LMFDB - Embedded newform 169.8.a.d.1.4

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			1.4
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	$-0.465621\pi$
$2$	2.43912	0.215590	0.107795	+	0.994173i	$0.465621\pi$
$3$	−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	$-0.838167\pi$
$5$	−488.312	−1.74704	−0.873518	+	0.486791i	$0.838167\pi$
$6$	−125.492	−0.237184
$7$	616.243	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$7$	616.243	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$8$	−609.904	−0.421160
$9$	460.056	0.210360
$10$	−1191.05	−0.376644
$11$	4625.43	1.04780	0.523899	−	0.851780i	$-0.324477\pi$
$11$	4625.43	1.04780	0.523899	+	0.851780i	$0.324477\pi$
$12$	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	$-0.608995\pi$
$19$	−20077.3	−0.671533	−0.335766	+	0.941945i	$0.608995\pi$
$20$	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	$-0.482446\pi$
$31$	18285.7	0.110242	0.0551208	+	0.998480i	$0.482446\pi$
$32$	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	$-0.757617\pi$
$37$	−446035.	−1.44765	−0.723824	+	0.689985i	$0.757617\pi$
$38$	−48970.9	−0.144776
$39$	0	0
$40$	297823.	0.735781
$41$	−101639.	−0.230312	−0.115156	−	0.993347i	$-0.536737\pi$
$41$	−101639.	−0.230312	−0.115156	+	0.993347i	$0.536737\pi$
$42$	−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	$-0.398770\pi$
$47$	445129.	0.625379	0.312690	+	0.949855i	$0.398770\pi$
$48$	−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	$-0.208556\pi$
$59$	2.50176e6	1.58585	0.792927	+	0.609317i	$0.208556\pi$
$60$	−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	$-0.277085\pi$
$67$	3.17309e6	1.28890	0.644452	+	0.764645i	$0.277085\pi$
$68$	−2.38441e6	−0.919601
$69$	3.67188e6	1.34560
$70$	−733977.	−0.255764
$71$	−1.85875e6	−0.616333	−0.308167	−	0.951332i	$-0.599715\pi$
$71$	−1.85875e6	−0.616333	−0.308167	+	0.951332i	$0.599715\pi$
$72$	−280590.	−0.0885950
$73$	4.25734e6	1.28088	0.640440	−	0.768008i	$-0.278752\pi$
$73$	4.25734e6	1.28088	0.640440	+	0.768008i	$0.278752\pi$
$74$	−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	$-0.465550\pi$
$83$	1.12536e6	0.216032	0.108016	+	0.994149i	$0.465550\pi$
$84$	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	$-0.705476\pi$
$89$	−8.00227e6	−1.20323	−0.601614	+	0.798787i	$0.705476\pi$
$90$	−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	$-0.495096\pi$
$97$	276982.	0.0308141	0.0154071	+	0.999881i	$0.495096\pi$
$98$	−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	$-0.271780\pi$
$109$	1.77688e7	1.31421	0.657107	+	0.753798i	$0.271780\pi$
$110$	−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	$-0.544344\pi$
$137$	−8.35854e6	−0.277721	−0.138860	+	0.990312i	$0.544344\pi$
$138$	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	$-0.470302\pi$
$149$	7.52366e6	0.186328	0.0931638	+	0.995651i	$0.470302\pi$
$150$	−2.01192e7	−0.486735
$151$	5.79965e7	1.37083	0.685413	−	0.728155i	$-0.259622\pi$
$151$	5.79965e7	1.37083	0.685413	+	0.728155i	$0.259622\pi$
$152$	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	$-0.598440\pi$
$163$	−3.36561e7	−0.608705	−0.304352	+	0.952560i	$0.598440\pi$
$164$	1.24051e7	0.219608
$165$	1.16207e8	2.01390
$166$	2.74489e6	0.0465743
$167$	−7.76344e7	−1.28987	−0.644936	−	0.764237i	$-0.723116\pi$
$167$	−7.76344e7	−1.28987	−0.644936	+	0.764237i	$0.723116\pi$
$168$	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	$-0.0547326\pi$
$193$	1.96802e8	1.97051	0.985253	+	0.171102i	$0.0547326\pi$
$194$	675593.	0.00664322
$195$	0	0
$196$	5.41646e7	0.513830
$197$	7.08567e7	0.660312	0.330156	−	0.943926i	$-0.392899\pi$
$197$	7.08567e7	0.660312	0.330156	+	0.943926i	$0.392899\pi$
$198$	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	$-0.541836\pi$
$223$	−4.34061e7	−0.262110	−0.131055	+	0.991375i	$0.541836\pi$
$224$	6.93546e7	0.412294
$225$	7.37577e7	0.431687
$226$	−4.38387e7	−0.252626
$227$	1.30814e8	0.742272	0.371136	−	0.928579i	$-0.378968\pi$
$227$	1.30814e8	0.742272	0.371136	+	0.928579i	$0.378968\pi$
$228$	−1.26074e8	−0.704457
$229$	1.82131e7	0.100221	0.0501105	−	0.998744i	$-0.484043\pi$
$229$	1.82131e7	0.100221	0.0501105	+	0.998744i	$0.484043\pi$
$230$	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	$-0.444478\pi$
$239$	7.32549e7	0.347092	0.173546	+	0.984826i	$0.444478\pi$
$240$	3.55116e8	1.65818
$241$	−2.20158e8	−1.01315	−0.506576	−	0.862195i	$-0.669089\pi$
$241$	−2.20158e8	−1.01315	−0.506576	+	0.862195i	$0.669089\pi$
$242$	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	$-0.668834\pi$
$271$	−3.31493e8	−1.01177	−0.505886	+	0.862601i	$0.668834\pi$
$272$	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	$-0.724266\pi$
$281$	−4.81806e8	−1.29539	−0.647694	+	0.761901i	$0.724266\pi$
$282$	−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	$-0.584257\pi$
$293$	−2.25288e8	−0.523240	−0.261620	+	0.965171i	$0.584257\pi$
$294$	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	$-0.561590\pi$
$307$	−1.94968e8	−0.384572	−0.192286	+	0.981339i	$0.561590\pi$
$308$	−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	$-0.470364\pi$
$317$	1.05459e8	0.185941	0.0929707	+	0.995669i	$0.470364\pi$
$318$	−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	$-0.334958\pi$
$331$	6.53937e8	0.991146	0.495573	+	0.868566i	$0.334958\pi$
$332$	−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	$-0.652566\pi$
$349$	−7.32436e8	−0.922318	−0.461159	+	0.887317i	$0.652566\pi$
$350$	2.40980e8	0.300430
$351$	0	0
$352$	5.20566e8	0.636175
$353$	−1.08614e9	−1.31424	−0.657118	−	0.753788i	$-0.728225\pi$
$353$	−1.08614e9	−1.31424	−0.657118	+	0.753788i	$0.728225\pi$
$354$	−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	$-0.371902\pi$
$359$	6.86701e8	0.783316	0.391658	+	0.920111i	$0.371902\pi$
$360$	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	$-0.520916\pi$
$379$	−1.39184e8	−0.131326	−0.0656631	+	0.997842i	$0.520916\pi$
$380$	−1.19658e9	−1.11866
$381$	1.03596e9	0.959631
$382$	−1.47923e8	−0.135773
$383$	−9.91073e8	−0.901385	−0.450692	−	0.892679i	$-0.648823\pi$
$383$	−9.91073e8	−0.901385	−0.450692	+	0.892679i	$0.648823\pi$
$384$	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	$-0.370648\pi$
$397$	9.85599e8	0.790558	0.395279	+	0.918561i	$0.370648\pi$
$398$	−3.08233e8	−0.245069
$399$	6.36558e8	0.501687
$400$	2.26615e9	1.77043
$401$	2.69423e8	0.208655	0.104328	−	0.994543i	$-0.466731\pi$
$401$	2.69423e8	0.208655	0.104328	+	0.994543i	$0.466731\pi$
$402$	−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	$-0.539322\pi$
$409$	−3.40992e8	−0.246441	−0.123220	+	0.992379i	$0.539322\pi$
$410$	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	$-0.202274\pi$
$421$	2.46436e9	1.60959	0.804797	+	0.593551i	$0.202274\pi$
$422$	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	$-0.583754\pi$
$431$	−8.64638e8	−0.520192	−0.260096	+	0.965583i	$0.583754\pi$
$432$	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	$-0.444252\pi$
$449$	6.68424e8	0.348490	0.174245	+	0.984702i	$0.444252\pi$
$450$	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	$-0.586760\pi$
$457$	−1.09854e9	−0.538406	−0.269203	+	0.963083i	$0.586760\pi$
$458$	4.44239e7	0.0216066
$459$	1.73580e9	0.837830
$460$	−4.25347e9	−2.03747
$461$	−1.96364e9	−0.933489	−0.466745	−	0.884392i	$-0.654573\pi$
$461$	−1.96364e9	−0.933489	−0.466745	+	0.884392i	$0.654573\pi$
$462$	−3.57700e8	−0.168761
$463$	4.36528e6	0.00204399	0.00102199	−	0.999999i	$-0.499675\pi$
$463$	4.36528e6	0.00204399	0.00102199	+	0.999999i	$0.499675\pi$
$464$	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	$-0.659913\pi$
$479$	−2.31639e9	−0.963026	−0.481513	+	0.876439i	$0.659913\pi$
$480$	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	$-0.631627\pi$
$487$	−2.04846e9	−0.803668	−0.401834	+	0.915713i	$0.631627\pi$
$488$	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	$-0.214121\pi$
$499$	4.34184e9	1.56431	0.782153	+	0.623086i	$0.214121\pi$
$500$	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	$-0.247844\pi$
$509$	4.23593e9	1.42376	0.711880	+	0.702301i	$0.247844\pi$
$510$	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	$-0.332318\pi$
$541$	3.70324e9	1.00552	0.502761	+	0.864426i	$0.332318\pi$
$542$	−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	$-0.714962\pi$
$557$	−5.09926e9	−1.25030	−0.625149	+	0.780505i	$0.714962\pi$
$558$	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	$-0.269097\pi$
$577$	6.12272e9	1.32687	0.663436	+	0.748233i	$0.269097\pi$
$578$	−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	$-0.548556\pi$
$587$	−1.48927e9	−0.303907	−0.151953	+	0.988388i	$0.548556\pi$
$588$	−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	$-0.318051\pi$
$593$	5.49424e9	1.08197	0.540987	+	0.841031i	$0.318051\pi$
$594$	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	$-0.744803\pi$
$613$	−7.93265e9	−1.39093	−0.695467	+	0.718558i	$0.744803\pi$
$614$	−4.75550e8	−0.0829099
$615$	−2.55352e9	−0.442666
$616$	−1.73846e9	−0.299663
$617$	8.87840e9	1.52173	0.760863	−	0.648912i	$-0.224776\pi$
$617$	8.87840e9	1.52173	0.760863	+	0.648912i	$0.224776\pi$
$618$	−8.87376e8	−0.151234
$619$	−7.70576e9	−1.30586	−0.652932	−	0.757416i	$-0.726461\pi$
$619$	−7.70576e9	−1.30586	−0.652932	+	0.757416i	$0.726461\pi$
$620$	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	$-0.151808\pi$
$631$	1.12137e10	1.77683	0.888413	+	0.459045i	$0.151808\pi$
$632$	−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	$-0.485607\pi$
$643$	6.09443e8	0.0904054	0.0452027	+	0.998978i	$0.485607\pi$
$644$	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	$-0.470674\pi$
$661$	1.36622e9	0.183999	0.0919993	+	0.995759i	$0.470674\pi$
$662$	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	$-0.477763\pi$
$683$	1.16246e9	0.139606	0.0698032	+	0.997561i	$0.477763\pi$
$684$	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	$-0.303793\pi$
$691$	1.00279e10	1.15621	0.578104	+	0.815963i	$0.303793\pi$
$692$	−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	$-0.448341\pi$
$709$	3.06676e9	0.323160	0.161580	+	0.986860i	$0.448341\pi$
$710$	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	$-0.696035\pi$
$733$	−1.23188e10	−1.15532	−0.577661	+	0.816277i	$0.696035\pi$
$734$	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	$-0.556438\pi$
$739$	−3.87017e9	−0.352756	−0.176378	+	0.984323i	$0.556438\pi$
$740$	−2.65831e10	−2.41154
$741$	0	0
$742$	3.05996e8	0.0274980
$743$	−1.30810e10	−1.16998	−0.584990	−	0.811040i	$-0.698902\pi$
$743$	−1.30810e10	−1.16998	−0.584990	+	0.811040i	$0.698902\pi$
$744$	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	$-0.745148\pi$
$761$	−1.69293e10	−1.39249	−0.696247	+	0.717803i	$0.745148\pi$
$762$	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	$-0.320169\pi$
$769$	1.35031e10	1.07076	0.535380	+	0.844611i	$0.320169\pi$
$770$	−3.39496e9	−0.267989
$771$	5.72840e7	0.00450135
$772$	−2.40198e10	−1.87892
$773$	1.01595e10	0.791126	0.395563	−	0.918439i	$-0.370549\pi$
$773$	1.01595e10	0.791126	0.395563	+	0.918439i	$0.370549\pi$
$774$	−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	$-0.219268\pi$
$787$	2.11128e10	1.54396	0.771978	+	0.635650i	$0.219268\pi$
$788$	−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	$-0.772301\pi$
$811$	−2.29338e10	−1.50974	−0.754872	+	0.655872i	$0.772301\pi$
$812$	−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	$-0.359000\pi$
$821$	1.35927e10	0.857242	0.428621	+	0.903484i	$0.359000\pi$
$822$	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	$-0.340110\pi$
$827$	1.56621e10	0.962900	0.481450	+	0.876473i	$0.340110\pi$
$828$	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	$-0.647214\pi$
$839$	−1.52652e10	−0.892348	−0.446174	+	0.894946i	$0.647214\pi$
$840$	−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	$-0.539580\pi$
$853$	−4.49630e9	−0.248047	−0.124023	+	0.992279i	$0.539580\pi$
$854$	−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	$-0.595525\pi$
$863$	−1.11634e10	−0.591231	−0.295616	+	0.955307i	$0.595525\pi$
$864$	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	$-0.512405\pi$
$877$	−1.55652e9	−0.0779210	−0.0389605	+	0.999241i	$0.512405\pi$
$878$	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	$-0.820481\pi$
$929$	−4.13059e10	−1.69027	−0.845136	+	0.534551i	$0.820481\pi$
$930$	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	$-0.480227\pi$
$941$	3.17351e9	0.124159	0.0620793	+	0.998071i	$0.480227\pi$
$942$	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	$-0.262666\pi$
$947$	3.54610e10	1.35683	0.678417	+	0.734677i	$0.262666\pi$
$948$	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	$-0.271734\pi$
$967$	3.69599e10	1.31443	0.657216	+	0.753703i	$0.271734\pi$
$968$	−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	$-0.407780\pi$
$977$	1.66549e10	0.571363	0.285681	+	0.958325i	$0.407780\pi$
$978$	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	$-0.894370\pi$
$983$	−5.63122e10	−1.89088	−0.945442	+	0.325790i	$0.894370\pi$
$984$	−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.1.4		6
13.5	odd	4		13.8.b.a.12.3	✓	6
13.8	odd	4		13.8.b.a.12.4	yes	6
13.12	even	2	inner	169.8.a.d.1.3		6
39.5	even	4		117.8.b.b.64.4		6
39.8	even	4		117.8.b.b.64.3		6
52.31	even	4		208.8.f.a.129.6		6
52.47	even	4		208.8.f.a.129.5		6

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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