5.95734e15	0.903347
$753$	−5.88741e15	−0.886242
$754$	6.42546e14	0.0960200
$755$	0	0
$756$	1.60764e15	0.236766
$757$	−7.37364e15	−1.07809	−0.539045	−	0.842277i	$-0.681215\pi$
$757$	−7.37364e15	−1.07809	−0.539045	+	0.842277i	$0.681215\pi$
$758$	1.14270e16	1.65864
$759$	−2.37333e15	−0.342001
$760$	0	0
$761$	6.20983e15	0.881991	0.440996	−	0.897509i	$-0.354625\pi$
$761$	6.20983e15	0.881991	0.440996	+	0.897509i	$0.354625\pi$
$762$	4.74368e15	0.668902
$763$	−1.92796e15	−0.269907
$764$	1.36703e16	1.90005
$765$	0	0
$766$	5.01866e15	0.687591
$767$	−4.25484e14	−0.0578774
$768$	−8.40308e15	−1.13488
$769$	−8.86195e15	−1.18832	−0.594162	−	0.804346i	$-0.702516\pi$
$769$	−8.86195e15	−1.18832	−0.594162	+	0.804346i	$0.702516\pi$
$770$	0	0
$771$	−1.89659e15	−0.250712
$772$	1.98943e16	2.61116
$773$	2.95047e15	0.384507	0.192254	−	0.981345i	$-0.438420\pi$
$773$	2.95047e15	0.384507	0.192254	+	0.981345i	$0.438420\pi$
$774$	−4.41213e15	−0.570918
$775$	0	0
$776$	−6.06184e15	−0.773330
$777$	−4.48413e15	−0.568019
$778$	3.11142e15	0.391355
$779$	−1.75375e16	−2.19034
$780$	0	0
$781$	−9.45980e15	−1.16493
$782$	−3.68384e15	−0.450468
$783$	1.54269e14	0.0187322
$784$	−4.62081e15	−0.557160
$785$	0	0
$786$	1.85544e15	0.220609
$787$	−1.35545e16	−1.60038	−0.800190	−	0.599746i	$-0.795268\pi$
$787$	−1.35545e16	−1.60038	−0.800190	+	0.599746i	$0.795268\pi$
$788$	−2.56603e16	−3.00862
$789$	−4.03272e15	−0.469543
$790$	0	0
$791$	9.98429e15	1.14643
$792$	−5.84027e15	−0.665954
$793$	−3.79308e15	−0.429526
$794$	8.31105e15	0.934638
$795$	0	0
$796$	−1.52784e16	−1.69456
$797$	1.59922e16	1.76152	0.880762	−	0.473558i	$-0.157031\pi$
$797$	1.59922e16	1.76152	0.880762	+	0.473558i	$0.157031\pi$
$798$	1.02662e16	1.12303
$799$	−4.79846e15	−0.521308
$800$	0	0
$801$	−1.50414e15	−0.161180
$802$	−2.66327e15	−0.283437
$803$	−8.91121e15	−0.941892
$804$	−5.19051e15	−0.544881
$805$	0	0
$806$	−3.04426e15	−0.315238
$807$	6.93123e15	0.712863
$808$	−1.44377e15	−0.147481
$809$	9.45859e15	0.959643	0.479821	−	0.877366i	$-0.340701\pi$
$809$	9.45859e15	0.959643	0.479821	+	0.877366i	$0.340701\pi$
$810$	0	0
$811$	−1.07996e16	−1.08092	−0.540461	−	0.841369i	$-0.681750\pi$
$811$	−1.07996e16	−1.08092	−0.540461	+	0.841369i	$0.681750\pi$
$812$	1.20456e15	0.119749
$813$	5.68134e15	0.560987
$814$	3.30717e16	3.24356
$815$	0	0
$816$	−2.87003e15	−0.277709
$817$	−1.86909e16	−1.79642
$818$	−4.16012e15	−0.397158
$819$	−1.25598e15	−0.119102
$820$	0	0
$821$	−1.10411e16	−1.03306	−0.516529	−	0.856270i	$-0.672776\pi$
$821$	−1.10411e16	−1.03306	−0.516529	+	0.856270i	$0.672776\pi$
$822$	−2.93815e15	−0.273073
$823$	−1.26104e16	−1.16420	−0.582101	−	0.813117i	$-0.697769\pi$
$823$	−1.26104e16	−1.16420	−0.582101	+	0.813117i	$0.697769\pi$
$824$	7.53224e15	0.690756
$825$	0	0
$826$	−1.20239e15	−0.108807
$827$	7.86172e14	0.0706704	0.0353352	−	0.999376i	$-0.488750\pi$
$827$	7.86172e14	0.0706704	0.0353352	+	0.999376i	$0.488750\pi$
$828$	−3.64927e15	−0.325866
$829$	−5.92589e15	−0.525659	−0.262829	−	0.964842i	$-0.584656\pi$
$829$	−5.92589e15	−0.525659	−0.262829	+	0.964842i	$0.584656\pi$
$830$	0	0
$831$	7.37848e15	0.645894
$832$	7.13777e15	0.620704
$833$	3.72192e15	0.321528
$834$	2.13472e16	1.83201
$835$	0	0
$836$	−5.02283e16	−4.25416
$837$	−7.30896e14	−0.0614986
$838$	7.90044e15	0.660403
$839$	2.10209e16	1.74566	0.872831	−	0.488022i	$-0.162282\pi$
$839$	2.10209e16	1.74566	0.872831	+	0.488022i	$0.162282\pi$
$840$	0	0
$841$	−1.20849e16	−0.990526
$842$	1.23184e16	1.00309
$843$	3.88189e15	0.314045
$844$	7.24437e14	0.0582260
$845$	0	0
$846$	−7.16547e15	−0.568470
$847$	3.37348e15	0.265901
$848$	−1.45222e16	−1.13725
$849$	−6.97520e15	−0.542705
$850$	0	0
$851$	1.01788e16	0.781775
$852$	−1.45456e16	−1.10997
$853$	1.98378e16	1.50409	0.752046	−	0.659110i	$-0.229067\pi$
$853$	1.98378e16	1.50409	0.752046	+	0.659110i	$0.229067\pi$
$854$	−1.07190e16	−0.807491
$855$	0	0
$856$	3.49818e16	2.60157
$857$	−7.96639e15	−0.588663	−0.294332	−	0.955703i	$-0.595097\pi$
$857$	−7.96639e15	−0.588663	−0.294332	+	0.955703i	$0.595097\pi$
$858$	9.26318e15	0.680111
$859$	−1.59749e16	−1.16540	−0.582701	−	0.812687i	$-0.698004\pi$
$859$	−1.59749e16	−1.16540	−0.582701	+	0.812687i	$0.698004\pi$
$860$	0	0
$861$	6.06324e15	0.436704
$862$	−4.00274e16	−2.86462
$863$	4.02926e15	0.286527	0.143264	−	0.989685i	$-0.454240\pi$
$863$	4.02926e15	0.286527	0.143264	+	0.989685i	$0.454240\pi$
$864$	2.71029e14	0.0191509
$865$	0	0
$866$	−5.84413e15	−0.407729
$867$	−6.01635e15	−0.417089
$868$	−5.70699e15	−0.393142
$869$	2.37310e15	0.162445
$870$	0	0
$871$	4.05511e15	0.274096
$872$	1.07694e16	0.723352
$873$	2.30837e15	0.154073
$874$	−2.33037e16	−1.54565
$875$	0	0
$876$	−1.37020e16	−0.897455
$877$	−9.76958e15	−0.635884	−0.317942	−	0.948110i	$-0.602992\pi$
$877$	−9.76958e15	−0.635884	−0.317942	+	0.948110i	$0.602992\pi$
$878$	2.87301e16	1.85831
$879$	−1.17066e16	−0.752476
$880$	0	0
$881$	1.62583e16	1.03207	0.516033	−	0.856569i	$-0.327408\pi$
$881$	1.62583e16	1.03207	0.516033	+	0.856569i	$0.327408\pi$
$882$	5.55789e15	0.350617
$883$	−1.66061e16	−1.04108	−0.520540	−	0.853837i	$-0.674269\pi$
$883$	−1.66061e16	−1.04108	−0.520540	+	0.853837i	$0.674269\pi$
$884$	9.53818e15	0.594262
$885$	0	0
$886$	−8.67733e15	−0.533950
$887$	1.91314e16	1.16995	0.584975	−	0.811051i	$-0.301104\pi$
$887$	1.91314e16	1.16995	0.584975	+	0.811051i	$0.301104\pi$
$888$	2.50478e16	1.52229
$889$	−6.94758e15	−0.419637
$890$	0	0
$891$	2.22400e15	0.132680
$892$	−8.98286e15	−0.532608
$893$	−3.03547e16	−1.78872
$894$	2.62433e16	1.53696
$895$	0	0
$896$	1.90971e16	1.10477
$897$	2.85101e15	0.163923
$898$	6.88985e15	0.393722
$899$	−5.47641e14	−0.0311041
$900$	0	0
$901$	1.16972e16	0.656287
$902$	−4.47180e16	−2.49372
$903$	6.46201e15	0.358166
$904$	−5.57710e16	−3.07243
$905$	0	0
$906$	1.32413e16	0.720651
$907$	1.07252e16	0.580182	0.290091	−	0.956999i	$-0.406314\pi$
$907$	1.07252e16	0.580182	0.290091	+	0.956999i	$0.406314\pi$
$908$	1.67836e15	0.0902430
$909$	5.49792e14	0.0293831
$910$	0	0
$911$	−2.82249e16	−1.49032	−0.745162	−	0.666883i	$-0.767628\pi$
$911$	−2.82249e16	−1.49032	−0.745162	+	0.666883i	$0.767628\pi$
$912$	−1.81556e16	−0.952881
$913$	−5.59284e15	−0.291771
$914$	−7.34833e14	−0.0381052
$915$	0	0
$916$	7.31516e16	3.74799
$917$	−2.71748e15	−0.138399
$918$	3.45205e15	0.174760
$919$	1.83551e16	0.923679	0.461839	−	0.886964i	$-0.347190\pi$
$919$	1.83551e16	0.923679	0.461839	+	0.886964i	$0.347190\pi$
$920$	0	0
$921$	−6.25361e15	−0.310959
$922$	5.43764e16	2.68776
$923$	1.13638e16	0.558359
$924$	1.73654e16	0.848184
$925$	0	0
$926$	−1.46594e16	−0.707549
$927$	−2.86831e15	−0.137622
$928$	2.03075e14	0.00968596
$929$	−1.10125e16	−0.522155	−0.261078	−	0.965318i	$-0.584078\pi$
$929$	−1.10125e16	−0.522155	−0.261078	+	0.965318i	$0.584078\pi$
$930$	0	0
$931$	2.35446e16	1.10323
$932$	7.57317e13	0.00352769
$933$	8.40884e15	0.389392
$934$	2.49605e16	1.14907
$935$	0	0
$936$	7.01574e15	0.319195
$937$	−8.36357e15	−0.378289	−0.189145	−	0.981949i	$-0.560571\pi$
$937$	−8.36357e15	−0.378289	−0.189145	+	0.981949i	$0.560571\pi$
$938$	1.14595e16	0.515289
$939$	−3.06340e15	−0.136944
$940$	0	0
$941$	2.22942e16	0.985028	0.492514	−	0.870305i	$-0.336078\pi$
$941$	2.22942e16	0.985028	0.492514	+	0.870305i	$0.336078\pi$
$942$	4.03961e16	1.77443
$943$	−1.37633e16	−0.601045
$944$	2.12642e15	0.0923214
$945$	0	0
$946$	−4.76590e16	−2.04524
$947$	3.78038e16	1.61291	0.806457	−	0.591293i	$-0.201382\pi$
$947$	3.78038e16	1.61291	0.806457	+	0.591293i	$0.201382\pi$
$948$	3.64891e15	0.154781
$949$	1.07048e16	0.451454
$950$	0	0
$951$	1.98833e16	0.828888
$952$	1.32768e16	0.550290
$953$	−4.79568e15	−0.197624	−0.0988119	−	0.995106i	$-0.531504\pi$
$953$	−4.79568e15	−0.197624	−0.0988119	+	0.995106i	$0.531504\pi$
$954$	1.74672e16	0.715661
$955$	0	0
$956$	−7.11355e16	−2.88116
$957$	1.66638e15	0.0671057
$958$	1.17298e16	0.469657
$959$	4.30321e15	0.171313
$960$	0	0
$961$	−2.28139e16	−0.897884
$962$	−3.97280e16	−1.55466
$963$	−1.33212e16	−0.518321
$964$	−3.59251e15	−0.138987
$965$	0	0
$966$	8.05680e15	0.308169
$967$	−3.37420e16	−1.28329	−0.641645	−	0.767002i	$-0.721748\pi$
$967$	−3.37420e16	−1.28329	−0.641645	+	0.767002i	$0.721748\pi$
$968$	−1.88439e16	−0.712616
$969$	1.46238e16	0.549893
$970$	0	0
$971$	3.58587e16	1.33318	0.666590	−	0.745425i	$-0.267753\pi$
$971$	3.58587e16	1.33318	0.666590	+	0.745425i	$0.267753\pi$
$972$	3.41966e15	0.126421
$973$	−3.12651e16	−1.14932
$974$	1.37706e16	0.503360
$975$	0	0
$976$	1.89565e16	0.685147
$977$	1.20023e16	0.431366	0.215683	−	0.976463i	$-0.430802\pi$
$977$	1.20023e16	0.431366	0.215683	+	0.976463i	$0.430802\pi$
$978$	−3.10020e15	−0.110797
$979$	−1.62474e16	−0.577405
$980$	0	0
$981$	−4.10102e15	−0.144116
$982$	6.71547e16	2.34673
$983$	4.42687e15	0.153834	0.0769170	−	0.997038i	$-0.475492\pi$
$983$	4.42687e15	0.153834	0.0769170	+	0.997038i	$0.475492\pi$
$984$	−3.38685e16	−1.17037
$985$	0	0
$986$	2.58653e15	0.0883884
$987$	1.04945e16	0.356631
$988$	6.03378e16	2.03904
$989$	−1.46684e16	−0.492951
$990$	0	0
$991$	−3.79167e16	−1.26016	−0.630080	−	0.776530i	$-0.716978\pi$
$991$	−3.79167e16	−1.26016	−0.630080	+	0.776530i	$0.716978\pi$
$992$	−9.62130e14	−0.0317994
$993$	−8.26479e15	−0.271651
$994$	3.21135e16	1.04969
$995$	0	0
$996$	−8.59964e15	−0.278006
$997$	−5.76003e13	−0.00185183	−0.000925915	−	1.00000i	$-0.500295\pi$
$997$	−5.76003e13	−0.00185183	−0.000925915		1.00000i	$0.500295\pi$
$998$	−4.68943e16	−1.49935
$999$	−9.53830e15	−0.303292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.12.a.a.1.1		1
3.2	odd	2		225.12.a.f.1.1		1
5.2	odd	4		75.12.b.a.49.1		2
5.3	odd	4		75.12.b.a.49.2		2
5.4	even	2		3.12.a.a.1.1	✓	1
15.2	even	4		225.12.b.a.199.2		2
15.8	even	4		225.12.b.a.199.1		2
15.14	odd	2		9.12.a.a.1.1		1
20.19	odd	2		48.12.a.f.1.1		1
35.34	odd	2		147.12.a.c.1.1		1
40.19	odd	2		192.12.a.g.1.1		1
40.29	even	2		192.12.a.q.1.1		1
45.4	even	6		81.12.c.a.55.1		2
45.14	odd	6		81.12.c.e.55.1		2
45.29	odd	6		81.12.c.e.28.1		2
45.34	even	6		81.12.c.a.28.1		2
60.59	even	2		144.12.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.12.a.a.1.1	✓	1	5.4	even	2
9.12.a.a.1.1		1	15.14	odd	2
48.12.a.f.1.1		1	20.19	odd	2
75.12.a.a.1.1		1	1.1	even	1	trivial
75.12.b.a.49.1		2	5.2	odd	4
75.12.b.a.49.2		2	5.3	odd	4
81.12.c.a.28.1		2	45.34	even	6
81.12.c.a.55.1		2	45.4	even	6
81.12.c.e.28.1		2	45.29	odd	6
81.12.c.e.55.1		2	45.14	odd	6
144.12.a.l.1.1		1	60.59	even	2
147.12.a.c.1.1		1	35.34	odd	2
192.12.a.g.1.1		1	40.19	odd	2
192.12.a.q.1.1		1	40.29	even	2
225.12.a.f.1.1		1	3.2	odd	2
225.12.b.a.199.1		2	15.8	even	4
225.12.b.a.199.2		2	15.2	even	4

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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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