LMFDB - Embedded newform 65.6.a.c.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [65,6,Mod(1,65)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$10.4249482878$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1878612.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} + 16x + 49 $ x^4 - 2x^3 - 16x^2 + 16*x + 49
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.24074$ 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)

$f(q)$	$=$	\(q+7.55727 q^{2} -14.4912 q^{3} +25.1123 q^{4} -25.0000 q^{5} -109.514 q^{6} +26.5515 q^{7} -52.0524 q^{8} -33.0063 q^{9} -188.932 q^{10} -532.748 q^{11} -363.906 q^{12} -169.000 q^{13} +200.657 q^{14} +362.279 q^{15} -1196.97 q^{16} -210.286 q^{17} -249.437 q^{18} -72.2469 q^{19} -627.807 q^{20} -384.762 q^{21} -4026.12 q^{22} +766.119 q^{23} +754.300 q^{24} +625.000 q^{25} -1277.18 q^{26} +3999.65 q^{27} +666.769 q^{28} +7441.26 q^{29} +2737.84 q^{30} +281.855 q^{31} -7380.12 q^{32} +7720.13 q^{33} -1589.19 q^{34} -663.788 q^{35} -828.863 q^{36} -10253.6 q^{37} -545.989 q^{38} +2449.01 q^{39} +1301.31 q^{40} -817.696 q^{41} -2907.75 q^{42} -5039.97 q^{43} -13378.5 q^{44} +825.157 q^{45} +5789.77 q^{46} -19743.1 q^{47} +17345.4 q^{48} -16102.0 q^{49} +4723.29 q^{50} +3047.29 q^{51} -4243.97 q^{52} +32689.8 q^{53} +30226.4 q^{54} +13318.7 q^{55} -1382.07 q^{56} +1046.94 q^{57} +56235.6 q^{58} -517.778 q^{59} +9097.65 q^{60} -15538.7 q^{61} +2130.06 q^{62} -876.367 q^{63} -17470.6 q^{64} +4225.00 q^{65} +58343.1 q^{66} +19642.0 q^{67} -5280.75 q^{68} -11102.0 q^{69} -5016.42 q^{70} -4456.07 q^{71} +1718.06 q^{72} -49182.4 q^{73} -77489.1 q^{74} -9056.97 q^{75} -1814.28 q^{76} -14145.3 q^{77} +18507.8 q^{78} +5454.34 q^{79} +29924.2 q^{80} -49939.1 q^{81} -6179.54 q^{82} -108095. q^{83} -9662.25 q^{84} +5257.15 q^{85} -38088.4 q^{86} -107833. q^{87} +27730.8 q^{88} -33432.3 q^{89} +6235.93 q^{90} -4487.21 q^{91} +19239.0 q^{92} -4084.41 q^{93} -149204. q^{94} +1806.17 q^{95} +106946. q^{96} +19554.2 q^{97} -121687. q^{98} +17584.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 9 q^{2} - 4 q^{3} + 93 q^{4} - 100 q^{5} - 74 q^{6} - 136 q^{7} - 447 q^{8} + 644 q^{9} + 225 q^{10} - 516 q^{11} - 710 q^{12} - 676 q^{13} + 1604 q^{14} + 100 q^{15} - 719 q^{16} + 344 q^{17} - 6905 q^{18}+ \cdots + 290668 q^{99}+O(q^{100}) $ 4 * q - 9 * q^2 - 4 * q^3 + 93 * q^4 - 100 * q^5 - 74 * q^6 - 136 * q^7 - 447 * q^8 + 644 * q^9 + 225 * q^10 - 516 * q^11 - 710 * q^12 - 676 * q^13 + 1604 * q^14 + 100 * q^15 - 719 * q^16 + 344 * q^17 - 6905 * q^18 - 5012 * q^19 - 2325 * q^20 - 4448 * q^21 - 9318 * q^22 + 708 * q^23 - 2094 * q^24 + 2500 * q^25 + 1521 * q^26 - 4264 * q^27 - 7268 * q^28 + 5416 * q^29 + 1850 * q^30 - 10124 * q^31 + 9497 * q^32 - 16256 * q^33 + 3022 * q^34 + 3400 * q^35 + 36205 * q^36 + 1448 * q^37 + 11922 * q^38 + 676 * q^39 + 11175 * q^40 + 21576 * q^41 + 33128 * q^42 - 25740 * q^43 + 24710 * q^44 - 16100 * q^45 + 28754 * q^46 - 51096 * q^47 + 43282 * q^48 - 40412 * q^49 - 5625 * q^50 + 52984 * q^51 - 15717 * q^52 + 39448 * q^53 + 60340 * q^54 + 12900 * q^55 + 30188 * q^56 - 56160 * q^57 + 1374 * q^58 + 37764 * q^59 + 17750 * q^60 + 95016 * q^61 + 83102 * q^62 - 43592 * q^63 - 142095 * q^64 + 16900 * q^65 + 198500 * q^66 - 50592 * q^67 - 47110 * q^68 - 177648 * q^69 - 40100 * q^70 - 89308 * q^71 - 99183 * q^72 - 108312 * q^73 - 16766 * q^74 - 2500 * q^75 - 3442 * q^76 + 6208 * q^77 + 12506 * q^78 - 25464 * q^79 + 17975 * q^80 - 106348 * q^81 - 206426 * q^82 + 20584 * q^83 - 213448 * q^84 - 8600 * q^85 + 363666 * q^86 - 128312 * q^87 + 20558 * q^88 - 148328 * q^89 + 172625 * q^90 + 22984 * q^91 - 173474 * q^92 - 10496 * q^93 + 63916 * q^94 + 125300 * q^95 + 85154 * q^96 - 77816 * q^97 - 83505 * q^98 + 290668 * 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 $

$2$	7.55727	1.33595	0.667974	−	0.744184i	$-0.267162\pi$
$2$	7.55727	1.33595	0.667974	+	0.744184i	$0.267162\pi$
$3$	−14.4912	−0.929608	−0.464804	−	0.885414i	$-0.653875\pi$
$3$	−14.4912	−0.929608	−0.464804	+	0.885414i	$0.653875\pi$
$4$	25.1123	0.784758
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−109.514	−1.24191
$7$	26.5515	0.204807	0.102403	−	0.994743i	$-0.467347\pi$
$7$	26.5515	0.204807	0.102403	+	0.994743i	$0.467347\pi$
$8$	−52.0524	−0.287552
$9$	−33.0063	−0.135828
$10$	−188.932	−0.597454
$11$	−532.748	−1.32752	−0.663758	−	0.747947i	$-0.731040\pi$
$11$	−532.748	−1.32752	−0.663758	+	0.747947i	$0.731040\pi$
$12$	−363.906	−0.729518
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	200.657	0.273611
$15$	362.279	0.415733
$16$	−1196.97	−1.16891
$17$	−210.286	−0.176477	−0.0882384	−	0.996099i	$-0.528124\pi$
$17$	−210.286	−0.176477	−0.0882384	+	0.996099i	$0.528124\pi$
$18$	−249.437	−0.181460
$19$	−72.2469	−0.0459129	−0.0229565	−	0.999736i	$-0.507308\pi$
$19$	−72.2469	−0.0459129	−0.0229565	+	0.999736i	$0.507308\pi$
$20$	−627.807	−0.350955
$21$	−384.762	−0.190390
$22$	−4026.12	−1.77349
$23$	766.119	0.301979	0.150990	−	0.988535i	$-0.451754\pi$
$23$	766.119	0.301979	0.150990	+	0.988535i	$0.451754\pi$
$24$	754.300	0.267311
$25$	625.000	0.200000
$26$	−1277.18	−0.370525
$27$	3999.65	1.05588
$28$	666.769	0.160724
$29$	7441.26	1.64305	0.821527	−	0.570170i	$-0.193123\pi$
$29$	7441.26	1.64305	0.821527	+	0.570170i	$0.193123\pi$
$30$	2737.84	0.555398
$31$	281.855	0.0526771	0.0263386	−	0.999653i	$-0.491615\pi$
$31$	281.855	0.0526771	0.0263386	+	0.999653i	$0.491615\pi$
$32$	−7380.12	−1.27406
$33$	7720.13	1.23407
$34$	−1589.19	−0.235764
$35$	−663.788	−0.0915924
$36$	−828.863	−0.106592
$37$	−10253.6	−1.23132	−0.615661	−	0.788011i	$-0.711111\pi$
$37$	−10253.6	−1.23132	−0.615661	+	0.788011i	$0.711111\pi$
$38$	−545.989	−0.0613373
$39$	2449.01	0.257827
$40$	1301.31	0.128597
$41$	−817.696	−0.0759682	−0.0379841	−	0.999278i	$-0.512094\pi$
$41$	−817.696	−0.0759682	−0.0379841	+	0.999278i	$0.512094\pi$
$42$	−2907.75	−0.254351
$43$	−5039.97	−0.415678	−0.207839	−	0.978163i	$-0.566643\pi$
$43$	−5039.97	−0.415678	−0.207839	+	0.978163i	$0.566643\pi$
$44$	−13378.5	−1.04178
$45$	825.157	0.0607443
$46$	5789.77	0.403428
$47$	−19743.1	−1.30368	−0.651839	−	0.758357i	$-0.726002\pi$
$47$	−19743.1	−1.30368	−0.651839	+	0.758357i	$0.726002\pi$
$48$	17345.4	1.08663
$49$	−16102.0	−0.958054
$50$	4723.29	0.267190
$51$	3047.29	0.164054
$52$	−4243.97	−0.217653
$53$	32689.8	1.59854	0.799269	−	0.600973i	$-0.205220\pi$
$53$	32689.8	1.59854	0.799269	+	0.600973i	$0.205220\pi$
$54$	30226.4	1.41060
$55$	13318.7	0.593683
$56$	−1382.07	−0.0588926
$57$	1046.94	0.0426811
$58$	56235.6	2.19504
$59$	−517.778	−0.0193648	−0.00968241	−	0.999953i	$-0.503082\pi$
$59$	−517.778	−0.0193648	−0.00968241	+	0.999953i	$0.503082\pi$
$60$	9097.65	0.326250
$61$	−15538.7	−0.534674	−0.267337	−	0.963603i	$-0.586144\pi$
$61$	−15538.7	−0.534674	−0.267337	+	0.963603i	$0.586144\pi$
$62$	2130.06	0.0703739
$63$	−876.367	−0.0278186
$64$	−17470.6	−0.533160
$65$	4225.00	0.124035
$66$	58343.1	1.64865
$67$	19642.0	0.534563	0.267282	−	0.963618i	$-0.413875\pi$
$67$	19642.0	0.534563	0.267282	+	0.963618i	$0.413875\pi$
$68$	−5280.75	−0.138492
$69$	−11102.0	−0.280722
$70$	−5016.42	−0.122363
$71$	−4456.07	−0.104907	−0.0524537	−	0.998623i	$-0.516704\pi$
$71$	−4456.07	−0.104907	−0.0524537	+	0.998623i	$0.516704\pi$
$72$	1718.06	0.0390577
$73$	−49182.4	−1.08020	−0.540098	−	0.841602i	$-0.681613\pi$
$73$	−49182.4	−1.08020	−0.540098	+	0.841602i	$0.681613\pi$
$74$	−77489.1	−1.64498
$75$	−9056.97	−0.185922
$76$	−1814.28	−0.0360306
$77$	−14145.3	−0.271884
$78$	18507.8	0.344444
$79$	5454.34	0.0983273	0.0491636	−	0.998791i	$-0.484344\pi$
$79$	5454.34	0.0983273	0.0491636	+	0.998791i	$0.484344\pi$
$80$	29924.2	0.522754
$81$	−49939.1	−0.845722
$82$	−6179.54	−0.101490
$83$	−108095.	−1.72231	−0.861154	−	0.508344i	$-0.830258\pi$
$83$	−108095.	−1.72231	−0.861154	+	0.508344i	$0.830258\pi$
$84$	−9662.25	−0.149410
$85$	5257.15	0.0789229
$86$	−38088.4	−0.555324
$87$	−107833.	−1.52740
$88$	27730.8	0.381730
$89$	−33432.3	−0.447396	−0.223698	−	0.974659i	$-0.571813\pi$
$89$	−33432.3	−0.447396	−0.223698	+	0.974659i	$0.571813\pi$
$90$	6235.93	0.0811513
$91$	−4487.21	−0.0568032
$92$	19239.0	0.236981
$93$	−4084.41	−0.0489691
$94$	−149204.	−1.74165
$95$	1806.17	0.0205329
$96$	106946.	1.18437
$97$	19554.2	0.211014	0.105507	−	0.994419i	$-0.466353\pi$
$97$	19554.2	0.211014	0.105507	+	0.994419i	$0.466353\pi$
$98$	−121687.	−1.27991
$99$	17584.0	0.180314
$100$	15695.2	0.156952
$101$	86925.1	0.847894	0.423947	−	0.905687i	$-0.360644\pi$
$101$	86925.1	0.847894	0.423947	+	0.905687i	$0.360644\pi$
$102$	23029.1	0.219168
$103$	−128981.	−1.19793	−0.598966	−	0.800774i	$-0.704422\pi$
$103$	−128981.	−1.19793	−0.598966	+	0.800774i	$0.704422\pi$
$104$	8796.86	0.0797525
$105$	9619.06	0.0851450
$106$	247046.	2.13557
$107$	−172004.	−1.45238	−0.726189	−	0.687495i	$-0.758710\pi$
$107$	−172004.	−1.45238	−0.726189	+	0.687495i	$0.758710\pi$
$108$	100440.	0.828607
$109$	199792.	1.61069	0.805345	−	0.592806i	$-0.201980\pi$
$109$	199792.	1.61069	0.805345	+	0.592806i	$0.201980\pi$
$110$	100653.	0.793130
$111$	148586.	1.14465
$112$	−31781.3	−0.239401
$113$	−183485.	−1.35177	−0.675886	−	0.737006i	$-0.736239\pi$
$113$	−183485.	−1.35177	−0.675886	+	0.737006i	$0.736239\pi$
$114$	7912.01	0.0570197
$115$	−19153.0	−0.135049
$116$	186867.	1.28940
$117$	5578.06	0.0376720
$118$	−3912.98	−0.0258704
$119$	−5583.41	−0.0361437
$120$	−18857.5	−0.119545
$121$	122769.	0.762300
$122$	−117430.	−0.714296
$123$	11849.4	0.0706207
$124$	7078.03	0.0413388
$125$	−15625.0	−0.0894427
$126$	−6622.94	−0.0371642
$127$	−193041.	−1.06204	−0.531020	−	0.847359i	$-0.678191\pi$
$127$	−193041.	−1.06204	−0.531020	+	0.847359i	$0.678191\pi$
$128$	104134.	0.561782
$129$	73035.0	0.386418
$130$	31929.4	0.165704
$131$	107072.	0.545127	0.272563	−	0.962138i	$-0.412129\pi$
$131$	107072.	0.545127	0.272563	+	0.962138i	$0.412129\pi$
$132$	193870.	0.968447
$133$	−1918.27	−0.00940328
$134$	148440.	0.714149
$135$	−99991.3	−0.472202
$136$	10945.9	0.0507462
$137$	201648.	0.917892	0.458946	−	0.888464i	$-0.348227\pi$
$137$	201648.	0.917892	0.458946	+	0.888464i	$0.348227\pi$
$138$	−83900.4	−0.375030
$139$	−157480.	−0.691336	−0.345668	−	0.938357i	$-0.612348\pi$
$139$	−157480.	−0.691336	−0.345668	+	0.938357i	$0.612348\pi$
$140$	−16669.2	−0.0718779
$141$	286100.	1.21191
$142$	−33675.7	−0.140151
$143$	90034.4	0.368187
$144$	39507.4	0.158772
$145$	−186032.	−0.734796
$146$	−371684.	−1.44309
$147$	233337.	0.890615
$148$	−257491.	−0.966290
$149$	462386.	1.70624	0.853118	−	0.521718i	$-0.174709\pi$
$149$	462386.	1.70624	0.853118	+	0.521718i	$0.174709\pi$
$150$	−68446.0	−0.248382
$151$	28498.2	0.101713	0.0508563	−	0.998706i	$-0.483805\pi$
$151$	28498.2	0.101713	0.0508563	+	0.998706i	$0.483805\pi$
$152$	3760.63	0.0132024
$153$	6940.76	0.0239706
$154$	−106900.	−0.363224
$155$	−7046.39	−0.0235579
$156$	61500.1	0.202332
$157$	50435.7	0.163301	0.0816505	−	0.996661i	$-0.473981\pi$
$157$	50435.7	0.163301	0.0816505	+	0.996661i	$0.473981\pi$
$158$	41219.9	0.131360
$159$	−473714.	−1.48602
$160$	184503.	0.569775
$161$	20341.6	0.0618474
$162$	−377403.	−1.12984
$163$	55500.3	0.163616	0.0818081	−	0.996648i	$-0.473931\pi$
$163$	55500.3	0.163616	0.0818081	+	0.996648i	$0.473931\pi$
$164$	−20534.2	−0.0596167
$165$	−193003.	−0.551893
$166$	−816903.	−2.30091
$167$	−448378.	−1.24409	−0.622046	−	0.782981i	$-0.713698\pi$
$167$	−448378.	−1.24409	−0.622046	+	0.782981i	$0.713698\pi$
$168$	20027.8	0.0547470
$169$	28561.0	0.0769231
$170$	39729.7	0.105437
$171$	2384.60	0.00623628
$172$	−126565.	−0.326207
$173$	655681.	1.66563	0.832813	−	0.553554i	$-0.186729\pi$
$173$	655681.	1.66563	0.832813	+	0.553554i	$0.186729\pi$
$174$	−814919.	−2.04052
$175$	16594.7	0.0409614
$176$	637681.	1.55175
$177$	7503.20	0.0180017
$178$	−252657.	−0.597697
$179$	189556.	0.442186	0.221093	−	0.975253i	$-0.429038\pi$
$179$	189556.	0.442186	0.221093	+	0.975253i	$0.429038\pi$
$180$	20721.6	0.0476696
$181$	−418433.	−0.949358	−0.474679	−	0.880159i	$-0.657436\pi$
$181$	−418433.	−0.949358	−0.474679	+	0.880159i	$0.657436\pi$
$182$	−33911.0	−0.0758861
$183$	225173.	0.497037
$184$	−39878.4	−0.0868346
$185$	256340.	0.550664
$186$	−30867.0	−0.0654202
$187$	112029.	0.234276
$188$	−495794.	−1.02307
$189$	106197.	0.216250
$190$	13649.7	0.0274309
$191$	868610.	1.72283	0.861413	−	0.507906i	$-0.169580\pi$
$191$	868610.	1.72283	0.861413	+	0.507906i	$0.169580\pi$
$192$	253169.	0.495630
$193$	30232.2	0.0584220	0.0292110	−	0.999573i	$-0.490701\pi$
$193$	30232.2	0.0584220	0.0292110	+	0.999573i	$0.490701\pi$
$194$	147776.	0.281904
$195$	−61225.1	−0.115304
$196$	−404358.	−0.751841
$197$	−480183.	−0.881539	−0.440769	−	0.897620i	$-0.645294\pi$
$197$	−480183.	−0.881539	−0.440769	+	0.897620i	$0.645294\pi$
$198$	132887.	0.240891
$199$	477273.	0.854347	0.427174	−	0.904170i	$-0.359509\pi$
$199$	477273.	0.854347	0.427174	+	0.904170i	$0.359509\pi$
$200$	−32532.8	−0.0575104
$201$	−284636.	−0.496934
$202$	656916.	1.13274
$203$	197577.	0.336509
$204$	76524.3	0.128743
$205$	20442.4	0.0339740
$206$	−974743.	−1.60038
$207$	−25286.8	−0.0410173
$208$	202287.	0.324198
$209$	38489.4	0.0609502
$210$	72693.8	0.113749
$211$	−1.06139e6	−1.64123	−0.820617	−	0.571479i	$-0.806370\pi$
$211$	−1.06139e6	−1.64123	−0.820617	+	0.571479i	$0.806370\pi$
$212$	820916.	1.25447
$213$	64573.6	0.0975228
$214$	−1.29988e6	−1.94030
$215$	125999.	0.185897
$216$	−208192.	−0.303619
$217$	7483.69	0.0107886
$218$	1.50988e6	2.15180
$219$	712710.	1.00416
$220$	334463.	0.465898
$221$	35538.3	0.0489459
$222$	1.12291e6	1.52919
$223$	−1.36296e6	−1.83535	−0.917677	−	0.397327i	$-0.869938\pi$
$223$	−1.36296e6	−1.83535	−0.917677	+	0.397327i	$0.869938\pi$
$224$	−195953.	−0.260935
$225$	−20628.9	−0.0271657
$226$	−1.38664e6	−1.80590
$227$	−277762.	−0.357773	−0.178887	−	0.983870i	$-0.557250\pi$
$227$	−277762.	−0.357773	−0.178887	+	0.983870i	$0.557250\pi$
$228$	26291.1	0.0334943
$229$	565012.	0.711981	0.355991	−	0.934489i	$-0.384143\pi$
$229$	565012.	0.711981	0.355991	+	0.934489i	$0.384143\pi$
$230$	−144744.	−0.180419
$231$	204981.	0.252746
$232$	−387336.	−0.472463
$233$	1.44268e6	1.74093	0.870464	−	0.492233i	$-0.163819\pi$
$233$	1.44268e6	1.74093	0.870464	+	0.492233i	$0.163819\pi$
$234$	42154.9	0.0503279
$235$	493577.	0.583023
$236$	−13002.6	−0.0151967
$237$	−79039.6	−0.0914059
$238$	−42195.3	−0.0482861
$239$	969954.	1.09839	0.549195	−	0.835694i	$-0.314934\pi$
$239$	969954.	1.09839	0.549195	+	0.835694i	$0.314934\pi$
$240$	−433636.	−0.485956
$241$	−1.01312e6	−1.12362	−0.561810	−	0.827266i	$-0.689895\pi$
$241$	−1.01312e6	−1.12362	−0.561810	+	0.827266i	$0.689895\pi$
$242$	927799.	1.01839
$243$	−248240.	−0.269685
$244$	−390211.	−0.419590
$245$	402550.	0.428455
$246$	89548.7	0.0943456
$247$	12209.7	0.0127340
$248$	−14671.3	−0.0151474
$249$	1.56642e6	1.60107
$250$	−118082.	−0.119491
$251$	−1.21251e6	−1.21479	−0.607395	−	0.794400i	$-0.707786\pi$
$251$	−1.21251e6	−1.21479	−0.607395	+	0.794400i	$0.707786\pi$
$252$	−22007.6	−0.0218309
$253$	−408148.	−0.400882
$254$	−1.45886e6	−1.41883
$255$	−76182.1	−0.0733673
$256$	1.34603e6	1.28367
$257$	−48654.6	−0.0459506	−0.0229753	−	0.999736i	$-0.507314\pi$
$257$	−48654.6	−0.0459506	−0.0229753	+	0.999736i	$0.507314\pi$
$258$	551945.	0.516234
$259$	−272248.	−0.252183
$260$	106099.	0.0973373
$261$	−245609.	−0.223173
$262$	809172.	0.728262
$263$	−609097.	−0.542996	−0.271498	−	0.962439i	$-0.587519\pi$
$263$	−609097.	−0.542996	−0.271498	+	0.962439i	$0.587519\pi$
$264$	−401852.	−0.354859
$265$	−817246.	−0.714888
$266$	−14496.8	−0.0125623
$267$	484473.	0.415903
$268$	493256.	0.419503
$269$	1.19518e6	1.00705	0.503527	−	0.863980i	$-0.332035\pi$
$269$	1.19518e6	1.00705	0.503527	+	0.863980i	$0.332035\pi$
$270$	−755661.	−0.630837
$271$	−1.07304e6	−0.887553	−0.443776	−	0.896138i	$-0.646362\pi$
$271$	−1.07304e6	−0.887553	−0.443776	+	0.896138i	$0.646362\pi$
$272$	251705.	0.206286
$273$	65024.8	0.0528047
$274$	1.52390e6	1.22626
$275$	−332967.	−0.265503
$276$	−278795.	−0.220299
$277$	524566.	0.410772	0.205386	−	0.978681i	$-0.434155\pi$
$277$	524566.	0.410772	0.205386	+	0.978681i	$0.434155\pi$
$278$	−1.19012e6	−0.923589
$279$	−9303.00	−0.00715505
$280$	34551.8	0.0263376
$281$	942346.	0.711942	0.355971	−	0.934497i	$-0.384150\pi$
$281$	942346.	0.711942	0.355971	+	0.934497i	$0.384150\pi$
$282$	2.16214e6	1.61905
$283$	1.05353e6	0.781951	0.390976	−	0.920401i	$-0.372138\pi$
$283$	1.05353e6	0.781951	0.390976	+	0.920401i	$0.372138\pi$
$284$	−111902.	−0.0823270
$285$	−26173.5	−0.0190875
$286$	680414.	0.491879
$287$	−21711.1	−0.0155588
$288$	243590.	0.173053
$289$	−1.37564e6	−0.968856
$290$	−1.40589e6	−0.981650
$291$	−283363.	−0.196160
$292$	−1.23508e6	−0.847693
$293$	−1.09067e6	−0.742206	−0.371103	−	0.928592i	$-0.621020\pi$
$293$	−1.09067e6	−0.742206	−0.371103	+	0.928592i	$0.621020\pi$
$294$	1.76339e6	1.18982
$295$	12944.4	0.00866021
$296$	533724.	0.354069
$297$	−2.13081e6	−1.40169
$298$	3.49437e6	2.27944
$299$	−129474.	−0.0837539
$300$	−227441.	−0.145904
$301$	−133819.	−0.0851336
$302$	215368.	0.135883
$303$	−1.25964e6	−0.788209
$304$	86477.1	0.0536682
$305$	388466.	0.239113
$306$	52453.2	0.0320234
$307$	1.21527e6	0.735912	0.367956	−	0.929843i	$-0.380058\pi$
$307$	1.21527e6	0.735912	0.367956	+	0.929843i	$0.380058\pi$
$308$	−355220.	−0.213364
$309$	1.86908e6	1.11361
$310$	−53251.4	−0.0314722
$311$	−706897.	−0.414434	−0.207217	−	0.978295i	$-0.566441\pi$
$311$	−706897.	−0.414434	−0.207217	+	0.978295i	$0.566441\pi$
$312$	−127477.	−0.0741386
$313$	−2.44785e6	−1.41229	−0.706146	−	0.708066i	$-0.749568\pi$
$313$	−2.44785e6	−1.41229	−0.706146	+	0.708066i	$0.749568\pi$
$314$	381156.	0.218162
$315$	21909.2	0.0124408
$316$	136971.	0.0771632
$317$	−1.77908e6	−0.994368	−0.497184	−	0.867645i	$-0.665633\pi$
$317$	−1.77908e6	−0.994368	−0.497184	+	0.867645i	$0.665633\pi$
$318$	−3.57998e6	−1.98524
$319$	−3.96432e6	−2.18118
$320$	436764.	0.238436
$321$	2.49254e6	1.35014
$322$	153727.	0.0826249
$323$	15192.5	0.00810257
$324$	−1.25408e6	−0.663688
$325$	−105625.	−0.0554700
$326$	419430.	0.218583
$327$	−2.89522e6	−1.49731
$328$	42563.1	0.0218448
$329$	−524209.	−0.267002
$330$	−1.45858e6	−0.737301
$331$	108028.	0.0541960	0.0270980	−	0.999633i	$-0.491373\pi$
$331$	108028.	0.0541960	0.0270980	+	0.999633i	$0.491373\pi$
$332$	−2.71451e6	−1.35160
$333$	338433.	0.167248
$334$	−3.38851e6	−1.66204
$335$	−491050.	−0.239064
$336$	460548.	0.222549
$337$	1.58459e6	0.760051	0.380026	−	0.924976i	$-0.375915\pi$
$337$	1.58459e6	0.760051	0.380026	+	0.924976i	$0.375915\pi$
$338$	215843.	0.102765
$339$	2.65890e6	1.25662
$340$	132019.	0.0619354
$341$	−150158.	−0.0699298
$342$	18021.1	0.00833135
$343$	−873785.	−0.401023
$344$	262343.	0.119529
$345$	277549.	0.125543
$346$	4.95516e6	2.22519
$347$	−2.45682e6	−1.09534	−0.547672	−	0.836693i	$-0.684486\pi$
$347$	−2.45682e6	−1.09534	−0.547672	+	0.836693i	$0.684486\pi$
$348$	−2.70792e6	−1.19864
$349$	−314003.	−0.137997	−0.0689986	−	0.997617i	$-0.521980\pi$
$349$	−314003.	−0.137997	−0.0689986	+	0.997617i	$0.521980\pi$
$350$	125411.	0.0547223
$351$	−675941.	−0.292847
$352$	3.93174e6	1.69133
$353$	−61151.3	−0.0261197	−0.0130599	−	0.999915i	$-0.504157\pi$
$353$	−61151.3	−0.0261197	−0.0130599	+	0.999915i	$0.504157\pi$
$354$	56703.7	0.0240493
$355$	111402.	0.0469160
$356$	−839562.	−0.351097
$357$	80910.1	0.0335994
$358$	1.43253e6	0.590738
$359$	2.81224e6	1.15164	0.575820	−	0.817577i	$-0.304683\pi$
$359$	2.81224e6	1.15164	0.575820	+	0.817577i	$0.304683\pi$
$360$	−42951.5	−0.0174671
$361$	−2.47088e6	−0.997892
$362$	−3.16221e6	−1.26829
$363$	−1.77907e6	−0.708640
$364$	−112684.	−0.0445768
$365$	1.22956e6	0.483078
$366$	1.70169e6	0.664016
$367$	4.31282e6	1.67146	0.835729	−	0.549141i	$-0.185045\pi$
$367$	4.31282e6	1.67146	0.835729	+	0.549141i	$0.185045\pi$
$368$	−917019.	−0.352987
$369$	26989.1	0.0103186
$370$	1.93723e6	0.735658
$371$	867965.	0.327392
$372$	−102569.	−0.0384289
$373$	3.14923e6	1.17201	0.586006	−	0.810307i	$-0.300700\pi$
$373$	3.14923e6	1.17201	0.586006	+	0.810307i	$0.300700\pi$
$374$	846635.	0.312981
$375$	226424.	0.0831467
$376$	1.02768e6	0.374875
$377$	−1.25757e6	−0.455701
$378$	802558.	0.288899
$379$	−1.39282e6	−0.498076	−0.249038	−	0.968494i	$-0.580114\pi$
$379$	−1.39282e6	−0.498076	−0.249038	+	0.968494i	$0.580114\pi$
$380$	45357.1	0.0161134
$381$	2.79739e6	0.987281
$382$	6.56431e6	2.30161
$383$	−698424.	−0.243289	−0.121644	−	0.992574i	$-0.538817\pi$
$383$	−698424.	−0.243289	−0.121644	+	0.992574i	$0.538817\pi$
$384$	−1.50902e6	−0.522237
$385$	353632.	0.121590
$386$	228473.	0.0780488
$387$	166351.	0.0564608
$388$	491051.	0.165595
$389$	5.50773e6	1.84543	0.922717	−	0.385479i	$-0.125964\pi$
$389$	5.50773e6	1.84543	0.922717	+	0.385479i	$0.125964\pi$
$390$	−462695.	−0.154040
$391$	−161104.	−0.0532923
$392$	838149.	0.275490
$393$	−1.55160e6	−0.506755
$394$	−3.62887e6	−1.17769
$395$	−136358.	−0.0439733
$396$	441575.	0.141503
$397$	1.05208e6	0.335022	0.167511	−	0.985870i	$-0.446427\pi$
$397$	1.05208e6	0.335022	0.167511	+	0.985870i	$0.446427\pi$
$398$	3.60688e6	1.14136
$399$	27797.9	0.00874137
$400$	−748104.	−0.233783
$401$	2.03732e6	0.632700	0.316350	−	0.948643i	$-0.397543\pi$
$401$	2.03732e6	0.632700	0.316350	+	0.948643i	$0.397543\pi$
$402$	−2.15107e6	−0.663879
$403$	−47633.6	−0.0146100
$404$	2.18289e6	0.665392
$405$	1.24848e6	0.378218
$406$	1.49314e6	0.449558
$407$	5.46257e6	1.63460
$408$	−158619.	−0.0471741
$409$	5.53708e6	1.63671	0.818356	−	0.574711i	$-0.194886\pi$
$409$	5.53708e6	1.63671	0.818356	+	0.574711i	$0.194886\pi$
$410$	154489.	0.0453875
$411$	−2.92211e6	−0.853280
$412$	−3.23900e6	−0.940087
$413$	−13747.8	−0.00396605
$414$	−191099.	−0.0547970
$415$	2.70238e6	0.770239
$416$	1.24724e6	0.353359
$417$	2.28207e6	0.642671
$418$	290874.	0.0814263
$419$	−4.95515e6	−1.37886	−0.689432	−	0.724350i	$-0.742140\pi$
$419$	−4.95515e6	−1.37886	−0.689432	+	0.724350i	$0.742140\pi$
$420$	241556.	0.0668183
$421$	−3.53335e6	−0.971587	−0.485794	−	0.874074i	$-0.661469\pi$
$421$	−3.53335e6	−0.971587	−0.485794	+	0.874074i	$0.661469\pi$
$422$	−8.02124e6	−2.19260
$423$	651646.	0.177076
$424$	−1.70159e6	−0.459663
$425$	−131429.	−0.0352954
$426$	488000.	0.130285
$427$	−412575.	−0.109505
$428$	−4.31942e6	−1.13977
$429$	−1.30470e6	−0.342270
$430$	952210.	0.248348
$431$	−6.97008e6	−1.80736	−0.903680	−	0.428208i	$-0.859145\pi$
$431$	−6.97008e6	−1.80736	−0.903680	+	0.428208i	$0.859145\pi$
$432$	−4.78745e6	−1.23423
$433$	6.19127e6	1.58694	0.793469	−	0.608611i	$-0.208273\pi$
$433$	6.19127e6	1.58694	0.793469	+	0.608611i	$0.208273\pi$
$434$	56556.2	0.0144131
$435$	2.69581e6	0.683072
$436$	5.01723e6	1.26400
$437$	−55349.7	−0.0138647
$438$	5.38614e6	1.34150
$439$	−7.67941e6	−1.90181	−0.950903	−	0.309488i	$-0.899842\pi$
$439$	−7.67941e6	−1.90181	−0.950903	+	0.309488i	$0.899842\pi$
$440$	−693271.	−0.170715
$441$	531468.	0.130131
$442$	268572.	0.0653892
$443$	−5.75772e6	−1.39393	−0.696966	−	0.717104i	$-0.745467\pi$
$443$	−5.75772e6	−1.39393	−0.696966	+	0.717104i	$0.745467\pi$
$444$	3.73134e6	0.898271
$445$	835808.	0.200081
$446$	−1.03002e7	−2.45194
$447$	−6.70051e6	−1.58613
$448$	−463870.	−0.109195
$449$	−5.88895e6	−1.37855	−0.689274	−	0.724500i	$-0.742071\pi$
$449$	−5.88895e6	−1.37855	−0.689274	+	0.724500i	$0.742071\pi$
$450$	−155898.	−0.0362919
$451$	435626.	0.100849
$452$	−4.60771e6	−1.06081
$453$	−412972.	−0.0945529
$454$	−2.09912e6	−0.477967
$455$	112180.	0.0254032
$456$	−54495.8	−0.0122730
$457$	−6.89569e6	−1.54450	−0.772249	−	0.635320i	$-0.780868\pi$
$457$	−6.89569e6	−1.54450	−0.772249	+	0.635320i	$0.780868\pi$
$458$	4.26994e6	0.951171
$459$	−841070.	−0.186338
$460$	−480975.	−0.105981
$461$	5.06460e6	1.10992	0.554962	−	0.831876i	$-0.312733\pi$
$461$	5.06460e6	1.10992	0.554962	+	0.831876i	$0.312733\pi$
$462$	1.54910e6	0.337656
$463$	−6.76624e6	−1.46688	−0.733441	−	0.679753i	$-0.762087\pi$
$463$	−6.76624e6	−1.46688	−0.733441	+	0.679753i	$0.762087\pi$
$464$	−8.90695e6	−1.92059
$465$	102110.	0.0218996
$466$	1.09027e7	2.32579
$467$	1.09528e6	0.232398	0.116199	−	0.993226i	$-0.462929\pi$
$467$	1.09528e6	0.232398	0.116199	+	0.993226i	$0.462929\pi$
$468$	140078.	0.0295634
$469$	521526.	0.109482
$470$	3.73009e6	0.778888
$471$	−730872.	−0.151806
$472$	26951.6	0.00556839
$473$	2.68503e6	0.551819
$474$	−597324.	−0.122114
$475$	−45154.3	−0.00918259
$476$	−140212.	−0.0283640
$477$	−1.07897e6	−0.217127
$478$	7.33020e6	1.46739
$479$	386228.	0.0769139	0.0384569	−	0.999260i	$-0.487756\pi$
$479$	386228.	0.0769139	0.0384569	+	0.999260i	$0.487756\pi$
$480$	−2.67366e6	−0.529667
$481$	1.73286e6	0.341507
$482$	−7.65643e6	−1.50110
$483$	−294774.	−0.0574938
$484$	3.08301e6	0.598221
$485$	−488855.	−0.0943683
$486$	−1.87602e6	−0.360285
$487$	3.84993e6	0.735581	0.367791	−	0.929909i	$-0.380114\pi$
$487$	3.84993e6	0.735581	0.367791	+	0.929909i	$0.380114\pi$
$488$	808825.	0.153746
$489$	−804264.	−0.152099
$490$	3.04218e6	0.572394
$491$	5.23475e6	0.979924	0.489962	−	0.871744i	$-0.337011\pi$
$491$	5.23475e6	0.979924	0.489962	+	0.871744i	$0.337011\pi$
$492$	297564.	0.0554202
$493$	−1.56479e6	−0.289961
$494$	92272.1	0.0170119
$495$	−439601.	−0.0806391
$496$	−337372.	−0.0615750
$497$	−118315.	−0.0214857
$498$	1.18379e7	2.13895
$499$	4.55781e6	0.819418	0.409709	−	0.912216i	$-0.365630\pi$
$499$	4.55781e6	0.819418	0.409709	+	0.912216i	$0.365630\pi$
$500$	−392379.	−0.0701909
$501$	6.49751e6	1.15652
$502$	−9.16327e6	−1.62290
$503$	6.80611e6	1.19944	0.599720	−	0.800210i	$-0.295278\pi$
$503$	6.80611e6	1.19944	0.599720	+	0.800210i	$0.295278\pi$
$504$	45617.1	0.00799928
$505$	−2.17313e6	−0.379190
$506$	−3.08449e6	−0.535558
$507$	−413882.	−0.0715083
$508$	−4.84771e6	−0.833445
$509$	1.12204e7	1.91962	0.959808	−	0.280656i	$-0.0905521\pi$
$509$	1.12204e7	1.91962	0.959808	+	0.280656i	$0.0905521\pi$
$510$	−575729.	−0.0980150
$511$	−1.30587e6	−0.221231
$512$	6.83999e6	1.15314
$513$	−288962.	−0.0484784
$514$	−367696.	−0.0613877
$515$	3.22452e6	0.535732
$516$	1.83407e6	0.303244
$517$	1.05181e7	1.73065
$518$	−2.05745e6	−0.336903
$519$	−9.50158e6	−1.54838
$520$	−219922.	−0.0356664
$521$	−8.41292e6	−1.35785	−0.678926	−	0.734207i	$-0.737554\pi$
$521$	−8.41292e6	−1.35785	−0.678926	+	0.734207i	$0.737554\pi$
$522$	−1.85613e6	−0.298148
$523$	−8.26955e6	−1.32199	−0.660994	−	0.750391i	$-0.729865\pi$
$523$	−8.26955e6	−1.32199	−0.660994	+	0.750391i	$0.729865\pi$
$524$	2.68882e6	0.427793
$525$	−240476.	−0.0380780
$526$	−4.60311e6	−0.725415
$527$	−59270.2	−0.00929630
$528$	−9.24074e6	−1.44252
$529$	−5.84940e6	−0.908809
$530$	−6.17615e6	−0.955054
$531$	17089.9	0.00263029
$532$	−48172.0	−0.00737930
$533$	138191.	0.0210698
$534$	3.66129e6	0.555624
$535$	4.30011e6	0.649523
$536$	−1.02242e6	−0.153715
$537$	−2.74689e6	−0.411060
$538$	9.03229e6	1.34537
$539$	8.57831e6	1.27183
$540$	−2.51101e6	−0.370564
$541$	−8.90138e6	−1.30757	−0.653784	−	0.756681i	$-0.726819\pi$
$541$	−8.90138e6	−1.30757	−0.653784	+	0.756681i	$0.726819\pi$
$542$	−8.10928e6	−1.18572
$543$	6.06358e6	0.882531
$544$	1.55193e6	0.224841
$545$	−4.99480e6	−0.720323
$546$	491410.	0.0705444
$547$	2.63555e6	0.376619	0.188310	−	0.982110i	$-0.439699\pi$
$547$	2.63555e6	0.376619	0.188310	+	0.982110i	$0.439699\pi$
$548$	5.06383e6	0.720324
$549$	512873.	0.0726239
$550$	−2.51632e6	−0.354699
$551$	−537608.	−0.0754374
$552$	577884.	0.0807222
$553$	144821.	0.0201381
$554$	3.96429e6	0.548770
$555$	−3.71466e6	−0.511902
$556$	−3.95469e6	−0.542531
$557$	1.16688e7	1.59364	0.796819	−	0.604218i	$-0.206515\pi$
$557$	1.16688e7	1.59364	0.796819	+	0.604218i	$0.206515\pi$
$558$	−70305.3	−0.00955878
$559$	851755.	0.115288
$560$	794532.	0.107063
$561$	−1.62343e6	−0.217785
$562$	7.12156e6	0.951118
$563$	4.77448e6	0.634826	0.317413	−	0.948287i	$-0.397186\pi$
$563$	4.77448e6	0.634826	0.317413	+	0.948287i	$0.397186\pi$
$564$	7.18462e6	0.951056
$565$	4.58711e6	0.604531
$566$	7.96179e6	1.04465
$567$	−1.32596e6	−0.173210
$568$	231949.	0.0301663
$569$	8.02617e6	1.03927	0.519634	−	0.854389i	$-0.326068\pi$
$569$	8.02617e6	1.03927	0.519634	+	0.854389i	$0.326068\pi$
$570$	−197800.	−0.0255000
$571$	1.13893e7	1.46187	0.730933	−	0.682449i	$-0.239085\pi$
$571$	1.13893e7	1.46187	0.730933	+	0.682449i	$0.239085\pi$
$572$	2.26097e6	0.288938
$573$	−1.25872e7	−1.60155
$574$	−164076.	−0.0207858
$575$	478825.	0.0603958
$576$	576639.	0.0724182
$577$	4.99875e6	0.625060	0.312530	−	0.949908i	$-0.398824\pi$
$577$	4.99875e6	0.625060	0.312530	+	0.949908i	$0.398824\pi$
$578$	−1.03961e7	−1.29434
$579$	−438100.	−0.0543096
$580$	−4.67168e6	−0.576637
$581$	−2.87009e6	−0.352740
$582$	−2.14145e6	−0.262060
$583$	−1.74154e7	−2.12209
$584$	2.56006e6	0.310612
$585$	−139452.	−0.0168474
$586$	−8.24248e6	−0.991549
$587$	6.86767e6	0.822649	0.411324	−	0.911489i	$-0.365066\pi$
$587$	6.86767e6	0.822649	0.411324	+	0.911489i	$0.365066\pi$
$588$	5.85962e6	0.698918
$589$	−20363.2	−0.00241856
$590$	97824.6	0.0115696
$591$	6.95841e6	0.819486
$592$	1.22732e7	1.43931
$593$	−1.58920e7	−1.85585	−0.927926	−	0.372765i	$-0.878410\pi$
$593$	−1.58920e7	−1.85585	−0.927926	+	0.372765i	$0.878410\pi$
$594$	−1.61031e7	−1.87259
$595$	139585.	0.0161639
$596$	1.16116e7	1.33898
$597$	−6.91624e6	−0.794208
$598$	−978471.	−0.111891
$599$	1.59662e7	1.81817	0.909085	−	0.416610i	$-0.136782\pi$
$599$	1.59662e7	1.81817	0.909085	+	0.416610i	$0.136782\pi$
$600$	471438.	0.0534621
$601$	7.05052e6	0.796223	0.398112	−	0.917337i	$-0.369666\pi$
$601$	7.05052e6	0.796223	0.398112	+	0.917337i	$0.369666\pi$
$602$	−1.01130e6	−0.113734
$603$	−648310.	−0.0726089
$604$	715654.	0.0798198
$605$	−3.06923e6	−0.340911
$606$	−9.51947e6	−1.05301
$607$	2.47889e6	0.273078	0.136539	−	0.990635i	$-0.456402\pi$
$607$	2.47889e6	0.273078	0.136539	+	0.990635i	$0.456402\pi$
$608$	533190.	0.0584956
$609$	−2.86312e6	−0.312821
$610$	2.93574e6	0.319443
$611$	3.33658e6	0.361575
$612$	174298.	0.0188111
$613$	−4.89084e6	−0.525693	−0.262846	−	0.964838i	$-0.584661\pi$
$613$	−4.89084e6	−0.525693	−0.262846	+	0.964838i	$0.584661\pi$
$614$	9.18410e6	0.983141
$615$	−296234.	−0.0315825
$616$	736296.	0.0781809
$617$	2.68229e6	0.283657	0.141828	−	0.989891i	$-0.454702\pi$
$617$	2.68229e6	0.283657	0.141828	+	0.989891i	$0.454702\pi$
$618$	1.41252e7	1.48772
$619$	−1.01074e7	−1.06026	−0.530128	−	0.847917i	$-0.677856\pi$
$619$	−1.01074e7	−1.06026	−0.530128	+	0.847917i	$0.677856\pi$
$620$	−176951.	−0.0184873
$621$	3.06421e6	0.318852
$622$	−5.34221e6	−0.553662
$623$	−887679.	−0.0916296
$624$	−2.93138e6	−0.301377
$625$	390625.	0.0400000
$626$	−1.84991e7	−1.88675
$627$	−557756.	−0.0566598
$628$	1.26656e6	0.128152
$629$	2.15618e6	0.217300
$630$	165574.	0.0166203
$631$	6.78829e6	0.678714	0.339357	−	0.940658i	$-0.389791\pi$
$631$	6.78829e6	0.678714	0.339357	+	0.940658i	$0.389791\pi$
$632$	−283911.	−0.0282742
$633$	1.53808e7	1.52570
$634$	−1.34450e7	−1.32842
$635$	4.82603e6	0.474959
$636$	−1.18960e7	−1.16616
$637$	2.72124e6	0.265716
$638$	−2.99594e7	−2.91395
$639$	147078.	0.0142494
$640$	−2.60335e6	−0.251236
$641$	−6.16985e6	−0.593102	−0.296551	−	0.955017i	$-0.595837\pi$
$641$	−6.16985e6	−0.593102	−0.296551	+	0.955017i	$0.595837\pi$
$642$	1.88368e7	1.80372
$643$	−3.69735e6	−0.352665	−0.176333	−	0.984331i	$-0.556423\pi$
$643$	−3.69735e6	−0.352665	−0.176333	+	0.984331i	$0.556423\pi$
$644$	510825.	0.0485352
$645$	−1.82588e6	−0.172811
$646$	114814.	0.0108246
$647$	1.58526e7	1.48881	0.744406	−	0.667727i	$-0.232733\pi$
$647$	1.58526e7	1.48881	0.744406	+	0.667727i	$0.232733\pi$
$648$	2.59945e6	0.243189
$649$	275845.	0.0257071
$650$	−798236.	−0.0741051
$651$	−108447.	−0.0100292
$652$	1.39374e6	0.128399
$653$	3.85176e6	0.353489	0.176745	−	0.984257i	$-0.443443\pi$
$653$	3.85176e6	0.353489	0.176745	+	0.984257i	$0.443443\pi$
$654$	−2.18799e7	−2.00033
$655$	−2.67680e6	−0.243788
$656$	978754.	0.0888002
$657$	1.62333e6	0.146721
$658$	−3.96159e6	−0.356701
$659$	2.64699e6	0.237432	0.118716	−	0.992928i	$-0.462122\pi$
$659$	2.64699e6	0.237432	0.118716	+	0.992928i	$0.462122\pi$
$660$	−4.84675e6	−0.433103
$661$	−1.38401e7	−1.23207	−0.616034	−	0.787719i	$-0.711262\pi$
$661$	−1.38401e7	−1.23207	−0.616034	+	0.787719i	$0.711262\pi$
$662$	816397.	0.0724030
$663$	−514991.	−0.0455005
$664$	5.62661e6	0.495253
$665$	47956.6	0.00420528
$666$	2.55763e6	0.223435
$667$	5.70090e6	0.496168
$668$	−1.12598e7	−0.976312
$669$	1.97508e7	1.70616
$670$	−3.71100e6	−0.319377
$671$	8.27818e6	0.709788
$672$	2.83959e6	0.242568
$673$	−1.64904e7	−1.40344	−0.701721	−	0.712452i	$-0.747585\pi$
$673$	−1.64904e7	−1.40344	−0.701721	+	0.712452i	$0.747585\pi$
$674$	1.19752e7	1.01539
$675$	2.49978e6	0.211175
$676$	717231.	0.0603660
$677$	−9.13772e6	−0.766242	−0.383121	−	0.923698i	$-0.625151\pi$
$677$	−9.13772e6	−0.766242	−0.383121	+	0.923698i	$0.625151\pi$
$678$	2.00940e7	1.67878
$679$	519194.	0.0432171
$680$	−273647.	−0.0226944
$681$	4.02509e6	0.332589
$682$	−1.13478e6	−0.0934226
$683$	1.34711e7	1.10498	0.552488	−	0.833521i	$-0.313679\pi$
$683$	1.34711e7	1.10498	0.552488	+	0.833521i	$0.313679\pi$
$684$	59882.8	0.00489397
$685$	−5.04119e6	−0.410494
$686$	−6.60342e6	−0.535746
$687$	−8.18767e6	−0.661864
$688$	6.03268e6	0.485891
$689$	−5.52458e6	−0.443355
$690$	2.09751e6	0.167719
$691$	−1.06017e7	−0.844659	−0.422329	−	0.906442i	$-0.638787\pi$
$691$	−1.06017e7	−0.844659	−0.422329	+	0.906442i	$0.638787\pi$
$692$	1.64656e7	1.30711
$693$	466883.	0.0369296
$694$	−1.85669e7	−1.46332
$695$	3.93701e6	0.309175
$696$	5.61295e6	0.439206
$697$	171950.	0.0134066
$698$	−2.37301e6	−0.184357
$699$	−2.09061e7	−1.61838
$700$	416731.	0.0321448
$701$	−2.46173e7	−1.89211	−0.946053	−	0.324011i	$-0.894969\pi$
$701$	−2.46173e7	−1.89211	−0.946053	+	0.324011i	$0.894969\pi$
$702$	−5.10827e6	−0.391229
$703$	740790.	0.0565336
$704$	9.30741e6	0.707778
$705$	−7.15251e6	−0.541983
$706$	−462136.	−0.0348946
$707$	2.30799e6	0.173654
$708$	188422.	0.0141270
$709$	−1.27065e7	−0.949317	−0.474658	−	0.880170i	$-0.657428\pi$
$709$	−1.27065e7	−0.949317	−0.474658	+	0.880170i	$0.657428\pi$
$710$	841893.	0.0626774
$711$	−180027.	−0.0133556
$712$	1.74023e6	0.128649
$713$	215935.	0.0159074
$714$	611459.	0.0448871
$715$	−2.25086e6	−0.164658
$716$	4.76018e6	0.347009
$717$	−1.40558e7	−1.02107
$718$	2.12529e7	1.53853
$719$	−4.89738e6	−0.353299	−0.176649	−	0.984274i	$-0.556526\pi$
$719$	−4.89738e6	−0.353299	−0.176649	+	0.984274i	$0.556526\pi$
$720$	−987686.	−0.0710048
$721$	−3.42464e6	−0.245345
$722$	−1.86731e7	−1.33313
$723$	1.46813e7	1.04453
$724$	−1.05078e7	−0.745016
$725$	4.65079e6	0.328611
$726$	−1.34449e7	−0.946707
$727$	−1.16983e7	−0.820892	−0.410446	−	0.911885i	$-0.634627\pi$
$727$	−1.16983e7	−0.820892	−0.410446	+	0.911885i	$0.634627\pi$
$728$	233570.	0.0163339
$729$	1.57325e7	1.09642
$730$	9.29211e6	0.645368
$731$	1.05983e6	0.0733575
$732$	5.65461e6	0.390054
$733$	955290.	0.0656713	0.0328356	−	0.999461i	$-0.489546\pi$
$733$	955290.	0.0656713	0.0328356	+	0.999461i	$0.489546\pi$
$734$	3.25931e7	2.23298
$735$	−5.83342e6	−0.398295
$736$	−5.65405e6	−0.384738
$737$	−1.04642e7	−0.709642
$738$	203964.	0.0137852
$739$	−2.92010e6	−0.196692	−0.0983461	−	0.995152i	$-0.531355\pi$
$739$	−2.92010e6	−0.196692	−0.0983461	+	0.995152i	$0.531355\pi$
$740$	6.43727e6	0.432138
$741$	−176933.	−0.0118376
$742$	6.55944e6	0.437378
$743$	−1.66149e7	−1.10415	−0.552073	−	0.833796i	$-0.686163\pi$
$743$	−1.66149e7	−1.10415	−0.552073	+	0.833796i	$0.686163\pi$
$744$	212604.	0.0140812
$745$	−1.15597e7	−0.763052
$746$	2.37995e7	1.56575
$747$	3.56782e6	0.233938
$748$	2.81331e6	0.183850
$749$	−4.56697e6	−0.297457
$750$	1.71115e6	0.111080
$751$	2.21810e6	0.143509	0.0717547	−	0.997422i	$-0.477140\pi$
$751$	2.21810e6	0.143509	0.0717547	+	0.997422i	$0.477140\pi$
$752$	2.36318e7	1.52389
$753$	1.75707e7	1.12928
$754$	−9.50382e6	−0.608793
$755$	−712455.	−0.0454873
$756$	2.66684e6	0.169704
$757$	−1.10448e7	−0.700515	−0.350257	−	0.936654i	$-0.613906\pi$
$757$	−1.10448e7	−0.700515	−0.350257	+	0.936654i	$0.613906\pi$
$758$	−1.05259e7	−0.665404
$759$	5.91454e6	0.372663
$760$	−94015.7	−0.00590427
$761$	−2.91615e7	−1.82536	−0.912680	−	0.408674i	$-0.865991\pi$
$761$	−2.91615e7	−1.82536	−0.912680	+	0.408674i	$0.865991\pi$
$762$	2.11406e7	1.31896
$763$	5.30478e6	0.329880
$764$	2.18128e7	1.35200
$765$	−173519.	−0.0107200
$766$	−5.27818e6	−0.325021
$767$	87504.5	0.00537083
$768$	−1.95055e7	−1.19331
$769$	−1.13504e7	−0.692144	−0.346072	−	0.938208i	$-0.612485\pi$
$769$	−1.13504e7	−0.692144	−0.346072	+	0.938208i	$0.612485\pi$
$770$	2.67249e6	0.162438
$771$	705062.	0.0427161
$772$	759199.	0.0458472
$773$	−4.87794e6	−0.293621	−0.146811	−	0.989165i	$-0.546901\pi$
$773$	−4.87794e6	−0.293621	−0.146811	+	0.989165i	$0.546901\pi$
$774$	1.25716e6	0.0754288
$775$	176160.	0.0105354
$776$	−1.01784e6	−0.0606774
$777$	3.94519e6	0.234431
$778$	4.16234e7	2.46540
$779$	59076.0	0.00348793
$780$	−1.53750e6	−0.0904856
$781$	2.37396e6	0.139266
$782$	−1.21751e6	−0.0711958
$783$	2.97625e7	1.73486
$784$	1.92736e7	1.11988
$785$	−1.26089e6	−0.0730305
$786$	−1.17258e7	−0.676998
$787$	−2.27090e7	−1.30696	−0.653478	−	0.756946i	$-0.726691\pi$
$787$	−2.27090e7	−1.30696	−0.653478	+	0.756946i	$0.726691\pi$
$788$	−1.20585e7	−0.691795
$789$	8.82652e6	0.504774
$790$	−1.03050e6	−0.0587461
$791$	−4.87179e6	−0.276852
$792$	−915292.	−0.0518497
$793$	2.62603e6	0.148292
$794$	7.95086e6	0.447572
$795$	1.18428e7	0.664566
$796$	1.19854e7	0.670456
$797$	−2.03178e7	−1.13300	−0.566500	−	0.824062i	$-0.691703\pi$
$797$	−2.03178e7	−1.13300	−0.566500	+	0.824062i	$0.691703\pi$
$798$	210076.	0.0116780
$799$	4.15169e6	0.230069
$800$	−4.61257e6	−0.254811
$801$	1.10348e6	0.0607690
$802$	1.53966e7	0.845255
$803$	2.62018e7	1.43398
$804$	−7.14785e6	−0.389973
$805$	−508541.	−0.0276590
$806$	−359980.	−0.0195182
$807$	−1.73195e7	−0.936166
$808$	−4.52466e6	−0.243813
$809$	1.28692e6	0.0691321	0.0345661	−	0.999402i	$-0.488995\pi$
$809$	1.28692e6	0.0691321	0.0345661	+	0.999402i	$0.488995\pi$
$810$	9.43507e6	0.505280
$811$	1.70128e7	0.908286	0.454143	−	0.890929i	$-0.349946\pi$
$811$	1.70128e7	0.908286	0.454143	+	0.890929i	$0.349946\pi$
$812$	4.96160e6	0.264078
$813$	1.55496e7	0.825076
$814$	4.12821e7	2.18374
$815$	−1.38751e6	−0.0731714
$816$	−3.64750e6	−0.191765
$817$	364122.	0.0190850
$818$	4.18452e7	2.18656
$819$	148106.	0.00771548
$820$	513355.	0.0266614
$821$	−9.80056e6	−0.507450	−0.253725	−	0.967276i	$-0.581656\pi$
$821$	−9.80056e6	−0.507450	−0.253725	+	0.967276i	$0.581656\pi$
$822$	−2.20831e7	−1.13994
$823$	2.17586e7	1.11977	0.559887	−	0.828569i	$-0.310844\pi$
$823$	2.17586e7	1.11977	0.559887	+	0.828569i	$0.310844\pi$
$824$	6.71377e6	0.344468
$825$	4.82508e6	0.246814
$826$	−103896.	−0.00529843
$827$	8.35521e6	0.424809	0.212404	−	0.977182i	$-0.431871\pi$
$827$	8.35521e6	0.424809	0.212404	+	0.977182i	$0.431871\pi$
$828$	−635008.	−0.0321887
$829$	−2.68021e7	−1.35451	−0.677255	−	0.735748i	$-0.736831\pi$
$829$	−2.68021e7	−1.35451	−0.677255	+	0.735748i	$0.736831\pi$
$830$	2.04226e7	1.02900
$831$	−7.60158e6	−0.381857
$832$	2.95253e6	0.147872
$833$	3.38603e6	0.169074
$834$	1.72462e7	0.858576
$835$	1.12094e7	0.556375
$836$	966555.	0.0478312
$837$	1.12732e6	0.0556205
$838$	−3.74474e7	−1.84209
$839$	−3.73402e7	−1.83135	−0.915675	−	0.401919i	$-0.868343\pi$
$839$	−3.73402e7	−1.83135	−0.915675	+	0.401919i	$0.868343\pi$
$840$	−500695.	−0.0244836
$841$	3.48613e7	1.69963
$842$	−2.67025e7	−1.29799
$843$	−1.36557e7	−0.661827
$844$	−2.66540e7	−1.28797
$845$	−714025.	−0.0344010
$846$	4.92466e6	0.236565
$847$	3.25971e6	0.156124
$848$	−3.91287e7	−1.86855
$849$	−1.52668e7	−0.726908
$850$	−993241.	−0.0471528
$851$	−7.85547e6	−0.371833
$852$	1.62159e6	0.0765318
$853$	−7.51191e6	−0.353490	−0.176745	−	0.984257i	$-0.556557\pi$
$853$	−7.51191e6	−0.353490	−0.176745	+	0.984257i	$0.556557\pi$
$854$	−3.11794e6	−0.146293
$855$	−59615.1	−0.00278895
$856$	8.95324e6	0.417634
$857$	−3.45535e7	−1.60709	−0.803545	−	0.595243i	$-0.797056\pi$
$857$	−3.45535e7	−1.60709	−0.803545	+	0.595243i	$0.797056\pi$
$858$	−9.85998e6	−0.457254
$859$	3.29796e7	1.52498	0.762488	−	0.647003i	$-0.223978\pi$
$859$	3.29796e7	1.52498	0.762488	+	0.647003i	$0.223978\pi$
$860$	3.16413e6	0.145884
$861$	314619.	0.0144636
$862$	−5.26748e7	−2.41454
$863$	−3.10299e7	−1.41825	−0.709126	−	0.705081i	$-0.750910\pi$
$863$	−3.10299e7	−1.41825	−0.709126	+	0.705081i	$0.750910\pi$
$864$	−2.95179e7	−1.34524
$865$	−1.63920e7	−0.744891
$866$	4.67890e7	2.12007
$867$	1.99346e7	0.900657
$868$	187932.	0.00846647
$869$	−2.90578e6	−0.130531
$870$	2.03730e7	0.912550
$871$	−3.31950e6	−0.148261
$872$	−1.03997e7	−0.463157
$873$	−645412.	−0.0286617
$874$	−418293.	−0.0185226
$875$	−414868.	−0.0183185
$876$	1.78978e7	0.788022
$877$	6.39006e6	0.280547	0.140274	−	0.990113i	$-0.455202\pi$
$877$	6.39006e6	0.280547	0.140274	+	0.990113i	$0.455202\pi$
$878$	−5.80353e7	−2.54072
$879$	1.58051e7	0.689961
$880$	−1.59420e7	−0.693964
$881$	3.21882e7	1.39719	0.698597	−	0.715515i	$-0.253808\pi$
$881$	3.21882e7	1.39719	0.698597	+	0.715515i	$0.253808\pi$
$882$	4.01644e6	0.173848
$883$	2.21100e7	0.954305	0.477153	−	0.878820i	$-0.341669\pi$
$883$	2.21100e7	0.954305	0.477153	+	0.878820i	$0.341669\pi$
$884$	892447.	0.0384107
$885$	−187580.	−0.00805060
$886$	−4.35126e7	−1.86222
$887$	−2.19233e7	−0.935614	−0.467807	−	0.883831i	$-0.654956\pi$
$887$	−2.19233e7	−0.935614	−0.467807	+	0.883831i	$0.654956\pi$
$888$	−7.73428e6	−0.329145
$889$	−5.12554e6	−0.217513
$890$	6.31643e6	0.267298
$891$	2.66049e7	1.12271
$892$	−3.42269e7	−1.44031
$893$	1.42638e6	0.0598557
$894$	−5.06375e7	−2.11899
$895$	−4.73890e6	−0.197752
$896$	2.76492e6	0.115057
$897$	1.87623e6	0.0778583
$898$	−4.45044e7	−1.84167
$899$	2.09736e6	0.0865514
$900$	−518039.	−0.0213185
$901$	−6.87421e6	−0.282105
$902$	3.29214e6	0.134729
$903$	1.93919e6	0.0791409
$904$	9.55082e6	0.388704
$905$	1.04608e7	0.424566
$906$	−3.12094e6	−0.126318
$907$	1.61222e7	0.650738	0.325369	−	0.945587i	$-0.394512\pi$
$907$	1.61222e7	0.650738	0.325369	+	0.945587i	$0.394512\pi$
$908$	−6.97523e6	−0.280765
$909$	−2.86907e6	−0.115168
$910$	847775.	0.0339373
$911$	2.05000e7	0.818385	0.409192	−	0.912448i	$-0.365810\pi$
$911$	2.05000e7	0.818385	0.409192	+	0.912448i	$0.365810\pi$
$912$	−1.25315e6	−0.0498904
$913$	5.75874e7	2.28639
$914$	−5.21126e7	−2.06337
$915$	−5.62933e6	−0.222282
$916$	1.41887e7	0.558733
$917$	2.84292e6	0.111646
$918$	−6.35619e6	−0.248937
$919$	−3.26676e7	−1.27593	−0.637967	−	0.770064i	$-0.720224\pi$
$919$	−3.26676e7	−1.27593	−0.637967	+	0.770064i	$0.720224\pi$
$920$	996960.	0.0388336
$921$	−1.76106e7	−0.684110
$922$	3.82746e7	1.48280
$923$	753076.	0.0290961
$924$	5.14754e6	0.198345
$925$	−6.40849e6	−0.246264
$926$	−5.11343e7	−1.95968
$927$	4.25718e6	0.162713
$928$	−5.49174e7	−2.09334
$929$	1.19596e7	0.454651	0.227326	−	0.973819i	$-0.427002\pi$
$929$	1.19596e7	0.454651	0.227326	+	0.973819i	$0.427002\pi$
$930$	771675.	0.0292568
$931$	1.16332e6	0.0439871
$932$	3.62290e7	1.36621
$933$	1.02438e7	0.385261
$934$	8.27730e6	0.310471
$935$	−2.80073e6	−0.104771
$936$	−290352.	−0.0108327
$937$	−5.90932e6	−0.219882	−0.109941	−	0.993938i	$-0.535066\pi$
$937$	−5.90932e6	−0.219882	−0.109941	+	0.993938i	$0.535066\pi$
$938$	3.94131e6	0.146263
$939$	3.54722e7	1.31288
$940$	1.23948e7	0.457532
$941$	1.51152e7	0.556466	0.278233	−	0.960514i	$-0.410251\pi$
$941$	1.51152e7	0.556466	0.278233	+	0.960514i	$0.410251\pi$
$942$	−5.52339e6	−0.202805
$943$	−626452.	−0.0229408
$944$	619763.	0.0226358
$945$	−2.65492e6	−0.0967101
$946$	2.02915e7	0.737202
$947$	−2.44913e6	−0.0887435	−0.0443718	−	0.999015i	$-0.514129\pi$
$947$	−2.44913e6	−0.0887435	−0.0443718	+	0.999015i	$0.514129\pi$
$948$	−1.98486e6	−0.0717315
$949$	8.31182e6	0.299592
$950$	−341243.	−0.0122675
$951$	2.57809e7	0.924372
$952$	290630.	0.0103932
$953$	2.21998e7	0.791801	0.395900	−	0.918293i	$-0.370433\pi$
$953$	2.21998e7	0.791801	0.395900	+	0.918293i	$0.370433\pi$
$954$	−8.15407e6	−0.290070
$955$	−2.17152e7	−0.770471
$956$	2.43577e7	0.861971
$957$	5.74476e7	2.02764
$958$	2.91882e6	0.102753
$959$	5.35405e6	0.187991
$960$	−6.32922e6	−0.221652
$961$	−2.85497e7	−0.997225
$962$	1.30957e7	0.456236
$963$	5.67722e6	0.197274
$964$	−2.54418e7	−0.881770
$965$	−755805.	−0.0261271
$966$	−2.22768e6	−0.0768088
$967$	−4.81604e7	−1.65624	−0.828121	−	0.560549i	$-0.810590\pi$
$967$	−4.81604e7	−1.65624	−0.828121	+	0.560549i	$0.810590\pi$
$968$	−6.39044e6	−0.219201
$969$	−220157.	−0.00753222
$970$	−3.69441e6	−0.126071
$971$	4.34935e7	1.48039	0.740194	−	0.672393i	$-0.234733\pi$
$971$	4.34935e7	1.48039	0.740194	+	0.672393i	$0.234733\pi$
$972$	−6.23388e6	−0.211638
$973$	−4.18134e6	−0.141590
$974$	2.90950e7	0.982699
$975$	1.53063e6	0.0515654
$976$	1.85992e7	0.624987
$977$	2.55776e7	0.857281	0.428640	−	0.903475i	$-0.358993\pi$
$977$	2.55776e7	0.857281	0.428640	+	0.903475i	$0.358993\pi$
$978$	−6.07803e6	−0.203196
$979$	1.78110e7	0.593925
$980$	1.01090e7	0.336233
$981$	−6.59440e6	−0.218777
$982$	3.95604e7	1.30913
$983$	−3.90936e7	−1.29039	−0.645197	−	0.764016i	$-0.723225\pi$
$983$	−3.90936e7	−1.29039	−0.645197	+	0.764016i	$0.723225\pi$
$984$	−616788.	−0.0203071
$985$	1.20046e7	0.394236
$986$	−1.18256e7	−0.387373
$987$	7.59640e6	0.248207
$988$	306614.	0.00999308
$989$	−3.86122e6	−0.125526
$990$	−3.32218e6	−0.107730
$991$	2.39821e7	0.775716	0.387858	−	0.921719i	$-0.373215\pi$
$991$	2.39821e7	0.775716	0.387858	+	0.921719i	$0.373215\pi$
$992$	−2.08013e6	−0.0671136
$993$	−1.56545e6	−0.0503810
$994$	−894141.	−0.0287039
$995$	−1.19318e7	−0.382076
$996$	3.93364e7	1.25645
$997$	537946.	0.0171396	0.00856981	−	0.999963i	$-0.497272\pi$
$997$	537946.	0.0171396	0.00856981	+	0.999963i	$0.497272\pi$
$998$	3.44446e7	1.09470
$999$	−4.10108e7	−1.30012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.6.a.c.1.4	✓	4
3.2	odd	2		585.6.a.g.1.1		4
4.3	odd	2		1040.6.a.o.1.3		4
5.2	odd	4		325.6.b.e.274.6		8
5.3	odd	4		325.6.b.e.274.3		8
5.4	even	2		325.6.a.e.1.1		4
13.12	even	2		845.6.a.f.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.c.1.4	✓	4	1.1	even	1	trivial
325.6.a.e.1.1		4	5.4	even	2
325.6.b.e.274.3		8	5.3	odd	4
325.6.b.e.274.6		8	5.2	odd	4
585.6.a.g.1.1		4	3.2	odd	2
845.6.a.f.1.1		4	13.12	even	2
1040.6.a.o.1.3		4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+7.55727 q^{2} -14.4912 q^{3} +25.1123 q^{4} -25.0000 q^{5} -109.514 q^{6} +26.5515 q^{7} -52.0524 q^{8} -33.0063 q^{9} -188.932 q^{10} -532.748 q^{11} -363.906 q^{12} -169.000 q^{13} +200.657 q^{14} +362.279 q^{15} -1196.97 q^{16} -210.286 q^{17} -249.437 q^{18} -72.2469 q^{19} -627.807 q^{20} -384.762 q^{21} -4026.12 q^{22} +766.119 q^{23} +754.300 q^{24} +625.000 q^{25} -1277.18 q^{26} +3999.65 q^{27} +666.769 q^{28} +7441.26 q^{29} +2737.84 q^{30} +281.855 q^{31} -7380.12 q^{32} +7720.13 q^{33} -1589.19 q^{34} -663.788 q^{35} -828.863 q^{36} -10253.6 q^{37} -545.989 q^{38} +2449.01 q^{39} +1301.31 q^{40} -817.696 q^{41} -2907.75 q^{42} -5039.97 q^{43} -13378.5 q^{44} +825.157 q^{45} +5789.77 q^{46} -19743.1 q^{47} +17345.4 q^{48} -16102.0 q^{49} +4723.29 q^{50} +3047.29 q^{51} -4243.97 q^{52} +32689.8 q^{53} +30226.4 q^{54} +13318.7 q^{55} -1382.07 q^{56} +1046.94 q^{57} +56235.6 q^{58} -517.778 q^{59} +9097.65 q^{60} -15538.7 q^{61} +2130.06 q^{62} -876.367 q^{63} -17470.6 q^{64} +4225.00 q^{65} +58343.1 q^{66} +19642.0 q^{67} -5280.75 q^{68} -11102.0 q^{69} -5016.42 q^{70} -4456.07 q^{71} +1718.06 q^{72} -49182.4 q^{73} -77489.1 q^{74} -9056.97 q^{75} -1814.28 q^{76} -14145.3 q^{77} +18507.8 q^{78} +5454.34 q^{79} +29924.2 q^{80} -49939.1 q^{81} -6179.54 q^{82} -108095. q^{83} -9662.25 q^{84} +5257.15 q^{85} -38088.4 q^{86} -107833. q^{87} +27730.8 q^{88} -33432.3 q^{89} +6235.93 q^{90} -4487.21 q^{91} +19239.0 q^{92} -4084.41 q^{93} -149204. q^{94} +1806.17 q^{95} +106946. q^{96} +19554.2 q^{97} -121687. q^{98} +17584.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 9 q^{2} - 4 q^{3} + 93 q^{4} - 100 q^{5} - 74 q^{6} - 136 q^{7} - 447 q^{8} + 644 q^{9} + 225 q^{10} - 516 q^{11} - 710 q^{12} - 676 q^{13} + 1604 q^{14} + 100 q^{15} - 719 q^{16} + 344 q^{17} - 6905 q^{18}+ \cdots + 290668 q^{99}+O(q^{100}) \) 4 * q - 9 * q^2 - 4 * q^3 + 93 * q^4 - 100 * q^5 - 74 * q^6 - 136 * q^7 - 447 * q^8 + 644 * q^9 + 225 * q^10 - 516 * q^11 - 710 * q^12 - 676 * q^13 + 1604 * q^14 + 100 * q^15 - 719 * q^16 + 344 * q^17 - 6905 * q^18 - 5012 * q^19 - 2325 * q^20 - 4448 * q^21 - 9318 * q^22 + 708 * q^23 - 2094 * q^24 + 2500 * q^25 + 1521 * q^26 - 4264 * q^27 - 7268 * q^28 + 5416 * q^29 + 1850 * q^30 - 10124 * q^31 + 9497 * q^32 - 16256 * q^33 + 3022 * q^34 + 3400 * q^35 + 36205 * q^36 + 1448 * q^37 + 11922 * q^38 + 676 * q^39 + 11175 * q^40 + 21576 * q^41 + 33128 * q^42 - 25740 * q^43 + 24710 * q^44 - 16100 * q^45 + 28754 * q^46 - 51096 * q^47 + 43282 * q^48 - 40412 * q^49 - 5625 * q^50 + 52984 * q^51 - 15717 * q^52 + 39448 * q^53 + 60340 * q^54 + 12900 * q^55 + 30188 * q^56 - 56160 * q^57 + 1374 * q^58 + 37764 * q^59 + 17750 * q^60 + 95016 * q^61 + 83102 * q^62 - 43592 * q^63 - 142095 * q^64 + 16900 * q^65 + 198500 * q^66 - 50592 * q^67 - 47110 * q^68 - 177648 * q^69 - 40100 * q^70 - 89308 * q^71 - 99183 * q^72 - 108312 * q^73 - 16766 * q^74 - 2500 * q^75 - 3442 * q^76 + 6208 * q^77 + 12506 * q^78 - 25464 * q^79 + 17975 * q^80 - 106348 * q^81 - 206426 * q^82 + 20584 * q^83 - 213448 * q^84 - 8600 * q^85 + 363666 * q^86 - 128312 * q^87 + 20558 * q^88 - 148328 * q^89 + 172625 * q^90 + 22984 * q^91 - 173474 * q^92 - 10496 * q^93 + 63916 * q^94 + 125300 * q^95 + 85154 * q^96 - 77816 * q^97 - 83505 * q^98 + 290668 * q^99

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