LMFDB - Embedded newform 150.12.c.g.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,12,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, 12, names="a")

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

chi := DirichletCharacter("150.49");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
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:	$115.251477084$
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			150.12.c.g.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-32.0000i q^{2} +243.000i q^{3} -1024.00 q^{4} +7776.00 q^{6} -22876.0i q^{7} +32768.0i q^{8} -59049.0 q^{9} -259128. q^{11} -248832. i q^{12} +2.32346e6i q^{13} -732032. q^{14} +1.04858e6 q^{16} -1.91827e6i q^{17} +1.88957e6i q^{18} +1.61374e6 q^{19} +5.55887e6 q^{21} +8.29210e6i q^{22} -3.28492e7i q^{23} -7.96262e6 q^{24} +7.43508e7 q^{26} -1.43489e7i q^{27} +2.34250e7i q^{28} -1.52590e8 q^{29} +5.68101e7 q^{31} -3.35544e7i q^{32} -6.29681e7i q^{33} -6.13845e7 q^{34} +6.04662e7 q^{36} +5.71517e8i q^{37} -5.16397e7i q^{38} -5.64601e8 q^{39} +7.34063e8 q^{41} -1.77884e8i q^{42} +5.80697e8i q^{43} +2.65347e8 q^{44} -1.05117e9 q^{46} +4.78847e8i q^{47} +2.54804e8i q^{48} +1.45402e9 q^{49} +4.66139e8 q^{51} -2.37923e9i q^{52} +2.55379e9i q^{53} -4.59165e8 q^{54} +7.49601e8 q^{56} +3.92139e8i q^{57} +4.88289e9i q^{58} -6.31792e9 q^{59} -7.86138e8 q^{61} -1.81792e9i q^{62} +1.35080e9i q^{63} -1.07374e9 q^{64} -2.01498e9 q^{66} -2.12877e10i q^{67} +1.96430e9i q^{68} +7.98236e9 q^{69} +1.17195e10 q^{71} -1.93492e9i q^{72} -2.31756e10i q^{73} +1.82886e10 q^{74} -1.65247e9 q^{76} +5.92781e9i q^{77} +1.80672e10i q^{78} -2.69897e9 q^{79} +3.48678e9 q^{81} -2.34900e10i q^{82} -3.26587e10i q^{83} -5.69228e9 q^{84} +1.85823e10 q^{86} -3.70794e10i q^{87} -8.49111e9i q^{88} -7.45563e10 q^{89} +5.31515e10 q^{91} +3.36376e10i q^{92} +1.38048e10i q^{93} +1.53231e10 q^{94} +8.15373e9 q^{96} +4.14349e10i q^{97} -4.65285e10i q^{98} +1.53012e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2048 q^{4} + 15552 q^{6} - 118098 q^{9} - 518256 q^{11} - 1464064 q^{14} + 2097152 q^{16} + 3227480 q^{19} + 11117736 q^{21} - 15925248 q^{24} + 148701568 q^{26} - 305180580 q^{29} + 113620144 q^{31}+ \cdots + 30602498544 q^{99}+O(q^{100}) $ 2 * q - 2048 * q^4 + 15552 * q^6 - 118098 * q^9 - 518256 * q^11 - 1464064 * q^14 + 2097152 * q^16 + 3227480 * q^19 + 11117736 * q^21 - 15925248 * q^24 + 148701568 * q^26 - 305180580 * q^29 + 113620144 * q^31 - 122769024 * q^34 + 120932352 * q^36 - 1129202532 * q^39 + 1468126164 * q^41 + 530694144 * q^44 - 2102349312 * q^46 + 2908030734 * q^49 + 932277276 * q^51 - 918330048 * q^54 + 1499201536 * q^56 - 12635840880 * q^59 - 1572276356 * q^61 - 2147483648 * q^64 - 4029958656 * q^66 + 15964715088 * q^69 + 23438960064 * q^71 + 36577110656 * q^74 - 3304939520 * q^76 - 5397939280 * q^79 + 6973568802 * q^81 - 11384561664 * q^84 + 37164583168 * q^86 - 149112660420 * q^89 + 106303033424 * q^91 + 30646181376 * q^94 + 16307453952 * q^96 + 30602498544 * 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 $

$2$	− 32.0000i	− 0.707107i
$2$	− 32.0000i	− 0.707107i
$3$	243.000i	0.577350i
$3$	243.000i	0.577350i
$4$	−1024.00	−0.500000
$5$	0	0
$5$	0	0
$6$	7776.00	0.408248
$7$	− 22876.0i	− 0.514447i	−0.966352	−	0.257224i	$-0.917192\pi$
$7$	− 22876.0i	− 0.514447i	0.966352	−	0.257224i	$-0.0828077\pi$
$8$	32768.0i	0.353553i
$9$	−59049.0	−0.333333
$10$	0	0
$11$	−259128.	−0.485126	−0.242563	−	0.970136i	$-0.577988\pi$
$11$	−259128.	−0.485126	−0.242563	+	0.970136i	$0.577988\pi$
$12$	− 248832.i	− 0.288675i
$13$	2.32346e6i	1.73559i	0.496922	+	0.867795i	$0.334463\pi$
$13$	2.32346e6i	1.73559i	−0.496922	+	0.867795i	$0.665537\pi$
$14$	−732032.	−0.363769
$15$	0	0
$16$	1.04858e6	0.250000
$17$	− 1.91827e6i	− 0.327672i	−0.986488	−	0.163836i	$-0.947613\pi$
$17$	− 1.91827e6i	− 0.327672i	0.986488	−	0.163836i	$-0.0523868\pi$
$18$	1.88957e6i	0.235702i
$19$	1.61374e6	0.149516	0.0747582	−	0.997202i	$-0.476182\pi$
$19$	1.61374e6	0.149516	0.0747582	+	0.997202i	$0.476182\pi$
$20$	0	0
$21$	5.55887e6	0.297016
$22$	8.29210e6i	0.343036i
$23$	− 3.28492e7i	− 1.06420i	−0.846683	−	0.532098i	$-0.821404\pi$
$23$	− 3.28492e7i	− 1.06420i	0.846683	−	0.532098i	$-0.178596\pi$
$24$	−7.96262e6	−0.204124
$25$	0	0
$26$	7.43508e7	1.22725
$27$	− 1.43489e7i	− 0.192450i
$28$	2.34250e7i	0.257224i
$29$	−1.52590e8	−1.38146	−0.690729	−	0.723113i	$-0.742710\pi$
$29$	−1.52590e8	−1.38146	−0.690729	+	0.723113i	$0.742710\pi$
$30$	0	0
$31$	5.68101e7	0.356399	0.178199	−	0.983994i	$-0.442973\pi$
$31$	5.68101e7	0.356399	0.178199	+	0.983994i	$0.442973\pi$
$32$	− 3.35544e7i	− 0.176777i
$33$	− 6.29681e7i	− 0.280088i
$34$	−6.13845e7	−0.231699
$35$	0	0
$36$	6.04662e7	0.166667
$37$	5.71517e8i	1.35494i	0.735551	+	0.677470i	$0.236924\pi$
$37$	5.71517e8i	1.35494i	−0.735551	+	0.677470i	$0.763076\pi$
$38$	− 5.16397e7i	− 0.105724i
$39$	−5.64601e8	−1.00204
$40$	0	0
$41$	7.34063e8	0.989515	0.494757	−	0.869031i	$-0.335257\pi$
$41$	7.34063e8	0.989515	0.494757	+	0.869031i	$0.335257\pi$
$42$	− 1.77884e8i	− 0.210022i
$43$	5.80697e8i	0.602383i	0.953564	+	0.301192i	$0.0973844\pi$
$43$	5.80697e8i	0.602383i	−0.953564	+	0.301192i	$0.902616\pi$
$44$	2.65347e8	0.242563
$45$	0	0
$46$	−1.05117e9	−0.752501
$47$	4.78847e8i	0.304550i	0.988338	+	0.152275i	$0.0486599\pi$
$47$	4.78847e8i	0.304550i	−0.988338	+	0.152275i	$0.951340\pi$
$48$	2.54804e8i	0.144338i
$49$	1.45402e9	0.735344
$50$	0	0
$51$	4.66139e8	0.189182
$52$	− 2.37923e9i	− 0.867795i
$53$	2.55379e9i	0.838818i	0.907797	+	0.419409i	$0.137763\pi$
$53$	2.55379e9i	0.838818i	−0.907797	+	0.419409i	$0.862237\pi$
$54$	−4.59165e8	−0.136083
$55$	0	0
$56$	7.49601e8	0.181885
$57$	3.92139e8i	0.0863233i
$58$	4.88289e9i	0.976839i
$59$	−6.31792e9	−1.15050	−0.575252	−	0.817976i	$-0.695096\pi$
$59$	−6.31792e9	−1.15050	−0.575252	+	0.817976i	$0.695096\pi$
$60$	0	0
$61$	−7.86138e8	−0.119175	−0.0595874	−	0.998223i	$-0.518979\pi$
$61$	−7.86138e8	−0.119175	−0.0595874	+	0.998223i	$0.518979\pi$
$62$	− 1.81792e9i	− 0.252012i
$63$	1.35080e9i	0.171482i
$64$	−1.07374e9	−0.125000
$65$	0	0
$66$	−2.01498e9	−0.198052
$67$	− 2.12877e10i	− 1.92627i	−0.269020	−	0.963135i	$-0.586700\pi$
$67$	− 2.12877e10i	− 1.92627i	0.269020	−	0.963135i	$-0.413300\pi$
$68$	1.96430e9i	0.163836i
$69$	7.98236e9	0.614414
$70$	0	0
$71$	1.17195e10	0.770881	0.385441	−	0.922733i	$-0.374049\pi$
$71$	1.17195e10	0.770881	0.385441	+	0.922733i	$0.374049\pi$
$72$	− 1.93492e9i	− 0.117851i
$73$	− 2.31756e10i	− 1.30845i	−0.756302	−	0.654223i	$-0.772996\pi$
$73$	− 2.31756e10i	− 1.30845i	0.756302	−	0.654223i	$-0.227004\pi$
$74$	1.82886e10	0.958087
$75$	0	0
$76$	−1.65247e9	−0.0747582
$77$	5.92781e9i	0.249572i
$78$	1.80672e10i	0.708552i
$79$	−2.69897e9	−0.0986845	−0.0493423	−	0.998782i	$-0.515713\pi$
$79$	−2.69897e9	−0.0986845	−0.0493423	+	0.998782i	$0.515713\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	− 2.34900e10i	− 0.699692i
$83$	− 3.26587e10i	− 0.910059i	−0.890476	−	0.455029i	$-0.849629\pi$
$83$	− 3.26587e10i	− 0.910059i	0.890476	−	0.455029i	$-0.150371\pi$
$84$	−5.69228e9	−0.148508
$85$	0	0
$86$	1.85823e10	0.425949
$87$	− 3.70794e10i	− 0.797586i
$88$	− 8.49111e9i	− 0.171518i
$89$	−7.45563e10	−1.41527	−0.707635	−	0.706578i	$-0.750238\pi$
$89$	−7.45563e10	−1.41527	−0.707635	+	0.706578i	$0.750238\pi$
$90$	0	0
$91$	5.31515e10	0.892870
$92$	3.36376e10i	0.532098i
$93$	1.38048e10i	0.205767i
$94$	1.53231e10	0.215349
$95$	0	0
$96$	8.15373e9	0.102062
$97$	4.14349e10i	0.489916i	0.969534	+	0.244958i	$0.0787741\pi$
$97$	4.14349e10i	0.489916i	−0.969534	+	0.244958i	$0.921226\pi$
$98$	− 4.65285e10i	− 0.519967i
$99$	1.53012e10	0.161709
$100$	0	0
$101$	1.16600e11	1.10391	0.551954	−	0.833875i	$-0.313882\pi$
$101$	1.16600e11	1.10391	0.551954	+	0.833875i	$0.313882\pi$
$102$	− 1.49164e10i	− 0.133772i
$103$	− 4.30231e10i	− 0.365676i	−0.983143	−	0.182838i	$-0.941472\pi$
$103$	− 4.30231e10i	− 0.365676i	0.983143	−	0.182838i	$-0.0585284\pi$
$104$	−7.61352e10	−0.613624
$105$	0	0
$106$	8.17212e10	0.593134
$107$	− 2.15323e11i	− 1.48416i	−0.670313	−	0.742078i	$-0.733840\pi$
$107$	− 2.15323e11i	− 1.48416i	0.670313	−	0.742078i	$-0.266160\pi$
$108$	1.46933e10i	0.0962250i
$109$	2.56069e11	1.59409	0.797044	−	0.603922i	$-0.206396\pi$
$109$	2.56069e11	1.59409	0.797044	+	0.603922i	$0.206396\pi$
$110$	0	0
$111$	−1.38879e11	−0.782275
$112$	− 2.39872e10i	− 0.128612i
$113$	− 1.32713e11i	− 0.677613i	−0.940856	−	0.338807i	$-0.889977\pi$
$113$	− 1.32713e11i	− 0.677613i	0.940856	−	0.338807i	$-0.110023\pi$
$114$	1.25484e10	0.0610398
$115$	0	0
$116$	1.56252e11	0.690729
$117$	− 1.37198e11i	− 0.578530i
$118$	2.02173e11i	0.813529i
$119$	−4.38823e10	−0.168570
$120$	0	0
$121$	−2.18164e11	−0.764653
$122$	2.51564e10i	0.0842694i
$123$	1.78377e11i	0.571296i
$124$	−5.81735e10	−0.178199
$125$	0	0
$126$	4.32258e10	0.121256
$127$	− 5.99930e11i	− 1.61132i	−0.592382	−	0.805658i	$-0.701812\pi$
$127$	− 5.99930e11i	− 1.61132i	0.592382	−	0.805658i	$-0.298188\pi$
$128$	3.43597e10i	0.0883883i
$129$	−1.41109e11	−0.347786
$130$	0	0
$131$	−2.86450e11	−0.648719	−0.324360	−	0.945934i	$-0.605149\pi$
$131$	−2.86450e11	−0.648719	−0.324360	+	0.945934i	$0.605149\pi$
$132$	6.44793e10i	0.140044i
$133$	− 3.69159e10i	− 0.0769183i
$134$	−6.81206e11	−1.36208
$135$	0	0
$136$	6.28577e10	0.115850
$137$	− 2.68219e11i	− 0.474817i	−0.971410	−	0.237408i	$-0.923702\pi$
$137$	− 2.68219e11i	− 0.474817i	0.971410	−	0.237408i	$-0.0762980\pi$
$138$	− 2.55435e11i	− 0.434456i
$139$	−1.00413e12	−1.64138	−0.820692	−	0.571371i	$-0.806412\pi$
$139$	−1.00413e12	−1.64138	−0.820692	+	0.571371i	$0.806412\pi$
$140$	0	0
$141$	−1.16360e11	−0.175832
$142$	− 3.75023e11i	− 0.545095i
$143$	− 6.02074e11i	− 0.841980i
$144$	−6.19174e10	−0.0833333
$145$	0	0
$146$	−7.41620e11	−0.925211
$147$	3.53326e11i	0.424551i
$148$	− 5.85234e11i	− 0.677470i
$149$	−6.60408e11	−0.736695	−0.368348	−	0.929688i	$-0.620076\pi$
$149$	−6.60408e11	−0.736695	−0.368348	+	0.929688i	$0.620076\pi$
$150$	0	0
$151$	1.86615e11	0.193453	0.0967263	−	0.995311i	$-0.469163\pi$
$151$	1.86615e11	0.193453	0.0967263	+	0.995311i	$0.469163\pi$
$152$	5.28790e10i	0.0528620i
$153$	1.13272e11i	0.109224i
$154$	1.89690e11	0.176474
$155$	0	0
$156$	5.78152e11	0.501022
$157$	4.52456e11i	0.378554i	0.981924	+	0.189277i	$0.0606144\pi$
$157$	4.52456e11i	0.378554i	−0.981924	+	0.189277i	$0.939386\pi$
$158$	8.63670e10i	0.0697805i
$159$	−6.20570e11	−0.484292
$160$	0	0
$161$	−7.51458e11	−0.547473
$162$	− 1.11577e11i	− 0.0785674i
$163$	− 1.51667e12i	− 1.03243i	−0.856460	−	0.516213i	$-0.827341\pi$
$163$	− 1.51667e12i	− 1.03243i	0.856460	−	0.516213i	$-0.172659\pi$
$164$	−7.51681e11	−0.494757
$165$	0	0
$166$	−1.04508e12	−0.643509
$167$	− 2.35061e12i	− 1.40036i	−0.713965	−	0.700181i	$-0.753103\pi$
$167$	− 2.35061e12i	− 1.40036i	0.713965	−	0.700181i	$-0.246897\pi$
$168$	1.82153e11i	0.105011i
$169$	−3.60632e12	−2.01227
$170$	0	0
$171$	−9.52897e10	−0.0498388
$172$	− 5.94633e11i	− 0.301192i
$173$	− 9.01111e11i	− 0.442104i	−0.975262	−	0.221052i	$-0.929051\pi$
$173$	− 9.01111e11i	− 0.442104i	0.975262	−	0.221052i	$-0.0709491\pi$
$174$	−1.18654e12	−0.563978
$175$	0	0
$176$	−2.71715e11	−0.121282
$177$	− 1.53525e12i	− 0.664244i
$178$	2.38580e12i	1.00075i
$179$	9.77006e11	0.397379	0.198690	−	0.980062i	$-0.436331\pi$
$179$	9.77006e11	0.397379	0.198690	+	0.980062i	$0.436331\pi$
$180$	0	0
$181$	3.83020e12	1.46551	0.732756	−	0.680491i	$-0.238234\pi$
$181$	3.83020e12	1.46551	0.732756	+	0.680491i	$0.238234\pi$
$182$	− 1.70085e12i	− 0.631354i
$183$	− 1.91032e11i	− 0.0688057i
$184$	1.07640e12	0.376250
$185$	0	0
$186$	4.41755e11	0.145499
$187$	4.97076e11i	0.158962i
$188$	− 4.90339e11i	− 0.152275i
$189$	−3.28246e11	−0.0990054
$190$	0	0
$191$	−2.49694e12	−0.710764	−0.355382	−	0.934721i	$-0.615649\pi$
$191$	−2.49694e12	−0.710764	−0.355382	+	0.934721i	$0.615649\pi$
$192$	− 2.60919e11i	− 0.0721688i
$193$	− 5.19794e12i	− 1.39722i	−0.715500	−	0.698612i	$-0.753801\pi$
$193$	− 5.19794e12i	− 1.39722i	0.715500	−	0.698612i	$-0.246199\pi$
$194$	1.32592e12	0.346423
$195$	0	0
$196$	−1.48891e12	−0.367672
$197$	− 5.70399e12i	− 1.36967i	−0.728700	−	0.684833i	$-0.759875\pi$
$197$	− 5.70399e12i	− 1.36967i	0.728700	−	0.684833i	$-0.240125\pi$
$198$	− 4.89640e11i	− 0.114345i
$199$	6.39817e11	0.145333	0.0726664	−	0.997356i	$-0.476849\pi$
$199$	6.39817e11	0.145333	0.0726664	+	0.997356i	$0.476849\pi$
$200$	0	0
$201$	5.17291e12	1.11213
$202$	− 3.73122e12i	− 0.780581i
$203$	3.49066e12i	0.710688i
$204$	−4.77326e11	−0.0945909
$205$	0	0
$206$	−1.37674e12	−0.258572
$207$	1.93971e12i	0.354732i
$208$	2.43633e12i	0.433897i
$209$	−4.18165e11	−0.0725343
$210$	0	0
$211$	2.47025e12	0.406618	0.203309	−	0.979115i	$-0.434830\pi$
$211$	2.47025e12	0.406618	0.203309	+	0.979115i	$0.434830\pi$
$212$	− 2.61508e12i	− 0.419409i
$213$	2.84783e12i	0.445068i
$214$	−6.89034e12	−1.04946
$215$	0	0
$216$	4.70185e11	0.0680414
$217$	− 1.29959e12i	− 0.183348i
$218$	− 8.19422e12i	− 1.12719i
$219$	5.63168e12	0.755432
$220$	0	0
$221$	4.45702e12	0.568705
$222$	4.44412e12i	0.553152i
$223$	5.26711e12i	0.639581i	0.947488	+	0.319790i	$0.103613\pi$
$223$	5.26711e12i	0.639581i	−0.947488	+	0.319790i	$0.896387\pi$
$224$	−7.67591e11	−0.0909423
$225$	0	0
$226$	−4.24682e12	−0.479145
$227$	− 1.14447e13i	− 1.26027i	−0.776488	−	0.630133i	$-0.783001\pi$
$227$	− 1.14447e13i	− 1.26027i	0.776488	−	0.630133i	$-0.216999\pi$
$228$	− 4.01550e11i	− 0.0431617i
$229$	−3.62497e12	−0.380372	−0.190186	−	0.981748i	$-0.560909\pi$
$229$	−3.62497e12	−0.380372	−0.190186	+	0.981748i	$0.560909\pi$
$230$	0	0
$231$	−1.44046e12	−0.144090
$232$	− 5.00008e12i	− 0.488419i
$233$	− 1.44062e13i	− 1.37433i	−0.726501	−	0.687165i	$-0.758855\pi$
$233$	− 1.44062e13i	− 1.37433i	0.726501	−	0.687165i	$-0.241145\pi$
$234$	−4.39034e12	−0.409082
$235$	0	0
$236$	6.46955e12	0.575252
$237$	− 6.55850e11i	− 0.0569755i
$238$	1.40423e12i	0.119197i
$239$	1.33198e13	1.10487	0.552433	−	0.833557i	$-0.313699\pi$
$239$	1.33198e13	1.10487	0.552433	+	0.833557i	$0.313699\pi$
$240$	0	0
$241$	1.13769e13	0.901428	0.450714	−	0.892668i	$-0.351169\pi$
$241$	1.13769e13	0.901428	0.450714	+	0.892668i	$0.351169\pi$
$242$	6.98126e12i	0.540691i
$243$	8.47289e11i	0.0641500i
$244$	8.05005e11	0.0595874
$245$	0	0
$246$	5.70807e12	0.403968
$247$	3.74946e12i	0.259499i
$248$	1.86155e12i	0.126006i
$249$	7.93607e12	0.525423
$250$	0	0
$251$	−2.83662e12	−0.179719	−0.0898597	−	0.995954i	$-0.528642\pi$
$251$	−2.83662e12	−0.179719	−0.0898597	+	0.995954i	$0.528642\pi$
$252$	− 1.38322e12i	− 0.0857412i
$253$	8.51215e12i	0.516269i
$254$	−1.91978e13	−1.13937
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	3.01443e13i	1.67715i	0.544784	+	0.838576i	$0.316612\pi$
$257$	3.01443e13i	1.67715i	−0.544784	+	0.838576i	$0.683388\pi$
$258$	4.51550e12i	0.245922i
$259$	1.30740e13	0.697045
$260$	0	0
$261$	9.01030e12	0.460486
$262$	9.16641e12i	0.458714i
$263$	− 3.80917e13i	− 1.86670i	−0.358971	−	0.933349i	$-0.616872\pi$
$263$	− 3.80917e13i	− 1.86670i	0.358971	−	0.933349i	$-0.383128\pi$
$264$	2.06334e12	0.0990259
$265$	0	0
$266$	−1.18131e12	−0.0543894
$267$	− 1.81172e13i	− 0.817107i
$268$	2.17986e13i	0.963135i
$269$	−2.00988e12	−0.0870026	−0.0435013	−	0.999053i	$-0.513851\pi$
$269$	−2.00988e12	−0.0870026	−0.0435013	+	0.999053i	$0.513851\pi$
$270$	0	0
$271$	−3.06081e13	−1.27205	−0.636027	−	0.771667i	$-0.719423\pi$
$271$	−3.06081e13	−1.27205	−0.636027	+	0.771667i	$0.719423\pi$
$272$	− 2.01145e12i	− 0.0819181i
$273$	1.29158e13i	0.515498i
$274$	−8.58300e12	−0.335746
$275$	0	0
$276$	−8.17393e12	−0.307207
$277$	2.77790e13i	1.02348i	0.859142	+	0.511738i	$0.170998\pi$
$277$	2.77790e13i	1.02348i	−0.859142	+	0.511738i	$0.829002\pi$
$278$	3.21323e13i	1.16063i
$279$	−3.35458e12	−0.118800
$280$	0	0
$281$	−3.95175e13	−1.34557	−0.672783	−	0.739840i	$-0.734901\pi$
$281$	−3.95175e13	−1.34557	−0.672783	+	0.739840i	$0.734901\pi$
$282$	3.72351e12i	0.124332i
$283$	3.57563e13i	1.17092i	0.810701	+	0.585460i	$0.199086\pi$
$283$	3.57563e13i	1.17092i	−0.810701	+	0.585460i	$0.800914\pi$
$284$	−1.20007e13	−0.385441
$285$	0	0
$286$	−1.92664e13	−0.595370
$287$	− 1.67924e13i	− 0.509053i
$288$	1.98136e12i	0.0589256i
$289$	3.05922e13	0.892631
$290$	0	0
$291$	−1.00687e13	−0.282853
$292$	2.37318e13i	0.654223i
$293$	− 1.16731e13i	− 0.315801i	−0.987455	−	0.157901i	$-0.949527\pi$
$293$	− 1.16731e13i	− 0.315801i	0.987455	−	0.157901i	$-0.0504726\pi$
$294$	1.13064e13	0.300203
$295$	0	0
$296$	−1.87275e13	−0.479043
$297$	3.71820e12i	0.0933625i
$298$	2.11331e13i	0.520922i
$299$	7.63239e13	1.84701
$300$	0	0
$301$	1.32840e13	0.309894
$302$	− 5.97169e12i	− 0.136792i
$303$	2.83339e13i	0.637342i
$304$	1.69213e12	0.0373791
$305$	0	0
$306$	3.62469e12	0.0772331
$307$	8.98114e13i	1.87962i	0.341695	+	0.939811i	$0.388999\pi$
$307$	8.98114e13i	1.87962i	−0.341695	+	0.939811i	$0.611001\pi$
$308$	− 6.07008e12i	− 0.124786i
$309$	1.04546e13	0.211123
$310$	0	0
$311$	6.75879e13	1.31731	0.658653	−	0.752446i	$-0.271126\pi$
$311$	6.75879e13	1.31731	0.658653	+	0.752446i	$0.271126\pi$
$312$	− 1.85009e13i	− 0.354276i
$313$	− 9.87171e13i	− 1.85737i	−0.370869	−	0.928685i	$-0.620940\pi$
$313$	− 9.87171e13i	− 1.85737i	0.370869	−	0.928685i	$-0.379060\pi$
$314$	1.44786e13	0.267678
$315$	0	0
$316$	2.76374e12	0.0493423
$317$	1.10523e14i	1.93921i	0.244674	+	0.969605i	$0.421319\pi$
$317$	1.10523e14i	1.93921i	−0.244674	+	0.969605i	$0.578681\pi$
$318$	1.98583e13i	0.342446i
$319$	3.95404e13	0.670182
$320$	0	0
$321$	5.23235e13	0.856878
$322$	2.40467e13i	0.387122i
$323$	− 3.09558e12i	− 0.0489924i
$324$	−3.57047e12	−0.0555556
$325$	0	0
$326$	−4.85334e13	−0.730036
$327$	6.22249e13i	0.920347i
$328$	2.40538e13i	0.349846i
$329$	1.09541e13	0.156675
$330$	0	0
$331$	−5.00155e13	−0.691912	−0.345956	−	0.938251i	$-0.612445\pi$
$331$	−5.00155e13	−0.691912	−0.345956	+	0.938251i	$0.612445\pi$
$332$	3.34425e13i	0.455029i
$333$	− 3.37475e13i	− 0.451647i
$334$	−7.52196e13	−0.990205
$335$	0	0
$336$	5.82890e12	0.0742541
$337$	− 3.62478e12i	− 0.0454274i	−0.999742	−	0.0227137i	$-0.992769\pi$
$337$	− 3.62478e12i	− 0.0454274i	0.999742	−	0.0227137i	$-0.00723061\pi$
$338$	1.15402e14i	1.42289i
$339$	3.22493e13	0.391220
$340$	0	0
$341$	−1.47211e13	−0.172898
$342$	3.04927e12i	0.0352413i
$343$	− 7.84954e13i	− 0.892743i
$344$	−1.90283e13	−0.212975
$345$	0	0
$346$	−2.88355e13	−0.312615
$347$	6.56058e12i	0.0700051i	0.999387	+	0.0350026i	$0.0111439\pi$
$347$	6.56058e12i	0.0700051i	−0.999387	+	0.0350026i	$0.988856\pi$
$348$	3.79693e13i	0.398793i
$349$	8.14353e13	0.841924	0.420962	−	0.907078i	$-0.361693\pi$
$349$	8.14353e13	0.841924	0.420962	+	0.907078i	$0.361693\pi$
$350$	0	0
$351$	3.33391e13	0.334014
$352$	8.69489e12i	0.0857590i
$353$	− 1.03792e14i	− 1.00787i	−0.863742	−	0.503934i	$-0.831885\pi$
$353$	− 1.03792e14i	− 1.00787i	0.863742	−	0.503934i	$-0.168115\pi$
$354$	−4.91281e13	−0.469691
$355$	0	0
$356$	7.63457e13	0.707635
$357$	− 1.06634e13i	− 0.0973240i
$358$	− 3.12642e13i	− 0.280990i
$359$	1.38258e14	1.22369	0.611845	−	0.790978i	$-0.290428\pi$
$359$	1.38258e14	1.22369	0.611845	+	0.790978i	$0.290428\pi$
$360$	0	0
$361$	−1.13886e14	−0.977645
$362$	− 1.22566e14i	− 1.03627i
$363$	− 5.30139e13i	− 0.441472i
$364$	−5.44272e13	−0.446435
$365$	0	0
$366$	−6.11301e12	−0.0486529
$367$	1.16357e14i	0.912282i	0.889908	+	0.456141i	$0.150769\pi$
$367$	1.16357e14i	0.912282i	−0.889908	+	0.456141i	$0.849231\pi$
$368$	− 3.44449e13i	− 0.266049i
$369$	−4.33457e13	−0.329838
$370$	0	0
$371$	5.84205e13	0.431527
$372$	− 1.41362e13i	− 0.102883i
$373$	6.72186e13i	0.482049i	0.970519	+	0.241024i	$0.0774833\pi$
$373$	6.72186e13i	0.482049i	−0.970519	+	0.241024i	$0.922517\pi$
$374$	1.59064e13	0.112403
$375$	0	0
$376$	−1.56908e13	−0.107675
$377$	− 3.54538e14i	− 2.39765i
$378$	1.05039e13i	0.0700074i
$379$	1.32840e14	0.872598	0.436299	−	0.899802i	$-0.356289\pi$
$379$	1.32840e14	0.872598	0.436299	+	0.899802i	$0.356289\pi$
$380$	0	0
$381$	1.45783e14	0.930293
$382$	7.99022e13i	0.502586i
$383$	2.69810e14i	1.67288i	0.548060	+	0.836439i	$0.315367\pi$
$383$	2.69810e14i	1.67288i	−0.548060	+	0.836439i	$0.684633\pi$
$384$	−8.34942e12	−0.0510310
$385$	0	0
$386$	−1.66334e14	−0.987987
$387$	− 3.42896e13i	− 0.200794i
$388$	− 4.24293e13i	− 0.244958i
$389$	−9.81084e13	−0.558449	−0.279224	−	0.960226i	$-0.590077\pi$
$389$	−9.81084e13	−0.558449	−0.279224	+	0.960226i	$0.590077\pi$
$390$	0	0
$391$	−6.30135e13	−0.348708
$392$	4.76452e13i	0.259983i
$393$	− 6.96074e13i	− 0.374538i
$394$	−1.82528e14	−0.968500
$395$	0	0
$396$	−1.56685e13	−0.0808543
$397$	− 6.44136e13i	− 0.327816i	−0.986476	−	0.163908i	$-0.947590\pi$
$397$	− 6.44136e13i	− 0.327816i	0.986476	−	0.163908i	$-0.0524100\pi$
$398$	− 2.04741e13i	− 0.102766i
$399$	8.97057e12	0.0444088
$400$	0	0
$401$	2.05799e14	0.991172	0.495586	−	0.868559i	$-0.334953\pi$
$401$	2.05799e14	0.991172	0.495586	+	0.868559i	$0.334953\pi$
$402$	− 1.65533e14i	− 0.786396i
$403$	1.31996e14i	0.618562i
$404$	−1.19399e14	−0.551954
$405$	0	0
$406$	1.11701e14	0.502532
$407$	− 1.48096e14i	− 0.657316i
$408$	1.52744e13i	0.0668858i
$409$	−5.95615e13	−0.257328	−0.128664	−	0.991688i	$-0.541069\pi$
$409$	−5.95615e13	−0.257328	−0.128664	+	0.991688i	$0.541069\pi$
$410$	0	0
$411$	6.51772e13	0.274136
$412$	4.40556e13i	0.182838i
$413$	1.44529e14i	0.591874i
$414$	6.20708e13	0.250834
$415$	0	0
$416$	7.79624e13	0.306812
$417$	− 2.44004e14i	− 0.947653i
$418$	1.33813e13i	0.0512895i
$419$	8.26291e13	0.312576	0.156288	−	0.987712i	$-0.450047\pi$
$419$	8.26291e13	0.312576	0.156288	+	0.987712i	$0.450047\pi$
$420$	0	0
$421$	−1.55708e14	−0.573800	−0.286900	−	0.957961i	$-0.592625\pi$
$421$	−1.55708e14	−0.573800	−0.286900	+	0.957961i	$0.592625\pi$
$422$	− 7.90479e13i	− 0.287523i
$423$	− 2.82754e13i	− 0.101517i
$424$	−8.36825e13	−0.296567
$425$	0	0
$426$	9.11307e13	0.314711
$427$	1.79837e13i	0.0613092i
$428$	2.20491e14i	0.742078i
$429$	1.46304e14	0.486117
$430$	0	0
$431$	4.40261e14	1.42589	0.712944	−	0.701221i	$-0.247361\pi$
$431$	4.40261e14	1.42589	0.712944	+	0.701221i	$0.247361\pi$
$432$	− 1.50459e13i	− 0.0481125i
$433$	− 3.21850e14i	− 1.01618i	−0.861304	−	0.508089i	$-0.830352\pi$
$433$	− 3.21850e14i	− 1.01618i	0.861304	−	0.508089i	$-0.169648\pi$
$434$	−4.15868e13	−0.129647
$435$	0	0
$436$	−2.62215e14	−0.797044
$437$	− 5.30101e13i	− 0.159115i
$438$	− 1.80214e14i	− 0.534171i
$439$	−6.53476e14	−1.91282	−0.956411	−	0.292025i	$-0.905671\pi$
$439$	−6.53476e14	−1.91282	−0.956411	+	0.292025i	$0.905671\pi$
$440$	0	0
$441$	−8.58582e13	−0.245115
$442$	− 1.42625e14i	− 0.402135i
$443$	− 5.50129e14i	− 1.53195i	−0.642872	−	0.765974i	$-0.722257\pi$
$443$	− 5.50129e14i	− 1.53195i	0.642872	−	0.765974i	$-0.277743\pi$
$444$	1.42212e14	0.391137
$445$	0	0
$446$	1.68547e14	0.452252
$447$	− 1.60479e14i	− 0.425331i
$448$	2.45629e13i	0.0643059i
$449$	−2.56147e13	−0.0662421	−0.0331210	−	0.999451i	$-0.510545\pi$
$449$	−2.56147e13	−0.0662421	−0.0331210	+	0.999451i	$0.510545\pi$
$450$	0	0
$451$	−1.90216e14	−0.480039
$452$	1.35898e14i	0.338807i
$453$	4.53476e13i	0.111690i
$454$	−3.66230e14	−0.891142
$455$	0	0
$456$	−1.28496e13	−0.0305199
$457$	1.73882e13i	0.0408052i	0.999792	+	0.0204026i	$0.00649480\pi$
$457$	1.73882e13i	0.0408052i	−0.999792	+	0.0204026i	$0.993505\pi$
$458$	1.15999e14i	0.268964i
$459$	−2.75250e13	−0.0630606
$460$	0	0
$461$	7.61820e14	1.70411	0.852055	−	0.523453i	$-0.175356\pi$
$461$	7.61820e14	1.70411	0.852055	+	0.523453i	$0.175356\pi$
$462$	4.60947e13i	0.101887i
$463$	− 3.88545e14i	− 0.848684i	−0.905502	−	0.424342i	$-0.860505\pi$
$463$	− 3.88545e14i	− 0.848684i	0.905502	−	0.424342i	$-0.139495\pi$
$464$	−1.60003e14	−0.345365
$465$	0	0
$466$	−4.60998e14	−0.971799
$467$	1.22217e14i	0.254618i	0.991863	+	0.127309i	$0.0406340\pi$
$467$	1.22217e14i	0.254618i	−0.991863	+	0.127309i	$0.959366\pi$
$468$	1.40491e14i	0.289265i
$469$	−4.86977e14	−0.990964
$470$	0	0
$471$	−1.09947e14	−0.218558
$472$	− 2.07026e14i	− 0.406765i
$473$	− 1.50475e14i	− 0.292232i
$474$	−2.09872e13	−0.0402878
$475$	0	0
$476$	4.49354e13	0.0842851
$477$	− 1.50799e14i	− 0.279606i
$478$	− 4.26234e14i	− 0.781259i
$479$	−4.35912e14	−0.789866	−0.394933	−	0.918710i	$-0.629232\pi$
$479$	−4.35912e14	−0.789866	−0.394933	+	0.918710i	$0.629232\pi$
$480$	0	0
$481$	−1.32790e15	−2.35162
$482$	− 3.64061e14i	− 0.637406i
$483$	− 1.82604e14i	− 0.316084i
$484$	2.23400e14	0.382326
$485$	0	0
$486$	2.71132e13	0.0453609
$487$	6.66191e14i	1.10202i	0.834499	+	0.551010i	$0.185757\pi$
$487$	6.66191e14i	1.10202i	−0.834499	+	0.551010i	$0.814243\pi$
$488$	− 2.57602e13i	− 0.0421347i
$489$	3.68551e14	0.596072
$490$	0	0
$491$	2.23817e13	0.0353953	0.0176976	−	0.999843i	$-0.494366\pi$
$491$	2.23817e13	0.0353953	0.0176976	+	0.999843i	$0.494366\pi$
$492$	− 1.82658e14i	− 0.285648i
$493$	2.92709e14i	0.452666i
$494$	1.19983e14	0.183494
$495$	0	0
$496$	5.95697e13	0.0890997
$497$	− 2.68095e14i	− 0.396578i
$498$	− 2.53954e14i	− 0.371530i
$499$	4.87506e13	0.0705386	0.0352693	−	0.999378i	$-0.488771\pi$
$499$	4.87506e13	0.0705386	0.0352693	+	0.999378i	$0.488771\pi$
$500$	0	0
$501$	5.71199e14	0.808499
$502$	9.07717e13i	0.127081i
$503$	− 2.64333e14i	− 0.366039i	−0.983109	−	0.183019i	$-0.941413\pi$
$503$	− 2.64333e14i	− 0.366039i	0.983109	−	0.183019i	$-0.0585871\pi$
$504$	−4.42632e13	−0.0606282
$505$	0	0
$506$	2.72389e14	0.365058
$507$	− 8.76335e14i	− 1.16179i
$508$	6.14329e14i	0.805658i
$509$	−8.09106e13	−0.104968	−0.0524841	−	0.998622i	$-0.516714\pi$
$509$	−8.09106e13	−0.104968	−0.0524841	+	0.998622i	$0.516714\pi$
$510$	0	0
$511$	−5.30166e14	−0.673127
$512$	− 3.51844e13i	− 0.0441942i
$513$	− 2.31554e13i	− 0.0287744i
$514$	9.64617e14	1.18593
$515$	0	0
$516$	1.44496e14	0.173893
$517$	− 1.24083e14i	− 0.147745i
$518$	− 4.18369e14i	− 0.492885i
$519$	2.18970e14	0.255249
$520$	0	0
$521$	−1.10833e14	−0.126491	−0.0632456	−	0.997998i	$-0.520145\pi$
$521$	−1.10833e14	−0.126491	−0.0632456	+	0.997998i	$0.520145\pi$
$522$	− 2.88330e14i	− 0.325613i
$523$	− 6.91114e14i	− 0.772308i	−0.922434	−	0.386154i	$-0.873803\pi$
$523$	− 6.91114e14i	− 0.772308i	0.922434	−	0.386154i	$-0.126197\pi$
$524$	2.93325e14	0.324360
$525$	0	0
$526$	−1.21894e15	−1.31995
$527$	− 1.08977e14i	− 0.116782i
$528$	− 6.60268e13i	− 0.0700219i
$529$	−1.26261e14	−0.132514
$530$	0	0
$531$	3.73067e14	0.383501
$532$	3.78019e13i	0.0384591i
$533$	1.70557e15i	1.71739i
$534$	−5.79750e14	−0.577782
$535$	0	0
$536$	6.97555e14	0.681039
$537$	2.37412e14i	0.229427i
$538$	6.43161e13i	0.0615201i
$539$	−3.76776e14	−0.356735
$540$	0	0
$541$	−2.02808e15	−1.88148	−0.940741	−	0.339125i	$-0.889869\pi$
$541$	−2.02808e15	−1.88148	−0.940741	+	0.339125i	$0.889869\pi$
$542$	9.79460e14i	0.899478i
$543$	9.30739e14i	0.846114i
$544$	−6.43663e13	−0.0579248
$545$	0	0
$546$	4.13306e14	0.364512
$547$	2.06348e15i	1.80165i	0.434180	+	0.900826i	$0.357038\pi$
$547$	2.06348e15i	1.80165i	−0.434180	+	0.900826i	$0.642962\pi$
$548$	2.74656e14i	0.237408i
$549$	4.64207e13	0.0397250
$550$	0	0
$551$	−2.46241e14	−0.206551
$552$	2.61566e14i	0.217228i
$553$	6.17416e13i	0.0507680i
$554$	8.88927e14	0.723707
$555$	0	0
$556$	1.02823e15	0.820692
$557$	− 2.11828e15i	− 1.67409i	−0.547130	−	0.837047i	$-0.684280\pi$
$557$	− 2.11828e15i	− 1.67409i	0.547130	−	0.837047i	$-0.315720\pi$
$558$	1.07346e14i	0.0840040i
$559$	−1.34923e15	−1.04549
$560$	0	0
$561$	−1.20790e14	−0.0917770
$562$	1.26456e15i	0.951458i
$563$	1.16211e15i	0.865870i	0.901425	+	0.432935i	$0.142522\pi$
$563$	1.16211e15i	0.865870i	−0.901425	+	0.432935i	$0.857478\pi$
$564$	1.19152e14	0.0879159
$565$	0	0
$566$	1.14420e15	0.827965
$567$	− 7.97637e13i	− 0.0571608i
$568$	3.84024e14i	0.272548i
$569$	−1.19782e15	−0.841927	−0.420964	−	0.907077i	$-0.638308\pi$
$569$	−1.19782e15	−0.841927	−0.420964	+	0.907077i	$0.638308\pi$
$570$	0	0
$571$	1.46355e15	1.00904	0.504522	−	0.863399i	$-0.331669\pi$
$571$	1.46355e15	1.00904	0.504522	+	0.863399i	$0.331669\pi$
$572$	6.16524e14i	0.420990i
$573$	− 6.06757e14i	− 0.410360i
$574$	−5.37358e14	−0.359955
$575$	0	0
$576$	6.34034e13	0.0416667
$577$	1.97111e15i	1.28305i	0.767101	+	0.641526i	$0.221698\pi$
$577$	1.97111e15i	1.28305i	−0.767101	+	0.641526i	$0.778302\pi$
$578$	− 9.78949e14i	− 0.631185i
$579$	1.26310e15	0.806688
$580$	0	0
$581$	−7.47101e14	−0.468177
$582$	3.22198e14i	0.200007i
$583$	− 6.61758e14i	− 0.406932i
$584$	7.59419e14	0.462606
$585$	0	0
$586$	−3.73539e14	−0.223305
$587$	3.58593e14i	0.212369i	0.994346	+	0.106185i	$0.0338635\pi$
$587$	3.58593e14i	0.212369i	−0.994346	+	0.106185i	$0.966137\pi$
$588$	− 3.61806e14i	− 0.212276i
$589$	9.16767e13	0.0532874
$590$	0	0
$591$	1.38607e15	0.790777
$592$	5.99279e14i	0.338735i
$593$	1.43925e15i	0.806003i	0.915199	+	0.403001i	$0.132033\pi$
$593$	1.43925e15i	0.806003i	−0.915199	+	0.403001i	$0.867967\pi$
$594$	1.18983e14	0.0660173
$595$	0	0
$596$	6.76258e14	0.368348
$597$	1.55475e14i	0.0839079i
$598$	− 2.44236e15i	− 1.30603i
$599$	3.41497e15	1.80942	0.904710	−	0.426028i	$-0.140088\pi$
$599$	3.41497e15	1.80942	0.904710	+	0.426028i	$0.140088\pi$
$600$	0	0
$601$	2.49987e15	1.30049	0.650245	−	0.759725i	$-0.274666\pi$
$601$	2.49987e15	1.30049	0.650245	+	0.759725i	$0.274666\pi$
$602$	− 4.25089e14i	− 0.219128i
$603$	1.25702e15i	0.642090i
$604$	−1.91094e14	−0.0967263
$605$	0	0
$606$	9.06685e14	0.450669
$607$	− 2.34097e13i	− 0.0115308i	−0.999983	−	0.00576538i	$-0.998165\pi$
$607$	− 2.34097e13i	− 0.0115308i	0.999983	−	0.00576538i	$-0.00183519\pi$
$608$	− 5.41481e13i	− 0.0264310i
$609$	−8.48229e14	−0.410316
$610$	0	0
$611$	−1.11258e15	−0.528573
$612$	− 1.15990e14i	− 0.0546121i
$613$	1.82595e15i	0.852032i	0.904716	+	0.426016i	$0.140083\pi$
$613$	1.82595e15i	0.852032i	−0.904716	+	0.426016i	$0.859917\pi$
$614$	2.87397e15	1.32909
$615$	0	0
$616$	−1.94243e14	−0.0882369
$617$	− 9.58777e14i	− 0.431667i	−0.976430	−	0.215834i	$-0.930753\pi$
$617$	− 9.58777e14i	− 0.431667i	0.976430	−	0.215834i	$-0.0692469\pi$
$618$	− 3.34548e14i	− 0.149287i
$619$	6.16431e13	0.0272638	0.0136319	−	0.999907i	$-0.495661\pi$
$619$	6.16431e13	0.0272638	0.0136319	+	0.999907i	$0.495661\pi$
$620$	0	0
$621$	−4.71350e14	−0.204805
$622$	− 2.16281e15i	− 0.931477i
$623$	1.70555e15i	0.728082i
$624$	−5.92027e14	−0.250511
$625$	0	0
$626$	−3.15895e15	−1.31336
$627$	− 1.01614e14i	− 0.0418777i
$628$	− 4.63315e14i	− 0.189277i
$629$	1.09632e15	0.443976
$630$	0	0
$631$	−3.38937e15	−1.34883	−0.674416	−	0.738352i	$-0.735604\pi$
$631$	−3.38937e15	−1.34883	−0.674416	+	0.738352i	$0.735604\pi$
$632$	− 8.84398e13i	− 0.0348902i
$633$	6.00270e14i	0.234761i
$634$	3.53672e15	1.37123
$635$	0	0
$636$	6.35464e14	0.242146
$637$	3.37835e15i	1.27626i
$638$	− 1.26529e15i	− 0.473890i
$639$	−6.92024e14	−0.256960
$640$	0	0
$641$	2.53363e15	0.924749	0.462375	−	0.886685i	$-0.346998\pi$
$641$	2.53363e15	0.924749	0.462375	+	0.886685i	$0.346998\pi$
$642$	− 1.67435e15i	− 0.605904i
$643$	4.33651e15i	1.55589i	0.628331	+	0.777946i	$0.283739\pi$
$643$	4.33651e15i	1.55589i	−0.628331	+	0.777946i	$0.716261\pi$
$644$	7.69493e14	0.273736
$645$	0	0
$646$	−9.90586e13	−0.0346428
$647$	− 1.86323e15i	− 0.646090i	−0.946384	−	0.323045i	$-0.895294\pi$
$647$	− 1.86323e15i	− 0.646090i	0.946384	−	0.323045i	$-0.104706\pi$
$648$	1.14255e14i	0.0392837i
$649$	1.63715e15	0.558139
$650$	0	0
$651$	3.15800e14	0.105856
$652$	1.55307e15i	0.516213i
$653$	− 5.87157e15i	− 1.93523i	−0.252440	−	0.967613i	$-0.581233\pi$
$653$	− 5.87157e15i	− 1.93523i	0.252440	−	0.967613i	$-0.418767\pi$
$654$	1.99120e15	0.650783
$655$	0	0
$656$	7.69721e14	0.247379
$657$	1.36850e15i	0.436149i
$658$	− 3.50531e14i	− 0.110786i
$659$	1.57270e14	0.0492920	0.0246460	−	0.999696i	$-0.492154\pi$
$659$	1.57270e14	0.0492920	0.0246460	+	0.999696i	$0.492154\pi$
$660$	0	0
$661$	−4.86756e15	−1.50039	−0.750193	−	0.661219i	$-0.770039\pi$
$661$	−4.86756e15	−1.50039	−0.750193	+	0.661219i	$0.770039\pi$
$662$	1.60050e15i	0.489256i
$663$	1.08306e15i	0.328342i
$664$	1.07016e15	0.321754
$665$	0	0
$666$	−1.07992e15	−0.319362
$667$	5.01247e15i	1.47014i
$668$	2.40703e15i	0.700181i
$669$	−1.27991e15	−0.369262
$670$	0	0
$671$	2.03710e14	0.0578148
$672$	− 1.86525e14i	− 0.0525056i
$673$	5.94031e14i	0.165854i	0.996556	+	0.0829270i	$0.0264268\pi$
$673$	5.94031e14i	0.165854i	−0.996556	+	0.0829270i	$0.973573\pi$
$674$	−1.15993e14	−0.0321220
$675$	0	0
$676$	3.69287e15	1.00614
$677$	3.03615e15i	0.820513i	0.911970	+	0.410257i	$0.134561\pi$
$677$	3.03615e15i	0.820513i	−0.911970	+	0.410257i	$0.865439\pi$
$678$	− 1.03198e15i	− 0.276635i
$679$	9.47864e14	0.252036
$680$	0	0
$681$	2.78106e15	0.727615
$682$	4.71075e14i	0.122258i
$683$	− 2.38176e15i	− 0.613174i	−0.951843	−	0.306587i	$-0.900813\pi$
$683$	− 2.38176e15i	− 0.613174i	0.951843	−	0.306587i	$-0.0991870\pi$
$684$	9.75767e13	0.0249194
$685$	0	0
$686$	−2.51185e15	−0.631265
$687$	− 8.80867e14i	− 0.219608i
$688$	6.08905e14i	0.150596i
$689$	−5.93363e15	−1.45584
$690$	0	0
$691$	1.02015e15	0.246339	0.123170	−	0.992386i	$-0.460694\pi$
$691$	1.02015e15	0.246339	0.123170	+	0.992386i	$0.460694\pi$
$692$	9.22737e14i	0.221052i
$693$	− 3.50031e14i	− 0.0831906i
$694$	2.09938e14	0.0495011
$695$	0	0
$696$	1.21502e15	0.281989
$697$	− 1.40813e15i	− 0.324237i
$698$	− 2.60593e15i	− 0.595330i
$699$	3.50070e15	0.793470
$700$	0	0
$701$	−9.47724e14	−0.211462	−0.105731	−	0.994395i	$-0.533718\pi$
$701$	−9.47724e14	−0.211462	−0.105731	+	0.994395i	$0.533718\pi$
$702$	− 1.06685e15i	− 0.236184i
$703$	9.22280e14i	0.202586i
$704$	2.78237e14	0.0606408
$705$	0	0
$706$	−3.32135e15	−0.712671
$707$	− 2.66735e15i	− 0.567902i
$708$	1.57210e15i	0.332122i
$709$	−1.41385e15	−0.296380	−0.148190	−	0.988959i	$-0.547345\pi$
$709$	−1.41385e15	−0.296380	−0.148190	+	0.988959i	$0.547345\pi$
$710$	0	0
$711$	1.59371e14	0.0328948
$712$	− 2.44306e15i	− 0.500374i
$713$	− 1.86617e15i	− 0.379278i
$714$	−3.41228e14	−0.0688185
$715$	0	0
$716$	−1.00045e15	−0.198690
$717$	3.23672e15i	0.637895i
$718$	− 4.42426e15i	− 0.865280i
$719$	−7.85082e15	−1.52372	−0.761862	−	0.647739i	$-0.775715\pi$
$719$	−7.85082e15	−1.52372	−0.761862	+	0.647739i	$0.775715\pi$
$720$	0	0
$721$	−9.84196e14	−0.188121
$722$	3.64436e15i	0.691299i
$723$	2.76459e15i	0.520440i
$724$	−3.92213e15	−0.732756
$725$	0	0
$726$	−1.69645e15	−0.312168
$727$	− 6.17476e15i	− 1.12767i	−0.825889	−	0.563833i	$-0.809326\pi$
$727$	− 6.17476e15i	− 1.12767i	0.825889	−	0.563833i	$-0.190674\pi$
$728$	1.74167e15i	0.315677i
$729$	−2.05891e14	−0.0370370
$730$	0	0
$731$	1.11393e15	0.197384
$732$	1.95616e14i	0.0344028i
$733$	7.25819e15i	1.26694i	0.773767	+	0.633471i	$0.218370\pi$
$733$	7.25819e15i	1.26694i	−0.773767	+	0.633471i	$0.781630\pi$
$734$	3.72342e15	0.645081
$735$	0	0
$736$	−1.10224e15	−0.188125
$737$	5.51623e15i	0.934483i
$738$	1.38706e15i	0.233231i
$739$	−1.04737e16	−1.74806	−0.874030	−	0.485872i	$-0.838502\pi$
$739$	−1.04737e16	−1.74806	−0.874030	+	0.485872i	$0.838502\pi$
$740$	0	0
$741$	−9.11120e14	−0.149822
$742$	− 1.86945e15i	− 0.305136i
$743$	− 2.62290e15i	− 0.424956i	−0.977166	−	0.212478i	$-0.931847\pi$
$743$	− 2.62290e15i	− 0.424956i	0.977166	−	0.212478i	$-0.0681534\pi$
$744$	−4.52357e14	−0.0727496
$745$	0	0
$746$	2.15099e15	0.340860
$747$	1.92846e15i	0.303353i
$748$	− 5.09006e14i	− 0.0794812i
$749$	−4.92573e15	−0.763520
$750$	0	0
$751$	3.09275e15	0.472417	0.236208	−	0.971702i	$-0.424095\pi$
$751$	3.09275e15	0.472417	0.236208	+	0.971702i	$0.424095\pi$
$752$	5.02107e14i	0.0761374i
$753$	− 6.89298e14i	− 0.103761i
$754$	−1.13452e16	−1.69539
$755$	0	0
$756$	3.36123e14	0.0495027
$757$	− 5.54759e15i	− 0.811105i	−0.914072	−	0.405553i	$-0.867079\pi$
$757$	− 5.54759e15i	− 0.811105i	0.914072	−	0.405553i	$-0.132921\pi$
$758$	− 4.25089e15i	− 0.617020i
$759$	−2.06845e15	−0.298068
$760$	0	0
$761$	1.79365e13	0.00254754	0.00127377	−	0.999999i	$-0.499595\pi$
$761$	1.79365e13	0.00254754	0.00127377	+	0.999999i	$0.499595\pi$
$762$	− 4.66506e15i	− 0.657817i
$763$	− 5.85784e15i	− 0.820074i
$764$	2.55687e15	0.355382
$765$	0	0
$766$	8.63391e15	1.18290
$767$	− 1.46794e16i	− 1.99680i
$768$	2.67181e14i	0.0360844i
$769$	1.07234e16	1.43794	0.718968	−	0.695043i	$-0.244615\pi$
$769$	1.07234e16	1.43794	0.718968	+	0.695043i	$0.244615\pi$
$770$	0	0
$771$	−7.32506e15	−0.968305
$772$	5.32269e15i	0.698612i
$773$	− 6.56995e15i	− 0.856199i	−0.903732	−	0.428100i	$-0.859183\pi$
$773$	− 6.56995e15i	− 0.856199i	0.903732	−	0.428100i	$-0.140817\pi$
$774$	−1.09727e15	−0.141983
$775$	0	0
$776$	−1.35774e15	−0.173211
$777$	3.17699e15i	0.402439i
$778$	3.13947e15i	0.394883i
$779$	1.18459e15	0.147949
$780$	0	0
$781$	−3.03685e15	−0.373975
$782$	2.01643e15i	0.246574i
$783$	2.18950e15i	0.265862i
$784$	1.52465e15	0.183836
$785$	0	0
$786$	−2.22744e15	−0.264839
$787$	− 1.38290e16i	− 1.63279i	−0.577495	−	0.816394i	$-0.695970\pi$
$787$	− 1.38290e16i	− 1.63279i	0.577495	−	0.816394i	$-0.304030\pi$
$788$	5.84088e15i	0.684833i
$789$	9.25629e15	1.07774
$790$	0	0
$791$	−3.03594e15	−0.348596
$792$	5.01391e14i	0.0571727i
$793$	− 1.82656e15i	− 0.206839i
$794$	−2.06124e15	−0.231801
$795$	0	0
$796$	−6.55172e14	−0.0726664
$797$	− 5.66909e15i	− 0.624442i	−0.950009	−	0.312221i	$-0.898927\pi$
$797$	− 5.66909e15i	− 0.624442i	0.950009	−	0.312221i	$-0.101073\pi$
$798$	− 2.87058e14i	− 0.0314018i
$799$	9.18555e14	0.0997925
$800$	0	0
$801$	4.40248e15	0.471757
$802$	− 6.58557e15i	− 0.700865i
$803$	6.00546e15i	0.634761i
$804$	−5.29706e15	−0.556066
$805$	0	0
$806$	4.22387e15	0.437389
$807$	− 4.88400e14i	− 0.0502310i
$808$	3.82077e15i	0.390290i
$809$	−1.22726e16	−1.24514	−0.622572	−	0.782563i	$-0.713912\pi$
$809$	−1.22726e16	−1.24514	−0.622572	+	0.782563i	$0.713912\pi$
$810$	0	0
$811$	2.48630e15	0.248851	0.124426	−	0.992229i	$-0.460291\pi$
$811$	2.48630e15	0.248851	0.124426	+	0.992229i	$0.460291\pi$
$812$	− 3.57443e15i	− 0.355344i
$813$	− 7.43777e15i	− 0.734420i
$814$	−4.73908e15	−0.464793
$815$	0	0
$816$	4.88782e14	0.0472954
$817$	9.37093e14i	0.0900661i
$818$	1.90597e15i	0.181959i
$819$	−3.13854e15	−0.297623
$820$	0	0
$821$	−1.90122e16	−1.77887	−0.889435	−	0.457062i	$-0.848902\pi$
$821$	−1.90122e16	−1.77887	−0.889435	+	0.457062i	$0.848902\pi$
$822$	− 2.08567e15i	− 0.193843i
$823$	− 2.42079e15i	− 0.223490i	−0.993737	−	0.111745i	$-0.964356\pi$
$823$	− 2.42079e15i	− 0.223490i	0.993737	−	0.111745i	$-0.0356439\pi$
$824$	1.40978e15	0.129286
$825$	0	0
$826$	4.62492e15	0.418518
$827$	− 1.83310e16i	− 1.64781i	−0.566731	−	0.823903i	$-0.691792\pi$
$827$	− 1.83310e16i	− 1.64781i	0.566731	−	0.823903i	$-0.308208\pi$
$828$	− 1.98627e15i	− 0.177366i
$829$	7.71702e15	0.684541	0.342271	−	0.939601i	$-0.388804\pi$
$829$	7.71702e15	0.684541	0.342271	+	0.939601i	$0.388804\pi$
$830$	0	0
$831$	−6.75029e15	−0.590904
$832$	− 2.49480e15i	− 0.216949i
$833$	− 2.78919e15i	− 0.240952i
$834$	−7.80814e15	−0.670092
$835$	0	0
$836$	4.28201e14	0.0362671
$837$	− 8.15162e14i	− 0.0685889i
$838$	− 2.64413e15i	− 0.221025i
$839$	1.33884e16	1.11183	0.555914	−	0.831240i	$-0.312368\pi$
$839$	1.33884e16	1.11183	0.555914	+	0.831240i	$0.312368\pi$
$840$	0	0
$841$	1.10833e16	0.908428
$842$	4.98267e15i	0.405738i
$843$	− 9.60275e15i	− 0.776862i
$844$	−2.52953e15	−0.203309
$845$	0	0
$846$	−9.04813e14	−0.0717831
$847$	4.99073e15i	0.393374i
$848$	2.67784e15i	0.209704i
$849$	−8.68878e15	−0.676031
$850$	0	0
$851$	1.87739e16	1.44192
$852$	− 2.91618e15i	− 0.222534i
$853$	− 3.99518e15i	− 0.302912i	−0.988464	−	0.151456i	$-0.951604\pi$
$853$	− 3.99518e15i	− 0.302912i	0.988464	−	0.151456i	$-0.0483963\pi$
$854$	5.75478e14	0.0433522
$855$	0	0
$856$	7.05570e15	0.524728
$857$	1.50656e16i	1.11324i	0.830766	+	0.556622i	$0.187903\pi$
$857$	1.50656e16i	1.11324i	−0.830766	+	0.556622i	$0.812097\pi$
$858$	− 4.68173e15i	− 0.343737i
$859$	2.66635e16	1.94516	0.972579	−	0.232572i	$-0.0747142\pi$
$859$	2.66635e16	1.94516	0.972579	+	0.232572i	$0.0747142\pi$
$860$	0	0
$861$	4.08056e15	0.293902
$862$	− 1.40884e16i	− 1.00826i
$863$	− 1.36614e16i	− 0.971486i	−0.874102	−	0.485743i	$-0.838549\pi$
$863$	− 1.36614e16i	− 0.971486i	0.874102	−	0.485743i	$-0.161451\pi$
$864$	−4.81469e14	−0.0340207
$865$	0	0
$866$	−1.02992e16	−0.718547
$867$	7.43389e15i	0.515361i
$868$	1.33078e15i	0.0916742i
$869$	6.99379e14	0.0478744
$870$	0	0
$871$	4.94611e16	3.34321
$872$	8.39088e15i	0.563595i
$873$	− 2.44669e15i	− 0.163305i
$874$	−1.69632e15	−0.112511
$875$	0	0
$876$	−5.76684e15	−0.377716
$877$	− 2.44903e16i	− 1.59403i	−0.603958	−	0.797016i	$-0.706410\pi$
$877$	− 2.44903e16i	− 1.59403i	0.603958	−	0.797016i	$-0.293590\pi$
$878$	2.09112e16i	1.35257i
$879$	2.83656e15	0.182328
$880$	0	0
$881$	9.28368e15	0.589322	0.294661	−	0.955602i	$-0.404793\pi$
$881$	9.28368e15	0.589322	0.294661	+	0.955602i	$0.404793\pi$
$882$	2.74746e15i	0.173322i
$883$	− 1.88185e16i	− 1.17978i	−0.807484	−	0.589890i	$-0.799171\pi$
$883$	− 1.88185e16i	− 1.17978i	0.807484	−	0.589890i	$-0.200829\pi$
$884$	−4.56399e15	−0.284352
$885$	0	0
$886$	−1.76041e16	−1.08325
$887$	− 6.50887e14i	− 0.0398039i	−0.999802	−	0.0199020i	$-0.993665\pi$
$887$	− 6.50887e14i	− 0.0398039i	0.999802	−	0.0199020i	$-0.00633541\pi$
$888$	− 4.55078e15i	− 0.276576i
$889$	−1.37240e16	−0.828937
$890$	0	0
$891$	−9.03523e14	−0.0539029
$892$	− 5.39352e15i	− 0.319790i
$893$	7.72734e14i	0.0455352i
$894$	−5.13533e15	−0.300755
$895$	0	0
$896$	7.86013e14	0.0454711
$897$	1.85467e16i	1.06637i
$898$	8.19670e14i	0.0468402i
$899$	−8.66867e15	−0.492350
$900$	0	0
$901$	4.89884e15	0.274857
$902$	6.08692e15i	0.339439i
$903$	3.22802e15i	0.178918i
$904$	4.34874e15	0.239573
$905$	0	0
$906$	1.45112e15	0.0789767
$907$	2.01411e16i	1.08954i	0.838585	+	0.544770i	$0.183383\pi$
$907$	2.01411e16i	1.08954i	−0.838585	+	0.544770i	$0.816617\pi$
$908$	1.17194e16i	0.630133i
$909$	−6.88514e15	−0.367969
$910$	0	0
$911$	−4.85179e15	−0.256184	−0.128092	−	0.991762i	$-0.540885\pi$
$911$	−4.85179e15	−0.256184	−0.128092	+	0.991762i	$0.540885\pi$
$912$	4.11187e14i	0.0215808i
$913$	8.46279e15i	0.441493i
$914$	5.56422e14	0.0288536
$915$	0	0
$916$	3.71197e15	0.190186
$917$	6.55283e15i	0.333732i
$918$	8.80801e14i	0.0445906i
$919$	−7.50130e15	−0.377486	−0.188743	−	0.982026i	$-0.560441\pi$
$919$	−7.50130e15	−0.377486	−0.188743	+	0.982026i	$0.560441\pi$
$920$	0	0
$921$	−2.18242e16	−1.08520
$922$	− 2.43782e16i	− 1.20499i
$923$	2.72298e16i	1.33793i
$924$	1.47503e15	0.0720452
$925$	0	0
$926$	−1.24335e16	−0.600111
$927$	2.54047e15i	0.121892i
$928$	5.12008e15i	0.244210i
$929$	6.04589e15	0.286664	0.143332	−	0.989675i	$-0.454218\pi$
$929$	6.04589e15	0.286664	0.143332	+	0.989675i	$0.454218\pi$
$930$	0	0
$931$	2.34640e15	0.109946
$932$	1.47519e16i	0.687165i
$933$	1.64239e16i	0.760547i
$934$	3.91094e15	0.180042
$935$	0	0
$936$	4.49571e15	0.204541
$937$	1.17484e16i	0.531389i	0.964057	+	0.265694i	$0.0856012\pi$
$937$	1.17484e16i	0.531389i	−0.964057	+	0.265694i	$0.914399\pi$
$938$	1.55833e16i	0.700717i
$939$	2.39883e16	1.07235
$940$	0	0
$941$	−5.89103e15	−0.260284	−0.130142	−	0.991495i	$-0.541543\pi$
$941$	−5.89103e15	−0.260284	−0.130142	+	0.991495i	$0.541543\pi$
$942$	3.51830e15i	0.154544i
$943$	− 2.41134e16i	− 1.05304i
$944$	−6.62482e15	−0.287626
$945$	0	0
$946$	−4.81519e15	−0.206639
$947$	2.96672e15i	0.126576i	0.997995	+	0.0632881i	$0.0201587\pi$
$947$	2.96672e15i	0.126576i	−0.997995	+	0.0632881i	$0.979841\pi$
$948$	6.71590e14i	0.0284878i
$949$	5.38477e16	2.27093
$950$	0	0
$951$	−2.68570e16	−1.11960
$952$	− 1.43793e15i	− 0.0595985i
$953$	− 1.10371e15i	− 0.0454826i	−0.999741	−	0.0227413i	$-0.992761\pi$
$953$	− 1.10371e15i	− 0.0454826i	0.999741	−	0.0227413i	$-0.00723940\pi$
$954$	−4.82556e15	−0.197711
$955$	0	0
$956$	−1.36395e16	−0.552433
$957$	9.60832e15i	0.386930i
$958$	1.39492e16i	0.558520i
$959$	−6.13577e15	−0.244268
$960$	0	0
$961$	−2.21811e16	−0.872980
$962$	4.24928e16i	1.66285i
$963$	1.27146e16i	0.494719i
$964$	−1.16500e16	−0.450714
$965$	0	0
$966$	−5.84334e15	−0.223505
$967$	− 1.84533e15i	− 0.0701826i	−0.999384	−	0.0350913i	$-0.988828\pi$
$967$	− 1.84533e15i	− 0.0701826i	0.999384	−	0.0350913i	$-0.0111722\pi$
$968$	− 7.14881e15i	− 0.270346i
$969$	7.52227e14	0.0282858
$970$	0	0
$971$	−1.46125e16	−0.543273	−0.271636	−	0.962400i	$-0.587565\pi$
$971$	−1.46125e16	−0.543273	−0.271636	+	0.962400i	$0.587565\pi$
$972$	− 8.67624e14i	− 0.0320750i
$973$	2.29706e16i	0.844405i
$974$	2.13181e16	0.779245
$975$	0	0
$976$	−8.24326e14	−0.0297937
$977$	1.87225e16i	0.672891i	0.941703	+	0.336445i	$0.109225\pi$
$977$	1.87225e16i	0.672891i	−0.941703	+	0.336445i	$0.890775\pi$
$978$	− 1.17936e16i	− 0.421486i
$979$	1.93196e16	0.686585
$980$	0	0
$981$	−1.51206e16	−0.531362
$982$	− 7.16215e14i	− 0.0250282i
$983$	− 2.79209e16i	− 0.970253i	−0.874444	−	0.485126i	$-0.838774\pi$
$983$	− 2.79209e16i	− 0.970253i	0.874444	−	0.485126i	$-0.161226\pi$
$984$	−5.84507e15	−0.201984
$985$	0	0
$986$	9.36668e15	0.320083
$987$	2.66184e15i	0.0904562i
$988$	− 3.83945e15i	− 0.129750i
$989$	1.90754e16	0.641054
$990$	0	0
$991$	2.46391e16	0.818878	0.409439	−	0.912337i	$-0.365724\pi$
$991$	2.46391e16	0.818878	0.409439	+	0.912337i	$0.365724\pi$
$992$	− 1.90623e15i	− 0.0630030i
$993$	− 1.21538e16i	− 0.399475i
$994$	−8.57903e15	−0.280423
$995$	0	0
$996$	−8.12653e15	−0.262711
$997$	4.80148e16i	1.54366i	0.635828	+	0.771831i	$0.280659\pi$
$997$	4.80148e16i	1.54366i	−0.635828	+	0.771831i	$0.719341\pi$
$998$	− 1.56002e15i	− 0.0498783i
$999$	8.20065e15	0.260758

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.12.c.g.49.1		2
5.2	odd	4		150.12.a.h.1.1		1
5.3	odd	4		30.12.a.a.1.1	✓	1
5.4	even	2	inner	150.12.c.g.49.2		2
15.8	even	4		90.12.a.j.1.1		1
20.3	even	4		240.12.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.12.a.a.1.1	✓	1	5.3	odd	4
90.12.a.j.1.1		1	15.8	even	4
150.12.a.h.1.1		1	5.2	odd	4
150.12.c.g.49.1		2	1.1	even	1	trivial
150.12.c.g.49.2		2	5.4	even	2	inner
240.12.a.d.1.1		1	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-32.0000i q^{2} +243.000i q^{3} -1024.00 q^{4} +7776.00 q^{6} -22876.0i q^{7} +32768.0i q^{8} -59049.0 q^{9} -259128. q^{11} -248832. i q^{12} +2.32346e6i q^{13} -732032. q^{14} +1.04858e6 q^{16} -1.91827e6i q^{17} +1.88957e6i q^{18} +1.61374e6 q^{19} +5.55887e6 q^{21} +8.29210e6i q^{22} -3.28492e7i q^{23} -7.96262e6 q^{24} +7.43508e7 q^{26} -1.43489e7i q^{27} +2.34250e7i q^{28} -1.52590e8 q^{29} +5.68101e7 q^{31} -3.35544e7i q^{32} -6.29681e7i q^{33} -6.13845e7 q^{34} +6.04662e7 q^{36} +5.71517e8i q^{37} -5.16397e7i q^{38} -5.64601e8 q^{39} +7.34063e8 q^{41} -1.77884e8i q^{42} +5.80697e8i q^{43} +2.65347e8 q^{44} -1.05117e9 q^{46} +4.78847e8i q^{47} +2.54804e8i q^{48} +1.45402e9 q^{49} +4.66139e8 q^{51} -2.37923e9i q^{52} +2.55379e9i q^{53} -4.59165e8 q^{54} +7.49601e8 q^{56} +3.92139e8i q^{57} +4.88289e9i q^{58} -6.31792e9 q^{59} -7.86138e8 q^{61} -1.81792e9i q^{62} +1.35080e9i q^{63} -1.07374e9 q^{64} -2.01498e9 q^{66} -2.12877e10i q^{67} +1.96430e9i q^{68} +7.98236e9 q^{69} +1.17195e10 q^{71} -1.93492e9i q^{72} -2.31756e10i q^{73} +1.82886e10 q^{74} -1.65247e9 q^{76} +5.92781e9i q^{77} +1.80672e10i q^{78} -2.69897e9 q^{79} +3.48678e9 q^{81} -2.34900e10i q^{82} -3.26587e10i q^{83} -5.69228e9 q^{84} +1.85823e10 q^{86} -3.70794e10i q^{87} -8.49111e9i q^{88} -7.45563e10 q^{89} +5.31515e10 q^{91} +3.36376e10i q^{92} +1.38048e10i q^{93} +1.53231e10 q^{94} +8.15373e9 q^{96} +4.14349e10i q^{97} -4.65285e10i q^{98} +1.53012e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2048 q^{4} + 15552 q^{6} - 118098 q^{9} - 518256 q^{11} - 1464064 q^{14} + 2097152 q^{16} + 3227480 q^{19} + 11117736 q^{21} - 15925248 q^{24} + 148701568 q^{26} - 305180580 q^{29} + 113620144 q^{31}+ \cdots + 30602498544 q^{99}+O(q^{100}) \) 2 * q - 2048 * q^4 + 15552 * q^6 - 118098 * q^9 - 518256 * q^11 - 1464064 * q^14 + 2097152 * q^16 + 3227480 * q^19 + 11117736 * q^21 - 15925248 * q^24 + 148701568 * q^26 - 305180580 * q^29 + 113620144 * q^31 - 122769024 * q^34 + 120932352 * q^36 - 1129202532 * q^39 + 1468126164 * q^41 + 530694144 * q^44 - 2102349312 * q^46 + 2908030734 * q^49 + 932277276 * q^51 - 918330048 * q^54 + 1499201536 * q^56 - 12635840880 * q^59 - 1572276356 * q^61 - 2147483648 * q^64 - 4029958656 * q^66 + 15964715088 * q^69 + 23438960064 * q^71 + 36577110656 * q^74 - 3304939520 * q^76 - 5397939280 * q^79 + 6973568802 * q^81 - 11384561664 * q^84 + 37164583168 * q^86 - 149112660420 * q^89 + 106303033424 * q^91 + 30646181376 * q^94 + 16307453952 * q^96 + 30602498544 * q^99

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