LMFDB - Embedded newform 50.12.b.f.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,12,Mod(49,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("50.49");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	50.b (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:	$38.4171590280$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1969})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 985x^{2} + 242064 $ x^4 + 985*x^2 + 242064
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$21.6867i$ of defining polynomial
Character	$\chi$	$=$	50.49
Dual form			50.12.b.f.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-32.0000i q^{2} -141.734i q^{3} -1024.00 q^{4} -4535.49 q^{6} -85586.9i q^{7} +32768.0i q^{8} +157058. q^{9} +767235. q^{11} +145136. i q^{12} +220960. i q^{13} -2.73878e6 q^{14} +1.04858e6 q^{16} +930719. i q^{17} -5.02587e6i q^{18} +1.77341e7 q^{19} -1.21306e7 q^{21} -2.45515e7i q^{22} -3.99596e7i q^{23} +4.64434e6 q^{24} +7.07072e6 q^{26} -4.73683e7i q^{27} +8.76410e7i q^{28} -7.68554e7 q^{29} -2.96314e7 q^{31} -3.35544e7i q^{32} -1.08743e8i q^{33} +2.97830e7 q^{34} -1.60828e8 q^{36} -5.40911e7i q^{37} -5.67491e8i q^{38} +3.13176e7 q^{39} +1.26006e8 q^{41} +3.88179e8i q^{42} -2.88676e8i q^{43} -7.85648e8 q^{44} -1.27871e9 q^{46} +1.57008e9i q^{47} -1.48619e8i q^{48} -5.34780e9 q^{49} +1.31915e8 q^{51} -2.26263e8i q^{52} -4.09006e9i q^{53} -1.51579e9 q^{54} +2.80451e9 q^{56} -2.51353e9i q^{57} +2.45937e9i q^{58} -3.77882e9 q^{59} -9.64103e9 q^{61} +9.48205e8i q^{62} -1.34422e10i q^{63} -1.07374e9 q^{64} -3.47979e9 q^{66} +1.63819e10i q^{67} -9.53056e8i q^{68} -5.66364e9 q^{69} +1.03471e10 q^{71} +5.14649e9i q^{72} +4.27149e9i q^{73} -1.73091e9 q^{74} -1.81597e10 q^{76} -6.56653e10i q^{77} -1.00216e9i q^{78} +1.96636e10 q^{79} +2.11087e10 q^{81} -4.03220e9i q^{82} -1.35791e10i q^{83} +1.24217e10 q^{84} -9.23762e9 q^{86} +1.08930e10i q^{87} +2.51407e10i q^{88} -2.25058e10 q^{89} +1.89113e10 q^{91} +4.09186e10i q^{92} +4.19978e9i q^{93} +5.02424e10 q^{94} -4.75581e9 q^{96} +1.08976e11i q^{97} +1.71130e11i q^{98} +1.20501e11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4096 q^{4} + 38656 q^{6} - 443828 q^{9} + 843168 q^{11} - 901888 q^{14} + 4194304 q^{16} + 57794800 q^{19} - 130893632 q^{21} - 39583744 q^{24} + 110753536 q^{26} - 116452440 q^{29} + 82826768 q^{31}+ \cdots + 502985449824 q^{99}+O(q^{100}) $ 4 * q - 4096 * q^4 + 38656 * q^6 - 443828 * q^9 + 843168 * q^11 - 901888 * q^14 + 4194304 * q^16 + 57794800 * q^19 - 130893632 * q^21 - 39583744 * q^24 + 110753536 * q^26 - 116452440 * q^29 + 82826768 * q^31 + 404712192 * q^34 + 454479872 * q^36 - 2188831696 * q^39 - 1570542072 * q^41 - 863404032 * q^44 - 2895108864 * q^46 - 16963997892 * q^49 - 7779524112 * q^51 - 12664092160 * q^54 + 923533312 * q^56 - 17354204880 * q^59 + 2230997528 * q^61 - 4294967296 * q^64 - 23456437248 * q^66 - 3457605696 * q^69 + 27599665968 * q^71 - 24144348928 * q^74 - 59181875200 * q^76 + 25273860640 * q^79 + 132427867844 * q^81 + 134035079168 * q^84 - 92991183104 * q^86 - 26417480040 * q^89 - 178029806192 * q^91 + 82440175872 * q^94 + 40533753856 * q^96 + 502985449824 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$-1$

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
$3$	− 141.734i	− 0.336750i	−0.985723	−	0.168375i	$-0.946148\pi$
$3$	− 141.734i	− 0.336750i	0.985723	−	0.168375i	$-0.0538520\pi$
$4$	−1024.00	−0.500000
$5$	0	0
$5$	0	0
$6$	−4535.49	−0.238118
$7$	− 85586.9i	− 1.92472i	−0.271772	−	0.962362i	$-0.587610\pi$
$7$	− 85586.9i	− 1.92472i	0.271772	−	0.962362i	$-0.412390\pi$
$8$	32768.0i	0.353553i
$9$	157058.	0.886599
$10$	0	0
$11$	767235.	1.43638	0.718188	−	0.695849i	$-0.244972\pi$
$11$	767235.	1.43638	0.718188	+	0.695849i	$0.244972\pi$
$12$	145136.i	0.168375i
$13$	220960.i	0.165054i	0.996589	+	0.0825268i	$0.0262990\pi$
$13$	220960.i	0.165054i	−0.996589	+	0.0825268i	$0.973701\pi$
$14$	−2.73878e6	−1.36098
$15$	0	0
$16$	1.04858e6	0.250000
$17$	930719.i	0.158983i	0.996836	+	0.0794913i	$0.0253296\pi$
$17$	930719.i	0.158983i	−0.996836	+	0.0794913i	$0.974670\pi$
$18$	− 5.02587e6i	− 0.626920i
$19$	1.77341e7	1.64310	0.821551	−	0.570135i	$-0.193109\pi$
$19$	1.77341e7	1.64310	0.821551	+	0.570135i	$0.193109\pi$
$20$	0	0
$21$	−1.21306e7	−0.648151
$22$	− 2.45515e7i	− 1.01567i
$23$	− 3.99596e7i	− 1.29455i	−0.762257	−	0.647274i	$-0.775909\pi$
$23$	− 3.99596e7i	− 1.29455i	0.762257	−	0.647274i	$-0.224091\pi$
$24$	4.64434e6	0.119059
$25$	0	0
$26$	7.07072e6	0.116711
$27$	− 4.73683e7i	− 0.635312i
$28$	8.76410e7i	0.962362i
$29$	−7.68554e7	−0.695801	−0.347901	−	0.937531i	$-0.613105\pi$
$29$	−7.68554e7	−0.695801	−0.347901	+	0.937531i	$0.613105\pi$
$30$	0	0
$31$	−2.96314e7	−0.185893	−0.0929465	−	0.995671i	$-0.529629\pi$
$31$	−2.96314e7	−0.185893	−0.0929465	+	0.995671i	$0.529629\pi$
$32$	− 3.35544e7i	− 0.176777i
$33$	− 1.08743e8i	− 0.483700i
$34$	2.97830e7	0.112418
$35$	0	0
$36$	−1.60828e8	−0.443300
$37$	− 5.40911e7i	− 0.128238i	−0.997942	−	0.0641189i	$-0.979576\pi$
$37$	− 5.40911e7i	− 0.128238i	0.997942	−	0.0641189i	$-0.0204237\pi$
$38$	− 5.67491e8i	− 1.16185i
$39$	3.13176e7	0.0555818
$40$	0	0
$41$	1.26006e8	0.169856	0.0849280	−	0.996387i	$-0.472934\pi$
$41$	1.26006e8	0.169856	0.0849280	+	0.996387i	$0.472934\pi$
$42$	3.88179e8i	0.458312i
$43$	− 2.88676e8i	− 0.299457i	−0.988727	−	0.149728i	$-0.952160\pi$
$43$	− 2.88676e8i	− 0.299457i	0.988727	−	0.149728i	$-0.0478399\pi$
$44$	−7.85648e8	−0.718188
$45$	0	0
$46$	−1.27871e9	−0.915384
$47$	1.57008e9i	0.998579i	0.866435	+	0.499290i	$0.166406\pi$
$47$	1.57008e9i	0.998579i	−0.866435	+	0.499290i	$0.833594\pi$
$48$	− 1.48619e8i	− 0.0841875i
$49$	−5.34780e9	−2.70456
$50$	0	0
$51$	1.31915e8	0.0535374
$52$	− 2.26263e8i	− 0.0825268i
$53$	− 4.09006e9i	− 1.34342i	−0.740813	−	0.671711i	$-0.765560\pi$
$53$	− 4.09006e9i	− 1.34342i	0.740813	−	0.671711i	$-0.234440\pi$
$54$	−1.51579e9	−0.449234
$55$	0	0
$56$	2.80451e9	0.680492
$57$	− 2.51353e9i	− 0.553315i
$58$	2.45937e9i	0.492006i
$59$	−3.77882e9	−0.688129	−0.344065	−	0.938946i	$-0.611804\pi$
$59$	−3.77882e9	−0.688129	−0.344065	+	0.938946i	$0.611804\pi$
$60$	0	0
$61$	−9.64103e9	−1.46154	−0.730768	−	0.682626i	$-0.760838\pi$
$61$	−9.64103e9	−1.46154	−0.730768	+	0.682626i	$0.760838\pi$
$62$	9.48205e8i	0.131446i
$63$	− 1.34422e10i	− 1.70646i
$64$	−1.07374e9	−0.125000
$65$	0	0
$66$	−3.47979e9	−0.342028
$67$	1.63819e10i	1.48236i	0.671308	+	0.741178i	$0.265733\pi$
$67$	1.63819e10i	1.48236i	−0.671308	+	0.741178i	$0.734267\pi$
$68$	− 9.53056e8i	− 0.0794913i
$69$	−5.66364e9	−0.435939
$70$	0	0
$71$	1.03471e10	0.680609	0.340304	−	0.940315i	$-0.389470\pi$
$71$	1.03471e10	0.680609	0.340304	+	0.940315i	$0.389470\pi$
$72$	5.14649e9i	0.313460i
$73$	4.27149e9i	0.241159i	0.992704	+	0.120580i	$0.0384753\pi$
$73$	4.27149e9i	0.241159i	−0.992704	+	0.120580i	$0.961525\pi$
$74$	−1.73091e9	−0.0906778
$75$	0	0
$76$	−1.81597e10	−0.821551
$77$	− 6.56653e10i	− 2.76463i
$78$	− 1.00216e9i	− 0.0393023i
$79$	1.96636e10	0.718977	0.359489	−	0.933149i	$-0.382951\pi$
$79$	1.96636e10	0.718977	0.359489	+	0.933149i	$0.382951\pi$
$80$	0	0
$81$	2.11087e10	0.672658
$82$	− 4.03220e9i	− 0.120106i
$83$	− 1.35791e10i	− 0.378391i	−0.981939	−	0.189196i	$-0.939412\pi$
$83$	− 1.35791e10i	− 0.378391i	0.981939	−	0.189196i	$-0.0605880\pi$
$84$	1.24217e10	0.324075
$85$	0	0
$86$	−9.23762e9	−0.211748
$87$	1.08930e10i	0.234311i
$88$	2.51407e10i	0.507836i
$89$	−2.25058e10	−0.427218	−0.213609	−	0.976919i	$-0.568522\pi$
$89$	−2.25058e10	−0.427218	−0.213609	+	0.976919i	$0.568522\pi$
$90$	0	0
$91$	1.89113e10	0.317683
$92$	4.09186e10i	0.647274i
$93$	4.19978e9i	0.0625994i
$94$	5.02424e10	0.706102
$95$	0	0
$96$	−4.75581e9	−0.0595296
$97$	1.08976e11i	1.28851i	0.764811	+	0.644255i	$0.222832\pi$
$97$	1.08976e11i	1.28851i	−0.764811	+	0.644255i	$0.777168\pi$
$98$	1.71130e11i	1.91241i
$99$	1.20501e11	1.27349
$100$	0	0
$101$	1.63516e11	1.54807	0.774037	−	0.633140i	$-0.218234\pi$
$101$	1.63516e11	1.54807	0.774037	+	0.633140i	$0.218234\pi$
$102$	− 4.22127e9i	− 0.0378566i
$103$	− 7.69876e10i	− 0.654359i	−0.944962	−	0.327180i	$-0.893902\pi$
$103$	− 7.69876e10i	− 0.654359i	0.944962	−	0.327180i	$-0.106098\pi$
$104$	−7.24042e9	−0.0583553
$105$	0	0
$106$	−1.30882e11	−0.949943
$107$	− 1.19891e11i	− 0.826376i	−0.910646	−	0.413188i	$-0.864415\pi$
$107$	− 1.19891e11i	− 0.826376i	0.910646	−	0.413188i	$-0.135585\pi$
$108$	4.85052e10i	0.317656i
$109$	7.33426e10	0.456573	0.228287	−	0.973594i	$-0.426688\pi$
$109$	7.33426e10	0.456573	0.228287	+	0.973594i	$0.426688\pi$
$110$	0	0
$111$	−7.66655e9	−0.0431841
$112$	− 8.97444e10i	− 0.481181i
$113$	2.18835e11i	1.11734i	0.829391	+	0.558669i	$0.188688\pi$
$113$	2.18835e11i	1.11734i	−0.829391	+	0.558669i	$0.811312\pi$
$114$	−8.04329e10	−0.391253
$115$	0	0
$116$	7.86999e10	0.347901
$117$	3.47036e10i	0.146337i
$118$	1.20922e11i	0.486581i
$119$	7.96574e10	0.305997
$120$	0	0
$121$	3.03337e11	1.06318
$122$	3.08513e11i	1.03346i
$123$	− 1.78594e10i	− 0.0571990i
$124$	3.03426e10	0.0929465
$125$	0	0
$126$	−4.30149e11	−1.20665
$127$	− 3.93053e11i	− 1.05568i	−0.849345	−	0.527838i	$-0.823003\pi$
$127$	− 3.93053e11i	− 1.05568i	0.849345	−	0.527838i	$-0.176997\pi$
$128$	3.43597e10i	0.0883883i
$129$	−4.09152e10	−0.100842
$130$	0	0
$131$	−2.63175e11	−0.596008	−0.298004	−	0.954565i	$-0.596321\pi$
$131$	−2.63175e11	−0.596008	−0.298004	+	0.954565i	$0.596321\pi$
$132$	1.11353e11i	0.241850i
$133$	− 1.51781e12i	− 3.16252i
$134$	5.24220e11	1.04818
$135$	0	0
$136$	−3.04978e10	−0.0562088
$137$	− 5.20521e11i	− 0.921457i	−0.887541	−	0.460728i	$-0.847588\pi$
$137$	− 5.20521e11i	− 0.921457i	0.887541	−	0.460728i	$-0.152412\pi$
$138$	1.81237e11i	0.308255i
$139$	−6.82675e11	−1.11592	−0.557960	−	0.829868i	$-0.688416\pi$
$139$	−6.82675e11	−1.11592	−0.557960	+	0.829868i	$0.688416\pi$
$140$	0	0
$141$	2.22533e11	0.336272
$142$	− 3.31107e11i	− 0.481263i
$143$	1.69528e11i	0.237079i
$144$	1.64688e11	0.221650
$145$	0	0
$146$	1.36688e11	0.170525
$147$	7.57966e11i	0.910761i
$148$	5.53893e10i	0.0641189i
$149$	−1.01566e12	−1.13299	−0.566493	−	0.824066i	$-0.691700\pi$
$149$	−1.01566e12	−1.13299	−0.566493	+	0.824066i	$0.691700\pi$
$150$	0	0
$151$	5.34040e11	0.553605	0.276803	−	0.960927i	$-0.410725\pi$
$151$	5.34040e11	0.553605	0.276803	+	0.960927i	$0.410725\pi$
$152$	5.81111e11i	0.580924i
$153$	1.46177e11i	0.140954i
$154$	−2.10129e12	−1.95489
$155$	0	0
$156$	−3.20692e10	−0.0277909
$157$	7.29675e11i	0.610494i	0.952273	+	0.305247i	$0.0987390\pi$
$157$	7.29675e11i	0.610494i	−0.952273	+	0.305247i	$0.901261\pi$
$158$	− 6.29237e11i	− 0.508394i
$159$	−5.79701e11	−0.452397
$160$	0	0
$161$	−3.42002e12	−2.49165
$162$	− 6.75479e11i	− 0.475641i
$163$	− 2.12171e11i	− 0.144429i	−0.997389	−	0.0722143i	$-0.976993\pi$
$163$	− 2.12171e11i	− 0.144429i	0.997389	−	0.0722143i	$-0.0230066\pi$
$164$	−1.29030e11	−0.0849280
$165$	0	0
$166$	−4.34531e11	−0.267563
$167$	2.36613e11i	0.140961i	0.997513	+	0.0704804i	$0.0224532\pi$
$167$	2.36613e11i	0.140961i	−0.997513	+	0.0704804i	$0.977547\pi$
$168$	− 3.97495e11i	− 0.229156i
$169$	1.74334e12	0.972757
$170$	0	0
$171$	2.78529e12	1.45677
$172$	2.95604e11i	0.149728i
$173$	− 7.36983e11i	− 0.361579i	−0.983522	−	0.180790i	$-0.942135\pi$
$173$	− 7.36983e11i	− 0.361579i	0.983522	−	0.180790i	$-0.0578653\pi$
$174$	3.48577e11	0.165683
$175$	0	0
$176$	8.04504e11	0.359094
$177$	5.35588e11i	0.231727i
$178$	7.20185e11i	0.302089i
$179$	−1.35344e11	−0.0550487	−0.0275244	−	0.999621i	$-0.508762\pi$
$179$	−1.35344e11	−0.0550487	−0.0275244	+	0.999621i	$0.508762\pi$
$180$	0	0
$181$	−1.35881e12	−0.519907	−0.259953	−	0.965621i	$-0.583707\pi$
$181$	−1.35881e12	−0.519907	−0.259953	+	0.965621i	$0.583707\pi$
$182$	− 6.05161e11i	− 0.224636i
$183$	1.36646e12i	0.492172i
$184$	1.30940e12	0.457692
$185$	0	0
$186$	1.34393e11	0.0442645
$187$	7.14080e11i	0.228359i
$188$	− 1.60776e12i	− 0.499290i
$189$	−4.05411e12	−1.22280
$190$	0	0
$191$	1.10648e12	0.314964	0.157482	−	0.987522i	$-0.449662\pi$
$191$	1.10648e12	0.314964	0.157482	+	0.987522i	$0.449662\pi$
$192$	1.52186e11i	0.0420938i
$193$	6.17025e12i	1.65858i	0.558816	+	0.829292i	$0.311256\pi$
$193$	6.17025e12i	1.65858i	−0.558816	+	0.829292i	$0.688744\pi$
$194$	3.48724e12	0.911114
$195$	0	0
$196$	5.47615e12	1.35228
$197$	− 2.00333e12i	− 0.481047i	−0.970643	−	0.240523i	$-0.922681\pi$
$197$	− 2.00333e12i	− 0.481047i	0.970643	−	0.240523i	$-0.0773191\pi$
$198$	− 3.85602e12i	− 0.900494i
$199$	7.73897e12	1.75789	0.878944	−	0.476925i	$-0.158249\pi$
$199$	7.73897e12	1.75789	0.878944	+	0.476925i	$0.158249\pi$
$200$	0	0
$201$	2.32187e12	0.499183
$202$	− 5.23250e12i	− 1.09465i
$203$	6.57782e12i	1.33922i
$204$	−1.35081e11	−0.0267687
$205$	0	0
$206$	−2.46360e12	−0.462702
$207$	− 6.27599e12i	− 1.14775i
$208$	2.31693e11i	0.0412634i
$209$	1.36062e13	2.36011
$210$	0	0
$211$	−9.01879e12	−1.48455	−0.742275	−	0.670096i	$-0.766253\pi$
$211$	−9.01879e12	−1.48455	−0.742275	+	0.670096i	$0.766253\pi$
$212$	4.18822e12i	0.671711i
$213$	− 1.46654e12i	− 0.229195i
$214$	−3.83653e12	−0.584336
$215$	0	0
$216$	1.55217e12	0.224617
$217$	2.53606e12i	0.357792i
$218$	− 2.34696e12i	− 0.322846i
$219$	6.05416e11	0.0812103
$220$	0	0
$221$	−2.05652e11	−0.0262407
$222$	2.45330e11i	0.0305358i
$223$	2.06620e12i	0.250897i	0.992100	+	0.125448i	$0.0400370\pi$
$223$	2.06620e12i	0.250897i	−0.992100	+	0.125448i	$0.959963\pi$
$224$	−2.87182e12	−0.340246
$225$	0	0
$226$	7.00271e12	0.790077
$227$	1.56806e13i	1.72672i	0.504591	+	0.863359i	$0.331643\pi$
$227$	1.56806e13i	1.72672i	−0.504591	+	0.863359i	$0.668357\pi$
$228$	2.57385e12i	0.276657i
$229$	−6.02309e12	−0.632010	−0.316005	−	0.948758i	$-0.602342\pi$
$229$	−6.02309e12	−0.632010	−0.316005	+	0.948758i	$0.602342\pi$
$230$	0	0
$231$	−9.30701e12	−0.930989
$232$	− 2.51840e12i	− 0.246003i
$233$	1.43129e13i	1.36544i	0.730682	+	0.682718i	$0.239202\pi$
$233$	1.43129e13i	1.36544i	−0.730682	+	0.682718i	$0.760798\pi$
$234$	1.11052e12	0.103476
$235$	0	0
$236$	3.86951e12	0.344065
$237$	− 2.78701e12i	− 0.242116i
$238$	− 2.54904e12i	− 0.216373i
$239$	−2.22886e13	−1.84881	−0.924407	−	0.381407i	$-0.875440\pi$
$239$	−2.22886e13	−1.84881	−0.924407	+	0.381407i	$0.875440\pi$
$240$	0	0
$241$	2.06638e13	1.63726	0.818629	−	0.574323i	$-0.194735\pi$
$241$	2.06638e13	1.63726	0.818629	+	0.574323i	$0.194735\pi$
$242$	− 9.70679e12i	− 0.751781i
$243$	− 1.13830e13i	− 0.861830i
$244$	9.87241e12	0.730768
$245$	0	0
$246$	−5.71501e11	−0.0404458
$247$	3.91853e12i	0.271200i
$248$	− 9.70962e11i	− 0.0657231i
$249$	−1.92462e12	−0.127423
$250$	0	0
$251$	−1.23384e13	−0.781721	−0.390861	−	0.920450i	$-0.627823\pi$
$251$	−1.23384e13	−0.781721	−0.390861	+	0.920450i	$0.627823\pi$
$252$	1.37648e13i	0.853229i
$253$	− 3.06584e13i	− 1.85946i
$254$	−1.25777e13	−0.746475
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	− 1.19160e13i	− 0.662975i	−0.943460	−	0.331487i	$-0.892450\pi$
$257$	− 1.19160e13i	− 0.662975i	0.943460	−	0.331487i	$-0.107550\pi$
$258$	1.30929e12i	0.0713061i
$259$	−4.62949e12	−0.246822
$260$	0	0
$261$	−1.20708e13	−0.616897
$262$	8.42159e12i	0.421441i
$263$	1.54645e12i	0.0757845i	0.999282	+	0.0378923i	$0.0120644\pi$
$263$	1.54645e12i	0.0757845i	−0.999282	+	0.0378923i	$0.987936\pi$
$264$	3.56330e12	0.171014
$265$	0	0
$266$	−4.85699e13	−2.23624
$267$	3.18984e12i	0.143866i
$268$	− 1.67751e13i	− 0.741178i
$269$	−1.43195e13	−0.619855	−0.309927	−	0.950760i	$-0.600305\pi$
$269$	−1.43195e13	−0.619855	−0.309927	+	0.950760i	$0.600305\pi$
$270$	0	0
$271$	2.26776e13	0.942468	0.471234	−	0.882008i	$-0.343809\pi$
$271$	2.26776e13	0.942468	0.471234	+	0.882008i	$0.343809\pi$
$272$	9.75929e11i	0.0397456i
$273$	− 2.68038e12i	− 0.106980i
$274$	−1.66567e13	−0.651568
$275$	0	0
$276$	5.79957e12	0.217970
$277$	− 9.25283e12i	− 0.340907i	−0.985366	−	0.170454i	$-0.945477\pi$
$277$	− 9.25283e12i	− 0.340907i	0.985366	−	0.170454i	$-0.0545233\pi$
$278$	2.18456e13i	0.789074i
$279$	−4.65386e12	−0.164813
$280$	0	0
$281$	−2.26511e12	−0.0771265	−0.0385633	−	0.999256i	$-0.512278\pi$
$281$	−2.26511e12	−0.0771265	−0.0385633	+	0.999256i	$0.512278\pi$
$282$	− 7.12107e12i	− 0.237780i
$283$	− 3.60409e13i	− 1.18024i	−0.807315	−	0.590121i	$-0.799080\pi$
$283$	− 3.60409e13i	− 1.18024i	0.807315	−	0.590121i	$-0.200920\pi$
$284$	−1.05954e13	−0.340304
$285$	0	0
$286$	5.42490e12	0.167640
$287$	− 1.07845e13i	− 0.326926i
$288$	− 5.27001e12i	− 0.156730i
$289$	3.34057e13	0.974725
$290$	0	0
$291$	1.54457e13	0.433905
$292$	− 4.37401e12i	− 0.120580i
$293$	1.63348e13i	0.441918i	0.975283	+	0.220959i	$0.0709188\pi$
$293$	1.63348e13i	0.441918i	−0.975283	+	0.220959i	$0.929081\pi$
$294$	2.42549e13	0.644005
$295$	0	0
$296$	1.77246e12	0.0453389
$297$	− 3.63426e13i	− 0.912548i
$298$	3.25012e13i	0.801142i
$299$	8.82948e12	0.213670
$300$	0	0
$301$	−2.47069e13	−0.576371
$302$	− 1.70893e13i	− 0.391458i
$303$	− 2.31758e13i	− 0.521314i
$304$	1.85956e13	0.410776
$305$	0	0
$306$	4.67767e12	0.0996694
$307$	5.61322e13i	1.17476i	0.809310	+	0.587382i	$0.199842\pi$
$307$	5.61322e13i	1.17476i	−0.809310	+	0.587382i	$0.800158\pi$
$308$	6.72412e13i	1.38231i
$309$	−1.09118e13	−0.220355
$310$	0	0
$311$	6.92340e13	1.34939	0.674694	−	0.738097i	$-0.264275\pi$
$311$	6.92340e13	1.34939	0.674694	+	0.738097i	$0.264275\pi$
$312$	1.02621e12i	0.0196511i
$313$	− 3.23822e13i	− 0.609273i	−0.952469	−	0.304637i	$-0.901465\pi$
$313$	− 3.23822e13i	− 0.609273i	0.952469	−	0.304637i	$-0.0985350\pi$
$314$	2.33496e13	0.431684
$315$	0	0
$316$	−2.01356e13	−0.359489
$317$	1.84023e13i	0.322885i	0.986882	+	0.161442i	$0.0516146\pi$
$317$	1.84023e13i	0.322885i	−0.986882	+	0.161442i	$0.948385\pi$
$318$	1.85504e13i	0.319893i
$319$	−5.89661e13	−0.999433
$320$	0	0
$321$	−1.69927e13	−0.278282
$322$	1.09441e14i	1.76186i
$323$	1.65055e13i	0.261224i
$324$	−2.16153e13	−0.336329
$325$	0	0
$326$	−6.78946e12	−0.102126
$327$	− 1.03951e13i	− 0.153751i
$328$	4.12897e12i	0.0600532i
$329$	1.34378e14	1.92199
$330$	0	0
$331$	6.74629e13	0.933277	0.466639	−	0.884448i	$-0.345465\pi$
$331$	6.74629e13	0.933277	0.466639	+	0.884448i	$0.345465\pi$
$332$	1.39050e13i	0.189196i
$333$	− 8.49546e12i	− 0.113696i
$334$	7.57163e12	0.0996744
$335$	0	0
$336$	−1.27198e13	−0.162038
$337$	− 1.07897e14i	− 1.35222i	−0.736802	−	0.676109i	$-0.763665\pi$
$337$	− 1.07897e14i	− 1.35222i	0.736802	−	0.676109i	$-0.236335\pi$
$338$	− 5.57868e13i	− 0.687843i
$339$	3.10163e13	0.376264
$340$	0	0
$341$	−2.27342e13	−0.267012
$342$	− 8.91293e13i	− 1.03009i
$343$	2.88468e14i	3.28081i
$344$	9.45933e12	0.105874
$345$	0	0
$346$	−2.35834e13	−0.255675
$347$	1.29756e13i	0.138458i	0.997601	+	0.0692288i	$0.0220538\pi$
$347$	1.29756e13i	0.138458i	−0.997601	+	0.0692288i	$0.977946\pi$
$348$	− 1.11545e13i	− 0.117156i
$349$	7.76358e13	0.802643	0.401321	−	0.915937i	$-0.368551\pi$
$349$	7.76358e13	0.802643	0.401321	+	0.915937i	$0.368551\pi$
$350$	0	0
$351$	1.04665e13	0.104861
$352$	− 2.57441e13i	− 0.253918i
$353$	− 4.21031e13i	− 0.408839i	−0.978883	−	0.204420i	$-0.934469\pi$
$353$	− 4.21031e13i	− 0.408839i	0.978883	−	0.204420i	$-0.0655307\pi$
$354$	1.71388e13	0.163856
$355$	0	0
$356$	2.30459e13	0.213609
$357$	− 1.12902e13i	− 0.103045i
$358$	4.33101e12i	0.0389253i
$359$	−1.84963e14	−1.63707	−0.818533	−	0.574460i	$-0.805212\pi$
$359$	−1.84963e14	−1.63707	−0.818533	+	0.574460i	$0.805212\pi$
$360$	0	0
$361$	1.98008e14	1.69978
$362$	4.34818e13i	0.367630i
$363$	− 4.29933e13i	− 0.358025i
$364$	−1.93652e13	−0.158841
$365$	0	0
$366$	4.37268e13	0.348018
$367$	1.68959e14i	1.32470i	0.749194	+	0.662351i	$0.230441\pi$
$367$	1.68959e14i	1.32470i	−0.749194	+	0.662351i	$0.769559\pi$
$368$	− 4.19007e13i	− 0.323637i
$369$	1.97903e13	0.150594
$370$	0	0
$371$	−3.50056e14	−2.58572
$372$	− 4.30058e12i	− 0.0312997i
$373$	− 1.87187e14i	− 1.34238i	−0.741284	−	0.671191i	$-0.765783\pi$
$373$	− 1.87187e14i	− 1.34238i	0.741284	−	0.671191i	$-0.234217\pi$
$374$	2.28505e13	0.161474
$375$	0	0
$376$	−5.14483e13	−0.353051
$377$	− 1.69820e13i	− 0.114845i
$378$	1.29732e14i	0.864651i
$379$	7.47613e13	0.491090	0.245545	−	0.969385i	$-0.421033\pi$
$379$	7.47613e13	0.491090	0.245545	+	0.969385i	$0.421033\pi$
$380$	0	0
$381$	−5.57090e13	−0.355499
$382$	− 3.54074e13i	− 0.222713i
$383$	− 1.55521e14i	− 0.964264i	−0.876099	−	0.482132i	$-0.839863\pi$
$383$	− 1.55521e14i	− 0.964264i	0.876099	−	0.482132i	$-0.160137\pi$
$384$	4.86995e12	0.0297648
$385$	0	0
$386$	1.97448e14	1.17280
$387$	− 4.53390e13i	− 0.265498i
$388$	− 1.11592e14i	− 0.644255i
$389$	8.76448e13	0.498889	0.249444	−	0.968389i	$-0.419752\pi$
$389$	8.76448e13	0.498889	0.249444	+	0.968389i	$0.419752\pi$
$390$	0	0
$391$	3.71912e13	0.205811
$392$	− 1.75237e14i	− 0.956206i
$393$	3.73009e13i	0.200706i
$394$	−6.41064e13	−0.340151
$395$	0	0
$396$	−1.23393e14	−0.636745
$397$	− 5.19767e13i	− 0.264521i	−0.991215	−	0.132261i	$-0.957776\pi$
$397$	− 5.19767e13i	− 0.264521i	0.991215	−	0.132261i	$-0.0422236\pi$
$398$	− 2.47647e14i	− 1.24301i
$399$	−2.15125e14	−1.06498
$400$	0	0
$401$	−3.40753e14	−1.64114	−0.820570	−	0.571546i	$-0.806344\pi$
$401$	−3.40753e14	−1.64114	−0.820570	+	0.571546i	$0.806344\pi$
$402$	− 7.42999e13i	− 0.352976i
$403$	− 6.54735e12i	− 0.0306823i
$404$	−1.67440e14	−0.774037
$405$	0	0
$406$	2.10490e14	0.946975
$407$	− 4.15005e13i	− 0.184198i
$408$	4.32258e12i	0.0189283i
$409$	−3.34635e14	−1.44575	−0.722875	−	0.690979i	$-0.757180\pi$
$409$	−3.34635e14	−1.44575	−0.722875	+	0.690979i	$0.757180\pi$
$410$	0	0
$411$	−7.37756e13	−0.310301
$412$	7.88353e13i	0.327180i
$413$	3.23417e14i	1.32446i
$414$	−2.00832e14	−0.811579
$415$	0	0
$416$	7.41419e12	0.0291776
$417$	9.67584e13i	0.375786i
$418$	− 4.35399e14i	− 1.66885i
$419$	−1.37830e14	−0.521396	−0.260698	−	0.965420i	$-0.583953\pi$
$419$	−1.37830e14	−0.521396	−0.260698	+	0.965420i	$0.583953\pi$
$420$	0	0
$421$	4.35453e14	1.60468	0.802342	−	0.596864i	$-0.203587\pi$
$421$	4.35453e14	1.60468	0.802342	+	0.596864i	$0.203587\pi$
$422$	2.88601e14i	1.04974i
$423$	2.46594e14i	0.885340i
$424$	1.34023e14	0.474972
$425$	0	0
$426$	−4.69292e13	−0.162065
$427$	8.25146e14i	2.81305i
$428$	1.22769e14i	0.413188i
$429$	2.40279e13	0.0798365
$430$	0	0
$431$	1.51185e14	0.489647	0.244824	−	0.969568i	$-0.421270\pi$
$431$	1.51185e14	0.489647	0.244824	+	0.969568i	$0.421270\pi$
$432$	− 4.96693e13i	− 0.158828i
$433$	− 3.75044e13i	− 0.118413i	−0.998246	−	0.0592065i	$-0.981143\pi$
$433$	− 3.75044e13i	− 0.118413i	0.998246	−	0.0592065i	$-0.0188570\pi$
$434$	8.11539e13	0.252997
$435$	0	0
$436$	−7.51028e13	−0.228287
$437$	− 7.08648e14i	− 2.12707i
$438$	− 1.93733e13i	− 0.0574244i
$439$	1.61934e14	0.474004	0.237002	−	0.971509i	$-0.423835\pi$
$439$	1.61934e14	0.474004	0.237002	+	0.971509i	$0.423835\pi$
$440$	0	0
$441$	−8.39917e14	−2.39786
$442$	6.58085e12i	0.0185549i
$443$	− 1.29342e14i	− 0.360179i	−0.983650	−	0.180090i	$-0.942361\pi$
$443$	− 1.29342e14i	− 0.360179i	0.983650	−	0.180090i	$-0.0576388\pi$
$444$	7.85055e12	0.0215920
$445$	0	0
$446$	6.61183e13	0.177411
$447$	1.43954e14i	0.381533i
$448$	9.18983e13i	0.240590i
$449$	−1.30255e13	−0.0336852	−0.0168426	−	0.999858i	$-0.505361\pi$
$449$	−1.30255e13	−0.0336852	−0.0168426	+	0.999858i	$0.505361\pi$
$450$	0	0
$451$	9.66764e13	0.243977
$452$	− 2.24087e14i	− 0.558669i
$453$	− 7.56916e13i	− 0.186427i
$454$	5.01780e14	1.22097
$455$	0	0
$456$	8.23633e13	0.195626
$457$	− 3.45724e14i	− 0.811318i	−0.914025	−	0.405659i	$-0.867042\pi$
$457$	− 3.45724e14i	− 0.811318i	0.914025	−	0.405659i	$-0.132958\pi$
$458$	1.92739e14i	0.446899i
$459$	4.40866e13	0.101004
$460$	0	0
$461$	2.96945e14	0.664234	0.332117	−	0.943238i	$-0.392237\pi$
$461$	2.96945e14	0.664234	0.332117	+	0.943238i	$0.392237\pi$
$462$	2.97824e14i	0.658308i
$463$	4.55258e14i	0.994402i	0.867635	+	0.497201i	$0.165639\pi$
$463$	4.55258e14i	0.994402i	−0.867635	+	0.497201i	$0.834361\pi$
$464$	−8.05887e13	−0.173950
$465$	0	0
$466$	4.58014e14	0.965509
$467$	3.66499e14i	0.763537i	0.924258	+	0.381768i	$0.124685\pi$
$467$	3.66499e14i	0.763537i	−0.924258	+	0.381768i	$0.875315\pi$
$468$	− 3.55365e13i	− 0.0731683i
$469$	1.40208e15	2.85313
$470$	0	0
$471$	1.03420e14	0.205584
$472$	− 1.23824e14i	− 0.243290i
$473$	− 2.21482e14i	− 0.430132i
$474$	−8.91843e13	−0.171202
$475$	0	0
$476$	−8.15691e13	−0.152999
$477$	− 6.42379e14i	− 1.19108i
$478$	7.13234e14i	1.30731i
$479$	−5.78377e14	−1.04801	−0.524005	−	0.851715i	$-0.675563\pi$
$479$	−5.78377e14	−1.04801	−0.524005	+	0.851715i	$0.675563\pi$
$480$	0	0
$481$	1.19520e13	0.0211661
$482$	− 6.61242e14i	− 1.15772i
$483$	4.84734e14i	0.839062i
$484$	−3.10617e14	−0.531589
$485$	0	0
$486$	−3.64255e14	−0.609406
$487$	4.75203e14i	0.786086i	0.919520	+	0.393043i	$0.128578\pi$
$487$	4.75203e14i	0.786086i	−0.919520	+	0.393043i	$0.871422\pi$
$488$	− 3.15917e14i	− 0.516731i
$489$	−3.00718e13	−0.0486363
$490$	0	0
$491$	7.59641e14	1.20132	0.600662	−	0.799503i	$-0.294904\pi$
$491$	7.59641e14	1.20132	0.600662	+	0.799503i	$0.294904\pi$
$492$	1.82880e13i	0.0285995i
$493$	− 7.15307e13i	− 0.110620i
$494$	1.25393e14	0.191767
$495$	0	0
$496$	−3.10708e13	−0.0464732
$497$	− 8.85576e14i	− 1.30998i
$498$	6.15879e13i	0.0901019i
$499$	4.91039e14	0.710499	0.355249	−	0.934772i	$-0.384396\pi$
$499$	4.91039e14	0.710499	0.355249	+	0.934772i	$0.384396\pi$
$500$	0	0
$501$	3.35362e13	0.0474686
$502$	3.94828e14i	0.552760i
$503$	− 4.25544e14i	− 0.589278i	−0.955609	−	0.294639i	$-0.904801\pi$
$503$	− 4.25544e14i	− 0.589278i	0.955609	−	0.294639i	$-0.0951995\pi$
$504$	4.40472e14	0.603324
$505$	0	0
$506$	−9.81069e14	−1.31484
$507$	− 2.47090e14i	− 0.327576i
$508$	4.02486e14i	0.527838i
$509$	1.41466e15	1.83529	0.917644	−	0.397404i	$-0.130089\pi$
$509$	1.41466e15	1.83529	0.917644	+	0.397404i	$0.130089\pi$
$510$	0	0
$511$	3.65584e14	0.464165
$512$	− 3.51844e13i	− 0.0441942i
$513$	− 8.40035e14i	− 1.04388i
$514$	−3.81311e14	−0.468794
$515$	0	0
$516$	4.18972e13	0.0504210
$517$	1.20462e15i	1.43434i
$518$	1.48144e14i	0.174530i
$519$	−1.04456e14	−0.121762
$520$	0	0
$521$	6.27267e14	0.715888	0.357944	−	0.933743i	$-0.383478\pi$
$521$	6.27267e14	0.715888	0.357944	+	0.933743i	$0.383478\pi$
$522$	3.86265e14i	0.436212i
$523$	2.23774e14i	0.250063i	0.992153	+	0.125032i	$0.0399032\pi$
$523$	2.23774e14i	0.250063i	−0.992153	+	0.125032i	$0.960097\pi$
$524$	2.69491e14	0.298004
$525$	0	0
$526$	4.94865e13	0.0535877
$527$	− 2.75785e13i	− 0.0295537i
$528$	− 1.14026e14i	− 0.120925i
$529$	−6.43961e14	−0.675855
$530$	0	0
$531$	−5.93495e14	−0.610095
$532$	1.55424e15i	1.58126i
$533$	2.78424e13i	0.0280354i
$534$	1.02075e14	0.101728
$535$	0	0
$536$	−5.36802e14	−0.524092
$537$	1.91829e13i	0.0185377i
$538$	4.58224e14i	0.438303i
$539$	−4.10302e15	−3.88477
$540$	0	0
$541$	5.47756e14	0.508162	0.254081	−	0.967183i	$-0.418227\pi$
$541$	5.47756e14	0.508162	0.254081	+	0.967183i	$0.418227\pi$
$542$	− 7.25684e14i	− 0.666425i
$543$	1.92589e14i	0.175079i
$544$	3.12297e13	0.0281044
$545$	0	0
$546$	−8.57720e13	−0.0756460
$547$	− 4.33518e14i	− 0.378509i	−0.981928	−	0.189255i	$-0.939393\pi$
$547$	− 4.33518e14i	− 0.378509i	0.981928	−	0.189255i	$-0.0606071\pi$
$548$	5.33013e14i	0.460728i
$549$	−1.51420e15	−1.29580
$550$	0	0
$551$	−1.36296e15	−1.14327
$552$	− 1.85586e14i	− 0.154128i
$553$	− 1.68295e15i	− 1.38383i
$554$	−2.96091e14	−0.241058
$555$	0	0
$556$	6.99060e14	0.557960
$557$	7.36061e14i	0.581715i	0.956766	+	0.290857i	$0.0939405\pi$
$557$	7.36061e14i	0.581715i	−0.956766	+	0.290857i	$0.906059\pi$
$558$	1.48924e14i	0.116540i
$559$	6.37858e13	0.0494264
$560$	0	0
$561$	1.01209e14	0.0768998
$562$	7.24834e13i	0.0545367i
$563$	− 9.03199e14i	− 0.672957i	−0.941691	−	0.336478i	$-0.890764\pi$
$563$	− 9.03199e14i	− 0.672957i	0.941691	−	0.336478i	$-0.109236\pi$
$564$	−2.27874e14	−0.168136
$565$	0	0
$566$	−1.15331e15	−0.834557
$567$	− 1.80663e15i	− 1.29468i
$568$	3.39054e14i	0.240632i
$569$	2.25786e15	1.58701	0.793503	−	0.608566i	$-0.208255\pi$
$569$	2.25786e15	1.58701	0.793503	+	0.608566i	$0.208255\pi$
$570$	0	0
$571$	9.15100e14	0.630913	0.315457	−	0.948940i	$-0.397842\pi$
$571$	9.15100e14	0.630913	0.315457	+	0.948940i	$0.397842\pi$
$572$	− 1.73597e14i	− 0.118540i
$573$	− 1.56826e14i	− 0.106064i
$574$	−3.45104e14	−0.231172
$575$	0	0
$576$	−1.68640e14	−0.110825
$577$	− 2.84484e14i	− 0.185179i	−0.995704	−	0.0925893i	$-0.970486\pi$
$577$	− 2.84484e14i	− 0.185179i	0.995704	−	0.0925893i	$-0.0295144\pi$
$578$	− 1.06898e15i	− 0.689234i
$579$	8.74535e14	0.558528
$580$	0	0
$581$	−1.16219e15	−0.728299
$582$	− 4.94261e14i	− 0.306817i
$583$	− 3.13804e15i	− 1.92966i
$584$	−1.39968e14	−0.0852626
$585$	0	0
$586$	5.22714e14	0.312484
$587$	1.99879e15i	1.18374i	0.806032	+	0.591871i	$0.201611\pi$
$587$	1.99879e15i	1.18374i	−0.806032	+	0.591871i	$0.798389\pi$
$588$	− 7.76157e14i	− 0.455380i
$589$	−5.25486e14	−0.305441
$590$	0	0
$591$	−2.83940e14	−0.161992
$592$	− 5.67186e13i	− 0.0320595i
$593$	8.52303e14i	0.477302i	0.971105	+	0.238651i	$0.0767052\pi$
$593$	8.52303e14i	0.477302i	−0.971105	+	0.238651i	$0.923295\pi$
$594$	−1.16296e15	−0.645269
$595$	0	0
$596$	1.04004e15	0.566493
$597$	− 1.09688e15i	− 0.591969i
$598$	− 2.82543e14i	− 0.151087i
$599$	−1.08984e14	−0.0577449	−0.0288725	−	0.999583i	$-0.509192\pi$
$599$	−1.08984e14	−0.0577449	−0.0288725	+	0.999583i	$0.509192\pi$
$600$	0	0
$601$	3.21315e14	0.167156	0.0835778	−	0.996501i	$-0.473365\pi$
$601$	3.21315e14	0.167156	0.0835778	+	0.996501i	$0.473365\pi$
$602$	7.90620e14i	0.407556i
$603$	2.57291e15i	1.31426i
$604$	−5.46856e14	−0.276803
$605$	0	0
$606$	−7.41624e14	−0.368625
$607$	3.08680e15i	1.52044i	0.649664	+	0.760222i	$0.274910\pi$
$607$	3.08680e15i	1.52044i	−0.649664	+	0.760222i	$0.725090\pi$
$608$	− 5.95058e14i	− 0.290462i
$609$	9.32301e14	0.450984
$610$	0	0
$611$	−3.46924e14	−0.164819
$612$	− 1.49685e14i	− 0.0704769i
$613$	8.70867e14i	0.406368i	0.979141	+	0.203184i	$0.0651289\pi$
$613$	8.70867e14i	0.406368i	−0.979141	+	0.203184i	$0.934871\pi$
$614$	1.79623e15	0.830684
$615$	0	0
$616$	2.15172e15	0.977444
$617$	− 3.06108e15i	− 1.37818i	−0.724675	−	0.689091i	$-0.758010\pi$
$617$	− 3.06108e15i	− 1.37818i	0.724675	−	0.689091i	$-0.241990\pi$
$618$	3.49177e14i	0.155815i
$619$	1.21317e15	0.536566	0.268283	−	0.963340i	$-0.413544\pi$
$619$	1.21317e15	0.536566	0.268283	+	0.963340i	$0.413544\pi$
$620$	0	0
$621$	−1.89282e15	−0.822442
$622$	− 2.21549e15i	− 0.954162i
$623$	1.92620e15i	0.822276i
$624$	3.28389e13	0.0138955
$625$	0	0
$626$	−1.03623e15	−0.430821
$627$	− 1.92847e15i	− 0.794768i
$628$	− 7.47187e14i	− 0.305247i
$629$	5.03436e13	0.0203876
$630$	0	0
$631$	2.95354e15	1.17539	0.587693	−	0.809084i	$-0.300036\pi$
$631$	2.95354e15	1.17539	0.587693	+	0.809084i	$0.300036\pi$
$632$	6.44338e14i	0.254197i
$633$	1.27827e15i	0.499922i
$634$	5.88875e14	0.228314
$635$	0	0
$636$	5.93614e14	0.226199
$637$	− 1.18165e15i	− 0.446398i
$638$	1.88691e15i	0.706706i
$639$	1.62510e15	0.603427
$640$	0	0
$641$	−1.24519e15	−0.454483	−0.227241	−	0.973838i	$-0.572971\pi$
$641$	−1.24519e15	−0.454483	−0.227241	+	0.973838i	$0.572971\pi$
$642$	5.43767e14i	0.196775i
$643$	2.82493e14i	0.101356i	0.998715	+	0.0506778i	$0.0161382\pi$
$643$	2.82493e14i	0.101356i	−0.998715	+	0.0506778i	$0.983862\pi$
$644$	3.50210e15	1.24582
$645$	0	0
$646$	5.28175e14	0.184714
$647$	3.45042e15i	1.19646i	0.801324	+	0.598230i	$0.204129\pi$
$647$	3.45042e15i	1.19646i	−0.801324	+	0.598230i	$0.795871\pi$
$648$	6.91691e14i	0.237821i
$649$	−2.89924e15	−0.988413
$650$	0	0
$651$	3.59446e14	0.120487
$652$	2.17263e14i	0.0722143i
$653$	3.55303e15i	1.17105i	0.810653	+	0.585527i	$0.199112\pi$
$653$	3.55303e15i	1.17105i	−0.810653	+	0.585527i	$0.800888\pi$
$654$	−3.32645e14	−0.108718
$655$	0	0
$656$	1.32127e14	0.0424640
$657$	6.70874e14i	0.213812i
$658$	− 4.30010e15i	− 1.35905i
$659$	−1.76602e14	−0.0553511	−0.0276756	−	0.999617i	$-0.508811\pi$
$659$	−1.76602e14	−0.0553511	−0.0276756	+	0.999617i	$0.508811\pi$
$660$	0	0
$661$	3.46137e15	1.06694	0.533470	−	0.845819i	$-0.320888\pi$
$661$	3.46137e15	1.06694	0.533470	+	0.845819i	$0.320888\pi$
$662$	− 2.15881e15i	− 0.659927i
$663$	2.91479e13i	0.00883654i
$664$	4.44960e14	0.133782
$665$	0	0
$666$	−2.71855e14	−0.0803949
$667$	3.07111e15i	0.900748i
$668$	− 2.42292e14i	− 0.0704804i
$669$	2.92851e14	0.0844895
$670$	0	0
$671$	−7.39693e15	−2.09932
$672$	4.07035e14i	0.114578i
$673$	6.92295e15i	1.93290i	0.256864	+	0.966448i	$0.417311\pi$
$673$	6.92295e15i	1.93290i	−0.256864	+	0.966448i	$0.582689\pi$
$674$	−3.45272e15	−0.956162
$675$	0	0
$676$	−1.78518e15	−0.486379
$677$	3.89173e15i	1.05173i	0.850568	+	0.525865i	$0.176258\pi$
$677$	3.89173e15i	1.05173i	−0.850568	+	0.525865i	$0.823742\pi$
$678$	− 9.92523e14i	− 0.266058i
$679$	9.32695e15	2.48002
$680$	0	0
$681$	2.22248e15	0.581472
$682$	7.27496e14i	0.188806i
$683$	4.79007e15i	1.23318i	0.787283	+	0.616592i	$0.211487\pi$
$683$	4.79007e15i	1.23318i	−0.787283	+	0.616592i	$0.788513\pi$
$684$	−2.85214e15	−0.728387
$685$	0	0
$686$	9.23099e15	2.31988
$687$	8.53677e14i	0.212829i
$688$	− 3.02698e14i	− 0.0748641i
$689$	9.03740e14	0.221737
$690$	0	0
$691$	1.60990e15	0.388749	0.194374	−	0.980927i	$-0.437732\pi$
$691$	1.60990e15	0.388749	0.194374	+	0.980927i	$0.437732\pi$
$692$	7.54670e14i	0.180790i
$693$	− 1.03133e16i	− 2.45112i
$694$	4.15220e14	0.0979043
$695$	0	0
$696$	−3.56943e14	−0.0828415
$697$	1.17276e14i	0.0270041i
$698$	− 2.48435e15i	− 0.567554i
$699$	2.02863e15	0.459810
$700$	0	0
$701$	−2.46356e15	−0.549686	−0.274843	−	0.961489i	$-0.588626\pi$
$701$	−2.46356e15	−0.549686	−0.274843	+	0.961489i	$0.588626\pi$
$702$	− 3.34928e14i	− 0.0741477i
$703$	− 9.59257e14i	− 0.210708i
$704$	−8.23812e14	−0.179547
$705$	0	0
$706$	−1.34730e15	−0.289093
$707$	− 1.39948e16i	− 2.97962i
$708$	− 5.48442e14i	− 0.115864i
$709$	−6.21323e15	−1.30246	−0.651228	−	0.758882i	$-0.725746\pi$
$709$	−6.21323e15	−1.30246	−0.651228	+	0.758882i	$0.725746\pi$
$710$	0	0
$711$	3.08834e15	0.637445
$712$	− 7.37470e14i	− 0.151044i
$713$	1.18406e15i	0.240647i
$714$	−3.61285e14	−0.0728635
$715$	0	0
$716$	1.38592e14	0.0275244
$717$	3.15905e15i	0.622588i
$718$	5.91883e15i	1.15758i
$719$	−7.12825e15	−1.38348	−0.691742	−	0.722145i	$-0.743156\pi$
$719$	−7.12825e15	−1.38348	−0.691742	+	0.722145i	$0.743156\pi$
$720$	0	0
$721$	−6.58914e15	−1.25946
$722$	− 6.33627e15i	− 1.20193i
$723$	− 2.92877e15i	− 0.551346i
$724$	1.39142e15	0.259953
$725$	0	0
$726$	−1.37578e15	−0.253162
$727$	− 5.59273e15i	− 1.02137i	−0.859767	−	0.510686i	$-0.829391\pi$
$727$	− 5.59273e15i	− 1.02137i	0.859767	−	0.510686i	$-0.170609\pi$
$728$	6.19685e14i	0.112318i
$729$	2.12599e15	0.382437
$730$	0	0
$731$	2.68676e14	0.0476084
$732$	− 1.39926e15i	− 0.246086i
$733$	− 2.75558e15i	− 0.480995i	−0.970650	−	0.240498i	$-0.922689\pi$
$733$	− 2.75558e15i	− 0.480995i	0.970650	−	0.240498i	$-0.0773106\pi$
$734$	5.40669e15	0.936705
$735$	0	0
$736$	−1.34082e15	−0.228846
$737$	1.25688e16i	2.12922i
$738$	− 6.33291e14i	− 0.106486i
$739$	−1.99004e15	−0.332138	−0.166069	−	0.986114i	$-0.553107\pi$
$739$	−1.99004e15	−0.332138	−0.166069	+	0.986114i	$0.553107\pi$
$740$	0	0
$741$	5.55389e14	0.0913266
$742$	1.12018e16i	1.82838i
$743$	− 7.01045e15i	− 1.13581i	−0.823093	−	0.567907i	$-0.807753\pi$
$743$	− 7.01045e15i	− 1.13581i	0.823093	−	0.567907i	$-0.192247\pi$
$744$	−1.37618e14	−0.0221322
$745$	0	0
$746$	−5.98997e15	−0.949208
$747$	− 2.13271e15i	− 0.335481i
$748$	− 7.31218e14i	− 0.114179i
$749$	−1.02611e16	−1.59054
$750$	0	0
$751$	−9.37589e15	−1.43216	−0.716082	−	0.698016i	$-0.754066\pi$
$751$	−9.37589e15	−1.43216	−0.716082	+	0.698016i	$0.754066\pi$
$752$	1.64634e15i	0.249645i
$753$	1.74877e15i	0.263245i
$754$	−5.43423e14	−0.0812074
$755$	0	0
$756$	4.15141e15	0.611400
$757$	− 1.88289e15i	− 0.275294i	−0.990481	−	0.137647i	$-0.956046\pi$
$757$	− 1.88289e15i	− 0.275294i	0.990481	−	0.137647i	$-0.0439540\pi$
$758$	− 2.39236e15i	− 0.347253i
$759$	−4.34534e15	−0.626173
$760$	0	0
$761$	6.71695e15	0.954018	0.477009	−	0.878898i	$-0.341721\pi$
$761$	6.71695e15	0.954018	0.477009	+	0.878898i	$0.341721\pi$
$762$	1.78269e15i	0.251375i
$763$	− 6.27717e15i	− 0.878777i
$764$	−1.13304e15	−0.157482
$765$	0	0
$766$	−4.97667e15	−0.681837
$767$	− 8.34968e14i	− 0.113578i
$768$	− 1.55838e14i	− 0.0210469i
$769$	5.72271e15	0.767374	0.383687	−	0.923463i	$-0.374654\pi$
$769$	5.72271e15	0.767374	0.383687	+	0.923463i	$0.374654\pi$
$770$	0	0
$771$	−1.68890e15	−0.223257
$772$	− 6.31833e15i	− 0.829292i
$773$	− 3.24936e15i	− 0.423458i	−0.977328	−	0.211729i	$-0.932091\pi$
$773$	− 3.24936e15i	− 0.423458i	0.977328	−	0.211729i	$-0.0679094\pi$
$774$	−1.45085e15	−0.187735
$775$	0	0
$776$	−3.57094e15	−0.455557
$777$	6.56157e14i	0.0831174i
$778$	− 2.80463e15i	− 0.352767i
$779$	2.23461e15	0.279091
$780$	0	0
$781$	7.93865e15	0.977611
$782$	− 1.19012e15i	− 0.145530i
$783$	3.64051e15i	0.442051i
$784$	−5.60757e15	−0.676140
$785$	0	0
$786$	1.19363e15	0.141920
$787$	− 1.29754e16i	− 1.53201i	−0.642838	−	0.766003i	$-0.722243\pi$
$787$	− 1.29754e16i	− 1.53201i	0.642838	−	0.766003i	$-0.277757\pi$
$788$	2.05140e15i	0.240523i
$789$	2.19185e14	0.0255204
$790$	0	0
$791$	1.87294e16	2.15057
$792$	3.94857e15i	0.450247i
$793$	− 2.13028e15i	− 0.241232i
$794$	−1.66325e15	−0.187045
$795$	0	0
$796$	−7.92470e15	−0.878944
$797$	1.21467e16i	1.33795i	0.743286	+	0.668974i	$0.233266\pi$
$797$	1.21467e16i	1.33795i	−0.743286	+	0.668974i	$0.766734\pi$
$798$	6.88401e15i	0.753053i
$799$	−1.46130e15	−0.158757
$800$	0	0
$801$	−3.53473e15	−0.378771
$802$	1.09041e16i	1.16046i
$803$	3.27724e15i	0.346395i
$804$	−2.37760e15	−0.249592
$805$	0	0
$806$	−2.09515e14	−0.0216957
$807$	2.02956e15i	0.208736i
$808$	5.35808e15i	0.547327i
$809$	−3.06008e15	−0.310468	−0.155234	−	0.987878i	$-0.549613\pi$
$809$	−3.06008e15	−0.310468	−0.155234	+	0.987878i	$0.549613\pi$
$810$	0	0
$811$	1.63765e15	0.163910	0.0819552	−	0.996636i	$-0.473884\pi$
$811$	1.63765e15	0.163910	0.0819552	+	0.996636i	$0.473884\pi$
$812$	− 6.73568e15i	− 0.669612i
$813$	− 3.21419e15i	− 0.317376i
$814$	−1.32802e15	−0.130248
$815$	0	0
$816$	1.38323e14	0.0133843
$817$	− 5.11941e15i	− 0.492038i
$818$	1.07083e16i	1.02230i
$819$	2.97018e15	0.281657
$820$	0	0
$821$	1.07044e15	0.100156	0.0500778	−	0.998745i	$-0.484053\pi$
$821$	1.07044e15	0.100156	0.0500778	+	0.998745i	$0.484053\pi$
$822$	2.36082e15i	0.219416i
$823$	− 3.24575e15i	− 0.299651i	−0.988712	−	0.149825i	$-0.952129\pi$
$823$	− 3.24575e15i	− 0.299651i	0.988712	−	0.149825i	$-0.0478712\pi$
$824$	2.52273e15	0.231351
$825$	0	0
$826$	1.03494e16	0.936533
$827$	2.05440e16i	1.84674i	0.383913	+	0.923369i	$0.374576\pi$
$827$	2.05440e16i	1.84674i	−0.383913	+	0.923369i	$0.625424\pi$
$828$	6.42662e15i	0.573873i
$829$	−5.85872e15	−0.519700	−0.259850	−	0.965649i	$-0.583673\pi$
$829$	−5.85872e15	−0.519700	−0.259850	+	0.965649i	$0.583673\pi$
$830$	0	0
$831$	−1.31144e15	−0.114800
$832$	− 2.37254e14i	− 0.0206317i
$833$	− 4.97730e15i	− 0.429978i
$834$	3.09627e15	0.265721
$835$	0	0
$836$	−1.39328e16	−1.18006
$837$	1.40359e15i	0.118100i
$838$	4.41057e15i	0.368683i
$839$	2.25383e16	1.87168	0.935838	−	0.352430i	$-0.114645\pi$
$839$	2.25383e16	1.87168	0.935838	+	0.352430i	$0.114645\pi$
$840$	0	0
$841$	−6.29376e15	−0.515861
$842$	− 1.39345e16i	− 1.13468i
$843$	3.21043e14i	0.0259724i
$844$	9.23524e15	0.742275
$845$	0	0
$846$	7.89100e15	0.626030
$847$	− 2.59617e16i	− 2.04632i
$848$	− 4.28874e15i	− 0.335856i
$849$	−5.10823e15	−0.397446
$850$	0	0
$851$	−2.16146e15	−0.166010
$852$	1.50173e15i	0.114597i
$853$	1.84471e16i	1.39865i	0.714803	+	0.699326i	$0.246516\pi$
$853$	1.84471e16i	1.39865i	−0.714803	+	0.699326i	$0.753484\pi$
$854$	2.64047e16	1.98913
$855$	0	0
$856$	3.92860e15	0.292168
$857$	1.53957e15i	0.113764i	0.998381	+	0.0568820i	$0.0181159\pi$
$857$	1.53957e15i	0.113764i	−0.998381	+	0.0568820i	$0.981884\pi$
$858$	− 7.68894e14i	− 0.0564529i
$859$	2.14973e16	1.56827	0.784136	−	0.620589i	$-0.213107\pi$
$859$	2.14973e16	1.56827	0.784136	+	0.620589i	$0.213107\pi$
$860$	0	0
$861$	−1.52853e15	−0.110092
$862$	− 4.83792e15i	− 0.346233i
$863$	− 1.16923e16i	− 0.831462i	−0.909488	−	0.415731i	$-0.863526\pi$
$863$	− 1.16923e16i	− 0.831462i	0.909488	−	0.415731i	$-0.136474\pi$
$864$	−1.58942e15	−0.112308
$865$	0	0
$866$	−1.20014e15	−0.0837306
$867$	− 4.73472e15i	− 0.328239i
$868$	− 2.59693e15i	− 0.178896i
$869$	1.50866e16	1.03272
$870$	0	0
$871$	−3.61974e15	−0.244668
$872$	2.40329e15i	0.161423i
$873$	1.71156e16i	1.14239i
$874$	−2.26767e16	−1.50407
$875$	0	0
$876$	−6.19946e14	−0.0406052
$877$	− 1.13573e16i	− 0.739225i	−0.929186	−	0.369613i	$-0.879490\pi$
$877$	− 1.13573e16i	− 0.739225i	0.929186	−	0.369613i	$-0.120510\pi$
$878$	− 5.18188e15i	− 0.335172i
$879$	2.31520e15	0.148816
$880$	0	0
$881$	1.12847e16	0.716346	0.358173	−	0.933655i	$-0.383400\pi$
$881$	1.12847e16	0.716346	0.358173	+	0.933655i	$0.383400\pi$
$882$	2.68773e16i	1.69554i
$883$	− 5.36286e15i	− 0.336211i	−0.985769	−	0.168106i	$-0.946235\pi$
$883$	− 5.36286e15i	− 0.336211i	0.985769	−	0.168106i	$-0.0537650\pi$
$884$	2.10587e14	0.0131203
$885$	0	0
$886$	−4.13894e15	−0.254685
$887$	− 8.12685e15i	− 0.496984i	−0.968634	−	0.248492i	$-0.920065\pi$
$887$	− 8.12685e15i	− 0.496984i	0.968634	−	0.248492i	$-0.0799350\pi$
$888$	− 2.51218e14i	− 0.0152679i
$889$	−3.36402e16	−2.03188
$890$	0	0
$891$	1.61953e16	0.966190
$892$	− 2.11579e15i	− 0.125448i
$893$	2.78439e16i	1.64077i
$894$	4.60653e15	0.269785
$895$	0	0
$896$	2.94075e15	0.170123
$897$	− 1.25144e15i	− 0.0719534i
$898$	4.16815e14i	0.0238190i
$899$	2.27733e15	0.129345
$900$	0	0
$901$	3.80670e15	0.213581
$902$	− 3.09364e15i	− 0.172518i
$903$	3.50181e15i	0.194093i
$904$	−7.17077e15	−0.395039
$905$	0	0
$906$	−2.42213e15	−0.131823
$907$	1.18191e16i	0.639361i	0.947525	+	0.319680i	$0.103576\pi$
$907$	1.18191e16i	0.639361i	−0.947525	+	0.319680i	$0.896424\pi$
$908$	− 1.60570e16i	− 0.863359i
$909$	2.56815e16	1.37252
$910$	0	0
$911$	1.50443e16	0.794366	0.397183	−	0.917739i	$-0.369988\pi$
$911$	1.50443e16	0.794366	0.397183	+	0.917739i	$0.369988\pi$
$912$	− 2.63563e15i	− 0.138329i
$913$	− 1.04183e16i	− 0.543513i
$914$	−1.10632e16	−0.573688
$915$	0	0
$916$	6.16764e15	0.316005
$917$	2.25243e16i	1.14715i
$918$	− 1.41077e15i	− 0.0714203i
$919$	3.98208e15	0.200389	0.100195	−	0.994968i	$-0.468053\pi$
$919$	3.98208e15	0.200389	0.100195	+	0.994968i	$0.468053\pi$
$920$	0	0
$921$	7.95585e15	0.395602
$922$	− 9.50225e15i	− 0.469685i
$923$	2.28629e15i	0.112337i
$924$	9.53038e15	0.465494
$925$	0	0
$926$	1.45682e16	0.703148
$927$	− 1.20916e16i	− 0.580154i
$928$	2.57884e15i	0.123001i
$929$	−4.06546e16	−1.92763	−0.963814	−	0.266575i	$-0.914108\pi$
$929$	−4.06546e16	−1.92763	−0.963814	+	0.266575i	$0.914108\pi$
$930$	0	0
$931$	−9.48384e16	−4.44387
$932$	− 1.46564e16i	− 0.682718i
$933$	− 9.81282e15i	− 0.454407i
$934$	1.17280e16	0.539902
$935$	0	0
$936$	−1.13717e15	−0.0517378
$937$	1.18458e15i	0.0535792i	0.999641	+	0.0267896i	$0.00852841\pi$
$937$	1.18458e15i	0.0535792i	−0.999641	+	0.0267896i	$0.991472\pi$
$938$	− 4.48664e16i	− 2.01746i
$939$	−4.58966e15	−0.205173
$940$	0	0
$941$	−1.02223e16	−0.451652	−0.225826	−	0.974168i	$-0.572508\pi$
$941$	−1.02223e16	−0.451652	−0.225826	+	0.974168i	$0.572508\pi$
$942$	− 3.30944e15i	− 0.145370i
$943$	− 5.03516e15i	− 0.219887i
$944$	−3.96238e15	−0.172032
$945$	0	0
$946$	−7.08743e15	−0.304150
$947$	1.68932e16i	0.720755i	0.932807	+	0.360378i	$0.117352\pi$
$947$	1.68932e16i	0.720755i	−0.932807	+	0.360378i	$0.882648\pi$
$948$	2.85390e15i	0.121058i
$949$	−9.43829e14	−0.0398042
$950$	0	0
$951$	2.60824e15	0.108731
$952$	2.61021e15i	0.108186i
$953$	− 3.76078e16i	− 1.54977i	−0.632103	−	0.774884i	$-0.717808\pi$
$953$	− 3.76078e16i	− 1.54977i	0.632103	−	0.774884i	$-0.282192\pi$
$954$	−2.05561e16	−0.842219
$955$	0	0
$956$	2.28235e16	0.924407
$957$	8.35751e15i	0.336559i
$958$	1.85081e16i	0.741055i
$959$	−4.45498e16	−1.77355
$960$	0	0
$961$	−2.45305e16	−0.965444
$962$	− 3.82463e14i	− 0.0149667i
$963$	− 1.88300e16i	− 0.732664i
$964$	−2.11598e16	−0.818629
$965$	0	0
$966$	1.55115e16	0.593306
$967$	3.52262e16i	1.33974i	0.742479	+	0.669869i	$0.233650\pi$
$967$	3.52262e16i	1.33974i	−0.742479	+	0.669869i	$0.766350\pi$
$968$	9.93976e15i	0.375890i
$969$	2.33939e15	0.0879674
$970$	0	0
$971$	3.56726e16	1.32626	0.663131	−	0.748504i	$-0.269227\pi$
$971$	3.56726e16	1.32626	0.663131	+	0.748504i	$0.269227\pi$
$972$	1.16562e16i	0.430915i
$973$	5.84281e16i	2.14784i
$974$	1.52065e16	0.555847
$975$	0	0
$976$	−1.01094e16	−0.365384
$977$	4.79343e16i	1.72277i	0.507957	+	0.861383i	$0.330401\pi$
$977$	4.79343e16i	1.72277i	−0.507957	+	0.861383i	$0.669599\pi$
$978$	9.62298e14i	0.0343911i
$979$	−1.72672e16	−0.613646
$980$	0	0
$981$	1.15191e16	0.404798
$982$	− 2.43085e16i	− 0.849464i
$983$	2.05131e16i	0.712833i	0.934327	+	0.356416i	$0.116002\pi$
$983$	2.05131e16i	0.712833i	−0.934327	+	0.356416i	$0.883998\pi$
$984$	5.85217e14	0.0202229
$985$	0	0
$986$	−2.28898e15	−0.0782203
$987$	− 1.90460e16i	− 0.647230i
$988$	− 4.01257e15i	− 0.135600i
$989$	−1.15354e16	−0.387661
$990$	0	0
$991$	−2.11314e16	−0.702301	−0.351150	−	0.936319i	$-0.614209\pi$
$991$	−2.11314e16	−0.702301	−0.351150	+	0.936319i	$0.614209\pi$
$992$	9.94265e14i	0.0328615i
$993$	− 9.56179e15i	− 0.314281i
$994$	−2.83384e16	−0.926298
$995$	0	0
$996$	1.97081e15	0.0637116
$997$	− 4.57566e16i	− 1.47106i	−0.677491	−	0.735531i	$-0.736933\pi$
$997$	− 4.57566e16i	− 1.47106i	0.677491	−	0.735531i	$-0.263067\pi$
$998$	− 1.57133e16i	− 0.502398i
$999$	−2.56220e15	−0.0814711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.12.b.f.49.1		4
5.2	odd	4		10.12.a.d.1.1	✓	2
5.3	odd	4		50.12.a.f.1.2		2
5.4	even	2	inner	50.12.b.f.49.4		4
15.2	even	4		90.12.a.l.1.2		2
20.7	even	4		80.12.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.12.a.d.1.1	✓	2	5.2	odd	4
50.12.a.f.1.2		2	5.3	odd	4
50.12.b.f.49.1		4	1.1	even	1	trivial
50.12.b.f.49.4		4	5.4	even	2	inner
80.12.a.g.1.2		2	20.7	even	4
90.12.a.l.1.2		2	15.2	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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-32.0000i q^{2} -141.734i q^{3} -1024.00 q^{4} -4535.49 q^{6} -85586.9i q^{7} +32768.0i q^{8} +157058. q^{9} +767235. q^{11} +145136. i q^{12} +220960. i q^{13} -2.73878e6 q^{14} +1.04858e6 q^{16} +930719. i q^{17} -5.02587e6i q^{18} +1.77341e7 q^{19} -1.21306e7 q^{21} -2.45515e7i q^{22} -3.99596e7i q^{23} +4.64434e6 q^{24} +7.07072e6 q^{26} -4.73683e7i q^{27} +8.76410e7i q^{28} -7.68554e7 q^{29} -2.96314e7 q^{31} -3.35544e7i q^{32} -1.08743e8i q^{33} +2.97830e7 q^{34} -1.60828e8 q^{36} -5.40911e7i q^{37} -5.67491e8i q^{38} +3.13176e7 q^{39} +1.26006e8 q^{41} +3.88179e8i q^{42} -2.88676e8i q^{43} -7.85648e8 q^{44} -1.27871e9 q^{46} +1.57008e9i q^{47} -1.48619e8i q^{48} -5.34780e9 q^{49} +1.31915e8 q^{51} -2.26263e8i q^{52} -4.09006e9i q^{53} -1.51579e9 q^{54} +2.80451e9 q^{56} -2.51353e9i q^{57} +2.45937e9i q^{58} -3.77882e9 q^{59} -9.64103e9 q^{61} +9.48205e8i q^{62} -1.34422e10i q^{63} -1.07374e9 q^{64} -3.47979e9 q^{66} +1.63819e10i q^{67} -9.53056e8i q^{68} -5.66364e9 q^{69} +1.03471e10 q^{71} +5.14649e9i q^{72} +4.27149e9i q^{73} -1.73091e9 q^{74} -1.81597e10 q^{76} -6.56653e10i q^{77} -1.00216e9i q^{78} +1.96636e10 q^{79} +2.11087e10 q^{81} -4.03220e9i q^{82} -1.35791e10i q^{83} +1.24217e10 q^{84} -9.23762e9 q^{86} +1.08930e10i q^{87} +2.51407e10i q^{88} -2.25058e10 q^{89} +1.89113e10 q^{91} +4.09186e10i q^{92} +4.19978e9i q^{93} +5.02424e10 q^{94} -4.75581e9 q^{96} +1.08976e11i q^{97} +1.71130e11i q^{98} +1.20501e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4096 q^{4} + 38656 q^{6} - 443828 q^{9} + 843168 q^{11} - 901888 q^{14} + 4194304 q^{16} + 57794800 q^{19} - 130893632 q^{21} - 39583744 q^{24} + 110753536 q^{26} - 116452440 q^{29} + 82826768 q^{31}+ \cdots + 502985449824 q^{99}+O(q^{100}) \) 4 * q - 4096 * q^4 + 38656 * q^6 - 443828 * q^9 + 843168 * q^11 - 901888 * q^14 + 4194304 * q^16 + 57794800 * q^19 - 130893632 * q^21 - 39583744 * q^24 + 110753536 * q^26 - 116452440 * q^29 + 82826768 * q^31 + 404712192 * q^34 + 454479872 * q^36 - 2188831696 * q^39 - 1570542072 * q^41 - 863404032 * q^44 - 2895108864 * q^46 - 16963997892 * q^49 - 7779524112 * q^51 - 12664092160 * q^54 + 923533312 * q^56 - 17354204880 * q^59 + 2230997528 * q^61 - 4294967296 * q^64 - 23456437248 * q^66 - 3457605696 * q^69 + 27599665968 * q^71 - 24144348928 * q^74 - 59181875200 * q^76 + 25273860640 * q^79 + 132427867844 * q^81 + 134035079168 * q^84 - 92991183104 * q^86 - 26417480040 * q^89 - 178029806192 * q^91 + 82440175872 * q^94 + 40533753856 * q^96 + 502985449824 * q^99

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