LMFDB - Embedded newform 169.10.a.f.1.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,10,Mod(1,169)] 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("169.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$87.0410563117$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 7679 x^{18} + 24599364 x^{16} - 42662336000 x^{14} + 43527566862400 x^{12} + \cdots + 25\!\cdots\!36 $ x^20 - 7679x^18 + 24599364x^16 - 42662336000x^14 + 43527566862400x^12 - 26625453181634304x^10 + 9550565219960082432x^8 - 1876223091685443895296x^6 + 172106193081772189679616x^4 - 4841822934658519419322368*x^2 + 25162285487879223389454336
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{10}\cdot 13^{12} $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$36.6943$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+36.6943 q^{2} -130.026 q^{3} +834.471 q^{4} -1417.57 q^{5} -4771.23 q^{6} +7507.92 q^{7} +11832.9 q^{8} -2776.14 q^{9} -52016.6 q^{10} +42852.6 q^{11} -108503. q^{12} +275498. q^{14} +184321. q^{15} +6948.87 q^{16} -535226. q^{17} -101868. q^{18} +418977. q^{19} -1.18292e6 q^{20} -976228. q^{21} +1.57245e6 q^{22} +294693. q^{23} -1.53858e6 q^{24} +56368.4 q^{25} +2.92028e6 q^{27} +6.26514e6 q^{28} +1.62070e6 q^{29} +6.76353e6 q^{30} -3.25108e6 q^{31} -5.80344e6 q^{32} -5.57197e6 q^{33} -1.96397e7 q^{34} -1.06430e7 q^{35} -2.31661e6 q^{36} +2.00820e7 q^{37} +1.53741e7 q^{38} -1.67738e7 q^{40} +3.44352e7 q^{41} -3.58220e7 q^{42} +3.13771e6 q^{43} +3.57593e7 q^{44} +3.93536e6 q^{45} +1.08136e7 q^{46} +5.82844e7 q^{47} -903537. q^{48} +1.60153e7 q^{49} +2.06840e6 q^{50} +6.95935e7 q^{51} -2.43962e7 q^{53} +1.07158e8 q^{54} -6.07464e7 q^{55} +8.88401e7 q^{56} -5.44780e7 q^{57} +5.94706e7 q^{58} +1.61913e8 q^{59} +1.53811e8 q^{60} +8.43506e7 q^{61} -1.19296e8 q^{62} -2.08430e7 q^{63} -2.16511e8 q^{64} -2.04459e8 q^{66} -4.89466e7 q^{67} -4.46631e8 q^{68} -3.83179e7 q^{69} -3.90536e8 q^{70} +3.31943e7 q^{71} -3.28496e7 q^{72} +2.22708e8 q^{73} +7.36896e8 q^{74} -7.32937e6 q^{75} +3.49624e8 q^{76} +3.21734e8 q^{77} +1.28556e7 q^{79} -9.85048e6 q^{80} -3.25071e8 q^{81} +1.26358e9 q^{82} +1.32887e8 q^{83} -8.14634e8 q^{84} +7.58718e8 q^{85} +1.15136e8 q^{86} -2.10734e8 q^{87} +5.07069e8 q^{88} -5.85065e8 q^{89} +1.44405e8 q^{90} +2.45913e8 q^{92} +4.22726e8 q^{93} +2.13870e9 q^{94} -5.93927e8 q^{95} +7.54600e8 q^{96} +1.03264e9 q^{97} +5.87670e8 q^{98} -1.18965e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100}) $ 20 * q + 326 * q^3 + 5118 * q^4 + 129526 * q^9 + 88390 * q^10 + 427652 * q^12 + 473556 * q^14 + 1189618 * q^16 - 99312 * q^17 - 5073532 * q^22 + 6252378 * q^23 + 1529274 * q^25 + 18052718 * q^27 + 5424828 * q^29 + 30045516 * q^30 - 15098160 * q^35 + 54078886 * q^36 + 39021096 * q^38 + 134360674 * q^40 + 121600068 * q^42 + 53242058 * q^43 + 82771076 * q^48 + 14055298 * q^49 + 10208934 * q^51 - 36429786 * q^53 - 131454160 * q^55 + 127272672 * q^56 + 536425792 * q^61 - 890759796 * q^62 - 343337066 * q^64 - 445758060 * q^66 + 336594090 * q^68 + 1083885582 * q^69 + 1083869262 * q^74 - 1218826882 * q^75 + 1470187374 * q^77 + 1556703616 * q^79 + 2139127516 * q^81 + 3719322754 * q^82 + 1288246146 * q^87 - 4109160928 * q^88 + 6489878934 * q^90 + 10148843820 * q^92 + 4894712828 * q^94 + 9251202540 * q^95

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 $

$2$	36.6943	1.62167	0.810837	−	0.585272i	$-0.199012\pi$
$2$	36.6943	1.62167	0.810837	+	0.585272i	$0.199012\pi$
$3$	−130.026	−0.926800	−0.463400	−	0.886149i	$-0.653371\pi$
$3$	−130.026	−0.926800	−0.463400	+	0.886149i	$0.653371\pi$
$4$	834.471	1.62983
$5$	−1417.57	−1.01433	−0.507164	−	0.861850i	$-0.669306\pi$
$5$	−1417.57	−1.01433	−0.507164	+	0.861850i	$0.669306\pi$
$6$	−4771.23	−1.50297
$7$	7507.92	1.18189	0.590947	−	0.806710i	$-0.298754\pi$
$7$	7507.92	1.18189	0.590947	+	0.806710i	$0.298754\pi$
$8$	11832.9	1.02137
$9$	−2776.14	−0.141042
$10$	−52016.6	−1.64491
$11$	42852.6	0.882491	0.441245	−	0.897386i	$-0.354537\pi$
$11$	42852.6	0.882491	0.441245	+	0.897386i	$0.354537\pi$
$12$	−108503.	−1.51052
$13$	0	0
$13$	0	0
$14$	275498.	1.91665
$15$	184321.	0.940078
$16$	6948.87	0.0265078
$17$	−535226.	−1.55424	−0.777118	−	0.629354i	$-0.783319\pi$
$17$	−535226.	−1.55424	−0.777118	+	0.629354i	$0.783319\pi$
$18$	−101868.	−0.228725
$19$	418977.	0.737563	0.368781	−	0.929516i	$-0.379775\pi$
$19$	418977.	0.737563	0.368781	+	0.929516i	$0.379775\pi$
$20$	−1.18292e6	−1.65318
$21$	−976228.	−1.09538
$22$	1.57245e6	1.43111
$23$	294693.	0.219581	0.109791	−	0.993955i	$-0.464982\pi$
$23$	294693.	0.219581	0.109791	+	0.993955i	$0.464982\pi$
$24$	−1.53858e6	−0.946608
$25$	56368.4	0.0288606
$26$	0	0
$27$	2.92028e6	1.05752
$28$	6.26514e6	1.92628
$29$	1.62070e6	0.425513	0.212756	−	0.977105i	$-0.431756\pi$
$29$	1.62070e6	0.425513	0.212756	+	0.977105i	$0.431756\pi$
$30$	6.76353e6	1.52450
$31$	−3.25108e6	−0.632265	−0.316133	−	0.948715i	$-0.602384\pi$
$31$	−3.25108e6	−0.632265	−0.316133	+	0.948715i	$0.602384\pi$
$32$	−5.80344e6	−0.978386
$33$	−5.57197e6	−0.817892
$34$	−1.96397e7	−2.52047
$35$	−1.06430e7	−1.19883
$36$	−2.31661e6	−0.229875
$37$	2.00820e7	1.76157	0.880785	−	0.473516i	$-0.157015\pi$
$37$	2.00820e7	1.76157	0.880785	+	0.473516i	$0.157015\pi$
$38$	1.53741e7	1.19609
$39$	0	0
$40$	−1.67738e7	−1.03601
$41$	3.44352e7	1.90316	0.951580	−	0.307401i	$-0.0994593\pi$
$41$	3.44352e7	1.90316	0.951580	+	0.307401i	$0.0994593\pi$
$42$	−3.58220e7	−1.77635
$43$	3.13771e6	0.139960	0.0699801	−	0.997548i	$-0.477706\pi$
$43$	3.13771e6	0.139960	0.0699801	+	0.997548i	$0.477706\pi$
$44$	3.57593e7	1.43831
$45$	3.93536e6	0.143063
$46$	1.08136e7	0.356089
$47$	5.82844e7	1.74226	0.871128	−	0.491056i	$-0.163389\pi$
$47$	5.82844e7	1.74226	0.871128	+	0.491056i	$0.163389\pi$
$48$	−903537.	−0.0245675
$49$	1.60153e7	0.396874
$50$	2.06840e6	0.0468025
$51$	6.95935e7	1.44047
$52$	0	0
$53$	−2.43962e7	−0.424699	−0.212349	−	0.977194i	$-0.568111\pi$
$53$	−2.43962e7	−0.424699	−0.212349	+	0.977194i	$0.568111\pi$
$54$	1.07158e8	1.71495
$55$	−6.07464e7	−0.895135
$56$	8.88401e7	1.20716
$57$	−5.44780e7	−0.683573
$58$	5.94706e7	0.690043
$59$	1.61913e8	1.73959	0.869795	−	0.493412i	$-0.164251\pi$
$59$	1.61913e8	1.73959	0.869795	+	0.493412i	$0.164251\pi$
$60$	1.53811e8	1.53216
$61$	8.43506e7	0.780017	0.390008	−	0.920811i	$-0.372472\pi$
$61$	8.43506e7	0.780017	0.390008	+	0.920811i	$0.372472\pi$
$62$	−1.19296e8	−1.02533
$63$	−2.08430e7	−0.166697
$64$	−2.16511e8	−1.61313
$65$	0	0
$66$	−2.04459e8	−1.32635
$67$	−4.89466e7	−0.296746	−0.148373	−	0.988931i	$-0.547404\pi$
$67$	−4.89466e7	−0.296746	−0.148373	+	0.988931i	$0.547404\pi$
$68$	−4.46631e8	−2.53314
$69$	−3.83179e7	−0.203508
$70$	−3.90536e8	−1.94411
$71$	3.31943e7	0.155025	0.0775125	−	0.996991i	$-0.475302\pi$
$71$	3.31943e7	0.155025	0.0775125	+	0.996991i	$0.475302\pi$
$72$	−3.28496e7	−0.144057
$73$	2.22708e8	0.917875	0.458938	−	0.888468i	$-0.348230\pi$
$73$	2.22708e8	0.917875	0.458938	+	0.888468i	$0.348230\pi$
$74$	7.36896e8	2.85669
$75$	−7.32937e6	−0.0267480
$76$	3.49624e8	1.20210
$77$	3.21734e8	1.04301
$78$	0	0
$79$	1.28556e7	0.0371338	0.0185669	−	0.999828i	$-0.494090\pi$
$79$	1.28556e7	0.0371338	0.0185669	+	0.999828i	$0.494090\pi$
$80$	−9.85048e6	−0.0268876
$81$	−3.25071e8	−0.839065
$82$	1.26358e9	3.08631
$83$	1.32887e8	0.307348	0.153674	−	0.988122i	$-0.450889\pi$
$83$	1.32887e8	0.307348	0.153674	+	0.988122i	$0.450889\pi$
$84$	−8.14634e8	−1.78528
$85$	7.58718e8	1.57651
$86$	1.15136e8	0.226970
$87$	−2.10734e8	−0.394365
$88$	5.07069e8	0.901352
$89$	−5.85065e8	−0.988437	−0.494218	−	0.869338i	$-0.664546\pi$
$89$	−5.85065e8	−0.988437	−0.494218	+	0.869338i	$0.664546\pi$
$90$	1.44405e8	0.232002
$91$	0	0
$92$	2.45913e8	0.357879
$93$	4.22726e8	0.585983
$94$	2.13870e9	2.82537
$95$	−5.93927e8	−0.748130
$96$	7.54600e8	0.906768
$97$	1.03264e9	1.18434	0.592170	−	0.805813i	$-0.298271\pi$
$97$	1.03264e9	1.18434	0.592170	+	0.805813i	$0.298271\pi$
$98$	5.87670e8	0.643600
$99$	−1.18965e8	−0.124469
$100$	4.70378e7	0.0470378
$101$	7.35672e8	0.703458	0.351729	−	0.936102i	$-0.385594\pi$
$101$	7.35672e8	0.703458	0.351729	+	0.936102i	$0.385594\pi$
$102$	2.55369e9	2.33597
$103$	1.16039e8	0.101587	0.0507934	−	0.998709i	$-0.483825\pi$
$103$	1.16039e8	0.101587	0.0507934	+	0.998709i	$0.483825\pi$
$104$	0	0
$105$	1.38387e9	1.11107
$106$	−8.95201e8	−0.688723
$107$	−4.21394e8	−0.310786	−0.155393	−	0.987853i	$-0.549664\pi$
$107$	−4.21394e8	−0.310786	−0.155393	+	0.987853i	$0.549664\pi$
$108$	2.43689e9	1.72357
$109$	1.30101e9	0.882796	0.441398	−	0.897311i	$-0.354483\pi$
$109$	1.30101e9	0.882796	0.441398	+	0.897311i	$0.354483\pi$
$110$	−2.22905e9	−1.45162
$111$	−2.61119e9	−1.63262
$112$	5.21716e7	0.0313295
$113$	−1.02461e9	−0.591160	−0.295580	−	0.955318i	$-0.595513\pi$
$113$	−1.02461e9	−0.591160	−0.295580	+	0.955318i	$0.595513\pi$
$114$	−1.99903e9	−1.10853
$115$	−4.17747e8	−0.222727
$116$	1.35243e9	0.693512
$117$	0	0
$118$	5.94128e9	2.82105
$119$	−4.01844e9	−1.83694
$120$	2.18104e9	0.960171
$121$	−5.21602e8	−0.221210
$122$	3.09519e9	1.26493
$123$	−4.47749e9	−1.76385
$124$	−2.71293e9	−1.03048
$125$	2.68878e9	0.985054
$126$	−7.64821e8	−0.270329
$127$	−4.76239e9	−1.62446	−0.812228	−	0.583339i	$-0.801746\pi$
$127$	−4.76239e9	−1.62446	−0.812228	+	0.583339i	$0.801746\pi$
$128$	−4.97335e9	−1.63759
$129$	−4.07985e8	−0.129715
$130$	0	0
$131$	−3.55676e9	−1.05520	−0.527599	−	0.849494i	$-0.676908\pi$
$131$	−3.55676e9	−1.05520	−0.527599	+	0.849494i	$0.676908\pi$
$132$	−4.64965e9	−1.33302
$133$	3.14565e9	0.871721
$134$	−1.79606e9	−0.481226
$135$	−4.13969e9	−1.07267
$136$	−6.33325e9	−1.58746
$137$	5.28532e9	1.28183	0.640913	−	0.767614i	$-0.278556\pi$
$137$	5.28532e9	1.28183	0.640913	+	0.767614i	$0.278556\pi$
$138$	−1.40605e9	−0.330023
$139$	9.34790e8	0.212397	0.106198	−	0.994345i	$-0.466132\pi$
$139$	9.34790e8	0.212397	0.106198	+	0.994345i	$0.466132\pi$
$140$	−8.88126e9	−1.95388
$141$	−7.57851e9	−1.61472
$142$	1.21804e9	0.251400
$143$	0	0
$144$	−1.92910e7	−0.00373873
$145$	−2.29745e9	−0.431609
$146$	8.17213e9	1.48849
$147$	−2.08241e9	−0.367823
$148$	1.67579e10	2.87105
$149$	−6.55190e9	−1.08900	−0.544502	−	0.838760i	$-0.683281\pi$
$149$	−6.55190e9	−1.08900	−0.544502	+	0.838760i	$0.683281\pi$
$150$	−2.68946e8	−0.0433765
$151$	−1.99924e8	−0.0312946	−0.0156473	−	0.999878i	$-0.504981\pi$
$151$	−1.99924e8	−0.0312946	−0.0156473	+	0.999878i	$0.504981\pi$
$152$	4.95769e9	0.753327
$153$	1.48586e9	0.219213
$154$	1.18058e10	1.69142
$155$	4.60861e9	0.641324
$156$	0	0
$157$	−6.66154e9	−0.875036	−0.437518	−	0.899210i	$-0.644142\pi$
$157$	−6.66154e9	−0.875036	−0.437518	+	0.899210i	$0.644142\pi$
$158$	4.71725e8	0.0602188
$159$	3.17215e9	0.393610
$160$	8.22675e9	0.992404
$161$	2.21253e9	0.259522
$162$	−1.19282e10	−1.36069
$163$	−5.88103e9	−0.652543	−0.326271	−	0.945276i	$-0.605792\pi$
$163$	−5.88103e9	−0.652543	−0.326271	+	0.945276i	$0.605792\pi$
$164$	2.87352e10	3.10182
$165$	7.89863e9	0.829611
$166$	4.87618e9	0.498418
$167$	5.12094e9	0.509478	0.254739	−	0.967010i	$-0.418010\pi$
$167$	5.12094e9	0.509478	0.254739	+	0.967010i	$0.418010\pi$
$168$	−1.15516e10	−1.11879
$169$	0	0
$170$	2.78406e10	2.55658
$171$	−1.16314e9	−0.104028
$172$	2.61833e9	0.228111
$173$	1.31542e10	1.11650	0.558249	−	0.829674i	$-0.311474\pi$
$173$	1.31542e10	1.11650	0.558249	+	0.829674i	$0.311474\pi$
$174$	−7.73274e9	−0.639532
$175$	4.23209e8	0.0341102
$176$	2.97777e8	0.0233929
$177$	−2.10529e10	−1.61225
$178$	−2.14685e10	−1.60292
$179$	1.03360e10	0.752513	0.376256	−	0.926516i	$-0.377211\pi$
$179$	1.03360e10	0.752513	0.376256	+	0.926516i	$0.377211\pi$
$180$	3.28394e9	0.233168
$181$	2.73183e9	0.189191	0.0945956	−	0.995516i	$-0.469844\pi$
$181$	2.73183e9	0.189191	0.0945956	+	0.995516i	$0.469844\pi$
$182$	0	0
$183$	−1.09678e10	−0.722919
$184$	3.48706e9	0.224274
$185$	−2.84676e10	−1.78681
$186$	1.55116e10	0.950274
$187$	−2.29358e10	−1.37160
$188$	4.86366e10	2.83957
$189$	2.19252e10	1.24987
$190$	−2.17937e10	−1.21322
$191$	1.01627e10	0.552536	0.276268	−	0.961081i	$-0.410902\pi$
$191$	1.01627e10	0.552536	0.276268	+	0.961081i	$0.410902\pi$
$192$	2.81521e10	1.49505
$193$	1.45341e10	0.754015	0.377007	−	0.926210i	$-0.376953\pi$
$193$	1.45341e10	0.754015	0.377007	+	0.926210i	$0.376953\pi$
$194$	3.78920e10	1.92061
$195$	0	0
$196$	1.33643e10	0.646836
$197$	2.45916e9	0.116329	0.0581645	−	0.998307i	$-0.481475\pi$
$197$	2.45916e9	0.116329	0.0581645	+	0.998307i	$0.481475\pi$
$198$	−4.36533e9	−0.201848
$199$	2.50102e10	1.13052	0.565259	−	0.824913i	$-0.308776\pi$
$199$	2.50102e10	1.13052	0.565259	+	0.824913i	$0.308776\pi$
$200$	6.66998e8	0.0294774
$201$	6.36434e9	0.275024
$202$	2.69950e10	1.14078
$203$	1.21681e10	0.502911
$204$	5.80738e10	2.34771
$205$	−4.88142e10	−1.93043
$206$	4.25798e9	0.164741
$207$	−8.18109e8	−0.0309703
$208$	0	0
$209$	1.79543e10	0.650892
$210$	5.07800e10	1.80180
$211$	−2.95626e10	−1.02677	−0.513383	−	0.858160i	$-0.671608\pi$
$211$	−2.95626e10	−1.02677	−0.513383	+	0.858160i	$0.671608\pi$
$212$	−2.03579e10	−0.692185
$213$	−4.31614e9	−0.143677
$214$	−1.54628e10	−0.503994
$215$	−4.44791e9	−0.141965
$216$	3.45552e10	1.08012
$217$	−2.44088e10	−0.747271
$218$	4.77395e10	1.43161
$219$	−2.89580e10	−0.850687
$220$	−5.06911e10	−1.45891
$221$	0	0
$222$	−9.58159e10	−2.64758
$223$	−4.59229e10	−1.24353	−0.621767	−	0.783203i	$-0.713585\pi$
$223$	−4.59229e10	−1.24353	−0.621767	+	0.783203i	$0.713585\pi$
$224$	−4.35717e10	−1.15635
$225$	−1.56486e8	−0.00407057
$226$	−3.75973e10	−0.958669
$227$	6.10441e9	0.152590	0.0762952	−	0.997085i	$-0.475691\pi$
$227$	6.10441e9	0.152590	0.0762952	+	0.997085i	$0.475691\pi$
$228$	−4.54604e10	−1.11410
$229$	2.74503e10	0.659610	0.329805	−	0.944049i	$-0.393017\pi$
$229$	2.74503e10	0.659610	0.329805	+	0.944049i	$0.393017\pi$
$230$	−1.53289e10	−0.361191
$231$	−4.18339e10	−0.966662
$232$	1.91775e10	0.434607
$233$	2.55024e10	0.566864	0.283432	−	0.958992i	$-0.408527\pi$
$233$	2.55024e10	0.566864	0.283432	+	0.958992i	$0.408527\pi$
$234$	0	0
$235$	−8.26220e10	−1.76722
$236$	1.35112e11	2.83523
$237$	−1.67156e9	−0.0344155
$238$	−1.47454e11	−2.97892
$239$	2.55166e10	0.505863	0.252931	−	0.967484i	$-0.418605\pi$
$239$	2.55166e10	0.505863	0.252931	+	0.967484i	$0.418605\pi$
$240$	1.28082e9	0.0249195
$241$	7.55461e10	1.44256	0.721282	−	0.692641i	$-0.243553\pi$
$241$	7.55461e10	1.44256	0.721282	+	0.692641i	$0.243553\pi$
$242$	−1.91398e10	−0.358731
$243$	−1.52121e10	−0.279873
$244$	7.03881e10	1.27129
$245$	−2.27027e10	−0.402560
$246$	−1.64298e11	−2.86039
$247$	0	0
$248$	−3.84695e10	−0.645779
$249$	−1.72788e10	−0.284850
$250$	9.86628e10	1.59744
$251$	3.66333e10	0.582565	0.291283	−	0.956637i	$-0.405918\pi$
$251$	3.66333e10	0.582565	0.291283	+	0.956637i	$0.405918\pi$
$252$	−1.73929e10	−0.271688
$253$	1.26284e10	0.193778
$254$	−1.74753e11	−2.63434
$255$	−9.86534e10	−1.46110
$256$	−7.16401e10	−1.04250
$257$	−2.81000e10	−0.401797	−0.200899	−	0.979612i	$-0.564386\pi$
$257$	−2.81000e10	−0.401797	−0.200899	+	0.979612i	$0.564386\pi$
$258$	−1.49707e10	−0.210355
$259$	1.50774e11	2.08199
$260$	0	0
$261$	−4.49930e9	−0.0600154
$262$	−1.30513e11	−1.71119
$263$	−4.07904e10	−0.525723	−0.262861	−	0.964834i	$-0.584666\pi$
$263$	−4.07904e10	−0.525723	−0.262861	+	0.964834i	$0.584666\pi$
$264$	−6.59323e10	−0.835373
$265$	3.45832e10	0.430784
$266$	1.15427e11	1.41365
$267$	7.60739e10	0.916083
$268$	−4.08445e10	−0.483645
$269$	8.05413e10	0.937849	0.468925	−	0.883238i	$-0.344642\pi$
$269$	8.05413e10	0.937849	0.468925	+	0.883238i	$0.344642\pi$
$270$	−1.51903e11	−1.73952
$271$	−1.65101e11	−1.85947	−0.929733	−	0.368235i	$-0.879962\pi$
$271$	−1.65101e11	−1.85947	−0.929733	+	0.368235i	$0.879962\pi$
$272$	−3.71922e9	−0.0411995
$273$	0	0
$274$	1.93941e11	2.07870
$275$	2.41553e9	0.0254692
$276$	−3.19752e10	−0.331682
$277$	−9.50226e10	−0.969768	−0.484884	−	0.874578i	$-0.661138\pi$
$277$	−9.50226e10	−0.969768	−0.484884	+	0.874578i	$0.661138\pi$
$278$	3.43015e10	0.344438
$279$	9.02544e9	0.0891763
$280$	−1.25937e11	−1.22445
$281$	−1.86053e11	−1.78016	−0.890078	−	0.455809i	$-0.849350\pi$
$281$	−1.86053e11	−1.78016	−0.890078	+	0.455809i	$0.849350\pi$
$282$	−2.78088e11	−2.61855
$283$	1.03792e11	0.961885	0.480942	−	0.876752i	$-0.340295\pi$
$283$	1.03792e11	0.961885	0.480942	+	0.876752i	$0.340295\pi$
$284$	2.76997e10	0.252664
$285$	7.72262e10	0.693367
$286$	0	0
$287$	2.58537e11	2.24933
$288$	1.61111e10	0.137994
$289$	1.67879e11	1.41565
$290$	−8.43035e10	−0.699930
$291$	−1.34271e11	−1.09765
$292$	1.85844e11	1.49598
$293$	−1.07967e11	−0.855825	−0.427913	−	0.903820i	$-0.640751\pi$
$293$	−1.07967e11	−0.855825	−0.427913	+	0.903820i	$0.640751\pi$
$294$	−7.64126e10	−0.596488
$295$	−2.29522e11	−1.76452
$296$	2.37628e11	1.79922
$297$	1.25142e11	0.933250
$298$	−2.40417e11	−1.76601
$299$	0	0
$300$	−6.11615e9	−0.0435946
$301$	2.35577e10	0.165418
$302$	−7.33607e9	−0.0507496
$303$	−9.56568e10	−0.651965
$304$	2.91142e9	0.0195512
$305$	−1.19573e11	−0.791193
$306$	5.45227e10	0.355493
$307$	−1.18337e11	−0.760326	−0.380163	−	0.924920i	$-0.624132\pi$
$307$	−1.18337e11	−0.760326	−0.380163	+	0.924920i	$0.624132\pi$
$308$	2.68478e11	1.69993
$309$	−1.50882e10	−0.0941506
$310$	1.69110e11	1.04002
$311$	1.19220e11	0.722647	0.361323	−	0.932441i	$-0.382325\pi$
$311$	1.19220e11	0.722647	0.361323	+	0.932441i	$0.382325\pi$
$312$	0	0
$313$	−3.12948e11	−1.84299	−0.921495	−	0.388389i	$-0.873032\pi$
$313$	−3.12948e11	−1.84299	−0.921495	+	0.388389i	$0.873032\pi$
$314$	−2.44440e11	−1.41902
$315$	2.95464e10	0.169086
$316$	1.07276e10	0.0605216
$317$	2.96387e10	0.164851	0.0824257	−	0.996597i	$-0.473733\pi$
$317$	2.96387e10	0.164851	0.0824257	+	0.996597i	$0.473733\pi$
$318$	1.16400e11	0.638308
$319$	6.94514e10	0.375511
$320$	3.06918e11	1.63624
$321$	5.47923e10	0.288036
$322$	8.11874e10	0.420859
$323$	−2.24247e11	−1.14635
$324$	−2.71262e11	−1.36753
$325$	0	0
$326$	−2.15800e11	−1.05821
$327$	−1.69165e11	−0.818175
$328$	4.07467e11	1.94384
$329$	4.37595e11	2.05916
$330$	2.89835e11	1.34536
$331$	−3.51155e11	−1.60795	−0.803976	−	0.594662i	$-0.797286\pi$
$331$	−3.51155e11	−1.60795	−0.803976	+	0.594662i	$0.797286\pi$
$332$	1.10890e11	0.500923
$333$	−5.57505e10	−0.248456
$334$	1.87909e11	0.826207
$335$	6.93850e10	0.300998
$336$	−6.78368e9	−0.0290361
$337$	5.10205e10	0.215481	0.107741	−	0.994179i	$-0.465638\pi$
$337$	5.10205e10	0.215481	0.107741	+	0.994179i	$0.465638\pi$
$338$	0	0
$339$	1.33226e11	0.547887
$340$	6.33129e11	2.56943
$341$	−1.39317e11	−0.557968
$342$	−4.26805e10	−0.168699
$343$	−1.82730e11	−0.712831
$344$	3.71280e10	0.142952
$345$	5.43181e10	0.206423
$346$	4.82685e11	1.81059
$347$	4.01637e11	1.48714	0.743569	−	0.668659i	$-0.233131\pi$
$347$	4.01637e11	1.48714	0.743569	+	0.668659i	$0.233131\pi$
$348$	−1.75852e11	−0.642747
$349$	1.22048e10	0.0440368	0.0220184	−	0.999758i	$-0.492991\pi$
$349$	1.22048e10	0.0440368	0.0220184	+	0.999758i	$0.492991\pi$
$350$	1.55294e10	0.0553156
$351$	0	0
$352$	−2.48692e11	−0.863417
$353$	1.09056e11	0.373822	0.186911	−	0.982377i	$-0.440152\pi$
$353$	1.09056e11	0.373822	0.186911	+	0.982377i	$0.440152\pi$
$354$	−7.72523e11	−2.61455
$355$	−4.70552e10	−0.157246
$356$	−4.88220e11	−1.61098
$357$	5.22503e11	1.70248
$358$	3.79272e11	1.22033
$359$	1.39991e11	0.444812	0.222406	−	0.974954i	$-0.428609\pi$
$359$	1.39991e11	0.444812	0.222406	+	0.974954i	$0.428609\pi$
$360$	4.65665e10	0.146121
$361$	−1.47146e11	−0.456001
$362$	1.00243e11	0.306806
$363$	6.78220e10	0.205017
$364$	0	0
$365$	−3.15704e11	−0.931026
$366$	−4.02456e11	−1.17234
$367$	−5.14356e11	−1.48002	−0.740008	−	0.672599i	$-0.765178\pi$
$367$	−5.14356e11	−1.48002	−0.740008	+	0.672599i	$0.765178\pi$
$368$	2.04779e9	0.00582062
$369$	−9.55969e10	−0.268426
$370$	−1.04460e12	−2.89762
$371$	−1.83165e11	−0.501949
$372$	3.52752e11	0.955051
$373$	−2.90587e11	−0.777297	−0.388649	−	0.921386i	$-0.627058\pi$
$373$	−2.90587e11	−0.777297	−0.388649	+	0.921386i	$0.627058\pi$
$374$	−8.41614e11	−2.22429
$375$	−3.49612e11	−0.912947
$376$	6.89671e11	1.77949
$377$	0	0
$378$	8.04531e11	2.02689
$379$	4.46749e10	0.111221	0.0556106	−	0.998453i	$-0.482289\pi$
$379$	4.46749e10	0.111221	0.0556106	+	0.998453i	$0.482289\pi$
$380$	−4.95615e11	−1.21932
$381$	6.19236e11	1.50555
$382$	3.72914e11	0.896033
$383$	9.54757e10	0.226725	0.113362	−	0.993554i	$-0.463838\pi$
$383$	9.54757e10	0.226725	0.113362	+	0.993554i	$0.463838\pi$
$384$	6.46667e11	1.51771
$385$	−4.56079e11	−1.05795
$386$	5.33318e11	1.22277
$387$	−8.71071e9	−0.0197403
$388$	8.61709e11	1.93027
$389$	2.40025e11	0.531476	0.265738	−	0.964045i	$-0.414384\pi$
$389$	2.40025e11	0.531476	0.265738	+	0.964045i	$0.414384\pi$
$390$	0	0
$391$	−1.57728e11	−0.341281
$392$	1.89507e11	0.405356
$393$	4.62473e11	0.977957
$394$	9.02370e10	0.188648
$395$	−1.82236e10	−0.0376658
$396$	−9.92727e10	−0.202862
$397$	3.20260e11	0.647061	0.323531	−	0.946218i	$-0.395130\pi$
$397$	3.20260e11	0.647061	0.323531	+	0.946218i	$0.395130\pi$
$398$	9.17730e11	1.83333
$399$	−4.09017e11	−0.807911
$400$	3.91697e8	0.000765032 0
$401$	1.37703e10	0.0265946	0.0132973	−	0.999912i	$-0.495767\pi$
$401$	1.37703e10	0.0265946	0.0132973	+	0.999912i	$0.495767\pi$
$402$	2.33535e11	0.446000
$403$	0	0
$404$	6.13897e11	1.14651
$405$	4.60809e11	0.851086
$406$	4.46500e11	0.815558
$407$	8.60568e11	1.55457
$408$	8.23490e11	1.47125
$409$	4.52919e11	0.800323	0.400161	−	0.916445i	$-0.368954\pi$
$409$	4.52919e11	0.800323	0.400161	+	0.916445i	$0.368954\pi$
$410$	−1.79120e12	−3.13052
$411$	−6.87231e11	−1.18800
$412$	9.68314e10	0.165569
$413$	1.21563e12	2.05601
$414$	−3.00199e10	−0.0502237
$415$	−1.88376e11	−0.311751
$416$	0	0
$417$	−1.21547e11	−0.196849
$418$	6.58819e11	1.05553
$419$	8.66707e11	1.37375	0.686877	−	0.726774i	$-0.258981\pi$
$419$	8.66707e11	1.37375	0.686877	+	0.726774i	$0.258981\pi$
$420$	1.15480e12	1.81086
$421$	9.07071e11	1.40725	0.703626	−	0.710570i	$-0.251563\pi$
$421$	9.07071e11	1.40725	0.703626	+	0.710570i	$0.251563\pi$
$422$	−1.08478e12	−1.66508
$423$	−1.61806e11	−0.245732
$424$	−2.88677e11	−0.433776
$425$	−3.01698e10	−0.0448562
$426$	−1.58378e11	−0.232997
$427$	6.33298e11	0.921897
$428$	−3.51641e11	−0.506527
$429$	0	0
$430$	−1.63213e11	−0.230222
$431$	1.18566e11	0.165506	0.0827528	−	0.996570i	$-0.473629\pi$
$431$	1.18566e11	0.165506	0.0827528	+	0.996570i	$0.473629\pi$
$432$	2.02927e10	0.0280325
$433$	−2.65940e11	−0.363570	−0.181785	−	0.983338i	$-0.558187\pi$
$433$	−2.65940e11	−0.363570	−0.181785	+	0.983338i	$0.558187\pi$
$434$	−8.95664e11	−1.21183
$435$	2.98730e11	0.400015
$436$	1.08565e12	1.43880
$437$	1.23470e11	0.161955
$438$	−1.06259e12	−1.37954
$439$	−7.76288e10	−0.0997545	−0.0498772	−	0.998755i	$-0.515883\pi$
$439$	−7.76288e10	−0.0997545	−0.0498772	+	0.998755i	$0.515883\pi$
$440$	−7.18803e11	−0.914267
$441$	−4.44607e10	−0.0559761
$442$	0	0
$443$	1.22924e12	1.51642	0.758212	−	0.652008i	$-0.226073\pi$
$443$	1.22924e12	1.51642	0.758212	+	0.652008i	$0.226073\pi$
$444$	−2.17897e12	−2.66089
$445$	8.29368e11	1.00260
$446$	−1.68511e12	−2.01661
$447$	8.51920e11	1.00929
$448$	−1.62555e12	−1.90655
$449$	7.73914e11	0.898637	0.449319	−	0.893372i	$-0.351667\pi$
$449$	7.73914e11	0.898637	0.449319	+	0.893372i	$0.351667\pi$
$450$	−5.74216e9	−0.00660114
$451$	1.47564e12	1.67952
$452$	−8.55006e11	−0.963488
$453$	2.59954e10	0.0290038
$454$	2.23997e11	0.247452
$455$	0	0
$456$	−6.44631e11	−0.698183
$457$	−3.75223e11	−0.402407	−0.201204	−	0.979549i	$-0.564485\pi$
$457$	−3.75223e11	−0.402407	−0.201204	+	0.979549i	$0.564485\pi$
$458$	1.00727e12	1.06967
$459$	−1.56301e12	−1.64363
$460$	−3.48598e11	−0.363007
$461$	−9.94424e11	−1.02546	−0.512728	−	0.858551i	$-0.671365\pi$
$461$	−9.94424e11	−1.02546	−0.512728	+	0.858551i	$0.671365\pi$
$462$	−1.53507e12	−1.56761
$463$	1.76404e12	1.78400	0.891999	−	0.452038i	$-0.149303\pi$
$463$	1.76404e12	1.78400	0.891999	+	0.452038i	$0.149303\pi$
$464$	1.12621e10	0.0112794
$465$	−5.99241e11	−0.594379
$466$	9.35792e11	0.919269
$467$	7.36402e10	0.0716455	0.0358227	−	0.999358i	$-0.488595\pi$
$467$	7.36402e10	0.0716455	0.0358227	+	0.999358i	$0.488595\pi$
$468$	0	0
$469$	−3.67487e11	−0.350723
$470$	−3.03176e12	−2.86585
$471$	8.66176e11	0.810983
$472$	1.91589e12	1.77677
$473$	1.34459e11	0.123514
$474$	−6.13368e10	−0.0558108
$475$	2.36170e10	0.0212865
$476$	−3.35327e12	−2.99390
$477$	6.77273e10	0.0599005
$478$	9.36315e11	0.820345
$479$	−1.72474e12	−1.49697	−0.748485	−	0.663151i	$-0.769219\pi$
$479$	−1.72474e12	−1.49697	−0.748485	+	0.663151i	$0.769219\pi$
$480$	−1.06969e12	−0.919760
$481$	0	0
$482$	2.77211e12	2.33937
$483$	−2.87688e11	−0.240524
$484$	−4.35262e11	−0.360534
$485$	−1.46384e12	−1.20131
$486$	−5.58197e11	−0.453863
$487$	1.91462e12	1.54242	0.771210	−	0.636580i	$-0.219652\pi$
$487$	1.91462e12	1.54242	0.771210	+	0.636580i	$0.219652\pi$
$488$	9.98108e11	0.796688
$489$	7.64688e11	0.604776
$490$	−8.33061e11	−0.652821
$491$	−1.79635e12	−1.39484	−0.697418	−	0.716664i	$-0.745668\pi$
$491$	−1.79635e12	−1.39484	−0.697418	+	0.716664i	$0.745668\pi$
$492$	−3.73633e12	−2.87477
$493$	−8.67443e11	−0.661348
$494$	0	0
$495$	1.68640e11	0.126252
$496$	−2.25913e10	−0.0167600
$497$	2.49221e11	0.183223
$498$	−6.34032e11	−0.461933
$499$	−1.29654e12	−0.936122	−0.468061	−	0.883696i	$-0.655047\pi$
$499$	−1.29654e12	−0.936122	−0.468061	+	0.883696i	$0.655047\pi$
$500$	2.24371e12	1.60547
$501$	−6.65857e11	−0.472184
$502$	1.34423e12	0.944731
$503$	1.08278e12	0.754193	0.377097	−	0.926174i	$-0.376922\pi$
$503$	1.08278e12	0.754193	0.377097	+	0.926174i	$0.376922\pi$
$504$	−2.46633e11	−0.170260
$505$	−1.04286e12	−0.713537
$506$	4.63389e11	0.314245
$507$	0	0
$508$	−3.97408e12	−2.64758
$509$	1.45339e12	0.959734	0.479867	−	0.877341i	$-0.340685\pi$
$509$	1.45339e12	0.959734	0.479867	+	0.877341i	$0.340685\pi$
$510$	−3.62002e12	−2.36944
$511$	1.67208e12	1.08483
$512$	−8.24265e10	−0.0530093
$513$	1.22353e12	0.779985
$514$	−1.03111e12	−0.651584
$515$	−1.64493e11	−0.103042
$516$	−3.40452e11	−0.211413
$517$	2.49764e12	1.53752
$518$	5.53256e12	3.37631
$519$	−1.71040e12	−1.03477
$520$	0	0
$521$	−9.18754e11	−0.546298	−0.273149	−	0.961972i	$-0.588065\pi$
$521$	−9.18754e11	−0.546298	−0.273149	+	0.961972i	$0.588065\pi$
$522$	−1.65099e11	−0.0973254
$523$	2.89678e12	1.69300	0.846501	−	0.532386i	$-0.178705\pi$
$523$	2.89678e12	1.69300	0.846501	+	0.532386i	$0.178705\pi$
$524$	−2.96802e12	−1.71979
$525$	−5.50284e10	−0.0316133
$526$	−1.49677e12	−0.852551
$527$	1.74006e12	0.982690
$528$	−3.87189e10	−0.0216806
$529$	−1.71431e12	−0.951784
$530$	1.26901e12	0.698590
$531$	−4.49492e11	−0.245356
$532$	2.62495e12	1.42075
$533$	0	0
$534$	2.79148e12	1.48559
$535$	5.97354e11	0.315239
$536$	−5.79177e11	−0.303089
$537$	−1.34395e12	−0.697428
$538$	2.95540e12	1.52089
$539$	6.86297e11	0.350238
$540$	−3.45445e12	−1.74827
$541$	6.88869e11	0.345739	0.172870	−	0.984945i	$-0.444696\pi$
$541$	6.88869e11	0.345739	0.172870	+	0.984945i	$0.444696\pi$
$542$	−6.05827e12	−3.01545
$543$	−3.55211e11	−0.175342
$544$	3.10615e12	1.52064
$545$	−1.84426e12	−0.895445
$546$	0	0
$547$	−3.12007e12	−1.49012	−0.745061	−	0.666996i	$-0.767580\pi$
$547$	−3.12007e12	−1.49012	−0.745061	+	0.666996i	$0.767580\pi$
$548$	4.41045e12	2.08915
$549$	−2.34169e11	−0.110015
$550$	8.86362e10	0.0413028
$551$	6.79037e11	0.313842
$552$	−4.53410e11	−0.207857
$553$	9.65185e10	0.0438882
$554$	−3.48679e12	−1.57265
$555$	3.70154e12	1.65601
$556$	7.80055e11	0.346169
$557$	−2.65417e12	−1.16837	−0.584185	−	0.811621i	$-0.698586\pi$
$557$	−2.65417e12	−1.16837	−0.584185	+	0.811621i	$0.698586\pi$
$558$	3.31182e11	0.144615
$559$	0	0
$560$	−7.39567e10	−0.0317783
$561$	2.98226e12	1.27120
$562$	−6.82708e12	−2.88683
$563$	1.37050e12	0.574900	0.287450	−	0.957796i	$-0.407193\pi$
$563$	1.37050e12	0.574900	0.287450	+	0.957796i	$0.407193\pi$
$564$	−6.32405e12	−2.63172
$565$	1.45245e12	0.599630
$566$	3.80856e12	1.55986
$567$	−2.44061e12	−0.991686
$568$	3.92784e11	0.158338
$569$	2.71253e12	1.08485	0.542425	−	0.840104i	$-0.317506\pi$
$569$	2.71253e12	1.08485	0.542425	+	0.840104i	$0.317506\pi$
$570$	2.83376e12	1.12441
$571$	3.89232e12	1.53231	0.766153	−	0.642658i	$-0.222168\pi$
$571$	3.89232e12	1.53231	0.766153	+	0.642658i	$0.222168\pi$
$572$	0	0
$573$	−1.32142e12	−0.512090
$574$	9.48683e12	3.64769
$575$	1.66114e10	0.00633724
$576$	6.01064e11	0.227520
$577$	4.61600e12	1.73370	0.866851	−	0.498567i	$-0.166140\pi$
$577$	4.61600e12	1.73370	0.866851	+	0.498567i	$0.166140\pi$
$578$	6.16021e12	2.29573
$579$	−1.88982e12	−0.698821
$580$	−1.91716e12	−0.703448
$581$	9.97703e11	0.363252
$582$	−4.92697e12	−1.78003
$583$	−1.04544e12	−0.374793
$584$	2.63528e12	0.937493
$585$	0	0
$586$	−3.96176e12	−1.38787
$587$	−1.49574e12	−0.519979	−0.259990	−	0.965611i	$-0.583719\pi$
$587$	−1.49574e12	−0.519979	−0.259990	+	0.965611i	$0.583719\pi$
$588$	−1.73771e12	−0.599487
$589$	−1.36213e12	−0.466335
$590$	−8.42215e12	−2.86147
$591$	−3.19755e11	−0.107814
$592$	1.39547e11	0.0466954
$593$	2.12351e12	0.705192	0.352596	−	0.935776i	$-0.385299\pi$
$593$	2.12351e12	0.705192	0.352596	+	0.935776i	$0.385299\pi$
$594$	4.59198e12	1.51343
$595$	5.69640e12	1.86326
$596$	−5.46738e12	−1.77489
$597$	−3.25198e12	−1.04776
$598$	0	0
$599$	1.24442e12	0.394954	0.197477	−	0.980308i	$-0.436725\pi$
$599$	1.24442e12	0.394954	0.197477	+	0.980308i	$0.436725\pi$
$600$	−8.67274e10	−0.0273197
$601$	7.25714e11	0.226898	0.113449	−	0.993544i	$-0.463810\pi$
$601$	7.25714e11	0.226898	0.113449	+	0.993544i	$0.463810\pi$
$602$	8.64432e11	0.268254
$603$	1.35882e11	0.0418539
$604$	−1.66831e11	−0.0510047
$605$	7.39405e11	0.224379
$606$	−3.51006e12	−1.05727
$607$	−3.16893e12	−0.947467	−0.473734	−	0.880668i	$-0.657094\pi$
$607$	−3.16893e12	−0.947467	−0.473734	+	0.880668i	$0.657094\pi$
$608$	−2.43151e12	−0.721621
$609$	−1.58218e12	−0.466098
$610$	−4.38763e12	−1.28306
$611$	0	0
$612$	1.23991e12	0.357280
$613$	8.87008e11	0.253720	0.126860	−	0.991921i	$-0.459510\pi$
$613$	8.87008e11	0.253720	0.126860	+	0.991921i	$0.459510\pi$
$614$	−4.34231e12	−1.23300
$615$	6.34713e12	1.78912
$616$	3.80703e12	1.06530
$617$	1.91841e12	0.532914	0.266457	−	0.963847i	$-0.414147\pi$
$617$	1.91841e12	0.532914	0.266457	+	0.963847i	$0.414147\pi$
$618$	−5.53650e11	−0.152682
$619$	3.56519e12	0.976057	0.488028	−	0.872828i	$-0.337716\pi$
$619$	3.56519e12	0.976057	0.488028	+	0.872828i	$0.337716\pi$
$620$	3.84576e12	1.04525
$621$	8.60587e11	0.232211
$622$	4.37468e12	1.17190
$623$	−4.39262e12	−1.16823
$624$	0	0
$625$	−3.92161e12	−1.02803
$626$	−1.14834e13	−2.98873
$627$	−2.33453e12	−0.603247
$628$	−5.55886e12	−1.42616
$629$	−1.07484e13	−2.73790
$630$	1.08418e12	0.274202
$631$	1.21408e12	0.304870	0.152435	−	0.988314i	$-0.451289\pi$
$631$	1.21408e12	0.304870	0.152435	+	0.988314i	$0.451289\pi$
$632$	1.52118e11	0.0379274
$633$	3.84392e12	0.951607
$634$	1.08757e12	0.267335
$635$	6.75100e12	1.64773
$636$	2.64707e12	0.641517
$637$	0	0
$638$	2.54847e12	0.608957
$639$	−9.21521e10	−0.0218651
$640$	7.05005e12	1.66105
$641$	−2.78119e12	−0.650684	−0.325342	−	0.945596i	$-0.605479\pi$
$641$	−2.78119e12	−0.650684	−0.325342	+	0.945596i	$0.605479\pi$
$642$	2.01057e12	0.467101
$643$	−5.06784e12	−1.16916	−0.584580	−	0.811336i	$-0.698741\pi$
$643$	−5.06784e12	−1.16916	−0.584580	+	0.811336i	$0.698741\pi$
$644$	1.84630e12	0.422975
$645$	5.78345e11	0.131574
$646$	−8.22860e12	−1.85900
$647$	−6.59226e12	−1.47899	−0.739494	−	0.673163i	$-0.764935\pi$
$647$	−6.59226e12	−1.47899	−0.739494	+	0.673163i	$0.764935\pi$
$648$	−3.84651e12	−0.856998
$649$	6.93839e12	1.53517
$650$	0	0
$651$	3.17379e12	0.692570
$652$	−4.90755e12	−1.06353
$653$	−8.90755e11	−0.191712	−0.0958560	−	0.995395i	$-0.530559\pi$
$653$	−8.90755e11	−0.191712	−0.0958560	+	0.995395i	$0.530559\pi$
$654$	−6.20740e12	−1.32681
$655$	5.04195e12	1.07032
$656$	2.39286e11	0.0504487
$657$	−6.18270e11	−0.129459
$658$	1.60572e13	3.33929
$659$	−1.31355e12	−0.271308	−0.135654	−	0.990756i	$-0.543314\pi$
$659$	−1.31355e12	−0.271308	−0.135654	+	0.990756i	$0.543314\pi$
$660$	6.59118e12	1.35212
$661$	−4.44228e12	−0.905106	−0.452553	−	0.891737i	$-0.649487\pi$
$661$	−4.44228e12	−0.905106	−0.452553	+	0.891737i	$0.649487\pi$
$662$	−1.28854e13	−2.60757
$663$	0	0
$664$	1.57243e12	0.313917
$665$	−4.45916e12	−0.884211
$666$	−2.04573e12	−0.402915
$667$	4.77610e11	0.0934345
$668$	4.27328e12	0.830361
$669$	5.97119e12	1.15251
$670$	2.54603e12	0.488121
$671$	3.61464e12	0.688358
$672$	5.66548e12	1.07170
$673$	5.51945e12	1.03712	0.518559	−	0.855042i	$-0.326469\pi$
$673$	5.51945e12	1.03712	0.518559	+	0.855042i	$0.326469\pi$
$674$	1.87216e12	0.349441
$675$	1.64611e11	0.0305206
$676$	0	0
$677$	−5.16749e12	−0.945432	−0.472716	−	0.881215i	$-0.656726\pi$
$677$	−5.16749e12	−0.945432	−0.472716	+	0.881215i	$0.656726\pi$
$678$	4.88864e12	0.888494
$679$	7.75299e12	1.39977
$680$	8.97780e12	1.61020
$681$	−7.93734e11	−0.141421
$682$	−5.11214e12	−0.904843
$683$	2.52311e12	0.443652	0.221826	−	0.975086i	$-0.428798\pi$
$683$	2.52311e12	0.443652	0.221826	+	0.975086i	$0.428798\pi$
$684$	−9.70605e11	−0.169547
$685$	−7.49229e12	−1.30019
$686$	−6.70515e12	−1.15598
$687$	−3.56926e12	−0.611326
$688$	2.18035e10	0.00371004
$689$	0	0
$690$	1.99317e12	0.334751
$691$	−1.13796e12	−0.189878	−0.0949392	−	0.995483i	$-0.530266\pi$
$691$	−1.13796e12	−0.189878	−0.0949392	+	0.995483i	$0.530266\pi$
$692$	1.09768e13	1.81970
$693$	−8.93179e11	−0.147109
$694$	1.47378e13	2.41165
$695$	−1.32513e12	−0.215440
$696$	−2.49359e12	−0.402794
$697$	−1.84306e13	−2.95796
$698$	4.47846e11	0.0714134
$699$	−3.31598e12	−0.525370
$700$	3.53156e11	0.0555937
$701$	1.08353e13	1.69476	0.847382	−	0.530984i	$-0.178178\pi$
$701$	1.08353e13	1.69476	0.847382	+	0.530984i	$0.178178\pi$
$702$	0	0
$703$	8.41391e12	1.29927
$704$	−9.27805e12	−1.42357
$705$	1.07430e13	1.63786
$706$	4.00175e12	0.606218
$707$	5.52337e12	0.831413
$708$	−1.75681e13	−2.62769
$709$	−3.85848e12	−0.573466	−0.286733	−	0.958011i	$-0.592569\pi$
$709$	−3.85848e12	−0.573466	−0.286733	+	0.958011i	$0.592569\pi$
$710$	−1.72666e12	−0.255002
$711$	−3.56888e10	−0.00523744
$712$	−6.92299e12	−1.00956
$713$	−9.58070e11	−0.138833
$714$	1.91729e13	2.76087
$715$	0	0
$716$	8.62509e12	1.22646
$717$	−3.31784e12	−0.468834
$718$	5.13688e12	0.721339
$719$	3.09231e12	0.431522	0.215761	−	0.976446i	$-0.430777\pi$
$719$	3.09231e12	0.431522	0.215761	+	0.976446i	$0.430777\pi$
$720$	2.73463e10	0.00379230
$721$	8.71214e11	0.120065
$722$	−5.39942e12	−0.739486
$723$	−9.82298e12	−1.33697
$724$	2.27964e12	0.308349
$725$	9.13564e10	0.0122806
$726$	2.48868e12	0.332471
$727$	5.87291e11	0.0779737	0.0389869	−	0.999240i	$-0.487587\pi$
$727$	5.87291e11	0.0779737	0.0389869	+	0.999240i	$0.487587\pi$
$728$	0	0
$729$	8.37634e12	1.09845
$730$	−1.15845e13	−1.50982
$731$	−1.67938e12	−0.217531
$732$	−9.15232e12	−1.17823
$733$	−2.05752e12	−0.263255	−0.131627	−	0.991299i	$-0.542020\pi$
$733$	−2.05752e12	−0.263255	−0.131627	+	0.991299i	$0.542020\pi$
$734$	−1.88739e13	−2.40010
$735$	2.95195e12	0.373093
$736$	−1.71023e12	−0.214835
$737$	−2.09749e12	−0.261876
$738$	−3.50786e12	−0.435300
$739$	3.58288e11	0.0441909	0.0220954	−	0.999756i	$-0.492966\pi$
$739$	3.58288e11	0.0441909	0.0220954	+	0.999756i	$0.492966\pi$
$740$	−2.37554e13	−2.91219
$741$	0	0
$742$	−6.72110e12	−0.813997
$743$	8.58825e12	1.03384	0.516922	−	0.856033i	$-0.327078\pi$
$743$	8.58825e12	1.03384	0.516922	+	0.856033i	$0.327078\pi$
$744$	5.00205e12	0.598508
$745$	9.28776e12	1.10461
$746$	−1.06629e13	−1.26052
$747$	−3.68912e11	−0.0433491
$748$	−1.91393e13	−2.23547
$749$	−3.16379e12	−0.367316
$750$	−1.28288e13	−1.48050
$751$	−2.96471e12	−0.340097	−0.170049	−	0.985436i	$-0.554392\pi$
$751$	−2.96471e12	−0.340097	−0.170049	+	0.985436i	$0.554392\pi$
$752$	4.05011e11	0.0461834
$753$	−4.76330e12	−0.539921
$754$	0	0
$755$	2.83406e11	0.0317429
$756$	1.82960e13	2.03708
$757$	−1.18699e13	−1.31376	−0.656881	−	0.753994i	$-0.728125\pi$
$757$	−1.18699e13	−1.31376	−0.656881	+	0.753994i	$0.728125\pi$
$758$	1.63931e12	0.180364
$759$	−1.64202e12	−0.179594
$760$	−7.02785e12	−0.764120
$761$	−5.09033e12	−0.550193	−0.275097	−	0.961417i	$-0.588710\pi$
$761$	−5.09033e12	−0.550193	−0.275097	+	0.961417i	$0.588710\pi$
$762$	2.27224e13	2.44150
$763$	9.76786e12	1.04337
$764$	8.48051e12	0.900537
$765$	−2.10631e12	−0.222354
$766$	3.50341e12	0.367673
$767$	0	0
$768$	9.31510e12	0.966189
$769$	4.36770e12	0.450385	0.225192	−	0.974314i	$-0.427699\pi$
$769$	4.36770e12	0.450385	0.225192	+	0.974314i	$0.427699\pi$
$770$	−1.67355e13	−1.71566
$771$	3.65374e12	0.372385
$772$	1.21283e13	1.22891
$773$	1.44562e13	1.45629	0.728144	−	0.685424i	$-0.240383\pi$
$773$	1.44562e13	1.45629	0.728144	+	0.685424i	$0.240383\pi$
$774$	−3.19634e11	−0.0320124
$775$	−1.83258e11	−0.0182476
$776$	1.22191e13	1.20965
$777$	−1.96046e13	−1.92959
$778$	8.80756e12	0.861881
$779$	1.44276e13	1.40370
$780$	0	0
$781$	1.42246e12	0.136808
$782$	−5.78770e12	−0.553446
$783$	4.73291e12	0.449987
$784$	1.11288e11	0.0105203
$785$	9.44317e12	0.887574
$786$	1.69701e13	1.58593
$787$	1.48558e13	1.38042	0.690209	−	0.723610i	$-0.257518\pi$
$787$	1.48558e13	1.38042	0.690209	+	0.723610i	$0.257518\pi$
$788$	2.05210e12	0.189596
$789$	5.30383e12	0.487240
$790$	−6.68702e11	−0.0610816
$791$	−7.69268e12	−0.698688
$792$	−1.40769e12	−0.127129
$793$	0	0
$794$	1.17517e13	1.04932
$795$	−4.49673e12	−0.399250
$796$	2.08703e13	1.84255
$797$	3.06631e11	0.0269187	0.0134594	−	0.999909i	$-0.495716\pi$
$797$	3.06631e11	0.0269187	0.0134594	+	0.999909i	$0.495716\pi$
$798$	−1.50086e13	−1.31017
$799$	−3.11953e13	−2.70788
$800$	−3.27130e11	−0.0282368
$801$	1.62422e12	0.139412
$802$	5.05291e11	0.0431278
$803$	9.54364e12	0.810017
$804$	5.31086e12	0.448242
$805$	−3.13641e12	−0.263240
$806$	0	0
$807$	−1.04725e13	−0.869199
$808$	8.70510e12	0.718493
$809$	−9.13599e12	−0.749872	−0.374936	−	0.927051i	$-0.622335\pi$
$809$	−9.13599e12	−0.749872	−0.374936	+	0.927051i	$0.622335\pi$
$810$	1.69091e13	1.38018
$811$	1.24574e13	1.01119	0.505594	−	0.862771i	$-0.331273\pi$
$811$	1.24574e13	1.01119	0.505594	+	0.862771i	$0.331273\pi$
$812$	1.01539e13	0.819658
$813$	2.14675e13	1.72335
$814$	3.15779e13	2.52100
$815$	8.33674e12	0.661892
$816$	4.83596e11	0.0381836
$817$	1.31463e12	0.103229
$818$	1.66195e13	1.29786
$819$	0	0
$820$	−4.07340e13	−3.14626
$821$	−1.41959e12	−0.109048	−0.0545241	−	0.998512i	$-0.517364\pi$
$821$	−1.41959e12	−0.109048	−0.0545241	+	0.998512i	$0.517364\pi$
$822$	−2.52175e13	−1.92654
$823$	−6.90254e12	−0.524457	−0.262228	−	0.965006i	$-0.584457\pi$
$823$	−6.90254e12	−0.524457	−0.262228	+	0.965006i	$0.584457\pi$
$824$	1.37308e12	0.103758
$825$	−3.14083e11	−0.0236049
$826$	4.46066e13	3.33418
$827$	3.79690e12	0.282263	0.141132	−	0.989991i	$-0.454926\pi$
$827$	3.79690e12	0.282263	0.141132	+	0.989991i	$0.454926\pi$
$828$	−6.82689e11	−0.0504761
$829$	−8.19356e12	−0.602528	−0.301264	−	0.953541i	$-0.597409\pi$
$829$	−8.19356e12	−0.602528	−0.301264	+	0.953541i	$0.597409\pi$
$830$	−6.91231e12	−0.505559
$831$	1.23554e13	0.898781
$832$	0	0
$833$	−8.57180e12	−0.616836
$834$	−4.46009e12	−0.319225
$835$	−7.25927e12	−0.516778
$836$	1.49823e13	1.06084
$837$	−9.49405e12	−0.668632
$838$	3.18032e13	2.22778
$839$	8.11293e11	0.0565261	0.0282630	−	0.999601i	$-0.491002\pi$
$839$	8.11293e11	0.0565261	0.0282630	+	0.999601i	$0.491002\pi$
$840$	1.63751e13	1.13482
$841$	−1.18805e13	−0.818939
$842$	3.32843e13	2.28211
$843$	2.41918e13	1.64985
$844$	−2.46691e13	−1.67345
$845$	0	0
$846$	−5.93734e12	−0.398497
$847$	−3.91615e12	−0.261447
$848$	−1.69526e11	−0.0112578
$849$	−1.34956e13	−0.891474
$850$	−1.10706e12	−0.0727422
$851$	5.91804e12	0.386807
$852$	−3.60169e12	−0.234169
$853$	−1.79193e13	−1.15891	−0.579455	−	0.815004i	$-0.696734\pi$
$853$	−1.79193e13	−1.15891	−0.579455	+	0.815004i	$0.696734\pi$
$854$	2.32384e13	1.49502
$855$	1.64883e12	0.105518
$856$	−4.98629e12	−0.317428
$857$	−1.89206e13	−1.19818	−0.599090	−	0.800681i	$-0.704471\pi$
$857$	−1.89206e13	−1.19818	−0.599090	+	0.800681i	$0.704471\pi$
$858$	0	0
$859$	−2.52359e13	−1.58143	−0.790716	−	0.612184i	$-0.790291\pi$
$859$	−2.52359e13	−1.58143	−0.790716	+	0.612184i	$0.790291\pi$
$860$	−3.71165e12	−0.231379
$861$	−3.36166e13	−2.08468
$862$	4.35070e12	0.268396
$863$	−2.98253e13	−1.83036	−0.915179	−	0.403048i	$-0.867951\pi$
$863$	−2.98253e13	−1.83036	−0.915179	+	0.403048i	$0.867951\pi$
$864$	−1.69477e13	−1.03466
$865$	−1.86470e13	−1.13249
$866$	−9.75848e12	−0.589592
$867$	−2.18287e13	−1.31203
$868$	−2.03685e13	−1.21792
$869$	5.50894e11	0.0327702
$870$	1.09617e13	0.648695
$871$	0	0
$872$	1.53946e13	0.901665
$873$	−2.86676e12	−0.167042
$874$	4.53063e12	0.262638
$875$	2.01871e13	1.16423
$876$	−2.41646e13	−1.38647
$877$	−5.87014e12	−0.335081	−0.167541	−	0.985865i	$-0.553583\pi$
$877$	−5.87014e12	−0.335081	−0.167541	+	0.985865i	$0.553583\pi$
$878$	−2.84853e12	−0.161769
$879$	1.40385e13	0.793179
$880$	−4.22119e11	−0.0237281
$881$	2.07929e13	1.16285	0.581426	−	0.813599i	$-0.302495\pi$
$881$	2.07929e13	1.16285	0.581426	+	0.813599i	$0.302495\pi$
$882$	−1.63145e12	−0.0907750
$883$	2.40223e13	1.32981	0.664907	−	0.746926i	$-0.268471\pi$
$883$	2.40223e13	1.32981	0.664907	+	0.746926i	$0.268471\pi$
$884$	0	0
$885$	2.98439e13	1.63535
$886$	4.51062e13	2.45914
$887$	−2.33740e12	−0.126788	−0.0633938	−	0.997989i	$-0.520192\pi$
$887$	−2.33740e12	−0.126788	−0.0633938	+	0.997989i	$0.520192\pi$
$888$	−3.08979e13	−1.66752
$889$	−3.57557e13	−1.91994
$890$	3.04331e13	1.62589
$891$	−1.39301e13	−0.740467
$892$	−3.83213e13	−2.02674
$893$	2.44198e13	1.28502
$894$	3.12606e13	1.63674
$895$	−1.46520e13	−0.763294
$896$	−3.73395e13	−1.93545
$897$	0	0
$898$	2.83982e13	1.45730
$899$	−5.26903e12	−0.269037
$900$	−1.30583e11	−0.00663432
$901$	1.30575e13	0.660082
$902$	5.41475e13	2.72364
$903$	−3.06312e12	−0.153309
$904$	−1.21240e13	−0.603795
$905$	−3.87256e12	−0.191902
$906$	9.53883e11	0.0470347
$907$	−1.84460e13	−0.905041	−0.452520	−	0.891754i	$-0.649475\pi$
$907$	−1.84460e13	−0.905041	−0.452520	+	0.891754i	$0.649475\pi$
$908$	5.09395e12	0.248696
$909$	−2.04233e12	−0.0992175
$910$	0	0
$911$	1.63941e13	0.788595	0.394298	−	0.918983i	$-0.370988\pi$
$911$	1.63941e13	0.788595	0.394298	+	0.918983i	$0.370988\pi$
$912$	−3.78561e11	−0.0181200
$913$	5.69454e12	0.271231
$914$	−1.37685e13	−0.652574
$915$	1.55476e13	0.733277
$916$	2.29065e13	1.07505
$917$	−2.67039e13	−1.24713
$918$	−5.73536e13	−2.66544
$919$	−1.11733e13	−0.516729	−0.258365	−	0.966047i	$-0.583184\pi$
$919$	−1.11733e13	−0.516729	−0.258365	+	0.966047i	$0.583184\pi$
$920$	−4.94314e12	−0.227488
$921$	1.53870e13	0.704670
$922$	−3.64897e13	−1.66296
$923$	0	0
$924$	−3.49092e13	−1.57549
$925$	1.13199e12	0.0508400
$926$	6.47302e13	2.89306
$927$	−3.22141e11	−0.0143281
$928$	−9.40565e12	−0.416316
$929$	2.32482e12	0.102404	0.0512022	−	0.998688i	$-0.483695\pi$
$929$	2.32482e12	0.102404	0.0512022	+	0.998688i	$0.483695\pi$
$930$	−2.19887e13	−0.963889
$931$	6.71004e12	0.292719
$932$	2.12810e13	0.923890
$933$	−1.55017e13	−0.669749
$934$	2.70217e12	0.116186
$935$	3.25131e13	1.39125
$936$	0	0
$937$	−3.73094e12	−0.158121	−0.0790606	−	0.996870i	$-0.525192\pi$
$937$	−3.73094e12	−0.158121	−0.0790606	+	0.996870i	$0.525192\pi$
$938$	−1.34847e13	−0.568758
$939$	4.06915e13	1.70808
$940$	−6.89457e13	−2.88026
$941$	1.79721e12	0.0747213	0.0373607	−	0.999302i	$-0.488105\pi$
$941$	1.79721e12	0.0747213	0.0373607	+	0.999302i	$0.488105\pi$
$942$	3.17837e13	1.31515
$943$	1.01478e13	0.417898
$944$	1.12511e12	0.0461128
$945$	−3.10805e13	−1.26778
$946$	4.93388e12	0.200299
$947$	−3.01107e13	−1.21660	−0.608298	−	0.793709i	$-0.708147\pi$
$947$	−3.01107e13	−1.21660	−0.608298	+	0.793709i	$0.708147\pi$
$948$	−1.39487e12	−0.0560914
$949$	0	0
$950$	8.66611e11	0.0345198
$951$	−3.85381e12	−0.152784
$952$	−4.75496e13	−1.87620
$953$	−2.29753e13	−0.902286	−0.451143	−	0.892452i	$-0.648983\pi$
$953$	−2.29753e13	−0.902286	−0.451143	+	0.892452i	$0.648983\pi$
$954$	2.48520e12	0.0971392
$955$	−1.44063e13	−0.560452
$956$	2.12929e13	0.824469
$957$	−9.03051e12	−0.348024
$958$	−6.32880e13	−2.42760
$959$	3.96818e13	1.51498
$960$	−3.99075e13	−1.51647
$961$	−1.58701e13	−0.600240
$962$	0	0
$963$	1.16985e12	0.0438340
$964$	6.30410e13	2.35113
$965$	−2.06030e13	−0.764818
$966$	−1.05565e13	−0.390052
$967$	−9.41605e11	−0.0346298	−0.0173149	−	0.999850i	$-0.505512\pi$
$967$	−9.41605e11	−0.0346298	−0.0173149	+	0.999850i	$0.505512\pi$
$968$	−6.17204e12	−0.225938
$969$	2.91581e13	1.06243
$970$	−5.37145e13	−1.94813
$971$	−1.86297e13	−0.672542	−0.336271	−	0.941765i	$-0.609166\pi$
$971$	−1.86297e13	−0.672542	−0.336271	+	0.941765i	$0.609166\pi$
$972$	−1.26941e13	−0.456145
$973$	7.01833e12	0.251030
$974$	7.02557e13	2.50130
$975$	0	0
$976$	5.86141e11	0.0206766
$977$	−1.37438e13	−0.482592	−0.241296	−	0.970452i	$-0.577573\pi$
$977$	−1.37438e13	−0.482592	−0.241296	+	0.970452i	$0.577573\pi$
$978$	2.80597e13	0.980750
$979$	−2.50716e13	−0.872286
$980$	−1.89448e13	−0.656103
$981$	−3.61178e12	−0.124512
$982$	−6.59156e13	−2.26197
$983$	2.97895e13	1.01759	0.508794	−	0.860888i	$-0.330091\pi$
$983$	2.97895e13	1.01759	0.508794	+	0.860888i	$0.330091\pi$
$984$	−5.29814e13	−1.80155
$985$	−3.48602e12	−0.117996
$986$	−3.18302e13	−1.07249
$987$	−5.68989e13	−1.90843
$988$	0	0
$989$	9.24661e11	0.0307326
$990$	6.18814e12	0.204740
$991$	−2.95962e13	−0.974774	−0.487387	−	0.873186i	$-0.662050\pi$
$991$	−2.95962e13	−0.974774	−0.487387	+	0.873186i	$0.662050\pi$
$992$	1.88674e13	0.618600
$993$	4.56594e13	1.49025
$994$	9.14497e12	0.297128
$995$	−3.54536e13	−1.14672
$996$	−1.44186e13	−0.464256
$997$	1.29524e13	0.415167	0.207584	−	0.978217i	$-0.433440\pi$
$997$	1.29524e13	0.415167	0.207584	+	0.978217i	$0.433440\pi$
$998$	−4.75755e13	−1.51809
$999$	5.86452e13	1.86289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.10.a.f.1.18		20
13.2	odd	12		13.10.e.a.4.9	✓	20
13.7	odd	12		13.10.e.a.10.9	yes	20
13.12	even	2	inner	169.10.a.f.1.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.e.a.4.9	✓	20	13.2	odd	12
13.10.e.a.10.9	yes	20	13.7	odd	12
169.10.a.f.1.3		20	13.12	even	2	inner
169.10.a.f.1.18		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+36.6943 q^{2} -130.026 q^{3} +834.471 q^{4} -1417.57 q^{5} -4771.23 q^{6} +7507.92 q^{7} +11832.9 q^{8} -2776.14 q^{9} -52016.6 q^{10} +42852.6 q^{11} -108503. q^{12} +275498. q^{14} +184321. q^{15} +6948.87 q^{16} -535226. q^{17} -101868. q^{18} +418977. q^{19} -1.18292e6 q^{20} -976228. q^{21} +1.57245e6 q^{22} +294693. q^{23} -1.53858e6 q^{24} +56368.4 q^{25} +2.92028e6 q^{27} +6.26514e6 q^{28} +1.62070e6 q^{29} +6.76353e6 q^{30} -3.25108e6 q^{31} -5.80344e6 q^{32} -5.57197e6 q^{33} -1.96397e7 q^{34} -1.06430e7 q^{35} -2.31661e6 q^{36} +2.00820e7 q^{37} +1.53741e7 q^{38} -1.67738e7 q^{40} +3.44352e7 q^{41} -3.58220e7 q^{42} +3.13771e6 q^{43} +3.57593e7 q^{44} +3.93536e6 q^{45} +1.08136e7 q^{46} +5.82844e7 q^{47} -903537. q^{48} +1.60153e7 q^{49} +2.06840e6 q^{50} +6.95935e7 q^{51} -2.43962e7 q^{53} +1.07158e8 q^{54} -6.07464e7 q^{55} +8.88401e7 q^{56} -5.44780e7 q^{57} +5.94706e7 q^{58} +1.61913e8 q^{59} +1.53811e8 q^{60} +8.43506e7 q^{61} -1.19296e8 q^{62} -2.08430e7 q^{63} -2.16511e8 q^{64} -2.04459e8 q^{66} -4.89466e7 q^{67} -4.46631e8 q^{68} -3.83179e7 q^{69} -3.90536e8 q^{70} +3.31943e7 q^{71} -3.28496e7 q^{72} +2.22708e8 q^{73} +7.36896e8 q^{74} -7.32937e6 q^{75} +3.49624e8 q^{76} +3.21734e8 q^{77} +1.28556e7 q^{79} -9.85048e6 q^{80} -3.25071e8 q^{81} +1.26358e9 q^{82} +1.32887e8 q^{83} -8.14634e8 q^{84} +7.58718e8 q^{85} +1.15136e8 q^{86} -2.10734e8 q^{87} +5.07069e8 q^{88} -5.85065e8 q^{89} +1.44405e8 q^{90} +2.45913e8 q^{92} +4.22726e8 q^{93} +2.13870e9 q^{94} -5.93927e8 q^{95} +7.54600e8 q^{96} +1.03264e9 q^{97} +5.87670e8 q^{98} -1.18965e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100}) \) 20 * q + 326 * q^3 + 5118 * q^4 + 129526 * q^9 + 88390 * q^10 + 427652 * q^12 + 473556 * q^14 + 1189618 * q^16 - 99312 * q^17 - 5073532 * q^22 + 6252378 * q^23 + 1529274 * q^25 + 18052718 * q^27 + 5424828 * q^29 + 30045516 * q^30 - 15098160 * q^35 + 54078886 * q^36 + 39021096 * q^38 + 134360674 * q^40 + 121600068 * q^42 + 53242058 * q^43 + 82771076 * q^48 + 14055298 * q^49 + 10208934 * q^51 - 36429786 * q^53 - 131454160 * q^55 + 127272672 * q^56 + 536425792 * q^61 - 890759796 * q^62 - 343337066 * q^64 - 445758060 * q^66 + 336594090 * q^68 + 1083885582 * q^69 + 1083869262 * q^74 - 1218826882 * q^75 + 1470187374 * q^77 + 1556703616 * q^79 + 2139127516 * q^81 + 3719322754 * q^82 + 1288246146 * q^87 - 4109160928 * q^88 + 6489878934 * q^90 + 10148843820 * q^92 + 4894712828 * q^94 + 9251202540 * q^95

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