LMFDB - Embedded newform 50.8.b.c.49.1

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)

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);

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")

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,-192] 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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000i q^{2} -12.0000i q^{3} -64.0000 q^{4} -96.0000 q^{6} +1016.00i q^{7} +512.000i q^{8} +2043.00 q^{9} +1092.00 q^{11} +768.000i q^{12} -1382.00i q^{13} +8128.00 q^{14} +4096.00 q^{16} +14706.0i q^{17} -16344.0i q^{18} +39940.0 q^{19} +12192.0 q^{21} -8736.00i q^{22} -68712.0i q^{23} +6144.00 q^{24} -11056.0 q^{26} -50760.0i q^{27} -65024.0i q^{28} +102570. q^{29} +227552. q^{31} -32768.0i q^{32} -13104.0i q^{33} +117648. q^{34} -130752. q^{36} +160526. i q^{37} -319520. i q^{38} -16584.0 q^{39} +10842.0 q^{41} -97536.0i q^{42} +630748. i q^{43} -69888.0 q^{44} -549696. q^{46} +472656. i q^{47} -49152.0i q^{48} -208713. q^{49} +176472. q^{51} +88448.0i q^{52} +1.49402e6i q^{53} -406080. q^{54} -520192. q^{56} -479280. i q^{57} -820560. i q^{58} -2.64066e6 q^{59} +827702. q^{61} -1.82042e6i q^{62} +2.07569e6i q^{63} -262144. q^{64} -104832. q^{66} -126004. i q^{67} -941184. i q^{68} -824544. q^{69} -1.41473e6 q^{71} +1.04602e6i q^{72} -980282. i q^{73} +1.28421e6 q^{74} -2.55616e6 q^{76} +1.10947e6i q^{77} +132672. i q^{78} +3.56680e6 q^{79} +3.85892e6 q^{81} -86736.0i q^{82} -5.67289e6i q^{83} -780288. q^{84} +5.04598e6 q^{86} -1.23084e6i q^{87} +559104. i q^{88} +1.19512e7 q^{89} +1.40411e6 q^{91} +4.39757e6i q^{92} -2.73062e6i q^{93} +3.78125e6 q^{94} -393216. q^{96} +8.68215e6i q^{97} +1.66970e6i q^{98} +2.23096e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 192 q^{6} + 4086 q^{9} + 2184 q^{11} + 16256 q^{14} + 8192 q^{16} + 79880 q^{19} + 24384 q^{21} + 12288 q^{24} - 22112 q^{26} + 205140 q^{29} + 455104 q^{31} + 235296 q^{34} - 261504 q^{36}+ \cdots + 4461912 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 192 * q^6 + 4086 * q^9 + 2184 * q^11 + 16256 * q^14 + 8192 * q^16 + 79880 * q^19 + 24384 * q^21 + 12288 * q^24 - 22112 * q^26 + 205140 * q^29 + 455104 * q^31 + 235296 * q^34 - 261504 * q^36 - 33168 * q^39 + 21684 * q^41 - 139776 * q^44 - 1099392 * q^46 - 417426 * q^49 + 352944 * q^51 - 812160 * q^54 - 1040384 * q^56 - 5281320 * q^59 + 1655404 * q^61 - 524288 * q^64 - 209664 * q^66 - 1649088 * q^69 - 2829456 * q^71 + 2568416 * q^74 - 5112320 * q^76 + 7133600 * q^79 + 7717842 * q^81 - 1560576 * q^84 + 10091968 * q^86 + 23902380 * q^89 + 2808224 * q^91 + 7562496 * q^94 - 786432 * q^96 + 4461912 * 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
$3$	− 12.0000i	− 0.256600i	−0.991735	−	0.128300i	$-0.959048\pi$
$3$	− 12.0000i	− 0.256600i	0.991735	−	0.128300i	$-0.0409521\pi$
$4$	−64.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−96.0000	−0.181444
$7$	1016.00i	1.11957i	0.828638	+	0.559784i	$0.189116\pi$
$7$	1016.00i	1.11957i	−0.828638	+	0.559784i	$0.810884\pi$
$8$	512.000i	0.353553i
$9$	2043.00	0.934156
$10$	0	0
$11$	1092.00	0.247371	0.123685	−	0.992321i	$-0.460529\pi$
$11$	1092.00	0.247371	0.123685	+	0.992321i	$0.460529\pi$
$12$	768.000i	0.128300i
$13$	− 1382.00i	− 0.174464i	−0.996188	−	0.0872321i	$-0.972198\pi$
$13$	− 1382.00i	− 0.174464i	0.996188	−	0.0872321i	$-0.0278022\pi$
$14$	8128.00	0.791654
$15$	0	0
$16$	4096.00	0.250000
$17$	14706.0i	0.725978i	0.931793	+	0.362989i	$0.118244\pi$
$17$	14706.0i	0.725978i	−0.931793	+	0.362989i	$0.881756\pi$
$18$	− 16344.0i	− 0.660548i
$19$	39940.0	1.33589	0.667945	−	0.744211i	$-0.267174\pi$
$19$	39940.0	1.33589	0.667945	+	0.744211i	$0.267174\pi$
$20$	0	0
$21$	12192.0	0.287281
$22$	− 8736.00i	− 0.174917i
$23$	− 68712.0i	− 1.17757i	−0.808291	−	0.588783i	$-0.799607\pi$
$23$	− 68712.0i	− 1.17757i	0.808291	−	0.588783i	$-0.200393\pi$
$24$	6144.00	0.0907218
$25$	0	0
$26$	−11056.0	−0.123365
$27$	− 50760.0i	− 0.496305i
$28$	− 65024.0i	− 0.559784i
$29$	102570.	0.780957	0.390479	−	0.920612i	$-0.372310\pi$
$29$	102570.	0.780957	0.390479	+	0.920612i	$0.372310\pi$
$30$	0	0
$31$	227552.	1.37188	0.685938	−	0.727660i	$-0.259392\pi$
$31$	227552.	1.37188	0.685938	+	0.727660i	$0.259392\pi$
$32$	− 32768.0i	− 0.176777i
$33$	− 13104.0i	− 0.0634753i
$34$	117648.	0.513344
$35$	0	0
$36$	−130752.	−0.467078
$37$	160526.i	0.521002i	0.965474	+	0.260501i	$0.0838877\pi$
$37$	160526.i	0.521002i	−0.965474	+	0.260501i	$0.916112\pi$
$38$	− 319520.i	− 0.944616i
$39$	−16584.0	−0.0447675
$40$	0	0
$41$	10842.0	0.0245678	0.0122839	−	0.999925i	$-0.496090\pi$
$41$	10842.0	0.0245678	0.0122839	+	0.999925i	$0.496090\pi$
$42$	− 97536.0i	− 0.203139i
$43$	630748.i	1.20981i	0.796299	+	0.604904i	$0.206788\pi$
$43$	630748.i	1.20981i	−0.796299	+	0.604904i	$0.793212\pi$
$44$	−69888.0	−0.123685
$45$	0	0
$46$	−549696.	−0.832665
$47$	472656.i	0.664053i	0.943270	+	0.332026i	$0.107732\pi$
$47$	472656.i	0.664053i	−0.943270	+	0.332026i	$0.892268\pi$
$48$	− 49152.0i	− 0.0641500i
$49$	−208713.	−0.253433
$50$	0	0
$51$	176472.	0.186286
$52$	88448.0i	0.0872321i
$53$	1.49402e6i	1.37845i	0.724548	+	0.689224i	$0.242048\pi$
$53$	1.49402e6i	1.37845i	−0.724548	+	0.689224i	$0.757952\pi$
$54$	−406080.	−0.350940
$55$	0	0
$56$	−520192.	−0.395827
$57$	− 479280.i	− 0.342789i
$58$	− 820560.i	− 0.552220i
$59$	−2.64066e6	−1.67390	−0.836952	−	0.547277i	$-0.815665\pi$
$59$	−2.64066e6	−1.67390	−0.836952	+	0.547277i	$0.815665\pi$
$60$	0	0
$61$	827702.	0.466895	0.233448	−	0.972369i	$-0.424999\pi$
$61$	827702.	0.466895	0.233448	+	0.972369i	$0.424999\pi$
$62$	− 1.82042e6i	− 0.970063i
$63$	2.07569e6i	1.04585i
$64$	−262144.	−0.125000
$65$	0	0
$66$	−104832.	−0.0448838
$67$	− 126004.i	− 0.0511826i	−0.999672	−	0.0255913i	$-0.991853\pi$
$67$	− 126004.i	− 0.0511826i	0.999672	−	0.0255913i	$-0.00814686\pi$
$68$	− 941184.i	− 0.362989i
$69$	−824544.	−0.302164
$70$	0	0
$71$	−1.41473e6	−0.469104	−0.234552	−	0.972104i	$-0.575362\pi$
$71$	−1.41473e6	−0.469104	−0.234552	+	0.972104i	$0.575362\pi$
$72$	1.04602e6i	0.330274i
$73$	− 980282.i	− 0.294931i	−0.989067	−	0.147466i	$-0.952888\pi$
$73$	− 980282.i	− 0.294931i	0.989067	−	0.147466i	$-0.0471116\pi$
$74$	1.28421e6	0.368404
$75$	0	0
$76$	−2.55616e6	−0.667945
$77$	1.10947e6i	0.276948i
$78$	132672.i	0.0316554i
$79$	3.56680e6	0.813924	0.406962	−	0.913445i	$-0.366588\pi$
$79$	3.56680e6	0.813924	0.406962	+	0.913445i	$0.366588\pi$
$80$	0	0
$81$	3.85892e6	0.806805
$82$	− 86736.0i	− 0.0173720i
$83$	− 5.67289e6i	− 1.08901i	−0.838758	−	0.544504i	$-0.816718\pi$
$83$	− 5.67289e6i	− 1.08901i	0.838758	−	0.544504i	$-0.183282\pi$
$84$	−780288.	−0.143641
$85$	0	0
$86$	5.04598e6	0.855463
$87$	− 1.23084e6i	− 0.200394i
$88$	559104.i	0.0874587i
$89$	1.19512e7	1.79699	0.898496	−	0.438982i	$-0.144661\pi$
$89$	1.19512e7	1.79699	0.898496	+	0.438982i	$0.144661\pi$
$90$	0	0
$91$	1.40411e6	0.195325
$92$	4.39757e6i	0.588783i
$93$	− 2.73062e6i	− 0.352023i
$94$	3.78125e6	0.469556
$95$	0	0
$96$	−393216.	−0.0453609
$97$	8.68215e6i	0.965886i	0.875652	+	0.482943i	$0.160432\pi$
$97$	8.68215e6i	0.965886i	−0.875652	+	0.482943i	$0.839568\pi$
$98$	1.66970e6i	0.179204i
$99$	2.23096e6	0.231083
$100$	0	0
$101$	−1.00795e7	−0.973455	−0.486727	−	0.873554i	$-0.661810\pi$
$101$	−1.00795e7	−0.973455	−0.486727	+	0.873554i	$0.661810\pi$
$102$	− 1.41178e6i	− 0.131724i
$103$	− 3.74799e6i	− 0.337962i	−0.985619	−	0.168981i	$-0.945952\pi$
$103$	− 3.74799e6i	− 0.337962i	0.985619	−	0.168981i	$-0.0540477\pi$
$104$	707584.	0.0616824
$105$	0	0
$106$	1.19521e7	0.974710
$107$	− 1.79856e7i	− 1.41932i	−0.704543	−	0.709661i	$-0.748848\pi$
$107$	− 1.79856e7i	− 1.41932i	0.704543	−	0.709661i	$-0.251152\pi$
$108$	3.24864e6i	0.248152i
$109$	−1.22570e7	−0.906552	−0.453276	−	0.891370i	$-0.649745\pi$
$109$	−1.22570e7	−0.906552	−0.453276	+	0.891370i	$0.649745\pi$
$110$	0	0
$111$	1.92631e6	0.133689
$112$	4.16154e6i	0.279892i
$113$	− 1.65950e7i	− 1.08194i	−0.841043	−	0.540968i	$-0.818058\pi$
$113$	− 1.65950e7i	− 1.08194i	0.841043	−	0.540968i	$-0.181942\pi$
$114$	−3.83424e6	−0.242389
$115$	0	0
$116$	−6.56448e6	−0.390479
$117$	− 2.82343e6i	− 0.162977i
$118$	2.11253e7i	1.18363i
$119$	−1.49413e7	−0.812782
$120$	0	0
$121$	−1.82947e7	−0.938808
$122$	− 6.62162e6i	− 0.330145i
$123$	− 130104.i	− 0.00630410i
$124$	−1.45633e7	−0.685938
$125$	0	0
$126$	1.66055e7	0.739529
$127$	1.16826e6i	0.0506087i	0.999680	+	0.0253043i	$0.00805548\pi$
$127$	1.16826e6i	0.0506087i	−0.999680	+	0.0253043i	$0.991945\pi$
$128$	2.09715e6i	0.0883883i
$129$	7.56898e6	0.310437
$130$	0	0
$131$	−7.92383e6	−0.307954	−0.153977	−	0.988074i	$-0.549208\pi$
$131$	−7.92383e6	−0.307954	−0.153977	+	0.988074i	$0.549208\pi$
$132$	838656.i	0.0317377i
$133$	4.05790e7i	1.49562i
$134$	−1.00803e6	−0.0361916
$135$	0	0
$136$	−7.52947e6	−0.256672
$137$	− 315654.i	− 0.0104879i	−0.999986	−	0.00524396i	$-0.998331\pi$
$137$	− 315654.i	− 0.0104879i	0.999986	−	0.00524396i	$-0.00166921\pi$
$138$	6.59635e6i	0.213662i
$139$	−3.92038e7	−1.23816	−0.619079	−	0.785329i	$-0.712494\pi$
$139$	−3.92038e7	−1.23816	−0.619079	+	0.785329i	$0.712494\pi$
$140$	0	0
$141$	5.67187e6	0.170396
$142$	1.13178e7i	0.331706i
$143$	− 1.50914e6i	− 0.0431573i
$144$	8.36813e6	0.233539
$145$	0	0
$146$	−7.84226e6	−0.208548
$147$	2.50456e6i	0.0650309i
$148$	− 1.02737e7i	− 0.260501i
$149$	2.18860e7	0.542020	0.271010	−	0.962577i	$-0.412642\pi$
$149$	2.18860e7	0.542020	0.271010	+	0.962577i	$0.412642\pi$
$150$	0	0
$151$	−2.94154e7	−0.695274	−0.347637	−	0.937629i	$-0.613016\pi$
$151$	−2.94154e7	−0.695274	−0.347637	+	0.937629i	$0.613016\pi$
$152$	2.04493e7i	0.472308i
$153$	3.00444e7i	0.678177i
$154$	8.87578e6	0.195832
$155$	0	0
$156$	1.06138e6	0.0223838
$157$	6.05550e7i	1.24882i	0.781095	+	0.624412i	$0.214661\pi$
$157$	6.05550e7i	1.24882i	−0.781095	+	0.624412i	$0.785339\pi$
$158$	− 2.85344e7i	− 0.575531i
$159$	1.79282e7	0.353710
$160$	0	0
$161$	6.98114e7	1.31837
$162$	− 3.08714e7i	− 0.570497i
$163$	− 5.70853e7i	− 1.03245i	−0.856454	−	0.516223i	$-0.827337\pi$
$163$	− 5.70853e7i	− 1.03245i	0.856454	−	0.516223i	$-0.172663\pi$
$164$	−693888.	−0.0122839
$165$	0	0
$166$	−4.53831e7	−0.770045
$167$	− 8.77265e7i	− 1.45755i	−0.684754	−	0.728775i	$-0.740090\pi$
$167$	− 8.77265e7i	− 1.45755i	0.684754	−	0.728775i	$-0.259910\pi$
$168$	6.24230e6i	0.101569i
$169$	6.08386e7	0.969562
$170$	0	0
$171$	8.15974e7	1.24793
$172$	− 4.03679e7i	− 0.604904i
$173$	− 8.56954e6i	− 0.125833i	−0.998019	−	0.0629167i	$-0.979960\pi$
$173$	− 8.56954e6i	− 0.125833i	0.998019	−	0.0629167i	$-0.0200403\pi$
$174$	−9.84672e6	−0.141700
$175$	0	0
$176$	4.47283e6	0.0618427
$177$	3.16879e7i	0.429524i
$178$	− 9.56095e7i	− 1.27067i
$179$	−1.88041e7	−0.245056	−0.122528	−	0.992465i	$-0.539100\pi$
$179$	−1.88041e7	−0.245056	−0.122528	+	0.992465i	$0.539100\pi$
$180$	0	0
$181$	−5.99625e7	−0.751631	−0.375816	−	0.926694i	$-0.622637\pi$
$181$	−5.99625e7	−0.751631	−0.375816	+	0.926694i	$0.622637\pi$
$182$	− 1.12329e7i	− 0.138115i
$183$	− 9.93242e6i	− 0.119805i
$184$	3.51805e7	0.416332
$185$	0	0
$186$	−2.18450e7	−0.248918
$187$	1.60590e7i	0.179586i
$188$	− 3.02500e7i	− 0.332026i
$189$	5.15722e7	0.555647
$190$	0	0
$191$	9.39861e7	0.975993	0.487997	−	0.872845i	$-0.337728\pi$
$191$	9.39861e7	0.975993	0.487997	+	0.872845i	$0.337728\pi$
$192$	3.14573e6i	0.0320750i
$193$	3.51946e7i	0.352391i	0.984355	+	0.176196i	$0.0563791\pi$
$193$	3.51946e7i	0.352391i	−0.984355	+	0.176196i	$0.943621\pi$
$194$	6.94572e7	0.682985
$195$	0	0
$196$	1.33576e7	0.126717
$197$	1.02985e8i	0.959718i	0.877346	+	0.479859i	$0.159312\pi$
$197$	1.02985e8i	0.959718i	−0.877346	+	0.479859i	$0.840688\pi$
$198$	− 1.78476e7i	− 0.163400i
$199$	−8.36376e7	−0.752342	−0.376171	−	0.926550i	$-0.622760\pi$
$199$	−8.36376e7	−0.752342	−0.376171	+	0.926550i	$0.622760\pi$
$200$	0	0
$201$	−1.51205e6	−0.0131335
$202$	8.06363e7i	0.688337i
$203$	1.04211e8i	0.874335i
$204$	−1.12942e7	−0.0931430
$205$	0	0
$206$	−2.99839e7	−0.238975
$207$	− 1.40379e8i	− 1.10003i
$208$	− 5.66067e6i	− 0.0436160i
$209$	4.36145e7	0.330460
$210$	0	0
$211$	−9.74010e7	−0.713797	−0.356899	−	0.934143i	$-0.616166\pi$
$211$	−9.74010e7	−0.713797	−0.356899	+	0.934143i	$0.616166\pi$
$212$	− 9.56172e7i	− 0.689224i
$213$	1.69767e7i	0.120372i
$214$	−1.43885e8	−1.00361
$215$	0	0
$216$	2.59891e7	0.175470
$217$	2.31193e8i	1.53591i
$218$	9.80562e7i	0.641029i
$219$	−1.17634e7	−0.0756794
$220$	0	0
$221$	2.03237e7	0.126657
$222$	− 1.54105e7i	− 0.0945325i
$223$	1.46457e7i	0.0884390i	0.999022	+	0.0442195i	$0.0140801\pi$
$223$	1.46457e7i	0.0884390i	−0.999022	+	0.0442195i	$0.985920\pi$
$224$	3.32923e7	0.197914
$225$	0	0
$226$	−1.32760e8	−0.765045
$227$	− 1.84541e8i	− 1.04713i	−0.851985	−	0.523567i	$-0.824601\pi$
$227$	− 1.84541e8i	− 1.04713i	0.851985	−	0.523567i	$-0.175399\pi$
$228$	3.06739e7i	0.171395i
$229$	8.75461e6	0.0481740	0.0240870	−	0.999710i	$-0.492332\pi$
$229$	8.75461e6	0.0481740	0.0240870	+	0.999710i	$0.492332\pi$
$230$	0	0
$231$	1.33137e7	0.0710650
$232$	5.25158e7i	0.276110i
$233$	1.19556e8i	0.619193i	0.950868	+	0.309597i	$0.100194\pi$
$233$	1.19556e8i	0.619193i	−0.950868	+	0.309597i	$0.899806\pi$
$234$	−2.25874e7	−0.115242
$235$	0	0
$236$	1.69002e8	0.836952
$237$	− 4.28016e7i	− 0.208853i
$238$	1.19530e8i	0.574723i
$239$	−3.96209e8	−1.87729	−0.938646	−	0.344883i	$-0.887919\pi$
$239$	−3.96209e8	−1.87729	−0.938646	+	0.344883i	$0.887919\pi$
$240$	0	0
$241$	−2.56606e8	−1.18089	−0.590443	−	0.807080i	$-0.701047\pi$
$241$	−2.56606e8	−1.18089	−0.590443	+	0.807080i	$0.701047\pi$
$242$	1.46358e8i	0.663837i
$243$	− 1.57319e8i	− 0.703331i
$244$	−5.29729e7	−0.233448
$245$	0	0
$246$	−1.04083e6	−0.00445767
$247$	− 5.51971e7i	− 0.233065i
$248$	1.16507e8i	0.485031i
$249$	−6.80747e7	−0.279440
$250$	0	0
$251$	−7.34775e7	−0.293290	−0.146645	−	0.989189i	$-0.546847\pi$
$251$	−7.34775e7	−0.293290	−0.146645	+	0.989189i	$0.546847\pi$
$252$	− 1.32844e8i	− 0.522926i
$253$	− 7.50335e7i	− 0.291295i
$254$	9.34605e6	0.0357857
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	− 2.02701e8i	− 0.744886i	−0.928055	−	0.372443i	$-0.878520\pi$
$257$	− 2.02701e8i	− 0.744886i	0.928055	−	0.372443i	$-0.121480\pi$
$258$	− 6.05518e7i	− 0.219512i
$259$	−1.63094e8	−0.583297
$260$	0	0
$261$	2.09551e8	0.729536
$262$	6.33906e7i	0.217756i
$263$	− 1.54254e8i	− 0.522867i	−0.965221	−	0.261434i	$-0.915805\pi$
$263$	− 1.54254e8i	− 0.522867i	0.965221	−	0.261434i	$-0.0841953\pi$
$264$	6.70925e6	0.0224419
$265$	0	0
$266$	3.24632e8	1.05756
$267$	− 1.43414e8i	− 0.461108i
$268$	8.06426e6i	0.0255913i
$269$	6.24018e8	1.95463	0.977315	−	0.211793i	$-0.0679302\pi$
$269$	6.24018e8	1.95463	0.977315	+	0.211793i	$0.0679302\pi$
$270$	0	0
$271$	−3.87983e8	−1.18419	−0.592094	−	0.805869i	$-0.701698\pi$
$271$	−3.87983e8	−1.18419	−0.592094	+	0.805869i	$0.701698\pi$
$272$	6.02358e7i	0.181494i
$273$	− 1.68493e7i	− 0.0501203i
$274$	−2.52523e6	−0.00741608
$275$	0	0
$276$	5.27708e7	0.151082
$277$	4.53952e8i	1.28331i	0.766994	+	0.641654i	$0.221752\pi$
$277$	4.53952e8i	1.28331i	−0.766994	+	0.641654i	$0.778248\pi$
$278$	3.13630e8i	0.875510i
$279$	4.64889e8	1.28155
$280$	0	0
$281$	3.33770e8	0.897377	0.448689	−	0.893688i	$-0.351891\pi$
$281$	3.33770e8	0.897377	0.448689	+	0.893688i	$0.351891\pi$
$282$	− 4.53750e7i	− 0.120488i
$283$	− 5.37695e8i	− 1.41021i	−0.709104	−	0.705104i	$-0.750900\pi$
$283$	− 5.37695e8i	− 1.41021i	0.709104	−	0.705104i	$-0.249100\pi$
$284$	9.05426e7	0.234552
$285$	0	0
$286$	−1.20732e7	−0.0305168
$287$	1.10155e7i	0.0275053i
$288$	− 6.69450e7i	− 0.165137i
$289$	1.94072e8	0.472956
$290$	0	0
$291$	1.04186e8	0.247847
$292$	6.27380e7i	0.147466i
$293$	− 3.35600e8i	− 0.779445i	−0.920932	−	0.389722i	$-0.872571\pi$
$293$	− 3.35600e8i	− 0.779445i	0.920932	−	0.389722i	$-0.127429\pi$
$294$	2.00364e7	0.0459838
$295$	0	0
$296$	−8.21893e7	−0.184202
$297$	− 5.54299e7i	− 0.122771i
$298$	− 1.75088e8i	− 0.383266i
$299$	−9.49600e7	−0.205443
$300$	0	0
$301$	−6.40840e8	−1.35446
$302$	2.35324e8i	0.491633i
$303$	1.20954e8i	0.249789i
$304$	1.63594e8	0.333972
$305$	0	0
$306$	2.40355e8	0.479543
$307$	2.15029e8i	0.424143i	0.977254	+	0.212072i	$0.0680210\pi$
$307$	2.15029e8i	0.424143i	−0.977254	+	0.212072i	$0.931979\pi$
$308$	− 7.10062e7i	− 0.138474i
$309$	−4.49759e7	−0.0867212
$310$	0	0
$311$	7.92062e8	1.49313	0.746565	−	0.665313i	$-0.231702\pi$
$311$	7.92062e8	1.49313	0.746565	+	0.665313i	$0.231702\pi$
$312$	− 8.49101e6i	− 0.0158277i
$313$	1.18457e8i	0.218352i	0.994022	+	0.109176i	$0.0348212\pi$
$313$	1.18457e8i	0.218352i	−0.994022	+	0.109176i	$0.965179\pi$
$314$	4.84440e8	0.883051
$315$	0	0
$316$	−2.28275e8	−0.406962
$317$	− 5.07310e7i	− 0.0894470i	−0.998999	−	0.0447235i	$-0.985759\pi$
$317$	− 5.07310e7i	− 0.0894470i	0.998999	−	0.0447235i	$-0.0142407\pi$
$318$	− 1.43426e8i	− 0.250111i
$319$	1.12006e8	0.193186
$320$	0	0
$321$	−2.15827e8	−0.364198
$322$	− 5.58491e8i	− 0.932225i
$323$	5.87358e8i	0.969826i
$324$	−2.46971e8	−0.403402
$325$	0	0
$326$	−4.56682e8	−0.730050
$327$	1.47084e8i	0.232621i
$328$	5.55110e6i	0.00868602i
$329$	−4.80218e8	−0.743453
$330$	0	0
$331$	2.73757e8	0.414923	0.207461	−	0.978243i	$-0.433480\pi$
$331$	2.73757e8	0.414923	0.207461	+	0.978243i	$0.433480\pi$
$332$	3.63065e8i	0.544504i
$333$	3.27955e8i	0.486697i
$334$	−7.01812e8	−1.03064
$335$	0	0
$336$	4.99384e7	0.0718203
$337$	− 9.18512e7i	− 0.130732i	−0.997861	−	0.0653658i	$-0.979179\pi$
$337$	− 9.18512e7i	− 0.130732i	0.997861	−	0.0653658i	$-0.0208214\pi$
$338$	− 4.86709e8i	− 0.685584i
$339$	−1.99140e8	−0.277625
$340$	0	0
$341$	2.48487e8	0.339362
$342$	− 6.52779e8i	− 0.882419i
$343$	6.24667e8i	0.835833i
$344$	−3.22943e8	−0.427732
$345$	0	0
$346$	−6.85563e7	−0.0889777
$347$	− 1.36700e9i	− 1.75637i	−0.478318	−	0.878187i	$-0.658753\pi$
$347$	− 1.36700e9i	− 1.75637i	0.478318	−	0.878187i	$-0.341247\pi$
$348$	7.87738e7i	0.100197i
$349$	−1.13143e9	−1.42475	−0.712377	−	0.701797i	$-0.752381\pi$
$349$	−1.13143e9	−1.42475	−0.712377	+	0.701797i	$0.752381\pi$
$350$	0	0
$351$	−7.01503e7	−0.0865874
$352$	− 3.57827e7i	− 0.0437294i
$353$	4.48395e7i	0.0542562i	0.999632	+	0.0271281i	$0.00863620\pi$
$353$	4.48395e7i	0.0542562i	−0.999632	+	0.0271281i	$0.991364\pi$
$354$	2.53503e8	0.303719
$355$	0	0
$356$	−7.64876e8	−0.898496
$357$	1.79296e8i	0.208560i
$358$	1.50432e8i	0.173281i
$359$	−3.98281e8	−0.454317	−0.227158	−	0.973858i	$-0.572943\pi$
$359$	−3.98281e8	−0.454317	−0.227158	+	0.973858i	$0.572943\pi$
$360$	0	0
$361$	7.01332e8	0.784600
$362$	4.79700e8i	0.531483i
$363$	2.19536e8i	0.240898i
$364$	−8.98632e7	−0.0976623
$365$	0	0
$366$	−7.94594e7	−0.0847152
$367$	1.63472e9i	1.72628i	0.504964	+	0.863140i	$0.331506\pi$
$367$	1.63472e9i	1.72628i	−0.504964	+	0.863140i	$0.668494\pi$
$368$	− 2.81444e8i	− 0.294391i
$369$	2.21502e7	0.0229501
$370$	0	0
$371$	−1.51792e9	−1.54327
$372$	1.74760e8i	0.176012i
$373$	1.54633e9i	1.54284i	0.636325	+	0.771421i	$0.280454\pi$
$373$	1.54633e9i	1.54284i	−0.636325	+	0.771421i	$0.719546\pi$
$374$	1.28472e8	0.126986
$375$	0	0
$376$	−2.42000e8	−0.234778
$377$	− 1.41752e8i	− 0.136249i
$378$	− 4.12577e8i	− 0.392902i
$379$	1.05688e9	0.997216	0.498608	−	0.866828i	$-0.333845\pi$
$379$	1.05688e9	0.997216	0.498608	+	0.866828i	$0.333845\pi$
$380$	0	0
$381$	1.40191e7	0.0129862
$382$	− 7.51889e8i	− 0.690132i
$383$	− 2.24910e8i	− 0.204556i	−0.994756	−	0.102278i	$-0.967387\pi$
$383$	− 2.24910e8i	− 0.204556i	0.994756	−	0.102278i	$-0.0326132\pi$
$384$	2.51658e7	0.0226805
$385$	0	0
$386$	2.81556e8	0.249178
$387$	1.28862e9i	1.13015i
$388$	− 5.55657e8i	− 0.482943i
$389$	−1.01788e9	−0.876746	−0.438373	−	0.898793i	$-0.644445\pi$
$389$	−1.01788e9	−0.876746	−0.438373	+	0.898793i	$0.644445\pi$
$390$	0	0
$391$	1.01048e9	0.854887
$392$	− 1.06861e8i	− 0.0896021i
$393$	9.50859e7i	0.0790210i
$394$	8.23883e8	0.678623
$395$	0	0
$396$	−1.42781e8	−0.115541
$397$	− 1.47565e9i	− 1.18363i	−0.806072	−	0.591817i	$-0.798411\pi$
$397$	− 1.47565e9i	− 1.18363i	0.806072	−	0.591817i	$-0.201589\pi$
$398$	6.69100e8i	0.531986i
$399$	4.86948e8	0.383776
$400$	0	0
$401$	2.74912e8	0.212906	0.106453	−	0.994318i	$-0.466051\pi$
$401$	2.74912e8	0.212906	0.106453	+	0.994318i	$0.466051\pi$
$402$	1.20964e7i	0.00928676i
$403$	− 3.14477e8i	− 0.239343i
$404$	6.45090e8	0.486727
$405$	0	0
$406$	8.33689e8	0.618248
$407$	1.75294e8i	0.128881i
$408$	9.03537e7i	0.0658620i
$409$	1.63427e9	1.18112	0.590558	−	0.806995i	$-0.298908\pi$
$409$	1.63427e9	1.18112	0.590558	+	0.806995i	$0.298908\pi$
$410$	0	0
$411$	−3.78785e6	−0.00269120
$412$	2.39871e8i	0.168981i
$413$	− 2.68291e9i	− 1.87405i
$414$	−1.12303e9	−0.777839
$415$	0	0
$416$	−4.52854e7	−0.0308412
$417$	4.70445e8i	0.317712i
$418$	− 3.48916e8i	− 0.233670i
$419$	1.11280e9	0.739039	0.369519	−	0.929223i	$-0.379522\pi$
$419$	1.11280e9	0.739039	0.369519	+	0.929223i	$0.379522\pi$
$420$	0	0
$421$	9.22528e8	0.602549	0.301274	−	0.953537i	$-0.402588\pi$
$421$	9.22528e8	0.602549	0.301274	+	0.953537i	$0.402588\pi$
$422$	7.79208e8i	0.504731i
$423$	9.65636e8i	0.620329i
$424$	−7.64937e8	−0.487355
$425$	0	0
$426$	1.35814e8	0.0851159
$427$	8.40945e8i	0.522721i
$428$	1.15108e9i	0.709661i
$429$	−1.81097e7	−0.0110742
$430$	0	0
$431$	−9.81508e8	−0.590505	−0.295252	−	0.955419i	$-0.595404\pi$
$431$	−9.81508e8	−0.590505	−0.295252	+	0.955419i	$0.595404\pi$
$432$	− 2.07913e8i	− 0.124076i
$433$	− 2.84998e9i	− 1.68707i	−0.537071	−	0.843537i	$-0.680469\pi$
$433$	− 2.84998e9i	− 1.68707i	0.537071	−	0.843537i	$-0.319531\pi$
$434$	1.84954e9	1.08605
$435$	0	0
$436$	7.84450e8	0.453276
$437$	− 2.74436e9i	− 1.57310i
$438$	9.41071e7i	0.0535134i
$439$	1.05622e9	0.595838	0.297919	−	0.954591i	$-0.403708\pi$
$439$	1.05622e9	0.595838	0.297919	+	0.954591i	$0.403708\pi$
$440$	0	0
$441$	−4.26401e8	−0.236746
$442$	− 1.62590e8i	− 0.0895601i
$443$	− 1.82325e9i	− 0.996401i	−0.867062	−	0.498201i	$-0.833994\pi$
$443$	− 1.82325e9i	− 0.996401i	0.867062	−	0.498201i	$-0.166006\pi$
$444$	−1.23284e8	−0.0668446
$445$	0	0
$446$	1.17166e8	0.0625358
$447$	− 2.62633e8i	− 0.139082i
$448$	− 2.66338e8i	− 0.139946i
$449$	−1.84846e9	−0.963713	−0.481856	−	0.876250i	$-0.660037\pi$
$449$	−1.84846e9	−0.963713	−0.481856	+	0.876250i	$0.660037\pi$
$450$	0	0
$451$	1.18395e7	0.00607735
$452$	1.06208e9i	0.540968i
$453$	3.52985e8i	0.178407i
$454$	−1.47633e9	−0.740435
$455$	0	0
$456$	2.45391e8	0.121194
$457$	− 2.98066e9i	− 1.46085i	−0.682993	−	0.730425i	$-0.739322\pi$
$457$	− 2.98066e9i	− 1.46085i	0.682993	−	0.730425i	$-0.260678\pi$
$458$	− 7.00369e7i	− 0.0340642i
$459$	7.46477e8	0.360306
$460$	0	0
$461$	−2.52781e9	−1.20169	−0.600843	−	0.799367i	$-0.705168\pi$
$461$	−2.52781e9	−1.20169	−0.600843	+	0.799367i	$0.705168\pi$
$462$	− 1.06509e8i	− 0.0502505i
$463$	8.90291e8i	0.416868i	0.978036	+	0.208434i	$0.0668366\pi$
$463$	8.90291e8i	0.416868i	−0.978036	+	0.208434i	$0.933163\pi$
$464$	4.20127e8	0.195239
$465$	0	0
$466$	9.56450e8	0.437836
$467$	2.65667e9i	1.20706i	0.797341	+	0.603529i	$0.206239\pi$
$467$	2.65667e9i	1.20706i	−0.797341	+	0.603529i	$0.793761\pi$
$468$	1.80699e8i	0.0814884i
$469$	1.28020e8	0.0573024
$470$	0	0
$471$	7.26660e8	0.320448
$472$	− 1.35202e9i	− 0.591814i
$473$	6.88777e8i	0.299271i
$474$	−3.42413e8	−0.147681
$475$	0	0
$476$	9.56243e8	0.406391
$477$	3.05228e9i	1.28769i
$478$	3.16967e9i	1.32745i
$479$	−1.30093e9	−0.540855	−0.270428	−	0.962740i	$-0.587165\pi$
$479$	−1.30093e9	−0.540855	−0.270428	+	0.962740i	$0.587165\pi$
$480$	0	0
$481$	2.21847e8	0.0908962
$482$	2.05285e9i	0.835012i
$483$	− 8.37737e8i	− 0.338293i
$484$	1.17086e9	0.469404
$485$	0	0
$486$	−1.25855e9	−0.497330
$487$	− 1.07447e9i	− 0.421542i	−0.977535	−	0.210771i	$-0.932402\pi$
$487$	− 1.07447e9i	− 0.421542i	0.977535	−	0.210771i	$-0.0675975\pi$
$488$	4.23783e8i	0.165072i
$489$	−6.85024e8	−0.264926
$490$	0	0
$491$	−7.83344e8	−0.298653	−0.149327	−	0.988788i	$-0.547711\pi$
$491$	−7.83344e8	−0.298653	−0.149327	+	0.988788i	$0.547711\pi$
$492$	8.32666e6i	0.00315205i
$493$	1.50839e9i	0.566958i
$494$	−4.41577e8	−0.164802
$495$	0	0
$496$	9.32053e8	0.342969
$497$	− 1.43736e9i	− 0.525193i
$498$	5.44598e8i	0.197594i
$499$	6.23188e8	0.224526	0.112263	−	0.993679i	$-0.464190\pi$
$499$	6.23188e8	0.224526	0.112263	+	0.993679i	$0.464190\pi$
$500$	0	0
$501$	−1.05272e9	−0.374007
$502$	5.87820e8i	0.207387i
$503$	2.70927e9i	0.949215i	0.880198	+	0.474607i	$0.157410\pi$
$503$	2.70927e9i	0.949215i	−0.880198	+	0.474607i	$0.842590\pi$
$504$	−1.06275e9	−0.369764
$505$	0	0
$506$	−6.00268e8	−0.205977
$507$	− 7.30063e8i	− 0.248790i
$508$	− 7.47684e7i	− 0.0253043i
$509$	−3.49943e9	−1.17621	−0.588106	−	0.808784i	$-0.700126\pi$
$509$	−3.49943e9	−1.17621	−0.588106	+	0.808784i	$0.700126\pi$
$510$	0	0
$511$	9.95967e8	0.330196
$512$	− 1.34218e8i	− 0.0441942i
$513$	− 2.02735e9i	− 0.663008i
$514$	−1.62161e9	−0.526714
$515$	0	0
$516$	−4.84414e8	−0.155218
$517$	5.16140e8i	0.164267i
$518$	1.30476e9i	0.412453i
$519$	−1.02835e8	−0.0322889
$520$	0	0
$521$	−1.37683e9	−0.426530	−0.213265	−	0.976994i	$-0.568410\pi$
$521$	−1.37683e9	−0.426530	−0.213265	+	0.976994i	$0.568410\pi$
$522$	− 1.67640e9i	− 0.515860i
$523$	2.86154e9i	0.874669i	0.899299	+	0.437334i	$0.144077\pi$
$523$	2.86154e9i	0.874669i	−0.899299	+	0.437334i	$0.855923\pi$
$524$	5.07125e8	0.153977
$525$	0	0
$526$	−1.23403e9	−0.369723
$527$	3.34638e9i	0.995951i
$528$	− 5.36740e7i	− 0.0158688i
$529$	−1.31651e9	−0.386661
$530$	0	0
$531$	−5.39487e9	−1.56369
$532$	− 2.59706e9i	− 0.747810i
$533$	− 1.49836e7i	− 0.00428620i
$534$	−1.14731e9	−0.326053
$535$	0	0
$536$	6.45140e7	0.0180958
$537$	2.25649e8i	0.0628815i
$538$	− 4.99215e9i	− 1.38213i
$539$	−2.27915e8	−0.0626919
$540$	0	0
$541$	5.34467e9	1.45121	0.725605	−	0.688111i	$-0.241560\pi$
$541$	5.34467e9	1.45121	0.725605	+	0.688111i	$0.241560\pi$
$542$	3.10387e9i	0.837347i
$543$	7.19550e8i	0.192869i
$544$	4.81886e8	0.128336
$545$	0	0
$546$	−1.34795e8	−0.0354404
$547$	− 3.37135e9i	− 0.880740i	−0.897816	−	0.440370i	$-0.854847\pi$
$547$	− 3.37135e9i	− 0.880740i	0.897816	−	0.440370i	$-0.145153\pi$
$548$	2.02019e7i	0.00524396i
$549$	1.69100e9	0.436153
$550$	0	0
$551$	4.09665e9	1.04327
$552$	− 4.22167e8i	− 0.106831i
$553$	3.62387e9i	0.911244i
$554$	3.63162e9	0.907436
$555$	0	0
$556$	2.50904e9	0.619079
$557$	− 5.61106e9i	− 1.37579i	−0.725811	−	0.687894i	$-0.758535\pi$
$557$	− 5.61106e9i	− 1.37579i	0.725811	−	0.687894i	$-0.241465\pi$
$558$	− 3.71911e9i	− 0.906190i
$559$	8.71694e8	0.211068
$560$	0	0
$561$	1.92707e8	0.0460817
$562$	− 2.67016e9i	− 0.634542i
$563$	− 6.69690e9i	− 1.58159i	−0.612081	−	0.790795i	$-0.709667\pi$
$563$	− 6.69690e9i	− 1.58159i	0.612081	−	0.790795i	$-0.290333\pi$
$564$	−3.63000e8	−0.0851980
$565$	0	0
$566$	−4.30156e9	−0.997168
$567$	3.92066e9i	0.903273i
$568$	− 7.24341e8i	− 0.165853i
$569$	−1.96850e9	−0.447964	−0.223982	−	0.974593i	$-0.571906\pi$
$569$	−1.96850e9	−0.447964	−0.223982	+	0.974593i	$0.571906\pi$
$570$	0	0
$571$	1.02926e9	0.231365	0.115682	−	0.993286i	$-0.463094\pi$
$571$	1.02926e9	0.231365	0.115682	+	0.993286i	$0.463094\pi$
$572$	9.65852e7i	0.0215787i
$573$	− 1.12783e9i	− 0.250440i
$574$	8.81238e7	0.0194492
$575$	0	0
$576$	−5.35560e8	−0.116770
$577$	3.31179e9i	0.717708i	0.933394	+	0.358854i	$0.116832\pi$
$577$	3.31179e9i	0.717708i	−0.933394	+	0.358854i	$0.883168\pi$
$578$	− 1.55258e9i	− 0.334431i
$579$	4.22335e8	0.0904236
$580$	0	0
$581$	5.76366e9	1.21922
$582$	− 8.33486e8i	− 0.175254i
$583$	1.63147e9i	0.340988i
$584$	5.01904e8	0.104274
$585$	0	0
$586$	−2.68480e9	−0.551151
$587$	− 5.59411e8i	− 0.114156i	−0.998370	−	0.0570778i	$-0.981822\pi$
$587$	− 5.59411e8i	− 0.114156i	0.998370	−	0.0570778i	$-0.0181783\pi$
$588$	− 1.60292e8i	− 0.0325155i
$589$	9.08843e9	1.83267
$590$	0	0
$591$	1.23582e9	0.246264
$592$	6.57514e8i	0.130250i
$593$	3.02459e9i	0.595628i	0.954624	+	0.297814i	$0.0962575\pi$
$593$	3.02459e9i	0.595628i	−0.954624	+	0.297814i	$0.903742\pi$
$594$	−4.43439e8	−0.0868124
$595$	0	0
$596$	−1.40071e9	−0.271010
$597$	1.00365e9i	0.193051i
$598$	7.59680e8i	0.145270i
$599$	5.63246e9	1.07079	0.535395	−	0.844602i	$-0.320163\pi$
$599$	5.63246e9	1.07079	0.535395	+	0.844602i	$0.320163\pi$
$600$	0	0
$601$	3.40792e8	0.0640366	0.0320183	−	0.999487i	$-0.489807\pi$
$601$	3.40792e8	0.0640366	0.0320183	+	0.999487i	$0.489807\pi$
$602$	5.12672e9i	0.957749i
$603$	− 2.57426e8i	− 0.0478126i
$604$	1.88259e9	0.347637
$605$	0	0
$606$	9.67636e8	0.176627
$607$	3.85420e9i	0.699477i	0.936847	+	0.349739i	$0.113730\pi$
$607$	3.85420e9i	0.699477i	−0.936847	+	0.349739i	$0.886270\pi$
$608$	− 1.30875e9i	− 0.236154i
$609$	1.25053e9	0.224355
$610$	0	0
$611$	6.53211e8	0.115853
$612$	− 1.92284e9i	− 0.339088i
$613$	− 9.22245e9i	− 1.61709i	−0.588434	−	0.808545i	$-0.700255\pi$
$613$	− 9.22245e9i	− 1.61709i	0.588434	−	0.808545i	$-0.299745\pi$
$614$	1.72023e9	0.299915
$615$	0	0
$616$	−5.68050e8	−0.0979160
$617$	6.53611e9i	1.12027i	0.828402	+	0.560133i	$0.189250\pi$
$617$	6.53611e9i	1.12027i	−0.828402	+	0.560133i	$0.810750\pi$
$618$	3.59807e8i	0.0613211i
$619$	−1.36559e9	−0.231420	−0.115710	−	0.993283i	$-0.536914\pi$
$619$	−1.36559e9	−0.231420	−0.115710	+	0.993283i	$0.536914\pi$
$620$	0	0
$621$	−3.48782e9	−0.584431
$622$	− 6.33649e9i	− 1.05580i
$623$	1.21424e10i	2.01186i
$624$	−6.79281e7	−0.0111919
$625$	0	0
$626$	9.47659e8	0.154398
$627$	− 5.23374e8i	− 0.0847960i
$628$	− 3.87552e9i	− 0.624412i
$629$	−2.36070e9	−0.378236
$630$	0	0
$631$	1.54079e9	0.244141	0.122070	−	0.992521i	$-0.461047\pi$
$631$	1.54079e9	0.244141	0.122070	+	0.992521i	$0.461047\pi$
$632$	1.82620e9i	0.287766i
$633$	1.16881e9i	0.183160i
$634$	−4.05848e8	−0.0632486
$635$	0	0
$636$	−1.14741e9	−0.176855
$637$	2.88441e8i	0.0442150i
$638$	− 8.96052e8i	− 0.136603i
$639$	−2.89029e9	−0.438216
$640$	0	0
$641$	−4.54018e9	−0.680879	−0.340440	−	0.940266i	$-0.610576\pi$
$641$	−4.54018e9	−0.680879	−0.340440	+	0.940266i	$0.610576\pi$
$642$	1.72661e9i	0.257527i
$643$	− 1.14054e10i	− 1.69189i	−0.533272	−	0.845944i	$-0.679038\pi$
$643$	− 1.14054e10i	− 1.69189i	0.533272	−	0.845944i	$-0.320962\pi$
$644$	−4.46793e9	−0.659183
$645$	0	0
$646$	4.69886e9	0.685770
$647$	− 1.26393e10i	− 1.83468i	−0.398109	−	0.917338i	$-0.630334\pi$
$647$	− 1.26393e10i	− 1.83468i	0.398109	−	0.917338i	$-0.369666\pi$
$648$	1.97577e9i	0.285248i
$649$	−2.88360e9	−0.414075
$650$	0	0
$651$	2.77431e9	0.394114
$652$	3.65346e9i	0.516223i
$653$	1.05004e10i	1.47575i	0.674940	+	0.737873i	$0.264170\pi$
$653$	1.05004e10i	1.47575i	−0.674940	+	0.737873i	$0.735830\pi$
$654$	1.17667e9	0.164488
$655$	0	0
$656$	4.44088e7	0.00614194
$657$	− 2.00272e9i	− 0.275512i
$658$	3.84175e9i	0.525700i
$659$	−9.64818e9	−1.31325	−0.656624	−	0.754219i	$-0.728016\pi$
$659$	−9.64818e9	−1.31325	−0.656624	+	0.754219i	$0.728016\pi$
$660$	0	0
$661$	−6.58299e9	−0.886580	−0.443290	−	0.896378i	$-0.646189\pi$
$661$	−6.58299e9	−0.886580	−0.443290	+	0.896378i	$0.646189\pi$
$662$	− 2.19006e9i	− 0.293395i
$663$	− 2.43884e8i	− 0.0325002i
$664$	2.90452e9	0.385023
$665$	0	0
$666$	2.62364e9	0.344147
$667$	− 7.04779e9i	− 0.919629i
$668$	5.61450e9i	0.728775i
$669$	1.75749e8	0.0226935
$670$	0	0
$671$	9.03851e8	0.115496
$672$	− 3.99507e8i	− 0.0507846i
$673$	8.54649e9i	1.08077i	0.841416	+	0.540387i	$0.181722\pi$
$673$	8.54649e9i	1.08077i	−0.841416	+	0.540387i	$0.818278\pi$
$674$	−7.34810e8	−0.0924411
$675$	0	0
$676$	−3.89367e9	−0.484781
$677$	8.71305e9i	1.07922i	0.841915	+	0.539610i	$0.181428\pi$
$677$	8.71305e9i	1.07922i	−0.841915	+	0.539610i	$0.818572\pi$
$678$	1.59312e9i	0.196311i
$679$	−8.82106e9	−1.08138
$680$	0	0
$681$	−2.21449e9	−0.268695
$682$	− 1.98789e9i	− 0.239965i
$683$	− 1.46109e10i	− 1.75470i	−0.479849	−	0.877351i	$-0.659308\pi$
$683$	− 1.46109e10i	− 1.75470i	0.479849	−	0.877351i	$-0.340692\pi$
$684$	−5.22223e9	−0.623965
$685$	0	0
$686$	4.99734e9	0.591023
$687$	− 1.05055e8i	− 0.0123615i
$688$	2.58354e9i	0.302452i
$689$	2.06473e9	0.240490
$690$	0	0
$691$	−1.47348e10	−1.69891	−0.849454	−	0.527662i	$-0.823069\pi$
$691$	−1.47348e10	−1.69891	−0.849454	+	0.527662i	$0.823069\pi$
$692$	5.48451e8i	0.0629167i
$693$	2.26665e9i	0.258713i
$694$	−1.09360e10	−1.24194
$695$	0	0
$696$	6.30190e8	0.0708499
$697$	1.59442e8i	0.0178357i
$698$	9.05146e9i	1.00745i
$699$	1.43467e9	0.158885
$700$	0	0
$701$	1.31502e9	0.144185	0.0720923	−	0.997398i	$-0.477032\pi$
$701$	1.31502e9	0.144185	0.0720923	+	0.997398i	$0.477032\pi$
$702$	5.61203e8i	0.0612265i
$703$	6.41141e9i	0.696001i
$704$	−2.86261e8	−0.0309213
$705$	0	0
$706$	3.58716e8	0.0383649
$707$	− 1.02408e10i	− 1.08985i
$708$	− 2.02803e9i	− 0.214762i
$709$	−6.64028e8	−0.0699721	−0.0349860	−	0.999388i	$-0.511139\pi$
$709$	−6.64028e8	−0.0699721	−0.0349860	+	0.999388i	$0.511139\pi$
$710$	0	0
$711$	7.28697e9	0.760332
$712$	6.11901e9i	0.635333i
$713$	− 1.56356e10i	− 1.61547i
$714$	1.43436e9	0.147474
$715$	0	0
$716$	1.20346e9	0.122528
$717$	4.75451e9i	0.481713i
$718$	3.18624e9i	0.321250i
$719$	−4.95034e9	−0.496689	−0.248344	−	0.968672i	$-0.579886\pi$
$719$	−4.95034e9	−0.496689	−0.248344	+	0.968672i	$0.579886\pi$
$720$	0	0
$721$	3.80796e9	0.378372
$722$	− 5.61065e9i	− 0.554796i
$723$	3.07928e9i	0.303015i
$724$	3.83760e9	0.375816
$725$	0	0
$726$	1.75629e9	0.170341
$727$	8.81101e9i	0.850463i	0.905085	+	0.425231i	$0.139807\pi$
$727$	8.81101e9i	0.850463i	−0.905085	+	0.425231i	$0.860193\pi$
$728$	7.18905e8i	0.0690577i
$729$	6.55163e9	0.626330
$730$	0	0
$731$	−9.27578e9	−0.878293
$732$	6.35675e8i	0.0599027i
$733$	1.49414e8i	0.0140129i	0.999975	+	0.00700643i	$0.00223023\pi$
$733$	1.49414e8i	0.0140129i	−0.999975	+	0.00700643i	$0.997770\pi$
$734$	1.30777e10	1.22066
$735$	0	0
$736$	−2.25155e9	−0.208166
$737$	− 1.37596e8i	− 0.0126611i
$738$	− 1.77202e8i	− 0.0162282i
$739$	4.70806e9	0.429127	0.214564	−	0.976710i	$-0.431167\pi$
$739$	4.70806e9	0.429127	0.214564	+	0.976710i	$0.431167\pi$
$740$	0	0
$741$	−6.62365e8	−0.0598045
$742$	1.21434e10i	1.09125i
$743$	− 1.69676e9i	− 0.151761i	−0.997117	−	0.0758805i	$-0.975823\pi$
$743$	− 1.69676e9i	− 0.151761i	0.997117	−	0.0758805i	$-0.0241768\pi$
$744$	1.39808e9	0.124459
$745$	0	0
$746$	1.23707e10	1.09095
$747$	− 1.15897e10i	− 1.01730i
$748$	− 1.02777e9i	− 0.0897928i
$749$	1.82733e10	1.58903
$750$	0	0
$751$	1.06650e10	0.918800	0.459400	−	0.888229i	$-0.348064\pi$
$751$	1.06650e10	0.918800	0.459400	+	0.888229i	$0.348064\pi$
$752$	1.93600e9i	0.166013i
$753$	8.81731e8i	0.0752581i
$754$	−1.13401e9	−0.0963427
$755$	0	0
$756$	−3.30062e9	−0.277824
$757$	6.22876e9i	0.521874i	0.965356	+	0.260937i	$0.0840315\pi$
$757$	6.22876e9i	0.521874i	−0.965356	+	0.260937i	$0.915968\pi$
$758$	− 8.45506e9i	− 0.705138i
$759$	−9.00402e8	−0.0747464
$760$	0	0
$761$	−8.38334e9	−0.689558	−0.344779	−	0.938684i	$-0.612046\pi$
$761$	−8.38334e9	−0.689558	−0.344779	+	0.938684i	$0.612046\pi$
$762$	− 1.12153e8i	− 0.00918263i
$763$	− 1.24531e10i	− 1.01495i
$764$	−6.01511e9	−0.487997
$765$	0	0
$766$	−1.79928e9	−0.144643
$767$	3.64939e9i	0.292036i
$768$	− 2.01327e8i	− 0.0160375i
$769$	1.18649e10	0.940852	0.470426	−	0.882439i	$-0.344100\pi$
$769$	1.18649e10	0.940852	0.470426	+	0.882439i	$0.344100\pi$
$770$	0	0
$771$	−2.43241e9	−0.191138
$772$	− 2.25245e9i	− 0.176196i
$773$	− 5.56680e9i	− 0.433488i	−0.976228	−	0.216744i	$-0.930456\pi$
$773$	− 5.56680e9i	− 0.433488i	0.976228	−	0.216744i	$-0.0695438\pi$
$774$	1.03089e10	0.799136
$775$	0	0
$776$	−4.44526e9	−0.341492
$777$	1.95713e9i	0.149674i
$778$	8.14306e9i	0.619953i
$779$	4.33029e8	0.0328198
$780$	0	0
$781$	−1.54488e9	−0.116042
$782$	− 8.08383e9i	− 0.604496i
$783$	− 5.20645e9i	− 0.387593i
$784$	−8.54888e8	−0.0633583
$785$	0	0
$786$	7.60687e8	0.0558763
$787$	1.34611e8i	0.00984395i	0.999988	+	0.00492198i	$0.00156672\pi$
$787$	1.34611e8i	0.00984395i	−0.999988	+	0.00492198i	$0.998433\pi$
$788$	− 6.59106e9i	− 0.479859i
$789$	−1.85105e9	−0.134168
$790$	0	0
$791$	1.68605e10	1.21130
$792$	1.14225e9i	0.0817001i
$793$	− 1.14388e9i	− 0.0814565i
$794$	−1.18052e10	−0.836955
$795$	0	0
$796$	5.35280e9	0.376171
$797$	− 7.41548e9i	− 0.518842i	−0.965764	−	0.259421i	$-0.916468\pi$
$797$	− 7.41548e9i	− 0.518842i	0.965764	−	0.259421i	$-0.0835317\pi$
$798$	− 3.89559e9i	− 0.271371i
$799$	−6.95088e9	−0.482088
$800$	0	0
$801$	2.44163e10	1.67867
$802$	− 2.19930e9i	− 0.150548i
$803$	− 1.07047e9i	− 0.0729574i
$804$	9.67711e7	0.00656673
$805$	0	0
$806$	−2.51581e9	−0.169241
$807$	− 7.48822e9i	− 0.501558i
$808$	− 5.16072e9i	− 0.344168i
$809$	1.41542e10	0.939863	0.469932	−	0.882703i	$-0.344279\pi$
$809$	1.41542e10	0.939863	0.469932	+	0.882703i	$0.344279\pi$
$810$	0	0
$811$	−2.63708e10	−1.73600	−0.868001	−	0.496563i	$-0.834595\pi$
$811$	−2.63708e10	−1.73600	−0.868001	+	0.496563i	$0.834595\pi$
$812$	− 6.66951e9i	− 0.437168i
$813$	4.65580e9i	0.303863i
$814$	1.40236e9	0.0911324
$815$	0	0
$816$	7.22829e8	0.0465715
$817$	2.51921e10i	1.61617i
$818$	− 1.30742e10i	− 0.835176i
$819$	2.86860e9	0.182464
$820$	0	0
$821$	8.06264e9	0.508483	0.254241	−	0.967141i	$-0.418174\pi$
$821$	8.06264e9	0.508483	0.254241	+	0.967141i	$0.418174\pi$
$822$	3.03028e7i	0.00190297i
$823$	2.34202e10i	1.46451i	0.681033	+	0.732253i	$0.261531\pi$
$823$	2.34202e10i	1.46451i	−0.681033	+	0.732253i	$0.738469\pi$
$824$	1.91897e9	0.119488
$825$	0	0
$826$	−2.14633e10	−1.32515
$827$	5.55722e9i	0.341655i	0.985301	+	0.170828i	$0.0546442\pi$
$827$	5.55722e9i	0.341655i	−0.985301	+	0.170828i	$0.945356\pi$
$828$	8.98423e9i	0.550015i
$829$	−2.84256e10	−1.73288	−0.866440	−	0.499281i	$-0.833597\pi$
$829$	−2.84256e10	−1.73288	−0.866440	+	0.499281i	$0.833597\pi$
$830$	0	0
$831$	5.44743e9	0.329297
$832$	3.62283e8i	0.0218080i
$833$	− 3.06933e9i	− 0.183987i
$834$	3.76356e9	0.224656
$835$	0	0
$836$	−2.79133e9	−0.165230
$837$	− 1.15505e10i	− 0.680868i
$838$	− 8.90238e9i	− 0.522579i
$839$	−1.04036e10	−0.608156	−0.304078	−	0.952647i	$-0.598348\pi$
$839$	−1.04036e10	−0.608156	−0.304078	+	0.952647i	$0.598348\pi$
$840$	0	0
$841$	−6.72927e9	−0.390105
$842$	− 7.38023e9i	− 0.426066i
$843$	− 4.00524e9i	− 0.230267i
$844$	6.23367e9	0.356899
$845$	0	0
$846$	7.72509e9	0.438639
$847$	− 1.85874e10i	− 1.05106i
$848$	6.11950e9i	0.344612i
$849$	−6.45234e9	−0.361860
$850$	0	0
$851$	1.10301e10	0.613514
$852$	− 1.08651e9i	− 0.0601860i
$853$	1.80580e10i	0.996205i	0.867118	+	0.498102i	$0.165970\pi$
$853$	1.80580e10i	0.996205i	−0.867118	+	0.498102i	$0.834030\pi$
$854$	6.72756e9	0.369620
$855$	0	0
$856$	9.20861e9	0.501806
$857$	− 6.34034e9i	− 0.344096i	−0.985089	−	0.172048i	$-0.944962\pi$
$857$	− 6.34034e9i	− 0.344096i	0.985089	−	0.172048i	$-0.0550384\pi$
$858$	1.44878e8i	0.00783062i
$859$	−1.21489e10	−0.653973	−0.326987	−	0.945029i	$-0.606033\pi$
$859$	−1.21489e10	−0.653973	−0.326987	+	0.945029i	$0.606033\pi$
$860$	0	0
$861$	1.32186e8	0.00705786
$862$	7.85206e9i	0.417550i
$863$	2.87111e10i	1.52059i	0.649578	+	0.760295i	$0.274946\pi$
$863$	2.87111e10i	1.52059i	−0.649578	+	0.760295i	$0.725054\pi$
$864$	−1.66330e9	−0.0877351
$865$	0	0
$866$	−2.27998e10	−1.19294
$867$	− 2.32887e9i	− 0.121361i
$868$	− 1.47963e10i	− 0.767954i
$869$	3.89495e9	0.201341
$870$	0	0
$871$	−1.74138e8	−0.00892953
$872$	− 6.27560e9i	− 0.320514i
$873$	1.77376e10i	0.902289i
$874$	−2.19549e10	−1.11235
$875$	0	0
$876$	7.52857e8	0.0378397
$877$	2.46021e10i	1.23161i	0.787898	+	0.615806i	$0.211169\pi$
$877$	2.46021e10i	1.23161i	−0.787898	+	0.615806i	$0.788831\pi$
$878$	− 8.44975e9i	− 0.421321i
$879$	−4.02720e9	−0.200006
$880$	0	0
$881$	−1.25378e10	−0.617738	−0.308869	−	0.951105i	$-0.599951\pi$
$881$	−1.25378e10	−0.617738	−0.308869	+	0.951105i	$0.599951\pi$
$882$	3.41121e9i	0.167405i
$883$	− 1.93097e10i	− 0.943873i	−0.881633	−	0.471937i	$-0.843555\pi$
$883$	− 1.93097e10i	− 0.943873i	0.881633	−	0.471937i	$-0.156445\pi$
$884$	−1.30072e9	−0.0633286
$885$	0	0
$886$	−1.45860e10	−0.704562
$887$	3.20268e10i	1.54092i	0.637486	+	0.770462i	$0.279974\pi$
$887$	3.20268e10i	1.54092i	−0.637486	+	0.770462i	$0.720026\pi$
$888$	9.86272e8i	0.0472663i
$889$	−1.18695e9	−0.0566599
$890$	0	0
$891$	4.21394e9	0.199580
$892$	− 9.37327e8i	− 0.0442195i
$893$	1.88779e10i	0.887101i
$894$	−2.10106e9	−0.0983461
$895$	0	0
$896$	−2.13071e9	−0.0989568
$897$	1.13952e9i	0.0527167i
$898$	1.47877e10i	0.681448i
$899$	2.33400e10	1.07138
$900$	0	0
$901$	−2.19710e10	−1.00072
$902$	− 9.47157e7i	− 0.00429733i
$903$	7.69008e9i	0.347555i
$904$	8.49662e9	0.382522
$905$	0	0
$906$	2.82388e9	0.126153
$907$	2.33703e9i	0.104002i	0.998647	+	0.0520008i	$0.0165598\pi$
$907$	2.33703e9i	0.104002i	−0.998647	+	0.0520008i	$0.983440\pi$
$908$	1.18106e10i	0.523567i
$909$	−2.05925e10	−0.909359
$910$	0	0
$911$	2.20343e10	0.965573	0.482786	−	0.875738i	$-0.339625\pi$
$911$	2.20343e10	0.965573	0.482786	+	0.875738i	$0.339625\pi$
$912$	− 1.96313e9i	− 0.0856973i
$913$	− 6.19480e9i	− 0.269389i
$914$	−2.38453e10	−1.03298
$915$	0	0
$916$	−5.60295e8	−0.0240870
$917$	− 8.05061e9i	− 0.344775i
$918$	− 5.97181e9i	− 0.254775i
$919$	1.43277e10	0.608938	0.304469	−	0.952522i	$-0.401521\pi$
$919$	1.43277e10	0.608938	0.304469	+	0.952522i	$0.401521\pi$
$920$	0	0
$921$	2.58035e9	0.108835
$922$	2.02225e10i	0.849720i
$923$	1.95515e9i	0.0818418i
$924$	−8.52074e8	−0.0355325
$925$	0	0
$926$	7.12233e9	0.294770
$927$	− 7.65715e9i	− 0.315710i
$928$	− 3.36101e9i	− 0.138055i
$929$	−1.31280e10	−0.537208	−0.268604	−	0.963251i	$-0.586562\pi$
$929$	−1.31280e10	−0.537208	−0.268604	+	0.963251i	$0.586562\pi$
$930$	0	0
$931$	−8.33600e9	−0.338558
$932$	− 7.65160e9i	− 0.309597i
$933$	− 9.50474e9i	− 0.383137i
$934$	2.12533e10	0.853519
$935$	0	0
$936$	1.44559e9	0.0576210
$937$	− 3.87626e10i	− 1.53930i	−0.638463	−	0.769652i	$-0.720429\pi$
$937$	− 3.87626e10i	− 1.53930i	0.638463	−	0.769652i	$-0.279571\pi$
$938$	− 1.02416e9i	− 0.0405189i
$939$	1.42149e9	0.0560291
$940$	0	0
$941$	2.06279e10	0.807035	0.403517	−	0.914972i	$-0.367788\pi$
$941$	2.06279e10	0.807035	0.403517	+	0.914972i	$0.367788\pi$
$942$	− 5.81328e9i	− 0.226591i
$943$	− 7.44976e8i	− 0.0289302i
$944$	−1.08161e10	−0.418476
$945$	0	0
$946$	5.51021e9	0.211617
$947$	− 2.11705e10i	− 0.810040i	−0.914308	−	0.405020i	$-0.867264\pi$
$947$	− 2.11705e10i	− 0.810040i	0.914308	−	0.405020i	$-0.132736\pi$
$948$	2.73930e9i	0.104427i
$949$	−1.35475e9	−0.0514550
$950$	0	0
$951$	−6.08771e8	−0.0229521
$952$	− 7.64994e9i	− 0.287362i
$953$	− 2.14876e10i	− 0.804196i	−0.915597	−	0.402098i	$-0.868281\pi$
$953$	− 2.14876e10i	− 0.804196i	0.915597	−	0.402098i	$-0.131719\pi$
$954$	2.44182e10	0.910531
$955$	0	0
$956$	2.53574e10	0.938646
$957$	− 1.34408e9i	− 0.0495715i
$958$	1.04075e10i	0.382442i
$959$	3.20704e8	0.0117419
$960$	0	0
$961$	2.42673e10	0.882043
$962$	− 1.77478e9i	− 0.0642733i
$963$	− 3.67445e10i	− 1.32587i
$964$	1.64228e10	0.590443
$965$	0	0
$966$	−6.70189e9	−0.239209
$967$	3.92625e10i	1.39632i	0.715941	+	0.698161i	$0.245998\pi$
$967$	3.92625e10i	1.39632i	−0.715941	+	0.698161i	$0.754002\pi$
$968$	− 9.36689e9i	− 0.331919i
$969$	7.04829e9	0.248857
$970$	0	0
$971$	−5.62647e10	−1.97228	−0.986140	−	0.165917i	$-0.946941\pi$
$971$	−5.62647e10	−1.97228	−0.986140	+	0.165917i	$0.946941\pi$
$972$	1.00684e10i	0.351665i
$973$	− 3.98310e10i	− 1.38620i
$974$	−8.59573e9	−0.298076
$975$	0	0
$976$	3.39027e9	0.116724
$977$	− 8.43437e9i	− 0.289349i	−0.989479	−	0.144674i	$-0.953787\pi$
$977$	− 8.43437e9i	− 0.289349i	0.989479	−	0.144674i	$-0.0462135\pi$
$978$	5.48019e9i	0.187331i
$979$	1.30507e10	0.444523
$980$	0	0
$981$	−2.50411e10	−0.846861
$982$	6.26675e9i	0.211180i
$983$	2.24230e10i	0.752932i	0.926430	+	0.376466i	$0.122861\pi$
$983$	2.24230e10i	0.752932i	−0.926430	+	0.376466i	$0.877139\pi$
$984$	6.66132e7	0.00222883
$985$	0	0
$986$	1.20672e10	0.400900
$987$	5.76262e9i	0.190770i
$988$	3.53261e9i	0.116532i
$989$	4.33400e10	1.42463
$990$	0	0
$991$	3.46728e10	1.13170	0.565849	−	0.824509i	$-0.308548\pi$
$991$	3.46728e10	1.13170	0.565849	+	0.824509i	$0.308548\pi$
$992$	− 7.45642e9i	− 0.242516i
$993$	− 3.28508e9i	− 0.106469i
$994$	−1.14989e10	−0.371368
$995$	0	0
$996$	4.35678e9	0.139720
$997$	− 2.96474e10i	− 0.947444i	−0.880674	−	0.473722i	$-0.842910\pi$
$997$	− 2.96474e10i	− 0.947444i	0.880674	−	0.473722i	$-0.157090\pi$
$998$	− 4.98550e9i	− 0.158764i
$999$	8.14830e9	0.258576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.8.b.c.49.1		2
3.2	odd	2		450.8.c.g.199.2		2
4.3	odd	2		400.8.c.j.49.2		2
5.2	odd	4		50.8.a.g.1.1		1
5.3	odd	4		2.8.a.a.1.1	✓	1
5.4	even	2	inner	50.8.b.c.49.2		2
15.2	even	4		450.8.a.c.1.1		1
15.8	even	4		18.8.a.b.1.1		1
15.14	odd	2		450.8.c.g.199.1		2
20.3	even	4		16.8.a.b.1.1		1
20.7	even	4		400.8.a.l.1.1		1
20.19	odd	2		400.8.c.j.49.1		2
35.3	even	12		98.8.c.e.79.1		2
35.13	even	4		98.8.a.a.1.1		1
35.18	odd	12		98.8.c.d.79.1		2
35.23	odd	12		98.8.c.d.67.1		2
35.33	even	12		98.8.c.e.67.1		2
40.3	even	4		64.8.a.e.1.1		1
40.13	odd	4		64.8.a.c.1.1		1
45.13	odd	12		162.8.c.l.55.1		2
45.23	even	12		162.8.c.a.55.1		2
45.38	even	12		162.8.c.a.109.1		2
45.43	odd	12		162.8.c.l.109.1		2
55.43	even	4		242.8.a.e.1.1		1
60.23	odd	4		144.8.a.i.1.1		1
65.8	even	4		338.8.b.d.337.1		2
65.18	even	4		338.8.b.d.337.2		2
65.38	odd	4		338.8.a.d.1.1		1
80.3	even	4		256.8.b.f.129.1		2
80.13	odd	4		256.8.b.b.129.2		2
80.43	even	4		256.8.b.f.129.2		2
80.53	odd	4		256.8.b.b.129.1		2
85.33	odd	4		578.8.a.b.1.1		1
120.53	even	4		576.8.a.g.1.1		1
120.83	odd	4		576.8.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.8.a.a.1.1	✓	1	5.3	odd	4
16.8.a.b.1.1		1	20.3	even	4
18.8.a.b.1.1		1	15.8	even	4
50.8.a.g.1.1		1	5.2	odd	4
50.8.b.c.49.1		2	1.1	even	1	trivial
50.8.b.c.49.2		2	5.4	even	2	inner
64.8.a.c.1.1		1	40.13	odd	4
64.8.a.e.1.1		1	40.3	even	4
98.8.a.a.1.1		1	35.13	even	4
98.8.c.d.67.1		2	35.23	odd	12
98.8.c.d.79.1		2	35.18	odd	12
98.8.c.e.67.1		2	35.33	even	12
98.8.c.e.79.1		2	35.3	even	12
144.8.a.i.1.1		1	60.23	odd	4
162.8.c.a.55.1		2	45.23	even	12
162.8.c.a.109.1		2	45.38	even	12
162.8.c.l.55.1		2	45.13	odd	12
162.8.c.l.109.1		2	45.43	odd	12
242.8.a.e.1.1		1	55.43	even	4
256.8.b.b.129.1		2	80.53	odd	4
256.8.b.b.129.2		2	80.13	odd	4
256.8.b.f.129.1		2	80.3	even	4
256.8.b.f.129.2		2	80.43	even	4
338.8.a.d.1.1		1	65.38	odd	4
338.8.b.d.337.1		2	65.8	even	4
338.8.b.d.337.2		2	65.18	even	4
400.8.a.l.1.1		1	20.7	even	4
400.8.c.j.49.1		2	20.19	odd	2
400.8.c.j.49.2		2	4.3	odd	2
450.8.a.c.1.1		1	15.2	even	4
450.8.c.g.199.1		2	15.14	odd	2
450.8.c.g.199.2		2	3.2	odd	2
576.8.a.f.1.1		1	120.83	odd	4
576.8.a.g.1.1		1	120.53	even	4
578.8.a.b.1.1		1	85.33	odd	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} -12.0000i q^{3} -64.0000 q^{4} -96.0000 q^{6} +1016.00i q^{7} +512.000i q^{8} +2043.00 q^{9} +1092.00 q^{11} +768.000i q^{12} -1382.00i q^{13} +8128.00 q^{14} +4096.00 q^{16} +14706.0i q^{17} -16344.0i q^{18} +39940.0 q^{19} +12192.0 q^{21} -8736.00i q^{22} -68712.0i q^{23} +6144.00 q^{24} -11056.0 q^{26} -50760.0i q^{27} -65024.0i q^{28} +102570. q^{29} +227552. q^{31} -32768.0i q^{32} -13104.0i q^{33} +117648. q^{34} -130752. q^{36} +160526. i q^{37} -319520. i q^{38} -16584.0 q^{39} +10842.0 q^{41} -97536.0i q^{42} +630748. i q^{43} -69888.0 q^{44} -549696. q^{46} +472656. i q^{47} -49152.0i q^{48} -208713. q^{49} +176472. q^{51} +88448.0i q^{52} +1.49402e6i q^{53} -406080. q^{54} -520192. q^{56} -479280. i q^{57} -820560. i q^{58} -2.64066e6 q^{59} +827702. q^{61} -1.82042e6i q^{62} +2.07569e6i q^{63} -262144. q^{64} -104832. q^{66} -126004. i q^{67} -941184. i q^{68} -824544. q^{69} -1.41473e6 q^{71} +1.04602e6i q^{72} -980282. i q^{73} +1.28421e6 q^{74} -2.55616e6 q^{76} +1.10947e6i q^{77} +132672. i q^{78} +3.56680e6 q^{79} +3.85892e6 q^{81} -86736.0i q^{82} -5.67289e6i q^{83} -780288. q^{84} +5.04598e6 q^{86} -1.23084e6i q^{87} +559104. i q^{88} +1.19512e7 q^{89} +1.40411e6 q^{91} +4.39757e6i q^{92} -2.73062e6i q^{93} +3.78125e6 q^{94} -393216. q^{96} +8.68215e6i q^{97} +1.66970e6i q^{98} +2.23096e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 192 q^{6} + 4086 q^{9} + 2184 q^{11} + 16256 q^{14} + 8192 q^{16} + 79880 q^{19} + 24384 q^{21} + 12288 q^{24} - 22112 q^{26} + 205140 q^{29} + 455104 q^{31} + 235296 q^{34} - 261504 q^{36}+ \cdots + 4461912 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 192 * q^6 + 4086 * q^9 + 2184 * q^11 + 16256 * q^14 + 8192 * q^16 + 79880 * q^19 + 24384 * q^21 + 12288 * q^24 - 22112 * q^26 + 205140 * q^29 + 455104 * q^31 + 235296 * q^34 - 261504 * q^36 - 33168 * q^39 + 21684 * q^41 - 139776 * q^44 - 1099392 * q^46 - 417426 * q^49 + 352944 * q^51 - 812160 * q^54 - 1040384 * q^56 - 5281320 * q^59 + 1655404 * q^61 - 524288 * q^64 - 209664 * q^66 - 1649088 * q^69 - 2829456 * q^71 + 2568416 * q^74 - 5112320 * q^76 + 7133600 * q^79 + 7717842 * q^81 - 1560576 * q^84 + 10091968 * q^86 + 23902380 * q^89 + 2808224 * q^91 + 7562496 * q^94 - 786432 * q^96 + 4461912 * q^99

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