LMFDB - Embedded newform 150.8.c.c.49.1

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000i q^{2} -27.0000i q^{3} -64.0000 q^{4} -216.000 q^{6} +713.000i q^{7} +512.000i q^{8} -729.000 q^{9} +3810.00 q^{11} +1728.00i q^{12} +391.000i q^{13} +5704.00 q^{14} +4096.00 q^{16} -4182.00i q^{17} +5832.00i q^{18} +1561.00 q^{19} +19251.0 q^{21} -30480.0i q^{22} -114150. i q^{23} +13824.0 q^{24} +3128.00 q^{26} +19683.0i q^{27} -45632.0i q^{28} +83214.0 q^{29} -83167.0 q^{31} -32768.0i q^{32} -102870. i q^{33} -33456.0 q^{34} +46656.0 q^{36} -231334. i q^{37} -12488.0i q^{38} +10557.0 q^{39} -124656. q^{41} -154008. i q^{42} -193757. i q^{43} -243840. q^{44} -913200. q^{46} +319290. i q^{47} -110592. i q^{48} +315174. q^{49} -112914. q^{51} -25024.0i q^{52} -1.64543e6i q^{53} +157464. q^{54} -365056. q^{56} -42147.0i q^{57} -665712. i q^{58} +38610.0 q^{59} -1.97390e6 q^{61} +665336. i q^{62} -519777. i q^{63} -262144. q^{64} -822960. q^{66} +4.40975e6i q^{67} +267648. i q^{68} -3.08205e6 q^{69} +124080. q^{71} -373248. i q^{72} -3.96763e6i q^{73} -1.85067e6 q^{74} -99904.0 q^{76} +2.71653e6i q^{77} -84456.0i q^{78} -7.10799e6 q^{79} +531441. q^{81} +997248. i q^{82} -8.11769e6i q^{83} -1.23206e6 q^{84} -1.55006e6 q^{86} -2.24678e6i q^{87} +1.95072e6i q^{88} -6.72787e6 q^{89} -278783. q^{91} +7.30560e6i q^{92} +2.24551e6i q^{93} +2.55432e6 q^{94} -884736. q^{96} -1.42687e7i q^{97} -2.52139e6i q^{98} -2.77749e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 432 q^{6} - 1458 q^{9} + 7620 q^{11} + 11408 q^{14} + 8192 q^{16} + 3122 q^{19} + 38502 q^{21} + 27648 q^{24} + 6256 q^{26} + 166428 q^{29} - 166334 q^{31} - 66912 q^{34} + 93312 q^{36}+ \cdots - 5554980 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 432 * q^6 - 1458 * q^9 + 7620 * q^11 + 11408 * q^14 + 8192 * q^16 + 3122 * q^19 + 38502 * q^21 + 27648 * q^24 + 6256 * q^26 + 166428 * q^29 - 166334 * q^31 - 66912 * q^34 + 93312 * q^36 + 21114 * q^39 - 249312 * q^41 - 487680 * q^44 - 1826400 * q^46 + 630348 * q^49 - 225828 * q^51 + 314928 * q^54 - 730112 * q^56 + 77220 * q^59 - 3947810 * q^61 - 524288 * q^64 - 1645920 * q^66 - 6164100 * q^69 + 248160 * q^71 - 3701344 * q^74 - 199808 * q^76 - 14215984 * q^79 + 1062882 * q^81 - 2464128 * q^84 - 3100112 * q^86 - 13455744 * q^89 - 557566 * q^91 + 5108640 * q^94 - 1769472 * q^96 - 5554980 * 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
$7$	713.000i	0.785681i	0.919607	+	0.392841i	$0.128508\pi$
$7$	713.000i	0.785681i	−0.919607	+	0.392841i	$0.871492\pi$
$8$	512.000i	0.353553i
$9$	−729.000	−0.333333
$10$	0	0
$11$	3810.00	0.863079	0.431540	−	0.902094i	$-0.357971\pi$
$11$	3810.00	0.863079	0.431540	+	0.902094i	$0.357971\pi$
$12$	1728.00i	0.288675i
$13$	391.000i	0.0493600i	0.999695	+	0.0246800i	$0.00785668\pi$
$13$	391.000i	0.0493600i	−0.999695	+	0.0246800i	$0.992143\pi$
$14$	5704.00	0.555561
$15$	0	0
$16$	4096.00	0.250000
$17$	− 4182.00i	− 0.206449i	−0.994658	−	0.103225i	$-0.967084\pi$
$17$	− 4182.00i	− 0.206449i	0.994658	−	0.103225i	$-0.0329160\pi$
$18$	5832.00i	0.235702i
$19$	1561.00	0.0522114	0.0261057	−	0.999659i	$-0.491689\pi$
$19$	1561.00	0.0522114	0.0261057	+	0.999659i	$0.491689\pi$
$20$	0	0
$21$	19251.0	0.453613
$22$	− 30480.0i	− 0.610289i
$23$	− 114150.i	− 1.95627i	−0.207974	−	0.978134i	$-0.566687\pi$
$23$	− 114150.i	− 1.95627i	0.207974	−	0.978134i	$-0.433313\pi$
$24$	13824.0	0.204124
$25$	0	0
$26$	3128.00	0.0349028
$27$	19683.0i	0.192450i
$28$	− 45632.0i	− 0.392841i
$29$	83214.0	0.633583	0.316791	−	0.948495i	$-0.397394\pi$
$29$	83214.0	0.633583	0.316791	+	0.948495i	$0.397394\pi$
$30$	0	0
$31$	−83167.0	−0.501401	−0.250700	−	0.968065i	$-0.580661\pi$
$31$	−83167.0	−0.501401	−0.250700	+	0.968065i	$0.580661\pi$
$32$	− 32768.0i	− 0.176777i
$33$	− 102870.i	− 0.498299i
$34$	−33456.0	−0.145981
$35$	0	0
$36$	46656.0	0.166667
$37$	− 231334.i	− 0.750816i	−0.926860	−	0.375408i	$-0.877503\pi$
$37$	− 231334.i	− 0.750816i	0.926860	−	0.375408i	$-0.122497\pi$
$38$	− 12488.0i	− 0.0369190i
$39$	10557.0	0.0284980
$40$	0	0
$41$	−124656.	−0.282468	−0.141234	−	0.989976i	$-0.545107\pi$
$41$	−124656.	−0.282468	−0.141234	+	0.989976i	$0.545107\pi$
$42$	− 154008.i	− 0.320753i
$43$	− 193757.i	− 0.371636i	−0.982584	−	0.185818i	$-0.940507\pi$
$43$	− 193757.i	− 0.371636i	0.982584	−	0.185818i	$-0.0594935\pi$
$44$	−243840.	−0.431540
$45$	0	0
$46$	−913200.	−1.38329
$47$	319290.i	0.448583i	0.974522	+	0.224292i	$0.0720068\pi$
$47$	319290.i	0.448583i	−0.974522	+	0.224292i	$0.927993\pi$
$48$	− 110592.i	− 0.144338i
$49$	315174.	0.382705
$50$	0	0
$51$	−112914.	−0.119193
$52$	− 25024.0i	− 0.0246800i
$53$	− 1.64543e6i	− 1.51815i	−0.651006	−	0.759073i	$-0.725653\pi$
$53$	− 1.64543e6i	− 1.51815i	0.651006	−	0.759073i	$-0.274347\pi$
$54$	157464.	0.136083
$55$	0	0
$56$	−365056.	−0.277780
$57$	− 42147.0i	− 0.0301443i
$58$	− 665712.i	− 0.448011i
$59$	38610.0	0.0244747	0.0122374	−	0.999925i	$-0.496105\pi$
$59$	38610.0	0.0244747	0.0122374	+	0.999925i	$0.496105\pi$
$60$	0	0
$61$	−1.97390e6	−1.11345	−0.556726	−	0.830696i	$-0.687943\pi$
$61$	−1.97390e6	−1.11345	−0.556726	+	0.830696i	$0.687943\pi$
$62$	665336.i	0.354544i
$63$	− 519777.i	− 0.261894i
$64$	−262144.	−0.125000
$65$	0	0
$66$	−822960.	−0.352351
$67$	4.40975e6i	1.79123i	0.444825	+	0.895617i	$0.353266\pi$
$67$	4.40975e6i	1.79123i	−0.444825	+	0.895617i	$0.646734\pi$
$68$	267648.i	0.103225i
$69$	−3.08205e6	−1.12945
$70$	0	0
$71$	124080.	0.0411432	0.0205716	−	0.999788i	$-0.493451\pi$
$71$	124080.	0.0411432	0.0205716	+	0.999788i	$0.493451\pi$
$72$	− 373248.i	− 0.117851i
$73$	− 3.96763e6i	− 1.19372i	−0.802346	−	0.596859i	$-0.796415\pi$
$73$	− 3.96763e6i	− 1.19372i	0.802346	−	0.596859i	$-0.203585\pi$
$74$	−1.85067e6	−0.530907
$75$	0	0
$76$	−99904.0	−0.0261057
$77$	2.71653e6i	0.678105i
$78$	− 84456.0i	− 0.0201511i
$79$	−7.10799e6	−1.62200	−0.811002	−	0.585043i	$-0.801078\pi$
$79$	−7.10799e6	−1.62200	−0.811002	+	0.585043i	$0.801078\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	997248.i	0.199735i
$83$	− 8.11769e6i	− 1.55833i	−0.626819	−	0.779165i	$-0.715643\pi$
$83$	− 8.11769e6i	− 1.55833i	0.626819	−	0.779165i	$-0.284357\pi$
$84$	−1.23206e6	−0.226807
$85$	0	0
$86$	−1.55006e6	−0.262786
$87$	− 2.24678e6i	− 0.365799i
$88$	1.95072e6i	0.305145i
$89$	−6.72787e6	−1.01161	−0.505805	−	0.862648i	$-0.668804\pi$
$89$	−6.72787e6	−1.01161	−0.505805	+	0.862648i	$0.668804\pi$
$90$	0	0
$91$	−278783.	−0.0387812
$92$	7.30560e6i	0.978134i
$93$	2.24551e6i	0.289484i
$94$	2.55432e6	0.317196
$95$	0	0
$96$	−884736.	−0.102062
$97$	− 1.42687e7i	− 1.58739i	−0.608318	−	0.793693i	$-0.708156\pi$
$97$	− 1.42687e7i	− 1.58739i	0.608318	−	0.793693i	$-0.291844\pi$
$98$	− 2.52139e6i	− 0.270613i
$99$	−2.77749e6	−0.287693
$100$	0	0
$101$	6.93119e6	0.669396	0.334698	−	0.942326i	$-0.391366\pi$
$101$	6.93119e6	0.669396	0.334698	+	0.942326i	$0.391366\pi$
$102$	903312.i	0.0842825i
$103$	− 1.34707e7i	− 1.21467i	−0.794444	−	0.607337i	$-0.792238\pi$
$103$	− 1.34707e7i	− 1.21467i	0.794444	−	0.607337i	$-0.207762\pi$
$104$	−200192.	−0.0174514
$105$	0	0
$106$	−1.31634e7	−1.07349
$107$	1.05625e7i	0.833533i	0.909014	+	0.416767i	$0.136837\pi$
$107$	1.05625e7i	0.833533i	−0.909014	+	0.416767i	$0.863163\pi$
$108$	− 1.25971e6i	− 0.0962250i
$109$	−6.74796e6	−0.499091	−0.249546	−	0.968363i	$-0.580281\pi$
$109$	−6.74796e6	−0.499091	−0.249546	+	0.968363i	$0.580281\pi$
$110$	0	0
$111$	−6.24602e6	−0.433484
$112$	2.92045e6i	0.196420i
$113$	− 730764.i	− 0.0476434i	−0.999716	−	0.0238217i	$-0.992417\pi$
$113$	− 730764.i	− 0.0476434i	0.999716	−	0.0238217i	$-0.00758340\pi$
$114$	−337176.	−0.0213152
$115$	0	0
$116$	−5.32570e6	−0.316791
$117$	− 285039.i	− 0.0164533i
$118$	− 308880.i	− 0.0173062i
$119$	2.98177e6	0.162203
$120$	0	0
$121$	−4.97107e6	−0.255095
$122$	1.57912e7i	0.787330i
$123$	3.36571e6i	0.163083i
$124$	5.32269e6	0.250700
$125$	0	0
$126$	−4.15822e6	−0.185187
$127$	− 3.53961e7i	− 1.53335i	−0.642034	−	0.766676i	$-0.721909\pi$
$127$	− 3.53961e7i	− 1.53335i	0.642034	−	0.766676i	$-0.278091\pi$
$128$	2.09715e6i	0.0883883i
$129$	−5.23144e6	−0.214564
$130$	0	0
$131$	2.58059e7	1.00293	0.501464	−	0.865178i	$-0.332795\pi$
$131$	2.58059e7	1.00293	0.501464	+	0.865178i	$0.332795\pi$
$132$	6.58368e6i	0.249149i
$133$	1.11299e6i	0.0410215i
$134$	3.52780e7	1.26659
$135$	0	0
$136$	2.14118e6	0.0729907
$137$	1.09054e7i	0.362344i	0.983451	+	0.181172i	$0.0579891\pi$
$137$	1.09054e7i	0.362344i	−0.983451	+	0.181172i	$0.942011\pi$
$138$	2.46564e7i	0.798643i
$139$	5.32412e7	1.68150	0.840748	−	0.541426i	$-0.182115\pi$
$139$	5.32412e7	1.68150	0.840748	+	0.541426i	$0.182115\pi$
$140$	0	0
$141$	8.62083e6	0.258990
$142$	− 992640.i	− 0.0290926i
$143$	1.48971e6i	0.0426016i
$144$	−2.98598e6	−0.0833333
$145$	0	0
$146$	−3.17411e7	−0.844086
$147$	− 8.50970e6i	− 0.220955i
$148$	1.48054e7i	0.375408i
$149$	3.39941e7	0.841884	0.420942	−	0.907088i	$-0.361700\pi$
$149$	3.39941e7	0.841884	0.420942	+	0.907088i	$0.361700\pi$
$150$	0	0
$151$	3.13410e7	0.740786	0.370393	−	0.928875i	$-0.379223\pi$
$151$	3.13410e7	0.740786	0.370393	+	0.928875i	$0.379223\pi$
$152$	799232.i	0.0184595i
$153$	3.04868e6i	0.0688163i
$154$	2.17322e7	0.479493
$155$	0	0
$156$	−675648.	−0.0142490
$157$	− 6.83485e7i	− 1.40955i	−0.709431	−	0.704775i	$-0.751048\pi$
$157$	− 6.83485e7i	− 1.40955i	0.709431	−	0.704775i	$-0.248952\pi$
$158$	5.68639e7i	1.14693i
$159$	−4.44266e7	−0.876502
$160$	0	0
$161$	8.13890e7	1.53700
$162$	− 4.25153e6i	− 0.0785674i
$163$	− 4.62777e7i	− 0.836980i	−0.908221	−	0.418490i	$-0.862560\pi$
$163$	− 4.62777e7i	− 0.836980i	0.908221	−	0.418490i	$-0.137440\pi$
$164$	7.97798e6	0.141234
$165$	0	0
$166$	−6.49416e7	−1.10191
$167$	− 3.99639e7i	− 0.663988i	−0.943282	−	0.331994i	$-0.892279\pi$
$167$	− 3.99639e7i	− 0.663988i	0.943282	−	0.331994i	$-0.107721\pi$
$168$	9.85651e6i	0.160377i
$169$	6.25956e7	0.997564
$170$	0	0
$171$	−1.13797e6	−0.0174038
$172$	1.24004e7i	0.185818i
$173$	9.07312e7i	1.33228i	0.745827	+	0.666139i	$0.232054\pi$
$173$	9.07312e7i	1.33228i	−0.745827	+	0.666139i	$0.767946\pi$
$174$	−1.79742e7	−0.258659
$175$	0	0
$176$	1.56058e7	0.215770
$177$	− 1.04247e6i	− 0.0141305i
$178$	5.38230e7i	0.715316i
$179$	−1.30570e8	−1.70160	−0.850802	−	0.525486i	$-0.823883\pi$
$179$	−1.30570e8	−1.70160	−0.850802	+	0.525486i	$0.823883\pi$
$180$	0	0
$181$	1.11281e8	1.39490	0.697452	−	0.716632i	$-0.254317\pi$
$181$	1.11281e8	1.39490	0.697452	+	0.716632i	$0.254317\pi$
$182$	2.23026e6i	0.0274225i
$183$	5.32954e7i	0.642852i
$184$	5.84448e7	0.691645
$185$	0	0
$186$	1.79641e7	0.204696
$187$	− 1.59334e7i	− 0.178182i
$188$	− 2.04346e7i	− 0.224292i
$189$	−1.40340e7	−0.151204
$190$	0	0
$191$	1.39416e8	1.44776	0.723880	−	0.689926i	$-0.242357\pi$
$191$	1.39416e8	1.44776	0.723880	+	0.689926i	$0.242357\pi$
$192$	7.07789e6i	0.0721688i
$193$	2.37893e7i	0.238194i	0.992883	+	0.119097i	$0.0379999\pi$
$193$	2.37893e7i	0.238194i	−0.992883	+	0.119097i	$0.962000\pi$
$194$	−1.14149e8	−1.12245
$195$	0	0
$196$	−2.01711e7	−0.191352
$197$	4.67913e7i	0.436047i	0.975943	+	0.218024i	$0.0699610\pi$
$197$	4.67913e7i	0.436047i	−0.975943	+	0.218024i	$0.930039\pi$
$198$	2.22199e7i	0.203430i
$199$	1.16854e8	1.05113	0.525566	−	0.850753i	$-0.323854\pi$
$199$	1.16854e8	1.05113	0.525566	+	0.850753i	$0.323854\pi$
$200$	0	0
$201$	1.19063e8	1.03417
$202$	− 5.54495e7i	− 0.473334i
$203$	5.93316e7i	0.497794i
$204$	7.22650e6	0.0595967
$205$	0	0
$206$	−1.07766e8	−0.858904
$207$	8.32154e7i	0.652090i
$208$	1.60154e6i	0.0123400i
$209$	5.94741e6	0.0450626
$210$	0	0
$211$	−1.83300e8	−1.34330	−0.671651	−	0.740867i	$-0.734415\pi$
$211$	−1.83300e8	−1.34330	−0.671651	+	0.740867i	$0.734415\pi$
$212$	1.05307e8i	0.759073i
$213$	− 3.35016e6i	− 0.0237540i
$214$	8.44999e7	0.589397
$215$	0	0
$216$	−1.00777e7	−0.0680414
$217$	− 5.92981e7i	− 0.393941i
$218$	5.39837e7i	0.352911i
$219$	−1.07126e8	−0.689193
$220$	0	0
$221$	1.63516e6	0.0101903
$222$	4.99681e7i	0.306519i
$223$	− 1.91650e8i	− 1.15729i	−0.815580	−	0.578644i	$-0.803582\pi$
$223$	− 1.91650e8i	− 1.15729i	0.815580	−	0.578644i	$-0.196418\pi$
$224$	2.33636e7	0.138890
$225$	0	0
$226$	−5.84611e6	−0.0336890
$227$	− 1.11119e8i	− 0.630521i	−0.949005	−	0.315260i	$-0.897908\pi$
$227$	− 1.11119e8i	− 0.630521i	0.949005	−	0.315260i	$-0.102092\pi$
$228$	2.69741e6i	0.0150721i
$229$	8.92220e7	0.490962	0.245481	−	0.969401i	$-0.421054\pi$
$229$	8.92220e7	0.490962	0.245481	+	0.969401i	$0.421054\pi$
$230$	0	0
$231$	7.33463e7	0.391504
$232$	4.26056e7i	0.224005i
$233$	5.04671e7i	0.261374i	0.991424	+	0.130687i	$0.0417183\pi$
$233$	5.04671e7i	0.261374i	−0.991424	+	0.130687i	$0.958282\pi$
$234$	−2.28031e6	−0.0116343
$235$	0	0
$236$	−2.47104e6	−0.0122374
$237$	1.91916e8i	0.936465i
$238$	− 2.38541e7i	− 0.114695i
$239$	−8.16563e7	−0.386899	−0.193449	−	0.981110i	$-0.561967\pi$
$239$	−8.16563e7	−0.386899	−0.193449	+	0.981110i	$0.561967\pi$
$240$	0	0
$241$	−2.40323e8	−1.10595	−0.552976	−	0.833197i	$-0.686508\pi$
$241$	−2.40323e8	−1.10595	−0.552976	+	0.833197i	$0.686508\pi$
$242$	3.97686e7i	0.180379i
$243$	− 1.43489e7i	− 0.0641500i
$244$	1.26330e8	0.556726
$245$	0	0
$246$	2.69257e7	0.115317
$247$	610351.i	0.00257715i
$248$	− 4.25815e7i	− 0.177272i
$249$	−2.19178e8	−0.899702
$250$	0	0
$251$	−3.23741e8	−1.29223	−0.646115	−	0.763240i	$-0.723607\pi$
$251$	−3.23741e8	−1.29223	−0.646115	+	0.763240i	$0.723607\pi$
$252$	3.32657e7i	0.130947i
$253$	− 4.34912e8i	− 1.68841i
$254$	−2.83169e8	−1.08424
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	− 8.10385e7i	− 0.297800i	−0.988852	−	0.148900i	$-0.952427\pi$
$257$	− 8.10385e7i	− 0.297800i	0.988852	−	0.148900i	$-0.0475733\pi$
$258$	4.18515e7i	0.151720i
$259$	1.64941e8	0.589902
$260$	0	0
$261$	−6.06630e7	−0.211194
$262$	− 2.06448e8i	− 0.709178i
$263$	− 2.11263e8i	− 0.716106i	−0.933701	−	0.358053i	$-0.883441\pi$
$263$	− 2.11263e8i	− 0.716106i	0.933701	−	0.358053i	$-0.116559\pi$
$264$	5.26694e7	0.176175
$265$	0	0
$266$	8.90394e6	0.0290066
$267$	1.81653e8i	0.584053i
$268$	− 2.82224e8i	− 0.895617i
$269$	−4.20832e8	−1.31818	−0.659092	−	0.752062i	$-0.729059\pi$
$269$	−4.20832e8	−1.31818	−0.659092	+	0.752062i	$0.729059\pi$
$270$	0	0
$271$	−2.13859e8	−0.652731	−0.326366	−	0.945244i	$-0.605824\pi$
$271$	−2.13859e8	−0.652731	−0.326366	+	0.945244i	$0.605824\pi$
$272$	− 1.71295e7i	− 0.0516123i
$273$	7.52714e6i	0.0223903i
$274$	8.72435e7	0.256216
$275$	0	0
$276$	1.97251e8	0.564726
$277$	− 2.94664e8i	− 0.833007i	−0.909134	−	0.416503i	$-0.863255\pi$
$277$	− 2.94664e8i	− 0.833007i	0.909134	−	0.416503i	$-0.136745\pi$
$278$	− 4.25930e8i	− 1.18900i
$279$	6.06287e7	0.167134
$280$	0	0
$281$	−2.81001e8	−0.755503	−0.377752	−	0.925907i	$-0.623303\pi$
$281$	−2.81001e8	−0.755503	−0.377752	+	0.925907i	$0.623303\pi$
$282$	− 6.89666e7i	− 0.183133i
$283$	− 5.81465e8i	− 1.52500i	−0.646986	−	0.762502i	$-0.723971\pi$
$283$	− 5.81465e8i	− 1.52500i	0.646986	−	0.762502i	$-0.276029\pi$
$284$	−7.94112e6	−0.0205716
$285$	0	0
$286$	1.19177e7	0.0301239
$287$	− 8.88797e7i	− 0.221930i
$288$	2.38879e7i	0.0589256i
$289$	3.92850e8	0.957379
$290$	0	0
$291$	−3.85254e8	−0.916478
$292$	2.53929e8i	0.596859i
$293$	7.92049e8i	1.83957i	0.392427	+	0.919783i	$0.371635\pi$
$293$	7.92049e8i	1.83957i	−0.392427	+	0.919783i	$0.628365\pi$
$294$	−6.80776e7	−0.156239
$295$	0	0
$296$	1.18443e8	0.265453
$297$	7.49922e7i	0.166100i
$298$	− 2.71953e8i	− 0.595302i
$299$	4.46326e7	0.0965614
$300$	0	0
$301$	1.38149e8	0.291987
$302$	− 2.50728e8i	− 0.523815i
$303$	− 1.87142e8i	− 0.386476i
$304$	6.39386e6	0.0130528
$305$	0	0
$306$	2.43894e7	0.0486605
$307$	2.03330e8i	0.401067i	0.979687	+	0.200533i	$0.0642675\pi$
$307$	2.03330e8i	0.401067i	−0.979687	+	0.200533i	$0.935732\pi$
$308$	− 1.73858e8i	− 0.339053i
$309$	−3.63709e8	−0.701292
$310$	0	0
$311$	3.29428e8	0.621011	0.310505	−	0.950572i	$-0.399502\pi$
$311$	3.29428e8	0.621011	0.310505	+	0.950572i	$0.399502\pi$
$312$	5.40518e6i	0.0100756i
$313$	− 1.61495e8i	− 0.297684i	−0.988861	−	0.148842i	$-0.952445\pi$
$313$	− 1.61495e8i	− 0.297684i	0.988861	−	0.148842i	$-0.0475545\pi$
$314$	−5.46788e8	−0.996702
$315$	0	0
$316$	4.54911e8	0.811002
$317$	6.39147e8i	1.12692i	0.826143	+	0.563460i	$0.190530\pi$
$317$	6.39147e8i	1.12692i	−0.826143	+	0.563460i	$0.809470\pi$
$318$	3.55412e8i	0.619780i
$319$	3.17045e8	0.546832
$320$	0	0
$321$	2.85187e8	0.481241
$322$	− 6.51112e8i	− 1.08683i
$323$	− 6.52810e6i	− 0.0107790i
$324$	−3.40122e7	−0.0555556
$325$	0	0
$326$	−3.70221e8	−0.591834
$327$	1.82195e8i	0.288151i
$328$	− 6.38239e7i	− 0.0998676i
$329$	−2.27654e8	−0.352443
$330$	0	0
$331$	−5.39401e8	−0.817548	−0.408774	−	0.912636i	$-0.634044\pi$
$331$	−5.39401e8	−0.817548	−0.408774	+	0.912636i	$0.634044\pi$
$332$	5.19532e8i	0.779165i
$333$	1.68642e8i	0.250272i
$334$	−3.19711e8	−0.469510
$335$	0	0
$336$	7.88521e7	0.113403
$337$	7.72336e8i	1.09926i	0.835407	+	0.549632i	$0.185232\pi$
$337$	7.72336e8i	1.09926i	−0.835407	+	0.549632i	$0.814768\pi$
$338$	− 5.00765e8i	− 0.705384i
$339$	−1.97306e7	−0.0275069
$340$	0	0
$341$	−3.16866e8	−0.432749
$342$	9.10375e6i	0.0123063i
$343$	8.11905e8i	1.08637i
$344$	9.92036e7	0.131393
$345$	0	0
$346$	7.25849e8	0.942063
$347$	5.21028e8i	0.669435i	0.942319	+	0.334717i	$0.108641\pi$
$347$	5.21028e8i	0.669435i	−0.942319	+	0.334717i	$0.891359\pi$
$348$	1.43794e8i	0.182900i
$349$	−5.22374e8	−0.657798	−0.328899	−	0.944365i	$-0.606678\pi$
$349$	−5.22374e8	−0.657798	−0.328899	+	0.944365i	$0.606678\pi$
$350$	0	0
$351$	−7.69605e6	−0.00949933
$352$	− 1.24846e8i	− 0.152572i
$353$	1.02256e9i	1.23731i	0.785662	+	0.618656i	$0.212323\pi$
$353$	1.02256e9i	1.23731i	−0.785662	+	0.618656i	$0.787677\pi$
$354$	−8.33976e6	−0.00999176
$355$	0	0
$356$	4.30584e8	0.505805
$357$	− 8.05077e7i	− 0.0936480i
$358$	1.04456e9i	1.20322i
$359$	1.17896e9	1.34484	0.672418	−	0.740171i	$-0.265256\pi$
$359$	1.17896e9	1.34484	0.672418	+	0.740171i	$0.265256\pi$
$360$	0	0
$361$	−8.91435e8	−0.997274
$362$	− 8.90244e8i	− 0.986346i
$363$	1.34219e8i	0.147279i
$364$	1.78421e7	0.0193906
$365$	0	0
$366$	4.26363e8	0.454565
$367$	− 7.85348e8i	− 0.829337i	−0.909973	−	0.414668i	$-0.863898\pi$
$367$	− 7.85348e8i	− 0.829337i	0.909973	−	0.414668i	$-0.136102\pi$
$368$	− 4.67558e8i	− 0.489067i
$369$	9.08742e7	0.0941561
$370$	0	0
$371$	1.17319e9	1.19278
$372$	− 1.43713e8i	− 0.144742i
$373$	7.41463e8i	0.739791i	0.929073	+	0.369895i	$0.120606\pi$
$373$	7.41463e8i	0.739791i	−0.929073	+	0.369895i	$0.879394\pi$
$374$	−1.27467e8	−0.125994
$375$	0	0
$376$	−1.63476e8	−0.158598
$377$	3.25367e7i	0.0312736i
$378$	1.12272e8i	0.106918i
$379$	4.13198e8	0.389871	0.194936	−	0.980816i	$-0.437550\pi$
$379$	4.13198e8	0.389871	0.194936	+	0.980816i	$0.437550\pi$
$380$	0	0
$381$	−9.55694e8	−0.885282
$382$	− 1.11533e9i	− 1.02372i
$383$	2.07784e9i	1.88980i	0.327361	+	0.944899i	$0.393841\pi$
$383$	2.07784e9i	1.88980i	−0.327361	+	0.944899i	$0.606159\pi$
$384$	5.66231e7	0.0510310
$385$	0	0
$386$	1.90314e8	0.168428
$387$	1.41249e8i	0.123879i
$388$	9.13195e8i	0.793693i
$389$	1.62801e9	1.40227	0.701137	−	0.713027i	$-0.252676\pi$
$389$	1.62801e9	1.40227	0.701137	+	0.713027i	$0.252676\pi$
$390$	0	0
$391$	−4.77375e8	−0.403870
$392$	1.61369e8i	0.135307i
$393$	− 6.96760e8i	− 0.579041i
$394$	3.74331e8	0.308332
$395$	0	0
$396$	1.77759e8	0.143847
$397$	6.54372e8i	0.524877i	0.964949	+	0.262439i	$0.0845268\pi$
$397$	6.54372e8i	0.524877i	−0.964949	+	0.262439i	$0.915473\pi$
$398$	− 9.34830e8i	− 0.743262i
$399$	3.00508e7	0.0236838
$400$	0	0
$401$	1.91730e9	1.48486	0.742430	−	0.669924i	$-0.233673\pi$
$401$	1.91730e9	1.48486	0.742430	+	0.669924i	$0.233673\pi$
$402$	− 9.52507e8i	− 0.731268i
$403$	− 3.25183e7i	− 0.0247491i
$404$	−4.43596e8	−0.334698
$405$	0	0
$406$	4.74653e8	0.351994
$407$	− 8.81383e8i	− 0.648013i
$408$	− 5.78120e7i	− 0.0421412i
$409$	−1.46590e9	−1.05943	−0.529717	−	0.848175i	$-0.677702\pi$
$409$	−1.46590e9	−1.05943	−0.529717	+	0.848175i	$0.677702\pi$
$410$	0	0
$411$	2.94447e8	0.209199
$412$	8.62124e8i	0.607337i
$413$	2.75289e7i	0.0192293i
$414$	6.65723e8	0.461097
$415$	0	0
$416$	1.28123e7	0.00872570
$417$	− 1.43751e9i	− 0.970813i
$418$	− 4.75793e7i	− 0.0318640i
$419$	1.87001e9	1.24192	0.620961	−	0.783841i	$-0.286742\pi$
$419$	1.87001e9	1.24192	0.620961	+	0.783841i	$0.286742\pi$
$420$	0	0
$421$	1.79884e9	1.17491	0.587456	−	0.809256i	$-0.300129\pi$
$421$	1.79884e9	1.17491	0.587456	+	0.809256i	$0.300129\pi$
$422$	1.46640e9i	0.949859i
$423$	− 2.32762e8i	− 0.149528i
$424$	8.42459e8	0.536745
$425$	0	0
$426$	−2.68013e7	−0.0167966
$427$	− 1.40739e9i	− 0.874819i
$428$	− 6.75999e8i	− 0.416767i
$429$	4.02222e7	0.0245960
$430$	0	0
$431$	3.61421e7	0.0217442	0.0108721	−	0.999941i	$-0.496539\pi$
$431$	3.61421e7	0.0217442	0.0108721	+	0.999941i	$0.496539\pi$
$432$	8.06216e7i	0.0481125i
$433$	1.54727e9i	0.915924i	0.888972	+	0.457962i	$0.151420\pi$
$433$	1.54727e9i	0.915924i	−0.888972	+	0.457962i	$0.848580\pi$
$434$	−4.74385e8	−0.278559
$435$	0	0
$436$	4.31870e8	0.249546
$437$	− 1.78188e8i	− 0.102140i
$438$	8.57009e8i	0.487333i
$439$	5.66134e8	0.319370	0.159685	−	0.987168i	$-0.448952\pi$
$439$	5.66134e8	0.319370	0.159685	+	0.987168i	$0.448952\pi$
$440$	0	0
$441$	−2.29762e8	−0.127568
$442$	− 1.30813e7i	− 0.00720564i
$443$	− 9.51833e8i	− 0.520173i	−0.965585	−	0.260087i	$-0.916249\pi$
$443$	− 9.51833e8i	− 0.520173i	0.965585	−	0.260087i	$-0.0837511\pi$
$444$	3.99745e8	0.216742
$445$	0	0
$446$	−1.53320e9	−0.818326
$447$	− 9.17842e8i	− 0.486062i
$448$	− 1.86909e8i	− 0.0982102i
$449$	1.12013e9	0.583989	0.291995	−	0.956420i	$-0.405681\pi$
$449$	1.12013e9	0.583989	0.291995	+	0.956420i	$0.405681\pi$
$450$	0	0
$451$	−4.74939e8	−0.243792
$452$	4.67689e7i	0.0238217i
$453$	− 8.46206e8i	− 0.427693i
$454$	−8.88955e8	−0.445845
$455$	0	0
$456$	2.15793e7	0.0106576
$457$	2.91044e9i	1.42644i	0.700943	+	0.713218i	$0.252763\pi$
$457$	2.91044e9i	1.42644i	−0.700943	+	0.713218i	$0.747237\pi$
$458$	− 7.13776e8i	− 0.347162i
$459$	8.23143e7	0.0397311
$460$	0	0
$461$	4.59640e8	0.218507	0.109253	−	0.994014i	$-0.465154\pi$
$461$	4.59640e8	0.218507	0.109253	+	0.994014i	$0.465154\pi$
$462$	− 5.86770e8i	− 0.276835i
$463$	− 5.60051e8i	− 0.262237i	−0.991367	−	0.131118i	$-0.958143\pi$
$463$	− 5.60051e8i	− 0.262237i	0.991367	−	0.131118i	$-0.0418568\pi$
$464$	3.40845e8	0.158396
$465$	0	0
$466$	4.03736e8	0.184819
$467$	− 3.70796e9i	− 1.68471i	−0.538921	−	0.842356i	$-0.681168\pi$
$467$	− 3.70796e9i	− 1.68471i	0.538921	−	0.842356i	$-0.318832\pi$
$468$	1.82425e7i	0.00822666i
$469$	−3.14415e9	−1.40734
$470$	0	0
$471$	−1.84541e9	−0.813804
$472$	1.97683e7i	0.00865312i
$473$	− 7.38214e8i	− 0.320751i
$474$	1.53533e9	0.662181
$475$	0	0
$476$	−1.90833e8	−0.0811016
$477$	1.19952e9i	0.506048i
$478$	6.53251e8i	0.273579i
$479$	3.96322e9	1.64768	0.823842	−	0.566820i	$-0.191826\pi$
$479$	3.96322e9	1.64768	0.823842	+	0.566820i	$0.191826\pi$
$480$	0	0
$481$	9.04516e7	0.0370603
$482$	1.92259e9i	0.782026i
$483$	− 2.19750e9i	− 0.887389i
$484$	3.18149e8	0.127547
$485$	0	0
$486$	−1.14791e8	−0.0453609
$487$	1.33008e9i	0.521825i	0.965362	+	0.260913i	$0.0840235\pi$
$487$	1.33008e9i	0.521825i	−0.965362	+	0.260913i	$0.915977\pi$
$488$	− 1.01064e9i	− 0.393665i
$489$	−1.24950e9	−0.483231
$490$	0	0
$491$	−2.55785e9	−0.975191	−0.487596	−	0.873070i	$-0.662126\pi$
$491$	−2.55785e9	−0.975191	−0.487596	+	0.873070i	$0.662126\pi$
$492$	− 2.15406e8i	− 0.0815416i
$493$	− 3.48001e8i	− 0.130803i
$494$	4.88281e6	0.00182232
$495$	0	0
$496$	−3.40652e8	−0.125350
$497$	8.84690e7i	0.0323254i
$498$	1.75342e9i	0.636185i
$499$	−4.31855e9	−1.55592	−0.777959	−	0.628316i	$-0.783745\pi$
$499$	−4.31855e9	−1.55592	−0.777959	+	0.628316i	$0.783745\pi$
$500$	0	0
$501$	−1.07903e9	−0.383354
$502$	2.58993e9i	0.913745i
$503$	− 6.29536e8i	− 0.220563i	−0.993900	−	0.110281i	$-0.964825\pi$
$503$	− 6.29536e8i	− 0.220563i	0.993900	−	0.110281i	$-0.0351752\pi$
$504$	2.66126e8	0.0925934
$505$	0	0
$506$	−3.47929e9	−1.19389
$507$	− 1.69008e9i	− 0.575944i
$508$	2.26535e9i	0.766676i
$509$	−5.79723e9	−1.94853	−0.974267	−	0.225397i	$-0.927632\pi$
$509$	−5.79723e9	−1.94853	−0.974267	+	0.225397i	$0.927632\pi$
$510$	0	0
$511$	2.82892e9	0.937882
$512$	− 1.34218e8i	− 0.0441942i
$513$	3.07252e7i	0.0100481i
$514$	−6.48308e8	−0.210577
$515$	0	0
$516$	3.34812e8	0.107282
$517$	1.21649e9i	0.387163i
$518$	− 1.31953e9i	− 0.417124i
$519$	2.44974e9	0.769191
$520$	0	0
$521$	1.39059e9	0.430790	0.215395	−	0.976527i	$-0.430896\pi$
$521$	1.39059e9	0.430790	0.215395	+	0.976527i	$0.430896\pi$
$522$	4.85304e8i	0.149337i
$523$	− 1.35994e9i	− 0.415684i	−0.978162	−	0.207842i	$-0.933356\pi$
$523$	− 1.35994e9i	− 0.415684i	0.978162	−	0.207842i	$-0.0666440\pi$
$524$	−1.65158e9	−0.501464
$525$	0	0
$526$	−1.69010e9	−0.506363
$527$	3.47804e8i	0.103514i
$528$	− 4.21356e8i	− 0.124575i
$529$	−9.62540e9	−2.82699
$530$	0	0
$531$	−2.81467e7	−0.00815824
$532$	− 7.12316e7i	− 0.0205108i
$533$	− 4.87405e7i	− 0.0139426i
$534$	1.45322e9	0.412988
$535$	0	0
$536$	−2.25779e9	−0.633297
$537$	3.52540e9i	0.982421i
$538$	3.36666e9i	0.932097i
$539$	1.20081e9	0.330305
$540$	0	0
$541$	6.15055e8	0.167003	0.0835013	−	0.996508i	$-0.473390\pi$
$541$	6.15055e8	0.167003	0.0835013	+	0.996508i	$0.473390\pi$
$542$	1.71087e9i	0.461551i
$543$	− 3.00457e9i	− 0.805348i
$544$	−1.37036e8	−0.0364954
$545$	0	0
$546$	6.02171e7	0.0158324
$547$	3.38341e8i	0.0883890i	0.999023	+	0.0441945i	$0.0140721\pi$
$547$	3.38341e8i	0.0883890i	−0.999023	+	0.0441945i	$0.985928\pi$
$548$	− 6.97948e8i	− 0.181172i
$549$	1.43898e9	0.371151
$550$	0	0
$551$	1.29897e8	0.0330802
$552$	− 1.57801e9i	− 0.399322i
$553$	− 5.06800e9i	− 1.27438i
$554$	−2.35732e9	−0.589025
$555$	0	0
$556$	−3.40744e9	−0.840748
$557$	1.09577e9i	0.268674i	0.990936	+	0.134337i	$0.0428905\pi$
$557$	1.09577e9i	0.268674i	−0.990936	+	0.134337i	$0.957110\pi$
$558$	− 4.85030e8i	− 0.118181i
$559$	7.57590e7	0.0183439
$560$	0	0
$561$	−4.30202e8	−0.102873
$562$	2.24801e9i	0.534222i
$563$	− 7.36019e8i	− 0.173824i	−0.996216	−	0.0869119i	$-0.972300\pi$
$563$	− 7.36019e8i	− 0.173824i	0.996216	−	0.0869119i	$-0.0276999\pi$
$564$	−5.51733e8	−0.129495
$565$	0	0
$566$	−4.65172e9	−1.07834
$567$	3.78917e8i	0.0872979i
$568$	6.35290e7i	0.0145463i
$569$	−9.08142e8	−0.206662	−0.103331	−	0.994647i	$-0.532950\pi$
$569$	−9.08142e8	−0.206662	−0.103331	+	0.994647i	$0.532950\pi$
$570$	0	0
$571$	1.02179e9	0.229685	0.114843	−	0.993384i	$-0.463364\pi$
$571$	1.02179e9	0.229685	0.114843	+	0.993384i	$0.463364\pi$
$572$	− 9.53414e7i	− 0.0213008i
$573$	− 3.76424e9i	− 0.835865i
$574$	−7.11038e8	−0.156928
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	9.10202e8i	0.197253i	0.995125	+	0.0986263i	$0.0314448\pi$
$577$	9.10202e8i	0.197253i	−0.995125	+	0.0986263i	$0.968555\pi$
$578$	− 3.14280e9i	− 0.676969i
$579$	6.42310e8	0.137521
$580$	0	0
$581$	5.78792e9	1.22435
$582$	3.08203e9i	0.648048i
$583$	− 6.26908e9i	− 1.31028i
$584$	2.03143e9	0.422043
$585$	0	0
$586$	6.33639e9	1.30077
$587$	− 7.80594e9i	− 1.59291i	−0.604697	−	0.796456i	$-0.706706\pi$
$587$	− 7.80594e9i	− 1.59291i	0.604697	−	0.796456i	$-0.293294\pi$
$588$	5.44621e8i	0.110477i
$589$	−1.29824e8	−0.0261788
$590$	0	0
$591$	1.26337e9	0.251752
$592$	− 9.47544e8i	− 0.187704i
$593$	5.90086e9i	1.16205i	0.813886	+	0.581024i	$0.197348\pi$
$593$	5.90086e9i	1.16205i	−0.813886	+	0.581024i	$0.802652\pi$
$594$	5.99938e8	0.117450
$595$	0	0
$596$	−2.17562e9	−0.420942
$597$	− 3.15505e9i	− 0.606871i
$598$	− 3.57061e8i	− 0.0682792i
$599$	3.05005e9	0.579847	0.289924	−	0.957050i	$-0.406370\pi$
$599$	3.05005e9	0.579847	0.289924	+	0.957050i	$0.406370\pi$
$600$	0	0
$601$	6.80081e8	0.127791	0.0638954	−	0.997957i	$-0.479648\pi$
$601$	6.80081e8	0.127791	0.0638954	+	0.997957i	$0.479648\pi$
$602$	− 1.10519e9i	− 0.206466i
$603$	− 3.21471e9i	− 0.597078i
$604$	−2.00582e9	−0.370393
$605$	0	0
$606$	−1.49714e9	−0.273280
$607$	7.79549e9i	1.41476i	0.706833	+	0.707381i	$0.250123\pi$
$607$	7.79549e9i	1.41476i	−0.706833	+	0.707381i	$0.749877\pi$
$608$	− 5.11508e7i	− 0.00922976i
$609$	1.60195e9	0.287402
$610$	0	0
$611$	−1.24842e8	−0.0221421
$612$	− 1.95115e8i	− 0.0344082i
$613$	9.21168e9i	1.61520i	0.589728	+	0.807602i	$0.299235\pi$
$613$	9.21168e9i	1.61520i	−0.589728	+	0.807602i	$0.700765\pi$
$614$	1.62664e9	0.283597
$615$	0	0
$616$	−1.39086e9	−0.239746
$617$	− 5.26863e9i	− 0.903024i	−0.892265	−	0.451512i	$-0.850885\pi$
$617$	− 5.26863e9i	− 0.903024i	0.892265	−	0.451512i	$-0.149115\pi$
$618$	2.90967e9i	0.495888i
$619$	5.84901e9	0.991208	0.495604	−	0.868548i	$-0.334947\pi$
$619$	5.84901e9	0.991208	0.495604	+	0.868548i	$0.334947\pi$
$620$	0	0
$621$	2.24681e9	0.376484
$622$	− 2.63542e9i	− 0.439121i
$623$	− 4.79697e9i	− 0.794802i
$624$	4.32415e7	0.00712450
$625$	0	0
$626$	−1.29196e9	−0.210494
$627$	− 1.60580e8i	− 0.0260169i
$628$	4.37430e9i	0.704775i
$629$	−9.67439e8	−0.155005
$630$	0	0
$631$	2.65927e9	0.421366	0.210683	−	0.977554i	$-0.432431\pi$
$631$	2.65927e9	0.421366	0.210683	+	0.977554i	$0.432431\pi$
$632$	− 3.63929e9i	− 0.573465i
$633$	4.94910e9i	0.775556i
$634$	5.11318e9	0.796853
$635$	0	0
$636$	2.84330e9	0.438251
$637$	1.23233e8i	0.0188903i
$638$	− 2.53636e9i	− 0.386669i
$639$	−9.04543e7	−0.0137144
$640$	0	0
$641$	4.97659e8	0.0746327	0.0373163	−	0.999304i	$-0.488119\pi$
$641$	4.97659e8	0.0746327	0.0373163	+	0.999304i	$0.488119\pi$
$642$	− 2.28150e9i	− 0.340288i
$643$	− 2.59642e9i	− 0.385156i	−0.981282	−	0.192578i	$-0.938315\pi$
$643$	− 2.59642e9i	− 0.385156i	0.981282	−	0.192578i	$-0.0616849\pi$
$644$	−5.20889e9	−0.768502
$645$	0	0
$646$	−5.22248e7	−0.00762190
$647$	− 4.44711e9i	− 0.645525i	−0.946480	−	0.322762i	$-0.895389\pi$
$647$	− 4.44711e9i	− 0.645525i	0.946480	−	0.322762i	$-0.104611\pi$
$648$	2.72098e8i	0.0392837i
$649$	1.47104e8	0.0211236
$650$	0	0
$651$	−1.60105e9	−0.227442
$652$	2.96177e9i	0.418490i
$653$	1.83971e9i	0.258556i	0.991608	+	0.129278i	$0.0412659\pi$
$653$	1.83971e9i	0.258556i	−0.991608	+	0.129278i	$0.958734\pi$
$654$	1.45756e9	0.203753
$655$	0	0
$656$	−5.10591e8	−0.0706171
$657$	2.89241e9i	0.397906i
$658$	1.82123e9i	0.249215i
$659$	3.41580e9	0.464936	0.232468	−	0.972604i	$-0.425320\pi$
$659$	3.41580e9	0.464936	0.232468	+	0.972604i	$0.425320\pi$
$660$	0	0
$661$	−4.71690e9	−0.635260	−0.317630	−	0.948215i	$-0.602887\pi$
$661$	−4.71690e9	−0.635260	−0.317630	+	0.948215i	$0.602887\pi$
$662$	4.31521e9i	0.578094i
$663$	− 4.41494e7i	− 0.00588338i
$664$	4.15626e9	0.550953
$665$	0	0
$666$	1.34914e9	0.176969
$667$	− 9.49888e9i	− 1.23946i
$668$	2.55769e9i	0.331994i
$669$	−5.17454e9	−0.668160
$670$	0	0
$671$	−7.52058e9	−0.960998
$672$	− 6.30817e8i	− 0.0801883i
$673$	5.83430e9i	0.737796i	0.929470	+	0.368898i	$0.120265\pi$
$673$	5.83430e9i	0.737796i	−0.929470	+	0.368898i	$0.879735\pi$
$674$	6.17869e9	0.777297
$675$	0	0
$676$	−4.00612e9	−0.498782
$677$	1.34655e8i	0.0166787i	0.999965	+	0.00833935i	$0.00265453\pi$
$677$	1.34655e8i	0.0166787i	−0.999965	+	0.00833935i	$0.997345\pi$
$678$	1.57845e8i	0.0194503i
$679$	1.01736e10	1.24718
$680$	0	0
$681$	−3.00022e9	−0.364031
$682$	2.53493e9i	0.305999i
$683$	1.33536e10i	1.60371i	0.597521	+	0.801853i	$0.296152\pi$
$683$	1.33536e10i	1.60371i	−0.597521	+	0.801853i	$0.703848\pi$
$684$	7.28300e7	0.00870190
$685$	0	0
$686$	6.49524e9	0.768176
$687$	− 2.40899e9i	− 0.283457i
$688$	− 7.93629e8i	− 0.0929090i
$689$	6.43362e8	0.0749356
$690$	0	0
$691$	−3.26642e9	−0.376616	−0.188308	−	0.982110i	$-0.560300\pi$
$691$	−3.26642e9	−0.376616	−0.188308	+	0.982110i	$0.560300\pi$
$692$	− 5.80679e9i	− 0.666139i
$693$	− 1.98035e9i	− 0.226035i
$694$	4.16823e9	0.473362
$695$	0	0
$696$	1.15035e9	0.129330
$697$	5.21311e8i	0.0583153i
$698$	4.17899e9i	0.465134i
$699$	1.36261e9	0.150904
$700$	0	0
$701$	7.82925e9	0.858434	0.429217	−	0.903201i	$-0.358789\pi$
$701$	7.82925e9	0.858434	0.429217	+	0.903201i	$0.358789\pi$
$702$	6.15684e7i	0.00671704i
$703$	− 3.61112e8i	− 0.0392011i
$704$	−9.98769e8	−0.107885
$705$	0	0
$706$	8.18052e9	0.874912
$707$	4.94194e9i	0.525932i
$708$	6.67181e7i	0.00706524i
$709$	3.26880e9	0.344450	0.172225	−	0.985058i	$-0.444904\pi$
$709$	3.26880e9	0.344450	0.172225	+	0.985058i	$0.444904\pi$
$710$	0	0
$711$	5.18173e9	0.540668
$712$	− 3.44467e9i	− 0.357658i
$713$	9.49351e9i	0.980875i
$714$	−6.44061e8	−0.0662191
$715$	0	0
$716$	8.35649e9	0.850802
$717$	2.20472e9i	0.223376i
$718$	− 9.43170e9i	− 0.950943i
$719$	4.69765e9	0.471335	0.235668	−	0.971834i	$-0.424272\pi$
$719$	4.69765e9	0.471335	0.235668	+	0.971834i	$0.424272\pi$
$720$	0	0
$721$	9.60460e9	0.954346
$722$	7.13148e9i	0.705179i
$723$	6.48873e9i	0.638522i
$724$	−7.12195e9	−0.697452
$725$	0	0
$726$	1.07375e9	0.104142
$727$	− 1.10503e10i	− 1.06661i	−0.845924	−	0.533304i	$-0.820950\pi$
$727$	− 1.10503e10i	− 1.06661i	0.845924	−	0.533304i	$-0.179050\pi$
$728$	− 1.42737e8i	− 0.0137112i
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	−8.10292e8	−0.0767239
$732$	− 3.41091e9i	− 0.321426i
$733$	1.52446e10i	1.42972i	0.699268	+	0.714860i	$0.253509\pi$
$733$	1.52446e10i	1.42972i	−0.699268	+	0.714860i	$0.746491\pi$
$734$	−6.28279e9	−0.586430
$735$	0	0
$736$	−3.74047e9	−0.345823
$737$	1.68012e10i	1.54598i
$738$	− 7.26994e8i	− 0.0665784i
$739$	1.54738e10	1.41039	0.705197	−	0.709012i	$-0.250859\pi$
$739$	1.54738e10	1.41039	0.705197	+	0.709012i	$0.250859\pi$
$740$	0	0
$741$	1.64795e7	0.00148792
$742$	− 9.38552e9i	− 0.843422i
$743$	− 1.25550e9i	− 0.112294i	−0.998423	−	0.0561471i	$-0.982118\pi$
$743$	− 1.25550e9i	− 0.112294i	0.998423	−	0.0561471i	$-0.0178816\pi$
$744$	−1.14970e9	−0.102348
$745$	0	0
$746$	5.93171e9	0.523111
$747$	5.91780e9i	0.519443i
$748$	1.01974e9i	0.0890909i
$749$	−7.53105e9	−0.654891
$750$	0	0
$751$	−1.75101e10	−1.50851	−0.754257	−	0.656580i	$-0.772003\pi$
$751$	−1.75101e10	−1.50851	−0.754257	+	0.656580i	$0.772003\pi$
$752$	1.30781e9i	0.112146i
$753$	8.74101e9i	0.746069i
$754$	2.60293e8	0.0221138
$755$	0	0
$756$	8.98175e8	0.0756022
$757$	1.55877e9i	0.130601i	0.997866	+	0.0653005i	$0.0208006\pi$
$757$	1.55877e9i	0.130601i	−0.997866	+	0.0653005i	$0.979199\pi$
$758$	− 3.30559e9i	− 0.275681i
$759$	−1.17426e10	−0.974807
$760$	0	0
$761$	8.14586e9	0.670025	0.335012	−	0.942214i	$-0.391260\pi$
$761$	8.14586e9	0.670025	0.335012	+	0.942214i	$0.391260\pi$
$762$	7.64555e9i	0.625989i
$763$	− 4.81130e9i	− 0.392127i
$764$	−8.92264e9	−0.723880
$765$	0	0
$766$	1.66227e10	1.33629
$767$	1.50965e7i	0.00120807i
$768$	− 4.52985e8i	− 0.0360844i
$769$	−1.61934e10	−1.28409	−0.642046	−	0.766666i	$-0.721914\pi$
$769$	−1.61934e10	−1.28409	−0.642046	+	0.766666i	$0.721914\pi$
$770$	0	0
$771$	−2.18804e9	−0.171935
$772$	− 1.52251e9i	− 0.119097i
$773$	3.85870e9i	0.300478i	0.988650	+	0.150239i	$0.0480044\pi$
$773$	3.85870e9i	0.300478i	−0.988650	+	0.150239i	$0.951996\pi$
$774$	1.12999e9	0.0875955
$775$	0	0
$776$	7.30556e9	0.561226
$777$	− 4.45341e9i	− 0.340580i
$778$	− 1.30241e10i	− 0.991557i
$779$	−1.94588e8	−0.0147481
$780$	0	0
$781$	4.72745e8	0.0355098
$782$	3.81900e9i	0.285579i
$783$	1.63790e9i	0.121933i
$784$	1.29095e9	0.0956762
$785$	0	0
$786$	−5.57408e9	−0.409444
$787$	− 9.76950e9i	− 0.714431i	−0.934022	−	0.357216i	$-0.883726\pi$
$787$	− 9.76950e9i	− 0.714431i	0.934022	−	0.357216i	$-0.116274\pi$
$788$	− 2.99465e9i	− 0.218024i
$789$	−5.70409e9	−0.413444
$790$	0	0
$791$	5.21035e8	0.0374325
$792$	− 1.42207e9i	− 0.101715i
$793$	− 7.71797e8i	− 0.0549600i
$794$	5.23498e9	0.371144
$795$	0	0
$796$	−7.47864e9	−0.525566
$797$	1.86704e10i	1.30632i	0.757221	+	0.653159i	$0.226557\pi$
$797$	1.86704e10i	1.30632i	−0.757221	+	0.653159i	$0.773443\pi$
$798$	− 2.40406e8i	− 0.0167470i
$799$	1.33527e9	0.0926095
$800$	0	0
$801$	4.90462e9	0.337203
$802$	− 1.53384e10i	− 1.04995i
$803$	− 1.51167e10i	− 1.03027i
$804$	−7.62005e9	−0.517085
$805$	0	0
$806$	−2.60146e8	−0.0175003
$807$	1.13625e10i	0.761054i
$808$	3.54877e9i	0.236667i
$809$	7.97985e9	0.529877	0.264938	−	0.964265i	$-0.414648\pi$
$809$	7.97985e9	0.529877	0.264938	+	0.964265i	$0.414648\pi$
$810$	0	0
$811$	−1.70481e10	−1.12228	−0.561141	−	0.827720i	$-0.689638\pi$
$811$	−1.70481e10	−1.12228	−0.561141	+	0.827720i	$0.689638\pi$
$812$	− 3.79722e9i	− 0.248897i
$813$	5.77419e9i	0.376855i
$814$	−7.05106e9	−0.458215
$815$	0	0
$816$	−4.62496e8	−0.0297983
$817$	− 3.02455e8i	− 0.0194036i
$818$	1.17272e10i	0.749132i
$819$	2.03233e8	0.0129271
$820$	0	0
$821$	−6.64028e9	−0.418779	−0.209390	−	0.977832i	$-0.567148\pi$
$821$	−6.64028e9	−0.418779	−0.209390	+	0.977832i	$0.567148\pi$
$822$	− 2.35557e9i	− 0.147926i
$823$	− 2.99586e10i	− 1.87336i	−0.350180	−	0.936682i	$-0.613880\pi$
$823$	− 2.99586e10i	− 1.87336i	0.350180	−	0.936682i	$-0.386120\pi$
$824$	6.89699e9	0.429452
$825$	0	0
$826$	2.20231e8	0.0135972
$827$	3.07056e9i	0.188777i	0.995535	+	0.0943883i	$0.0300895\pi$
$827$	3.07056e9i	0.188777i	−0.995535	+	0.0943883i	$0.969910\pi$
$828$	− 5.32578e9i	− 0.326045i
$829$	2.10442e10	1.28289	0.641447	−	0.767167i	$-0.278334\pi$
$829$	2.10442e10	1.28289	0.641447	+	0.767167i	$0.278334\pi$
$830$	0	0
$831$	−7.95594e9	−0.480937
$832$	− 1.02498e8i	− 0.00617000i
$833$	− 1.31806e9i	− 0.0790091i
$834$	−1.15001e10	−0.686468
$835$	0	0
$836$	−3.80634e8	−0.0225313
$837$	− 1.63698e9i	− 0.0964946i
$838$	− 1.49601e10i	− 0.878172i
$839$	−1.16048e10	−0.678377	−0.339189	−	0.940718i	$-0.610152\pi$
$839$	−1.16048e10	−0.678377	−0.339189	+	0.940718i	$0.610152\pi$
$840$	0	0
$841$	−1.03253e10	−0.598573
$842$	− 1.43907e10i	− 0.830789i
$843$	7.58704e9i	0.436190i
$844$	1.17312e10	0.671651
$845$	0	0
$846$	−1.86210e9	−0.105732
$847$	− 3.54437e9i	− 0.200423i
$848$	− 6.73967e9i	− 0.379536i
$849$	−1.56995e10	−0.880461
$850$	0	0
$851$	−2.64068e10	−1.46880
$852$	2.14410e8i	0.0118770i
$853$	− 3.08015e10i	− 1.69922i	−0.527411	−	0.849610i	$-0.676837\pi$
$853$	− 3.08015e10i	− 1.69922i	0.527411	−	0.849610i	$-0.323163\pi$
$854$	−1.12592e10	−0.618590
$855$	0	0
$856$	−5.40799e9	−0.294698
$857$	2.03442e10i	1.10410i	0.833811	+	0.552050i	$0.186154\pi$
$857$	2.03442e10i	1.10410i	−0.833811	+	0.552050i	$0.813846\pi$
$858$	− 3.21777e8i	− 0.0173920i
$859$	5.51078e9	0.296645	0.148323	−	0.988939i	$-0.452613\pi$
$859$	5.51078e9	0.296645	0.148323	+	0.988939i	$0.452613\pi$
$860$	0	0
$861$	−2.39975e9	−0.128131
$862$	− 2.89137e8i	− 0.0153755i
$863$	− 2.54712e10i	− 1.34900i	−0.738274	−	0.674501i	$-0.764359\pi$
$863$	− 2.54712e10i	− 1.34900i	0.738274	−	0.674501i	$-0.235641\pi$
$864$	6.44973e8	0.0340207
$865$	0	0
$866$	1.23782e10	0.647656
$867$	− 1.06069e10i	− 0.552743i
$868$	3.79508e9i	0.196971i
$869$	−2.70814e10	−1.39992
$870$	0	0
$871$	−1.72421e9	−0.0884153
$872$	− 3.45496e9i	− 0.176455i
$873$	1.04019e10i	0.529129i
$874$	−1.42551e9	−0.0722235
$875$	0	0
$876$	6.85607e9	0.344597
$877$	− 1.83885e9i	− 0.0920549i	−0.998940	−	0.0460275i	$-0.985344\pi$
$877$	− 1.83885e9i	− 0.0920549i	0.998940	−	0.0460275i	$-0.0146562\pi$
$878$	− 4.52907e9i	− 0.225828i
$879$	2.13853e10	1.06207
$880$	0	0
$881$	−3.40718e10	−1.67873	−0.839363	−	0.543571i	$-0.817072\pi$
$881$	−3.40718e10	−1.67873	−0.839363	+	0.543571i	$0.817072\pi$
$882$	1.83809e9i	0.0902044i
$883$	2.54935e10i	1.24614i	0.782165	+	0.623071i	$0.214115\pi$
$883$	2.54935e10i	1.24614i	−0.782165	+	0.623071i	$0.785885\pi$
$884$	−1.04650e8	−0.00509516
$885$	0	0
$886$	−7.61467e9	−0.367818
$887$	2.29613e10i	1.10475i	0.833596	+	0.552374i	$0.186278\pi$
$887$	2.29613e10i	1.10475i	−0.833596	+	0.552374i	$0.813722\pi$
$888$	− 3.19796e9i	− 0.153260i
$889$	2.52374e10	1.20473
$890$	0	0
$891$	2.02479e9	0.0958977
$892$	1.22656e10i	0.578644i
$893$	4.98412e8i	0.0234211i
$894$	−7.34273e9	−0.343698
$895$	0	0
$896$	−1.49527e9	−0.0694451
$897$	− 1.20508e9i	− 0.0557497i
$898$	− 8.96101e9i	− 0.412943i
$899$	−6.92066e9	−0.317679
$900$	0	0
$901$	−6.88118e9	−0.313420
$902$	3.79951e9i	0.172387i
$903$	− 3.73002e9i	− 0.168579i
$904$	3.74151e8	0.0168445
$905$	0	0
$906$	−6.76965e9	−0.302425
$907$	5.32481e9i	0.236962i	0.992956	+	0.118481i	$0.0378025\pi$
$907$	5.32481e9i	0.236962i	−0.992956	+	0.118481i	$0.962198\pi$
$908$	7.11164e9i	0.315260i
$909$	−5.05284e9	−0.223132
$910$	0	0
$911$	2.29502e10	1.00571	0.502853	−	0.864372i	$-0.332284\pi$
$911$	2.29502e10	1.00571	0.502853	+	0.864372i	$0.332284\pi$
$912$	− 1.72634e8i	− 0.00753607i
$913$	− 3.09284e10i	− 1.34496i
$914$	2.32835e10	1.00864
$915$	0	0
$916$	−5.71021e9	−0.245481
$917$	1.83996e10i	0.787982i
$918$	− 6.58514e8i	− 0.0280942i
$919$	2.49309e10	1.05958	0.529790	−	0.848129i	$-0.322271\pi$
$919$	2.49309e10	1.05958	0.529790	+	0.848129i	$0.322271\pi$
$920$	0	0
$921$	5.48991e9	0.231556
$922$	− 3.67712e9i	− 0.154507i
$923$	4.85153e7i	0.00203083i
$924$	−4.69416e9	−0.195752
$925$	0	0
$926$	−4.48040e9	−0.185429
$927$	9.82013e9i	0.404891i
$928$	− 2.72676e9i	− 0.112003i
$929$	−1.04115e10	−0.426049	−0.213024	−	0.977047i	$-0.568331\pi$
$929$	−1.04115e10	−0.426049	−0.213024	+	0.977047i	$0.568331\pi$
$930$	0	0
$931$	4.91987e8	0.0199816
$932$	− 3.22989e9i	− 0.130687i
$933$	− 8.89456e9i	− 0.358541i
$934$	−2.96637e10	−1.19127
$935$	0	0
$936$	1.45940e8	0.00581713
$937$	− 7.28154e9i	− 0.289158i	−0.989493	−	0.144579i	$-0.953817\pi$
$937$	− 7.28154e9i	− 0.289158i	0.989493	−	0.144579i	$-0.0461827\pi$
$938$	2.51532e10i	0.995139i
$939$	−4.36038e9	−0.171868
$940$	0	0
$941$	2.30278e10	0.900927	0.450464	−	0.892795i	$-0.351259\pi$
$941$	2.30278e10	0.900927	0.450464	+	0.892795i	$0.351259\pi$
$942$	1.47633e10i	0.575446i
$943$	1.42295e10i	0.552584i
$944$	1.58147e8	0.00611868
$945$	0	0
$946$	−5.90571e9	−0.226805
$947$	− 4.69600e10i	− 1.79681i	−0.439164	−	0.898407i	$-0.644725\pi$
$947$	− 4.69600e10i	− 1.79681i	0.439164	−	0.898407i	$-0.355275\pi$
$948$	− 1.22826e10i	− 0.468232i
$949$	1.55134e9	0.0589219
$950$	0	0
$951$	1.72570e10	0.650628
$952$	1.52666e9i	0.0573475i
$953$	− 3.43809e10i	− 1.28674i	−0.765553	−	0.643372i	$-0.777535\pi$
$953$	− 3.43809e10i	− 1.28674i	0.765553	−	0.643372i	$-0.222465\pi$
$954$	9.59614e9	0.357830
$955$	0	0
$956$	5.22600e9	0.193449
$957$	− 8.56022e9i	− 0.315714i
$958$	− 3.17058e10i	− 1.16509i
$959$	−7.77558e9	−0.284687
$960$	0	0
$961$	−2.05959e10	−0.748597
$962$	− 7.23613e8i	− 0.0262056i
$963$	− 7.70005e9i	− 0.277844i
$964$	1.53807e10	0.552976
$965$	0	0
$966$	−1.75800e10	−0.627479
$967$	1.14483e10i	0.407146i	0.979060	+	0.203573i	$0.0652554\pi$
$967$	1.14483e10i	0.407146i	−0.979060	+	0.203573i	$0.934745\pi$
$968$	− 2.54519e9i	− 0.0901895i
$969$	−1.76259e8	−0.00622325
$970$	0	0
$971$	1.85255e8	0.00649385	0.00324693	−	0.999995i	$-0.498966\pi$
$971$	1.85255e8	0.00649385	0.00324693	+	0.999995i	$0.498966\pi$
$972$	9.18330e8i	0.0320750i
$973$	3.79610e10i	1.32112i
$974$	1.06406e10	0.368986
$975$	0	0
$976$	−8.08511e9	−0.278363
$977$	− 3.06849e10i	− 1.05267i	−0.850276	−	0.526337i	$-0.823565\pi$
$977$	− 3.06849e10i	− 1.05267i	0.850276	−	0.526337i	$-0.176435\pi$
$978$	9.99598e9i	0.341696i
$979$	−2.56332e10	−0.873099
$980$	0	0
$981$	4.91927e9	0.166364
$982$	2.04628e10i	0.689564i
$983$	1.00287e9i	0.0336751i	0.999858	+	0.0168376i	$0.00535982\pi$
$983$	1.00287e9i	0.0336751i	−0.999858	+	0.0168376i	$0.994640\pi$
$984$	−1.72324e9	−0.0576586
$985$	0	0
$986$	−2.78401e9	−0.0924914
$987$	6.14665e9i	0.203483i
$988$	− 3.90625e7i	− 0.00128858i
$989$	−2.21174e10	−0.727020
$990$	0	0
$991$	4.82282e10	1.57414	0.787070	−	0.616864i	$-0.211597\pi$
$991$	4.82282e10	1.57414	0.787070	+	0.616864i	$0.211597\pi$
$992$	2.72522e9i	0.0886360i
$993$	1.45638e10i	0.472012i
$994$	7.07752e8	0.0228575
$995$	0	0
$996$	1.40274e10	0.449851
$997$	− 4.68836e10i	− 1.49826i	−0.662421	−	0.749132i	$-0.730471\pi$
$997$	− 4.68836e10i	− 1.49826i	0.662421	−	0.749132i	$-0.269529\pi$
$998$	3.45484e10i	1.10020i
$999$	4.55335e9	0.144495

Twists

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

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

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} -27.0000i q^{3} -64.0000 q^{4} -216.000 q^{6} +713.000i q^{7} +512.000i q^{8} -729.000 q^{9} +3810.00 q^{11} +1728.00i q^{12} +391.000i q^{13} +5704.00 q^{14} +4096.00 q^{16} -4182.00i q^{17} +5832.00i q^{18} +1561.00 q^{19} +19251.0 q^{21} -30480.0i q^{22} -114150. i q^{23} +13824.0 q^{24} +3128.00 q^{26} +19683.0i q^{27} -45632.0i q^{28} +83214.0 q^{29} -83167.0 q^{31} -32768.0i q^{32} -102870. i q^{33} -33456.0 q^{34} +46656.0 q^{36} -231334. i q^{37} -12488.0i q^{38} +10557.0 q^{39} -124656. q^{41} -154008. i q^{42} -193757. i q^{43} -243840. q^{44} -913200. q^{46} +319290. i q^{47} -110592. i q^{48} +315174. q^{49} -112914. q^{51} -25024.0i q^{52} -1.64543e6i q^{53} +157464. q^{54} -365056. q^{56} -42147.0i q^{57} -665712. i q^{58} +38610.0 q^{59} -1.97390e6 q^{61} +665336. i q^{62} -519777. i q^{63} -262144. q^{64} -822960. q^{66} +4.40975e6i q^{67} +267648. i q^{68} -3.08205e6 q^{69} +124080. q^{71} -373248. i q^{72} -3.96763e6i q^{73} -1.85067e6 q^{74} -99904.0 q^{76} +2.71653e6i q^{77} -84456.0i q^{78} -7.10799e6 q^{79} +531441. q^{81} +997248. i q^{82} -8.11769e6i q^{83} -1.23206e6 q^{84} -1.55006e6 q^{86} -2.24678e6i q^{87} +1.95072e6i q^{88} -6.72787e6 q^{89} -278783. q^{91} +7.30560e6i q^{92} +2.24551e6i q^{93} +2.55432e6 q^{94} -884736. q^{96} -1.42687e7i q^{97} -2.52139e6i q^{98} -2.77749e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 432 q^{6} - 1458 q^{9} + 7620 q^{11} + 11408 q^{14} + 8192 q^{16} + 3122 q^{19} + 38502 q^{21} + 27648 q^{24} + 6256 q^{26} + 166428 q^{29} - 166334 q^{31} - 66912 q^{34} + 93312 q^{36}+ \cdots - 5554980 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 432 * q^6 - 1458 * q^9 + 7620 * q^11 + 11408 * q^14 + 8192 * q^16 + 3122 * q^19 + 38502 * q^21 + 27648 * q^24 + 6256 * q^26 + 166428 * q^29 - 166334 * q^31 - 66912 * q^34 + 93312 * q^36 + 21114 * q^39 - 249312 * q^41 - 487680 * q^44 - 1826400 * q^46 + 630348 * q^49 - 225828 * q^51 + 314928 * q^54 - 730112 * q^56 + 77220 * q^59 - 3947810 * q^61 - 524288 * q^64 - 1645920 * q^66 - 6164100 * q^69 + 248160 * q^71 - 3701344 * q^74 - 199808 * q^76 - 14215984 * q^79 + 1062882 * q^81 - 2464128 * q^84 - 3100112 * q^86 - 13455744 * q^89 - 557566 * q^91 + 5108640 * q^94 - 1769472 * q^96 - 5554980 * q^99

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