LMFDB - Embedded newform 62.6.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [62,6,Mod(1,62)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(62, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("62.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 62 = 2 \cdot 31 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	62.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,8] 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:	$9.94379682840$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	62.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} +1.29150 q^{3} +16.0000 q^{4} -65.5830 q^{5} +5.16601 q^{6} -195.288 q^{7} +64.0000 q^{8} -241.332 q^{9} -262.332 q^{10} +309.984 q^{11} +20.6640 q^{12} -173.129 q^{13} -781.150 q^{14} -84.7006 q^{15} +256.000 q^{16} -852.110 q^{17} -965.328 q^{18} +1712.40 q^{19} -1049.33 q^{20} -252.214 q^{21} +1239.94 q^{22} -4592.57 q^{23} +82.6562 q^{24} +1176.13 q^{25} -692.518 q^{26} -625.516 q^{27} -3124.60 q^{28} +5710.27 q^{29} -338.802 q^{30} +961.000 q^{31} +1024.00 q^{32} +400.345 q^{33} -3408.44 q^{34} +12807.5 q^{35} -3861.31 q^{36} -8135.04 q^{37} +6849.59 q^{38} -223.597 q^{39} -4197.31 q^{40} -7341.41 q^{41} -1008.86 q^{42} +7371.18 q^{43} +4959.75 q^{44} +15827.3 q^{45} -18370.3 q^{46} +5723.32 q^{47} +330.625 q^{48} +21330.2 q^{49} +4704.52 q^{50} -1100.50 q^{51} -2770.07 q^{52} -178.889 q^{53} -2502.06 q^{54} -20329.7 q^{55} -12498.4 q^{56} +2211.57 q^{57} +22841.1 q^{58} +43351.1 q^{59} -1355.21 q^{60} +8809.09 q^{61} +3844.00 q^{62} +47129.1 q^{63} +4096.00 q^{64} +11354.3 q^{65} +1601.38 q^{66} -32810.7 q^{67} -13633.8 q^{68} -5931.32 q^{69} +51230.2 q^{70} -75149.7 q^{71} -15445.2 q^{72} -72572.8 q^{73} -32540.2 q^{74} +1518.98 q^{75} +27398.4 q^{76} -60536.1 q^{77} -894.388 q^{78} -90109.7 q^{79} -16789.2 q^{80} +57835.8 q^{81} -29365.6 q^{82} -93947.8 q^{83} -4035.43 q^{84} +55883.9 q^{85} +29484.7 q^{86} +7374.82 q^{87} +19839.0 q^{88} +26373.1 q^{89} +63309.1 q^{90} +33810.0 q^{91} -73481.2 q^{92} +1241.13 q^{93} +22893.3 q^{94} -112304. q^{95} +1322.50 q^{96} +76137.4 q^{97} +85320.9 q^{98} -74809.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} - 8 q^{3} + 32 q^{4} - 110 q^{5} - 32 q^{6} - 126 q^{7} + 128 q^{8} - 398 q^{9} - 440 q^{10} - 396 q^{11} - 128 q^{12} - 632 q^{13} - 504 q^{14} + 328 q^{15} + 512 q^{16} - 720 q^{17} - 1592 q^{18}+ \cdots + 35796 q^{99}+O(q^{100}) $ 2 * q + 8 * q^2 - 8 * q^3 + 32 * q^4 - 110 * q^5 - 32 * q^6 - 126 * q^7 + 128 * q^8 - 398 * q^9 - 440 * q^10 - 396 * q^11 - 128 * q^12 - 632 * q^13 - 504 * q^14 + 328 * q^15 + 512 * q^16 - 720 * q^17 - 1592 * q^18 - 438 * q^19 - 1760 * q^20 - 896 * q^21 - 1584 * q^22 - 560 * q^23 - 512 * q^24 + 24 * q^25 - 2528 * q^26 + 3088 * q^27 - 2016 * q^28 + 4404 * q^29 + 1312 * q^30 + 1922 * q^31 + 2048 * q^32 + 6960 * q^33 - 2880 * q^34 + 9730 * q^35 - 6368 * q^36 - 8100 * q^37 - 1752 * q^38 + 4040 * q^39 - 7040 * q^40 + 1742 * q^41 - 3584 * q^42 - 8572 * q^43 - 6336 * q^44 + 22786 * q^45 - 2240 * q^46 + 26792 * q^47 - 2048 * q^48 + 9324 * q^49 + 96 * q^50 - 2328 * q^51 - 10112 * q^52 + 33656 * q^53 + 12352 * q^54 + 11028 * q^55 - 8064 * q^56 + 22192 * q^57 + 17616 * q^58 + 9478 * q^59 + 5248 * q^60 + 26328 * q^61 + 7688 * q^62 + 36274 * q^63 + 8192 * q^64 + 31736 * q^65 + 27840 * q^66 - 60224 * q^67 - 11520 * q^68 - 43400 * q^69 + 38920 * q^70 - 152490 * q^71 - 25472 * q^72 - 32860 * q^73 - 32400 * q^74 + 12224 * q^75 - 7008 * q^76 - 109452 * q^77 + 16160 * q^78 - 163276 * q^79 - 28160 * q^80 + 61402 * q^81 + 6968 * q^82 - 130324 * q^83 - 14336 * q^84 + 50016 * q^85 - 34288 * q^86 + 19512 * q^87 - 25344 * q^88 - 39908 * q^89 + 91144 * q^90 + 2016 * q^91 - 8960 * q^92 - 7688 * q^93 + 107168 * q^94 - 16790 * q^95 - 8192 * q^96 + 11034 * q^97 + 37296 * q^98 + 35796 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	1.29150	0.0828499	0.0414250	−	0.999142i	$-0.486810\pi$
$3$	1.29150	0.0828499	0.0414250	+	0.999142i	$0.486810\pi$
$4$	16.0000	0.500000
$5$	−65.5830	−1.17318	−0.586592	−	0.809882i	$-0.699531\pi$
$5$	−65.5830	−1.17318	−0.586592	+	0.809882i	$0.699531\pi$
$6$	5.16601	0.0585837
$7$	−195.288	−1.50636	−0.753181	−	0.657813i	$-0.771482\pi$
$7$	−195.288	−1.50636	−0.753181	+	0.657813i	$0.771482\pi$
$8$	64.0000	0.353553
$9$	−241.332	−0.993136
$10$	−262.332	−0.829567
$11$	309.984	0.772428	0.386214	−	0.922409i	$-0.373783\pi$
$11$	309.984	0.772428	0.386214	+	0.922409i	$0.373783\pi$
$12$	20.6640	0.0414250
$13$	−173.129	−0.284127	−0.142064	−	0.989858i	$-0.545374\pi$
$13$	−173.129	−0.284127	−0.142064	+	0.989858i	$0.545374\pi$
$14$	−781.150	−1.06516
$15$	−84.7006	−0.0971983
$16$	256.000	0.250000
$17$	−852.110	−0.715111	−0.357555	−	0.933892i	$-0.616390\pi$
$17$	−852.110	−0.715111	−0.357555	+	0.933892i	$0.616390\pi$
$18$	−965.328	−0.702253
$19$	1712.40	1.08823	0.544115	−	0.839011i	$-0.316865\pi$
$19$	1712.40	1.08823	0.544115	+	0.839011i	$0.316865\pi$
$20$	−1049.33	−0.586592
$21$	−252.214	−0.124802
$22$	1239.94	0.546189
$23$	−4592.57	−1.81024	−0.905121	−	0.425154i	$-0.860220\pi$
$23$	−4592.57	−1.81024	−0.905121	+	0.425154i	$0.860220\pi$
$24$	82.6562	0.0292919
$25$	1176.13	0.376362
$26$	−692.518	−0.200908
$27$	−625.516	−0.165131
$28$	−3124.60	−0.753181
$29$	5710.27	1.26084	0.630422	−	0.776253i	$-0.282882\pi$
$29$	5710.27	1.26084	0.630422	+	0.776253i	$0.282882\pi$
$30$	−338.802	−0.0687295
$31$	961.000	0.179605
$31$	961.000	0.179605
$32$	1024.00	0.176777
$33$	400.345	0.0639956
$34$	−3408.44	−0.505660
$35$	12807.5	1.76724
$36$	−3861.31	−0.496568
$37$	−8135.04	−0.976912	−0.488456	−	0.872589i	$-0.662440\pi$
$37$	−8135.04	−0.976912	−0.488456	+	0.872589i	$0.662440\pi$
$38$	6849.59	0.769495
$39$	−223.597	−0.0235399
$40$	−4197.31	−0.414783
$41$	−7341.41	−0.682056	−0.341028	−	0.940053i	$-0.610775\pi$
$41$	−7341.41	−0.682056	−0.341028	+	0.940053i	$0.610775\pi$
$42$	−1008.86	−0.0882484
$43$	7371.18	0.607947	0.303974	−	0.952680i	$-0.401687\pi$
$43$	7371.18	0.607947	0.303974	+	0.952680i	$0.401687\pi$
$44$	4959.75	0.386214
$45$	15827.3	1.16513
$46$	−18370.3	−1.28003
$47$	5723.32	0.377923	0.188962	−	0.981984i	$-0.439488\pi$
$47$	5723.32	0.377923	0.188962	+	0.981984i	$0.439488\pi$
$48$	330.625	0.0207125
$49$	21330.2	1.26913
$50$	4704.52	0.266128
$51$	−1100.50	−0.0592469
$52$	−2770.07	−0.142064
$53$	−178.889	−0.00874772	−0.00437386	−	0.999990i	$-0.501392\pi$
$53$	−178.889	−0.00874772	−0.00437386	+	0.999990i	$0.501392\pi$
$54$	−2502.06	−0.116765
$55$	−20329.7	−0.906200
$56$	−12498.4	−0.532580
$57$	2211.57	0.0901598
$58$	22841.1	0.891551
$59$	43351.1	1.62132	0.810662	−	0.585514i	$-0.199107\pi$
$59$	43351.1	1.62132	0.810662	+	0.585514i	$0.199107\pi$
$60$	−1355.21	−0.0485991
$61$	8809.09	0.303114	0.151557	−	0.988448i	$-0.451571\pi$
$61$	8809.09	0.303114	0.151557	+	0.988448i	$0.451571\pi$
$62$	3844.00	0.127000
$63$	47129.1	1.49602
$64$	4096.00	0.125000
$65$	11354.3	0.333333
$66$	1601.38	0.0452517
$67$	−32810.7	−0.892952	−0.446476	−	0.894796i	$-0.647321\pi$
$67$	−32810.7	−0.892952	−0.446476	+	0.894796i	$0.647321\pi$
$68$	−13633.8	−0.357555
$69$	−5931.32	−0.149978
$70$	51230.2	1.24963
$71$	−75149.7	−1.76922	−0.884609	−	0.466334i	$-0.845574\pi$
$71$	−75149.7	−1.76922	−0.884609	+	0.466334i	$0.845574\pi$
$72$	−15445.2	−0.351127
$73$	−72572.8	−1.59392	−0.796961	−	0.604031i	$-0.793560\pi$
$73$	−72572.8	−1.59392	−0.796961	+	0.604031i	$0.793560\pi$
$74$	−32540.2	−0.690781
$75$	1518.98	0.0311815
$76$	27398.4	0.544115
$77$	−60536.1	−1.16356
$78$	−894.388	−0.0166452
$79$	−90109.7	−1.62444	−0.812220	−	0.583351i	$-0.801741\pi$
$79$	−90109.7	−1.62444	−0.812220	+	0.583351i	$0.801741\pi$
$80$	−16789.2	−0.293296
$81$	57835.8	0.979455
$82$	−29365.6	−0.482286
$83$	−93947.8	−1.49689	−0.748447	−	0.663194i	$-0.769200\pi$
$83$	−93947.8	−1.49689	−0.748447	+	0.663194i	$0.769200\pi$
$84$	−4035.43	−0.0624010
$85$	55883.9	0.838957
$86$	29484.7	0.429884
$87$	7374.82	0.104461
$88$	19839.0	0.273094
$89$	26373.1	0.352928	0.176464	−	0.984307i	$-0.443534\pi$
$89$	26373.1	0.352928	0.176464	+	0.984307i	$0.443534\pi$
$90$	63309.1	0.823872
$91$	33810.0	0.427998
$92$	−73481.2	−0.905121
$93$	1241.13	0.0148803
$94$	22893.3	0.267232
$95$	−112304.	−1.27669
$96$	1322.50	0.0146459
$97$	76137.4	0.821616	0.410808	−	0.911722i	$-0.365247\pi$
$97$	76137.4	0.821616	0.410808	+	0.911722i	$0.365247\pi$
$98$	85320.9	0.897409
$99$	−74809.1	−0.767126
$100$	18818.1	0.188181
$101$	51708.1	0.504377	0.252188	−	0.967678i	$-0.418850\pi$
$101$	51708.1	0.504377	0.252188	+	0.967678i	$0.418850\pi$
$102$	−4402.01	−0.0418939
$103$	−143290.	−1.33083	−0.665414	−	0.746474i	$-0.731745\pi$
$103$	−143290.	−1.33083	−0.665414	+	0.746474i	$0.731745\pi$
$104$	−11080.3	−0.100454
$105$	16541.0	0.146416
$106$	−715.558	−0.00618557
$107$	−96262.8	−0.812829	−0.406414	−	0.913689i	$-0.633221\pi$
$107$	−96262.8	−0.812829	−0.406414	+	0.913689i	$0.633221\pi$
$108$	−10008.3	−0.0825656
$109$	−96625.4	−0.778977	−0.389489	−	0.921031i	$-0.627348\pi$
$109$	−96625.4	−0.778977	−0.389489	+	0.921031i	$0.627348\pi$
$110$	−81318.8	−0.640780
$111$	−10506.4	−0.0809371
$112$	−49993.6	−0.376591
$113$	224729.	1.65563	0.827815	−	0.561002i	$-0.189584\pi$
$113$	224729.	1.65563	0.827815	+	0.561002i	$0.189584\pi$
$114$	8846.27	0.0637526
$115$	301195.	2.12375
$116$	91364.3	0.630422
$117$	41781.7	0.282177
$118$	173404.	1.14645
$119$	166406.	1.07722
$120$	−5420.84	−0.0343648
$121$	−64960.8	−0.403355
$122$	35236.4	0.214334
$123$	−9481.45	−0.0565083
$124$	15376.0	0.0898027
$125$	127813.	0.731643
$126$	188517.	1.05785
$127$	−27227.6	−0.149796	−0.0748979	−	0.997191i	$-0.523863\pi$
$127$	−27227.6	−0.149796	−0.0748979	+	0.997191i	$0.523863\pi$
$128$	16384.0	0.0883883
$129$	9519.90	0.0503684
$130$	45417.4	0.235702
$131$	−9919.84	−0.0505041	−0.0252520	−	0.999681i	$-0.508039\pi$
$131$	−9919.84	−0.0505041	−0.0252520	+	0.999681i	$0.508039\pi$
$132$	6405.53	0.0319978
$133$	−334410.	−1.63927
$134$	−131243.	−0.631412
$135$	41023.2	0.193729
$136$	−54535.0	−0.252830
$137$	306512.	1.39523	0.697616	−	0.716471i	$-0.254244\pi$
$137$	306512.	1.39523	0.697616	+	0.716471i	$0.254244\pi$
$138$	−23725.3	−0.106051
$139$	244113.	1.07165	0.535825	−	0.844329i	$-0.320001\pi$
$139$	244113.	1.07165	0.535825	+	0.844329i	$0.320001\pi$
$140$	204921.	0.883621
$141$	7391.68	0.0313109
$142$	−300599.	−1.25103
$143$	−53667.4	−0.219468
$144$	−61781.0	−0.248284
$145$	−374496.	−1.47920
$146$	−290291.	−1.12707
$147$	27548.1	0.105147
$148$	−130161.	−0.488456
$149$	−265675.	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$149$	−265675.	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$150$	6075.90	0.0220487
$151$	−37210.4	−0.132807	−0.0664036	−	0.997793i	$-0.521152\pi$
$151$	−37210.4	−0.132807	−0.0664036	+	0.997793i	$0.521152\pi$
$152$	109594.	0.384748
$153$	205641.	0.710202
$154$	−242144.	−0.822759
$155$	−63025.3	−0.210710
$156$	−3577.55	−0.0117700
$157$	352648.	1.14181	0.570903	−	0.821017i	$-0.306593\pi$
$157$	352648.	1.14181	0.570903	+	0.821017i	$0.306593\pi$
$158$	−360439.	−1.14865
$159$	−231.036	−0.000724748 0
$160$	−67157.0	−0.207392
$161$	896873.	2.72688
$162$	231343.	0.692579
$163$	−590432.	−1.74061	−0.870305	−	0.492514i	$-0.836078\pi$
$163$	−590432.	−1.74061	−0.870305	+	0.492514i	$0.836078\pi$
$164$	−117463.	−0.341028
$165$	−26255.9	−0.0750786
$166$	−375791.	−1.05846
$167$	68521.1	0.190122	0.0950611	−	0.995471i	$-0.469695\pi$
$167$	68521.1	0.190122	0.0950611	+	0.995471i	$0.469695\pi$
$168$	−16141.7	−0.0441242
$169$	−341319.	−0.919272
$170$	223536.	0.593232
$171$	−413257.	−1.08076
$172$	117939.	0.303974
$173$	99010.0	0.251515	0.125758	−	0.992061i	$-0.459864\pi$
$173$	99010.0	0.251515	0.125758	+	0.992061i	$0.459864\pi$
$174$	29499.3	0.0738650
$175$	−229684.	−0.566937
$176$	79356.0	0.193107
$177$	55988.1	0.134327
$178$	105492.	0.249558
$179$	54258.8	0.126572	0.0632860	−	0.997995i	$-0.479842\pi$
$179$	54258.8	0.126572	0.0632860	+	0.997995i	$0.479842\pi$
$180$	253236.	0.582566
$181$	−557488.	−1.26485	−0.632425	−	0.774622i	$-0.717940\pi$
$181$	−557488.	−1.26485	−0.632425	+	0.774622i	$0.717940\pi$
$182$	135240.	0.302640
$183$	11377.0	0.0251130
$184$	−293925.	−0.640017
$185$	533520.	1.14610
$186$	4964.54	0.0105220
$187$	−264141.	−0.552371
$188$	91573.1	0.188962
$189$	122156.	0.248747
$190$	−449217.	−0.902760
$191$	−138605.	−0.274912	−0.137456	−	0.990508i	$-0.543893\pi$
$191$	−138605.	−0.274912	−0.137456	+	0.990508i	$0.543893\pi$
$192$	5289.99	0.0103562
$193$	243571.	0.470688	0.235344	−	0.971912i	$-0.424378\pi$
$193$	243571.	0.470688	0.235344	+	0.971912i	$0.424378\pi$
$194$	304550.	0.580970
$195$	14664.2	0.0276166
$196$	341284.	0.634564
$197$	528257.	0.969794	0.484897	−	0.874571i	$-0.338857\pi$
$197$	528257.	0.969794	0.484897	+	0.874571i	$0.338857\pi$
$198$	−299237.	−0.542440
$199$	−114329.	−0.204656	−0.102328	−	0.994751i	$-0.532629\pi$
$199$	−114329.	−0.204656	−0.102328	+	0.994751i	$0.532629\pi$
$200$	75272.4	0.133064
$201$	−42375.1	−0.0739810
$202$	206832.	0.356648
$203$	−1.11514e6	−1.89929
$204$	−17608.0	−0.0296234
$205$	481472.	0.800177
$206$	−573159.	−0.941038
$207$	1.10834e6	1.79782
$208$	−44321.1	−0.0710318
$209$	530817.	0.840579
$210$	66163.9	0.103532
$211$	−202926.	−0.313785	−0.156892	−	0.987616i	$-0.550148\pi$
$211$	−202926.	−0.313785	−0.156892	+	0.987616i	$0.550148\pi$
$212$	−2862.23	−0.00437386
$213$	−97056.0	−0.146580
$214$	−385051.	−0.574757
$215$	−483424.	−0.713234
$216$	−40033.0	−0.0583827
$217$	−187671.	−0.270551
$218$	−386501.	−0.550820
$219$	−93728.0	−0.132056
$220$	−325275.	−0.453100
$221$	147525.	0.203182
$222$	−42025.7	−0.0572312
$223$	241672.	0.325435	0.162717	−	0.986673i	$-0.447974\pi$
$223$	241672.	0.325435	0.162717	+	0.986673i	$0.447974\pi$
$224$	−199974.	−0.266290
$225$	−283838.	−0.373778
$226$	898916.	1.17071
$227$	−214659.	−0.276493	−0.138246	−	0.990398i	$-0.544147\pi$
$227$	−214659.	−0.276493	−0.138246	+	0.990398i	$0.544147\pi$
$228$	35385.1	0.0450799
$229$	963826.	1.21453	0.607267	−	0.794498i	$-0.292266\pi$
$229$	963826.	1.21453	0.607267	+	0.794498i	$0.292266\pi$
$230$	1.20478e6	1.50172
$231$	−78182.5	−0.0964006
$232$	365457.	0.445776
$233$	1.50308e6	1.81382	0.906909	−	0.421327i	$-0.138435\pi$
$233$	1.50308e6	1.81382	0.906909	+	0.421327i	$0.138435\pi$
$234$	167127.	0.199529
$235$	−375353.	−0.443374
$236$	693618.	0.810662
$237$	−116377.	−0.134585
$238$	665626.	0.761707
$239$	−501423.	−0.567819	−0.283909	−	0.958851i	$-0.591632\pi$
$239$	−501423.	−0.567819	−0.283909	+	0.958851i	$0.591632\pi$
$240$	−21683.4	−0.0242996
$241$	−1.28032e6	−1.41996	−0.709979	−	0.704223i	$-0.751296\pi$
$241$	−1.28032e6	−1.41996	−0.709979	+	0.704223i	$0.751296\pi$
$242$	−259843.	−0.285215
$243$	226696.	0.246279
$244$	140945.	0.151557
$245$	−1.39890e6	−1.48892
$246$	−37925.8	−0.0399574
$247$	−296467.	−0.309196
$248$	61504.0	0.0635001
$249$	−121334.	−0.124018
$250$	511251.	0.517349
$251$	1.79494e6	1.79832	0.899160	−	0.437621i	$-0.144179\pi$
$251$	1.79494e6	1.79832	0.899160	+	0.437621i	$0.144179\pi$
$252$	754066.	0.748011
$253$	−1.42363e6	−1.39828
$254$	−108910.	−0.105922
$255$	72174.2	0.0695075
$256$	65536.0	0.0625000
$257$	−764457.	−0.721972	−0.360986	−	0.932571i	$-0.617560\pi$
$257$	−764457.	−0.721972	−0.360986	+	0.932571i	$0.617560\pi$
$258$	38079.6	0.0356158
$259$	1.58867e6	1.47158
$260$	181670.	0.166667
$261$	−1.37807e6	−1.25219
$262$	−39679.4	−0.0357118
$263$	−192800.	−0.171877	−0.0859385	−	0.996300i	$-0.527389\pi$
$263$	−192800.	−0.171877	−0.0859385	+	0.996300i	$0.527389\pi$
$264$	25622.1	0.0226259
$265$	11732.1	0.0102627
$266$	−1.33764e6	−1.15914
$267$	34060.9	0.0292401
$268$	−524971.	−0.446476
$269$	1.19600e6	1.00775	0.503873	−	0.863778i	$-0.331908\pi$
$269$	1.19600e6	1.00775	0.503873	+	0.863778i	$0.331908\pi$
$270$	164093.	0.136987
$271$	498025.	0.411934	0.205967	−	0.978559i	$-0.433966\pi$
$271$	498025.	0.411934	0.205967	+	0.978559i	$0.433966\pi$
$272$	−218140.	−0.178778
$273$	43665.7	0.0354596
$274$	1.22605e6	0.986579
$275$	364582.	0.290712
$276$	−94901.2	−0.0749892
$277$	−196572.	−0.153929	−0.0769646	−	0.997034i	$-0.524523\pi$
$277$	−196572.	−0.153929	−0.0769646	+	0.997034i	$0.524523\pi$
$278$	976450.	0.757771
$279$	−231920.	−0.178372
$280$	819683.	0.624814
$281$	−1.51427e6	−1.14403	−0.572015	−	0.820243i	$-0.693838\pi$
$281$	−1.51427e6	−1.14403	−0.572015	+	0.820243i	$0.693838\pi$
$282$	29566.7	0.0221402
$283$	185214.	0.137470	0.0687350	−	0.997635i	$-0.478104\pi$
$283$	185214.	0.137470	0.0687350	+	0.997635i	$0.478104\pi$
$284$	−1.20239e6	−0.884609
$285$	−145041.	−0.105774
$286$	−214670.	−0.155187
$287$	1.43369e6	1.02742
$288$	−247124.	−0.175563
$289$	−693766.	−0.488617
$290$	−1.49799e6	−1.04595
$291$	98331.6	0.0680708
$292$	−1.16117e6	−0.796961
$293$	1.94301e6	1.32223	0.661113	−	0.750286i	$-0.270084\pi$
$293$	1.94301e6	1.32223	0.661113	+	0.750286i	$0.270084\pi$
$294$	110192.	0.0743503
$295$	−2.84310e6	−1.90211
$296$	−520643.	−0.345391
$297$	−193900.	−0.127552
$298$	−1.06270e6	−0.693219
$299$	795110.	0.514339
$300$	24303.6	0.0155908
$301$	−1.43950e6	−0.915789
$302$	−148841.	−0.0939088
$303$	66781.1	0.0417876
$304$	438374.	0.272058
$305$	−577727.	−0.355609
$306$	822565.	0.502189
$307$	−1.31431e6	−0.795886	−0.397943	−	0.917410i	$-0.630276\pi$
$307$	−1.31431e6	−0.795886	−0.397943	+	0.917410i	$0.630276\pi$
$308$	−968577.	−0.581778
$309$	−185059.	−0.110259
$310$	−252101.	−0.148995
$311$	1.53845e6	0.901953	0.450976	−	0.892536i	$-0.351076\pi$
$311$	1.53845e6	0.901953	0.450976	+	0.892536i	$0.351076\pi$
$312$	−14310.2	−0.00832261
$313$	−2.73153e6	−1.57596	−0.787980	−	0.615701i	$-0.788873\pi$
$313$	−2.73153e6	−1.57596	−0.787980	+	0.615701i	$0.788873\pi$
$314$	1.41059e6	0.807379
$315$	−3.09087e6	−1.75511
$316$	−1.44176e6	−0.812220
$317$	1.57517e6	0.880397	0.440199	−	0.897900i	$-0.354908\pi$
$317$	1.57517e6	0.880397	0.440199	+	0.897900i	$0.354908\pi$
$318$	−924.145	−0.000512474 0
$319$	1.77009e6	0.973911
$320$	−268628.	−0.146648
$321$	−124324.	−0.0673428
$322$	3.58749e6	1.92820
$323$	−1.45915e6	−0.778205
$324$	925373.	0.489727
$325$	−203623.	−0.106935
$326$	−2.36173e6	−1.23080
$327$	−124792.	−0.0645382
$328$	−469850.	−0.241143
$329$	−1.11769e6	−0.569289
$330$	−105023.	−0.0530886
$331$	1.17314e6	0.588546	0.294273	−	0.955721i	$-0.404922\pi$
$331$	1.17314e6	0.588546	0.294273	+	0.955721i	$0.404922\pi$
$332$	−1.50316e6	−0.748447
$333$	1.96325e6	0.970206
$334$	274084.	0.134437
$335$	2.15182e6	1.04760
$336$	−64566.9	−0.0312005
$337$	3.02853e6	1.45264	0.726319	−	0.687358i	$-0.241230\pi$
$337$	3.02853e6	1.45264	0.726319	+	0.687358i	$0.241230\pi$
$338$	−1.36528e6	−0.650023
$339$	290238.	0.137169
$340$	894143.	0.419478
$341$	297895.	0.138732
$342$	−1.65303e6	−0.764213
$343$	−883331.	−0.405404
$344$	471756.	0.214942
$345$	388994.	0.175952
$346$	396040.	0.177848
$347$	585540.	0.261055	0.130528	−	0.991445i	$-0.458333\pi$
$347$	585540.	0.261055	0.130528	+	0.991445i	$0.458333\pi$
$348$	117997.	0.0522304
$349$	683086.	0.300201	0.150100	−	0.988671i	$-0.452040\pi$
$349$	683086.	0.300201	0.150100	+	0.988671i	$0.452040\pi$
$350$	−918735.	−0.400885
$351$	108295.	0.0469182
$352$	317424.	0.136547
$353$	−15814.1	−0.00675472	−0.00337736	−	0.999994i	$-0.501075\pi$
$353$	−15814.1	−0.00675472	−0.00337736	+	0.999994i	$0.501075\pi$
$354$	223952.	0.0949833
$355$	4.92854e6	2.07562
$356$	421970.	0.176464
$357$	214914.	0.0892473
$358$	217035.	0.0895000
$359$	−2.80905e6	−1.15033	−0.575166	−	0.818037i	$-0.695063\pi$
$359$	−2.80905e6	−1.15033	−0.575166	+	0.818037i	$0.695063\pi$
$360$	1.01295e6	0.411936
$361$	456209.	0.184245
$362$	−2.22995e6	−0.894384
$363$	−83897.0	−0.0334180
$364$	540960.	0.213999
$365$	4.75955e6	1.86996
$366$	45507.9	0.0177576
$367$	1.21884e6	0.472371	0.236185	−	0.971708i	$-0.424103\pi$
$367$	1.21884e6	0.472371	0.236185	+	0.971708i	$0.424103\pi$
$368$	−1.17570e6	−0.452560
$369$	1.77172e6	0.677374
$370$	2.13408e6	0.810414
$371$	34934.9	0.0131772
$372$	19858.1	0.00744014
$373$	−4.15198e6	−1.54519	−0.772596	−	0.634897i	$-0.781042\pi$
$373$	−4.15198e6	−1.54519	−0.772596	+	0.634897i	$0.781042\pi$
$374$	−1.05656e6	−0.390586
$375$	165070.	0.0606165
$376$	366293.	0.133616
$377$	−988615.	−0.358240
$378$	488622.	0.175891
$379$	3.61806e6	1.29383	0.646915	−	0.762562i	$-0.276059\pi$
$379$	3.61806e6	1.29383	0.646915	+	0.762562i	$0.276059\pi$
$380$	−1.79687e6	−0.638347
$381$	−35164.5	−0.0124106
$382$	−554419.	−0.194392
$383$	−2.06786e6	−0.720318	−0.360159	−	0.932891i	$-0.617277\pi$
$383$	−2.06786e6	−0.720318	−0.360159	+	0.932891i	$0.617277\pi$
$384$	21160.0	0.00732297
$385$	3.97014e6	1.36507
$386$	974286.	0.332827
$387$	−1.77890e6	−0.603774
$388$	1.21820e6	0.410808
$389$	−5.52964e6	−1.85278	−0.926388	−	0.376570i	$-0.877104\pi$
$389$	−5.52964e6	−1.85278	−0.926388	+	0.376570i	$0.877104\pi$
$390$	58656.7	0.0195279
$391$	3.91338e6	1.29452
$392$	1.36513e6	0.448704
$393$	−12811.5	−0.00418426
$394$	2.11303e6	0.685748
$395$	5.90966e6	1.90577
$396$	−1.19695e6	−0.383563
$397$	−59019.4	−0.0187940	−0.00939699	−	0.999956i	$-0.502991\pi$
$397$	−59019.4	−0.0187940	−0.00939699	+	0.999956i	$0.502991\pi$
$398$	−457317.	−0.144714
$399$	−431892.	−0.135813
$400$	301089.	0.0940904
$401$	4.45947e6	1.38491	0.692456	−	0.721460i	$-0.256529\pi$
$401$	4.45947e6	1.38491	0.692456	+	0.721460i	$0.256529\pi$
$402$	−169500.	−0.0523125
$403$	−166377.	−0.0510307
$404$	827329.	0.252188
$405$	−3.79305e6	−1.14908
$406$	−4.46058e6	−1.34300
$407$	−2.52173e6	−0.754594
$408$	−70432.1	−0.0209469
$409$	4.44468e6	1.31381	0.656905	−	0.753974i	$-0.271865\pi$
$409$	4.44468e6	1.31381	0.656905	+	0.753974i	$0.271865\pi$
$410$	1.92589e6	0.565811
$411$	395862.	0.115595
$412$	−2.29264e6	−0.665414
$413$	−8.46593e6	−2.44230
$414$	4.43334e6	1.27125
$415$	6.16138e6	1.75613
$416$	−177285.	−0.0502270
$417$	315272.	0.0887862
$418$	2.12327e6	0.594379
$419$	−3.69305e6	−1.02766	−0.513831	−	0.857892i	$-0.671774\pi$
$419$	−3.69305e6	−1.02766	−0.513831	+	0.857892i	$0.671774\pi$
$420$	264656.	0.0732079
$421$	−4.14595e6	−1.14004	−0.570018	−	0.821632i	$-0.693064\pi$
$421$	−4.14595e6	−1.14004	−0.570018	+	0.821632i	$0.693064\pi$
$422$	−811705.	−0.221879
$423$	−1.38122e6	−0.375329
$424$	−11448.9	−0.00309279
$425$	−1.00219e6	−0.269140
$426$	−388224.	−0.103647
$427$	−1.72031e6	−0.456600
$428$	−1.54020e6	−0.406414
$429$	−69311.6	−0.0181829
$430$	−1.93370e6	−0.504333
$431$	−1.82807e6	−0.474022	−0.237011	−	0.971507i	$-0.576168\pi$
$431$	−1.82807e6	−0.474022	−0.237011	+	0.971507i	$0.576168\pi$
$432$	−160132.	−0.0412828
$433$	−6.74348e6	−1.72848	−0.864240	−	0.503080i	$-0.832200\pi$
$433$	−6.74348e6	−1.72848	−0.864240	+	0.503080i	$0.832200\pi$
$434$	−750685.	−0.191308
$435$	−483663.	−0.122552
$436$	−1.54601e6	−0.389489
$437$	−7.86432e6	−1.96996
$438$	−374912.	−0.0933779
$439$	476092.	0.117904	0.0589521	−	0.998261i	$-0.481224\pi$
$439$	476092.	0.117904	0.0589521	+	0.998261i	$0.481224\pi$
$440$	−1.30110e6	−0.320390
$441$	−5.14767e6	−1.26042
$442$	590101.	0.143672
$443$	4.07935e6	0.987602	0.493801	−	0.869575i	$-0.335607\pi$
$443$	4.07935e6	0.987602	0.493801	+	0.869575i	$0.335607\pi$
$444$	−168103.	−0.0404685
$445$	−1.72963e6	−0.414050
$446$	966688.	0.230117
$447$	−343120.	−0.0812228
$448$	−799898.	−0.188295
$449$	737645.	0.172676	0.0863379	−	0.996266i	$-0.472484\pi$
$449$	737645.	0.172676	0.0863379	+	0.996266i	$0.472484\pi$
$450$	−1.13535e6	−0.264301
$451$	−2.27572e6	−0.526839
$452$	3.59566e6	0.827815
$453$	−48057.3	−0.0110031
$454$	−858635.	−0.195510
$455$	−2.21736e6	−0.502121
$456$	141540.	0.0318763
$457$	5.13171e6	1.14940	0.574700	−	0.818364i	$-0.305119\pi$
$457$	5.13171e6	1.14940	0.574700	+	0.818364i	$0.305119\pi$
$458$	3.85530e6	0.858806
$459$	533008.	0.118087
$460$	4.81912e6	1.06187
$461$	−5.79486e6	−1.26996	−0.634981	−	0.772528i	$-0.718992\pi$
$461$	−5.79486e6	−1.26996	−0.634981	+	0.772528i	$0.718992\pi$
$462$	−312730.	−0.0681655
$463$	−5.86480e6	−1.27145	−0.635727	−	0.771914i	$-0.719300\pi$
$463$	−5.86480e6	−1.27145	−0.635727	+	0.771914i	$0.719300\pi$
$464$	1.46183e6	0.315211
$465$	−81397.3	−0.0174573
$466$	6.01234e6	1.28256
$467$	6.82248e6	1.44760	0.723802	−	0.690007i	$-0.242393\pi$
$467$	6.82248e6	1.44760	0.723802	+	0.690007i	$0.242393\pi$
$468$	668507.	0.141088
$469$	6.40752e6	1.34511
$470$	−1.50141e6	−0.313512
$471$	455446.	0.0945986
$472$	2.77447e6	0.573225
$473$	2.28495e6	0.469595
$474$	−465508.	−0.0951658
$475$	2.01400e6	0.409568
$476$	2.66250e6	0.538608
$477$	43171.7	0.00868768
$478$	−2.00569e6	−0.401509
$479$	2.38213e6	0.474380	0.237190	−	0.971463i	$-0.423774\pi$
$479$	2.38213e6	0.474380	0.237190	+	0.971463i	$0.423774\pi$
$480$	−86733.4	−0.0171824
$481$	1.40841e6	0.277567
$482$	−5.12128e6	−1.00406
$483$	1.15831e6	0.225922
$484$	−1.03937e6	−0.201678
$485$	−4.99332e6	−0.963907
$486$	906782.	0.174146
$487$	−7.51429e6	−1.43571	−0.717853	−	0.696195i	$-0.754875\pi$
$487$	−7.51429e6	−1.43571	−0.717853	+	0.696195i	$0.754875\pi$
$488$	563782.	0.107167
$489$	−762545.	−0.144209
$490$	−5.59560e6	−1.05283
$491$	−9.46802e6	−1.77237	−0.886187	−	0.463328i	$-0.846655\pi$
$491$	−9.46802e6	−1.77237	−0.886187	+	0.463328i	$0.846655\pi$
$492$	−151703.	−0.0282541
$493$	−4.86577e6	−0.901643
$494$	−1.18587e6	−0.218634
$495$	4.90621e6	0.899980
$496$	246016.	0.0449013
$497$	1.46758e7	2.66508
$498$	−485335.	−0.0876937
$499$	−4.96201e6	−0.892085	−0.446042	−	0.895012i	$-0.647167\pi$
$499$	−4.96201e6	−0.892085	−0.446042	+	0.895012i	$0.647167\pi$
$500$	2.04500e6	0.365821
$501$	88495.1	0.0157516
$502$	7.17978e6	1.27160
$503$	−2.37337e6	−0.418259	−0.209129	−	0.977888i	$-0.567063\pi$
$503$	−2.37337e6	−0.418259	−0.209129	+	0.977888i	$0.567063\pi$
$504$	3.01627e6	0.528924
$505$	−3.39117e6	−0.591727
$506$	−5.69450e6	−0.988734
$507$	−440815.	−0.0761616
$508$	−435641.	−0.0748979
$509$	8.01464e6	1.37116	0.685582	−	0.727995i	$-0.259548\pi$
$509$	8.01464e6	1.37116	0.685582	+	0.727995i	$0.259548\pi$
$510$	288697.	0.0491492
$511$	1.41726e7	2.40102
$512$	262144.	0.0441942
$513$	−1.07113e6	−0.179701
$514$	−3.05783e6	−0.510511
$515$	9.39737e6	1.56131
$516$	152318.	0.0251842
$517$	1.77414e6	0.291918
$518$	6.35469e6	1.04057
$519$	127872.	0.0208380
$520$	726678.	0.117851
$521$	−1.11852e7	−1.80530	−0.902651	−	0.430373i	$-0.858382\pi$
$521$	−1.11852e7	−1.80530	−0.902651	+	0.430373i	$0.858382\pi$
$522$	−5.51228e6	−0.885432
$523$	−1.01033e7	−1.61514	−0.807570	−	0.589772i	$-0.799218\pi$
$523$	−1.01033e7	−1.61514	−0.807570	+	0.589772i	$0.799218\pi$
$524$	−158717.	−0.0252520
$525$	−296637.	−0.0469707
$526$	−771200.	−0.121535
$527$	−818877.	−0.128438
$528$	102488.	0.0159989
$529$	1.46554e7	2.27698
$530$	46928.4	0.00725682
$531$	−1.04620e7	−1.61020
$532$	−5.35056e6	−0.819635
$533$	1.27101e6	0.193790
$534$	136244.	0.0206759
$535$	6.31320e6	0.953598
$536$	−2.09988e6	−0.315706
$537$	70075.4	0.0104865
$538$	4.78401e6	0.712584
$539$	6.61204e6	0.980310
$540$	656372.	0.0968647
$541$	−2.72151e6	−0.399777	−0.199888	−	0.979819i	$-0.564058\pi$
$541$	−2.72151e6	−0.399777	−0.199888	+	0.979819i	$0.564058\pi$
$542$	1.99210e6	0.291281
$543$	−719997.	−0.104793
$544$	−872560.	−0.126415
$545$	6.33698e6	0.913884
$546$	174663.	0.0250737
$547$	−5.18727e6	−0.741260	−0.370630	−	0.928781i	$-0.620858\pi$
$547$	−5.18727e6	−0.741260	−0.370630	+	0.928781i	$0.620858\pi$
$548$	4.90420e6	0.697616
$549$	−2.12592e6	−0.301034
$550$	1.45833e6	0.205565
$551$	9.77825e6	1.37209
$552$	−379605.	−0.0530254
$553$	1.75973e7	2.44700
$554$	−786286.	−0.108844
$555$	689043.	0.0949541
$556$	3.90580e6	0.535825
$557$	−1.08410e7	−1.48058	−0.740292	−	0.672286i	$-0.765313\pi$
$557$	−1.08410e7	−1.48058	−0.740292	+	0.672286i	$0.765313\pi$
$558$	−927680.	−0.126128
$559$	−1.27617e6	−0.172734
$560$	3.27873e6	0.441810
$561$	−341138.	−0.0457639
$562$	−6.05708e6	−0.808951
$563$	−1.06532e7	−1.41647	−0.708236	−	0.705975i	$-0.750509\pi$
$563$	−1.06532e7	−1.41647	−0.708236	+	0.705975i	$0.750509\pi$
$564$	118267.	0.0156555
$565$	−1.47384e7	−1.94236
$566$	740856.	0.0972060
$567$	−1.12946e7	−1.47541
$568$	−4.80958e6	−0.625513
$569$	1.08514e7	1.40509	0.702544	−	0.711640i	$-0.252047\pi$
$569$	1.08514e7	1.40509	0.702544	+	0.711640i	$0.252047\pi$
$570$	−580165.	−0.0747936
$571$	9.81386e6	1.25965	0.629825	−	0.776737i	$-0.283127\pi$
$571$	9.81386e6	1.25965	0.629825	+	0.776737i	$0.283127\pi$
$572$	−858678.	−0.109734
$573$	−179008.	−0.0227765
$574$	5.73475e6	0.726498
$575$	−5.40147e6	−0.681306
$576$	−988496.	−0.124142
$577$	6.82499e6	0.853419	0.426709	−	0.904389i	$-0.359673\pi$
$577$	6.82499e6	0.853419	0.426709	+	0.904389i	$0.359673\pi$
$578$	−2.77506e6	−0.345504
$579$	314573.	0.0389965
$580$	−5.99194e6	−0.739601
$581$	1.83468e7	2.25487
$582$	393327.	0.0481333
$583$	−55452.9	−0.00675698
$584$	−4.64466e6	−0.563536
$585$	−2.74017e6	−0.331045
$586$	7.77204e6	0.934955
$587$	4.01700e6	0.481179	0.240589	−	0.970627i	$-0.422659\pi$
$587$	4.01700e6	0.481179	0.240589	+	0.970627i	$0.422659\pi$
$588$	440769.	0.0525736
$589$	1.64561e6	0.195452
$590$	−1.13724e7	−1.34500
$591$	682245.	0.0803474
$592$	−2.08257e6	−0.244228
$593$	5.65794e6	0.660727	0.330363	−	0.943854i	$-0.392829\pi$
$593$	5.65794e6	0.660727	0.330363	+	0.943854i	$0.392829\pi$
$594$	−775601.	−0.0901928
$595$	−1.09134e7	−1.26377
$596$	−4.25081e6	−0.490180
$597$	−147656.	−0.0169557
$598$	3.18044e6	0.363692
$599$	8.75561e6	0.997056	0.498528	−	0.866874i	$-0.333874\pi$
$599$	8.75561e6	0.997056	0.498528	+	0.866874i	$0.333874\pi$
$600$	97214.4	0.0110243
$601$	−4.88494e6	−0.551662	−0.275831	−	0.961206i	$-0.588953\pi$
$601$	−4.88494e6	−0.551662	−0.275831	+	0.961206i	$0.588953\pi$
$602$	−5.75800e6	−0.647561
$603$	7.91826e6	0.886823
$604$	−595366.	−0.0664036
$605$	4.26032e6	0.473210
$606$	267124.	0.0295483
$607$	−1.43042e7	−1.57577	−0.787884	−	0.615823i	$-0.788824\pi$
$607$	−1.43042e7	−1.57577	−0.787884	+	0.615823i	$0.788824\pi$
$608$	1.75350e6	0.192374
$609$	−1.44021e6	−0.157356
$610$	−2.31091e6	−0.251454
$611$	−990875.	−0.107378
$612$	3.29026e6	0.355101
$613$	−301354.	−0.0323911	−0.0161956	−	0.999869i	$-0.505155\pi$
$613$	−301354.	−0.0323911	−0.0161956	+	0.999869i	$0.505155\pi$
$614$	−5.25723e6	−0.562776
$615$	621822.	0.0662946
$616$	−3.87431e6	−0.411379
$617$	−7.57845e6	−0.801433	−0.400717	−	0.916202i	$-0.631239\pi$
$617$	−7.57845e6	−0.801433	−0.400717	+	0.916202i	$0.631239\pi$
$618$	−740236.	−0.0779649
$619$	3.84826e6	0.403681	0.201840	−	0.979418i	$-0.435308\pi$
$619$	3.84826e6	0.403681	0.201840	+	0.979418i	$0.435308\pi$
$620$	−1.00840e6	−0.105355
$621$	2.87273e6	0.298927
$622$	6.15382e6	0.637777
$623$	−5.15034e6	−0.531638
$624$	−57240.9	−0.00588498
$625$	−1.20577e7	−1.23471
$626$	−1.09261e7	−1.11437
$627$	685551.	0.0696419
$628$	5.64237e6	0.570903
$629$	6.93195e6	0.698600
$630$	−1.23635e7	−1.24105
$631$	6.96320e6	0.696202	0.348101	−	0.937457i	$-0.386827\pi$
$631$	6.96320e6	0.696202	0.348101	+	0.937457i	$0.386827\pi$
$632$	−5.76702e6	−0.574326
$633$	−262080.	−0.0259971
$634$	6.30067e6	0.622535
$635$	1.78567e6	0.175738
$636$	−3696.58	−0.000362374 0
$637$	−3.69289e6	−0.360594
$638$	7.08037e6	0.688659
$639$	1.81360e7	1.75707
$640$	−1.07451e6	−0.103696
$641$	−4.32890e6	−0.416134	−0.208067	−	0.978115i	$-0.566717\pi$
$641$	−4.32890e6	−0.416134	−0.208067	+	0.978115i	$0.566717\pi$
$642$	−497295.	−0.0476186
$643$	−1.45356e7	−1.38645	−0.693225	−	0.720721i	$-0.743811\pi$
$643$	−1.45356e7	−1.38645	−0.693225	+	0.720721i	$0.743811\pi$
$644$	1.43500e7	1.36344
$645$	−624344.	−0.0590914
$646$	−5.83661e6	−0.550274
$647$	−1.29585e7	−1.21701	−0.608504	−	0.793551i	$-0.708230\pi$
$647$	−1.29585e7	−1.21701	−0.608504	+	0.793551i	$0.708230\pi$
$648$	3.70149e6	0.346290
$649$	1.34382e7	1.25236
$650$	−814491.	−0.0756141
$651$	−242378.	−0.0224151
$652$	−9.44692e6	−0.870305
$653$	−1.48643e7	−1.36415	−0.682076	−	0.731281i	$-0.738923\pi$
$653$	−1.48643e7	−1.36415	−0.682076	+	0.731281i	$0.738923\pi$
$654$	−499168.	−0.0456354
$655$	650573.	0.0592506
$656$	−1.87940e6	−0.170514
$657$	1.75142e7	1.58298
$658$	−4.47077e6	−0.402548
$659$	1.37201e6	0.123067	0.0615336	−	0.998105i	$-0.480401\pi$
$659$	1.37201e6	0.123067	0.0615336	+	0.998105i	$0.480401\pi$
$660$	−420094.	−0.0375393
$661$	6.98650e6	0.621951	0.310975	−	0.950418i	$-0.399344\pi$
$661$	6.98650e6	0.621951	0.310975	+	0.950418i	$0.399344\pi$
$662$	4.69257e6	0.416165
$663$	190529.	0.0168336
$664$	−6.01266e6	−0.529232
$665$	2.19316e7	1.92317
$666$	7.85298e6	0.686039
$667$	−2.62248e7	−2.28243
$668$	1.09634e6	0.0950611
$669$	312120.	0.0269623
$670$	8.60729e6	0.740763
$671$	2.73068e6	0.234134
$672$	−258268.	−0.0220621
$673$	−1.98838e7	−1.69224	−0.846118	−	0.532995i	$-0.821066\pi$
$673$	−1.98838e7	−1.69224	−0.846118	+	0.532995i	$0.821066\pi$
$674$	1.21141e7	1.02717
$675$	−735689.	−0.0621491
$676$	−5.46111e6	−0.459636
$677$	−1.56443e6	−0.131185	−0.0655925	−	0.997846i	$-0.520894\pi$
$677$	−1.56443e6	−0.131185	−0.0655925	+	0.997846i	$0.520894\pi$
$678$	1.16095e6	0.0969930
$679$	−1.48687e7	−1.23765
$680$	3.57657e6	0.296616
$681$	−277232.	−0.0229074
$682$	1.19158e6	0.0980984
$683$	−6.99860e6	−0.574063	−0.287032	−	0.957921i	$-0.592668\pi$
$683$	−6.99860e6	−0.574063	−0.287032	+	0.957921i	$0.592668\pi$
$684$	−6.61211e6	−0.540380
$685$	−2.01020e7	−1.63687
$686$	−3.53332e6	−0.286664
$687$	1.24478e6	0.100624
$688$	1.88702e6	0.151987
$689$	30971.0	0.00248546
$690$	1.55598e6	0.124417
$691$	−3.27317e6	−0.260779	−0.130390	−	0.991463i	$-0.541623\pi$
$691$	−3.27317e6	−0.260779	−0.130390	+	0.991463i	$0.541623\pi$
$692$	1.58416e6	0.125758
$693$	1.46093e7	1.15557
$694$	2.34216e6	0.184594
$695$	−1.60096e7	−1.25724
$696$	471989.	0.0369325
$697$	6.25569e6	0.487745
$698$	2.73234e6	0.212274
$699$	1.94124e6	0.150275
$700$	−3.67494e6	−0.283469
$701$	−1.71951e7	−1.32163	−0.660815	−	0.750549i	$-0.729789\pi$
$701$	−1.71951e7	−1.32163	−0.660815	+	0.750549i	$0.729789\pi$
$702$	433181.	0.0331762
$703$	−1.39304e7	−1.06311
$704$	1.26970e6	0.0965535
$705$	−484769.	−0.0367335
$706$	−63256.4	−0.00477631
$707$	−1.00979e7	−0.759774
$708$	895809.	0.0671633
$709$	326145.	0.0243666	0.0121833	−	0.999926i	$-0.496122\pi$
$709$	326145.	0.0243666	0.0121833	+	0.999926i	$0.496122\pi$
$710$	1.97142e7	1.46768
$711$	2.17464e7	1.61329
$712$	1.68788e6	0.124779
$713$	−4.41346e6	−0.325129
$714$	859657.	0.0631073
$715$	3.51967e6	0.257476
$716$	868141.	0.0632860
$717$	−647590.	−0.0470438
$718$	−1.12362e7	−0.813408
$719$	4.12077e6	0.297274	0.148637	−	0.988892i	$-0.452511\pi$
$719$	4.12077e6	0.297274	0.148637	+	0.988892i	$0.452511\pi$
$720$	4.05178e6	0.291283
$721$	2.79827e7	2.00471
$722$	1.82484e6	0.130281
$723$	−1.65354e6	−0.117643
$724$	−8.91981e6	−0.632425
$725$	6.71602e6	0.474534
$726$	−335588.	−0.0236301
$727$	2.06821e7	1.45130	0.725652	−	0.688062i	$-0.241538\pi$
$727$	2.06821e7	1.45130	0.725652	+	0.688062i	$0.241538\pi$
$728$	2.16384e6	0.151320
$729$	−1.37613e7	−0.959051
$730$	1.90382e7	1.32226
$731$	−6.28105e6	−0.434750
$732$	182031.	0.0125565
$733$	7.82914e6	0.538213	0.269106	−	0.963110i	$-0.413272\pi$
$733$	7.82914e6	0.538213	0.269106	+	0.963110i	$0.413272\pi$
$734$	4.87538e6	0.334017
$735$	−1.80668e6	−0.123357
$736$	−4.70280e6	−0.320009
$737$	−1.01708e7	−0.689741
$738$	7.08687e6	0.478976
$739$	2.44001e7	1.64354	0.821770	−	0.569819i	$-0.192987\pi$
$739$	2.44001e7	1.64354	0.821770	+	0.569819i	$0.192987\pi$
$740$	8.53633e6	0.573049
$741$	−382887.	−0.0256168
$742$	139740.	0.00931772
$743$	4.47508e6	0.297392	0.148696	−	0.988883i	$-0.452492\pi$
$743$	4.47508e6	0.297392	0.148696	+	0.988883i	$0.452492\pi$
$744$	79432.6	0.00526098
$745$	1.74238e7	1.15014
$746$	−1.66079e7	−1.09262
$747$	2.26726e7	1.48662
$748$	−4.22625e6	−0.276186
$749$	1.87989e7	1.22441
$750$	660282.	0.0428624
$751$	−2.22861e7	−1.44190	−0.720948	−	0.692990i	$-0.756293\pi$
$751$	−2.22861e7	−1.44190	−0.720948	+	0.692990i	$0.756293\pi$
$752$	1.46517e6	0.0944808
$753$	2.31818e6	0.148991
$754$	−3.95446e6	−0.253314
$755$	2.44037e6	0.155807
$756$	1.95449e6	0.124374
$757$	4.47201e6	0.283637	0.141819	−	0.989893i	$-0.454705\pi$
$757$	4.47201e6	0.283637	0.141819	+	0.989893i	$0.454705\pi$
$758$	1.44722e7	0.914876
$759$	−1.83862e6	−0.115848
$760$	−7.18747e6	−0.451380
$761$	2.92597e7	1.83150	0.915752	−	0.401743i	$-0.131596\pi$
$761$	2.92597e7	1.83150	0.915752	+	0.401743i	$0.131596\pi$
$762$	−140658.	−0.00877560
$763$	1.88697e7	1.17342
$764$	−2.21767e6	−0.137456
$765$	−1.34866e7	−0.833198
$766$	−8.27144e6	−0.509341
$767$	−7.50535e6	−0.460662
$768$	84639.9	0.00517812
$769$	−9.16244e6	−0.558721	−0.279361	−	0.960186i	$-0.590122\pi$
$769$	−9.16244e6	−0.558721	−0.279361	+	0.960186i	$0.590122\pi$
$770$	1.58805e7	0.965248
$771$	−987298.	−0.0598153
$772$	3.89714e6	0.235344
$773$	1.74646e7	1.05126	0.525630	−	0.850714i	$-0.323830\pi$
$773$	1.74646e7	1.05126	0.525630	+	0.850714i	$0.323830\pi$
$774$	−7.11561e6	−0.426933
$775$	1.13026e6	0.0675966
$776$	4.87279e6	0.290485
$777$	2.05177e6	0.121921
$778$	−2.21186e7	−1.31011
$779$	−1.25714e7	−0.742234
$780$	234627.	0.0138083
$781$	−2.32952e7	−1.36659
$782$	1.56535e7	0.915366
$783$	−3.57186e6	−0.208205
$784$	5.46054e6	0.317282
$785$	−2.31277e7	−1.33955
$786$	−51246.0	−0.00295872
$787$	3.32773e7	1.91519	0.957595	−	0.288118i	$-0.0930296\pi$
$787$	3.32773e7	1.91519	0.957595	+	0.288118i	$0.0930296\pi$
$788$	8.45211e6	0.484897
$789$	−249002.	−0.0142400
$790$	2.36387e7	1.34758
$791$	−4.38868e7	−2.49398
$792$	−4.78778e6	−0.271220
$793$	−1.52511e6	−0.0861230
$794$	−236078.	−0.0132894
$795$	15152.0	0.000850263 0
$796$	−1.82927e6	−0.102328
$797$	−3.67811e6	−0.205106	−0.102553	−	0.994728i	$-0.532701\pi$
$797$	−3.67811e6	−0.205106	−0.102553	+	0.994728i	$0.532701\pi$
$798$	−1.72757e6	−0.0960345
$799$	−4.87690e6	−0.270257
$800$	1.20436e6	0.0665320
$801$	−6.36467e6	−0.350506
$802$	1.78379e7	0.979280
$803$	−2.24964e7	−1.23119
$804$	−678001.	−0.0369905
$805$	−5.88196e7	−3.19913
$806$	−665510.	−0.0360842
$807$	1.54464e6	0.0834917
$808$	3.30932e6	0.178324
$809$	−1.05623e7	−0.567400	−0.283700	−	0.958913i	$-0.591562\pi$
$809$	−1.05623e7	−0.567400	−0.283700	+	0.958913i	$0.591562\pi$
$810$	−1.51722e7	−0.812523
$811$	5.67117e6	0.302775	0.151388	−	0.988474i	$-0.451626\pi$
$811$	5.67117e6	0.302775	0.151388	+	0.988474i	$0.451626\pi$
$812$	−1.78423e7	−0.949644
$813$	643200.	0.0341287
$814$	−1.00869e7	−0.533578
$815$	3.87223e7	2.04206
$816$	−281729.	−0.0148117
$817$	1.26224e7	0.661587
$818$	1.77787e7	0.929003
$819$	−8.15944e6	−0.425060
$820$	7.70355e6	0.400089
$821$	−1.77771e7	−0.920458	−0.460229	−	0.887800i	$-0.652233\pi$
$821$	−1.77771e7	−0.920458	−0.460229	+	0.887800i	$0.652233\pi$
$822$	1.58345e6	0.0817380
$823$	−1.91512e7	−0.985592	−0.492796	−	0.870145i	$-0.664025\pi$
$823$	−1.91512e7	−0.985592	−0.492796	+	0.870145i	$0.664025\pi$
$824$	−9.17055e6	−0.470519
$825$	470859.	0.0240855
$826$	−3.38637e7	−1.72697
$827$	1.30294e7	0.662463	0.331232	−	0.943549i	$-0.392536\pi$
$827$	1.30294e7	0.662463	0.331232	+	0.943549i	$0.392536\pi$
$828$	1.77334e7	0.898908
$829$	8.55334e6	0.432264	0.216132	−	0.976364i	$-0.430656\pi$
$829$	8.55334e6	0.432264	0.216132	+	0.976364i	$0.430656\pi$
$830$	2.46455e7	1.24177
$831$	−253873.	−0.0127530
$832$	−709138.	−0.0355159
$833$	−1.81757e7	−0.907567
$834$	1.26109e6	0.0627813
$835$	−4.49382e6	−0.223048
$836$	8.49306e6	0.420290
$837$	−601121.	−0.0296584
$838$	−1.47722e7	−0.726666
$839$	−3.93700e7	−1.93091	−0.965453	−	0.260579i	$-0.916087\pi$
$839$	−3.93700e7	−1.93091	−0.965453	+	0.260579i	$0.916087\pi$
$840$	1.05862e6	0.0517658
$841$	1.20960e7	0.589728
$842$	−1.65838e7	−0.806127
$843$	−1.95568e6	−0.0947828
$844$	−3.24682e6	−0.156892
$845$	2.23847e7	1.07848
$846$	−5.52488e6	−0.265398
$847$	1.26860e7	0.607599
$848$	−45795.7	−0.00218693
$849$	239204.	0.0113894
$850$	−4.00877e6	−0.190311
$851$	3.73608e7	1.76845
$852$	−1.55290e6	−0.0732898
$853$	3.45441e7	1.62555	0.812777	−	0.582574i	$-0.197955\pi$
$853$	3.45441e7	1.62555	0.812777	+	0.582574i	$0.197955\pi$
$854$	−6.88123e6	−0.322865
$855$	2.71026e7	1.26793
$856$	−6.16082e6	−0.287378
$857$	1.99270e7	0.926810	0.463405	−	0.886147i	$-0.346628\pi$
$857$	1.99270e7	0.926810	0.463405	+	0.886147i	$0.346628\pi$
$858$	−277246.	−0.0128572
$859$	−1.02562e7	−0.474248	−0.237124	−	0.971479i	$-0.576205\pi$
$859$	−1.02562e7	−0.474248	−0.237124	+	0.971479i	$0.576205\pi$
$860$	−7.73479e6	−0.356617
$861$	1.85161e6	0.0851220
$862$	−7.31226e6	−0.335184
$863$	−2.78227e7	−1.27166	−0.635831	−	0.771828i	$-0.719343\pi$
$863$	−2.78227e7	−1.27166	−0.635831	+	0.771828i	$0.719343\pi$
$864$	−640528.	−0.0291913
$865$	−6.49337e6	−0.295074
$866$	−2.69739e7	−1.22222
$867$	−896001.	−0.0404819
$868$	−3.00274e6	−0.135275
$869$	−2.79326e7	−1.25476
$870$	−1.93465e6	−0.0866572
$871$	5.68049e6	0.253712
$872$	−6.18402e6	−0.275410
$873$	−1.83744e7	−0.815976
$874$	−3.14573e7	−1.39297
$875$	−2.49602e7	−1.10212
$876$	−1.49965e6	−0.0660282
$877$	1.05256e7	0.462112	0.231056	−	0.972940i	$-0.425782\pi$
$877$	1.05256e7	0.462112	0.231056	+	0.972940i	$0.425782\pi$
$878$	1.90437e6	0.0833708
$879$	2.50940e6	0.109546
$880$	−5.20440e6	−0.226550
$881$	−1.08572e7	−0.471280	−0.235640	−	0.971840i	$-0.575719\pi$
$881$	−1.08572e7	−0.471280	−0.235640	+	0.971840i	$0.575719\pi$
$882$	−2.05907e7	−0.891249
$883$	2.05110e6	0.0885290	0.0442645	−	0.999020i	$-0.485906\pi$
$883$	2.05110e6	0.0885290	0.0442645	+	0.999020i	$0.485906\pi$
$884$	2.36040e6	0.101591
$885$	−3.67186e6	−0.157590
$886$	1.63174e7	0.698340
$887$	1.58350e7	0.675785	0.337892	−	0.941185i	$-0.390286\pi$
$887$	1.58350e7	0.675785	0.337892	+	0.941185i	$0.390286\pi$
$888$	−672411.	−0.0286156
$889$	5.31721e6	0.225647
$890$	−6.91851e6	−0.292777
$891$	1.79282e7	0.756558
$892$	3.86675e6	0.162717
$893$	9.80061e6	0.411267
$894$	−1.37248e6	−0.0574332
$895$	−3.55846e6	−0.148492
$896$	−3.19959e6	−0.133145
$897$	1.02689e6	0.0426129
$898$	2.95058e6	0.122100
$899$	5.48757e6	0.226454
$900$	−4.54141e6	−0.186889
$901$	152433.	0.00625559
$902$	−9.10289e6	−0.372531
$903$	−1.85912e6	−0.0758730
$904$	1.43827e7	0.585353
$905$	3.65617e7	1.48390
$906$	−192229.	−0.00778034
$907$	−1.39215e7	−0.561912	−0.280956	−	0.959721i	$-0.590652\pi$
$907$	−1.39215e7	−0.561912	−0.280956	+	0.959721i	$0.590652\pi$
$908$	−3.43454e6	−0.138246
$909$	−1.24788e7	−0.500914
$910$	−8.86945e6	−0.355053
$911$	−381409.	−0.0152263	−0.00761316	−	0.999971i	$-0.502423\pi$
$911$	−381409.	−0.0152263	−0.00761316	+	0.999971i	$0.502423\pi$
$912$	566161.	0.0225400
$913$	−2.91223e7	−1.15624
$914$	2.05268e7	0.812749
$915$	−746136.	−0.0294622
$916$	1.54212e7	0.607267
$917$	1.93722e6	0.0760775
$918$	2.13203e6	0.0835002
$919$	−2.12996e7	−0.831922	−0.415961	−	0.909382i	$-0.636555\pi$
$919$	−2.12996e7	−0.831922	−0.415961	+	0.909382i	$0.636555\pi$
$920$	1.92765e7	0.750858
$921$	−1.69743e6	−0.0659391
$922$	−2.31794e7	−0.897998
$923$	1.30106e7	0.502682
$924$	−1.25092e6	−0.0482003
$925$	−9.56787e6	−0.367672
$926$	−2.34592e7	−0.899054
$927$	3.45804e7	1.32169
$928$	5.84731e6	0.222888
$929$	1.77700e7	0.675534	0.337767	−	0.941230i	$-0.390328\pi$
$929$	1.77700e7	0.675534	0.337767	+	0.941230i	$0.390328\pi$
$930$	−325589.	−0.0123442
$931$	3.65259e7	1.38110
$932$	2.40494e7	0.906909
$933$	1.98692e6	0.0747267
$934$	2.72899e7	1.02361
$935$	1.73231e7	0.648033
$936$	2.67403e6	0.0997645
$937$	1.58592e7	0.590109	0.295054	−	0.955480i	$-0.404662\pi$
$937$	1.58592e7	0.590109	0.295054	+	0.955480i	$0.404662\pi$
$938$	2.56301e7	0.951136
$939$	−3.52778e6	−0.130568
$940$	−6.00564e6	−0.221687
$941$	3.83125e7	1.41048	0.705239	−	0.708970i	$-0.250840\pi$
$941$	3.83125e7	1.41048	0.705239	+	0.708970i	$0.250840\pi$
$942$	1.82178e6	0.0668913
$943$	3.37160e7	1.23469
$944$	1.10979e7	0.405331
$945$	−8.01133e6	−0.291827
$946$	9.13980e6	0.332054
$947$	−3.35059e7	−1.21408	−0.607039	−	0.794672i	$-0.707643\pi$
$947$	−3.35059e7	−1.21408	−0.607039	+	0.794672i	$0.707643\pi$
$948$	−1.86203e6	−0.0672924
$949$	1.25645e7	0.452876
$950$	8.05602e6	0.289609
$951$	2.03433e6	0.0729409
$952$	1.06500e7	0.380853
$953$	1.08332e7	0.386388	0.193194	−	0.981161i	$-0.438115\pi$
$953$	1.08332e7	0.386388	0.193194	+	0.981161i	$0.438115\pi$
$954$	172687.	0.00614312
$955$	9.09011e6	0.322523
$956$	−8.02277e6	−0.283909
$957$	2.28608e6	0.0806885
$958$	9.52852e6	0.335438
$959$	−5.98581e7	−2.10173
$960$	−346934.	−0.0121498
$961$	923521.	0.0322581
$962$	5.63366e6	0.196270
$963$	2.32313e7	0.807249
$964$	−2.04851e7	−0.709979
$965$	−1.59741e7	−0.552204
$966$	4.63325e6	0.159751
$967$	5.10890e7	1.75696	0.878479	−	0.477781i	$-0.158559\pi$
$967$	5.10890e7	1.75696	0.878479	+	0.477781i	$0.158559\pi$
$968$	−4.15749e6	−0.142608
$969$	−1.88450e6	−0.0644742
$970$	−1.99733e7	−0.681585
$971$	2.87512e7	0.978607	0.489304	−	0.872113i	$-0.337251\pi$
$971$	2.87512e7	0.978607	0.489304	+	0.872113i	$0.337251\pi$
$972$	3.62713e6	0.123139
$973$	−4.76722e7	−1.61429
$974$	−3.00572e7	−1.01520
$975$	−262979.	−0.00885952
$976$	2.25513e6	0.0757786
$977$	2.66635e6	0.0893676	0.0446838	−	0.999001i	$-0.485772\pi$
$977$	2.66635e6	0.0893676	0.0446838	+	0.999001i	$0.485772\pi$
$978$	−3.05018e6	−0.101971
$979$	8.17525e6	0.272612
$980$	−2.23824e7	−0.744461
$981$	2.33188e7	0.773630
$982$	−3.78721e7	−1.25326
$983$	1.08397e7	0.357794	0.178897	−	0.983868i	$-0.442747\pi$
$983$	1.08397e7	0.357794	0.178897	+	0.983868i	$0.442747\pi$
$984$	−606813.	−0.0199787
$985$	−3.46447e7	−1.13775
$986$	−1.94631e7	−0.637558
$987$	−1.44350e6	−0.0471656
$988$	−4.74347e6	−0.154598
$989$	−3.38527e7	−1.10053
$990$	1.96248e7	0.636382
$991$	3.30389e7	1.06867	0.534333	−	0.845274i	$-0.320563\pi$
$991$	3.30389e7	1.06867	0.534333	+	0.845274i	$0.320563\pi$
$992$	984064.	0.0317500
$993$	1.51512e6	0.0487610
$994$	5.87032e7	1.88450
$995$	7.49805e6	0.240099
$996$	−1.94134e6	−0.0620088
$997$	−4.66511e7	−1.48636	−0.743181	−	0.669091i	$-0.766684\pi$
$997$	−4.66511e7	−1.48636	−0.743181	+	0.669091i	$0.766684\pi$
$998$	−1.98480e7	−0.630799
$999$	5.08860e6	0.161319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	62.6.a.b.1.2	✓	2
3.2	odd	2		558.6.a.c.1.2		2
4.3	odd	2		496.6.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.6.a.b.1.2	✓	2	1.1	even	1	trivial
496.6.a.a.1.1		2	4.3	odd	2
558.6.a.c.1.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 62 = 2 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	62.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +1.29150 q^{3} +16.0000 q^{4} -65.5830 q^{5} +5.16601 q^{6} -195.288 q^{7} +64.0000 q^{8} -241.332 q^{9} -262.332 q^{10} +309.984 q^{11} +20.6640 q^{12} -173.129 q^{13} -781.150 q^{14} -84.7006 q^{15} +256.000 q^{16} -852.110 q^{17} -965.328 q^{18} +1712.40 q^{19} -1049.33 q^{20} -252.214 q^{21} +1239.94 q^{22} -4592.57 q^{23} +82.6562 q^{24} +1176.13 q^{25} -692.518 q^{26} -625.516 q^{27} -3124.60 q^{28} +5710.27 q^{29} -338.802 q^{30} +961.000 q^{31} +1024.00 q^{32} +400.345 q^{33} -3408.44 q^{34} +12807.5 q^{35} -3861.31 q^{36} -8135.04 q^{37} +6849.59 q^{38} -223.597 q^{39} -4197.31 q^{40} -7341.41 q^{41} -1008.86 q^{42} +7371.18 q^{43} +4959.75 q^{44} +15827.3 q^{45} -18370.3 q^{46} +5723.32 q^{47} +330.625 q^{48} +21330.2 q^{49} +4704.52 q^{50} -1100.50 q^{51} -2770.07 q^{52} -178.889 q^{53} -2502.06 q^{54} -20329.7 q^{55} -12498.4 q^{56} +2211.57 q^{57} +22841.1 q^{58} +43351.1 q^{59} -1355.21 q^{60} +8809.09 q^{61} +3844.00 q^{62} +47129.1 q^{63} +4096.00 q^{64} +11354.3 q^{65} +1601.38 q^{66} -32810.7 q^{67} -13633.8 q^{68} -5931.32 q^{69} +51230.2 q^{70} -75149.7 q^{71} -15445.2 q^{72} -72572.8 q^{73} -32540.2 q^{74} +1518.98 q^{75} +27398.4 q^{76} -60536.1 q^{77} -894.388 q^{78} -90109.7 q^{79} -16789.2 q^{80} +57835.8 q^{81} -29365.6 q^{82} -93947.8 q^{83} -4035.43 q^{84} +55883.9 q^{85} +29484.7 q^{86} +7374.82 q^{87} +19839.0 q^{88} +26373.1 q^{89} +63309.1 q^{90} +33810.0 q^{91} -73481.2 q^{92} +1241.13 q^{93} +22893.3 q^{94} -112304. q^{95} +1322.50 q^{96} +76137.4 q^{97} +85320.9 q^{98} -74809.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} - 8 q^{3} + 32 q^{4} - 110 q^{5} - 32 q^{6} - 126 q^{7} + 128 q^{8} - 398 q^{9} - 440 q^{10} - 396 q^{11} - 128 q^{12} - 632 q^{13} - 504 q^{14} + 328 q^{15} + 512 q^{16} - 720 q^{17} - 1592 q^{18}+ \cdots + 35796 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 - 8 * q^3 + 32 * q^4 - 110 * q^5 - 32 * q^6 - 126 * q^7 + 128 * q^8 - 398 * q^9 - 440 * q^10 - 396 * q^11 - 128 * q^12 - 632 * q^13 - 504 * q^14 + 328 * q^15 + 512 * q^16 - 720 * q^17 - 1592 * q^18 - 438 * q^19 - 1760 * q^20 - 896 * q^21 - 1584 * q^22 - 560 * q^23 - 512 * q^24 + 24 * q^25 - 2528 * q^26 + 3088 * q^27 - 2016 * q^28 + 4404 * q^29 + 1312 * q^30 + 1922 * q^31 + 2048 * q^32 + 6960 * q^33 - 2880 * q^34 + 9730 * q^35 - 6368 * q^36 - 8100 * q^37 - 1752 * q^38 + 4040 * q^39 - 7040 * q^40 + 1742 * q^41 - 3584 * q^42 - 8572 * q^43 - 6336 * q^44 + 22786 * q^45 - 2240 * q^46 + 26792 * q^47 - 2048 * q^48 + 9324 * q^49 + 96 * q^50 - 2328 * q^51 - 10112 * q^52 + 33656 * q^53 + 12352 * q^54 + 11028 * q^55 - 8064 * q^56 + 22192 * q^57 + 17616 * q^58 + 9478 * q^59 + 5248 * q^60 + 26328 * q^61 + 7688 * q^62 + 36274 * q^63 + 8192 * q^64 + 31736 * q^65 + 27840 * q^66 - 60224 * q^67 - 11520 * q^68 - 43400 * q^69 + 38920 * q^70 - 152490 * q^71 - 25472 * q^72 - 32860 * q^73 - 32400 * q^74 + 12224 * q^75 - 7008 * q^76 - 109452 * q^77 + 16160 * q^78 - 163276 * q^79 - 28160 * q^80 + 61402 * q^81 + 6968 * q^82 - 130324 * q^83 - 14336 * q^84 + 50016 * q^85 - 34288 * q^86 + 19512 * q^87 - 25344 * q^88 - 39908 * q^89 + 91144 * q^90 + 2016 * q^91 - 8960 * q^92 - 7688 * q^93 + 107168 * q^94 - 16790 * q^95 - 8192 * q^96 + 11034 * q^97 + 37296 * q^98 + 35796 * q^99

Introduction

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