LMFDB - Embedded newform 150.8.c.e.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:	no (minimal twist has level 30)
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.e.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000i q^{2} -27.0000i q^{3} -64.0000 q^{4} -216.000 q^{6} -512.000i q^{7} +512.000i q^{8} -729.000 q^{9} +5460.00 q^{11} +1728.00i q^{12} +10166.0i q^{13} -4096.00 q^{14} +4096.00 q^{16} +9918.00i q^{17} +5832.00i q^{18} +12436.0 q^{19} -13824.0 q^{21} -43680.0i q^{22} +33600.0i q^{23} +13824.0 q^{24} +81328.0 q^{26} +19683.0i q^{27} +32768.0i q^{28} +187914. q^{29} -42592.0 q^{31} -32768.0i q^{32} -147420. i q^{33} +79344.0 q^{34} +46656.0 q^{36} +544066. i q^{37} -99488.0i q^{38} +274482. q^{39} +374394. q^{41} +110592. i q^{42} -540532. i q^{43} -349440. q^{44} +268800. q^{46} -1.33836e6i q^{47} -110592. i q^{48} +561399. q^{49} +267786. q^{51} -650624. i q^{52} +1.30822e6i q^{53} +157464. q^{54} +262144. q^{56} -335772. i q^{57} -1.50331e6i q^{58} -262740. q^{59} -976330. q^{61} +340736. i q^{62} +373248. i q^{63} -262144. q^{64} -1.17936e6 q^{66} -3.55917e6i q^{67} -634752. i q^{68} +907200. q^{69} -2.67372e6 q^{71} -373248. i q^{72} -3.03213e6i q^{73} +4.35253e6 q^{74} -795904. q^{76} -2.79552e6i q^{77} -2.19586e6i q^{78} +5.47581e6 q^{79} +531441. q^{81} -2.99515e6i q^{82} +2.23156e6i q^{83} +884736. q^{84} -4.32426e6 q^{86} -5.07368e6i q^{87} +2.79552e6i q^{88} +1.00507e7 q^{89} +5.20499e6 q^{91} -2.15040e6i q^{92} +1.14998e6i q^{93} -1.07069e7 q^{94} -884736. q^{96} -5.72755e6i q^{97} -4.49119e6i q^{98} -3.98034e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 432 q^{6} - 1458 q^{9} + 10920 q^{11} - 8192 q^{14} + 8192 q^{16} + 24872 q^{19} - 27648 q^{21} + 27648 q^{24} + 162656 q^{26} + 375828 q^{29} - 85184 q^{31} + 158688 q^{34} + 93312 q^{36}+ \cdots - 7960680 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 432 * q^6 - 1458 * q^9 + 10920 * q^11 - 8192 * q^14 + 8192 * q^16 + 24872 * q^19 - 27648 * q^21 + 27648 * q^24 + 162656 * q^26 + 375828 * q^29 - 85184 * q^31 + 158688 * q^34 + 93312 * q^36 + 548964 * q^39 + 748788 * q^41 - 698880 * q^44 + 537600 * q^46 + 1122798 * q^49 + 535572 * q^51 + 314928 * q^54 + 524288 * q^56 - 525480 * q^59 - 1952660 * q^61 - 524288 * q^64 - 2358720 * q^66 + 1814400 * q^69 - 5347440 * q^71 + 8705056 * q^74 - 1591808 * q^76 + 10951616 * q^79 + 1062882 * q^81 + 1769472 * q^84 - 8648512 * q^86 + 20101356 * q^89 + 10409984 * q^91 - 21413760 * q^94 - 1769472 * q^96 - 7960680 * 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
$7$	− 512.000i	− 0.564192i	−0.959386	−	0.282096i	$-0.908970\pi$
$7$	− 512.000i	− 0.564192i	0.959386	−	0.282096i	$-0.0910296\pi$
$8$	512.000i	0.353553i
$9$	−729.000	−0.333333
$10$	0	0
$11$	5460.00	1.23685	0.618427	−	0.785842i	$-0.287770\pi$
$11$	5460.00	1.23685	0.618427	+	0.785842i	$0.287770\pi$
$12$	1728.00i	0.288675i
$13$	10166.0i	1.28336i	0.766973	+	0.641680i	$0.221762\pi$
$13$	10166.0i	1.28336i	−0.766973	+	0.641680i	$0.778238\pi$
$14$	−4096.00	−0.398944
$15$	0	0
$16$	4096.00	0.250000
$17$	9918.00i	0.489613i	0.969572	+	0.244806i	$0.0787244\pi$
$17$	9918.00i	0.489613i	−0.969572	+	0.244806i	$0.921276\pi$
$18$	5832.00i	0.235702i
$19$	12436.0	0.415952	0.207976	−	0.978134i	$-0.433312\pi$
$19$	12436.0	0.415952	0.207976	+	0.978134i	$0.433312\pi$
$20$	0	0
$21$	−13824.0	−0.325736
$22$	− 43680.0i	− 0.874587i
$23$	33600.0i	0.575827i	0.957656	+	0.287913i	$0.0929615\pi$
$23$	33600.0i	0.575827i	−0.957656	+	0.287913i	$0.907038\pi$
$24$	13824.0	0.204124
$25$	0	0
$26$	81328.0	0.907472
$27$	19683.0i	0.192450i
$28$	32768.0i	0.282096i
$29$	187914.	1.43076	0.715379	−	0.698737i	$-0.246254\pi$
$29$	187914.	1.43076	0.715379	+	0.698737i	$0.246254\pi$
$30$	0	0
$31$	−42592.0	−0.256781	−0.128390	−	0.991724i	$-0.540981\pi$
$31$	−42592.0	−0.256781	−0.128390	+	0.991724i	$0.540981\pi$
$32$	− 32768.0i	− 0.176777i
$33$	− 147420.i	− 0.714098i
$34$	79344.0	0.346209
$35$	0	0
$36$	46656.0	0.166667
$37$	544066.i	1.76582i	0.469546	+	0.882908i	$0.344418\pi$
$37$	544066.i	1.76582i	−0.469546	+	0.882908i	$0.655582\pi$
$38$	− 99488.0i	− 0.294122i
$39$	274482.	0.740948
$40$	0	0
$41$	374394.	0.848370	0.424185	−	0.905576i	$-0.360561\pi$
$41$	374394.	0.848370	0.424185	+	0.905576i	$0.360561\pi$
$42$	110592.i	0.230330i
$43$	− 540532.i	− 1.03677i	−0.855148	−	0.518384i	$-0.826534\pi$
$43$	− 540532.i	− 1.03677i	0.855148	−	0.518384i	$-0.173466\pi$
$44$	−349440.	−0.618427
$45$	0	0
$46$	268800.	0.407171
$47$	− 1.33836e6i	− 1.88031i	−0.340741	−	0.940157i	$-0.610678\pi$
$47$	− 1.33836e6i	− 1.88031i	0.340741	−	0.940157i	$-0.389322\pi$
$48$	− 110592.i	− 0.144338i
$49$	561399.	0.681688
$50$	0	0
$51$	267786.	0.282678
$52$	− 650624.i	− 0.641680i
$53$	1.30822e6i	1.20702i	0.797354	+	0.603512i	$0.206232\pi$
$53$	1.30822e6i	1.20702i	−0.797354	+	0.603512i	$0.793768\pi$
$54$	157464.	0.136083
$55$	0	0
$56$	262144.	0.199472
$57$	− 335772.i	− 0.240150i
$58$	− 1.50331e6i	− 1.01170i
$59$	−262740.	−0.166550	−0.0832749	−	0.996527i	$-0.526538\pi$
$59$	−262740.	−0.166550	−0.0832749	+	0.996527i	$0.526538\pi$
$60$	0	0
$61$	−976330.	−0.550734	−0.275367	−	0.961339i	$-0.588799\pi$
$61$	−976330.	−0.550734	−0.275367	+	0.961339i	$0.588799\pi$
$62$	340736.i	0.181571i
$63$	373248.i	0.188064i
$64$	−262144.	−0.125000
$65$	0	0
$66$	−1.17936e6	−0.504943
$67$	− 3.55917e6i	− 1.44573i	−0.690989	−	0.722865i	$-0.742825\pi$
$67$	− 3.55917e6i	− 1.44573i	0.690989	−	0.722865i	$-0.257175\pi$
$68$	− 634752.i	− 0.244806i
$69$	907200.	0.332454
$70$	0	0
$71$	−2.67372e6	−0.886567	−0.443284	−	0.896381i	$-0.646187\pi$
$71$	−2.67372e6	−0.886567	−0.443284	+	0.896381i	$0.646187\pi$
$72$	− 373248.i	− 0.117851i
$73$	− 3.03213e6i	− 0.912259i	−0.889913	−	0.456130i	$-0.849235\pi$
$73$	− 3.03213e6i	− 0.912259i	0.889913	−	0.456130i	$-0.150765\pi$
$74$	4.35253e6	1.24862
$75$	0	0
$76$	−795904.	−0.207976
$77$	− 2.79552e6i	− 0.697823i
$78$	− 2.19586e6i	− 0.523929i
$79$	5.47581e6	1.24955	0.624775	−	0.780805i	$-0.285191\pi$
$79$	5.47581e6	1.24955	0.624775	+	0.780805i	$0.285191\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	− 2.99515e6i	− 0.599888i
$83$	2.23156e6i	0.428385i	0.976791	+	0.214193i	$0.0687120\pi$
$83$	2.23156e6i	0.428385i	−0.976791	+	0.214193i	$0.931288\pi$
$84$	884736.	0.162868
$85$	0	0
$86$	−4.32426e6	−0.733106
$87$	− 5.07368e6i	− 0.826048i
$88$	2.79552e6i	0.437294i
$89$	1.00507e7	1.51123	0.755615	−	0.655016i	$-0.227338\pi$
$89$	1.00507e7	1.51123	0.755615	+	0.655016i	$0.227338\pi$
$90$	0	0
$91$	5.20499e6	0.724061
$92$	− 2.15040e6i	− 0.287913i
$93$	1.14998e6i	0.148252i
$94$	−1.07069e7	−1.32958
$95$	0	0
$96$	−884736.	−0.102062
$97$	− 5.72755e6i	− 0.637189i	−0.947891	−	0.318594i	$-0.896789\pi$
$97$	− 5.72755e6i	− 0.637189i	0.947891	−	0.318594i	$-0.103211\pi$
$98$	− 4.49119e6i	− 0.482026i
$99$	−3.98034e6	−0.412284
$100$	0	0
$101$	−1.33358e7	−1.28793	−0.643966	−	0.765054i	$-0.722712\pi$
$101$	−1.33358e7	−1.28793	−0.643966	+	0.765054i	$0.722712\pi$
$102$	− 2.14229e6i	− 0.199884i
$103$	− 2.71019e6i	− 0.244382i	−0.992507	−	0.122191i	$-0.961008\pi$
$103$	− 2.71019e6i	− 0.244382i	0.992507	−	0.122191i	$-0.0389921\pi$
$104$	−5.20499e6	−0.453736
$105$	0	0
$106$	1.04658e7	0.853495
$107$	− 1.13195e7i	− 0.893274i	−0.894715	−	0.446637i	$-0.852622\pi$
$107$	− 1.13195e7i	− 0.893274i	0.894715	−	0.446637i	$-0.147378\pi$
$108$	− 1.25971e6i	− 0.0962250i
$109$	2.19732e7	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$109$	2.19732e7	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$110$	0	0
$111$	1.46898e7	1.01949
$112$	− 2.09715e6i	− 0.141048i
$113$	3.68359e6i	0.240158i	0.992764	+	0.120079i	$0.0383147\pi$
$113$	3.68359e6i	0.240158i	−0.992764	+	0.120079i	$0.961685\pi$
$114$	−2.68618e6	−0.169812
$115$	0	0
$116$	−1.20265e7	−0.715379
$117$	− 7.41101e6i	− 0.427787i
$118$	2.10192e6i	0.117769i
$119$	5.07802e6	0.276236
$120$	0	0
$121$	1.03244e7	0.529806
$122$	7.81064e6i	0.389428i
$123$	− 1.01086e7i	− 0.489807i
$124$	2.72589e6	0.128390
$125$	0	0
$126$	2.98598e6	0.132981
$127$	2.67953e7i	1.16077i	0.814342	+	0.580385i	$0.197098\pi$
$127$	2.67953e7i	1.16077i	−0.814342	+	0.580385i	$0.802902\pi$
$128$	2.09715e6i	0.0883883i
$129$	−1.45944e7	−0.598579
$130$	0	0
$131$	1.48085e7	0.575523	0.287762	−	0.957702i	$-0.407089\pi$
$131$	1.48085e7	0.575523	0.287762	+	0.957702i	$0.407089\pi$
$132$	9.43488e6i	0.357049i
$133$	− 6.36723e6i	− 0.234677i
$134$	−2.84734e7	−1.02229
$135$	0	0
$136$	−5.07802e6	−0.173104
$137$	5.76532e7i	1.91559i	0.287458	+	0.957793i	$0.407190\pi$
$137$	5.76532e7i	1.91559i	−0.287458	+	0.957793i	$0.592810\pi$
$138$	− 7.25760e6i	− 0.235080i
$139$	8.37800e6	0.264599	0.132300	−	0.991210i	$-0.457764\pi$
$139$	8.37800e6	0.264599	0.132300	+	0.991210i	$0.457764\pi$
$140$	0	0
$141$	−3.61357e7	−1.08560
$142$	2.13898e7i	0.626898i
$143$	5.55064e7i	1.58733i
$144$	−2.98598e6	−0.0833333
$145$	0	0
$146$	−2.42571e7	−0.645065
$147$	− 1.51578e7i	− 0.393572i
$148$	− 3.48202e7i	− 0.882908i
$149$	−5.56477e6	−0.137815	−0.0689073	−	0.997623i	$-0.521951\pi$
$149$	−5.56477e6	−0.137815	−0.0689073	+	0.997623i	$0.521951\pi$
$150$	0	0
$151$	6.62933e7	1.56693	0.783466	−	0.621434i	$-0.213450\pi$
$151$	6.62933e7	1.56693	0.783466	+	0.621434i	$0.213450\pi$
$152$	6.36723e6i	0.147061i
$153$	− 7.23022e6i	− 0.163204i
$154$	−2.23642e7	−0.493435
$155$	0	0
$156$	−1.75668e7	−0.370474
$157$	6.42791e7i	1.32563i	0.748785	+	0.662813i	$0.230637\pi$
$157$	6.42791e7i	1.32563i	−0.748785	+	0.662813i	$0.769363\pi$
$158$	− 4.38065e7i	− 0.883565i
$159$	3.53220e7	0.696876
$160$	0	0
$161$	1.72032e7	0.324877
$162$	− 4.25153e6i	− 0.0785674i
$163$	8.48552e7i	1.53469i	0.641232	+	0.767347i	$0.278424\pi$
$163$	8.48552e7i	1.53469i	−0.641232	+	0.767347i	$0.721576\pi$
$164$	−2.39612e7	−0.424185
$165$	0	0
$166$	1.78524e7	0.302914
$167$	− 6.42144e7i	− 1.06690i	−0.845831	−	0.533451i	$-0.820895\pi$
$167$	− 6.42144e7i	− 1.06690i	0.845831	−	0.533451i	$-0.179105\pi$
$168$	− 7.07789e6i	− 0.115165i
$169$	−4.05990e7	−0.647012
$170$	0	0
$171$	−9.06584e6	−0.138651
$172$	3.45940e7i	0.518384i
$173$	1.10058e8i	1.61607i	0.589134	+	0.808036i	$0.299469\pi$
$173$	1.10058e8i	1.61607i	−0.589134	+	0.808036i	$0.700531\pi$
$174$	−4.05894e7	−0.584104
$175$	0	0
$176$	2.23642e7	0.309213
$177$	7.09398e6i	0.0961576i
$178$	− 8.04054e7i	− 1.06860i
$179$	−8.68778e7	−1.13220	−0.566100	−	0.824337i	$-0.691548\pi$
$179$	−8.68778e7	−1.13220	−0.566100	+	0.824337i	$0.691548\pi$
$180$	0	0
$181$	1.29730e8	1.62617	0.813083	−	0.582148i	$-0.197787\pi$
$181$	1.29730e8	1.62617	0.813083	+	0.582148i	$0.197787\pi$
$182$	− 4.16399e7i	− 0.511988i
$183$	2.63609e7i	0.317967i
$184$	−1.72032e7	−0.203586
$185$	0	0
$186$	9.19987e6	0.104830
$187$	5.41523e7i	0.605579i
$188$	8.56550e7i	0.940157i
$189$	1.00777e7	0.108579
$190$	0	0
$191$	−1.42024e8	−1.47484	−0.737418	−	0.675437i	$-0.763955\pi$
$191$	−1.42024e8	−1.47484	−0.737418	+	0.675437i	$0.763955\pi$
$192$	7.07789e6i	0.0721688i
$193$	− 2.58910e7i	− 0.259238i	−0.991564	−	0.129619i	$-0.958625\pi$
$193$	− 2.58910e7i	− 0.259238i	0.991564	−	0.129619i	$-0.0413754\pi$
$194$	−4.58204e7	−0.450561
$195$	0	0
$196$	−3.59295e7	−0.340844
$197$	3.59130e7i	0.334673i	0.985900	+	0.167336i	$0.0535166\pi$
$197$	3.59130e7i	0.334673i	−0.985900	+	0.167336i	$0.946483\pi$
$198$	3.18427e7i	0.291529i
$199$	1.75453e8	1.57824	0.789122	−	0.614237i	$-0.210536\pi$
$199$	1.75453e8	1.57824	0.789122	+	0.614237i	$0.210536\pi$
$200$	0	0
$201$	−9.60976e7	−0.834693
$202$	1.06686e8i	0.910706i
$203$	− 9.62120e7i	− 0.807222i
$204$	−1.71383e7	−0.141339
$205$	0	0
$206$	−2.16815e7	−0.172804
$207$	− 2.44944e7i	− 0.191942i
$208$	4.16399e7i	0.320840i
$209$	6.79006e7	0.514472
$210$	0	0
$211$	−9.18337e7	−0.672998	−0.336499	−	0.941684i	$-0.609243\pi$
$211$	−9.18337e7	−0.672998	−0.336499	+	0.941684i	$0.609243\pi$
$212$	− 8.37262e7i	− 0.603512i
$213$	7.21904e7i	0.511860i
$214$	−9.05561e7	−0.631640
$215$	0	0
$216$	−1.00777e7	−0.0680414
$217$	2.18071e7i	0.144873i
$218$	− 1.75786e8i	− 1.14917i
$219$	−8.18676e7	−0.526693
$220$	0	0
$221$	−1.00826e8	−0.628349
$222$	− 1.17518e8i	− 0.720892i
$223$	1.15780e7i	0.0699146i	0.999389	+	0.0349573i	$0.0111295\pi$
$223$	1.15780e7i	0.0699146i	−0.999389	+	0.0349573i	$0.988870\pi$
$224$	−1.67772e7	−0.0997360
$225$	0	0
$226$	2.94687e7	0.169817
$227$	2.99769e7i	0.170097i	0.996377	+	0.0850485i	$0.0271045\pi$
$227$	2.99769e7i	0.170097i	−0.996377	+	0.0850485i	$0.972895\pi$
$228$	2.14894e7i	0.120075i
$229$	−1.19721e8	−0.658787	−0.329394	−	0.944193i	$-0.606844\pi$
$229$	−1.19721e8	−0.658787	−0.329394	+	0.944193i	$0.606844\pi$
$230$	0	0
$231$	−7.54790e7	−0.402888
$232$	9.62120e7i	0.505849i
$233$	− 1.27607e8i	− 0.660890i	−0.943825	−	0.330445i	$-0.892801\pi$
$233$	− 1.27607e8i	− 0.660890i	0.943825	−	0.330445i	$-0.107199\pi$
$234$	−5.92881e7	−0.302491
$235$	0	0
$236$	1.68154e7	0.0832749
$237$	− 1.47847e8i	− 0.721428i
$238$	− 4.06241e7i	− 0.195328i
$239$	1.64949e8	0.781549	0.390775	−	0.920486i	$-0.372207\pi$
$239$	1.64949e8	0.781549	0.390775	+	0.920486i	$0.372207\pi$
$240$	0	0
$241$	3.98628e8	1.83446	0.917230	−	0.398357i	$-0.130420\pi$
$241$	3.98628e8	1.83446	0.917230	+	0.398357i	$0.130420\pi$
$242$	− 8.25954e7i	− 0.374630i
$243$	− 1.43489e7i	− 0.0641500i
$244$	6.24851e7	0.275367
$245$	0	0
$246$	−8.08691e7	−0.346346
$247$	1.26424e8i	0.533816i
$248$	− 2.18071e7i	− 0.0907856i
$249$	6.02520e7	0.247328
$250$	0	0
$251$	2.42512e8	0.967999	0.484000	−	0.875068i	$-0.339184\pi$
$251$	2.42512e8	0.967999	0.484000	+	0.875068i	$0.339184\pi$
$252$	− 2.38879e7i	− 0.0940320i
$253$	1.83456e8i	0.712213i
$254$	2.14363e8	0.820788
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	− 3.22956e8i	− 1.18680i	−0.804908	−	0.593399i	$-0.797786\pi$
$257$	− 3.22956e8i	− 1.18680i	0.804908	−	0.593399i	$-0.202214\pi$
$258$	1.16755e8i	0.423259i
$259$	2.78562e8	0.996259
$260$	0	0
$261$	−1.36989e8	−0.476919
$262$	− 1.18468e8i	− 0.406956i
$263$	− 3.37426e8i	− 1.14376i	−0.820338	−	0.571879i	$-0.806215\pi$
$263$	− 3.37426e8i	− 1.14376i	0.820338	−	0.571879i	$-0.193785\pi$
$264$	7.54790e7	0.252472
$265$	0	0
$266$	−5.09379e7	−0.165941
$267$	− 2.71368e8i	− 0.872509i
$268$	2.27787e8i	0.722865i
$269$	−2.55886e8	−0.801517	−0.400759	−	0.916184i	$-0.631253\pi$
$269$	−2.55886e8	−0.801517	−0.400759	+	0.916184i	$0.631253\pi$
$270$	0	0
$271$	4.85679e8	1.48237	0.741185	−	0.671301i	$-0.234264\pi$
$271$	4.85679e8	1.48237	0.741185	+	0.671301i	$0.234264\pi$
$272$	4.06241e7i	0.122403i
$273$	− 1.40535e8i	− 0.418037i
$274$	4.61226e8	1.35452
$275$	0	0
$276$	−5.80608e7	−0.166227
$277$	− 2.92732e8i	− 0.827545i	−0.910380	−	0.413772i	$-0.864211\pi$
$277$	− 2.92732e8i	− 0.827545i	0.910380	−	0.413772i	$-0.135789\pi$
$278$	− 6.70240e7i	− 0.187100i
$279$	3.10496e7	0.0855935
$280$	0	0
$281$	−1.86225e8	−0.500685	−0.250343	−	0.968157i	$-0.580543\pi$
$281$	−1.86225e8	−0.500685	−0.250343	+	0.968157i	$0.580543\pi$
$282$	2.89086e8i	0.767635i
$283$	− 6.58251e8i	− 1.72639i	−0.504870	−	0.863196i	$-0.668459\pi$
$283$	− 6.58251e8i	− 1.72639i	0.504870	−	0.863196i	$-0.331541\pi$
$284$	1.71118e8	0.443284
$285$	0	0
$286$	4.44051e8	1.12241
$287$	− 1.91690e8i	− 0.478644i
$288$	2.38879e7i	0.0589256i
$289$	3.11972e8	0.760279
$290$	0	0
$291$	−1.54644e8	−0.367881
$292$	1.94057e8i	0.456130i
$293$	1.61426e8i	0.374918i	0.982272	+	0.187459i	$0.0600252\pi$
$293$	1.61426e8i	0.374918i	−0.982272	+	0.187459i	$0.939975\pi$
$294$	−1.21262e8	−0.278298
$295$	0	0
$296$	−2.78562e8	−0.624310
$297$	1.07469e8i	0.238033i
$298$	4.45181e7i	0.0974496i
$299$	−3.41578e8	−0.738993
$300$	0	0
$301$	−2.76752e8	−0.584936
$302$	− 5.30346e8i	− 1.10799i
$303$	3.60066e8i	0.743588i
$304$	5.09379e7	0.103988
$305$	0	0
$306$	−5.78418e7	−0.115403
$307$	7.15488e8i	1.41130i	0.708562	+	0.705648i	$0.249344\pi$
$307$	7.15488e8i	1.41130i	−0.708562	+	0.705648i	$0.750656\pi$
$308$	1.78913e8i	0.348911i
$309$	−7.31752e7	−0.141094
$310$	0	0
$311$	−6.86692e7	−0.129449	−0.0647247	−	0.997903i	$-0.520617\pi$
$311$	−6.86692e7	−0.129449	−0.0647247	+	0.997903i	$0.520617\pi$
$312$	1.40535e8i	0.261965i
$313$	9.27771e7i	0.171015i	0.996338	+	0.0855077i	$0.0272512\pi$
$313$	9.27771e7i	0.171015i	−0.996338	+	0.0855077i	$0.972749\pi$
$314$	5.14233e8	0.937359
$315$	0	0
$316$	−3.50452e8	−0.624775
$317$	− 3.90749e8i	− 0.688954i	−0.938795	−	0.344477i	$-0.888056\pi$
$317$	− 3.90749e8i	− 0.688954i	0.938795	−	0.344477i	$-0.111944\pi$
$318$	− 2.82576e8i	− 0.492765i
$319$	1.02601e9	1.76964
$320$	0	0
$321$	−3.05627e8	−0.515732
$322$	− 1.37626e8i	− 0.229723i
$323$	1.23340e8i	0.203655i
$324$	−3.40122e7	−0.0555556
$325$	0	0
$326$	6.78842e8	1.08519
$327$	− 5.93277e8i	− 0.938297i
$328$	1.91690e8i	0.299944i
$329$	−6.85240e8	−1.06086
$330$	0	0
$331$	−4.80269e8	−0.727925	−0.363962	−	0.931414i	$-0.618576\pi$
$331$	−4.80269e8	−0.727925	−0.363962	+	0.931414i	$0.618576\pi$
$332$	− 1.42820e8i	− 0.214193i
$333$	− 3.96624e8i	− 0.588605i
$334$	−5.13715e8	−0.754414
$335$	0	0
$336$	−5.66231e7	−0.0814341
$337$	2.30504e8i	0.328075i	0.986454	+	0.164038i	$0.0524519\pi$
$337$	2.30504e8i	0.328075i	−0.986454	+	0.164038i	$0.947548\pi$
$338$	3.24792e8i	0.457507i
$339$	9.94568e7	0.138655
$340$	0	0
$341$	−2.32552e8	−0.317600
$342$	7.25268e7i	0.0980408i
$343$	− 7.09090e8i	− 0.948794i
$344$	2.76752e8	0.366553
$345$	0	0
$346$	8.80464e8	1.14273
$347$	8.36727e8i	1.07505i	0.843246	+	0.537527i	$0.180641\pi$
$347$	8.36727e8i	1.07505i	−0.843246	+	0.537527i	$0.819359\pi$
$348$	3.24715e8i	0.413024i
$349$	−1.51659e9	−1.90976	−0.954881	−	0.296988i	$-0.904018\pi$
$349$	−1.51659e9	−1.90976	−0.954881	+	0.296988i	$0.904018\pi$
$350$	0	0
$351$	−2.00097e8	−0.246983
$352$	− 1.78913e8i	− 0.218647i
$353$	2.33016e7i	0.0281952i	0.999901	+	0.0140976i	$0.00448755\pi$
$353$	2.33016e7i	0.0281952i	−0.999901	+	0.0140976i	$0.995512\pi$
$354$	5.67518e7	0.0679937
$355$	0	0
$356$	−6.43243e8	−0.755615
$357$	− 1.37106e8i	− 0.159485i
$358$	6.95022e8i	0.800586i
$359$	−1.63162e9	−1.86118	−0.930590	−	0.366063i	$-0.880705\pi$
$359$	−1.63162e9	−1.86118	−0.930590	+	0.366063i	$0.880705\pi$
$360$	0	0
$361$	−7.39218e8	−0.826984
$362$	− 1.03784e9i	− 1.14987i
$363$	− 2.78760e8i	− 0.305884i
$364$	−3.33119e8	−0.362031
$365$	0	0
$366$	2.10887e8	0.224836
$367$	− 3.47807e8i	− 0.367289i	−0.982993	−	0.183644i	$-0.941211\pi$
$367$	− 3.47807e8i	− 0.367289i	0.982993	−	0.183644i	$-0.0587895\pi$
$368$	1.37626e8i	0.143957i
$369$	−2.72933e8	−0.282790
$370$	0	0
$371$	6.69810e8	0.680993
$372$	− 7.35990e7i	− 0.0741262i
$373$	5.00769e7i	0.0499639i	0.999688	+	0.0249820i	$0.00795283\pi$
$373$	5.00769e7i	0.0499639i	−0.999688	+	0.0249820i	$0.992047\pi$
$374$	4.33218e8	0.428209
$375$	0	0
$376$	6.85240e8	0.664791
$377$	1.91033e9i	1.83618i
$378$	− 8.06216e7i	− 0.0767768i
$379$	−1.40154e9	−1.32241	−0.661207	−	0.750204i	$-0.729955\pi$
$379$	−1.40154e9	−1.32241	−0.661207	+	0.750204i	$0.729955\pi$
$380$	0	0
$381$	7.23474e8	0.670170
$382$	1.13619e9i	1.04287i
$383$	1.06830e9i	0.971623i	0.874064	+	0.485811i	$0.161476\pi$
$383$	1.06830e9i	0.971623i	−0.874064	+	0.485811i	$0.838524\pi$
$384$	5.66231e7	0.0510310
$385$	0	0
$386$	−2.07128e8	−0.183309
$387$	3.94048e8i	0.345590i
$388$	3.66563e8i	0.318594i
$389$	2.13713e8	0.184080	0.0920401	−	0.995755i	$-0.470661\pi$
$389$	2.13713e8	0.184080	0.0920401	+	0.995755i	$0.470661\pi$
$390$	0	0
$391$	−3.33245e8	−0.281932
$392$	2.87436e8i	0.241013i
$393$	− 3.99831e8i	− 0.332278i
$394$	2.87304e8	0.236649
$395$	0	0
$396$	2.54742e8	0.206142
$397$	− 9.69088e8i	− 0.777314i	−0.921383	−	0.388657i	$-0.872939\pi$
$397$	− 9.69088e8i	− 0.777314i	0.921383	−	0.388657i	$-0.127061\pi$
$398$	− 1.40362e9i	− 1.11599i
$399$	−1.71915e8	−0.135491
$400$	0	0
$401$	−2.21767e9	−1.71748	−0.858738	−	0.512415i	$-0.828751\pi$
$401$	−2.21767e9	−1.71748	−0.858738	+	0.512415i	$0.828751\pi$
$402$	7.68781e8i	0.590217i
$403$	− 4.32990e8i	− 0.329542i
$404$	8.53489e8	0.643966
$405$	0	0
$406$	−7.69696e8	−0.570792
$407$	2.97060e9i	2.18406i
$408$	1.37106e8i	0.0999418i
$409$	1.04837e9	0.757674	0.378837	−	0.925463i	$-0.376324\pi$
$409$	1.04837e9	0.757674	0.378837	+	0.925463i	$0.376324\pi$
$410$	0	0
$411$	1.55664e9	1.10596
$412$	1.73452e8i	0.122191i
$413$	1.34523e8i	0.0939661i
$414$	−1.95955e8	−0.135724
$415$	0	0
$416$	3.33119e8	0.226868
$417$	− 2.26206e8i	− 0.152766i
$418$	− 5.43204e8i	− 0.363786i
$419$	2.21021e8	0.146786	0.0733930	−	0.997303i	$-0.476617\pi$
$419$	2.21021e8	0.146786	0.0733930	+	0.997303i	$0.476617\pi$
$420$	0	0
$421$	1.51339e9	0.988472	0.494236	−	0.869328i	$-0.335448\pi$
$421$	1.51339e9	0.988472	0.494236	+	0.869328i	$0.335448\pi$
$422$	7.34670e8i	0.475881i
$423$	9.75664e8i	0.626771i
$424$	−6.69810e8	−0.426747
$425$	0	0
$426$	5.77524e8	0.361940
$427$	4.99881e8i	0.310720i
$428$	7.24449e8i	0.446637i
$429$	1.49867e9	0.916444
$430$	0	0
$431$	1.46801e9	0.883201	0.441601	−	0.897212i	$-0.354411\pi$
$431$	1.46801e9	0.883201	0.441601	+	0.897212i	$0.354411\pi$
$432$	8.06216e7i	0.0481125i
$433$	3.97508e8i	0.235309i	0.993055	+	0.117654i	$0.0375375\pi$
$433$	3.97508e8i	0.235309i	−0.993055	+	0.117654i	$0.962463\pi$
$434$	1.74457e8	0.102441
$435$	0	0
$436$	−1.40629e9	−0.812589
$437$	4.17850e8i	0.239516i
$438$	6.54941e8i	0.372428i
$439$	−1.22071e9	−0.688629	−0.344315	−	0.938854i	$-0.611889\pi$
$439$	−1.22071e9	−0.688629	−0.344315	+	0.938854i	$0.611889\pi$
$440$	0	0
$441$	−4.09260e8	−0.227229
$442$	8.06611e8i	0.444310i
$443$	3.24551e9i	1.77366i	0.462095	+	0.886830i	$0.347098\pi$
$443$	3.24551e9i	1.77366i	−0.462095	+	0.886830i	$0.652902\pi$
$444$	−9.40146e8	−0.509747
$445$	0	0
$446$	9.26243e7	0.0494371
$447$	1.50249e8i	0.0795672i
$448$	1.34218e8i	0.0705240i
$449$	7.04211e8	0.367147	0.183574	−	0.983006i	$-0.441233\pi$
$449$	7.04211e8	0.367147	0.183574	+	0.983006i	$0.441233\pi$
$450$	0	0
$451$	2.04419e9	1.04931
$452$	− 2.35750e8i	− 0.120079i
$453$	− 1.78992e9i	− 0.904669i
$454$	2.39815e8	0.120277
$455$	0	0
$456$	1.71915e8	0.0849058
$457$	− 3.70970e9i	− 1.81816i	−0.416621	−	0.909080i	$-0.636786\pi$
$457$	− 3.70970e9i	− 1.81816i	0.416621	−	0.909080i	$-0.363214\pi$
$458$	9.57766e8i	0.465833i
$459$	−1.95216e8	−0.0942261
$460$	0	0
$461$	1.12514e9	0.534878	0.267439	−	0.963575i	$-0.413823\pi$
$461$	1.12514e9	0.534878	0.267439	+	0.963575i	$0.413823\pi$
$462$	6.03832e8i	0.284885i
$463$	− 7.64328e7i	− 0.0357887i	−0.999840	−	0.0178944i	$-0.994304\pi$
$463$	− 7.64328e7i	− 0.0357887i	0.999840	−	0.0178944i	$-0.00569626\pi$
$464$	7.69696e8	0.357689
$465$	0	0
$466$	−1.02086e9	−0.467320
$467$	6.41328e7i	0.0291388i	0.999894	+	0.0145694i	$0.00463774\pi$
$467$	6.41328e7i	0.0291388i	−0.999894	+	0.0145694i	$0.995362\pi$
$468$	4.74305e8i	0.213893i
$469$	−1.82230e9	−0.815669
$470$	0	0
$471$	1.73553e9	0.765350
$472$	− 1.34523e8i	− 0.0588843i
$473$	− 2.95130e9i	− 1.28233i
$474$	−1.18277e9	−0.510126
$475$	0	0
$476$	−3.24993e8	−0.138118
$477$	− 9.53694e8i	− 0.402341i
$478$	− 1.31959e9i	− 0.552639i
$479$	−1.01386e9	−0.421505	−0.210753	−	0.977539i	$-0.567591\pi$
$479$	−1.01386e9	−0.421505	−0.210753	+	0.977539i	$0.567591\pi$
$480$	0	0
$481$	−5.53097e9	−2.26618
$482$	− 3.18903e9i	− 1.29716i
$483$	− 4.64486e8i	− 0.187568i
$484$	−6.60763e8	−0.264903
$485$	0	0
$486$	−1.14791e8	−0.0453609
$487$	2.79009e8i	0.109463i	0.998501	+	0.0547315i	$0.0174303\pi$
$487$	2.79009e8i	0.109463i	−0.998501	+	0.0547315i	$0.982570\pi$
$488$	− 4.99881e8i	− 0.194714i
$489$	2.29109e9	0.886056
$490$	0	0
$491$	1.77339e9	0.676112	0.338056	−	0.941126i	$-0.390231\pi$
$491$	1.77339e9	0.676112	0.338056	+	0.941126i	$0.390231\pi$
$492$	6.46953e8i	0.244903i
$493$	1.86373e9i	0.700518i
$494$	1.01140e9	0.377465
$495$	0	0
$496$	−1.74457e8	−0.0641951
$497$	1.36894e9i	0.500194i
$498$	− 4.82016e8i	− 0.174888i
$499$	4.66125e9	1.67939	0.839693	−	0.543061i	$-0.182735\pi$
$499$	4.66125e9	1.67939	0.839693	+	0.543061i	$0.182735\pi$
$500$	0	0
$501$	−1.73379e9	−0.615977
$502$	− 1.94010e9i	− 0.684479i
$503$	− 2.59549e9i	− 0.909352i	−0.890657	−	0.454676i	$-0.849755\pi$
$503$	− 2.59549e9i	− 0.909352i	0.890657	−	0.454676i	$-0.150245\pi$
$504$	−1.91103e8	−0.0664906
$505$	0	0
$506$	1.46765e9	0.503611
$507$	1.09617e9i	0.373553i
$508$	− 1.71490e9i	− 0.580385i
$509$	−9.18100e8	−0.308587	−0.154293	−	0.988025i	$-0.549310\pi$
$509$	−9.18100e8	−0.308587	−0.154293	+	0.988025i	$0.549310\pi$
$510$	0	0
$511$	−1.55245e9	−0.514689
$512$	− 1.34218e8i	− 0.0441942i
$513$	2.44778e8i	0.0800500i
$514$	−2.58365e9	−0.839193
$515$	0	0
$516$	9.34039e8	0.299289
$517$	− 7.30745e9i	− 2.32567i
$518$	− 2.22849e9i	− 0.704462i
$519$	2.97157e9	0.933039
$520$	0	0
$521$	−5.35323e8	−0.165838	−0.0829189	−	0.996556i	$-0.526424\pi$
$521$	−5.35323e8	−0.165838	−0.0829189	+	0.996556i	$0.526424\pi$
$522$	1.09591e9i	0.337233i
$523$	5.20020e8i	0.158951i	0.996837	+	0.0794756i	$0.0253246\pi$
$523$	5.20020e8i	0.158951i	−0.996837	+	0.0794756i	$0.974675\pi$
$524$	−9.47747e8	−0.287762
$525$	0	0
$526$	−2.69941e9	−0.808758
$527$	− 4.22427e8i	− 0.125723i
$528$	− 6.03832e8i	− 0.178524i
$529$	2.27587e9	0.668424
$530$	0	0
$531$	1.91537e8	0.0555166
$532$	4.07503e8i	0.117338i
$533$	3.80609e9i	1.08876i
$534$	−2.17095e9	−0.616957
$535$	0	0
$536$	1.82230e9	0.511143
$537$	2.34570e9i	0.653676i
$538$	2.04708e9i	0.566758i
$539$	3.06524e9	0.843148
$540$	0	0
$541$	−3.93186e9	−1.06760	−0.533798	−	0.845612i	$-0.679236\pi$
$541$	−3.93186e9	−1.06760	−0.533798	+	0.845612i	$0.679236\pi$
$542$	− 3.88543e9i	− 1.04819i
$543$	− 3.50271e9i	− 0.938867i
$544$	3.24993e8	0.0865522
$545$	0	0
$546$	−1.12428e9	−0.295597
$547$	− 3.11516e9i	− 0.813814i	−0.913470	−	0.406907i	$-0.866607\pi$
$547$	− 3.11516e9i	− 0.813814i	0.913470	−	0.406907i	$-0.133393\pi$
$548$	− 3.68981e9i	− 0.957793i
$549$	7.11745e8	0.183578
$550$	0	0
$551$	2.33690e9	0.595126
$552$	4.64486e8i	0.117540i
$553$	− 2.80361e9i	− 0.704986i
$554$	−2.34186e9	−0.585163
$555$	0	0
$556$	−5.36192e8	−0.132300
$557$	− 4.14733e9i	− 1.01689i	−0.861093	−	0.508447i	$-0.830220\pi$
$557$	− 4.14733e9i	− 1.01689i	0.861093	−	0.508447i	$-0.169780\pi$
$558$	− 2.48397e8i	− 0.0605238i
$559$	5.49505e9	1.33055
$560$	0	0
$561$	1.46211e9	0.349631
$562$	1.48980e9i	0.354038i
$563$	− 5.15380e8i	− 0.121716i	−0.998146	−	0.0608580i	$-0.980616\pi$
$563$	− 5.15380e8i	− 0.121716i	0.998146	−	0.0608580i	$-0.0193837\pi$
$564$	2.31269e9	0.542800
$565$	0	0
$566$	−5.26601e9	−1.22074
$567$	− 2.72098e8i	− 0.0626880i
$568$	− 1.36894e9i	− 0.313449i
$569$	4.82868e8	0.109884	0.0549421	−	0.998490i	$-0.482503\pi$
$569$	4.82868e8	0.109884	0.0549421	+	0.998490i	$0.482503\pi$
$570$	0	0
$571$	−2.23974e9	−0.503466	−0.251733	−	0.967797i	$-0.581001\pi$
$571$	−2.23974e9	−0.503466	−0.251733	+	0.967797i	$0.581001\pi$
$572$	− 3.55241e9i	− 0.793664i
$573$	3.83463e9i	0.851496i
$574$	−1.53352e9	−0.338452
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	− 6.93164e9i	− 1.50218i	−0.660202	−	0.751088i	$-0.729529\pi$
$577$	− 6.93164e9i	− 1.50218i	0.660202	−	0.751088i	$-0.270471\pi$
$578$	− 2.49578e9i	− 0.537599i
$579$	−6.99057e8	−0.149671
$580$	0	0
$581$	1.14256e9	0.241691
$582$	1.23715e9i	0.260131i
$583$	7.14289e9i	1.49291i
$584$	1.55245e9	0.322532
$585$	0	0
$586$	1.29141e9	0.265107
$587$	4.17248e9i	0.851454i	0.904852	+	0.425727i	$0.139982\pi$
$587$	4.17248e9i	0.851454i	−0.904852	+	0.425727i	$0.860018\pi$
$588$	9.70097e8i	0.196786i
$589$	−5.29674e8	−0.106808
$590$	0	0
$591$	9.69652e8	0.193223
$592$	2.22849e9i	0.441454i
$593$	6.27101e9i	1.23494i	0.786594	+	0.617470i	$0.211842\pi$
$593$	6.27101e9i	1.23494i	−0.786594	+	0.617470i	$0.788158\pi$
$594$	8.59753e8	0.168314
$595$	0	0
$596$	3.56145e8	0.0689073
$597$	− 4.73722e9i	− 0.911199i
$598$	2.73262e9i	0.522547i
$599$	2.10159e9	0.399535	0.199767	−	0.979843i	$-0.435981\pi$
$599$	2.10159e9	0.399535	0.199767	+	0.979843i	$0.435981\pi$
$600$	0	0
$601$	−2.41110e9	−0.453059	−0.226529	−	0.974004i	$-0.572738\pi$
$601$	−2.41110e9	−0.453059	−0.226529	+	0.974004i	$0.572738\pi$
$602$	2.21402e9i	0.413612i
$603$	2.59464e9i	0.481910i
$604$	−4.24277e9	−0.783466
$605$	0	0
$606$	2.88052e9	0.525796
$607$	5.94932e9i	1.07971i	0.841758	+	0.539855i	$0.181521\pi$
$607$	5.94932e9i	1.07971i	−0.841758	+	0.539855i	$0.818479\pi$
$608$	− 4.07503e8i	− 0.0735306i
$609$	−2.59772e9	−0.466050
$610$	0	0
$611$	1.36058e10	2.41312
$612$	4.62734e8i	0.0816022i
$613$	5.45618e9i	0.956702i	0.878169	+	0.478351i	$0.158765\pi$
$613$	5.45618e9i	0.956702i	−0.878169	+	0.478351i	$0.841235\pi$
$614$	5.72391e9	0.997937
$615$	0	0
$616$	1.43131e9	0.246718
$617$	4.11503e9i	0.705301i	0.935755	+	0.352651i	$0.114720\pi$
$617$	4.11503e9i	0.705301i	−0.935755	+	0.352651i	$0.885280\pi$
$618$	5.85401e8i	0.0997687i
$619$	−2.78751e9	−0.472388	−0.236194	−	0.971706i	$-0.575900\pi$
$619$	−2.78751e9	−0.472388	−0.236194	+	0.971706i	$0.575900\pi$
$620$	0	0
$621$	−6.61349e8	−0.110818
$622$	5.49353e8i	0.0915346i
$623$	− 5.14595e9i	− 0.852623i
$624$	1.12428e9	0.185237
$625$	0	0
$626$	7.42217e8	0.120926
$627$	− 1.83332e9i	− 0.297030i
$628$	− 4.11386e9i	− 0.662813i
$629$	−5.39605e9	−0.864567
$630$	0	0
$631$	4.87155e8	0.0771906	0.0385953	−	0.999255i	$-0.487712\pi$
$631$	4.87155e8	0.0771906	0.0385953	+	0.999255i	$0.487712\pi$
$632$	2.80361e9i	0.441782i
$633$	2.47951e9i	0.388555i
$634$	−3.12599e9	−0.487164
$635$	0	0
$636$	−2.26061e9	−0.348438
$637$	5.70718e9i	0.874850i
$638$	− 8.20808e9i	− 1.25132i
$639$	1.94914e9	0.295522
$640$	0	0
$641$	−2.39918e9	−0.359798	−0.179899	−	0.983685i	$-0.557577\pi$
$641$	−2.39918e9	−0.359798	−0.179899	+	0.983685i	$0.557577\pi$
$642$	2.44502e9i	0.364678i
$643$	− 1.02984e10i	− 1.52767i	−0.645410	−	0.763836i	$-0.723313\pi$
$643$	− 1.02984e10i	− 1.52767i	0.645410	−	0.763836i	$-0.276687\pi$
$644$	−1.10100e9	−0.162438
$645$	0	0
$646$	9.86722e8	0.144006
$647$	8.49896e9i	1.23367i	0.787091	+	0.616837i	$0.211586\pi$
$647$	8.49896e9i	1.23367i	−0.787091	+	0.616837i	$0.788414\pi$
$648$	2.72098e8i	0.0392837i
$649$	−1.43456e9	−0.205998
$650$	0	0
$651$	5.88792e8	0.0836427
$652$	− 5.43073e9i	− 0.767347i
$653$	− 3.78543e9i	− 0.532009i	−0.963972	−	0.266004i	$-0.914296\pi$
$653$	− 3.78543e9i	− 0.532009i	0.963972	−	0.266004i	$-0.0857035\pi$
$654$	−4.74621e9	−0.663476
$655$	0	0
$656$	1.53352e9	0.212093
$657$	2.21043e9i	0.304086i
$658$	5.48192e9i	0.750140i
$659$	7.69061e9	1.04680	0.523398	−	0.852089i	$-0.324664\pi$
$659$	7.69061e9	1.04680	0.523398	+	0.852089i	$0.324664\pi$
$660$	0	0
$661$	5.78185e9	0.778685	0.389343	−	0.921093i	$-0.372702\pi$
$661$	5.78185e9	0.778685	0.389343	+	0.921093i	$0.372702\pi$
$662$	3.84215e9i	0.514720i
$663$	2.72231e9i	0.362778i
$664$	−1.14256e9	−0.151457
$665$	0	0
$666$	−3.17299e9	−0.416207
$667$	6.31391e9i	0.823869i
$668$	4.10972e9i	0.533451i
$669$	3.12607e8	0.0403652
$670$	0	0
$671$	−5.33076e9	−0.681178
$672$	4.52985e8i	0.0575826i
$673$	− 1.07125e10i	− 1.35468i	−0.735670	−	0.677340i	$-0.763133\pi$
$673$	− 1.07125e10i	− 1.35468i	0.735670	−	0.677340i	$-0.236867\pi$
$674$	1.84403e9	0.231984
$675$	0	0
$676$	2.59834e9	0.323506
$677$	− 1.24253e8i	− 0.0153903i	−0.999970	−	0.00769516i	$-0.997551\pi$
$677$	− 1.24253e8i	− 0.0153903i	0.999970	−	0.00769516i	$-0.00244947\pi$
$678$	− 7.95655e8i	− 0.0980439i
$679$	−2.93251e9	−0.359497
$680$	0	0
$681$	8.09377e8	0.0982055
$682$	1.86042e9i	0.224577i
$683$	− 5.48973e8i	− 0.0659294i	−0.999457	−	0.0329647i	$-0.989505\pi$
$683$	− 5.48973e8i	− 0.0659294i	0.999457	−	0.0329647i	$-0.0104949\pi$
$684$	5.80214e8	0.0693253
$685$	0	0
$686$	−5.67272e9	−0.670899
$687$	3.23246e9i	0.380351i
$688$	− 2.21402e9i	− 0.259192i
$689$	−1.32994e10	−1.54905
$690$	0	0
$691$	3.86163e9	0.445243	0.222622	−	0.974905i	$-0.428539\pi$
$691$	3.86163e9	0.445243	0.222622	+	0.974905i	$0.428539\pi$
$692$	− 7.04372e9i	− 0.808036i
$693$	2.03793e9i	0.232608i
$694$	6.69381e9	0.760178
$695$	0	0
$696$	2.59772e9	0.292052
$697$	3.71324e9i	0.415373i
$698$	1.21327e10i	1.35041i
$699$	−3.44539e9	−0.381565
$700$	0	0
$701$	−1.24129e10	−1.36100	−0.680501	−	0.732748i	$-0.738238\pi$
$701$	−1.24129e10	−1.36100	−0.680501	+	0.732748i	$0.738238\pi$
$702$	1.60078e9i	0.174643i
$703$	6.76600e9i	0.734495i
$704$	−1.43131e9	−0.154607
$705$	0	0
$706$	1.86413e8	0.0199370
$707$	6.82791e9i	0.726641i
$708$	− 4.54015e8i	− 0.0480788i
$709$	−8.68970e9	−0.915678	−0.457839	−	0.889035i	$-0.651376\pi$
$709$	−8.68970e9	−0.915678	−0.457839	+	0.889035i	$0.651376\pi$
$710$	0	0
$711$	−3.99186e9	−0.416516
$712$	5.14595e9i	0.534300i
$713$	− 1.43109e9i	− 0.147861i
$714$	−1.09685e9	−0.112773
$715$	0	0
$716$	5.56018e9	0.566100
$717$	− 4.45362e9i	− 0.451228i
$718$	1.30530e10i	1.31605i
$719$	−1.34874e10	−1.35325	−0.676625	−	0.736328i	$-0.736558\pi$
$719$	−1.34874e10	−1.35325	−0.676625	+	0.736328i	$0.736558\pi$
$720$	0	0
$721$	−1.38762e9	−0.137879
$722$	5.91374e9i	0.584766i
$723$	− 1.07630e10i	− 1.05913i
$724$	−8.30271e9	−0.813083
$725$	0	0
$726$	−2.23008e9	−0.216293
$727$	4.09956e9i	0.395700i	0.980232	+	0.197850i	$0.0633960\pi$
$727$	4.09956e9i	0.395700i	−0.980232	+	0.197850i	$0.936604\pi$
$728$	2.66496e9i	0.255994i
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	5.36100e9	0.507615
$732$	− 1.68710e9i	− 0.158983i
$733$	9.50716e9i	0.891635i	0.895124	+	0.445818i	$0.147087\pi$
$733$	9.50716e9i	0.891635i	−0.895124	+	0.445818i	$0.852913\pi$
$734$	−2.78246e9	−0.259712
$735$	0	0
$736$	1.10100e9	0.101793
$737$	− 1.94331e10i	− 1.78816i
$738$	2.18347e9i	0.199963i
$739$	−2.75490e9	−0.251102	−0.125551	−	0.992087i	$-0.540070\pi$
$739$	−2.75490e9	−0.251102	−0.125551	+	0.992087i	$0.540070\pi$
$740$	0	0
$741$	3.41346e9	0.308199
$742$	− 5.35848e9i	− 0.481535i
$743$	7.88700e9i	0.705425i	0.935732	+	0.352712i	$0.114741\pi$
$743$	7.88700e9i	0.705425i	−0.935732	+	0.352712i	$0.885259\pi$
$744$	−5.88792e8	−0.0524151
$745$	0	0
$746$	4.00615e8	0.0353298
$747$	− 1.62680e9i	− 0.142795i
$748$	− 3.46575e9i	− 0.302790i
$749$	−5.79559e9	−0.503978
$750$	0	0
$751$	1.54231e10	1.32871	0.664356	−	0.747416i	$-0.268706\pi$
$751$	1.54231e10	1.32871	0.664356	+	0.747416i	$0.268706\pi$
$752$	− 5.48192e9i	− 0.470079i
$753$	− 6.54782e9i	− 0.558875i
$754$	1.52827e10	1.29837
$755$	0	0
$756$	−6.44973e8	−0.0542894
$757$	2.28786e10i	1.91687i	0.285307	+	0.958436i	$0.407904\pi$
$757$	2.28786e10i	1.91687i	−0.285307	+	0.958436i	$0.592096\pi$
$758$	1.12123e10i	0.935087i
$759$	4.95331e9	0.411197
$760$	0	0
$761$	−2.06723e10	−1.70037	−0.850183	−	0.526488i	$-0.823509\pi$
$761$	−2.06723e10	−1.70037	−0.850183	+	0.526488i	$0.823509\pi$
$762$	− 5.78779e9i	− 0.473882i
$763$	− 1.12503e10i	− 0.916912i
$764$	9.08950e9	0.737418
$765$	0	0
$766$	8.54640e9	0.687041
$767$	− 2.67101e9i	− 0.213743i
$768$	− 4.52985e8i	− 0.0360844i
$769$	7.74699e9	0.614315	0.307157	−	0.951659i	$-0.400622\pi$
$769$	7.74699e9	0.614315	0.307157	+	0.951659i	$0.400622\pi$
$770$	0	0
$771$	−8.71980e9	−0.685198
$772$	1.65702e9i	0.129619i
$773$	1.55352e10i	1.20973i	0.796327	+	0.604866i	$0.206773\pi$
$773$	1.55352e10i	1.20973i	−0.796327	+	0.604866i	$0.793227\pi$
$774$	3.15238e9	0.244369
$775$	0	0
$776$	2.93251e9	0.225280
$777$	− 7.52117e9i	− 0.575191i
$778$	− 1.70970e9i	− 0.130164i
$779$	4.65596e9	0.352881
$780$	0	0
$781$	−1.45985e10	−1.09655
$782$	2.66596e9i	0.199356i
$783$	3.69871e9i	0.275349i
$784$	2.29949e9	0.170422
$785$	0	0
$786$	−3.19864e9	−0.234956
$787$	− 1.21211e10i	− 0.886399i	−0.896423	−	0.443199i	$-0.853843\pi$
$787$	− 1.21211e10i	− 0.886399i	0.896423	−	0.443199i	$-0.146157\pi$
$788$	− 2.29843e9i	− 0.167336i
$789$	−9.11051e9	−0.660349
$790$	0	0
$791$	1.88600e9	0.135495
$792$	− 2.03793e9i	− 0.145765i
$793$	− 9.92537e9i	− 0.706790i
$794$	−7.75270e9	−0.549644
$795$	0	0
$796$	−1.12290e10	−0.789122
$797$	− 8.20343e9i	− 0.573973i	−0.957935	−	0.286987i	$-0.907346\pi$
$797$	− 8.20343e9i	− 0.573973i	0.957935	−	0.286987i	$-0.0926535\pi$
$798$	1.37532e9i	0.0958064i
$799$	1.32739e10	0.920626
$800$	0	0
$801$	−7.32694e9	−0.503743
$802$	1.77413e10i	1.21444i
$803$	− 1.65555e10i	− 1.12833i
$804$	6.15025e9	0.417346
$805$	0	0
$806$	−3.46392e9	−0.233021
$807$	6.90891e9i	0.462756i
$808$	− 6.82791e9i	− 0.455353i
$809$	−9.91739e9	−0.658533	−0.329267	−	0.944237i	$-0.606801\pi$
$809$	−9.91739e9	−0.658533	−0.329267	+	0.944237i	$0.606801\pi$
$810$	0	0
$811$	2.13417e10	1.40494	0.702469	−	0.711715i	$-0.252081\pi$
$811$	2.13417e10	1.40494	0.702469	+	0.711715i	$0.252081\pi$
$812$	6.15757e9i	0.403611i
$813$	− 1.31133e10i	− 0.855847i
$814$	2.37648e10	1.54436
$815$	0	0
$816$	1.09685e9	0.0706695
$817$	− 6.72206e9i	− 0.431246i
$818$	− 8.38695e9i	− 0.535756i
$819$	−3.79444e9	−0.241354
$820$	0	0
$821$	1.81759e9	0.114629	0.0573145	−	0.998356i	$-0.481746\pi$
$821$	1.81759e9	0.114629	0.0573145	+	0.998356i	$0.481746\pi$
$822$	− 1.24531e10i	− 0.782035i
$823$	9.68482e9i	0.605609i	0.953053	+	0.302804i	$0.0979228\pi$
$823$	9.68482e9i	0.605609i	−0.953053	+	0.302804i	$0.902077\pi$
$824$	1.38762e9	0.0864022
$825$	0	0
$826$	1.07618e9	0.0664440
$827$	− 1.22370e10i	− 0.752326i	−0.926553	−	0.376163i	$-0.877243\pi$
$827$	− 1.22370e10i	− 0.752326i	0.926553	−	0.376163i	$-0.122757\pi$
$828$	1.56764e9i	0.0959711i
$829$	1.30073e9	0.0792951	0.0396476	−	0.999214i	$-0.487376\pi$
$829$	1.30073e9	0.0792951	0.0396476	+	0.999214i	$0.487376\pi$
$830$	0	0
$831$	−7.90377e9	−0.477783
$832$	− 2.66496e9i	− 0.160420i
$833$	5.56796e9i	0.333763i
$834$	−1.80965e9	−0.108022
$835$	0	0
$836$	−4.34564e9	−0.257236
$837$	− 8.38338e8i	− 0.0494174i
$838$	− 1.76817e9i	− 0.103793i
$839$	1.65720e10	0.968743	0.484372	−	0.874862i	$-0.339048\pi$
$839$	1.65720e10	0.968743	0.484372	+	0.874862i	$0.339048\pi$
$840$	0	0
$841$	1.80618e10	1.04707
$842$	− 1.21071e10i	− 0.698956i
$843$	5.02806e9i	0.289071i
$844$	5.87736e9	0.336499
$845$	0	0
$846$	7.80532e9	0.443194
$847$	− 5.28611e9i	− 0.298912i
$848$	5.35848e9i	0.301756i
$849$	−1.77728e10	−0.996732
$850$	0	0
$851$	−1.82806e10	−1.01680
$852$	− 4.62019e9i	− 0.255930i
$853$	− 2.87728e10i	− 1.58730i	−0.608372	−	0.793652i	$-0.708177\pi$
$853$	− 2.87728e10i	− 1.58730i	0.608372	−	0.793652i	$-0.291823\pi$
$854$	3.99905e9	0.219712
$855$	0	0
$856$	5.79559e9	0.315820
$857$	2.85737e9i	0.155072i	0.996990	+	0.0775361i	$0.0247053\pi$
$857$	2.85737e9i	0.155072i	−0.996990	+	0.0775361i	$0.975295\pi$
$858$	− 1.19894e10i	− 0.648024i
$859$	−1.38088e10	−0.743327	−0.371663	−	0.928368i	$-0.621212\pi$
$859$	−1.38088e10	−0.743327	−0.371663	+	0.928368i	$0.621212\pi$
$860$	0	0
$861$	−5.17562e9	−0.276345
$862$	− 1.17441e10i	− 0.624518i
$863$	− 2.72992e9i	− 0.144581i	−0.997384	−	0.0722905i	$-0.976969\pi$
$863$	− 2.72992e9i	− 0.144581i	0.997384	−	0.0722905i	$-0.0230309\pi$
$864$	6.44973e8	0.0340207
$865$	0	0
$866$	3.18006e9	0.166388
$867$	− 8.42324e9i	− 0.438947i
$868$	− 1.39565e9i	− 0.0724367i
$869$	2.98979e10	1.54551
$870$	0	0
$871$	3.61825e10	1.85539
$872$	1.12503e10i	0.574587i
$873$	4.17539e9i	0.212396i
$874$	3.34280e9	0.169364
$875$	0	0
$876$	5.23953e9	0.263347
$877$	− 1.52412e10i	− 0.762992i	−0.924370	−	0.381496i	$-0.875409\pi$
$877$	− 1.52412e10i	− 0.762992i	0.924370	−	0.381496i	$-0.124591\pi$
$878$	9.76565e9i	0.486935i
$879$	4.35850e9	0.216459
$880$	0	0
$881$	−1.89703e10	−0.934672	−0.467336	−	0.884080i	$-0.654786\pi$
$881$	−1.89703e10	−0.934672	−0.467336	+	0.884080i	$0.654786\pi$
$882$	3.27408e9i	0.160675i
$883$	1.03565e10i	0.506232i	0.967436	+	0.253116i	$0.0814554\pi$
$883$	1.03565e10i	0.506232i	−0.967436	+	0.253116i	$0.918545\pi$
$884$	6.45289e9	0.314175
$885$	0	0
$886$	2.59641e10	1.25417
$887$	4.04503e9i	0.194621i	0.995254	+	0.0973103i	$0.0310239\pi$
$887$	4.04503e9i	0.194621i	−0.995254	+	0.0973103i	$0.968976\pi$
$888$	7.52117e9i	0.360446i
$889$	1.37192e10	0.654897
$890$	0	0
$891$	2.90167e9	0.137428
$892$	− 7.40995e8i	− 0.0349573i
$893$	− 1.66438e10i	− 0.782120i
$894$	1.20199e9	0.0562625
$895$	0	0
$896$	1.07374e9	0.0498680
$897$	9.22260e9i	0.426658i
$898$	− 5.63369e9i	− 0.259612i
$899$	−8.00363e9	−0.367391
$900$	0	0
$901$	−1.29749e10	−0.590975
$902$	− 1.63535e10i	− 0.741974i
$903$	7.47231e9i	0.337713i
$904$	−1.88600e9	−0.0849085
$905$	0	0
$906$	−1.43194e10	−0.639697
$907$	3.30880e10i	1.47247i	0.676728	+	0.736233i	$0.263397\pi$
$907$	3.30880e10i	1.47247i	−0.676728	+	0.736233i	$0.736603\pi$
$908$	− 1.91852e9i	− 0.0850485i
$909$	9.72177e9	0.429311
$910$	0	0
$911$	−2.13526e10	−0.935700	−0.467850	−	0.883808i	$-0.654971\pi$
$911$	−2.13526e10	−0.935700	−0.467850	+	0.883808i	$0.654971\pi$
$912$	− 1.37532e9i	− 0.0600375i
$913$	1.21843e10i	0.529850i
$914$	−2.96776e10	−1.28563
$915$	0	0
$916$	7.66213e9	0.329394
$917$	− 7.58197e9i	− 0.324705i
$918$	1.56173e9i	0.0666279i
$919$	−4.49811e10	−1.91173	−0.955863	−	0.293813i	$-0.905076\pi$
$919$	−4.49811e10	−1.91173	−0.955863	+	0.293813i	$0.905076\pi$
$920$	0	0
$921$	1.93182e10	0.814813
$922$	− 9.00115e9i	− 0.378216i
$923$	− 2.71810e10i	− 1.13778i
$924$	4.83066e9	0.201444
$925$	0	0
$926$	−6.11462e8	−0.0253065
$927$	1.97573e9i	0.0814608i
$928$	− 6.15757e9i	− 0.252925i
$929$	2.15112e10	0.880258	0.440129	−	0.897935i	$-0.354933\pi$
$929$	2.15112e10	0.880258	0.440129	+	0.897935i	$0.354933\pi$
$930$	0	0
$931$	6.98156e9	0.283549
$932$	8.16685e9i	0.330445i
$933$	1.85407e9i	0.0747377i
$934$	5.13062e8	0.0206042
$935$	0	0
$936$	3.79444e9	0.151245
$937$	3.31276e10i	1.31553i	0.753222	+	0.657766i	$0.228498\pi$
$937$	3.31276e10i	1.31553i	−0.753222	+	0.657766i	$0.771502\pi$
$938$	1.45784e10i	0.576765i
$939$	2.50498e9	0.0987358
$940$	0	0
$941$	1.55361e10	0.607824	0.303912	−	0.952700i	$-0.401707\pi$
$941$	1.55361e10	0.607824	0.303912	+	0.952700i	$0.401707\pi$
$942$	− 1.38843e10i	− 0.541184i
$943$	1.25796e10i	0.488514i
$944$	−1.07618e9	−0.0416375
$945$	0	0
$946$	−2.36104e10	−0.906745
$947$	− 5.05218e10i	− 1.93310i	−0.256482	−	0.966549i	$-0.582564\pi$
$947$	− 5.05218e10i	− 1.93310i	0.256482	−	0.966549i	$-0.417436\pi$
$948$	9.46220e9i	0.360714i
$949$	3.08247e10	1.17076
$950$	0	0
$951$	−1.05502e10	−0.397768
$952$	2.59994e9i	0.0976640i
$953$	− 7.93237e9i	− 0.296878i	−0.988922	−	0.148439i	$-0.952575\pi$
$953$	− 7.93237e9i	− 0.296878i	0.988922	−	0.148439i	$-0.0474248\pi$
$954$	−7.62955e9	−0.284498
$955$	0	0
$956$	−1.05567e10	−0.390775
$957$	− 2.77023e10i	− 1.02170i
$958$	8.11087e9i	0.298049i
$959$	2.95185e10	1.08076
$960$	0	0
$961$	−2.56985e10	−0.934064
$962$	4.42478e10i	1.60243i
$963$	8.25193e9i	0.297758i
$964$	−2.55122e10	−0.917230
$965$	0	0
$966$	−3.71589e9	−0.132630
$967$	− 2.50000e10i	− 0.889095i	−0.895755	−	0.444547i	$-0.853365\pi$
$967$	− 2.50000e10i	− 0.889095i	0.895755	−	0.444547i	$-0.146635\pi$
$968$	5.28611e9i	0.187315i
$969$	3.33019e9	0.117581
$970$	0	0
$971$	2.23544e10	0.783603	0.391801	−	0.920050i	$-0.371852\pi$
$971$	2.23544e10	0.783603	0.391801	+	0.920050i	$0.371852\pi$
$972$	9.18330e8i	0.0320750i
$973$	− 4.28953e9i	− 0.149285i
$974$	2.23208e9	0.0774020
$975$	0	0
$976$	−3.99905e9	−0.137684
$977$	1.84311e9i	0.0632297i	0.999500	+	0.0316148i	$0.0100650\pi$
$977$	1.84311e9i	0.0632297i	−0.999500	+	0.0316148i	$0.989935\pi$
$978$	− 1.83287e10i	− 0.626536i
$979$	5.48767e10	1.86917
$980$	0	0
$981$	−1.60185e10	−0.541726
$982$	− 1.41871e10i	− 0.478083i
$983$	1.25816e10i	0.422473i	0.977435	+	0.211236i	$0.0677490\pi$
$983$	1.25816e10i	0.422473i	−0.977435	+	0.211236i	$0.932251\pi$
$984$	5.17562e9	0.173173
$985$	0	0
$986$	1.49098e10	0.495341
$987$	1.85015e10i	0.612487i
$988$	− 8.09116e9i	− 0.266908i
$989$	1.81619e10	0.596999
$990$	0	0
$991$	−2.30746e10	−0.753142	−0.376571	−	0.926388i	$-0.622897\pi$
$991$	−2.30746e10	−0.753142	−0.376571	+	0.926388i	$0.622897\pi$
$992$	1.39565e9i	0.0453928i
$993$	1.29673e10i	0.420267i
$994$	1.09516e10	0.353691
$995$	0	0
$996$	−3.85613e9	−0.123664
$997$	5.91619e9i	0.189064i	0.995522	+	0.0945320i	$0.0301355\pi$
$997$	5.91619e9i	0.189064i	−0.995522	+	0.0945320i	$0.969865\pi$
$998$	− 3.72900e10i	− 1.18751i
$999$	−1.07089e10	−0.339832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.8.c.e.49.1		2
3.2	odd	2		450.8.c.c.199.2		2
5.2	odd	4		30.8.a.e.1.1	✓	1
5.3	odd	4		150.8.a.g.1.1		1
5.4	even	2	inner	150.8.c.e.49.2		2
15.2	even	4		90.8.a.b.1.1		1
15.8	even	4		450.8.a.r.1.1		1
15.14	odd	2		450.8.c.c.199.1		2
20.7	even	4		240.8.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.a.e.1.1	✓	1	5.2	odd	4
90.8.a.b.1.1		1	15.2	even	4
150.8.a.g.1.1		1	5.3	odd	4
150.8.c.e.49.1		2	1.1	even	1	trivial
150.8.c.e.49.2		2	5.4	even	2	inner
240.8.a.l.1.1		1	20.7	even	4
450.8.a.r.1.1		1	15.8	even	4
450.8.c.c.199.1		2	15.14	odd	2
450.8.c.c.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)

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} -27.0000i q^{3} -64.0000 q^{4} -216.000 q^{6} -512.000i q^{7} +512.000i q^{8} -729.000 q^{9} +5460.00 q^{11} +1728.00i q^{12} +10166.0i q^{13} -4096.00 q^{14} +4096.00 q^{16} +9918.00i q^{17} +5832.00i q^{18} +12436.0 q^{19} -13824.0 q^{21} -43680.0i q^{22} +33600.0i q^{23} +13824.0 q^{24} +81328.0 q^{26} +19683.0i q^{27} +32768.0i q^{28} +187914. q^{29} -42592.0 q^{31} -32768.0i q^{32} -147420. i q^{33} +79344.0 q^{34} +46656.0 q^{36} +544066. i q^{37} -99488.0i q^{38} +274482. q^{39} +374394. q^{41} +110592. i q^{42} -540532. i q^{43} -349440. q^{44} +268800. q^{46} -1.33836e6i q^{47} -110592. i q^{48} +561399. q^{49} +267786. q^{51} -650624. i q^{52} +1.30822e6i q^{53} +157464. q^{54} +262144. q^{56} -335772. i q^{57} -1.50331e6i q^{58} -262740. q^{59} -976330. q^{61} +340736. i q^{62} +373248. i q^{63} -262144. q^{64} -1.17936e6 q^{66} -3.55917e6i q^{67} -634752. i q^{68} +907200. q^{69} -2.67372e6 q^{71} -373248. i q^{72} -3.03213e6i q^{73} +4.35253e6 q^{74} -795904. q^{76} -2.79552e6i q^{77} -2.19586e6i q^{78} +5.47581e6 q^{79} +531441. q^{81} -2.99515e6i q^{82} +2.23156e6i q^{83} +884736. q^{84} -4.32426e6 q^{86} -5.07368e6i q^{87} +2.79552e6i q^{88} +1.00507e7 q^{89} +5.20499e6 q^{91} -2.15040e6i q^{92} +1.14998e6i q^{93} -1.07069e7 q^{94} -884736. q^{96} -5.72755e6i q^{97} -4.49119e6i q^{98} -3.98034e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 432 q^{6} - 1458 q^{9} + 10920 q^{11} - 8192 q^{14} + 8192 q^{16} + 24872 q^{19} - 27648 q^{21} + 27648 q^{24} + 162656 q^{26} + 375828 q^{29} - 85184 q^{31} + 158688 q^{34} + 93312 q^{36}+ \cdots - 7960680 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 432 * q^6 - 1458 * q^9 + 10920 * q^11 - 8192 * q^14 + 8192 * q^16 + 24872 * q^19 - 27648 * q^21 + 27648 * q^24 + 162656 * q^26 + 375828 * q^29 - 85184 * q^31 + 158688 * q^34 + 93312 * q^36 + 548964 * q^39 + 748788 * q^41 - 698880 * q^44 + 537600 * q^46 + 1122798 * q^49 + 535572 * q^51 + 314928 * q^54 + 524288 * q^56 - 525480 * q^59 - 1952660 * q^61 - 524288 * q^64 - 2358720 * q^66 + 1814400 * q^69 - 5347440 * q^71 + 8705056 * q^74 - 1591808 * q^76 + 10951616 * q^79 + 1062882 * q^81 + 1769472 * q^84 - 8648512 * q^86 + 20101356 * q^89 + 10409984 * q^91 - 21413760 * q^94 - 1769472 * q^96 - 7960680 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more