LMFDB - Embedded newform 169.8.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,8,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, 8); 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, 8, names="a")

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0] 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:	$52.7930693068$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 449x^{4} + 37224x^{2} - 205776 $ x^6 - 449x^4 + 37224x^2 - 205776
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-10.0583$ 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)

$f(q)$	$=$	\(q-10.0583 q^{2} +50.8656 q^{3} -26.8297 q^{4} -8.88672 q^{5} -511.623 q^{6} +510.452 q^{7} +1557.33 q^{8} +400.306 q^{9} +89.3858 q^{10} -4065.46 q^{11} -1364.71 q^{12} -5134.30 q^{14} -452.028 q^{15} -12230.0 q^{16} +1791.54 q^{17} -4026.41 q^{18} +19743.0 q^{19} +238.428 q^{20} +25964.4 q^{21} +40891.8 q^{22} +11749.0 q^{23} +79214.5 q^{24} -78046.0 q^{25} -90881.2 q^{27} -13695.3 q^{28} -183054. q^{29} +4546.66 q^{30} +247198. q^{31} -76325.0 q^{32} -206792. q^{33} -18020.0 q^{34} -4536.25 q^{35} -10740.1 q^{36} +203775. q^{37} -198582. q^{38} -13839.6 q^{40} -419322. q^{41} -261159. q^{42} -17481.7 q^{43} +109075. q^{44} -3557.41 q^{45} -118175. q^{46} +1.17985e6 q^{47} -622084. q^{48} -562982. q^{49} +785014. q^{50} +91127.9 q^{51} +1.32587e6 q^{53} +914115. q^{54} +36128.6 q^{55} +794942. q^{56} +1.00424e6 q^{57} +1.84123e6 q^{58} -401485. q^{59} +12127.8 q^{60} -3.28655e6 q^{61} -2.48640e6 q^{62} +204337. q^{63} +2.33314e6 q^{64} +2.07999e6 q^{66} -2.10629e6 q^{67} -48066.5 q^{68} +597618. q^{69} +45627.1 q^{70} -4.22986e6 q^{71} +623408. q^{72} -3.29726e6 q^{73} -2.04964e6 q^{74} -3.96986e6 q^{75} -529699. q^{76} -2.07522e6 q^{77} +3.79387e6 q^{79} +108684. q^{80} -5.49819e6 q^{81} +4.21768e6 q^{82} -4.36639e6 q^{83} -696617. q^{84} -15920.9 q^{85} +175837. q^{86} -9.31117e6 q^{87} -6.33127e6 q^{88} -8.72168e6 q^{89} +35781.6 q^{90} -315221. q^{92} +1.25738e7 q^{93} -1.18673e7 q^{94} -175451. q^{95} -3.88231e6 q^{96} +9.21482e6 q^{97} +5.66267e6 q^{98} -1.62743e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 56 q^{3} + 130 q^{4} - 1150 q^{9} + 406 q^{10} - 1898 q^{12} + 9558 q^{14} + 7778 q^{16} - 13152 q^{17} - 125080 q^{22} - 27264 q^{23} + 18262 q^{25} + 194560 q^{27} + 42924 q^{29} + 60114 q^{30}+ \cdots + 22075632 q^{95}+O(q^{100}) $ 6 * q - 56 * q^3 + 130 * q^4 - 1150 * q^9 + 406 * q^10 - 1898 * q^12 + 9558 * q^14 + 7778 * q^16 - 13152 * q^17 - 125080 * q^22 - 27264 * q^23 + 18262 * q^25 + 194560 * q^27 + 42924 * q^29 + 60114 * q^30 - 546720 * q^35 - 747772 * q^36 + 243492 * q^38 + 792058 * q^40 - 1138134 * q^42 + 1005576 * q^43 - 2807426 * q^48 - 3246846 * q^49 - 297984 * q^51 + 1705524 * q^53 - 5320576 * q^55 + 2282730 * q^56 - 8237572 * q^61 - 9625800 * q^62 - 5070862 * q^64 + 9289056 * q^66 - 16801086 * q^68 + 6017400 * q^69 - 28760826 * q^74 - 20426072 * q^75 - 4093560 * q^77 + 15147200 * q^79 - 21318442 * q^81 + 9130240 * q^82 - 36474240 * q^87 - 38008000 * q^88 - 4769676 * q^90 + 36462348 * q^92 - 34229770 * q^94 + 22075632 * 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 $

$2$	−10.0583	−0.889041	−0.444520	−	0.895769i	$-0.646626\pi$
$2$	−10.0583	−0.889041	−0.444520	+	0.895769i	$0.646626\pi$
$3$	50.8656	1.08768	0.543838	−	0.839190i	$-0.316971\pi$
$3$	50.8656	1.08768	0.543838	+	0.839190i	$0.316971\pi$
$4$	−26.8297	−0.209607
$5$	−8.88672	−0.0317941	−0.0158971	−	0.999874i	$-0.505060\pi$
$5$	−8.88672	−0.0317941	−0.0158971	+	0.999874i	$0.505060\pi$
$6$	−511.623	−0.966988
$7$	510.452	0.562486	0.281243	−	0.959637i	$-0.409253\pi$
$7$	510.452	0.562486	0.281243	+	0.959637i	$0.409253\pi$
$8$	1557.33	1.07539
$9$	400.306	0.183039
$10$	89.3858	0.0282663
$11$	−4065.46	−0.920949	−0.460474	−	0.887673i	$-0.652321\pi$
$11$	−4065.46	−0.920949	−0.460474	+	0.887673i	$0.652321\pi$
$12$	−1364.71	−0.227984
$13$	0	0
$13$	0	0
$14$	−5134.30	−0.500073
$15$	−452.028	−0.0345817
$16$	−12230.0	−0.746458
$17$	1791.54	0.0884415	0.0442207	−	0.999022i	$-0.485920\pi$
$17$	1791.54	0.0884415	0.0442207	+	0.999022i	$0.485920\pi$
$18$	−4026.41	−0.162729
$19$	19743.0	0.660353	0.330176	−	0.943919i	$-0.392892\pi$
$19$	19743.0	0.660353	0.330176	+	0.943919i	$0.392892\pi$
$20$	238.428	0.00666426
$21$	25964.4	0.611802
$22$	40891.8	0.818761
$23$	11749.0	0.201350	0.100675	−	0.994919i	$-0.467900\pi$
$23$	11749.0	0.201350	0.100675	+	0.994919i	$0.467900\pi$
$24$	79214.5	1.16968
$25$	−78046.0	−0.998989
$26$	0	0
$27$	−90881.2	−0.888589
$28$	−13695.3	−0.117901
$29$	−183054.	−1.39376	−0.696879	−	0.717189i	$-0.745429\pi$
$29$	−183054.	−1.39376	−0.696879	+	0.717189i	$0.745429\pi$
$30$	4546.66	0.0307445
$31$	247198.	1.49032	0.745158	−	0.666888i	$-0.232374\pi$
$31$	247198.	1.49032	0.745158	+	0.666888i	$0.232374\pi$
$32$	−76325.0	−0.411758
$33$	−206792.	−1.00169
$34$	−18020.0	−0.0786281
$35$	−4536.25	−0.0178837
$36$	−10740.1	−0.0383662
$37$	203775.	0.661372	0.330686	−	0.943741i	$-0.392720\pi$
$37$	203775.	0.661372	0.330686	+	0.943741i	$0.392720\pi$
$38$	−198582.	−0.587081
$39$	0	0
$40$	−13839.6	−0.0341911
$41$	−419322.	−0.950175	−0.475088	−	0.879938i	$-0.657584\pi$
$41$	−419322.	−0.950175	−0.475088	+	0.879938i	$0.657584\pi$
$42$	−261159.	−0.543917
$43$	−17481.7	−0.0335309	−0.0167654	−	0.999859i	$-0.505337\pi$
$43$	−17481.7	−0.0335309	−0.0167654	+	0.999859i	$0.505337\pi$
$44$	109075.	0.193037
$45$	−3557.41	−0.00581956
$46$	−118175.	−0.179009
$47$	1.17985e6	1.65761	0.828806	−	0.559537i	$-0.189021\pi$
$47$	1.17985e6	1.65761	0.828806	+	0.559537i	$0.189021\pi$
$48$	−622084.	−0.811905
$49$	−562982.	−0.683610
$50$	785014.	0.888142
$51$	91127.9	0.0961957
$52$	0	0
$53$	1.32587e6	1.22330	0.611651	−	0.791127i	$-0.290506\pi$
$53$	1.32587e6	1.22330	0.611651	+	0.791127i	$0.290506\pi$
$54$	914115.	0.789992
$55$	36128.6	0.0292807
$56$	794942.	0.604891
$57$	1.00424e6	0.718250
$58$	1.84123e6	1.23911
$59$	−401485.	−0.254499	−0.127250	−	0.991871i	$-0.540615\pi$
$59$	−401485.	−0.254499	−0.127250	+	0.991871i	$0.540615\pi$
$60$	12127.8	0.00724855
$61$	−3.28655e6	−1.85390	−0.926949	−	0.375188i	$-0.877578\pi$
$61$	−3.28655e6	−1.85390	−0.926949	+	0.375188i	$0.877578\pi$
$62$	−2.48640e6	−1.32495
$63$	204337.	0.102957
$64$	2.33314e6	1.11253
$65$	0	0
$66$	2.07999e6	0.890546
$67$	−2.10629e6	−0.855572	−0.427786	−	0.903880i	$-0.640706\pi$
$67$	−2.10629e6	−0.855572	−0.427786	+	0.903880i	$0.640706\pi$
$68$	−48066.5	−0.0185379
$69$	597618.	0.219004
$70$	45627.1	0.0158994
$71$	−4.22986e6	−1.40256	−0.701280	−	0.712886i	$-0.747388\pi$
$71$	−4.22986e6	−1.40256	−0.701280	+	0.712886i	$0.747388\pi$
$72$	623408.	0.196838
$73$	−3.29726e6	−0.992027	−0.496014	−	0.868315i	$-0.665203\pi$
$73$	−3.29726e6	−0.992027	−0.496014	+	0.868315i	$0.665203\pi$
$74$	−2.04964e6	−0.587986
$75$	−3.96986e6	−1.08658
$76$	−529699.	−0.138414
$77$	−2.07522e6	−0.518021
$78$	0	0
$79$	3.79387e6	0.865741	0.432870	−	0.901456i	$-0.357501\pi$
$79$	3.79387e6	0.865741	0.432870	+	0.901456i	$0.357501\pi$
$80$	108684.	0.0237330
$81$	−5.49819e6	−1.14954
$82$	4.21768e6	0.844745
$83$	−4.36639e6	−0.838203	−0.419102	−	0.907939i	$-0.637655\pi$
$83$	−4.36639e6	−0.838203	−0.419102	+	0.907939i	$0.637655\pi$
$84$	−696617.	−0.128238
$85$	−15920.9	−0.00281192
$86$	175837.	0.0298103
$87$	−9.31117e6	−1.51596
$88$	−6.33127e6	−0.990379
$89$	−8.72168e6	−1.31140	−0.655700	−	0.755022i	$-0.727626\pi$
$89$	−8.72168e6	−1.31140	−0.655700	+	0.755022i	$0.727626\pi$
$90$	35781.6	0.00517382
$91$	0	0
$92$	−315221.	−0.0422044
$93$	1.25738e7	1.62098
$94$	−1.18673e7	−1.47368
$95$	−175451.	−0.0209953
$96$	−3.88231e6	−0.447859
$97$	9.21482e6	1.02515	0.512573	−	0.858644i	$-0.328692\pi$
$97$	9.21482e6	1.02515	0.512573	+	0.858644i	$0.328692\pi$
$98$	5.66267e6	0.607757
$99$	−1.62743e6	−0.168569
$100$	2.09395e6	0.209395
$101$	−2.06993e6	−0.199908	−0.0999539	−	0.994992i	$-0.531870\pi$
$101$	−2.06993e6	−0.199908	−0.0999539	+	0.994992i	$0.531870\pi$
$102$	−916596.	−0.0855219
$103$	−1.74931e7	−1.57738	−0.788688	−	0.614793i	$-0.789239\pi$
$103$	−1.74931e7	−1.57738	−0.788688	+	0.614793i	$0.789239\pi$
$104$	0	0
$105$	−230739.	−0.0194517
$106$	−1.33360e7	−1.08757
$107$	−1.55162e7	−1.22445	−0.612226	−	0.790683i	$-0.709726\pi$
$107$	−1.55162e7	−1.22445	−0.612226	+	0.790683i	$0.709726\pi$
$108$	2.43831e6	0.186254
$109$	−1.59529e7	−1.17990	−0.589952	−	0.807438i	$-0.700853\pi$
$109$	−1.59529e7	−1.17990	−0.589952	+	0.807438i	$0.700853\pi$
$110$	−363394.	−0.0260318
$111$	1.03651e7	0.719358
$112$	−6.24281e6	−0.419872
$113$	−1.65661e7	−1.08006	−0.540028	−	0.841647i	$-0.681586\pi$
$113$	−1.65661e7	−1.08006	−0.540028	+	0.841647i	$0.681586\pi$
$114$	−1.01010e7	−0.638553
$115$	−104410.	−0.00640176
$116$	4.91129e6	0.292141
$117$	0	0
$118$	4.03827e6	0.226260
$119$	914497.	0.0497471
$120$	−703957.	−0.0371888
$121$	−2.95919e6	−0.151853
$122$	3.30573e7	1.64819
$123$	−2.13290e7	−1.03348
$124$	−6.63223e6	−0.312380
$125$	1.38785e6	0.0635561
$126$	−2.05529e6	−0.0915327
$127$	−1.98781e6	−0.0861118	−0.0430559	−	0.999073i	$-0.513709\pi$
$127$	−1.98781e6	−0.0861118	−0.0430559	+	0.999073i	$0.513709\pi$
$128$	−1.36979e7	−0.577325
$129$	−889218.	−0.0364707
$130$	0	0
$131$	1.58779e7	0.617081	0.308541	−	0.951211i	$-0.400159\pi$
$131$	1.58779e7	0.617081	0.308541	+	0.951211i	$0.400159\pi$
$132$	5.54816e6	0.209962
$133$	1.00779e7	0.371439
$134$	2.11858e7	0.760638
$135$	807636.	0.0282519
$136$	2.79002e6	0.0951090
$137$	−4.95975e7	−1.64793	−0.823963	−	0.566643i	$-0.808242\pi$
$137$	−4.95975e7	−1.64793	−0.823963	+	0.566643i	$0.808242\pi$
$138$	−6.01105e6	−0.194703
$139$	−5.71630e6	−0.180536	−0.0902678	−	0.995918i	$-0.528772\pi$
$139$	−5.71630e6	−0.180536	−0.0902678	+	0.995918i	$0.528772\pi$
$140$	121706.	0.00374855
$141$	6.00135e7	1.80294
$142$	4.25454e7	1.24693
$143$	0	0
$144$	−4.89573e6	−0.136631
$145$	1.62675e6	0.0443133
$146$	3.31650e7	0.881953
$147$	−2.86364e7	−0.743546
$148$	−5.46722e6	−0.138628
$149$	1.51998e7	0.376431	0.188216	−	0.982128i	$-0.439730\pi$
$149$	1.51998e7	0.376431	0.188216	+	0.982128i	$0.439730\pi$
$150$	3.99302e7	0.966011
$151$	6.47181e7	1.52970	0.764850	−	0.644208i	$-0.222813\pi$
$151$	6.47181e7	1.52970	0.764850	+	0.644208i	$0.222813\pi$
$152$	3.07464e7	0.710137
$153$	717165.	0.0161882
$154$	2.08733e7	0.460541
$155$	−2.19678e6	−0.0473833
$156$	0	0
$157$	2.86572e6	0.0590997	0.0295499	−	0.999563i	$-0.490593\pi$
$157$	2.86572e6	0.0590997	0.0295499	+	0.999563i	$0.490593\pi$
$158$	−3.81601e7	−0.769679
$159$	6.74409e7	1.33056
$160$	678279.	0.0130915
$161$	5.99728e6	0.113257
$162$	5.53027e7	1.02198
$163$	−3.56517e7	−0.644797	−0.322399	−	0.946604i	$-0.604489\pi$
$163$	−3.56517e7	−0.644797	−0.322399	+	0.946604i	$0.604489\pi$
$164$	1.12503e7	0.199163
$165$	1.83770e6	0.0318480
$166$	4.39187e7	0.745197
$167$	−2.20447e7	−0.366266	−0.183133	−	0.983088i	$-0.558624\pi$
$167$	−2.20447e7	−0.366266	−0.183133	+	0.983088i	$0.558624\pi$
$168$	4.04352e7	0.657926
$169$	0	0
$170$	160138.	0.00249991
$171$	7.90325e6	0.120870
$172$	469029.	0.00702830
$173$	1.96908e7	0.289136	0.144568	−	0.989495i	$-0.453821\pi$
$173$	1.96908e7	0.289136	0.144568	+	0.989495i	$0.453821\pi$
$174$	9.36550e7	1.34775
$175$	−3.98387e7	−0.561917
$176$	4.97205e7	0.687450
$177$	−2.04217e7	−0.276813
$178$	8.77257e7	1.16589
$179$	7.90229e7	1.02983	0.514917	−	0.857240i	$-0.327823\pi$
$179$	7.90229e7	1.02983	0.514917	+	0.857240i	$0.327823\pi$
$180$	95444.0	0.00121982
$181$	−4.94136e7	−0.619400	−0.309700	−	0.950834i	$-0.600229\pi$
$181$	−4.94136e7	−0.619400	−0.309700	+	0.950834i	$0.600229\pi$
$182$	0	0
$183$	−1.67172e8	−2.01644
$184$	1.82970e7	0.216530
$185$	−1.81090e6	−0.0210277
$186$	−1.26472e8	−1.44112
$187$	−7.28345e6	−0.0814501
$188$	−3.16549e7	−0.347446
$189$	−4.63905e7	−0.499819
$190$	1.76475e6	0.0186657
$191$	1.64464e8	1.70787	0.853935	−	0.520379i	$-0.174209\pi$
$191$	1.64464e8	1.70787	0.853935	+	0.520379i	$0.174209\pi$
$192$	1.18676e8	1.21007
$193$	−8.78627e7	−0.879740	−0.439870	−	0.898062i	$-0.644975\pi$
$193$	−8.78627e7	−0.879740	−0.439870	+	0.898062i	$0.644975\pi$
$194$	−9.26859e7	−0.911397
$195$	0	0
$196$	1.51046e7	0.143289
$197$	−1.98762e7	−0.185225	−0.0926127	−	0.995702i	$-0.529522\pi$
$197$	−1.98762e7	−0.185225	−0.0926127	+	0.995702i	$0.529522\pi$
$198$	1.63692e7	0.149865
$199$	2.80120e7	0.251975	0.125988	−	0.992032i	$-0.459790\pi$
$199$	2.80120e7	0.251975	0.125988	+	0.992032i	$0.459790\pi$
$200$	−1.21543e8	−1.07430
$201$	−1.07138e8	−0.930585
$202$	2.08200e7	0.177726
$203$	−9.34405e7	−0.783969
$204$	−2.44493e6	−0.0201633
$205$	3.72640e6	0.0302100
$206$	1.75951e8	1.40235
$207$	4.70318e6	0.0368549
$208$	0	0
$209$	−8.02645e7	−0.608151
$210$	2.32085e6	0.0172934
$211$	2.39701e7	0.175663	0.0878315	−	0.996135i	$-0.472006\pi$
$211$	2.39701e7	0.175663	0.0878315	+	0.996135i	$0.472006\pi$
$212$	−3.55725e7	−0.256413
$213$	−2.15154e8	−1.52553
$214$	1.56067e8	1.08859
$215$	155355.	0.00106608
$216$	−1.41532e8	−0.955579
$217$	1.26182e8	0.838281
$218$	1.60460e8	1.04898
$219$	−1.67717e8	−1.07900
$220$	−969319.	−0.00613744
$221$	0	0
$222$	−1.04256e8	−0.639538
$223$	1.88776e8	1.13993	0.569967	−	0.821668i	$-0.306956\pi$
$223$	1.88776e8	1.13993	0.569967	+	0.821668i	$0.306956\pi$
$224$	−3.89602e7	−0.231608
$225$	−3.12423e7	−0.182854
$226$	1.66628e8	0.960213
$227$	−766382.	−0.00434865	−0.00217433	−	0.999998i	$-0.500692\pi$
$227$	−766382.	−0.00434865	−0.00217433	+	0.999998i	$0.500692\pi$
$228$	−2.69434e7	−0.150550
$229$	1.06420e8	0.585597	0.292798	−	0.956174i	$-0.405414\pi$
$229$	1.06420e8	0.585597	0.292798	+	0.956174i	$0.405414\pi$
$230$	1.05019e6	0.00569142
$231$	−1.05557e8	−0.563439
$232$	−2.85076e8	−1.49883
$233$	2.05485e7	0.106423	0.0532113	−	0.998583i	$-0.483054\pi$
$233$	2.05485e7	0.106423	0.0532113	+	0.998583i	$0.483054\pi$
$234$	0	0
$235$	−1.04850e7	−0.0527023
$236$	1.07717e7	0.0533448
$237$	1.92977e8	0.941645
$238$	−9.19832e6	−0.0442272
$239$	−6.47473e7	−0.306781	−0.153391	−	0.988166i	$-0.549019\pi$
$239$	−6.47473e7	−0.306781	−0.153391	+	0.988166i	$0.549019\pi$
$240$	5.52829e6	0.0258138
$241$	4.01781e8	1.84897	0.924486	−	0.381217i	$-0.124495\pi$
$241$	4.01781e8	1.84897	0.924486	+	0.381217i	$0.124495\pi$
$242$	2.97646e7	0.135004
$243$	−8.09115e7	−0.361733
$244$	8.81770e7	0.388589
$245$	5.00306e6	0.0217348
$246$	2.14535e8	0.918808
$247$	0	0
$248$	3.84968e8	1.60267
$249$	−2.22099e8	−0.911694
$250$	−1.39595e7	−0.0565039
$251$	2.77091e8	1.10602	0.553011	−	0.833174i	$-0.313479\pi$
$251$	2.77091e8	1.10602	0.553011	+	0.833174i	$0.313479\pi$
$252$	−5.48229e6	−0.0215804
$253$	−4.77650e7	−0.185433
$254$	1.99941e7	0.0765569
$255$	−809828.	−0.00305846
$256$	−1.60863e8	−0.599263
$257$	−4.67270e8	−1.71713	−0.858563	−	0.512708i	$-0.828642\pi$
$257$	−4.67270e8	−1.71713	−0.858563	+	0.512708i	$0.828642\pi$
$258$	8.94407e6	0.0324240
$259$	1.04017e8	0.372012
$260$	0	0
$261$	−7.32778e7	−0.255112
$262$	−1.59705e8	−0.548610
$263$	7.58768e7	0.257196	0.128598	−	0.991697i	$-0.458952\pi$
$263$	7.58768e7	0.257196	0.128598	+	0.991697i	$0.458952\pi$
$264$	−3.22044e8	−1.07721
$265$	−1.17826e7	−0.0388938
$266$	−1.01367e8	−0.330225
$267$	−4.43633e8	−1.42638
$268$	5.65111e7	0.179334
$269$	−1.93468e8	−0.606005	−0.303002	−	0.952990i	$-0.597989\pi$
$269$	−1.93468e8	−0.606005	−0.303002	+	0.952990i	$0.597989\pi$
$270$	−8.12349e6	−0.0251171
$271$	4.04769e8	1.23542	0.617710	−	0.786406i	$-0.288060\pi$
$271$	4.04769e8	1.23542	0.617710	+	0.786406i	$0.288060\pi$
$272$	−2.19105e7	−0.0660179
$273$	0	0
$274$	4.98869e8	1.46507
$275$	3.17293e8	0.920018
$276$	−1.60339e7	−0.0459047
$277$	4.66835e8	1.31973	0.659864	−	0.751385i	$-0.270614\pi$
$277$	4.66835e8	1.31973	0.659864	+	0.751385i	$0.270614\pi$
$278$	5.74965e7	0.160503
$279$	9.89546e7	0.272786
$280$	−7.06443e6	−0.0192320
$281$	−2.83566e8	−0.762397	−0.381199	−	0.924493i	$-0.624489\pi$
$281$	−2.83566e8	−0.762397	−0.381199	+	0.924493i	$0.624489\pi$
$282$	−6.03637e8	−1.60289
$283$	−3.54304e8	−0.929229	−0.464615	−	0.885513i	$-0.653807\pi$
$283$	−3.54304e8	−0.929229	−0.464615	+	0.885513i	$0.653807\pi$
$284$	1.13486e8	0.293986
$285$	−8.92441e6	−0.0228361
$286$	0	0
$287$	−2.14044e8	−0.534460
$288$	−3.05533e7	−0.0753676
$289$	−4.07129e8	−0.992178
$290$	−1.63625e7	−0.0393963
$291$	4.68717e8	1.11503
$292$	8.84645e7	0.207936
$293$	−5.68237e8	−1.31975	−0.659877	−	0.751374i	$-0.729392\pi$
$293$	−5.68237e8	−1.31975	−0.659877	+	0.751374i	$0.729392\pi$
$294$	2.88035e8	0.661042
$295$	3.56788e6	0.00809158
$296$	3.17346e8	0.711232
$297$	3.69474e8	0.818345
$298$	−1.52885e8	−0.334663
$299$	0	0
$300$	1.06510e8	0.227754
$301$	−8.92359e6	−0.0188606
$302$	−6.50957e8	−1.35997
$303$	−1.05288e8	−0.217435
$304$	−2.41457e8	−0.492926
$305$	2.92067e7	0.0589430
$306$	−7.21350e6	−0.0143920
$307$	6.06885e8	1.19708	0.598538	−	0.801094i	$-0.295749\pi$
$307$	6.06885e8	1.19708	0.598538	+	0.801094i	$0.295749\pi$
$308$	5.56775e7	0.108581
$309$	−8.89794e8	−1.71567
$310$	2.20959e7	0.0421256
$311$	−5.70878e8	−1.07617	−0.538086	−	0.842890i	$-0.680853\pi$
$311$	−5.70878e8	−1.07617	−0.538086	+	0.842890i	$0.680853\pi$
$312$	0	0
$313$	1.19008e8	0.219367	0.109684	−	0.993967i	$-0.465016\pi$
$313$	1.19008e8	0.219367	0.109684	+	0.993967i	$0.465016\pi$
$314$	−2.88244e7	−0.0525421
$315$	−1.81589e6	−0.00327342
$316$	−1.01788e8	−0.181465
$317$	7.11011e8	1.25363	0.626814	−	0.779169i	$-0.284358\pi$
$317$	7.11011e8	1.25363	0.626814	+	0.779169i	$0.284358\pi$
$318$	−6.78344e8	−1.18292
$319$	7.44201e8	1.28358
$320$	−2.07340e7	−0.0353718
$321$	−7.89240e8	−1.33181
$322$	−6.03228e7	−0.100690
$323$	3.53705e7	0.0584026
$324$	1.47515e8	0.240950
$325$	0	0
$326$	3.58597e8	0.573251
$327$	−8.11453e8	−1.28335
$328$	−6.53022e8	−1.02181
$329$	6.02254e8	0.932383
$330$	−1.84843e7	−0.0283141
$331$	1.03380e9	1.56689	0.783447	−	0.621459i	$-0.213460\pi$
$331$	1.03380e9	1.56689	0.783447	+	0.621459i	$0.213460\pi$
$332$	1.17149e8	0.175693
$333$	8.15724e7	0.121057
$334$	2.21733e8	0.325625
$335$	1.87180e7	0.0272021
$336$	−3.17544e8	−0.456685
$337$	−8.03307e8	−1.14334	−0.571672	−	0.820482i	$-0.693705\pi$
$337$	−8.03307e8	−1.14334	−0.571672	+	0.820482i	$0.693705\pi$
$338$	0	0
$339$	−8.42645e8	−1.17475
$340$	427154.	0.000589397 0
$341$	−1.00497e9	−1.37250
$342$	−7.94936e7	−0.107459
$343$	−7.07754e8	−0.947007
$344$	−2.72248e7	−0.0360588
$345$	−5.31087e6	−0.00696303
$346$	−1.98057e8	−0.257054
$347$	3.48273e8	0.447473	0.223737	−	0.974650i	$-0.428174\pi$
$347$	3.48273e8	0.447473	0.223737	+	0.974650i	$0.428174\pi$
$348$	2.49816e8	0.317755
$349$	−6.90910e8	−0.870027	−0.435013	−	0.900424i	$-0.643256\pi$
$349$	−6.90910e8	−0.870027	−0.435013	+	0.900424i	$0.643256\pi$
$350$	4.00712e8	0.499567
$351$	0	0
$352$	3.10296e8	0.379208
$353$	3.16345e8	0.382780	0.191390	−	0.981514i	$-0.438700\pi$
$353$	3.16345e8	0.382780	0.191390	+	0.981514i	$0.438700\pi$
$354$	2.05409e8	0.246098
$355$	3.75896e7	0.0445932
$356$	2.34000e8	0.274878
$357$	4.65164e7	0.0541087
$358$	−7.94840e8	−0.915565
$359$	−5.14441e8	−0.586820	−0.293410	−	0.955987i	$-0.594790\pi$
$359$	−5.14441e8	−0.586820	−0.293410	+	0.955987i	$0.594790\pi$
$360$	−5.54006e6	−0.00625829
$361$	−5.04085e8	−0.563934
$362$	4.97019e8	0.550672
$363$	−1.50521e8	−0.165167
$364$	0	0
$365$	2.93019e7	0.0315406
$366$	1.68148e9	1.79270
$367$	1.47338e9	1.55591	0.777953	−	0.628323i	$-0.216258\pi$
$367$	1.47338e9	1.55591	0.777953	+	0.628323i	$0.216258\pi$
$368$	−1.43690e8	−0.150300
$369$	−1.67857e8	−0.173919
$370$	1.82146e7	0.0186945
$371$	6.76791e8	0.688091
$372$	−3.37352e8	−0.339768
$373$	−9.69375e8	−0.967188	−0.483594	−	0.875292i	$-0.660669\pi$
$373$	−9.69375e8	−0.967188	−0.483594	+	0.875292i	$0.660669\pi$
$374$	7.32595e7	0.0724124
$375$	7.05937e7	0.0691284
$376$	1.83741e9	1.78258
$377$	0	0
$378$	4.66612e8	0.444359
$379$	−3.46921e8	−0.327335	−0.163668	−	0.986516i	$-0.552332\pi$
$379$	−3.46921e8	−0.327335	−0.163668	+	0.986516i	$0.552332\pi$
$380$	4.70729e6	0.00440076
$381$	−1.01111e8	−0.0936617
$382$	−1.65424e9	−1.51837
$383$	−1.83888e9	−1.67247	−0.836234	−	0.548373i	$-0.815247\pi$
$383$	−1.83888e9	−1.67247	−0.836234	+	0.548373i	$0.815247\pi$
$384$	−6.96753e8	−0.627942
$385$	1.84419e7	0.0164700
$386$	8.83754e8	0.782124
$387$	−6.99804e6	−0.00613745
$388$	−2.47231e8	−0.214878
$389$	−1.00176e9	−0.862861	−0.431431	−	0.902146i	$-0.641991\pi$
$389$	−1.00176e9	−0.862861	−0.431431	+	0.902146i	$0.641991\pi$
$390$	0	0
$391$	2.10488e7	0.0178077
$392$	−8.76749e8	−0.735147
$393$	8.07636e8	0.671184
$394$	1.99921e8	0.164673
$395$	−3.37151e7	−0.0275255
$396$	4.36633e7	0.0353333
$397$	1.39793e9	1.12129	0.560647	−	0.828055i	$-0.310552\pi$
$397$	1.39793e9	1.12129	0.560647	+	0.828055i	$0.310552\pi$
$398$	−2.81754e8	−0.224016
$399$	5.12616e8	0.404005
$400$	9.54501e8	0.745704
$401$	−1.21136e9	−0.938144	−0.469072	−	0.883160i	$-0.655411\pi$
$401$	−1.21136e9	−0.938144	−0.469072	+	0.883160i	$0.655411\pi$
$402$	1.07763e9	0.827328
$403$	0	0
$404$	5.55354e7	0.0419020
$405$	4.88609e7	0.0365485
$406$	9.39857e8	0.696981
$407$	−8.28441e8	−0.609089
$408$	1.41916e8	0.103448
$409$	1.45644e9	1.05260	0.526299	−	0.850300i	$-0.323579\pi$
$409$	1.45644e9	1.05260	0.526299	+	0.850300i	$0.323579\pi$
$410$	−3.74814e7	−0.0268579
$411$	−2.52281e9	−1.79241
$412$	4.69333e8	0.330629
$413$	−2.04939e8	−0.143152
$414$	−4.73062e7	−0.0327655
$415$	3.88029e7	0.0266499
$416$	0	0
$417$	−2.90763e8	−0.196364
$418$	8.07328e8	0.540671
$419$	1.35223e9	0.898051	0.449026	−	0.893519i	$-0.351771\pi$
$419$	1.35223e9	0.898051	0.449026	+	0.893519i	$0.351771\pi$
$420$	6.19064e6	0.00407721
$421$	1.35611e9	0.885742	0.442871	−	0.896585i	$-0.353960\pi$
$421$	1.35611e9	0.885742	0.442871	+	0.896585i	$0.353960\pi$
$422$	−2.41099e8	−0.156172
$423$	4.72299e8	0.303407
$424$	2.06481e9	1.31553
$425$	−1.39823e8	−0.0883521
$426$	2.16409e9	1.35626
$427$	−1.67763e9	−1.04279
$428$	4.16294e8	0.256654
$429$	0	0
$430$	−1.56262e6	−0.000947793 0
$431$	−1.27281e9	−0.765764	−0.382882	−	0.923797i	$-0.625068\pi$
$431$	−1.27281e9	−0.765764	−0.382882	+	0.923797i	$0.625068\pi$
$432$	1.11147e9	0.663295
$433$	3.39003e8	0.200676	0.100338	−	0.994953i	$-0.468008\pi$
$433$	3.39003e8	0.200676	0.100338	+	0.994953i	$0.468008\pi$
$434$	−1.26919e9	−0.745266
$435$	8.27458e7	0.0481985
$436$	4.28011e8	0.247316
$437$	2.31960e8	0.132962
$438$	1.68696e9	0.959279
$439$	1.40195e9	0.790870	0.395435	−	0.918494i	$-0.370594\pi$
$439$	1.40195e9	0.790870	0.395435	+	0.918494i	$0.370594\pi$
$440$	5.62642e7	0.0314882
$441$	−2.25365e8	−0.125127
$442$	0	0
$443$	−3.31716e9	−1.81282	−0.906408	−	0.422404i	$-0.861187\pi$
$443$	−3.31716e9	−1.81282	−0.906408	+	0.422404i	$0.861187\pi$
$444$	−2.78093e8	−0.150782
$445$	7.75071e7	0.0416948
$446$	−1.89877e9	−1.01345
$447$	7.73146e8	0.409435
$448$	1.19096e9	0.625781
$449$	−8.90007e8	−0.464014	−0.232007	−	0.972714i	$-0.574529\pi$
$449$	−8.90007e8	−0.464014	−0.232007	+	0.972714i	$0.574529\pi$
$450$	3.14246e8	0.162564
$451$	1.70474e9	0.875063
$452$	4.44463e8	0.226387
$453$	3.29192e9	1.66382
$454$	7.70853e6	0.00386613
$455$	0	0
$456$	1.56393e9	0.772399
$457$	1.66381e9	0.815448	0.407724	−	0.913105i	$-0.366323\pi$
$457$	1.66381e9	0.815448	0.407724	+	0.913105i	$0.366323\pi$
$458$	−1.07041e9	−0.520619
$459$	−1.62818e8	−0.0785881
$460$	2.80128e6	0.00134185
$461$	−2.68236e7	−0.0127516	−0.00637580	−	0.999980i	$-0.502029\pi$
$461$	−2.68236e7	−0.0127516	−0.00637580	+	0.999980i	$0.502029\pi$
$462$	1.06173e9	0.500920
$463$	3.55847e9	1.66621	0.833105	−	0.553115i	$-0.186561\pi$
$463$	3.55847e9	1.66621	0.833105	+	0.553115i	$0.186561\pi$
$464$	2.23875e9	1.04038
$465$	−1.11740e8	−0.0515376
$466$	−2.06684e8	−0.0946141
$467$	−2.59791e9	−1.18036	−0.590180	−	0.807272i	$-0.700943\pi$
$467$	−2.59791e9	−1.18036	−0.590180	+	0.807272i	$0.700943\pi$
$468$	0	0
$469$	−1.07516e9	−0.481247
$470$	1.05461e8	0.0468545
$471$	1.45767e8	0.0642813
$472$	−6.25244e8	−0.273686
$473$	7.10713e7	0.0308802
$474$	−1.94103e9	−0.837161
$475$	−1.54086e9	−0.659685
$476$	−2.45356e7	−0.0104273
$477$	5.30752e8	0.223912
$478$	6.51250e8	0.272741
$479$	2.90167e9	1.20635	0.603176	−	0.797608i	$-0.293902\pi$
$479$	2.90167e9	1.20635	0.603176	+	0.797608i	$0.293902\pi$
$480$	3.45011e7	0.0142393
$481$	0	0
$482$	−4.04126e9	−1.64381
$483$	3.05055e8	0.123187
$484$	7.93941e7	0.0318295
$485$	−8.18896e7	−0.0325936
$486$	8.13836e8	0.321595
$487$	1.15313e9	0.452406	0.226203	−	0.974080i	$-0.427369\pi$
$487$	1.15313e9	0.452406	0.226203	+	0.974080i	$0.427369\pi$
$488$	−5.11824e9	−1.99366
$489$	−1.81344e9	−0.701331
$490$	−5.03226e7	−0.0193231
$491$	3.78661e9	1.44366	0.721831	−	0.692069i	$-0.243301\pi$
$491$	3.78661e9	1.44366	0.721831	+	0.692069i	$0.243301\pi$
$492$	5.72251e8	0.216625
$493$	−3.27950e8	−0.123266
$494$	0	0
$495$	1.44625e7	0.00535951
$496$	−3.02322e9	−1.11246
$497$	−2.15914e9	−0.788920
$498$	2.23395e9	0.810533
$499$	1.98153e9	0.713918	0.356959	−	0.934120i	$-0.383814\pi$
$499$	1.98153e9	0.713918	0.356959	+	0.934120i	$0.383814\pi$
$500$	−3.72355e7	−0.0133218
$501$	−1.12132e9	−0.398379
$502$	−2.78707e9	−0.983299
$503$	2.32772e9	0.815536	0.407768	−	0.913086i	$-0.366307\pi$
$503$	2.32772e9	0.815536	0.407768	+	0.913086i	$0.366307\pi$
$504$	3.18220e8	0.110719
$505$	1.83949e7	0.00635589
$506$	4.80437e8	0.164858
$507$	0	0
$508$	5.33324e7	0.0180496
$509$	−1.44356e9	−0.485203	−0.242601	−	0.970126i	$-0.578001\pi$
$509$	−1.44356e9	−0.485203	−0.242601	+	0.970126i	$0.578001\pi$
$510$	8.14553e6	0.00271909
$511$	−1.68309e9	−0.558001
$512$	3.37135e9	1.11009
$513$	−1.79427e9	−0.586782
$514$	4.69997e9	1.52660
$515$	1.55456e8	0.0501513
$516$	2.38574e7	0.00764451
$517$	−4.79662e9	−1.52657
$518$	−1.04624e9	−0.330734
$519$	1.00158e9	0.314486
$520$	0	0
$521$	2.02069e9	0.625990	0.312995	−	0.949755i	$-0.398668\pi$
$521$	2.02069e9	0.625990	0.312995	+	0.949755i	$0.398668\pi$
$522$	7.37053e8	0.226805
$523$	3.62640e9	1.10846	0.554230	−	0.832364i	$-0.313013\pi$
$523$	3.62640e9	1.10846	0.554230	+	0.832364i	$0.313013\pi$
$524$	−4.25998e8	−0.129344
$525$	−2.02642e9	−0.611184
$526$	−7.63195e8	−0.228658
$527$	4.42865e8	0.131806
$528$	2.52906e9	0.747723
$529$	−3.26679e9	−0.959458
$530$	1.18513e8	0.0345782
$531$	−1.60717e8	−0.0465833
$532$	−2.70386e8	−0.0778562
$533$	0	0
$534$	4.46222e9	1.26811
$535$	1.37888e8	0.0389304
$536$	−3.28019e9	−0.920073
$537$	4.01955e9	1.12013
$538$	1.94597e9	0.538763
$539$	2.28878e9	0.629569
$540$	−2.16686e7	−0.00592179
$541$	7.29172e9	1.97988	0.989941	−	0.141481i	$-0.0451865\pi$
$541$	7.29172e9	1.97988	0.989941	+	0.141481i	$0.0451865\pi$
$542$	−4.07131e9	−1.09834
$543$	−2.51345e9	−0.673706
$544$	−1.36740e8	−0.0364165
$545$	1.41769e8	0.0375140
$546$	0	0
$547$	−2.20432e9	−0.575862	−0.287931	−	0.957651i	$-0.592967\pi$
$547$	−2.20432e9	−0.575862	−0.287931	+	0.957651i	$0.592967\pi$
$548$	1.33068e9	0.345417
$549$	−1.31562e9	−0.339335
$550$	−3.19144e9	−0.817933
$551$	−3.61405e9	−0.920372
$552$	9.30689e8	0.235515
$553$	1.93659e9	0.486967
$554$	−4.69559e9	−1.17329
$555$	−9.21122e7	−0.0228713
$556$	1.53366e8	0.0378415
$557$	−6.48577e8	−0.159026	−0.0795131	−	0.996834i	$-0.525337\pi$
$557$	−6.48577e8	−0.159026	−0.0795131	+	0.996834i	$0.525337\pi$
$558$	−9.95320e8	−0.242517
$559$	0	0
$560$	5.54782e7	0.0133495
$561$	−3.70477e8	−0.0885913
$562$	2.85220e9	0.677802
$563$	7.79259e8	0.184036	0.0920180	−	0.995757i	$-0.470668\pi$
$563$	7.79259e8	0.184036	0.0920180	+	0.995757i	$0.470668\pi$
$564$	−1.61014e9	−0.377909
$565$	1.47218e8	0.0343394
$566$	3.56371e9	0.826123
$567$	−2.80656e9	−0.646598
$568$	−6.58729e9	−1.50830
$569$	−4.16166e9	−0.947051	−0.473525	−	0.880780i	$-0.657019\pi$
$569$	−4.16166e9	−0.947051	−0.473525	+	0.880780i	$0.657019\pi$
$570$	8.97648e7	0.0203022
$571$	8.48385e9	1.90707	0.953535	−	0.301282i	$-0.0974145\pi$
$571$	8.48385e9	1.90707	0.953535	+	0.301282i	$0.0974145\pi$
$572$	0	0
$573$	8.36557e9	1.85761
$574$	2.15292e9	0.475157
$575$	−9.16960e8	−0.201147
$576$	9.33969e8	0.203636
$577$	−4.94714e9	−1.07211	−0.536054	−	0.844184i	$-0.680086\pi$
$577$	−4.94714e9	−1.07211	−0.536054	+	0.844184i	$0.680086\pi$
$578$	4.09505e9	0.882087
$579$	−4.46919e9	−0.956872
$580$	−4.36453e7	−0.00928837
$581$	−2.22883e9	−0.471478
$582$	−4.71452e9	−0.991304
$583$	−5.39026e9	−1.12660
$584$	−5.13493e9	−1.06682
$585$	0	0
$586$	5.71553e9	1.17332
$587$	6.91524e9	1.41115	0.705577	−	0.708634i	$-0.250688\pi$
$587$	6.91524e9	1.41115	0.705577	+	0.708634i	$0.250688\pi$
$588$	7.68305e8	0.155852
$589$	4.88043e9	0.984134
$590$	−3.58870e7	−0.00719375
$591$	−1.01101e9	−0.201465
$592$	−2.49217e9	−0.493686
$593$	−4.73029e9	−0.931529	−0.465765	−	0.884909i	$-0.654221\pi$
$593$	−4.73029e9	−0.931529	−0.465765	+	0.884909i	$0.654221\pi$
$594$	−3.71630e9	−0.727542
$595$	−8.12688e6	−0.00158166
$596$	−4.07805e8	−0.0789025
$597$	1.42484e9	0.274067
$598$	0	0
$599$	−5.05854e9	−0.961682	−0.480841	−	0.876808i	$-0.659669\pi$
$599$	−5.05854e9	−0.961682	−0.480841	+	0.876808i	$0.659669\pi$
$600$	−6.18238e9	−1.16849
$601$	−2.81440e9	−0.528840	−0.264420	−	0.964408i	$-0.585181\pi$
$601$	−2.81440e9	−0.528840	−0.264420	+	0.964408i	$0.585181\pi$
$602$	8.97565e7	0.0167679
$603$	−8.43160e8	−0.156603
$604$	−1.73636e9	−0.320635
$605$	2.62975e7	0.00482804
$606$	1.05902e9	0.193308
$607$	−3.54047e9	−0.642541	−0.321271	−	0.946987i	$-0.604110\pi$
$607$	−3.54047e9	−0.642541	−0.321271	+	0.946987i	$0.604110\pi$
$608$	−1.50689e9	−0.271905
$609$	−4.75290e9	−0.852704
$610$	−2.93771e8	−0.0524028
$611$	0	0
$612$	−1.92413e7	−0.00339316
$613$	−1.90580e8	−0.0334169	−0.0167085	−	0.999860i	$-0.505319\pi$
$613$	−1.90580e8	−0.0334169	−0.0167085	+	0.999860i	$0.505319\pi$
$614$	−6.10426e9	−1.06425
$615$	1.89545e8	0.0328587
$616$	−3.23181e9	−0.557074
$617$	−9.31051e9	−1.59579	−0.797894	−	0.602798i	$-0.794053\pi$
$617$	−9.31051e9	−1.59579	−0.797894	+	0.602798i	$0.794053\pi$
$618$	8.94986e9	1.52530
$619$	−8.75671e9	−1.48396	−0.741982	−	0.670419i	$-0.766114\pi$
$619$	−8.75671e9	−1.48396	−0.741982	+	0.670419i	$0.766114\pi$
$620$	5.89388e7	0.00993185
$621$	−1.06776e9	−0.178918
$622$	5.74209e9	0.956761
$623$	−4.45200e9	−0.737644
$624$	0	0
$625$	6.08501e9	0.996968
$626$	−1.19702e9	−0.195026
$627$	−4.08270e9	−0.661471
$628$	−7.68864e7	−0.0123877
$629$	3.65072e8	0.0584927
$630$	1.82648e7	0.00291020
$631$	−2.59032e9	−0.410440	−0.205220	−	0.978716i	$-0.565791\pi$
$631$	−2.59032e9	−0.410440	−0.205220	+	0.978716i	$0.565791\pi$
$632$	5.90831e9	0.931009
$633$	1.21925e9	0.191065
$634$	−7.15159e9	−1.11453
$635$	1.76652e7	0.00273785
$636$	−1.80942e9	−0.278894
$637$	0	0
$638$	−7.48543e9	−1.14115
$639$	−1.69324e9	−0.256723
$640$	1.21730e8	0.0183555
$641$	−4.49376e9	−0.673918	−0.336959	−	0.941519i	$-0.609398\pi$
$641$	−4.49376e9	−0.673918	−0.336959	+	0.941519i	$0.609398\pi$
$642$	7.93845e9	1.18403
$643$	−5.61571e9	−0.833041	−0.416521	−	0.909126i	$-0.636751\pi$
$643$	−5.61571e9	−0.833041	−0.416521	+	0.909126i	$0.636751\pi$
$644$	−1.60905e8	−0.0237394
$645$	7.90224e6	0.00115955
$646$	−3.55769e8	−0.0519223
$647$	1.45087e9	0.210602	0.105301	−	0.994440i	$-0.466419\pi$
$647$	1.45087e9	0.210602	0.105301	+	0.994440i	$0.466419\pi$
$648$	−8.56250e9	−1.23620
$649$	1.63222e9	0.234381
$650$	0	0
$651$	6.41834e9	0.911779
$652$	9.56522e8	0.135154
$653$	−9.22610e9	−1.29665	−0.648324	−	0.761365i	$-0.724530\pi$
$653$	−9.22610e9	−1.29665	−0.648324	+	0.761365i	$0.724530\pi$
$654$	8.16187e9	1.14095
$655$	−1.41102e8	−0.0196195
$656$	5.12829e9	0.709266
$657$	−1.31991e9	−0.181579
$658$	−6.05768e9	−0.828926
$659$	4.26023e9	0.579875	0.289937	−	0.957046i	$-0.406366\pi$
$659$	4.26023e9	0.579875	0.289937	+	0.957046i	$0.406366\pi$
$660$	−4.93050e7	−0.00667555
$661$	1.14507e9	0.154216	0.0771078	−	0.997023i	$-0.475431\pi$
$661$	1.14507e9	0.154216	0.0771078	+	0.997023i	$0.475431\pi$
$662$	−1.03983e10	−1.39303
$663$	0	0
$664$	−6.79992e9	−0.901395
$665$	−8.95592e7	−0.0118096
$666$	−8.20484e8	−0.107624
$667$	−2.15070e9	−0.280634
$668$	5.91452e8	0.0767718
$669$	9.60219e9	1.23988
$670$	−1.88272e8	−0.0241838
$671$	1.33613e10	1.70735
$672$	−1.98173e9	−0.251914
$673$	5.46085e9	0.690570	0.345285	−	0.938498i	$-0.387782\pi$
$673$	5.46085e9	0.690570	0.345285	+	0.938498i	$0.387782\pi$
$674$	8.07994e9	1.01648
$675$	7.09292e9	0.887691
$676$	0	0
$677$	−8.92821e9	−1.10587	−0.552935	−	0.833225i	$-0.686492\pi$
$677$	−8.92821e9	−1.10587	−0.552935	+	0.833225i	$0.686492\pi$
$678$	8.47561e9	1.04440
$679$	4.70372e9	0.576630
$680$	−2.47942e7	−0.00302391
$681$	−3.89824e7	−0.00472992
$682$	1.01084e10	1.22021
$683$	8.50973e9	1.02198	0.510991	−	0.859586i	$-0.329279\pi$
$683$	8.50973e9	1.02198	0.510991	+	0.859586i	$0.329279\pi$
$684$	−2.12041e8	−0.0253352
$685$	4.40759e8	0.0523944
$686$	7.11884e9	0.841927
$687$	5.41311e9	0.636940
$688$	2.13801e8	0.0250294
$689$	0	0
$690$	5.34185e7	0.00619042
$691$	−1.11787e9	−0.128889	−0.0644447	−	0.997921i	$-0.520528\pi$
$691$	−1.11787e9	−0.128889	−0.0644447	+	0.997921i	$0.520528\pi$
$692$	−5.28298e8	−0.0606049
$693$	−8.30724e8	−0.0948179
$694$	−3.50306e9	−0.397822
$695$	5.07991e7	0.00573997
$696$	−1.45006e10	−1.63024
$697$	−7.51233e8	−0.0840349
$698$	6.94942e9	0.773489
$699$	1.04521e9	0.115753
$700$	1.06886e9	0.117782
$701$	−3.84858e9	−0.421976	−0.210988	−	0.977489i	$-0.567668\pi$
$701$	−3.84858e9	−0.421976	−0.210988	+	0.977489i	$0.567668\pi$
$702$	0	0
$703$	4.02314e9	0.436739
$704$	−9.48529e9	−1.02458
$705$	−5.33324e8	−0.0573230
$706$	−3.18191e9	−0.340307
$707$	−1.05660e9	−0.112445
$708$	5.47908e8	0.0580219
$709$	−5.36316e9	−0.565144	−0.282572	−	0.959246i	$-0.591188\pi$
$709$	−5.36316e9	−0.565144	−0.282572	+	0.959246i	$0.591188\pi$
$710$	−3.78089e8	−0.0396451
$711$	1.51871e9	0.158464
$712$	−1.35825e10	−1.41027
$713$	2.90432e9	0.300076
$714$	−4.67878e8	−0.0481048
$715$	0	0
$716$	−2.12016e9	−0.215860
$717$	−3.29341e9	−0.333679
$718$	5.17443e9	0.521707
$719$	−6.62611e9	−0.664825	−0.332413	−	0.943134i	$-0.607863\pi$
$719$	−6.62611e9	−0.664825	−0.332413	+	0.943134i	$0.607863\pi$
$720$	4.35070e7	0.00434406
$721$	−8.92936e9	−0.887252
$722$	5.07026e9	0.501360
$723$	2.04368e10	2.01108
$724$	1.32575e9	0.129830
$725$	1.42867e10	1.39235
$726$	1.51399e9	0.146840
$727$	9.52174e8	0.0919064	0.0459532	−	0.998944i	$-0.485367\pi$
$727$	9.52174e8	0.0919064	0.0459532	+	0.998944i	$0.485367\pi$
$728$	0	0
$729$	7.90894e9	0.756087
$730$	−2.94728e8	−0.0280409
$731$	−3.13193e7	−0.00296552
$732$	4.48517e9	0.422659
$733$	1.46510e10	1.37405	0.687027	−	0.726632i	$-0.258916\pi$
$733$	1.46510e10	1.37405	0.687027	+	0.726632i	$0.258916\pi$
$734$	−1.48198e10	−1.38326
$735$	2.54484e8	0.0236404
$736$	−8.96740e8	−0.0829076
$737$	8.56305e9	0.787938
$738$	1.68836e9	0.154621
$739$	1.21913e10	1.11121	0.555604	−	0.831447i	$-0.312487\pi$
$739$	1.21913e10	1.11121	0.555604	+	0.831447i	$0.312487\pi$
$740$	4.85857e7	0.00440755
$741$	0	0
$742$	−6.80739e9	−0.611740
$743$	3.89555e9	0.348424	0.174212	−	0.984708i	$-0.444262\pi$
$743$	3.89555e9	0.348424	0.174212	+	0.984708i	$0.444262\pi$
$744$	1.95816e10	1.74319
$745$	−1.35076e8	−0.0119683
$746$	9.75031e9	0.859870
$747$	−1.74789e9	−0.153424
$748$	1.95413e8	0.0170725
$749$	−7.92027e9	−0.688737
$750$	−7.10056e8	−0.0614580
$751$	1.53353e10	1.32115	0.660574	−	0.750761i	$-0.270313\pi$
$751$	1.53353e10	1.32115	0.660574	+	0.750761i	$0.270313\pi$
$752$	−1.44295e10	−1.23734
$753$	1.40944e10	1.20299
$754$	0	0
$755$	−5.75132e8	−0.0486355
$756$	1.24464e9	0.104765
$757$	1.71312e10	1.43533	0.717665	−	0.696389i	$-0.245211\pi$
$757$	1.71312e10	1.43533	0.717665	+	0.696389i	$0.245211\pi$
$758$	3.48945e9	0.291014
$759$	−2.42959e9	−0.201691
$760$	−2.73235e8	−0.0225782
$761$	−9.75251e9	−0.802177	−0.401088	−	0.916039i	$-0.631368\pi$
$761$	−9.75251e9	−0.802177	−0.401088	+	0.916039i	$0.631368\pi$
$762$	1.01701e9	0.0832691
$763$	−8.14318e9	−0.663679
$764$	−4.41252e9	−0.357981
$765$	−6.37325e6	−0.000514690 0
$766$	1.84961e10	1.48689
$767$	0	0
$768$	−8.18241e9	−0.651804
$769$	−6.98230e9	−0.553677	−0.276839	−	0.960916i	$-0.589287\pi$
$769$	−6.98230e9	−0.553677	−0.276839	+	0.960916i	$0.589287\pi$
$770$	−1.85495e8	−0.0146425
$771$	−2.37680e10	−1.86768
$772$	2.35733e9	0.184399
$773$	1.86799e10	1.45461	0.727305	−	0.686314i	$-0.240773\pi$
$773$	1.86799e10	1.45461	0.727305	+	0.686314i	$0.240773\pi$
$774$	7.03887e7	0.00545644
$775$	−1.92928e10	−1.48881
$776$	1.43505e10	1.10243
$777$	5.29091e9	0.404629
$778$	1.00761e10	0.767119
$779$	−8.27868e9	−0.627451
$780$	0	0
$781$	1.71963e10	1.29169
$782$	−2.11716e8	−0.0158318
$783$	1.66362e10	1.23848
$784$	6.88525e9	0.510286
$785$	−2.54669e7	−0.00187902
$786$	−8.12348e9	−0.596710
$787$	−5.17067e9	−0.378125	−0.189062	−	0.981965i	$-0.560545\pi$
$787$	−5.17067e9	−0.378125	−0.189062	+	0.981965i	$0.560545\pi$
$788$	5.33271e8	0.0388245
$789$	3.85952e9	0.279746
$790$	3.39118e8	0.0244713
$791$	−8.45620e9	−0.607516
$792$	−2.53444e9	−0.181278
$793$	0	0
$794$	−1.40609e10	−0.996877
$795$	−5.99329e8	−0.0423039
$796$	−7.51552e8	−0.0528157
$797$	8.89261e9	0.622193	0.311097	−	0.950378i	$-0.399304\pi$
$797$	8.89261e9	0.622193	0.311097	+	0.950378i	$0.399304\pi$
$798$	−5.15607e9	−0.359177
$799$	2.11374e9	0.146602
$800$	5.95686e9	0.411342
$801$	−3.49134e9	−0.240037
$802$	1.21843e10	0.834048
$803$	1.34049e10	0.913606
$804$	2.87447e9	0.195057
$805$	−5.32962e7	−0.00360090
$806$	0	0
$807$	−9.84086e9	−0.659137
$808$	−3.22356e9	−0.214979
$809$	−1.93280e10	−1.28342	−0.641709	−	0.766948i	$-0.721774\pi$
$809$	−1.93280e10	−1.28342	−0.641709	+	0.766948i	$0.721774\pi$
$810$	−4.91460e8	−0.0324931
$811$	−1.58565e9	−0.104384	−0.0521922	−	0.998637i	$-0.516621\pi$
$811$	−1.58565e9	−0.104384	−0.0521922	+	0.998637i	$0.516621\pi$
$812$	2.50698e9	0.164325
$813$	2.05888e10	1.34374
$814$	8.33274e9	0.541505
$815$	3.16827e8	0.0205008
$816$	−1.11449e9	−0.0718061
$817$	−3.45142e8	−0.0221422
$818$	−1.46494e10	−0.935802
$819$	0	0
$820$	−9.99780e7	−0.00633222
$821$	1.92167e10	1.21193	0.605967	−	0.795490i	$-0.292786\pi$
$821$	1.92167e10	1.21193	0.605967	+	0.795490i	$0.292786\pi$
$822$	2.53753e10	1.59353
$823$	−7.27465e9	−0.454897	−0.227448	−	0.973790i	$-0.573038\pi$
$823$	−7.27465e9	−0.454897	−0.227448	+	0.973790i	$0.573038\pi$
$824$	−2.72425e10	−1.69629
$825$	1.61393e10	1.00068
$826$	2.06134e9	0.127268
$827$	7.39476e9	0.454626	0.227313	−	0.973822i	$-0.427006\pi$
$827$	7.39476e9	0.454626	0.227313	+	0.973822i	$0.427006\pi$
$828$	−1.26185e8	−0.00772504
$829$	1.20575e10	0.735048	0.367524	−	0.930014i	$-0.380206\pi$
$829$	1.20575e10	0.735048	0.367524	+	0.930014i	$0.380206\pi$
$830$	−3.90293e8	−0.0236929
$831$	2.37458e10	1.43544
$832$	0	0
$833$	−1.00861e9	−0.0604594
$834$	2.92459e9	0.174576
$835$	1.95905e8	0.0116451
$836$	2.15347e9	0.127473
$837$	−2.24656e10	−1.32428
$838$	−1.36012e10	−0.798404
$839$	1.74685e9	0.102115	0.0510575	−	0.998696i	$-0.483741\pi$
$839$	1.74685e9	0.102115	0.0510575	+	0.998696i	$0.483741\pi$
$840$	−3.59336e8	−0.0209182
$841$	1.62591e10	0.942562
$842$	−1.36402e10	−0.787461
$843$	−1.44237e10	−0.829241
$844$	−6.43109e8	−0.0368202
$845$	0	0
$846$	−4.75055e9	−0.269741
$847$	−1.51052e9	−0.0854153
$848$	−1.62153e10	−0.913144
$849$	−1.80218e10	−1.01070
$850$	1.40639e9	0.0785486
$851$	2.39415e9	0.133167
$852$	5.77251e9	0.319762
$853$	2.10759e9	0.116269	0.0581346	−	0.998309i	$-0.481485\pi$
$853$	2.10759e9	0.116269	0.0581346	+	0.998309i	$0.481485\pi$
$854$	1.68741e10	0.927084
$855$	−7.02340e7	−0.00384296
$856$	−2.41638e10	−1.31676
$857$	−2.74086e10	−1.48749	−0.743744	−	0.668464i	$-0.766952\pi$
$857$	−2.74086e10	−1.48749	−0.743744	+	0.668464i	$0.766952\pi$
$858$	0	0
$859$	2.95561e9	0.159100	0.0795502	−	0.996831i	$-0.474652\pi$
$859$	2.95561e9	0.159100	0.0795502	+	0.996831i	$0.474652\pi$
$860$	−4.16813e6	−0.000223459 0
$861$	−1.08874e10	−0.581320
$862$	1.28024e10	0.680795
$863$	2.24829e10	1.19073	0.595366	−	0.803454i	$-0.297007\pi$
$863$	2.24829e10	1.19073	0.595366	+	0.803454i	$0.297007\pi$
$864$	6.93651e9	0.365883
$865$	−1.74987e8	−0.00919283
$866$	−3.40981e9	−0.178409
$867$	−2.07088e10	−1.07917
$868$	−3.38543e9	−0.175709
$869$	−1.54238e10	−0.797303
$870$	−8.32286e8	−0.0428504
$871$	0	0
$872$	−2.48439e10	−1.26886
$873$	3.68875e9	0.187642
$874$	−2.33314e9	−0.118209
$875$	7.08430e8	0.0357494
$876$	4.49980e9	0.226167
$877$	−3.27881e10	−1.64141	−0.820707	−	0.571350i	$-0.806420\pi$
$877$	−3.27881e10	−1.64141	−0.820707	+	0.571350i	$0.806420\pi$
$878$	−1.41012e10	−0.703116
$879$	−2.89037e10	−1.43546
$880$	−4.41852e8	−0.0218569
$881$	3.78138e10	1.86310	0.931548	−	0.363619i	$-0.118459\pi$
$881$	3.78138e10	1.86310	0.931548	+	0.363619i	$0.118459\pi$
$882$	2.26680e9	0.111243
$883$	−8.05584e9	−0.393775	−0.196888	−	0.980426i	$-0.563083\pi$
$883$	−8.05584e9	−0.393775	−0.196888	+	0.980426i	$0.563083\pi$
$884$	0	0
$885$	1.81482e8	0.00880102
$886$	3.33652e10	1.61167
$887$	−3.15643e10	−1.51867	−0.759335	−	0.650700i	$-0.774476\pi$
$887$	−3.15643e10	−1.51867	−0.759335	+	0.650700i	$0.774476\pi$
$888$	1.61420e10	0.773590
$889$	−1.01468e9	−0.0484367
$890$	−7.79594e8	−0.0370684
$891$	2.23527e10	1.05866
$892$	−5.06479e9	−0.238938
$893$	2.32937e10	1.09461
$894$	−7.77657e9	−0.364004
$895$	−7.02255e8	−0.0327427
$896$	−6.99213e9	−0.324737
$897$	0	0
$898$	8.95200e9	0.412527
$899$	−4.52506e10	−2.07714
$900$	8.38220e8	0.0383274
$901$	2.37535e9	0.108191
$902$	−1.71468e10	−0.777967
$903$	−4.53903e8	−0.0205143
$904$	−2.57989e10	−1.16148
$905$	4.39125e8	0.0196933
$906$	−3.31113e10	−1.47920
$907$	−3.74494e10	−1.66656	−0.833278	−	0.552854i	$-0.813539\pi$
$907$	−3.74494e10	−1.66656	−0.833278	+	0.552854i	$0.813539\pi$
$908$	2.05618e7	0.000911507 0
$909$	−8.28603e8	−0.0365909
$910$	0	0
$911$	4.69150e9	0.205588	0.102794	−	0.994703i	$-0.467222\pi$
$911$	4.69150e9	0.205588	0.102794	+	0.994703i	$0.467222\pi$
$912$	−1.22818e10	−0.536144
$913$	1.77514e10	0.771942
$914$	−1.67351e10	−0.724966
$915$	1.48561e9	0.0641109
$916$	−2.85521e9	−0.122745
$917$	8.10488e9	0.347099
$918$	1.63768e9	0.0698680
$919$	−3.13816e10	−1.33374	−0.666870	−	0.745174i	$-0.732366\pi$
$919$	−3.13816e10	−1.33374	−0.666870	+	0.745174i	$0.732366\pi$
$920$	−1.62601e8	−0.00688438
$921$	3.08695e10	1.30203
$922$	2.69801e8	0.0113367
$923$	0	0
$924$	2.83207e9	0.118101
$925$	−1.59039e10	−0.660703
$926$	−3.57923e10	−1.48133
$927$	−7.00257e9	−0.288721
$928$	1.39716e10	0.573891
$929$	2.99605e10	1.22601	0.613005	−	0.790079i	$-0.289961\pi$
$929$	2.99605e10	1.22601	0.613005	+	0.790079i	$0.289961\pi$
$930$	1.12392e9	0.0458190
$931$	−1.11150e10	−0.451424
$932$	−5.51309e8	−0.0223069
$933$	−2.90380e10	−1.17053
$934$	2.61307e10	1.04939
$935$	6.47260e7	0.00258963
$936$	0	0
$937$	3.17700e10	1.26162	0.630810	−	0.775937i	$-0.282723\pi$
$937$	3.17700e10	1.26162	0.630810	+	0.775937i	$0.282723\pi$
$938$	1.08143e10	0.427848
$939$	6.05341e9	0.238600
$940$	2.81308e8	0.0110468
$941$	3.63923e8	0.0142379	0.00711895	−	0.999975i	$-0.497734\pi$
$941$	3.63923e8	0.0142379	0.00711895	+	0.999975i	$0.497734\pi$
$942$	−1.46617e9	−0.0571487
$943$	−4.92660e9	−0.191318
$944$	4.91015e9	0.189973
$945$	4.12259e8	0.0158913
$946$	−7.14860e8	−0.0274538
$947$	8.10602e9	0.310158	0.155079	−	0.987902i	$-0.450437\pi$
$947$	8.10602e9	0.310158	0.155079	+	0.987902i	$0.450437\pi$
$948$	−5.17752e9	−0.197375
$949$	0	0
$950$	1.54986e10	0.586487
$951$	3.61660e10	1.36354
$952$	1.42417e9	0.0534975
$953$	3.07889e10	1.15231	0.576155	−	0.817340i	$-0.304552\pi$
$953$	3.07889e10	1.15231	0.576155	+	0.817340i	$0.304552\pi$
$954$	−5.33848e9	−0.199067
$955$	−1.46155e9	−0.0543002
$956$	1.73715e9	0.0643034
$957$	3.78542e10	1.39612
$958$	−2.91860e10	−1.07250
$959$	−2.53171e10	−0.926936
$960$	−1.05465e9	−0.0384731
$961$	3.35940e10	1.22104
$962$	0	0
$963$	−6.21122e9	−0.224122
$964$	−1.07797e10	−0.387557
$965$	7.80812e8	0.0279705
$966$	−3.06835e9	−0.109518
$967$	2.71301e10	0.964846	0.482423	−	0.875938i	$-0.339757\pi$
$967$	2.71301e10	0.964846	0.482423	+	0.875938i	$0.339757\pi$
$968$	−4.60844e9	−0.163301
$969$	1.79914e9	0.0635231
$970$	8.23674e8	0.0289770
$971$	−2.35980e10	−0.827197	−0.413598	−	0.910459i	$-0.635728\pi$
$971$	−2.35980e10	−0.827197	−0.413598	+	0.910459i	$0.635728\pi$
$972$	2.17083e9	0.0758217
$973$	−2.91789e9	−0.101549
$974$	−1.15986e10	−0.402207
$975$	0	0
$976$	4.01944e10	1.38386
$977$	1.46911e9	0.0503992	0.0251996	−	0.999682i	$-0.491978\pi$
$977$	1.46911e9	0.0503992	0.0251996	+	0.999682i	$0.491978\pi$
$978$	1.82402e10	0.623511
$979$	3.54576e10	1.20773
$980$	−1.34231e8	−0.00455575
$981$	−6.38603e9	−0.215968
$982$	−3.80871e10	−1.28347
$983$	3.36024e10	1.12832	0.564161	−	0.825665i	$-0.309200\pi$
$983$	3.36024e10	1.12832	0.564161	+	0.825665i	$0.309200\pi$
$984$	−3.32164e10	−1.11140
$985$	1.76634e8	0.00588908
$986$	3.29863e9	0.109589
$987$	3.06340e10	1.01413
$988$	0	0
$989$	−2.05392e8	−0.00675146
$990$	−1.45469e8	−0.00476482
$991$	−2.03355e10	−0.663737	−0.331869	−	0.943326i	$-0.607679\pi$
$991$	−2.03355e10	−0.663737	−0.331869	+	0.943326i	$0.607679\pi$
$992$	−1.88673e10	−0.613649
$993$	5.25849e10	1.70427
$994$	2.17174e10	0.701382
$995$	−2.48935e8	−0.00801133
$996$	5.95884e9	0.191097
$997$	8.04557e9	0.257113	0.128556	−	0.991702i	$-0.458966\pi$
$997$	8.04557e9	0.257113	0.128556	+	0.991702i	$0.458966\pi$
$998$	−1.99309e10	−0.634702
$999$	−1.85193e10	−0.587688

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.d.1.2		6
13.5	odd	4		13.8.b.a.12.5	yes	6
13.8	odd	4		13.8.b.a.12.2	✓	6
13.12	even	2	inner	169.8.a.d.1.5		6
39.5	even	4		117.8.b.b.64.2		6
39.8	even	4		117.8.b.b.64.5		6
52.31	even	4		208.8.f.a.129.2		6
52.47	even	4		208.8.f.a.129.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.b.a.12.2	✓	6	13.8	odd	4
13.8.b.a.12.5	yes	6	13.5	odd	4
117.8.b.b.64.2		6	39.5	even	4
117.8.b.b.64.5		6	39.8	even	4
169.8.a.d.1.2		6	1.1	even	1	trivial
169.8.a.d.1.5		6	13.12	even	2	inner
208.8.f.a.129.1		6	52.47	even	4
208.8.f.a.129.2		6	52.31	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q-10.0583 q^{2} +50.8656 q^{3} -26.8297 q^{4} -8.88672 q^{5} -511.623 q^{6} +510.452 q^{7} +1557.33 q^{8} +400.306 q^{9} +89.3858 q^{10} -4065.46 q^{11} -1364.71 q^{12} -5134.30 q^{14} -452.028 q^{15} -12230.0 q^{16} +1791.54 q^{17} -4026.41 q^{18} +19743.0 q^{19} +238.428 q^{20} +25964.4 q^{21} +40891.8 q^{22} +11749.0 q^{23} +79214.5 q^{24} -78046.0 q^{25} -90881.2 q^{27} -13695.3 q^{28} -183054. q^{29} +4546.66 q^{30} +247198. q^{31} -76325.0 q^{32} -206792. q^{33} -18020.0 q^{34} -4536.25 q^{35} -10740.1 q^{36} +203775. q^{37} -198582. q^{38} -13839.6 q^{40} -419322. q^{41} -261159. q^{42} -17481.7 q^{43} +109075. q^{44} -3557.41 q^{45} -118175. q^{46} +1.17985e6 q^{47} -622084. q^{48} -562982. q^{49} +785014. q^{50} +91127.9 q^{51} +1.32587e6 q^{53} +914115. q^{54} +36128.6 q^{55} +794942. q^{56} +1.00424e6 q^{57} +1.84123e6 q^{58} -401485. q^{59} +12127.8 q^{60} -3.28655e6 q^{61} -2.48640e6 q^{62} +204337. q^{63} +2.33314e6 q^{64} +2.07999e6 q^{66} -2.10629e6 q^{67} -48066.5 q^{68} +597618. q^{69} +45627.1 q^{70} -4.22986e6 q^{71} +623408. q^{72} -3.29726e6 q^{73} -2.04964e6 q^{74} -3.96986e6 q^{75} -529699. q^{76} -2.07522e6 q^{77} +3.79387e6 q^{79} +108684. q^{80} -5.49819e6 q^{81} +4.21768e6 q^{82} -4.36639e6 q^{83} -696617. q^{84} -15920.9 q^{85} +175837. q^{86} -9.31117e6 q^{87} -6.33127e6 q^{88} -8.72168e6 q^{89} +35781.6 q^{90} -315221. q^{92} +1.25738e7 q^{93} -1.18673e7 q^{94} -175451. q^{95} -3.88231e6 q^{96} +9.21482e6 q^{97} +5.66267e6 q^{98} -1.62743e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 56 q^{3} + 130 q^{4} - 1150 q^{9} + 406 q^{10} - 1898 q^{12} + 9558 q^{14} + 7778 q^{16} - 13152 q^{17} - 125080 q^{22} - 27264 q^{23} + 18262 q^{25} + 194560 q^{27} + 42924 q^{29} + 60114 q^{30}+ \cdots + 22075632 q^{95}+O(q^{100}) \) 6 * q - 56 * q^3 + 130 * q^4 - 1150 * q^9 + 406 * q^10 - 1898 * q^12 + 9558 * q^14 + 7778 * q^16 - 13152 * q^17 - 125080 * q^22 - 27264 * q^23 + 18262 * q^25 + 194560 * q^27 + 42924 * q^29 + 60114 * q^30 - 546720 * q^35 - 747772 * q^36 + 243492 * q^38 + 792058 * q^40 - 1138134 * q^42 + 1005576 * q^43 - 2807426 * q^48 - 3246846 * q^49 - 297984 * q^51 + 1705524 * q^53 - 5320576 * q^55 + 2282730 * q^56 - 8237572 * q^61 - 9625800 * q^62 - 5070862 * q^64 + 9289056 * q^66 - 16801086 * q^68 + 6017400 * q^69 - 28760826 * q^74 - 20426072 * q^75 - 4093560 * q^77 + 15147200 * q^79 - 21318442 * q^81 + 9130240 * q^82 - 36474240 * q^87 - 38008000 * q^88 - 4769676 * q^90 + 36462348 * q^92 - 34229770 * q^94 + 22075632 * q^95

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