LMFDB - Embedded newform 51.6.a.c.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [51,6,Mod(1,51)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(51, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("51.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 51 = 3 \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	51.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4] 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:	$8.17957481046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 117x^{2} + 299x + 1026 $ x^4 - x^3 - 117x^2 + 299x + 1026
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$8.91396$ of defining polynomial
Character	$\chi$	$=$	51.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+9.91396 q^{2} +9.00000 q^{3} +66.2866 q^{4} -19.7859 q^{5} +89.2257 q^{6} +33.0231 q^{7} +339.916 q^{8} +81.0000 q^{9} -196.156 q^{10} +107.108 q^{11} +596.580 q^{12} -770.526 q^{13} +327.390 q^{14} -178.073 q^{15} +1248.75 q^{16} -289.000 q^{17} +803.031 q^{18} +795.202 q^{19} -1311.54 q^{20} +297.208 q^{21} +1061.87 q^{22} -2887.76 q^{23} +3059.25 q^{24} -2733.52 q^{25} -7638.96 q^{26} +729.000 q^{27} +2188.99 q^{28} +1903.63 q^{29} -1765.41 q^{30} +9799.76 q^{31} +1502.69 q^{32} +963.976 q^{33} -2865.13 q^{34} -653.390 q^{35} +5369.22 q^{36} -7375.94 q^{37} +7883.60 q^{38} -6934.73 q^{39} -6725.54 q^{40} +5183.08 q^{41} +2946.51 q^{42} -13657.6 q^{43} +7099.85 q^{44} -1602.66 q^{45} -28629.1 q^{46} +9454.30 q^{47} +11238.7 q^{48} -15716.5 q^{49} -27100.0 q^{50} -2601.00 q^{51} -51075.5 q^{52} +5277.19 q^{53} +7227.28 q^{54} -2119.23 q^{55} +11225.1 q^{56} +7156.82 q^{57} +18872.5 q^{58} +24244.2 q^{59} -11803.8 q^{60} +46903.0 q^{61} +97154.5 q^{62} +2674.87 q^{63} -25062.2 q^{64} +15245.5 q^{65} +9556.82 q^{66} +19204.9 q^{67} -19156.8 q^{68} -25989.8 q^{69} -6477.69 q^{70} +82934.5 q^{71} +27533.2 q^{72} -25898.3 q^{73} -73124.8 q^{74} -24601.7 q^{75} +52711.3 q^{76} +3537.05 q^{77} -68750.7 q^{78} +72070.9 q^{79} -24707.5 q^{80} +6561.00 q^{81} +51384.8 q^{82} +25916.9 q^{83} +19700.9 q^{84} +5718.12 q^{85} -135401. q^{86} +17132.7 q^{87} +36407.9 q^{88} +59658.6 q^{89} -15888.7 q^{90} -25445.1 q^{91} -191420. q^{92} +88197.9 q^{93} +93729.6 q^{94} -15733.8 q^{95} +13524.2 q^{96} -103143. q^{97} -155813. q^{98} +8675.78 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{2} + 36 q^{3} + 113 q^{4} + 146 q^{5} + 45 q^{6} + 60 q^{7} - 153 q^{8} + 324 q^{9} - 536 q^{10} + 1114 q^{11} + 1017 q^{12} - 166 q^{13} + 1738 q^{14} + 1314 q^{15} + 2745 q^{16} - 1156 q^{17}+ \cdots + 90234 q^{99}+O(q^{100}) $ 4 * q + 5 * q^2 + 36 * q^3 + 113 * q^4 + 146 * q^5 + 45 * q^6 + 60 * q^7 - 153 * q^8 + 324 * q^9 - 536 * q^10 + 1114 * q^11 + 1017 * q^12 - 166 * q^13 + 1738 * q^14 + 1314 * q^15 + 2745 * q^16 - 1156 * q^17 + 405 * q^18 + 3454 * q^19 + 5748 * q^20 + 540 * q^21 - 6368 * q^22 + 998 * q^23 - 1377 * q^24 + 3150 * q^25 + 2890 * q^26 + 2916 * q^27 - 13382 * q^28 + 3048 * q^29 - 4824 * q^30 - 7808 * q^31 - 8881 * q^32 + 10026 * q^33 - 1445 * q^34 - 14080 * q^35 + 9153 * q^36 - 1700 * q^37 - 26836 * q^38 - 1494 * q^39 - 55648 * q^40 - 1386 * q^41 + 15642 * q^42 - 27654 * q^43 + 35196 * q^44 + 11826 * q^45 - 43646 * q^46 + 38156 * q^47 + 24705 * q^48 - 17012 * q^49 - 42601 * q^50 - 10404 * q^51 - 79378 * q^52 + 52648 * q^53 + 3645 * q^54 + 38782 * q^55 + 78510 * q^56 + 31086 * q^57 - 75320 * q^58 + 50028 * q^59 + 51732 * q^60 + 13076 * q^61 + 83438 * q^62 + 4860 * q^63 - 32847 * q^64 + 59250 * q^65 - 57312 * q^66 + 7880 * q^67 - 32657 * q^68 + 8982 * q^69 + 132272 * q^70 + 106836 * q^71 - 12393 * q^72 + 52488 * q^73 - 164192 * q^74 + 28350 * q^75 + 178608 * q^76 - 8520 * q^77 + 26010 * q^78 + 37688 * q^79 + 17368 * q^80 + 26244 * q^81 + 104254 * q^82 + 9812 * q^83 - 120438 * q^84 - 42194 * q^85 - 145316 * q^86 + 27432 * q^87 - 162784 * q^88 + 20876 * q^89 - 43416 * q^90 + 65984 * q^91 - 302994 * q^92 - 70272 * q^93 - 125700 * q^94 + 81658 * q^95 - 79929 * q^96 - 1996 * q^97 - 321967 * q^98 + 90234 * 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 $

$2$	9.91396	1.75256	0.876279	−	0.481805i	$-0.160019\pi$
$2$	9.91396	1.75256	0.876279	+	0.481805i	$0.160019\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	66.2866	2.07146
$5$	−19.7859	−0.353940	−0.176970	−	0.984216i	$-0.556630\pi$
$5$	−19.7859	−0.353940	−0.176970	+	0.984216i	$0.556630\pi$
$6$	89.2257	1.01184
$7$	33.0231	0.254726	0.127363	−	0.991856i	$-0.459349\pi$
$7$	33.0231	0.254726	0.127363	+	0.991856i	$0.459349\pi$
$8$	339.916	1.87779
$9$	81.0000	0.333333
$10$	−196.156	−0.620301
$11$	107.108	0.266896	0.133448	−	0.991056i	$-0.457395\pi$
$11$	107.108	0.266896	0.133448	+	0.991056i	$0.457395\pi$
$12$	596.580	1.19596
$13$	−770.526	−1.26453	−0.632264	−	0.774753i	$-0.717874\pi$
$13$	−770.526	−1.26453	−0.632264	+	0.774753i	$0.717874\pi$
$14$	327.390	0.446421
$15$	−178.073	−0.204348
$16$	1248.75	1.21948
$17$	−289.000	−0.242536
$17$	−289.000	−0.242536
$18$	803.031	0.584186
$19$	795.202	0.505351	0.252676	−	0.967551i	$-0.418689\pi$
$19$	795.202	0.505351	0.252676	+	0.967551i	$0.418689\pi$
$20$	−1311.54	−0.733172
$21$	297.208	0.147066
$22$	1061.87	0.467750
$23$	−2887.76	−1.13826	−0.569129	−	0.822248i	$-0.692720\pi$
$23$	−2887.76	−1.13826	−0.569129	+	0.822248i	$0.692720\pi$
$24$	3059.25	1.08414
$25$	−2733.52	−0.874726
$26$	−7638.96	−2.21616
$27$	729.000	0.192450
$28$	2188.99	0.527653
$29$	1903.63	0.420327	0.210163	−	0.977666i	$-0.432600\pi$
$29$	1903.63	0.420327	0.210163	+	0.977666i	$0.432600\pi$
$30$	−1765.41	−0.358131
$31$	9799.76	1.83152	0.915759	−	0.401727i	$-0.131590\pi$
$31$	9799.76	1.83152	0.915759	+	0.401727i	$0.131590\pi$
$32$	1502.69	0.259414
$33$	963.976	0.154092
$34$	−2865.13	−0.425058
$35$	−653.390	−0.0901576
$36$	5369.22	0.690486
$37$	−7375.94	−0.885754	−0.442877	−	0.896582i	$-0.646042\pi$
$37$	−7375.94	−0.885754	−0.442877	+	0.896582i	$0.646042\pi$
$38$	7883.60	0.885657
$39$	−6934.73	−0.730076
$40$	−6725.54	−0.664626
$41$	5183.08	0.481535	0.240768	−	0.970583i	$-0.422601\pi$
$41$	5183.08	0.481535	0.240768	+	0.970583i	$0.422601\pi$
$42$	2946.51	0.257741
$43$	−13657.6	−1.12643	−0.563215	−	0.826311i	$-0.690436\pi$
$43$	−13657.6	−1.12643	−0.563215	+	0.826311i	$0.690436\pi$
$44$	7099.85	0.552863
$45$	−1602.66	−0.117980
$46$	−28629.1	−1.99486
$47$	9454.30	0.624288	0.312144	−	0.950035i	$-0.398953\pi$
$47$	9454.30	0.624288	0.312144	+	0.950035i	$0.398953\pi$
$48$	11238.7	0.704066
$49$	−15716.5	−0.935115
$50$	−27100.0	−1.53301
$51$	−2601.00	−0.140028
$52$	−51075.5	−2.61942
$53$	5277.19	0.258056	0.129028	−	0.991641i	$-0.458814\pi$
$53$	5277.19	0.258056	0.129028	+	0.991641i	$0.458814\pi$
$54$	7227.28	0.337280
$55$	−2119.23	−0.0944652
$56$	11225.1	0.478321
$57$	7156.82	0.291765
$58$	18872.5	0.736647
$59$	24244.2	0.906730	0.453365	−	0.891325i	$-0.350223\pi$
$59$	24244.2	0.906730	0.453365	+	0.891325i	$0.350223\pi$
$60$	−11803.8	−0.423297
$61$	46903.0	1.61390	0.806948	−	0.590622i	$-0.201118\pi$
$61$	46903.0	1.61390	0.806948	+	0.590622i	$0.201118\pi$
$62$	97154.5	3.20984
$63$	2674.87	0.0849085
$64$	−25062.2	−0.764839
$65$	15245.5	0.447568
$66$	9556.82	0.270056
$67$	19204.9	0.522668	0.261334	−	0.965248i	$-0.415838\pi$
$67$	19204.9	0.522668	0.261334	+	0.965248i	$0.415838\pi$
$68$	−19156.8	−0.502402
$69$	−25989.8	−0.657174
$70$	−6477.69	−0.158006
$71$	82934.5	1.95249	0.976246	−	0.216663i	$-0.0695174\pi$
$71$	82934.5	1.95249	0.976246	+	0.216663i	$0.0695174\pi$
$72$	27533.2	0.625930
$73$	−25898.3	−0.568805	−0.284403	−	0.958705i	$-0.591795\pi$
$73$	−25898.3	−0.568805	−0.284403	+	0.958705i	$0.591795\pi$
$74$	−73124.8	−1.55233
$75$	−24601.7	−0.505023
$76$	52711.3	1.04681
$77$	3537.05	0.0679852
$78$	−68750.7	−1.27950
$79$	72070.9	1.29925	0.649624	−	0.760256i	$-0.274926\pi$
$79$	72070.9	1.29925	0.649624	+	0.760256i	$0.274926\pi$
$80$	−24707.5	−0.431622
$81$	6561.00	0.111111
$82$	51384.8	0.843918
$83$	25916.9	0.412941	0.206470	−	0.978453i	$-0.433802\pi$
$83$	25916.9	0.412941	0.206470	+	0.978453i	$0.433802\pi$
$84$	19700.9	0.304641
$85$	5718.12	0.0858432
$86$	−135401.	−1.97413
$87$	17132.7	0.242676
$88$	36407.9	0.501174
$89$	59658.6	0.798359	0.399180	−	0.916873i	$-0.369295\pi$
$89$	59658.6	0.798359	0.399180	+	0.916873i	$0.369295\pi$
$90$	−15888.7	−0.206767
$91$	−25445.1	−0.322108
$92$	−191420.	−2.35785
$93$	88197.9	1.05743
$94$	93729.6	1.09410
$95$	−15733.8	−0.178864
$96$	13524.2	0.149773
$97$	−103143.	−1.11304	−0.556520	−	0.830834i	$-0.687864\pi$
$97$	−103143.	−1.11304	−0.556520	+	0.830834i	$0.687864\pi$
$98$	−155813.	−1.63884
$99$	8675.78	0.0889653
$100$	−181196.	−1.81196
$101$	87667.0	0.855131	0.427565	−	0.903984i	$-0.359371\pi$
$101$	87667.0	0.855131	0.427565	+	0.903984i	$0.359371\pi$
$102$	−25786.2	−0.245407
$103$	−199325.	−1.85126	−0.925632	−	0.378424i	$-0.876466\pi$
$103$	−199325.	−1.85126	−0.925632	+	0.378424i	$0.876466\pi$
$104$	−261914.	−2.37452
$105$	−5880.51	−0.0520525
$106$	52317.9	0.452257
$107$	36092.0	0.304756	0.152378	−	0.988322i	$-0.451307\pi$
$107$	36092.0	0.304756	0.152378	+	0.988322i	$0.451307\pi$
$108$	48323.0	0.398652
$109$	94952.7	0.765493	0.382747	−	0.923853i	$-0.374978\pi$
$109$	94952.7	0.765493	0.382747	+	0.923853i	$0.374978\pi$
$110$	−21010.0	−0.165556
$111$	−66383.5	−0.511390
$112$	41237.4	0.310632
$113$	−143218.	−1.05512	−0.527559	−	0.849518i	$-0.676893\pi$
$113$	−143218.	−1.05512	−0.527559	+	0.849518i	$0.676893\pi$
$114$	70952.4	0.511335
$115$	57136.8	0.402876
$116$	126185.	0.870689
$117$	−62412.6	−0.421510
$118$	240356.	1.58910
$119$	−9543.67	−0.0617800
$120$	−60529.9	−0.383722
$121$	−149579.	−0.928767
$122$	464994.	2.82845
$123$	46647.7	0.278015
$124$	649593.	3.79391
$125$	115916.	0.663541
$126$	26518.5	0.148807
$127$	−178178.	−0.980268	−0.490134	−	0.871647i	$-0.663052\pi$
$127$	−178178.	−0.980268	−0.490134	+	0.871647i	$0.663052\pi$
$128$	−296552.	−1.59984
$129$	−122919.	−0.650344
$130$	151143.	0.784388
$131$	−147067.	−0.748750	−0.374375	−	0.927277i	$-0.622143\pi$
$131$	−147067.	−0.748750	−0.374375	+	0.927277i	$0.622143\pi$
$132$	63898.7	0.319196
$133$	26260.0	0.128726
$134$	190397.	0.916005
$135$	−14423.9	−0.0681159
$136$	−98235.8	−0.455431
$137$	400754.	1.82422	0.912109	−	0.409949i	$-0.134453\pi$
$137$	400754.	1.82422	0.912109	+	0.409949i	$0.134453\pi$
$138$	−257662.	−1.15174
$139$	−102960.	−0.451995	−0.225997	−	0.974128i	$-0.572564\pi$
$139$	−102960.	−0.451995	−0.225997	+	0.974128i	$0.572564\pi$
$140$	−43311.0	−0.186758
$141$	85088.7	0.360433
$142$	822210.	3.42186
$143$	−82529.8	−0.337497
$144$	101148.	0.406493
$145$	−37664.9	−0.148771
$146$	−256755.	−0.996864
$147$	−141448.	−0.539889
$148$	−488926.	−1.83480
$149$	−506487.	−1.86897	−0.934485	−	0.356002i	$-0.884140\pi$
$149$	−506487.	−1.86897	−0.934485	+	0.356002i	$0.884140\pi$
$150$	−243900.	−0.885082
$151$	−370254.	−1.32147	−0.660735	−	0.750619i	$-0.729755\pi$
$151$	−370254.	−1.32147	−0.660735	+	0.750619i	$0.729755\pi$
$152$	270302.	0.948944
$153$	−23409.0	−0.0808452
$154$	35066.2	0.119148
$155$	−193897.	−0.648248
$156$	−459680.	−1.51232
$157$	−109544.	−0.354684	−0.177342	−	0.984149i	$-0.556750\pi$
$157$	−109544.	−0.354684	−0.177342	+	0.984149i	$0.556750\pi$
$158$	714508.	2.27701
$159$	47494.8	0.148989
$160$	−29732.0	−0.0918172
$161$	−95362.6	−0.289944
$162$	65045.5	0.194729
$163$	−463726.	−1.36708	−0.683538	−	0.729915i	$-0.739559\pi$
$163$	−463726.	−1.36708	−0.683538	+	0.729915i	$0.739559\pi$
$164$	343569.	0.997480
$165$	−19073.1	−0.0545395
$166$	256939.	0.723702
$167$	405951.	1.12637	0.563187	−	0.826329i	$-0.309575\pi$
$167$	405951.	1.12637	0.563187	+	0.826329i	$0.309575\pi$
$168$	101026.	0.276159
$169$	222417.	0.599033
$170$	56689.2	0.150445
$171$	64411.4	0.168450
$172$	−905318.	−2.33335
$173$	408480.	1.03766	0.518831	−	0.854877i	$-0.326367\pi$
$173$	408480.	1.03766	0.518831	+	0.854877i	$0.326367\pi$
$174$	169852.	0.425303
$175$	−90269.2	−0.222815
$176$	133751.	0.325474
$177$	218198.	0.523501
$178$	591453.	1.39917
$179$	228012.	0.531893	0.265947	−	0.963988i	$-0.414315\pi$
$179$	228012.	0.531893	0.265947	+	0.963988i	$0.414315\pi$
$180$	−106235.	−0.244391
$181$	−120860.	−0.274212	−0.137106	−	0.990556i	$-0.543780\pi$
$181$	−120860.	−0.274212	−0.137106	+	0.990556i	$0.543780\pi$
$182$	−252262.	−0.564512
$183$	422127.	0.931783
$184$	−981596.	−2.13741
$185$	145939.	0.313504
$186$	874390.	1.85320
$187$	−30954.3	−0.0647317
$188$	626694.	1.29319
$189$	24073.8	0.0490219
$190$	−155984.	−0.313470
$191$	−624470.	−1.23859	−0.619296	−	0.785158i	$-0.712582\pi$
$191$	−624470.	−1.23859	−0.619296	+	0.785158i	$0.712582\pi$
$192$	−225560.	−0.441580
$193$	579212.	1.11929	0.559647	−	0.828731i	$-0.310937\pi$
$193$	579212.	1.11929	0.559647	+	0.828731i	$0.310937\pi$
$194$	−1.02256e6	−1.95067
$195$	137210.	0.258403
$196$	−1.04179e6	−1.93705
$197$	−20203.4	−0.0370901	−0.0185451	−	0.999828i	$-0.505903\pi$
$197$	−20203.4	−0.0370901	−0.0185451	+	0.999828i	$0.505903\pi$
$198$	86011.3	0.155917
$199$	−3814.68	−0.00682851	−0.00341426	−	0.999994i	$-0.501087\pi$
$199$	−3814.68	−0.00682851	−0.00341426	+	0.999994i	$0.501087\pi$
$200$	−929168.	−1.64255
$201$	172844.	0.301762
$202$	869127.	1.49867
$203$	62863.7	0.107068
$204$	−172412.	−0.290062
$205$	−102552.	−0.170435
$206$	−1.97610e6	−3.24445
$207$	−233908.	−0.379420
$208$	−962190.	−1.54206
$209$	85172.8	0.134876
$210$	−58299.2	−0.0912251
$211$	−47633.5	−0.0736557	−0.0368278	−	0.999322i	$-0.511725\pi$
$211$	−47633.5	−0.0736557	−0.0368278	+	0.999322i	$0.511725\pi$
$212$	349807.	0.534551
$213$	746411.	1.12727
$214$	357815.	0.534102
$215$	270228.	0.398689
$216$	247799.	0.361381
$217$	323618.	0.466535
$218$	941358.	1.34157
$219$	−233085.	−0.328400
$220$	−140477.	−0.195681
$221$	222682.	0.306693
$222$	−658123.	−0.896241
$223$	780856.	1.05150	0.525749	−	0.850639i	$-0.323785\pi$
$223$	780856.	1.05150	0.525749	+	0.850639i	$0.323785\pi$
$224$	49623.4	0.0660795
$225$	−221415.	−0.291575
$226$	−1.41986e6	−1.84916
$227$	−1.01316e6	−1.30500	−0.652502	−	0.757787i	$-0.726281\pi$
$227$	−1.01316e6	−1.30500	−0.652502	+	0.757787i	$0.726281\pi$
$228$	474401.	0.604378
$229$	785820.	0.990226	0.495113	−	0.868829i	$-0.335127\pi$
$229$	785820.	0.990226	0.495113	+	0.868829i	$0.335127\pi$
$230$	566452.	0.706063
$231$	31833.4	0.0392513
$232$	647074.	0.789286
$233$	−704506.	−0.850148	−0.425074	−	0.905158i	$-0.639752\pi$
$233$	−704506.	−0.850148	−0.425074	+	0.905158i	$0.639752\pi$
$234$	−618756.	−0.738720
$235$	−187062.	−0.220961
$236$	1.60707e6	1.87825
$237$	648638.	0.750121
$238$	−94615.6	−0.108273
$239$	−272819.	−0.308944	−0.154472	−	0.987997i	$-0.549368\pi$
$239$	−272819.	−0.308944	−0.154472	+	0.987997i	$0.549368\pi$
$240$	−222368.	−0.249197
$241$	517668.	0.574128	0.287064	−	0.957911i	$-0.407321\pi$
$241$	517668.	0.574128	0.287064	+	0.957911i	$0.407321\pi$
$242$	−1.48292e6	−1.62772
$243$	59049.0	0.0641500
$244$	3.10904e6	3.34312
$245$	310964.	0.330975
$246$	462464.	0.487236
$247$	−612724.	−0.639032
$248$	3.33110e6	3.43921
$249$	233252.	0.238412
$250$	1.14919e6	1.16289
$251$	−1.62806e6	−1.63112	−0.815561	−	0.578671i	$-0.803572\pi$
$251$	−1.62806e6	−1.63112	−0.815561	+	0.578671i	$0.803572\pi$
$252$	177308.	0.175884
$253$	−309303.	−0.303797
$254$	−1.76645e6	−1.71798
$255$	51463.0	0.0495616
$256$	−2.13801e6	−2.03897
$257$	1.11957e6	1.05735	0.528677	−	0.848823i	$-0.322688\pi$
$257$	1.11957e6	1.05735	0.528677	+	0.848823i	$0.322688\pi$
$258$	−1.21861e6	−1.13977
$259$	−243576.	−0.225624
$260$	1.01057e6	0.927118
$261$	154194.	0.140109
$262$	−1.45802e6	−1.31223
$263$	428448.	0.381952	0.190976	−	0.981595i	$-0.438835\pi$
$263$	428448.	0.381952	0.190976	+	0.981595i	$0.438835\pi$
$264$	327671.	0.289353
$265$	−104414.	−0.0913363
$266$	260341.	0.225600
$267$	536928.	0.460933
$268$	1.27303e6	1.08268
$269$	309879.	0.261103	0.130551	−	0.991442i	$-0.458325\pi$
$269$	309879.	0.261103	0.130551	+	0.991442i	$0.458325\pi$
$270$	−142998.	−0.119377
$271$	−2.12339e6	−1.75633	−0.878167	−	0.478355i	$-0.841233\pi$
$271$	−2.12339e6	−1.75633	−0.878167	+	0.478355i	$0.841233\pi$
$272$	−360887.	−0.295767
$273$	−229006.	−0.185969
$274$	3.97306e6	3.19705
$275$	−292783.	−0.233461
$276$	−1.72278e6	−1.36131
$277$	960814.	0.752385	0.376192	−	0.926542i	$-0.377233\pi$
$277$	960814.	0.752385	0.376192	+	0.926542i	$0.377233\pi$
$278$	−1.02075e6	−0.792146
$279$	793781.	0.610506
$280$	−222098.	−0.169297
$281$	−298743.	−0.225700	−0.112850	−	0.993612i	$-0.535998\pi$
$281$	−298743.	−0.225700	−0.112850	+	0.993612i	$0.535998\pi$
$282$	843566.	0.631679
$283$	−472806.	−0.350927	−0.175464	−	0.984486i	$-0.556142\pi$
$283$	−472806.	−0.350927	−0.175464	+	0.984486i	$0.556142\pi$
$284$	5.49745e6	4.04451
$285$	−141604.	−0.103267
$286$	−818197.	−0.591484
$287$	171161.	0.122659
$288$	121718.	0.0864715
$289$	83521.0	0.0588235
$290$	−373409.	−0.260729
$291$	−928289.	−0.642614
$292$	−1.71671e6	−1.17826
$293$	2.12593e6	1.44670	0.723352	−	0.690479i	$-0.242600\pi$
$293$	2.12593e6	1.44670	0.723352	+	0.690479i	$0.242600\pi$
$294$	−1.40231e6	−0.946186
$295$	−479693.	−0.320928
$296$	−2.50720e6	−1.66326
$297$	78082.0	0.0513641
$298$	−5.02129e6	−3.27548
$299$	2.22509e6	1.43936
$300$	−1.63076e6	−1.04613
$301$	−451017.	−0.286930
$302$	−3.67068e6	−2.31595
$303$	789003.	0.493710
$304$	993005.	0.616265
$305$	−928016.	−0.571223
$306$	−232076.	−0.141686
$307$	−524663.	−0.317712	−0.158856	−	0.987302i	$-0.550781\pi$
$307$	−524663.	−0.317712	−0.158856	+	0.987302i	$0.550781\pi$
$308$	234459.	0.140828
$309$	−1.79392e6	−1.06883
$310$	−1.92229e6	−1.13609
$311$	−1.31043e6	−0.768266	−0.384133	−	0.923278i	$-0.625500\pi$
$311$	−1.31043e6	−0.768266	−0.384133	+	0.923278i	$0.625500\pi$
$312$	−2.35723e6	−1.37093
$313$	−1.30589e6	−0.753437	−0.376719	−	0.926328i	$-0.622948\pi$
$313$	−1.30589e6	−0.753437	−0.376719	+	0.926328i	$0.622948\pi$
$314$	−1.08602e6	−0.621604
$315$	−52924.6	−0.0300525
$316$	4.77733e6	2.69134
$317$	−2.36368e6	−1.32111	−0.660557	−	0.750776i	$-0.729680\pi$
$317$	−2.36368e6	−1.32111	−0.660557	+	0.750776i	$0.729680\pi$
$318$	470861.	0.261111
$319$	203895.	0.112183
$320$	495878.	0.270707
$321$	324828.	0.175951
$322$	−945421.	−0.508143
$323$	−229813.	−0.122566
$324$	434907.	0.230162
$325$	2.10625e6	1.10612
$326$	−4.59736e6	−2.39588
$327$	854575.	0.441958
$328$	1.76181e6	0.904222
$329$	312210.	0.159022
$330$	−189090.	−0.0955836
$331$	3.88315e6	1.94811	0.974057	−	0.226304i	$-0.0726642\pi$
$331$	3.88315e6	1.94811	0.974057	+	0.226304i	$0.0726642\pi$
$332$	1.71794e6	0.855389
$333$	−597451.	−0.295251
$334$	4.02458e6	1.97404
$335$	−379986.	−0.184993
$336$	371137.	0.179344
$337$	811480.	0.389227	0.194614	−	0.980880i	$-0.437655\pi$
$337$	811480.	0.389227	0.194614	+	0.980880i	$0.437655\pi$
$338$	2.20503e6	1.04984
$339$	−1.28896e6	−0.609173
$340$	379035.	0.177820
$341$	1.04964e6	0.488825
$342$	638572.	0.295219
$343$	−1.07403e6	−0.492923
$344$	−4.64245e6	−2.11520
$345$	514231.	0.232600
$346$	4.04966e6	1.81856
$347$	−959058.	−0.427584	−0.213792	−	0.976879i	$-0.568581\pi$
$347$	−959058.	−0.427584	−0.213792	+	0.976879i	$0.568581\pi$
$348$	1.13567e6	0.502693
$349$	2.98884e6	1.31353	0.656763	−	0.754097i	$-0.271925\pi$
$349$	2.98884e6	1.31353	0.656763	+	0.754097i	$0.271925\pi$
$350$	−894926.	−0.390496
$351$	−561713.	−0.243359
$352$	160951.	0.0692366
$353$	1.91425e6	0.817641	0.408820	−	0.912615i	$-0.365940\pi$
$353$	1.91425e6	0.817641	0.408820	+	0.912615i	$0.365940\pi$
$354$	2.16321e6	0.917466
$355$	−1.64093e6	−0.691066
$356$	3.95457e6	1.65377
$357$	−85893.0	−0.0356687
$358$	2.26050e6	0.932174
$359$	1.63841e6	0.670943	0.335472	−	0.942050i	$-0.391104\pi$
$359$	1.63841e6	0.670943	0.335472	+	0.942050i	$0.391104\pi$
$360$	−544769.	−0.221542
$361$	−1.84375e6	−0.744620
$362$	−1.19820e6	−0.480572
$363$	−1.34621e6	−0.536224
$364$	−1.68667e6	−0.667232
$365$	512420.	0.201323
$366$	4.18495e6	1.63300
$367$	907163.	0.351577	0.175788	−	0.984428i	$-0.443753\pi$
$367$	907163.	0.351577	0.175788	+	0.984428i	$0.443753\pi$
$368$	−3.60607e6	−1.38808
$369$	419829.	0.160512
$370$	1.44684e6	0.549434
$371$	174269.	0.0657334
$372$	5.84634e6	2.19042
$373$	−1.81235e6	−0.674482	−0.337241	−	0.941418i	$-0.609494\pi$
$373$	−1.81235e6	−0.674482	−0.337241	+	0.941418i	$0.609494\pi$
$374$	−306880.	−0.113446
$375$	1.04324e6	0.383096
$376$	3.21367e6	1.17228
$377$	−1.46679e6	−0.531515
$378$	238667.	0.0859138
$379$	−1.83455e6	−0.656043	−0.328022	−	0.944670i	$-0.606382\pi$
$379$	−1.83455e6	−0.656043	−0.328022	+	0.944670i	$0.606382\pi$
$380$	−1.04294e6	−0.370510
$381$	−1.60360e6	−0.565958
$382$	−6.19097e6	−2.17070
$383$	4.14101e6	1.44248	0.721239	−	0.692686i	$-0.243573\pi$
$383$	4.14101e6	1.44248	0.721239	+	0.692686i	$0.243573\pi$
$384$	−2.66897e6	−0.923667
$385$	−69983.6	−0.0240627
$386$	5.74229e6	1.96163
$387$	−1.10627e6	−0.375476
$388$	−6.83701e6	−2.30562
$389$	1.95266e6	0.654262	0.327131	−	0.944979i	$-0.393918\pi$
$389$	1.95266e6	0.654262	0.327131	+	0.944979i	$0.393918\pi$
$390$	1.36029e6	0.452867
$391$	834562.	0.276068
$392$	−5.34229e6	−1.75595
$393$	−1.32360e6	−0.432291
$394$	−200295.	−0.0650026
$395$	−1.42598e6	−0.459856
$396$	575088.	0.184288
$397$	5.08273e6	1.61853	0.809265	−	0.587443i	$-0.199865\pi$
$397$	5.08273e6	1.61853	0.809265	+	0.587443i	$0.199865\pi$
$398$	−37818.6	−0.0119674
$399$	236340.	0.0743199
$400$	−3.41347e6	−1.06671
$401$	−3.61136e6	−1.12153	−0.560763	−	0.827976i	$-0.689492\pi$
$401$	−3.61136e6	−1.12153	−0.560763	+	0.827976i	$0.689492\pi$
$402$	1.71357e6	0.528856
$403$	−7.55097e6	−2.31601
$404$	5.81115e6	1.77137
$405$	−129815.	−0.0393267
$406$	623228.	0.187643
$407$	−790025.	−0.236404
$408$	−884122.	−0.262943
$409$	3.89265e6	1.15063	0.575317	−	0.817931i	$-0.304879\pi$
$409$	3.89265e6	1.15063	0.575317	+	0.817931i	$0.304879\pi$
$410$	−1.01669e6	−0.298697
$411$	3.60679e6	1.05321
$412$	−1.32126e7	−3.83482
$413$	800619.	0.230967
$414$	−2.31896e6	−0.664955
$415$	−512788.	−0.146156
$416$	−1.15786e6	−0.328037
$417$	−926644.	−0.260959
$418$	844400.	0.236378
$419$	−3.72664e6	−1.03701	−0.518504	−	0.855075i	$-0.673511\pi$
$419$	−3.72664e6	−1.03701	−0.518504	+	0.855075i	$0.673511\pi$
$420$	−389799.	−0.107825
$421$	−415385.	−0.114221	−0.0571104	−	0.998368i	$-0.518189\pi$
$421$	−415385.	−0.114221	−0.0571104	+	0.998368i	$0.518189\pi$
$422$	−472237.	−0.129086
$423$	765798.	0.208096
$424$	1.79380e6	0.484574
$425$	789987.	0.212152
$426$	7.39989e6	1.97561
$427$	1.54888e6	0.411101
$428$	2.39242e6	0.631289
$429$	−742768.	−0.194854
$430$	2.67903e6	0.698725
$431$	5.35696e6	1.38907	0.694537	−	0.719457i	$-0.255609\pi$
$431$	5.35696e6	1.38907	0.694537	+	0.719457i	$0.255609\pi$
$432$	910335.	0.234689
$433$	−3.58560e6	−0.919055	−0.459528	−	0.888163i	$-0.651981\pi$
$433$	−3.58560e6	−0.919055	−0.459528	+	0.888163i	$0.651981\pi$
$434$	3.20834e6	0.817628
$435$	−338984.	−0.0858928
$436$	6.29410e6	1.58569
$437$	−2.29635e6	−0.575221
$438$	−2.31079e6	−0.575540
$439$	−1.11636e6	−0.276466	−0.138233	−	0.990400i	$-0.544142\pi$
$439$	−1.11636e6	−0.276466	−0.138233	+	0.990400i	$0.544142\pi$
$440$	−720362.	−0.177386
$441$	−1.27303e6	−0.311705
$442$	2.20766e6	0.537498
$443$	80981.9	0.0196055	0.00980277	−	0.999952i	$-0.496880\pi$
$443$	80981.9	0.0196055	0.00980277	+	0.999952i	$0.496880\pi$
$444$	−4.40034e6	−1.05932
$445$	−1.18040e6	−0.282572
$446$	7.74138e6	1.84281
$447$	−4.55838e6	−1.07905
$448$	−827632.	−0.194824
$449$	1.95939e6	0.458674	0.229337	−	0.973347i	$-0.426344\pi$
$449$	1.95939e6	0.458674	0.229337	+	0.973347i	$0.426344\pi$
$450$	−2.19510e6	−0.511003
$451$	555151.	0.128520
$452$	−9.49343e6	−2.18563
$453$	−3.33229e6	−0.762951
$454$	−1.00444e7	−2.28709
$455$	503454.	0.114007
$456$	2.43272e6	0.547873
$457$	−5.18165e6	−1.16059	−0.580294	−	0.814407i	$-0.697062\pi$
$457$	−5.18165e6	−1.16059	−0.580294	+	0.814407i	$0.697062\pi$
$458$	7.79059e6	1.73543
$459$	−210681.	−0.0466760
$460$	3.78740e6	0.834540
$461$	−5.44215e6	−1.19267	−0.596333	−	0.802737i	$-0.703376\pi$
$461$	−5.44215e6	−1.19267	−0.596333	+	0.802737i	$0.703376\pi$
$462$	315595.	0.0687901
$463$	6.57477e6	1.42537	0.712686	−	0.701483i	$-0.247478\pi$
$463$	6.57477e6	1.42537	0.712686	+	0.701483i	$0.247478\pi$
$464$	2.37715e6	0.512579
$465$	−1.74507e6	−0.374266
$466$	−6.98444e6	−1.48993
$467$	9.13184e6	1.93761	0.968804	−	0.247826i	$-0.0797163\pi$
$467$	9.13184e6	1.93761	0.968804	+	0.247826i	$0.0797163\pi$
$468$	−4.13712e6	−0.873139
$469$	634206.	0.133137
$470$	−1.85452e6	−0.387246
$471$	−985900.	−0.204777
$472$	8.24101e6	1.70265
$473$	−1.46285e6	−0.300639
$474$	6.43057e6	1.31463
$475$	−2.17370e6	−0.442044
$476$	−632618.	−0.127975
$477$	427453.	0.0860186
$478$	−2.70472e6	−0.541442
$479$	−9.18219e6	−1.82855	−0.914277	−	0.405090i	$-0.867240\pi$
$479$	−9.18219e6	−1.82855	−0.914277	+	0.405090i	$0.867240\pi$
$480$	−267588.	−0.0530107
$481$	5.68335e6	1.12006
$482$	5.13214e6	1.00619
$483$	−858263.	−0.167399
$484$	−9.91507e6	−1.92390
$485$	2.04078e6	0.393950
$486$	585410.	0.112427
$487$	2.37828e6	0.454403	0.227202	−	0.973848i	$-0.427042\pi$
$487$	2.37828e6	0.454403	0.227202	+	0.973848i	$0.427042\pi$
$488$	1.59431e7	3.03056
$489$	−4.17353e6	−0.789281
$490$	3.08289e6	0.580053
$491$	272673.	0.0510433	0.0255216	−	0.999674i	$-0.491875\pi$
$491$	272673.	0.0510433	0.0255216	+	0.999674i	$0.491875\pi$
$492$	3.09212e6	0.575895
$493$	−550149.	−0.101944
$494$	−6.07452e6	−1.11994
$495$	−171658.	−0.0314884
$496$	1.22374e7	2.23350
$497$	2.73875e6	0.497350
$498$	2.31245e6	0.417830
$499$	2.27012e6	0.408129	0.204064	−	0.978957i	$-0.434585\pi$
$499$	2.27012e6	0.408129	0.204064	+	0.978957i	$0.434585\pi$
$500$	7.68367e6	1.37450
$501$	3.65356e6	0.650312
$502$	−1.61405e7	−2.85864
$503$	−8.21498e6	−1.44773	−0.723864	−	0.689943i	$-0.757636\pi$
$503$	−8.21498e6	−1.44773	−0.723864	+	0.689943i	$0.757636\pi$
$504$	909232.	0.159440
$505$	−1.73457e6	−0.302665
$506$	−3.06642e6	−0.532421
$507$	2.00175e6	0.345852
$508$	−1.18108e7	−2.03058
$509$	−1.17166e6	−0.200450	−0.100225	−	0.994965i	$-0.531956\pi$
$509$	−1.17166e6	−0.200450	−0.100225	+	0.994965i	$0.531956\pi$
$510$	510203.	0.0868595
$511$	−855241.	−0.144889
$512$	−1.17065e7	−1.97357
$513$	579702.	0.0972549
$514$	1.10994e7	1.85307
$515$	3.94382e6	0.655237
$516$	−8.14786e6	−1.34716
$517$	1.01263e6	0.166620
$518$	−2.41481e6	−0.395419
$519$	3.67632e6	0.599094
$520$	5.18220e6	0.840438
$521$	−2.07857e6	−0.335483	−0.167741	−	0.985831i	$-0.553647\pi$
$521$	−2.07857e6	−0.335483	−0.167741	+	0.985831i	$0.553647\pi$
$522$	1.52867e6	0.245549
$523$	−7.72810e6	−1.23543	−0.617716	−	0.786402i	$-0.711942\pi$
$523$	−7.72810e6	−1.23543	−0.617716	+	0.786402i	$0.711942\pi$
$524$	−9.74857e6	−1.55100
$525$	−812423.	−0.128642
$526$	4.24761e6	0.669392
$527$	−2.83213e6	−0.444209
$528$	1.20376e6	0.187912
$529$	1.90279e6	0.295633
$530$	−1.03516e6	−0.160072
$531$	1.96378e6	0.302243
$532$	1.74069e6	0.266650
$533$	−3.99370e6	−0.608915
$534$	5.32308e6	0.807811
$535$	−714113.	−0.107865
$536$	6.52807e6	0.981460
$537$	2.05211e6	0.307089
$538$	3.07213e6	0.457597
$539$	−1.68337e6	−0.249578
$540$	−956112.	−0.141099
$541$	7.18332e6	1.05519	0.527597	−	0.849495i	$-0.323093\pi$
$541$	7.18332e6	1.05519	0.527597	+	0.849495i	$0.323093\pi$
$542$	−2.10512e7	−3.07807
$543$	−1.08774e6	−0.158316
$544$	−434277.	−0.0629172
$545$	−1.87872e6	−0.270939
$546$	−2.27036e6	−0.325921
$547$	1.10835e7	1.58383	0.791914	−	0.610633i	$-0.209085\pi$
$547$	1.10835e7	1.58383	0.791914	+	0.610633i	$0.209085\pi$
$548$	2.65646e7	3.77879
$549$	3.79914e6	0.537965
$550$	−2.90264e6	−0.409153
$551$	1.51377e6	0.212413
$552$	−8.83436e6	−1.23403
$553$	2.38000e6	0.330952
$554$	9.52548e6	1.31860
$555$	1.31345e6	0.181002
$556$	−6.82490e6	−0.936287
$557$	−8.48471e6	−1.15877	−0.579387	−	0.815052i	$-0.696708\pi$
$557$	−8.48471e6	−1.15877	−0.579387	+	0.815052i	$0.696708\pi$
$558$	7.86951e6	1.06995
$559$	1.05235e7	1.42440
$560$	−815918.	−0.109945
$561$	−278589.	−0.0373729
$562$	−2.96173e6	−0.395553
$563$	4.60288e6	0.612010	0.306005	−	0.952030i	$-0.401008\pi$
$563$	4.60288e6	0.612010	0.306005	+	0.952030i	$0.401008\pi$
$564$	5.64024e6	0.746621
$565$	2.83369e6	0.373449
$566$	−4.68738e6	−0.615020
$567$	216664.	0.0283028
$568$	2.81908e7	3.66637
$569$	9.86745e6	1.27769	0.638843	−	0.769337i	$-0.279413\pi$
$569$	9.86745e6	1.27769	0.638843	+	0.769337i	$0.279413\pi$
$570$	−1.40386e6	−0.180982
$571$	−4.53976e6	−0.582697	−0.291349	−	0.956617i	$-0.594104\pi$
$571$	−4.53976e6	−0.582697	−0.291349	+	0.956617i	$0.594104\pi$
$572$	−5.47062e6	−0.699112
$573$	−5.62023e6	−0.715101
$574$	1.69689e6	0.214968
$575$	7.89374e6	0.995665
$576$	−2.03004e6	−0.254946
$577$	−9.67427e6	−1.20970	−0.604851	−	0.796338i	$-0.706767\pi$
$577$	−9.67427e6	−1.20970	−0.604851	+	0.796338i	$0.706767\pi$
$578$	828024.	0.103092
$579$	5.21291e6	0.646225
$580$	−2.49668e6	−0.308172
$581$	855856.	0.105187
$582$	−9.20302e6	−1.12622
$583$	565232.	0.0688740
$584$	−8.80325e6	−1.06810
$585$	1.23489e6	0.149189
$586$	2.10764e7	2.53543
$587$	5.58946e6	0.669537	0.334768	−	0.942300i	$-0.391342\pi$
$587$	5.58946e6	0.669537	0.334768	+	0.942300i	$0.391342\pi$
$588$	−9.37613e6	−1.11836
$589$	7.79279e6	0.925561
$590$	−4.75566e6	−0.562446
$591$	−181830.	−0.0214140
$592$	−9.21067e6	−1.08016
$593$	−1.44514e7	−1.68762	−0.843809	−	0.536644i	$-0.819692\pi$
$593$	−1.44514e7	−1.68762	−0.843809	+	0.536644i	$0.819692\pi$
$594$	774102.	0.0900186
$595$	188830.	0.0218664
$596$	−3.35733e7	−3.87149
$597$	−34332.2	−0.00394244
$598$	2.20595e7	2.52256
$599$	7.07091e6	0.805208	0.402604	−	0.915374i	$-0.368105\pi$
$599$	7.07091e6	0.805208	0.402604	+	0.915374i	$0.368105\pi$
$600$	−8.36251e6	−0.948328
$601$	1.28581e7	1.45208	0.726040	−	0.687653i	$-0.241359\pi$
$601$	1.28581e7	1.45208	0.726040	+	0.687653i	$0.241359\pi$
$602$	−4.47136e6	−0.502862
$603$	1.55560e6	0.174223
$604$	−2.45429e7	−2.73737
$605$	2.95955e6	0.328728
$606$	7.82214e6	0.865255
$607$	−3.87881e6	−0.427294	−0.213647	−	0.976911i	$-0.568534\pi$
$607$	−3.87881e6	−0.427294	−0.213647	+	0.976911i	$0.568534\pi$
$608$	1.19494e6	0.131095
$609$	565773.	0.0618157
$610$	−9.20031e6	−1.00110
$611$	−7.28478e6	−0.789430
$612$	−1.55170e6	−0.167467
$613$	−6.15434e6	−0.661500	−0.330750	−	0.943718i	$-0.607302\pi$
$613$	−6.15434e6	−0.661500	−0.330750	+	0.943718i	$0.607302\pi$
$614$	−5.20149e6	−0.556809
$615$	−922965.	−0.0984006
$616$	1.20230e6	0.127662
$617$	−1.76097e7	−1.86226	−0.931129	−	0.364690i	$-0.881175\pi$
$617$	−1.76097e7	−1.86226	−0.931129	+	0.364690i	$0.881175\pi$
$618$	−1.77849e7	−1.87318
$619$	−5.14543e6	−0.539753	−0.269876	−	0.962895i	$-0.586983\pi$
$619$	−5.14543e6	−0.539753	−0.269876	+	0.962895i	$0.586983\pi$
$620$	−1.28528e7	−1.34282
$621$	−2.10517e6	−0.219058
$622$	−1.29915e7	−1.34643
$623$	1.97011e6	0.203362
$624$	−8.65971e6	−0.890311
$625$	6.24875e6	0.639872
$626$	−1.29466e7	−1.32044
$627$	766555.	0.0778708
$628$	−7.26133e6	−0.734712
$629$	2.13165e6	0.214827
$630$	−524693.	−0.0526688
$631$	−887224.	−0.0887074	−0.0443537	−	0.999016i	$-0.514123\pi$
$631$	−887224.	−0.0887074	−0.0443537	+	0.999016i	$0.514123\pi$
$632$	2.44981e7	2.43971
$633$	−428701.	−0.0425251
$634$	−2.34334e7	−2.31533
$635$	3.52541e6	0.346956
$636$	3.14827e6	0.308623
$637$	1.21099e7	1.18248
$638$	2.02140e6	0.196608
$639$	6.71770e6	0.650831
$640$	5.86754e6	0.566248
$641$	−8.85717e6	−0.851431	−0.425716	−	0.904857i	$-0.639978\pi$
$641$	−8.85717e6	−0.851431	−0.425716	+	0.904857i	$0.639978\pi$
$642$	3.22034e6	0.308364
$643$	3.94541e6	0.376327	0.188163	−	0.982138i	$-0.439747\pi$
$643$	3.94541e6	0.376327	0.188163	+	0.982138i	$0.439747\pi$
$644$	−6.32127e6	−0.600606
$645$	2.43205e6	0.230183
$646$	−2.27836e6	−0.214803
$647$	−7.27121e6	−0.682882	−0.341441	−	0.939903i	$-0.610915\pi$
$647$	−7.27121e6	−0.682882	−0.341441	+	0.939903i	$0.610915\pi$
$648$	2.23019e6	0.208643
$649$	2.59676e6	0.242003
$650$	2.08812e7	1.93853
$651$	2.91256e6	0.269354
$652$	−3.07388e7	−2.83184
$653$	1.14492e7	1.05074	0.525368	−	0.850875i	$-0.323928\pi$
$653$	1.14492e7	1.05074	0.525368	+	0.850875i	$0.323928\pi$
$654$	8.47222e6	0.774556
$655$	2.90985e6	0.265013
$656$	6.47234e6	0.587222
$657$	−2.09776e6	−0.189602
$658$	3.09524e6	0.278695
$659$	1.47106e7	1.31952	0.659762	−	0.751474i	$-0.270657\pi$
$659$	1.47106e7	1.31952	0.659762	+	0.751474i	$0.270657\pi$
$660$	−1.26429e6	−0.112976
$661$	3.15987e6	0.281297	0.140649	−	0.990060i	$-0.455081\pi$
$661$	3.15987e6	0.281297	0.140649	+	0.990060i	$0.455081\pi$
$662$	3.84974e7	3.41418
$663$	2.00414e6	0.177069
$664$	8.80958e6	0.775416
$665$	−519577.	−0.0455613
$666$	−5.92311e6	−0.517445
$667$	−5.49721e6	−0.478441
$668$	2.69091e7	2.33324
$669$	7.02770e6	0.607083
$670$	−3.76717e6	−0.324211
$671$	5.02370e6	0.430742
$672$	446611.	0.0381510
$673$	−1.00433e7	−0.854746	−0.427373	−	0.904075i	$-0.640561\pi$
$673$	−1.00433e7	−0.854746	−0.427373	+	0.904075i	$0.640561\pi$
$674$	8.04498e6	0.682143
$675$	−1.99274e6	−0.168341
$676$	1.47433e7	1.24087
$677$	1.41644e7	1.18776	0.593878	−	0.804555i	$-0.297596\pi$
$677$	1.41644e7	1.18776	0.593878	+	0.804555i	$0.297596\pi$
$678$	−1.27787e7	−1.06761
$679$	−3.40610e6	−0.283520
$680$	1.94368e6	0.161195
$681$	−9.11841e6	−0.753444
$682$	1.04061e7	0.856693
$683$	7.75823e6	0.636372	0.318186	−	0.948028i	$-0.396926\pi$
$683$	7.75823e6	0.636372	0.318186	+	0.948028i	$0.396926\pi$
$684$	4.26961e6	0.348938
$685$	−7.92927e6	−0.645664
$686$	−1.06478e7	−0.863876
$687$	7.07238e6	0.571707
$688$	−1.70549e7	−1.37366
$689$	−4.06621e6	−0.326319
$690$	5.09807e6	0.407646
$691$	−1.04746e7	−0.834529	−0.417265	−	0.908785i	$-0.637011\pi$
$691$	−1.04746e7	−0.834529	−0.417265	+	0.908785i	$0.637011\pi$
$692$	2.70768e7	2.14947
$693$	286501.	0.0226617
$694$	−9.50807e6	−0.749365
$695$	2.03716e6	0.159979
$696$	5.82367e6	0.455694
$697$	−1.49791e6	−0.116789
$698$	2.96312e7	2.30203
$699$	−6.34055e6	−0.490833
$700$	−5.98364e6	−0.461552
$701$	−2.12786e7	−1.63549	−0.817745	−	0.575581i	$-0.804776\pi$
$701$	−2.12786e7	−1.63549	−0.817745	+	0.575581i	$0.804776\pi$
$702$	−5.56880e6	−0.426500
$703$	−5.86536e6	−0.447617
$704$	−2.68438e6	−0.204132
$705$	−1.68355e6	−0.127572
$706$	1.89778e7	1.43296
$707$	2.89503e6	0.217824
$708$	1.44636e7	1.08441
$709$	−1.36868e7	−1.02256	−0.511279	−	0.859415i	$-0.670828\pi$
$709$	−1.36868e7	−1.02256	−0.511279	+	0.859415i	$0.670828\pi$
$710$	−1.62681e7	−1.21113
$711$	5.83774e6	0.433083
$712$	2.02789e7	1.49915
$713$	−2.82993e7	−2.08474
$714$	−851540.	−0.0625114
$715$	1.63292e6	0.119454
$716$	1.51141e7	1.10179
$717$	−2.45537e6	−0.178369
$718$	1.62431e7	1.17587
$719$	6.23644e6	0.449898	0.224949	−	0.974371i	$-0.427778\pi$
$719$	6.23644e6	0.449898	0.224949	+	0.974371i	$0.427778\pi$
$720$	−2.00131e6	−0.143874
$721$	−6.58232e6	−0.471564
$722$	−1.82789e7	−1.30499
$723$	4.65901e6	0.331473
$724$	−8.01140e6	−0.568018
$725$	−5.20360e6	−0.367671
$726$	−1.33463e7	−0.939763
$727$	−6.49977e6	−0.456102	−0.228051	−	0.973649i	$-0.573235\pi$
$727$	−6.49977e6	−0.456102	−0.228051	+	0.973649i	$0.573235\pi$
$728$	−8.64921e6	−0.604851
$729$	531441.	0.0370370
$730$	5.08011e6	0.352831
$731$	3.94705e6	0.273199
$732$	2.79813e7	1.93015
$733$	1.80568e7	1.24131	0.620657	−	0.784082i	$-0.286866\pi$
$733$	1.80568e7	1.24131	0.620657	+	0.784082i	$0.286866\pi$
$734$	8.99357e6	0.616158
$735$	2.79868e6	0.191088
$736$	−4.33940e6	−0.295281
$737$	2.05701e6	0.139498
$738$	4.16217e6	0.281306
$739$	−1.04958e7	−0.706977	−0.353489	−	0.935439i	$-0.615005\pi$
$739$	−1.04958e7	−0.706977	−0.353489	+	0.935439i	$0.615005\pi$
$740$	9.67383e6	0.649410
$741$	−5.51451e6	−0.368945
$742$	1.72770e6	0.115201
$743$	−2.83337e6	−0.188292	−0.0941460	−	0.995558i	$-0.530012\pi$
$743$	−2.83337e6	−0.188292	−0.0941460	+	0.995558i	$0.530012\pi$
$744$	2.99799e7	1.98563
$745$	1.00213e7	0.661504
$746$	−1.79676e7	−1.18207
$747$	2.09927e6	0.137647
$748$	−2.05186e6	−0.134089
$749$	1.19187e6	0.0776291
$750$	1.03427e7	0.671397
$751$	−2.88029e7	−1.86353	−0.931763	−	0.363066i	$-0.881730\pi$
$751$	−2.88029e7	−1.86353	−0.931763	+	0.363066i	$0.881730\pi$
$752$	1.18060e7	0.761305
$753$	−1.46526e7	−0.941729
$754$	−1.45417e7	−0.931511
$755$	7.32580e6	0.467722
$756$	1.59577e6	0.101547
$757$	2.34715e6	0.148868	0.0744340	−	0.997226i	$-0.476285\pi$
$757$	2.34715e6	0.148868	0.0744340	+	0.997226i	$0.476285\pi$
$758$	−1.81877e7	−1.14975
$759$	−2.78373e6	−0.175397
$760$	−5.34816e6	−0.335870
$761$	−2.63740e7	−1.65088	−0.825439	−	0.564492i	$-0.809072\pi$
$761$	−2.63740e7	−1.65088	−0.825439	+	0.564492i	$0.809072\pi$
$762$	−1.58980e7	−0.991874
$763$	3.13563e6	0.194991
$764$	−4.13940e7	−2.56569
$765$	463167.	0.0286144
$766$	4.10538e7	2.52803
$767$	−1.86808e7	−1.14659
$768$	−1.92421e7	−1.17720
$769$	4.56258e6	0.278224	0.139112	−	0.990277i	$-0.455575\pi$
$769$	4.56258e6	0.278224	0.139112	+	0.990277i	$0.455575\pi$
$770$	−693815.	−0.0421713
$771$	1.00762e7	0.610463
$772$	3.83940e7	2.31857
$773$	−1.90736e7	−1.14811	−0.574054	−	0.818817i	$-0.694630\pi$
$773$	−1.90736e7	−1.14811	−0.574054	+	0.818817i	$0.694630\pi$
$774$	−1.09675e7	−0.658044
$775$	−2.67878e7	−1.60208
$776$	−3.50600e7	−2.09006
$777$	−2.19219e6	−0.130264
$778$	1.93586e7	1.14663
$779$	4.12160e6	0.243345
$780$	9.09517e6	0.535272
$781$	8.88298e6	0.521112
$782$	8.27381e6	0.483826
$783$	1.38774e6	0.0808919
$784$	−1.96259e7	−1.14035
$785$	2.16743e6	0.125537
$786$	−1.31221e7	−0.757615
$787$	1.99947e7	1.15074	0.575371	−	0.817893i	$-0.304858\pi$
$787$	1.99947e7	1.15074	0.575371	+	0.817893i	$0.304858\pi$
$788$	−1.33921e6	−0.0768306
$789$	3.85603e6	0.220520
$790$	−1.41372e7	−0.805925
$791$	−4.72950e6	−0.268766
$792$	2.94904e6	0.167058
$793$	−3.61399e7	−2.04082
$794$	5.03900e7	2.83657
$795$	−939725.	−0.0527331
$796$	−252863.	−0.0141450
$797$	2.94054e7	1.63976	0.819881	−	0.572534i	$-0.194040\pi$
$797$	2.94054e7	1.63976	0.819881	+	0.572534i	$0.194040\pi$
$798$	2.34307e6	0.130250
$799$	−2.73229e6	−0.151412
$800$	−4.10763e6	−0.226917
$801$	4.83235e6	0.266120
$802$	−3.58028e7	−1.96554
$803$	−2.77392e6	−0.151812
$804$	1.14573e7	0.625088
$805$	1.88683e6	0.102623
$806$	−7.48600e7	−4.05894
$807$	2.78891e6	0.150748
$808$	2.97994e7	1.60576
$809$	1.09983e7	0.590820	0.295410	−	0.955371i	$-0.404544\pi$
$809$	1.09983e7	0.590820	0.295410	+	0.955371i	$0.404544\pi$
$810$	−1.28698e6	−0.0689223
$811$	−5.50304e6	−0.293799	−0.146900	−	0.989151i	$-0.546929\pi$
$811$	−5.50304e6	−0.293799	−0.146900	+	0.989151i	$0.546929\pi$
$812$	4.16702e6	0.221787
$813$	−1.91105e7	−1.01402
$814$	−7.83228e6	−0.414312
$815$	9.17522e6	0.483863
$816$	−3.24799e6	−0.170761
$817$	−1.08606e7	−0.569243
$818$	3.85916e7	2.01655
$819$	−2.06106e6	−0.107369
$820$	−6.79781e6	−0.353048
$821$	3.29829e7	1.70778	0.853889	−	0.520455i	$-0.174238\pi$
$821$	3.29829e7	1.70778	0.853889	+	0.520455i	$0.174238\pi$
$822$	3.57575e7	1.84581
$823$	1.69682e7	0.873246	0.436623	−	0.899645i	$-0.356174\pi$
$823$	1.69682e7	0.873246	0.436623	+	0.899645i	$0.356174\pi$
$824$	−6.77538e7	−3.47629
$825$	−2.63505e6	−0.134789
$826$	7.93730e6	0.404784
$827$	−1.65421e7	−0.841060	−0.420530	−	0.907279i	$-0.638156\pi$
$827$	−1.65421e7	−0.841060	−0.420530	+	0.907279i	$0.638156\pi$
$828$	−1.55050e7	−0.785951
$829$	7.93594e6	0.401063	0.200531	−	0.979687i	$-0.435733\pi$
$829$	7.93594e6	0.401063	0.200531	+	0.979687i	$0.435733\pi$
$830$	−5.08376e6	−0.256148
$831$	8.64733e6	0.434390
$832$	1.93111e7	0.967161
$833$	4.54206e6	0.226799
$834$	−9.18671e6	−0.457346
$835$	−8.03210e6	−0.398669
$836$	5.64582e6	0.279390
$837$	7.14403e6	0.352476
$838$	−3.69457e7	−1.81742
$839$	2.58192e7	1.26630	0.633151	−	0.774028i	$-0.281761\pi$
$839$	2.58192e7	1.26630	0.633151	+	0.774028i	$0.281761\pi$
$840$	−1.99888e6	−0.0977437
$841$	−1.68873e7	−0.823325
$842$	−4.11811e6	−0.200178
$843$	−2.68869e6	−0.130308
$844$	−3.15746e6	−0.152575
$845$	−4.40071e6	−0.212022
$846$	7.59210e6	0.364700
$847$	−4.93955e6	−0.236581
$848$	6.58987e6	0.314693
$849$	−4.25525e6	−0.202608
$850$	7.83190e6	0.371809
$851$	2.12999e7	1.00822
$852$	4.94770e7	2.33510
$853$	−2.00221e7	−0.942188	−0.471094	−	0.882083i	$-0.656141\pi$
$853$	−2.00221e7	−0.942188	−0.471094	+	0.882083i	$0.656141\pi$
$854$	1.53555e7	0.720477
$855$	−1.27444e6	−0.0596214
$856$	1.22683e7	0.572268
$857$	−5.27660e6	−0.245416	−0.122708	−	0.992443i	$-0.539158\pi$
$857$	−5.27660e6	−0.245416	−0.122708	+	0.992443i	$0.539158\pi$
$858$	−7.36377e6	−0.341493
$859$	3.13305e7	1.44872	0.724360	−	0.689422i	$-0.242136\pi$
$859$	3.13305e7	1.44872	0.724360	+	0.689422i	$0.242136\pi$
$860$	1.79125e7	0.825867
$861$	1.54045e6	0.0708174
$862$	5.31087e7	2.43443
$863$	2.26971e7	1.03739	0.518697	−	0.854958i	$-0.326417\pi$
$863$	2.26971e7	1.03739	0.518697	+	0.854958i	$0.326417\pi$
$864$	1.09546e6	0.0499243
$865$	−8.08214e6	−0.367270
$866$	−3.55475e7	−1.61070
$867$	751689.	0.0339618
$868$	2.14516e7	0.966406
$869$	7.71939e6	0.346764
$870$	−3.36068e6	−0.150532
$871$	−1.47979e7	−0.660928
$872$	3.22760e7	1.43744
$873$	−8.35460e6	−0.371014
$874$	−2.27659e7	−1.00811
$875$	3.82790e6	0.169021
$876$	−1.54504e7	−0.680267
$877$	3.11361e7	1.36699	0.683494	−	0.729956i	$-0.260460\pi$
$877$	3.11361e7	1.36699	0.683494	+	0.729956i	$0.260460\pi$
$878$	−1.10675e7	−0.484523
$879$	1.91334e7	0.835255
$880$	−2.64638e6	−0.115198
$881$	1.35953e7	0.590134	0.295067	−	0.955477i	$-0.404658\pi$
$881$	1.35953e7	0.590134	0.295067	+	0.955477i	$0.404658\pi$
$882$	−1.26208e7	−0.546281
$883$	−1.82777e7	−0.788897	−0.394448	−	0.918918i	$-0.629064\pi$
$883$	−1.82777e7	−0.788897	−0.394448	+	0.918918i	$0.629064\pi$
$884$	1.47608e7	0.635302
$885$	−4.31724e6	−0.185288
$886$	802852.	0.0343598
$887$	2.27419e7	0.970549	0.485274	−	0.874362i	$-0.338720\pi$
$887$	2.27419e7	0.970549	0.485274	+	0.874362i	$0.338720\pi$
$888$	−2.25648e7	−0.960284
$889$	−5.88398e6	−0.249699
$890$	−1.17024e7	−0.495223
$891$	702738.	0.0296551
$892$	5.17603e7	2.17814
$893$	7.51808e6	0.315485
$894$	−4.51916e7	−1.89110
$895$	−4.51141e6	−0.188259
$896$	−9.79307e6	−0.407520
$897$	2.00258e7	0.831015
$898$	1.94253e7	0.803852
$899$	1.86551e7	0.769836
$900$	−1.46769e7	−0.603986
$901$	−1.52511e6	−0.0625877
$902$	5.50375e6	0.225238
$903$	−4.05915e6	−0.165659
$904$	−4.86821e7	−1.98129
$905$	2.39132e6	0.0970546
$906$	−3.30361e7	−1.33712
$907$	−1.38636e7	−0.559574	−0.279787	−	0.960062i	$-0.590264\pi$
$907$	−1.38636e7	−0.559574	−0.279787	+	0.960062i	$0.590264\pi$
$908$	−6.71587e7	−2.70326
$909$	7.10103e6	0.285044
$910$	4.99122e6	0.199804
$911$	−4.02480e6	−0.160675	−0.0803375	−	0.996768i	$-0.525600\pi$
$911$	−4.02480e6	−0.160675	−0.0803375	+	0.996768i	$0.525600\pi$
$912$	8.93704e6	0.355801
$913$	2.77592e6	0.110212
$914$	−5.13707e7	−2.03400
$915$	−8.35214e6	−0.329796
$916$	5.20894e7	2.05121
$917$	−4.85660e6	−0.190726
$918$	−2.08868e6	−0.0818024
$919$	−2.42089e7	−0.945553	−0.472777	−	0.881182i	$-0.656748\pi$
$919$	−2.42089e7	−0.945553	−0.472777	+	0.881182i	$0.656748\pi$
$920$	1.94217e7	0.756516
$921$	−4.72196e6	−0.183431
$922$	−5.39533e7	−2.09021
$923$	−6.39032e7	−2.46898
$924$	2.11013e6	0.0813073
$925$	2.01623e7	0.774792
$926$	6.51820e7	2.49805
$927$	−1.61453e7	−0.617088
$928$	2.86056e6	0.109039
$929$	4.11227e7	1.56330	0.781650	−	0.623718i	$-0.214379\pi$
$929$	4.11227e7	1.56330	0.781650	+	0.623718i	$0.214379\pi$
$930$	−1.73006e7	−0.655923
$931$	−1.24978e7	−0.472562
$932$	−4.66993e7	−1.76105
$933$	−1.17938e7	−0.443559
$934$	9.05327e7	3.39577
$935$	612458.	0.0229112
$936$	−2.12151e7	−0.791507
$937$	1.77193e7	0.659321	0.329661	−	0.944100i	$-0.393066\pi$
$937$	1.77193e7	0.659321	0.329661	+	0.944100i	$0.393066\pi$
$938$	6.28749e6	0.233330
$939$	−1.17530e7	−0.434997
$940$	−1.23997e7	−0.457710
$941$	3.59509e7	1.32354	0.661768	−	0.749709i	$-0.269806\pi$
$941$	3.59509e7	1.32354	0.661768	+	0.749709i	$0.269806\pi$
$942$	−9.77418e6	−0.358883
$943$	−1.49675e7	−0.548112
$944$	3.02749e7	1.10574
$945$	−476321.	−0.0173508
$946$	−1.45026e7	−0.526888
$947$	2.73018e7	0.989272	0.494636	−	0.869100i	$-0.335301\pi$
$947$	2.73018e7	0.989272	0.494636	+	0.869100i	$0.335301\pi$
$948$	4.29960e7	1.55384
$949$	1.99553e7	0.719271
$950$	−2.15500e7	−0.774708
$951$	−2.12731e7	−0.762746
$952$	−3.24405e6	−0.116010
$953$	1.59245e6	0.0567982	0.0283991	−	0.999597i	$-0.490959\pi$
$953$	1.59245e6	0.0567982	0.0283991	+	0.999597i	$0.490959\pi$
$954$	4.23775e6	0.150752
$955$	1.23557e7	0.438387
$956$	−1.80843e7	−0.639965
$957$	1.83505e6	0.0647692
$958$	−9.10319e7	−3.20464
$959$	1.32341e7	0.464675
$960$	4.46291e6	0.156293
$961$	6.74062e7	2.35446
$962$	5.63445e7	1.96297
$963$	2.92346e6	0.101585
$964$	3.43144e7	1.18928
$965$	−1.14602e7	−0.396164
$966$	−8.50879e6	−0.293376
$967$	4.77330e6	0.164154	0.0820772	−	0.996626i	$-0.473845\pi$
$967$	4.77330e6	0.164154	0.0820772	+	0.996626i	$0.473845\pi$
$968$	−5.08443e7	−1.74403
$969$	−2.06832e6	−0.0707634
$970$	2.02322e7	0.690420
$971$	7.08746e6	0.241236	0.120618	−	0.992699i	$-0.461512\pi$
$971$	7.08746e6	0.241236	0.120618	+	0.992699i	$0.461512\pi$
$972$	3.91416e6	0.132884
$973$	−3.40007e6	−0.115135
$974$	2.35782e7	0.796367
$975$	1.89562e7	0.638617
$976$	5.85698e7	1.96811
$977$	−4.51020e7	−1.51168	−0.755839	−	0.654757i	$-0.772771\pi$
$977$	−4.51020e7	−1.51168	−0.755839	+	0.654757i	$0.772771\pi$
$978$	−4.13763e7	−1.38326
$979$	6.38994e6	0.213079
$980$	2.06128e7	0.685600
$981$	7.69117e6	0.255164
$982$	2.70327e6	0.0894563
$983$	−3.08113e7	−1.01701	−0.508505	−	0.861059i	$-0.669802\pi$
$983$	−3.08113e7	−1.01701	−0.508505	+	0.861059i	$0.669802\pi$
$984$	1.58563e7	0.522053
$985$	399741.	0.0131277
$986$	−5.45415e6	−0.178663
$987$	2.80989e6	0.0918114
$988$	−4.06154e7	−1.32373
$989$	3.94399e7	1.28217
$990$	−1.70181e6	−0.0551852
$991$	−8.88730e6	−0.287466	−0.143733	−	0.989617i	$-0.545911\pi$
$991$	−8.88730e6	−0.287466	−0.143733	+	0.989617i	$0.545911\pi$
$992$	1.47260e7	0.475122
$993$	3.49484e7	1.12474
$994$	2.71519e7	0.871634
$995$	75476.9	0.00241689
$996$	1.54615e7	0.493859
$997$	5.93598e7	1.89127	0.945636	−	0.325226i	$-0.105440\pi$
$997$	5.93598e7	1.89127	0.945636	+	0.325226i	$0.105440\pi$
$998$	2.25059e7	0.715269
$999$	−5.37706e6	−0.170463

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	51.6.a.c.1.4	✓	4
3.2	odd	2		153.6.a.e.1.1		4
4.3	odd	2		816.6.a.o.1.1		4
17.16	even	2		867.6.a.f.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.6.a.c.1.4	✓	4	1.1	even	1	trivial
153.6.a.e.1.1		4	3.2	odd	2
816.6.a.o.1.1		4	4.3	odd	2
867.6.a.f.1.4		4	17.16	even	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:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	51.a (trivial)

\(f(q)\)	\(=\)	\(q+9.91396 q^{2} +9.00000 q^{3} +66.2866 q^{4} -19.7859 q^{5} +89.2257 q^{6} +33.0231 q^{7} +339.916 q^{8} +81.0000 q^{9} -196.156 q^{10} +107.108 q^{11} +596.580 q^{12} -770.526 q^{13} +327.390 q^{14} -178.073 q^{15} +1248.75 q^{16} -289.000 q^{17} +803.031 q^{18} +795.202 q^{19} -1311.54 q^{20} +297.208 q^{21} +1061.87 q^{22} -2887.76 q^{23} +3059.25 q^{24} -2733.52 q^{25} -7638.96 q^{26} +729.000 q^{27} +2188.99 q^{28} +1903.63 q^{29} -1765.41 q^{30} +9799.76 q^{31} +1502.69 q^{32} +963.976 q^{33} -2865.13 q^{34} -653.390 q^{35} +5369.22 q^{36} -7375.94 q^{37} +7883.60 q^{38} -6934.73 q^{39} -6725.54 q^{40} +5183.08 q^{41} +2946.51 q^{42} -13657.6 q^{43} +7099.85 q^{44} -1602.66 q^{45} -28629.1 q^{46} +9454.30 q^{47} +11238.7 q^{48} -15716.5 q^{49} -27100.0 q^{50} -2601.00 q^{51} -51075.5 q^{52} +5277.19 q^{53} +7227.28 q^{54} -2119.23 q^{55} +11225.1 q^{56} +7156.82 q^{57} +18872.5 q^{58} +24244.2 q^{59} -11803.8 q^{60} +46903.0 q^{61} +97154.5 q^{62} +2674.87 q^{63} -25062.2 q^{64} +15245.5 q^{65} +9556.82 q^{66} +19204.9 q^{67} -19156.8 q^{68} -25989.8 q^{69} -6477.69 q^{70} +82934.5 q^{71} +27533.2 q^{72} -25898.3 q^{73} -73124.8 q^{74} -24601.7 q^{75} +52711.3 q^{76} +3537.05 q^{77} -68750.7 q^{78} +72070.9 q^{79} -24707.5 q^{80} +6561.00 q^{81} +51384.8 q^{82} +25916.9 q^{83} +19700.9 q^{84} +5718.12 q^{85} -135401. q^{86} +17132.7 q^{87} +36407.9 q^{88} +59658.6 q^{89} -15888.7 q^{90} -25445.1 q^{91} -191420. q^{92} +88197.9 q^{93} +93729.6 q^{94} -15733.8 q^{95} +13524.2 q^{96} -103143. q^{97} -155813. q^{98} +8675.78 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{2} + 36 q^{3} + 113 q^{4} + 146 q^{5} + 45 q^{6} + 60 q^{7} - 153 q^{8} + 324 q^{9} - 536 q^{10} + 1114 q^{11} + 1017 q^{12} - 166 q^{13} + 1738 q^{14} + 1314 q^{15} + 2745 q^{16} - 1156 q^{17}+ \cdots + 90234 q^{99}+O(q^{100}) \) 4 * q + 5 * q^2 + 36 * q^3 + 113 * q^4 + 146 * q^5 + 45 * q^6 + 60 * q^7 - 153 * q^8 + 324 * q^9 - 536 * q^10 + 1114 * q^11 + 1017 * q^12 - 166 * q^13 + 1738 * q^14 + 1314 * q^15 + 2745 * q^16 - 1156 * q^17 + 405 * q^18 + 3454 * q^19 + 5748 * q^20 + 540 * q^21 - 6368 * q^22 + 998 * q^23 - 1377 * q^24 + 3150 * q^25 + 2890 * q^26 + 2916 * q^27 - 13382 * q^28 + 3048 * q^29 - 4824 * q^30 - 7808 * q^31 - 8881 * q^32 + 10026 * q^33 - 1445 * q^34 - 14080 * q^35 + 9153 * q^36 - 1700 * q^37 - 26836 * q^38 - 1494 * q^39 - 55648 * q^40 - 1386 * q^41 + 15642 * q^42 - 27654 * q^43 + 35196 * q^44 + 11826 * q^45 - 43646 * q^46 + 38156 * q^47 + 24705 * q^48 - 17012 * q^49 - 42601 * q^50 - 10404 * q^51 - 79378 * q^52 + 52648 * q^53 + 3645 * q^54 + 38782 * q^55 + 78510 * q^56 + 31086 * q^57 - 75320 * q^58 + 50028 * q^59 + 51732 * q^60 + 13076 * q^61 + 83438 * q^62 + 4860 * q^63 - 32847 * q^64 + 59250 * q^65 - 57312 * q^66 + 7880 * q^67 - 32657 * q^68 + 8982 * q^69 + 132272 * q^70 + 106836 * q^71 - 12393 * q^72 + 52488 * q^73 - 164192 * q^74 + 28350 * q^75 + 178608 * q^76 - 8520 * q^77 + 26010 * q^78 + 37688 * q^79 + 17368 * q^80 + 26244 * q^81 + 104254 * q^82 + 9812 * q^83 - 120438 * q^84 - 42194 * q^85 - 145316 * q^86 + 27432 * q^87 - 162784 * q^88 + 20876 * q^89 - 43416 * q^90 + 65984 * q^91 - 302994 * q^92 - 70272 * q^93 - 125700 * q^94 + 81658 * q^95 - 79929 * q^96 - 1996 * q^97 - 321967 * q^98 + 90234 * q^99

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