LMFDB - Embedded newform 75.8.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,8,Mod(1,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.4288769113$
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} - 81x^{2} - 150x + 400 $ x^4 - 81x^2 - 150x + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.26440$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-3.02250 q^{2} +27.0000 q^{3} -118.865 q^{4} -81.6074 q^{6} -1505.28 q^{7} +746.147 q^{8} +729.000 q^{9} +1596.37 q^{11} -3209.34 q^{12} -956.100 q^{13} +4549.69 q^{14} +12959.4 q^{16} +32470.6 q^{17} -2203.40 q^{18} -39168.1 q^{19} -40642.5 q^{21} -4825.04 q^{22} +59361.5 q^{23} +20146.0 q^{24} +2889.81 q^{26} +19683.0 q^{27} +178924. q^{28} +66150.5 q^{29} -19664.1 q^{31} -134677. q^{32} +43102.1 q^{33} -98142.2 q^{34} -86652.2 q^{36} +376045. q^{37} +118386. q^{38} -25814.7 q^{39} +385003. q^{41} +122842. q^{42} +466410. q^{43} -189752. q^{44} -179420. q^{46} +468903. q^{47} +349905. q^{48} +1.44232e6 q^{49} +876706. q^{51} +113646. q^{52} +1.60516e6 q^{53} -59491.8 q^{54} -1.12316e6 q^{56} -1.05754e6 q^{57} -199940. q^{58} -2.04044e6 q^{59} -378667. q^{61} +59434.6 q^{62} -1.09735e6 q^{63} -1.25175e6 q^{64} -130276. q^{66} -4644.40 q^{67} -3.85960e6 q^{68} +1.60276e6 q^{69} -2.79333e6 q^{71} +543941. q^{72} -2.01174e6 q^{73} -1.13660e6 q^{74} +4.65570e6 q^{76} -2.40299e6 q^{77} +78024.8 q^{78} +1.76767e6 q^{79} +531441. q^{81} -1.16367e6 q^{82} -3.06625e6 q^{83} +4.83095e6 q^{84} -1.40972e6 q^{86} +1.78606e6 q^{87} +1.19113e6 q^{88} -6.14397e6 q^{89} +1.43920e6 q^{91} -7.05598e6 q^{92} -530930. q^{93} -1.41726e6 q^{94} -3.63627e6 q^{96} +3.02969e6 q^{97} -4.35940e6 q^{98} +1.16376e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 9 q^{2} + 108 q^{3} + 333 q^{4} + 243 q^{6} + 1188 q^{7} + 2043 q^{8} + 2916 q^{9} + 5376 q^{11} + 8991 q^{12} + 8424 q^{13} + 6762 q^{14} + 43265 q^{16} + 4896 q^{17} + 6561 q^{18} + 15232 q^{19}+ \cdots + 3919104 q^{99}+O(q^{100}) $ 4 * q + 9 * q^2 + 108 * q^3 + 333 * q^4 + 243 * q^6 + 1188 * q^7 + 2043 * q^8 + 2916 * q^9 + 5376 * q^11 + 8991 * q^12 + 8424 * q^13 + 6762 * q^14 + 43265 * q^16 + 4896 * q^17 + 6561 * q^18 + 15232 * q^19 + 32076 * q^21 - 44118 * q^22 + 110016 * q^23 + 55161 * q^24 - 396762 * q^26 + 78732 * q^27 + 674514 * q^28 - 120036 * q^29 + 116864 * q^31 + 621963 * q^32 + 145152 * q^33 - 92374 * q^34 + 242757 * q^36 + 663768 * q^37 + 1888776 * q^38 + 227448 * q^39 + 253824 * q^41 + 182574 * q^42 + 1092960 * q^43 - 1289286 * q^44 + 331204 * q^46 + 132840 * q^47 + 1168155 * q^48 + 1633580 * q^49 + 132192 * q^51 - 779274 * q^52 + 1994616 * q^53 + 177147 * q^54 + 352830 * q^56 + 411264 * q^57 - 742608 * q^58 - 545712 * q^59 - 3216760 * q^61 + 476856 * q^62 + 866052 * q^63 + 284297 * q^64 - 1191186 * q^66 + 2013336 * q^67 - 7540038 * q^68 + 2970432 * q^69 - 690912 * q^71 + 1489347 * q^72 - 5498064 * q^73 - 11980638 * q^74 + 6557832 * q^76 - 795744 * q^77 - 10712574 * q^78 + 7190080 * q^79 + 2125764 * q^81 - 26456814 * q^82 - 13158432 * q^83 + 18211878 * q^84 - 9918804 * q^86 - 3240972 * q^87 - 22119642 * q^88 - 22889448 * q^89 + 12037824 * q^91 - 17368812 * q^92 + 3155328 * q^93 + 22864448 * q^94 + 16793001 * q^96 + 17873136 * q^97 - 20819979 * q^98 + 3919104 * 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 $

$2$	−3.02250	−0.267153	−0.133577	−	0.991038i	$-0.542646\pi$
$2$	−3.02250	−0.267153	−0.133577	+	0.991038i	$0.542646\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−118.865	−0.928629
$5$	0	0
$5$	0	0
$6$	−81.6074	−0.154241
$7$	−1505.28	−1.65872	−0.829361	−	0.558714i	$-0.811295\pi$
$7$	−1505.28	−1.65872	−0.829361	+	0.558714i	$0.811295\pi$
$8$	746.147	0.515240
$9$	729.000	0.333333
$10$	0	0
$11$	1596.37	0.361627	0.180813	−	0.983517i	$-0.442127\pi$
$11$	1596.37	0.361627	0.180813	+	0.983517i	$0.442127\pi$
$12$	−3209.34	−0.536144
$13$	−956.100	−0.120698	−0.0603492	−	0.998177i	$-0.519221\pi$
$13$	−956.100	−0.120698	−0.0603492	+	0.998177i	$0.519221\pi$
$14$	4549.69	0.443133
$15$	0	0
$16$	12959.4	0.790981
$17$	32470.6	1.60295	0.801473	−	0.598031i	$-0.204050\pi$
$17$	32470.6	1.60295	0.801473	+	0.598031i	$0.204050\pi$
$18$	−2203.40	−0.0890511
$19$	−39168.1	−1.31007	−0.655036	−	0.755598i	$-0.727347\pi$
$19$	−39168.1	−1.31007	−0.655036	+	0.755598i	$0.727347\pi$
$20$	0	0
$21$	−40642.5	−0.957663
$22$	−4825.04	−0.0966098
$23$	59361.5	1.01732	0.508660	−	0.860968i	$-0.330141\pi$
$23$	59361.5	1.01732	0.508660	+	0.860968i	$0.330141\pi$
$24$	20146.0	0.297474
$25$	0	0
$26$	2889.81	0.0322450
$27$	19683.0	0.192450
$28$	178924.	1.54034
$29$	66150.5	0.503663	0.251831	−	0.967771i	$-0.418967\pi$
$29$	66150.5	0.503663	0.251831	+	0.967771i	$0.418967\pi$
$30$	0	0
$31$	−19664.1	−0.118552	−0.0592758	−	0.998242i	$-0.518879\pi$
$31$	−19664.1	−0.118552	−0.0592758	+	0.998242i	$0.518879\pi$
$32$	−134677.	−0.726553
$33$	43102.1	0.208785
$34$	−98142.2	−0.428232
$35$	0	0
$36$	−86652.2	−0.309543
$37$	376045.	1.22049	0.610245	−	0.792213i	$-0.291071\pi$
$37$	376045.	1.22049	0.610245	+	0.792213i	$0.291071\pi$
$38$	118386.	0.349990
$39$	−25814.7	−0.0696853
$40$	0	0
$41$	385003.	0.872409	0.436205	−	0.899848i	$-0.356322\pi$
$41$	385003.	0.872409	0.436205	+	0.899848i	$0.356322\pi$
$42$	122842.	0.255843
$43$	466410.	0.894600	0.447300	−	0.894384i	$-0.352386\pi$
$43$	466410.	0.894600	0.447300	+	0.894384i	$0.352386\pi$
$44$	−189752.	−0.335817
$45$	0	0
$46$	−179420.	−0.271780
$47$	468903.	0.658779	0.329390	−	0.944194i	$-0.393157\pi$
$47$	468903.	0.658779	0.329390	+	0.944194i	$0.393157\pi$
$48$	349905.	0.456673
$49$	1.44232e6	1.75136
$50$	0	0
$51$	876706.	0.925461
$52$	113646.	0.112084
$53$	1.60516e6	1.48100	0.740498	−	0.672058i	$-0.234590\pi$
$53$	1.60516e6	1.48100	0.740498	+	0.672058i	$0.234590\pi$
$54$	−59491.8	−0.0514137
$55$	0	0
$56$	−1.12316e6	−0.854639
$57$	−1.05754e6	−0.756371
$58$	−199940.	−0.134555
$59$	−2.04044e6	−1.29342	−0.646712	−	0.762734i	$-0.723856\pi$
$59$	−2.04044e6	−1.29342	−0.646712	+	0.762734i	$0.723856\pi$
$60$	0	0
$61$	−378667.	−0.213601	−0.106800	−	0.994280i	$-0.534061\pi$
$61$	−378667.	−0.213601	−0.106800	+	0.994280i	$0.534061\pi$
$62$	59434.6	0.0316715
$63$	−1.09735e6	−0.552907
$64$	−1.25175e6	−0.596880
$65$	0	0
$66$	−130276.	−0.0557777
$67$	−4644.40	−0.00188655	−0.000943274	−	1.00000i	$-0.500300\pi$
$67$	−4644.40	−0.00188655	−0.000943274		1.00000i	$0.500300\pi$
$68$	−3.85960e6	−1.48854
$69$	1.60276e6	0.587350
$70$	0	0
$71$	−2.79333e6	−0.926229	−0.463115	−	0.886298i	$-0.653268\pi$
$71$	−2.79333e6	−0.926229	−0.463115	+	0.886298i	$0.653268\pi$
$72$	543941.	0.171747
$73$	−2.01174e6	−0.605259	−0.302630	−	0.953108i	$-0.597864\pi$
$73$	−2.01174e6	−0.605259	−0.302630	+	0.953108i	$0.597864\pi$
$74$	−1.13660e6	−0.326058
$75$	0	0
$76$	4.65570e6	1.21657
$77$	−2.40299e6	−0.599838
$78$	78024.8	0.0186167
$79$	1.76767e6	0.403372	0.201686	−	0.979450i	$-0.435358\pi$
$79$	1.76767e6	0.403372	0.201686	+	0.979450i	$0.435358\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	−1.16367e6	−0.233067
$83$	−3.06625e6	−0.588619	−0.294310	−	0.955710i	$-0.595090\pi$
$83$	−3.06625e6	−0.588619	−0.294310	+	0.955710i	$0.595090\pi$
$84$	4.83095e6	0.889314
$85$	0	0
$86$	−1.40972e6	−0.238995
$87$	1.78606e6	0.290790
$88$	1.19113e6	0.186324
$89$	−6.14397e6	−0.923813	−0.461907	−	0.886929i	$-0.652834\pi$
$89$	−6.14397e6	−0.923813	−0.461907	+	0.886929i	$0.652834\pi$
$90$	0	0
$91$	1.43920e6	0.200205
$92$	−7.05598e6	−0.944713
$93$	−530930.	−0.0684459
$94$	−1.41726e6	−0.175995
$95$	0	0
$96$	−3.63627e6	−0.419476
$97$	3.02969e6	0.337053	0.168526	−	0.985697i	$-0.446099\pi$
$97$	3.02969e6	0.337053	0.168526	+	0.985697i	$0.446099\pi$
$98$	−4.35940e6	−0.467881
$99$	1.16376e6	0.120542
$100$	0	0
$101$	7.17509e6	0.692951	0.346475	−	0.938059i	$-0.387378\pi$
$101$	7.17509e6	0.692951	0.346475	+	0.938059i	$0.387378\pi$
$102$	−2.64984e6	−0.247240
$103$	−1.45694e7	−1.31374	−0.656872	−	0.754002i	$-0.728121\pi$
$103$	−1.45694e7	−1.31374	−0.656872	+	0.754002i	$0.728121\pi$
$104$	−713391.	−0.0621886
$105$	0	0
$106$	−4.85160e6	−0.395653
$107$	1.64442e7	1.29769	0.648843	−	0.760923i	$-0.275253\pi$
$107$	1.64442e7	1.29769	0.648843	+	0.760923i	$0.275253\pi$
$108$	−2.33961e6	−0.178715
$109$	2.34832e7	1.73686	0.868430	−	0.495812i	$-0.165129\pi$
$109$	2.34832e7	1.73686	0.868430	+	0.495812i	$0.165129\pi$
$110$	0	0
$111$	1.01532e7	0.704650
$112$	−1.95075e7	−1.31202
$113$	1.25747e7	0.819826	0.409913	−	0.912125i	$-0.365559\pi$
$113$	1.25747e7	0.819826	0.409913	+	0.912125i	$0.365559\pi$
$114$	3.19641e6	0.202067
$115$	0	0
$116$	−7.86294e6	−0.467716
$117$	−696997.	−0.0402328
$118$	6.16721e6	0.345543
$119$	−4.88772e7	−2.65884
$120$	0	0
$121$	−1.69388e7	−0.869226
$122$	1.14452e6	0.0570642
$123$	1.03951e7	0.503686
$124$	2.33736e6	0.110091
$125$	0	0
$126$	3.31673e6	0.147711
$127$	2.41070e7	1.04431	0.522155	−	0.852850i	$-0.325128\pi$
$127$	2.41070e7	1.04431	0.522155	+	0.852850i	$0.325128\pi$
$128$	2.10220e7	0.886012
$129$	1.25931e7	0.516497
$130$	0	0
$131$	−2.08040e7	−0.808533	−0.404266	−	0.914641i	$-0.632473\pi$
$131$	−2.08040e7	−0.808533	−0.404266	+	0.914641i	$0.632473\pi$
$132$	−5.12331e6	−0.193884
$133$	5.89589e7	2.17304
$134$	14037.7	0.000503998 0
$135$	0	0
$136$	2.42278e7	0.825901
$137$	9.94308e6	0.330369	0.165184	−	0.986263i	$-0.447178\pi$
$137$	9.94308e6	0.330369	0.165184	+	0.986263i	$0.447178\pi$
$138$	−4.84434e6	−0.156912
$139$	1.47414e7	0.465571	0.232785	−	0.972528i	$-0.425216\pi$
$139$	1.47414e7	0.465571	0.232785	+	0.972528i	$0.425216\pi$
$140$	0	0
$141$	1.26604e7	0.380347
$142$	8.44284e6	0.247445
$143$	−1.52629e6	−0.0436478
$144$	9.44743e6	0.263660
$145$	0	0
$146$	6.08047e6	0.161697
$147$	3.89425e7	1.01115
$148$	−4.46985e7	−1.13338
$149$	−2.21899e7	−0.549545	−0.274772	−	0.961509i	$-0.588602\pi$
$149$	−2.21899e7	−0.549545	−0.274772	+	0.961509i	$0.588602\pi$
$150$	0	0
$151$	4.90759e7	1.15998	0.579988	−	0.814625i	$-0.303057\pi$
$151$	4.90759e7	1.15998	0.579988	+	0.814625i	$0.303057\pi$
$152$	−2.92252e7	−0.675001
$153$	2.36711e7	0.534315
$154$	7.26302e6	0.160249
$155$	0	0
$156$	3.06845e6	0.0647118
$157$	−6.21368e6	−0.128145	−0.0640723	−	0.997945i	$-0.520409\pi$
$157$	−6.21368e6	−0.128145	−0.0640723	+	0.997945i	$0.520409\pi$
$158$	−5.34277e6	−0.107762
$159$	4.33394e7	0.855054
$160$	0	0
$161$	−8.93555e7	−1.68745
$162$	−1.60628e6	−0.0296837
$163$	1.54205e7	0.278895	0.139448	−	0.990229i	$-0.455467\pi$
$163$	1.54205e7	0.278895	0.139448	+	0.990229i	$0.455467\pi$
$164$	−4.57632e7	−0.810145
$165$	0	0
$166$	9.26773e6	0.157252
$167$	7.74611e6	0.128699	0.0643496	−	0.997927i	$-0.479503\pi$
$167$	7.74611e6	0.128699	0.0643496	+	0.997927i	$0.479503\pi$
$168$	−3.03253e7	−0.493426
$169$	−6.18344e7	−0.985432
$170$	0	0
$171$	−2.85536e7	−0.436691
$172$	−5.54397e7	−0.830751
$173$	1.00086e7	0.146965	0.0734824	−	0.997297i	$-0.476589\pi$
$173$	1.00086e7	0.146965	0.0734824	+	0.997297i	$0.476589\pi$
$174$	−5.39837e6	−0.0776855
$175$	0	0
$176$	2.06881e7	0.286040
$177$	−5.50918e7	−0.746758
$178$	1.85701e7	0.246800
$179$	1.09231e8	1.42350	0.711751	−	0.702431i	$-0.247902\pi$
$179$	1.09231e8	1.42350	0.711751	+	0.702431i	$0.247902\pi$
$180$	0	0
$181$	−1.27824e7	−0.160228	−0.0801138	−	0.996786i	$-0.525528\pi$
$181$	−1.27824e7	−0.160228	−0.0801138	+	0.996786i	$0.525528\pi$
$182$	−4.34996e6	−0.0534854
$183$	−1.02240e7	−0.123322
$184$	4.42924e7	0.524164
$185$	0	0
$186$	1.60473e6	0.0182855
$187$	5.18352e7	0.579668
$188$	−5.57359e7	−0.611762
$189$	−2.96284e7	−0.319221
$190$	0	0
$191$	1.54482e8	1.60421	0.802103	−	0.597185i	$-0.203714\pi$
$191$	1.54482e8	1.60421	0.802103	+	0.597185i	$0.203714\pi$
$192$	−3.37972e7	−0.344609
$193$	−1.51365e8	−1.51556	−0.757782	−	0.652507i	$-0.773717\pi$
$193$	−1.51365e8	−1.51556	−0.757782	+	0.652507i	$0.773717\pi$
$194$	−9.15724e6	−0.0900448
$195$	0	0
$196$	−1.71440e8	−1.62636
$197$	1.23829e8	1.15396	0.576981	−	0.816758i	$-0.304231\pi$
$197$	1.23829e8	1.15396	0.576981	+	0.816758i	$0.304231\pi$
$198$	−3.51745e6	−0.0322033
$199$	4.77920e7	0.429902	0.214951	−	0.976625i	$-0.431041\pi$
$199$	4.77920e7	0.429902	0.214951	+	0.976625i	$0.431041\pi$
$200$	0	0
$201$	−125399.	−0.00108920
$202$	−2.16867e7	−0.185124
$203$	−9.95748e7	−0.835436
$204$	−1.04209e8	−0.859410
$205$	0	0
$206$	4.40359e7	0.350971
$207$	4.32745e7	0.339107
$208$	−1.23905e7	−0.0954701
$209$	−6.25270e7	−0.473757
$210$	0	0
$211$	6.28256e7	0.460413	0.230207	−	0.973142i	$-0.426060\pi$
$211$	6.28256e7	0.460413	0.230207	+	0.973142i	$0.426060\pi$
$212$	−1.90797e8	−1.37530
$213$	−7.54200e7	−0.534759
$214$	−4.97025e7	−0.346681
$215$	0	0
$216$	1.46864e7	0.0991580
$217$	2.95999e7	0.196644
$218$	−7.09779e7	−0.464008
$219$	−5.43169e7	−0.349447
$220$	0	0
$221$	−3.10451e7	−0.193473
$222$	−3.06881e7	−0.188250
$223$	5.17882e6	0.0312726	0.0156363	−	0.999878i	$-0.495023\pi$
$223$	5.17882e6	0.0312726	0.0156363	+	0.999878i	$0.495023\pi$
$224$	2.02726e8	1.20515
$225$	0	0
$226$	−3.80069e7	−0.219019
$227$	−2.43478e8	−1.38156	−0.690778	−	0.723067i	$-0.742732\pi$
$227$	−2.43478e8	−1.38156	−0.690778	+	0.723067i	$0.742732\pi$
$228$	1.25704e8	0.702388
$229$	−1.09734e8	−0.603832	−0.301916	−	0.953335i	$-0.597626\pi$
$229$	−1.09734e8	−0.603832	−0.301916	+	0.953335i	$0.597626\pi$
$230$	0	0
$231$	−6.48806e7	−0.346317
$232$	4.93580e7	0.259507
$233$	1.85771e8	0.962129	0.481064	−	0.876685i	$-0.340250\pi$
$233$	1.85771e8	0.962129	0.481064	+	0.876685i	$0.340250\pi$
$234$	2.10667e6	0.0107483
$235$	0	0
$236$	2.42535e8	1.20111
$237$	4.77270e7	0.232887
$238$	1.47731e8	0.710318
$239$	3.06316e8	1.45137	0.725684	−	0.688028i	$-0.241523\pi$
$239$	3.06316e8	1.45137	0.725684	+	0.688028i	$0.241523\pi$
$240$	0	0
$241$	−5.78177e7	−0.266073	−0.133037	−	0.991111i	$-0.542473\pi$
$241$	−5.78177e7	−0.266073	−0.133037	+	0.991111i	$0.542473\pi$
$242$	5.11973e7	0.232217
$243$	1.43489e7	0.0641500
$244$	4.50101e7	0.198356
$245$	0	0
$246$	−3.14191e7	−0.134561
$247$	3.74486e7	0.158124
$248$	−1.46723e7	−0.0610826
$249$	−8.27888e7	−0.339840
$250$	0	0
$251$	−3.90109e8	−1.55714	−0.778571	−	0.627557i	$-0.784055\pi$
$251$	−3.90109e8	−1.55714	−0.778571	+	0.627557i	$0.784055\pi$
$252$	1.30436e8	0.513446
$253$	9.47632e7	0.367890
$254$	−7.28632e7	−0.278991
$255$	0	0
$256$	9.66848e7	0.360179
$257$	5.32200e8	1.95573	0.977865	−	0.209237i	$-0.0670980\pi$
$257$	5.32200e8	1.95573	0.977865	+	0.209237i	$0.0670980\pi$
$258$	−3.80625e7	−0.137984
$259$	−5.66053e8	−2.02445
$260$	0	0
$261$	4.82237e7	0.167888
$262$	6.28800e7	0.216002
$263$	3.12888e8	1.06058	0.530291	−	0.847816i	$-0.322083\pi$
$263$	3.12888e8	1.06058	0.530291	+	0.847816i	$0.322083\pi$
$264$	3.21605e7	0.107574
$265$	0	0
$266$	−1.78203e8	−0.580536
$267$	−1.65887e8	−0.533364
$268$	552054.	0.00175190
$269$	2.95534e8	0.925710	0.462855	−	0.886434i	$-0.346825\pi$
$269$	2.95534e8	0.925710	0.462855	+	0.886434i	$0.346825\pi$
$270$	0	0
$271$	3.96274e8	1.20949	0.604746	−	0.796419i	$-0.293275\pi$
$271$	3.96274e8	1.20949	0.604746	+	0.796419i	$0.293275\pi$
$272$	4.20800e8	1.26790
$273$	3.88583e7	0.115588
$274$	−3.00529e7	−0.0882591
$275$	0	0
$276$	−1.90511e8	−0.545430
$277$	−4.00402e8	−1.13192	−0.565961	−	0.824432i	$-0.691495\pi$
$277$	−4.00402e8	−1.13192	−0.565961	+	0.824432i	$0.691495\pi$
$278$	−4.45557e7	−0.124379
$279$	−1.43351e7	−0.0395172
$280$	0	0
$281$	3.58146e8	0.962916	0.481458	−	0.876469i	$-0.340107\pi$
$281$	3.58146e8	0.962916	0.481458	+	0.876469i	$0.340107\pi$
$282$	−3.82659e7	−0.101611
$283$	3.95688e8	1.03777	0.518884	−	0.854845i	$-0.326348\pi$
$283$	3.95688e8	1.03777	0.518884	+	0.854845i	$0.326348\pi$
$284$	3.32028e8	0.860123
$285$	0	0
$286$	4.61322e6	0.0116606
$287$	−5.79536e8	−1.44708
$288$	−9.81793e7	−0.242184
$289$	6.44000e8	1.56943
$290$	0	0
$291$	8.18017e7	0.194597
$292$	2.39124e8	0.562061
$293$	1.93280e8	0.448900	0.224450	−	0.974486i	$-0.427941\pi$
$293$	1.93280e8	0.448900	0.224450	+	0.974486i	$0.427941\pi$
$294$	−1.17704e8	−0.270131
$295$	0	0
$296$	2.80585e8	0.628845
$297$	3.14214e7	0.0695951
$298$	6.70688e7	0.146813
$299$	−5.67555e7	−0.122789
$300$	0	0
$301$	−7.02077e8	−1.48389
$302$	−1.48332e8	−0.309892
$303$	1.93727e8	0.400075
$304$	−5.07597e8	−1.03624
$305$	0	0
$306$	−7.15457e7	−0.142744
$307$	−2.11267e8	−0.416722	−0.208361	−	0.978052i	$-0.566813\pi$
$307$	−2.11267e8	−0.416722	−0.208361	+	0.978052i	$0.566813\pi$
$308$	2.85630e8	0.557027
$309$	−3.93373e8	−0.758490
$310$	0	0
$311$	2.59442e8	0.489079	0.244540	−	0.969639i	$-0.421363\pi$
$311$	2.59442e8	0.489079	0.244540	+	0.969639i	$0.421363\pi$
$312$	−1.92616e7	−0.0359046
$313$	−9.29772e8	−1.71384	−0.856922	−	0.515446i	$-0.827626\pi$
$313$	−9.29772e8	−1.71384	−0.856922	+	0.515446i	$0.827626\pi$
$314$	1.87808e7	0.0342343
$315$	0	0
$316$	−2.10113e8	−0.374583
$317$	−1.63510e8	−0.288295	−0.144147	−	0.989556i	$-0.546044\pi$
$317$	−1.63510e8	−0.288295	−0.144147	+	0.989556i	$0.546044\pi$
$318$	−1.30993e8	−0.228431
$319$	1.05601e8	0.182138
$320$	0	0
$321$	4.43993e8	0.749219
$322$	2.70077e8	0.450808
$323$	−1.27181e9	−2.09997
$324$	−6.31695e7	−0.103181
$325$	0	0
$326$	−4.66083e7	−0.0745079
$327$	6.34047e8	1.00278
$328$	2.87269e8	0.449500
$329$	−7.05828e8	−1.09273
$330$	0	0
$331$	−2.57948e8	−0.390962	−0.195481	−	0.980707i	$-0.562627\pi$
$331$	−2.57948e8	−0.390962	−0.195481	+	0.980707i	$0.562627\pi$
$332$	3.64469e8	0.546609
$333$	2.74137e8	0.406830
$334$	−2.34126e7	−0.0343825
$335$	0	0
$336$	−5.26704e8	−0.757493
$337$	2.34282e8	0.333452	0.166726	−	0.986003i	$-0.446680\pi$
$337$	2.34282e8	0.333452	0.166726	+	0.986003i	$0.446680\pi$
$338$	1.86894e8	0.263261
$339$	3.39516e8	0.473327
$340$	0	0
$341$	−3.13912e7	−0.0428715
$342$	8.63030e7	0.116663
$343$	−9.31426e8	−1.24629
$344$	3.48011e8	0.460933
$345$	0	0
$346$	−3.02511e7	−0.0392622
$347$	−6.73974e8	−0.865944	−0.432972	−	0.901407i	$-0.642535\pi$
$347$	−6.73974e8	−0.865944	−0.432972	+	0.901407i	$0.642535\pi$
$348$	−2.12299e8	−0.270036
$349$	8.77464e8	1.10494	0.552472	−	0.833531i	$-0.313685\pi$
$349$	8.77464e8	1.10494	0.552472	+	0.833531i	$0.313685\pi$
$350$	0	0
$351$	−1.88189e7	−0.0232284
$352$	−2.14994e8	−0.262741
$353$	−6.75088e8	−0.816863	−0.408431	−	0.912789i	$-0.633924\pi$
$353$	−6.75088e8	−0.816863	−0.408431	+	0.912789i	$0.633924\pi$
$354$	1.66515e8	0.199499
$355$	0	0
$356$	7.30300e8	0.857880
$357$	−1.31969e9	−1.53508
$358$	−3.30149e8	−0.380294
$359$	−7.40244e8	−0.844393	−0.422196	−	0.906504i	$-0.638741\pi$
$359$	−7.40244e8	−0.844393	−0.422196	+	0.906504i	$0.638741\pi$
$360$	0	0
$361$	6.40271e8	0.716290
$362$	3.86348e7	0.0428054
$363$	−4.57346e8	−0.501848
$364$	−1.71069e8	−0.185916
$365$	0	0
$366$	3.09020e7	0.0329460
$367$	−1.05986e9	−1.11922	−0.559611	−	0.828755i	$-0.689049\pi$
$367$	−1.05986e9	−1.11922	−0.559611	+	0.828755i	$0.689049\pi$
$368$	7.69291e8	0.804680
$369$	2.80667e8	0.290803
$370$	0	0
$371$	−2.41622e9	−2.45656
$372$	6.31088e7	0.0635608
$373$	1.28239e9	1.27950	0.639748	−	0.768585i	$-0.279039\pi$
$373$	1.28239e9	1.27950	0.639748	+	0.768585i	$0.279039\pi$
$374$	−1.56672e8	−0.154860
$375$	0	0
$376$	3.49870e8	0.339429
$377$	−6.32465e7	−0.0607913
$378$	8.95516e7	0.0852810
$379$	−2.97355e8	−0.280568	−0.140284	−	0.990111i	$-0.544802\pi$
$379$	−2.97355e8	−0.280568	−0.140284	+	0.990111i	$0.544802\pi$
$380$	0	0
$381$	6.50888e8	0.602933
$382$	−4.66920e8	−0.428569
$383$	−8.10508e8	−0.737160	−0.368580	−	0.929596i	$-0.620156\pi$
$383$	−8.10508e8	−0.737160	−0.368580	+	0.929596i	$0.620156\pi$
$384$	5.67594e8	0.511539
$385$	0	0
$386$	4.57500e8	0.404888
$387$	3.40013e8	0.298200
$388$	−3.60123e8	−0.312997
$389$	−3.38899e8	−0.291908	−0.145954	−	0.989291i	$-0.546625\pi$
$389$	−3.38899e8	−0.291908	−0.145954	+	0.989291i	$0.546625\pi$
$390$	0	0
$391$	1.92750e9	1.63071
$392$	1.07618e9	0.902368
$393$	−5.61708e8	−0.466807
$394$	−3.74273e8	−0.308285
$395$	0	0
$396$	−1.38329e8	−0.111939
$397$	1.00265e9	0.804235	0.402117	−	0.915588i	$-0.368274\pi$
$397$	1.00265e9	0.804235	0.402117	+	0.915588i	$0.368274\pi$
$398$	−1.44451e8	−0.114850
$399$	1.59189e9	1.25461
$400$	0	0
$401$	−1.09336e9	−0.846755	−0.423377	−	0.905953i	$-0.639156\pi$
$401$	−1.09336e9	−0.846755	−0.423377	+	0.905953i	$0.639156\pi$
$402$	379017.	0.000290983 0
$403$	1.88008e7	0.0143090
$404$	−8.52863e8	−0.643494
$405$	0	0
$406$	3.00964e8	0.223190
$407$	6.00309e8	0.441362
$408$	6.54151e8	0.476834
$409$	1.09511e9	0.791453	0.395726	−	0.918368i	$-0.370493\pi$
$409$	1.09511e9	0.791453	0.395726	+	0.918368i	$0.370493\pi$
$410$	0	0
$411$	2.68463e8	0.190738
$412$	1.73178e9	1.21998
$413$	3.07142e9	2.14543
$414$	−1.30797e8	−0.0905935
$415$	0	0
$416$	1.28764e8	0.0876938
$417$	3.98016e8	0.268797
$418$	1.88988e8	0.126566
$419$	−1.75196e9	−1.16352	−0.581762	−	0.813359i	$-0.697636\pi$
$419$	−1.75196e9	−1.16352	−0.581762	+	0.813359i	$0.697636\pi$
$420$	0	0
$421$	−4.58026e8	−0.299160	−0.149580	−	0.988750i	$-0.547792\pi$
$421$	−4.58026e8	−0.299160	−0.149580	+	0.988750i	$0.547792\pi$
$422$	−1.89890e8	−0.123001
$423$	3.41830e8	0.219593
$424$	1.19769e9	0.763068
$425$	0	0
$426$	2.27957e8	0.142863
$427$	5.69999e8	0.354304
$428$	−1.95463e9	−1.20507
$429$	−4.12099e7	−0.0252000
$430$	0	0
$431$	−7.60129e8	−0.457317	−0.228658	−	0.973507i	$-0.573434\pi$
$431$	−7.60129e8	−0.457317	−0.228658	+	0.973507i	$0.573434\pi$
$432$	2.55081e8	0.152224
$433$	2.48461e9	1.47079	0.735396	−	0.677638i	$-0.236996\pi$
$433$	2.48461e9	1.47079	0.735396	+	0.677638i	$0.236996\pi$
$434$	−8.94656e7	−0.0525342
$435$	0	0
$436$	−2.79132e9	−1.61290
$437$	−2.32508e9	−1.33276
$438$	1.64173e8	0.0933558
$439$	1.69934e8	0.0958637	0.0479318	−	0.998851i	$-0.484737\pi$
$439$	1.69934e8	0.0958637	0.0479318	+	0.998851i	$0.484737\pi$
$440$	0	0
$441$	1.05145e9	0.583785
$442$	9.38337e7	0.0516870
$443$	1.53246e9	0.837481	0.418741	−	0.908106i	$-0.362472\pi$
$443$	1.53246e9	0.837481	0.418741	+	0.908106i	$0.362472\pi$
$444$	−1.20686e9	−0.654359
$445$	0	0
$446$	−1.56530e7	−0.00835458
$447$	−5.99127e8	−0.317280
$448$	1.88423e9	0.990057
$449$	−3.45325e9	−1.80039	−0.900194	−	0.435489i	$-0.856575\pi$
$449$	−3.45325e9	−1.80039	−0.900194	+	0.435489i	$0.856575\pi$
$450$	0	0
$451$	6.14609e8	0.315486
$452$	−1.49468e9	−0.761315
$453$	1.32505e9	0.669713
$454$	7.35910e8	0.369087
$455$	0	0
$456$	−7.89080e8	−0.389712
$457$	−7.56279e8	−0.370659	−0.185330	−	0.982676i	$-0.559335\pi$
$457$	−7.56279e8	−0.370659	−0.185330	+	0.982676i	$0.559335\pi$
$458$	3.31670e8	0.161316
$459$	6.39118e8	0.308487
$460$	0	0
$461$	−8.78272e8	−0.417518	−0.208759	−	0.977967i	$-0.566942\pi$
$461$	−8.78272e8	−0.417518	−0.208759	+	0.977967i	$0.566942\pi$
$462$	1.96101e8	0.0925196
$463$	−3.54973e9	−1.66212	−0.831058	−	0.556185i	$-0.812264\pi$
$463$	−3.54973e9	−1.66212	−0.831058	+	0.556185i	$0.812264\pi$
$464$	8.57273e8	0.398388
$465$	0	0
$466$	−5.61493e8	−0.257036
$467$	3.03363e9	1.37833	0.689165	−	0.724605i	$-0.257978\pi$
$467$	3.03363e9	1.37833	0.689165	+	0.724605i	$0.257978\pi$
$468$	8.28482e7	0.0373613
$469$	6.99111e6	0.00312926
$470$	0	0
$471$	−1.67769e8	−0.0739843
$472$	−1.52246e9	−0.666423
$473$	7.44566e8	0.323511
$474$	−1.44255e8	−0.0622165
$475$	0	0
$476$	5.80977e9	2.46908
$477$	1.17017e9	0.493666
$478$	−9.25840e8	−0.387738
$479$	3.00343e9	1.24866	0.624328	−	0.781162i	$-0.285373\pi$
$479$	3.00343e9	1.24866	0.624328	+	0.781162i	$0.285373\pi$
$480$	0	0
$481$	−3.59537e8	−0.147311
$482$	1.74754e8	0.0710823
$483$	−2.41260e9	−0.974249
$484$	2.01342e9	0.807189
$485$	0	0
$486$	−4.33695e7	−0.0171379
$487$	3.28649e8	0.128938	0.0644690	−	0.997920i	$-0.479465\pi$
$487$	3.28649e8	0.128938	0.0644690	+	0.997920i	$0.479465\pi$
$488$	−2.82541e8	−0.110056
$489$	4.16353e8	0.161020
$490$	0	0
$491$	−6.97275e8	−0.265839	−0.132919	−	0.991127i	$-0.542435\pi$
$491$	−6.97275e8	−0.265839	−0.132919	+	0.991127i	$0.542435\pi$
$492$	−1.23561e9	−0.467737
$493$	2.14794e9	0.807344
$494$	−1.13188e8	−0.0422433
$495$	0	0
$496$	−2.54835e8	−0.0937721
$497$	4.20474e9	1.53636
$498$	2.50229e8	0.0907893
$499$	−3.18226e9	−1.14652	−0.573262	−	0.819372i	$-0.694322\pi$
$499$	−3.18226e9	−1.14652	−0.573262	+	0.819372i	$0.694322\pi$
$500$	0	0
$501$	2.09145e8	0.0743046
$502$	1.17910e9	0.415996
$503$	−4.07550e9	−1.42788	−0.713942	−	0.700205i	$-0.753092\pi$
$503$	−4.07550e9	−1.42788	−0.713942	+	0.700205i	$0.753092\pi$
$504$	−8.18782e8	−0.284880
$505$	0	0
$506$	−2.86421e8	−0.0982830
$507$	−1.66953e9	−0.568939
$508$	−2.86546e9	−0.969777
$509$	4.18725e8	0.140740	0.0703699	−	0.997521i	$-0.477582\pi$
$509$	4.18725e8	0.140740	0.0703699	+	0.997521i	$0.477582\pi$
$510$	0	0
$511$	3.02822e9	1.00396
$512$	−2.98305e9	−0.982235
$513$	−7.70946e8	−0.252124
$514$	−1.60857e9	−0.522480
$515$	0	0
$516$	−1.49687e9	−0.479634
$517$	7.48544e8	0.238232
$518$	1.71089e9	0.540839
$519$	2.70233e8	0.0848502
$520$	0	0
$521$	−2.02201e9	−0.626400	−0.313200	−	0.949687i	$-0.601401\pi$
$521$	−2.02201e9	−0.626400	−0.313200	+	0.949687i	$0.601401\pi$
$522$	−1.45756e8	−0.0448518
$523$	−1.87087e8	−0.0571857	−0.0285928	−	0.999591i	$-0.509103\pi$
$523$	−1.87087e8	−0.0571857	−0.0285928	+	0.999591i	$0.509103\pi$
$524$	2.47286e9	0.750827
$525$	0	0
$526$	−9.45704e8	−0.283338
$527$	−6.38504e8	−0.190032
$528$	5.58579e8	0.165145
$529$	1.18962e8	0.0349392
$530$	0	0
$531$	−1.48748e9	−0.431141
$532$	−7.00812e9	−2.01795
$533$	−3.68101e8	−0.105298
$534$	5.01393e8	0.142490
$535$	0	0
$536$	−3.46541e6	−0.000972025 0
$537$	2.94922e9	0.821860
$538$	−8.93251e8	−0.247307
$539$	2.30248e9	0.633337
$540$	0	0
$541$	3.94897e9	1.07224	0.536122	−	0.844140i	$-0.319889\pi$
$541$	3.94897e9	1.07224	0.536122	+	0.844140i	$0.319889\pi$
$542$	−1.19774e9	−0.323120
$543$	−3.45125e8	−0.0925075
$544$	−4.37303e9	−1.16463
$545$	0	0
$546$	−1.17449e8	−0.0308798
$547$	−1.71845e9	−0.448932	−0.224466	−	0.974482i	$-0.572064\pi$
$547$	−1.71845e9	−0.448932	−0.224466	+	0.974482i	$0.572064\pi$
$548$	−1.18188e9	−0.306790
$549$	−2.76048e8	−0.0712003
$550$	0	0
$551$	−2.59099e9	−0.659835
$552$	1.19589e9	0.302626
$553$	−2.66083e9	−0.669082
$554$	1.21021e9	0.302397
$555$	0	0
$556$	−1.75222e9	−0.432342
$557$	−3.22406e9	−0.790516	−0.395258	−	0.918570i	$-0.629345\pi$
$557$	−3.22406e9	−0.790516	−0.395258	+	0.918570i	$0.629345\pi$
$558$	4.33278e7	0.0105572
$559$	−4.45935e8	−0.107977
$560$	0	0
$561$	1.39955e9	0.334671
$562$	−1.08250e9	−0.257246
$563$	−5.69907e9	−1.34594	−0.672968	−	0.739671i	$-0.734981\pi$
$563$	−5.69907e9	−1.34594	−0.672968	+	0.739671i	$0.734981\pi$
$564$	−1.50487e9	−0.353201
$565$	0	0
$566$	−1.19596e9	−0.277243
$567$	−7.99966e8	−0.184302
$568$	−2.08424e9	−0.477230
$569$	3.70804e9	0.843823	0.421912	−	0.906637i	$-0.361359\pi$
$569$	3.70804e9	0.843823	0.421912	+	0.906637i	$0.361359\pi$
$570$	0	0
$571$	−3.34811e9	−0.752615	−0.376308	−	0.926495i	$-0.622806\pi$
$571$	−3.34811e9	−0.752615	−0.376308	+	0.926495i	$0.622806\pi$
$572$	1.81422e8	0.0405326
$573$	4.17101e9	0.926189
$574$	1.75164e9	0.386593
$575$	0	0
$576$	−9.12524e8	−0.198960
$577$	8.37348e9	1.81464	0.907320	−	0.420441i	$-0.138124\pi$
$577$	8.37348e9	1.81464	0.907320	+	0.420441i	$0.138124\pi$
$578$	−1.94649e9	−0.419280
$579$	−4.08685e9	−0.875012
$580$	0	0
$581$	4.61556e9	0.976355
$582$	−2.47245e8	−0.0519874
$583$	2.56244e9	0.535568
$584$	−1.50105e9	−0.311854
$585$	0	0
$586$	−5.84188e8	−0.119925
$587$	−9.32230e8	−0.190235	−0.0951174	−	0.995466i	$-0.530323\pi$
$587$	−9.32230e8	−0.190235	−0.0951174	+	0.995466i	$0.530323\pi$
$588$	−4.62889e9	−0.938979
$589$	7.70205e8	0.155311
$590$	0	0
$591$	3.34339e9	0.666240
$592$	4.87334e9	0.965385
$593$	4.69858e9	0.925283	0.462642	−	0.886545i	$-0.346902\pi$
$593$	4.69858e9	0.925283	0.462642	+	0.886545i	$0.346902\pi$
$594$	−9.49712e7	−0.0185926
$595$	0	0
$596$	2.63759e9	0.510323
$597$	1.29038e9	0.248204
$598$	1.71543e8	0.0328035
$599$	−2.66487e9	−0.506620	−0.253310	−	0.967385i	$-0.581519\pi$
$599$	−2.66487e9	−0.506620	−0.253310	+	0.967385i	$0.581519\pi$
$600$	0	0
$601$	−1.21738e9	−0.228753	−0.114376	−	0.993437i	$-0.536487\pi$
$601$	−1.21738e9	−0.228753	−0.114376	+	0.993437i	$0.536487\pi$
$602$	2.12203e9	0.396427
$603$	−3.38577e6	−0.000628849 0
$604$	−5.83339e9	−1.07719
$605$	0	0
$606$	−5.85540e8	−0.106881
$607$	−2.74917e9	−0.498932	−0.249466	−	0.968384i	$-0.580255\pi$
$607$	−2.74917e9	−0.498932	−0.249466	+	0.968384i	$0.580255\pi$
$608$	5.27503e9	0.951837
$609$	−2.68852e9	−0.482339
$610$	0	0
$611$	−4.48318e8	−0.0795136
$612$	−2.81365e9	−0.496181
$613$	7.89654e9	1.38460	0.692301	−	0.721609i	$-0.256597\pi$
$613$	7.89654e9	1.38460	0.692301	+	0.721609i	$0.256597\pi$
$614$	6.38552e8	0.111329
$615$	0	0
$616$	−1.79298e9	−0.309060
$617$	3.64303e9	0.624403	0.312202	−	0.950016i	$-0.398934\pi$
$617$	3.64303e9	0.624403	0.312202	+	0.950016i	$0.398934\pi$
$618$	1.18897e9	0.202633
$619$	7.22253e9	1.22397	0.611987	−	0.790868i	$-0.290370\pi$
$619$	7.22253e9	1.22397	0.611987	+	0.790868i	$0.290370\pi$
$620$	0	0
$621$	1.16841e9	0.195783
$622$	−7.84163e8	−0.130659
$623$	9.24838e9	1.53235
$624$	−3.34544e8	−0.0551197
$625$	0	0
$626$	2.81023e9	0.457859
$627$	−1.68823e9	−0.273524
$628$	7.38586e8	0.118999
$629$	1.22104e10	1.95638
$630$	0	0
$631$	−1.55879e9	−0.246993	−0.123496	−	0.992345i	$-0.539411\pi$
$631$	−1.55879e9	−0.246993	−0.123496	+	0.992345i	$0.539411\pi$
$632$	1.31894e9	0.207833
$633$	1.69629e9	0.265820
$634$	4.94208e8	0.0770189
$635$	0	0
$636$	−5.15152e9	−0.794028
$637$	−1.37900e9	−0.211386
$638$	−3.19178e8	−0.0486588
$639$	−2.03634e9	−0.308743
$640$	0	0
$641$	3.18649e9	0.477869	0.238935	−	0.971036i	$-0.423202\pi$
$641$	3.18649e9	0.477869	0.238935	+	0.971036i	$0.423202\pi$
$642$	−1.34197e9	−0.200156
$643$	1.65144e9	0.244976	0.122488	−	0.992470i	$-0.460913\pi$
$643$	1.65144e9	0.244976	0.122488	+	0.992470i	$0.460913\pi$
$644$	1.06212e10	1.56701
$645$	0	0
$646$	3.84405e9	0.561015
$647$	3.95797e9	0.574523	0.287262	−	0.957852i	$-0.407255\pi$
$647$	3.95797e9	0.574523	0.287262	+	0.957852i	$0.407255\pi$
$648$	3.96533e8	0.0572489
$649$	−3.25730e9	−0.467736
$650$	0	0
$651$	7.99197e8	0.113533
$652$	−1.83295e9	−0.258990
$653$	−2.09998e9	−0.295134	−0.147567	−	0.989052i	$-0.547144\pi$
$653$	−2.09998e9	−0.295134	−0.147567	+	0.989052i	$0.547144\pi$
$654$	−1.91640e9	−0.267895
$655$	0	0
$656$	4.98942e9	0.690059
$657$	−1.46656e9	−0.201753
$658$	2.13336e9	0.291927
$659$	−1.18536e10	−1.61344	−0.806718	−	0.590937i	$-0.798758\pi$
$659$	−1.18536e10	−1.61344	−0.806718	+	0.590937i	$0.798758\pi$
$660$	0	0
$661$	−1.14873e10	−1.54709	−0.773543	−	0.633744i	$-0.781517\pi$
$661$	−1.14873e10	−1.54709	−0.773543	+	0.633744i	$0.781517\pi$
$662$	7.79648e8	0.104447
$663$	−8.38218e8	−0.111702
$664$	−2.28787e9	−0.303280
$665$	0	0
$666$	−8.28578e8	−0.108686
$667$	3.92679e9	0.512386
$668$	−9.20738e8	−0.119514
$669$	1.39828e8	0.0180552
$670$	0	0
$671$	−6.04494e8	−0.0772437
$672$	5.47359e9	0.695793
$673$	−1.99478e9	−0.252257	−0.126128	−	0.992014i	$-0.540255\pi$
$673$	−1.99478e9	−0.252257	−0.126128	+	0.992014i	$0.540255\pi$
$674$	−7.08115e8	−0.0890829
$675$	0	0
$676$	7.34992e9	0.915101
$677$	−3.21860e9	−0.398664	−0.199332	−	0.979932i	$-0.563877\pi$
$677$	−3.21860e9	−0.398664	−0.199332	+	0.979932i	$0.563877\pi$
$678$	−1.02619e9	−0.126451
$679$	−4.56053e9	−0.559076
$680$	0	0
$681$	−6.57389e9	−0.797642
$682$	9.48799e7	0.0114533
$683$	−1.87259e9	−0.224890	−0.112445	−	0.993658i	$-0.535868\pi$
$683$	−1.87259e9	−0.224890	−0.112445	+	0.993658i	$0.535868\pi$
$684$	3.39401e9	0.405524
$685$	0	0
$686$	2.81523e9	0.332950
$687$	−2.96281e9	−0.348622
$688$	6.04442e9	0.707611
$689$	−1.53470e9	−0.178754
$690$	0	0
$691$	3.08223e8	0.0355380	0.0177690	−	0.999842i	$-0.494344\pi$
$691$	3.08223e8	0.0355380	0.0177690	+	0.999842i	$0.494344\pi$
$692$	−1.18967e9	−0.136476
$693$	−1.75178e9	−0.199946
$694$	2.03708e9	0.231340
$695$	0	0
$696$	1.33267e9	0.149827
$697$	1.25013e10	1.39842
$698$	−2.65213e9	−0.295190
$699$	5.01583e9	0.555485
$700$	0	0
$701$	−1.39953e10	−1.53450	−0.767251	−	0.641346i	$-0.778376\pi$
$701$	−1.39953e10	−1.53450	−0.767251	+	0.641346i	$0.778376\pi$
$702$	5.68801e7	0.00620555
$703$	−1.47290e10	−1.59893
$704$	−1.99826e9	−0.215848
$705$	0	0
$706$	2.04045e9	0.218228
$707$	−1.08005e10	−1.14941
$708$	6.54845e9	0.693462
$709$	−1.02356e10	−1.07858	−0.539290	−	0.842120i	$-0.681307\pi$
$709$	−1.02356e10	−1.07858	−0.539290	+	0.842120i	$0.681307\pi$
$710$	0	0
$711$	1.28863e9	0.134457
$712$	−4.58430e9	−0.475985
$713$	−1.16729e9	−0.120605
$714$	3.98874e9	0.410102
$715$	0	0
$716$	−1.29836e10	−1.32191
$717$	8.27054e9	0.837948
$718$	2.23739e9	0.225582
$719$	9.63078e9	0.966297	0.483148	−	0.875538i	$-0.339493\pi$
$719$	9.63078e9	0.966297	0.483148	+	0.875538i	$0.339493\pi$
$720$	0	0
$721$	2.19309e10	2.17913
$722$	−1.93522e9	−0.191359
$723$	−1.56108e9	−0.153617
$724$	1.51937e9	0.148792
$725$	0	0
$726$	1.38233e9	0.134070
$727$	−1.62740e10	−1.57081	−0.785407	−	0.618980i	$-0.787546\pi$
$727$	−1.62740e10	−1.57081	−0.785407	+	0.618980i	$0.787546\pi$
$728$	1.07385e9	0.103154
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.51446e10	1.43399
$732$	1.21527e9	0.114521
$733$	1.62212e10	1.52132	0.760659	−	0.649152i	$-0.224876\pi$
$733$	1.62212e10	1.52132	0.760659	+	0.649152i	$0.224876\pi$
$734$	3.20342e9	0.299004
$735$	0	0
$736$	−7.99461e9	−0.739137
$737$	−7.41420e6	−0.000682226 0
$738$	−8.48315e8	−0.0776890
$739$	1.78796e10	1.62968	0.814841	−	0.579684i	$-0.196824\pi$
$739$	1.78796e10	1.62968	0.814841	+	0.579684i	$0.196824\pi$
$740$	0	0
$741$	1.01111e9	0.0912927
$742$	7.30301e9	0.656278
$743$	−1.10072e10	−0.984499	−0.492249	−	0.870454i	$-0.663825\pi$
$743$	−1.10072e10	−0.984499	−0.492249	+	0.870454i	$0.663825\pi$
$744$	−3.96152e8	−0.0352660
$745$	0	0
$746$	−3.87602e9	−0.341822
$747$	−2.23530e9	−0.196206
$748$	−6.16137e9	−0.538296
$749$	−2.47531e10	−2.15250
$750$	0	0
$751$	−8.82495e9	−0.760278	−0.380139	−	0.924929i	$-0.624124\pi$
$751$	−8.82495e9	−0.760278	−0.380139	+	0.924929i	$0.624124\pi$
$752$	6.07671e9	0.521082
$753$	−1.05329e10	−0.899016
$754$	1.91162e8	0.0162406
$755$	0	0
$756$	3.52176e9	0.296438
$757$	−1.26747e10	−1.06194	−0.530972	−	0.847389i	$-0.678173\pi$
$757$	−1.26747e10	−1.06194	−0.530972	+	0.847389i	$0.678173\pi$
$758$	8.98754e8	0.0749547
$759$	2.55861e9	0.212401
$760$	0	0
$761$	−4.76538e9	−0.391969	−0.195984	−	0.980607i	$-0.562790\pi$
$761$	−4.76538e9	−0.391969	−0.195984	+	0.980607i	$0.562790\pi$
$762$	−1.96731e9	−0.161076
$763$	−3.53487e10	−2.88097
$764$	−1.83624e10	−1.48971
$765$	0	0
$766$	2.44976e9	0.196935
$767$	1.95086e9	0.156114
$768$	2.61049e9	0.207949
$769$	−3.33620e9	−0.264552	−0.132276	−	0.991213i	$-0.542228\pi$
$769$	−3.33620e9	−0.264552	−0.132276	+	0.991213i	$0.542228\pi$
$770$	0	0
$771$	1.43694e10	1.12914
$772$	1.79919e10	1.40740
$773$	1.22441e9	0.0953453	0.0476726	−	0.998863i	$-0.484820\pi$
$773$	1.22441e9	0.0953453	0.0476726	+	0.998863i	$0.484820\pi$
$774$	−1.02769e9	−0.0796651
$775$	0	0
$776$	2.26060e9	0.173663
$777$	−1.52834e10	−1.16882
$778$	1.02432e9	0.0779843
$779$	−1.50798e10	−1.14292
$780$	0	0
$781$	−4.45921e9	−0.334949
$782$	−5.82587e9	−0.435649
$783$	1.30204e9	0.0969300
$784$	1.86916e10	1.38529
$785$	0	0
$786$	1.69776e9	0.124709
$787$	−1.72256e10	−1.25969	−0.629844	−	0.776721i	$-0.716881\pi$
$787$	−1.72256e10	−1.25969	−0.629844	+	0.776721i	$0.716881\pi$
$788$	−1.47189e10	−1.07160
$789$	8.44799e9	0.612327
$790$	0	0
$791$	−1.89284e10	−1.35986
$792$	8.68334e8	0.0621082
$793$	3.62043e8	0.0257813
$794$	−3.03051e9	−0.214854
$795$	0	0
$796$	−5.68077e9	−0.399219
$797$	1.01065e10	0.707127	0.353564	−	0.935411i	$-0.384970\pi$
$797$	1.01065e10	0.707127	0.353564	+	0.935411i	$0.384970\pi$
$798$	−4.81148e9	−0.335173
$799$	1.52255e10	1.05599
$800$	0	0
$801$	−4.47895e9	−0.307938
$802$	3.30467e9	0.226213
$803$	−3.21149e9	−0.218878
$804$	1.49055e7	0.00101146
$805$	0	0
$806$	−5.68254e7	−0.00382270
$807$	7.97943e9	0.534459
$808$	5.35367e9	0.357036
$809$	9.46505e9	0.628497	0.314248	−	0.949341i	$-0.398247\pi$
$809$	9.46505e9	0.628497	0.314248	+	0.949341i	$0.398247\pi$
$810$	0	0
$811$	1.67478e10	1.10251	0.551257	−	0.834335i	$-0.314148\pi$
$811$	1.67478e10	1.10251	0.551257	+	0.834335i	$0.314148\pi$
$812$	1.18359e10	0.775810
$813$	1.06994e10	0.698300
$814$	−1.81443e9	−0.117911
$815$	0	0
$816$	1.13616e10	0.732022
$817$	−1.82684e10	−1.17199
$818$	−3.30996e9	−0.211439
$819$	1.04917e9	0.0667350
$820$	0	0
$821$	−3.34978e9	−0.211259	−0.105630	−	0.994406i	$-0.533686\pi$
$821$	−3.34978e9	−0.211259	−0.105630	+	0.994406i	$0.533686\pi$
$822$	−8.11429e8	−0.0509564
$823$	6.69632e9	0.418732	0.209366	−	0.977837i	$-0.432860\pi$
$823$	6.69632e9	0.418732	0.209366	+	0.977837i	$0.432860\pi$
$824$	−1.08709e10	−0.676893
$825$	0	0
$826$	−9.28336e9	−0.573159
$827$	−5.81000e9	−0.357196	−0.178598	−	0.983922i	$-0.557156\pi$
$827$	−5.81000e9	−0.357196	−0.178598	+	0.983922i	$0.557156\pi$
$828$	−5.14381e9	−0.314904
$829$	6.84048e9	0.417009	0.208505	−	0.978021i	$-0.433140\pi$
$829$	6.84048e9	0.417009	0.208505	+	0.978021i	$0.433140\pi$
$830$	0	0
$831$	−1.08108e10	−0.653516
$832$	1.19680e9	0.0720424
$833$	4.68329e10	2.80733
$834$	−1.20300e9	−0.0718101
$835$	0	0
$836$	7.43224e9	0.439945
$837$	−3.87048e8	−0.0228153
$838$	5.29529e9	0.310839
$839$	−1.04085e8	−0.00608445	−0.00304223	−	0.999995i	$-0.500968\pi$
$839$	−1.04085e8	−0.00608445	−0.00304223	+	0.999995i	$0.500968\pi$
$840$	0	0
$841$	−1.28740e10	−0.746324
$842$	1.38438e9	0.0799215
$843$	9.66995e9	0.555940
$844$	−7.46773e9	−0.427553
$845$	0	0
$846$	−1.03318e9	−0.0586651
$847$	2.54975e10	1.44180
$848$	2.08020e10	1.17144
$849$	1.06836e10	0.599155
$850$	0	0
$851$	2.23226e10	1.24163
$852$	8.96476e9	0.496593
$853$	6.47246e9	0.357066	0.178533	−	0.983934i	$-0.442865\pi$
$853$	6.47246e9	0.357066	0.178533	+	0.983934i	$0.442865\pi$
$854$	−1.72282e9	−0.0946536
$855$	0	0
$856$	1.22698e10	0.668619
$857$	−2.27261e10	−1.23337	−0.616684	−	0.787211i	$-0.711524\pi$
$857$	−2.27261e10	−1.23337	−0.616684	+	0.787211i	$0.711524\pi$
$858$	1.24557e8	0.00673228
$859$	−1.08363e10	−0.583315	−0.291657	−	0.956523i	$-0.594207\pi$
$859$	−1.08363e10	−0.583315	−0.291657	+	0.956523i	$0.594207\pi$
$860$	0	0
$861$	−1.56475e10	−0.835474
$862$	2.29749e9	0.122174
$863$	−3.23218e10	−1.71182	−0.855910	−	0.517125i	$-0.827002\pi$
$863$	−3.23218e10	−1.71182	−0.855910	+	0.517125i	$0.827002\pi$
$864$	−2.65084e9	−0.139825
$865$	0	0
$866$	−7.50973e9	−0.392927
$867$	1.73880e10	0.906114
$868$	−3.51838e9	−0.182610
$869$	2.82186e9	0.145870
$870$	0	0
$871$	4.44051e6	0.000227703 0
$872$	1.75219e10	0.894899
$873$	2.20865e9	0.112351
$874$	7.02754e9	0.356052
$875$	0	0
$876$	6.45635e9	0.324506
$877$	4.21367e9	0.210941	0.105471	−	0.994422i	$-0.466365\pi$
$877$	4.21367e9	0.210941	0.105471	+	0.994422i	$0.466365\pi$
$878$	−5.13624e8	−0.0256103
$879$	5.21856e9	0.259173
$880$	0	0
$881$	5.03021e9	0.247840	0.123920	−	0.992292i	$-0.460453\pi$
$881$	5.03021e9	0.247840	0.123920	+	0.992292i	$0.460453\pi$
$882$	−3.17800e9	−0.155960
$883$	2.81606e10	1.37651	0.688255	−	0.725469i	$-0.258377\pi$
$883$	2.81606e10	1.37651	0.688255	+	0.725469i	$0.258377\pi$
$884$	3.69016e9	0.179665
$885$	0	0
$886$	−4.63184e9	−0.223736
$887$	−1.59422e10	−0.767038	−0.383519	−	0.923533i	$-0.625288\pi$
$887$	−1.59422e10	−0.767038	−0.383519	+	0.923533i	$0.625288\pi$
$888$	7.57580e9	0.363064
$889$	−3.62877e10	−1.73222
$890$	0	0
$891$	8.48379e8	0.0401807
$892$	−6.15578e8	−0.0290406
$893$	−1.83660e10	−0.863049
$894$	1.81086e9	0.0847624
$895$	0	0
$896$	−3.16440e10	−1.46965
$897$	−1.53240e9	−0.0708922
$898$	1.04374e10	0.480980
$899$	−1.30079e9	−0.0597101
$900$	0	0
$901$	5.21206e10	2.37396
$902$	−1.85765e9	−0.0842833
$903$	−1.89561e10	−0.856725
$904$	9.38255e9	0.422407
$905$	0	0
$906$	−4.00496e9	−0.178916
$907$	2.20381e10	0.980729	0.490364	−	0.871517i	$-0.336864\pi$
$907$	2.20381e10	0.980729	0.490364	+	0.871517i	$0.336864\pi$
$908$	2.89408e10	1.28295
$909$	5.23064e9	0.230984
$910$	0	0
$911$	1.50904e10	0.661283	0.330641	−	0.943756i	$-0.392735\pi$
$911$	1.50904e10	0.661283	0.330641	+	0.943756i	$0.392735\pi$
$912$	−1.37051e10	−0.598275
$913$	−4.89489e9	−0.212860
$914$	2.28585e9	0.0990229
$915$	0	0
$916$	1.30434e10	0.560736
$917$	3.13158e10	1.34113
$918$	−1.93173e9	−0.0824134
$919$	9.21105e9	0.391476	0.195738	−	0.980656i	$-0.437290\pi$
$919$	9.21105e9	0.391476	0.195738	+	0.980656i	$0.437290\pi$
$920$	0	0
$921$	−5.70420e9	−0.240595
$922$	2.65457e9	0.111541
$923$	2.67071e9	0.111794
$924$	7.71201e9	0.321600
$925$	0	0
$926$	1.07290e10	0.444040
$927$	−1.06211e10	−0.437915
$928$	−8.90892e9	−0.365938
$929$	4.08831e10	1.67297	0.836487	−	0.547987i	$-0.184606\pi$
$929$	4.08831e10	1.67297	0.836487	+	0.547987i	$0.184606\pi$
$930$	0	0
$931$	−5.64929e10	−2.29440
$932$	−2.20816e10	−0.893461
$933$	7.00494e9	0.282370
$934$	−9.16912e9	−0.368225
$935$	0	0
$936$	−5.20062e8	−0.0207295
$937$	−3.38015e10	−1.34229	−0.671147	−	0.741324i	$-0.734198\pi$
$937$	−3.38015e10	−1.34229	−0.671147	+	0.741324i	$0.734198\pi$
$938$	−2.11306e7	−0.000835992 0
$939$	−2.51038e10	−0.989488
$940$	0	0
$941$	2.16815e10	0.848252	0.424126	−	0.905603i	$-0.360581\pi$
$941$	2.16815e10	0.848252	0.424126	+	0.905603i	$0.360581\pi$
$942$	5.07082e8	0.0197652
$943$	2.28543e10	0.887519
$944$	−2.64429e10	−1.02307
$945$	0	0
$946$	−2.25045e9	−0.0864271
$947$	1.47060e10	0.562692	0.281346	−	0.959606i	$-0.409219\pi$
$947$	1.47060e10	0.562692	0.281346	+	0.959606i	$0.409219\pi$
$948$	−5.67305e9	−0.216266
$949$	1.92342e9	0.0730538
$950$	0	0
$951$	−4.41476e9	−0.166447
$952$	−3.64696e10	−1.36994
$953$	1.26663e10	0.474050	0.237025	−	0.971504i	$-0.423828\pi$
$953$	1.26663e10	0.474050	0.237025	+	0.971504i	$0.423828\pi$
$954$	−3.53682e9	−0.131884
$955$	0	0
$956$	−3.64102e10	−1.34778
$957$	2.85123e9	0.105157
$958$	−9.07785e9	−0.333583
$959$	−1.49671e10	−0.547989
$960$	0	0
$961$	−2.71259e10	−0.985945
$962$	1.08670e9	0.0393547
$963$	1.19878e10	0.432562
$964$	6.87247e9	0.247083
$965$	0	0
$966$	7.29207e9	0.260274
$967$	2.68713e10	0.955644	0.477822	−	0.878457i	$-0.341426\pi$
$967$	2.68713e10	0.955644	0.477822	+	0.878457i	$0.341426\pi$
$968$	−1.26388e10	−0.447860
$969$	−3.43389e10	−1.21242
$970$	0	0
$971$	1.85571e9	0.0650494	0.0325247	−	0.999471i	$-0.489645\pi$
$971$	1.85571e9	0.0650494	0.0325247	+	0.999471i	$0.489645\pi$
$972$	−1.70558e9	−0.0595716
$973$	−2.21898e10	−0.772252
$974$	−9.93340e8	−0.0344462
$975$	0	0
$976$	−4.90731e9	−0.168954
$977$	3.13360e10	1.07501	0.537506	−	0.843260i	$-0.319367\pi$
$977$	3.13360e10	1.07501	0.537506	+	0.843260i	$0.319367\pi$
$978$	−1.25843e9	−0.0430171
$979$	−9.80808e9	−0.334075
$980$	0	0
$981$	1.71193e10	0.578953
$982$	2.10751e9	0.0710198
$983$	−2.78355e9	−0.0934679	−0.0467339	−	0.998907i	$-0.514881\pi$
$983$	−2.78355e9	−0.0934679	−0.0467339	+	0.998907i	$0.514881\pi$
$984$	7.75625e9	0.259519
$985$	0	0
$986$	−6.49215e9	−0.215685
$987$	−1.90574e10	−0.630889
$988$	−4.45132e9	−0.146838
$989$	2.76868e10	0.910094
$990$	0	0
$991$	2.20283e10	0.718991	0.359495	−	0.933147i	$-0.382949\pi$
$991$	2.20283e10	0.718991	0.359495	+	0.933147i	$0.382949\pi$
$992$	2.64829e9	0.0861341
$993$	−6.96461e9	−0.225722
$994$	−1.27088e10	−0.410443
$995$	0	0
$996$	9.84065e9	0.315585
$997$	5.02880e10	1.60706	0.803528	−	0.595266i	$-0.202953\pi$
$997$	5.02880e10	1.60706	0.803528	+	0.595266i	$0.202953\pi$
$998$	9.61836e9	0.306298
$999$	7.40170e9	0.234883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.8.a.j.1.2		4
3.2	odd	2		225.8.a.z.1.3		4
5.2	odd	4		15.8.b.a.4.4	✓	8
5.3	odd	4		15.8.b.a.4.5	yes	8
5.4	even	2		75.8.a.i.1.3		4
15.2	even	4		45.8.b.d.19.5		8
15.8	even	4		45.8.b.d.19.4		8
15.14	odd	2		225.8.a.bb.1.2		4
20.3	even	4		240.8.f.e.49.1		8
20.7	even	4		240.8.f.e.49.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.b.a.4.4	✓	8	5.2	odd	4
15.8.b.a.4.5	yes	8	5.3	odd	4
45.8.b.d.19.4		8	15.8	even	4
45.8.b.d.19.5		8	15.2	even	4
75.8.a.i.1.3		4	5.4	even	2
75.8.a.j.1.2		4	1.1	even	1	trivial
225.8.a.z.1.3		4	3.2	odd	2
225.8.a.bb.1.2		4	15.14	odd	2
240.8.f.e.49.1		8	20.3	even	4
240.8.f.e.49.5		8	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-3.02250 q^{2} +27.0000 q^{3} -118.865 q^{4} -81.6074 q^{6} -1505.28 q^{7} +746.147 q^{8} +729.000 q^{9} +1596.37 q^{11} -3209.34 q^{12} -956.100 q^{13} +4549.69 q^{14} +12959.4 q^{16} +32470.6 q^{17} -2203.40 q^{18} -39168.1 q^{19} -40642.5 q^{21} -4825.04 q^{22} +59361.5 q^{23} +20146.0 q^{24} +2889.81 q^{26} +19683.0 q^{27} +178924. q^{28} +66150.5 q^{29} -19664.1 q^{31} -134677. q^{32} +43102.1 q^{33} -98142.2 q^{34} -86652.2 q^{36} +376045. q^{37} +118386. q^{38} -25814.7 q^{39} +385003. q^{41} +122842. q^{42} +466410. q^{43} -189752. q^{44} -179420. q^{46} +468903. q^{47} +349905. q^{48} +1.44232e6 q^{49} +876706. q^{51} +113646. q^{52} +1.60516e6 q^{53} -59491.8 q^{54} -1.12316e6 q^{56} -1.05754e6 q^{57} -199940. q^{58} -2.04044e6 q^{59} -378667. q^{61} +59434.6 q^{62} -1.09735e6 q^{63} -1.25175e6 q^{64} -130276. q^{66} -4644.40 q^{67} -3.85960e6 q^{68} +1.60276e6 q^{69} -2.79333e6 q^{71} +543941. q^{72} -2.01174e6 q^{73} -1.13660e6 q^{74} +4.65570e6 q^{76} -2.40299e6 q^{77} +78024.8 q^{78} +1.76767e6 q^{79} +531441. q^{81} -1.16367e6 q^{82} -3.06625e6 q^{83} +4.83095e6 q^{84} -1.40972e6 q^{86} +1.78606e6 q^{87} +1.19113e6 q^{88} -6.14397e6 q^{89} +1.43920e6 q^{91} -7.05598e6 q^{92} -530930. q^{93} -1.41726e6 q^{94} -3.63627e6 q^{96} +3.02969e6 q^{97} -4.35940e6 q^{98} +1.16376e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 9 q^{2} + 108 q^{3} + 333 q^{4} + 243 q^{6} + 1188 q^{7} + 2043 q^{8} + 2916 q^{9} + 5376 q^{11} + 8991 q^{12} + 8424 q^{13} + 6762 q^{14} + 43265 q^{16} + 4896 q^{17} + 6561 q^{18} + 15232 q^{19}+ \cdots + 3919104 q^{99}+O(q^{100}) \) 4 * q + 9 * q^2 + 108 * q^3 + 333 * q^4 + 243 * q^6 + 1188 * q^7 + 2043 * q^8 + 2916 * q^9 + 5376 * q^11 + 8991 * q^12 + 8424 * q^13 + 6762 * q^14 + 43265 * q^16 + 4896 * q^17 + 6561 * q^18 + 15232 * q^19 + 32076 * q^21 - 44118 * q^22 + 110016 * q^23 + 55161 * q^24 - 396762 * q^26 + 78732 * q^27 + 674514 * q^28 - 120036 * q^29 + 116864 * q^31 + 621963 * q^32 + 145152 * q^33 - 92374 * q^34 + 242757 * q^36 + 663768 * q^37 + 1888776 * q^38 + 227448 * q^39 + 253824 * q^41 + 182574 * q^42 + 1092960 * q^43 - 1289286 * q^44 + 331204 * q^46 + 132840 * q^47 + 1168155 * q^48 + 1633580 * q^49 + 132192 * q^51 - 779274 * q^52 + 1994616 * q^53 + 177147 * q^54 + 352830 * q^56 + 411264 * q^57 - 742608 * q^58 - 545712 * q^59 - 3216760 * q^61 + 476856 * q^62 + 866052 * q^63 + 284297 * q^64 - 1191186 * q^66 + 2013336 * q^67 - 7540038 * q^68 + 2970432 * q^69 - 690912 * q^71 + 1489347 * q^72 - 5498064 * q^73 - 11980638 * q^74 + 6557832 * q^76 - 795744 * q^77 - 10712574 * q^78 + 7190080 * q^79 + 2125764 * q^81 - 26456814 * q^82 - 13158432 * q^83 + 18211878 * q^84 - 9918804 * q^86 - 3240972 * q^87 - 22119642 * q^88 - 22889448 * q^89 + 12037824 * q^91 - 17368812 * q^92 + 3155328 * q^93 + 22864448 * q^94 + 16793001 * q^96 + 17873136 * q^97 - 20819979 * q^98 + 3919104 * q^99

Introduction

L-functions

Modular forms

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