LMFDB - Embedded newform 75.14.b.g.49.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,14,Mod(49,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.49"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	75.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-41504] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$80.4231967139$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3303x^{6} + 3427971x^{4} + 1167506350x^{2} + 119802515625 $ x^8 + 3303x^6 + 3427971x^4 + 1167506350*x^2 + 119802515625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{2}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.7
Root			$38.8667i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.14.b.g.49.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+143.467i q^{2} +729.000i q^{3} -12390.7 q^{4} -104587. q^{6} +115746. i q^{7} -602380. i q^{8} -531441. q^{9} -1.68912e6 q^{11} -9.03285e6i q^{12} +8.38266e6i q^{13} -1.66058e7 q^{14} -1.50834e7 q^{16} -1.18185e8i q^{17} -7.62442e7i q^{18} -2.93887e6 q^{19} -8.43791e7 q^{21} -2.42333e8i q^{22} -1.14328e9i q^{23} +4.39135e8 q^{24} -1.20263e9 q^{26} -3.87420e8i q^{27} -1.43418e9i q^{28} +2.58117e9 q^{29} -5.28510e9 q^{31} -7.09866e9i q^{32} -1.23137e9i q^{33} +1.69556e10 q^{34} +6.58495e9 q^{36} -2.25904e9i q^{37} -4.21631e8i q^{38} -6.11096e9 q^{39} +3.08318e10 q^{41} -1.21056e10i q^{42} +1.84280e10i q^{43} +2.09294e10 q^{44} +1.64023e11 q^{46} -8.91497e10i q^{47} -1.09958e10i q^{48} +8.34918e10 q^{49} +8.61570e10 q^{51} -1.03867e11i q^{52} +2.29652e11i q^{53} +5.55820e10 q^{54} +6.97232e10 q^{56} -2.14244e9i q^{57} +3.70312e11i q^{58} +1.48583e11 q^{59} -8.88553e10 q^{61} -7.58237e11i q^{62} -6.15124e10i q^{63} +8.94860e11 q^{64} +1.76661e11 q^{66} -1.15668e12i q^{67} +1.46440e12i q^{68} +8.33451e11 q^{69} -1.39371e12 q^{71} +3.20129e11i q^{72} -2.23868e12i q^{73} +3.24097e11 q^{74} +3.64148e10 q^{76} -1.95509e11i q^{77} -8.76720e11i q^{78} -4.29509e11 q^{79} +2.82430e11 q^{81} +4.42335e12i q^{82} +4.77368e12i q^{83} +1.04552e12 q^{84} -2.64380e12 q^{86} +1.88167e12i q^{87} +1.01749e12i q^{88} -4.52557e12 q^{89} -9.70262e11 q^{91} +1.41661e13i q^{92} -3.85284e12i q^{93} +1.27900e13 q^{94} +5.17493e12 q^{96} +1.54548e13i q^{97} +1.19783e13i q^{98} +8.97667e11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 41504 q^{4} + 75816 q^{6} - 4251528 q^{9} - 11748208 q^{11} + 128644584 q^{14} + 12392960 q^{16} + 407861624 q^{19} - 435936168 q^{21} - 618098688 q^{24} - 7083351544 q^{26} + 17010752528 q^{29} - 4110176552 q^{31}+ \cdots + 6243479407728 q^{99}+O(q^{100}) $ 8 * q - 41504 * q^4 + 75816 * q^6 - 4251528 * q^9 - 11748208 * q^11 + 128644584 * q^14 + 12392960 * q^16 + 407861624 * q^19 - 435936168 * q^21 - 618098688 * q^24 - 7083351544 * q^26 + 17010752528 * q^29 - 4110176552 * q^31 + 11940660912 * q^34 + 22056927264 * q^36 + 46721569176 * q^39 + 67258917088 * q^41 + 14163760448 * q^44 + 271062871248 * q^46 - 31472829840 * q^49 + 98282858544 * q^51 - 40291730856 * q^54 - 1219622814720 * q^56 - 723761174864 * q^59 - 721947748520 * q^61 + 1750438395904 * q^64 - 62193066192 * q^66 + 841332496464 * q^69 - 7767788474144 * q^71 + 4692855762224 * q^74 - 8686082155616 * q^76 + 12081014646080 * q^79 + 2259436291848 * q^81 + 17223728111136 * q^84 + 7312090231832 * q^86 + 8511047028864 * q^89 + 30847013172728 * q^91 + 80024185437456 * q^94 + 1701377851392 * q^96 + 6243479407728 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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 $

$2$	143.467i	1.58510i	0.609807	+	0.792550i	$0.291247\pi$
$2$	143.467i	1.58510i	−0.609807	+	0.792550i	$0.708753\pi$
$3$	729.000i	0.577350i
$3$	729.000i	0.577350i
$4$	−12390.7	−1.51254
$5$	0	0
$5$	0	0
$6$	−104587.	−0.915158
$7$	115746.i	0.371852i	0.982564	+	0.185926i	$0.0595285\pi$
$7$	115746.i	0.371852i	−0.982564	+	0.185926i	$0.940472\pi$
$8$	− 602380.i	− 0.812429i
$9$	−531441.	−0.333333
$10$	0	0
$11$	−1.68912e6	−0.287480	−0.143740	−	0.989615i	$-0.545913\pi$
$11$	−1.68912e6	−0.287480	−0.143740	+	0.989615i	$0.545913\pi$
$12$	− 9.03285e6i	− 0.873266i
$13$	8.38266e6i	0.481670i	0.970566	+	0.240835i	$0.0774213\pi$
$13$	8.38266e6i	0.481670i	−0.970566	+	0.240835i	$0.922579\pi$
$14$	−1.66058e7	−0.589422
$15$	0	0
$16$	−1.50834e7	−0.224760
$17$	− 1.18185e8i	− 1.18753i	−0.804638	−	0.593765i	$-0.797641\pi$
$17$	− 1.18185e8i	− 1.18753i	0.804638	−	0.593765i	$-0.202359\pi$
$18$	− 7.62442e7i	− 0.528367i
$19$	−2.93887e6	−0.0143312	−0.00716560	−	0.999974i	$-0.502281\pi$
$19$	−2.93887e6	−0.0143312	−0.00716560	+	0.999974i	$0.502281\pi$
$20$	0	0
$21$	−8.43791e7	−0.214689
$22$	− 2.42333e8i	− 0.455685i
$23$	− 1.14328e9i	− 1.61036i	−0.593033	−	0.805178i	$-0.702070\pi$
$23$	− 1.14328e9i	− 1.61036i	0.593033	−	0.805178i	$-0.297930\pi$
$24$	4.39135e8	0.469056
$25$	0	0
$26$	−1.20263e9	−0.763496
$27$	− 3.87420e8i	− 0.192450i
$28$	− 1.43418e9i	− 0.562441i
$29$	2.58117e9	0.805804	0.402902	−	0.915243i	$-0.368002\pi$
$29$	2.58117e9	0.805804	0.402902	+	0.915243i	$0.368002\pi$
$30$	0	0
$31$	−5.28510e9	−1.06955	−0.534776	−	0.844994i	$-0.679604\pi$
$31$	−5.28510e9	−1.06955	−0.534776	+	0.844994i	$0.679604\pi$
$32$	− 7.09866e9i	− 1.16870i
$33$	− 1.23137e9i	− 0.165977i
$34$	1.69556e10	1.88235
$35$	0	0
$36$	6.58495e9	0.504180
$37$	− 2.25904e9i	− 0.144748i	−0.997378	−	0.0723740i	$-0.976942\pi$
$37$	− 2.25904e9i	− 0.144748i	0.997378	−	0.0723740i	$-0.0230575\pi$
$38$	− 4.21631e8i	− 0.0227164i
$39$	−6.11096e9	−0.278093
$40$	0	0
$41$	3.08318e10	1.01369	0.506844	−	0.862038i	$-0.330812\pi$
$41$	3.08318e10	1.01369	0.506844	+	0.862038i	$0.330812\pi$
$42$	− 1.21056e10i	− 0.340303i
$43$	1.84280e10i	0.444562i	0.974983	+	0.222281i	$0.0713503\pi$
$43$	1.84280e10i	0.444562i	−0.974983	+	0.222281i	$0.928650\pi$
$44$	2.09294e10	0.434826
$45$	0	0
$46$	1.64023e11	2.55257
$47$	− 8.91497e10i	− 1.20638i	−0.797598	−	0.603190i	$-0.793896\pi$
$47$	− 8.91497e10i	− 1.20638i	0.797598	−	0.603190i	$-0.206104\pi$
$48$	− 1.09958e10i	− 0.129765i
$49$	8.34918e10	0.861726
$50$	0	0
$51$	8.61570e10	0.685621
$52$	− 1.03867e11i	− 0.728546i
$53$	2.29652e11i	1.42324i	0.702567	+	0.711618i	$0.252037\pi$
$53$	2.29652e11i	1.42324i	−0.702567	+	0.711618i	$0.747963\pi$
$54$	5.55820e10	0.305053
$55$	0	0
$56$	6.97232e10	0.302103
$57$	− 2.14244e9i	− 0.00827412i
$58$	3.70312e11i	1.27728i
$59$	1.48583e11	0.458596	0.229298	−	0.973356i	$-0.426357\pi$
$59$	1.48583e11	0.458596	0.229298	+	0.973356i	$0.426357\pi$
$60$	0	0
$61$	−8.88553e10	−0.220821	−0.110410	−	0.993886i	$-0.535216\pi$
$61$	−8.88553e10	−0.220821	−0.110410	+	0.993886i	$0.535216\pi$
$62$	− 7.58237e11i	− 1.69535i
$63$	− 6.15124e10i	− 0.123951i
$64$	8.94860e11	1.62774
$65$	0	0
$66$	1.76661e11	0.263090
$67$	− 1.15668e12i	− 1.56217i	−0.624425	−	0.781085i	$-0.714667\pi$
$67$	− 1.15668e12i	− 1.56217i	0.624425	−	0.781085i	$-0.285333\pi$
$68$	1.46440e12i	1.79619i
$69$	8.33451e11	0.929739
$70$	0	0
$71$	−1.39371e12	−1.29120	−0.645601	−	0.763675i	$-0.723393\pi$
$71$	−1.39371e12	−1.29120	−0.645601	+	0.763675i	$0.723393\pi$
$72$	3.20129e11i	0.270810i
$73$	− 2.23868e12i	− 1.73138i	−0.500578	−	0.865691i	$-0.666879\pi$
$73$	− 2.23868e12i	− 1.73138i	0.500578	−	0.865691i	$-0.333121\pi$
$74$	3.24097e11	0.229440
$75$	0	0
$76$	3.64148e10	0.0216765
$77$	− 1.95509e11i	− 0.106900i
$78$	− 8.76720e11i	− 0.440804i
$79$	−4.29509e11	−0.198791	−0.0993953	−	0.995048i	$-0.531691\pi$
$79$	−4.29509e11	−0.198791	−0.0993953	+	0.995048i	$0.531691\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	4.42335e12i	1.60680i
$83$	4.77368e12i	1.60267i	0.598213	+	0.801337i	$0.295878\pi$
$83$	4.77368e12i	1.60267i	−0.598213	+	0.801337i	$0.704122\pi$
$84$	1.04552e12	0.324726
$85$	0	0
$86$	−2.64380e12	−0.704676
$87$	1.88167e12i	0.465231i
$88$	1.01749e12i	0.233557i
$89$	−4.52557e12	−0.965246	−0.482623	−	0.875828i	$-0.660316\pi$
$89$	−4.52557e12	−0.965246	−0.482623	+	0.875828i	$0.660316\pi$
$90$	0	0
$91$	−9.70262e11	−0.179110
$92$	1.41661e13i	2.43573i
$93$	− 3.85284e12i	− 0.617507i
$94$	1.27900e13	1.91223
$95$	0	0
$96$	5.17493e12	0.674747
$97$	1.54548e13i	1.88385i	0.335822	+	0.941925i	$0.390986\pi$
$97$	1.54548e13i	1.88385i	−0.335822	+	0.941925i	$0.609014\pi$
$98$	1.19783e13i	1.36592i
$99$	8.97667e11	0.0958267
$100$	0	0
$101$	−4.38671e12	−0.411197	−0.205599	−	0.978636i	$-0.565914\pi$
$101$	−4.38671e12	−0.411197	−0.205599	+	0.978636i	$0.565914\pi$
$102$	1.23607e13i	1.08678i
$103$	− 9.72298e12i	− 0.802338i	−0.916004	−	0.401169i	$-0.868604\pi$
$103$	− 9.72298e12i	− 0.802338i	0.916004	−	0.401169i	$-0.131396\pi$
$104$	5.04954e12	0.391323
$105$	0	0
$106$	−3.29474e13	−2.25597
$107$	7.70349e12i	0.496241i	0.968729	+	0.248121i	$0.0798129\pi$
$107$	7.70349e12i	0.496241i	−0.968729	+	0.248121i	$0.920187\pi$
$108$	4.80043e12i	0.291089i
$109$	1.04685e13	0.597877	0.298938	−	0.954272i	$-0.403367\pi$
$109$	1.04685e13	0.597877	0.298938	+	0.954272i	$0.403367\pi$
$110$	0	0
$111$	1.64684e12	0.0835703
$112$	− 1.74585e12i	− 0.0835776i
$113$	− 5.48831e12i	− 0.247987i	−0.992283	−	0.123993i	$-0.960430\pi$
$113$	− 5.48831e12i	− 0.247987i	0.992283	−	0.123993i	$-0.0395702\pi$
$114$	3.07369e11	0.0131153
$115$	0	0
$116$	−3.19826e13	−1.21881
$117$	− 4.45489e12i	− 0.160557i
$118$	2.13167e13i	0.726920i
$119$	1.36795e13	0.441586
$120$	0	0
$121$	−3.16696e13	−0.917355
$122$	− 1.27478e13i	− 0.350023i
$123$	2.24764e13i	0.585253i
$124$	6.54863e13	1.61774
$125$	0	0
$126$	8.82498e12	0.196474
$127$	− 3.76054e13i	− 0.795291i	−0.917539	−	0.397645i	$-0.869827\pi$
$127$	− 3.76054e13i	− 0.795291i	0.917539	−	0.397645i	$-0.130173\pi$
$128$	7.02304e13i	1.41143i
$129$	−1.34340e13	−0.256668
$130$	0	0
$131$	9.13894e13	1.57991	0.789955	−	0.613165i	$-0.210104\pi$
$131$	9.13894e13	1.57991	0.789955	+	0.613165i	$0.210104\pi$
$132$	1.52576e13i	0.251047i
$133$	− 3.40164e11i	− 0.00532909i
$134$	1.65946e14	2.47620
$135$	0	0
$136$	−7.11923e13	−0.964784
$137$	8.99178e13i	1.16188i	0.813946	+	0.580941i	$0.197315\pi$
$137$	8.99178e13i	1.16188i	−0.813946	+	0.580941i	$0.802685\pi$
$138$	1.19573e14i	1.47373i
$139$	3.27383e13	0.384999	0.192499	−	0.981297i	$-0.438341\pi$
$139$	3.27383e13	0.384999	0.192499	+	0.981297i	$0.438341\pi$
$140$	0	0
$141$	6.49901e13	0.696503
$142$	− 1.99952e14i	− 2.04668i
$143$	− 1.41593e13i	− 0.138471i
$144$	8.01595e12	0.0749201
$145$	0	0
$146$	3.21176e14	2.74441
$147$	6.08655e13i	0.497518i
$148$	2.79912e13i	0.218937i
$149$	5.86656e13	0.439211	0.219605	−	0.975589i	$-0.429523\pi$
$149$	5.86656e13	0.439211	0.219605	+	0.975589i	$0.429523\pi$
$150$	0	0
$151$	−9.17724e13	−0.630031	−0.315015	−	0.949087i	$-0.602010\pi$
$151$	−9.17724e13	−0.630031	−0.315015	+	0.949087i	$0.602010\pi$
$152$	1.77032e12i	0.0116431i
$153$	6.28084e13i	0.395844i
$154$	2.80491e13	0.169447
$155$	0	0
$156$	7.57193e13	0.420626
$157$	− 9.62156e13i	− 0.512741i	−0.966579	−	0.256370i	$-0.917473\pi$
$157$	− 9.62156e13i	− 0.512741i	0.966579	−	0.256370i	$-0.0825267\pi$
$158$	− 6.16202e13i	− 0.315103i
$159$	−1.67416e14	−0.821705
$160$	0	0
$161$	1.32331e14	0.598814
$162$	4.05193e13i	0.176122i
$163$	− 3.61390e14i	− 1.50923i	−0.656166	−	0.754617i	$-0.727823\pi$
$163$	− 3.61390e14i	− 1.50923i	0.656166	−	0.754617i	$-0.272177\pi$
$164$	−3.82029e14	−1.53324
$165$	0	0
$166$	−6.84864e14	−2.54040
$167$	1.39666e14i	0.498234i	0.968473	+	0.249117i	$0.0801404\pi$
$167$	1.39666e14i	0.498234i	−0.968473	+	0.249117i	$0.919860\pi$
$168$	5.08282e13i	0.174419i
$169$	2.32606e14	0.767994
$170$	0	0
$171$	1.56184e12	0.00477707
$172$	− 2.28336e14i	− 0.672419i
$173$	− 4.50349e14i	− 1.27717i	−0.769551	−	0.638586i	$-0.779520\pi$
$173$	− 4.50349e14i	− 1.27717i	0.769551	−	0.638586i	$-0.220480\pi$
$174$	−2.69958e14	−0.737438
$175$	0	0
$176$	2.54777e13	0.0646142
$177$	1.08317e14i	0.264770i
$178$	− 6.49269e14i	− 1.53001i
$179$	4.63543e13	0.105328	0.0526642	−	0.998612i	$-0.483229\pi$
$179$	4.63543e13	0.105328	0.0526642	+	0.998612i	$0.483229\pi$
$180$	0	0
$181$	−5.73683e14	−1.21272	−0.606361	−	0.795189i	$-0.707371\pi$
$181$	−5.73683e14	−1.21272	−0.606361	+	0.795189i	$0.707371\pi$
$182$	− 1.39200e14i	− 0.283907i
$183$	− 6.47755e13i	− 0.127491i
$184$	−6.88689e14	−1.30830
$185$	0	0
$186$	5.52755e14	0.978810
$187$	1.99629e14i	0.341392i
$188$	1.10463e15i	1.82470i
$189$	4.48425e13	0.0715629
$190$	0	0
$191$	−2.45056e14	−0.365214	−0.182607	−	0.983186i	$-0.558454\pi$
$191$	−2.45056e14	−0.365214	−0.182607	+	0.983186i	$0.558454\pi$
$192$	6.52353e14i	0.939776i
$193$	− 1.07891e15i	− 1.50267i	−0.659921	−	0.751335i	$-0.729410\pi$
$193$	− 1.07891e15i	− 1.50267i	0.659921	−	0.751335i	$-0.270590\pi$
$194$	−2.21725e15	−2.98609
$195$	0	0
$196$	−1.03452e15	−1.30340
$197$	5.37451e14i	0.655101i	0.944834	+	0.327550i	$0.106223\pi$
$197$	5.37451e14i	0.655101i	−0.944834	+	0.327550i	$0.893777\pi$
$198$	1.28786e14i	0.151895i
$199$	−4.43570e14	−0.506310	−0.253155	−	0.967426i	$-0.581468\pi$
$199$	−4.43570e14	−0.506310	−0.253155	+	0.967426i	$0.581468\pi$
$200$	0	0
$201$	8.43222e14	0.901919
$202$	− 6.29348e14i	− 0.651789i
$203$	2.98761e14i	0.299640i
$204$	−1.06755e15	−1.03703
$205$	0	0
$206$	1.39492e15	1.27179
$207$	6.07586e14i	0.536785i
$208$	− 1.26439e14i	− 0.108260i
$209$	4.96411e12	0.00411994
$210$	0	0
$211$	1.83905e15	1.43469	0.717345	−	0.696718i	$-0.245357\pi$
$211$	1.83905e15	1.43469	0.717345	+	0.696718i	$0.245357\pi$
$212$	− 2.84555e15i	− 2.15270i
$213$	− 1.01602e15i	− 0.745475i
$214$	−1.10519e15	−0.786592
$215$	0	0
$216$	−2.33374e14	−0.156352
$217$	− 6.11731e14i	− 0.397715i
$218$	1.50188e15i	0.947695i
$219$	1.63200e15	0.999614
$220$	0	0
$221$	9.90706e14	0.571999
$222$	2.36267e14i	0.132467i
$223$	− 2.71677e15i	− 1.47935i	−0.672963	−	0.739676i	$-0.734979\pi$
$223$	− 2.71677e15i	− 1.47935i	0.672963	−	0.739676i	$-0.265021\pi$
$224$	8.21644e14	0.434582
$225$	0	0
$226$	7.87390e14	0.393084
$227$	− 1.47916e15i	− 0.717542i	−0.933426	−	0.358771i	$-0.883196\pi$
$227$	− 1.47916e15i	− 0.717542i	0.933426	−	0.358771i	$-0.116804\pi$
$228$	2.65464e13i	0.0125150i
$229$	3.13274e15	1.43547	0.717735	−	0.696316i	$-0.245179\pi$
$229$	3.13274e15	1.43547	0.717735	+	0.696316i	$0.245179\pi$
$230$	0	0
$231$	1.42526e14	0.0617188
$232$	− 1.55484e15i	− 0.654658i
$233$	− 2.51709e14i	− 0.103059i	−0.998671	−	0.0515294i	$-0.983590\pi$
$233$	− 2.51709e14i	− 0.103059i	0.998671	−	0.0515294i	$-0.0164096\pi$
$234$	6.39129e14	0.254499
$235$	0	0
$236$	−1.84105e15	−0.693645
$237$	− 3.13112e14i	− 0.114772i
$238$	1.96255e15i	0.699957i
$239$	4.54846e15	1.57862	0.789311	−	0.613993i	$-0.210438\pi$
$239$	4.54846e15	1.57862	0.789311	+	0.613993i	$0.210438\pi$
$240$	0	0
$241$	−9.11615e14	−0.299710	−0.149855	−	0.988708i	$-0.547881\pi$
$241$	−9.11615e14	−0.299710	−0.149855	+	0.988708i	$0.547881\pi$
$242$	− 4.54354e15i	− 1.45410i
$243$	2.05891e14i	0.0641500i
$244$	1.10098e15	0.334000
$245$	0	0
$246$	−3.22462e15	−0.927685
$247$	− 2.46356e13i	− 0.00690292i
$248$	3.18364e15i	0.868936i
$249$	−3.48001e15	−0.925305
$250$	0	0
$251$	5.57048e15	1.40609	0.703046	−	0.711145i	$-0.251823\pi$
$251$	5.57048e15	1.40609	0.703046	+	0.711145i	$0.251823\pi$
$252$	7.62183e14i	0.187480i
$253$	1.93114e15i	0.462945i
$254$	5.39513e15	1.26062
$255$	0	0
$256$	−2.74505e15	−0.609523
$257$	− 4.51207e14i	− 0.0976811i	−0.998807	−	0.0488405i	$-0.984447\pi$
$257$	− 4.51207e14i	− 0.0976811i	0.998807	−	0.0488405i	$-0.0155526\pi$
$258$	− 1.92733e15i	− 0.406845i
$259$	2.61475e14	0.0538248
$260$	0	0
$261$	−1.37174e15	−0.268601
$262$	1.31113e16i	2.50431i
$263$	− 7.11310e15i	− 1.32540i	−0.748885	−	0.662700i	$-0.769411\pi$
$263$	− 7.11310e15i	− 1.32540i	0.748885	−	0.662700i	$-0.230589\pi$
$264$	−7.41751e14	−0.134844
$265$	0	0
$266$	4.88023e13	0.00844713
$267$	− 3.29914e15i	− 0.557285i
$268$	1.43322e16i	2.36285i
$269$	1.34023e15	0.215671	0.107835	−	0.994169i	$-0.465608\pi$
$269$	1.34023e15	0.215671	0.107835	+	0.994169i	$0.465608\pi$
$270$	0	0
$271$	6.00564e15	0.920998	0.460499	−	0.887660i	$-0.347670\pi$
$271$	6.00564e15	0.920998	0.460499	+	0.887660i	$0.347670\pi$
$272$	1.78264e15i	0.266910i
$273$	− 7.07321e14i	− 0.103409i
$274$	−1.29002e16	−1.84170
$275$	0	0
$276$	−1.03271e16	−1.40627
$277$	1.09407e16i	1.45521i	0.685994	+	0.727607i	$0.259368\pi$
$277$	1.09407e16i	1.45521i	−0.685994	+	0.727607i	$0.740632\pi$
$278$	4.69686e15i	0.610262i
$279$	2.80872e15	0.356518
$280$	0	0
$281$	−2.82786e15	−0.342662	−0.171331	−	0.985213i	$-0.554807\pi$
$281$	−2.82786e15	−0.342662	−0.171331	+	0.985213i	$0.554807\pi$
$282$	9.32393e15i	1.10403i
$283$	5.47390e15i	0.633410i	0.948524	+	0.316705i	$0.102576\pi$
$283$	5.47390e15i	0.633410i	−0.948524	+	0.316705i	$0.897424\pi$
$284$	1.72691e16	1.95300
$285$	0	0
$286$	2.03139e15	0.219490
$287$	3.56867e15i	0.376942i
$288$	3.77252e15i	0.389566i
$289$	−4.06315e15	−0.410229
$290$	0	0
$291$	−1.12665e16	−1.08764
$292$	2.77389e16i	2.61879i
$293$	− 1.00494e16i	− 0.927897i	−0.885862	−	0.463948i	$-0.846432\pi$
$293$	− 1.00494e16i	− 0.927897i	0.885862	−	0.463948i	$-0.153568\pi$
$294$	−8.73218e15	−0.788615
$295$	0	0
$296$	−1.36080e15	−0.117597
$297$	6.54400e14i	0.0553256i
$298$	8.41657e15i	0.696193i
$299$	9.58373e15	0.775661
$300$	0	0
$301$	−2.13297e15	−0.165311
$302$	− 1.31663e16i	− 0.998662i
$303$	− 3.19791e15i	− 0.237405i
$304$	4.43283e13	0.00322109
$305$	0	0
$306$	−9.01093e15	−0.627452
$307$	− 4.72392e15i	− 0.322035i	−0.986952	−	0.161018i	$-0.948522\pi$
$307$	− 4.72392e15i	− 0.322035i	0.986952	−	0.161018i	$-0.0514776\pi$
$308$	2.42251e15i	0.161691i
$309$	7.08805e15	0.463230
$310$	0	0
$311$	−3.07381e16	−1.92635	−0.963174	−	0.268878i	$-0.913347\pi$
$311$	−3.07381e16	−1.92635	−0.963174	+	0.268878i	$0.913347\pi$
$312$	3.68112e15i	0.225930i
$313$	− 1.74281e16i	− 1.04764i	−0.851829	−	0.523819i	$-0.824507\pi$
$313$	− 1.74281e16i	− 1.04764i	0.851829	−	0.523819i	$-0.175493\pi$
$314$	1.38038e16	0.812745
$315$	0	0
$316$	5.32193e15	0.300679
$317$	− 2.05312e16i	− 1.13639i	−0.822892	−	0.568197i	$-0.807641\pi$
$317$	− 2.05312e16i	− 1.13639i	0.822892	−	0.568197i	$-0.192359\pi$
$318$	− 2.40187e16i	− 1.30248i
$319$	−4.35990e15	−0.231653
$320$	0	0
$321$	−5.61584e15	−0.286505
$322$	1.89850e16i	0.949180i
$323$	3.47331e14i	0.0170187i
$324$	−3.49951e15	−0.168060
$325$	0	0
$326$	5.18475e16	2.39229
$327$	7.63153e15i	0.345184i
$328$	− 1.85725e16i	− 0.823549i
$329$	1.03188e16	0.448594
$330$	0	0
$331$	1.08020e16	0.451463	0.225731	−	0.974190i	$-0.427523\pi$
$331$	1.08020e16	0.451463	0.225731	+	0.974190i	$0.427523\pi$
$332$	− 5.91494e16i	− 2.42411i
$333$	1.20055e15i	0.0482493i
$334$	−2.00374e16	−0.789750
$335$	0	0
$336$	1.27272e15	0.0482535
$337$	− 2.49305e16i	− 0.927120i	−0.886065	−	0.463560i	$-0.846572\pi$
$337$	− 2.49305e16i	− 0.927120i	0.886065	−	0.463560i	$-0.153428\pi$
$338$	3.33713e16i	1.21735i
$339$	4.00098e15	0.143175
$340$	0	0
$341$	8.92717e15	0.307475
$342$	2.24072e14i	0.00757213i
$343$	2.08784e16i	0.692286i
$344$	1.11006e16	0.361175
$345$	0	0
$346$	6.46101e16	2.02444
$347$	− 4.84184e16i	− 1.48891i	−0.667672	−	0.744455i	$-0.732709\pi$
$347$	− 4.84184e16i	− 1.48891i	0.667672	−	0.744455i	$-0.267291\pi$
$348$	− 2.33153e16i	− 0.703681i
$349$	−3.58064e15	−0.106071	−0.0530353	−	0.998593i	$-0.516890\pi$
$349$	−3.58064e15	−0.106071	−0.0530353	+	0.998593i	$0.516890\pi$
$350$	0	0
$351$	3.24761e15	0.0926975
$352$	1.19905e16i	0.335977i
$353$	6.71086e16i	1.84605i	0.384744	+	0.923023i	$0.374290\pi$
$353$	6.71086e16i	1.84605i	−0.384744	+	0.923023i	$0.625710\pi$
$354$	−1.55399e16	−0.419687
$355$	0	0
$356$	5.60751e16	1.45997
$357$	9.97235e15i	0.254950i
$358$	6.65031e15i	0.166956i
$359$	−6.13471e16	−1.51245	−0.756224	−	0.654313i	$-0.772958\pi$
$359$	−6.13471e16	−1.51245	−0.756224	+	0.654313i	$0.772958\pi$
$360$	0	0
$361$	−4.20443e16	−0.999795
$362$	− 8.23045e16i	− 1.92229i
$363$	− 2.30871e16i	− 0.529635i
$364$	1.20223e16	0.270911
$365$	0	0
$366$	9.29314e15	0.202086
$367$	1.34842e16i	0.288067i	0.989573	+	0.144034i	$0.0460074\pi$
$367$	1.34842e16i	0.288067i	−0.989573	+	0.144034i	$0.953993\pi$
$368$	1.72446e16i	0.361944i
$369$	−1.63853e16	−0.337896
$370$	0	0
$371$	−2.65814e16	−0.529233
$372$	4.77395e16i	0.934004i
$373$	− 3.69623e16i	− 0.710643i	−0.934744	−	0.355322i	$-0.884371\pi$
$373$	− 3.69623e16i	− 0.710643i	0.934744	−	0.355322i	$-0.115629\pi$
$374$	−2.86401e16	−0.541140
$375$	0	0
$376$	−5.37020e16	−0.980097
$377$	2.16371e16i	0.388132i
$378$	6.43341e15i	0.113434i
$379$	6.08164e14	0.0105406	0.00527031	−	0.999986i	$-0.498322\pi$
$379$	6.08164e14	0.0105406	0.00527031	+	0.999986i	$0.498322\pi$
$380$	0	0
$381$	2.74143e16	0.459161
$382$	− 3.51573e16i	− 0.578901i
$383$	− 2.22248e16i	− 0.359788i	−0.983686	−	0.179894i	$-0.942425\pi$
$383$	− 2.22248e16i	− 0.359788i	0.983686	−	0.179894i	$-0.0575754\pi$
$384$	−5.11980e16	−0.814892
$385$	0	0
$386$	1.54788e17	2.38188
$387$	− 9.79338e15i	− 0.148187i
$388$	− 1.91496e17i	− 2.84940i
$389$	1.28354e15	0.0187818	0.00939088	−	0.999956i	$-0.497011\pi$
$389$	1.28354e15	0.0187818	0.00939088	+	0.999956i	$0.497011\pi$
$390$	0	0
$391$	−1.35119e17	−1.91235
$392$	− 5.02938e16i	− 0.700091i
$393$	6.66228e16i	0.912161i
$394$	−7.71064e16	−1.03840
$395$	0	0
$396$	−1.11228e16	−0.144942
$397$	− 9.82778e16i	− 1.25985i	−0.776658	−	0.629923i	$-0.783087\pi$
$397$	− 9.82778e16i	− 1.25985i	0.776658	−	0.629923i	$-0.216913\pi$
$398$	− 6.36375e16i	− 0.802552i
$399$	2.47980e14	0.00307675
$400$	0	0
$401$	−1.22578e17	−1.47222	−0.736110	−	0.676862i	$-0.763339\pi$
$401$	−1.22578e17	−1.47222	−0.736110	+	0.676862i	$0.763339\pi$
$402$	1.20974e17i	1.42963i
$403$	− 4.43032e16i	− 0.515172i
$404$	5.43546e16	0.621953
$405$	0	0
$406$	−4.28623e16	−0.474959
$407$	3.81579e15i	0.0416122i
$408$	− 5.18992e16i	− 0.557018i
$409$	−1.53819e17	−1.62483	−0.812417	−	0.583077i	$-0.801849\pi$
$409$	−1.53819e17	−1.62483	−0.812417	+	0.583077i	$0.801849\pi$
$410$	0	0
$411$	−6.55501e16	−0.670813
$412$	1.20475e17i	1.21357i
$413$	1.71979e16i	0.170530i
$414$	−8.71685e16	−0.850858
$415$	0	0
$416$	5.95057e16	0.562927
$417$	2.38662e16i	0.222279i
$418$	7.12185e14i	0.00653051i
$419$	6.70532e16	0.605381	0.302690	−	0.953089i	$-0.402115\pi$
$419$	6.70532e16	0.605381	0.302690	+	0.953089i	$0.402115\pi$
$420$	0	0
$421$	−1.75428e17	−1.53555	−0.767776	−	0.640719i	$-0.778636\pi$
$421$	−1.75428e17	−1.53555	−0.767776	+	0.640719i	$0.778636\pi$
$422$	2.63843e17i	2.27413i
$423$	4.73778e16i	0.402126i
$424$	1.38338e17	1.15628
$425$	0	0
$426$	1.45765e17	1.18165
$427$	− 1.02847e16i	− 0.0821125i
$428$	− 9.54519e16i	− 0.750585i
$429$	1.03221e16	0.0799461
$430$	0	0
$431$	1.32859e17	0.998360	0.499180	−	0.866498i	$-0.333635\pi$
$431$	1.32859e17	0.998360	0.499180	+	0.866498i	$0.333635\pi$
$432$	5.84362e15i	0.0432552i
$433$	7.83879e16i	0.571581i	0.958292	+	0.285790i	$0.0922561\pi$
$433$	7.83879e16i	0.571581i	−0.958292	+	0.285790i	$0.907744\pi$
$434$	8.77631e16	0.630418
$435$	0	0
$436$	−1.29712e17	−0.904314
$437$	3.35996e15i	0.0230783i
$438$	2.34137e17i	1.58449i
$439$	5.39578e16	0.359778	0.179889	−	0.983687i	$-0.442426\pi$
$439$	5.39578e16	0.359778	0.179889	+	0.983687i	$0.442426\pi$
$440$	0	0
$441$	−4.43710e16	−0.287242
$442$	1.42133e17i	0.906675i
$443$	− 3.24769e16i	− 0.204150i	−0.994777	−	0.102075i	$-0.967452\pi$
$443$	− 3.24769e16i	− 0.204150i	0.994777	−	0.102075i	$-0.0325482\pi$
$444$	−2.04056e16	−0.126403
$445$	0	0
$446$	3.89767e17	2.34492
$447$	4.27672e16i	0.253578i
$448$	1.03577e17i	0.605278i
$449$	1.12233e17	0.646424	0.323212	−	0.946327i	$-0.395237\pi$
$449$	1.12233e17	0.646424	0.323212	+	0.946327i	$0.395237\pi$
$450$	0	0
$451$	−5.20787e16	−0.291415
$452$	6.80042e16i	0.375090i
$453$	− 6.69021e16i	− 0.363748i
$454$	2.12211e17	1.13738
$455$	0	0
$456$	−1.29056e15	−0.00672214
$457$	− 2.41097e17i	− 1.23805i	−0.785372	−	0.619024i	$-0.787529\pi$
$457$	− 2.41097e17i	− 1.23805i	0.785372	−	0.619024i	$-0.212471\pi$
$458$	4.49445e17i	2.27536i
$459$	−4.57873e16	−0.228540
$460$	0	0
$461$	−9.87829e16	−0.479320	−0.239660	−	0.970857i	$-0.577036\pi$
$461$	−9.87829e16	−0.479320	−0.239660	+	0.970857i	$0.577036\pi$
$462$	2.04478e16i	0.0978304i
$463$	− 2.02766e17i	− 0.956576i	−0.878203	−	0.478288i	$-0.841258\pi$
$463$	− 2.02766e17i	− 0.956576i	0.878203	−	0.478288i	$-0.158742\pi$
$464$	−3.89328e16	−0.181113
$465$	0	0
$466$	3.61119e16	0.163358
$467$	− 1.35876e17i	− 0.606152i	−0.952966	−	0.303076i	$-0.901986\pi$
$467$	− 1.35876e17i	− 0.606152i	0.952966	−	0.303076i	$-0.0980136\pi$
$468$	5.51994e16i	0.242849i
$469$	1.33882e17	0.580896
$470$	0	0
$471$	7.01412e16	0.296031
$472$	− 8.95032e16i	− 0.372576i
$473$	− 3.11271e16i	− 0.127803i
$474$	4.49212e16	0.181925
$475$	0	0
$476$	−1.69499e17	−0.667916
$477$	− 1.22046e17i	− 0.474412i
$478$	6.52554e17i	2.50227i
$479$	1.81495e17	0.686568	0.343284	−	0.939232i	$-0.388461\pi$
$479$	1.81495e17	0.686568	0.343284	+	0.939232i	$0.388461\pi$
$480$	0	0
$481$	1.89368e16	0.0697208
$482$	− 1.30787e17i	− 0.475070i
$483$	9.64689e16i	0.345725i
$484$	3.92410e17	1.38754
$485$	0	0
$486$	−2.95386e16	−0.101684
$487$	− 5.39647e17i	− 1.83304i	−0.399992	−	0.916518i	$-0.630987\pi$
$487$	− 5.39647e17i	− 1.83304i	0.399992	−	0.916518i	$-0.369013\pi$
$488$	5.35246e16i	0.179401i
$489$	2.63453e17	0.871356
$490$	0	0
$491$	−3.53623e17	−1.13897	−0.569483	−	0.822003i	$-0.692857\pi$
$491$	−3.53623e17	−1.13897	−0.569483	+	0.822003i	$0.692857\pi$
$492$	− 2.78499e17i	− 0.885219i
$493$	− 3.05056e17i	− 0.956917i
$494$	3.53439e15	0.0109418
$495$	0	0
$496$	7.97174e16	0.240393
$497$	− 1.61317e17i	− 0.480136i
$498$	− 4.99266e17i	− 1.46670i
$499$	1.25995e17	0.365344	0.182672	−	0.983174i	$-0.441525\pi$
$499$	1.25995e17	0.365344	0.182672	+	0.983174i	$0.441525\pi$
$500$	0	0
$501$	−1.01816e17	−0.287655
$502$	7.99179e17i	2.22879i
$503$	− 1.25319e16i	− 0.0345006i	−0.999851	−	0.0172503i	$-0.994509\pi$
$503$	− 1.25319e16i	− 0.0345006i	0.999851	−	0.0172503i	$-0.00549121\pi$
$504$	−3.70538e16	−0.100701
$505$	0	0
$506$	−2.77054e17	−0.733815
$507$	1.69570e17i	0.443401i
$508$	4.65959e17i	1.20291i
$509$	3.72187e17	0.948628	0.474314	−	0.880356i	$-0.342696\pi$
$509$	3.72187e17	0.948628	0.474314	+	0.880356i	$0.342696\pi$
$510$	0	0
$511$	2.59119e17	0.643818
$512$	1.81504e17i	0.445279i
$513$	1.13858e15i	0.00275804i
$514$	6.47332e16	0.154834
$515$	0	0
$516$	1.66457e17	0.388221
$517$	1.50585e17i	0.346810i
$518$	3.75131e16i	0.0853177i
$519$	3.28304e17	0.737375
$520$	0	0
$521$	−4.42287e17	−0.968855	−0.484428	−	0.874831i	$-0.660972\pi$
$521$	−4.42287e17	−0.968855	−0.484428	+	0.874831i	$0.660972\pi$
$522$	− 1.96799e17i	− 0.425760i
$523$	− 7.13084e17i	− 1.52363i	−0.647794	−	0.761816i	$-0.724308\pi$
$523$	− 7.13084e17i	− 1.52363i	0.647794	−	0.761816i	$-0.275692\pi$
$524$	−1.13238e18	−2.38968
$525$	0	0
$526$	1.02049e18	2.10089
$527$	6.24620e17i	1.27013i
$528$	1.85732e16i	0.0373050i
$529$	−8.03054e17	−1.59325
$530$	0	0
$531$	−7.89629e16	−0.152865
$532$	4.21488e15i	0.00806046i
$533$	2.58453e17i	0.488264i
$534$	4.73317e17	0.883352
$535$	0	0
$536$	−6.96762e17	−1.26915
$537$	3.37923e16i	0.0608114i
$538$	1.92279e17i	0.341859i
$539$	−1.41028e17	−0.247729
$540$	0	0
$541$	1.13125e18	1.93988	0.969941	−	0.243342i	$-0.0782437\pi$
$541$	1.13125e18	1.93988	0.969941	+	0.243342i	$0.0782437\pi$
$542$	8.61610e17i	1.45987i
$543$	− 4.18215e17i	− 0.700166i
$544$	−8.38957e17	−1.38786
$545$	0	0
$546$	1.01477e17	0.163914
$547$	7.05433e17i	1.12600i	0.826457	+	0.563000i	$0.190353\pi$
$547$	7.05433e17i	1.12600i	−0.826457	+	0.563000i	$0.809647\pi$
$548$	− 1.11415e18i	− 1.75739i
$549$	4.72214e16	0.0736069
$550$	0	0
$551$	−7.58573e15	−0.0115481
$552$	− 5.02054e17i	− 0.755347i
$553$	− 4.97140e16i	− 0.0739206i
$554$	−1.56963e18	−2.30666
$555$	0	0
$556$	−4.05651e17	−0.582327
$557$	− 6.45619e17i	− 0.916047i	−0.888940	−	0.458023i	$-0.848558\pi$
$557$	− 6.45619e17i	− 0.916047i	0.888940	−	0.458023i	$-0.151442\pi$
$558$	4.02958e17i	0.565116i
$559$	−1.54475e17	−0.214133
$560$	0	0
$561$	−1.45529e17	−0.197103
$562$	− 4.05704e17i	− 0.543154i
$563$	− 7.17316e17i	− 0.949306i	−0.880173	−	0.474653i	$-0.842574\pi$
$563$	− 7.17316e17i	− 0.949306i	0.880173	−	0.474653i	$-0.157426\pi$
$564$	−8.05276e17	−1.05349
$565$	0	0
$566$	−7.85323e17	−1.00402
$567$	3.26902e16i	0.0413169i
$568$	8.39544e17i	1.04901i
$569$	1.04468e17	0.129048	0.0645241	−	0.997916i	$-0.479447\pi$
$569$	1.04468e17	0.129048	0.0645241	+	0.997916i	$0.479447\pi$
$570$	0	0
$571$	5.56964e17	0.672500	0.336250	−	0.941773i	$-0.390841\pi$
$571$	5.56964e17	0.672500	0.336250	+	0.941773i	$0.390841\pi$
$572$	1.75444e17i	0.209443i
$573$	− 1.78645e17i	− 0.210856i
$574$	−5.11986e17	−0.597490
$575$	0	0
$576$	−4.75565e17	−0.542580
$577$	3.32481e17i	0.375080i	0.982257	+	0.187540i	$0.0600514\pi$
$577$	3.32481e17i	0.375080i	−0.982257	+	0.187540i	$0.939949\pi$
$578$	− 5.82927e17i	− 0.650254i
$579$	7.86527e17	0.867566
$580$	0	0
$581$	−5.52536e17	−0.595958
$582$	− 1.61637e18i	− 1.72402i
$583$	− 3.87909e17i	− 0.409152i
$584$	−1.34853e18	−1.40663
$585$	0	0
$586$	1.44175e18	1.47081
$587$	8.00524e17i	0.807657i	0.914835	+	0.403828i	$0.132321\pi$
$587$	8.00524e17i	0.807657i	−0.914835	+	0.403828i	$0.867679\pi$
$588$	− 7.54169e17i	− 0.752516i
$589$	1.55322e16	0.0153280
$590$	0	0
$591$	−3.91802e17	−0.378223
$592$	3.40740e16i	0.0325336i
$593$	− 1.73961e18i	− 1.64284i	−0.570322	−	0.821421i	$-0.693182\pi$
$593$	− 1.73961e18i	− 1.64284i	0.570322	−	0.821421i	$-0.306818\pi$
$594$	−9.38846e16	−0.0876966
$595$	0	0
$596$	−7.26910e17	−0.664324
$597$	− 3.23362e17i	− 0.292318i
$598$	1.37495e18i	1.22950i
$599$	1.43596e18	1.27019	0.635095	−	0.772434i	$-0.280961\pi$
$599$	1.43596e18	1.27019	0.635095	+	0.772434i	$0.280961\pi$
$600$	0	0
$601$	−1.23937e18	−1.07279	−0.536396	−	0.843966i	$-0.680215\pi$
$601$	−1.23937e18	−1.07279	−0.536396	+	0.843966i	$0.680215\pi$
$602$	− 3.06011e17i	− 0.262035i
$603$	6.14709e17i	0.520723i
$604$	1.13713e18	0.952948
$605$	0	0
$606$	4.58795e17	0.376310
$607$	− 2.40600e18i	− 1.95240i	−0.216875	−	0.976199i	$-0.569586\pi$
$607$	− 2.40600e18i	− 1.95240i	0.216875	−	0.976199i	$-0.430414\pi$
$608$	2.08621e16i	0.0167488i
$609$	−2.17797e17	−0.172997
$610$	0	0
$611$	7.47312e17	0.581077
$612$	− 7.78243e17i	− 0.598730i
$613$	7.71007e17i	0.586902i	0.955974	+	0.293451i	$0.0948037\pi$
$613$	7.71007e17i	0.586902i	−0.955974	+	0.293451i	$0.905196\pi$
$614$	6.77726e17	0.510458
$615$	0	0
$616$	−1.17771e17	−0.0868487
$617$	6.35119e17i	0.463449i	0.972781	+	0.231724i	$0.0744367\pi$
$617$	6.35119e17i	0.463449i	−0.972781	+	0.231724i	$0.925563\pi$
$618$	1.01690e18i	0.734266i
$619$	−1.75343e18	−1.25285	−0.626426	−	0.779481i	$-0.715483\pi$
$619$	−1.75343e18	−1.25285	−0.626426	+	0.779481i	$0.715483\pi$
$620$	0	0
$621$	−4.42930e17	−0.309913
$622$	− 4.40990e18i	− 3.05345i
$623$	− 5.23818e17i	− 0.358928i
$624$	9.21741e16	0.0625042
$625$	0	0
$626$	2.50035e18	1.66061
$627$	3.61884e15i	0.00237865i
$628$	1.19218e18i	0.775541i
$629$	−2.66985e17	−0.171893
$630$	0	0
$631$	1.21896e18	0.768775	0.384387	−	0.923172i	$-0.374413\pi$
$631$	1.21896e18	0.768775	0.384387	+	0.923172i	$0.374413\pi$
$632$	2.58727e17i	0.161503i
$633$	1.34067e18i	0.828318i
$634$	2.94555e18	1.80130
$635$	0	0
$636$	2.07441e18	1.24286
$637$	6.99883e17i	0.415068i
$638$	− 6.25501e17i	− 0.367193i
$639$	7.40676e17	0.430400
$640$	0	0
$641$	1.09626e18	0.624216	0.312108	−	0.950047i	$-0.398965\pi$
$641$	1.09626e18	0.624216	0.312108	+	0.950047i	$0.398965\pi$
$642$	− 8.05687e17i	− 0.454139i
$643$	− 2.51721e18i	− 1.40459i	−0.711888	−	0.702293i	$-0.752160\pi$
$643$	− 2.51721e18i	− 1.40459i	0.711888	−	0.702293i	$-0.247840\pi$
$644$	−1.63967e18	−0.905731
$645$	0	0
$646$	−4.98305e16	−0.0269764
$647$	6.31684e17i	0.338549i	0.985569	+	0.169275i	$0.0541425\pi$
$647$	6.31684e17i	0.338549i	−0.985569	+	0.169275i	$0.945858\pi$
$648$	− 1.70130e17i	− 0.0902699i
$649$	−2.50974e17	−0.131837
$650$	0	0
$651$	4.45952e17	0.229621
$652$	4.47789e18i	2.28278i
$653$	2.66002e18i	1.34261i	0.741183	+	0.671303i	$0.234265\pi$
$653$	2.66002e18i	1.34261i	−0.741183	+	0.671303i	$0.765735\pi$
$654$	−1.09487e18	−0.547152
$655$	0	0
$656$	−4.65049e17	−0.227837
$657$	1.18973e18i	0.577128i
$658$	1.48040e18i	0.711067i
$659$	1.04314e18	0.496121	0.248061	−	0.968745i	$-0.420207\pi$
$659$	1.04314e18	0.496121	0.248061	+	0.968745i	$0.420207\pi$
$660$	0	0
$661$	8.68858e17	0.405172	0.202586	−	0.979264i	$-0.435065\pi$
$661$	8.68858e17	0.405172	0.202586	+	0.979264i	$0.435065\pi$
$662$	1.54973e18i	0.715613i
$663$	7.22224e17i	0.330243i
$664$	2.87557e18	1.30206
$665$	0	0
$666$	−1.72239e17	−0.0764800
$667$	− 2.95100e18i	− 1.29763i
$668$	− 1.73056e18i	− 0.753599i
$669$	1.98053e18	0.854105
$670$	0	0
$671$	1.50087e17	0.0634815
$672$	5.98979e17i	0.250906i
$673$	− 2.71903e18i	− 1.12802i	−0.825769	−	0.564009i	$-0.809258\pi$
$673$	− 2.71903e18i	− 1.12802i	0.825769	−	0.564009i	$-0.190742\pi$
$674$	3.57670e18	1.46958
$675$	0	0
$676$	−2.88216e18	−1.16162
$677$	2.17548e18i	0.868418i	0.900812	+	0.434209i	$0.142972\pi$
$677$	2.17548e18i	0.868418i	−0.900812	+	0.434209i	$0.857028\pi$
$678$	5.74007e17i	0.226947i
$679$	−1.78883e18	−0.700514
$680$	0	0
$681$	1.07831e18	0.414273
$682$	1.28075e18i	0.487379i
$683$	4.67971e18i	1.76394i	0.471305	+	0.881970i	$0.343783\pi$
$683$	4.67971e18i	1.76394i	−0.471305	+	0.881970i	$0.656217\pi$
$684$	−1.93523e16	−0.00722551
$685$	0	0
$686$	−2.99536e18	−1.09734
$687$	2.28377e18i	0.828769i
$688$	− 2.77957e17i	− 0.0999200i
$689$	−1.92509e18	−0.685530
$690$	0	0
$691$	−3.58472e18	−1.25270	−0.626352	−	0.779541i	$-0.715453\pi$
$691$	−3.58472e18	−1.25270	−0.626352	+	0.779541i	$0.715453\pi$
$692$	5.58015e18i	1.93177i
$693$	1.03902e17i	0.0356334i
$694$	6.94643e18	2.36007
$695$	0	0
$696$	1.13348e18	0.377967
$697$	− 3.64386e18i	− 1.20379i
$698$	− 5.13703e17i	− 0.168132i
$699$	1.83496e17	0.0595010
$700$	0	0
$701$	−5.81898e18	−1.85217	−0.926083	−	0.377321i	$-0.876845\pi$
$701$	−5.81898e18	−1.85217	−0.926083	+	0.377321i	$0.876845\pi$
$702$	4.65925e17i	0.146935i
$703$	6.63903e15i	0.00207441i
$704$	−1.51152e18	−0.467943
$705$	0	0
$706$	−9.62786e18	−2.92617
$707$	− 5.07746e17i	− 0.152905i
$708$	− 1.34212e18i	− 0.400476i
$709$	−6.35434e18	−1.87876	−0.939378	−	0.342884i	$-0.888596\pi$
$709$	−6.35434e18	−1.87876	−0.939378	+	0.342884i	$0.888596\pi$
$710$	0	0
$711$	2.28258e17	0.0662635
$712$	2.72611e18i	0.784193i
$713$	6.04235e18i	1.72236i
$714$	−1.43070e18	−0.404120
$715$	0	0
$716$	−5.74364e17	−0.159313
$717$	3.31583e18i	0.911418i
$718$	− 8.80128e18i	− 2.39738i
$719$	−2.59269e18	−0.699863	−0.349932	−	0.936775i	$-0.613795\pi$
$719$	−2.59269e18	−0.699863	−0.349932	+	0.936775i	$0.613795\pi$
$720$	0	0
$721$	1.12540e18	0.298351
$722$	− 6.03197e18i	− 1.58477i
$723$	− 6.64568e17i	− 0.173038i
$724$	7.10836e18	1.83429
$725$	0	0
$726$	3.31224e18	0.839525
$727$	7.34447e18i	1.84496i	0.386046	+	0.922479i	$0.373840\pi$
$727$	7.34447e18i	1.84496i	−0.386046	+	0.922479i	$0.626160\pi$
$728$	5.84466e17i	0.145514i
$729$	−1.50095e17	−0.0370370
$730$	0	0
$731$	2.17791e18	0.527932
$732$	8.02617e17i	0.192835i
$733$	4.76931e18i	1.13574i	0.823118	+	0.567871i	$0.192233\pi$
$733$	4.76931e18i	1.13574i	−0.823118	+	0.567871i	$0.807767\pi$
$734$	−1.93453e18	−0.456616
$735$	0	0
$736$	−8.11576e18	−1.88202
$737$	1.95378e18i	0.449093i
$738$	− 2.35075e18i	− 0.535599i
$739$	2.70323e18	0.610513	0.305256	−	0.952270i	$-0.401258\pi$
$739$	2.70323e18	0.610513	0.305256	+	0.952270i	$0.401258\pi$
$740$	0	0
$741$	1.79593e16	0.00398540
$742$	− 3.81354e18i	− 0.838887i
$743$	1.89279e18i	0.412738i	0.978474	+	0.206369i	$0.0661647\pi$
$743$	1.89279e18i	0.412738i	−0.978474	+	0.206369i	$0.933835\pi$
$744$	−2.32087e18	−0.501680
$745$	0	0
$746$	5.30287e18	1.12644
$747$	− 2.53693e18i	− 0.534225i
$748$	− 2.47355e18i	− 0.516369i
$749$	−8.91650e17	−0.184528
$750$	0	0
$751$	7.31039e18	1.48690	0.743448	−	0.668793i	$-0.233189\pi$
$751$	7.31039e18	1.48690	0.743448	+	0.668793i	$0.233189\pi$
$752$	1.34468e18i	0.271146i
$753$	4.06088e18i	0.811807i
$754$	−3.10420e18	−0.615228
$755$	0	0
$756$	−5.55632e17	−0.108242
$757$	− 7.31767e18i	− 1.41335i	−0.707539	−	0.706675i	$-0.750195\pi$
$757$	− 7.31767e18i	− 1.41335i	0.707539	−	0.706675i	$-0.249805\pi$
$758$	8.72514e16i	0.0167079i
$759$	−1.40780e18	−0.267282
$760$	0	0
$761$	4.77770e18	0.891700	0.445850	−	0.895108i	$-0.352901\pi$
$761$	4.77770e18	0.891700	0.445850	+	0.895108i	$0.352901\pi$
$762$	3.93305e18i	0.727817i
$763$	1.21169e18i	0.222322i
$764$	3.03642e18	0.552401
$765$	0	0
$766$	3.18852e18	0.570299
$767$	1.24552e18i	0.220892i
$768$	− 2.00114e18i	− 0.351908i
$769$	−2.59264e18	−0.452086	−0.226043	−	0.974117i	$-0.572579\pi$
$769$	−2.59264e18	−0.452086	−0.226043	+	0.974117i	$0.572579\pi$
$770$	0	0
$771$	3.28930e17	0.0563962
$772$	1.33685e19i	2.27285i
$773$	5.76567e18i	0.972038i	0.873948	+	0.486019i	$0.161551\pi$
$773$	5.76567e18i	0.972038i	−0.873948	+	0.486019i	$0.838449\pi$
$774$	1.40503e18	0.234892
$775$	0	0
$776$	9.30964e18	1.53049
$777$	1.90616e17i	0.0310758i
$778$	1.84145e17i	0.0297710i
$779$	−9.06109e16	−0.0145274
$780$	0	0
$781$	2.35415e18	0.371195
$782$	− 1.93851e19i	− 3.03126i
$783$	− 9.99998e17i	− 0.155077i
$784$	−1.25934e18	−0.193682
$785$	0	0
$786$	−9.55817e18	−1.44587
$787$	− 1.72894e18i	− 0.259385i	−0.991554	−	0.129692i	$-0.958601\pi$
$787$	− 1.72894e18i	− 0.259385i	0.991554	−	0.129692i	$-0.0413990\pi$
$788$	− 6.65941e18i	− 0.990867i
$789$	5.18545e18	0.765220
$790$	0	0
$791$	6.35251e17	0.0922143
$792$	− 5.40737e17i	− 0.0778524i
$793$	− 7.44844e17i	− 0.106363i
$794$	1.40996e19	1.99698
$795$	0	0
$796$	5.49615e18	0.765815
$797$	− 1.06264e19i	− 1.46861i	−0.678820	−	0.734305i	$-0.737508\pi$
$797$	− 1.06264e19i	− 1.46861i	0.678820	−	0.734305i	$-0.262492\pi$
$798$	3.55768e16i	0.00487695i
$799$	−1.05362e19	−1.43261
$800$	0	0
$801$	2.40507e18	0.321749
$802$	− 1.75858e19i	− 2.33361i
$803$	3.78139e18i	0.497738i
$804$	−1.04481e19	−1.36419
$805$	0	0
$806$	6.35604e18	0.816599
$807$	9.77031e17i	0.124517i
$808$	2.64247e18i	0.334069i
$809$	8.95164e18	1.12263	0.561316	−	0.827602i	$-0.310295\pi$
$809$	8.95164e18	1.12263	0.561316	+	0.827602i	$0.310295\pi$
$810$	0	0
$811$	5.95858e18	0.735373	0.367686	−	0.929950i	$-0.380150\pi$
$811$	5.95858e18	0.735373	0.367686	+	0.929950i	$0.380150\pi$
$812$	− 3.70187e18i	− 0.453217i
$813$	4.37811e18i	0.531739i
$814$	−5.47439e17	−0.0659595
$815$	0	0
$816$	−1.29954e18	−0.154101
$817$	− 5.41575e16i	− 0.00637111i
$818$	− 2.20680e19i	− 2.57552i
$819$	5.15637e17	0.0597034
$820$	0	0
$821$	2.78763e18	0.317691	0.158846	−	0.987303i	$-0.449223\pi$
$821$	2.78763e18	0.317691	0.158846	+	0.987303i	$0.449223\pi$
$822$	− 9.40426e18i	− 1.06331i
$823$	− 3.20816e18i	− 0.359880i	−0.983678	−	0.179940i	$-0.942410\pi$
$823$	− 3.20816e18i	− 0.359880i	0.983678	−	0.179940i	$-0.0575903\pi$
$824$	−5.85692e18	−0.651842
$825$	0	0
$826$	−2.46733e18	−0.270307
$827$	1.18081e18i	0.128349i	0.997939	+	0.0641747i	$0.0204415\pi$
$827$	1.18081e18i	0.128349i	−0.997939	+	0.0641747i	$0.979559\pi$
$828$	− 7.52844e18i	− 0.811910i
$829$	−3.64568e18	−0.390099	−0.195049	−	0.980793i	$-0.562487\pi$
$829$	−3.64568e18	−0.390099	−0.195049	+	0.980793i	$0.562487\pi$
$830$	0	0
$831$	−7.97578e18	−0.840169
$832$	7.50130e18i	0.784034i
$833$	− 9.86749e18i	− 1.02333i
$834$	−3.42401e18	−0.352335
$835$	0	0
$836$	−6.15090e16	−0.00623157
$837$	2.04756e18i	0.205836i
$838$	9.61992e18i	0.959589i
$839$	6.16574e18	0.610285	0.305142	−	0.952307i	$-0.401296\pi$
$839$	6.16574e18	0.610285	0.305142	+	0.952307i	$0.401296\pi$
$840$	0	0
$841$	−3.59820e18	−0.350680
$842$	− 2.51680e19i	− 2.43400i
$843$	− 2.06151e18i	− 0.197836i
$844$	−2.27872e19	−2.17003
$845$	0	0
$846$	−6.79715e18	−0.637411
$847$	− 3.66564e18i	− 0.341120i
$848$	− 3.46393e18i	− 0.319887i
$849$	−3.99047e18	−0.365699
$850$	0	0
$851$	−2.58271e18	−0.233096
$852$	1.25892e19i	1.12756i
$853$	− 8.43377e18i	− 0.749640i	−0.927098	−	0.374820i	$-0.877704\pi$
$853$	− 8.43377e18i	− 0.749640i	0.927098	−	0.374820i	$-0.122296\pi$
$854$	1.47551e18	0.130157
$855$	0	0
$856$	4.64042e18	0.403161
$857$	9.57885e18i	0.825920i	0.910749	+	0.412960i	$0.135505\pi$
$857$	9.57885e18i	0.825920i	−0.910749	+	0.412960i	$0.864495\pi$
$858$	1.48088e18i	0.126723i
$859$	−1.27645e19	−1.08404	−0.542022	−	0.840364i	$-0.682341\pi$
$859$	−1.27645e19	−1.08404	−0.542022	+	0.840364i	$0.682341\pi$
$860$	0	0
$861$	−2.60156e18	−0.217627
$862$	1.90608e19i	1.58250i
$863$	6.90133e18i	0.568673i	0.958725	+	0.284337i	$0.0917733\pi$
$863$	6.90133e18i	0.568673i	−0.958725	+	0.284337i	$0.908227\pi$
$864$	−2.75017e18	−0.224916
$865$	0	0
$866$	−1.12461e19	−0.906013
$867$	− 2.96204e18i	− 0.236846i
$868$	7.57980e18i	0.601561i
$869$	7.25491e17	0.0571483
$870$	0	0
$871$	9.69608e18	0.752451
$872$	− 6.30601e18i	− 0.485733i
$873$	− 8.21330e18i	− 0.627950i
$874$	−4.82043e17	−0.0365815
$875$	0	0
$876$	−2.02216e19	−1.51196
$877$	2.53540e19i	1.88169i	0.338835	+	0.940846i	$0.389967\pi$
$877$	2.53540e19i	1.88169i	−0.338835	+	0.940846i	$0.610033\pi$
$878$	7.74115e18i	0.570284i
$879$	7.32600e18	0.535721
$880$	0	0
$881$	2.27879e18	0.164195	0.0820976	−	0.996624i	$-0.473838\pi$
$881$	2.27879e18	0.164195	0.0820976	+	0.996624i	$0.473838\pi$
$882$	− 6.36576e18i	− 0.455307i
$883$	1.55074e19i	1.10102i	0.834830	+	0.550508i	$0.185566\pi$
$883$	1.55074e19i	1.10102i	−0.834830	+	0.550508i	$0.814434\pi$
$884$	−1.22756e19	−0.865171
$885$	0	0
$886$	4.65936e18	0.323599
$887$	5.37007e18i	0.370234i	0.982716	+	0.185117i	$0.0592665\pi$
$887$	5.37007e18i	0.370234i	−0.982716	+	0.185117i	$0.940734\pi$
$888$	− 9.92022e17i	− 0.0678949i
$889$	4.35269e18	0.295730
$890$	0	0
$891$	−4.77057e17	−0.0319422
$892$	3.36628e19i	2.23758i
$893$	2.62000e17i	0.0172889i
$894$	−6.13568e18	−0.401947
$895$	0	0
$896$	−8.12892e18	−0.524844
$897$	6.98654e18i	0.447828i
$898$	1.61016e19i	1.02465i
$899$	−1.36417e19	−0.861850
$900$	0	0
$901$	2.71414e19	1.69014
$902$	− 7.47156e18i	− 0.461922i
$903$	− 1.55494e18i	− 0.0954426i
$904$	−3.30604e18	−0.201472
$905$	0	0
$906$	9.59823e18	0.576578
$907$	2.21370e19i	1.32029i	0.751136	+	0.660147i	$0.229506\pi$
$907$	2.21370e19i	1.32029i	−0.751136	+	0.660147i	$0.770494\pi$
$908$	1.83279e19i	1.08531i
$909$	2.33128e18	0.137066
$910$	0	0
$911$	−5.15063e18	−0.298532	−0.149266	−	0.988797i	$-0.547691\pi$
$911$	−5.15063e18	−0.298532	−0.149266	+	0.988797i	$0.547691\pi$
$912$	3.23153e16i	0.00185970i
$913$	− 8.06331e18i	− 0.460737i
$914$	3.45895e19	1.96243
$915$	0	0
$916$	−3.88170e19	−2.17121
$917$	1.05780e19i	0.587492i
$918$	− 6.56897e18i	− 0.362259i
$919$	1.11627e19	0.611247	0.305624	−	0.952152i	$-0.401135\pi$
$919$	1.11627e19	0.611247	0.305624	+	0.952152i	$0.401135\pi$
$920$	0	0
$921$	3.44374e18	0.185927
$922$	− 1.41721e19i	− 0.759770i
$923$	− 1.16830e19i	− 0.621934i
$924$	−1.76601e18	−0.0933522
$925$	0	0
$926$	2.90902e19	1.51627
$927$	5.16719e18i	0.267446i
$928$	− 1.83228e19i	− 0.941740i
$929$	−4.48353e18	−0.228833	−0.114416	−	0.993433i	$-0.536500\pi$
$929$	−4.48353e18	−0.228833	−0.114416	+	0.993433i	$0.536500\pi$
$930$	0	0
$931$	−2.45372e17	−0.0123496
$932$	3.11886e18i	0.155881i
$933$	− 2.24081e19i	− 1.11218i
$934$	1.94936e19	0.960812
$935$	0	0
$936$	−2.68353e18	−0.130441
$937$	− 1.27997e19i	− 0.617864i	−0.951084	−	0.308932i	$-0.900028\pi$
$937$	− 1.27997e19i	− 0.617864i	0.951084	−	0.308932i	$-0.0999716\pi$
$938$	1.92076e19i	0.920778i
$939$	1.27051e19	0.604855
$940$	0	0
$941$	−2.16104e19	−1.01468	−0.507342	−	0.861745i	$-0.669372\pi$
$941$	−2.16104e19	−1.01468	−0.507342	+	0.861745i	$0.669372\pi$
$942$	1.00629e19i	0.469239i
$943$	− 3.52494e19i	− 1.63240i
$944$	−2.24113e18	−0.103074
$945$	0	0
$946$	4.46570e18	0.202580
$947$	− 1.44301e19i	− 0.650120i	−0.945693	−	0.325060i	$-0.894616\pi$
$947$	− 1.44301e19i	− 0.650120i	0.945693	−	0.325060i	$-0.105384\pi$
$948$	3.87969e18i	0.173597i
$949$	1.87661e19	0.833956
$950$	0	0
$951$	1.49672e19	0.656098
$952$	− 8.24025e18i	− 0.358757i
$953$	2.49670e19i	1.07960i	0.841794	+	0.539799i	$0.181500\pi$
$953$	2.49670e19i	1.07960i	−0.841794	+	0.539799i	$0.818500\pi$
$954$	1.75096e19	0.751990
$955$	0	0
$956$	−5.63588e19	−2.38773
$957$	− 3.17837e18i	− 0.133745i
$958$	2.60385e19i	1.08828i
$959$	−1.04077e19	−0.432048
$960$	0	0
$961$	3.51474e18	0.143943
$962$	2.71680e18i	0.110514i
$963$	− 4.09395e18i	− 0.165414i
$964$	1.12956e19	0.453323
$965$	0	0
$966$	−1.38401e19	−0.548009
$967$	− 2.66065e19i	− 1.04644i	−0.852196	−	0.523222i	$-0.824730\pi$
$967$	− 2.66065e19i	− 1.04644i	0.852196	−	0.523222i	$-0.175270\pi$
$968$	1.90771e19i	0.745286i
$969$	−2.53205e17	−0.00982578
$970$	0	0
$971$	1.94068e19	0.743069	0.371534	−	0.928419i	$-0.378832\pi$
$971$	1.94068e19	0.743069	0.371534	+	0.928419i	$0.378832\pi$
$972$	− 2.55114e18i	− 0.0970296i
$973$	3.78934e18i	0.143163i
$974$	7.74214e19	2.90555
$975$	0	0
$976$	1.34024e18	0.0496317
$977$	− 3.01511e19i	− 1.10915i	−0.832135	−	0.554573i	$-0.812882\pi$
$977$	− 3.01511e19i	− 1.10915i	0.832135	−	0.554573i	$-0.187118\pi$
$978$	3.77968e19i	1.38119i
$979$	7.64422e18	0.277489
$980$	0	0
$981$	−5.56339e18	−0.199292
$982$	− 5.07332e19i	− 1.80538i
$983$	1.32822e19i	0.469540i	0.972051	+	0.234770i	$0.0754338\pi$
$983$	1.32822e19i	0.469540i	−0.972051	+	0.234770i	$0.924566\pi$
$984$	1.35393e19	0.475476
$985$	0	0
$986$	4.37654e19	1.51681
$987$	7.52237e18i	0.258996i
$988$	3.05253e17i	0.0104409i
$989$	2.10683e19	0.715904
$990$	0	0
$991$	−2.75507e19	−0.923960	−0.461980	−	0.886890i	$-0.652861\pi$
$991$	−2.75507e19	−0.923960	−0.461980	+	0.886890i	$0.652861\pi$
$992$	3.75172e19i	1.24998i
$993$	7.87465e18i	0.260652i
$994$	2.31437e19	0.761063
$995$	0	0
$996$	4.31199e19	1.39956
$997$	3.62751e19i	1.16974i	0.811127	+	0.584870i	$0.198855\pi$
$997$	3.62751e19i	1.16974i	−0.811127	+	0.584870i	$0.801145\pi$
$998$	1.80762e19i	0.579106i
$999$	−8.75198e17	−0.0278568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.14.b.g.49.7		8
5.2	odd	4		75.14.a.h.1.1	yes	4
5.3	odd	4		75.14.a.g.1.4	✓	4
5.4	even	2	inner	75.14.b.g.49.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.14.a.g.1.4	✓	4	5.3	odd	4
75.14.a.h.1.1	yes	4	5.2	odd	4
75.14.b.g.49.2		8	5.4	even	2	inner
75.14.b.g.49.7		8	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+143.467i q^{2} +729.000i q^{3} -12390.7 q^{4} -104587. q^{6} +115746. i q^{7} -602380. i q^{8} -531441. q^{9} -1.68912e6 q^{11} -9.03285e6i q^{12} +8.38266e6i q^{13} -1.66058e7 q^{14} -1.50834e7 q^{16} -1.18185e8i q^{17} -7.62442e7i q^{18} -2.93887e6 q^{19} -8.43791e7 q^{21} -2.42333e8i q^{22} -1.14328e9i q^{23} +4.39135e8 q^{24} -1.20263e9 q^{26} -3.87420e8i q^{27} -1.43418e9i q^{28} +2.58117e9 q^{29} -5.28510e9 q^{31} -7.09866e9i q^{32} -1.23137e9i q^{33} +1.69556e10 q^{34} +6.58495e9 q^{36} -2.25904e9i q^{37} -4.21631e8i q^{38} -6.11096e9 q^{39} +3.08318e10 q^{41} -1.21056e10i q^{42} +1.84280e10i q^{43} +2.09294e10 q^{44} +1.64023e11 q^{46} -8.91497e10i q^{47} -1.09958e10i q^{48} +8.34918e10 q^{49} +8.61570e10 q^{51} -1.03867e11i q^{52} +2.29652e11i q^{53} +5.55820e10 q^{54} +6.97232e10 q^{56} -2.14244e9i q^{57} +3.70312e11i q^{58} +1.48583e11 q^{59} -8.88553e10 q^{61} -7.58237e11i q^{62} -6.15124e10i q^{63} +8.94860e11 q^{64} +1.76661e11 q^{66} -1.15668e12i q^{67} +1.46440e12i q^{68} +8.33451e11 q^{69} -1.39371e12 q^{71} +3.20129e11i q^{72} -2.23868e12i q^{73} +3.24097e11 q^{74} +3.64148e10 q^{76} -1.95509e11i q^{77} -8.76720e11i q^{78} -4.29509e11 q^{79} +2.82430e11 q^{81} +4.42335e12i q^{82} +4.77368e12i q^{83} +1.04552e12 q^{84} -2.64380e12 q^{86} +1.88167e12i q^{87} +1.01749e12i q^{88} -4.52557e12 q^{89} -9.70262e11 q^{91} +1.41661e13i q^{92} -3.85284e12i q^{93} +1.27900e13 q^{94} +5.17493e12 q^{96} +1.54548e13i q^{97} +1.19783e13i q^{98} +8.97667e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 41504 q^{4} + 75816 q^{6} - 4251528 q^{9} - 11748208 q^{11} + 128644584 q^{14} + 12392960 q^{16} + 407861624 q^{19} - 435936168 q^{21} - 618098688 q^{24} - 7083351544 q^{26} + 17010752528 q^{29} - 4110176552 q^{31}+ \cdots + 6243479407728 q^{99}+O(q^{100}) \) 8 * q - 41504 * q^4 + 75816 * q^6 - 4251528 * q^9 - 11748208 * q^11 + 128644584 * q^14 + 12392960 * q^16 + 407861624 * q^19 - 435936168 * q^21 - 618098688 * q^24 - 7083351544 * q^26 + 17010752528 * q^29 - 4110176552 * q^31 + 11940660912 * q^34 + 22056927264 * q^36 + 46721569176 * q^39 + 67258917088 * q^41 + 14163760448 * q^44 + 271062871248 * q^46 - 31472829840 * q^49 + 98282858544 * q^51 - 40291730856 * q^54 - 1219622814720 * q^56 - 723761174864 * q^59 - 721947748520 * q^61 + 1750438395904 * q^64 - 62193066192 * q^66 + 841332496464 * q^69 - 7767788474144 * q^71 + 4692855762224 * q^74 - 8686082155616 * q^76 + 12081014646080 * q^79 + 2259436291848 * q^81 + 17223728111136 * q^84 + 7312090231832 * q^86 + 8511047028864 * q^89 + 30847013172728 * q^91 + 80024185437456 * q^94 + 1701377851392 * q^96 + 6243479407728 * q^99

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