LMFDB - Embedded newform 294.10.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,10,Mod(1,294)] 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("294.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-48,-243,768,-361] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$151.420535832$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 373x - 378 $ x^3 - 373*x - 378
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3\cdot 5\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$19.8013$ of defining polynomial
Character	$\chi$	$=$	294.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-16.0000 q^{2} -81.0000 q^{3} +256.000 q^{4} -472.452 q^{5} +1296.00 q^{6} -4096.00 q^{8} +6561.00 q^{9} +7559.24 q^{10} -18360.6 q^{11} -20736.0 q^{12} -4756.67 q^{13} +38268.6 q^{15} +65536.0 q^{16} -262780. q^{17} -104976. q^{18} +716930. q^{19} -120948. q^{20} +293770. q^{22} +259822. q^{23} +331776. q^{24} -1.72991e6 q^{25} +76106.8 q^{26} -531441. q^{27} +4.40930e6 q^{29} -612298. q^{30} -6.02490e6 q^{31} -1.04858e6 q^{32} +1.48721e6 q^{33} +4.20449e6 q^{34} +1.67962e6 q^{36} -4.56879e6 q^{37} -1.14709e7 q^{38} +385291. q^{39} +1.93516e6 q^{40} +1.47923e7 q^{41} -2.14447e7 q^{43} -4.70032e6 q^{44} -3.09976e6 q^{45} -4.15715e6 q^{46} +3.80870e7 q^{47} -5.30842e6 q^{48} +2.76786e7 q^{50} +2.12852e7 q^{51} -1.21771e6 q^{52} +3.17480e6 q^{53} +8.50306e6 q^{54} +8.67452e6 q^{55} -5.80713e7 q^{57} -7.05487e7 q^{58} +5.35841e7 q^{59} +9.79677e6 q^{60} -6.13901e7 q^{61} +9.63983e7 q^{62} +1.67772e7 q^{64} +2.24730e6 q^{65} -2.37954e7 q^{66} +2.35128e8 q^{67} -6.72718e7 q^{68} -2.10456e7 q^{69} -2.74423e8 q^{71} -2.68739e7 q^{72} +2.33403e8 q^{73} +7.31007e7 q^{74} +1.40123e8 q^{75} +1.83534e8 q^{76} -6.16465e6 q^{78} -5.50930e7 q^{79} -3.09626e7 q^{80} +4.30467e7 q^{81} -2.36676e8 q^{82} +9.49452e7 q^{83} +1.24151e8 q^{85} +3.43116e8 q^{86} -3.57153e8 q^{87} +7.52051e7 q^{88} +6.98706e8 q^{89} +4.95962e7 q^{90} +6.65144e7 q^{92} +4.88017e8 q^{93} -6.09392e8 q^{94} -3.38715e8 q^{95} +8.49347e7 q^{96} +1.52006e9 q^{97} -1.20464e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 48 q^{2} - 243 q^{3} + 768 q^{4} - 361 q^{5} + 3888 q^{6} - 12288 q^{8} + 19683 q^{9} + 5776 q^{10} - 37799 q^{11} - 62208 q^{12} - 220586 q^{13} + 29241 q^{15} + 196608 q^{16} + 781816 q^{17} - 314928 q^{18}+ \cdots - 247999239 q^{99}+O(q^{100}) $ 3 * q - 48 * q^2 - 243 * q^3 + 768 * q^4 - 361 * q^5 + 3888 * q^6 - 12288 * q^8 + 19683 * q^9 + 5776 * q^10 - 37799 * q^11 - 62208 * q^12 - 220586 * q^13 + 29241 * q^15 + 196608 * q^16 + 781816 * q^17 - 314928 * q^18 + 620154 * q^19 - 92416 * q^20 + 604784 * q^22 - 1204784 * q^23 + 995328 * q^24 + 4020946 * q^25 + 3529376 * q^26 - 1594323 * q^27 - 5280535 * q^29 - 467856 * q^30 - 12838457 * q^31 - 3145728 * q^32 + 3061719 * q^33 - 12509056 * q^34 + 5038848 * q^36 - 8014770 * q^37 - 9922464 * q^38 + 17867466 * q^39 + 1478656 * q^40 + 6503028 * q^41 - 22543488 * q^43 - 9676544 * q^44 - 2368521 * q^45 + 19276544 * q^46 + 21473082 * q^47 - 15925248 * q^48 - 64335136 * q^50 - 63327096 * q^51 - 56470016 * q^52 - 13685715 * q^53 + 25509168 * q^54 + 305107391 * q^55 - 50232474 * q^57 + 84488560 * q^58 + 92528141 * q^59 + 7485696 * q^60 + 7516114 * q^61 + 205415312 * q^62 + 50331648 * q^64 - 79637616 * q^65 - 48987504 * q^66 + 137325404 * q^67 + 200144896 * q^68 + 97587504 * q^69 + 8190346 * q^71 - 80621568 * q^72 - 394016164 * q^73 + 128236320 * q^74 - 325696626 * q^75 + 158759424 * q^76 - 285879456 * q^78 - 35090117 * q^79 - 23658496 * q^80 + 129140163 * q^81 - 104048448 * q^82 + 956948005 * q^83 - 411899320 * q^85 + 360695808 * q^86 + 427723335 * q^87 + 154824704 * q^88 + 855499686 * q^89 + 37896336 * q^90 - 308424704 * q^92 + 1039915017 * q^93 - 343569312 * q^94 + 2705508088 * q^95 + 254803968 * q^96 + 2010569795 * q^97 - 247999239 * q^99

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 $

$2$	−16.0000	−0.707107
$2$	−16.0000	−0.707107
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	256.000	0.500000
$5$	−472.452	−0.338059	−0.169030	−	0.985611i	$-0.554063\pi$
$5$	−472.452	−0.338059	−0.169030	+	0.985611i	$0.554063\pi$
$6$	1296.00	0.408248
$7$	0	0
$7$	0	0
$8$	−4096.00	−0.353553
$9$	6561.00	0.333333
$10$	7559.24	0.239044
$11$	−18360.6	−0.378112	−0.189056	−	0.981966i	$-0.560543\pi$
$11$	−18360.6	−0.378112	−0.189056	+	0.981966i	$0.560543\pi$
$12$	−20736.0	−0.288675
$13$	−4756.67	−0.0461911	−0.0230956	−	0.999733i	$-0.507352\pi$
$13$	−4756.67	−0.0461911	−0.0230956	+	0.999733i	$0.507352\pi$
$14$	0	0
$15$	38268.6	0.195179
$16$	65536.0	0.250000
$17$	−262780.	−0.763085	−0.381543	−	0.924351i	$-0.624607\pi$
$17$	−262780.	−0.763085	−0.381543	+	0.924351i	$0.624607\pi$
$18$	−104976.	−0.235702
$19$	716930.	1.26208	0.631038	−	0.775752i	$-0.282629\pi$
$19$	716930.	1.26208	0.631038	+	0.775752i	$0.282629\pi$
$20$	−120948.	−0.169030
$21$	0	0
$22$	293770.	0.267366
$23$	259822.	0.193598	0.0967989	−	0.995304i	$-0.469140\pi$
$23$	259822.	0.193598	0.0967989	+	0.995304i	$0.469140\pi$
$24$	331776.	0.204124
$25$	−1.72991e6	−0.885716
$26$	76106.8	0.0326620
$27$	−531441.	−0.192450
$28$	0	0
$29$	4.40930e6	1.15765	0.578826	−	0.815451i	$-0.303511\pi$
$29$	4.40930e6	1.15765	0.578826	+	0.815451i	$0.303511\pi$
$30$	−612298.	−0.138012
$31$	−6.02490e6	−1.17171	−0.585857	−	0.810414i	$-0.699242\pi$
$31$	−6.02490e6	−1.17171	−0.585857	+	0.810414i	$0.699242\pi$
$32$	−1.04858e6	−0.176777
$33$	1.48721e6	0.218303
$34$	4.20449e6	0.539583
$35$	0	0
$36$	1.67962e6	0.166667
$37$	−4.56879e6	−0.400769	−0.200384	−	0.979717i	$-0.564219\pi$
$37$	−4.56879e6	−0.400769	−0.200384	+	0.979717i	$0.564219\pi$
$38$	−1.14709e7	−0.892423
$39$	385291.	0.0266684
$40$	1.93516e6	0.119522
$41$	1.47923e7	0.817537	0.408769	−	0.912638i	$-0.365958\pi$
$41$	1.47923e7	0.817537	0.408769	+	0.912638i	$0.365958\pi$
$42$	0	0
$43$	−2.14447e7	−0.956561	−0.478281	−	0.878207i	$-0.658740\pi$
$43$	−2.14447e7	−0.956561	−0.478281	+	0.878207i	$0.658740\pi$
$44$	−4.70032e6	−0.189056
$45$	−3.09976e6	−0.112686
$46$	−4.15715e6	−0.136894
$47$	3.80870e7	1.13851	0.569254	−	0.822162i	$-0.307232\pi$
$47$	3.80870e7	1.13851	0.569254	+	0.822162i	$0.307232\pi$
$48$	−5.30842e6	−0.144338
$49$	0	0
$50$	2.76786e7	0.626296
$51$	2.12852e7	0.440567
$52$	−1.21771e6	−0.0230956
$53$	3.17480e6	0.0552682	0.0276341	−	0.999618i	$-0.491203\pi$
$53$	3.17480e6	0.0552682	0.0276341	+	0.999618i	$0.491203\pi$
$54$	8.50306e6	0.136083
$55$	8.67452e6	0.127824
$56$	0	0
$57$	−5.80713e7	−0.728660
$58$	−7.05487e7	−0.818584
$59$	5.35841e7	0.575708	0.287854	−	0.957674i	$-0.407058\pi$
$59$	5.35841e7	0.575708	0.287854	+	0.957674i	$0.407058\pi$
$60$	9.79677e6	0.0975893
$61$	−6.13901e7	−0.567693	−0.283847	−	0.958870i	$-0.591611\pi$
$61$	−6.13901e7	−0.567693	−0.283847	+	0.958870i	$0.591611\pi$
$62$	9.63983e7	0.828527
$63$	0	0
$64$	1.67772e7	0.125000
$65$	2.24730e6	0.0156153
$66$	−2.37954e7	−0.154364
$67$	2.35128e8	1.42550	0.712750	−	0.701419i	$-0.247450\pi$
$67$	2.35128e8	1.42550	0.712750	+	0.701419i	$0.247450\pi$
$68$	−6.72718e7	−0.381543
$69$	−2.10456e7	−0.111774
$70$	0	0
$71$	−2.74423e8	−1.28162	−0.640808	−	0.767701i	$-0.721400\pi$
$71$	−2.74423e8	−1.28162	−0.640808	+	0.767701i	$0.721400\pi$
$72$	−2.68739e7	−0.117851
$73$	2.33403e8	0.961950	0.480975	−	0.876734i	$-0.340283\pi$
$73$	2.33403e8	0.961950	0.480975	+	0.876734i	$0.340283\pi$
$74$	7.31007e7	0.283386
$75$	1.40123e8	0.511368
$76$	1.83534e8	0.631038
$77$	0	0
$78$	−6.16465e6	−0.0188574
$79$	−5.50930e7	−0.159138	−0.0795691	−	0.996829i	$-0.525354\pi$
$79$	−5.50930e7	−0.159138	−0.0795691	+	0.996829i	$0.525354\pi$
$80$	−3.09626e7	−0.0845148
$81$	4.30467e7	0.111111
$82$	−2.36676e8	−0.578086
$83$	9.49452e7	0.219595	0.109797	−	0.993954i	$-0.464980\pi$
$83$	9.49452e7	0.219595	0.109797	+	0.993954i	$0.464980\pi$
$84$	0	0
$85$	1.24151e8	0.257968
$86$	3.43116e8	0.676391
$87$	−3.57153e8	−0.668371
$88$	7.52051e7	0.133683
$89$	6.98706e8	1.18043	0.590214	−	0.807247i	$-0.299043\pi$
$89$	6.98706e8	1.18043	0.590214	+	0.807247i	$0.299043\pi$
$90$	4.95962e7	0.0796813
$91$	0	0
$92$	6.65144e7	0.0967989
$93$	4.88017e8	0.676490
$94$	−6.09392e8	−0.805047
$95$	−3.38715e8	−0.426657
$96$	8.49347e7	0.102062
$97$	1.52006e9	1.74336	0.871681	−	0.490073i	$-0.163030\pi$
$97$	1.52006e9	1.74336	0.871681	+	0.490073i	$0.163030\pi$
$98$	0	0
$99$	−1.20464e8	−0.126037
$100$	−4.42858e8	−0.442858
$101$	−2.64704e8	−0.253113	−0.126557	−	0.991959i	$-0.540393\pi$
$101$	−2.64704e8	−0.253113	−0.126557	+	0.991959i	$0.540393\pi$
$102$	−3.40564e8	−0.311528
$103$	−1.51768e9	−1.32866	−0.664328	−	0.747441i	$-0.731282\pi$
$103$	−1.51768e9	−1.32866	−0.664328	+	0.747441i	$0.731282\pi$
$104$	1.94833e7	0.0163310
$105$	0	0
$106$	−5.07969e7	−0.0390805
$107$	2.24545e9	1.65606	0.828031	−	0.560682i	$-0.189461\pi$
$107$	2.24545e9	1.65606	0.828031	+	0.560682i	$0.189461\pi$
$108$	−1.36049e8	−0.0962250
$109$	9.65239e8	0.654961	0.327481	−	0.944858i	$-0.393800\pi$
$109$	9.65239e8	0.654961	0.327481	+	0.944858i	$0.393800\pi$
$110$	−1.38792e8	−0.0903854
$111$	3.70072e8	0.231384
$112$	0	0
$113$	1.82803e9	1.05470	0.527352	−	0.849647i	$-0.323185\pi$
$113$	1.82803e9	1.05470	0.527352	+	0.849647i	$0.323185\pi$
$114$	9.29141e8	0.515240
$115$	−1.22753e8	−0.0654475
$116$	1.12878e9	0.578826
$117$	−3.12085e7	−0.0153970
$118$	−8.57346e8	−0.407087
$119$	0	0
$120$	−1.56748e8	−0.0690061
$121$	−2.02083e9	−0.857031
$122$	9.82241e8	0.401420
$123$	−1.19817e9	−0.472005
$124$	−1.54237e9	−0.585857
$125$	1.74006e9	0.637484
$126$	0	0
$127$	1.83082e9	0.624495	0.312248	−	0.950001i	$-0.398918\pi$
$127$	1.83082e9	0.624495	0.312248	+	0.950001i	$0.398918\pi$
$128$	−2.68435e8	−0.0883883
$129$	1.73702e9	0.552271
$130$	−3.59568e7	−0.0110417
$131$	−3.01836e9	−0.895468	−0.447734	−	0.894167i	$-0.647769\pi$
$131$	−3.01836e9	−0.895468	−0.447734	+	0.894167i	$0.647769\pi$
$132$	3.80726e8	0.109152
$133$	0	0
$134$	−3.76204e9	−1.00798
$135$	2.51081e8	0.0650595
$136$	1.07635e9	0.269791
$137$	−2.38667e9	−0.578829	−0.289414	−	0.957204i	$-0.593461\pi$
$137$	−2.38667e9	−0.578829	−0.289414	+	0.957204i	$0.593461\pi$
$138$	3.36729e8	0.0790360
$139$	−2.56337e8	−0.0582432	−0.0291216	−	0.999576i	$-0.509271\pi$
$139$	−2.56337e8	−0.0582432	−0.0291216	+	0.999576i	$0.509271\pi$
$140$	0	0
$141$	−3.08505e9	−0.657318
$142$	4.39077e9	0.906239
$143$	8.73356e7	0.0174654
$144$	4.29982e8	0.0833333
$145$	−2.08318e9	−0.391355
$146$	−3.73444e9	−0.680202
$147$	0	0
$148$	−1.16961e9	−0.200384
$149$	5.29448e8	0.0880005	0.0440003	−	0.999032i	$-0.485990\pi$
$149$	5.29448e8	0.0880005	0.0440003	+	0.999032i	$0.485990\pi$
$150$	−2.24197e9	−0.361592
$151$	−3.47450e9	−0.543871	−0.271935	−	0.962315i	$-0.587664\pi$
$151$	−3.47450e9	−0.543871	−0.271935	+	0.962315i	$0.587664\pi$
$152$	−2.93655e9	−0.446211
$153$	−1.72410e9	−0.254362
$154$	0	0
$155$	2.84648e9	0.396109
$156$	9.86344e7	0.0133342
$157$	5.50302e8	0.0722857	0.0361429	−	0.999347i	$-0.488493\pi$
$157$	5.50302e8	0.0722857	0.0361429	+	0.999347i	$0.488493\pi$
$158$	8.81488e8	0.112528
$159$	−2.57159e8	−0.0319091
$160$	4.95402e8	0.0597610
$161$	0	0
$162$	−6.88748e8	−0.0785674
$163$	−6.16114e8	−0.0683624	−0.0341812	−	0.999416i	$-0.510882\pi$
$163$	−6.16114e8	−0.0683624	−0.0341812	+	0.999416i	$0.510882\pi$
$164$	3.78682e9	0.408769
$165$	−7.02636e8	−0.0737994
$166$	−1.51912e9	−0.155277
$167$	1.61177e10	1.60354	0.801771	−	0.597632i	$-0.203891\pi$
$167$	1.61177e10	1.60354	0.801771	+	0.597632i	$0.203891\pi$
$168$	0	0
$169$	−1.05819e10	−0.997866
$170$	−1.98642e9	−0.182411
$171$	4.70378e9	0.420692
$172$	−5.48985e9	−0.478281
$173$	−3.58098e9	−0.303945	−0.151973	−	0.988385i	$-0.548563\pi$
$173$	−3.58098e9	−0.303945	−0.151973	+	0.988385i	$0.548563\pi$
$174$	5.71445e9	0.472610
$175$	0	0
$176$	−1.20328e9	−0.0945280
$177$	−4.34031e9	−0.332385
$178$	−1.11793e10	−0.834688
$179$	−2.53160e10	−1.84313	−0.921566	−	0.388223i	$-0.873089\pi$
$179$	−2.53160e10	−1.84313	−0.921566	+	0.388223i	$0.873089\pi$
$180$	−7.93538e8	−0.0563432
$181$	−8.79219e9	−0.608896	−0.304448	−	0.952529i	$-0.598472\pi$
$181$	−8.79219e9	−0.608896	−0.304448	+	0.952529i	$0.598472\pi$
$182$	0	0
$183$	4.97260e9	0.327758
$184$	−1.06423e9	−0.0684471
$185$	2.15854e9	0.135484
$186$	−7.80826e9	−0.478351
$187$	4.82482e9	0.288532
$188$	9.75027e9	0.569254
$189$	0	0
$190$	5.41944e9	0.301692
$191$	−2.24566e9	−0.122094	−0.0610468	−	0.998135i	$-0.519444\pi$
$191$	−2.24566e9	−0.122094	−0.0610468	+	0.998135i	$0.519444\pi$
$192$	−1.35895e9	−0.0721688
$193$	−2.56013e10	−1.32817	−0.664085	−	0.747657i	$-0.731179\pi$
$193$	−2.56013e10	−1.32817	−0.664085	+	0.747657i	$0.731179\pi$
$194$	−2.43209e10	−1.23274
$195$	−1.82031e8	−0.00901552
$196$	0	0
$197$	−3.17764e10	−1.50317	−0.751583	−	0.659638i	$-0.770710\pi$
$197$	−3.17764e10	−1.50317	−0.751583	+	0.659638i	$0.770710\pi$
$198$	1.92743e9	0.0891219
$199$	−2.47156e10	−1.11720	−0.558602	−	0.829436i	$-0.688662\pi$
$199$	−2.47156e10	−1.11720	−0.558602	+	0.829436i	$0.688662\pi$
$200$	7.08573e9	0.313148
$201$	−1.90453e10	−0.823012
$202$	4.23527e9	0.178978
$203$	0	0
$204$	5.44902e9	0.220284
$205$	−6.98864e9	−0.276376
$206$	2.42829e10	0.939502
$207$	1.70469e9	0.0645326
$208$	−3.11733e8	−0.0115478
$209$	−1.31633e10	−0.477206
$210$	0	0
$211$	−3.48794e10	−1.21143	−0.605715	−	0.795681i	$-0.707113\pi$
$211$	−3.48794e10	−1.21143	−0.605715	+	0.795681i	$0.707113\pi$
$212$	8.12750e8	0.0276341
$213$	2.22283e10	0.739941
$214$	−3.59272e10	−1.17101
$215$	1.01316e10	0.323374
$216$	2.17678e9	0.0680414
$217$	0	0
$218$	−1.54438e10	−0.463127
$219$	−1.89056e10	−0.555382
$220$	2.22068e9	0.0639122
$221$	1.24996e9	0.0352477
$222$	−5.92116e9	−0.163613
$223$	−1.39097e9	−0.0376656	−0.0188328	−	0.999823i	$-0.505995\pi$
$223$	−1.39097e9	−0.0376656	−0.0188328	+	0.999823i	$0.505995\pi$
$224$	0	0
$225$	−1.13500e10	−0.295239
$226$	−2.92485e10	−0.745789
$227$	3.13351e10	0.783277	0.391638	−	0.920119i	$-0.371908\pi$
$227$	3.13351e10	0.783277	0.391638	+	0.920119i	$0.371908\pi$
$228$	−1.48663e10	−0.364330
$229$	−3.94499e10	−0.947952	−0.473976	−	0.880538i	$-0.657182\pi$
$229$	−3.94499e10	−0.947952	−0.473976	+	0.880538i	$0.657182\pi$
$230$	1.96405e9	0.0462784
$231$	0	0
$232$	−1.80605e10	−0.409292
$233$	−8.37335e10	−1.86122	−0.930610	−	0.366012i	$-0.880723\pi$
$233$	−8.37335e10	−1.86122	−0.930610	+	0.366012i	$0.880723\pi$
$234$	4.99337e8	0.0108873
$235$	−1.79943e10	−0.384883
$236$	1.37175e10	0.287854
$237$	4.46253e9	0.0918785
$238$	0	0
$239$	−1.03880e9	−0.0205941	−0.0102970	−	0.999947i	$-0.503278\pi$
$239$	−1.03880e9	−0.0205941	−0.0102970	+	0.999947i	$0.503278\pi$
$240$	2.50797e9	0.0487947
$241$	1.04164e11	1.98903	0.994515	−	0.104596i	$-0.0333548\pi$
$241$	1.04164e11	1.98903	0.994515	+	0.104596i	$0.0333548\pi$
$242$	3.23334e10	0.606013
$243$	−3.48678e9	−0.0641500
$244$	−1.57159e10	−0.283847
$245$	0	0
$246$	1.91708e10	0.333758
$247$	−3.41020e9	−0.0582967
$248$	2.46780e10	0.414264
$249$	−7.69056e9	−0.126783
$250$	−2.78410e10	−0.450769
$251$	1.03278e11	1.64238	0.821192	−	0.570653i	$-0.193310\pi$
$251$	1.03278e11	1.64238	0.821192	+	0.570653i	$0.193310\pi$
$252$	0	0
$253$	−4.77049e9	−0.0732016
$254$	−2.92931e10	−0.441585
$255$	−1.00563e10	−0.148938
$256$	4.29497e9	0.0625000
$257$	9.63771e9	0.137808	0.0689040	−	0.997623i	$-0.478050\pi$
$257$	9.63771e9	0.137808	0.0689040	+	0.997623i	$0.478050\pi$
$258$	−2.77924e10	−0.390515
$259$	0	0
$260$	5.75309e8	0.00780767
$261$	2.89294e10	0.385884
$262$	4.82937e10	0.633191
$263$	−1.46526e11	−1.88849	−0.944243	−	0.329249i	$-0.893204\pi$
$263$	−1.46526e11	−1.88849	−0.944243	+	0.329249i	$0.893204\pi$
$264$	−6.09162e9	−0.0771818
$265$	−1.49994e9	−0.0186839
$266$	0	0
$267$	−5.65952e10	−0.681520
$268$	6.01927e10	0.712750
$269$	−4.14925e10	−0.483153	−0.241576	−	0.970382i	$-0.577664\pi$
$269$	−4.14925e10	−0.483153	−0.241576	+	0.970382i	$0.577664\pi$
$270$	−4.01729e9	−0.0460040
$271$	−1.19541e11	−1.34634	−0.673170	−	0.739488i	$-0.735068\pi$
$271$	−1.19541e11	−1.34634	−0.673170	+	0.739488i	$0.735068\pi$
$272$	−1.72216e10	−0.190771
$273$	0	0
$274$	3.81867e10	0.409294
$275$	3.17623e10	0.334900
$276$	−5.38766e9	−0.0558869
$277$	−6.18736e10	−0.631461	−0.315731	−	0.948849i	$-0.602250\pi$
$277$	−6.18736e10	−0.631461	−0.315731	+	0.948849i	$0.602250\pi$
$278$	4.10139e9	0.0411841
$279$	−3.95293e10	−0.390572
$280$	0	0
$281$	−1.12291e11	−1.07440	−0.537199	−	0.843456i	$-0.680518\pi$
$281$	−1.12291e11	−1.07440	−0.537199	+	0.843456i	$0.680518\pi$
$282$	4.93607e10	0.464794
$283$	−1.19513e11	−1.10759	−0.553793	−	0.832655i	$-0.686820\pi$
$283$	−1.19513e11	−1.10759	−0.553793	+	0.832655i	$0.686820\pi$
$284$	−7.02523e10	−0.640808
$285$	2.74359e10	0.246330
$286$	−1.39737e9	−0.0123499
$287$	0	0
$288$	−6.87971e9	−0.0589256
$289$	−4.95343e10	−0.417701
$290$	3.33309e10	0.276730
$291$	−1.23125e11	−1.00653
$292$	5.97510e10	0.480975
$293$	1.77567e11	1.40753	0.703765	−	0.710433i	$-0.251501\pi$
$293$	1.77567e11	1.40753	0.703765	+	0.710433i	$0.251501\pi$
$294$	0	0
$295$	−2.53159e10	−0.194623
$296$	1.87138e10	0.141693
$297$	9.75759e9	0.0727677
$298$	−8.47117e9	−0.0622258
$299$	−1.23589e9	−0.00894249
$300$	3.58715e10	0.255684
$301$	0	0
$302$	5.55920e10	0.384575
$303$	2.14410e10	0.146135
$304$	4.69847e10	0.315519
$305$	2.90039e10	0.191914
$306$	2.75856e10	0.179861
$307$	1.52250e11	0.978218	0.489109	−	0.872223i	$-0.337322\pi$
$307$	1.52250e11	0.978218	0.489109	+	0.872223i	$0.337322\pi$
$308$	0	0
$309$	1.22932e11	0.767100
$310$	−4.55436e10	−0.280091
$311$	−1.52678e11	−0.925456	−0.462728	−	0.886500i	$-0.653129\pi$
$311$	−1.52678e11	−0.925456	−0.462728	+	0.886500i	$0.653129\pi$
$312$	−1.57815e9	−0.00942872
$313$	−2.49886e11	−1.47161	−0.735806	−	0.677192i	$-0.763196\pi$
$313$	−2.49886e11	−1.47161	−0.735806	+	0.677192i	$0.763196\pi$
$314$	−8.80483e9	−0.0511137
$315$	0	0
$316$	−1.41038e10	−0.0795691
$317$	−1.23490e11	−0.686855	−0.343428	−	0.939179i	$-0.611588\pi$
$317$	−1.23490e11	−0.686855	−0.343428	+	0.939179i	$0.611588\pi$
$318$	4.11455e9	0.0225632
$319$	−8.09575e10	−0.437722
$320$	−7.92643e9	−0.0422574
$321$	−1.81882e11	−0.956128
$322$	0	0
$323$	−1.88395e11	−0.963072
$324$	1.10200e10	0.0555556
$325$	8.22864e9	0.0409122
$326$	9.85783e9	0.0483395
$327$	−7.81843e10	−0.378142
$328$	−6.05892e10	−0.289043
$329$	0	0
$330$	1.12422e10	0.0521841
$331$	4.13629e10	0.189402	0.0947011	−	0.995506i	$-0.469810\pi$
$331$	4.13629e10	0.189402	0.0947011	+	0.995506i	$0.469810\pi$
$332$	2.43060e10	0.109797
$333$	−2.99759e10	−0.133590
$334$	−2.57884e11	−1.13388
$335$	−1.11087e11	−0.481903
$336$	0	0
$337$	−3.30223e11	−1.39468	−0.697338	−	0.716743i	$-0.745632\pi$
$337$	−3.30223e11	−1.39468	−0.697338	+	0.716743i	$0.745632\pi$
$338$	1.69310e11	0.705598
$339$	−1.48071e11	−0.608934
$340$	3.17827e10	0.128984
$341$	1.10621e11	0.443039
$342$	−7.52605e10	−0.297474
$343$	0	0
$344$	8.78377e10	0.338195
$345$	9.94303e9	0.0377861
$346$	5.72958e10	0.214922
$347$	5.33493e11	1.97536	0.987679	−	0.156491i	$-0.0500182\pi$
$347$	5.33493e11	1.97536	0.987679	+	0.156491i	$0.0500182\pi$
$348$	−9.14312e10	−0.334186
$349$	−8.45986e10	−0.305245	−0.152623	−	0.988285i	$-0.548772\pi$
$349$	−8.45986e10	−0.305245	−0.152623	+	0.988285i	$0.548772\pi$
$350$	0	0
$351$	2.52789e9	0.00888948
$352$	1.92525e10	0.0668414
$353$	1.67815e11	0.575233	0.287616	−	0.957746i	$-0.407137\pi$
$353$	1.67815e11	0.575233	0.287616	+	0.957746i	$0.407137\pi$
$354$	6.94450e10	0.235032
$355$	1.29652e11	0.433262
$356$	1.78869e11	0.590214
$357$	0	0
$358$	4.05056e11	1.30329
$359$	7.66901e10	0.243677	0.121838	−	0.992550i	$-0.461121\pi$
$359$	7.66901e10	0.243677	0.121838	+	0.992550i	$0.461121\pi$
$360$	1.26966e10	0.0398407
$361$	1.91301e11	0.592837
$362$	1.40675e11	0.430555
$363$	1.63688e11	0.494807
$364$	0	0
$365$	−1.10272e11	−0.325196
$366$	−7.95615e10	−0.231760
$367$	−2.67084e11	−0.768511	−0.384255	−	0.923227i	$-0.625542\pi$
$367$	−2.67084e11	−0.768511	−0.384255	+	0.923227i	$0.625542\pi$
$368$	1.70277e10	0.0483994
$369$	9.70521e10	0.272512
$370$	−3.45366e10	−0.0958014
$371$	0	0
$372$	1.24932e11	0.338245
$373$	5.70594e11	1.52629	0.763146	−	0.646226i	$-0.223654\pi$
$373$	5.70594e11	1.52629	0.763146	+	0.646226i	$0.223654\pi$
$374$	−7.71971e10	−0.204023
$375$	−1.40945e11	−0.368051
$376$	−1.56004e11	−0.402523
$377$	−2.09736e10	−0.0534733
$378$	0	0
$379$	1.64654e10	0.0409918	0.0204959	−	0.999790i	$-0.493476\pi$
$379$	1.64654e10	0.0409918	0.0204959	+	0.999790i	$0.493476\pi$
$380$	−8.67111e10	−0.213328
$381$	−1.48297e11	−0.360553
$382$	3.59305e10	0.0863332
$383$	2.76757e11	0.657209	0.328605	−	0.944468i	$-0.393422\pi$
$383$	2.76757e11	0.657209	0.328605	+	0.944468i	$0.393422\pi$
$384$	2.17433e10	0.0510310
$385$	0	0
$386$	4.09620e11	0.939158
$387$	−1.40699e11	−0.318854
$388$	3.89135e11	0.871681
$389$	3.84246e11	0.850817	0.425409	−	0.905001i	$-0.360130\pi$
$389$	3.84246e11	0.850817	0.425409	+	0.905001i	$0.360130\pi$
$390$	2.91250e9	0.00637493
$391$	−6.82761e10	−0.147732
$392$	0	0
$393$	2.44487e11	0.516998
$394$	5.08423e11	1.06290
$395$	2.60288e10	0.0537982
$396$	−3.08388e10	−0.0630187
$397$	3.53752e11	0.714730	0.357365	−	0.933965i	$-0.383675\pi$
$397$	3.53752e11	0.714730	0.357365	+	0.933965i	$0.383675\pi$
$398$	3.95450e11	0.789983
$399$	0	0
$400$	−1.13372e11	−0.221429
$401$	−3.49189e11	−0.674389	−0.337195	−	0.941435i	$-0.609478\pi$
$401$	−3.49189e11	−0.674389	−0.337195	+	0.941435i	$0.609478\pi$
$402$	3.04725e11	0.581958
$403$	2.86585e10	0.0541228
$404$	−6.77643e10	−0.126557
$405$	−2.03375e10	−0.0375621
$406$	0	0
$407$	8.38860e10	0.151536
$408$	−8.71843e10	−0.155764
$409$	−1.01214e12	−1.78848	−0.894240	−	0.447588i	$-0.852283\pi$
$409$	−1.01214e12	−1.78848	−0.894240	+	0.447588i	$0.852283\pi$
$410$	1.11818e11	0.195427
$411$	1.93320e11	0.334187
$412$	−3.88526e11	−0.664328
$413$	0	0
$414$	−2.72751e10	−0.0456314
$415$	−4.48571e10	−0.0742360
$416$	4.98773e9	0.00816551
$417$	2.07633e10	0.0336267
$418$	2.10613e11	0.337436
$419$	5.26641e11	0.834741	0.417371	−	0.908736i	$-0.362952\pi$
$419$	5.26641e11	0.834741	0.417371	+	0.908736i	$0.362952\pi$
$420$	0	0
$421$	−4.25683e10	−0.0660414	−0.0330207	−	0.999455i	$-0.510513\pi$
$421$	−4.25683e10	−0.0660414	−0.0330207	+	0.999455i	$0.510513\pi$
$422$	5.58071e11	0.856611
$423$	2.49889e11	0.379503
$424$	−1.30040e10	−0.0195403
$425$	4.54588e11	0.675877
$426$	−3.55652e11	−0.523217
$427$	0	0
$428$	5.74836e11	0.828031
$429$	−7.07418e9	−0.0100837
$430$	−1.62106e11	−0.228660
$431$	2.81899e11	0.393501	0.196751	−	0.980454i	$-0.436961\pi$
$431$	2.81899e11	0.393501	0.196751	+	0.980454i	$0.436961\pi$
$432$	−3.48285e10	−0.0481125
$433$	5.58015e10	0.0762870	0.0381435	−	0.999272i	$-0.487856\pi$
$433$	5.58015e10	0.0762870	0.0381435	+	0.999272i	$0.487856\pi$
$434$	0	0
$435$	1.68738e11	0.225949
$436$	2.47101e11	0.327481
$437$	1.86274e11	0.244335
$438$	3.02490e11	0.392715
$439$	7.59167e11	0.975544	0.487772	−	0.872971i	$-0.337810\pi$
$439$	7.59167e11	0.975544	0.487772	+	0.872971i	$0.337810\pi$
$440$	−3.55308e10	−0.0451927
$441$	0	0
$442$	−1.99994e10	−0.0249239
$443$	−1.45876e12	−1.79957	−0.899783	−	0.436338i	$-0.856275\pi$
$443$	−1.45876e12	−1.79957	−0.899783	+	0.436338i	$0.856275\pi$
$444$	9.47385e10	0.115692
$445$	−3.30105e11	−0.399055
$446$	2.22555e10	0.0266336
$447$	−4.28853e10	−0.0508071
$448$	0	0
$449$	9.76798e11	1.13422	0.567109	−	0.823643i	$-0.308062\pi$
$449$	9.76798e11	1.13422	0.567109	+	0.823643i	$0.308062\pi$
$450$	1.81599e11	0.208765
$451$	−2.71595e11	−0.309121
$452$	4.67976e11	0.527352
$453$	2.81434e11	0.314004
$454$	−5.01362e11	−0.553860
$455$	0	0
$456$	2.37860e11	0.257620
$457$	−4.92561e9	−0.00528247	−0.00264123	−	0.999997i	$-0.500841\pi$
$457$	−4.92561e9	−0.00528247	−0.00264123	+	0.999997i	$0.500841\pi$
$458$	6.31198e11	0.670303
$459$	1.39652e11	0.146856
$460$	−3.14249e10	−0.0327238
$461$	1.45140e11	0.149670	0.0748348	−	0.997196i	$-0.476157\pi$
$461$	1.45140e11	0.149670	0.0748348	+	0.997196i	$0.476157\pi$
$462$	0	0
$463$	−1.06638e12	−1.07844	−0.539221	−	0.842164i	$-0.681281\pi$
$463$	−1.06638e12	−1.07844	−0.539221	+	0.842164i	$0.681281\pi$
$464$	2.88968e11	0.289413
$465$	−2.30565e11	−0.228694
$466$	1.33974e12	1.31608
$467$	1.42939e12	1.39068	0.695338	−	0.718683i	$-0.255255\pi$
$467$	1.42939e12	1.39068	0.695338	+	0.718683i	$0.255255\pi$
$468$	−7.98939e9	−0.00769852
$469$	0	0
$470$	2.87909e11	0.272154
$471$	−4.45744e10	−0.0417342
$472$	−2.19481e11	−0.203543
$473$	3.93739e11	0.361687
$474$	−7.14005e10	−0.0649679
$475$	−1.24023e12	−1.11784
$476$	0	0
$477$	2.08299e10	0.0184227
$478$	1.66208e10	0.0145622
$479$	3.78834e10	0.0328806	0.0164403	−	0.999865i	$-0.494767\pi$
$479$	3.78834e10	0.0328806	0.0164403	+	0.999865i	$0.494767\pi$
$480$	−4.01276e10	−0.0345030
$481$	2.17323e10	0.0185120
$482$	−1.66663e12	−1.40646
$483$	0	0
$484$	−5.17334e11	−0.428516
$485$	−7.18155e11	−0.589360
$486$	5.57886e10	0.0453609
$487$	2.41311e12	1.94400	0.972002	−	0.234971i	$-0.0754994\pi$
$487$	2.41311e12	1.94400	0.972002	+	0.234971i	$0.0754994\pi$
$488$	2.51454e11	0.200710
$489$	4.99053e10	0.0394690
$490$	0	0
$491$	−8.41135e11	−0.653129	−0.326564	−	0.945175i	$-0.605891\pi$
$491$	−8.41135e11	−0.653129	−0.326564	+	0.945175i	$0.605891\pi$
$492$	−3.06733e11	−0.236003
$493$	−1.15868e12	−0.883387
$494$	5.45633e10	0.0412220
$495$	5.69135e10	0.0426081
$496$	−3.94848e11	−0.292929
$497$	0	0
$498$	1.23049e11	0.0896491
$499$	−1.38372e12	−0.999072	−0.499536	−	0.866293i	$-0.666496\pi$
$499$	−1.38372e12	−0.999072	−0.499536	+	0.866293i	$0.666496\pi$
$500$	4.45455e11	0.318742
$501$	−1.30554e12	−0.925805
$502$	−1.65244e12	−1.16134
$503$	−2.11862e11	−0.147570	−0.0737850	−	0.997274i	$-0.523508\pi$
$503$	−2.11862e11	−0.147570	−0.0737850	+	0.997274i	$0.523508\pi$
$504$	0	0
$505$	1.25060e11	0.0855673
$506$	7.63279e10	0.0517614
$507$	8.57132e11	0.576118
$508$	4.68690e11	0.312248
$509$	1.43085e12	0.944849	0.472425	−	0.881371i	$-0.343379\pi$
$509$	1.43085e12	0.944849	0.472425	+	0.881371i	$0.343379\pi$
$510$	1.60900e11	0.105315
$511$	0	0
$512$	−6.87195e10	−0.0441942
$513$	−3.81006e11	−0.242887
$514$	−1.54203e11	−0.0974450
$515$	7.17031e11	0.449165
$516$	4.44678e11	0.276135
$517$	−6.99301e11	−0.430484
$518$	0	0
$519$	2.90060e11	0.175483
$520$	−9.20495e9	−0.00552085
$521$	−1.69681e12	−1.00894	−0.504469	−	0.863430i	$-0.668312\pi$
$521$	−1.69681e12	−1.00894	−0.504469	+	0.863430i	$0.668312\pi$
$522$	−4.62870e11	−0.272861
$523$	−2.75371e12	−1.60939	−0.804694	−	0.593690i	$-0.797670\pi$
$523$	−2.75371e12	−1.60939	−0.804694	+	0.593690i	$0.797670\pi$
$524$	−7.72700e11	−0.447734
$525$	0	0
$526$	2.34442e12	1.33536
$527$	1.58323e12	0.894118
$528$	9.74659e10	0.0545758
$529$	−1.73365e12	−0.962520
$530$	2.39991e10	0.0132115
$531$	3.51565e11	0.191903
$532$	0	0
$533$	−7.03620e10	−0.0377630
$534$	9.05523e11	0.481907
$535$	−1.06087e12	−0.559847
$536$	−9.63083e11	−0.503990
$537$	2.05060e12	1.06413
$538$	6.63880e11	0.341641
$539$	0	0
$540$	6.42766e10	0.0325298
$541$	−1.25714e12	−0.630954	−0.315477	−	0.948933i	$-0.602164\pi$
$541$	−1.25714e12	−0.630954	−0.315477	+	0.948933i	$0.602164\pi$
$542$	1.91265e12	0.952006
$543$	7.12167e11	0.351546
$544$	2.75545e11	0.134896
$545$	−4.56029e11	−0.221416
$546$	0	0
$547$	−8.97132e11	−0.428463	−0.214231	−	0.976783i	$-0.568725\pi$
$547$	−8.97132e11	−0.428463	−0.214231	+	0.976783i	$0.568725\pi$
$548$	−6.10988e11	−0.289414
$549$	−4.02780e11	−0.189231
$550$	−5.08197e11	−0.236810
$551$	3.16116e12	1.46105
$552$	8.62026e10	0.0395180
$553$	0	0
$554$	9.89978e11	0.446511
$555$	−1.74842e11	−0.0782215
$556$	−6.56223e10	−0.0291216
$557$	−4.13377e12	−1.81969	−0.909846	−	0.414947i	$-0.863800\pi$
$557$	−4.13377e12	−1.81969	−0.909846	+	0.414947i	$0.863800\pi$
$558$	6.32469e11	0.276176
$559$	1.02006e11	0.0441846
$560$	0	0
$561$	−3.90810e11	−0.166584
$562$	1.79665e12	0.759714
$563$	−2.83409e11	−0.118885	−0.0594424	−	0.998232i	$-0.518932\pi$
$563$	−2.83409e11	−0.118885	−0.0594424	+	0.998232i	$0.518932\pi$
$564$	−7.89772e11	−0.328659
$565$	−8.63658e11	−0.356553
$566$	1.91221e12	0.783181
$567$	0	0
$568$	1.12404e12	0.453120
$569$	1.97500e12	0.789881	0.394940	−	0.918707i	$-0.370765\pi$
$569$	1.97500e12	0.789881	0.394940	+	0.918707i	$0.370765\pi$
$570$	−4.38975e11	−0.174182
$571$	2.67328e12	1.05240	0.526200	−	0.850361i	$-0.323616\pi$
$571$	2.67328e12	1.05240	0.526200	+	0.850361i	$0.323616\pi$
$572$	2.23579e10	0.00873271
$573$	1.81898e11	0.0704908
$574$	0	0
$575$	−4.49469e11	−0.171473
$576$	1.10075e11	0.0416667
$577$	−1.15919e12	−0.435377	−0.217688	−	0.976018i	$-0.569852\pi$
$577$	−1.15919e12	−0.435377	−0.217688	+	0.976018i	$0.569852\pi$
$578$	7.92549e11	0.295359
$579$	2.07370e12	0.766819
$580$	−5.33295e11	−0.195678
$581$	0	0
$582$	1.97000e12	0.711725
$583$	−5.82914e10	−0.0208976
$584$	−9.56017e11	−0.340101
$585$	1.47445e10	0.00520511
$586$	−2.84107e12	−0.995274
$587$	8.46356e10	0.0294226	0.0147113	−	0.999892i	$-0.495317\pi$
$587$	8.46356e10	0.0294226	0.0147113	+	0.999892i	$0.495317\pi$
$588$	0	0
$589$	−4.31943e12	−1.47879
$590$	4.05055e11	0.137620
$591$	2.57389e12	0.867854
$592$	−2.99421e11	−0.100192
$593$	−2.25259e12	−0.748058	−0.374029	−	0.927417i	$-0.622024\pi$
$593$	−2.25259e12	−0.748058	−0.374029	+	0.927417i	$0.622024\pi$
$594$	−1.56121e11	−0.0514545
$595$	0	0
$596$	1.35539e11	0.0440003
$597$	2.00196e12	0.645018
$598$	1.97742e10	0.00632330
$599$	−4.00748e12	−1.27189	−0.635947	−	0.771733i	$-0.719390\pi$
$599$	−4.00748e12	−1.27189	−0.635947	+	0.771733i	$0.719390\pi$
$600$	−5.73944e11	−0.180796
$601$	−4.08150e11	−0.127610	−0.0638050	−	0.997962i	$-0.520324\pi$
$601$	−4.08150e11	−0.127610	−0.0638050	+	0.997962i	$0.520324\pi$
$602$	0	0
$603$	1.54267e12	0.475166
$604$	−8.89472e11	−0.271935
$605$	9.54748e11	0.289727
$606$	−3.43057e11	−0.103333
$607$	−4.96845e12	−1.48550	−0.742749	−	0.669570i	$-0.766478\pi$
$607$	−4.96845e12	−1.48550	−0.742749	+	0.669570i	$0.766478\pi$
$608$	−7.51756e11	−0.223106
$609$	0	0
$610$	−4.64062e11	−0.135704
$611$	−1.81167e11	−0.0525890
$612$	−4.41370e11	−0.127181
$613$	3.04885e12	0.872094	0.436047	−	0.899924i	$-0.356378\pi$
$613$	3.04885e12	0.872094	0.436047	+	0.899924i	$0.356378\pi$
$614$	−2.43601e12	−0.691705
$615$	5.66080e11	0.159566
$616$	0	0
$617$	4.69271e12	1.30359	0.651794	−	0.758396i	$-0.274017\pi$
$617$	4.69271e12	1.30359	0.651794	+	0.758396i	$0.274017\pi$
$618$	−1.96691e12	−0.542422
$619$	1.02865e12	0.281616	0.140808	−	0.990037i	$-0.455030\pi$
$619$	1.02865e12	0.281616	0.140808	+	0.990037i	$0.455030\pi$
$620$	7.28698e11	0.198055
$621$	−1.38080e11	−0.0372579
$622$	2.44285e12	0.654396
$623$	0	0
$624$	2.52504e10	0.00666711
$625$	2.55664e12	0.670209
$626$	3.99818e12	1.04059
$627$	1.06623e12	0.275515
$628$	1.40877e11	0.0361429
$629$	1.20059e12	0.305821
$630$	0	0
$631$	−5.01733e12	−1.25991	−0.629956	−	0.776631i	$-0.716927\pi$
$631$	−5.01733e12	−1.25991	−0.629956	+	0.776631i	$0.716927\pi$
$632$	2.25661e11	0.0562639
$633$	2.82524e12	0.699420
$634$	1.97584e12	0.485680
$635$	−8.64976e11	−0.211117
$636$	−6.58327e10	−0.0159546
$637$	0	0
$638$	1.29532e12	0.309516
$639$	−1.80049e12	−0.427205
$640$	1.26823e11	0.0298805
$641$	2.42403e12	0.567122	0.283561	−	0.958954i	$-0.408484\pi$
$641$	2.42403e12	0.567122	0.283561	+	0.958954i	$0.408484\pi$
$642$	2.91010e12	0.676085
$643$	4.20461e12	0.970011	0.485006	−	0.874511i	$-0.338818\pi$
$643$	4.20461e12	0.970011	0.485006	+	0.874511i	$0.338818\pi$
$644$	0	0
$645$	−8.20661e11	−0.186700
$646$	3.01432e12	0.680994
$647$	−5.50408e12	−1.23485	−0.617427	−	0.786628i	$-0.711825\pi$
$647$	−5.50408e12	−1.23485	−0.617427	+	0.786628i	$0.711825\pi$
$648$	−1.76319e11	−0.0392837
$649$	−9.83838e11	−0.217682
$650$	−1.31658e11	−0.0289293
$651$	0	0
$652$	−1.57725e11	−0.0341812
$653$	−6.17678e12	−1.32939	−0.664696	−	0.747114i	$-0.731439\pi$
$653$	−6.17678e12	−1.32939	−0.664696	+	0.747114i	$0.731439\pi$
$654$	1.25095e12	0.267387
$655$	1.42603e12	0.302721
$656$	9.69426e11	0.204384
$657$	1.53135e12	0.320650
$658$	0	0
$659$	−4.17021e12	−0.861338	−0.430669	−	0.902510i	$-0.641722\pi$
$659$	−4.17021e12	−0.861338	−0.430669	+	0.902510i	$0.641722\pi$
$660$	−1.79875e11	−0.0368997
$661$	−3.73958e12	−0.761932	−0.380966	−	0.924589i	$-0.624408\pi$
$661$	−3.73958e12	−0.761932	−0.380966	+	0.924589i	$0.624408\pi$
$662$	−6.61807e11	−0.133928
$663$	−1.01247e11	−0.0203503
$664$	−3.88896e11	−0.0776384
$665$	0	0
$666$	4.79614e11	0.0944621
$667$	1.14563e12	0.224119
$668$	4.12614e12	0.801771
$669$	1.12668e11	0.0217462
$670$	1.77738e12	0.340757
$671$	1.12716e12	0.214652
$672$	0	0
$673$	−8.78180e12	−1.65012	−0.825060	−	0.565044i	$-0.808859\pi$
$673$	−8.78180e12	−1.65012	−0.825060	+	0.565044i	$0.808859\pi$
$674$	5.28357e12	0.986184
$675$	9.19347e11	0.170456
$676$	−2.70896e12	−0.498933
$677$	−1.07791e13	−1.97212	−0.986060	−	0.166392i	$-0.946788\pi$
$677$	−1.07791e13	−1.97212	−0.986060	+	0.166392i	$0.946788\pi$
$678$	2.36913e12	0.430581
$679$	0	0
$680$	−5.08524e11	−0.0912055
$681$	−2.53815e12	−0.452225
$682$	−1.76993e12	−0.313276
$683$	−5.91946e12	−1.04085	−0.520426	−	0.853907i	$-0.674227\pi$
$683$	−5.91946e12	−1.04085	−0.520426	+	0.853907i	$0.674227\pi$
$684$	1.20417e12	0.210346
$685$	1.12759e12	0.195678
$686$	0	0
$687$	3.19544e12	0.547300
$688$	−1.40540e12	−0.239140
$689$	−1.51015e10	−0.00255290
$690$	−1.59088e11	−0.0267188
$691$	−7.43466e12	−1.24054	−0.620268	−	0.784390i	$-0.712976\pi$
$691$	−7.43466e12	−1.24054	−0.620268	+	0.784390i	$0.712976\pi$
$692$	−9.16732e11	−0.151973
$693$	0	0
$694$	−8.53589e12	−1.39679
$695$	1.21107e11	0.0196896
$696$	1.46290e12	0.236305
$697$	−3.88712e12	−0.623851
$698$	1.35358e12	0.215841
$699$	6.78242e12	1.07458
$700$	0	0
$701$	3.36697e12	0.526634	0.263317	−	0.964709i	$-0.415184\pi$
$701$	3.36697e12	0.526634	0.263317	+	0.964709i	$0.415184\pi$
$702$	−4.04463e10	−0.00628581
$703$	−3.27551e12	−0.505801
$704$	−3.08040e11	−0.0472640
$705$	1.45754e12	0.222213
$706$	−2.68503e12	−0.406751
$707$	0	0
$708$	−1.11112e12	−0.166192
$709$	1.17165e13	1.74136	0.870682	−	0.491846i	$-0.163678\pi$
$709$	1.17165e13	1.74136	0.870682	+	0.491846i	$0.163678\pi$
$710$	−2.07443e12	−0.306363
$711$	−3.61465e11	−0.0530461
$712$	−2.86190e12	−0.417344
$713$	−1.56540e12	−0.226841
$714$	0	0
$715$	−4.12619e10	−0.00590435
$716$	−6.48089e12	−0.921566
$717$	8.41430e10	0.0118900
$718$	−1.22704e12	−0.172305
$719$	1.28507e13	1.79328	0.896639	−	0.442763i	$-0.146002\pi$
$719$	1.28507e13	1.79328	0.896639	+	0.442763i	$0.146002\pi$
$720$	−2.03146e11	−0.0281716
$721$	0	0
$722$	−3.06082e12	−0.419199
$723$	−8.43729e12	−1.14837
$724$	−2.25080e12	−0.304448
$725$	−7.62770e12	−1.02535
$726$	−2.61900e12	−0.349882
$727$	1.03159e13	1.36963	0.684817	−	0.728715i	$-0.259882\pi$
$727$	1.03159e13	1.36963	0.684817	+	0.728715i	$0.259882\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	1.76434e12	0.229949
$731$	5.63526e12	0.729938
$732$	1.27298e12	0.163879
$733$	−1.05772e13	−1.35333	−0.676666	−	0.736290i	$-0.736576\pi$
$733$	−1.05772e13	−1.35333	−0.676666	+	0.736290i	$0.736576\pi$
$734$	4.27334e12	0.543419
$735$	0	0
$736$	−2.72443e11	−0.0342236
$737$	−4.31709e12	−0.538998
$738$	−1.55283e12	−0.192695
$739$	−7.25690e12	−0.895058	−0.447529	−	0.894269i	$-0.647696\pi$
$739$	−7.25690e12	−0.895058	−0.447529	+	0.894269i	$0.647696\pi$
$740$	5.52586e11	0.0677418
$741$	2.76226e11	0.0336576
$742$	0	0
$743$	7.60678e12	0.915696	0.457848	−	0.889031i	$-0.348620\pi$
$743$	7.60678e12	0.915696	0.457848	+	0.889031i	$0.348620\pi$
$744$	−1.99892e12	−0.239175
$745$	−2.50139e11	−0.0297494
$746$	−9.12951e12	−1.07925
$747$	6.22935e11	0.0731982
$748$	1.23515e12	0.144266
$749$	0	0
$750$	2.25512e12	0.260252
$751$	−3.30167e11	−0.0378751	−0.0189375	−	0.999821i	$-0.506028\pi$
$751$	−3.30167e11	−0.0378751	−0.0189375	+	0.999821i	$0.506028\pi$
$752$	2.49607e12	0.284627
$753$	−8.36549e12	−0.948230
$754$	3.35577e11	0.0378113
$755$	1.64153e12	0.183861
$756$	0	0
$757$	−9.46308e12	−1.04737	−0.523686	−	0.851911i	$-0.675443\pi$
$757$	−9.46308e12	−1.04737	−0.523686	+	0.851911i	$0.675443\pi$
$758$	−2.63447e11	−0.0289856
$759$	3.86410e11	0.0422630
$760$	1.38738e12	0.150846
$761$	−1.04314e13	−1.12748	−0.563742	−	0.825951i	$-0.690639\pi$
$761$	−1.04314e13	−1.12748	−0.563742	+	0.825951i	$0.690639\pi$
$762$	2.37274e12	0.254949
$763$	0	0
$764$	−5.74888e11	−0.0610468
$765$	8.14556e11	0.0859893
$766$	−4.42811e12	−0.464717
$767$	−2.54882e11	−0.0265926
$768$	−3.47892e11	−0.0360844
$769$	−5.48400e12	−0.565496	−0.282748	−	0.959194i	$-0.591246\pi$
$769$	−5.48400e12	−0.565496	−0.282748	+	0.959194i	$0.591246\pi$
$770$	0	0
$771$	−7.80654e11	−0.0795635
$772$	−6.55392e12	−0.664085
$773$	1.34594e13	1.35587	0.677935	−	0.735122i	$-0.262875\pi$
$773$	1.34594e13	1.35587	0.677935	+	0.735122i	$0.262875\pi$
$774$	2.25118e12	0.225464
$775$	1.04226e13	1.03781
$776$	−6.22616e12	−0.616372
$777$	0	0
$778$	−6.14794e12	−0.601619
$779$	1.06050e13	1.03179
$780$	−4.66001e10	−0.00450776
$781$	5.03858e12	0.484594
$782$	1.09242e12	0.104462
$783$	−2.34328e12	−0.222790
$784$	0	0
$785$	−2.59991e11	−0.0244369
$786$	−3.91179e12	−0.365573
$787$	4.95097e12	0.460049	0.230025	−	0.973185i	$-0.426119\pi$
$787$	4.95097e12	0.460049	0.230025	+	0.973185i	$0.426119\pi$
$788$	−8.13477e12	−0.751583
$789$	1.18686e13	1.09032
$790$	−4.16461e11	−0.0380410
$791$	0	0
$792$	4.93421e11	0.0445609
$793$	2.92013e11	0.0262224
$794$	−5.66004e12	−0.505391
$795$	1.21495e11	0.0107872
$796$	−6.32720e12	−0.558602
$797$	−1.45639e13	−1.27854	−0.639269	−	0.768983i	$-0.720763\pi$
$797$	−1.45639e13	−1.27854	−0.639269	+	0.768983i	$0.720763\pi$
$798$	0	0
$799$	−1.00085e13	−0.868779
$800$	1.81395e12	0.156574
$801$	4.58421e12	0.393476
$802$	5.58702e12	0.476865
$803$	−4.28542e12	−0.363725
$804$	−4.87561e12	−0.411506
$805$	0	0
$806$	−4.58535e11	−0.0382706
$807$	3.36089e12	0.278948
$808$	1.08423e12	0.0894890
$809$	−4.00456e12	−0.328690	−0.164345	−	0.986403i	$-0.552551\pi$
$809$	−4.00456e12	−0.328690	−0.164345	+	0.986403i	$0.552551\pi$
$810$	3.25400e11	0.0265604
$811$	−1.23992e13	−1.00646	−0.503232	−	0.864151i	$-0.667856\pi$
$811$	−1.23992e13	−1.00646	−0.503232	+	0.864151i	$0.667856\pi$
$812$	0	0
$813$	9.68281e12	0.777309
$814$	−1.34218e12	−0.107152
$815$	2.91085e11	0.0231105
$816$	1.39495e12	0.110142
$817$	−1.53744e13	−1.20725
$818$	1.61942e13	1.26465
$819$	0	0
$820$	−1.78909e12	−0.138188
$821$	−1.78994e13	−1.37497	−0.687486	−	0.726198i	$-0.741286\pi$
$821$	−1.78994e13	−1.37497	−0.687486	+	0.726198i	$0.741286\pi$
$822$	−3.09313e12	−0.236306
$823$	7.20978e12	0.547801	0.273900	−	0.961758i	$-0.411686\pi$
$823$	7.20978e12	0.547801	0.273900	+	0.961758i	$0.411686\pi$
$824$	6.21642e12	0.469751
$825$	−2.57275e12	−0.193355
$826$	0	0
$827$	1.78205e13	1.32478	0.662391	−	0.749159i	$-0.269542\pi$
$827$	1.78205e13	1.32478	0.662391	+	0.749159i	$0.269542\pi$
$828$	4.36401e11	0.0322663
$829$	2.01652e12	0.148288	0.0741442	−	0.997248i	$-0.476377\pi$
$829$	2.01652e12	0.148288	0.0741442	+	0.997248i	$0.476377\pi$
$830$	7.17713e11	0.0524928
$831$	5.01177e12	0.364574
$832$	−7.98038e10	−0.00577389
$833$	0	0
$834$	−3.32213e11	−0.0237777
$835$	−7.61487e12	−0.542092
$836$	−3.36980e12	−0.238603
$837$	3.20188e12	0.225497
$838$	−8.42626e12	−0.590251
$839$	−2.75875e13	−1.92214	−0.961068	−	0.276312i	$-0.910888\pi$
$839$	−2.75875e13	−1.92214	−0.961068	+	0.276312i	$0.910888\pi$
$840$	0	0
$841$	4.93474e12	0.340159
$842$	6.81092e11	0.0466983
$843$	9.09554e12	0.620304
$844$	−8.92914e12	−0.605715
$845$	4.99943e12	0.337338
$846$	−3.99822e12	−0.268349
$847$	0	0
$848$	2.08064e11	0.0138171
$849$	9.68057e12	0.639465
$850$	−7.27340e12	−0.477917
$851$	−1.18707e12	−0.0775879
$852$	5.69043e12	0.369971
$853$	−7.06363e12	−0.456833	−0.228416	−	0.973564i	$-0.573355\pi$
$853$	−7.06363e12	−0.456833	−0.228416	+	0.973564i	$0.573355\pi$
$854$	0	0
$855$	−2.22231e12	−0.142219
$856$	−9.19737e12	−0.585506
$857$	2.82446e13	1.78863	0.894317	−	0.447435i	$-0.147662\pi$
$857$	2.82446e13	1.78863	0.894317	+	0.447435i	$0.147662\pi$
$858$	1.13187e11	0.00713023
$859$	−2.30014e13	−1.44140	−0.720702	−	0.693245i	$-0.756180\pi$
$859$	−2.30014e13	−1.44140	−0.720702	+	0.693245i	$0.756180\pi$
$860$	2.59369e12	0.161687
$861$	0	0
$862$	−4.51039e12	−0.278247
$863$	6.34466e12	0.389367	0.194684	−	0.980866i	$-0.437632\pi$
$863$	6.34466e12	0.389367	0.194684	+	0.980866i	$0.437632\pi$
$864$	5.57256e11	0.0340207
$865$	1.69184e12	0.102751
$866$	−8.92825e11	−0.0539431
$867$	4.01228e12	0.241160
$868$	0	0
$869$	1.01154e12	0.0601721
$870$	−2.69980e12	−0.159770
$871$	−1.11843e12	−0.0658454
$872$	−3.95362e12	−0.231564
$873$	9.97311e12	0.581121
$874$	−2.98039e12	−0.172771
$875$	0	0
$876$	−4.83983e12	−0.277691
$877$	2.72979e12	0.155823	0.0779113	−	0.996960i	$-0.475175\pi$
$877$	2.72979e12	0.155823	0.0779113	+	0.996960i	$0.475175\pi$
$878$	−1.21467e13	−0.689814
$879$	−1.43829e13	−0.812638
$880$	5.68493e11	0.0319561
$881$	1.05136e13	0.587974	0.293987	−	0.955809i	$-0.405018\pi$
$881$	1.05136e13	0.587974	0.293987	+	0.955809i	$0.405018\pi$
$882$	0	0
$883$	2.11154e13	1.16890	0.584448	−	0.811431i	$-0.301311\pi$
$883$	2.11154e13	1.16890	0.584448	+	0.811431i	$0.301311\pi$
$884$	3.19990e11	0.0176239
$885$	2.05059e12	0.112366
$886$	2.33402e13	1.27249
$887$	2.73751e13	1.48491	0.742453	−	0.669898i	$-0.233662\pi$
$887$	2.73751e13	1.48491	0.742453	+	0.669898i	$0.233662\pi$
$888$	−1.51582e12	−0.0818066
$889$	0	0
$890$	5.28168e12	0.282174
$891$	−7.90365e11	−0.0420125
$892$	−3.56088e11	−0.0188328
$893$	2.73057e13	1.43688
$894$	6.86165e11	0.0359261
$895$	1.19606e13	0.623088
$896$	0	0
$897$	1.00107e11	0.00516295
$898$	−1.56288e13	−0.802013
$899$	−2.65655e13	−1.35644
$900$	−2.90559e12	−0.147619
$901$	−8.34276e11	−0.0421744
$902$	4.34553e12	0.218581
$903$	0	0
$904$	−7.48762e12	−0.372894
$905$	4.15389e12	0.205843
$906$	−4.50295e12	−0.222034
$907$	1.48687e13	0.729526	0.364763	−	0.931100i	$-0.381150\pi$
$907$	1.48687e13	0.729526	0.364763	+	0.931100i	$0.381150\pi$
$908$	8.02179e12	0.391638
$909$	−1.73672e12	−0.0843710
$910$	0	0
$911$	3.26369e12	0.156992	0.0784958	−	0.996914i	$-0.474988\pi$
$911$	3.26369e12	0.156992	0.0784958	+	0.996914i	$0.474988\pi$
$912$	−3.80576e12	−0.182165
$913$	−1.74325e12	−0.0830314
$914$	7.88097e10	0.00373527
$915$	−2.34931e12	−0.110802
$916$	−1.00992e13	−0.473976
$917$	0	0
$918$	−2.23444e12	−0.103843
$919$	3.86599e13	1.78789	0.893944	−	0.448178i	$-0.147927\pi$
$919$	3.86599e13	1.78789	0.893944	+	0.448178i	$0.147927\pi$
$920$	5.02798e11	0.0231392
$921$	−1.23323e13	−0.564774
$922$	−2.32224e12	−0.105832
$923$	1.30534e12	0.0591992
$924$	0	0
$925$	7.90362e12	0.354967
$926$	1.70620e13	0.762573
$927$	−9.95750e12	−0.442885
$928$	−4.62348e12	−0.204646
$929$	1.32204e13	0.582337	0.291169	−	0.956672i	$-0.405956\pi$
$929$	1.32204e13	0.582337	0.291169	+	0.956672i	$0.405956\pi$
$930$	3.68903e12	0.161711
$931$	0	0
$932$	−2.14358e13	−0.930610
$933$	1.23670e13	0.534312
$934$	−2.28703e13	−0.983356
$935$	−2.27950e12	−0.0975408
$936$	1.27830e11	0.00544367
$937$	4.49241e12	0.190393	0.0951966	−	0.995458i	$-0.469652\pi$
$937$	4.49241e12	0.190393	0.0951966	+	0.995458i	$0.469652\pi$
$938$	0	0
$939$	2.02408e13	0.849635
$940$	−4.60654e12	−0.192442
$941$	−2.52572e13	−1.05010	−0.525052	−	0.851070i	$-0.675954\pi$
$941$	−2.52572e13	−1.05010	−0.525052	+	0.851070i	$0.675954\pi$
$942$	7.13191e11	0.0295105
$943$	3.84336e12	0.158273
$944$	3.51169e12	0.143927
$945$	0	0
$946$	−6.29982e12	−0.255752
$947$	−2.28345e13	−0.922605	−0.461303	−	0.887243i	$-0.652618\pi$
$947$	−2.28345e13	−0.922605	−0.461303	+	0.887243i	$0.652618\pi$
$948$	1.14241e12	0.0459392
$949$	−1.11022e12	−0.0444336
$950$	1.98436e13	0.790433
$951$	1.00027e13	0.396556
$952$	0	0
$953$	2.31698e13	0.909923	0.454961	−	0.890511i	$-0.349653\pi$
$953$	2.31698e13	0.909923	0.454961	+	0.890511i	$0.349653\pi$
$954$	−3.33278e11	−0.0130268
$955$	1.06097e12	0.0412749
$956$	−2.65934e11	−0.0102970
$957$	6.55755e12	0.252719
$958$	−6.06135e11	−0.0232501
$959$	0	0
$960$	6.42041e11	0.0243973
$961$	9.85974e12	0.372915
$962$	−3.47716e11	−0.0130899
$963$	1.47324e13	0.552021
$964$	2.66660e13	0.994515
$965$	1.20954e13	0.449000
$966$	0	0
$967$	3.31795e13	1.22026	0.610128	−	0.792303i	$-0.291118\pi$
$967$	3.31795e13	1.22026	0.610128	+	0.792303i	$0.291118\pi$
$968$	8.27734e12	0.303006
$969$	1.52600e13	0.556030
$970$	1.14905e13	0.416740
$971$	1.08978e13	0.393415	0.196707	−	0.980462i	$-0.436975\pi$
$971$	1.08978e13	0.393415	0.196707	+	0.980462i	$0.436975\pi$
$972$	−8.92617e11	−0.0320750
$973$	0	0
$974$	−3.86098e13	−1.37462
$975$	−6.66520e11	−0.0236207
$976$	−4.02326e12	−0.141923
$977$	5.33310e13	1.87264	0.936320	−	0.351149i	$-0.114209\pi$
$977$	5.33310e13	1.87264	0.936320	+	0.351149i	$0.114209\pi$
$978$	−7.98484e11	−0.0279088
$979$	−1.28287e13	−0.446334
$980$	0	0
$981$	6.33293e12	0.218320
$982$	1.34582e13	0.461832
$983$	2.91131e13	0.994484	0.497242	−	0.867612i	$-0.334346\pi$
$983$	2.91131e13	0.994484	0.497242	+	0.867612i	$0.334346\pi$
$984$	4.90772e12	0.166879
$985$	1.50128e13	0.508159
$986$	1.85388e13	0.624649
$987$	0	0
$988$	−8.73012e11	−0.0291484
$989$	−5.57181e12	−0.185188
$990$	−9.10617e11	−0.0301285
$991$	3.38220e13	1.11396	0.556978	−	0.830527i	$-0.311961\pi$
$991$	3.38220e13	1.11396	0.556978	+	0.830527i	$0.311961\pi$
$992$	6.31756e12	0.207132
$993$	−3.35040e12	−0.109351
$994$	0	0
$995$	1.16769e13	0.377681
$996$	−1.96878e12	−0.0633915
$997$	−2.18098e13	−0.699074	−0.349537	−	0.936923i	$-0.613661\pi$
$997$	−2.18098e13	−0.699074	−0.349537	+	0.936923i	$0.613661\pi$
$998$	2.21396e13	0.706451
$999$	2.42804e12	0.0771280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.10.a.r.1.2		3
7.2	even	3		42.10.e.d.25.2	✓	6
7.4	even	3		42.10.e.d.37.2	yes	6
7.6	odd	2		294.10.a.u.1.2		3
21.2	odd	6		126.10.g.d.109.2		6
21.11	odd	6		126.10.g.d.37.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.10.e.d.25.2	✓	6	7.2	even	3
42.10.e.d.37.2	yes	6	7.4	even	3
126.10.g.d.37.2		6	21.11	odd	6
126.10.g.d.109.2		6	21.2	odd	6
294.10.a.r.1.2		3	1.1	even	1	trivial
294.10.a.u.1.2		3	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

\(f(q)\)	\(=\)	\(q-16.0000 q^{2} -81.0000 q^{3} +256.000 q^{4} -472.452 q^{5} +1296.00 q^{6} -4096.00 q^{8} +6561.00 q^{9} +7559.24 q^{10} -18360.6 q^{11} -20736.0 q^{12} -4756.67 q^{13} +38268.6 q^{15} +65536.0 q^{16} -262780. q^{17} -104976. q^{18} +716930. q^{19} -120948. q^{20} +293770. q^{22} +259822. q^{23} +331776. q^{24} -1.72991e6 q^{25} +76106.8 q^{26} -531441. q^{27} +4.40930e6 q^{29} -612298. q^{30} -6.02490e6 q^{31} -1.04858e6 q^{32} +1.48721e6 q^{33} +4.20449e6 q^{34} +1.67962e6 q^{36} -4.56879e6 q^{37} -1.14709e7 q^{38} +385291. q^{39} +1.93516e6 q^{40} +1.47923e7 q^{41} -2.14447e7 q^{43} -4.70032e6 q^{44} -3.09976e6 q^{45} -4.15715e6 q^{46} +3.80870e7 q^{47} -5.30842e6 q^{48} +2.76786e7 q^{50} +2.12852e7 q^{51} -1.21771e6 q^{52} +3.17480e6 q^{53} +8.50306e6 q^{54} +8.67452e6 q^{55} -5.80713e7 q^{57} -7.05487e7 q^{58} +5.35841e7 q^{59} +9.79677e6 q^{60} -6.13901e7 q^{61} +9.63983e7 q^{62} +1.67772e7 q^{64} +2.24730e6 q^{65} -2.37954e7 q^{66} +2.35128e8 q^{67} -6.72718e7 q^{68} -2.10456e7 q^{69} -2.74423e8 q^{71} -2.68739e7 q^{72} +2.33403e8 q^{73} +7.31007e7 q^{74} +1.40123e8 q^{75} +1.83534e8 q^{76} -6.16465e6 q^{78} -5.50930e7 q^{79} -3.09626e7 q^{80} +4.30467e7 q^{81} -2.36676e8 q^{82} +9.49452e7 q^{83} +1.24151e8 q^{85} +3.43116e8 q^{86} -3.57153e8 q^{87} +7.52051e7 q^{88} +6.98706e8 q^{89} +4.95962e7 q^{90} +6.65144e7 q^{92} +4.88017e8 q^{93} -6.09392e8 q^{94} -3.38715e8 q^{95} +8.49347e7 q^{96} +1.52006e9 q^{97} -1.20464e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 48 q^{2} - 243 q^{3} + 768 q^{4} - 361 q^{5} + 3888 q^{6} - 12288 q^{8} + 19683 q^{9} + 5776 q^{10} - 37799 q^{11} - 62208 q^{12} - 220586 q^{13} + 29241 q^{15} + 196608 q^{16} + 781816 q^{17} - 314928 q^{18}+ \cdots - 247999239 q^{99}+O(q^{100}) \) 3 * q - 48 * q^2 - 243 * q^3 + 768 * q^4 - 361 * q^5 + 3888 * q^6 - 12288 * q^8 + 19683 * q^9 + 5776 * q^10 - 37799 * q^11 - 62208 * q^12 - 220586 * q^13 + 29241 * q^15 + 196608 * q^16 + 781816 * q^17 - 314928 * q^18 + 620154 * q^19 - 92416 * q^20 + 604784 * q^22 - 1204784 * q^23 + 995328 * q^24 + 4020946 * q^25 + 3529376 * q^26 - 1594323 * q^27 - 5280535 * q^29 - 467856 * q^30 - 12838457 * q^31 - 3145728 * q^32 + 3061719 * q^33 - 12509056 * q^34 + 5038848 * q^36 - 8014770 * q^37 - 9922464 * q^38 + 17867466 * q^39 + 1478656 * q^40 + 6503028 * q^41 - 22543488 * q^43 - 9676544 * q^44 - 2368521 * q^45 + 19276544 * q^46 + 21473082 * q^47 - 15925248 * q^48 - 64335136 * q^50 - 63327096 * q^51 - 56470016 * q^52 - 13685715 * q^53 + 25509168 * q^54 + 305107391 * q^55 - 50232474 * q^57 + 84488560 * q^58 + 92528141 * q^59 + 7485696 * q^60 + 7516114 * q^61 + 205415312 * q^62 + 50331648 * q^64 - 79637616 * q^65 - 48987504 * q^66 + 137325404 * q^67 + 200144896 * q^68 + 97587504 * q^69 + 8190346 * q^71 - 80621568 * q^72 - 394016164 * q^73 + 128236320 * q^74 - 325696626 * q^75 + 158759424 * q^76 - 285879456 * q^78 - 35090117 * q^79 - 23658496 * q^80 + 129140163 * q^81 - 104048448 * q^82 + 956948005 * q^83 - 411899320 * q^85 + 360695808 * q^86 + 427723335 * q^87 + 154824704 * q^88 + 855499686 * q^89 + 37896336 * q^90 - 308424704 * q^92 + 1039915017 * q^93 - 343569312 * q^94 + 2705508088 * q^95 + 254803968 * q^96 + 2010569795 * q^97 - 247999239 * q^99

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