LMFDB - Embedded newform 105.4.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [105,4,Mod(1,105)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(105, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("105.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 105 = 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	105.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:	$6.19520055060$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.53113$ of defining polynomial
Character	$\chi$	$=$	105.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.53113 q^{2} +3.00000 q^{3} +12.5311 q^{4} +5.00000 q^{5} +13.5934 q^{6} -7.00000 q^{7} +20.5311 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.53113 q^{2} +3.00000 q^{3} +12.5311 q^{4} +5.00000 q^{5} +13.5934 q^{6} -7.00000 q^{7} +20.5311 q^{8} +9.00000 q^{9} +22.6556 q^{10} -19.0623 q^{11} +37.5934 q^{12} -2.93774 q^{13} -31.7179 q^{14} +15.0000 q^{15} -7.21984 q^{16} -6.49806 q^{17} +40.7802 q^{18} -5.43580 q^{19} +62.6556 q^{20} -21.0000 q^{21} -86.3735 q^{22} +49.3774 q^{23} +61.5934 q^{24} +25.0000 q^{25} -13.3113 q^{26} +27.0000 q^{27} -87.7179 q^{28} -291.494 q^{29} +67.9669 q^{30} +244.307 q^{31} -196.963 q^{32} -57.1868 q^{33} -29.4436 q^{34} -35.0000 q^{35} +112.780 q^{36} -193.121 q^{37} -24.6303 q^{38} -8.81323 q^{39} +102.656 q^{40} +315.113 q^{41} -95.1537 q^{42} -300.996 q^{43} -238.872 q^{44} +45.0000 q^{45} +223.735 q^{46} +86.5058 q^{47} -21.6595 q^{48} +49.0000 q^{49} +113.278 q^{50} -19.4942 q^{51} -36.8132 q^{52} +509.677 q^{53} +122.340 q^{54} -95.3113 q^{55} -143.718 q^{56} -16.3074 q^{57} -1320.80 q^{58} -83.3852 q^{59} +187.967 q^{60} -5.25291 q^{61} +1106.99 q^{62} -63.0000 q^{63} -834.706 q^{64} -14.6887 q^{65} -259.121 q^{66} +205.992 q^{67} -81.4281 q^{68} +148.132 q^{69} -158.590 q^{70} +1004.31 q^{71} +184.780 q^{72} -1007.29 q^{73} -875.055 q^{74} +75.0000 q^{75} -68.1168 q^{76} +133.436 q^{77} -39.9339 q^{78} -863.237 q^{79} -36.0992 q^{80} +81.0000 q^{81} +1427.82 q^{82} +1334.72 q^{83} -263.154 q^{84} -32.4903 q^{85} -1363.85 q^{86} -874.483 q^{87} -391.370 q^{88} +326.249 q^{89} +203.901 q^{90} +20.5642 q^{91} +618.755 q^{92} +732.922 q^{93} +391.969 q^{94} -27.1790 q^{95} -590.889 q^{96} +1526.77 q^{97} +222.025 q^{98} -171.560 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 3 q^{6} - 14 q^{7} + 33 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 + 3 * q^6 - 14 * q^7 + 33 * q^8 + 18 * q^9	\( 2 q + q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 3 q^{6} - 14 q^{7} + 33 q^{8} + 18 q^{9} + 5 q^{10} - 22 q^{11} + 51 q^{12} - 22 q^{13} - 7 q^{14} + 30 q^{15} - 87 q^{16} + 116 q^{17} + 9 q^{18} + 102 q^{19} + 85 q^{20} - 42 q^{21} - 76 q^{22} + 260 q^{23} + 99 q^{24} + 50 q^{25} + 54 q^{26} + 54 q^{27} - 119 q^{28} - 196 q^{29} + 15 q^{30} + 150 q^{31} - 15 q^{32} - 66 q^{33} - 462 q^{34} - 70 q^{35} + 153 q^{36} - 96 q^{37} - 404 q^{38} - 66 q^{39} + 165 q^{40} - 176 q^{41} - 21 q^{42} - 344 q^{43} - 252 q^{44} + 90 q^{45} - 520 q^{46} + 560 q^{47} - 261 q^{48} + 98 q^{49} + 25 q^{50} + 348 q^{51} - 122 q^{52} + 326 q^{53} + 27 q^{54} - 110 q^{55} - 231 q^{56} + 306 q^{57} - 1658 q^{58} - 844 q^{59} + 255 q^{60} - 204 q^{61} + 1440 q^{62} - 126 q^{63} - 839 q^{64} - 110 q^{65} - 228 q^{66} - 104 q^{67} + 466 q^{68} + 780 q^{69} - 35 q^{70} + 1670 q^{71} + 297 q^{72} - 386 q^{73} - 1218 q^{74} + 150 q^{75} + 412 q^{76} + 154 q^{77} + 162 q^{78} - 888 q^{79} - 435 q^{80} + 162 q^{81} + 3162 q^{82} + 928 q^{83} - 357 q^{84} + 580 q^{85} - 1212 q^{86} - 588 q^{87} - 428 q^{88} + 588 q^{89} + 45 q^{90} + 154 q^{91} + 1560 q^{92} + 450 q^{93} - 1280 q^{94} + 510 q^{95} - 45 q^{96} + 522 q^{97} + 49 q^{98} - 198 q^{99}+O(q^{100}) \) 2 * q + q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 + 3 * q^6 - 14 * q^7 + 33 * q^8 + 18 * q^9 + 5 * q^10 - 22 * q^11 + 51 * q^12 - 22 * q^13 - 7 * q^14 + 30 * q^15 - 87 * q^16 + 116 * q^17 + 9 * q^18 + 102 * q^19 + 85 * q^20 - 42 * q^21 - 76 * q^22 + 260 * q^23 + 99 * q^24 + 50 * q^25 + 54 * q^26 + 54 * q^27 - 119 * q^28 - 196 * q^29 + 15 * q^30 + 150 * q^31 - 15 * q^32 - 66 * q^33 - 462 * q^34 - 70 * q^35 + 153 * q^36 - 96 * q^37 - 404 * q^38 - 66 * q^39 + 165 * q^40 - 176 * q^41 - 21 * q^42 - 344 * q^43 - 252 * q^44 + 90 * q^45 - 520 * q^46 + 560 * q^47 - 261 * q^48 + 98 * q^49 + 25 * q^50 + 348 * q^51 - 122 * q^52 + 326 * q^53 + 27 * q^54 - 110 * q^55 - 231 * q^56 + 306 * q^57 - 1658 * q^58 - 844 * q^59 + 255 * q^60 - 204 * q^61 + 1440 * q^62 - 126 * q^63 - 839 * q^64 - 110 * q^65 - 228 * q^66 - 104 * q^67 + 466 * q^68 + 780 * q^69 - 35 * q^70 + 1670 * q^71 + 297 * q^72 - 386 * q^73 - 1218 * q^74 + 150 * q^75 + 412 * q^76 + 154 * q^77 + 162 * q^78 - 888 * q^79 - 435 * q^80 + 162 * q^81 + 3162 * q^82 + 928 * q^83 - 357 * q^84 + 580 * q^85 - 1212 * q^86 - 588 * q^87 - 428 * q^88 + 588 * q^89 + 45 * q^90 + 154 * q^91 + 1560 * q^92 + 450 * q^93 - 1280 * q^94 + 510 * q^95 - 45 * q^96 + 522 * q^97 + 49 * q^98 - 198 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.53113	1.60200	0.800998	−	0.598667i	$-0.204303\pi$
$2$	4.53113	1.60200	0.800998	+	0.598667i	$0.204303\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	12.5311	1.56639
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	13.5934	0.924913
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	20.5311	0.907356
$9$	9.00000	0.333333
$10$	22.6556	0.716434
$11$	−19.0623	−0.522499	−0.261249	−	0.965271i	$-0.584135\pi$
$11$	−19.0623	−0.522499	−0.261249	+	0.965271i	$0.584135\pi$
$12$	37.5934	0.904356
$13$	−2.93774	−0.0626756	−0.0313378	−	0.999509i	$-0.509977\pi$
$13$	−2.93774	−0.0626756	−0.0313378	+	0.999509i	$0.509977\pi$
$14$	−31.7179	−0.605498
$15$	15.0000	0.258199
$16$	−7.21984	−0.112810
$17$	−6.49806	−0.0927066	−0.0463533	−	0.998925i	$-0.514760\pi$
$17$	−6.49806	−0.0927066	−0.0463533	+	0.998925i	$0.514760\pi$
$18$	40.7802	0.533999
$19$	−5.43580	−0.0656347	−0.0328173	−	0.999461i	$-0.510448\pi$
$19$	−5.43580	−0.0656347	−0.0328173	+	0.999461i	$0.510448\pi$
$20$	62.6556	0.700511
$21$	−21.0000	−0.218218
$22$	−86.3735	−0.837041
$23$	49.3774	0.447648	0.223824	−	0.974630i	$-0.428146\pi$
$23$	49.3774	0.447648	0.223824	+	0.974630i	$0.428146\pi$
$24$	61.5934	0.523862
$25$	25.0000	0.200000
$26$	−13.3113	−0.100406
$27$	27.0000	0.192450
$28$	−87.7179	−0.592040
$29$	−291.494	−1.86652	−0.933261	−	0.359200i	$-0.883050\pi$
$29$	−291.494	−1.86652	−0.933261	+	0.359200i	$0.883050\pi$
$30$	67.9669	0.413634
$31$	244.307	1.41545	0.707724	−	0.706489i	$-0.249722\pi$
$31$	244.307	1.41545	0.707724	+	0.706489i	$0.249722\pi$
$32$	−196.963	−1.08808
$33$	−57.1868	−0.301665
$34$	−29.4436	−0.148516
$35$	−35.0000	−0.169031
$36$	112.780	0.522130
$37$	−193.121	−0.858077	−0.429038	−	0.903286i	$-0.641147\pi$
$37$	−193.121	−0.858077	−0.429038	+	0.903286i	$0.641147\pi$
$38$	−24.6303	−0.105147
$39$	−8.81323	−0.0361858
$40$	102.656	0.405782
$41$	315.113	1.20030	0.600151	−	0.799887i	$-0.295107\pi$
$41$	315.113	1.20030	0.600151	+	0.799887i	$0.295107\pi$
$42$	−95.1537	−0.349584
$43$	−300.996	−1.06748	−0.533738	−	0.845650i	$-0.679213\pi$
$43$	−300.996	−1.06748	−0.533738	+	0.845650i	$0.679213\pi$
$44$	−238.872	−0.818437
$45$	45.0000	0.149071
$46$	223.735	0.717130
$47$	86.5058	0.268472	0.134236	−	0.990949i	$-0.457142\pi$
$47$	86.5058	0.268472	0.134236	+	0.990949i	$0.457142\pi$
$48$	−21.6595	−0.0651309
$49$	49.0000	0.142857
$50$	113.278	0.320399
$51$	−19.4942	−0.0535242
$52$	−36.8132	−0.0981745
$53$	509.677	1.32093	0.660467	−	0.750855i	$-0.270358\pi$
$53$	509.677	1.32093	0.660467	+	0.750855i	$0.270358\pi$
$54$	122.340	0.308304
$55$	−95.3113	−0.233669
$56$	−143.718	−0.342948
$57$	−16.3074	−0.0378942
$58$	−1320.80	−2.99016
$59$	−83.3852	−0.183997	−0.0919985	−	0.995759i	$-0.529326\pi$
$59$	−83.3852	−0.183997	−0.0919985	+	0.995759i	$0.529326\pi$
$60$	187.967	0.404440
$61$	−5.25291	−0.0110257	−0.00551283	−	0.999985i	$-0.501755\pi$
$61$	−5.25291	−0.0110257	−0.00551283	+	0.999985i	$0.501755\pi$
$62$	1106.99	2.26754
$63$	−63.0000	−0.125988
$64$	−834.706	−1.63029
$65$	−14.6887	−0.0280294
$66$	−259.121	−0.483266
$67$	205.992	0.375611	0.187806	−	0.982206i	$-0.439862\pi$
$67$	205.992	0.375611	0.187806	+	0.982206i	$0.439862\pi$
$68$	−81.4281	−0.145215
$69$	148.132	0.258450
$70$	−158.590	−0.270787
$71$	1004.31	1.67872	0.839362	−	0.543573i	$-0.182929\pi$
$71$	1004.31	1.67872	0.839362	+	0.543573i	$0.182929\pi$
$72$	184.780	0.302452
$73$	−1007.29	−1.61499	−0.807494	−	0.589876i	$-0.799177\pi$
$73$	−1007.29	−1.61499	−0.807494	+	0.589876i	$0.799177\pi$
$74$	−875.055	−1.37464
$75$	75.0000	0.115470
$76$	−68.1168	−0.102810
$77$	133.436	0.197486
$78$	−39.9339	−0.0579695
$79$	−863.237	−1.22939	−0.614695	−	0.788765i	$-0.710721\pi$
$79$	−863.237	−1.22939	−0.614695	+	0.788765i	$0.710721\pi$
$80$	−36.0992	−0.0504502
$81$	81.0000	0.111111
$82$	1427.82	1.92288
$83$	1334.72	1.76512	0.882560	−	0.470200i	$-0.155818\pi$
$83$	1334.72	1.76512	0.882560	+	0.470200i	$0.155818\pi$
$84$	−263.154	−0.341815
$85$	−32.4903	−0.0414596
$86$	−1363.85	−1.71009
$87$	−874.483	−1.07764
$88$	−391.370	−0.474093
$89$	326.249	0.388565	0.194283	−	0.980946i	$-0.437762\pi$
$89$	326.249	0.388565	0.194283	+	0.980946i	$0.437762\pi$
$90$	203.901	0.238811
$91$	20.5642	0.0236892
$92$	618.755	0.701192
$93$	732.922	0.817210
$94$	391.969	0.430091
$95$	−27.1790	−0.0293527
$96$	−590.889	−0.628202
$97$	1526.77	1.59815	0.799075	−	0.601232i	$-0.205323\pi$
$97$	1526.77	1.59815	0.799075	+	0.601232i	$0.205323\pi$
$98$	222.025	0.228857
$99$	−171.560	−0.174166
$100$	313.278	0.313278
$101$	96.8716	0.0954365	0.0477182	−	0.998861i	$-0.484805\pi$
$101$	96.8716	0.0954365	0.0477182	+	0.998861i	$0.484805\pi$
$102$	−88.3307	−0.0857455
$103$	1321.99	1.26466	0.632329	−	0.774700i	$-0.282099\pi$
$103$	1321.99	1.26466	0.632329	+	0.774700i	$0.282099\pi$
$104$	−60.3152	−0.0568691
$105$	−105.000	−0.0975900
$106$	2309.41	2.11613
$107$	1745.71	1.57724	0.788619	−	0.614883i	$-0.210797\pi$
$107$	1745.71	1.57724	0.788619	+	0.614883i	$0.210797\pi$
$108$	338.340	0.301452
$109$	476.856	0.419032	0.209516	−	0.977805i	$-0.432811\pi$
$109$	476.856	0.419032	0.209516	+	0.977805i	$0.432811\pi$
$110$	−431.868	−0.374336
$111$	−579.362	−0.495411
$112$	50.5389	0.0426382
$113$	1641.65	1.36666	0.683332	−	0.730108i	$-0.260530\pi$
$113$	1641.65	1.36666	0.683332	+	0.730108i	$0.260530\pi$
$114$	−73.8910	−0.0607064
$115$	246.887	0.200194
$116$	−3652.75	−2.92370
$117$	−26.4397	−0.0208919
$118$	−377.829	−0.294763
$119$	45.4864	0.0350398
$120$	307.967	0.234278
$121$	−967.630	−0.726995
$122$	−23.8016	−0.0176631
$123$	945.339	0.692994
$124$	3061.45	2.21715
$125$	125.000	0.0894427
$126$	−285.461	−0.201833
$127$	−844.016	−0.589719	−0.294859	−	0.955541i	$-0.595273\pi$
$127$	−844.016	−0.589719	−0.294859	+	0.955541i	$0.595273\pi$
$128$	−2206.46	−1.52363
$129$	−902.988	−0.616308
$130$	−66.5564	−0.0449030
$131$	−2796.20	−1.86493	−0.932463	−	0.361265i	$-0.882345\pi$
$131$	−2796.20	−1.86493	−0.932463	+	0.361265i	$0.882345\pi$
$132$	−716.615	−0.472525
$133$	38.0506	0.0248076
$134$	933.377	0.601728
$135$	135.000	0.0860663
$136$	−133.413	−0.0841179
$137$	−2057.13	−1.28287	−0.641433	−	0.767179i	$-0.721660\pi$
$137$	−2057.13	−1.28287	−0.641433	+	0.767179i	$0.721660\pi$
$138$	671.206	0.414035
$139$	−1745.12	−1.06489	−0.532444	−	0.846465i	$-0.678726\pi$
$139$	−1745.12	−1.06489	−0.532444	+	0.846465i	$0.678726\pi$
$140$	−438.590	−0.264768
$141$	259.517	0.155002
$142$	4550.65	2.68931
$143$	56.0000	0.0327479
$144$	−64.9786	−0.0376033
$145$	−1457.47	−0.834734
$146$	−4564.15	−2.58720
$147$	147.000	0.0824786
$148$	−2420.02	−1.34408
$149$	−1173.57	−0.645254	−0.322627	−	0.946526i	$-0.604566\pi$
$149$	−1173.57	−0.645254	−0.322627	+	0.946526i	$0.604566\pi$
$150$	339.835	0.184983
$151$	1540.07	0.829994	0.414997	−	0.909823i	$-0.363783\pi$
$151$	1540.07	0.829994	0.414997	+	0.909823i	$0.363783\pi$
$152$	−111.603	−0.0595540
$153$	−58.4826	−0.0309022
$154$	604.615	0.316372
$155$	1221.54	0.633008
$156$	−110.440	−0.0566811
$157$	−2544.53	−1.29348	−0.646738	−	0.762712i	$-0.723867\pi$
$157$	−2544.53	−1.29348	−0.646738	+	0.762712i	$0.723867\pi$
$158$	−3911.44	−1.96948
$159$	1529.03	0.762642
$160$	−984.815	−0.486603
$161$	−345.642	−0.169195
$162$	367.021	0.178000
$163$	−594.708	−0.285774	−0.142887	−	0.989739i	$-0.545639\pi$
$163$	−594.708	−0.285774	−0.142887	+	0.989739i	$0.545639\pi$
$164$	3948.72	1.88014
$165$	−285.934	−0.134909
$166$	6047.81	2.82772
$167$	928.498	0.430236	0.215118	−	0.976588i	$-0.430986\pi$
$167$	928.498	0.430236	0.215118	+	0.976588i	$0.430986\pi$
$168$	−431.154	−0.198001
$169$	−2188.37	−0.996072
$170$	−147.218	−0.0664182
$171$	−48.9222	−0.0218782
$172$	−3771.82	−1.67209
$173$	−315.642	−0.138716	−0.0693578	−	0.997592i	$-0.522095\pi$
$173$	−315.642	−0.138716	−0.0693578	+	0.997592i	$0.522095\pi$
$174$	−3962.39	−1.72637
$175$	−175.000	−0.0755929
$176$	137.626	0.0589431
$177$	−250.156	−0.106231
$178$	1478.28	0.622480
$179$	−1445.49	−0.603581	−0.301791	−	0.953374i	$-0.597584\pi$
$179$	−1445.49	−0.603581	−0.301791	+	0.953374i	$0.597584\pi$
$180$	563.901	0.233504
$181$	−1843.81	−0.757180	−0.378590	−	0.925564i	$-0.623591\pi$
$181$	−1843.81	−0.757180	−0.378590	+	0.925564i	$0.623591\pi$
$182$	93.1790	0.0379499
$183$	−15.7587	−0.00636567
$184$	1013.77	0.406176
$185$	−965.603	−0.383744
$186$	3320.97	1.30917
$187$	123.868	0.0484391
$188$	1084.02	0.420532
$189$	−189.000	−0.0727393
$190$	−123.152	−0.0470229
$191$	−244.074	−0.0924637	−0.0462318	−	0.998931i	$-0.514721\pi$
$191$	−244.074	−0.0924637	−0.0462318	+	0.998931i	$0.514721\pi$
$192$	−2504.12	−0.941246
$193$	−1733.03	−0.646355	−0.323178	−	0.946338i	$-0.604751\pi$
$193$	−1733.03	−0.646355	−0.323178	+	0.946338i	$0.604751\pi$
$194$	6918.01	2.56023
$195$	−44.0661	−0.0161828
$196$	614.025	0.223770
$197$	358.230	0.129557	0.0647787	−	0.997900i	$-0.479366\pi$
$197$	358.230	0.129557	0.0647787	+	0.997900i	$0.479366\pi$
$198$	−777.362	−0.279014
$199$	−3203.63	−1.14120	−0.570601	−	0.821227i	$-0.693290\pi$
$199$	−3203.63	−1.14120	−0.570601	+	0.821227i	$0.693290\pi$
$200$	513.278	0.181471
$201$	617.977	0.216859
$202$	438.938	0.152889
$203$	2040.46	0.705479
$204$	−244.284	−0.0838398
$205$	1575.56	0.536791
$206$	5990.12	2.02598
$207$	444.397	0.149216
$208$	21.2100	0.00707044
$209$	103.619	0.0342940
$210$	−475.769	−0.156339
$211$	4943.16	1.61280	0.806401	−	0.591369i	$-0.201412\pi$
$211$	4943.16	1.61280	0.806401	+	0.591369i	$0.201412\pi$
$212$	6386.83	2.06910
$213$	3012.92	0.969211
$214$	7910.05	2.52673
$215$	−1504.98	−0.477390
$216$	554.340	0.174621
$217$	−1710.15	−0.534989
$218$	2160.70	0.671288
$219$	−3021.86	−0.932414
$220$	−1194.36	−0.366016
$221$	19.0896	0.00581044
$222$	−2625.16	−0.793646
$223$	3160.15	0.948965	0.474482	−	0.880265i	$-0.342635\pi$
$223$	3160.15	0.948965	0.474482	+	0.880265i	$0.342635\pi$
$224$	1378.74	0.411255
$225$	225.000	0.0666667
$226$	7438.51	2.18939
$227$	−3651.11	−1.06755	−0.533773	−	0.845628i	$-0.679226\pi$
$227$	−3651.11	−1.06755	−0.533773	+	0.845628i	$0.679226\pi$
$228$	−204.350	−0.0593571
$229$	−4083.70	−1.17842	−0.589210	−	0.807980i	$-0.700561\pi$
$229$	−4083.70	−1.17842	−0.589210	+	0.807980i	$0.700561\pi$
$230$	1118.68	0.320710
$231$	400.307	0.114019
$232$	−5984.70	−1.69360
$233$	−3682.51	−1.03540	−0.517702	−	0.855561i	$-0.673212\pi$
$233$	−3682.51	−1.03540	−0.517702	+	0.855561i	$0.673212\pi$
$234$	−119.802	−0.0334687
$235$	432.529	0.120064
$236$	−1044.91	−0.288211
$237$	−2589.71	−0.709789
$238$	206.105	0.0561336
$239$	2658.78	0.719591	0.359796	−	0.933031i	$-0.382846\pi$
$239$	2658.78	0.719591	0.359796	+	0.933031i	$0.382846\pi$
$240$	−108.298	−0.0291274
$241$	−4820.39	−1.28842	−0.644209	−	0.764850i	$-0.722813\pi$
$241$	−4820.39	−1.28842	−0.644209	+	0.764850i	$0.722813\pi$
$242$	−4384.46	−1.16464
$243$	243.000	0.0641500
$244$	−65.8249	−0.0172705
$245$	245.000	0.0638877
$246$	4283.45	1.11017
$247$	15.9690	0.00411369
$248$	5015.91	1.28432
$249$	4004.17	1.01909
$250$	566.391	0.143287
$251$	1672.27	0.420530	0.210265	−	0.977644i	$-0.432567\pi$
$251$	1672.27	0.420530	0.210265	+	0.977644i	$0.432567\pi$
$252$	−789.461	−0.197347
$253$	−941.245	−0.233896
$254$	−3824.34	−0.944727
$255$	−97.4709	−0.0239367
$256$	−3320.09	−0.810569
$257$	−3697.74	−0.897506	−0.448753	−	0.893656i	$-0.648132\pi$
$257$	−3697.74	−0.897506	−0.448753	+	0.893656i	$0.648132\pi$
$258$	−4091.56	−0.987322
$259$	1351.84	0.324323
$260$	−184.066	−0.0439050
$261$	−2623.45	−0.622174
$262$	−12670.0	−2.98760
$263$	7319.00	1.71600	0.858002	−	0.513646i	$-0.171706\pi$
$263$	7319.00	1.71600	0.858002	+	0.513646i	$0.171706\pi$
$264$	−1174.11	−0.273717
$265$	2548.39	0.590740
$266$	172.412	0.0397416
$267$	978.747	0.224338
$268$	2581.32	0.588354
$269$	−815.097	−0.184749	−0.0923743	−	0.995724i	$-0.529446\pi$
$269$	−815.097	−0.184749	−0.0923743	+	0.995724i	$0.529446\pi$
$270$	611.702	0.137878
$271$	−5106.02	−1.14453	−0.572267	−	0.820068i	$-0.693936\pi$
$271$	−5106.02	−1.14453	−0.572267	+	0.820068i	$0.693936\pi$
$272$	46.9150	0.0104582
$273$	61.6926	0.0136769
$274$	−9321.13	−2.05515
$275$	−476.556	−0.104500
$276$	1856.26	0.404833
$277$	1398.72	0.303398	0.151699	−	0.988427i	$-0.451526\pi$
$277$	1398.72	0.303398	0.151699	+	0.988427i	$0.451526\pi$
$278$	−7907.39	−1.70595
$279$	2198.77	0.471816
$280$	−718.590	−0.153371
$281$	−7102.38	−1.50780	−0.753901	−	0.656988i	$-0.771830\pi$
$281$	−7102.38	−1.50780	−0.753901	+	0.656988i	$0.771830\pi$
$282$	1175.91	0.248313
$283$	4465.18	0.937907	0.468953	−	0.883223i	$-0.344631\pi$
$283$	4465.18	0.937907	0.468953	+	0.883223i	$0.344631\pi$
$284$	12585.1	2.62954
$285$	−81.5371	−0.0169468
$286$	253.743	0.0524621
$287$	−2205.79	−0.453671
$288$	−1772.67	−0.362692
$289$	−4870.78	−0.991405
$290$	−6603.99	−1.33724
$291$	4580.32	0.922692
$292$	−12622.5	−2.52970
$293$	7590.61	1.51348	0.756738	−	0.653718i	$-0.226792\pi$
$293$	7590.61	1.51348	0.756738	+	0.653718i	$0.226792\pi$
$294$	666.076	0.132130
$295$	−416.926	−0.0822860
$296$	−3964.98	−0.778581
$297$	−514.681	−0.100555
$298$	−5317.61	−1.03369
$299$	−145.058	−0.0280566
$300$	939.835	0.180871
$301$	2106.97	0.403468
$302$	6978.26	1.32965
$303$	290.615	0.0551003
$304$	39.2456	0.00740425
$305$	−26.2645	−0.00493083
$306$	−264.992	−0.0495052
$307$	9480.12	1.76241	0.881203	−	0.472737i	$-0.156734\pi$
$307$	9480.12	1.76241	0.881203	+	0.472737i	$0.156734\pi$
$308$	1672.10	0.309340
$309$	3965.98	0.730151
$310$	5534.94	1.01408
$311$	7078.01	1.29054	0.645268	−	0.763956i	$-0.276746\pi$
$311$	7078.01	1.29054	0.645268	+	0.763956i	$0.276746\pi$
$312$	−180.945	−0.0328334
$313$	5593.84	1.01017	0.505084	−	0.863070i	$-0.331461\pi$
$313$	5593.84	1.01017	0.505084	+	0.863070i	$0.331461\pi$
$314$	−11529.6	−2.07214
$315$	−315.000	−0.0563436
$316$	−10817.3	−1.92571
$317$	3567.81	0.632139	0.316070	−	0.948736i	$-0.397637\pi$
$317$	3567.81	0.632139	0.316070	+	0.948736i	$0.397637\pi$
$318$	6928.24	1.22175
$319$	5556.54	0.975255
$320$	−4173.53	−0.729086
$321$	5237.14	0.910618
$322$	−1566.15	−0.271050
$323$	35.3222	0.00608477
$324$	1015.02	0.174043
$325$	−73.4436	−0.0125351
$326$	−2694.70	−0.457809
$327$	1430.57	0.241928
$328$	6469.62	1.08910
$329$	−605.541	−0.101473
$330$	−1295.60	−0.216123
$331$	−4389.67	−0.728936	−0.364468	−	0.931216i	$-0.618749\pi$
$331$	−4389.67	−0.728936	−0.364468	+	0.931216i	$0.618749\pi$
$332$	16725.6	2.76487
$333$	−1738.09	−0.286026
$334$	4207.14	0.689236
$335$	1029.96	0.167978
$336$	151.617	0.0246172
$337$	2348.83	0.379671	0.189835	−	0.981816i	$-0.439205\pi$
$337$	2348.83	0.379671	0.189835	+	0.981816i	$0.439205\pi$
$338$	−9915.79	−1.59570
$339$	4924.94	0.789044
$340$	−407.140	−0.0649420
$341$	−4657.05	−0.739570
$342$	−221.673	−0.0350488
$343$	−343.000	−0.0539949
$344$	−6179.79	−0.968581
$345$	740.661	0.115582
$346$	−1430.21	−0.222222
$347$	558.436	0.0863931	0.0431965	−	0.999067i	$-0.486246\pi$
$347$	558.436	0.0863931	0.0431965	+	0.999067i	$0.486246\pi$
$348$	−10958.3	−1.68800
$349$	3233.89	0.496006	0.248003	−	0.968759i	$-0.420226\pi$
$349$	3233.89	0.496006	0.248003	+	0.968759i	$0.420226\pi$
$350$	−792.948	−0.121100
$351$	−79.3190	−0.0120619
$352$	3754.56	0.568519
$353$	−7516.35	−1.13330	−0.566650	−	0.823959i	$-0.691761\pi$
$353$	−7516.35	−1.13330	−0.566650	+	0.823959i	$0.691761\pi$
$354$	−1133.49	−0.170181
$355$	5021.54	0.750748
$356$	4088.27	0.608646
$357$	136.459	0.0202302
$358$	−6549.70	−0.966935
$359$	−6577.76	−0.967021	−0.483511	−	0.875338i	$-0.660639\pi$
$359$	−6577.76	−0.967021	−0.483511	+	0.875338i	$0.660639\pi$
$360$	923.901	0.135261
$361$	−6829.45	−0.995692
$362$	−8354.56	−1.21300
$363$	−2902.89	−0.419731
$364$	257.693	0.0371065
$365$	−5036.44	−0.722245
$366$	−71.4048	−0.0101978
$367$	8307.17	1.18155	0.590777	−	0.806835i	$-0.298821\pi$
$367$	8307.17	1.18155	0.590777	+	0.806835i	$0.298821\pi$
$368$	−356.497	−0.0504992
$369$	2836.02	0.400101
$370$	−4375.27	−0.614756
$371$	−3567.74	−0.499266
$372$	9184.34	1.28007
$373$	−4551.09	−0.631760	−0.315880	−	0.948799i	$-0.602300\pi$
$373$	−4551.09	−0.631760	−0.315880	+	0.948799i	$0.602300\pi$
$374$	561.261	0.0775992
$375$	375.000	0.0516398
$376$	1776.06	0.243599
$377$	856.335	0.116985
$378$	−856.383	−0.116528
$379$	−1788.29	−0.242370	−0.121185	−	0.992630i	$-0.538669\pi$
$379$	−1788.29	−0.242370	−0.121185	+	0.992630i	$0.538669\pi$
$380$	−340.584	−0.0459778
$381$	−2532.05	−0.340474
$382$	−1105.93	−0.148126
$383$	−1358.47	−0.181240	−0.0906199	−	0.995886i	$-0.528885\pi$
$383$	−1358.47	−0.181240	−0.0906199	+	0.995886i	$0.528885\pi$
$384$	−6619.37	−0.879670
$385$	667.179	0.0883184
$386$	−7852.60	−1.03546
$387$	−2708.97	−0.355825
$388$	19132.2	2.50333
$389$	9722.54	1.26723	0.633615	−	0.773649i	$-0.281570\pi$
$389$	9722.54	1.26723	0.633615	+	0.773649i	$0.281570\pi$
$390$	−199.669	−0.0259247
$391$	−320.858	−0.0414999
$392$	1006.03	0.129622
$393$	−8388.61	−1.07672
$394$	1623.19	0.207551
$395$	−4316.19	−0.549800
$396$	−2149.84	−0.272812
$397$	−4788.04	−0.605302	−0.302651	−	0.953101i	$-0.597872\pi$
$397$	−4788.04	−0.605302	−0.302651	+	0.953101i	$0.597872\pi$
$398$	−14516.1	−1.82820
$399$	114.152	0.0143227
$400$	−180.496	−0.0225620
$401$	9681.41	1.20565	0.602826	−	0.797873i	$-0.294041\pi$
$401$	9681.41	1.20565	0.602826	+	0.797873i	$0.294041\pi$
$402$	2800.13	0.347408
$403$	−717.712	−0.0887141
$404$	1213.91	0.149491
$405$	405.000	0.0496904
$406$	9245.58	1.13017
$407$	3681.32	0.448344
$408$	−400.238	−0.0485655
$409$	11113.1	1.34353	0.671767	−	0.740763i	$-0.265536\pi$
$409$	11113.1	1.34353	0.671767	+	0.740763i	$0.265536\pi$
$410$	7139.09	0.859937
$411$	−6171.40	−0.740663
$412$	16566.1	1.98095
$413$	583.696	0.0695443
$414$	2013.62	0.239043
$415$	6673.62	0.789386
$416$	578.627	0.0681959
$417$	−5235.37	−0.614814
$418$	469.510	0.0549389
$419$	−1230.09	−0.143421	−0.0717107	−	0.997425i	$-0.522846\pi$
$419$	−1230.09	−0.143421	−0.0717107	+	0.997425i	$0.522846\pi$
$420$	−1315.77	−0.152864
$421$	−12356.5	−1.43044	−0.715222	−	0.698897i	$-0.753674\pi$
$421$	−12356.5	−1.43044	−0.715222	+	0.698897i	$0.753674\pi$
$422$	22398.1	2.58370
$423$	778.552	0.0894906
$424$	10464.2	1.19856
$425$	−162.452	−0.0185413
$426$	13651.9	1.55267
$427$	36.7703	0.00416731
$428$	21875.7	2.47057
$429$	168.000	0.0189070
$430$	−6819.26	−0.764777
$431$	−7375.27	−0.824256	−0.412128	−	0.911126i	$-0.635214\pi$
$431$	−7375.27	−0.824256	−0.412128	+	0.911126i	$0.635214\pi$
$432$	−194.936	−0.0217103
$433$	−690.067	−0.0765877	−0.0382939	−	0.999267i	$-0.512192\pi$
$433$	−690.067	−0.0765877	−0.0382939	+	0.999267i	$0.512192\pi$
$434$	−7748.92	−0.857051
$435$	−4372.41	−0.481934
$436$	5975.55	0.656369
$437$	−268.406	−0.0293812
$438$	−13692.5	−1.49372
$439$	8408.79	0.914191	0.457095	−	0.889418i	$-0.348890\pi$
$439$	8408.79	0.914191	0.457095	+	0.889418i	$0.348890\pi$
$440$	−1956.85	−0.212021
$441$	441.000	0.0476190
$442$	86.4976	0.00930830
$443$	6568.55	0.704473	0.352236	−	0.935911i	$-0.385421\pi$
$443$	6568.55	0.704473	0.352236	+	0.935911i	$0.385421\pi$
$444$	−7260.06	−0.776007
$445$	1631.25	0.173772
$446$	14319.0	1.52024
$447$	−3520.72	−0.372537
$448$	5842.94	0.616190
$449$	2954.55	0.310543	0.155271	−	0.987872i	$-0.450375\pi$
$449$	2954.55	0.310543	0.155271	+	0.987872i	$0.450375\pi$
$450$	1019.50	0.106800
$451$	−6006.76	−0.627156
$452$	20571.7	2.14073
$453$	4620.21	0.479197
$454$	−16543.7	−1.71020
$455$	102.821	0.0105941
$456$	−334.810	−0.0343835
$457$	−8144.84	−0.833697	−0.416849	−	0.908976i	$-0.636865\pi$
$457$	−8144.84	−0.833697	−0.416849	+	0.908976i	$0.636865\pi$
$458$	−18503.8	−1.88782
$459$	−175.448	−0.0178414
$460$	3093.77	0.313583
$461$	2495.26	0.252095	0.126048	−	0.992024i	$-0.459771\pi$
$461$	2495.26	0.252095	0.126048	+	0.992024i	$0.459771\pi$
$462$	1813.84	0.182657
$463$	−5755.66	−0.577728	−0.288864	−	0.957370i	$-0.593278\pi$
$463$	−5755.66	−0.577728	−0.288864	+	0.957370i	$0.593278\pi$
$464$	2104.54	0.210562
$465$	3664.61	0.365467
$466$	−16685.9	−1.65871
$467$	−4143.73	−0.410598	−0.205299	−	0.978699i	$-0.565817\pi$
$467$	−4143.73	−0.410598	−0.205299	+	0.978699i	$0.565817\pi$
$468$	−331.319	−0.0327248
$469$	−1441.95	−0.141968
$470$	1959.84	0.192342
$471$	−7633.60	−0.746789
$472$	−1711.99	−0.166951
$473$	5737.67	0.557755
$474$	−11734.3	−1.13708
$475$	−135.895	−0.0131269
$476$	569.996	0.0548860
$477$	4587.09	0.440312
$478$	12047.3	1.15278
$479$	−6765.96	−0.645396	−0.322698	−	0.946502i	$-0.604590\pi$
$479$	−6765.96	−0.645396	−0.322698	+	0.946502i	$0.604590\pi$
$480$	−2954.45	−0.280940
$481$	567.339	0.0537805
$482$	−21841.8	−2.06404
$483$	−1036.93	−0.0976848
$484$	−12125.5	−1.13876
$485$	7633.87	0.714714
$486$	1101.06	0.102768
$487$	6360.42	0.591824	0.295912	−	0.955215i	$-0.404377\pi$
$487$	6360.42	0.591824	0.295912	+	0.955215i	$0.404377\pi$
$488$	−107.848	−0.0100042
$489$	−1784.12	−0.164992
$490$	1110.13	0.102348
$491$	−7072.54	−0.650060	−0.325030	−	0.945704i	$-0.605374\pi$
$491$	−7072.54	−0.650060	−0.325030	+	0.945704i	$0.605374\pi$
$492$	11846.2	1.08550
$493$	1894.15	0.173039
$494$	72.3576	0.00659012
$495$	−857.802	−0.0778895
$496$	−1763.86	−0.159677
$497$	−7030.15	−0.634498
$498$	18143.4	1.63258
$499$	−18473.9	−1.65732	−0.828661	−	0.559751i	$-0.810897\pi$
$499$	−18473.9	−1.65732	−0.828661	+	0.559751i	$0.810897\pi$
$500$	1566.39	0.140102
$501$	2785.49	0.248397
$502$	7577.28	0.673687
$503$	11379.2	1.00869	0.504347	−	0.863501i	$-0.331733\pi$
$503$	11379.2	1.00869	0.504347	+	0.863501i	$0.331733\pi$
$504$	−1293.46	−0.114316
$505$	484.358	0.0426805
$506$	−4264.90	−0.374700
$507$	−6565.11	−0.575082
$508$	−10576.5	−0.923730
$509$	6064.48	0.528101	0.264051	−	0.964509i	$-0.414941\pi$
$509$	6064.48	0.528101	0.264051	+	0.964509i	$0.414941\pi$
$510$	−441.653	−0.0383465
$511$	7051.02	0.610408
$512$	2607.89	0.225105
$513$	−146.767	−0.0126314
$514$	−16755.0	−1.43780
$515$	6609.96	0.565572
$516$	−11315.5	−0.965379
$517$	−1649.00	−0.140276
$518$	6125.38	0.519563
$519$	−946.926	−0.0800875
$520$	−301.576	−0.0254326
$521$	2682.88	0.225603	0.112801	−	0.993618i	$-0.464018\pi$
$521$	2682.88	0.225603	0.112801	+	0.993618i	$0.464018\pi$
$522$	−11887.2	−0.996720
$523$	−4309.02	−0.360268	−0.180134	−	0.983642i	$-0.557653\pi$
$523$	−4309.02	−0.360268	−0.180134	+	0.983642i	$0.557653\pi$
$524$	−35039.6	−2.92120
$525$	−525.000	−0.0436436
$526$	33163.3	2.74903
$527$	−1587.52	−0.131221
$528$	412.879	0.0340308
$529$	−9728.87	−0.799611
$530$	11547.1	0.946363
$531$	−750.467	−0.0613323
$532$	476.817	0.0388584
$533$	−925.720	−0.0752296
$534$	4434.83	0.359389
$535$	8728.56	0.705362
$536$	4229.25	0.340813
$537$	−4336.47	−0.348478
$538$	−3693.31	−0.295966
$539$	−934.051	−0.0746427
$540$	1691.70	0.134813
$541$	4081.47	0.324355	0.162178	−	0.986762i	$-0.448148\pi$
$541$	4081.47	0.324355	0.162178	+	0.986762i	$0.448148\pi$
$542$	−23136.0	−1.83354
$543$	−5531.44	−0.437158
$544$	1279.88	0.100872
$545$	2384.28	0.187397
$546$	279.537	0.0219104
$547$	8844.82	0.691366	0.345683	−	0.938351i	$-0.387647\pi$
$547$	8844.82	0.691366	0.345683	+	0.938351i	$0.387647\pi$
$548$	−25778.2	−2.00947
$549$	−47.2762	−0.00367522
$550$	−2159.34	−0.167408
$551$	1584.51	0.122509
$552$	3041.32	0.234506
$553$	6042.66	0.464666
$554$	6337.80	0.486042
$555$	−2896.81	−0.221554
$556$	−21868.4	−1.66803
$557$	−11144.7	−0.847787	−0.423894	−	0.905712i	$-0.639337\pi$
$557$	−11144.7	−0.847787	−0.423894	+	0.905712i	$0.639337\pi$
$558$	9962.90	0.755848
$559$	884.249	0.0669047
$560$	252.694	0.0190684
$561$	371.603	0.0279663
$562$	−32181.8	−2.41549
$563$	−21857.5	−1.63621	−0.818104	−	0.575071i	$-0.804975\pi$
$563$	−21857.5	−1.63621	−0.818104	+	0.575071i	$0.804975\pi$
$564$	3252.05	0.242794
$565$	8208.23	0.611191
$566$	20232.3	1.50252
$567$	−567.000	−0.0419961
$568$	20619.6	1.52320
$569$	23496.4	1.73115	0.865573	−	0.500783i	$-0.166954\pi$
$569$	23496.4	1.73115	0.865573	+	0.500783i	$0.166954\pi$
$570$	−369.455	−0.0271487
$571$	11067.8	0.811164	0.405582	−	0.914059i	$-0.367069\pi$
$571$	11067.8	0.811164	0.405582	+	0.914059i	$0.367069\pi$
$572$	701.743	0.0512961
$573$	−732.222	−0.0533839
$574$	−9994.72	−0.726780
$575$	1234.44	0.0895296
$576$	−7512.36	−0.543429
$577$	20482.9	1.47784	0.738922	−	0.673791i	$-0.235335\pi$
$577$	20482.9	1.47784	0.738922	+	0.673791i	$0.235335\pi$
$578$	−22070.1	−1.58823
$579$	−5199.10	−0.373173
$580$	−18263.8	−1.30752
$581$	−9343.07	−0.667153
$582$	20754.0	1.47815
$583$	−9715.60	−0.690187
$584$	−20680.8	−1.46537
$585$	−132.198	−0.00934313
$586$	34394.1	2.42458
$587$	−23444.3	−1.64847	−0.824235	−	0.566248i	$-0.808394\pi$
$587$	−23444.3	−1.64847	−0.824235	+	0.566248i	$0.808394\pi$
$588$	1842.08	0.129194
$589$	−1328.01	−0.0929025
$590$	−1889.14	−0.131822
$591$	1074.69	0.0748000
$592$	1394.30	0.0967996
$593$	4404.69	0.305024	0.152512	−	0.988302i	$-0.451264\pi$
$593$	4404.69	0.305024	0.152512	+	0.988302i	$0.451264\pi$
$594$	−2332.09	−0.161089
$595$	227.432	0.0156703
$596$	−14706.2	−1.01072
$597$	−9610.89	−0.658874
$598$	−657.277	−0.0449466
$599$	3327.05	0.226945	0.113472	−	0.993541i	$-0.463803\pi$
$599$	3327.05	0.226945	0.113472	+	0.993541i	$0.463803\pi$
$600$	1539.83	0.104772
$601$	−14244.8	−0.966818	−0.483409	−	0.875395i	$-0.660602\pi$
$601$	−14244.8	−0.966818	−0.483409	+	0.875395i	$0.660602\pi$
$602$	9546.97	0.646354
$603$	1853.93	0.125204
$604$	19298.8	1.30010
$605$	−4838.15	−0.325122
$606$	1316.81	0.0882704
$607$	11446.5	0.765402	0.382701	−	0.923872i	$-0.374994\pi$
$607$	11446.5	0.765402	0.382701	+	0.923872i	$0.374994\pi$
$608$	1070.65	0.0714156
$609$	6121.38	0.407308
$610$	−119.008	−0.00789917
$611$	−254.132	−0.0168266
$612$	−732.852	−0.0484049
$613$	−19436.4	−1.28063	−0.640316	−	0.768111i	$-0.721197\pi$
$613$	−19436.4	−1.28063	−0.640316	+	0.768111i	$0.721197\pi$
$614$	42955.6	2.82337
$615$	4726.69	0.309917
$616$	2739.59	0.179190
$617$	−20530.1	−1.33956	−0.669781	−	0.742558i	$-0.733612\pi$
$617$	−20530.1	−1.33956	−0.669781	+	0.742558i	$0.733612\pi$
$618$	17970.4	1.16970
$619$	5833.35	0.378776	0.189388	−	0.981902i	$-0.439350\pi$
$619$	5833.35	0.378776	0.189388	+	0.981902i	$0.439350\pi$
$620$	15307.2	0.991538
$621$	1333.19	0.0861499
$622$	32071.4	2.06744
$623$	−2283.74	−0.146864
$624$	63.6301	0.00408212
$625$	625.000	0.0400000
$626$	25346.4	1.61829
$627$	310.856	0.0197997
$628$	−31885.9	−2.02609
$629$	1254.91	0.0795493
$630$	−1427.31	−0.0902622
$631$	24776.6	1.56314	0.781568	−	0.623820i	$-0.214420\pi$
$631$	24776.6	1.56314	0.781568	+	0.623820i	$0.214420\pi$
$632$	−17723.2	−1.11549
$633$	14829.5	0.931152
$634$	16166.2	1.01268
$635$	−4220.08	−0.263730
$636$	19160.5	1.19460
$637$	−143.949	−0.00895366
$638$	25177.4	1.56235
$639$	9038.77	0.559574
$640$	−11032.3	−0.681390
$641$	27219.4	1.67723	0.838613	−	0.544728i	$-0.183367\pi$
$641$	27219.4	1.67723	0.838613	+	0.544728i	$0.183367\pi$
$642$	23730.1	1.45881
$643$	7091.79	0.434950	0.217475	−	0.976066i	$-0.430218\pi$
$643$	7091.79	0.434950	0.217475	+	0.976066i	$0.430218\pi$
$644$	−4331.28	−0.265026
$645$	−4514.94	−0.275621
$646$	160.049	0.00974777
$647$	27773.0	1.68758	0.843792	−	0.536670i	$-0.180318\pi$
$647$	27773.0	1.68758	0.843792	+	0.536670i	$0.180318\pi$
$648$	1663.02	0.100817
$649$	1589.51	0.0961382
$650$	−332.782	−0.0200812
$651$	−5130.46	−0.308876
$652$	−7452.37	−0.447634
$653$	21380.4	1.28129	0.640643	−	0.767839i	$-0.278668\pi$
$653$	21380.4	1.28129	0.640643	+	0.767839i	$0.278668\pi$
$654$	6482.09	0.387568
$655$	−13981.0	−0.834020
$656$	−2275.06	−0.135406
$657$	−9065.59	−0.538329
$658$	−2743.78	−0.162559
$659$	−17232.3	−1.01863	−0.509315	−	0.860580i	$-0.670101\pi$
$659$	−17232.3	−1.01863	−0.509315	+	0.860580i	$0.670101\pi$
$660$	−3583.07	−0.211320
$661$	26577.7	1.56392	0.781962	−	0.623326i	$-0.214219\pi$
$661$	26577.7	1.56392	0.781962	+	0.623326i	$0.214219\pi$
$662$	−19890.1	−1.16775
$663$	57.2689	0.00335466
$664$	27403.4	1.60159
$665$	190.253	0.0110943
$666$	−7875.49	−0.458212
$667$	−14393.2	−0.835544
$668$	11635.1	0.673917
$669$	9480.44	0.547885
$670$	4666.89	0.269101
$671$	100.132	0.00576090
$672$	4136.22	0.237438
$673$	−31695.2	−1.81540	−0.907698	−	0.419624i	$-0.862162\pi$
$673$	−31695.2	−1.81540	−0.907698	+	0.419624i	$0.862162\pi$
$674$	10642.9	0.608231
$675$	675.000	0.0384900
$676$	−27422.7	−1.56024
$677$	20440.3	1.16039	0.580195	−	0.814477i	$-0.302976\pi$
$677$	20440.3	1.16039	0.580195	+	0.814477i	$0.302976\pi$
$678$	22315.5	1.26405
$679$	−10687.4	−0.604044
$680$	−667.063	−0.0376187
$681$	−10953.3	−0.616348
$682$	−21101.7	−1.18479
$683$	−22896.9	−1.28276	−0.641381	−	0.767223i	$-0.721638\pi$
$683$	−22896.9	−1.28276	−0.641381	+	0.767223i	$0.721638\pi$
$684$	−613.051	−0.0342699
$685$	−10285.7	−0.573715
$686$	−1554.18	−0.0864997
$687$	−12251.1	−0.680361
$688$	2173.14	0.120422
$689$	−1497.30	−0.0827904
$690$	3356.03	0.185162
$691$	−23764.0	−1.30829	−0.654143	−	0.756371i	$-0.726971\pi$
$691$	−23764.0	−1.30829	−0.654143	+	0.756371i	$0.726971\pi$
$692$	−3955.35	−0.217283
$693$	1200.92	0.0658287
$694$	2530.34	0.138401
$695$	−8725.62	−0.476233
$696$	−17954.1	−0.977800
$697$	−2047.62	−0.111276
$698$	14653.2	0.794600
$699$	−11047.5	−0.597791
$700$	−2192.95	−0.118408
$701$	26259.5	1.41485	0.707423	−	0.706791i	$-0.249858\pi$
$701$	26259.5	1.41485	0.707423	+	0.706791i	$0.249858\pi$
$702$	−359.405	−0.0193232
$703$	1049.77	0.0563196
$704$	15911.4	0.851822
$705$	1297.59	0.0693191
$706$	−34057.6	−1.81554
$707$	−678.101	−0.0360716
$708$	−3134.73	−0.166399
$709$	12783.0	0.677116	0.338558	−	0.940945i	$-0.390061\pi$
$709$	12783.0	0.677116	0.338558	+	0.940945i	$0.390061\pi$
$710$	22753.2	1.20270
$711$	−7769.14	−0.409797
$712$	6698.26	0.352567
$713$	12063.3	0.633623
$714$	618.315	0.0324087
$715$	280.000	0.0146453
$716$	−18113.6	−0.945444
$717$	7976.35	0.415456
$718$	−29804.7	−1.54916
$719$	−27609.0	−1.43205	−0.716025	−	0.698075i	$-0.754040\pi$
$719$	−27609.0	−1.43205	−0.716025	+	0.698075i	$0.754040\pi$
$720$	−324.893	−0.0168167
$721$	−9253.95	−0.477996
$722$	−30945.1	−1.59509
$723$	−14461.2	−0.743868
$724$	−23105.1	−1.18604
$725$	−7287.35	−0.373304
$726$	−13153.4	−0.672407
$727$	31306.2	1.59709	0.798544	−	0.601937i	$-0.205604\pi$
$727$	31306.2	1.59709	0.798544	+	0.601937i	$0.205604\pi$
$728$	422.206	0.0214945
$729$	729.000	0.0370370
$730$	−22820.8	−1.15703
$731$	1955.89	0.0989621
$732$	−197.475	−0.00997113
$733$	−15765.8	−0.794441	−0.397220	−	0.917723i	$-0.630025\pi$
$733$	−15765.8	−0.794441	−0.397220	+	0.917723i	$0.630025\pi$
$734$	37640.8	1.89285
$735$	735.000	0.0368856
$736$	−9725.53	−0.487076
$737$	−3926.68	−0.196256
$738$	12850.4	0.640959
$739$	3966.51	0.197443	0.0987216	−	0.995115i	$-0.468525\pi$
$739$	3966.51	0.197443	0.0987216	+	0.995115i	$0.468525\pi$
$740$	−12100.1	−0.601093
$741$	47.9070	0.00237504
$742$	−16165.9	−0.799823
$743$	8224.50	0.406094	0.203047	−	0.979169i	$-0.434916\pi$
$743$	8224.50	0.406094	0.203047	+	0.979169i	$0.434916\pi$
$744$	15047.7	0.741500
$745$	−5867.86	−0.288566
$746$	−20621.6	−1.01208
$747$	12012.5	0.588373
$748$	1552.20	0.0758745
$749$	−12220.0	−0.596140
$750$	1699.17	0.0827267
$751$	18929.2	0.919754	0.459877	−	0.887983i	$-0.347894\pi$
$751$	18929.2	0.919754	0.459877	+	0.887983i	$0.347894\pi$
$752$	−624.558	−0.0302863
$753$	5016.82	0.242793
$754$	3880.16	0.187410
$755$	7700.35	0.371185
$756$	−2368.38	−0.113938
$757$	−34906.8	−1.67597	−0.837984	−	0.545695i	$-0.816266\pi$
$757$	−34906.8	−1.67597	−0.837984	+	0.545695i	$0.816266\pi$
$758$	−8102.96	−0.388276
$759$	−2823.74	−0.135040
$760$	−558.016	−0.0266334
$761$	−13683.4	−0.651803	−0.325902	−	0.945404i	$-0.605668\pi$
$761$	−13683.4	−0.651803	−0.325902	+	0.945404i	$0.605668\pi$
$762$	−11473.0	−0.545438
$763$	−3337.99	−0.158379
$764$	−3058.52	−0.144834
$765$	−292.413	−0.0138199
$766$	−6155.42	−0.290345
$767$	244.964	0.0115321
$768$	−9960.28	−0.467982
$769$	41837.3	1.96189	0.980943	−	0.194294i	$-0.0622416\pi$
$769$	41837.3	1.96189	0.980943	+	0.194294i	$0.0622416\pi$
$770$	3023.07	0.141486
$771$	−11093.2	−0.518175
$772$	−21716.9	−1.01245
$773$	19640.0	0.913843	0.456921	−	0.889507i	$-0.348952\pi$
$773$	19640.0	0.913843	0.456921	+	0.889507i	$0.348952\pi$
$774$	−12274.7	−0.570031
$775$	6107.69	0.283090
$776$	31346.4	1.45009
$777$	4055.53	0.187248
$778$	44054.1	2.03010
$779$	−1712.89	−0.0787814
$780$	−552.198	−0.0253486
$781$	−19144.4	−0.877131
$782$	−1453.85	−0.0664827
$783$	−7870.34	−0.359212
$784$	−353.772	−0.0161157
$785$	−12722.7	−0.578460
$786$	−38009.9	−1.72489
$787$	24935.3	1.12941	0.564705	−	0.825293i	$-0.308990\pi$
$787$	24935.3	1.12941	0.564705	+	0.825293i	$0.308990\pi$
$788$	4489.02	0.202938
$789$	21957.0	0.990736
$790$	−19557.2	−0.880777
$791$	−11491.5	−0.516551
$792$	−3522.33	−0.158031
$793$	15.4317	0.000691041 0
$794$	−21695.2	−0.969692
$795$	7645.16	0.341064
$796$	−40145.1	−1.78757
$797$	−1168.33	−0.0519251	−0.0259625	−	0.999663i	$-0.508265\pi$
$797$	−1168.33	−0.0519251	−0.0259625	+	0.999663i	$0.508265\pi$
$798$	517.237	0.0229448
$799$	−562.120	−0.0248891
$800$	−4924.08	−0.217615
$801$	2936.24	0.129522
$802$	43867.7	1.93145
$803$	19201.2	0.843829
$804$	7743.95	0.339686
$805$	−1728.21	−0.0756663
$806$	−3252.05	−0.142120
$807$	−2445.29	−0.106665
$808$	1988.88	0.0865949
$809$	−35175.7	−1.52869	−0.764345	−	0.644807i	$-0.776938\pi$
$809$	−35175.7	−1.52869	−0.764345	+	0.644807i	$0.776938\pi$
$810$	1835.11	0.0796038
$811$	−15256.5	−0.660577	−0.330288	−	0.943880i	$-0.607146\pi$
$811$	−15256.5	−0.660577	−0.330288	+	0.943880i	$0.607146\pi$
$812$	25569.3	1.10506
$813$	−15318.1	−0.660797
$814$	16680.5	0.718245
$815$	−2973.54	−0.127802
$816$	140.745	0.00603806
$817$	1636.16	0.0700635
$818$	50354.7	2.15233
$819$	185.078	0.00789639
$820$	19743.6	0.840825
$821$	−15971.9	−0.678956	−0.339478	−	0.940614i	$-0.610250\pi$
$821$	−15971.9	−0.678956	−0.339478	+	0.940614i	$0.610250\pi$
$822$	−27963.4	−1.18654
$823$	2312.41	0.0979409	0.0489705	−	0.998800i	$-0.484406\pi$
$823$	2312.41	0.0979409	0.0489705	+	0.998800i	$0.484406\pi$
$824$	27142.0	1.14750
$825$	−1429.67	−0.0603330
$826$	2644.80	0.111410
$827$	10422.4	0.438238	0.219119	−	0.975698i	$-0.429682\pi$
$827$	10422.4	0.438238	0.219119	+	0.975698i	$0.429682\pi$
$828$	5568.79	0.233731
$829$	−13213.4	−0.553584	−0.276792	−	0.960930i	$-0.589271\pi$
$829$	−13213.4	−0.553584	−0.276792	+	0.960930i	$0.589271\pi$
$830$	30239.0	1.26459
$831$	4196.17	0.175167
$832$	2452.15	0.102179
$833$	−318.405	−0.0132438
$834$	−23722.2	−0.984929
$835$	4642.49	0.192407
$836$	1298.46	0.0537179
$837$	6596.30	0.272403
$838$	−5573.67	−0.229761
$839$	−10119.6	−0.416409	−0.208205	−	0.978085i	$-0.566762\pi$
$839$	−10119.6	−0.416409	−0.208205	+	0.978085i	$0.566762\pi$
$840$	−2155.77	−0.0885489
$841$	60579.9	2.48390
$842$	−55988.7	−2.29157
$843$	−21307.1	−0.870530
$844$	61943.4	2.52628
$845$	−10941.8	−0.445457
$846$	3527.72	0.143364
$847$	6773.41	0.274778
$848$	−3679.79	−0.149015
$849$	13395.5	0.541501
$850$	−736.089	−0.0297031
$851$	−9535.80	−0.384116
$852$	37755.3	1.51816
$853$	35378.1	1.42007	0.710037	−	0.704165i	$-0.248678\pi$
$853$	35378.1	1.42007	0.710037	+	0.704165i	$0.248678\pi$
$854$	166.611	0.00667602
$855$	−244.611	−0.00978424
$856$	35841.4	1.43112
$857$	6697.57	0.266960	0.133480	−	0.991052i	$-0.457385\pi$
$857$	6697.57	0.266960	0.133480	+	0.991052i	$0.457385\pi$
$858$	761.230	0.0302890
$859$	−24298.4	−0.965135	−0.482568	−	0.875859i	$-0.660296\pi$
$859$	−24298.4	−0.965135	−0.482568	+	0.875859i	$0.660296\pi$
$860$	−18859.1	−0.747779
$861$	−6617.37	−0.261927
$862$	−33418.3	−1.32045
$863$	24942.9	0.983853	0.491926	−	0.870637i	$-0.336293\pi$
$863$	24942.9	0.983853	0.491926	+	0.870637i	$0.336293\pi$
$864$	−5318.00	−0.209401
$865$	−1578.21	−0.0620355
$866$	−3126.78	−0.122693
$867$	−14612.3	−0.572388
$868$	−21430.1	−0.838002
$869$	16455.3	0.642355
$870$	−19812.0	−0.772056
$871$	−605.152	−0.0235417
$872$	9790.39	0.380212
$873$	13741.0	0.532716
$874$	−1216.18	−0.0470686
$875$	−875.000	−0.0338062
$876$	−37867.4	−1.46052
$877$	−16276.6	−0.626705	−0.313353	−	0.949637i	$-0.601452\pi$
$877$	−16276.6	−0.626705	−0.313353	+	0.949637i	$0.601452\pi$
$878$	38101.3	1.46453
$879$	22771.8	0.873806
$880$	688.132	0.0263602
$881$	26636.5	1.01862	0.509311	−	0.860582i	$-0.329900\pi$
$881$	26636.5	1.01862	0.509311	+	0.860582i	$0.329900\pi$
$882$	1998.23	0.0762855
$883$	−21788.3	−0.830392	−0.415196	−	0.909732i	$-0.636287\pi$
$883$	−21788.3	−0.830392	−0.415196	+	0.909732i	$0.636287\pi$
$884$	239.215	0.00910142
$885$	−1250.78	−0.0475078
$886$	29763.0	1.12856
$887$	−26813.2	−1.01499	−0.507496	−	0.861654i	$-0.669429\pi$
$887$	−26813.2	−1.01499	−0.507496	+	0.861654i	$0.669429\pi$
$888$	−11895.0	−0.449514
$889$	5908.11	0.222893
$890$	7391.38	0.278382
$891$	−1544.04	−0.0580554
$892$	39600.2	1.48645
$893$	−470.229	−0.0176211
$894$	−15952.8	−0.596803
$895$	−7227.45	−0.269930
$896$	15445.2	0.575879
$897$	−435.174	−0.0161985
$898$	13387.4	0.497488
$899$	−71214.2	−2.64196
$900$	2819.50	0.104426
$901$	−3311.91	−0.122459
$902$	−27217.4	−1.00470
$903$	6320.92	0.232942
$904$	33704.8	1.24005
$905$	−9219.07	−0.338621
$906$	20934.8	0.767672
$907$	15543.0	0.569014	0.284507	−	0.958674i	$-0.408170\pi$
$907$	15543.0	0.569014	0.284507	+	0.958674i	$0.408170\pi$
$908$	−45752.6	−1.67219
$909$	871.844	0.0318122
$910$	465.895	0.0169717
$911$	48711.1	1.77154	0.885768	−	0.464128i	$-0.153632\pi$
$911$	48711.1	1.77154	0.885768	+	0.464128i	$0.153632\pi$
$912$	117.737	0.00427485
$913$	−25442.8	−0.922273
$914$	−36905.3	−1.33558
$915$	−78.7936	−0.00284682
$916$	−51173.3	−1.84587
$917$	19573.4	0.704876
$918$	−794.976	−0.0285818
$919$	1030.47	0.0369883	0.0184941	−	0.999829i	$-0.494113\pi$
$919$	1030.47	0.0369883	0.0184941	+	0.999829i	$0.494113\pi$
$920$	5068.87	0.181648
$921$	28440.4	1.01753
$922$	11306.3	0.403855
$923$	−2950.40	−0.105215
$924$	5016.30	0.178598
$925$	−4828.02	−0.171615
$926$	−26079.6	−0.925518
$927$	11897.9	0.421553
$928$	57413.6	2.03092
$929$	879.756	0.0310698	0.0155349	−	0.999879i	$-0.495055\pi$
$929$	879.756	0.0310698	0.0155349	+	0.999879i	$0.495055\pi$
$930$	16604.8	0.585477
$931$	−266.354	−0.00937638
$932$	−46146.0	−1.62185
$933$	21234.0	0.745092
$934$	−18775.8	−0.657776
$935$	619.339	0.0216626
$936$	−542.836	−0.0189564
$937$	−18668.1	−0.650864	−0.325432	−	0.945565i	$-0.605510\pi$
$937$	−18668.1	−0.650864	−0.325432	+	0.945565i	$0.605510\pi$
$938$	−6533.64	−0.227432
$939$	16781.5	0.583221
$940$	5420.08	0.188067
$941$	29613.4	1.02590	0.512948	−	0.858420i	$-0.328553\pi$
$941$	29613.4	1.02590	0.512948	+	0.858420i	$0.328553\pi$
$942$	−34588.8	−1.19635
$943$	15559.5	0.537313
$944$	602.028	0.0207567
$945$	−945.000	−0.0325300
$946$	25998.1	0.893521
$947$	20738.9	0.711640	0.355820	−	0.934554i	$-0.384202\pi$
$947$	20738.9	0.711640	0.355820	+	0.934554i	$0.384202\pi$
$948$	−32452.0	−1.11181
$949$	2959.15	0.101220
$950$	−615.758	−0.0210293
$951$	10703.4	0.364966
$952$	933.888	0.0317936
$953$	−45776.5	−1.55598	−0.777988	−	0.628279i	$-0.783760\pi$
$953$	−45776.5	−1.55598	−0.777988	+	0.628279i	$0.783760\pi$
$954$	20784.7	0.705377
$955$	−1220.37	−0.0413510
$956$	33317.5	1.12716
$957$	16669.6	0.563064
$958$	−30657.4	−1.03392
$959$	14399.9	0.484878
$960$	−12520.6	−0.420938
$961$	29895.1	1.00349
$962$	2570.68	0.0861561
$963$	15711.4	0.525746
$964$	−60404.9	−2.01817
$965$	−8665.17	−0.289059
$966$	−4698.44	−0.156491
$967$	34461.0	1.14601	0.573005	−	0.819552i	$-0.305778\pi$
$967$	34461.0	1.14601	0.573005	+	0.819552i	$0.305778\pi$
$968$	−19866.5	−0.659643
$969$	105.967	0.00351304
$970$	34590.1	1.14497
$971$	−22762.8	−0.752309	−0.376154	−	0.926557i	$-0.622754\pi$
$971$	−22762.8	−0.752309	−0.376154	+	0.926557i	$0.622754\pi$
$972$	3045.06	0.100484
$973$	12215.9	0.402490
$974$	28819.9	0.948099
$975$	−220.331	−0.00723716
$976$	37.9251	0.00124381
$977$	4809.57	0.157494	0.0787470	−	0.996895i	$-0.474908\pi$
$977$	4809.57	0.157494	0.0787470	+	0.996895i	$0.474908\pi$
$978$	−8084.10	−0.264316
$979$	−6219.04	−0.203025
$980$	3070.13	0.100073
$981$	4291.70	0.139677
$982$	−32046.6	−1.04139
$983$	−27591.6	−0.895256	−0.447628	−	0.894220i	$-0.647731\pi$
$983$	−27591.6	−0.895256	−0.447628	+	0.894220i	$0.647731\pi$
$984$	19408.9	0.628793
$985$	1791.15	0.0579399
$986$	8582.63	0.277207
$987$	−1816.62	−0.0585853
$988$	200.109	0.00644365
$989$	−14862.4	−0.477854
$990$	−3886.81	−0.124779
$991$	−22263.4	−0.713643	−0.356822	−	0.934173i	$-0.616140\pi$
$991$	−22263.4	−0.713643	−0.356822	+	0.934173i	$0.616140\pi$
$992$	−48119.5	−1.54012
$993$	−13169.0	−0.420851
$994$	−31854.5	−1.01646
$995$	−16018.2	−0.510361
$996$	50176.8	1.59630
$997$	30378.2	0.964983	0.482491	−	0.875901i	$-0.339732\pi$
$997$	30378.2	0.964983	0.482491	+	0.875901i	$0.339732\pi$
$998$	−83707.4	−2.65502
$999$	−5214.26	−0.165137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.a.f.1.2	✓	2
3.2	odd	2		315.4.a.i.1.1		2
4.3	odd	2		1680.4.a.bg.1.2		2
5.2	odd	4		525.4.d.h.274.4		4
5.3	odd	4		525.4.d.h.274.1		4
5.4	even	2		525.4.a.k.1.1		2
7.6	odd	2		735.4.a.p.1.2		2
15.14	odd	2		1575.4.a.w.1.2		2
21.20	even	2		2205.4.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.f.1.2	✓	2	1.1	even	1	trivial
315.4.a.i.1.1		2	3.2	odd	2
525.4.a.k.1.1		2	5.4	even	2
525.4.d.h.274.1		4	5.3	odd	4
525.4.d.h.274.4		4	5.2	odd	4
735.4.a.p.1.2		2	7.6	odd	2
1575.4.a.w.1.2		2	15.14	odd	2
1680.4.a.bg.1.2		2	4.3	odd	2
2205.4.a.z.1.1		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)

\(f(q)\)	\(=\)	\(q+4.53113 q^{2} +3.00000 q^{3} +12.5311 q^{4} +5.00000 q^{5} +13.5934 q^{6} -7.00000 q^{7} +20.5311 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.53113 q^{2} +3.00000 q^{3} +12.5311 q^{4} +5.00000 q^{5} +13.5934 q^{6} -7.00000 q^{7} +20.5311 q^{8} +9.00000 q^{9} +22.6556 q^{10} -19.0623 q^{11} +37.5934 q^{12} -2.93774 q^{13} -31.7179 q^{14} +15.0000 q^{15} -7.21984 q^{16} -6.49806 q^{17} +40.7802 q^{18} -5.43580 q^{19} +62.6556 q^{20} -21.0000 q^{21} -86.3735 q^{22} +49.3774 q^{23} +61.5934 q^{24} +25.0000 q^{25} -13.3113 q^{26} +27.0000 q^{27} -87.7179 q^{28} -291.494 q^{29} +67.9669 q^{30} +244.307 q^{31} -196.963 q^{32} -57.1868 q^{33} -29.4436 q^{34} -35.0000 q^{35} +112.780 q^{36} -193.121 q^{37} -24.6303 q^{38} -8.81323 q^{39} +102.656 q^{40} +315.113 q^{41} -95.1537 q^{42} -300.996 q^{43} -238.872 q^{44} +45.0000 q^{45} +223.735 q^{46} +86.5058 q^{47} -21.6595 q^{48} +49.0000 q^{49} +113.278 q^{50} -19.4942 q^{51} -36.8132 q^{52} +509.677 q^{53} +122.340 q^{54} -95.3113 q^{55} -143.718 q^{56} -16.3074 q^{57} -1320.80 q^{58} -83.3852 q^{59} +187.967 q^{60} -5.25291 q^{61} +1106.99 q^{62} -63.0000 q^{63} -834.706 q^{64} -14.6887 q^{65} -259.121 q^{66} +205.992 q^{67} -81.4281 q^{68} +148.132 q^{69} -158.590 q^{70} +1004.31 q^{71} +184.780 q^{72} -1007.29 q^{73} -875.055 q^{74} +75.0000 q^{75} -68.1168 q^{76} +133.436 q^{77} -39.9339 q^{78} -863.237 q^{79} -36.0992 q^{80} +81.0000 q^{81} +1427.82 q^{82} +1334.72 q^{83} -263.154 q^{84} -32.4903 q^{85} -1363.85 q^{86} -874.483 q^{87} -391.370 q^{88} +326.249 q^{89} +203.901 q^{90} +20.5642 q^{91} +618.755 q^{92} +732.922 q^{93} +391.969 q^{94} -27.1790 q^{95} -590.889 q^{96} +1526.77 q^{97} +222.025 q^{98} -171.560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 3 q^{6} - 14 q^{7} + 33 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 + 3 * q^6 - 14 * q^7 + 33 * q^8 + 18 * q^9	\( 2 q + q^{2} + 6 q^{3} + 17 q^{4} + 10 q^{5} + 3 q^{6} - 14 q^{7} + 33 q^{8} + 18 q^{9} + 5 q^{10} - 22 q^{11} + 51 q^{12} - 22 q^{13} - 7 q^{14} + 30 q^{15} - 87 q^{16} + 116 q^{17} + 9 q^{18} + 102 q^{19} + 85 q^{20} - 42 q^{21} - 76 q^{22} + 260 q^{23} + 99 q^{24} + 50 q^{25} + 54 q^{26} + 54 q^{27} - 119 q^{28} - 196 q^{29} + 15 q^{30} + 150 q^{31} - 15 q^{32} - 66 q^{33} - 462 q^{34} - 70 q^{35} + 153 q^{36} - 96 q^{37} - 404 q^{38} - 66 q^{39} + 165 q^{40} - 176 q^{41} - 21 q^{42} - 344 q^{43} - 252 q^{44} + 90 q^{45} - 520 q^{46} + 560 q^{47} - 261 q^{48} + 98 q^{49} + 25 q^{50} + 348 q^{51} - 122 q^{52} + 326 q^{53} + 27 q^{54} - 110 q^{55} - 231 q^{56} + 306 q^{57} - 1658 q^{58} - 844 q^{59} + 255 q^{60} - 204 q^{61} + 1440 q^{62} - 126 q^{63} - 839 q^{64} - 110 q^{65} - 228 q^{66} - 104 q^{67} + 466 q^{68} + 780 q^{69} - 35 q^{70} + 1670 q^{71} + 297 q^{72} - 386 q^{73} - 1218 q^{74} + 150 q^{75} + 412 q^{76} + 154 q^{77} + 162 q^{78} - 888 q^{79} - 435 q^{80} + 162 q^{81} + 3162 q^{82} + 928 q^{83} - 357 q^{84} + 580 q^{85} - 1212 q^{86} - 588 q^{87} - 428 q^{88} + 588 q^{89} + 45 q^{90} + 154 q^{91} + 1560 q^{92} + 450 q^{93} - 1280 q^{94} + 510 q^{95} - 45 q^{96} + 522 q^{97} + 49 q^{98} - 198 q^{99}+O(q^{100}) \) 2 * q + q^2 + 6 * q^3 + 17 * q^4 + 10 * q^5 + 3 * q^6 - 14 * q^7 + 33 * q^8 + 18 * q^9 + 5 * q^10 - 22 * q^11 + 51 * q^12 - 22 * q^13 - 7 * q^14 + 30 * q^15 - 87 * q^16 + 116 * q^17 + 9 * q^18 + 102 * q^19 + 85 * q^20 - 42 * q^21 - 76 * q^22 + 260 * q^23 + 99 * q^24 + 50 * q^25 + 54 * q^26 + 54 * q^27 - 119 * q^28 - 196 * q^29 + 15 * q^30 + 150 * q^31 - 15 * q^32 - 66 * q^33 - 462 * q^34 - 70 * q^35 + 153 * q^36 - 96 * q^37 - 404 * q^38 - 66 * q^39 + 165 * q^40 - 176 * q^41 - 21 * q^42 - 344 * q^43 - 252 * q^44 + 90 * q^45 - 520 * q^46 + 560 * q^47 - 261 * q^48 + 98 * q^49 + 25 * q^50 + 348 * q^51 - 122 * q^52 + 326 * q^53 + 27 * q^54 - 110 * q^55 - 231 * q^56 + 306 * q^57 - 1658 * q^58 - 844 * q^59 + 255 * q^60 - 204 * q^61 + 1440 * q^62 - 126 * q^63 - 839 * q^64 - 110 * q^65 - 228 * q^66 - 104 * q^67 + 466 * q^68 + 780 * q^69 - 35 * q^70 + 1670 * q^71 + 297 * q^72 - 386 * q^73 - 1218 * q^74 + 150 * q^75 + 412 * q^76 + 154 * q^77 + 162 * q^78 - 888 * q^79 - 435 * q^80 + 162 * q^81 + 3162 * q^82 + 928 * q^83 - 357 * q^84 + 580 * q^85 - 1212 * q^86 - 588 * q^87 - 428 * q^88 + 588 * q^89 + 45 * q^90 + 154 * q^91 + 1560 * q^92 + 450 * q^93 - 1280 * q^94 + 510 * q^95 - 45 * q^96 + 522 * q^97 + 49 * q^98 - 198 * q^99

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