LMFDB - Embedded newform 169.8.a.d.1.3

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			1.3
Root			$-2.43912$ 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)

$f(q)$	$=$	\(q-2.43912 q^{2} -51.4496 q^{3} -122.051 q^{4} +488.312 q^{5} +125.492 q^{6} -616.243 q^{7} +609.904 q^{8} +460.056 q^{9} -1191.05 q^{10} -4625.43 q^{11} +6279.45 q^{12} +1503.09 q^{14} -25123.4 q^{15} +14134.9 q^{16} +19536.2 q^{17} -1122.13 q^{18} +20077.3 q^{19} -59598.8 q^{20} +31705.4 q^{21} +11282.0 q^{22} -71368.5 q^{23} -31379.3 q^{24} +160323. q^{25} +88850.5 q^{27} +75212.8 q^{28} +138095. q^{29} +61279.1 q^{30} -18285.7 q^{31} -112544. q^{32} +237976. q^{33} -47651.3 q^{34} -300918. q^{35} -56150.2 q^{36} +446035. q^{37} -48970.9 q^{38} +297823. q^{40} +101639. q^{41} -77333.4 q^{42} -103690. q^{43} +564537. q^{44} +224651. q^{45} +174076. q^{46} -445129. q^{47} -727232. q^{48} -443788. q^{49} -391048. q^{50} -1.00513e6 q^{51} +203578. q^{53} -216717. q^{54} -2.25865e6 q^{55} -375849. q^{56} -1.03297e6 q^{57} -336832. q^{58} -2.50176e6 q^{59} +3.06633e6 q^{60} -792320. q^{61} +44601.1 q^{62} -283506. q^{63} -1.53475e6 q^{64} -580454. q^{66} -3.17309e6 q^{67} -2.38441e6 q^{68} +3.67188e6 q^{69} +733977. q^{70} +1.85875e6 q^{71} +280590. q^{72} -4.25734e6 q^{73} -1.08793e6 q^{74} -8.24856e6 q^{75} -2.45045e6 q^{76} +2.85039e6 q^{77} +2.38970e6 q^{79} +6.90221e6 q^{80} -5.57746e6 q^{81} -247910. q^{82} -1.12536e6 q^{83} -3.86967e6 q^{84} +9.53976e6 q^{85} +252913. q^{86} -7.10495e6 q^{87} -2.82107e6 q^{88} +8.00227e6 q^{89} -547951. q^{90} +8.71057e6 q^{92} +940791. q^{93} +1.08572e6 q^{94} +9.80397e6 q^{95} +5.79036e6 q^{96} -276982. q^{97} +1.08245e6 q^{98} -2.12796e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.43912	−0.215590	−0.107795	−	0.994173i	$-0.534379\pi$
$2$	−2.43912	−0.215590	−0.107795	+	0.994173i	$0.534379\pi$
$3$	−51.4496	−1.10016	−0.550082	−	0.835111i	$-0.685403\pi$
$3$	−51.4496	−1.10016	−0.550082	+	0.835111i	$0.685403\pi$
$4$	−122.051	−0.953521
$5$	488.312	1.74704	0.873518	−	0.486791i	$-0.161833\pi$
$5$	488.312	1.74704	0.873518	+	0.486791i	$0.161833\pi$
$6$	125.492	0.237184
$7$	−616.243	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$7$	−616.243	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$8$	609.904	0.421160
$9$	460.056	0.210360
$10$	−1191.05	−0.376644
$11$	−4625.43	−1.04780	−0.523899	−	0.851780i	$-0.675523\pi$
$11$	−4625.43	−1.04780	−0.523899	+	0.851780i	$0.675523\pi$
$12$	6279.45	1.04903
$13$	0	0
$13$	0	0
$14$	1503.09	0.146399
$15$	−25123.4	−1.92203
$16$	14134.9	0.862723
$17$	19536.2	0.964427	0.482214	−	0.876054i	$-0.339833\pi$
$17$	19536.2	0.964427	0.482214	+	0.876054i	$0.339833\pi$
$18$	−1122.13	−0.0453514
$19$	20077.3	0.671533	0.335766	−	0.941945i	$-0.391005\pi$
$19$	20077.3	0.671533	0.335766	+	0.941945i	$0.391005\pi$
$20$	−59598.8	−1.66584
$21$	31705.4	0.747078
$22$	11282.0	0.225895
$23$	−71368.5	−1.22309	−0.611546	−	0.791209i	$-0.709452\pi$
$23$	−71368.5	−1.22309	−0.611546	+	0.791209i	$0.709452\pi$
$24$	−31379.3	−0.463344
$25$	160323.	2.05214
$26$	0	0
$27$	88850.5	0.868734
$28$	75212.8	0.647498
$29$	138095.	1.05144	0.525722	−	0.850656i	$-0.323795\pi$
$29$	138095.	1.05144	0.525722	+	0.850656i	$0.323795\pi$
$30$	61279.1	0.414370
$31$	−18285.7	−0.110242	−0.0551208	−	0.998480i	$-0.517554\pi$
$31$	−18285.7	−0.110242	−0.0551208	+	0.998480i	$0.517554\pi$
$32$	−112544.	−0.607154
$33$	237976.	1.15275
$34$	−47651.3	−0.207921
$35$	−300918.	−1.18634
$36$	−56150.2	−0.200582
$37$	446035.	1.44765	0.723824	−	0.689985i	$-0.242383\pi$
$37$	446035.	1.44765	0.723824	+	0.689985i	$0.242383\pi$
$38$	−48970.9	−0.144776
$39$	0	0
$40$	297823.	0.735781
$41$	101639.	0.230312	0.115156	−	0.993347i	$-0.463263\pi$
$41$	101639.	0.230312	0.115156	+	0.993347i	$0.463263\pi$
$42$	−77333.4	−0.161062
$43$	−103690.	−0.198883	−0.0994416	−	0.995043i	$-0.531706\pi$
$43$	−103690.	−0.198883	−0.0994416	+	0.995043i	$0.531706\pi$
$44$	564537.	0.999098
$45$	224651.	0.367506
$46$	174076.	0.263686
$47$	−445129.	−0.625379	−0.312690	−	0.949855i	$-0.601230\pi$
$47$	−445129.	−0.625379	−0.312690	+	0.949855i	$0.601230\pi$
$48$	−727232.	−0.949136
$49$	−443788.	−0.538877
$50$	−391048.	−0.442420
$51$	−1.00513e6	−1.06103
$52$	0	0
$53$	203578.	0.187830	0.0939149	−	0.995580i	$-0.470062\pi$
$53$	203578.	0.187830	0.0939149	+	0.995580i	$0.470062\pi$
$54$	−216717.	−0.187290
$55$	−2.25865e6	−1.83054
$56$	−375849.	−0.285993
$57$	−1.03297e6	−0.738796
$58$	−336832.	−0.226681
$59$	−2.50176e6	−1.58585	−0.792927	−	0.609317i	$-0.791444\pi$
$59$	−2.50176e6	−1.58585	−0.792927	+	0.609317i	$0.791444\pi$
$60$	3.06633e6	1.83269
$61$	−792320.	−0.446937	−0.223468	−	0.974711i	$-0.571738\pi$
$61$	−792320.	−0.446937	−0.223468	+	0.974711i	$0.571738\pi$
$62$	44601.1	0.0237670
$63$	−283506.	−0.142847
$64$	−1.53475e6	−0.731827
$65$	0	0
$66$	−580454.	−0.248521
$67$	−3.17309e6	−1.28890	−0.644452	−	0.764645i	$-0.722915\pi$
$67$	−3.17309e6	−1.28890	−0.644452	+	0.764645i	$0.722915\pi$
$68$	−2.38441e6	−0.919601
$69$	3.67188e6	1.34560
$70$	733977.	0.255764
$71$	1.85875e6	0.616333	0.308167	−	0.951332i	$-0.400285\pi$
$71$	1.85875e6	0.616333	0.308167	+	0.951332i	$0.400285\pi$
$72$	280590.	0.0885950
$73$	−4.25734e6	−1.28088	−0.640440	−	0.768008i	$-0.721248\pi$
$73$	−4.25734e6	−1.28088	−0.640440	+	0.768008i	$0.721248\pi$
$74$	−1.08793e6	−0.312098
$75$	−8.24856e6	−2.25769
$76$	−2.45045e6	−0.640320
$77$	2.85039e6	0.711519
$78$	0	0
$79$	2.38970e6	0.545317	0.272658	−	0.962111i	$-0.412097\pi$
$79$	2.38970e6	0.545317	0.272658	+	0.962111i	$0.412097\pi$
$80$	6.90221e6	1.50721
$81$	−5.57746e6	−1.16611
$82$	−247910.	−0.0496530
$83$	−1.12536e6	−0.216032	−0.108016	−	0.994149i	$-0.534450\pi$
$83$	−1.12536e6	−0.216032	−0.108016	+	0.994149i	$0.534450\pi$
$84$	−3.86967e6	−0.712354
$85$	9.53976e6	1.68489
$86$	252913.	0.0428773
$87$	−7.10495e6	−1.15676
$88$	−2.82107e6	−0.441291
$89$	8.00227e6	1.20323	0.601614	−	0.798787i	$-0.294524\pi$
$89$	8.00227e6	1.20323	0.601614	+	0.798787i	$0.294524\pi$
$90$	−547951.	−0.0792306
$91$	0	0
$92$	8.71057e6	1.16624
$93$	940791.	0.121284
$94$	1.08572e6	0.134826
$95$	9.80397e6	1.17319
$96$	5.79036e6	0.667969
$97$	−276982.	−0.0308141	−0.0154071	−	0.999881i	$-0.504904\pi$
$97$	−276982.	−0.0308141	−0.0154071	+	0.999881i	$0.504904\pi$
$98$	1.08245e6	0.116176
$99$	−2.12796e6	−0.220414
$100$	−1.95676e7	−1.95676
$101$	−1.32796e7	−1.28251	−0.641256	−	0.767327i	$-0.721586\pi$
$101$	−1.32796e7	−1.28251	−0.641256	+	0.767327i	$0.721586\pi$
$102$	2.45164e6	0.228747
$103$	7.07119e6	0.637621	0.318810	−	0.947819i	$-0.396717\pi$
$103$	7.07119e6	0.637621	0.318810	+	0.947819i	$0.396717\pi$
$104$	0	0
$105$	1.54821e7	1.30517
$106$	−496551.	−0.0404942
$107$	2.81264e6	0.221958	0.110979	−	0.993823i	$-0.464601\pi$
$107$	2.81264e6	0.221958	0.110979	+	0.993823i	$0.464601\pi$
$108$	−1.08443e7	−0.828356
$109$	−1.77688e7	−1.31421	−0.657107	−	0.753798i	$-0.728220\pi$
$109$	−1.77688e7	−1.31421	−0.657107	+	0.753798i	$0.728220\pi$
$110$	5.50913e6	0.394647
$111$	−2.29483e7	−1.59265
$112$	−8.71050e6	−0.585841
$113$	−1.79731e7	−1.17179	−0.585894	−	0.810388i	$-0.699257\pi$
$113$	−1.79731e7	−1.17179	−0.585894	+	0.810388i	$0.699257\pi$
$114$	2.51953e6	0.159277
$115$	−3.48501e7	−2.13679
$116$	−1.68546e7	−1.00257
$117$	0	0
$118$	6.10209e6	0.341894
$119$	−1.20391e7	−0.654904
$120$	−1.53229e7	−0.809480
$121$	1.90744e6	0.0978819
$122$	1.93256e6	0.0963551
$123$	−5.22928e6	−0.253381
$124$	2.23178e6	0.105118
$125$	4.01383e7	1.83812
$126$	691507.	0.0307964
$127$	−2.01354e7	−0.872263	−0.436131	−	0.899883i	$-0.643652\pi$
$127$	−2.01354e7	−0.872263	−0.436131	+	0.899883i	$0.643652\pi$
$128$	1.81491e7	0.764929
$129$	5.33482e6	0.218804
$130$	0	0
$131$	−2.41394e7	−0.938159	−0.469079	−	0.883156i	$-0.655414\pi$
$131$	−2.41394e7	−0.938159	−0.469079	+	0.883156i	$0.655414\pi$
$132$	−2.90452e7	−1.09917
$133$	−1.23725e7	−0.456011
$134$	7.73956e6	0.277875
$135$	4.33867e7	1.51771
$136$	1.19152e7	0.406178
$137$	8.35854e6	0.277721	0.138860	−	0.990312i	$-0.455656\pi$
$137$	8.35854e6	0.277721	0.138860	+	0.990312i	$0.455656\pi$
$138$	−8.95616e6	−0.290098
$139$	4.59389e7	1.45087	0.725435	−	0.688290i	$-0.241639\pi$
$139$	4.59389e7	1.45087	0.725435	+	0.688290i	$0.241639\pi$
$140$	3.67273e7	1.13120
$141$	2.29017e7	0.688020
$142$	−4.53371e6	−0.132875
$143$	0	0
$144$	6.50283e6	0.181482
$145$	6.74336e7	1.83691
$146$	1.03842e7	0.276145
$147$	2.28327e7	0.592852
$148$	−5.44389e7	−1.38036
$149$	−7.52366e6	−0.186328	−0.0931638	−	0.995651i	$-0.529698\pi$
$149$	−7.52366e6	−0.186328	−0.0931638	+	0.995651i	$0.529698\pi$
$150$	2.01192e7	0.486735
$151$	−5.79965e7	−1.37083	−0.685413	−	0.728155i	$-0.740378\pi$
$151$	−5.79965e7	−1.37083	−0.685413	+	0.728155i	$0.740378\pi$
$152$	1.22452e7	0.282822
$153$	8.98776e6	0.202876
$154$	−6.95244e6	−0.153396
$155$	−8.92912e6	−0.192596
$156$	0	0
$157$	−8.29958e7	−1.71162	−0.855810	−	0.517291i	$-0.826941\pi$
$157$	−8.29958e7	−1.71162	−0.855810	+	0.517291i	$0.826941\pi$
$158$	−5.82878e6	−0.117565
$159$	−1.04740e7	−0.206643
$160$	−5.49567e7	−1.06072
$161$	4.39803e7	0.830553
$162$	1.36041e7	0.251401
$163$	3.36561e7	0.608705	0.304352	−	0.952560i	$-0.401560\pi$
$163$	3.36561e7	0.608705	0.304352	+	0.952560i	$0.401560\pi$
$164$	−1.24051e7	−0.219608
$165$	1.16207e8	2.01390
$166$	2.74489e6	0.0465743
$167$	7.76344e7	1.28987	0.644936	−	0.764237i	$-0.276884\pi$
$167$	7.76344e7	1.28987	0.644936	+	0.764237i	$0.276884\pi$
$168$	1.93373e7	0.314639
$169$	0	0
$170$	−2.32687e7	−0.363245
$171$	9.23668e6	0.141263
$172$	1.26555e7	0.189639
$173$	668606.	0.00981768	0.00490884	−	0.999988i	$-0.498437\pi$
$173$	668606.	0.00981768	0.00490884	+	0.999988i	$0.498437\pi$
$174$	1.73298e7	0.249386
$175$	−9.87980e7	−1.39353
$176$	−6.53798e7	−0.903960
$177$	1.28714e8	1.74470
$178$	−1.95185e7	−0.259404
$179$	1.05607e8	1.37628	0.688142	−	0.725576i	$-0.258427\pi$
$179$	1.05607e8	1.37628	0.688142	+	0.725576i	$0.258427\pi$
$180$	−2.74188e7	−0.350425
$181$	1.42306e7	0.178381	0.0891904	−	0.996015i	$-0.471572\pi$
$181$	1.42306e7	0.178381	0.0891904	+	0.996015i	$0.471572\pi$
$182$	0	0
$183$	4.07645e7	0.491703
$184$	−4.35279e7	−0.515117
$185$	2.17804e8	2.52909
$186$	−2.29470e6	−0.0261476
$187$	−9.03635e7	−1.01053
$188$	5.43283e7	0.596312
$189$	−5.47534e7	−0.589923
$190$	−2.39131e7	−0.252929
$191$	−6.06460e7	−0.629775	−0.314887	−	0.949129i	$-0.601967\pi$
$191$	−6.06460e7	−0.629775	−0.314887	+	0.949129i	$0.601967\pi$
$192$	7.89623e7	0.805129
$193$	−1.96802e8	−1.97051	−0.985253	−	0.171102i	$-0.945267\pi$
$193$	−1.96802e8	−1.97051	−0.985253	+	0.171102i	$0.945267\pi$
$194$	675593.	0.00664322
$195$	0	0
$196$	5.41646e7	0.513830
$197$	−7.08567e7	−0.660312	−0.330156	−	0.943926i	$-0.607101\pi$
$197$	−7.08567e7	−0.660312	−0.330156	+	0.943926i	$0.607101\pi$
$198$	5.19035e6	0.0475192
$199$	−1.26370e8	−1.13674	−0.568368	−	0.822774i	$-0.692425\pi$
$199$	−1.26370e8	−1.13674	−0.568368	+	0.822774i	$0.692425\pi$
$200$	9.77818e7	0.864277
$201$	1.63254e8	1.41801
$202$	3.23907e7	0.276497
$203$	−8.51003e7	−0.713995
$204$	1.22677e8	1.01171
$205$	4.96315e7	0.402364
$206$	−1.72475e7	−0.137465
$207$	−3.28335e7	−0.257289
$208$	0	0
$209$	−9.28660e7	−0.703631
$210$	−3.77628e7	−0.281382
$211$	1.08405e8	0.794440	0.397220	−	0.917723i	$-0.369975\pi$
$211$	1.08405e8	0.794440	0.397220	+	0.917723i	$0.369975\pi$
$212$	−2.48468e7	−0.179100
$213$	−9.56316e7	−0.678067
$214$	−6.86038e6	−0.0478520
$215$	−5.06332e7	−0.347456
$216$	5.41903e7	0.365876
$217$	1.12684e7	0.0748607
$218$	4.33403e7	0.283331
$219$	2.19038e8	1.40918
$220$	2.75670e8	1.74546
$221$	0	0
$222$	5.59737e7	0.343359
$223$	4.34061e7	0.262110	0.131055	−	0.991375i	$-0.458164\pi$
$223$	4.34061e7	0.262110	0.131055	+	0.991375i	$0.458164\pi$
$224$	6.93546e7	0.412294
$225$	7.37577e7	0.431687
$226$	4.38387e7	0.252626
$227$	−1.30814e8	−0.742272	−0.371136	−	0.928579i	$-0.621032\pi$
$227$	−1.30814e8	−0.742272	−0.371136	+	0.928579i	$0.621032\pi$
$228$	1.26074e8	0.704457
$229$	−1.82131e7	−0.100221	−0.0501105	−	0.998744i	$-0.515957\pi$
$229$	−1.82131e7	−0.100221	−0.0501105	+	0.998744i	$0.515957\pi$
$230$	8.50036e7	0.460670
$231$	−1.46651e8	−0.782787
$232$	8.42250e7	0.442826
$233$	−1.76958e8	−0.916482	−0.458241	−	0.888828i	$-0.651520\pi$
$233$	−1.76958e8	−0.916482	−0.458241	+	0.888828i	$0.651520\pi$
$234$	0	0
$235$	−2.17362e8	−1.09256
$236$	3.05341e8	1.51214
$237$	−1.22949e8	−0.599937
$238$	2.93647e7	0.141191
$239$	−7.32549e7	−0.347092	−0.173546	−	0.984826i	$-0.555522\pi$
$239$	−7.32549e7	−0.347092	−0.173546	+	0.984826i	$0.555522\pi$
$240$	−3.55116e8	−1.65818
$241$	2.20158e8	1.01315	0.506576	−	0.862195i	$-0.330911\pi$
$241$	2.20158e8	1.01315	0.506576	+	0.862195i	$0.330911\pi$
$242$	−4.65248e6	−0.0211024
$243$	9.26418e7	0.414176
$244$	9.67031e7	0.426163
$245$	−2.16707e8	−0.941437
$246$	1.27549e7	0.0546264
$247$	0	0
$248$	−1.11525e7	−0.0464293
$249$	5.78992e7	0.237670
$250$	−9.79024e7	−0.396281
$251$	1.29959e8	0.518738	0.259369	−	0.965778i	$-0.416485\pi$
$251$	1.29959e8	0.518738	0.259369	+	0.965778i	$0.416485\pi$
$252$	3.46021e7	0.136207
$253$	3.30110e8	1.28155
$254$	4.91127e7	0.188051
$255$	−4.90817e8	−1.85365
$256$	1.52180e8	0.566916
$257$	−1.11340e6	−0.00409153	−0.00204577	−	0.999998i	$-0.500651\pi$
$257$	−1.11340e6	−0.00409153	−0.00204577	+	0.999998i	$0.500651\pi$
$258$	−1.30123e7	−0.0471720
$259$	−2.74866e8	−0.983040
$260$	0	0
$261$	6.35317e7	0.221181
$262$	5.88789e7	0.202258
$263$	−3.02037e8	−1.02380	−0.511900	−	0.859045i	$-0.671058\pi$
$263$	−3.02037e8	−1.02380	−0.511900	+	0.859045i	$0.671058\pi$
$264$	1.45143e8	0.485492
$265$	9.94093e7	0.328145
$266$	3.01780e7	0.0983115
$267$	−4.11713e8	−1.32375
$268$	3.87278e8	1.22900
$269$	−2.42963e8	−0.761040	−0.380520	−	0.924773i	$-0.624255\pi$
$269$	−2.42963e8	−0.761040	−0.380520	+	0.924773i	$0.624255\pi$
$270$	−1.05826e8	−0.327203
$271$	3.31493e8	1.01177	0.505886	−	0.862601i	$-0.331166\pi$
$271$	3.31493e8	1.01177	0.505886	+	0.862601i	$0.331166\pi$
$272$	2.76142e8	0.832034
$273$	0	0
$274$	−2.03875e7	−0.0598738
$275$	−7.41564e8	−2.15023
$276$	−4.48155e8	−1.28306
$277$	−2.92754e8	−0.827607	−0.413803	−	0.910366i	$-0.635800\pi$
$277$	−2.92754e8	−0.827607	−0.413803	+	0.910366i	$0.635800\pi$
$278$	−1.12051e8	−0.312793
$279$	−8.41245e6	−0.0231904
$280$	−1.83531e8	−0.499640
$281$	4.81806e8	1.29539	0.647694	−	0.761901i	$-0.275734\pi$
$281$	4.81806e8	1.29539	0.647694	+	0.761901i	$0.275734\pi$
$282$	−5.58601e7	−0.148330
$283$	−6.21029e8	−1.62877	−0.814385	−	0.580326i	$-0.802925\pi$
$283$	−6.21029e8	−1.62877	−0.814385	+	0.580326i	$0.802925\pi$
$284$	−2.26861e8	−0.587687
$285$	−5.04410e8	−1.29070
$286$	0	0
$287$	−6.26343e7	−0.156396
$288$	−5.17768e7	−0.127721
$289$	−2.86746e7	−0.0698804
$290$	−1.64479e8	−0.396020
$291$	1.42506e7	0.0339006
$292$	5.19612e8	1.22135
$293$	2.25288e8	0.523240	0.261620	−	0.965171i	$-0.415743\pi$
$293$	2.25288e8	0.523240	0.261620	+	0.965171i	$0.415743\pi$
$294$	−5.56918e7	−0.127813
$295$	−1.22164e9	−2.77054
$296$	2.72039e8	0.609691
$297$	−4.10972e8	−0.910258
$298$	1.83511e7	0.0401704
$299$	0	0
$300$	1.00674e9	2.15275
$301$	6.38983e7	0.135054
$302$	1.41460e8	0.295536
$303$	6.83232e8	1.41097
$304$	2.83789e8	0.579347
$305$	−3.86899e8	−0.780815
$306$	−2.19223e7	−0.0437381
$307$	1.94968e8	0.384572	0.192286	−	0.981339i	$-0.438410\pi$
$307$	1.94968e8	0.384572	0.192286	+	0.981339i	$0.438410\pi$
$308$	−3.47892e8	−0.678448
$309$	−3.63810e8	−0.701487
$310$	2.17792e7	0.0415218
$311$	−3.23645e8	−0.610110	−0.305055	−	0.952335i	$-0.598675\pi$
$311$	−3.23645e8	−0.610110	−0.305055	+	0.952335i	$0.598675\pi$
$312$	0	0
$313$	1.52942e8	0.281917	0.140959	−	0.990016i	$-0.454982\pi$
$313$	1.52942e8	0.281917	0.140959	+	0.990016i	$0.454982\pi$
$314$	2.02437e8	0.369008
$315$	−1.38439e8	−0.249559
$316$	−2.91665e8	−0.519971
$317$	−1.05459e8	−0.185941	−0.0929707	−	0.995669i	$-0.529636\pi$
$317$	−1.05459e8	−0.185941	−0.0929707	+	0.995669i	$0.529636\pi$
$318$	2.55473e7	0.0445503
$319$	−6.38751e8	−1.10170
$320$	−7.49437e8	−1.27853
$321$	−1.44709e8	−0.244190
$322$	−1.07273e8	−0.179059
$323$	3.92234e8	0.647644
$324$	6.80733e8	1.11191
$325$	0	0
$326$	−8.20913e7	−0.131231
$327$	9.14198e8	1.44585
$328$	6.19901e7	0.0969982
$329$	2.74308e8	0.424670
$330$	−2.83442e8	−0.434176
$331$	−6.53937e8	−0.991146	−0.495573	−	0.868566i	$-0.665042\pi$
$331$	−6.53937e8	−0.991146	−0.495573	+	0.868566i	$0.665042\pi$
$332$	1.37351e8	0.205991
$333$	2.05201e8	0.304526
$334$	−1.89360e8	−0.278083
$335$	−1.54946e9	−2.25176
$336$	4.48151e8	0.644521
$337$	6.58782e8	0.937642	0.468821	−	0.883293i	$-0.344679\pi$
$337$	6.58782e8	0.937642	0.468821	+	0.883293i	$0.344679\pi$
$338$	0	0
$339$	9.24709e8	1.28916
$340$	−1.16433e9	−1.60658
$341$	8.45792e7	0.115511
$342$	−2.25294e7	−0.0304550
$343$	7.80983e8	1.04499
$344$	−6.32411e7	−0.0837616
$345$	1.79302e9	2.35081
$346$	−1.63081e6	−0.00211659
$347$	−3.76864e8	−0.484208	−0.242104	−	0.970250i	$-0.577837\pi$
$347$	−3.76864e8	−0.484208	−0.242104	+	0.970250i	$0.577837\pi$
$348$	8.67164e8	1.10300
$349$	7.32436e8	0.922318	0.461159	−	0.887317i	$-0.347434\pi$
$349$	7.32436e8	0.922318	0.461159	+	0.887317i	$0.347434\pi$
$350$	2.40980e8	0.300430
$351$	0	0
$352$	5.20566e8	0.636175
$353$	1.08614e9	1.31424	0.657118	−	0.753788i	$-0.271775\pi$
$353$	1.08614e9	1.31424	0.657118	+	0.753788i	$0.271775\pi$
$354$	−3.13950e8	−0.376140
$355$	9.07647e8	1.07676
$356$	−9.76682e8	−1.14730
$357$	6.19404e8	0.720502
$358$	−2.57589e8	−0.296713
$359$	−6.86701e8	−0.783316	−0.391658	−	0.920111i	$-0.628098\pi$
$359$	−6.86701e8	−0.783316	−0.391658	+	0.920111i	$0.628098\pi$
$360$	1.37016e8	0.154779
$361$	−4.90775e8	−0.549044
$362$	−3.47102e7	−0.0384571
$363$	−9.81370e7	−0.107686
$364$	0	0
$365$	−2.07891e9	−2.23775
$366$	−9.94296e7	−0.106006
$367$	4.74836e8	0.501432	0.250716	−	0.968061i	$-0.419334\pi$
$367$	4.74836e8	0.501432	0.250716	+	0.968061i	$0.419334\pi$
$368$	−1.00878e9	−1.05519
$369$	4.67597e7	0.0484484
$370$	−5.31251e8	−0.545247
$371$	−1.25453e8	−0.127548
$372$	−1.14824e8	−0.115647
$373$	1.55637e9	1.55286	0.776431	−	0.630202i	$-0.217028\pi$
$373$	1.55637e9	1.55286	0.776431	+	0.630202i	$0.217028\pi$
$374$	2.20408e8	0.217859
$375$	−2.06510e9	−2.02224
$376$	−2.71486e8	−0.263385
$377$	0	0
$378$	1.33550e8	0.127181
$379$	1.39184e8	0.131326	0.0656631	−	0.997842i	$-0.479084\pi$
$379$	1.39184e8	0.131326	0.0656631	+	0.997842i	$0.479084\pi$
$380$	−1.19658e9	−1.11866
$381$	1.03596e9	0.959631
$382$	1.47923e8	0.135773
$383$	9.91073e8	0.901385	0.450692	−	0.892679i	$-0.351177\pi$
$383$	9.91073e8	0.901385	0.450692	+	0.892679i	$0.351177\pi$
$384$	−9.33765e8	−0.841547
$385$	1.39188e9	1.24305
$386$	4.80023e8	0.424822
$387$	−4.77034e7	−0.0418370
$388$	3.38058e7	0.0293819
$389$	−9.01206e8	−0.776248	−0.388124	−	0.921607i	$-0.626877\pi$
$389$	−9.01206e8	−0.776248	−0.388124	+	0.921607i	$0.626877\pi$
$390$	0	0
$391$	−1.39427e9	−1.17958
$392$	−2.70668e8	−0.226953
$393$	1.24196e9	1.03213
$394$	1.72828e8	0.142357
$395$	1.16692e9	0.952688
$396$	2.59719e8	0.210170
$397$	−9.85599e8	−0.790558	−0.395279	−	0.918561i	$-0.629352\pi$
$397$	−9.85599e8	−0.790558	−0.395279	+	0.918561i	$0.629352\pi$
$398$	3.08233e8	0.245069
$399$	6.36558e8	0.501687
$400$	2.26615e9	1.77043
$401$	−2.69423e8	−0.208655	−0.104328	−	0.994543i	$-0.533269\pi$
$401$	−2.69423e8	−0.208655	−0.104328	+	0.994543i	$0.533269\pi$
$402$	−3.98197e8	−0.305708
$403$	0	0
$404$	1.62079e9	1.22290
$405$	−2.72354e9	−2.03723
$406$	2.07570e8	0.153930
$407$	−2.06310e9	−1.51684
$408$	−6.13033e8	−0.446862
$409$	3.40992e8	0.246441	0.123220	−	0.992379i	$-0.460678\pi$
$409$	3.40992e8	0.246441	0.123220	+	0.992379i	$0.460678\pi$
$410$	−1.21057e8	−0.0867457
$411$	−4.30043e8	−0.305538
$412$	−8.63044e8	−0.607985
$413$	1.54169e9	1.07689
$414$	8.00850e7	0.0554689
$415$	−5.49526e8	−0.377416
$416$	0	0
$417$	−2.36353e9	−1.59619
$418$	2.26512e8	0.151696
$419$	−2.32463e9	−1.54385	−0.771923	−	0.635716i	$-0.780705\pi$
$419$	−2.32463e9	−1.54385	−0.771923	+	0.635716i	$0.780705\pi$
$420$	−1.88960e9	−1.24451
$421$	−2.46436e9	−1.60959	−0.804797	−	0.593551i	$-0.797726\pi$
$421$	−2.46436e9	−1.60959	−0.804797	+	0.593551i	$0.797726\pi$
$422$	−2.64413e8	−0.171273
$423$	−2.04784e8	−0.131555
$424$	1.24163e8	0.0791063
$425$	3.13211e9	1.97914
$426$	2.33257e8	0.146185
$427$	4.88261e8	0.303497
$428$	−3.43285e8	−0.211642
$429$	0	0
$430$	1.23500e8	0.0749081
$431$	8.64638e8	0.520192	0.260096	−	0.965583i	$-0.416246\pi$
$431$	8.64638e8	0.520192	0.260096	+	0.965583i	$0.416246\pi$
$432$	1.25589e9	0.749476
$433$	3.99758e8	0.236641	0.118320	−	0.992975i	$-0.462249\pi$
$433$	3.99758e8	0.236641	0.118320	+	0.992975i	$0.462249\pi$
$434$	−2.74851e7	−0.0161392
$435$	−3.46943e9	−2.02090
$436$	2.16870e9	1.25313
$437$	−1.43288e9	−0.821346
$438$	−5.34262e8	−0.303805
$439$	1.30846e8	0.0738130	0.0369065	−	0.999319i	$-0.488250\pi$
$439$	1.30846e8	0.0738130	0.0369065	+	0.999319i	$0.488250\pi$
$440$	−1.37756e9	−0.770951
$441$	−2.04168e8	−0.113358
$442$	0	0
$443$	−6.35046e8	−0.347050	−0.173525	−	0.984829i	$-0.555516\pi$
$443$	−6.35046e8	−0.347050	−0.173525	+	0.984829i	$0.555516\pi$
$444$	2.80085e9	1.51862
$445$	3.90760e9	2.10208
$446$	−1.05873e8	−0.0565083
$447$	3.87089e8	0.204991
$448$	9.45779e8	0.496955
$449$	−6.68424e8	−0.348490	−0.174245	−	0.984702i	$-0.555748\pi$
$449$	−6.68424e8	−0.348490	−0.174245	+	0.984702i	$0.555748\pi$
$450$	−1.79904e8	−0.0930673
$451$	−4.70124e8	−0.241321
$452$	2.19363e9	1.11732
$453$	2.98389e9	1.50813
$454$	3.19071e8	0.160026
$455$	0	0
$456$	−6.30011e8	−0.311151
$457$	1.09854e9	0.538406	0.269203	−	0.963083i	$-0.413240\pi$
$457$	1.09854e9	0.538406	0.269203	+	0.963083i	$0.413240\pi$
$458$	4.44239e7	0.0216066
$459$	1.73580e9	0.837830
$460$	4.25347e9	2.03747
$461$	1.96364e9	0.933489	0.466745	−	0.884392i	$-0.345427\pi$
$461$	1.96364e9	0.933489	0.466745	+	0.884392i	$0.345427\pi$
$462$	3.57700e8	0.168761
$463$	−4.36528e6	−0.00204399	−0.00102199	−	0.999999i	$-0.500325\pi$
$463$	−4.36528e6	−0.00204399	−0.00102199	+	0.999999i	$0.500325\pi$
$464$	1.95196e9	0.907106
$465$	4.59399e8	0.211887
$466$	4.31622e8	0.197584
$467$	1.77366e9	0.805861	0.402931	−	0.915230i	$-0.367992\pi$
$467$	1.77366e9	0.805861	0.402931	+	0.915230i	$0.367992\pi$
$468$	0	0
$469$	1.95539e9	0.875244
$470$	5.30172e8	0.235545
$471$	4.27009e9	1.88306
$472$	−1.52583e9	−0.667898
$473$	4.79612e8	0.208390
$474$	2.99888e8	0.129341
$475$	3.21885e9	1.37808
$476$	1.46937e9	0.624465
$477$	9.36571e7	0.0395118
$478$	1.78678e8	0.0748295
$479$	2.31639e9	0.963026	0.481513	−	0.876439i	$-0.340087\pi$
$479$	2.31639e9	0.963026	0.481513	+	0.876439i	$0.340087\pi$
$480$	2.82750e9	1.16697
$481$	0	0
$482$	−5.36992e8	−0.218425
$483$	−2.26277e9	−0.913744
$484$	−2.32804e8	−0.0933324
$485$	−1.35253e8	−0.0538334
$486$	−2.25965e8	−0.0892923
$487$	2.04846e9	0.803668	0.401834	−	0.915713i	$-0.368373\pi$
$487$	2.04846e9	0.803668	0.401834	+	0.915713i	$0.368373\pi$
$488$	−4.83239e8	−0.188232
$489$	−1.73159e9	−0.669675
$490$	5.28575e8	0.202965
$491$	−3.64385e9	−1.38923	−0.694616	−	0.719381i	$-0.744426\pi$
$491$	−3.64385e9	−1.38923	−0.694616	+	0.719381i	$0.744426\pi$
$492$	6.38237e8	0.241604
$493$	2.69786e9	1.01404
$494$	0	0
$495$	−1.03911e9	−0.385072
$496$	−2.58466e8	−0.0951080
$497$	−1.14544e9	−0.418528
$498$	−1.41223e8	−0.0512394
$499$	−4.34184e9	−1.56431	−0.782153	−	0.623086i	$-0.785879\pi$
$499$	−4.34184e9	−1.56431	−0.782153	+	0.623086i	$0.785879\pi$
$500$	−4.89891e9	−1.75269
$501$	−3.99425e9	−1.41907
$502$	−3.16985e8	−0.111835
$503$	1.29100e9	0.452311	0.226155	−	0.974091i	$-0.427384\pi$
$503$	1.29100e9	0.452311	0.226155	+	0.974091i	$0.427384\pi$
$504$	−1.72912e8	−0.0601613
$505$	−6.48460e9	−2.24060
$506$	−8.05179e8	−0.276290
$507$	0	0
$508$	2.45754e9	0.831721
$509$	−4.23593e9	−1.42376	−0.711880	−	0.702301i	$-0.752156\pi$
$509$	−4.23593e9	−1.42376	−0.711880	+	0.702301i	$0.752156\pi$
$510$	1.19716e9	0.399629
$511$	2.62356e9	0.869795
$512$	−2.69428e9	−0.887150
$513$	1.78388e9	0.583383
$514$	2.71572e6	0.000882094 0
$515$	3.45294e9	1.11395
$516$	−6.51118e8	−0.208634
$517$	2.05891e9	0.655272
$518$	6.70431e8	0.211934
$519$	−3.43995e7	−0.0108011
$520$	0	0
$521$	−2.70853e9	−0.839076	−0.419538	−	0.907738i	$-0.637808\pi$
$521$	−2.70853e9	−0.839076	−0.419538	+	0.907738i	$0.637808\pi$
$522$	−1.54962e8	−0.0476845
$523$	−2.07895e9	−0.635461	−0.317731	−	0.948181i	$-0.602921\pi$
$523$	−2.07895e9	−0.635461	−0.317731	+	0.948181i	$0.602921\pi$
$524$	2.94623e9	0.894554
$525$	5.08311e9	1.53311
$526$	7.36705e8	0.220721
$527$	−3.57233e8	−0.106320
$528$	3.36376e9	0.994504
$529$	1.68863e9	0.495953
$530$	−2.42471e8	−0.0707449
$531$	−1.15095e9	−0.333599
$532$	1.51007e9	0.434816
$533$	0	0
$534$	1.00422e9	0.285387
$535$	1.37345e9	0.387769
$536$	−1.93528e9	−0.542835
$537$	−5.43344e9	−1.51414
$538$	5.92617e8	0.164073
$539$	2.05271e9	0.564634
$540$	−5.29538e9	−1.44717
$541$	−3.70324e9	−1.00552	−0.502761	−	0.864426i	$-0.667682\pi$
$541$	−3.70324e9	−1.00552	−0.502761	+	0.864426i	$0.667682\pi$
$542$	−8.08553e8	−0.218128
$543$	−7.32158e8	−0.196248
$544$	−2.19869e9	−0.585556
$545$	−8.67672e9	−2.29598
$546$	0	0
$547$	2.64179e9	0.690149	0.345074	−	0.938575i	$-0.387854\pi$
$547$	2.64179e9	0.690149	0.345074	+	0.938575i	$0.387854\pi$
$548$	−1.02017e9	−0.264813
$549$	−3.64512e8	−0.0940174
$550$	1.80877e9	0.463567
$551$	2.77258e9	0.706080
$552$	2.23949e9	0.566713
$553$	−1.47264e9	−0.370303
$554$	7.14064e8	0.178424
$555$	−1.12059e10	−2.78242
$556$	−5.60687e9	−1.38344
$557$	5.09926e9	1.25030	0.625149	−	0.780505i	$-0.285038\pi$
$557$	5.09926e9	1.25030	0.625149	+	0.780505i	$0.285038\pi$
$558$	2.05190e7	0.00499961
$559$	0	0
$560$	−4.25344e9	−1.02349
$561$	4.64916e9	1.11174
$562$	−1.17518e9	−0.279273
$563$	1.20149e8	0.0283754	0.0141877	−	0.999899i	$-0.495484\pi$
$563$	1.20149e8	0.0283754	0.0141877	+	0.999899i	$0.495484\pi$
$564$	−2.79517e9	−0.656041
$565$	−8.77649e9	−2.04716
$566$	1.51477e9	0.351146
$567$	3.43707e9	0.791858
$568$	1.13366e9	0.259575
$569$	4.82795e9	1.09868	0.549338	−	0.835600i	$-0.314880\pi$
$569$	4.82795e9	1.09868	0.549338	+	0.835600i	$0.314880\pi$
$570$	1.23032e9	0.278263
$571$	−6.31138e9	−1.41872	−0.709362	−	0.704845i	$-0.751017\pi$
$571$	−6.31138e9	−1.41872	−0.709362	+	0.704845i	$0.751017\pi$
$572$	0	0
$573$	3.12021e9	0.692855
$574$	1.52773e8	0.0337174
$575$	−1.14420e10	−2.50995
$576$	−7.06072e8	−0.153947
$577$	−6.12272e9	−1.32687	−0.663436	−	0.748233i	$-0.730903\pi$
$577$	−6.12272e9	−1.32687	−0.663436	+	0.748233i	$0.730903\pi$
$578$	6.99410e7	0.0150655
$579$	1.01254e10	2.16788
$580$	−8.23032e9	−1.75153
$581$	6.93494e8	0.146699
$582$	−3.47589e7	−0.00730863
$583$	−9.41634e8	−0.196808
$584$	−2.59657e9	−0.539455
$585$	0	0
$586$	−5.49505e8	−0.112805
$587$	1.48927e9	0.303907	0.151953	−	0.988388i	$-0.451444\pi$
$587$	1.48927e9	0.303907	0.151953	+	0.988388i	$0.451444\pi$
$588$	−2.78675e9	−0.565297
$589$	−3.67127e8	−0.0740309
$590$	2.97972e9	0.597302
$591$	3.64555e9	0.726451
$592$	6.30464e9	1.24892
$593$	−5.49424e9	−1.08197	−0.540987	−	0.841031i	$-0.681949\pi$
$593$	−5.49424e9	−1.08197	−0.540987	+	0.841031i	$0.681949\pi$
$594$	1.00241e9	0.196243
$595$	−5.87881e9	−1.14414
$596$	9.18267e8	0.177667
$597$	6.50170e9	1.25060
$598$	0	0
$599$	4.52708e9	0.860645	0.430322	−	0.902675i	$-0.358400\pi$
$599$	4.52708e9	0.860645	0.430322	+	0.902675i	$0.358400\pi$
$600$	−5.03083e9	−0.950846
$601$	−2.93232e9	−0.550998	−0.275499	−	0.961301i	$-0.588843\pi$
$601$	−2.93232e9	−0.550998	−0.275499	+	0.961301i	$0.588843\pi$
$602$	−1.55856e8	−0.0291163
$603$	−1.45980e9	−0.271133
$604$	7.07851e9	1.30711
$605$	9.31425e8	0.171003
$606$	−1.66649e9	−0.304192
$607$	−1.02999e10	−1.86927	−0.934637	−	0.355604i	$-0.884275\pi$
$607$	−1.02999e10	−1.86927	−0.934637	+	0.355604i	$0.884275\pi$
$608$	−2.25958e9	−0.407724
$609$	4.37837e9	0.785511
$610$	9.43694e8	0.168336
$611$	0	0
$612$	−1.09696e9	−0.193447
$613$	7.93265e9	1.39093	0.695467	−	0.718558i	$-0.255197\pi$
$613$	7.93265e9	1.39093	0.695467	+	0.718558i	$0.255197\pi$
$614$	−4.75550e8	−0.0829099
$615$	−2.55352e9	−0.442666
$616$	1.73846e9	0.299663
$617$	−8.87840e9	−1.52173	−0.760863	−	0.648912i	$-0.775224\pi$
$617$	−8.87840e9	−1.52173	−0.760863	+	0.648912i	$0.775224\pi$
$618$	8.87376e8	0.151234
$619$	7.70576e9	1.30586	0.652932	−	0.757416i	$-0.273539\pi$
$619$	7.70576e9	1.30586	0.652932	+	0.757416i	$0.273539\pi$
$620$	1.08981e9	0.183644
$621$	−6.34112e9	−1.06254
$622$	7.89411e8	0.131534
$623$	−4.93134e9	−0.817065
$624$	0	0
$625$	7.07477e9	1.15913
$626$	−3.73044e8	−0.0607785
$627$	4.77792e9	0.774109
$628$	1.01297e10	1.63206
$629$	8.71384e9	1.39615
$630$	3.37671e8	0.0538024
$631$	−1.12137e10	−1.77683	−0.888413	−	0.459045i	$-0.848192\pi$
$631$	−1.12137e10	−1.77683	−0.888413	+	0.459045i	$0.848192\pi$
$632$	1.45749e9	0.229665
$633$	−5.57739e9	−0.874014
$634$	2.57227e8	0.0400871
$635$	−9.83235e9	−1.52387
$636$	1.27836e9	0.197039
$637$	0	0
$638$	1.55799e9	0.237516
$639$	8.55127e8	0.129652
$640$	8.86243e9	1.33636
$641$	9.37779e9	1.40636	0.703181	−	0.711011i	$-0.251762\pi$
$641$	9.37779e9	1.40636	0.703181	+	0.711011i	$0.251762\pi$
$642$	3.52963e8	0.0526450
$643$	−6.09443e8	−0.0904054	−0.0452027	−	0.998978i	$-0.514393\pi$
$643$	−6.09443e8	−0.0904054	−0.0452027	+	0.998978i	$0.514393\pi$
$644$	−5.36782e9	−0.791950
$645$	2.60505e9	0.382259
$646$	−9.56707e8	−0.139626
$647$	3.63441e9	0.527556	0.263778	−	0.964583i	$-0.415031\pi$
$647$	3.63441e9	0.527556	0.263778	+	0.964583i	$0.415031\pi$
$648$	−3.40172e9	−0.491118
$649$	1.15717e10	1.66166
$650$	0	0
$651$	−5.79755e8	−0.0823591
$652$	−4.10774e9	−0.580413
$653$	−3.71780e9	−0.522505	−0.261252	−	0.965271i	$-0.584135\pi$
$653$	−3.71780e9	−0.522505	−0.261252	+	0.965271i	$0.584135\pi$
$654$	−2.22984e9	−0.311711
$655$	−1.17875e10	−1.63900
$656$	1.43665e9	0.198696
$657$	−1.95862e9	−0.269445
$658$	−6.69070e8	−0.0915547
$659$	6.01769e9	0.819088	0.409544	−	0.912290i	$-0.365688\pi$
$659$	6.01769e9	0.819088	0.409544	+	0.912290i	$0.365688\pi$
$660$	−1.41831e10	−1.92029
$661$	−1.36622e9	−0.183999	−0.0919993	−	0.995759i	$-0.529326\pi$
$661$	−1.36622e9	−0.183999	−0.0919993	+	0.995759i	$0.529326\pi$
$662$	1.59503e9	0.213681
$663$	0	0
$664$	−6.86361e8	−0.0909839
$665$	−6.04162e9	−0.796669
$666$	−5.00511e8	−0.0656529
$667$	−9.85566e9	−1.28601
$668$	−9.47533e9	−1.22992
$669$	−2.23322e9	−0.288364
$670$	3.77932e9	0.485458
$671$	3.66482e9	0.468300
$672$	−3.56827e9	−0.453591
$673$	−1.13200e9	−0.143150	−0.0715751	−	0.997435i	$-0.522803\pi$
$673$	−1.13200e9	−0.143150	−0.0715751	+	0.997435i	$0.522803\pi$
$674$	−1.60685e9	−0.202146
$675$	1.42448e10	1.78276
$676$	0	0
$677$	2.50487e9	0.310259	0.155130	−	0.987894i	$-0.450420\pi$
$677$	2.50487e9	0.310259	0.155130	+	0.987894i	$0.450420\pi$
$678$	−2.25548e9	−0.277930
$679$	1.70688e8	0.0209247
$680$	5.81834e9	0.709608
$681$	6.73031e9	0.816620
$682$	−2.06299e8	−0.0249030
$683$	−1.16246e9	−0.139606	−0.0698032	−	0.997561i	$-0.522237\pi$
$683$	−1.16246e9	−0.139606	−0.0698032	+	0.997561i	$0.522237\pi$
$684$	−1.12734e9	−0.134698
$685$	4.08157e9	0.485188
$686$	−1.90491e9	−0.225290
$687$	9.37053e8	0.110259
$688$	−1.46565e9	−0.171581
$689$	0	0
$690$	−4.37340e9	−0.506812
$691$	−1.00279e10	−1.15621	−0.578104	−	0.815963i	$-0.696207\pi$
$691$	−1.00279e10	−1.15621	−0.578104	+	0.815963i	$0.696207\pi$
$692$	−8.16038e7	−0.00936136
$693$	1.31134e9	0.149675
$694$	9.19218e8	0.104390
$695$	2.24325e10	2.53472
$696$	−4.33334e9	−0.487181
$697$	1.98564e9	0.222119
$698$	−1.78650e9	−0.198843
$699$	9.10440e9	1.00828
$700$	1.20584e10	1.32876
$701$	−1.58959e10	−1.74290	−0.871449	−	0.490486i	$-0.836819\pi$
$701$	−1.58959e10	−1.74290	−0.871449	+	0.490486i	$0.836819\pi$
$702$	0	0
$703$	8.95516e9	0.972142
$704$	7.09889e9	0.766807
$705$	1.11832e10	1.20200
$706$	−2.64922e9	−0.283336
$707$	8.18348e9	0.870904
$708$	−1.57097e10	−1.66361
$709$	−3.06676e9	−0.323160	−0.161580	−	0.986860i	$-0.551659\pi$
$709$	−3.06676e9	−0.323160	−0.161580	+	0.986860i	$0.551659\pi$
$710$	−2.21386e9	−0.232138
$711$	1.09940e9	0.114713
$712$	4.88062e9	0.506751
$713$	1.30502e9	0.134836
$714$	−1.51080e9	−0.155333
$715$	0	0
$716$	−1.28894e10	−1.31232
$717$	3.76893e9	0.381857
$718$	1.67495e9	0.168875
$719$	−7.17747e9	−0.720145	−0.360073	−	0.932924i	$-0.617248\pi$
$719$	−7.17747e9	−0.720145	−0.360073	+	0.932924i	$0.617248\pi$
$720$	3.17541e9	0.317056
$721$	−4.35757e9	−0.432983
$722$	1.19706e9	0.118368
$723$	−1.13270e10	−1.11463
$724$	−1.73685e9	−0.170090
$725$	2.21399e10	2.15771
$726$	2.39368e8	0.0232160
$727$	1.33854e10	1.29200	0.645998	−	0.763339i	$-0.276442\pi$
$727$	1.33854e10	1.29200	0.645998	+	0.763339i	$0.276442\pi$
$728$	0	0
$729$	7.43152e9	0.710447
$730$	5.07072e9	0.482436
$731$	−2.02572e9	−0.191808
$732$	−4.97533e9	−0.468849
$733$	1.23188e10	1.15532	0.577661	−	0.816277i	$-0.303965\pi$
$733$	1.23188e10	1.15532	0.577661	+	0.816277i	$0.303965\pi$
$734$	−1.15818e9	−0.108104
$735$	1.11495e10	1.03574
$736$	8.03212e9	0.742605
$737$	1.46769e10	1.35051
$738$	−1.14053e8	−0.0104450
$739$	3.87017e9	0.352756	0.176378	−	0.984323i	$-0.443562\pi$
$739$	3.87017e9	0.352756	0.176378	+	0.984323i	$0.443562\pi$
$740$	−2.65831e10	−2.41154
$741$	0	0
$742$	3.05996e8	0.0274980
$743$	1.30810e10	1.16998	0.584990	−	0.811040i	$-0.301098\pi$
$743$	1.30810e10	1.16998	0.584990	+	0.811040i	$0.301098\pi$
$744$	5.73793e8	0.0510798
$745$	−3.67389e9	−0.325521
$746$	−3.79619e9	−0.334782
$747$	−5.17729e8	−0.0454444
$748$	1.10289e10	0.963557
$749$	−1.73327e9	−0.150723
$750$	5.03703e9	0.435974
$751$	−2.33046e8	−0.0200772	−0.0100386	−	0.999950i	$-0.503195\pi$
$751$	−2.33046e8	−0.0200772	−0.0100386	+	0.999950i	$0.503195\pi$
$752$	−6.29184e9	−0.539529
$753$	−6.68632e9	−0.570696
$754$	0	0
$755$	−2.83203e10	−2.39488
$756$	6.68269e9	0.562504
$757$	−8.89085e9	−0.744917	−0.372458	−	0.928049i	$-0.621485\pi$
$757$	−8.89085e9	−0.744917	−0.372458	+	0.928049i	$0.621485\pi$
$758$	−3.39486e8	−0.0283126
$759$	−1.69840e10	−1.40992
$760$	5.97948e9	0.494101
$761$	1.69293e10	1.39249	0.696247	−	0.717803i	$-0.254852\pi$
$761$	1.69293e10	1.39249	0.696247	+	0.717803i	$0.254852\pi$
$762$	−2.52683e9	−0.206887
$763$	1.09499e10	0.892430
$764$	7.40188e9	0.600503
$765$	4.38883e9	0.354433
$766$	−2.41735e9	−0.194330
$767$	0	0
$768$	−7.82961e9	−0.623700
$769$	−1.35031e10	−1.07076	−0.535380	−	0.844611i	$-0.679831\pi$
$769$	−1.35031e10	−1.07076	−0.535380	+	0.844611i	$0.679831\pi$
$770$	−3.39496e9	−0.267989
$771$	5.72840e7	0.00450135
$772$	2.40198e10	1.87892
$773$	−1.01595e10	−0.791126	−0.395563	−	0.918439i	$-0.629451\pi$
$773$	−1.01595e10	−0.791126	−0.395563	+	0.918439i	$0.629451\pi$
$774$	1.16354e8	0.00901964
$775$	−2.93162e9	−0.226231
$776$	−1.68932e8	−0.0129777
$777$	1.41417e10	1.08150
$778$	2.19815e9	0.167351
$779$	2.04063e9	0.154662
$780$	0	0
$781$	−8.59750e9	−0.645793
$782$	3.40080e9	0.254306
$783$	1.22698e10	0.913425
$784$	−6.27288e9	−0.464901
$785$	−4.05278e10	−2.99026
$786$	−3.02929e9	−0.222516
$787$	−2.11128e10	−1.54396	−0.771978	−	0.635650i	$-0.780732\pi$
$787$	−2.11128e10	−1.54396	−0.771978	+	0.635650i	$0.780732\pi$
$788$	8.64811e9	0.629622
$789$	1.55397e10	1.12635
$790$	−2.84626e9	−0.205390
$791$	1.10758e10	0.795715
$792$	−1.29785e9	−0.0928297
$793$	0	0
$794$	2.40400e9	0.170436
$795$	−5.11456e9	−0.361014
$796$	1.54236e10	1.08390
$797$	−1.50164e10	−1.05066	−0.525329	−	0.850899i	$-0.676058\pi$
$797$	−1.50164e10	−1.05066	−0.525329	+	0.850899i	$0.676058\pi$
$798$	−1.55264e9	−0.108159
$799$	−8.69614e9	−0.603133
$800$	−1.80435e10	−1.24596
$801$	3.68149e9	0.253111
$802$	6.57155e8	0.0449840
$803$	1.96920e10	1.34210
$804$	−1.99253e10	−1.35210
$805$	2.14761e10	1.45101
$806$	0	0
$807$	1.25004e10	0.837269
$808$	−8.09931e9	−0.540143
$809$	−1.73621e10	−1.15288	−0.576439	−	0.817140i	$-0.695558\pi$
$809$	−1.73621e10	−1.15288	−0.576439	+	0.817140i	$0.695558\pi$
$810$	6.64305e9	0.439207
$811$	2.29338e10	1.50974	0.754872	−	0.655872i	$-0.227699\pi$
$811$	2.29338e10	1.50974	0.754872	+	0.655872i	$0.227699\pi$
$812$	1.03865e10	0.680809
$813$	−1.70552e10	−1.11311
$814$	5.03216e9	0.327016
$815$	1.64346e10	1.06343
$816$	−1.42074e10	−0.915373
$817$	−2.08182e9	−0.133557
$818$	−8.31720e8	−0.0531301
$819$	0	0
$820$	−6.05756e9	−0.383662
$821$	−1.35927e10	−0.857242	−0.428621	−	0.903484i	$-0.641000\pi$
$821$	−1.35927e10	−0.857242	−0.428621	+	0.903484i	$0.641000\pi$
$822$	1.04893e9	0.0658710
$823$	1.74580e9	0.109168	0.0545841	−	0.998509i	$-0.482617\pi$
$823$	1.74580e9	0.109168	0.0545841	+	0.998509i	$0.482617\pi$
$824$	4.31275e9	0.268540
$825$	3.81531e10	2.36560
$826$	−3.76037e9	−0.232167
$827$	−1.56621e10	−0.962900	−0.481450	−	0.876473i	$-0.659890\pi$
$827$	−1.56621e10	−0.962900	−0.481450	+	0.876473i	$0.659890\pi$
$828$	4.00735e9	0.245330
$829$	−6.04053e9	−0.368243	−0.184122	−	0.982903i	$-0.558944\pi$
$829$	−6.04053e9	−0.368243	−0.184122	+	0.982903i	$0.558944\pi$
$830$	1.34036e9	0.0813671
$831$	1.50621e10	0.910503
$832$	0	0
$833$	−8.66994e9	−0.519707
$834$	5.76495e9	0.344124
$835$	3.79098e10	2.25345
$836$	1.13344e10	0.670927
$837$	−1.62469e9	−0.0957706
$838$	5.67005e9	0.332838
$839$	1.52652e10	0.892348	0.446174	−	0.894946i	$-0.352786\pi$
$839$	1.52652e10	0.892348	0.446174	+	0.894946i	$0.352786\pi$
$840$	9.44261e9	0.549686
$841$	1.82048e9	0.105536
$842$	6.01087e9	0.347012
$843$	−2.47887e10	−1.42514
$844$	−1.32309e10	−0.757515
$845$	0	0
$846$	4.99495e8	0.0283618
$847$	−1.17545e9	−0.0664677
$848$	2.87754e9	0.162045
$849$	3.19517e10	1.79191
$850$	−7.63960e9	−0.426682
$851$	−3.18328e10	−1.77061
$852$	1.16719e10	0.646551
$853$	4.49630e9	0.248047	0.124023	−	0.992279i	$-0.460420\pi$
$853$	4.49630e9	0.248047	0.124023	+	0.992279i	$0.460420\pi$
$854$	−1.19093e9	−0.0654309
$855$	4.51038e9	0.246792
$856$	1.71544e9	0.0934798
$857$	1.87369e10	1.01687	0.508435	−	0.861101i	$-0.330224\pi$
$857$	1.87369e10	1.01687	0.508435	+	0.861101i	$0.330224\pi$
$858$	0	0
$859$	5.28210e9	0.284335	0.142168	−	0.989843i	$-0.454593\pi$
$859$	5.28210e9	0.284335	0.142168	+	0.989843i	$0.454593\pi$
$860$	6.17981e9	0.331307
$861$	3.22251e9	0.172061
$862$	−2.10896e9	−0.112148
$863$	1.11634e10	0.591231	0.295616	−	0.955307i	$-0.404475\pi$
$863$	1.11634e10	0.591231	0.295616	+	0.955307i	$0.404475\pi$
$864$	−9.99962e9	−0.527455
$865$	3.26488e8	0.0171518
$866$	−9.75058e8	−0.0510174
$867$	1.47530e9	0.0768799
$868$	−1.37532e9	−0.0713813
$869$	−1.10534e10	−0.571382
$870$	8.46237e9	0.435687
$871$	0	0
$872$	−1.08373e10	−0.553494
$873$	−1.27427e8	−0.00648205
$874$	3.49498e9	0.177074
$875$	−2.47350e10	−1.24820
$876$	−2.67338e10	−1.34368
$877$	1.55652e9	0.0779210	0.0389605	−	0.999241i	$-0.487595\pi$
$877$	1.55652e9	0.0779210	0.0389605	+	0.999241i	$0.487595\pi$
$878$	−3.19148e8	−0.0159134
$879$	−1.15910e10	−0.575650
$880$	−3.19257e10	−1.57925
$881$	−3.77180e10	−1.85837	−0.929187	−	0.369611i	$-0.879491\pi$
$881$	−3.77180e10	−1.85837	−0.929187	+	0.369611i	$0.879491\pi$
$882$	4.97990e8	0.0244388
$883$	1.53832e10	0.751943	0.375972	−	0.926631i	$-0.377309\pi$
$883$	1.53832e10	0.751943	0.375972	+	0.926631i	$0.377309\pi$
$884$	0	0
$885$	6.28527e10	3.04805
$886$	1.54896e9	0.0748205
$887$	−1.58874e10	−0.764401	−0.382201	−	0.924079i	$-0.624834\pi$
$887$	−1.58874e10	−0.764401	−0.382201	+	0.924079i	$0.624834\pi$
$888$	−1.39963e10	−0.670759
$889$	1.24083e10	0.592319
$890$	−9.53112e9	−0.453188
$891$	2.57982e10	1.22185
$892$	−5.29774e9	−0.249927
$893$	−8.93698e9	−0.419963
$894$	−9.44157e8	−0.0441940
$895$	5.15692e10	2.40442
$896$	−1.11843e10	−0.519433
$897$	0	0
$898$	1.63037e9	0.0751309
$899$	−2.52517e9	−0.115913
$900$	−9.00218e9	−0.411622
$901$	3.97714e9	0.181148
$902$	1.14669e9	0.0520264
$903$	−3.28754e9	−0.148581
$904$	−1.09619e10	−0.493510
$905$	6.94896e9	0.311638
$906$	−7.27808e9	−0.325138
$907$	1.04545e10	0.465240	0.232620	−	0.972568i	$-0.425270\pi$
$907$	1.04545e10	0.465240	0.232620	+	0.972568i	$0.425270\pi$
$908$	1.59659e10	0.707772
$909$	−6.10938e9	−0.269789
$910$	0	0
$911$	−4.40778e10	−1.93155	−0.965774	−	0.259386i	$-0.916480\pi$
$911$	−4.40778e10	−1.93155	−0.965774	+	0.259386i	$0.916480\pi$
$912$	−1.46008e10	−0.637376
$913$	5.20527e9	0.226358
$914$	−2.67948e9	−0.116075
$915$	1.99058e10	0.859024
$916$	2.22292e9	0.0955628
$917$	1.48757e10	0.637067
$918$	−4.23384e9	−0.180628
$919$	−4.22709e10	−1.79654	−0.898270	−	0.439445i	$-0.855175\pi$
$919$	−4.22709e10	−1.79654	−0.898270	+	0.439445i	$0.855175\pi$
$920$	−2.12552e10	−0.899928
$921$	−1.00310e10	−0.423092
$922$	−4.78957e9	−0.201251
$923$	0	0
$924$	1.78989e10	0.746404
$925$	7.15097e10	2.97077
$926$	1.06475e7	0.000440664 0
$927$	3.25315e9	0.134130
$928$	−1.55419e10	−0.638389
$929$	4.13059e10	1.69027	0.845136	−	0.534551i	$-0.179519\pi$
$929$	4.13059e10	1.69027	0.845136	+	0.534551i	$0.179519\pi$
$930$	−1.12053e9	−0.0456808
$931$	−8.91006e9	−0.361873
$932$	2.15978e10	0.873885
$933$	1.66514e10	0.671221
$934$	−4.32617e9	−0.173736
$935$	−4.41255e10	−1.76542
$936$	0	0
$937$	3.77754e10	1.50010	0.750050	−	0.661382i	$-0.230030\pi$
$937$	3.77754e10	1.50010	0.750050	+	0.661382i	$0.230030\pi$
$938$	−4.76944e9	−0.188694
$939$	−7.86879e9	−0.310155
$940$	2.65291e10	1.04178
$941$	−3.17351e9	−0.124159	−0.0620793	−	0.998071i	$-0.519773\pi$
$941$	−3.17351e9	−0.124159	−0.0620793	+	0.998071i	$0.519773\pi$
$942$	−1.04153e10	−0.405969
$943$	−7.25382e9	−0.281693
$944$	−3.53620e10	−1.36815
$945$	−2.67367e10	−1.03062
$946$	−1.16983e9	−0.0449267
$947$	−3.54610e10	−1.35683	−0.678417	−	0.734677i	$-0.737334\pi$
$947$	−3.54610e10	−1.35683	−0.678417	+	0.734677i	$0.737334\pi$
$948$	1.50060e10	0.572053
$949$	0	0
$950$	−7.85118e9	−0.297100
$951$	5.42582e9	0.204566
$952$	−7.34267e9	−0.275819
$953$	−5.38620e9	−0.201585	−0.100792	−	0.994907i	$-0.532138\pi$
$953$	−5.38620e9	−0.201585	−0.100792	+	0.994907i	$0.532138\pi$
$954$	−2.28441e8	−0.00851834
$955$	−2.96141e10	−1.10024
$956$	8.94081e9	0.330959
$957$	3.28635e10	1.21205
$958$	−5.64997e9	−0.207619
$959$	−5.15089e9	−0.188589
$960$	3.85582e10	1.40659
$961$	−2.71782e10	−0.987847
$962$	0	0
$963$	1.29397e9	0.0466910
$964$	−2.68704e10	−0.966061
$965$	−9.61005e10	−3.44255
$966$	5.51916e9	0.196994
$967$	−3.69599e10	−1.31443	−0.657216	−	0.753703i	$-0.728266\pi$
$967$	−3.69599e10	−1.31443	−0.657216	+	0.753703i	$0.728266\pi$
$968$	1.16336e9	0.0412239
$969$	−2.01803e10	−0.712515
$970$	3.29900e8	0.0116060
$971$	−9.10466e9	−0.319151	−0.159576	−	0.987186i	$-0.551013\pi$
$971$	−9.10466e9	−0.319151	−0.159576	+	0.987186i	$0.551013\pi$
$972$	−1.13070e10	−0.394926
$973$	−2.83095e10	−0.985229
$974$	−4.99645e9	−0.173263
$975$	0	0
$976$	−1.11993e10	−0.385583
$977$	−1.66549e10	−0.571363	−0.285681	−	0.958325i	$-0.592220\pi$
$977$	−1.66549e10	−0.571363	−0.285681	+	0.958325i	$0.592220\pi$
$978$	4.22356e9	0.144375
$979$	−3.70139e10	−1.26074
$980$	2.64492e10	0.897680
$981$	−8.17466e9	−0.276457
$982$	8.88779e9	0.299505
$983$	5.63122e10	1.89088	0.945442	−	0.325790i	$-0.105630\pi$
$983$	5.63122e10	1.89088	0.945442	+	0.325790i	$0.105630\pi$
$984$	−3.18936e9	−0.106714
$985$	−3.46002e10	−1.15359
$986$	−6.58042e9	−0.218617
$987$	−1.41130e10	−0.467207
$988$	0	0
$989$	7.40022e9	0.243253
$990$	2.53451e9	0.0830177
$991$	3.57927e9	0.116825	0.0584127	−	0.998293i	$-0.481396\pi$
$991$	3.57927e9	0.116825	0.0584127	+	0.998293i	$0.481396\pi$
$992$	2.05795e9	0.0669337
$993$	3.36447e10	1.09042
$994$	2.79386e9	0.0902304
$995$	−6.17082e10	−1.98592
$996$	−7.06664e9	−0.226624
$997$	−1.48467e10	−0.474456	−0.237228	−	0.971454i	$-0.576239\pi$
$997$	−1.48467e10	−0.474456	−0.237228	+	0.971454i	$0.576239\pi$
$998$	1.05903e10	0.337249
$999$	3.96304e10	1.25762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.d.1.3		6
13.5	odd	4		13.8.b.a.12.4	yes	6
13.8	odd	4		13.8.b.a.12.3	✓	6
13.12	even	2	inner	169.8.a.d.1.4		6
39.5	even	4		117.8.b.b.64.3		6
39.8	even	4		117.8.b.b.64.4		6
52.31	even	4		208.8.f.a.129.5		6
52.47	even	4		208.8.f.a.129.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.b.a.12.3	✓	6	13.8	odd	4
13.8.b.a.12.4	yes	6	13.5	odd	4
117.8.b.b.64.3		6	39.5	even	4
117.8.b.b.64.4		6	39.8	even	4
169.8.a.d.1.3		6	1.1	even	1	trivial
169.8.a.d.1.4		6	13.12	even	2	inner
208.8.f.a.129.5		6	52.31	even	4
208.8.f.a.129.6		6	52.47	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)

\(f(q)\)	\(=\)	\(q-2.43912 q^{2} -51.4496 q^{3} -122.051 q^{4} +488.312 q^{5} +125.492 q^{6} -616.243 q^{7} +609.904 q^{8} +460.056 q^{9} -1191.05 q^{10} -4625.43 q^{11} +6279.45 q^{12} +1503.09 q^{14} -25123.4 q^{15} +14134.9 q^{16} +19536.2 q^{17} -1122.13 q^{18} +20077.3 q^{19} -59598.8 q^{20} +31705.4 q^{21} +11282.0 q^{22} -71368.5 q^{23} -31379.3 q^{24} +160323. q^{25} +88850.5 q^{27} +75212.8 q^{28} +138095. q^{29} +61279.1 q^{30} -18285.7 q^{31} -112544. q^{32} +237976. q^{33} -47651.3 q^{34} -300918. q^{35} -56150.2 q^{36} +446035. q^{37} -48970.9 q^{38} +297823. q^{40} +101639. q^{41} -77333.4 q^{42} -103690. q^{43} +564537. q^{44} +224651. q^{45} +174076. q^{46} -445129. q^{47} -727232. q^{48} -443788. q^{49} -391048. q^{50} -1.00513e6 q^{51} +203578. q^{53} -216717. q^{54} -2.25865e6 q^{55} -375849. q^{56} -1.03297e6 q^{57} -336832. q^{58} -2.50176e6 q^{59} +3.06633e6 q^{60} -792320. q^{61} +44601.1 q^{62} -283506. q^{63} -1.53475e6 q^{64} -580454. q^{66} -3.17309e6 q^{67} -2.38441e6 q^{68} +3.67188e6 q^{69} +733977. q^{70} +1.85875e6 q^{71} +280590. q^{72} -4.25734e6 q^{73} -1.08793e6 q^{74} -8.24856e6 q^{75} -2.45045e6 q^{76} +2.85039e6 q^{77} +2.38970e6 q^{79} +6.90221e6 q^{80} -5.57746e6 q^{81} -247910. q^{82} -1.12536e6 q^{83} -3.86967e6 q^{84} +9.53976e6 q^{85} +252913. q^{86} -7.10495e6 q^{87} -2.82107e6 q^{88} +8.00227e6 q^{89} -547951. q^{90} +8.71057e6 q^{92} +940791. q^{93} +1.08572e6 q^{94} +9.80397e6 q^{95} +5.79036e6 q^{96} -276982. q^{97} +1.08245e6 q^{98} -2.12796e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 56 q^{3} + 130 q^{4} - 1150 q^{9} + 406 q^{10} - 1898 q^{12} + 9558 q^{14} + 7778 q^{16} - 13152 q^{17} - 125080 q^{22} - 27264 q^{23} + 18262 q^{25} + 194560 q^{27} + 42924 q^{29} + 60114 q^{30}+ \cdots + 22075632 q^{95}+O(q^{100}) \) 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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