LMFDB - Embedded newform 65.6.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,6,Mod(1,65)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("65.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 65 = 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	65.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$10.4249482878$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 $ x^6 - 163x^4 - 8x^3 + 6120x^2 + 6624x - 19440
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.34530$ of defining polynomial
Character	$\chi$	$=$	65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.34530 q^{2} -19.6439 q^{3} -30.1902 q^{4} -25.0000 q^{5} -26.4269 q^{6} +48.6530 q^{7} -83.6645 q^{8} +142.882 q^{9} -33.6325 q^{10} +283.440 q^{11} +593.052 q^{12} +169.000 q^{13} +65.4530 q^{14} +491.097 q^{15} +853.531 q^{16} +789.522 q^{17} +192.219 q^{18} +83.3795 q^{19} +754.754 q^{20} -955.734 q^{21} +381.312 q^{22} -1935.13 q^{23} +1643.49 q^{24} +625.000 q^{25} +227.356 q^{26} +1966.71 q^{27} -1468.84 q^{28} +222.752 q^{29} +660.673 q^{30} -2789.48 q^{31} +3825.52 q^{32} -5567.86 q^{33} +1062.15 q^{34} -1216.33 q^{35} -4313.62 q^{36} +8369.56 q^{37} +112.170 q^{38} -3319.81 q^{39} +2091.61 q^{40} +1218.51 q^{41} -1285.75 q^{42} +5638.89 q^{43} -8557.10 q^{44} -3572.04 q^{45} -2603.34 q^{46} +17776.2 q^{47} -16766.7 q^{48} -14439.9 q^{49} +840.813 q^{50} -15509.3 q^{51} -5102.14 q^{52} +10834.4 q^{53} +2645.82 q^{54} -7086.00 q^{55} -4070.53 q^{56} -1637.90 q^{57} +299.668 q^{58} -5363.36 q^{59} -14826.3 q^{60} -18670.7 q^{61} -3752.69 q^{62} +6951.63 q^{63} -22166.5 q^{64} -4225.00 q^{65} -7490.44 q^{66} +13985.1 q^{67} -23835.8 q^{68} +38013.5 q^{69} -1636.32 q^{70} +50969.5 q^{71} -11954.1 q^{72} +42394.9 q^{73} +11259.6 q^{74} -12277.4 q^{75} -2517.24 q^{76} +13790.2 q^{77} -4466.15 q^{78} +106279. q^{79} -21338.3 q^{80} -73354.1 q^{81} +1639.26 q^{82} +75513.6 q^{83} +28853.8 q^{84} -19738.1 q^{85} +7586.01 q^{86} -4375.71 q^{87} -23713.8 q^{88} +77017.8 q^{89} -4805.47 q^{90} +8222.36 q^{91} +58422.0 q^{92} +54796.2 q^{93} +23914.4 q^{94} -2084.49 q^{95} -75148.0 q^{96} +126702. q^{97} -19426.0 q^{98} +40498.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 38 q^{3} + 134 q^{4} - 150 q^{5} + 318 q^{6} + 220 q^{7} + 24 q^{8} + 518 q^{9} - 170 q^{11} + 2238 q^{12} + 1014 q^{13} - 1440 q^{14} - 950 q^{15} + 3506 q^{16} + 728 q^{17} + 7788 q^{18} + 1218 q^{19}+ \cdots - 32270 q^{99}+O(q^{100}) $ 6 * q + 38 * q^3 + 134 * q^4 - 150 * q^5 + 318 * q^6 + 220 * q^7 + 24 * q^8 + 518 * q^9 - 170 * q^11 + 2238 * q^12 + 1014 * q^13 - 1440 * q^14 - 950 * q^15 + 3506 * q^16 + 728 * q^17 + 7788 * q^18 + 1218 * q^19 - 3350 * q^20 - 396 * q^21 + 5154 * q^22 + 8954 * q^23 + 12618 * q^24 + 3750 * q^25 + 13112 * q^27 + 13212 * q^28 + 8364 * q^29 - 7950 * q^30 + 2862 * q^31 - 29672 * q^32 - 3772 * q^33 - 21484 * q^34 - 5500 * q^35 - 2138 * q^36 + 13840 * q^37 - 27706 * q^38 + 6422 * q^39 - 600 * q^40 + 2248 * q^41 - 47028 * q^42 + 3822 * q^43 - 73482 * q^44 - 12950 * q^45 - 39886 * q^46 - 10860 * q^47 + 13962 * q^48 + 85902 * q^49 - 55028 * q^51 + 22646 * q^52 + 6584 * q^53 + 45504 * q^54 + 4250 * q^55 - 188800 * q^56 + 11692 * q^57 - 88500 * q^58 - 28874 * q^59 - 55950 * q^60 + 972 * q^61 + 123466 * q^62 - 125236 * q^63 + 14022 * q^64 - 25350 * q^65 - 158040 * q^66 + 60040 * q^67 + 3504 * q^68 + 153088 * q^69 + 36000 * q^70 - 74174 * q^71 + 298668 * q^72 + 48960 * q^73 + 226112 * q^74 + 23750 * q^75 - 93422 * q^76 + 231588 * q^77 + 53742 * q^78 + 214508 * q^79 - 87650 * q^80 + 116006 * q^81 + 58732 * q^82 + 123260 * q^83 - 235268 * q^84 - 18200 * q^85 - 304666 * q^86 + 228144 * q^87 + 127622 * q^88 + 94028 * q^89 - 194700 * q^90 + 37180 * q^91 + 430298 * q^92 + 192892 * q^93 - 125920 * q^94 - 30450 * q^95 - 385858 * q^96 + 246284 * q^97 - 4736 * q^98 - 32270 * q^99

Coefficient data

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

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

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

$2$	1.34530	0.237818	0.118909	−	0.992905i	$-0.462060\pi$
$2$	1.34530	0.237818	0.118909	+	0.992905i	$0.462060\pi$
$3$	−19.6439	−1.26015	−0.630077	−	0.776532i	$-0.716977\pi$
$3$	−19.6439	−1.26015	−0.630077	+	0.776532i	$0.716977\pi$
$4$	−30.1902	−0.943443
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−26.4269	−0.299687
$7$	48.6530	0.375288	0.187644	−	0.982237i	$-0.439915\pi$
$7$	48.6530	0.375288	0.187644	+	0.982237i	$0.439915\pi$
$8$	−83.6645	−0.462185
$9$	142.882	0.587990
$10$	−33.6325	−0.106355
$11$	283.440	0.706284	0.353142	−	0.935570i	$-0.385113\pi$
$11$	283.440	0.706284	0.353142	+	0.935570i	$0.385113\pi$
$12$	593.052	1.18888
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	65.4530	0.0892502
$15$	491.097	0.563558
$16$	853.531	0.833527
$17$	789.522	0.662586	0.331293	−	0.943528i	$-0.392515\pi$
$17$	789.522	0.662586	0.331293	+	0.943528i	$0.392515\pi$
$18$	192.219	0.139835
$19$	83.3795	0.0529877	0.0264939	−	0.999649i	$-0.491566\pi$
$19$	83.3795	0.0529877	0.0264939	+	0.999649i	$0.491566\pi$
$20$	754.754	0.421920
$21$	−955.734	−0.472921
$22$	381.312	0.167967
$23$	−1935.13	−0.762767	−0.381383	−	0.924417i	$-0.624552\pi$
$23$	−1935.13	−0.762767	−0.381383	+	0.924417i	$0.624552\pi$
$24$	1643.49	0.582425
$25$	625.000	0.200000
$26$	227.356	0.0659588
$27$	1966.71	0.519196
$28$	−1468.84	−0.354063
$29$	222.752	0.0491843	0.0245922	−	0.999698i	$-0.492171\pi$
$29$	222.752	0.0491843	0.0245922	+	0.999698i	$0.492171\pi$
$30$	660.673	0.134024
$31$	−2789.48	−0.521338	−0.260669	−	0.965428i	$-0.583943\pi$
$31$	−2789.48	−0.521338	−0.260669	+	0.965428i	$0.583943\pi$
$32$	3825.52	0.660413
$33$	−5567.86	−0.890027
$34$	1062.15	0.157575
$35$	−1216.33	−0.167834
$36$	−4313.62	−0.554735
$37$	8369.56	1.00507	0.502537	−	0.864555i	$-0.332400\pi$
$37$	8369.56	1.00507	0.502537	+	0.864555i	$0.332400\pi$
$38$	112.170	0.0126014
$39$	−3319.81	−0.349504
$40$	2091.61	0.206696
$41$	1218.51	0.113206	0.0566029	−	0.998397i	$-0.481973\pi$
$41$	1218.51	0.113206	0.0566029	+	0.998397i	$0.481973\pi$
$42$	−1285.75	−0.112469
$43$	5638.89	0.465075	0.232537	−	0.972587i	$-0.425297\pi$
$43$	5638.89	0.465075	0.232537	+	0.972587i	$0.425297\pi$
$44$	−8557.10	−0.666338
$45$	−3572.04	−0.262957
$46$	−2603.34	−0.181399
$47$	17776.2	1.17380	0.586902	−	0.809658i	$-0.300348\pi$
$47$	17776.2	1.17380	0.586902	+	0.809658i	$0.300348\pi$
$48$	−16766.7	−1.05037
$49$	−14439.9	−0.859159
$50$	840.813	0.0475636
$51$	−15509.3	−0.834961
$52$	−5102.14	−0.261664
$53$	10834.4	0.529803	0.264902	−	0.964275i	$-0.414661\pi$
$53$	10834.4	0.529803	0.264902	+	0.964275i	$0.414661\pi$
$54$	2645.82	0.123474
$55$	−7086.00	−0.315860
$56$	−4070.53	−0.173453
$57$	−1637.90	−0.0667727
$58$	299.668	0.0116969
$59$	−5363.36	−0.200589	−0.100295	−	0.994958i	$-0.531979\pi$
$59$	−5363.36	−0.200589	−0.100295	+	0.994958i	$0.531979\pi$
$60$	−14826.3	−0.531685
$61$	−18670.7	−0.642445	−0.321222	−	0.947004i	$-0.604094\pi$
$61$	−18670.7	−0.642445	−0.321222	+	0.947004i	$0.604094\pi$
$62$	−3752.69	−0.123983
$63$	6951.63	0.220666
$64$	−22166.5	−0.676469
$65$	−4225.00	−0.124035
$66$	−7490.44	−0.211664
$67$	13985.1	0.380607	0.190304	−	0.981725i	$-0.439053\pi$
$67$	13985.1	0.380607	0.190304	+	0.981725i	$0.439053\pi$
$68$	−23835.8	−0.625112
$69$	38013.5	0.961204
$70$	−1636.32	−0.0399139
$71$	50969.5	1.19995	0.599976	−	0.800018i	$-0.295177\pi$
$71$	50969.5	1.19995	0.599976	+	0.800018i	$0.295177\pi$
$72$	−11954.1	−0.271761
$73$	42394.9	0.931122	0.465561	−	0.885016i	$-0.345853\pi$
$73$	42394.9	0.931122	0.465561	+	0.885016i	$0.345853\pi$
$74$	11259.6	0.239025
$75$	−12277.4	−0.252031
$76$	−2517.24	−0.0499909
$77$	13790.2	0.265060
$78$	−4466.15	−0.0831183
$79$	106279.	1.91593	0.957964	−	0.286888i	$-0.0926208\pi$
$79$	106279.	1.91593	0.957964	+	0.286888i	$0.0926208\pi$
$80$	−21338.3	−0.372765
$81$	−73354.1	−1.24226
$82$	1639.26	0.0269224
$83$	75513.6	1.20318	0.601589	−	0.798806i	$-0.294535\pi$
$83$	75513.6	1.20318	0.601589	+	0.798806i	$0.294535\pi$
$84$	28853.8	0.446174
$85$	−19738.1	−0.296317
$86$	7586.01	0.110603
$87$	−4375.71	−0.0619799
$88$	−23713.8	−0.326434
$89$	77017.8	1.03066	0.515331	−	0.856991i	$-0.327669\pi$
$89$	77017.8	1.03066	0.515331	+	0.856991i	$0.327669\pi$
$90$	−4805.47	−0.0625359
$91$	8222.36	0.104086
$92$	58422.0	0.719627
$93$	54796.2	0.656966
$94$	23914.4	0.279151
$95$	−2084.49	−0.0236968
$96$	−75148.0	−0.832222
$97$	126702.	1.36727	0.683637	−	0.729822i	$-0.260397\pi$
$97$	126702.	1.36727	0.683637	+	0.729822i	$0.260397\pi$
$98$	−19426.0	−0.204323
$99$	40498.4	0.415288
$100$	−18868.9	−0.188689
$101$	−177513.	−1.73151	−0.865757	−	0.500464i	$-0.833163\pi$
$101$	−177513.	−1.73151	−0.865757	+	0.500464i	$0.833163\pi$
$102$	−20864.6	−0.198569
$103$	149599.	1.38943	0.694714	−	0.719286i	$-0.255531\pi$
$103$	149599.	1.38943	0.694714	+	0.719286i	$0.255531\pi$
$104$	−14139.3	−0.128187
$105$	23893.3	0.211497
$106$	14575.5	0.125997
$107$	−111466.	−0.941199	−0.470600	−	0.882347i	$-0.655962\pi$
$107$	−111466.	−0.941199	−0.470600	+	0.882347i	$0.655962\pi$
$108$	−59375.3	−0.489832
$109$	−85200.4	−0.686872	−0.343436	−	0.939176i	$-0.611591\pi$
$109$	−85200.4	−0.686872	−0.343436	+	0.939176i	$0.611591\pi$
$110$	−9532.80	−0.0751171
$111$	−164411.	−1.26655
$112$	41526.9	0.312813
$113$	−204137.	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$113$	−204137.	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$114$	−2203.46	−0.0158797
$115$	48378.4	0.341120
$116$	−6724.92	−0.0464026
$117$	24147.0	0.163079
$118$	−7215.34	−0.0477037
$119$	38412.7	0.248661
$120$	−41087.4	−0.260468
$121$	−80712.8	−0.501163
$122$	−25117.7	−0.152785
$123$	−23936.2	−0.142657
$124$	84214.9	0.491852
$125$	−15625.0	−0.0894427
$126$	9352.03	0.0524783
$127$	272404.	1.49866	0.749331	−	0.662196i	$-0.230375\pi$
$127$	272404.	1.49866	0.749331	+	0.662196i	$0.230375\pi$
$128$	−152237.	−0.821289
$129$	−110770.	−0.586066
$130$	−5683.90	−0.0294977
$131$	42029.3	0.213980	0.106990	−	0.994260i	$-0.465879\pi$
$131$	42029.3	0.213980	0.106990	+	0.994260i	$0.465879\pi$
$132$	168095.	0.839689
$133$	4056.66	0.0198857
$134$	18814.1	0.0905152
$135$	−49167.8	−0.232191
$136$	−66055.0	−0.306237
$137$	−56817.3	−0.258630	−0.129315	−	0.991604i	$-0.541278\pi$
$137$	−56817.3	−0.258630	−0.129315	+	0.991604i	$0.541278\pi$
$138$	51139.6	0.228591
$139$	−136297.	−0.598342	−0.299171	−	0.954200i	$-0.596710\pi$
$139$	−136297.	−0.598342	−0.299171	+	0.954200i	$0.596710\pi$
$140$	36721.1	0.158342
$141$	−349194.	−1.47917
$142$	68569.3	0.285370
$143$	47901.3	0.195888
$144$	121954.	0.490106
$145$	−5568.80	−0.0219959
$146$	57033.9	0.221437
$147$	283655.	1.08267
$148$	−252678.	−0.948230
$149$	−398843.	−1.47176	−0.735879	−	0.677113i	$-0.763231\pi$
$149$	−398843.	−1.47176	−0.735879	+	0.677113i	$0.763231\pi$
$150$	−16516.8	−0.0599375
$151$	−16131.0	−0.0575730	−0.0287865	−	0.999586i	$-0.509164\pi$
$151$	−16131.0	−0.0575730	−0.0287865	+	0.999586i	$0.509164\pi$
$152$	−6975.90	−0.0244901
$153$	112808.	0.389594
$154$	18552.0	0.0630360
$155$	69737.0	0.233149
$156$	100226.	0.329737
$157$	9495.80	0.0307456	0.0153728	−	0.999882i	$-0.495107\pi$
$157$	9495.80	0.0307456	0.0153728	+	0.999882i	$0.495107\pi$
$158$	142977.	0.455642
$159$	−212829.	−0.667634
$160$	−95638.0	−0.295346
$161$	−94150.2	−0.286257
$162$	−98683.3	−0.295431
$163$	439257.	1.29494	0.647469	−	0.762091i	$-0.275827\pi$
$163$	439257.	1.29494	0.647469	+	0.762091i	$0.275827\pi$
$164$	−36787.0	−0.106803
$165$	139196.	0.398032
$166$	101588.	0.286137
$167$	−233417.	−0.647651	−0.323825	−	0.946117i	$-0.604969\pi$
$167$	−233417.	−0.647651	−0.323825	+	0.946117i	$0.604969\pi$
$168$	79961.0	0.218577
$169$	28561.0	0.0769231
$170$	−26553.6	−0.0704696
$171$	11913.4	0.0311563
$172$	−170239.	−0.438771
$173$	121696.	0.309144	0.154572	−	0.987982i	$-0.450600\pi$
$173$	121696.	0.309144	0.154572	+	0.987982i	$0.450600\pi$
$174$	−5886.65	−0.0147399
$175$	30408.1	0.0750576
$176$	241925.	0.588706
$177$	105357.	0.252773
$178$	103612.	0.245110
$179$	−280901.	−0.655271	−0.327636	−	0.944804i	$-0.606252\pi$
$179$	−280901.	−0.655271	−0.327636	+	0.944804i	$0.606252\pi$
$180$	107841.	0.248085
$181$	324458.	0.736142	0.368071	−	0.929798i	$-0.380018\pi$
$181$	324458.	0.736142	0.368071	+	0.929798i	$0.380018\pi$
$182$	11061.6	0.0247536
$183$	366765.	0.809580
$184$	161902.	0.352540
$185$	−209239.	−0.449483
$186$	73717.4	0.156238
$187$	223782.	0.467974
$188$	−536668.	−1.10742
$189$	95686.5	0.194848
$190$	−2804.26	−0.00563553
$191$	803448.	1.59358	0.796791	−	0.604256i	$-0.206529\pi$
$191$	803448.	1.59358	0.796791	+	0.604256i	$0.206529\pi$
$192$	435436.	0.852456
$193$	−62706.7	−0.121177	−0.0605886	−	0.998163i	$-0.519298\pi$
$193$	−62706.7	−0.121177	−0.0605886	+	0.998163i	$0.519298\pi$
$194$	170453.	0.325162
$195$	82995.4	0.156303
$196$	435942.	0.810567
$197$	−367605.	−0.674864	−0.337432	−	0.941350i	$-0.609558\pi$
$197$	−367605.	−0.674864	−0.337432	+	0.941350i	$0.609558\pi$
$198$	54482.5	0.0987629
$199$	860523.	1.54039	0.770193	−	0.637810i	$-0.220160\pi$
$199$	860523.	1.54039	0.770193	+	0.637810i	$0.220160\pi$
$200$	−52290.3	−0.0924371
$201$	−274721.	−0.479624
$202$	−238808.	−0.411785
$203$	10837.6	0.0184583
$204$	468228.	0.787738
$205$	−30462.7	−0.0506272
$206$	201256.	0.330431
$207$	−276495.	−0.448499
$208$	144247.	0.231179
$209$	23633.1	0.0374244
$210$	32143.7	0.0502977
$211$	−457688.	−0.707723	−0.353861	−	0.935298i	$-0.615132\pi$
$211$	−457688.	−0.707723	−0.353861	+	0.935298i	$0.615132\pi$
$212$	−327092.	−0.499839
$213$	−1.00124e6	−1.51213
$214$	−149955.	−0.223834
$215$	−140972.	−0.207988
$216$	−164544.	−0.239965
$217$	−135717.	−0.195652
$218$	−114620.	−0.163350
$219$	−832800.	−1.17336
$220$	213927.	0.297996
$221$	133429.	0.183768
$222$	−221182.	−0.301208
$223$	−925567.	−1.24637	−0.623183	−	0.782076i	$-0.714161\pi$
$223$	−925567.	−1.24637	−0.623183	+	0.782076i	$0.714161\pi$
$224$	186123.	0.247845
$225$	89301.0	0.117598
$226$	−274625.	−0.357659
$227$	−1.54248e6	−1.98681	−0.993403	−	0.114679i	$-0.963416\pi$
$227$	−1.54248e6	−1.98681	−0.993403	+	0.114679i	$0.963416\pi$
$228$	49448.3	0.0629962
$229$	1.09184e6	1.37584	0.687922	−	0.725784i	$-0.258523\pi$
$229$	1.09184e6	1.37584	0.687922	+	0.725784i	$0.258523\pi$
$230$	65083.5	0.0811243
$231$	−270893.	−0.334017
$232$	−18636.4	−0.0227323
$233$	141596.	0.170868	0.0854341	−	0.996344i	$-0.472772\pi$
$233$	141596.	0.170868	0.0854341	+	0.996344i	$0.472772\pi$
$234$	32485.0	0.0387831
$235$	−444406.	−0.524941
$236$	161921.	0.189244
$237$	−2.08773e6	−2.41437
$238$	51676.6	0.0591359
$239$	−791678.	−0.896507	−0.448253	−	0.893906i	$-0.647954\pi$
$239$	−791678.	−0.896507	−0.448253	+	0.893906i	$0.647954\pi$
$240$	419167.	0.469741
$241$	1.39653e6	1.54884	0.774421	−	0.632671i	$-0.218041\pi$
$241$	1.39653e6	1.54884	0.774421	+	0.632671i	$0.218041\pi$
$242$	−108583.	−0.119186
$243$	963047.	1.04624
$244$	563671.	0.606110
$245$	360997.	0.384228
$246$	−32201.4	−0.0339264
$247$	14091.1	0.0146961
$248$	233380.	0.240955
$249$	−1.48338e6	−1.51619
$250$	−21020.3	−0.0212711
$251$	927565.	0.929309	0.464654	−	0.885492i	$-0.346179\pi$
$251$	927565.	0.929309	0.464654	+	0.885492i	$0.346179\pi$
$252$	−209871.	−0.208186
$253$	−548494.	−0.538730
$254$	366465.	0.356408
$255$	387732.	0.373406
$256$	504524.	0.481152
$257$	1.21697e6	1.14934	0.574670	−	0.818385i	$-0.305130\pi$
$257$	1.21697e6	1.14934	0.574670	+	0.818385i	$0.305130\pi$
$258$	−149019.	−0.139377
$259$	407205.	0.377193
$260$	127553.	0.117020
$261$	31827.2	0.0289199
$262$	56542.0	0.0508883
$263$	−188972.	−0.168464	−0.0842321	−	0.996446i	$-0.526844\pi$
$263$	−188972.	−0.168464	−0.0842321	+	0.996446i	$0.526844\pi$
$264$	465832.	0.411357
$265$	−270860.	−0.236935
$266$	5457.43	0.00472916
$267$	−1.51293e6	−1.29879
$268$	−422211.	−0.359081
$269$	−1.68629e6	−1.42086	−0.710432	−	0.703766i	$-0.751500\pi$
$269$	−1.68629e6	−1.42086	−0.710432	+	0.703766i	$0.751500\pi$
$270$	−66145.5	−0.0552193
$271$	1.71354e6	1.41733	0.708664	−	0.705546i	$-0.249298\pi$
$271$	1.71354e6	1.41733	0.708664	+	0.705546i	$0.249298\pi$
$272$	673882.	0.552283
$273$	−161519.	−0.131165
$274$	−76436.4	−0.0615069
$275$	177150.	0.141257
$276$	−1.14764e6	−0.906841
$277$	1.51635e6	1.18741	0.593706	−	0.804682i	$-0.297664\pi$
$277$	1.51635e6	1.18741	0.593706	+	0.804682i	$0.297664\pi$
$278$	−183360.	−0.142296
$279$	−398566.	−0.306542
$280$	101763.	0.0775704
$281$	2.01362e6	1.52129	0.760644	−	0.649169i	$-0.224883\pi$
$281$	2.01362e6	1.52129	0.760644	+	0.649169i	$0.224883\pi$
$282$	−469771.	−0.351774
$283$	451062.	0.334788	0.167394	−	0.985890i	$-0.446465\pi$
$283$	451062.	0.334788	0.167394	+	0.985890i	$0.446465\pi$
$284$	−1.53878e6	−1.13209
$285$	40947.4	0.0298617
$286$	64441.7	0.0465856
$287$	59284.1	0.0424848
$288$	546597.	0.388316
$289$	−796511.	−0.560980
$290$	−7491.71	−0.00523102
$291$	−2.48893e6	−1.72298
$292$	−1.27991e6	−0.878460
$293$	−1.60140e6	−1.08976	−0.544879	−	0.838514i	$-0.683425\pi$
$293$	−1.60140e6	−1.08976	−0.544879	+	0.838514i	$0.683425\pi$
$294$	381602.	0.257479
$295$	134084.	0.0897062
$296$	−700235.	−0.464531
$297$	557444.	0.366700
$298$	−536564.	−0.350010
$299$	−327038.	−0.211553
$300$	370657.	0.237777
$301$	274349.	0.174537
$302$	−21701.0	−0.0136919
$303$	3.48704e6	2.18198
$304$	71167.0	0.0441667
$305$	466767.	0.287310
$306$	151761.	0.0926524
$307$	3.03553e6	1.83818	0.919091	−	0.394046i	$-0.128925\pi$
$307$	3.03553e6	1.83818	0.919091	+	0.394046i	$0.128925\pi$
$308$	−416329.	−0.250069
$309$	−2.93871e6	−1.75089
$310$	93817.2	0.0554470
$311$	1.48427e6	0.870184	0.435092	−	0.900386i	$-0.356716\pi$
$311$	1.48427e6	0.870184	0.435092	+	0.900386i	$0.356716\pi$
$312$	277751.	0.161536
$313$	1.57466e6	0.908504	0.454252	−	0.890873i	$-0.349906\pi$
$313$	1.57466e6	0.908504	0.454252	+	0.890873i	$0.349906\pi$
$314$	12774.7	0.00731184
$315$	−173791.	−0.0986848
$316$	−3.20858e6	−1.80757
$317$	2.88960e6	1.61507	0.807533	−	0.589822i	$-0.200802\pi$
$317$	2.88960e6	1.61507	0.807533	+	0.589822i	$0.200802\pi$
$318$	−286319.	−0.158775
$319$	63136.8	0.0347381
$320$	554163.	0.302526
$321$	2.18962e6	1.18606
$322$	−126660.	−0.0680771
$323$	65830.0	0.0351089
$324$	2.21457e6	1.17200
$325$	105625.	0.0554700
$326$	590932.	0.307960
$327$	1.67367e6	0.865565
$328$	−101946.	−0.0523221
$329$	864868.	0.440514
$330$	187261.	0.0946591
$331$	2.93760e6	1.47374	0.736872	−	0.676032i	$-0.236302\pi$
$331$	2.93760e6	1.47374	0.736872	+	0.676032i	$0.236302\pi$
$332$	−2.27977e6	−1.13513
$333$	1.19586e6	0.590974
$334$	−314016.	−0.154023
$335$	−349626.	−0.170213
$336$	−815749.	−0.394192
$337$	−842611.	−0.404159	−0.202079	−	0.979369i	$-0.564770\pi$
$337$	−842611.	−0.404159	−0.202079	+	0.979369i	$0.564770\pi$
$338$	38423.1	0.0182937
$339$	4.01004e6	1.89517
$340$	595895.	0.279558
$341$	−790650.	−0.368212
$342$	16027.1	0.00740951
$343$	−1.52026e6	−0.697720
$344$	−471775.	−0.214951
$345$	−950339.	−0.429864
$346$	163718.	0.0735200
$347$	−719797.	−0.320912	−0.160456	−	0.987043i	$-0.551297\pi$
$347$	−719797.	−0.320912	−0.160456	+	0.987043i	$0.551297\pi$
$348$	132103.	0.0584744
$349$	−2.28139e6	−1.00262	−0.501309	−	0.865269i	$-0.667148\pi$
$349$	−2.28139e6	−1.00262	−0.501309	+	0.865269i	$0.667148\pi$
$350$	40908.1	0.0178500
$351$	332374.	0.143999
$352$	1.08430e6	0.466439
$353$	7769.01	0.00331840	0.00165920	−	0.999999i	$-0.499472\pi$
$353$	7769.01	0.00331840	0.00165920	+	0.999999i	$0.499472\pi$
$354$	141737.	0.0601140
$355$	−1.27424e6	−0.536635
$356$	−2.32518e6	−0.972371
$357$	−754573.	−0.313351
$358$	−377897.	−0.155835
$359$	309145.	0.126598	0.0632989	−	0.997995i	$-0.479838\pi$
$359$	309145.	0.126598	0.0632989	+	0.997995i	$0.479838\pi$
$360$	298853.	0.121535
$361$	−2.46915e6	−0.997192
$362$	436493.	0.175068
$363$	1.58551e6	0.631543
$364$	−248234.	−0.0981994
$365$	−1.05987e6	−0.416410
$366$	493409.	0.192533
$367$	−2.32781e6	−0.902156	−0.451078	−	0.892484i	$-0.648960\pi$
$367$	−2.32781e6	−0.902156	−0.451078	+	0.892484i	$0.648960\pi$
$368$	−1.65170e6	−0.635786
$369$	174103.	0.0665640
$370$	−281489.	−0.106895
$371$	527126.	0.198829
$372$	−1.65431e6	−0.619810
$373$	−1.60840e6	−0.598579	−0.299289	−	0.954162i	$-0.596750\pi$
$373$	−1.60840e6	−0.598579	−0.299289	+	0.954162i	$0.596750\pi$
$374$	301054.	0.111292
$375$	306935.	0.112712
$376$	−1.48724e6	−0.542515
$377$	37645.1	0.0136413
$378$	128727.	0.0463383
$379$	−473484.	−0.169320	−0.0846599	−	0.996410i	$-0.526980\pi$
$379$	−473484.	−0.169320	−0.0846599	+	0.996410i	$0.526980\pi$
$380$	62931.0	0.0223566
$381$	−5.35106e6	−1.88855
$382$	1.08088e6	0.378982
$383$	363362.	0.126574	0.0632868	−	0.997995i	$-0.479842\pi$
$383$	363362.	0.126574	0.0632868	+	0.997995i	$0.479842\pi$
$384$	2.99053e6	1.03495
$385$	−344755.	−0.118538
$386$	−84359.4	−0.0288181
$387$	805695.	0.273460
$388$	−3.82517e6	−1.28994
$389$	−901091.	−0.301922	−0.150961	−	0.988540i	$-0.548237\pi$
$389$	−901091.	−0.301922	−0.150961	+	0.988540i	$0.548237\pi$
$390$	111654.	0.0371716
$391$	−1.52783e6	−0.505398
$392$	1.20811e6	0.397091
$393$	−825618.	−0.269648
$394$	−494539.	−0.160495
$395$	−2.65697e6	−0.856829
$396$	−1.22265e6	−0.391801
$397$	−3.36685e6	−1.07213	−0.536064	−	0.844177i	$-0.680090\pi$
$397$	−3.36685e6	−1.07213	−0.536064	+	0.844177i	$0.680090\pi$
$398$	1.15766e6	0.366331
$399$	−79688.6	−0.0250590
$400$	533457.	0.166705
$401$	1.49612e6	0.464629	0.232315	−	0.972641i	$-0.425370\pi$
$401$	1.49612e6	0.464629	0.232315	+	0.972641i	$0.425370\pi$
$402$	−369582.	−0.114063
$403$	−471422.	−0.144593
$404$	5.35914e6	1.63358
$405$	1.83385e6	0.555555
$406$	14579.8	0.00438971
$407$	2.37227e6	0.709868
$408$	1.29758e6	0.385907
$409$	1.78884e6	0.528765	0.264382	−	0.964418i	$-0.414832\pi$
$409$	1.78884e6	0.528765	0.264382	+	0.964418i	$0.414832\pi$
$410$	−40981.5	−0.0120401
$411$	1.11611e6	0.325914
$412$	−4.51642e6	−1.31085
$413$	−260944.	−0.0752787
$414$	−371969.	−0.106661
$415$	−1.88784e6	−0.538077
$416$	646513.	0.183166
$417$	2.67740e6	0.754003
$418$	31793.6	0.00890018
$419$	3.06163e6	0.851956	0.425978	−	0.904733i	$-0.359930\pi$
$419$	3.06163e6	0.851956	0.425978	+	0.904733i	$0.359930\pi$
$420$	−721344.	−0.199535
$421$	−6.32711e6	−1.73980	−0.869901	−	0.493226i	$-0.835817\pi$
$421$	−6.32711e6	−1.73980	−0.869901	+	0.493226i	$0.835817\pi$
$422$	−615728.	−0.168309
$423$	2.53990e6	0.690185
$424$	−906453.	−0.244867
$425$	493452.	0.132517
$426$	−1.34697e6	−0.359611
$427$	−908386.	−0.241102
$428$	3.36517e6	0.887967
$429$	−940968.	−0.246849
$430$	−189650.	−0.0494632
$431$	−2.68796e6	−0.696995	−0.348498	−	0.937310i	$-0.613308\pi$
$431$	−2.68796e6	−0.696995	−0.348498	+	0.937310i	$0.613308\pi$
$432$	1.67865e6	0.432764
$433$	1.62239e6	0.415850	0.207925	−	0.978145i	$-0.433329\pi$
$433$	1.62239e6	0.415850	0.207925	+	0.978145i	$0.433329\pi$
$434$	−182580.	−0.0465295
$435$	109393.	0.0277182
$436$	2.57222e6	0.648024
$437$	−161351.	−0.0404172
$438$	−1.12037e6	−0.279045
$439$	6.21478e6	1.53909	0.769546	−	0.638592i	$-0.220483\pi$
$439$	6.21478e6	1.53909	0.769546	+	0.638592i	$0.220483\pi$
$440$	592846.	0.145986
$441$	−2.06319e6	−0.505177
$442$	179503.	0.0437034
$443$	−517107.	−0.125191	−0.0625953	−	0.998039i	$-0.519938\pi$
$443$	−517107.	−0.125191	−0.0625953	+	0.998039i	$0.519938\pi$
$444$	4.96358e6	1.19492
$445$	−1.92545e6	−0.460926
$446$	−1.24517e6	−0.296408
$447$	7.83482e6	1.85464
$448$	−1.07847e6	−0.253871
$449$	−7.08559e6	−1.65867	−0.829336	−	0.558750i	$-0.811281\pi$
$449$	−7.08559e6	−1.65867	−0.829336	+	0.558750i	$0.811281\pi$
$450$	120137.	0.0279669
$451$	345374.	0.0799555
$452$	6.16292e6	1.41886
$453$	316875.	0.0725509
$454$	−2.07510e6	−0.472498
$455$	−205559.	−0.0465488
$456$	137034.	0.0308614
$457$	−929335.	−0.208153	−0.104076	−	0.994569i	$-0.533189\pi$
$457$	−929335.	−0.208153	−0.104076	+	0.994569i	$0.533189\pi$
$458$	1.46885e6	0.327200
$459$	1.55276e6	0.344012
$460$	−1.46055e6	−0.321827
$461$	7.91310e6	1.73418	0.867091	−	0.498150i	$-0.165987\pi$
$461$	7.91310e6	1.73418	0.867091	+	0.498150i	$0.165987\pi$
$462$	−364433.	−0.0794351
$463$	−1.60654e6	−0.348289	−0.174144	−	0.984720i	$-0.555716\pi$
$463$	−1.60654e6	−0.348289	−0.174144	+	0.984720i	$0.555716\pi$
$464$	190126.	0.0409964
$465$	−1.36990e6	−0.293804
$466$	190489.	0.0406355
$467$	2.57909e6	0.547236	0.273618	−	0.961838i	$-0.411780\pi$
$467$	2.57909e6	0.547236	0.273618	+	0.961838i	$0.411780\pi$
$468$	−729002.	−0.153856
$469$	680415.	0.142837
$470$	−597860.	−0.124840
$471$	−186534.	−0.0387442
$472$	448723.	0.0927093
$473$	1.59829e6	0.328475
$474$	−2.80862e6	−0.574179
$475$	52112.2	0.0105975
$476$	−1.15968e6	−0.234597
$477$	1.54803e6	0.311519
$478$	−1.06504e6	−0.213205
$479$	1.57546e6	0.313739	0.156869	−	0.987619i	$-0.449860\pi$
$479$	1.57546e6	0.313739	0.156869	+	0.987619i	$0.449860\pi$
$480$	1.87870e6	0.372181
$481$	1.41446e6	0.278758
$482$	1.87875e6	0.368342
$483$	1.84947e6	0.360728
$484$	2.43673e6	0.472819
$485$	−3.16756e6	−0.611464
$486$	1.29559e6	0.248815
$487$	−3.52518e6	−0.673534	−0.336767	−	0.941588i	$-0.609333\pi$
$487$	−3.52518e6	−0.673534	−0.336767	+	0.941588i	$0.609333\pi$
$488$	1.56207e6	0.296929
$489$	−8.62870e6	−1.63182
$490$	485650.	0.0913761
$491$	−3.53510e6	−0.661756	−0.330878	−	0.943674i	$-0.607345\pi$
$491$	−3.53510e6	−0.661756	−0.330878	+	0.943674i	$0.607345\pi$
$492$	722639.	0.134589
$493$	175868.	0.0325888
$494$	18956.8	0.00349501
$495$	−1.01246e6	−0.185722
$496$	−2.38091e6	−0.434549
$497$	2.47982e6	0.450328
$498$	−1.99559e6	−0.360577
$499$	311968.	0.0560866	0.0280433	−	0.999607i	$-0.491072\pi$
$499$	311968.	0.0560866	0.0280433	+	0.999607i	$0.491072\pi$
$500$	471721.	0.0843841
$501$	4.58521e6	0.816140
$502$	1.24785e6	0.221006
$503$	6.52751e6	1.15034	0.575172	−	0.818032i	$-0.304935\pi$
$503$	6.52751e6	1.15034	0.575172	+	0.818032i	$0.304935\pi$
$504$	−581604.	−0.101989
$505$	4.43782e6	0.774357
$506$	−737890.	−0.128120
$507$	−561049.	−0.0969350
$508$	−8.22391e6	−1.41390
$509$	−6.28149e6	−1.07465	−0.537327	−	0.843374i	$-0.680566\pi$
$509$	−6.28149e6	−1.07465	−0.537327	+	0.843374i	$0.680566\pi$
$510$	521616.	0.0888026
$511$	2.06264e6	0.349439
$512$	5.55033e6	0.935716
$513$	163983.	0.0275110
$514$	1.63720e6	0.273334
$515$	−3.73998e6	−0.621371
$516$	3.34416e6	0.552920
$517$	5.03850e6	0.829038
$518$	547813.	0.0897031
$519$	−2.39058e6	−0.389569
$520$	353482.	0.0573270
$521$	−7.39635e6	−1.19378	−0.596888	−	0.802324i	$-0.703596\pi$
$521$	−7.39635e6	−1.19378	−0.596888	+	0.802324i	$0.703596\pi$
$522$	42817.1	0.00687767
$523$	6.26869e6	1.00213	0.501063	−	0.865411i	$-0.332943\pi$
$523$	6.26869e6	1.00213	0.501063	+	0.865411i	$0.332943\pi$
$524$	−1.26887e6	−0.201878
$525$	−597334.	−0.0945842
$526$	−254224.	−0.0400638
$527$	−2.20236e6	−0.345431
$528$	−4.75234e6	−0.741861
$529$	−2.69160e6	−0.418187
$530$	−364388.	−0.0563474
$531$	−766327.	−0.117944
$532$	−122471.	−0.0187610
$533$	205928.	0.0313977
$534$	−2.03534e6	−0.308876
$535$	2.78664e6	0.420917
$536$	−1.17005e6	−0.175911
$537$	5.51799e6	0.825743
$538$	−2.26857e6	−0.337907
$539$	−4.09284e6	−0.606810
$540$	1.48438e6	0.219059
$541$	−2.81963e6	−0.414189	−0.207095	−	0.978321i	$-0.566401\pi$
$541$	−2.81963e6	−0.414189	−0.207095	+	0.978321i	$0.566401\pi$
$542$	2.30522e6	0.337066
$543$	−6.37360e6	−0.927653
$544$	3.02033e6	0.437580
$545$	2.13001e6	0.307178
$546$	−217292.	−0.0311933
$547$	−9.17262e6	−1.31077	−0.655383	−	0.755297i	$-0.727493\pi$
$547$	−9.17262e6	−1.31077	−0.655383	+	0.755297i	$0.727493\pi$
$548$	1.71532e6	0.244003
$549$	−2.66770e6	−0.377751
$550$	238320.	0.0335934
$551$	18572.9	0.00260616
$552$	−3.18038e6	−0.444254
$553$	5.17079e6	0.719025
$554$	2.03995e6	0.282388
$555$	4.11026e6	0.566418
$556$	4.11483e6	0.564501
$557$	−4.27403e6	−0.583713	−0.291856	−	0.956462i	$-0.594273\pi$
$557$	−4.27403e6	−0.583713	−0.291856	+	0.956462i	$0.594273\pi$
$558$	−536191.	−0.0729010
$559$	952973.	0.128989
$560$	−1.03817e6	−0.139894
$561$	−4.39595e6	−0.589719
$562$	2.70892e6	0.361789
$563$	9.29935e6	1.23646	0.618232	−	0.785995i	$-0.287849\pi$
$563$	9.29935e6	1.23646	0.618232	+	0.785995i	$0.287849\pi$
$564$	1.05422e7	1.39552
$565$	5.10342e6	0.672574
$566$	606813.	0.0796185
$567$	−3.56890e6	−0.466205
$568$	−4.26433e6	−0.554601
$569$	−4.10069e6	−0.530977	−0.265489	−	0.964114i	$-0.585533\pi$
$569$	−4.10069e6	−0.530977	−0.265489	+	0.964114i	$0.585533\pi$
$570$	55086.5	0.00710164
$571$	−1.03048e7	−1.32267	−0.661334	−	0.750092i	$-0.730009\pi$
$571$	−1.03048e7	−1.32267	−0.661334	+	0.750092i	$0.730009\pi$
$572$	−1.44615e6	−0.184809
$573$	−1.57828e7	−2.00816
$574$	79755.0	0.0101036
$575$	−1.20946e6	−0.152553
$576$	−3.16719e6	−0.397757
$577$	2.17391e6	0.271833	0.135916	−	0.990720i	$-0.456602\pi$
$577$	2.17391e6	0.271833	0.135916	+	0.990720i	$0.456602\pi$
$578$	−1.07155e6	−0.133411
$579$	1.23180e6	0.152702
$580$	168123.	0.0207519
$581$	3.67396e6	0.451538
$582$	−3.34835e6	−0.409755
$583$	3.07090e6	0.374191
$584$	−3.54695e6	−0.430351
$585$	−603675.	−0.0729312
$586$	−2.15436e6	−0.259164
$587$	−1.56597e7	−1.87581	−0.937905	−	0.346892i	$-0.887237\pi$
$587$	−1.56597e7	−1.87581	−0.937905	+	0.346892i	$0.887237\pi$
$588$	−8.56360e6	−1.02144
$589$	−232585.	−0.0276245
$590$	180383.	0.0213337
$591$	7.22119e6	0.850433
$592$	7.14368e6	0.837757
$593$	−5.51737e6	−0.644310	−0.322155	−	0.946687i	$-0.604407\pi$
$593$	−5.51737e6	−0.644310	−0.322155	+	0.946687i	$0.604407\pi$
$594$	749930.	0.0872077
$595$	−960317.	−0.111204
$596$	1.20411e7	1.38852
$597$	−1.69040e7	−1.94113
$598$	−439964.	−0.0503112
$599$	−8.76025e6	−0.997584	−0.498792	−	0.866722i	$-0.666223\pi$
$599$	−8.76025e6	−0.997584	−0.498792	+	0.866722i	$0.666223\pi$
$600$	1.02718e6	0.116485
$601$	1.21014e7	1.36662	0.683310	−	0.730128i	$-0.260540\pi$
$601$	1.21014e7	1.36662	0.683310	+	0.730128i	$0.260540\pi$
$602$	369082.	0.0415080
$603$	1.99821e6	0.223794
$604$	486997.	0.0543168
$605$	2.01782e6	0.224127
$606$	4.69111e6	0.518913
$607$	−1.29407e7	−1.42556	−0.712779	−	0.701388i	$-0.752564\pi$
$607$	−1.29407e7	−1.42556	−0.712779	+	0.701388i	$0.752564\pi$
$608$	318970.	0.0349938
$609$	−212892.	−0.0232603
$610$	627942.	0.0683274
$611$	3.00418e6	0.325554
$612$	−3.40570e6	−0.367560
$613$	2.31350e6	0.248667	0.124333	−	0.992240i	$-0.460321\pi$
$613$	2.31350e6	0.248667	0.124333	+	0.992240i	$0.460321\pi$
$614$	4.08370e6	0.437152
$615$	598406.	0.0637981
$616$	−1.15375e6	−0.122507
$617$	8.53024e6	0.902086	0.451043	−	0.892502i	$-0.351052\pi$
$617$	8.53024e6	0.902086	0.451043	+	0.892502i	$0.351052\pi$
$618$	−3.95344e6	−0.416394
$619$	8.89321e6	0.932893	0.466447	−	0.884549i	$-0.345534\pi$
$619$	8.89321e6	0.932893	0.466447	+	0.884549i	$0.345534\pi$
$620$	−2.10537e6	−0.219963
$621$	−3.80585e6	−0.396025
$622$	1.99679e6	0.206945
$623$	3.74715e6	0.386795
$624$	−2.83357e6	−0.291321
$625$	390625.	0.0400000
$626$	2.11840e6	0.216059
$627$	−464245.	−0.0471605
$628$	−286680.	−0.0290067
$629$	6.60796e6	0.665948
$630$	−233801.	−0.0234690
$631$	1.01842e7	1.01824	0.509122	−	0.860695i	$-0.329970\pi$
$631$	1.01842e7	1.01824	0.509122	+	0.860695i	$0.329970\pi$
$632$	−8.89177e6	−0.885514
$633$	8.99076e6	0.891840
$634$	3.88739e6	0.384091
$635$	−6.81009e6	−0.670222
$636$	6.42535e6	0.629874
$637$	−2.44034e6	−0.238288
$638$	84938.0	0.00826133
$639$	7.28260e6	0.705561
$640$	3.80593e6	0.367292
$641$	−1.30240e7	−1.25198	−0.625991	−	0.779830i	$-0.715306\pi$
$641$	−1.30240e7	−1.25198	−0.625991	+	0.779830i	$0.715306\pi$
$642$	2.94569e6	0.282065
$643$	−4.52103e6	−0.431231	−0.215616	−	0.976478i	$-0.569176\pi$
$643$	−4.52103e6	−0.431231	−0.215616	+	0.976478i	$0.569176\pi$
$644$	2.84241e6	0.270067
$645$	2.76924e6	0.262097
$646$	88561.1	0.00834952
$647$	2.72009e6	0.255459	0.127730	−	0.991809i	$-0.459231\pi$
$647$	2.72009e6	0.255459	0.127730	+	0.991809i	$0.459231\pi$
$648$	6.13713e6	0.574153
$649$	−1.52019e6	−0.141673
$650$	142097.	0.0131918
$651$	2.66600e6	0.246552
$652$	−1.32612e7	−1.22170
$653$	1.83786e7	1.68667	0.843334	−	0.537389i	$-0.180589\pi$
$653$	1.83786e7	1.68667	0.843334	+	0.537389i	$0.180589\pi$
$654$	2.25158e6	0.205847
$655$	−1.05073e6	−0.0956949
$656$	1.04004e6	0.0943601
$657$	6.05745e6	0.547491
$658$	1.16351e6	0.104762
$659$	−1.39753e7	−1.25357	−0.626786	−	0.779192i	$-0.715630\pi$
$659$	−1.39753e7	−1.25357	−0.626786	+	0.779192i	$0.715630\pi$
$660$	−4.20236e6	−0.375521
$661$	9.05787e6	0.806348	0.403174	−	0.915123i	$-0.367907\pi$
$661$	9.05787e6	0.806348	0.403174	+	0.915123i	$0.367907\pi$
$662$	3.95195e6	0.350483
$663$	−2.62107e6	−0.231576
$664$	−6.31780e6	−0.556091
$665$	−101417.	−0.00889314
$666$	1.60879e6	0.140544
$667$	−431055.	−0.0375161
$668$	7.04689e6	0.611021
$669$	1.81817e7	1.57061
$670$	−470353.	−0.0404796
$671$	−5.29202e6	−0.453748
$672$	−3.65618e6	−0.312323
$673$	1.68758e7	1.43624	0.718121	−	0.695919i	$-0.245003\pi$
$673$	1.68758e7	1.43624	0.718121	+	0.695919i	$0.245003\pi$
$674$	−1.13356e6	−0.0961162
$675$	1.22919e6	0.103839
$676$	−862261.	−0.0725725
$677$	7.96769e6	0.668129	0.334065	−	0.942550i	$-0.391580\pi$
$677$	7.96769e6	0.668129	0.334065	+	0.942550i	$0.391580\pi$
$678$	5.39470e6	0.450706
$679$	6.16446e6	0.513122
$680$	1.65137e6	0.136954
$681$	3.03003e7	2.50368
$682$	−1.06366e6	−0.0875674
$683$	−1.29108e7	−1.05901	−0.529507	−	0.848306i	$-0.677623\pi$
$683$	−1.29108e7	−1.05901	−0.529507	+	0.848306i	$0.677623\pi$
$684$	−359667.	−0.0293941
$685$	1.42043e6	0.115663
$686$	−2.04520e6	−0.165930
$687$	−2.14479e7	−1.73378
$688$	4.81297e6	0.387652
$689$	1.83101e6	0.146941
$690$	−1.27849e6	−0.102229
$691$	−1.37987e7	−1.09936	−0.549682	−	0.835374i	$-0.685251\pi$
$691$	−1.37987e7	−1.09936	−0.549682	+	0.835374i	$0.685251\pi$
$692$	−3.67402e6	−0.291660
$693$	1.97037e6	0.155853
$694$	−968344.	−0.0763187
$695$	3.40743e6	0.267587
$696$	366092.	0.0286462
$697$	962040.	0.0750086
$698$	−3.06915e6	−0.238440
$699$	−2.78149e6	−0.215320
$700$	−918027.	−0.0708126
$701$	−1.54955e6	−0.119099	−0.0595497	−	0.998225i	$-0.518966\pi$
$701$	−1.54955e6	−0.119099	−0.0595497	+	0.998225i	$0.518966\pi$
$702$	447143.	0.0342455
$703$	697849.	0.0532566
$704$	−6.28288e6	−0.477779
$705$	8.72985e6	0.661507
$706$	10451.7	0.000789175 0
$707$	−8.63653e6	−0.649817
$708$	−3.18075e6	−0.238477
$709$	−1.43383e7	−1.07123	−0.535614	−	0.844463i	$-0.679920\pi$
$709$	−1.43383e7	−1.07123	−0.535614	+	0.844463i	$0.679920\pi$
$710$	−1.71423e6	−0.127621
$711$	1.51853e7	1.12655
$712$	−6.44366e6	−0.476357
$713$	5.39802e6	0.397659
$714$	−1.01513e6	−0.0745204
$715$	−1.19753e6	−0.0876037
$716$	8.48046e6	0.618211
$717$	1.55516e7	1.12974
$718$	415893.	0.0301072
$719$	−1.37368e7	−0.990980	−0.495490	−	0.868614i	$-0.665011\pi$
$719$	−1.37368e7	−0.990980	−0.495490	+	0.868614i	$0.665011\pi$
$720$	−3.04885e6	−0.219182
$721$	7.27845e6	0.521436
$722$	−3.32175e6	−0.237150
$723$	−2.74332e7	−1.95178
$724$	−9.79543e6	−0.694508
$725$	139220.	0.00983686
$726$	2.13299e6	0.150192
$727$	−1.61824e6	−0.113555	−0.0567776	−	0.998387i	$-0.518083\pi$
$727$	−1.61824e6	−0.113555	−0.0567776	+	0.998387i	$0.518083\pi$
$728$	−687920.	−0.0481071
$729$	−1.09293e6	−0.0761684
$730$	−1.42585e6	−0.0990298
$731$	4.45203e6	0.308152
$732$	−1.10727e7	−0.763792
$733$	−1.57329e7	−1.08156	−0.540779	−	0.841165i	$-0.681870\pi$
$733$	−1.57329e7	−1.08156	−0.540779	+	0.841165i	$0.681870\pi$
$734$	−3.13160e6	−0.214549
$735$	−7.09138e6	−0.484186
$736$	−7.40290e6	−0.503741
$737$	3.96392e6	0.268817
$738$	234220.	0.0158301
$739$	−1.62586e6	−0.109515	−0.0547574	−	0.998500i	$-0.517439\pi$
$739$	−1.62586e6	−0.109515	−0.0547574	+	0.998500i	$0.517439\pi$
$740$	6.31696e6	0.424062
$741$	−276804.	−0.0185194
$742$	709142.	0.0472850
$743$	1.87162e7	1.24379	0.621894	−	0.783101i	$-0.286363\pi$
$743$	1.87162e7	1.24379	0.621894	+	0.783101i	$0.286363\pi$
$744$	−4.58449e6	−0.303640
$745$	9.97108e6	0.658190
$746$	−2.16378e6	−0.142353
$747$	1.07895e7	0.707457
$748$	−6.75602e6	−0.441506
$749$	−5.42314e6	−0.353221
$750$	412921.	0.0268048
$751$	7.16867e6	0.463809	0.231904	−	0.972739i	$-0.425504\pi$
$751$	7.16867e6	0.463809	0.231904	+	0.972739i	$0.425504\pi$
$752$	1.51726e7	0.978396
$753$	−1.82210e7	−1.17107
$754$	50644.0	0.00324414
$755$	403275.	0.0257474
$756$	−2.88879e6	−0.183828
$757$	2.67712e7	1.69796	0.848982	−	0.528422i	$-0.177216\pi$
$757$	2.67712e7	1.69796	0.848982	+	0.528422i	$0.177216\pi$
$758$	−636979.	−0.0402673
$759$	1.07746e7	0.678883
$760$	174397.	0.0109523
$761$	−708017.	−0.0443182	−0.0221591	−	0.999754i	$-0.507054\pi$
$761$	−708017.	−0.0443182	−0.0221591	+	0.999754i	$0.507054\pi$
$762$	−7.19879e6	−0.449130
$763$	−4.14526e6	−0.257775
$764$	−2.42562e7	−1.50345
$765$	−2.82021e6	−0.174232
$766$	488832.	0.0301014
$767$	−906409.	−0.0556334
$768$	−9.91081e6	−0.606326
$769$	1.72011e7	1.04891	0.524456	−	0.851437i	$-0.324269\pi$
$769$	1.72011e7	1.04891	0.524456	+	0.851437i	$0.324269\pi$
$770$	−463799.	−0.0281905
$771$	−2.39061e7	−1.44835
$772$	1.89313e6	0.114324
$773$	7.85325e6	0.472716	0.236358	−	0.971666i	$-0.424046\pi$
$773$	7.85325e6	0.472716	0.236358	+	0.971666i	$0.424046\pi$
$774$	1.08390e6	0.0650336
$775$	−1.74343e6	−0.104268
$776$	−1.06005e7	−0.631934
$777$	−7.99907e6	−0.475321
$778$	−1.21224e6	−0.0718024
$779$	101599.	0.00599852
$780$	−2.50564e6	−0.147463
$781$	1.44468e7	0.847507
$782$	−2.05539e6	−0.120193
$783$	438089.	0.0255363
$784$	−1.23249e7	−0.716132
$785$	−237395.	−0.0137498
$786$	−1.11070e6	−0.0641272
$787$	2.05200e7	1.18097	0.590487	−	0.807047i	$-0.298936\pi$
$787$	2.05200e7	1.18097	0.590487	+	0.807047i	$0.298936\pi$
$788$	1.10981e7	0.636695
$789$	3.71214e6	0.212291
$790$	−3.57443e6	−0.203769
$791$	−9.93187e6	−0.564404
$792$	−3.38827e6	−0.191940
$793$	−3.15535e6	−0.178182
$794$	−4.52942e6	−0.254971
$795$	5.32073e6	0.298575
$796$	−2.59793e7	−1.45327
$797$	−2.57349e7	−1.43508	−0.717540	−	0.696517i	$-0.754732\pi$
$797$	−2.57349e7	−1.43508	−0.717540	+	0.696517i	$0.754732\pi$
$798$	−107205.	−0.00595948
$799$	1.40347e7	0.777745
$800$	2.39095e6	0.132083
$801$	1.10044e7	0.606019
$802$	2.01274e6	0.110497
$803$	1.20164e7	0.657636
$804$	8.29386e6	0.452498
$805$	2.35375e6	0.128018
$806$	−634205.	−0.0343868
$807$	3.31253e7	1.79051
$808$	1.48515e7	0.800280
$809$	−1.45330e7	−0.780699	−0.390350	−	0.920667i	$-0.627646\pi$
$809$	−1.45330e7	−0.780699	−0.390350	+	0.920667i	$0.627646\pi$
$810$	2.46708e6	0.132121
$811$	1.60281e6	0.0855717	0.0427858	−	0.999084i	$-0.486377\pi$
$811$	1.60281e6	0.0855717	0.0427858	+	0.999084i	$0.486377\pi$
$812$	−327188.	−0.0174143
$813$	−3.36605e7	−1.78605
$814$	3.19141e6	0.168819
$815$	−1.09814e7	−0.579114
$816$	−1.32377e7	−0.695962
$817$	470168.	0.0246432
$818$	2.40652e6	0.125750
$819$	1.17482e6	0.0612017
$820$	919674.	0.0477639
$821$	1.22945e6	0.0636579	0.0318289	−	0.999493i	$-0.489867\pi$
$821$	1.22945e6	0.0636579	0.0318289	+	0.999493i	$0.489867\pi$
$822$	1.50151e6	0.0775082
$823$	−260323.	−0.0133972	−0.00669859	−	0.999978i	$-0.502132\pi$
$823$	−260323.	−0.0133972	−0.00669859	+	0.999978i	$0.502132\pi$
$824$	−1.25161e7	−0.642173
$825$	−3.47991e6	−0.178005
$826$	−351048.	−0.0179026
$827$	1.33246e7	0.677471	0.338736	−	0.940882i	$-0.390001\pi$
$827$	1.33246e7	0.677471	0.338736	+	0.940882i	$0.390001\pi$
$828$	8.34744e6	0.423134
$829$	3.52026e7	1.77905	0.889526	−	0.456885i	$-0.151035\pi$
$829$	3.52026e7	1.77905	0.889526	+	0.456885i	$0.151035\pi$
$830$	−2.53971e6	−0.127964
$831$	−2.97871e7	−1.49632
$832$	−3.74614e6	−0.187619
$833$	−1.14006e7	−0.569267
$834$	3.60191e6	0.179315
$835$	5.83542e6	0.289638
$836$	−713486.	−0.0353077
$837$	−5.48610e6	−0.270676
$838$	4.11881e6	0.202610
$839$	−3.45100e7	−1.69255	−0.846273	−	0.532750i	$-0.821159\pi$
$839$	−3.45100e7	−1.69255	−0.846273	+	0.532750i	$0.821159\pi$
$840$	−1.99902e6	−0.0977507
$841$	−2.04615e7	−0.997581
$842$	−8.51186e6	−0.413756
$843$	−3.95553e7	−1.91706
$844$	1.38177e7	0.667696
$845$	−714025.	−0.0344010
$846$	3.41693e6	0.164138
$847$	−3.92692e6	−0.188081
$848$	9.24749e6	0.441605
$849$	−8.86059e6	−0.421885
$850$	663841.	0.0315149
$851$	−1.61962e7	−0.766637
$852$	3.02275e7	1.42660
$853$	−1.67507e7	−0.788245	−0.394122	−	0.919058i	$-0.628951\pi$
$853$	−1.67507e7	−0.788245	−0.394122	+	0.919058i	$0.628951\pi$
$854$	−1.22205e6	−0.0573383
$855$	−297835.	−0.0139335
$856$	9.32571e6	0.435008
$857$	−2.15120e6	−0.100053	−0.0500264	−	0.998748i	$-0.515931\pi$
$857$	−2.15120e6	−0.100053	−0.0500264	+	0.998748i	$0.515931\pi$
$858$	−1.26588e6	−0.0587051
$859$	1.66088e7	0.767991	0.383996	−	0.923335i	$-0.374548\pi$
$859$	1.66088e7	0.767991	0.383996	+	0.923335i	$0.374548\pi$
$860$	4.25598e6	0.196225
$861$	−1.16457e6	−0.0535374
$862$	−3.61612e6	−0.165758
$863$	−2.71816e7	−1.24236	−0.621181	−	0.783667i	$-0.713347\pi$
$863$	−2.71816e7	−1.24236	−0.621181	+	0.783667i	$0.713347\pi$
$864$	7.52369e6	0.342884
$865$	−3.04240e6	−0.138253
$866$	2.18261e6	0.0988965
$867$	1.56466e7	0.706922
$868$	4.09731e6	0.184586
$869$	3.01237e7	1.35319
$870$	147166.	0.00659189
$871$	2.36347e6	0.105562
$872$	7.12825e6	0.317462
$873$	1.81035e7	0.803944
$874$	−217065.	−0.00961194
$875$	−760204.	−0.0335668
$876$	2.51424e7	1.10700
$877$	1.52455e6	0.0669333	0.0334667	−	0.999440i	$-0.489345\pi$
$877$	1.52455e6	0.0669333	0.0334667	+	0.999440i	$0.489345\pi$
$878$	8.36075e6	0.366023
$879$	3.14577e7	1.37326
$880$	−6.04812e6	−0.263278
$881$	2.77728e7	1.20554	0.602769	−	0.797916i	$-0.294064\pi$
$881$	2.77728e7	1.20554	0.602769	+	0.797916i	$0.294064\pi$
$882$	−2.77562e6	−0.120140
$883$	3.15964e7	1.36376	0.681878	−	0.731466i	$-0.261164\pi$
$883$	3.15964e7	1.36376	0.681878	+	0.731466i	$0.261164\pi$
$884$	−4.02825e6	−0.173375
$885$	−2.63393e6	−0.113044
$886$	−695665.	−0.0297725
$887$	9.17500e6	0.391559	0.195780	−	0.980648i	$-0.437276\pi$
$887$	9.17500e6	0.391559	0.195780	+	0.980648i	$0.437276\pi$
$888$	1.37553e7	0.585381
$889$	1.32533e7	0.562430
$890$	−2.59030e6	−0.109616
$891$	−2.07915e7	−0.877386
$892$	2.79430e7	1.17588
$893$	1.48217e6	0.0621971
$894$	1.05402e7	0.441067
$895$	7.02253e6	0.293046
$896$	−7.40681e6	−0.308220
$897$	6.42429e6	0.266590
$898$	−9.53226e6	−0.394462
$899$	−621362.	−0.0256416
$900$	−2.69601e6	−0.110947
$901$	8.55399e6	0.351040
$902$	464632.	0.0190148
$903$	−5.38928e6	−0.219944
$904$	1.70790e7	0.695090
$905$	−8.11144e6	−0.329213
$906$	426292.	0.0172539
$907$	−1.42674e7	−0.575874	−0.287937	−	0.957649i	$-0.592969\pi$
$907$	−1.42674e7	−0.575874	−0.287937	+	0.957649i	$0.592969\pi$
$908$	4.65678e7	1.87444
$909$	−2.53633e7	−1.01811
$910$	−276539.	−0.0110701
$911$	3.34037e7	1.33352	0.666759	−	0.745273i	$-0.267681\pi$
$911$	3.34037e7	1.33352	0.666759	+	0.745273i	$0.267681\pi$
$912$	−1.39800e6	−0.0556568
$913$	2.14036e7	0.849785
$914$	−1.25024e6	−0.0495024
$915$	−9.16912e6	−0.362055
$916$	−3.29628e7	−1.29803
$917$	2.04485e6	0.0803042
$918$	2.08893e6	0.0818121
$919$	4.52117e7	1.76588	0.882942	−	0.469482i	$-0.155559\pi$
$919$	4.52117e7	1.76588	0.882942	+	0.469482i	$0.155559\pi$
$920$	−4.04755e6	−0.157660
$921$	−5.96295e7	−2.31639
$922$	1.06455e7	0.412419
$923$	8.61384e6	0.332807
$924$	8.17831e6	0.315125
$925$	5.23098e6	0.201015
$926$	−2.16128e6	−0.0828292
$927$	2.13750e7	0.816970
$928$	852142.	0.0324819
$929$	−517735.	−0.0196820	−0.00984098	−	0.999952i	$-0.503133\pi$
$929$	−517735.	−0.0196820	−0.00984098	+	0.999952i	$0.503133\pi$
$930$	−1.84293e6	−0.0698719
$931$	−1.20399e6	−0.0455249
$932$	−4.27481e6	−0.161204
$933$	−2.91567e7	−1.09657
$934$	3.46965e6	0.130142
$935$	−5.59455e6	−0.209284
$936$	−2.02025e6	−0.0753728
$937$	−2.58406e7	−0.961508	−0.480754	−	0.876856i	$-0.659637\pi$
$937$	−2.58406e7	−0.961508	−0.480754	+	0.876856i	$0.659637\pi$
$938$	915363.	0.0339693
$939$	−3.09325e7	−1.14486
$940$	1.34167e7	0.495251
$941$	4.58537e7	1.68811	0.844053	−	0.536259i	$-0.180163\pi$
$941$	4.58537e7	1.68811	0.844053	+	0.536259i	$0.180163\pi$
$942$	−250945.	−0.00921405
$943$	−2.35798e6	−0.0863496
$944$	−4.57780e6	−0.167196
$945$	−2.39216e6	−0.0871387
$946$	2.15018e6	0.0781172
$947$	604846.	0.0219164	0.0109582	−	0.999940i	$-0.496512\pi$
$947$	604846.	0.0219164	0.0109582	+	0.999940i	$0.496512\pi$
$948$	6.30289e7	2.27782
$949$	7.16474e6	0.258247
$950$	70106.5	0.00252028
$951$	−5.67630e7	−2.03523
$952$	−3.21378e6	−0.114927
$953$	3.20607e7	1.14351	0.571755	−	0.820424i	$-0.306263\pi$
$953$	3.20607e7	1.14351	0.571755	+	0.820424i	$0.306263\pi$
$954$	2.08257e6	0.0740848
$955$	−2.00862e7	−0.712671
$956$	2.39009e7	0.845803
$957$	−1.24025e6	−0.0437754
$958$	2.11947e6	0.0746127
$959$	−2.76434e6	−0.0970609
$960$	−1.08859e7	−0.381230
$961$	−2.08480e7	−0.728207
$962$	1.90287e6	0.0662935
$963$	−1.59264e7	−0.553416
$964$	−4.21614e7	−1.46124
$965$	1.56767e6	0.0541920
$966$	2.48810e6	0.0857877
$967$	−3.19660e7	−1.09931	−0.549657	−	0.835391i	$-0.685242\pi$
$967$	−3.19660e7	−1.09931	−0.549657	+	0.835391i	$0.685242\pi$
$968$	6.75280e6	0.231630
$969$	−1.29316e6	−0.0442427
$970$	−4.26132e6	−0.145417
$971$	−5.47040e7	−1.86196	−0.930982	−	0.365066i	$-0.881046\pi$
$971$	−5.47040e7	−1.86196	−0.930982	+	0.365066i	$0.881046\pi$
$972$	−2.90746e7	−0.987069
$973$	−6.63126e6	−0.224551
$974$	−4.74243e6	−0.160178
$975$	−2.07488e6	−0.0699008
$976$	−1.59360e7	−0.535495
$977$	−3.12938e7	−1.04887	−0.524436	−	0.851450i	$-0.675724\pi$
$977$	−3.12938e7	−1.04887	−0.524436	+	0.851450i	$0.675724\pi$
$978$	−1.16082e7	−0.388077
$979$	2.18299e7	0.727940
$980$	−1.08986e7	−0.362497
$981$	−1.21736e7	−0.403874
$982$	−4.75577e6	−0.157377
$983$	1.21142e7	0.399863	0.199932	−	0.979810i	$-0.435928\pi$
$983$	1.21142e7	0.399863	0.199932	+	0.979810i	$0.435928\pi$
$984$	2.00261e6	0.0659339
$985$	9.19013e6	0.301808
$986$	236595.	0.00775020
$987$	−1.69894e7	−0.555116
$988$	−425414.	−0.0138650
$989$	−1.09120e7	−0.354744
$990$	−1.36206e6	−0.0441681
$991$	−2.58041e7	−0.834650	−0.417325	−	0.908757i	$-0.637032\pi$
$991$	−2.58041e7	−0.834650	−0.417325	+	0.908757i	$0.637032\pi$
$992$	−1.06712e7	−0.344298
$993$	−5.77058e7	−1.85715
$994$	3.33610e6	0.107096
$995$	−2.15131e7	−0.688882
$996$	4.47835e7	1.43044
$997$	4.21820e7	1.34397	0.671984	−	0.740566i	$-0.265442\pi$
$997$	4.21820e7	1.34397	0.671984	+	0.740566i	$0.265442\pi$
$998$	419691.	0.0133384
$999$	1.64605e7	0.521831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.6.a.d.1.4	✓	6
3.2	odd	2		585.6.a.m.1.3		6
4.3	odd	2		1040.6.a.q.1.6		6
5.2	odd	4		325.6.b.g.274.7		12
5.3	odd	4		325.6.b.g.274.6		12
5.4	even	2		325.6.a.g.1.3		6
13.12	even	2		845.6.a.h.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.4	✓	6	1.1	even	1	trivial
325.6.a.g.1.3		6	5.4	even	2
325.6.b.g.274.6		12	5.3	odd	4
325.6.b.g.274.7		12	5.2	odd	4
585.6.a.m.1.3		6	3.2	odd	2
845.6.a.h.1.3		6	13.12	even	2
1040.6.a.q.1.6		6	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	65.a (trivial)

\(f(q)\)	\(=\)	\(q+1.34530 q^{2} -19.6439 q^{3} -30.1902 q^{4} -25.0000 q^{5} -26.4269 q^{6} +48.6530 q^{7} -83.6645 q^{8} +142.882 q^{9} -33.6325 q^{10} +283.440 q^{11} +593.052 q^{12} +169.000 q^{13} +65.4530 q^{14} +491.097 q^{15} +853.531 q^{16} +789.522 q^{17} +192.219 q^{18} +83.3795 q^{19} +754.754 q^{20} -955.734 q^{21} +381.312 q^{22} -1935.13 q^{23} +1643.49 q^{24} +625.000 q^{25} +227.356 q^{26} +1966.71 q^{27} -1468.84 q^{28} +222.752 q^{29} +660.673 q^{30} -2789.48 q^{31} +3825.52 q^{32} -5567.86 q^{33} +1062.15 q^{34} -1216.33 q^{35} -4313.62 q^{36} +8369.56 q^{37} +112.170 q^{38} -3319.81 q^{39} +2091.61 q^{40} +1218.51 q^{41} -1285.75 q^{42} +5638.89 q^{43} -8557.10 q^{44} -3572.04 q^{45} -2603.34 q^{46} +17776.2 q^{47} -16766.7 q^{48} -14439.9 q^{49} +840.813 q^{50} -15509.3 q^{51} -5102.14 q^{52} +10834.4 q^{53} +2645.82 q^{54} -7086.00 q^{55} -4070.53 q^{56} -1637.90 q^{57} +299.668 q^{58} -5363.36 q^{59} -14826.3 q^{60} -18670.7 q^{61} -3752.69 q^{62} +6951.63 q^{63} -22166.5 q^{64} -4225.00 q^{65} -7490.44 q^{66} +13985.1 q^{67} -23835.8 q^{68} +38013.5 q^{69} -1636.32 q^{70} +50969.5 q^{71} -11954.1 q^{72} +42394.9 q^{73} +11259.6 q^{74} -12277.4 q^{75} -2517.24 q^{76} +13790.2 q^{77} -4466.15 q^{78} +106279. q^{79} -21338.3 q^{80} -73354.1 q^{81} +1639.26 q^{82} +75513.6 q^{83} +28853.8 q^{84} -19738.1 q^{85} +7586.01 q^{86} -4375.71 q^{87} -23713.8 q^{88} +77017.8 q^{89} -4805.47 q^{90} +8222.36 q^{91} +58422.0 q^{92} +54796.2 q^{93} +23914.4 q^{94} -2084.49 q^{95} -75148.0 q^{96} +126702. q^{97} -19426.0 q^{98} +40498.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 38 q^{3} + 134 q^{4} - 150 q^{5} + 318 q^{6} + 220 q^{7} + 24 q^{8} + 518 q^{9} - 170 q^{11} + 2238 q^{12} + 1014 q^{13} - 1440 q^{14} - 950 q^{15} + 3506 q^{16} + 728 q^{17} + 7788 q^{18} + 1218 q^{19}+ \cdots - 32270 q^{99}+O(q^{100}) \) 6 * q + 38 * q^3 + 134 * q^4 - 150 * q^5 + 318 * q^6 + 220 * q^7 + 24 * q^8 + 518 * q^9 - 170 * q^11 + 2238 * q^12 + 1014 * q^13 - 1440 * q^14 - 950 * q^15 + 3506 * q^16 + 728 * q^17 + 7788 * q^18 + 1218 * q^19 - 3350 * q^20 - 396 * q^21 + 5154 * q^22 + 8954 * q^23 + 12618 * q^24 + 3750 * q^25 + 13112 * q^27 + 13212 * q^28 + 8364 * q^29 - 7950 * q^30 + 2862 * q^31 - 29672 * q^32 - 3772 * q^33 - 21484 * q^34 - 5500 * q^35 - 2138 * q^36 + 13840 * q^37 - 27706 * q^38 + 6422 * q^39 - 600 * q^40 + 2248 * q^41 - 47028 * q^42 + 3822 * q^43 - 73482 * q^44 - 12950 * q^45 - 39886 * q^46 - 10860 * q^47 + 13962 * q^48 + 85902 * q^49 - 55028 * q^51 + 22646 * q^52 + 6584 * q^53 + 45504 * q^54 + 4250 * q^55 - 188800 * q^56 + 11692 * q^57 - 88500 * q^58 - 28874 * q^59 - 55950 * q^60 + 972 * q^61 + 123466 * q^62 - 125236 * q^63 + 14022 * q^64 - 25350 * q^65 - 158040 * q^66 + 60040 * q^67 + 3504 * q^68 + 153088 * q^69 + 36000 * q^70 - 74174 * q^71 + 298668 * q^72 + 48960 * q^73 + 226112 * q^74 + 23750 * q^75 - 93422 * q^76 + 231588 * q^77 + 53742 * q^78 + 214508 * q^79 - 87650 * q^80 + 116006 * q^81 + 58732 * q^82 + 123260 * q^83 - 235268 * q^84 - 18200 * q^85 - 304666 * q^86 + 228144 * q^87 + 127622 * q^88 + 94028 * q^89 - 194700 * q^90 + 37180 * q^91 + 430298 * q^92 + 192892 * q^93 - 125920 * q^94 - 30450 * q^95 - 385858 * q^96 + 246284 * q^97 - 4736 * q^98 - 32270 * q^99

Introduction

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