LMFDB - Embedded newform 150.8.c.d.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,8,Mod(49,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("150.49");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	150.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$46.8577538226$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	150.49
Dual form			150.8.c.d.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.00000i q^{2} +27.0000i q^{3} -64.0000 q^{4} -216.000 q^{6} +1427.00i q^{7} -512.000i q^{8} -729.000 q^{9} +5070.00 q^{11} -1728.00i q^{12} +11899.0i q^{13} -11416.0 q^{14} +4096.00 q^{16} +19242.0i q^{17} -5832.00i q^{18} +43711.0 q^{19} -38529.0 q^{21} +40560.0i q^{22} +3150.00i q^{23} +13824.0 q^{24} -95192.0 q^{26} -19683.0i q^{27} -91328.0i q^{28} -140106. q^{29} +147563. q^{31} +32768.0i q^{32} +136890. i q^{33} -153936. q^{34} +46656.0 q^{36} +561674. i q^{37} +349688. i q^{38} -321273. q^{39} -270336. q^{41} -308232. i q^{42} -180683. i q^{43} -324480. q^{44} -25200.0 q^{46} +97470.0i q^{47} +110592. i q^{48} -1.21279e6 q^{49} -519534. q^{51} -761536. i q^{52} -2.13013e6i q^{53} +157464. q^{54} +730624. q^{56} +1.18020e6i q^{57} -1.12085e6i q^{58} +935070. q^{59} +135875. q^{61} +1.18050e6i q^{62} -1.04028e6i q^{63} -262144. q^{64} -1.09512e6 q^{66} -1.44343e6i q^{67} -1.23149e6i q^{68} -85050.0 q^{69} -2.68584e6 q^{71} +373248. i q^{72} -3.28047e6i q^{73} -4.49339e6 q^{74} -2.79750e6 q^{76} +7.23489e6i q^{77} -2.57018e6i q^{78} -5.67267e6 q^{79} +531441. q^{81} -2.16269e6i q^{82} -4.22791e6i q^{83} +2.46586e6 q^{84} +1.44546e6 q^{86} -3.78286e6i q^{87} -2.59584e6i q^{88} +1.18908e7 q^{89} -1.69799e7 q^{91} -201600. i q^{92} +3.98420e6i q^{93} -779760. q^{94} -884736. q^{96} +3.80853e6i q^{97} -9.70229e6i q^{98} -3.69603e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 432 q^{6} - 1458 q^{9} + 10140 q^{11} - 22832 q^{14} + 8192 q^{16} + 87422 q^{19} - 77058 q^{21} + 27648 q^{24} - 190384 q^{26} - 280212 q^{29} + 295126 q^{31} - 307872 q^{34} + 93312 q^{36}+ \cdots - 7392060 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 432 * q^6 - 1458 * q^9 + 10140 * q^11 - 22832 * q^14 + 8192 * q^16 + 87422 * q^19 - 77058 * q^21 + 27648 * q^24 - 190384 * q^26 - 280212 * q^29 + 295126 * q^31 - 307872 * q^34 + 93312 * q^36 - 642546 * q^39 - 540672 * q^41 - 648960 * q^44 - 50400 * q^46 - 2425572 * q^49 - 1039068 * q^51 + 314928 * q^54 + 1461248 * q^56 + 1870140 * q^59 + 271750 * q^61 - 524288 * q^64 - 2190240 * q^66 - 170100 * q^69 - 5371680 * q^71 - 8986784 * q^74 - 5595008 * q^76 - 11345344 * q^79 + 1062882 * q^81 + 4931712 * q^84 + 2890928 * q^86 + 23781696 * q^89 - 33959746 * q^91 - 1559520 * q^94 - 1769472 * q^96 - 7392060 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	8.00000i	0.707107i
$2$	8.00000i	0.707107i
$3$	27.0000i	0.577350i
$3$	27.0000i	0.577350i
$4$	−64.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−216.000	−0.408248
$7$	1427.00i	1.57246i	0.617931	+	0.786232i	$0.287971\pi$
$7$	1427.00i	1.57246i	−0.617931	+	0.786232i	$0.712029\pi$
$8$	− 512.000i	− 0.353553i
$9$	−729.000	−0.333333
$10$	0	0
$11$	5070.00	1.14851	0.574253	−	0.818678i	$-0.305292\pi$
$11$	5070.00	1.14851	0.574253	+	0.818678i	$0.305292\pi$
$12$	− 1728.00i	− 0.288675i
$13$	11899.0i	1.50213i	0.660226	+	0.751067i	$0.270461\pi$
$13$	11899.0i	1.50213i	−0.660226	+	0.751067i	$0.729539\pi$
$14$	−11416.0	−1.11190
$15$	0	0
$16$	4096.00	0.250000
$17$	19242.0i	0.949902i	0.880012	+	0.474951i	$0.157534\pi$
$17$	19242.0i	0.949902i	−0.880012	+	0.474951i	$0.842466\pi$
$18$	− 5832.00i	− 0.235702i
$19$	43711.0	1.46202	0.731010	−	0.682367i	$-0.239049\pi$
$19$	43711.0	1.46202	0.731010	+	0.682367i	$0.239049\pi$
$20$	0	0
$21$	−38529.0	−0.907863
$22$	40560.0i	0.812117i
$23$	3150.00i	0.0539838i	0.999636	+	0.0269919i	$0.00859283\pi$
$23$	3150.00i	0.0539838i	−0.999636	+	0.0269919i	$0.991407\pi$
$24$	13824.0	0.204124
$25$	0	0
$26$	−95192.0	−1.06217
$27$	− 19683.0i	− 0.192450i
$28$	− 91328.0i	− 0.786232i
$29$	−140106.	−1.06675	−0.533376	−	0.845878i	$-0.679077\pi$
$29$	−140106.	−1.06675	−0.533376	+	0.845878i	$0.679077\pi$
$30$	0	0
$31$	147563.	0.889634	0.444817	−	0.895621i	$-0.353269\pi$
$31$	147563.	0.889634	0.444817	+	0.895621i	$0.353269\pi$
$32$	32768.0i	0.176777i
$33$	136890.i	0.663091i
$34$	−153936.	−0.671682
$35$	0	0
$36$	46656.0	0.166667
$37$	561674.i	1.82296i	0.411339	+	0.911482i	$0.365061\pi$
$37$	561674.i	1.82296i	−0.411339	+	0.911482i	$0.634939\pi$
$38$	349688.i	1.03380i
$39$	−321273.	−0.867258
$40$	0	0
$41$	−270336.	−0.612577	−0.306288	−	0.951939i	$-0.599087\pi$
$41$	−270336.	−0.612577	−0.306288	+	0.951939i	$0.599087\pi$
$42$	− 308232.i	− 0.641956i
$43$	− 180683.i	− 0.346559i	−0.984873	−	0.173280i	$-0.944564\pi$
$43$	− 180683.i	− 0.346559i	0.984873	−	0.173280i	$-0.0554365\pi$
$44$	−324480.	−0.574253
$45$	0	0
$46$	−25200.0	−0.0381723
$47$	97470.0i	0.136939i	0.997653	+	0.0684697i	$0.0218116\pi$
$47$	97470.0i	0.136939i	−0.997653	+	0.0684697i	$0.978188\pi$
$48$	110592.i	0.144338i
$49$	−1.21279e6	−1.47264
$50$	0	0
$51$	−519534.	−0.548426
$52$	− 761536.i	− 0.751067i
$53$	− 2.13013e6i	− 1.96535i	−0.185324	−	0.982677i	$-0.559334\pi$
$53$	− 2.13013e6i	− 1.96535i	0.185324	−	0.982677i	$-0.440666\pi$
$54$	157464.	0.136083
$55$	0	0
$56$	730624.	0.555950
$57$	1.18020e6i	0.844097i
$58$	− 1.12085e6i	− 0.754308i
$59$	935070.	0.592737	0.296369	−	0.955074i	$-0.404224\pi$
$59$	935070.	0.592737	0.296369	+	0.955074i	$0.404224\pi$
$60$	0	0
$61$	135875.	0.0766452	0.0383226	−	0.999265i	$-0.487799\pi$
$61$	135875.	0.0766452	0.0383226	+	0.999265i	$0.487799\pi$
$62$	1.18050e6i	0.629066i
$63$	− 1.04028e6i	− 0.524155i
$64$	−262144.	−0.125000
$65$	0	0
$66$	−1.09512e6	−0.468876
$67$	− 1.44343e6i	− 0.586320i	−0.956063	−	0.293160i	$-0.905293\pi$
$67$	− 1.44343e6i	− 0.586320i	0.956063	−	0.293160i	$-0.0947069\pi$
$68$	− 1.23149e6i	− 0.474951i
$69$	−85050.0	−0.0311675
$70$	0	0
$71$	−2.68584e6	−0.890586	−0.445293	−	0.895385i	$-0.646900\pi$
$71$	−2.68584e6	−0.890586	−0.445293	+	0.895385i	$0.646900\pi$
$72$	373248.i	0.117851i
$73$	− 3.28047e6i	− 0.986974i	−0.869753	−	0.493487i	$-0.835722\pi$
$73$	− 3.28047e6i	− 0.986974i	0.869753	−	0.493487i	$-0.164278\pi$
$74$	−4.49339e6	−1.28903
$75$	0	0
$76$	−2.79750e6	−0.731010
$77$	7.23489e6i	1.80599i
$78$	− 2.57018e6i	− 0.613244i
$79$	−5.67267e6	−1.29447	−0.647236	−	0.762289i	$-0.724075\pi$
$79$	−5.67267e6	−1.29447	−0.647236	+	0.762289i	$0.724075\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	− 2.16269e6i	− 0.433157i
$83$	− 4.22791e6i	− 0.811619i	−0.913958	−	0.405809i	$-0.866990\pi$
$83$	− 4.22791e6i	− 0.811619i	0.913958	−	0.405809i	$-0.133010\pi$
$84$	2.46586e6	0.453931
$85$	0	0
$86$	1.44546e6	0.245055
$87$	− 3.78286e6i	− 0.615890i
$88$	− 2.59584e6i	− 0.406058i
$89$	1.18908e7	1.78792	0.893959	−	0.448148i	$-0.147916\pi$
$89$	1.18908e7	1.78792	0.893959	+	0.448148i	$0.147916\pi$
$90$	0	0
$91$	−1.69799e7	−2.36205
$92$	− 201600.i	− 0.0269919i
$93$	3.98420e6i	0.513631i
$94$	−779760.	−0.0968308
$95$	0	0
$96$	−884736.	−0.102062
$97$	3.80853e6i	0.423698i	0.977302	+	0.211849i	$0.0679485\pi$
$97$	3.80853e6i	0.423698i	−0.977302	+	0.211849i	$0.932052\pi$
$98$	− 9.70229e6i	− 1.04132i
$99$	−3.69603e6	−0.382836
$100$	0	0
$101$	−1.20593e7	−1.16465	−0.582327	−	0.812955i	$-0.697858\pi$
$101$	−1.20593e7	−1.16465	−0.582327	+	0.812955i	$0.697858\pi$
$102$	− 4.15627e6i	− 0.387796i
$103$	− 3.61951e6i	− 0.326377i	−0.986595	−	0.163188i	$-0.947822\pi$
$103$	− 3.61951e6i	− 0.326377i	0.986595	−	0.163188i	$-0.0521778\pi$
$104$	6.09229e6	0.531085
$105$	0	0
$106$	1.70411e7	1.38972
$107$	− 2.56928e6i	− 0.202754i	−0.994848	−	0.101377i	$-0.967675\pi$
$107$	− 2.56928e6i	− 0.202754i	0.994848	−	0.101377i	$-0.0323248\pi$
$108$	1.25971e6i	0.0962250i
$109$	−6.71514e6	−0.496664	−0.248332	−	0.968675i	$-0.579882\pi$
$109$	−6.71514e6	−0.496664	−0.248332	+	0.968675i	$0.579882\pi$
$110$	0	0
$111$	−1.51652e7	−1.05249
$112$	5.84499e6i	0.393116i
$113$	− 1.37565e7i	− 0.896876i	−0.893814	−	0.448438i	$-0.851980\pi$
$113$	− 1.37565e7i	− 0.896876i	0.893814	−	0.448438i	$-0.148020\pi$
$114$	−9.44158e6	−0.596867
$115$	0	0
$116$	8.96678e6	0.533376
$117$	− 8.67437e6i	− 0.500711i
$118$	7.48056e6i	0.419128i
$119$	−2.74583e7	−1.49369
$120$	0	0
$121$	6.21773e6	0.319068
$122$	1.08700e6i	0.0541964i
$123$	− 7.29907e6i	− 0.353671i
$124$	−9.44403e6	−0.444817
$125$	0	0
$126$	8.32226e6	0.370633
$127$	4.01769e7i	1.74046i	0.492647	+	0.870229i	$0.336029\pi$
$127$	4.01769e7i	1.74046i	−0.492647	+	0.870229i	$0.663971\pi$
$128$	− 2.09715e6i	− 0.0883883i
$129$	4.87844e6	0.200086
$130$	0	0
$131$	3.44456e7	1.33870	0.669351	−	0.742946i	$-0.266572\pi$
$131$	3.44456e7	1.33870	0.669351	+	0.742946i	$0.266572\pi$
$132$	− 8.76096e6i	− 0.331545i
$133$	6.23756e7i	2.29897i
$134$	1.15475e7	0.414591
$135$	0	0
$136$	9.85190e6	0.335841
$137$	2.78217e7i	0.924405i	0.886774	+	0.462203i	$0.152941\pi$
$137$	2.78217e7i	0.924405i	−0.886774	+	0.462203i	$0.847059\pi$
$138$	− 680400.i	− 0.0220388i
$139$	8.75852e6	0.276617	0.138308	−	0.990389i	$-0.455833\pi$
$139$	8.75852e6	0.276617	0.138308	+	0.990389i	$0.455833\pi$
$140$	0	0
$141$	−2.63169e6	−0.0790620
$142$	− 2.14867e7i	− 0.629739i
$143$	6.03279e7i	1.72521i
$144$	−2.98598e6	−0.0833333
$145$	0	0
$146$	2.62437e7	0.697896
$147$	− 3.27452e7i	− 0.850232i
$148$	− 3.59471e7i	− 0.911482i
$149$	7.68194e7	1.90247	0.951237	−	0.308460i	$-0.0998137\pi$
$149$	7.68194e7	1.90247	0.951237	+	0.308460i	$0.0998137\pi$
$150$	0	0
$151$	−1.42992e7	−0.337981	−0.168991	−	0.985618i	$-0.554051\pi$
$151$	−1.42992e7	−0.337981	−0.168991	+	0.985618i	$0.554051\pi$
$152$	− 2.23800e7i	− 0.516902i
$153$	− 1.40274e7i	− 0.316634i
$154$	−5.78791e7	−1.27702
$155$	0	0
$156$	2.05615e7	0.433629
$157$	− 6.38587e7i	− 1.31696i	−0.752600	−	0.658478i	$-0.771200\pi$
$157$	− 6.38587e7i	− 1.31696i	0.752600	−	0.658478i	$-0.228800\pi$
$158$	− 4.53814e7i	− 0.915330i
$159$	5.75136e7	1.13470
$160$	0	0
$161$	−4.49505e6	−0.0848875
$162$	4.25153e6i	0.0785674i
$163$	8.81015e7i	1.59341i	0.604371	+	0.796703i	$0.293425\pi$
$163$	8.81015e7i	1.59341i	−0.604371	+	0.796703i	$0.706575\pi$
$164$	1.73015e7	0.306288
$165$	0	0
$166$	3.38232e7	0.573901
$167$	6.22383e7i	1.03407i	0.855964	+	0.517035i	$0.172964\pi$
$167$	6.22383e7i	1.03407i	−0.855964	+	0.517035i	$0.827036\pi$
$168$	1.97268e7i	0.320978i
$169$	−7.88377e7	−1.25641
$170$	0	0
$171$	−3.18653e7	−0.487340
$172$	1.15637e7i	0.173280i
$173$	8.69882e7i	1.27732i	0.769490	+	0.638659i	$0.220510\pi$
$173$	8.69882e7i	1.27732i	−0.769490	+	0.638659i	$0.779490\pi$
$174$	3.02629e7	0.435500
$175$	0	0
$176$	2.07667e7	0.287127
$177$	2.52469e7i	0.342217i
$178$	9.51268e7i	1.26425i
$179$	1.03821e8	1.35300	0.676500	−	0.736443i	$-0.263496\pi$
$179$	1.03821e8	1.35300	0.676500	+	0.736443i	$0.263496\pi$
$180$	0	0
$181$	−1.21159e7	−0.151873	−0.0759365	−	0.997113i	$-0.524195\pi$
$181$	−1.21159e7	−0.151873	−0.0759365	+	0.997113i	$0.524195\pi$
$182$	− 1.35839e8i	− 1.67022i
$183$	3.66862e6i	0.0442511i
$184$	1.61280e6	0.0190861
$185$	0	0
$186$	−3.18736e7	−0.363192
$187$	9.75569e7i	1.09097i
$188$	− 6.23808e6i	− 0.0684697i
$189$	2.80876e7	0.302621
$190$	0	0
$191$	−8.64043e7	−0.897260	−0.448630	−	0.893718i	$-0.648088\pi$
$191$	−8.64043e7	−0.897260	−0.448630	+	0.893718i	$0.648088\pi$
$192$	− 7.07789e6i	− 0.0721688i
$193$	8.89367e7i	0.890493i	0.895408	+	0.445247i	$0.146884\pi$
$193$	8.89367e7i	0.890493i	−0.895408	+	0.445247i	$0.853116\pi$
$194$	−3.04682e7	−0.299600
$195$	0	0
$196$	7.76183e7	0.736322
$197$	− 1.32518e8i	− 1.23493i	−0.786599	−	0.617465i	$-0.788160\pi$
$197$	− 1.32518e8i	− 1.23493i	0.786599	−	0.617465i	$-0.211840\pi$
$198$	− 2.95682e7i	− 0.270706i
$199$	3.94534e7	0.354894	0.177447	−	0.984130i	$-0.443216\pi$
$199$	3.94534e7	0.354894	0.177447	+	0.984130i	$0.443216\pi$
$200$	0	0
$201$	3.89727e7	0.338512
$202$	− 9.64743e7i	− 0.823535i
$203$	− 1.99931e8i	− 1.67743i
$204$	3.32502e7	0.274213
$205$	0	0
$206$	2.89561e7	0.230783
$207$	− 2.29635e6i	− 0.0179946i
$208$	4.87383e7i	0.375534i
$209$	2.21615e8	1.67914
$210$	0	0
$211$	1.93707e8	1.41957	0.709786	−	0.704417i	$-0.248792\pi$
$211$	1.93707e8	1.41957	0.709786	+	0.704417i	$0.248792\pi$
$212$	1.36328e8i	0.982677i
$213$	− 7.25177e7i	− 0.514180i
$214$	2.05543e7	0.143369
$215$	0	0
$216$	−1.00777e7	−0.0680414
$217$	2.10572e8i	1.39892i
$218$	− 5.37212e7i	− 0.351194i
$219$	8.85726e7	0.569829
$220$	0	0
$221$	−2.28961e8	−1.42688
$222$	− 1.21322e8i	− 0.744222i
$223$	− 1.36236e8i	− 0.822671i	−0.911484	−	0.411335i	$-0.865062\pi$
$223$	− 1.36236e8i	− 0.822671i	0.911484	−	0.411335i	$-0.134938\pi$
$224$	−4.67599e7	−0.277975
$225$	0	0
$226$	1.10052e8	0.634187
$227$	− 2.86433e8i	− 1.62530i	−0.582755	−	0.812648i	$-0.698025\pi$
$227$	− 2.86433e8i	− 1.62530i	0.582755	−	0.812648i	$-0.301975\pi$
$228$	− 7.55326e7i	− 0.422049i
$229$	−2.84427e8	−1.56512	−0.782558	−	0.622578i	$-0.786085\pi$
$229$	−2.84427e8	−1.56512	−0.782558	+	0.622578i	$0.786085\pi$
$230$	0	0
$231$	−1.95342e8	−1.04269
$232$	7.17343e7i	0.377154i
$233$	3.24032e8i	1.67820i	0.543981	+	0.839098i	$0.316917\pi$
$233$	3.24032e8i	1.67820i	−0.543981	+	0.839098i	$0.683083\pi$
$234$	6.93950e7	0.354056
$235$	0	0
$236$	−5.98445e7	−0.296369
$237$	− 1.53162e8i	− 0.747364i
$238$	− 2.19667e8i	− 1.05620i
$239$	−2.53375e8	−1.20053	−0.600263	−	0.799802i	$-0.704938\pi$
$239$	−2.53375e8	−1.20053	−0.600263	+	0.799802i	$0.704938\pi$
$240$	0	0
$241$	1.07795e8	0.496067	0.248034	−	0.968751i	$-0.420216\pi$
$241$	1.07795e8	0.496067	0.248034	+	0.968751i	$0.420216\pi$
$242$	4.97418e7i	0.225615i
$243$	1.43489e7i	0.0641500i
$244$	−8.69600e6	−0.0383226
$245$	0	0
$246$	5.83926e7	0.250083
$247$	5.20117e8i	2.19615i
$248$	− 7.55523e7i	− 0.314533i
$249$	1.14153e8	0.468588
$250$	0	0
$251$	2.09909e8	0.837863	0.418932	−	0.908018i	$-0.362405\pi$
$251$	2.09909e8	0.837863	0.418932	+	0.908018i	$0.362405\pi$
$252$	6.65781e7i	0.262077i
$253$	1.59705e7i	0.0620007i
$254$	−3.21415e8	−1.23069
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	− 1.88661e8i	− 0.693293i	−0.937996	−	0.346646i	$-0.887320\pi$
$257$	− 1.88661e8i	− 0.693293i	0.937996	−	0.346646i	$-0.112680\pi$
$258$	3.90275e7i	0.141482i
$259$	−8.01509e8	−2.86655
$260$	0	0
$261$	1.02137e8	0.355584
$262$	2.75565e8i	0.946605i
$263$	− 4.38730e8i	− 1.48714i	−0.668657	−	0.743571i	$-0.733131\pi$
$263$	− 4.38730e8i	− 1.48714i	0.668657	−	0.743571i	$-0.266869\pi$
$264$	7.00877e7	0.234438
$265$	0	0
$266$	−4.99005e8	−1.62562
$267$	3.21053e8i	1.03226i
$268$	9.23797e7i	0.293160i
$269$	1.19910e8	0.375598	0.187799	−	0.982207i	$-0.439865\pi$
$269$	1.19910e8	0.375598	0.187799	+	0.982207i	$0.439865\pi$
$270$	0	0
$271$	−3.99299e8	−1.21872	−0.609362	−	0.792892i	$-0.708574\pi$
$271$	−3.99299e8	−1.21872	−0.609362	+	0.792892i	$0.708574\pi$
$272$	7.88152e7i	0.237476i
$273$	− 4.58457e8i	− 1.36373i
$274$	−2.22574e8	−0.653653
$275$	0	0
$276$	5.44320e6	0.0155838
$277$	− 9.45520e7i	− 0.267295i	−0.991029	−	0.133648i	$-0.957331\pi$
$277$	− 9.45520e7i	− 0.267295i	0.991029	−	0.133648i	$-0.0426691\pi$
$278$	7.00681e7i	0.195598i
$279$	−1.07573e8	−0.296545
$280$	0	0
$281$	−2.37829e8	−0.639430	−0.319715	−	0.947514i	$-0.603587\pi$
$281$	−2.37829e8	−0.639430	−0.319715	+	0.947514i	$0.603587\pi$
$282$	− 2.10535e7i	− 0.0559053i
$283$	5.34471e8i	1.40175i	0.713282	+	0.700877i	$0.247208\pi$
$283$	5.34471e8i	1.40175i	−0.713282	+	0.700877i	$0.752792\pi$
$284$	1.71894e8	0.445293
$285$	0	0
$286$	−4.82623e8	−1.21991
$287$	− 3.85769e8i	− 0.963255i
$288$	− 2.38879e7i	− 0.0589256i
$289$	4.00841e7	0.0976854
$290$	0	0
$291$	−1.02830e8	−0.244622
$292$	2.09950e8i	0.493487i
$293$	− 6.18845e8i	− 1.43729i	−0.695375	−	0.718647i	$-0.744762\pi$
$293$	− 6.18845e8i	− 1.43729i	0.695375	−	0.718647i	$-0.255238\pi$
$294$	2.61962e8	0.601205
$295$	0	0
$296$	2.87577e8	0.644515
$297$	− 9.97928e7i	− 0.221030i
$298$	6.14555e8i	1.34525i
$299$	−3.74818e7	−0.0810909
$300$	0	0
$301$	2.57835e8	0.544952
$302$	− 1.14394e8i	− 0.238989i
$303$	− 3.25601e8i	− 0.672413i
$304$	1.79040e8	0.365505
$305$	0	0
$306$	1.12219e8	0.223894
$307$	− 1.74921e8i	− 0.345030i	−0.985007	−	0.172515i	$-0.944811\pi$
$307$	− 1.74921e8i	− 0.345030i	0.985007	−	0.172515i	$-0.0551893\pi$
$308$	− 4.63033e8i	− 0.902993i
$309$	9.77267e7	0.188434
$310$	0	0
$311$	−4.03443e7	−0.0760537	−0.0380269	−	0.999277i	$-0.512107\pi$
$311$	−4.03443e7	−0.0760537	−0.0380269	+	0.999277i	$0.512107\pi$
$312$	1.64492e8i	0.306622i
$313$	− 8.33163e8i	− 1.53577i	−0.640590	−	0.767883i	$-0.721310\pi$
$313$	− 8.33163e8i	− 1.53577i	0.640590	−	0.767883i	$-0.278690\pi$
$314$	5.10870e8	0.931229
$315$	0	0
$316$	3.63051e8	0.647236
$317$	1.06497e9i	1.87771i	0.344307	+	0.938857i	$0.388114\pi$
$317$	1.06497e9i	1.87771i	−0.344307	+	0.938857i	$0.611886\pi$
$318$	4.60109e8i	0.802353i
$319$	−7.10337e8	−1.22517
$320$	0	0
$321$	6.93707e7	0.117060
$322$	− 3.59604e7i	− 0.0600246i
$323$	8.41087e8i	1.38878i
$324$	−3.40122e7	−0.0555556
$325$	0	0
$326$	−7.04812e8	−1.12671
$327$	− 1.81309e8i	− 0.286749i
$328$	1.38412e8i	0.216579i
$329$	−1.39090e8	−0.215332
$330$	0	0
$331$	1.92701e8	0.292070	0.146035	−	0.989279i	$-0.453349\pi$
$331$	1.92701e8	0.292070	0.146035	+	0.989279i	$0.453349\pi$
$332$	2.70586e8i	0.405809i
$333$	− 4.09460e8i	− 0.607655i
$334$	−4.97906e8	−0.731198
$335$	0	0
$336$	−1.57815e8	−0.226966
$337$	2.05520e8i	0.292516i	0.989247	+	0.146258i	$0.0467229\pi$
$337$	2.05520e8i	0.292516i	−0.989247	+	0.146258i	$0.953277\pi$
$338$	− 6.30701e8i	− 0.888414i
$339$	3.71425e8	0.517812
$340$	0	0
$341$	7.48144e8	1.02175
$342$	− 2.54923e8i	− 0.344601i
$343$	− 5.55450e8i	− 0.743217i
$344$	−9.25097e7	−0.122527
$345$	0	0
$346$	−6.95905e8	−0.903200
$347$	− 1.24873e9i	− 1.60441i	−0.597050	−	0.802204i	$-0.703660\pi$
$347$	− 1.24873e9i	− 1.60441i	0.597050	−	0.802204i	$-0.296340\pi$
$348$	2.42103e8i	0.307945i
$349$	9.75467e8	1.22835	0.614177	−	0.789168i	$-0.289488\pi$
$349$	9.75467e8	1.22835	0.614177	+	0.789168i	$0.289488\pi$
$350$	0	0
$351$	2.34208e8	0.289086
$352$	1.66134e8i	0.203029i
$353$	8.12177e8i	0.982742i	0.870950	+	0.491371i	$0.163504\pi$
$353$	8.12177e8i	0.982742i	−0.870950	+	0.491371i	$0.836496\pi$
$354$	−2.01975e8	−0.241984
$355$	0	0
$356$	−7.61014e8	−0.893959
$357$	− 7.41375e8i	− 0.862381i
$358$	8.30565e8i	0.956716i
$359$	8.19316e8	0.934589	0.467294	−	0.884102i	$-0.345229\pi$
$359$	8.19316e8	0.934589	0.467294	+	0.884102i	$0.345229\pi$
$360$	0	0
$361$	1.01678e9	1.13750
$362$	− 9.69271e7i	− 0.107390i
$363$	1.67879e8i	0.184214i
$364$	1.08671e9	1.18103
$365$	0	0
$366$	−2.93490e7	−0.0312903
$367$	− 4.44464e8i	− 0.469360i	−0.972073	−	0.234680i	$-0.924596\pi$
$367$	− 4.44464e8i	− 0.469360i	0.972073	−	0.234680i	$-0.0754042\pi$
$368$	1.29024e7i	0.0134959i
$369$	1.97075e8	0.204192
$370$	0	0
$371$	3.03970e9	3.09045
$372$	− 2.54989e8i	− 0.256815i
$373$	− 5.23822e8i	− 0.522640i	−0.965252	−	0.261320i	$-0.915842\pi$
$373$	− 5.23822e8i	− 0.522640i	0.965252	−	0.261320i	$-0.0841578\pi$
$374$	−7.80456e8	−0.771432
$375$	0	0
$376$	4.99046e7	0.0484154
$377$	− 1.66712e9i	− 1.60241i
$378$	2.24701e8i	0.213985i
$379$	6.01805e8	0.567830	0.283915	−	0.958849i	$-0.408367\pi$
$379$	6.01805e8	0.567830	0.283915	+	0.958849i	$0.408367\pi$
$380$	0	0
$381$	−1.08478e9	−1.00485
$382$	− 6.91234e8i	− 0.634459i
$383$	4.67737e8i	0.425408i	0.977117	+	0.212704i	$0.0682270\pi$
$383$	4.67737e8i	0.425408i	−0.977117	+	0.212704i	$0.931773\pi$
$384$	5.66231e7	0.0510310
$385$	0	0
$386$	−7.11494e8	−0.629674
$387$	1.31718e8i	0.115520i
$388$	− 2.43746e8i	− 0.211849i
$389$	1.11784e9	0.962841	0.481421	−	0.876490i	$-0.340121\pi$
$389$	1.11784e9	0.962841	0.481421	+	0.876490i	$0.340121\pi$
$390$	0	0
$391$	−6.06123e7	−0.0512793
$392$	6.20946e8i	0.520658i
$393$	9.30031e8i	0.772900i
$394$	1.06014e9	0.873227
$395$	0	0
$396$	2.36546e8	0.191418
$397$	− 1.01062e9i	− 0.810626i	−0.914178	−	0.405313i	$-0.867162\pi$
$397$	− 1.01062e9i	− 0.810626i	0.914178	−	0.405313i	$-0.132838\pi$
$398$	3.15627e8i	0.250948i
$399$	−1.68414e9	−1.32731
$400$	0	0
$401$	−4.30852e8	−0.333675	−0.166837	−	0.985984i	$-0.553355\pi$
$401$	−4.30852e8	−0.333675	−0.166837	+	0.985984i	$0.553355\pi$
$402$	3.11782e8i	0.239364i
$403$	1.75585e9i	1.33635i
$404$	7.71795e8	0.582327
$405$	0	0
$406$	1.59945e9	1.18612
$407$	2.84769e9i	2.09369i
$408$	2.66001e8i	0.193898i
$409$	−7.35181e8	−0.531328	−0.265664	−	0.964066i	$-0.585591\pi$
$409$	−7.35181e8	−0.531328	−0.265664	+	0.964066i	$0.585591\pi$
$410$	0	0
$411$	−7.51187e8	−0.533706
$412$	2.31649e8i	0.163188i
$413$	1.33434e9i	0.932058i
$414$	1.83708e7	0.0127241
$415$	0	0
$416$	−3.89906e8	−0.265542
$417$	2.36480e8i	0.159705i
$418$	1.77292e9i	1.18733i
$419$	7.97408e8	0.529580	0.264790	−	0.964306i	$-0.414697\pi$
$419$	7.97408e8	0.529580	0.264790	+	0.964306i	$0.414697\pi$
$420$	0	0
$421$	−2.16840e8	−0.141629	−0.0708144	−	0.997490i	$-0.522560\pi$
$421$	−2.16840e8	−0.141629	−0.0708144	+	0.997490i	$0.522560\pi$
$422$	1.54966e9i	1.00379i
$423$	− 7.10556e7i	− 0.0456465i
$424$	−1.09063e9	−0.694858
$425$	0	0
$426$	5.80141e8	0.363580
$427$	1.93894e8i	0.120522i
$428$	1.64434e8i	0.101377i
$429$	−1.62885e9	−0.996051
$430$	0	0
$431$	1.05196e9	0.632890	0.316445	−	0.948611i	$-0.397511\pi$
$431$	1.05196e9	0.632890	0.316445	+	0.948611i	$0.397511\pi$
$432$	− 8.06216e7i	− 0.0481125i
$433$	− 8.42421e8i	− 0.498680i	−0.968416	−	0.249340i	$-0.919786\pi$
$433$	− 8.42421e8i	− 0.498680i	0.968416	−	0.249340i	$-0.0802136\pi$
$434$	−1.68458e9	−0.989185
$435$	0	0
$436$	4.29769e8	0.248332
$437$	1.37690e8i	0.0789253i
$438$	7.08581e8i	0.402930i
$439$	−8.78820e8	−0.495763	−0.247882	−	0.968790i	$-0.579734\pi$
$439$	−8.78820e8	−0.495763	−0.247882	+	0.968790i	$0.579734\pi$
$440$	0	0
$441$	8.84121e8	0.490881
$442$	− 1.83168e9i	− 1.00896i
$443$	2.09919e9i	1.14720i	0.819136	+	0.573599i	$0.194453\pi$
$443$	2.09919e9i	1.14720i	−0.819136	+	0.573599i	$0.805547\pi$
$444$	9.70573e8	0.526245
$445$	0	0
$446$	1.08989e9	0.581716
$447$	2.07412e9i	1.09839i
$448$	− 3.74079e8i	− 0.196558i
$449$	1.30937e9	0.682656	0.341328	−	0.939944i	$-0.389123\pi$
$449$	1.30937e9	0.682656	0.341328	+	0.939944i	$0.389123\pi$
$450$	0	0
$451$	−1.37060e9	−0.703548
$452$	8.80414e8i	0.448438i
$453$	− 3.86079e8i	− 0.195134i
$454$	2.29146e9	1.14926
$455$	0	0
$456$	6.04261e8	0.298433
$457$	− 2.54163e9i	− 1.24568i	−0.782350	−	0.622839i	$-0.785979\pi$
$457$	− 2.54163e9i	− 1.24568i	0.782350	−	0.622839i	$-0.214021\pi$
$458$	− 2.27541e9i	− 1.10670i
$459$	3.78740e8	0.182809
$460$	0	0
$461$	5.86141e8	0.278644	0.139322	−	0.990247i	$-0.455508\pi$
$461$	5.86141e8	0.278644	0.139322	+	0.990247i	$0.455508\pi$
$462$	− 1.56274e9i	− 0.737291i
$463$	− 1.57279e9i	− 0.736439i	−0.929739	−	0.368220i	$-0.879967\pi$
$463$	− 1.57279e9i	− 0.736439i	0.929739	−	0.368220i	$-0.120033\pi$
$464$	−5.73874e8	−0.266688
$465$	0	0
$466$	−2.59226e9	−1.18666
$467$	1.58046e9i	0.718083i	0.933322	+	0.359042i	$0.116896\pi$
$467$	1.58046e9i	0.718083i	−0.933322	+	0.359042i	$0.883104\pi$
$468$	5.55160e8i	0.250356i
$469$	2.05978e9	0.921968
$470$	0	0
$471$	1.72419e9	0.760345
$472$	− 4.78756e8i	− 0.209564i
$473$	− 9.16063e8i	− 0.398026i
$474$	1.22530e9	0.528466
$475$	0	0
$476$	1.75733e9	0.746844
$477$	1.55287e9i	0.655118i
$478$	− 2.02700e9i	− 0.848901i
$479$	−3.70114e9	−1.53873	−0.769364	−	0.638811i	$-0.779427\pi$
$479$	−3.70114e9	−1.53873	−0.769364	+	0.638811i	$0.779427\pi$
$480$	0	0
$481$	−6.68336e9	−2.73834
$482$	8.62363e8i	0.350773i
$483$	− 1.21366e8i	− 0.0490098i
$484$	−3.97935e8	−0.159534
$485$	0	0
$486$	−1.14791e8	−0.0453609
$487$	− 9.78261e8i	− 0.383798i	−0.981415	−	0.191899i	$-0.938535\pi$
$487$	− 9.78261e8i	− 0.383798i	0.981415	−	0.191899i	$-0.0614647\pi$
$488$	− 6.95680e7i	− 0.0270982i
$489$	−2.37874e9	−0.919954
$490$	0	0
$491$	−8.54921e8	−0.325942	−0.162971	−	0.986631i	$-0.552108\pi$
$491$	−8.54921e8	−0.325942	−0.162971	+	0.986631i	$0.552108\pi$
$492$	4.67141e8i	0.176836i
$493$	− 2.69592e9i	− 1.01331i
$494$	−4.16094e9	−1.55291
$495$	0	0
$496$	6.04418e8	0.222409
$497$	− 3.83269e9i	− 1.40041i
$498$	9.13228e8i	0.331342i
$499$	2.00214e9	0.721345	0.360672	−	0.932693i	$-0.382547\pi$
$499$	2.00214e9	0.721345	0.360672	+	0.932693i	$0.382547\pi$
$500$	0	0
$501$	−1.68043e9	−0.597021
$502$	1.67927e9i	0.592459i
$503$	3.51483e9i	1.23145i	0.787962	+	0.615724i	$0.211136\pi$
$503$	3.51483e9i	1.23145i	−0.787962	+	0.615724i	$0.788864\pi$
$504$	−5.32625e8	−0.185317
$505$	0	0
$506$	−1.27764e8	−0.0438411
$507$	− 2.12862e9i	− 0.725387i
$508$	− 2.57132e9i	− 0.870229i
$509$	3.70703e9	1.24599	0.622995	−	0.782226i	$-0.285916\pi$
$509$	3.70703e9	1.24599	0.622995	+	0.782226i	$0.285916\pi$
$510$	0	0
$511$	4.68122e9	1.55198
$512$	1.34218e8i	0.0441942i
$513$	− 8.60364e8i	− 0.281366i
$514$	1.50929e9	0.490232
$515$	0	0
$516$	−3.12220e8	−0.100043
$517$	4.94173e8i	0.157276i
$518$	− 6.41207e9i	− 2.02696i
$519$	−2.34868e9	−0.737460
$520$	0	0
$521$	−1.31752e9	−0.408156	−0.204078	−	0.978955i	$-0.565420\pi$
$521$	−1.31752e9	−0.408156	−0.204078	+	0.978955i	$0.565420\pi$
$522$	8.17098e8i	0.251436i
$523$	4.45839e8i	0.136277i	0.997676	+	0.0681385i	$0.0217060\pi$
$523$	4.45839e8i	0.136277i	−0.997676	+	0.0681385i	$0.978294\pi$
$524$	−2.20452e9	−0.669351
$525$	0	0
$526$	3.50984e9	1.05157
$527$	2.83941e9i	0.845066i
$528$	5.60701e8i	0.165773i
$529$	3.39490e9	0.997086
$530$	0	0
$531$	−6.81666e8	−0.197579
$532$	− 3.99204e9i	− 1.14949i
$533$	− 3.21673e9i	− 0.920172i
$534$	−2.56842e9	−0.729915
$535$	0	0
$536$	−7.39038e8	−0.207295
$537$	2.80316e9i	0.781155i
$538$	9.59282e8i	0.265588i
$539$	−6.14883e9	−1.69134
$540$	0	0
$541$	−7.31894e9	−1.98727	−0.993637	−	0.112631i	$-0.964072\pi$
$541$	−7.31894e9	−1.98727	−0.993637	+	0.112631i	$0.964072\pi$
$542$	− 3.19439e9i	− 0.861768i
$543$	− 3.27129e8i	− 0.0876839i
$544$	−6.30522e8	−0.167921
$545$	0	0
$546$	3.66765e9	0.964304
$547$	5.21813e9i	1.36320i	0.731725	+	0.681600i	$0.238715\pi$
$547$	5.21813e9i	1.36320i	−0.731725	+	0.681600i	$0.761285\pi$
$548$	− 1.78059e9i	− 0.462203i
$549$	−9.90529e7	−0.0255484
$550$	0	0
$551$	−6.12417e9	−1.55961
$552$	4.35456e7i	0.0110194i
$553$	− 8.09490e9i	− 2.03551i
$554$	7.56416e8	0.189006
$555$	0	0
$556$	−5.60545e8	−0.138308
$557$	5.22742e7i	0.0128172i	0.999979	+	0.00640862i	$0.00203994\pi$
$557$	5.22742e7i	0.0128172i	−0.999979	+	0.00640862i	$0.997960\pi$
$558$	− 8.60587e8i	− 0.209689i
$559$	2.14995e9	0.520579
$560$	0	0
$561$	−2.63404e9	−0.629871
$562$	− 1.90263e9i	− 0.452145i
$563$	7.63499e9i	1.80314i	0.432636	+	0.901569i	$0.357584\pi$
$563$	7.63499e9i	1.80314i	−0.432636	+	0.901569i	$0.642416\pi$
$564$	1.68428e8	0.0395310
$565$	0	0
$566$	−4.27577e9	−0.991190
$567$	7.58366e8i	0.174718i
$568$	1.37515e9i	0.314870i
$569$	2.52617e9	0.574871	0.287435	−	0.957800i	$-0.407197\pi$
$569$	2.52617e9	0.574871	0.287435	+	0.957800i	$0.407197\pi$
$570$	0	0
$571$	−4.75335e9	−1.06850	−0.534248	−	0.845328i	$-0.679405\pi$
$571$	−4.75335e9	−1.06850	−0.534248	+	0.845328i	$0.679405\pi$
$572$	− 3.86099e9i	− 0.862606i
$573$	− 2.33291e9i	− 0.518033i
$574$	3.08616e9	0.681124
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	7.07337e9i	1.53289i	0.642309	+	0.766445i	$0.277976\pi$
$577$	7.07337e9i	1.53289i	−0.642309	+	0.766445i	$0.722024\pi$
$578$	3.20673e8i	0.0690740i
$579$	−2.40129e9	−0.514127
$580$	0	0
$581$	6.03322e9	1.27624
$582$	− 8.22642e8i	− 0.172974i
$583$	− 1.07998e10i	− 2.25722i
$584$	−1.67960e9	−0.348948
$585$	0	0
$586$	4.95076e9	1.01632
$587$	− 6.82995e8i	− 0.139375i	−0.997569	−	0.0696874i	$-0.977800\pi$
$587$	− 6.82995e8i	− 0.139375i	0.997569	−	0.0696874i	$-0.0222002\pi$
$588$	2.09569e9i	0.425116i
$589$	6.45013e9	1.30066
$590$	0	0
$591$	3.57798e9	0.712987
$592$	2.30062e9i	0.455741i
$593$	− 4.11466e9i	− 0.810294i	−0.914251	−	0.405147i	$-0.867220\pi$
$593$	− 4.11466e9i	− 0.810294i	0.914251	−	0.405147i	$-0.132780\pi$
$594$	7.98342e8	0.156292
$595$	0	0
$596$	−4.91644e9	−0.951237
$597$	1.06524e9i	0.204898i
$598$	− 2.99855e8i	− 0.0573399i
$599$	2.71835e8	0.0516787	0.0258394	−	0.999666i	$-0.491774\pi$
$599$	2.71835e8	0.0516787	0.0258394	+	0.999666i	$0.491774\pi$
$600$	0	0
$601$	−1.63646e9	−0.307500	−0.153750	−	0.988110i	$-0.549135\pi$
$601$	−1.63646e9	−0.307500	−0.153750	+	0.988110i	$0.549135\pi$
$602$	2.06268e9i	0.385340i
$603$	1.05226e9i	0.195440i
$604$	9.15149e8	0.168991
$605$	0	0
$606$	2.60481e9	0.475468
$607$	− 7.25840e8i	− 0.131729i	−0.997829	−	0.0658644i	$-0.979020\pi$
$607$	− 7.25840e8i	− 0.131729i	0.997829	−	0.0658644i	$-0.0209805\pi$
$608$	1.43232e9i	0.258451i
$609$	5.39814e9	0.968465
$610$	0	0
$611$	−1.15980e9	−0.205701
$612$	8.97755e8i	0.158317i
$613$	9.79507e9i	1.71750i	0.512397	+	0.858749i	$0.328758\pi$
$613$	9.79507e9i	1.71750i	−0.512397	+	0.858749i	$0.671242\pi$
$614$	1.39937e9	0.243973
$615$	0	0
$616$	3.70426e9	0.638512
$617$	− 7.54736e8i	− 0.129359i	−0.997906	−	0.0646796i	$-0.979397\pi$
$617$	− 7.54736e8i	− 0.129359i	0.997906	−	0.0646796i	$-0.0206025\pi$
$618$	7.81814e8i	0.133243i
$619$	−9.48066e9	−1.60665	−0.803325	−	0.595541i	$-0.796938\pi$
$619$	−9.48066e9	−1.60665	−0.803325	+	0.595541i	$0.796938\pi$
$620$	0	0
$621$	6.20014e7	0.0103892
$622$	− 3.22754e8i	− 0.0537781i
$623$	1.69682e10i	2.81144i
$624$	−1.31593e9	−0.216814
$625$	0	0
$626$	6.66531e9	1.08595
$627$	5.98360e9i	0.969451i
$628$	4.08696e9i	0.658478i
$629$	−1.08077e10	−1.73164
$630$	0	0
$631$	2.11281e8	0.0334779	0.0167389	−	0.999860i	$-0.494672\pi$
$631$	2.11281e8	0.0334779	0.0167389	+	0.999860i	$0.494672\pi$
$632$	2.90441e9i	0.457665i
$633$	5.23010e9i	0.819590i
$634$	−8.51975e9	−1.32774
$635$	0	0
$636$	−3.68087e9	−0.567349
$637$	− 1.44309e10i	− 2.21211i
$638$	− 5.68270e9i	− 0.866328i
$639$	1.95798e9	0.296862
$640$	0	0
$641$	7.52664e9	1.12875	0.564375	−	0.825518i	$-0.309117\pi$
$641$	7.52664e9	1.12875	0.564375	+	0.825518i	$0.309117\pi$
$642$	5.54965e8i	0.0827739i
$643$	3.52375e9i	0.522717i	0.965242	+	0.261358i	$0.0841704\pi$
$643$	3.52375e9i	0.522717i	−0.965242	+	0.261358i	$0.915830\pi$
$644$	2.87683e8	0.0424438
$645$	0	0
$646$	−6.72870e9	−0.982013
$647$	3.90185e9i	0.566377i	0.959064	+	0.283189i	$0.0913922\pi$
$647$	3.90185e9i	0.566377i	−0.959064	+	0.283189i	$0.908608\pi$
$648$	− 2.72098e8i	− 0.0392837i
$649$	4.74080e9	0.680763
$650$	0	0
$651$	−5.68545e9	−0.807666
$652$	− 5.63850e9i	− 0.796703i
$653$	8.20768e9i	1.15352i	0.816915	+	0.576759i	$0.195683\pi$
$653$	8.20768e9i	1.15352i	−0.816915	+	0.576759i	$0.804317\pi$
$654$	1.45047e9	0.202762
$655$	0	0
$656$	−1.10730e9	−0.153144
$657$	2.39146e9i	0.328991i
$658$	− 1.11272e9i	− 0.152263i
$659$	−4.49137e9	−0.611336	−0.305668	−	0.952138i	$-0.598880\pi$
$659$	−4.49137e9	−0.611336	−0.305668	+	0.952138i	$0.598880\pi$
$660$	0	0
$661$	9.78484e9	1.31780	0.658898	−	0.752232i	$-0.271023\pi$
$661$	9.78484e9	1.31780	0.658898	+	0.752232i	$0.271023\pi$
$662$	1.54161e9i	0.206525i
$663$	− 6.18194e9i	− 0.823810i
$664$	−2.16469e9	−0.286951
$665$	0	0
$666$	3.27568e9	0.429677
$667$	− 4.41334e8i	− 0.0575873i
$668$	− 3.98325e9i	− 0.517035i
$669$	3.67838e9	0.474969
$670$	0	0
$671$	6.88886e8	0.0880276
$672$	− 1.26252e9i	− 0.160489i
$673$	1.99688e9i	0.252522i	0.991997	+	0.126261i	$0.0402978\pi$
$673$	1.99688e9i	0.252522i	−0.991997	+	0.126261i	$0.959702\pi$
$674$	−1.64416e9	−0.206840
$675$	0	0
$676$	5.04561e9	0.628204
$677$	− 1.76852e9i	− 0.219053i	−0.993984	−	0.109526i	$-0.965067\pi$
$677$	− 1.76852e9i	− 0.219053i	0.993984	−	0.109526i	$-0.0349334\pi$
$678$	2.97140e9i	0.366148i
$679$	−5.43477e9	−0.666250
$680$	0	0
$681$	7.73369e9	0.938365
$682$	5.98516e9i	0.722487i
$683$	− 1.33982e10i	− 1.60906i	−0.593910	−	0.804531i	$-0.702417\pi$
$683$	− 1.33982e10i	− 1.60906i	0.593910	−	0.804531i	$-0.297583\pi$
$684$	2.03938e9	0.243670
$685$	0	0
$686$	4.44360e9	0.525533
$687$	− 7.67952e9i	− 0.903620i
$688$	− 7.40078e8i	− 0.0866399i
$689$	2.53464e10	2.95223
$690$	0	0
$691$	1.36908e10	1.57854	0.789270	−	0.614046i	$-0.210459\pi$
$691$	1.36908e10	1.57854	0.789270	+	0.614046i	$0.210459\pi$
$692$	− 5.56724e9i	− 0.638659i
$693$	− 5.27423e9i	− 0.601995i
$694$	9.98983e9	1.13449
$695$	0	0
$696$	−1.93683e9	−0.217750
$697$	− 5.20181e9i	− 0.581888i
$698$	7.80374e9i	0.868578i
$699$	−8.74887e9	−0.968907
$700$	0	0
$701$	−8.98742e9	−0.985422	−0.492711	−	0.870193i	$-0.663994\pi$
$701$	−8.98742e9	−0.985422	−0.492711	+	0.870193i	$0.663994\pi$
$702$	1.87366e9i	0.204415i
$703$	2.45513e10i	2.66521i
$704$	−1.32907e9	−0.143563
$705$	0	0
$706$	−6.49742e9	−0.694903
$707$	− 1.72086e10i	− 1.83138i
$708$	− 1.61580e9i	− 0.171108i
$709$	4.36499e9	0.459962	0.229981	−	0.973195i	$-0.426134\pi$
$709$	4.36499e9	0.459962	0.229981	+	0.973195i	$0.426134\pi$
$710$	0	0
$711$	4.13538e9	0.431491
$712$	− 6.08811e9i	− 0.632125i
$713$	4.64823e8i	0.0480258i
$714$	5.93100e9	0.609795
$715$	0	0
$716$	−6.64452e9	−0.676500
$717$	− 6.84114e9i	− 0.693125i
$718$	6.55452e9i	0.660854i
$719$	1.46045e10	1.46534	0.732668	−	0.680587i	$-0.238275\pi$
$719$	1.46045e10	1.46534	0.732668	+	0.680587i	$0.238275\pi$
$720$	0	0
$721$	5.16504e9	0.513216
$722$	8.13424e9i	0.804335i
$723$	2.91048e9i	0.286405i
$724$	7.75417e8	0.0759365
$725$	0	0
$726$	−1.34303e9	−0.130259
$727$	1.42461e10i	1.37507i	0.726150	+	0.687536i	$0.241308\pi$
$727$	1.42461e10i	1.37507i	−0.726150	+	0.687536i	$0.758692\pi$
$728$	8.69369e9i	0.835112i
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	3.47670e9	0.329198
$732$	− 2.34792e8i	− 0.0221256i
$733$	− 8.78749e9i	− 0.824140i	−0.911152	−	0.412070i	$-0.864806\pi$
$733$	− 8.78749e9i	− 0.824140i	0.911152	−	0.412070i	$-0.135194\pi$
$734$	3.55571e9	0.331887
$735$	0	0
$736$	−1.03219e8	−0.00954307
$737$	− 7.31821e9i	− 0.673393i
$738$	1.57660e9i	0.144386i
$739$	−1.59660e10	−1.45526	−0.727632	−	0.685968i	$-0.759379\pi$
$739$	−1.59660e10	−1.45526	−0.727632	+	0.685968i	$0.759379\pi$
$740$	0	0
$741$	−1.40432e10	−1.26795
$742$	2.43176e10i	2.18528i
$743$	− 4.01209e9i	− 0.358847i	−0.983772	−	0.179424i	$-0.942577\pi$
$743$	− 4.01209e9i	− 0.358847i	0.983772	−	0.179424i	$-0.0574232\pi$
$744$	2.03991e9	0.181596
$745$	0	0
$746$	4.19057e9	0.369562
$747$	3.08214e9i	0.270540i
$748$	− 6.24364e9i	− 0.545485i
$749$	3.66637e9	0.318823
$750$	0	0
$751$	7.41663e9	0.638950	0.319475	−	0.947595i	$-0.396493\pi$
$751$	7.41663e9	0.638950	0.319475	+	0.947595i	$0.396493\pi$
$752$	3.99237e8i	0.0342349i
$753$	5.66755e9i	0.483741i
$754$	1.33370e10	1.13307
$755$	0	0
$756$	−1.79761e9	−0.151310
$757$	1.51143e8i	0.0126634i	0.999980	+	0.00633171i	$0.00201546\pi$
$757$	1.51143e8i	0.0126634i	−0.999980	+	0.00633171i	$0.997985\pi$
$758$	4.81444e9i	0.401516i
$759$	−4.31204e8	−0.0357961
$760$	0	0
$761$	3.85268e9	0.316896	0.158448	−	0.987367i	$-0.449351\pi$
$761$	3.85268e9	0.316896	0.158448	+	0.987367i	$0.449351\pi$
$762$	− 8.67821e9i	− 0.710539i
$763$	− 9.58251e9i	− 0.780986i
$764$	5.52987e9	0.448630
$765$	0	0
$766$	−3.74189e9	−0.300809
$767$	1.11264e10i	0.890371i
$768$	4.52985e8i	0.0360844i
$769$	−1.56192e10	−1.23856	−0.619280	−	0.785170i	$-0.712576\pi$
$769$	−1.56192e10	−1.23856	−0.619280	+	0.785170i	$0.712576\pi$
$770$	0	0
$771$	5.09385e9	0.400273
$772$	− 5.69195e9i	− 0.445247i
$773$	− 1.44828e10i	− 1.12778i	−0.825850	−	0.563889i	$-0.809304\pi$
$773$	− 1.44828e10i	− 1.12778i	0.825850	−	0.563889i	$-0.190696\pi$
$774$	−1.05374e9	−0.0816848
$775$	0	0
$776$	1.94997e9	0.149800
$777$	− 2.16407e10i	− 1.65500i
$778$	8.94269e9i	0.680831i
$779$	−1.18167e10	−0.895599
$780$	0	0
$781$	−1.36172e10	−1.02284
$782$	− 4.84898e8i	− 0.0362599i
$783$	2.75771e9i	0.205297i
$784$	−4.96757e9	−0.368161
$785$	0	0
$786$	−7.44025e9	−0.546523
$787$	1.37729e10i	1.00720i	0.863938	+	0.503599i	$0.167991\pi$
$787$	1.37729e10i	1.00720i	−0.863938	+	0.503599i	$0.832009\pi$
$788$	8.48113e9i	0.617465i
$789$	1.18457e10	0.858601
$790$	0	0
$791$	1.96305e10	1.41031
$792$	1.89237e9i	0.135353i
$793$	1.61678e9i	0.115131i
$794$	8.08495e9	0.573199
$795$	0	0
$796$	−2.52502e9	−0.177447
$797$	− 6.88529e9i	− 0.481746i	−0.970557	−	0.240873i	$-0.922566\pi$
$797$	− 6.88529e9i	− 0.481746i	0.970557	−	0.240873i	$-0.0774338\pi$
$798$	− 1.34731e10i	− 0.938552i
$799$	−1.87552e9	−0.130079
$800$	0	0
$801$	−8.66843e9	−0.595973
$802$	− 3.44682e9i	− 0.235944i
$803$	− 1.66320e10i	− 1.13355i
$804$	−2.49425e9	−0.169256
$805$	0	0
$806$	−1.40468e10	−0.944942
$807$	3.23758e9i	0.216852i
$808$	6.17436e9i	0.411767i
$809$	1.28256e10	0.851644	0.425822	−	0.904807i	$-0.359985\pi$
$809$	1.28256e10	0.851644	0.425822	+	0.904807i	$0.359985\pi$
$810$	0	0
$811$	−7.36094e9	−0.484574	−0.242287	−	0.970205i	$-0.577898\pi$
$811$	−7.36094e9	−0.484574	−0.242287	+	0.970205i	$0.577898\pi$
$812$	1.27956e10i	0.838715i
$813$	− 1.07811e10i	− 0.703631i
$814$	−2.27815e10	−1.48046
$815$	0	0
$816$	−2.12801e9	−0.137107
$817$	− 7.89783e9i	− 0.506677i
$818$	− 5.88144e9i	− 0.375705i
$819$	1.23783e10	0.787351
$820$	0	0
$821$	−1.09919e10	−0.693219	−0.346610	−	0.938009i	$-0.612667\pi$
$821$	−1.09919e10	−0.693219	−0.346610	+	0.938009i	$0.612667\pi$
$822$	− 6.00950e9i	− 0.377387i
$823$	1.15788e10i	0.724041i	0.932170	+	0.362020i	$0.117913\pi$
$823$	1.15788e10i	0.724041i	−0.932170	+	0.362020i	$0.882087\pi$
$824$	−1.85319e9	−0.115392
$825$	0	0
$826$	−1.06748e10	−0.659064
$827$	4.97823e9i	0.306059i	0.988222	+	0.153030i	$0.0489030\pi$
$827$	4.97823e9i	0.306059i	−0.988222	+	0.153030i	$0.951097\pi$
$828$	1.46966e8i	0.00899729i
$829$	1.15589e10	0.704655	0.352328	−	0.935877i	$-0.385390\pi$
$829$	1.15589e10	0.704655	0.352328	+	0.935877i	$0.385390\pi$
$830$	0	0
$831$	2.55290e9	0.154323
$832$	− 3.11925e9i	− 0.187767i
$833$	− 2.33364e10i	− 1.39887i
$834$	−1.89184e9	−0.112928
$835$	0	0
$836$	−1.41833e10	−0.839570
$837$	− 2.90448e9i	− 0.171210i
$838$	6.37926e9i	0.374469i
$839$	−4.50868e9	−0.263562	−0.131781	−	0.991279i	$-0.542070\pi$
$839$	−4.50868e9	−0.263562	−0.131781	+	0.991279i	$0.542070\pi$
$840$	0	0
$841$	2.37981e9	0.137961
$842$	− 1.73472e9i	− 0.100147i
$843$	− 6.42139e9i	− 0.369175i
$844$	−1.23973e10	−0.709786
$845$	0	0
$846$	5.68445e8	0.0322769
$847$	8.87270e9i	0.501723i
$848$	− 8.72502e9i	− 0.491339i
$849$	−1.44307e10	−0.809303
$850$	0	0
$851$	−1.76927e9	−0.0984105
$852$	4.64113e9i	0.257090i
$853$	9.35143e9i	0.515889i	0.966160	+	0.257945i	$0.0830452\pi$
$853$	9.35143e9i	0.515889i	−0.966160	+	0.257945i	$0.916955\pi$
$854$	−1.55115e9	−0.0852219
$855$	0	0
$856$	−1.31547e9	−0.0716843
$857$	1.85745e9i	0.100806i	0.998729	+	0.0504029i	$0.0160505\pi$
$857$	1.85745e9i	0.100806i	−0.998729	+	0.0504029i	$0.983949\pi$
$858$	− 1.30308e10i	− 0.704315i
$859$	−3.34304e10	−1.79956	−0.899779	−	0.436346i	$-0.856272\pi$
$859$	−3.34304e10	−1.79956	−0.899779	+	0.436346i	$0.856272\pi$
$860$	0	0
$861$	1.04158e10	0.556135
$862$	8.41567e9i	0.447521i
$863$	− 1.34259e10i	− 0.711061i	−0.934665	−	0.355531i	$-0.884300\pi$
$863$	− 1.34259e10i	− 0.711061i	0.934665	−	0.355531i	$-0.115700\pi$
$864$	6.44973e8	0.0340207
$865$	0	0
$866$	6.73937e9	0.352620
$867$	1.08227e9i	0.0563987i
$868$	− 1.34766e10i	− 0.699459i
$869$	−2.87604e10	−1.48671
$870$	0	0
$871$	1.71754e10	0.880732
$872$	3.43815e9i	0.175597i
$873$	− 2.77642e9i	− 0.141233i
$874$	−1.10152e9	−0.0558086
$875$	0	0
$876$	−5.66865e9	−0.284915
$877$	3.08992e9i	0.154685i	0.997005	+	0.0773426i	$0.0246435\pi$
$877$	3.08992e9i	0.154685i	−0.997005	+	0.0773426i	$0.975356\pi$
$878$	− 7.03056e9i	− 0.350558i
$879$	1.67088e10	0.829822
$880$	0	0
$881$	8.72396e9	0.429831	0.214916	−	0.976633i	$-0.431052\pi$
$881$	8.72396e9	0.429831	0.214916	+	0.976633i	$0.431052\pi$
$882$	7.07297e9i	0.347106i
$883$	3.90723e9i	0.190988i	0.995430	+	0.0954942i	$0.0304431\pi$
$883$	3.90723e9i	0.190988i	−0.995430	+	0.0954942i	$0.969557\pi$
$884$	1.46535e10	0.713440
$885$	0	0
$886$	−1.67935e10	−0.811192
$887$	1.53854e10i	0.740246i	0.928983	+	0.370123i	$0.120684\pi$
$887$	1.53854e10i	0.740246i	−0.928983	+	0.370123i	$0.879316\pi$
$888$	7.76458e9i	0.372111i
$889$	−5.73325e10	−2.73681
$890$	0	0
$891$	2.69441e9	0.127612
$892$	8.71913e9i	0.411335i
$893$	4.26051e9i	0.200208i
$894$	−1.65930e10	−0.776682
$895$	0	0
$896$	2.99264e9	0.138988
$897$	− 1.01201e9i	− 0.0468178i
$898$	1.04750e10i	0.482711i
$899$	−2.06745e10	−0.949020
$900$	0	0
$901$	4.09880e10	1.86690
$902$	− 1.09648e10i	− 0.497484i
$903$	6.96154e9i	0.314628i
$904$	−7.04332e9	−0.317094
$905$	0	0
$906$	3.08863e9	0.137980
$907$	− 2.63986e10i	− 1.17478i	−0.809305	−	0.587388i	$-0.800156\pi$
$907$	− 2.63986e10i	− 1.17478i	0.809305	−	0.587388i	$-0.199844\pi$
$908$	1.83317e10i	0.812648i
$909$	8.79122e9	0.388218
$910$	0	0
$911$	−2.12340e10	−0.930503	−0.465251	−	0.885179i	$-0.654036\pi$
$911$	−2.12340e10	−0.930503	−0.465251	+	0.885179i	$0.654036\pi$
$912$	4.83409e9i	0.211024i
$913$	− 2.14355e10i	− 0.932149i
$914$	2.03331e10	0.880828
$915$	0	0
$916$	1.82033e10	0.782558
$917$	4.91538e10i	2.10506i
$918$	3.02992e9i	0.129265i
$919$	−9.50036e9	−0.403771	−0.201886	−	0.979409i	$-0.564707\pi$
$919$	−9.50036e9	−0.403771	−0.201886	+	0.979409i	$0.564707\pi$
$920$	0	0
$921$	4.72286e9	0.199203
$922$	4.68913e9i	0.197031i
$923$	− 3.19588e10i	− 1.33778i
$924$	1.25019e10	0.521343
$925$	0	0
$926$	1.25823e10	0.520741
$927$	2.63862e9i	0.108792i
$928$	− 4.59099e9i	− 0.188577i
$929$	4.29492e10	1.75752	0.878760	−	0.477264i	$-0.158371\pi$
$929$	4.29492e10	1.75752	0.878760	+	0.477264i	$0.158371\pi$
$930$	0	0
$931$	−5.30121e10	−2.15303
$932$	− 2.07381e10i	− 0.839098i
$933$	− 1.08930e9i	− 0.0439096i
$934$	−1.26437e10	−0.507761
$935$	0	0
$936$	−4.44128e9	−0.177028
$937$	1.52405e10i	0.605216i	0.953115	+	0.302608i	$0.0978573\pi$
$937$	1.52405e10i	0.605216i	−0.953115	+	0.302608i	$0.902143\pi$
$938$	1.64782e10i	0.651930i
$939$	2.24954e10	0.886675
$940$	0	0
$941$	−2.27448e10	−0.889853	−0.444926	−	0.895567i	$-0.646770\pi$
$941$	−2.27448e10	−0.889853	−0.444926	+	0.895567i	$0.646770\pi$
$942$	1.37935e10i	0.537645i
$943$	− 8.51558e8i	− 0.0330692i
$944$	3.83005e9	0.148184
$945$	0	0
$946$	7.32850e9	0.281447
$947$	− 8.80667e9i	− 0.336967i	−0.985705	−	0.168483i	$-0.946113\pi$
$947$	− 8.80667e9i	− 0.336967i	0.985705	−	0.168483i	$-0.0538869\pi$
$948$	9.80238e9i	0.373682i
$949$	3.90343e10	1.48257
$950$	0	0
$951$	−2.87542e10	−1.08410
$952$	1.40587e10i	0.528098i
$953$	1.37186e9i	0.0513434i	0.999670	+	0.0256717i	$0.00817246\pi$
$953$	1.37186e9i	0.0513434i	−0.999670	+	0.0256717i	$0.991828\pi$
$954$	−1.24229e10	−0.463239
$955$	0	0
$956$	1.62160e10	0.600263
$957$	− 1.91791e10i	− 0.707354i
$958$	− 2.96092e10i	− 1.08804i
$959$	−3.97016e10	−1.45359
$960$	0	0
$961$	−5.73778e9	−0.208551
$962$	− 5.34669e10i	− 1.93630i
$963$	1.87301e9i	0.0675846i
$964$	−6.89891e9	−0.248034
$965$	0	0
$966$	9.70931e8	0.0346552
$967$	2.51907e10i	0.895874i	0.894065	+	0.447937i	$0.147841\pi$
$967$	2.51907e10i	0.895874i	−0.894065	+	0.447937i	$0.852159\pi$
$968$	− 3.18348e9i	− 0.112808i
$969$	−2.27094e10	−0.801810
$970$	0	0
$971$	−5.11915e9	−0.179445	−0.0897223	−	0.995967i	$-0.528598\pi$
$971$	−5.11915e9	−0.179445	−0.0897223	+	0.995967i	$0.528598\pi$
$972$	− 9.18330e8i	− 0.0320750i
$973$	1.24984e10i	0.434970i
$974$	7.82608e9	0.271386
$975$	0	0
$976$	5.56544e8	0.0191613
$977$	5.06512e8i	0.0173764i	0.999962	+	0.00868818i	$0.00276557\pi$
$977$	5.06512e8i	0.0173764i	−0.999962	+	0.00868818i	$0.997234\pi$
$978$	− 1.90299e10i	− 0.650506i
$979$	6.02866e10	2.05344
$980$	0	0
$981$	4.89534e9	0.165555
$982$	− 6.83937e9i	− 0.230476i
$983$	1.00756e10i	0.338323i	0.985588	+	0.169162i	$0.0541060\pi$
$983$	1.00756e10i	0.338323i	−0.985588	+	0.169162i	$0.945894\pi$
$984$	−3.73712e9	−0.125042
$985$	0	0
$986$	2.15674e10	0.716519
$987$	− 3.75542e9i	− 0.124322i
$988$	− 3.32875e10i	− 1.09807i
$989$	5.69151e8	0.0187086
$990$	0	0
$991$	4.56127e10	1.48877	0.744386	−	0.667749i	$-0.232742\pi$
$991$	4.56127e10	1.48877	0.744386	+	0.667749i	$0.232742\pi$
$992$	4.83534e9i	0.157267i
$993$	5.20294e9i	0.168627i
$994$	3.06615e10	0.990243
$995$	0	0
$996$	−7.30582e9	−0.234294
$997$	3.05555e10i	0.976463i	0.872714	+	0.488232i	$0.162358\pi$
$997$	3.05555e10i	0.976463i	−0.872714	+	0.488232i	$0.837642\pi$
$998$	1.60171e10i	0.510068i
$999$	1.10554e10	0.350830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.8.c.d.49.2		2
3.2	odd	2		450.8.c.d.199.1		2
5.2	odd	4		150.8.a.f.1.1	✓	1
5.3	odd	4		150.8.a.l.1.1	yes	1
5.4	even	2	inner	150.8.c.d.49.1		2
15.2	even	4		450.8.a.o.1.1		1
15.8	even	4		450.8.a.m.1.1		1
15.14	odd	2		450.8.c.d.199.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.8.a.f.1.1	✓	1	5.2	odd	4
150.8.a.l.1.1	yes	1	5.3	odd	4
150.8.c.d.49.1		2	5.4	even	2	inner
150.8.c.d.49.2		2	1.1	even	1	trivial
450.8.a.m.1.1		1	15.8	even	4
450.8.a.o.1.1		1	15.2	even	4
450.8.c.d.199.1		2	3.2	odd	2
450.8.c.d.199.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} +27.0000i q^{3} -64.0000 q^{4} -216.000 q^{6} +1427.00i q^{7} -512.000i q^{8} -729.000 q^{9} +5070.00 q^{11} -1728.00i q^{12} +11899.0i q^{13} -11416.0 q^{14} +4096.00 q^{16} +19242.0i q^{17} -5832.00i q^{18} +43711.0 q^{19} -38529.0 q^{21} +40560.0i q^{22} +3150.00i q^{23} +13824.0 q^{24} -95192.0 q^{26} -19683.0i q^{27} -91328.0i q^{28} -140106. q^{29} +147563. q^{31} +32768.0i q^{32} +136890. i q^{33} -153936. q^{34} +46656.0 q^{36} +561674. i q^{37} +349688. i q^{38} -321273. q^{39} -270336. q^{41} -308232. i q^{42} -180683. i q^{43} -324480. q^{44} -25200.0 q^{46} +97470.0i q^{47} +110592. i q^{48} -1.21279e6 q^{49} -519534. q^{51} -761536. i q^{52} -2.13013e6i q^{53} +157464. q^{54} +730624. q^{56} +1.18020e6i q^{57} -1.12085e6i q^{58} +935070. q^{59} +135875. q^{61} +1.18050e6i q^{62} -1.04028e6i q^{63} -262144. q^{64} -1.09512e6 q^{66} -1.44343e6i q^{67} -1.23149e6i q^{68} -85050.0 q^{69} -2.68584e6 q^{71} +373248. i q^{72} -3.28047e6i q^{73} -4.49339e6 q^{74} -2.79750e6 q^{76} +7.23489e6i q^{77} -2.57018e6i q^{78} -5.67267e6 q^{79} +531441. q^{81} -2.16269e6i q^{82} -4.22791e6i q^{83} +2.46586e6 q^{84} +1.44546e6 q^{86} -3.78286e6i q^{87} -2.59584e6i q^{88} +1.18908e7 q^{89} -1.69799e7 q^{91} -201600. i q^{92} +3.98420e6i q^{93} -779760. q^{94} -884736. q^{96} +3.80853e6i q^{97} -9.70229e6i q^{98} -3.69603e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 432 q^{6} - 1458 q^{9} + 10140 q^{11} - 22832 q^{14} + 8192 q^{16} + 87422 q^{19} - 77058 q^{21} + 27648 q^{24} - 190384 q^{26} - 280212 q^{29} + 295126 q^{31} - 307872 q^{34} + 93312 q^{36}+ \cdots - 7392060 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 432 * q^6 - 1458 * q^9 + 10140 * q^11 - 22832 * q^14 + 8192 * q^16 + 87422 * q^19 - 77058 * q^21 + 27648 * q^24 - 190384 * q^26 - 280212 * q^29 + 295126 * q^31 - 307872 * q^34 + 93312 * q^36 - 642546 * q^39 - 540672 * q^41 - 648960 * q^44 - 50400 * q^46 - 2425572 * q^49 - 1039068 * q^51 + 314928 * q^54 + 1461248 * q^56 + 1870140 * q^59 + 271750 * q^61 - 524288 * q^64 - 2190240 * q^66 - 170100 * q^69 - 5371680 * q^71 - 8986784 * q^74 - 5595008 * q^76 - 11345344 * q^79 + 1062882 * q^81 + 4931712 * q^84 + 2890928 * q^86 + 23781696 * q^89 - 33959746 * q^91 - 1559520 * q^94 - 1769472 * q^96 - 7392060 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more