LMFDB - Embedded newform 31.6.a.b.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [31,6,Mod(1,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 31 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	31.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.97189841420$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 199x^{6} + 256x^{5} + 12633x^{4} - 18583x^{3} - 260319x^{2} + 410640x + 275908 $ x^8 - x^7 - 199x^6 + 256x^5 + 12633x^4 - 18583x^3 - 260319x^2 + 410640x + 275908
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 5\cdot 13 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.513172$ of defining polynomial
Character	$\chi$	$=$	31.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.51317 q^{2} +22.4697 q^{3} -29.7103 q^{4} +77.4836 q^{5} +34.0005 q^{6} +20.7014 q^{7} -93.3783 q^{8} +261.888 q^{9} +117.246 q^{10} -13.6402 q^{11} -667.582 q^{12} +703.908 q^{13} +31.3248 q^{14} +1741.03 q^{15} +809.433 q^{16} -1360.44 q^{17} +396.281 q^{18} -2516.75 q^{19} -2302.06 q^{20} +465.155 q^{21} -20.6399 q^{22} -3840.67 q^{23} -2098.18 q^{24} +2878.70 q^{25} +1065.13 q^{26} +424.401 q^{27} -615.046 q^{28} +1600.17 q^{29} +2634.48 q^{30} +961.000 q^{31} +4212.92 q^{32} -306.491 q^{33} -2058.58 q^{34} +1604.02 q^{35} -7780.76 q^{36} -7831.82 q^{37} -3808.27 q^{38} +15816.6 q^{39} -7235.28 q^{40} -1135.29 q^{41} +703.859 q^{42} +15289.3 q^{43} +405.254 q^{44} +20292.0 q^{45} -5811.60 q^{46} +29909.8 q^{47} +18187.7 q^{48} -16378.5 q^{49} +4355.97 q^{50} -30568.7 q^{51} -20913.3 q^{52} +17590.3 q^{53} +642.192 q^{54} -1056.89 q^{55} -1933.06 q^{56} -56550.6 q^{57} +2421.33 q^{58} -16403.3 q^{59} -51726.6 q^{60} +23461.6 q^{61} +1454.16 q^{62} +5421.45 q^{63} -19527.0 q^{64} +54541.3 q^{65} -463.773 q^{66} +12225.2 q^{67} +40419.1 q^{68} -86298.8 q^{69} +2427.16 q^{70} -50123.9 q^{71} -24454.6 q^{72} +63123.3 q^{73} -11850.9 q^{74} +64683.6 q^{75} +74773.3 q^{76} -282.371 q^{77} +23933.2 q^{78} -34426.9 q^{79} +62717.7 q^{80} -54102.5 q^{81} -1717.89 q^{82} +34019.7 q^{83} -13819.9 q^{84} -105412. q^{85} +23135.3 q^{86} +35955.3 q^{87} +1273.70 q^{88} +22185.6 q^{89} +30705.3 q^{90} +14571.9 q^{91} +114108. q^{92} +21593.4 q^{93} +45258.6 q^{94} -195007. q^{95} +94663.0 q^{96} -113665. q^{97} -24783.4 q^{98} -3572.20 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 7 q^{2} - 2 q^{3} + 149 q^{4} + 128 q^{5} + 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} + 574 q^{11} - 46 q^{12} - 122 q^{13} - 309 q^{14} - 524 q^{15} + 833 q^{16} + 1932 q^{17} - 6845 q^{18}+ \cdots - 180150 q^{99}+O(q^{100}) $ 8 * q + 7 * q^2 - 2 * q^3 + 149 * q^4 + 128 * q^5 + 72 * q^6 + 88 * q^7 + 924 * q^8 + 1512 * q^9 + 1581 * q^10 + 574 * q^11 - 46 * q^12 - 122 * q^13 - 309 * q^14 - 524 * q^15 + 833 * q^16 + 1932 * q^17 - 6845 * q^18 - 1796 * q^19 - 37 * q^20 + 996 * q^21 - 9000 * q^22 - 4136 * q^23 - 17468 * q^24 + 3308 * q^25 + 1524 * q^26 - 2624 * q^27 - 21245 * q^28 + 5050 * q^29 - 30450 * q^30 + 7688 * q^31 + 21045 * q^32 + 27920 * q^33 - 24738 * q^34 + 25700 * q^35 + 21637 * q^36 + 23674 * q^37 - 4289 * q^38 + 38652 * q^39 + 36606 * q^40 + 44828 * q^41 - 9994 * q^42 + 21058 * q^43 + 13168 * q^44 - 1344 * q^45 + 6198 * q^46 + 5348 * q^47 - 69588 * q^48 + 19356 * q^49 + 33242 * q^50 + 22300 * q^51 - 60386 * q^52 + 4926 * q^53 - 24476 * q^54 - 56892 * q^55 - 87652 * q^56 + 22072 * q^57 - 25002 * q^58 + 16944 * q^59 - 138556 * q^60 - 73682 * q^61 + 6727 * q^62 - 138784 * q^63 - 1300 * q^64 + 63316 * q^65 - 128796 * q^66 - 134768 * q^67 - 9524 * q^68 + 3992 * q^69 - 142737 * q^70 + 123724 * q^71 + 93232 * q^72 + 70792 * q^73 + 307468 * q^74 - 302902 * q^75 - 65405 * q^76 + 107932 * q^77 + 36820 * q^78 - 26036 * q^79 + 576783 * q^80 + 82528 * q^81 + 142335 * q^82 + 21882 * q^83 + 299144 * q^84 - 130464 * q^85 + 480478 * q^86 - 440244 * q^87 - 71982 * q^88 + 164392 * q^89 + 397969 * q^90 - 60028 * q^91 + 31012 * q^92 - 1922 * q^93 - 170388 * q^94 - 7404 * q^95 - 331010 * q^96 - 131948 * q^97 + 288768 * q^98 - 180150 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.51317	0.267494	0.133747	−	0.991016i	$-0.457299\pi$
$2$	1.51317	0.267494	0.133747	+	0.991016i	$0.457299\pi$
$3$	22.4697	1.44143	0.720716	−	0.693230i	$-0.243813\pi$
$3$	22.4697	1.44143	0.720716	+	0.693230i	$0.243813\pi$
$4$	−29.7103	−0.928447
$5$	77.4836	1.38607	0.693034	−	0.720905i	$-0.256274\pi$
$5$	77.4836	1.38607	0.693034	+	0.720905i	$0.256274\pi$
$6$	34.0005	0.385574
$7$	20.7014	0.159682	0.0798408	−	0.996808i	$-0.474559\pi$
$7$	20.7014	0.159682	0.0798408	+	0.996808i	$0.474559\pi$
$8$	−93.3783	−0.515847
$9$	261.888	1.07773
$10$	117.246	0.370764
$11$	−13.6402	−0.0339890	−0.0169945	−	0.999856i	$-0.505410\pi$
$11$	−13.6402	−0.0339890	−0.0169945	+	0.999856i	$0.505410\pi$
$12$	−667.582	−1.33829
$13$	703.908	1.15520	0.577600	−	0.816320i	$-0.303989\pi$
$13$	703.908	1.15520	0.577600	+	0.816320i	$0.303989\pi$
$14$	31.3248	0.0427138
$15$	1741.03	1.99792
$16$	809.433	0.790461
$17$	−1360.44	−1.14171	−0.570857	−	0.821049i	$-0.693389\pi$
$17$	−1360.44	−1.14171	−0.570857	+	0.821049i	$0.693389\pi$
$18$	396.281	0.288285
$19$	−2516.75	−1.59939	−0.799697	−	0.600404i	$-0.795007\pi$
$19$	−2516.75	−1.59939	−0.799697	+	0.600404i	$0.795007\pi$
$20$	−2302.06	−1.28689
$21$	465.155	0.230170
$22$	−20.6399	−0.00909184
$23$	−3840.67	−1.51387	−0.756934	−	0.653492i	$-0.773303\pi$
$23$	−3840.67	−1.51387	−0.756934	+	0.653492i	$0.773303\pi$
$24$	−2098.18	−0.743559
$25$	2878.70	0.921185
$26$	1065.13	0.309009
$27$	424.401	0.112038
$28$	−615.046	−0.148256
$29$	1600.17	0.353322	0.176661	−	0.984272i	$-0.443470\pi$
$29$	1600.17	0.353322	0.176661	+	0.984272i	$0.443470\pi$
$30$	2634.48	0.534432
$31$	961.000	0.179605
$31$	961.000	0.179605
$32$	4212.92	0.727290
$33$	−306.491	−0.0489929
$34$	−2058.58	−0.305401
$35$	1604.02	0.221330
$36$	−7780.76	−1.00061
$37$	−7831.82	−0.940500	−0.470250	−	0.882533i	$-0.655836\pi$
$37$	−7831.82	−0.940500	−0.470250	+	0.882533i	$0.655836\pi$
$38$	−3808.27	−0.427828
$39$	15816.6	1.66514
$40$	−7235.28	−0.714999
$41$	−1135.29	−0.105474	−0.0527371	−	0.998608i	$-0.516795\pi$
$41$	−1135.29	−0.105474	−0.0527371	+	0.998608i	$0.516795\pi$
$42$	703.859	0.0615691
$43$	15289.3	1.26100	0.630501	−	0.776189i	$-0.282850\pi$
$43$	15289.3	1.26100	0.630501	+	0.776189i	$0.282850\pi$
$44$	405.254	0.0315570
$45$	20292.0	1.49380
$46$	−5811.60	−0.404950
$47$	29909.8	1.97500	0.987502	−	0.157604i	$-0.0503770\pi$
$47$	29909.8	1.97500	0.987502	+	0.157604i	$0.0503770\pi$
$48$	18187.7	1.13940
$49$	−16378.5	−0.974502
$50$	4355.97	0.246411
$51$	−30568.7	−1.64570
$52$	−20913.3	−1.07254
$53$	17590.3	0.860169	0.430084	−	0.902789i	$-0.358484\pi$
$53$	17590.3	0.860169	0.430084	+	0.902789i	$0.358484\pi$
$54$	642.192	0.0299696
$55$	−1056.89	−0.0471111
$56$	−1933.06	−0.0823713
$57$	−56550.6	−2.30542
$58$	2421.33	0.0945114
$59$	−16403.3	−0.613481	−0.306741	−	0.951793i	$-0.599238\pi$
$59$	−16403.3	−0.613481	−0.306741	+	0.951793i	$0.599238\pi$
$60$	−51726.6	−1.85497
$61$	23461.6	0.807295	0.403648	−	0.914915i	$-0.367742\pi$
$61$	23461.6	0.807295	0.403648	+	0.914915i	$0.367742\pi$
$62$	1454.16	0.0480433
$63$	5421.45	0.172093
$64$	−19527.0	−0.595916
$65$	54541.3	1.60119
$66$	−463.773	−0.0131053
$67$	12225.2	0.332712	0.166356	−	0.986066i	$-0.446800\pi$
$67$	12225.2	0.332712	0.166356	+	0.986066i	$0.446800\pi$
$68$	40419.1	1.06002
$69$	−86298.8	−2.18214
$70$	2427.16	0.0592043
$71$	−50123.9	−1.18005	−0.590023	−	0.807387i	$-0.700881\pi$
$71$	−50123.9	−1.18005	−0.590023	+	0.807387i	$0.700881\pi$
$72$	−24454.6	−0.555942
$73$	63123.3	1.38638	0.693190	−	0.720755i	$-0.256205\pi$
$73$	63123.3	1.38638	0.693190	+	0.720755i	$0.256205\pi$
$74$	−11850.9	−0.251578
$75$	64683.6	1.32783
$76$	74773.3	1.48495
$77$	−282.371	−0.00542742
$78$	23933.2	0.445415
$79$	−34426.9	−0.620627	−0.310313	−	0.950634i	$-0.600434\pi$
$79$	−34426.9	−0.620627	−0.310313	+	0.950634i	$0.600434\pi$
$80$	62717.7	1.09563
$81$	−54102.5	−0.916231
$82$	−1717.89	−0.0282137
$83$	34019.7	0.542044	0.271022	−	0.962573i	$-0.412638\pi$
$83$	34019.7	0.542044	0.271022	+	0.962573i	$0.412638\pi$
$84$	−13819.9	−0.213701
$85$	−105412.	−1.58249
$86$	23135.3	0.337310
$87$	35955.3	0.509290
$88$	1273.70	0.0175331
$89$	22185.6	0.296890	0.148445	−	0.988921i	$-0.452573\pi$
$89$	22185.6	0.296890	0.148445	+	0.988921i	$0.452573\pi$
$90$	30705.3	0.399583
$91$	14571.9	0.184464
$92$	114108.	1.40555
$93$	21593.4	0.258889
$94$	45258.6	0.528301
$95$	−195007.	−2.21687
$96$	94663.0	1.04834
$97$	−113665.	−1.22658	−0.613292	−	0.789857i	$-0.710155\pi$
$97$	−113665.	−1.22658	−0.613292	+	0.789857i	$0.710155\pi$
$98$	−24783.4	−0.260673
$99$	−3572.20	−0.0366309
$100$	−85527.1	−0.855271
$101$	136887.	1.33524	0.667620	−	0.744502i	$-0.267313\pi$
$101$	136887.	1.33524	0.667620	+	0.744502i	$0.267313\pi$
$102$	−46255.7	−0.440215
$103$	−44100.8	−0.409594	−0.204797	−	0.978805i	$-0.565653\pi$
$103$	−44100.8	−0.409594	−0.204797	+	0.978805i	$0.565653\pi$
$104$	−65729.7	−0.595907
$105$	36041.9	0.319032
$106$	26617.2	0.230090
$107$	100756.	0.850769	0.425385	−	0.905013i	$-0.360139\pi$
$107$	100756.	0.850769	0.425385	+	0.905013i	$0.360139\pi$
$108$	−12609.1	−0.104022
$109$	149149.	1.20241	0.601206	−	0.799094i	$-0.294687\pi$
$109$	149149.	1.20241	0.601206	+	0.799094i	$0.294687\pi$
$110$	−1599.26	−0.0126019
$111$	−175979.	−1.35567
$112$	16756.4	0.126222
$113$	215378.	1.58674	0.793370	−	0.608740i	$-0.208325\pi$
$113$	215378.	1.58674	0.793370	+	0.608740i	$0.208325\pi$
$114$	−85570.7	−0.616685
$115$	−297589.	−2.09832
$116$	−47541.5	−0.328041
$117$	184345.	1.24499
$118$	−24821.0	−0.164102
$119$	−28163.1	−0.182311
$120$	−162575.	−1.03062
$121$	−160865.	−0.998845
$122$	35501.4	0.215946
$123$	−25509.6	−0.152034
$124$	−28551.6	−0.166754
$125$	−19084.0	−0.109243
$126$	8203.58	0.0460338
$127$	−347582.	−1.91226	−0.956131	−	0.292939i	$-0.905367\pi$
$127$	−347582.	−1.91226	−0.956131	+	0.292939i	$0.905367\pi$
$128$	−164361.	−0.886694
$129$	343545.	1.81765
$130$	82530.3	0.428307
$131$	−12743.3	−0.0648790	−0.0324395	−	0.999474i	$-0.510328\pi$
$131$	−12743.3	−0.0648790	−0.0324395	+	0.999474i	$0.510328\pi$
$132$	9105.94	0.0454873
$133$	−52100.2	−0.255394
$134$	18498.8	0.0889983
$135$	32884.1	0.155293
$136$	127036.	0.588950
$137$	54494.2	0.248056	0.124028	−	0.992279i	$-0.460419\pi$
$137$	54494.2	0.248056	0.124028	+	0.992279i	$0.460419\pi$
$138$	−130585.	−0.583708
$139$	292153.	1.28255	0.641273	−	0.767312i	$-0.278406\pi$
$139$	292153.	1.28255	0.641273	+	0.767312i	$0.278406\pi$
$140$	−47655.9	−0.205493
$141$	672063.	2.84684
$142$	−75846.0	−0.315655
$143$	−9601.43	−0.0392641
$144$	211980.	0.851902
$145$	123987.	0.489728
$146$	95516.4	0.370848
$147$	−368019.	−1.40468
$148$	232686.	0.873204
$149$	−205340.	−0.757720	−0.378860	−	0.925454i	$-0.623684\pi$
$149$	−205340.	−0.757720	−0.378860	+	0.925454i	$0.623684\pi$
$150$	97877.4	0.355185
$151$	−208992.	−0.745911	−0.372956	−	0.927849i	$-0.621656\pi$
$151$	−208992.	−0.745911	−0.372956	+	0.927849i	$0.621656\pi$
$152$	235010.	0.825043
$153$	−356283.	−1.23046
$154$	−427.276	−0.00145180
$155$	74461.7	0.248945
$156$	−469916.	−1.54600
$157$	414617.	1.34245	0.671225	−	0.741254i	$-0.265769\pi$
$157$	414617.	1.34245	0.671225	+	0.741254i	$0.265769\pi$
$158$	−52093.8	−0.166014
$159$	395249.	1.23988
$160$	326432.	1.00807
$161$	−79507.4	−0.241737
$162$	−81866.4	−0.245086
$163$	−74081.6	−0.218394	−0.109197	−	0.994020i	$-0.534828\pi$
$163$	−74081.6	−0.218394	−0.109197	+	0.994020i	$0.534828\pi$
$164$	33729.8	0.0979272
$165$	−23748.0	−0.0679074
$166$	51477.6	0.144993
$167$	−355478.	−0.986329	−0.493164	−	0.869936i	$-0.664160\pi$
$167$	−355478.	−0.986329	−0.493164	+	0.869936i	$0.664160\pi$
$168$	−43435.4	−0.118733
$169$	124193.	0.334488
$170$	−159506.	−0.423307
$171$	−659105.	−1.72371
$172$	−454249.	−1.17077
$173$	511596.	1.29961	0.649803	−	0.760102i	$-0.274851\pi$
$173$	511596.	1.29961	0.649803	+	0.760102i	$0.274851\pi$
$174$	54406.6	0.136232
$175$	59593.2	0.147096
$176$	−11040.8	−0.0268670
$177$	−368577.	−0.884291
$178$	33570.6	0.0794162
$179$	326430.	0.761477	0.380739	−	0.924683i	$-0.375670\pi$
$179$	326430.	0.761477	0.380739	+	0.924683i	$0.375670\pi$
$180$	−602881.	−1.38692
$181$	529461.	1.20126	0.600631	−	0.799527i	$-0.294916\pi$
$181$	529461.	1.20126	0.600631	+	0.799527i	$0.294916\pi$
$182$	22049.8	0.0493430
$183$	527174.	1.16366
$184$	358636.	0.780924
$185$	−606838.	−1.30360
$186$	32674.5	0.0692511
$187$	18556.7	0.0388058
$188$	−888628.	−1.83369
$189$	8785.71	0.0178905
$190$	−295078.	−0.592998
$191$	−227182.	−0.450598	−0.225299	−	0.974290i	$-0.572336\pi$
$191$	−227182.	−0.450598	−0.225299	+	0.974290i	$0.572336\pi$
$192$	−438765.	−0.858973
$193$	82697.5	0.159808	0.0799041	−	0.996803i	$-0.474539\pi$
$193$	82697.5	0.159808	0.0799041	+	0.996803i	$0.474539\pi$
$194$	−171995.	−0.328103
$195$	1.22553e6	2.30800
$196$	486609.	0.904773
$197$	−246100.	−0.451800	−0.225900	−	0.974150i	$-0.572532\pi$
$197$	−246100.	−0.451800	−0.225900	+	0.974150i	$0.572532\pi$
$198$	−5405.35	−0.00979852
$199$	−762519.	−1.36495	−0.682477	−	0.730907i	$-0.739097\pi$
$199$	−762519.	−1.36495	−0.682477	+	0.730907i	$0.739097\pi$
$200$	−268808.	−0.475191
$201$	274696.	0.479582
$202$	207134.	0.357168
$203$	33125.8	0.0564191
$204$	908206.	1.52795
$205$	−87966.1	−0.146194
$206$	−66732.0	−0.109564
$207$	−1.00583e6	−1.63154
$208$	569766.	0.913142
$209$	34328.9	0.0543618
$210$	54537.5	0.0853389
$211$	−233035.	−0.360343	−0.180171	−	0.983635i	$-0.557665\pi$
$211$	−233035.	−0.360343	−0.180171	+	0.983635i	$0.557665\pi$
$212$	−522613.	−0.798621
$213$	−1.12627e6	−1.70096
$214$	152461.	0.227575
$215$	1.18467e6	1.74783
$216$	−39629.9	−0.0577947
$217$	19894.1	0.0286797
$218$	225688.	0.321637
$219$	1.41836e6	1.99837
$220$	31400.5	0.0437402
$221$	−957626.	−1.31891
$222$	−266286.	−0.362632
$223$	−861370.	−1.15992	−0.579959	−	0.814646i	$-0.696931\pi$
$223$	−861370.	−1.15992	−0.579959	+	0.814646i	$0.696931\pi$
$224$	87213.4	0.116135
$225$	753897.	0.992786
$226$	325904.	0.424443
$227$	833699.	1.07385	0.536927	−	0.843629i	$-0.319585\pi$
$227$	833699.	1.07385	0.536927	+	0.843629i	$0.319585\pi$
$228$	1.68013e6	2.14046
$229$	362589.	0.456906	0.228453	−	0.973555i	$-0.426633\pi$
$229$	362589.	0.456906	0.228453	+	0.973555i	$0.426633\pi$
$230$	−450303.	−0.561288
$231$	−6344.80	−0.00782326
$232$	−149421.	−0.182260
$233$	−273542.	−0.330092	−0.165046	−	0.986286i	$-0.552777\pi$
$233$	−273542.	−0.330092	−0.165046	+	0.986286i	$0.552777\pi$
$234$	278945.	0.333027
$235$	2.31751e6	2.73749
$236$	487347.	0.569585
$237$	−773563.	−0.894591
$238$	−42615.6	−0.0487670
$239$	1.26051e6	1.42742	0.713712	−	0.700439i	$-0.247013\pi$
$239$	1.26051e6	1.42742	0.713712	+	0.700439i	$0.247013\pi$
$240$	1.40925e6	1.57928
$241$	−837791.	−0.929165	−0.464583	−	0.885530i	$-0.653796\pi$
$241$	−837791.	−0.929165	−0.464583	+	0.885530i	$0.653796\pi$
$242$	−243416.	−0.267185
$243$	−1.31880e6	−1.43272
$244$	−697050.	−0.749531
$245$	−1.26906e6	−1.35073
$246$	−38600.4	−0.0406681
$247$	−1.77156e6	−1.84762
$248$	−89736.6	−0.0926489
$249$	764412.	0.781320
$250$	−28877.4	−0.0292219
$251$	−1.42635e6	−1.42903	−0.714517	−	0.699619i	$-0.753353\pi$
$251$	−1.42635e6	−1.42903	−0.714517	+	0.699619i	$0.753353\pi$
$252$	−161073.	−0.159780
$253$	52387.5	0.0514549
$254$	−525951.	−0.511518
$255$	−2.36857e6	−2.28106
$256$	376157.	0.358731
$257$	763758.	0.721312	0.360656	−	0.932699i	$-0.382553\pi$
$257$	763758.	0.721312	0.360656	+	0.932699i	$0.382553\pi$
$258$	519843.	0.486209
$259$	−162130.	−0.150181
$260$	−1.62044e6	−1.48662
$261$	419064.	0.380785
$262$	−19282.8	−0.0173547
$263$	−571402.	−0.509393	−0.254696	−	0.967021i	$-0.581975\pi$
$263$	−571402.	−0.509393	−0.254696	+	0.967021i	$0.581975\pi$
$264$	28619.6	0.0252728
$265$	1.36296e6	1.19225
$266$	−78836.6	−0.0683162
$267$	498504.	0.427947
$268$	−363214.	−0.308905
$269$	−222030.	−0.187081	−0.0935407	−	0.995615i	$-0.529819\pi$
$269$	−222030.	−0.187081	−0.0935407	+	0.995615i	$0.529819\pi$
$270$	49759.3	0.0415398
$271$	−1.47266e6	−1.21809	−0.609043	−	0.793137i	$-0.708447\pi$
$271$	−1.47266e6	−1.21809	−0.609043	+	0.793137i	$0.708447\pi$
$272$	−1.10119e6	−0.902482
$273$	327426.	0.265893
$274$	82459.1	0.0663533
$275$	−39266.0	−0.0313102
$276$	2.56396e6	2.02600
$277$	483155.	0.378344	0.189172	−	0.981944i	$-0.439420\pi$
$277$	483155.	0.378344	0.189172	+	0.981944i	$0.439420\pi$
$278$	442078.	0.343073
$279$	251674.	0.193566
$280$	−149781.	−0.114172
$281$	1.92887e6	1.45726	0.728631	−	0.684907i	$-0.240157\pi$
$281$	1.92887e6	1.45726	0.728631	+	0.684907i	$0.240157\pi$
$282$	1.01695e6	0.761510
$283$	1.90674e6	1.41523	0.707614	−	0.706599i	$-0.249772\pi$
$283$	1.90674e6	1.41523	0.707614	+	0.706599i	$0.249772\pi$
$284$	1.48920e6	1.09561
$285$	−4.38174e6	−3.19547
$286$	−14528.6	−0.0105029
$287$	−23502.1	−0.0168423
$288$	1.10331e6	0.783821
$289$	430945.	0.303513
$290$	187613.	0.130999
$291$	−2.55402e6	−1.76804
$292$	−1.87541e6	−1.28718
$293$	2.19302e6	1.49236	0.746179	−	0.665746i	$-0.231887\pi$
$293$	2.19302e6	1.49236	0.746179	+	0.665746i	$0.231887\pi$
$294$	−556876.	−0.375742
$295$	−1.27099e6	−0.850327
$296$	731322.	0.485154
$297$	−5788.91	−0.00380808
$298$	−310715.	−0.202685
$299$	−2.70348e6	−1.74882
$300$	−1.92177e6	−1.23282
$301$	316510.	0.201359
$302$	−316241.	−0.199526
$303$	3.07582e6	1.92466
$304$	−2.03714e6	−1.26426
$305$	1.81789e6	1.11897
$306$	−539117.	−0.329139
$307$	735651.	0.445478	0.222739	−	0.974878i	$-0.428500\pi$
$307$	735651.	0.445478	0.222739	+	0.974878i	$0.428500\pi$
$308$	8389.34	0.00503908
$309$	−990931.	−0.590401
$310$	112673.	0.0665912
$311$	2.03955e6	1.19573	0.597865	−	0.801597i	$-0.296016\pi$
$311$	2.03955e6	1.19573	0.597865	+	0.801597i	$0.296016\pi$
$312$	−1.47693e6	−0.858959
$313$	646526.	0.373014	0.186507	−	0.982454i	$-0.440283\pi$
$313$	646526.	0.373014	0.186507	+	0.982454i	$0.440283\pi$
$314$	627387.	0.359097
$315$	420073.	0.238533
$316$	1.02283e6	0.576219
$317$	−2.74682e6	−1.53526	−0.767631	−	0.640892i	$-0.778565\pi$
$317$	−2.74682e6	−1.53526	−0.767631	+	0.640892i	$0.778565\pi$
$318$	598080.	0.331659
$319$	−21826.6	−0.0120091
$320$	−1.51302e6	−0.825980
$321$	2.26396e6	1.22633
$322$	−120308.	−0.0646630
$323$	3.42389e6	1.82605
$324$	1.60740e6	0.850672
$325$	2.02634e6	1.06415
$326$	−112098.	−0.0584191
$327$	3.35133e6	1.73320
$328$	106011.	0.0544086
$329$	619174.	0.315372
$330$	−35934.8	−0.0181648
$331$	−1.07651e6	−0.540068	−0.270034	−	0.962851i	$-0.587035\pi$
$331$	−1.07651e6	−0.540068	−0.270034	+	0.962851i	$0.587035\pi$
$332$	−1.01073e6	−0.503259
$333$	−2.05106e6	−1.01360
$334$	−537899.	−0.263837
$335$	947250.	0.461161
$336$	376512.	0.181941
$337$	−1.39604e6	−0.669613	−0.334806	−	0.942287i	$-0.608671\pi$
$337$	−1.39604e6	−0.669613	−0.334806	+	0.942287i	$0.608671\pi$
$338$	187926.	0.0894735
$339$	4.83949e6	2.28718
$340$	3.13182e6	1.46926
$341$	−13108.2	−0.00610461
$342$	−997339.	−0.461081
$343$	−686986.	−0.315292
$344$	−1.42769e6	−0.650484
$345$	−6.68674e6	−3.02459
$346$	774133.	0.347636
$347$	192447.	0.0858002	0.0429001	−	0.999079i	$-0.486340\pi$
$347$	192447.	0.0858002	0.0429001	+	0.999079i	$0.486340\pi$
$348$	−1.06824e6	−0.472849
$349$	−1.64570e6	−0.723247	−0.361623	−	0.932324i	$-0.617777\pi$
$349$	−1.64570e6	−0.723247	−0.361623	+	0.932324i	$0.617777\pi$
$350$	90174.8	0.0393473
$351$	298739.	0.129427
$352$	−57465.0	−0.0247199
$353$	1.39838e6	0.597292	0.298646	−	0.954364i	$-0.403465\pi$
$353$	1.39838e6	0.597292	0.298646	+	0.954364i	$0.403465\pi$
$354$	−557721.	−0.236542
$355$	−3.88378e6	−1.63562
$356$	−659141.	−0.275647
$357$	−632816.	−0.262789
$358$	493944.	0.203690
$359$	157908.	0.0646650	0.0323325	−	0.999477i	$-0.489706\pi$
$359$	157908.	0.0646650	0.0323325	+	0.999477i	$0.489706\pi$
$360$	−1.89483e6	−0.770574
$361$	3.85792e6	1.55806
$362$	801165.	0.321330
$363$	−3.61459e6	−1.43977
$364$	−432936.	−0.171265
$365$	4.89102e6	1.92162
$366$	797706.	0.311272
$367$	−1.67243e6	−0.648161	−0.324081	−	0.946029i	$-0.605055\pi$
$367$	−1.67243e6	−0.648161	−0.324081	+	0.946029i	$0.605055\pi$
$368$	−3.10877e6	−1.19665
$369$	−297318.	−0.113672
$370$	−918250.	−0.348704
$371$	364144.	0.137353
$372$	−641546.	−0.240365
$373$	−2.68053e6	−0.997583	−0.498791	−	0.866722i	$-0.666223\pi$
$373$	−2.68053e6	−0.997583	−0.498791	+	0.866722i	$0.666223\pi$
$374$	28079.4	0.0103803
$375$	−428812.	−0.157467
$376$	−2.79292e6	−1.01880
$377$	1.12637e6	0.408158
$378$	13294.3	0.00478559
$379$	−949913.	−0.339692	−0.169846	−	0.985471i	$-0.554327\pi$
$379$	−949913.	−0.339692	−0.169846	+	0.985471i	$0.554327\pi$
$380$	5.79370e6	2.05825
$381$	−7.81005e6	−2.75640
$382$	−343765.	−0.120532
$383$	−512479.	−0.178517	−0.0892584	−	0.996009i	$-0.528450\pi$
$383$	−512479.	−0.178517	−0.0892584	+	0.996009i	$0.528450\pi$
$384$	−3.69314e6	−1.27811
$385$	−21879.1	−0.00752278
$386$	125136.	0.0427477
$387$	4.00407e6	1.35902
$388$	3.37702e6	1.13882
$389$	2.75649e6	0.923597	0.461798	−	0.886985i	$-0.347204\pi$
$389$	2.75649e6	0.923597	0.461798	+	0.886985i	$0.347204\pi$
$390$	1.85443e6	0.617376
$391$	5.22501e6	1.72840
$392$	1.52939e6	0.502694
$393$	−286339.	−0.0935187
$394$	−372392.	−0.120854
$395$	−2.66752e6	−0.860231
$396$	106131.	0.0340098
$397$	−4.00659e6	−1.27585	−0.637923	−	0.770100i	$-0.720206\pi$
$397$	−4.00659e6	−1.27585	−0.637923	+	0.770100i	$0.720206\pi$
$398$	−1.15382e6	−0.365116
$399$	−1.17068e6	−0.368133
$400$	2.33012e6	0.728161
$401$	756703.	0.234998	0.117499	−	0.993073i	$-0.462512\pi$
$401$	756703.	0.234998	0.117499	+	0.993073i	$0.462512\pi$
$402$	415663.	0.128285
$403$	676455.	0.207480
$404$	−4.06696e6	−1.23970
$405$	−4.19206e6	−1.26996
$406$	50125.0	0.0150917
$407$	106828.	0.0319667
$408$	2.85446e6	0.848932
$409$	4.13510e6	1.22230	0.611150	−	0.791515i	$-0.290707\pi$
$409$	4.13510e6	1.22230	0.611150	+	0.791515i	$0.290707\pi$
$410$	−133108.	−0.0391061
$411$	1.22447e6	0.357555
$412$	1.31025e6	0.380286
$413$	−339572.	−0.0979617
$414$	−1.52199e6	−0.436425
$415$	2.63596e6	0.751310
$416$	2.96550e6	0.840166
$417$	6.56459e6	1.84870
$418$	51945.5	0.0145414
$419$	309961.	0.0862526	0.0431263	−	0.999070i	$-0.486268\pi$
$419$	309961.	0.0862526	0.0431263	+	0.999070i	$0.486268\pi$
$420$	−1.07081e6	−0.296204
$421$	986987.	0.271398	0.135699	−	0.990750i	$-0.456672\pi$
$421$	986987.	0.271398	0.135699	+	0.990750i	$0.456672\pi$
$422$	−352622.	−0.0963893
$423$	7.83300e6	2.12852
$424$	−1.64255e6	−0.443716
$425$	−3.91631e6	−1.05173
$426$	−1.70424e6	−0.454995
$427$	485688.	0.128910
$428$	−2.99349e6	−0.789894
$429$	−215741.	−0.0565966
$430$	1.79260e6	0.467534
$431$	1.55460e6	0.403111	0.201556	−	0.979477i	$-0.435400\pi$
$431$	1.55460e6	0.403111	0.201556	+	0.979477i	$0.435400\pi$
$432$	343524.	0.0885621
$433$	−6.73484e6	−1.72626	−0.863132	−	0.504978i	$-0.831501\pi$
$433$	−6.73484e6	−1.72626	−0.863132	+	0.504978i	$0.831501\pi$
$434$	30103.1	0.00767163
$435$	2.78595e6	0.705910
$436$	−4.43125e6	−1.11638
$437$	9.66600e6	2.42127
$438$	2.14623e6	0.534552
$439$	692966.	0.171613	0.0858065	−	0.996312i	$-0.472653\pi$
$439$	692966.	0.171613	0.0858065	+	0.996312i	$0.472653\pi$
$440$	98690.6	0.0243021
$441$	−4.28931e6	−1.05025
$442$	−1.44905e6	−0.352800
$443$	−583148.	−0.141179	−0.0705894	−	0.997505i	$-0.522488\pi$
$443$	−583148.	−0.141179	−0.0705894	+	0.997505i	$0.522488\pi$
$444$	5.22838e6	1.25866
$445$	1.71902e6	0.411510
$446$	−1.30340e6	−0.310271
$447$	−4.61394e6	−1.09220
$448$	−404236.	−0.0951569
$449$	−4.44718e6	−1.04104	−0.520522	−	0.853848i	$-0.674263\pi$
$449$	−4.44718e6	−1.04104	−0.520522	+	0.853848i	$0.674263\pi$
$450$	1.14078e6	0.265564
$451$	15485.5	0.00358496
$452$	−6.39896e6	−1.47320
$453$	−4.69599e6	−1.07518
$454$	1.26153e6	0.287249
$455$	1.12908e6	0.255680
$456$	5.28060e6	1.18924
$457$	−7.39409e6	−1.65613	−0.828065	−	0.560633i	$-0.810558\pi$
$457$	−7.39409e6	−1.65613	−0.828065	+	0.560633i	$0.810558\pi$
$458$	548660.	0.122219
$459$	−577373.	−0.127916
$460$	8.84146e6	1.94818
$461$	420559.	0.0921668	0.0460834	−	0.998938i	$-0.485326\pi$
$461$	420559.	0.0921668	0.0460834	+	0.998938i	$0.485326\pi$
$462$	−9600.77	−0.00209267
$463$	−8.68419e6	−1.88268	−0.941341	−	0.337458i	$-0.890433\pi$
$463$	−8.68419e6	−1.88268	−0.941341	+	0.337458i	$0.890433\pi$
$464$	1.29523e6	0.279287
$465$	1.67313e6	0.358838
$466$	−413917.	−0.0882974
$467$	5.35288e6	1.13578	0.567892	−	0.823103i	$-0.307759\pi$
$467$	5.35288e6	1.13578	0.567892	+	0.823103i	$0.307759\pi$
$468$	−5.47694e6	−1.15591
$469$	253079.	0.0531280
$470$	3.50680e6	0.732261
$471$	9.31632e6	1.93505
$472$	1.53171e6	0.316462
$473$	−208548.	−0.0428602
$474$	−1.17053e6	−0.239297
$475$	−7.24497e6	−1.47334
$476$	836734.	0.169266
$477$	4.60668e6	0.927027
$478$	1.90737e6	0.381827
$479$	−1.50802e6	−0.300310	−0.150155	−	0.988662i	$-0.547977\pi$
$479$	−1.50802e6	−0.300310	−0.150155	+	0.988662i	$0.547977\pi$
$480$	7.33483e6	1.45307
$481$	−5.51288e6	−1.08647
$482$	−1.26772e6	−0.248546
$483$	−1.78651e6	−0.348447
$484$	4.77935e6	0.927375
$485$	−8.80716e6	−1.70013
$486$	−1.99557e6	−0.383244
$487$	−3.00208e6	−0.573588	−0.286794	−	0.957992i	$-0.592590\pi$
$487$	−3.00208e6	−0.573588	−0.286794	+	0.957992i	$0.592590\pi$
$488$	−2.19080e6	−0.416441
$489$	−1.66459e6	−0.314801
$490$	−1.92031e6	−0.361310
$491$	−6.33984e6	−1.18679	−0.593396	−	0.804911i	$-0.702213\pi$
$491$	−6.33984e6	−1.18679	−0.593396	+	0.804911i	$0.702213\pi$
$492$	757898.	0.141155
$493$	−2.17694e6	−0.403393
$494$	−2.68067e6	−0.494227
$495$	−276787.	−0.0507729
$496$	777865.	0.141971
$497$	−1.03764e6	−0.188432
$498$	1.15669e6	0.208998
$499$	−5.20836e6	−0.936375	−0.468188	−	0.883629i	$-0.655093\pi$
$499$	−5.20836e6	−0.936375	−0.468188	+	0.883629i	$0.655093\pi$
$500$	566992.	0.101427
$501$	−7.98749e6	−1.42173
$502$	−2.15832e6	−0.382257
$503$	−4.63376e6	−0.816608	−0.408304	−	0.912846i	$-0.633880\pi$
$503$	−4.63376e6	−0.816608	−0.408304	+	0.912846i	$0.633880\pi$
$504$	−506246.	−0.0887738
$505$	1.06065e7	1.85073
$506$	79271.3	0.0137638
$507$	2.79058e6	0.482142
$508$	1.03268e7	1.77543
$509$	1.02585e6	0.175506	0.0877529	−	0.996142i	$-0.472031\pi$
$509$	1.02585e6	0.175506	0.0877529	+	0.996142i	$0.472031\pi$
$510$	−3.58406e6	−0.610168
$511$	1.30674e6	0.221380
$512$	5.82874e6	0.982652
$513$	−1.06811e6	−0.179194
$514$	1.15570e6	0.192946
$515$	−3.41708e6	−0.567725
$516$	−1.02068e7	−1.68759
$517$	−407975.	−0.0671285
$518$	−245330.	−0.0401723
$519$	1.14954e7	1.87330
$520$	−5.09297e6	−0.825968
$521$	−461300.	−0.0744542	−0.0372271	−	0.999307i	$-0.511852\pi$
$521$	−461300.	−0.0744542	−0.0372271	+	0.999307i	$0.511852\pi$
$522$	634117.	0.101857
$523$	1.69073e6	0.270284	0.135142	−	0.990826i	$-0.456851\pi$
$523$	1.69073e6	0.270284	0.135142	+	0.990826i	$0.456851\pi$
$524$	378608.	0.0602367
$525$	1.33904e6	0.212029
$526$	−864630.	−0.136259
$527$	−1.30738e6	−0.205058
$528$	−248084.	−0.0387270
$529$	8.31443e6	1.29179
$530$	2.06239e6	0.318920
$531$	−4.29582e6	−0.661165
$532$	1.54791e6	0.237120
$533$	−799138.	−0.121844
$534$	754322.	0.114473
$535$	7.80694e6	1.17922
$536$	−1.14157e6	−0.171628
$537$	7.33478e6	1.09762
$538$	−335969.	−0.0500430
$539$	223405.	0.0331224
$540$	−976997.	−0.144181
$541$	−1.30072e6	−0.191069	−0.0955344	−	0.995426i	$-0.530456\pi$
$541$	−1.30072e6	−0.191069	−0.0955344	+	0.995426i	$0.530456\pi$
$542$	−2.22838e6	−0.325830
$543$	1.18968e7	1.73154
$544$	−5.73143e6	−0.830358
$545$	1.15566e7	1.66662
$546$	495452.	0.0711246
$547$	8.08359e6	1.15514	0.577572	−	0.816340i	$-0.304000\pi$
$547$	8.08359e6	1.15514	0.577572	+	0.816340i	$0.304000\pi$
$548$	−1.61904e6	−0.230307
$549$	6.14429e6	0.870044
$550$	−59416.3	−0.00837526
$551$	−4.02722e6	−0.565101
$552$	8.05844e6	1.12565
$553$	−712686.	−0.0991027
$554$	731096.	0.101205
$555$	−1.36355e7	−1.87905
$556$	−8.67995e6	−1.19078
$557$	3.20915e6	0.438280	0.219140	−	0.975693i	$-0.429675\pi$
$557$	3.20915e6	0.438280	0.219140	+	0.975693i	$0.429675\pi$
$558$	380826.	0.0517775
$559$	1.07622e7	1.45671
$560$	1.29835e6	0.174953
$561$	416963.	0.0559359
$562$	2.91871e6	0.389808
$563$	−8.74751e6	−1.16309	−0.581545	−	0.813514i	$-0.697552\pi$
$563$	−8.74751e6	−1.16309	−0.581545	+	0.813514i	$0.697552\pi$
$564$	−1.99672e7	−2.64314
$565$	1.66883e7	2.19933
$566$	2.88523e6	0.378564
$567$	−1.12000e6	−0.146305
$568$	4.68048e6	0.608723
$569$	−7.46467e6	−0.966562	−0.483281	−	0.875465i	$-0.660555\pi$
$569$	−7.46467e6	−0.966562	−0.483281	+	0.875465i	$0.660555\pi$
$570$	−6.63032e6	−0.854767
$571$	−4.98994e6	−0.640479	−0.320240	−	0.947337i	$-0.603763\pi$
$571$	−4.98994e6	−0.640479	−0.320240	+	0.947337i	$0.603763\pi$
$572$	285262.	0.0364547
$573$	−5.10470e6	−0.649507
$574$	−35562.7	−0.00450521
$575$	−1.10562e7	−1.39455
$576$	−5.11387e6	−0.642235
$577$	7.68798e6	0.961331	0.480666	−	0.876904i	$-0.340395\pi$
$577$	7.68798e6	0.961331	0.480666	+	0.876904i	$0.340395\pi$
$578$	652094.	0.0811877
$579$	1.85819e6	0.230353
$580$	−3.68369e6	−0.454687
$581$	704255.	0.0865545
$582$	−3.86467e6	−0.472938
$583$	−239935.	−0.0292363
$584$	−5.89435e6	−0.715161
$585$	1.42837e7	1.72564
$586$	3.31841e6	0.399196
$587$	1.26155e7	1.51116	0.755580	−	0.655057i	$-0.227355\pi$
$587$	1.26155e7	1.51116	0.755580	+	0.655057i	$0.227355\pi$
$588$	1.09340e7	1.30417
$589$	−2.41859e6	−0.287260
$590$	−1.92322e6	−0.227457
$591$	−5.52980e6	−0.651240
$592$	−6.33933e6	−0.743429
$593$	7.73692e6	0.903507	0.451754	−	0.892143i	$-0.350799\pi$
$593$	7.73692e6	0.903507	0.451754	+	0.892143i	$0.350799\pi$
$594$	−8759.61	−0.00101864
$595$	−2.18218e6	−0.252695
$596$	6.10073e6	0.703503
$597$	−1.71336e7	−1.96749
$598$	−4.09083e6	−0.467798
$599$	−1.42113e7	−1.61833	−0.809163	−	0.587584i	$-0.800079\pi$
$599$	−1.42113e7	−1.61833	−0.809163	+	0.587584i	$0.800079\pi$
$600$	−6.04004e6	−0.684955
$601$	−448124.	−0.0506071	−0.0253036	−	0.999680i	$-0.508055\pi$
$601$	−448124.	−0.0506071	−0.0253036	+	0.999680i	$0.508055\pi$
$602$	478934.	0.0538622
$603$	3.20162e6	0.358573
$604$	6.20922e6	0.692539
$605$	−1.24644e7	−1.38447
$606$	4.65424e6	0.514834
$607$	4.64948e6	0.512192	0.256096	−	0.966651i	$-0.417564\pi$
$607$	4.64948e6	0.512192	0.256096	+	0.966651i	$0.417564\pi$
$608$	−1.06028e7	−1.16322
$609$	744326.	0.0813243
$610$	2.75077e6	0.299316
$611$	2.10537e7	2.28153
$612$	1.05853e7	1.14241
$613$	−2.36416e6	−0.254112	−0.127056	−	0.991896i	$-0.540553\pi$
$613$	−2.36416e6	−0.254112	−0.127056	+	0.991896i	$0.540553\pi$
$614$	1.11317e6	0.119162
$615$	−1.97657e6	−0.210729
$616$	26367.4	0.00279972
$617$	5.59685e6	0.591876	0.295938	−	0.955207i	$-0.404368\pi$
$617$	5.59685e6	0.591876	0.295938	+	0.955207i	$0.404368\pi$
$618$	−1.49945e6	−0.157929
$619$	−1.58165e7	−1.65914	−0.829569	−	0.558404i	$-0.811414\pi$
$619$	−1.58165e7	−1.65914	−0.829569	+	0.558404i	$0.811414\pi$
$620$	−2.21228e6	−0.231132
$621$	−1.62999e6	−0.169611
$622$	3.08619e6	0.319850
$623$	459273.	0.0474079
$624$	1.28025e7	1.31623
$625$	−1.04746e7	−1.07260
$626$	978304.	0.0997788
$627$	771360.	0.0783589
$628$	−1.23184e7	−1.24639
$629$	1.06547e7	1.07378
$630$	635643.	0.0638060
$631$	8.25092e6	0.824952	0.412476	−	0.910968i	$-0.364664\pi$
$631$	8.25092e6	0.824952	0.412476	+	0.910968i	$0.364664\pi$
$632$	3.21473e6	0.320148
$633$	−5.23624e6	−0.519409
$634$	−4.15642e6	−0.410673
$635$	−2.69319e7	−2.65053
$636$	−1.17430e7	−1.15116
$637$	−1.15289e7	−1.12574
$638$	−33027.4	−0.00321235
$639$	−1.31268e7	−1.27177
$640$	−1.27353e7	−1.22902
$641$	1.83243e7	1.76150	0.880751	−	0.473579i	$-0.157038\pi$
$641$	1.83243e7	1.76150	0.880751	+	0.473579i	$0.157038\pi$
$642$	3.42576e6	0.328034
$643$	1.37837e7	1.31474	0.657369	−	0.753569i	$-0.271669\pi$
$643$	1.37837e7	1.31474	0.657369	+	0.753569i	$0.271669\pi$
$644$	2.36219e6	0.224440
$645$	2.66191e7	2.51938
$646$	5.18093e6	0.488457
$647$	−5.84932e6	−0.549344	−0.274672	−	0.961538i	$-0.588569\pi$
$647$	−5.84932e6	−0.549344	−0.274672	+	0.961538i	$0.588569\pi$
$648$	5.05200e6	0.472635
$649$	223744.	0.0208516
$650$	3.06620e6	0.284654
$651$	447014.	0.0413398
$652$	2.20099e6	0.202768
$653$	−1.67161e7	−1.53410	−0.767048	−	0.641589i	$-0.778275\pi$
$653$	−1.67161e7	−1.53410	−0.767048	+	0.641589i	$0.778275\pi$
$654$	5.07114e6	0.463619
$655$	−987397.	−0.0899267
$656$	−918939.	−0.0833733
$657$	1.65312e7	1.49414
$658$	936917.	0.0843600
$659$	−1.31524e7	−1.17975	−0.589877	−	0.807493i	$-0.700824\pi$
$659$	−1.31524e7	−1.17975	−0.589877	+	0.807493i	$0.700824\pi$
$660$	705561.	0.0630485
$661$	1.26307e7	1.12441	0.562205	−	0.826998i	$-0.309953\pi$
$661$	1.26307e7	1.12441	0.562205	+	0.826998i	$0.309953\pi$
$662$	−1.62895e6	−0.144465
$663$	−2.15176e7	−1.90112
$664$	−3.17670e6	−0.279612
$665$	−4.03691e6	−0.353993
$666$	−3.10360e6	−0.271132
$667$	−6.14572e6	−0.534883
$668$	1.05614e7	0.915754
$669$	−1.93547e7	−1.67194
$670$	1.43335e6	0.123358
$671$	−320020.	−0.0274392
$672$	1.95966e6	0.167401
$673$	−8.15038e6	−0.693649	−0.346825	−	0.937930i	$-0.612740\pi$
$673$	−8.15038e6	−0.693649	−0.346825	+	0.937930i	$0.612740\pi$
$674$	−2.11245e6	−0.179117
$675$	1.22172e6	0.103208
$676$	−3.68982e6	−0.310555
$677$	−6.87475e6	−0.576481	−0.288241	−	0.957558i	$-0.593070\pi$
$677$	−6.87475e6	−0.576481	−0.288241	+	0.957558i	$0.593070\pi$
$678$	7.32298e6	0.611805
$679$	−2.35303e6	−0.195863
$680$	9.84318e6	0.816325
$681$	1.87330e7	1.54789
$682$	−19835.0	−0.00163294
$683$	4.71691e6	0.386906	0.193453	−	0.981110i	$-0.438031\pi$
$683$	4.71691e6	0.386906	0.193453	+	0.981110i	$0.438031\pi$
$684$	1.95822e7	1.60037
$685$	4.22241e6	0.343822
$686$	−1.03953e6	−0.0843385
$687$	8.14728e6	0.658598
$688$	1.23756e7	0.996773
$689$	1.23820e7	0.993668
$690$	−1.01182e7	−0.809058
$691$	1.74137e6	0.138738	0.0693692	−	0.997591i	$-0.477901\pi$
$691$	1.74137e6	0.138738	0.0693692	+	0.997591i	$0.477901\pi$
$692$	−1.51997e7	−1.20662
$693$	−73949.6	−0.00584928
$694$	291206.	0.0229510
$695$	2.26371e7	1.77770
$696$	−3.35745e6	−0.262716
$697$	1.54449e6	0.120421
$698$	−2.49022e6	−0.193464
$699$	−6.14642e6	−0.475805
$700$	−1.77053e6	−0.136571
$701$	1.57418e7	1.20993	0.604964	−	0.796253i	$-0.293188\pi$
$701$	1.57418e7	1.20993	0.604964	+	0.796253i	$0.293188\pi$
$702$	452044.	0.0346209
$703$	1.97107e7	1.50423
$704$	266352.	0.0202546
$705$	5.20739e7	3.94591
$706$	2.11598e6	0.159772
$707$	2.83376e6	0.213214
$708$	1.09505e7	0.821018
$709$	−2.30768e7	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$709$	−2.30768e7	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$710$	−5.87682e6	−0.437519
$711$	−9.01599e6	−0.668866
$712$	−2.07165e6	−0.153150
$713$	−3.69089e6	−0.271899
$714$	−957560.	−0.0702943
$715$	−743953.	−0.0544228
$716$	−9.69833e6	−0.706992
$717$	2.83234e7	2.05754
$718$	238943.	0.0172975
$719$	−7.67945e6	−0.553997	−0.276999	−	0.960870i	$-0.589340\pi$
$719$	−7.67945e6	−0.553997	−0.276999	+	0.960870i	$0.589340\pi$
$720$	1.64250e7	1.18079
$721$	−912949.	−0.0654046
$722$	5.83769e6	0.416772
$723$	−1.88249e7	−1.33933
$724$	−1.57304e7	−1.11531
$725$	4.60641e6	0.325475
$726$	−5.46949e6	−0.385128
$727$	−1.11931e7	−0.785445	−0.392722	−	0.919657i	$-0.628467\pi$
$727$	−1.11931e7	−0.785445	−0.392722	+	0.919657i	$0.628467\pi$
$728$	−1.36070e6	−0.0951554
$729$	−1.64861e7	−1.14894
$730$	7.40095e6	0.514020
$731$	−2.08002e7	−1.43970
$732$	−1.56625e7	−1.08040
$733$	−5.04784e6	−0.347013	−0.173506	−	0.984833i	$-0.555510\pi$
$733$	−5.04784e6	−0.347013	−0.173506	+	0.984833i	$0.555510\pi$
$734$	−2.53068e6	−0.173379
$735$	−2.85154e7	−1.94698
$736$	−1.61804e7	−1.10102
$737$	−166754.	−0.0113085
$738$	−449893.	−0.0304066
$739$	6.20023e6	0.417635	0.208817	−	0.977955i	$-0.433039\pi$
$739$	6.20023e6	0.417635	0.208817	+	0.977955i	$0.433039\pi$
$740$	1.80293e7	1.21032
$741$	−3.98064e7	−2.66322
$742$	551013.	0.0367411
$743$	2.87166e6	0.190836	0.0954181	−	0.995437i	$-0.469581\pi$
$743$	2.87166e6	0.190836	0.0954181	+	0.995437i	$0.469581\pi$
$744$	−2.01635e6	−0.133547
$745$	−1.59105e7	−1.05025
$746$	−4.05611e6	−0.266847
$747$	8.90933e6	0.584176
$748$	−551325.	−0.0360291
$749$	2.08579e6	0.135852
$750$	−648867.	−0.0421214
$751$	2.88744e7	1.86816	0.934079	−	0.357066i	$-0.116223\pi$
$751$	2.88744e7	1.86816	0.934079	+	0.357066i	$0.116223\pi$
$752$	2.42099e7	1.56117
$753$	−3.20497e7	−2.05985
$754$	1.70439e6	0.109180
$755$	−1.61934e7	−1.03388
$756$	−261026.	−0.0166104
$757$	−2.22743e6	−0.141275	−0.0706375	−	0.997502i	$-0.522503\pi$
$757$	−2.22743e6	−0.141275	−0.0706375	+	0.997502i	$0.522503\pi$
$758$	−1.43738e6	−0.0908655
$759$	1.17713e6	0.0741687
$760$	1.82094e7	1.14357
$761$	−1.04675e7	−0.655214	−0.327607	−	0.944814i	$-0.606242\pi$
$761$	−1.04675e7	−0.655214	−0.327607	+	0.944814i	$0.606242\pi$
$762$	−1.18180e7	−0.737318
$763$	3.08759e6	0.192003
$764$	6.74963e6	0.418357
$765$	−2.76061e7	−1.70550
$766$	−775469.	−0.0477521
$767$	−1.15464e7	−0.708694
$768$	8.45213e6	0.517086
$769$	1.71329e7	1.04476	0.522378	−	0.852714i	$-0.325045\pi$
$769$	1.71329e7	1.04476	0.522378	+	0.852714i	$0.325045\pi$
$770$	−33106.9	−0.00201229
$771$	1.71614e7	1.03972
$772$	−2.45697e6	−0.148373
$773$	1.01987e7	0.613898	0.306949	−	0.951726i	$-0.400692\pi$
$773$	1.01987e7	0.613898	0.306949	+	0.951726i	$0.400692\pi$
$774$	6.05885e6	0.363528
$775$	2.76643e6	0.165450
$776$	1.06138e7	0.632729
$777$	−3.64301e6	−0.216475
$778$	4.17104e6	0.247056
$779$	2.85723e6	0.168695
$780$	−3.64108e7	−2.14286
$781$	683699.	0.0401086
$782$	7.90634e6	0.462337
$783$	679113.	0.0395857
$784$	−1.32573e7	−0.770306
$785$	3.21260e7	1.86073
$786$	−433279.	−0.0250156
$787$	1.98443e7	1.14208	0.571042	−	0.820921i	$-0.306539\pi$
$787$	1.98443e7	1.14208	0.571042	+	0.820921i	$0.306539\pi$
$788$	7.31171e6	0.419473
$789$	−1.28392e7	−0.734255
$790$	−4.03642e6	−0.230106
$791$	4.45864e6	0.253373
$792$	333566.	0.0188959
$793$	1.65148e7	0.932588
$794$	−6.06265e6	−0.341281
$795$	3.06253e7	1.71855
$796$	2.26547e7	1.26729
$797$	−260118.	−0.0145052	−0.00725261	−	0.999974i	$-0.502309\pi$
$797$	−260118.	−0.0145052	−0.00725261	+	0.999974i	$0.502309\pi$
$798$	−1.77144e6	−0.0984732
$799$	−4.06905e7	−2.25489
$800$	1.21277e7	0.669969
$801$	5.81013e6	0.319967
$802$	1.14502e6	0.0628605
$803$	−861013.	−0.0471217
$804$	−8.16131e6	−0.445266
$805$	−6.16052e6	−0.335064
$806$	1.02359e6	0.0554996
$807$	−4.98894e6	−0.269665
$808$	−1.27823e7	−0.688780
$809$	−6.37528e6	−0.342474	−0.171237	−	0.985230i	$-0.554776\pi$
$809$	−6.37528e6	−0.342474	−0.171237	+	0.985230i	$0.554776\pi$
$810$	−6.34330e6	−0.339706
$811$	3.33348e7	1.77970	0.889848	−	0.456257i	$-0.150810\pi$
$811$	3.33348e7	1.77970	0.889848	+	0.456257i	$0.150810\pi$
$812$	−984177.	−0.0523821
$813$	−3.30902e7	−1.75579
$814$	161648.	0.00855087
$815$	−5.74011e6	−0.302709
$816$	−2.47433e7	−1.30087
$817$	−3.84792e7	−2.01684
$818$	6.25711e6	0.326957
$819$	3.81620e6	0.198802
$820$	2.61350e6	0.135734
$821$	1.71741e7	0.889232	0.444616	−	0.895721i	$-0.353340\pi$
$821$	1.71741e7	0.889232	0.444616	+	0.895721i	$0.353340\pi$
$822$	1.85283e6	0.0956438
$823$	−2.41387e7	−1.24227	−0.621134	−	0.783705i	$-0.713328\pi$
$823$	−2.41387e7	−1.24227	−0.621134	+	0.783705i	$0.713328\pi$
$824$	4.11806e6	0.211288
$825$	−882296.	−0.0451315
$826$	−513830.	−0.0262041
$827$	1.58197e7	0.804332	0.402166	−	0.915567i	$-0.368257\pi$
$827$	1.58197e7	0.804332	0.402166	+	0.915567i	$0.368257\pi$
$828$	2.98834e7	1.51479
$829$	−7.34033e6	−0.370962	−0.185481	−	0.982648i	$-0.559384\pi$
$829$	−7.34033e6	−0.370962	−0.185481	+	0.982648i	$0.559384\pi$
$830$	3.98867e6	0.200971
$831$	1.08563e7	0.545357
$832$	−1.37452e7	−0.688402
$833$	2.22819e7	1.11260
$834$	9.93335e6	0.494517
$835$	−2.75437e7	−1.36712
$836$	−1.01992e6	−0.0504721
$837$	407849.	0.0201227
$838$	469024.	0.0230720
$839$	3.72549e6	0.182717	0.0913584	−	0.995818i	$-0.470879\pi$
$839$	3.72549e6	0.182717	0.0913584	+	0.995818i	$0.470879\pi$
$840$	−3.36553e6	−0.164572
$841$	−1.79506e7	−0.875164
$842$	1.49348e6	0.0725972
$843$	4.33412e7	2.10054
$844$	6.92355e6	0.334559
$845$	9.62293e6	0.463624
$846$	1.18527e7	0.569364
$847$	−3.33013e6	−0.159497
$848$	1.42382e7	0.679930
$849$	4.28440e7	2.03996
$850$	−5.92605e6	−0.281331
$851$	3.00795e7	1.42379
$852$	3.34618e7	1.57925
$853$	2.22483e7	1.04695	0.523473	−	0.852042i	$-0.324636\pi$
$853$	2.22483e7	1.04695	0.523473	+	0.852042i	$0.324636\pi$
$854$	734929.	0.0344827
$855$	−5.10698e7	−2.38918
$856$	−9.40843e6	−0.438867
$857$	−3.64158e7	−1.69370	−0.846852	−	0.531828i	$-0.821505\pi$
$857$	−3.64158e7	−1.69370	−0.846852	+	0.531828i	$0.821505\pi$
$858$	−326454.	−0.0151392
$859$	−9.85579e6	−0.455731	−0.227865	−	0.973693i	$-0.573175\pi$
$859$	−9.85579e6	−0.455731	−0.227865	+	0.973693i	$0.573175\pi$
$860$	−3.51968e7	−1.62277
$861$	−528085.	−0.0242770
$862$	2.35237e6	0.107830
$863$	2.80815e7	1.28349	0.641745	−	0.766918i	$-0.278211\pi$
$863$	2.80815e7	1.28349	0.641745	+	0.766918i	$0.278211\pi$
$864$	1.78797e6	0.0814845
$865$	3.96403e7	1.80134
$866$	−1.01910e7	−0.461765
$867$	9.68321e6	0.437493
$868$	−591059.	−0.0266276
$869$	469590.	0.0210945
$870$	4.21562e6	0.188826
$871$	8.60540e6	0.384349
$872$	−1.39273e7	−0.620261
$873$	−2.97674e7	−1.32192
$874$	1.46263e7	0.647674
$875$	−395067.	−0.0174442
$876$	−4.21400e7	−1.85538
$877$	−3.47813e6	−0.152703	−0.0763514	−	0.997081i	$-0.524327\pi$
$877$	−3.47813e6	−0.152703	−0.0763514	+	0.997081i	$0.524327\pi$
$878$	1.04858e6	0.0459054
$879$	4.92764e7	2.15113
$880$	−855481.	−0.0372395
$881$	2.91173e7	1.26390	0.631948	−	0.775011i	$-0.282256\pi$
$881$	2.91173e7	1.26390	0.631948	+	0.775011i	$0.282256\pi$
$882$	−6.49047e6	−0.280934
$883$	−1.51390e7	−0.653424	−0.326712	−	0.945124i	$-0.605941\pi$
$883$	−1.51390e7	−0.653424	−0.326712	+	0.945124i	$0.605941\pi$
$884$	2.84514e7	1.22454
$885$	−2.85587e7	−1.22569
$886$	−882404.	−0.0377644
$887$	2.42213e6	0.103369	0.0516843	−	0.998663i	$-0.483541\pi$
$887$	2.42213e6	0.103369	0.0516843	+	0.998663i	$0.483541\pi$
$888$	1.64326e7	0.699317
$889$	−7.19543e6	−0.305353
$890$	2.60117e6	0.110076
$891$	737969.	0.0311418
$892$	2.55916e7	1.07692
$893$	−7.52753e7	−3.15881
$894$	−6.98168e6	−0.292157
$895$	2.52929e7	1.05546
$896$	−3.40251e6	−0.141589
$897$	−6.07464e7	−2.52081
$898$	−6.72935e6	−0.278473
$899$	1.53776e6	0.0634585
$900$	−2.23985e7	−0.921749
$901$	−2.39306e7	−0.982068
$902$	23432.3	0.000958955 0
$903$	7.11188e6	0.290245
$904$	−2.01117e7	−0.818515
$905$	4.10245e7	1.66503
$906$	−7.10584e6	−0.287604
$907$	3.02210e6	0.121981	0.0609903	−	0.998138i	$-0.480574\pi$
$907$	3.02210e6	0.121981	0.0609903	+	0.998138i	$0.480574\pi$
$908$	−2.47695e7	−0.997016
$909$	3.58491e7	1.43903
$910$	1.70850e6	0.0683928
$911$	−5.58822e6	−0.223089	−0.111544	−	0.993759i	$-0.535580\pi$
$911$	−5.58822e6	−0.223089	−0.111544	+	0.993759i	$0.535580\pi$
$912$	−4.57739e7	−1.82234
$913$	−464034.	−0.0184235
$914$	−1.11885e7	−0.443004
$915$	4.08474e7	1.61291
$916$	−1.07726e7	−0.424213
$917$	−263805.	−0.0103600
$918$	−873665.	−0.0342167
$919$	6.16660e6	0.240856	0.120428	−	0.992722i	$-0.461573\pi$
$919$	6.16660e6	0.240856	0.120428	+	0.992722i	$0.461573\pi$
$920$	2.77884e7	1.08241
$921$	1.65299e7	0.642126
$922$	636378.	0.0246540
$923$	−3.52826e7	−1.36319
$924$	188506.	0.00726349
$925$	−2.25455e7	−0.866374
$926$	−1.31407e7	−0.503605
$927$	−1.15494e7	−0.441430
$928$	6.74138e6	0.256968
$929$	1.72089e7	0.654203	0.327102	−	0.944989i	$-0.393928\pi$
$929$	1.72089e7	0.654203	0.327102	+	0.944989i	$0.393928\pi$
$930$	2.53174e6	0.0959867
$931$	4.12204e7	1.55861
$932$	8.12703e6	0.306473
$933$	4.58280e7	1.72356
$934$	8.09983e6	0.303815
$935$	1.43784e6	0.0537874
$936$	−1.72138e7	−0.642225
$937$	−4.03184e7	−1.50022	−0.750110	−	0.661313i	$-0.769999\pi$
$937$	−4.03184e7	−1.50022	−0.750110	+	0.661313i	$0.769999\pi$
$938$	382951.	0.0142114
$939$	1.45272e7	0.537674
$940$	−6.88541e7	−2.54162
$941$	−2.26902e7	−0.835343	−0.417671	−	0.908598i	$-0.637154\pi$
$941$	−2.26902e7	−0.835343	−0.417671	+	0.908598i	$0.637154\pi$
$942$	1.40972e7	0.517613
$943$	4.36027e6	0.159674
$944$	−1.32774e7	−0.484933
$945$	680748.	0.0247974
$946$	−315570.	−0.0114648
$947$	−1.86682e7	−0.676439	−0.338219	−	0.941067i	$-0.609825\pi$
$947$	−1.86682e7	−0.676439	−0.338219	+	0.941067i	$0.609825\pi$
$948$	2.29828e7	0.830581
$949$	4.44330e7	1.60155
$950$	−1.09629e7	−0.394108
$951$	−6.17203e7	−2.21298
$952$	2.62982e6	0.0940446
$953$	−1.87730e7	−0.669580	−0.334790	−	0.942293i	$-0.608665\pi$
$953$	−1.87730e7	−0.669580	−0.334790	+	0.942293i	$0.608665\pi$
$954$	6.97071e6	0.247974
$955$	−1.76028e7	−0.624560
$956$	−3.74503e7	−1.32529
$957$	−490437.	−0.0173103
$958$	−2.28190e6	−0.0803309
$959$	1.12811e6	0.0396099
$960$	−3.39971e7	−1.19059
$961$	923521.	0.0322581
$962$	−8.34194e6	−0.290623
$963$	2.63868e7	0.916897
$964$	2.48910e7	0.862681
$965$	6.40770e6	0.221505
$966$	−2.70329e6	−0.0932074
$967$	−352616.	−0.0121265	−0.00606325	−	0.999982i	$-0.501930\pi$
$967$	−352616.	−0.0121265	−0.00606325	+	0.999982i	$0.501930\pi$
$968$	1.50213e7	0.515251
$969$	7.69338e7	2.63213
$970$	−1.33267e7	−0.454773
$971$	−2.36047e6	−0.0803433	−0.0401716	−	0.999193i	$-0.512790\pi$
$971$	−2.36047e6	−0.0803433	−0.0401716	+	0.999193i	$0.512790\pi$
$972$	3.91819e7	1.33021
$973$	6.04798e6	0.204799
$974$	−4.54266e6	−0.153431
$975$	4.55313e7	1.53390
$976$	1.89906e7	0.638136
$977$	−2.51190e7	−0.841910	−0.420955	−	0.907081i	$-0.638305\pi$
$977$	−2.51190e7	−0.841910	−0.420955	+	0.907081i	$0.638305\pi$
$978$	−2.51881e6	−0.0842071
$979$	−302616.	−0.0100910
$980$	3.77042e7	1.25408
$981$	3.90602e7	1.29587
$982$	−9.59326e6	−0.317459
$983$	−5.09828e7	−1.68283	−0.841414	−	0.540391i	$-0.818276\pi$
$983$	−5.09828e7	−1.68283	−0.841414	+	0.540391i	$0.818276\pi$
$984$	2.38204e6	0.0784263
$985$	−1.90687e7	−0.626226
$986$	−3.29408e6	−0.107905
$987$	1.39127e7	0.454588
$988$	5.26335e7	1.71542
$989$	−5.87211e7	−1.90899
$990$	−418826.	−0.0135814
$991$	3.39948e7	1.09958	0.549792	−	0.835301i	$-0.314707\pi$
$991$	3.39948e7	1.09958	0.549792	+	0.835301i	$0.314707\pi$
$992$	4.04861e6	0.130625
$993$	−2.41889e7	−0.778472
$994$	−1.57012e6	−0.0504043
$995$	−5.90827e7	−1.89192
$996$	−2.27109e7	−0.725414
$997$	4.13726e7	1.31818	0.659090	−	0.752064i	$-0.270941\pi$
$997$	4.13726e7	1.31818	0.659090	+	0.752064i	$0.270941\pi$
$998$	−7.88115e6	−0.250474
$999$	−3.32383e6	−0.105372

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.6.a.b.1.5	✓	8
3.2	odd	2		279.6.a.f.1.4		8
4.3	odd	2		496.6.a.h.1.2		8
5.4	even	2		775.6.a.b.1.4		8
31.30	odd	2		961.6.a.c.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.6.a.b.1.5	✓	8	1.1	even	1	trivial
279.6.a.f.1.4		8	3.2	odd	2
496.6.a.h.1.2		8	4.3	odd	2
775.6.a.b.1.4		8	5.4	even	2
961.6.a.c.1.5		8	31.30	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	31.a (trivial)

\(f(q)\)	\(=\)	\(q+1.51317 q^{2} +22.4697 q^{3} -29.7103 q^{4} +77.4836 q^{5} +34.0005 q^{6} +20.7014 q^{7} -93.3783 q^{8} +261.888 q^{9} +117.246 q^{10} -13.6402 q^{11} -667.582 q^{12} +703.908 q^{13} +31.3248 q^{14} +1741.03 q^{15} +809.433 q^{16} -1360.44 q^{17} +396.281 q^{18} -2516.75 q^{19} -2302.06 q^{20} +465.155 q^{21} -20.6399 q^{22} -3840.67 q^{23} -2098.18 q^{24} +2878.70 q^{25} +1065.13 q^{26} +424.401 q^{27} -615.046 q^{28} +1600.17 q^{29} +2634.48 q^{30} +961.000 q^{31} +4212.92 q^{32} -306.491 q^{33} -2058.58 q^{34} +1604.02 q^{35} -7780.76 q^{36} -7831.82 q^{37} -3808.27 q^{38} +15816.6 q^{39} -7235.28 q^{40} -1135.29 q^{41} +703.859 q^{42} +15289.3 q^{43} +405.254 q^{44} +20292.0 q^{45} -5811.60 q^{46} +29909.8 q^{47} +18187.7 q^{48} -16378.5 q^{49} +4355.97 q^{50} -30568.7 q^{51} -20913.3 q^{52} +17590.3 q^{53} +642.192 q^{54} -1056.89 q^{55} -1933.06 q^{56} -56550.6 q^{57} +2421.33 q^{58} -16403.3 q^{59} -51726.6 q^{60} +23461.6 q^{61} +1454.16 q^{62} +5421.45 q^{63} -19527.0 q^{64} +54541.3 q^{65} -463.773 q^{66} +12225.2 q^{67} +40419.1 q^{68} -86298.8 q^{69} +2427.16 q^{70} -50123.9 q^{71} -24454.6 q^{72} +63123.3 q^{73} -11850.9 q^{74} +64683.6 q^{75} +74773.3 q^{76} -282.371 q^{77} +23933.2 q^{78} -34426.9 q^{79} +62717.7 q^{80} -54102.5 q^{81} -1717.89 q^{82} +34019.7 q^{83} -13819.9 q^{84} -105412. q^{85} +23135.3 q^{86} +35955.3 q^{87} +1273.70 q^{88} +22185.6 q^{89} +30705.3 q^{90} +14571.9 q^{91} +114108. q^{92} +21593.4 q^{93} +45258.6 q^{94} -195007. q^{95} +94663.0 q^{96} -113665. q^{97} -24783.4 q^{98} -3572.20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 7 q^{2} - 2 q^{3} + 149 q^{4} + 128 q^{5} + 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} + 574 q^{11} - 46 q^{12} - 122 q^{13} - 309 q^{14} - 524 q^{15} + 833 q^{16} + 1932 q^{17} - 6845 q^{18}+ \cdots - 180150 q^{99}+O(q^{100}) \) 8 * q + 7 * q^2 - 2 * q^3 + 149 * q^4 + 128 * q^5 + 72 * q^6 + 88 * q^7 + 924 * q^8 + 1512 * q^9 + 1581 * q^10 + 574 * q^11 - 46 * q^12 - 122 * q^13 - 309 * q^14 - 524 * q^15 + 833 * q^16 + 1932 * q^17 - 6845 * q^18 - 1796 * q^19 - 37 * q^20 + 996 * q^21 - 9000 * q^22 - 4136 * q^23 - 17468 * q^24 + 3308 * q^25 + 1524 * q^26 - 2624 * q^27 - 21245 * q^28 + 5050 * q^29 - 30450 * q^30 + 7688 * q^31 + 21045 * q^32 + 27920 * q^33 - 24738 * q^34 + 25700 * q^35 + 21637 * q^36 + 23674 * q^37 - 4289 * q^38 + 38652 * q^39 + 36606 * q^40 + 44828 * q^41 - 9994 * q^42 + 21058 * q^43 + 13168 * q^44 - 1344 * q^45 + 6198 * q^46 + 5348 * q^47 - 69588 * q^48 + 19356 * q^49 + 33242 * q^50 + 22300 * q^51 - 60386 * q^52 + 4926 * q^53 - 24476 * q^54 - 56892 * q^55 - 87652 * q^56 + 22072 * q^57 - 25002 * q^58 + 16944 * q^59 - 138556 * q^60 - 73682 * q^61 + 6727 * q^62 - 138784 * q^63 - 1300 * q^64 + 63316 * q^65 - 128796 * q^66 - 134768 * q^67 - 9524 * q^68 + 3992 * q^69 - 142737 * q^70 + 123724 * q^71 + 93232 * q^72 + 70792 * q^73 + 307468 * q^74 - 302902 * q^75 - 65405 * q^76 + 107932 * q^77 + 36820 * q^78 - 26036 * q^79 + 576783 * q^80 + 82528 * q^81 + 142335 * q^82 + 21882 * q^83 + 299144 * q^84 - 130464 * q^85 + 480478 * q^86 - 440244 * q^87 - 71982 * q^88 + 164392 * q^89 + 397969 * q^90 - 60028 * q^91 + 31012 * q^92 - 1922 * q^93 - 170388 * q^94 - 7404 * q^95 - 331010 * q^96 - 131948 * q^97 + 288768 * q^98 - 180150 * q^99

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