LMFDB - Embedded newform 50.8.b.a.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [50,8,Mod(49,50)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	50.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-128,0,-1392] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.6192512742$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	50.49
Dual form			50.8.b.a.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.00000i q^{2} +87.0000i q^{3} -64.0000 q^{4} -696.000 q^{6} -1366.00i q^{7} -512.000i q^{8} -5382.00 q^{9} -1083.00 q^{11} -5568.00i q^{12} -5468.00i q^{13} +10928.0 q^{14} +4096.00 q^{16} +25269.0i q^{17} -43056.0i q^{18} -33485.0 q^{19} +118842. q^{21} -8664.00i q^{22} -5838.00i q^{23} +44544.0 q^{24} +43744.0 q^{26} -277965. i q^{27} +87424.0i q^{28} -125280. q^{29} -73798.0 q^{31} +32768.0i q^{32} -94221.0i q^{33} -202152. q^{34} +344448. q^{36} -395926. i q^{37} -267880. i q^{38} +475716. q^{39} -22683.0 q^{41} +950736. i q^{42} -100148. i q^{43} +69312.0 q^{44} +46704.0 q^{46} +1.14524e6i q^{47} +356352. i q^{48} -1.04241e6 q^{49} -2.19840e6 q^{51} +349952. i q^{52} +354882. i q^{53} +2.22372e6 q^{54} -699392. q^{56} -2.91320e6i q^{57} -1.00224e6i q^{58} -1.09836e6 q^{59} -422998. q^{61} -590384. i q^{62} +7.35181e6i q^{63} -262144. q^{64} +753768. q^{66} +2.55858e6i q^{67} -1.61722e6i q^{68} +507906. q^{69} -2.28743e6 q^{71} +2.75558e6i q^{72} -6.37244e6i q^{73} +3.16741e6 q^{74} +2.14304e6 q^{76} +1.47938e6i q^{77} +3.80573e6i q^{78} +2.01925e6 q^{79} +1.24125e7 q^{81} -181464. i q^{82} -7.97298e6i q^{83} -7.60589e6 q^{84} +801184. q^{86} -1.08994e7i q^{87} +554496. i q^{88} -2.18594e6 q^{89} -7.46929e6 q^{91} +373632. i q^{92} -6.42043e6i q^{93} -9.16195e6 q^{94} -2.85082e6 q^{96} -5.82365e6i q^{97} -8.33930e6i q^{98} +5.82871e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 1392 q^{6} - 10764 q^{9} - 2166 q^{11} + 21856 q^{14} + 8192 q^{16} - 66970 q^{19} + 237684 q^{21} + 89088 q^{24} + 87488 q^{26} - 250560 q^{29} - 147596 q^{31} - 404304 q^{34} + 688896 q^{36}+ \cdots + 11657412 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 1392 * q^6 - 10764 * q^9 - 2166 * q^11 + 21856 * q^14 + 8192 * q^16 - 66970 * q^19 + 237684 * q^21 + 89088 * q^24 + 87488 * q^26 - 250560 * q^29 - 147596 * q^31 - 404304 * q^34 + 688896 * q^36 + 951432 * q^39 - 45366 * q^41 + 138624 * q^44 + 93408 * q^46 - 2084826 * q^49 - 4396806 * q^51 + 4447440 * q^54 - 1398784 * q^56 - 2196720 * q^59 - 845996 * q^61 - 524288 * q^64 + 1507536 * q^66 + 1015812 * q^69 - 4574856 * q^71 + 6334816 * q^74 + 4286080 * q^76 + 4038500 * q^79 + 24825042 * q^81 - 15211776 * q^84 + 1602368 * q^86 - 4371870 * q^89 - 14938576 * q^91 - 18323904 * q^94 - 5701632 * q^96 + 11657412 * 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 $

$2$	8.00000i	0.707107i
$2$	8.00000i	0.707107i
$3$	87.0000i	1.86035i	0.367115	+	0.930175i	$0.380345\pi$
$3$	87.0000i	1.86035i	−0.367115	+	0.930175i	$0.619655\pi$
$4$	−64.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−696.000	−1.31547
$7$	− 1366.00i	− 1.50525i	−0.658452	−	0.752623i	$-0.728788\pi$
$7$	− 1366.00i	− 1.50525i	0.658452	−	0.752623i	$-0.271212\pi$
$8$	− 512.000i	− 0.353553i
$9$	−5382.00	−2.46091
$10$	0	0
$11$	−1083.00	−0.245332	−0.122666	−	0.992448i	$-0.539144\pi$
$11$	−1083.00	−0.245332	−0.122666	+	0.992448i	$0.539144\pi$
$12$	− 5568.00i	− 0.930175i
$13$	− 5468.00i	− 0.690282i	−0.938551	−	0.345141i	$-0.887831\pi$
$13$	− 5468.00i	− 0.690282i	0.938551	−	0.345141i	$-0.112169\pi$
$14$	10928.0	1.06437
$15$	0	0
$16$	4096.00	0.250000
$17$	25269.0i	1.24743i	0.781651	+	0.623716i	$0.214378\pi$
$17$	25269.0i	1.24743i	−0.781651	+	0.623716i	$0.785622\pi$
$18$	− 43056.0i	− 1.74012i
$19$	−33485.0	−1.11999	−0.559993	−	0.828497i	$-0.689196\pi$
$19$	−33485.0	−1.11999	−0.559993	+	0.828497i	$0.689196\pi$
$20$	0	0
$21$	118842.	2.80029
$22$	− 8664.00i	− 0.173476i
$23$	− 5838.00i	− 0.100050i	−0.998748	−	0.0500250i	$-0.984070\pi$
$23$	− 5838.00i	− 0.100050i	0.998748	−	0.0500250i	$-0.0159301\pi$
$24$	44544.0	0.657733
$25$	0	0
$26$	43744.0	0.488103
$27$	− 277965.i	− 2.71780i
$28$	87424.0i	0.752623i
$29$	−125280.	−0.953869	−0.476935	−	0.878939i	$-0.658252\pi$
$29$	−125280.	−0.953869	−0.476935	+	0.878939i	$0.658252\pi$
$30$	0	0
$31$	−73798.0	−0.444917	−0.222458	−	0.974942i	$-0.571408\pi$
$31$	−73798.0	−0.444917	−0.222458	+	0.974942i	$0.571408\pi$
$32$	32768.0i	0.176777i
$33$	− 94221.0i	− 0.456403i
$34$	−202152.	−0.882068
$35$	0	0
$36$	344448.	1.23045
$37$	− 395926.i	− 1.28501i	−0.766280	−	0.642507i	$-0.777894\pi$
$37$	− 395926.i	− 1.28501i	0.766280	−	0.642507i	$-0.222106\pi$
$38$	− 267880.i	− 0.791950i
$39$	475716.	1.28417
$40$	0	0
$41$	−22683.0	−0.0513993	−0.0256996	−	0.999670i	$-0.508181\pi$
$41$	−22683.0	−0.0513993	−0.0256996	+	0.999670i	$0.508181\pi$
$42$	950736.i	1.98010i
$43$	− 100148.i	− 0.192089i	−0.995377	−	0.0960445i	$-0.969381\pi$
$43$	− 100148.i	− 0.192089i	0.995377	−	0.0960445i	$-0.0306191\pi$
$44$	69312.0	0.122666
$45$	0	0
$46$	46704.0	0.0707460
$47$	1.14524e6i	1.60900i	0.593954	+	0.804499i	$0.297566\pi$
$47$	1.14524e6i	1.60900i	−0.593954	+	0.804499i	$0.702434\pi$
$48$	356352.i	0.465088i
$49$	−1.04241e6	−1.26577
$50$	0	0
$51$	−2.19840e6	−2.32066
$52$	349952.i	0.345141i
$53$	354882.i	0.327430i	0.986508	+	0.163715i	$0.0523477\pi$
$53$	354882.i	0.327430i	−0.986508	+	0.163715i	$0.947652\pi$
$54$	2.22372e6	1.92177
$55$	0	0
$56$	−699392.	−0.532185
$57$	− 2.91320e6i	− 2.08357i
$58$	− 1.00224e6i	− 0.674487i
$59$	−1.09836e6	−0.696246	−0.348123	−	0.937449i	$-0.613181\pi$
$59$	−1.09836e6	−0.696246	−0.348123	+	0.937449i	$0.613181\pi$
$60$	0	0
$61$	−422998.	−0.238607	−0.119304	−	0.992858i	$-0.538066\pi$
$61$	−422998.	−0.238607	−0.119304	+	0.992858i	$0.538066\pi$
$62$	− 590384.i	− 0.314604i
$63$	7.35181e6i	3.70427i
$64$	−262144.	−0.125000
$65$	0	0
$66$	753768.	0.322726
$67$	2.55858e6i	1.03929i	0.854382	+	0.519645i	$0.173936\pi$
$67$	2.55858e6i	1.03929i	−0.854382	+	0.519645i	$0.826064\pi$
$68$	− 1.61722e6i	− 0.623716i
$69$	507906.	0.186128
$70$	0	0
$71$	−2.28743e6	−0.758478	−0.379239	−	0.925299i	$-0.623814\pi$
$71$	−2.28743e6	−0.758478	−0.379239	+	0.925299i	$0.623814\pi$
$72$	2.75558e6i	0.870061i
$73$	− 6.37244e6i	− 1.91724i	−0.284693	−	0.958619i	$-0.591892\pi$
$73$	− 6.37244e6i	− 1.91724i	0.284693	−	0.958619i	$-0.408108\pi$
$74$	3.16741e6	0.908642
$75$	0	0
$76$	2.14304e6	0.559993
$77$	1.47938e6i	0.369285i
$78$	3.80573e6i	0.908043i
$79$	2.01925e6	0.460782	0.230391	−	0.973098i	$-0.426000\pi$
$79$	2.01925e6	0.460782	0.230391	+	0.973098i	$0.426000\pi$
$80$	0	0
$81$	1.24125e7	2.59515
$82$	− 181464.i	− 0.0363448i
$83$	− 7.97298e6i	− 1.53055i	−0.643703	−	0.765275i	$-0.722603\pi$
$83$	− 7.97298e6i	− 1.53055i	0.643703	−	0.765275i	$-0.277397\pi$
$84$	−7.60589e6	−1.40014
$85$	0	0
$86$	801184.	0.135827
$87$	− 1.08994e7i	− 1.77453i
$88$	554496.i	0.0867379i
$89$	−2.18594e6	−0.328679	−0.164340	−	0.986404i	$-0.552549\pi$
$89$	−2.18594e6	−0.328679	−0.164340	+	0.986404i	$0.552549\pi$
$90$	0	0
$91$	−7.46929e6	−1.03904
$92$	373632.i	0.0500250i
$93$	− 6.42043e6i	− 0.827701i
$94$	−9.16195e6	−1.13773
$95$	0	0
$96$	−2.85082e6	−0.328867
$97$	− 5.82365e6i	− 0.647879i	−0.946078	−	0.323939i	$-0.894993\pi$
$97$	− 5.82365e6i	− 0.647879i	0.946078	−	0.323939i	$-0.105007\pi$
$98$	− 8.33930e6i	− 0.895032i
$99$	5.82871e6	0.603739
$100$	0	0
$101$	433062.	0.0418240	0.0209120	−	0.999781i	$-0.493343\pi$
$101$	433062.	0.0418240	0.0209120	+	0.999781i	$0.493343\pi$
$102$	− 1.75872e7i	− 1.64096i
$103$	1.04295e7i	0.940444i	0.882548	+	0.470222i	$0.155826\pi$
$103$	1.04295e7i	0.940444i	−0.882548	+	0.470222i	$0.844174\pi$
$104$	−2.79962e6	−0.244052
$105$	0	0
$106$	−2.83906e6	−0.231528
$107$	2.13164e7i	1.68217i	0.540901	+	0.841086i	$0.318083\pi$
$107$	2.13164e7i	1.68217i	−0.540901	+	0.841086i	$0.681917\pi$
$108$	1.77898e7i	1.35890i
$109$	−1.60304e7	−1.18564	−0.592819	−	0.805335i	$-0.701985\pi$
$109$	−1.60304e7	−1.18564	−0.592819	+	0.805335i	$0.701985\pi$
$110$	0	0
$111$	3.44456e7	2.39058
$112$	− 5.59514e6i	− 0.376312i
$113$	− 905463.i	− 0.0590332i	−0.999564	−	0.0295166i	$-0.990603\pi$
$113$	− 905463.i	− 0.0590332i	0.999564	−	0.0295166i	$-0.00939679\pi$
$114$	2.33056e7	1.47330
$115$	0	0
$116$	8.01792e6	0.476935
$117$	2.94288e7i	1.69872i
$118$	− 8.78688e6i	− 0.492320i
$119$	3.45175e7	1.87769
$120$	0	0
$121$	−1.83143e7	−0.939812
$122$	− 3.38398e6i	− 0.168721i
$123$	− 1.97342e6i	− 0.0956207i
$124$	4.72307e6	0.222458
$125$	0	0
$126$	−5.88145e7	−2.61931
$127$	1.60352e7i	0.694642i	0.937746	+	0.347321i	$0.112909\pi$
$127$	1.60352e7i	0.694642i	−0.937746	+	0.347321i	$0.887091\pi$
$128$	− 2.09715e6i	− 0.0883883i
$129$	8.71288e6	0.357353
$130$	0	0
$131$	3.83077e6	0.148880	0.0744401	−	0.997225i	$-0.476283\pi$
$131$	3.83077e6	0.148880	0.0744401	+	0.997225i	$0.476283\pi$
$132$	6.03014e6i	0.228202i
$133$	4.57405e7i	1.68586i
$134$	−2.04686e7	−0.734889
$135$	0	0
$136$	1.29377e7	0.441034
$137$	9.69343e6i	0.322074i	0.986948	+	0.161037i	$0.0514838\pi$
$137$	9.69343e6i	0.322074i	−0.986948	+	0.161037i	$0.948516\pi$
$138$	4.06325e6i	0.131612i
$139$	−4.49406e7	−1.41934	−0.709671	−	0.704533i	$-0.751156\pi$
$139$	−4.49406e7	−1.41934	−0.709671	+	0.704533i	$0.751156\pi$
$140$	0	0
$141$	−9.96362e7	−2.99330
$142$	− 1.82994e7i	− 0.536325i
$143$	5.92184e6i	0.169348i
$144$	−2.20447e7	−0.615226
$145$	0	0
$146$	5.09795e7	1.35569
$147$	− 9.06899e7i	− 2.35477i
$148$	2.53393e7i	0.642507i
$149$	4.94208e7	1.22393	0.611967	−	0.790883i	$-0.290379\pi$
$149$	4.94208e7	1.22393	0.611967	+	0.790883i	$0.290379\pi$
$150$	0	0
$151$	2.01078e7	0.475275	0.237638	−	0.971354i	$-0.423627\pi$
$151$	2.01078e7	0.475275	0.237638	+	0.971354i	$0.423627\pi$
$152$	1.71443e7i	0.395975i
$153$	− 1.35998e8i	− 3.06981i
$154$	−1.18350e7	−0.261124
$155$	0	0
$156$	−3.04458e7	−0.642084
$157$	7.86694e7i	1.62240i	0.584771	+	0.811199i	$0.301185\pi$
$157$	7.86694e7i	1.62240i	−0.584771	+	0.811199i	$0.698815\pi$
$158$	1.61540e7i	0.325822i
$159$	−3.08747e7	−0.609135
$160$	0	0
$161$	−7.97471e6	−0.150600
$162$	9.93002e7i	1.83505i
$163$	6.99749e7i	1.26557i	0.774328	+	0.632784i	$0.218088\pi$
$163$	6.99749e7i	1.26557i	−0.774328	+	0.632784i	$0.781912\pi$
$164$	1.45171e6	0.0256996
$165$	0	0
$166$	6.37839e7	1.08226
$167$	− 8.38918e7i	− 1.39384i	−0.717151	−	0.696918i	$-0.754554\pi$
$167$	− 8.38918e7i	− 1.39384i	0.717151	−	0.696918i	$-0.245446\pi$
$168$	− 6.08471e7i	− 0.990051i
$169$	3.28495e7	0.523510
$170$	0	0
$171$	1.80216e8	2.75618
$172$	6.40947e6i	0.0960445i
$173$	2.73668e7i	0.401849i	0.979607	+	0.200924i	$0.0643945\pi$
$173$	2.73668e7i	0.401849i	−0.979607	+	0.200924i	$0.935605\pi$
$174$	8.71949e7	1.25478
$175$	0	0
$176$	−4.43597e6	−0.0613330
$177$	− 9.55573e7i	− 1.29526i
$178$	− 1.74875e7i	− 0.232411i
$179$	7.97634e7	1.03948	0.519742	−	0.854323i	$-0.326028\pi$
$179$	7.97634e7	1.03948	0.519742	+	0.854323i	$0.326028\pi$
$180$	0	0
$181$	5.28714e7	0.662744	0.331372	−	0.943500i	$-0.392488\pi$
$181$	5.28714e7	0.662744	0.331372	+	0.943500i	$0.392488\pi$
$182$	− 5.97543e7i	− 0.734716i
$183$	− 3.68008e7i	− 0.443893i
$184$	−2.98906e6	−0.0353730
$185$	0	0
$186$	5.13634e7	0.585273
$187$	− 2.73663e7i	− 0.306035i
$188$	− 7.32956e7i	− 0.804499i
$189$	−3.79700e8	−4.09095
$190$	0	0
$191$	−1.03225e8	−1.07194	−0.535969	−	0.844238i	$-0.680054\pi$
$191$	−1.03225e8	−1.07194	−0.535969	+	0.844238i	$0.680054\pi$
$192$	− 2.28065e7i	− 0.232544i
$193$	4.99651e7i	0.500283i	0.968209	+	0.250142i	$0.0804772\pi$
$193$	4.99651e7i	0.500283i	−0.968209	+	0.250142i	$0.919523\pi$
$194$	4.65892e7	0.458120
$195$	0	0
$196$	6.67144e7	0.632883
$197$	− 3.33856e7i	− 0.311120i	−0.987826	−	0.155560i	$-0.950282\pi$
$197$	− 3.33856e7i	− 0.311120i	0.987826	−	0.155560i	$-0.0497182\pi$
$198$	4.66296e7i	0.426908i
$199$	−9.25107e7	−0.832158	−0.416079	−	0.909328i	$-0.636596\pi$
$199$	−9.25107e7	−0.832158	−0.416079	+	0.909328i	$0.636596\pi$
$200$	0	0
$201$	−2.22596e8	−1.93345
$202$	3.46450e6i	0.0295740i
$203$	1.71132e8i	1.43581i
$204$	1.40698e8	1.16033
$205$	0	0
$206$	−8.34359e7	−0.664994
$207$	3.14201e7i	0.246213i
$208$	− 2.23969e7i	− 0.172571i
$209$	3.62643e7	0.274768
$210$	0	0
$211$	5.61153e7	0.411238	0.205619	−	0.978632i	$-0.434079\pi$
$211$	5.61153e7	0.411238	0.205619	+	0.978632i	$0.434079\pi$
$212$	− 2.27124e7i	− 0.163715i
$213$	− 1.99006e8i	− 1.41104i
$214$	−1.70531e8	−1.18948
$215$	0	0
$216$	−1.42318e8	−0.960886
$217$	1.00808e8i	0.669709i
$218$	− 1.28243e8i	− 0.838373i
$219$	5.54403e8	3.56673
$220$	0	0
$221$	1.38171e8	0.861080
$222$	2.75564e8i	1.69039i
$223$	7.79936e7i	0.470968i	0.971878	+	0.235484i	$0.0756676\pi$
$223$	7.79936e7i	0.470968i	−0.971878	+	0.235484i	$0.924332\pi$
$224$	4.47611e7	0.266092
$225$	0	0
$226$	7.24370e6	0.0417428
$227$	1.31342e8i	0.745272i	0.927978	+	0.372636i	$0.121546\pi$
$227$	1.31342e8i	0.745272i	−0.927978	+	0.372636i	$0.878454\pi$
$228$	1.86444e8i	1.04178i
$229$	−2.28442e8	−1.25705	−0.628525	−	0.777790i	$-0.716341\pi$
$229$	−2.28442e8	−1.25705	−0.628525	+	0.777790i	$0.716341\pi$
$230$	0	0
$231$	−1.28706e8	−0.687000
$232$	6.41434e7i	0.337244i
$233$	− 3.77949e8i	− 1.95743i	−0.205216	−	0.978717i	$-0.565790\pi$
$233$	− 3.77949e8i	− 1.95743i	0.205216	−	0.978717i	$-0.434210\pi$
$234$	−2.35430e8	−1.20118
$235$	0	0
$236$	7.02950e7	0.348123
$237$	1.75675e8i	0.857216i
$238$	2.76140e8i	1.32773i
$239$	1.13186e8	0.536291	0.268146	−	0.963378i	$-0.413589\pi$
$239$	1.13186e8	0.536291	0.268146	+	0.963378i	$0.413589\pi$
$240$	0	0
$241$	−2.00236e8	−0.921473	−0.460736	−	0.887537i	$-0.652415\pi$
$241$	−2.00236e8	−0.921473	−0.460736	+	0.887537i	$0.652415\pi$
$242$	− 1.46514e8i	− 0.664548i
$243$	4.71980e8i	2.11009i
$244$	2.70719e7	0.119304
$245$	0	0
$246$	1.57874e7	0.0676140
$247$	1.83096e8i	0.773107i
$248$	3.77846e7i	0.157302i
$249$	6.93650e8	2.84736
$250$	0	0
$251$	−4.62823e8	−1.84738	−0.923692	−	0.383136i	$-0.874844\pi$
$251$	−4.62823e8	−1.84738	−0.923692	+	0.383136i	$0.874844\pi$
$252$	− 4.70516e8i	− 1.85213i
$253$	6.32255e6i	0.0245454i
$254$	−1.28282e8	−0.491186
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	− 4.87751e8i	− 1.79239i	−0.443660	−	0.896195i	$-0.646320\pi$
$257$	− 4.87751e8i	− 1.79239i	0.443660	−	0.896195i	$-0.353680\pi$
$258$	6.97030e7i	0.252687i
$259$	−5.40835e8	−1.93426
$260$	0	0
$261$	6.74257e8	2.34738
$262$	3.06462e7i	0.105274i
$263$	4.01821e8i	1.36203i	0.732268	+	0.681017i	$0.238462\pi$
$263$	4.01821e8i	1.36203i	−0.732268	+	0.681017i	$0.761538\pi$
$264$	−4.82412e7	−0.161363
$265$	0	0
$266$	−3.65924e8	−1.19208
$267$	− 1.90176e8i	− 0.611459i
$268$	− 1.63749e8i	− 0.519645i
$269$	−4.55386e8	−1.42642	−0.713209	−	0.700951i	$-0.752759\pi$
$269$	−4.55386e8	−1.42642	−0.713209	+	0.700951i	$0.752759\pi$
$270$	0	0
$271$	5.08459e8	1.55190	0.775949	−	0.630796i	$-0.217271\pi$
$271$	5.08459e8	1.55190	0.775949	+	0.630796i	$0.217271\pi$
$272$	1.03502e8i	0.311858i
$273$	− 6.49828e8i	− 1.93299i
$274$	−7.75474e7	−0.227741
$275$	0	0
$276$	−3.25060e7	−0.0930640
$277$	6.46827e8i	1.82856i	0.405085	+	0.914279i	$0.367242\pi$
$277$	6.46827e8i	1.82856i	−0.405085	+	0.914279i	$0.632758\pi$
$278$	− 3.59525e8i	− 1.00363i
$279$	3.97181e8	1.09490
$280$	0	0
$281$	1.89546e8	0.509617	0.254808	−	0.966992i	$-0.417988\pi$
$281$	1.89546e8	0.509617	0.254808	+	0.966992i	$0.417988\pi$
$282$	− 7.97090e8i	− 2.11658i
$283$	− 4.63384e8i	− 1.21531i	−0.794200	−	0.607657i	$-0.792110\pi$
$283$	− 4.63384e8i	− 1.21531i	0.794200	−	0.607657i	$-0.207890\pi$
$284$	1.46395e8	0.379239
$285$	0	0
$286$	−4.73748e7	−0.119747
$287$	3.09850e7i	0.0773686i
$288$	− 1.76357e8i	− 0.435031i
$289$	−2.28184e8	−0.556086
$290$	0	0
$291$	5.06657e8	1.20528
$292$	4.07836e8i	0.958619i
$293$	− 1.23173e8i	− 0.286075i	−0.989717	−	0.143038i	$-0.954313\pi$
$293$	− 1.23173e8i	− 0.286075i	0.989717	−	0.143038i	$-0.0456870\pi$
$294$	7.25519e8	1.66507
$295$	0	0
$296$	−2.02714e8	−0.454321
$297$	3.01036e8i	0.666762i
$298$	3.95366e8i	0.865452i
$299$	−3.19222e7	−0.0690627
$300$	0	0
$301$	−1.36802e8	−0.289141
$302$	1.60862e8i	0.336070i
$303$	3.76764e7i	0.0778073i
$304$	−1.37155e8	−0.279997
$305$	0	0
$306$	1.08798e9	2.17068
$307$	5.64202e8i	1.11289i	0.830886	+	0.556443i	$0.187834\pi$
$307$	5.64202e8i	1.11289i	−0.830886	+	0.556443i	$0.812166\pi$
$308$	− 9.46802e7i	− 0.184642i
$309$	−9.07366e8	−1.74956
$310$	0	0
$311$	5.51417e8	1.03949	0.519743	−	0.854323i	$-0.326028\pi$
$311$	5.51417e8	1.03949	0.519743	+	0.854323i	$0.326028\pi$
$312$	− 2.43567e8i	− 0.454022i
$313$	1.71416e8i	0.315970i	0.987442	+	0.157985i	$0.0504997\pi$
$313$	1.71416e8i	0.315970i	−0.987442	+	0.157985i	$0.949500\pi$
$314$	−6.29355e8	−1.14721
$315$	0	0
$316$	−1.29232e8	−0.230391
$317$	− 1.03349e9i	− 1.82222i	−0.412167	−	0.911108i	$-0.635228\pi$
$317$	− 1.03349e9i	− 1.82222i	0.412167	−	0.911108i	$-0.364772\pi$
$318$	− 2.46998e8i	− 0.430723i
$319$	1.35678e8	0.234015
$320$	0	0
$321$	−1.85453e9	−3.12943
$322$	− 6.37977e7i	− 0.106490i
$323$	− 8.46132e8i	− 1.39711i
$324$	−7.94401e8	−1.29757
$325$	0	0
$326$	−5.59799e8	−0.894892
$327$	− 1.39465e9i	− 2.20570i
$328$	1.16137e7i	0.0181724i
$329$	1.56440e9	2.42194
$330$	0	0
$331$	2.41233e7	0.0365627	0.0182813	−	0.999833i	$-0.494181\pi$
$331$	2.41233e7	0.0365627	0.0182813	+	0.999833i	$0.494181\pi$
$332$	5.10271e8i	0.765275i
$333$	2.13087e9i	3.16230i
$334$	6.71134e8	0.985591
$335$	0	0
$336$	4.86777e8	0.700072
$337$	− 8.75080e8i	− 1.24550i	−0.782422	−	0.622749i	$-0.786016\pi$
$337$	− 8.75080e8i	− 1.24550i	0.782422	−	0.622749i	$-0.213984\pi$
$338$	2.62796e8i	0.370178i
$339$	7.87753e7	0.109822
$340$	0	0
$341$	7.99232e7	0.109152
$342$	1.44173e9i	1.94891i
$343$	2.98976e8i	0.400044i
$344$	−5.12758e7	−0.0679137
$345$	0	0
$346$	−2.18934e8	−0.284150
$347$	− 8.32449e7i	− 0.106956i	−0.998569	−	0.0534779i	$-0.982969\pi$
$347$	− 8.32449e7i	− 0.106956i	0.998569	−	0.0534779i	$-0.0170307\pi$
$348$	6.97559e8i	0.887266i
$349$	−3.97607e8	−0.500685	−0.250343	−	0.968157i	$-0.580543\pi$
$349$	−3.97607e8	−0.500685	−0.250343	+	0.968157i	$0.580543\pi$
$350$	0	0
$351$	−1.51991e9	−1.87605
$352$	− 3.54877e7i	− 0.0433690i
$353$	2.93710e8i	0.355392i	0.984085	+	0.177696i	$0.0568643\pi$
$353$	2.93710e8i	0.355392i	−0.984085	+	0.177696i	$0.943136\pi$
$354$	7.64459e8	0.915888
$355$	0	0
$356$	1.39900e8	0.164340
$357$	3.00302e9i	3.49317i
$358$	6.38107e8i	0.735027i
$359$	1.71667e8	0.195820	0.0979098	−	0.995195i	$-0.468784\pi$
$359$	1.71667e8	0.195820	0.0979098	+	0.995195i	$0.468784\pi$
$360$	0	0
$361$	2.27373e8	0.254369
$362$	4.22971e8i	0.468631i
$363$	− 1.59334e9i	− 1.74838i
$364$	4.78034e8	0.519522
$365$	0	0
$366$	2.94407e8	0.313880
$367$	− 9.55463e8i	− 1.00898i	−0.863417	−	0.504490i	$-0.831680\pi$
$367$	− 9.55463e8i	− 1.00898i	0.863417	−	0.504490i	$-0.168320\pi$
$368$	− 2.39124e7i	− 0.0250125i
$369$	1.22080e8	0.126489
$370$	0	0
$371$	4.84769e8	0.492863
$372$	4.10907e8i	0.413851i
$373$	− 3.44673e8i	− 0.343896i	−0.985106	−	0.171948i	$-0.944994\pi$
$373$	− 3.44673e8i	− 0.343896i	0.985106	−	0.171948i	$-0.0550061\pi$
$374$	2.18931e8	0.216399
$375$	0	0
$376$	5.86365e8	0.568867
$377$	6.85031e8i	0.658439i
$378$	− 3.03760e9i	− 2.89274i
$379$	−1.45273e9	−1.37071	−0.685356	−	0.728208i	$-0.740353\pi$
$379$	−1.45273e9	−1.37071	−0.685356	+	0.728208i	$0.740353\pi$
$380$	0	0
$381$	−1.39506e9	−1.29228
$382$	− 8.25803e8i	− 0.757974i
$383$	1.87839e9i	1.70840i	0.519941	+	0.854202i	$0.325954\pi$
$383$	1.87839e9i	1.70840i	−0.519941	+	0.854202i	$0.674046\pi$
$384$	1.82452e8	0.164433
$385$	0	0
$386$	−3.99721e8	−0.353754
$387$	5.38997e8i	0.472713i
$388$	3.72713e8i	0.323939i
$389$	1.12632e9	0.970145	0.485072	−	0.874474i	$-0.338793\pi$
$389$	1.12632e9	0.970145	0.485072	+	0.874474i	$0.338793\pi$
$390$	0	0
$391$	1.47520e8	0.124805
$392$	5.33715e8i	0.447516i
$393$	3.33277e8i	0.276969i
$394$	2.67085e8	0.219995
$395$	0	0
$396$	−3.73037e8	−0.301869
$397$	− 1.45733e9i	− 1.16894i	−0.811415	−	0.584470i	$-0.801302\pi$
$397$	− 1.45733e9i	− 1.16894i	0.811415	−	0.584470i	$-0.198698\pi$
$398$	− 7.40085e8i	− 0.588425i
$399$	−3.97942e9	−3.13628
$400$	0	0
$401$	1.95849e9	1.51676	0.758380	−	0.651813i	$-0.225991\pi$
$401$	1.95849e9	1.51676	0.758380	+	0.651813i	$0.225991\pi$
$402$	− 1.78077e9i	− 1.36715i
$403$	4.03527e8i	0.307118i
$404$	−2.77160e7	−0.0209120
$405$	0	0
$406$	−1.36906e9	−1.01527
$407$	4.28788e8i	0.315255i
$408$	1.12558e9i	0.820478i
$409$	1.25594e9	0.907687	0.453844	−	0.891081i	$-0.350052\pi$
$409$	1.25594e9	0.907687	0.453844	+	0.891081i	$0.350052\pi$
$410$	0	0
$411$	−8.43328e8	−0.599170
$412$	− 6.67487e8i	− 0.470222i
$413$	1.50036e9i	1.04802i
$414$	−2.51361e8	−0.174099
$415$	0	0
$416$	1.79175e8	0.122026
$417$	− 3.90983e9i	− 2.64047i
$418$	2.90114e8i	0.194291i
$419$	4.46548e7	0.0296564	0.0148282	−	0.999890i	$-0.495280\pi$
$419$	4.46548e7	0.0296564	0.0148282	+	0.999890i	$0.495280\pi$
$420$	0	0
$421$	−5.83121e7	−0.0380865	−0.0190433	−	0.999819i	$-0.506062\pi$
$421$	−5.83121e7	−0.0380865	−0.0190433	+	0.999819i	$0.506062\pi$
$422$	4.48923e8i	0.290789i
$423$	− 6.16370e9i	− 3.95959i
$424$	1.81700e8	0.115764
$425$	0	0
$426$	1.59205e9	0.997753
$427$	5.77815e8i	0.359163i
$428$	− 1.36425e9i	− 0.841086i
$429$	−5.15200e8	−0.315047
$430$	0	0
$431$	2.15807e9	1.29836	0.649179	−	0.760635i	$-0.275112\pi$
$431$	2.15807e9	1.29836	0.649179	+	0.760635i	$0.275112\pi$
$432$	− 1.13854e9i	− 0.679449i
$433$	− 2.02867e9i	− 1.20089i	−0.799666	−	0.600446i	$-0.794990\pi$
$433$	− 2.02867e9i	− 1.20089i	0.799666	−	0.600446i	$-0.205010\pi$
$434$	−8.06465e8	−0.473556
$435$	0	0
$436$	1.02595e9	0.592819
$437$	1.95485e8i	0.112055i
$438$	4.43522e9i	2.52206i
$439$	2.51641e9	1.41956	0.709782	−	0.704422i	$-0.248794\pi$
$439$	2.51641e9	1.41956	0.709782	+	0.704422i	$0.248794\pi$
$440$	0	0
$441$	5.61027e9	3.11493
$442$	1.10537e9i	0.608876i
$443$	2.56720e9i	1.40296i	0.712687	+	0.701482i	$0.247478\pi$
$443$	2.56720e9i	1.40296i	−0.712687	+	0.701482i	$0.752522\pi$
$444$	−2.20452e9	−1.19529
$445$	0	0
$446$	−6.23949e8	−0.333025
$447$	4.29961e9i	2.27695i
$448$	3.58089e8i	0.188156i
$449$	−8.28489e8	−0.431941	−0.215970	−	0.976400i	$-0.569291\pi$
$449$	−8.28489e8	−0.431941	−0.215970	+	0.976400i	$0.569291\pi$
$450$	0	0
$451$	2.45657e7	0.0126099
$452$	5.79496e7i	0.0295166i
$453$	1.74938e9i	0.884179i
$454$	−1.05074e9	−0.526987
$455$	0	0
$456$	−1.49156e9	−0.736652
$457$	3.45538e8i	0.169352i	0.996409	+	0.0846758i	$0.0269855\pi$
$457$	3.45538e8i	0.169352i	−0.996409	+	0.0846758i	$0.973015\pi$
$458$	− 1.82754e9i	− 0.888868i
$459$	7.02390e9	3.39027
$460$	0	0
$461$	4.81628e8	0.228959	0.114480	−	0.993426i	$-0.463480\pi$
$461$	4.81628e8	0.228959	0.114480	+	0.993426i	$0.463480\pi$
$462$	− 1.02965e9i	− 0.485782i
$463$	− 1.18307e9i	− 0.553958i	−0.960876	−	0.276979i	$-0.910667\pi$
$463$	− 1.18307e9i	− 0.553958i	0.960876	−	0.276979i	$-0.0893333\pi$
$464$	−5.13147e8	−0.238467
$465$	0	0
$466$	3.02359e9	1.38411
$467$	1.01191e9i	0.459764i	0.973219	+	0.229882i	$0.0738340\pi$
$467$	1.01191e9i	0.459764i	−0.973219	+	0.229882i	$0.926166\pi$
$468$	− 1.88344e9i	− 0.849360i
$469$	3.49502e9	1.56439
$470$	0	0
$471$	−6.84424e9	−3.01823
$472$	5.62360e8i	0.246160i
$473$	1.08460e8i	0.0471256i
$474$	−1.40540e9	−0.606143
$475$	0	0
$476$	−2.20912e9	−0.938846
$477$	− 1.90997e9i	− 0.805774i
$478$	9.05489e8i	0.379215i
$479$	−3.78673e9	−1.57431	−0.787155	−	0.616755i	$-0.788447\pi$
$479$	−3.78673e9	−1.57431	−0.787155	+	0.616755i	$0.788447\pi$
$480$	0	0
$481$	−2.16492e9	−0.887023
$482$	− 1.60189e9i	− 0.651579i
$483$	− 6.93800e8i	− 0.280168i
$484$	1.17211e9	0.469906
$485$	0	0
$486$	−3.77584e9	−1.49206
$487$	− 7.20259e8i	− 0.282577i	−0.989968	−	0.141289i	$-0.954875\pi$
$487$	− 7.20259e8i	− 0.282577i	0.989968	−	0.141289i	$-0.0451246\pi$
$488$	2.16575e8i	0.0843605i
$489$	−6.08782e9	−2.35440
$490$	0	0
$491$	−1.79667e9	−0.684988	−0.342494	−	0.939520i	$-0.611272\pi$
$491$	−1.79667e9	−0.684988	−0.342494	+	0.939520i	$0.611272\pi$
$492$	1.26299e8i	0.0478103i
$493$	− 3.16570e9i	− 1.18989i
$494$	−1.46477e9	−0.546669
$495$	0	0
$496$	−3.02277e8	−0.111229
$497$	3.12463e9i	1.14170i
$498$	5.54920e9i	2.01339i
$499$	−2.54540e9	−0.917073	−0.458536	−	0.888676i	$-0.651626\pi$
$499$	−2.54540e9	−0.917073	−0.458536	+	0.888676i	$0.651626\pi$
$500$	0	0
$501$	7.29859e9	2.59303
$502$	− 3.70259e9i	− 1.30630i
$503$	1.31401e9i	0.460373i	0.973147	+	0.230187i	$0.0739336\pi$
$503$	1.31401e9i	0.460373i	−0.973147	+	0.230187i	$0.926066\pi$
$504$	3.76413e9	1.30966
$505$	0	0
$506$	−5.05804e7	−0.0173562
$507$	2.85791e9i	0.973913i
$508$	− 1.02625e9i	− 0.347321i
$509$	2.50386e9	0.841583	0.420792	−	0.907157i	$-0.361752\pi$
$509$	2.50386e9	0.841583	0.420792	+	0.907157i	$0.361752\pi$
$510$	0	0
$511$	−8.70476e9	−2.88591
$512$	1.34218e8i	0.0441942i
$513$	9.30766e9i	3.04389i
$514$	3.90201e9	1.26741
$515$	0	0
$516$	−5.57624e8	−0.178677
$517$	− 1.24030e9i	− 0.394739i
$518$	− 4.32668e9i	− 1.36773i
$519$	−2.38091e9	−0.747580
$520$	0	0
$521$	−1.22028e9	−0.378032	−0.189016	−	0.981974i	$-0.560530\pi$
$521$	−1.22028e9	−0.378032	−0.189016	+	0.981974i	$0.560530\pi$
$522$	5.39406e9i	1.65985i
$523$	− 1.93318e9i	− 0.590905i	−0.955357	−	0.295452i	$-0.904530\pi$
$523$	− 1.93318e9i	− 0.590905i	0.955357	−	0.295452i	$-0.0954703\pi$
$524$	−2.45169e8	−0.0744401
$525$	0	0
$526$	−3.21457e9	−0.963103
$527$	− 1.86480e9i	− 0.555003i
$528$	− 3.85929e8i	− 0.114101i
$529$	3.37074e9	0.989990
$530$	0	0
$531$	5.91137e9	1.71340
$532$	− 2.92739e9i	− 0.842928i
$533$	1.24031e8i	0.0354800i
$534$	1.52141e9	0.432367
$535$	0	0
$536$	1.30999e9	0.367445
$537$	6.93942e9i	1.93381i
$538$	− 3.64309e9i	− 1.00863i
$539$	1.12893e9	0.310533
$540$	0	0
$541$	−2.54920e9	−0.692172	−0.346086	−	0.938203i	$-0.612489\pi$
$541$	−2.54920e9	−0.692172	−0.346086	+	0.938203i	$0.612489\pi$
$542$	4.06767e9i	1.09736i
$543$	4.59981e9i	1.23294i
$544$	−8.28015e8	−0.220517
$545$	0	0
$546$	5.19862e9	1.36683
$547$	4.58291e9i	1.19725i	0.801028	+	0.598626i	$0.204287\pi$
$547$	4.58291e9i	1.19725i	−0.801028	+	0.598626i	$0.795713\pi$
$548$	− 6.20379e8i	− 0.161037i
$549$	2.27658e9	0.587190
$550$	0	0
$551$	4.19500e9	1.06832
$552$	− 2.60048e8i	− 0.0658062i
$553$	− 2.75830e9i	− 0.693590i
$554$	−5.17461e9	−1.29299
$555$	0	0
$556$	2.87620e9	0.709671
$557$	− 4.94207e9i	− 1.21176i	−0.795557	−	0.605879i	$-0.792822\pi$
$557$	− 4.94207e9i	− 1.21176i	0.795557	−	0.605879i	$-0.207178\pi$
$558$	3.17745e9i	0.774210i
$559$	−5.47609e8	−0.132596
$560$	0	0
$561$	2.38087e9	0.569332
$562$	1.51637e9i	0.360353i
$563$	− 7.82258e8i	− 0.184744i	−0.995725	−	0.0923721i	$-0.970555\pi$
$563$	− 7.82258e8i	− 0.184744i	0.995725	−	0.0923721i	$-0.0294449\pi$
$564$	6.37672e9	1.49665
$565$	0	0
$566$	3.70707e9	0.859356
$567$	− 1.69555e10i	− 3.90634i
$568$	1.17116e9i	0.268163i
$569$	−1.19628e9	−0.272232	−0.136116	−	0.990693i	$-0.543462\pi$
$569$	−1.19628e9	−0.272232	−0.136116	+	0.990693i	$0.543462\pi$
$570$	0	0
$571$	2.41071e9	0.541899	0.270949	−	0.962594i	$-0.412662\pi$
$571$	2.41071e9	0.541899	0.270949	+	0.962594i	$0.412662\pi$
$572$	− 3.78998e8i	− 0.0846741i
$573$	− 8.98060e9i	− 1.99418i
$574$	−2.47880e8	−0.0547078
$575$	0	0
$576$	1.41086e9	0.307613
$577$	− 6.45394e9i	− 1.39865i	−0.714803	−	0.699326i	$-0.753484\pi$
$577$	− 6.45394e9i	− 1.39865i	0.714803	−	0.699326i	$-0.246516\pi$
$578$	− 1.82547e9i	− 0.393212i
$579$	−4.34696e9	−0.930702
$580$	0	0
$581$	−1.08911e10	−2.30385
$582$	4.05326e9i	0.852263i
$583$	− 3.84337e8i	− 0.0803290i
$584$	−3.26269e9	−0.677846
$585$	0	0
$586$	9.85387e8	0.202286
$587$	5.27194e8i	0.107581i	0.998552	+	0.0537907i	$0.0171304\pi$
$587$	5.27194e8i	0.107581i	−0.998552	+	0.0537907i	$0.982870\pi$
$588$	5.80416e9i	1.17738i
$589$	2.47113e9	0.498301
$590$	0	0
$591$	2.90455e9	0.578792
$592$	− 1.62171e9i	− 0.321254i
$593$	8.99056e9i	1.77050i	0.465117	+	0.885249i	$0.346012\pi$
$593$	8.99056e9i	1.77050i	−0.465117	+	0.885249i	$0.653988\pi$
$594$	−2.40829e9	−0.471472
$595$	0	0
$596$	−3.16293e9	−0.611967
$597$	− 8.04843e9i	− 1.54811i
$598$	− 2.55377e8i	− 0.0488347i
$599$	−3.40720e9	−0.647745	−0.323872	−	0.946101i	$-0.604985\pi$
$599$	−3.40720e9	−0.647745	−0.323872	+	0.946101i	$0.604985\pi$
$600$	0	0
$601$	−4.00808e9	−0.753140	−0.376570	−	0.926388i	$-0.622897\pi$
$601$	−4.00808e9	−0.753140	−0.376570	+	0.926388i	$0.622897\pi$
$602$	− 1.09442e9i	− 0.204454i
$603$	− 1.37703e10i	− 2.55760i
$604$	−1.28690e9	−0.237638
$605$	0	0
$606$	−3.01411e8	−0.0550180
$607$	3.61151e9i	0.655434i	0.944776	+	0.327717i	$0.106279\pi$
$607$	3.61151e9i	0.655434i	−0.944776	+	0.327717i	$0.893721\pi$
$608$	− 1.09724e9i	− 0.197987i
$609$	−1.48885e10	−2.67111
$610$	0	0
$611$	6.26219e9	1.11066
$612$	8.70386e9i	1.53491i
$613$	− 4.16268e8i	− 0.0729897i	−0.999334	−	0.0364948i	$-0.988381\pi$
$613$	− 4.16268e8i	− 0.0729897i	0.999334	−	0.0364948i	$-0.0116192\pi$
$614$	−4.51362e9	−0.786929
$615$	0	0
$616$	7.57442e8	0.130562
$617$	− 2.32066e8i	− 0.0397752i	−0.999802	−	0.0198876i	$-0.993669\pi$
$617$	− 2.32066e8i	− 0.0397752i	0.999802	−	0.0198876i	$-0.00633084\pi$
$618$	− 7.25893e9i	− 1.23712i
$619$	1.50914e9	0.255748	0.127874	−	0.991790i	$-0.459185\pi$
$619$	1.50914e9	0.255748	0.127874	+	0.991790i	$0.459185\pi$
$620$	0	0
$621$	−1.62276e9	−0.271915
$622$	4.41133e9i	0.735027i
$623$	2.98599e9i	0.494743i
$624$	1.94853e9	0.321042
$625$	0	0
$626$	−1.37133e9	−0.223424
$627$	3.15499e9i	0.511166i
$628$	− 5.03484e9i	− 0.811199i
$629$	1.00047e10	1.60297
$630$	0	0
$631$	1.18512e10	1.87785	0.938923	−	0.344127i	$-0.111825\pi$
$631$	1.18512e10	1.87785	0.938923	+	0.344127i	$0.111825\pi$
$632$	− 1.03386e9i	− 0.162911i
$633$	4.88204e9i	0.765047i
$634$	8.26794e9	1.28850
$635$	0	0
$636$	1.97598e9	0.304567
$637$	5.69991e9i	0.873736i
$638$	1.08543e9i	0.165473i
$639$	1.23109e10	1.86654
$640$	0	0
$641$	−3.89475e9	−0.584086	−0.292043	−	0.956405i	$-0.594335\pi$
$641$	−3.89475e9	−0.584086	−0.292043	+	0.956405i	$0.594335\pi$
$642$	− 1.48362e10i	− 2.21284i
$643$	6.87187e8i	0.101938i	0.998700	+	0.0509690i	$0.0162310\pi$
$643$	6.87187e8i	0.101938i	−0.998700	+	0.0509690i	$0.983769\pi$
$644$	5.10381e8	0.0752999
$645$	0	0
$646$	6.76906e9	0.987904
$647$	− 2.24157e9i	− 0.325378i	−0.986677	−	0.162689i	$-0.947983\pi$
$647$	− 2.24157e9i	− 0.325378i	0.986677	−	0.162689i	$-0.0520167\pi$
$648$	− 6.35521e9i	− 0.917524i
$649$	1.18952e9	0.170811
$650$	0	0
$651$	−8.77030e9	−1.24589
$652$	− 4.47839e9i	− 0.632784i
$653$	− 3.44569e9i	− 0.484261i	−0.970244	−	0.242131i	$-0.922154\pi$
$653$	− 3.44569e9i	− 0.484261i	0.970244	−	0.242131i	$-0.0778463\pi$
$654$	1.11572e10	1.55967
$655$	0	0
$656$	−9.29096e7	−0.0128498
$657$	3.42965e10i	4.71814i
$658$	1.25152e10i	1.71257i
$659$	−7.59753e9	−1.03413	−0.517063	−	0.855947i	$-0.672975\pi$
$659$	−7.59753e9	−1.03413	−0.517063	+	0.855947i	$0.672975\pi$
$660$	0	0
$661$	−5.49901e8	−0.0740592	−0.0370296	−	0.999314i	$-0.511790\pi$
$661$	−5.49901e8	−0.0740592	−0.0370296	+	0.999314i	$0.511790\pi$
$662$	1.92986e8i	0.0258537i
$663$	1.20209e10i	1.60191i
$664$	−4.08217e9	−0.541131
$665$	0	0
$666$	−1.70470e10	−2.23608
$667$	7.31385e8i	0.0954345i
$668$	5.36907e9i	0.696918i
$669$	−6.78544e9	−0.876167
$670$	0	0
$671$	4.58107e8	0.0585380
$672$	3.89421e9i	0.495025i
$673$	− 7.29473e7i	− 0.00922478i	−0.999989	−	0.00461239i	$-0.998532\pi$
$673$	− 7.29473e7i	− 0.00922478i	0.999989	−	0.00461239i	$-0.00146818\pi$
$674$	7.00064e9	0.880700
$675$	0	0
$676$	−2.10237e9	−0.261755
$677$	− 3.97003e9i	− 0.491738i	−0.969303	−	0.245869i	$-0.920927\pi$
$677$	− 3.97003e9i	− 0.491738i	0.969303	−	0.245869i	$-0.0790733\pi$
$678$	6.30202e8i	0.0776562i
$679$	−7.95510e9	−0.975217
$680$	0	0
$681$	−1.14268e10	−1.38647
$682$	6.39386e8i	0.0771823i
$683$	1.18943e10i	1.42845i	0.699914	+	0.714227i	$0.253221\pi$
$683$	1.18943e10i	1.42845i	−0.699914	+	0.714227i	$0.746779\pi$
$684$	−1.15338e10	−1.37809
$685$	0	0
$686$	−2.39181e9	−0.282874
$687$	− 1.98745e10i	− 2.33855i
$688$	− 4.10206e8i	− 0.0480223i
$689$	1.94049e9	0.226019
$690$	0	0
$691$	2.08948e9	0.240916	0.120458	−	0.992718i	$-0.461564\pi$
$691$	2.08948e9	0.240916	0.120458	+	0.992718i	$0.461564\pi$
$692$	− 1.75147e9i	− 0.200924i
$693$	− 7.96201e9i	− 0.908775i
$694$	6.65959e8	0.0756292
$695$	0	0
$696$	−5.58047e9	−0.627391
$697$	− 5.73177e8i	− 0.0641171i
$698$	− 3.18085e9i	− 0.354038i
$699$	3.28815e10	3.64151
$700$	0	0
$701$	1.21818e10	1.33567	0.667834	−	0.744311i	$-0.267222\pi$
$701$	1.21818e10	1.33567	0.667834	+	0.744311i	$0.267222\pi$
$702$	− 1.21593e10i	− 1.32657i
$703$	1.32576e10i	1.43920i
$704$	2.83902e8	0.0306665
$705$	0	0
$706$	−2.34968e9	−0.251300
$707$	− 5.91563e8i	− 0.0629554i
$708$	6.11567e9i	0.647631i
$709$	−3.80726e9	−0.401191	−0.200595	−	0.979674i	$-0.564288\pi$
$709$	−3.80726e9	−0.401191	−0.200595	+	0.979674i	$0.564288\pi$
$710$	0	0
$711$	−1.08676e10	−1.13394
$712$	1.11920e9i	0.116206i
$713$	4.30833e8i	0.0445139i
$714$	−2.40241e10	−2.47004
$715$	0	0
$716$	−5.10486e9	−0.519742
$717$	9.84720e9i	0.997690i
$718$	1.37334e9i	0.138465i
$719$	−2.74426e9	−0.275343	−0.137671	−	0.990478i	$-0.543962\pi$
$719$	−2.74426e9	−0.275343	−0.137671	+	0.990478i	$0.543962\pi$
$720$	0	0
$721$	1.42467e10	1.41560
$722$	1.81899e9i	0.179866i
$723$	− 1.74205e10i	− 1.71426i
$724$	−3.38377e9	−0.331372
$725$	0	0
$726$	1.27467e10	1.23629
$727$	5.15574e8i	0.0497647i	0.999690	+	0.0248823i	$0.00792111\pi$
$727$	5.15574e8i	0.0497647i	−0.999690	+	0.0248823i	$0.992079\pi$
$728$	3.82428e9i	0.367358i
$729$	−1.39161e10	−1.33036
$730$	0	0
$731$	2.53064e9	0.239618
$732$	2.35525e9i	0.221947i
$733$	− 9.16791e9i	− 0.859818i	−0.902872	−	0.429909i	$-0.858546\pi$
$733$	− 9.16791e9i	− 0.859818i	0.902872	−	0.429909i	$-0.141454\pi$
$734$	7.64371e9	0.713457
$735$	0	0
$736$	1.91300e8	0.0176865
$737$	− 2.77094e9i	− 0.254971i
$738$	9.76639e8i	0.0894411i
$739$	7.95134e9	0.724744	0.362372	−	0.932034i	$-0.381967\pi$
$739$	7.95134e9	0.724744	0.362372	+	0.932034i	$0.381967\pi$
$740$	0	0
$741$	−1.59294e10	−1.43825
$742$	3.87815e9i	0.348507i
$743$	− 1.20333e10i	− 1.07628i	−0.842856	−	0.538139i	$-0.819128\pi$
$743$	− 1.20333e10i	− 1.07628i	0.842856	−	0.538139i	$-0.180872\pi$
$744$	−3.28726e9	−0.292637
$745$	0	0
$746$	2.75739e9	0.243171
$747$	4.29106e10i	3.76654i
$748$	1.75144e9i	0.153017i
$749$	2.91182e10	2.53208
$750$	0	0
$751$	−1.46650e10	−1.26341	−0.631704	−	0.775210i	$-0.717644\pi$
$751$	−1.46650e10	−1.26341	−0.631704	+	0.775210i	$0.717644\pi$
$752$	4.69092e9i	0.402250i
$753$	− 4.02656e10i	− 3.43678i
$754$	−5.48025e9	−0.465587
$755$	0	0
$756$	2.43008e10	2.04548
$757$	− 1.86757e10i	− 1.56474i	−0.622816	−	0.782368i	$-0.714012\pi$
$757$	− 1.86757e10i	− 1.56474i	0.622816	−	0.782368i	$-0.285988\pi$
$758$	− 1.16218e10i	− 0.969240i
$759$	−5.50062e8	−0.0456631
$760$	0	0
$761$	1.45502e10	1.19680	0.598400	−	0.801198i	$-0.295803\pi$
$761$	1.45502e10	1.19680	0.598400	+	0.801198i	$0.295803\pi$
$762$	− 1.11605e10i	− 0.913779i
$763$	2.18976e10i	1.78468i
$764$	6.60642e9	0.535969
$765$	0	0
$766$	−1.50271e10	−1.20802
$767$	6.00583e9i	0.480606i
$768$	1.45962e9i	0.116272i
$769$	−7.45506e9	−0.591166	−0.295583	−	0.955317i	$-0.595514\pi$
$769$	−7.45506e9	−0.591166	−0.295583	+	0.955317i	$0.595514\pi$
$770$	0	0
$771$	4.24344e10	3.33447
$772$	− 3.19776e9i	− 0.250142i
$773$	− 1.85671e10i	− 1.44582i	−0.690941	−	0.722911i	$-0.742803\pi$
$773$	− 1.85671e10i	− 1.44582i	0.690941	−	0.722911i	$-0.257197\pi$
$774$	−4.31197e9	−0.334259
$775$	0	0
$776$	−2.98171e9	−0.229060
$777$	− 4.70526e10i	− 3.59841i
$778$	9.01053e9i	0.685996i
$779$	7.59540e8	0.0575665
$780$	0	0
$781$	2.47728e9	0.186079
$782$	1.18016e9i	0.0882508i
$783$	3.48235e10i	2.59242i
$784$	−4.26972e9	−0.316442
$785$	0	0
$786$	−2.66622e9	−0.195847
$787$	− 2.17240e8i	− 0.0158865i	−0.999968	−	0.00794324i	$-0.997472\pi$
$787$	− 2.17240e8i	− 0.0158865i	0.999968	−	0.00794324i	$-0.00252844\pi$
$788$	2.13668e9i	0.155560i
$789$	−3.49584e10	−2.53386
$790$	0	0
$791$	−1.23686e9	−0.0888595
$792$	− 2.98430e9i	− 0.213454i
$793$	2.31295e9i	0.164706i
$794$	1.16587e10	0.826566
$795$	0	0
$796$	5.92068e9	0.416079
$797$	1.04021e10i	0.727807i	0.931437	+	0.363903i	$0.118556\pi$
$797$	1.04021e10i	0.727807i	−0.931437	+	0.363903i	$0.881444\pi$
$798$	− 3.18354e10i	− 2.21769i
$799$	−2.89392e10	−2.00712
$800$	0	0
$801$	1.17647e10	0.808848
$802$	1.56679e10i	1.07251i
$803$	6.90136e9i	0.470360i
$804$	1.42462e10	0.966723
$805$	0	0
$806$	−3.22822e9	−0.217165
$807$	− 3.96186e10i	− 2.65364i
$808$	− 2.21728e8i	− 0.0147870i
$809$	−1.54020e10	−1.02272	−0.511360	−	0.859367i	$-0.670858\pi$
$809$	−1.54020e10	−1.02272	−0.511360	+	0.859367i	$0.670858\pi$
$810$	0	0
$811$	9.37914e9	0.617433	0.308717	−	0.951154i	$-0.400101\pi$
$811$	9.37914e9	0.617433	0.308717	+	0.951154i	$0.400101\pi$
$812$	− 1.09525e10i	− 0.717904i
$813$	4.42359e10i	2.88707i
$814$	−3.43030e9	−0.222919
$815$	0	0
$816$	−9.00466e9	−0.580165
$817$	3.35346e9i	0.215137i
$818$	1.00475e10i	0.641832i
$819$	4.01997e10	2.55699
$820$	0	0
$821$	−1.49190e10	−0.940887	−0.470443	−	0.882430i	$-0.655906\pi$
$821$	−1.49190e10	−0.940887	−0.470443	+	0.882430i	$0.655906\pi$
$822$	− 6.74663e9i	− 0.423677i
$823$	− 1.72905e10i	− 1.08121i	−0.841278	−	0.540603i	$-0.818196\pi$
$823$	− 1.72905e10i	− 1.08121i	0.841278	−	0.540603i	$-0.181804\pi$
$824$	5.33990e9	0.332497
$825$	0	0
$826$	−1.20029e10	−0.741063
$827$	− 1.92633e8i	− 0.0118430i	−0.999982	−	0.00592149i	$-0.998115\pi$
$827$	− 1.92633e8i	− 0.0118430i	0.999982	−	0.00592149i	$-0.00188488\pi$
$828$	− 2.01089e9i	− 0.123107i
$829$	−1.55199e10	−0.946124	−0.473062	−	0.881029i	$-0.656851\pi$
$829$	−1.55199e10	−0.946124	−0.473062	+	0.881029i	$0.656851\pi$
$830$	0	0
$831$	−5.62739e10	−3.40176
$832$	1.43340e9i	0.0862853i
$833$	− 2.63407e10i	− 1.57896i
$834$	3.12787e10	1.86710
$835$	0	0
$836$	−2.32091e9	−0.137384
$837$	2.05133e10i	1.20919i
$838$	3.57238e8i	0.0209703i
$839$	−1.04108e10	−0.608580	−0.304290	−	0.952579i	$-0.598419\pi$
$839$	−1.04108e10	−0.608580	−0.304290	+	0.952579i	$0.598419\pi$
$840$	0	0
$841$	−1.55480e9	−0.0901339
$842$	− 4.66497e8i	− 0.0269312i
$843$	1.64905e10i	0.948066i
$844$	−3.59138e9	−0.205619
$845$	0	0
$846$	4.93096e10	2.79985
$847$	2.50173e10i	1.41465i
$848$	1.45360e9i	0.0818575i
$849$	4.03144e10	2.26091
$850$	0	0
$851$	−2.31142e9	−0.128566
$852$	1.27364e10i	0.705518i
$853$	1.54506e10i	0.852362i	0.904638	+	0.426181i	$0.140141\pi$
$853$	1.54506e10i	0.852362i	−0.904638	+	0.426181i	$0.859859\pi$
$854$	−4.62252e9	−0.253967
$855$	0	0
$856$	1.09140e10	0.594738
$857$	2.50311e9i	0.135846i	0.997691	+	0.0679231i	$0.0216373\pi$
$857$	2.50311e9i	0.135846i	−0.997691	+	0.0679231i	$0.978363\pi$
$858$	− 4.12160e9i	− 0.222772i
$859$	−2.05842e10	−1.10805	−0.554024	−	0.832500i	$-0.686909\pi$
$859$	−2.05842e10	−1.10805	−0.554024	+	0.832500i	$0.686909\pi$
$860$	0	0
$861$	−2.69569e9	−0.143933
$862$	1.72645e10i	0.918078i
$863$	− 1.91073e10i	− 1.01196i	−0.862546	−	0.505978i	$-0.831132\pi$
$863$	− 1.91073e10i	− 1.01196i	0.862546	−	0.505978i	$-0.168868\pi$
$864$	9.10836e9	0.480443
$865$	0	0
$866$	1.62294e10	0.849158
$867$	− 1.98520e10i	− 1.03452i
$868$	− 6.45172e9i	− 0.334855i
$869$	−2.18685e9	−0.113044
$870$	0	0
$871$	1.39903e10	0.717404
$872$	8.20758e9i	0.419187i
$873$	3.13429e10i	1.59437i
$874$	−1.56388e9	−0.0792345
$875$	0	0
$876$	−3.54818e10	−1.78337
$877$	3.33063e10i	1.66736i	0.552251	+	0.833678i	$0.313769\pi$
$877$	3.33063e10i	1.66736i	−0.552251	+	0.833678i	$0.686231\pi$
$878$	2.01312e10i	1.00378i
$879$	1.07161e10	0.532200
$880$	0	0
$881$	−9.31313e9	−0.458860	−0.229430	−	0.973325i	$-0.573686\pi$
$881$	−9.31313e9	−0.458860	−0.229430	+	0.973325i	$0.573686\pi$
$882$	4.48821e10i	2.20259i
$883$	− 4.38514e9i	− 0.214349i	−0.994240	−	0.107174i	$-0.965820\pi$
$883$	− 4.38514e9i	− 0.214349i	0.994240	−	0.107174i	$-0.0341803\pi$
$884$	−8.84294e9	−0.430540
$885$	0	0
$886$	−2.05376e10	−0.992045
$887$	− 1.78764e10i	− 0.860098i	−0.902806	−	0.430049i	$-0.858496\pi$
$887$	− 1.78764e10i	− 0.860098i	0.902806	−	0.430049i	$-0.141504\pi$
$888$	− 1.76361e10i	− 0.845197i
$889$	2.19041e10	1.04561
$890$	0	0
$891$	−1.34428e10	−0.636673
$892$	− 4.99159e9i	− 0.235484i
$893$	− 3.83485e10i	− 1.80206i
$894$	−3.43969e10	−1.61004
$895$	0	0
$896$	−2.86471e9	−0.133046
$897$	− 2.77723e9i	− 0.128481i
$898$	− 6.62791e9i	− 0.305428i
$899$	9.24541e9	0.424392
$900$	0	0
$901$	−8.96751e9	−0.408447
$902$	1.96526e8i	0.00891653i
$903$	− 1.19018e10i	− 0.537904i
$904$	−4.63597e8	−0.0208714
$905$	0	0
$906$	−1.39950e10	−0.625209
$907$	− 2.84742e9i	− 0.126714i	−0.997991	−	0.0633572i	$-0.979819\pi$
$907$	− 2.84742e9i	− 0.126714i	0.997991	−	0.0633572i	$-0.0201807\pi$
$908$	− 8.40592e9i	− 0.372636i
$909$	−2.33074e9	−0.102925
$910$	0	0
$911$	−3.84643e10	−1.68556	−0.842779	−	0.538260i	$-0.819082\pi$
$911$	−3.84643e10	−1.68556	−0.842779	+	0.538260i	$0.819082\pi$
$912$	− 1.19324e10i	− 0.520892i
$913$	8.63474e9i	0.375493i
$914$	−2.76431e9	−0.119750
$915$	0	0
$916$	1.46203e10	0.628525
$917$	− 5.23283e9i	− 0.224101i
$918$	5.61912e10i	2.39728i
$919$	−1.07375e10	−0.456352	−0.228176	−	0.973620i	$-0.573276\pi$
$919$	−1.07375e10	−0.456352	−0.228176	+	0.973620i	$0.573276\pi$
$920$	0	0
$921$	−4.90856e10	−2.07036
$922$	3.85303e9i	0.161899i
$923$	1.25077e10i	0.523564i
$924$	8.23718e9	0.343500
$925$	0	0
$926$	9.46456e9	0.391708
$927$	− 5.61315e10i	− 2.31434i
$928$	− 4.10518e9i	− 0.168622i
$929$	−5.34493e9	−0.218719	−0.109360	−	0.994002i	$-0.534880\pi$
$929$	−5.34493e9	−0.218719	−0.109360	+	0.994002i	$0.534880\pi$
$930$	0	0
$931$	3.49052e10	1.41764
$932$	2.41887e10i	0.978717i
$933$	4.79732e10i	1.93381i
$934$	−8.09531e9	−0.325102
$935$	0	0
$936$	1.50675e10	0.600588
$937$	− 1.48125e10i	− 0.588218i	−0.955772	−	0.294109i	$-0.904977\pi$
$937$	− 1.48125e10i	− 0.588218i	0.955772	−	0.294109i	$-0.0950229\pi$
$938$	2.79602e10i	1.10619i
$939$	−1.49132e10	−0.587814
$940$	0	0
$941$	2.18734e10	0.855763	0.427882	−	0.903835i	$-0.359260\pi$
$941$	2.18734e10	0.855763	0.427882	+	0.903835i	$0.359260\pi$
$942$	− 5.47539e10i	− 2.13421i
$943$	1.32423e8i	0.00514249i
$944$	−4.49888e9	−0.174061
$945$	0	0
$946$	−8.67682e8	−0.0333228
$947$	2.45163e10i	0.938060i	0.883182	+	0.469030i	$0.155396\pi$
$947$	2.45163e10i	0.938060i	−0.883182	+	0.469030i	$0.844604\pi$
$948$	− 1.12432e10i	− 0.428608i
$949$	−3.48445e10	−1.32344
$950$	0	0
$951$	8.99139e10	3.38996
$952$	− 1.76729e10i	− 0.663864i
$953$	− 1.34056e10i	− 0.501721i	−0.968023	−	0.250860i	$-0.919286\pi$
$953$	− 1.34056e10i	− 0.501721i	0.968023	−	0.250860i	$-0.0807135\pi$
$954$	1.52798e10	0.569768
$955$	0	0
$956$	−7.24391e9	−0.268146
$957$	1.18040e10i	0.435349i
$958$	− 3.02939e10i	− 1.11321i
$959$	1.32412e10	0.484800
$960$	0	0
$961$	−2.20665e10	−0.802049
$962$	− 1.73194e10i	− 0.627220i
$963$	− 1.14725e11i	− 4.13967i
$964$	1.28151e10	0.460736
$965$	0	0
$966$	5.55040e9	0.198109
$967$	− 4.64740e9i	− 0.165279i	−0.996580	−	0.0826395i	$-0.973665\pi$
$967$	− 4.64740e9i	− 0.165279i	0.996580	−	0.0826395i	$-0.0263350\pi$
$968$	9.37691e9i	0.332274i
$969$	7.36135e10	2.59911
$970$	0	0
$971$	−3.24860e10	−1.13875	−0.569375	−	0.822078i	$-0.692815\pi$
$971$	−3.24860e10	−1.13875	−0.569375	+	0.822078i	$0.692815\pi$
$972$	− 3.02067e10i	− 1.05505i
$973$	6.13889e10i	2.13646i
$974$	5.76207e9	0.199812
$975$	0	0
$976$	−1.73260e9	−0.0596518
$977$	− 6.06506e9i	− 0.208067i	−0.994574	−	0.104034i	$-0.966825\pi$
$977$	− 6.06506e9i	− 0.208067i	0.994574	−	0.104034i	$-0.0331750\pi$
$978$	− 4.87025e10i	− 1.66481i
$979$	2.36737e9	0.0806355
$980$	0	0
$981$	8.62758e10	2.91775
$982$	− 1.43734e10i	− 0.484360i
$983$	2.10842e10i	0.707979i	0.935249	+	0.353989i	$0.115175\pi$
$983$	2.10842e10i	0.707979i	−0.935249	+	0.353989i	$0.884825\pi$
$984$	−1.01039e9	−0.0338070
$985$	0	0
$986$	2.53256e10	0.841377
$987$	1.36103e11i	4.50566i
$988$	− 1.17181e10i	− 0.386553i
$989$	−5.84664e8	−0.0192185
$990$	0	0
$991$	4.26476e10	1.39199	0.695997	−	0.718045i	$-0.254963\pi$
$991$	4.26476e10	1.39199	0.695997	+	0.718045i	$0.254963\pi$
$992$	− 2.41821e9i	− 0.0786509i
$993$	2.09872e9i	0.0680194i
$994$	−2.49970e10	−0.807302
$995$	0	0
$996$	−4.43936e10	−1.42368
$997$	− 1.59226e10i	− 0.508841i	−0.967094	−	0.254420i	$-0.918115\pi$
$997$	− 1.59226e10i	− 0.508841i	0.967094	−	0.254420i	$-0.0818847\pi$
$998$	− 2.03632e10i	− 0.648468i
$999$	−1.10054e11	−3.49241

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.8.b.a.49.2		2
3.2	odd	2		450.8.c.l.199.1		2
4.3	odd	2		400.8.c.a.49.1		2
5.2	odd	4		50.8.a.d.1.1	✓	1
5.3	odd	4		50.8.a.e.1.1	yes	1
5.4	even	2	inner	50.8.b.a.49.1		2
15.2	even	4		450.8.a.z.1.1		1
15.8	even	4		450.8.a.a.1.1		1
15.14	odd	2		450.8.c.l.199.2		2
20.3	even	4		400.8.a.s.1.1		1
20.7	even	4		400.8.a.a.1.1		1
20.19	odd	2		400.8.c.a.49.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.8.a.d.1.1	✓	1	5.2	odd	4
50.8.a.e.1.1	yes	1	5.3	odd	4
50.8.b.a.49.1		2	5.4	even	2	inner
50.8.b.a.49.2		2	1.1	even	1	trivial
400.8.a.a.1.1		1	20.7	even	4
400.8.a.s.1.1		1	20.3	even	4
400.8.c.a.49.1		2	4.3	odd	2
400.8.c.a.49.2		2	20.19	odd	2
450.8.a.a.1.1		1	15.8	even	4
450.8.a.z.1.1		1	15.2	even	4
450.8.c.l.199.1		2	3.2	odd	2
450.8.c.l.199.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} +87.0000i q^{3} -64.0000 q^{4} -696.000 q^{6} -1366.00i q^{7} -512.000i q^{8} -5382.00 q^{9} -1083.00 q^{11} -5568.00i q^{12} -5468.00i q^{13} +10928.0 q^{14} +4096.00 q^{16} +25269.0i q^{17} -43056.0i q^{18} -33485.0 q^{19} +118842. q^{21} -8664.00i q^{22} -5838.00i q^{23} +44544.0 q^{24} +43744.0 q^{26} -277965. i q^{27} +87424.0i q^{28} -125280. q^{29} -73798.0 q^{31} +32768.0i q^{32} -94221.0i q^{33} -202152. q^{34} +344448. q^{36} -395926. i q^{37} -267880. i q^{38} +475716. q^{39} -22683.0 q^{41} +950736. i q^{42} -100148. i q^{43} +69312.0 q^{44} +46704.0 q^{46} +1.14524e6i q^{47} +356352. i q^{48} -1.04241e6 q^{49} -2.19840e6 q^{51} +349952. i q^{52} +354882. i q^{53} +2.22372e6 q^{54} -699392. q^{56} -2.91320e6i q^{57} -1.00224e6i q^{58} -1.09836e6 q^{59} -422998. q^{61} -590384. i q^{62} +7.35181e6i q^{63} -262144. q^{64} +753768. q^{66} +2.55858e6i q^{67} -1.61722e6i q^{68} +507906. q^{69} -2.28743e6 q^{71} +2.75558e6i q^{72} -6.37244e6i q^{73} +3.16741e6 q^{74} +2.14304e6 q^{76} +1.47938e6i q^{77} +3.80573e6i q^{78} +2.01925e6 q^{79} +1.24125e7 q^{81} -181464. i q^{82} -7.97298e6i q^{83} -7.60589e6 q^{84} +801184. q^{86} -1.08994e7i q^{87} +554496. i q^{88} -2.18594e6 q^{89} -7.46929e6 q^{91} +373632. i q^{92} -6.42043e6i q^{93} -9.16195e6 q^{94} -2.85082e6 q^{96} -5.82365e6i q^{97} -8.33930e6i q^{98} +5.82871e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 1392 q^{6} - 10764 q^{9} - 2166 q^{11} + 21856 q^{14} + 8192 q^{16} - 66970 q^{19} + 237684 q^{21} + 89088 q^{24} + 87488 q^{26} - 250560 q^{29} - 147596 q^{31} - 404304 q^{34} + 688896 q^{36}+ \cdots + 11657412 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 1392 * q^6 - 10764 * q^9 - 2166 * q^11 + 21856 * q^14 + 8192 * q^16 - 66970 * q^19 + 237684 * q^21 + 89088 * q^24 + 87488 * q^26 - 250560 * q^29 - 147596 * q^31 - 404304 * q^34 + 688896 * q^36 + 951432 * q^39 - 45366 * q^41 + 138624 * q^44 + 93408 * q^46 - 2084826 * q^49 - 4396806 * q^51 + 4447440 * q^54 - 1398784 * q^56 - 2196720 * q^59 - 845996 * q^61 - 524288 * q^64 + 1507536 * q^66 + 1015812 * q^69 - 4574856 * q^71 + 6334816 * q^74 + 4286080 * q^76 + 4038500 * q^79 + 24825042 * q^81 - 15211776 * q^84 + 1602368 * q^86 - 4371870 * q^89 - 14938576 * q^91 - 18323904 * q^94 - 5701632 * q^96 + 11657412 * q^99

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