LMFDB - Embedded newform 550.5.d.a.351.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,5,Mod(351,550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("550.351");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 550 = 2 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	550.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$56.8534796961$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{553})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 271x^{2} + 272x + 19602 $ x^4 - 2x^3 - 271x^2 + 272*x + 19602
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.4
Root			$-11.2580 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	550.351
Dual form			550.5.d.a.351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} +11.2580 q^{3} -8.00000 q^{4} +31.8424i q^{6} +33.9411i q^{7} -22.6274i q^{8} +45.7420 q^{9} +O(q^{10})$	\(q+2.82843i q^{2} +11.2580 q^{3} -8.00000 q^{4} +31.8424i q^{6} +33.9411i q^{7} -22.6274i q^{8} +45.7420 q^{9} +(2.03190 - 120.983i) q^{11} -90.0638 q^{12} +72.2602i q^{13} -96.0000 q^{14} +64.0000 q^{16} -517.873i q^{17} +129.378i q^{18} -411.672i q^{19} +382.108i q^{21} +(342.191 + 5.74709i) q^{22} +967.641 q^{23} -254.739i q^{24} -204.383 q^{26} -396.933 q^{27} -271.529i q^{28} -640.503i q^{29} +716.487 q^{31} +181.019i q^{32} +(22.8751 - 1362.02i) q^{33} +1464.77 q^{34} -365.936 q^{36} -1209.64 q^{37} +1164.38 q^{38} +813.504i q^{39} -966.781i q^{41} -1080.77 q^{42} +2720.75i q^{43} +(-16.2552 + 967.864i) q^{44} +2736.90i q^{46} +177.085 q^{47} +720.510 q^{48} +1249.00 q^{49} -5830.20i q^{51} -578.082i q^{52} +4023.28 q^{53} -1122.70i q^{54} +768.000 q^{56} -4634.59i q^{57} +1811.62 q^{58} -1988.02 q^{59} +2396.69i q^{61} +2026.53i q^{62} +1552.54i q^{63} -512.000 q^{64} +(3852.38 + 64.7006i) q^{66} +2007.13 q^{67} +4142.98i q^{68} +10893.7 q^{69} +4255.04 q^{71} -1035.02i q^{72} +1288.68i q^{73} -3421.37i q^{74} +3293.37i q^{76} +(4106.30 + 68.9651i) q^{77} -2300.94 q^{78} -1480.23i q^{79} -8173.77 q^{81} +2734.47 q^{82} -10124.2i q^{83} -3056.87i q^{84} -7695.45 q^{86} -7210.77i q^{87} +(-2737.53 - 45.9767i) q^{88} +5133.72 q^{89} -2452.59 q^{91} -7741.13 q^{92} +8066.19 q^{93} +500.871i q^{94} +2037.91i q^{96} -2239.34 q^{97} +3532.71i q^{98} +(92.9434 - 5534.00i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 32 q^{4} + 230 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 32 * q^4 + 230 * q^9	$ 4 q - 2 q^{3} - 32 q^{4} + 230 q^{9} - 180 q^{11} + 16 q^{12} - 384 q^{14} + 256 q^{16} + 240 q^{22} + 1566 q^{23} + 1440 q^{26} - 506 q^{27} + 4418 q^{31} + 2302 q^{33} + 1344 q^{34} - 1840 q^{36} + 382 q^{37} + 2400 q^{38} + 192 q^{42} + 1440 q^{44} - 5688 q^{47} - 128 q^{48} + 4996 q^{49} + 8568 q^{53} + 3072 q^{56} + 9504 q^{58} - 3390 q^{59} - 2048 q^{64} + 13152 q^{66} + 8734 q^{67} + 26314 q^{69} + 3522 q^{71} + 2880 q^{77} - 27264 q^{78} - 31096 q^{81} + 19968 q^{82} - 10464 q^{86} - 1920 q^{88} - 8766 q^{89} + 17280 q^{91} - 12528 q^{92} - 20458 q^{93} - 17282 q^{97} - 12562 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 32 * q^4 + 230 * q^9 - 180 * q^11 + 16 * q^12 - 384 * q^14 + 256 * q^16 + 240 * q^22 + 1566 * q^23 + 1440 * q^26 - 506 * q^27 + 4418 * q^31 + 2302 * q^33 + 1344 * q^34 - 1840 * q^36 + 382 * q^37 + 2400 * q^38 + 192 * q^42 + 1440 * q^44 - 5688 * q^47 - 128 * q^48 + 4996 * q^49 + 8568 * q^53 + 3072 * q^56 + 9504 * q^58 - 3390 * q^59 - 2048 * q^64 + 13152 * q^66 + 8734 * q^67 + 26314 * q^69 + 3522 * q^71 + 2880 * q^77 - 27264 * q^78 - 31096 * q^81 + 19968 * q^82 - 10464 * q^86 - 1920 * q^88 - 8766 * q^89 + 17280 * q^91 - 12528 * q^92 - 20458 * q^93 - 17282 * q^97 - 12562 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/550\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$-1$	$1$

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 $

$2$			2.82843i			0.707107i
$2$			2.82843i			0.707107i
$3$	11.2580			1.25089			0.625443	−	0.780270i	$-0.284918\pi$
$3$	11.2580			1.25089			0.625443	+	0.780270i	$0.284918\pi$
$4$	−8.00000			−0.500000
$5$		0			0
$5$		0			0
$6$			31.8424i			0.884510i
$7$			33.9411i			0.692676i	0.938110	+	0.346338i	$0.112575\pi$
$7$			33.9411i			0.692676i	−0.938110	+	0.346338i	$0.887425\pi$
$8$		−	22.6274i		−	0.353553i
$9$	45.7420			0.564716
$10$		0			0
$11$	2.03190	−	120.983i	0.0167926	−	0.999859i
$11$	2.03190	−	120.983i	0.0167926	−	0.999859i
$12$	−90.0638			−0.625443
$13$			72.2602i			0.427575i	0.976880	+	0.213788i	$0.0685801\pi$
$13$			72.2602i			0.427575i	−0.976880	+	0.213788i	$0.931420\pi$
$14$	−96.0000			−0.489796
$15$		0			0
$16$	64.0000			0.250000
$17$		−	517.873i		−	1.79195i	−0.444107	−	0.895974i	$-0.646479\pi$
$17$		−	517.873i		−	1.79195i	0.444107	−	0.895974i	$-0.353521\pi$
$18$			129.378i			0.399315i
$19$		−	411.672i		−	1.14036i	−0.821518	−	0.570182i	$-0.806873\pi$
$19$		−	411.672i		−	1.14036i	0.821518	−	0.570182i	$-0.193127\pi$
$20$		0			0
$21$			382.108i			0.866459i
$22$	342.191	+	5.74709i	0.707007	+	0.0118742i
$23$	967.641			1.82919			0.914594	−	0.404373i	$-0.132510\pi$
$23$	967.641			1.82919			0.914594	+	0.404373i	$0.132510\pi$
$24$		−	254.739i		−	0.442255i
$25$		0			0
$26$	−204.383			−0.302341
$27$	−396.933			−0.544490
$28$		−	271.529i		−	0.346338i
$29$		−	640.503i		−	0.761597i	−0.924658	−	0.380799i	$-0.875649\pi$
$29$		−	640.503i		−	0.761597i	0.924658	−	0.380799i	$-0.124351\pi$
$30$		0			0
$31$	716.487			0.745564			0.372782	−	0.927919i	$-0.378404\pi$
$31$	716.487			0.745564			0.372782	+	0.927919i	$0.378404\pi$
$32$			181.019i			0.176777i
$33$	22.8751	−	1362.02i	0.0210056	−	1.25071i
$34$	1464.77			1.26710
$35$		0			0
$36$	−365.936			−0.282358
$37$	−1209.64			−0.883590			−0.441795	−	0.897116i	$-0.645658\pi$
$37$	−1209.64			−0.883590			−0.441795	+	0.897116i	$0.645658\pi$
$38$	1164.38			0.806359
$39$			813.504i			0.534848i
$40$		0			0
$41$		−	966.781i		−	0.575122i	−0.957762	−	0.287561i	$-0.907156\pi$
$41$		−	966.781i		−	0.575122i	0.957762	−	0.287561i	$-0.0928445\pi$
$42$	−1080.77			−0.612679
$43$			2720.75i			1.47147i	0.677269	+	0.735736i	$0.263163\pi$
$43$			2720.75i			1.47147i	−0.677269	+	0.735736i	$0.736837\pi$
$44$	−16.2552	+	967.864i	−0.00839630	+	0.499929i
$45$		0			0
$46$			2736.90i			1.29343i
$47$	177.085			0.0801651			0.0400826	−	0.999196i	$-0.487238\pi$
$47$	177.085			0.0801651			0.0400826	+	0.999196i	$0.487238\pi$
$48$	720.510			0.312722
$49$	1249.00			0.520200
$50$		0			0
$51$		−	5830.20i		−	2.24152i
$52$		−	578.082i		−	0.213788i
$53$	4023.28			1.43228			0.716140	−	0.697956i	$-0.245907\pi$
$53$	4023.28			1.43228			0.716140	+	0.697956i	$0.245907\pi$
$54$		−	1122.70i		−	0.385013i
$55$		0			0
$56$	768.000			0.244898
$57$		−	4634.59i		−	1.42647i
$58$	1811.62			0.538531
$59$	−1988.02			−0.571107			−0.285554	−	0.958363i	$-0.592177\pi$
$59$	−1988.02			−0.571107			−0.285554	+	0.958363i	$0.592177\pi$
$60$		0			0
$61$			2396.69i			0.644097i	0.946723	+	0.322049i	$0.104371\pi$
$61$			2396.69i			0.644097i	−0.946723	+	0.322049i	$0.895629\pi$
$62$			2026.53i			0.527193i
$63$			1552.54i			0.391165i
$64$	−512.000			−0.125000
$65$		0			0
$66$	3852.38	+	64.7006i	0.884385	+	0.0148532i
$67$	2007.13			0.447122			0.223561	−	0.974690i	$-0.428232\pi$
$67$	2007.13			0.447122			0.223561	+	0.974690i	$0.428232\pi$
$68$			4142.98i			0.895974i
$69$	10893.7			2.28811
$70$		0			0
$71$	4255.04			0.844086			0.422043	−	0.906576i	$-0.361313\pi$
$71$	4255.04			0.844086			0.422043	+	0.906576i	$0.361313\pi$
$72$		−	1035.02i		−	0.199657i
$73$			1288.68i			0.241824i	0.992663	+	0.120912i	$0.0385819\pi$
$73$			1288.68i			0.241824i	−0.992663	+	0.120912i	$0.961418\pi$
$74$		−	3421.37i		−	0.624793i
$75$		0			0
$76$			3293.37i			0.570182i
$77$	4106.30	+	68.9651i	0.692578	+	0.0116318i
$78$	−2300.94			−0.378195
$79$		−	1480.23i		−	0.237178i	−0.992943	−	0.118589i	$-0.962163\pi$
$79$		−	1480.23i		−	0.237178i	0.992943	−	0.118589i	$-0.0378371\pi$
$80$		0			0
$81$	−8173.77			−1.24581
$82$	2734.47			0.406673
$83$		−	10124.2i		−	1.46963i	−0.678270	−	0.734813i	$-0.737270\pi$
$83$		−	10124.2i		−	1.46963i	0.678270	−	0.734813i	$-0.262730\pi$
$84$		−	3056.87i		−	0.433229i
$85$		0			0
$86$	−7695.45			−1.04049
$87$		−	7210.77i		−	0.952672i
$88$	−2737.53	−	45.9767i	−0.353504	−	0.00593708i
$89$	5133.72			0.648115			0.324058	−	0.946037i	$-0.394953\pi$
$89$	5133.72			0.648115			0.324058	+	0.946037i	$0.394953\pi$
$90$		0			0
$91$	−2452.59			−0.296171
$92$	−7741.13			−0.914594
$93$	8066.19			0.932615
$94$			500.871i			0.0566853i
$95$		0			0
$96$			2037.91i			0.221128i
$97$	−2239.34			−0.238000			−0.119000	−	0.992894i	$-0.537969\pi$
$97$	−2239.34			−0.238000			−0.119000	+	0.992894i	$0.537969\pi$
$98$			3532.71i			0.367837i
$99$	92.9434	−	5534.00i	0.00948305	−	0.564637i
$100$		0			0
$101$		−	15580.0i		−	1.52730i	−0.645629	−	0.763651i	$-0.723405\pi$
$101$		−	15580.0i		−	1.52730i	0.645629	−	0.763651i	$-0.276595\pi$
$102$	16490.3			1.58500
$103$	9119.47			0.859597			0.429799	−	0.902925i	$-0.358585\pi$
$103$	9119.47			0.859597			0.429799	+	0.902925i	$0.358585\pi$
$104$	1635.06			0.151171
$105$		0			0
$106$			11379.5i			1.01278i
$107$		−	11204.9i		−	0.978684i	−0.872092	−	0.489342i	$-0.837237\pi$
$107$		−	11204.9i		−	0.978684i	0.872092	−	0.489342i	$-0.162763\pi$
$108$	3175.47			0.272245
$109$		−	10286.3i		−	0.865776i	−0.901448	−	0.432888i	$-0.857495\pi$
$109$		−	10286.3i		−	0.865776i	0.901448	−	0.432888i	$-0.142505\pi$
$110$		0			0
$111$	−13618.0			−1.10527
$112$			2172.23i			0.173169i
$113$	−17026.9			−1.33345			−0.666727	−	0.745302i	$-0.732305\pi$
$113$	−17026.9			−1.33345			−0.666727	+	0.745302i	$0.732305\pi$
$114$	13108.6			1.00866
$115$		0			0
$116$			5124.03i			0.380799i
$117$			3305.33i			0.241459i
$118$		−	5622.98i		−	0.403834i
$119$	17577.2			1.24124
$120$		0			0
$121$	−14632.7	−	491.651i	−0.999436	−	0.0335805i
$122$	−6778.85			−0.455446
$123$		−	10884.0i		−	0.719413i
$124$	−5731.89			−0.372782
$125$		0			0
$126$	−4391.23			−0.276596
$127$			27813.0i			1.72441i	0.506558	+	0.862206i	$0.330918\pi$
$127$			27813.0i			1.72441i	−0.506558	+	0.862206i	$0.669082\pi$
$128$		−	1448.15i		−	0.0883883i
$129$			30630.1i			1.84064i
$130$		0			0
$131$		−	6310.74i		−	0.367738i	−0.982951	−	0.183869i	$-0.941138\pi$
$131$		−	6310.74i		−	0.367738i	0.982951	−	0.183869i	$-0.0588621\pi$
$132$	−183.001	+	10896.2i	−0.0105028	+	0.625355i
$133$	13972.6			0.789903
$134$			5677.02i			0.316163i
$135$		0			0
$136$	−11718.1			−0.633549
$137$	14207.1			0.756944			0.378472	−	0.925613i	$-0.376450\pi$
$137$	14207.1			0.756944			0.378472	+	0.925613i	$0.376450\pi$
$138$			30812.0i			1.61794i
$139$			27772.6i			1.43743i	0.695306	+	0.718714i	$0.255269\pi$
$139$			27772.6i			1.43743i	−0.695306	+	0.718714i	$0.744731\pi$
$140$		0			0
$141$	1993.62			0.100277
$142$			12035.1i			0.596859i
$143$	8742.26	+	146.826i	0.427515	+	0.00718010i
$144$	2927.49			0.141179
$145$		0			0
$146$	−3644.94			−0.170995
$147$	14061.2			0.650711
$148$	9677.08			0.441795
$149$		−	23436.8i		−	1.05567i	−0.849348	−	0.527833i	$-0.823005\pi$
$149$		−	23436.8i		−	1.05567i	0.849348	−	0.527833i	$-0.176995\pi$
$150$		0			0
$151$		−	6556.14i		−	0.287538i	−0.989611	−	0.143769i	$-0.954078\pi$
$151$		−	6556.14i		−	0.287538i	0.989611	−	0.143769i	$-0.0459222\pi$
$152$	−9315.06			−0.403180
$153$		−	23688.6i		−	1.01194i
$154$	−195.063	+	11614.4i	−0.00822494	+	0.489727i
$155$		0			0
$156$		−	6508.03i		−	0.267424i
$157$	−2315.17			−0.0939254			−0.0469627	−	0.998897i	$-0.514954\pi$
$157$	−2315.17			−0.0939254			−0.0469627	+	0.998897i	$0.514954\pi$
$158$	4186.72			0.167710
$159$	45293.9			1.79162
$160$		0			0
$161$			32842.8i			1.26704i
$162$		−	23118.9i		−	0.880922i
$163$	21728.0			0.817796			0.408898	−	0.912580i	$-0.365913\pi$
$163$	21728.0			0.817796			0.408898	+	0.912580i	$0.365913\pi$
$164$			7734.25i			0.287561i
$165$		0			0
$166$	28635.7			1.03918
$167$			26470.6i			0.949143i	0.880217	+	0.474571i	$0.157397\pi$
$167$			26470.6i			0.949143i	−0.880217	+	0.474571i	$0.842603\pi$
$168$	8646.13			0.306339
$169$	23339.5			0.817179
$170$		0			0
$171$		−	18830.7i		−	0.643982i
$172$		−	21766.0i		−	0.735736i
$173$		−	6136.89i		−	0.205048i	−0.994731	−	0.102524i	$-0.967308\pi$
$173$		−	6136.89i		−	0.205048i	0.994731	−	0.102524i	$-0.0326919\pi$
$174$	20395.1			0.673641
$175$		0			0
$176$	130.042	−	7742.91i	0.00419815	−	0.249965i
$177$	−22381.1			−0.714390
$178$			14520.4i			0.458287i
$179$	−8164.31			−0.254808			−0.127404	−	0.991851i	$-0.540665\pi$
$179$	−8164.31			−0.254808			−0.127404	+	0.991851i	$0.540665\pi$
$180$		0			0
$181$	−10130.6			−0.309228			−0.154614	−	0.987975i	$-0.549413\pi$
$181$	−10130.6			−0.309228			−0.154614	+	0.987975i	$0.549413\pi$
$182$		−	6936.98i		−	0.209425i
$183$			26981.8i			0.805692i
$184$		−	21895.2i		−	0.646716i
$185$		0			0
$186$			22814.6i			0.659459i
$187$	−62653.8	−	1052.27i	−1.79170	−	0.0300915i
$188$	−1416.68			−0.0400826
$189$		−	13472.4i		−	0.377155i
$190$		0			0
$191$	13853.6			0.379749			0.189875	−	0.981808i	$-0.439192\pi$
$191$	13853.6			0.379749			0.189875	+	0.981808i	$0.439192\pi$
$192$	−5764.08			−0.156361
$193$		−	55752.8i		−	1.49676i	−0.663270	−	0.748380i	$-0.730832\pi$
$193$		−	55752.8i		−	1.49676i	0.663270	−	0.748380i	$-0.269168\pi$
$194$		−	6333.81i		−	0.168291i
$195$		0			0
$196$	−9992.00			−0.260100
$197$			38577.9i			0.994046i	0.867737	+	0.497023i	$0.165574\pi$
$197$			38577.9i			0.994046i	−0.867737	+	0.497023i	$0.834426\pi$
$198$	15652.5	+	262.884i	0.399258	+	0.00670553i
$199$	−43741.2			−1.10455			−0.552275	−	0.833662i	$-0.686240\pi$
$199$	−43741.2			−1.10455			−0.552275	+	0.833662i	$0.686240\pi$
$200$		0			0
$201$	22596.2			0.559299
$202$	44066.9			1.07997
$203$	21739.4			0.527540
$204$			46641.6i			1.12076i
$205$		0			0
$206$			25793.7i			0.607827i
$207$	44261.8			1.03297
$208$			4624.66i			0.106894i
$209$	−49805.2	−	836.477i	−1.14020	−	0.0191497i
$210$		0			0
$211$		−	40460.8i		−	0.908803i	−0.890797	−	0.454402i	$-0.849853\pi$
$211$		−	40460.8i		−	0.908803i	0.890797	−	0.454402i	$-0.150147\pi$
$212$	−32186.2			−0.716140
$213$	47903.1			1.05586
$214$	31692.4			0.692034
$215$		0			0
$216$			8981.58i			0.192506i
$217$			24318.4i			0.516434i
$218$	29094.0			0.612196
$219$			14507.9i			0.302494i
$220$		0			0
$221$	37421.6			0.766193
$222$		−	38517.6i		−	0.781545i
$223$	−12067.6			−0.242667			−0.121334	−	0.992612i	$-0.538717\pi$
$223$	−12067.6			−0.242667			−0.121334	+	0.992612i	$0.538717\pi$
$224$	−6144.00			−0.122449
$225$		0			0
$226$		−	48159.3i		−	0.942894i
$227$			38616.4i			0.749411i	0.927144	+	0.374705i	$0.122256\pi$
$227$			38616.4i			0.749411i	−0.927144	+	0.374705i	$0.877744\pi$
$228$			37076.7i			0.713233i
$229$	74080.9			1.41265			0.706326	−	0.707887i	$-0.250351\pi$
$229$	74080.9			1.41265			0.706326	+	0.707887i	$0.250351\pi$
$230$		0			0
$231$	46228.6	+	776.408i	0.866337	+	0.0145501i
$232$	−14492.9			−0.269265
$233$		−	51719.5i		−	0.952670i	−0.879264	−	0.476335i	$-0.841965\pi$
$233$		−	51719.5i		−	0.952670i	0.879264	−	0.476335i	$-0.158035\pi$
$234$	−9348.89			−0.170737
$235$		0			0
$236$	15904.2			0.285554
$237$		−	16664.4i		−	0.296683i
$238$			49715.8i			0.877689i
$239$			16911.3i			0.296062i	0.988983	+	0.148031i	$0.0472935\pi$
$239$			16911.3i			0.296062i	−0.988983	+	0.148031i	$0.952706\pi$
$240$		0			0
$241$			59892.0i			1.03118i	0.856835	+	0.515590i	$0.172427\pi$
$241$			59892.0i			1.03118i	−0.856835	+	0.515590i	$0.827573\pi$
$242$	1390.60	−	41387.6i	0.0237450	−	0.706708i
$243$	−59868.5			−1.01388
$244$		−	19173.5i		−	0.322049i
$245$		0			0
$246$	30784.6			0.508702
$247$	29747.5			0.487592
$248$		−	16212.2i		−	0.263597i
$249$		−	113979.i		−	1.83833i
$250$		0			0
$251$	−19186.2			−0.304538			−0.152269	−	0.988339i	$-0.548658\pi$
$251$	−19186.2			−0.304538			−0.152269	+	0.988339i	$0.548658\pi$
$252$		−	12420.3i		−	0.195583i
$253$	1966.15	−	117068.i	0.0307168	−	1.82893i
$254$	−78667.1			−1.21934
$255$		0			0
$256$	4096.00			0.0625000
$257$	−30625.1			−0.463672			−0.231836	−	0.972755i	$-0.574473\pi$
$257$	−30625.1			−0.463672			−0.231836	+	0.972755i	$0.574473\pi$
$258$	−86635.1			−1.30153
$259$		−	41056.4i		−	0.612042i
$260$		0			0
$261$		−	29297.9i		−	0.430086i
$262$	17849.5			0.260030
$263$			73815.1i			1.06717i	0.845746	+	0.533585i	$0.179156\pi$
$263$			73815.1i			1.06717i	−0.845746	+	0.533585i	$0.820844\pi$
$264$	−30819.1	−	517.605i	−0.442193	−	0.00742661i
$265$		0			0
$266$			39520.5i			0.558546i
$267$	57795.3			0.810718
$268$	−16057.0			−0.223561
$269$	−87292.0			−1.20634			−0.603170	−	0.797612i	$-0.706096\pi$
$269$	−87292.0			−1.20634			−0.603170	+	0.797612i	$0.706096\pi$
$270$		0			0
$271$		−	4518.78i		−	0.0615294i	−0.999527	−	0.0307647i	$-0.990206\pi$
$271$		−	4518.78i		−	0.0615294i	0.999527	−	0.0307647i	$-0.00979425\pi$
$272$		−	33143.9i		−	0.447987i
$273$	−27611.2			−0.370477
$274$			40183.7i			0.535240i
$275$		0			0
$276$	−87149.4			−1.14405
$277$		−	4641.04i		−	0.0604861i	−0.999543	−	0.0302431i	$-0.990372\pi$
$277$		−	4641.04i		−	0.0604861i	0.999543	−	0.0302431i	$-0.00962813\pi$
$278$	−78552.6			−1.01642
$279$	32773.6			0.421032
$280$		0			0
$281$		−	58621.1i		−	0.742406i	−0.928552	−	0.371203i	$-0.878945\pi$
$281$		−	58621.1i		−	0.742406i	0.928552	−	0.371203i	$-0.121055\pi$
$282$			5638.80i			0.0709069i
$283$			90326.0i			1.12782i	0.825836	+	0.563910i	$0.190704\pi$
$283$			90326.0i			1.12782i	−0.825836	+	0.563910i	$0.809296\pi$
$284$	−34040.3			−0.422043
$285$		0			0
$286$	−415.286	+	24726.8i	−0.00507710	+	0.302299i
$287$	32813.6			0.398373
$288$			8280.19i			0.0998287i
$289$	−184671.			−2.21108
$290$		0			0
$291$	−25210.4			−0.297710
$292$		−	10309.4i		−	0.120912i
$293$			77687.4i			0.904931i	0.891782	+	0.452465i	$0.149455\pi$
$293$			77687.4i			0.904931i	−0.891782	+	0.452465i	$0.850545\pi$
$294$			39771.1i			0.460122i
$295$		0			0
$296$			27370.9i			0.312396i
$297$	−806.531	+	48022.2i	−0.00914341	+	0.544414i
$298$	66289.4			0.746469
$299$			69922.0i			0.782116i
$300$		0			0
$301$	−92345.3			−1.01925
$302$	18543.6			0.203320
$303$		−	175399.i		−	1.91048i
$304$		−	26347.0i		−	0.285091i
$305$		0			0
$306$	67001.3			0.715551
$307$		−	60487.2i		−	0.641781i	−0.947116	−	0.320890i	$-0.896018\pi$
$307$		−	60487.2i		−	0.641781i	0.947116	−	0.320890i	$-0.103982\pi$
$308$	−32850.4	−	551.721i	−0.346289	−	0.00581591i
$309$	102667.			1.07526
$310$		0			0
$311$	32106.2			0.331947			0.165973	−	0.986130i	$-0.446923\pi$
$311$	32106.2			0.331947			0.165973	+	0.986130i	$0.446923\pi$
$312$	18407.5			0.189097
$313$	−111475.			−1.13786			−0.568932	−	0.822384i	$-0.692643\pi$
$313$	−111475.			−1.13786			−0.568932	+	0.822384i	$0.692643\pi$
$314$		−	6548.28i		−	0.0664153i
$315$		0			0
$316$			11841.8i			0.118589i
$317$	−139267.			−1.38589			−0.692945	−	0.720990i	$-0.743687\pi$
$317$	−139267.			−1.38589			−0.692945	+	0.720990i	$0.743687\pi$
$318$			128111.i			1.26687i
$319$	−77490.0	−	1301.44i	−0.761490	−	0.0127892i
$320$		0			0
$321$		−	126145.i		−	1.22422i
$322$	−92893.5			−0.895929
$323$	−213194.			−2.04347
$324$	65390.2			0.622906
$325$		0			0
$326$			61456.1i			0.578269i
$327$		−	115803.i		−	1.08299i
$328$	−21875.7			−0.203336
$329$			6010.46i			0.0555285i
$330$		0			0
$331$	−60223.2			−0.549677			−0.274839	−	0.961490i	$-0.588624\pi$
$331$	−60223.2			−0.549677			−0.274839	+	0.961490i	$0.588624\pi$
$332$			80994.0i			0.734813i
$333$	−55331.2			−0.498978
$334$	−74870.3			−0.671145
$335$		0			0
$336$			24454.9i			0.216615i
$337$		−	85639.9i		−	0.754078i	−0.926197	−	0.377039i	$-0.876942\pi$
$337$		−	85639.9i		−	0.754078i	0.926197	−	0.377039i	$-0.123058\pi$
$338$			66014.0i			0.577833i
$339$	−191688.			−1.66800
$340$		0			0
$341$	1455.83	−	86682.7i	0.0125200	−	0.745459i
$342$	53261.2			0.455364
$343$			123885.i			1.05301i
$344$	61563.6			0.520244
$345$		0			0
$346$	17357.8			0.144991
$347$		−	99499.7i		−	0.826348i	−0.910652	−	0.413174i	$-0.864420\pi$
$347$		−	99499.7i		−	0.826348i	0.910652	−	0.413174i	$-0.135580\pi$
$348$			57686.2i			0.476336i
$349$		−	47515.3i		−	0.390106i	−0.980793	−	0.195053i	$-0.937512\pi$
$349$		−	47515.3i		−	0.390106i	0.980793	−	0.195053i	$-0.0624878\pi$
$350$		0			0
$351$		−	28682.5i		−	0.232811i
$352$	21900.3	+	367.814i	0.176752	+	0.00296854i
$353$	−167482.			−1.34406			−0.672030	−	0.740524i	$-0.734577\pi$
$353$	−167482.			−1.34406			−0.672030	+	0.740524i	$0.734577\pi$
$354$		−	63303.4i		−	0.505150i
$355$		0			0
$356$	−41069.8			−0.324058
$357$	197884.			1.55265
$358$		−	23092.2i		−	0.180177i
$359$		−	5063.91i		−	0.0392914i	−0.999807	−	0.0196457i	$-0.993746\pi$
$359$		−	5063.91i		−	0.0392914i	0.999807	−	0.0196457i	$-0.00625382\pi$
$360$		0			0
$361$	−39152.4			−0.300431
$362$		−	28653.7i		−	0.218657i
$363$	−164735.	−	5535.00i	−1.25018	−	0.0420053i
$364$	19620.8			0.148086
$365$		0			0
$366$	−76316.1			−0.569711
$367$	−264021.			−1.96023			−0.980115	−	0.198429i	$-0.936416\pi$
$367$	−264021.			−1.96023			−0.980115	+	0.198429i	$0.936416\pi$
$368$	61929.0			0.457297
$369$		−	44222.5i		−	0.324781i
$370$		0			0
$371$			136555.i			0.992106i
$372$	−64529.5			−0.466308
$373$		−	270510.i		−	1.94431i	−0.234339	−	0.972155i	$-0.575293\pi$
$373$		−	270510.i		−	1.94431i	0.234339	−	0.972155i	$-0.424707\pi$
$374$	2976.26	−	177212.i	0.0212779	−	1.26692i
$375$		0			0
$376$		−	4006.97i		−	0.0283426i
$377$	46282.9			0.325640
$378$	38105.6			0.266689
$379$	−38850.0			−0.270466			−0.135233	−	0.990814i	$-0.543178\pi$
$379$	−38850.0			−0.270466			−0.135233	+	0.990814i	$0.543178\pi$
$380$		0			0
$381$			313118.i			2.15704i
$382$			39184.0i			0.268523i
$383$	243070.			1.65704			0.828522	−	0.559956i	$-0.189182\pi$
$383$	243070.			1.65704			0.828522	+	0.559956i	$0.189182\pi$
$384$		−	16303.3i		−	0.110564i
$385$		0			0
$386$	157693.			1.05837
$387$			124453.i			0.830964i
$388$	17914.7			0.119000
$389$	−142544.			−0.941995			−0.470998	−	0.882135i	$-0.656106\pi$
$389$	−142544.			−0.941995			−0.470998	+	0.882135i	$0.656106\pi$
$390$		0			0
$391$		−	501115.i		−	3.27781i
$392$		−	28261.6i		−	0.183918i
$393$		−	71046.2i		−	0.459998i
$394$	−109115.			−0.702897
$395$		0			0
$396$	−743.547	+	44272.0i	−0.00474153	+	0.282318i
$397$	73860.0			0.468628			0.234314	−	0.972161i	$-0.424716\pi$
$397$	73860.0			0.468628			0.234314	+	0.972161i	$0.424716\pi$
$398$		−	123719.i		−	0.781034i
$399$	157303.			0.988079
$400$		0			0
$401$	225776.			1.40407			0.702034	−	0.712143i	$-0.252275\pi$
$401$	225776.			1.40407			0.702034	+	0.712143i	$0.252275\pi$
$402$			63911.8i			0.395484i
$403$			51773.5i			0.318785i
$404$			124640.i			0.763651i
$405$		0			0
$406$			61488.3i			0.373027i
$407$	−2457.86	+	146345.i	−0.0148378	+	0.883466i
$408$	−131922.			−0.792498
$409$			97972.9i			0.585679i	0.956162	+	0.292839i	$0.0946001\pi$
$409$			97972.9i			0.585679i	−0.956162	+	0.292839i	$0.905400\pi$
$410$		0			0
$411$	159943.			0.946851
$412$	−72955.7			−0.429799
$413$		−	67475.8i		−	0.395592i
$414$			125191.i			0.730422i
$415$		0			0
$416$	−13080.5			−0.0755854
$417$			312663.i			1.79806i
$418$	2365.91	−	140870.i	0.0135409	−	0.806246i
$419$	−64572.5			−0.367807			−0.183903	−	0.982944i	$-0.558873\pi$
$419$	−64572.5			−0.367807			−0.183903	+	0.982944i	$0.558873\pi$
$420$		0			0
$421$	203272.			1.14687			0.573434	−	0.819252i	$-0.305611\pi$
$421$	203272.			1.14687			0.573434	+	0.819252i	$0.305611\pi$
$422$	114441.			0.642621
$423$	8100.21			0.0452706
$424$		−	91036.3i		−	0.506388i
$425$		0			0
$426$			135491.i			0.746603i
$427$	−81346.2			−0.446151
$428$			89639.6i			0.489342i
$429$	98420.1	+	1652.96i	0.534773	+	0.00898149i
$430$		0			0
$431$			11113.4i			0.0598266i	0.999552	+	0.0299133i	$0.00952311\pi$
$431$			11113.4i			0.0598266i	−0.999552	+	0.0299133i	$0.990477\pi$
$432$	−25403.7			−0.136123
$433$	288512.			1.53882			0.769409	−	0.638756i	$-0.220551\pi$
$433$	288512.			1.53882			0.769409	+	0.638756i	$0.220551\pi$
$434$	−68782.7			−0.365174
$435$		0			0
$436$			82290.2i			0.432888i
$437$		−	398350.i		−	2.08594i
$438$	−41034.6			−0.213896
$439$			107467.i			0.557630i	0.960345	+	0.278815i	$0.0899417\pi$
$439$			107467.i			0.557630i	−0.960345	+	0.278815i	$0.910058\pi$
$440$		0			0
$441$	57131.8			0.293765
$442$			105844.i			0.541780i
$443$	130812.			0.666563			0.333281	−	0.942827i	$-0.391844\pi$
$443$	130812.			0.666563			0.333281	+	0.942827i	$0.391844\pi$
$444$	108944.			0.552636
$445$		0			0
$446$		−	34132.3i		−	0.171592i
$447$		−	263851.i		−	1.32052i
$448$		−	17377.9i		−	0.0865845i
$449$	−158898.			−0.788181			−0.394090	−	0.919072i	$-0.628940\pi$
$449$	−158898.			−0.788181			−0.394090	+	0.919072i	$0.628940\pi$
$450$		0			0
$451$	−116964.	−	1964.41i	−0.575041	−	0.00965780i
$452$	136215.			0.666727
$453$		−	73808.9i		−	0.359677i
$454$	−109224.			−0.529913
$455$		0			0
$456$	−104869.			−0.504332
$457$			291561.i			1.39604i	0.716079	+	0.698019i	$0.245935\pi$
$457$			291561.i			1.39604i	−0.716079	+	0.698019i	$0.754065\pi$
$458$			209532.i			0.998896i
$459$			205561.i			0.975698i
$460$		0			0
$461$		−	62731.9i		−	0.295180i	−0.989049	−	0.147590i	$-0.952848\pi$
$461$		−	62731.9i		−	0.295180i	0.989049	−	0.147590i	$-0.0471515\pi$
$462$	−2196.01	+	130754.i	−0.0102885	+	0.612593i
$463$	170883.			0.797146			0.398573	−	0.917137i	$-0.369506\pi$
$463$	170883.			0.797146			0.398573	+	0.917137i	$0.369506\pi$
$464$		−	40992.2i		−	0.190399i
$465$		0			0
$466$	146285.			0.673639
$467$	−94559.0			−0.433580			−0.216790	−	0.976218i	$-0.569559\pi$
$467$	−94559.0			−0.433580			−0.216790	+	0.976218i	$0.569559\pi$
$468$		−	26442.6i		−	0.120729i
$469$			68124.3i			0.309711i
$470$		0			0
$471$	−26064.1			−0.117490
$472$			44983.8i			0.201917i
$473$	329164.	+	5528.30i	1.47126	+	0.0247098i
$474$	47134.0			0.209786
$475$		0			0
$476$	−140618.			−0.620620
$477$	184033.			0.808832
$478$	−47832.5			−0.209347
$479$			154781.i			0.674600i	0.941397	+	0.337300i	$0.109514\pi$
$479$			154781.i			0.674600i	−0.941397	+	0.337300i	$0.890486\pi$
$480$		0			0
$481$		−	87408.6i		−	0.377802i
$482$	−169400.			−0.729155
$483$			369744.i			1.58492i
$484$	117062.	+	3933.21i	0.499718	+	0.0167902i
$485$		0			0
$486$		−	169334.i		−	0.716920i
$487$	−205330.			−0.865754			−0.432877	−	0.901453i	$-0.642502\pi$
$487$	−205330.			−0.865754			−0.432877	+	0.901453i	$0.642502\pi$
$488$	54230.8			0.227723
$489$	244614.			1.02297
$490$		0			0
$491$			180799.i			0.749950i	0.927035	+	0.374975i	$0.122349\pi$
$491$			180799.i			0.749950i	−0.927035	+	0.374975i	$0.877651\pi$
$492$			87071.9i			0.359706i
$493$	−331699.			−1.36474
$494$			84138.6i			0.344779i
$495$		0			0
$496$	45855.2			0.186391
$497$			144421.i			0.584678i
$498$	322380.			1.29990
$499$	198403.			0.796797			0.398399	−	0.917212i	$-0.369566\pi$
$499$	198403.			0.796797			0.398399	+	0.917212i	$0.369566\pi$
$500$		0			0
$501$			298006.i			1.18727i
$502$		−	54266.8i		−	0.215341i
$503$			278604.i			1.10116i	0.834781	+	0.550582i	$0.185594\pi$
$503$			278604.i			1.10116i	−0.834781	+	0.550582i	$0.814406\pi$
$504$	35129.9			0.138298
$505$		0			0
$506$	331118.	+	5561.12i	1.29325	+	0.0217201i
$507$	262755.			1.02220
$508$		−	222504.i		−	0.862206i
$509$	−195175.			−0.753336			−0.376668	−	0.926348i	$-0.622930\pi$
$509$	−195175.			−0.753336			−0.376668	+	0.926348i	$0.622930\pi$
$510$		0			0
$511$	−43739.2			−0.167506
$512$			11585.2i			0.0441942i
$513$			163406.i			0.620917i
$514$		−	86620.8i		−	0.327866i
$515$		0			0
$516$		−	245041.i		−	0.920322i
$517$	359.819	−	21424.2i	0.00134618	−	0.0801538i
$518$	116125.			0.432779
$519$		−	69089.0i		−	0.256492i
$520$		0			0
$521$	−2617.52			−0.00964304			−0.00482152	−	0.999988i	$-0.501535\pi$
$521$	−2617.52			−0.00964304			−0.00482152	+	0.999988i	$0.501535\pi$
$522$	82867.0			0.304117
$523$		−	309803.i		−	1.13261i	−0.824194	−	0.566307i	$-0.808372\pi$
$523$		−	309803.i		−	1.13261i	0.824194	−	0.566307i	$-0.191628\pi$
$524$			50486.0i			0.183869i
$525$		0			0
$526$	−208781.			−0.754603
$527$		−	371049.i		−	1.33601i
$528$	1464.01	−	87169.5i	0.00525141	−	0.312677i
$529$	656488.			2.34593
$530$		0			0
$531$	−90936.2			−0.322513
$532$	−111781.			−0.394952
$533$	69859.8			0.245908
$534$			163470.i			0.573264i
$535$		0			0
$536$		−	45416.2i		−	0.158081i
$537$	−91913.7			−0.318736
$538$		−	246899.i		−	0.853012i
$539$	2537.85	−	151108.i	0.00873551	−	0.520127i
$540$		0			0
$541$		−	197545.i		−	0.674948i	−0.941335	−	0.337474i	$-0.890427\pi$
$541$		−	197545.i		−	0.674948i	0.941335	−	0.337474i	$-0.109573\pi$
$542$	12781.0			0.0435079
$543$	−114050.			−0.386809
$544$	93745.0			0.316775
$545$		0			0
$546$		−	78096.4i		−	0.261966i
$547$			440541.i			1.47235i	0.676790	+	0.736176i	$0.263370\pi$
$547$			440541.i			1.47235i	−0.676790	+	0.736176i	$0.736630\pi$
$548$	−113657.			−0.378472
$549$			109629.i			0.363732i
$550$		0			0
$551$	−263677.			−0.868498
$552$		−	246496.i		−	0.808968i
$553$	50240.6			0.164288
$554$	13126.8			0.0427701
$555$		0			0
$556$		−	222180.i		−	0.718714i
$557$			422793.i			1.36275i	0.731933	+	0.681376i	$0.238618\pi$
$557$			422793.i			1.36275i	−0.731933	+	0.681376i	$0.761382\pi$
$558$			92697.6i			0.297715i
$559$	−196602.			−0.629165
$560$		0			0
$561$	−705355.	−	11846.4i	−2.24121	−	0.0376410i
$562$	165805.			0.524960
$563$			52520.6i			0.165696i	0.996562	+	0.0828482i	$0.0264017\pi$
$563$			52520.6i			0.165696i	−0.996562	+	0.0828482i	$0.973598\pi$
$564$	−15948.9			−0.0501387
$565$		0			0
$566$	−255481.			−0.797490
$567$		−	277427.i		−	0.862944i
$568$		−	96280.5i		−	0.298430i
$569$		−	140197.i		−	0.433025i	−0.976280	−	0.216513i	$-0.930532\pi$
$569$		−	140197.i		−	0.433025i	0.976280	−	0.216513i	$-0.0694683\pi$
$570$		0			0
$571$			207548.i			0.636569i	0.947995	+	0.318284i	$0.103107\pi$
$571$			207548.i			0.636569i	−0.947995	+	0.318284i	$0.896893\pi$
$572$	−69938.1	−	1174.61i	−0.213758	−	0.00359005i
$573$	155964.			0.475023
$574$			92810.9i			0.281693i
$575$		0			0
$576$	−23419.9			−0.0705895
$577$	273572.			0.821712			0.410856	−	0.911700i	$-0.365230\pi$
$577$	273572.			0.821712			0.410856	+	0.911700i	$0.365230\pi$
$578$		−	522329.i		−	1.56347i
$579$		−	627664.i		−	1.87228i
$580$		0			0
$581$	343628.			1.01797
$582$		−	71305.8i		−	0.210513i
$583$	8174.91	−	486748.i	0.0240517	−	1.43208i
$584$	29159.5			0.0854977
$585$		0			0
$586$	−219733.			−0.639883
$587$	576701.			1.67369			0.836843	−	0.547442i	$-0.184398\pi$
$587$	576701.			1.67369			0.836843	+	0.547442i	$0.184398\pi$
$588$	−112490.			−0.325355
$589$		−	294957.i		−	0.850214i
$590$		0			0
$591$			434309.i			1.24344i
$592$	−77416.7			−0.220898
$593$			671956.i			1.91087i	0.295202	+	0.955435i	$0.404613\pi$
$593$			671956.i			1.91087i	−0.295202	+	0.955435i	$0.595387\pi$
$594$	−135827.	−	2281.21i	−0.384959	−	0.00646536i
$595$		0			0
$596$			187495.i			0.527833i
$597$	−492438.			−1.38167
$598$	−197769.			−0.553040
$599$	582984.			1.62481			0.812406	−	0.583092i	$-0.198157\pi$
$599$	582984.			1.62481			0.812406	+	0.583092i	$0.198157\pi$
$600$		0			0
$601$			377092.i			1.04399i	0.852947	+	0.521997i	$0.174813\pi$
$601$			377092.i			1.04399i	−0.852947	+	0.521997i	$0.825187\pi$
$602$		−	261192.i		−	0.720721i
$603$	91810.2			0.252497
$604$			52449.2i			0.143769i
$605$		0			0
$606$	496104.			1.35091
$607$		−	465535.i		−	1.26350i	−0.775173	−	0.631750i	$-0.782337\pi$
$607$		−	465535.i		−	1.26350i	0.775173	−	0.631750i	$-0.217663\pi$
$608$	74520.5			0.201590
$609$	244742.			0.659893
$610$		0			0
$611$			12796.2i			0.0342766i
$612$			189508.i			0.505971i
$613$			623300.i			1.65873i	0.558705	+	0.829366i	$0.311298\pi$
$613$			623300.i			1.65873i	−0.558705	+	0.829366i	$0.688702\pi$
$614$	171084.			0.453807
$615$		0			0
$616$	1560.50	−	92914.9i	0.00411247	−	0.244863i
$617$	−197189.			−0.517980			−0.258990	−	0.965880i	$-0.583390\pi$
$617$	−197189.			−0.517980			−0.258990	+	0.965880i	$0.583390\pi$
$618$			290385.i			0.760322i
$619$	214900.			0.560861			0.280431	−	0.959874i	$-0.409523\pi$
$619$	214900.			0.560861			0.280431	+	0.959874i	$0.409523\pi$
$620$		0			0
$621$	−384089.			−0.995976
$622$			90810.1i			0.234722i
$623$			174244.i			0.448934i
$624$			52064.3i			0.133712i
$625$		0			0
$626$		−	315300.i		−	0.804592i
$627$	−560706.	−	9417.04i	−1.42626	−	0.0239541i
$628$	18521.3			0.0469627
$629$			626437.i			1.58335i
$630$		0			0
$631$	−40334.6			−0.101302			−0.0506511	−	0.998716i	$-0.516130\pi$
$631$	−40334.6			−0.101302			−0.0506511	+	0.998716i	$0.516130\pi$
$632$	−33493.8			−0.0838551
$633$		−	455507.i		−	1.13681i
$634$		−	393906.i		−	0.979973i
$635$		0			0
$636$	−362352.			−0.895810
$637$			90253.1i			0.222425i
$638$	3681.03	−	219175.i	0.00904333	−	0.538455i
$639$	194634.			0.476669
$640$		0			0
$641$	−19965.1			−0.0485909			−0.0242954	−	0.999705i	$-0.507734\pi$
$641$	−19965.1			−0.0485909			−0.0242954	+	0.999705i	$0.507734\pi$
$642$	356792.			0.865656
$643$	−21885.5			−0.0529340			−0.0264670	−	0.999650i	$-0.508426\pi$
$643$	−21885.5			−0.0529340			−0.0264670	+	0.999650i	$0.508426\pi$
$644$		−	262743.i		−	0.633518i
$645$		0			0
$646$		−	603002.i		−	1.44495i
$647$	345969.			0.826472			0.413236	−	0.910624i	$-0.364398\pi$
$647$	345969.			0.826472			0.413236	+	0.910624i	$0.364398\pi$
$648$			184951.i			0.440461i
$649$	−4039.47	+	240517.i	−0.00959037	+	0.571027i
$650$		0			0
$651$			273776.i			0.646000i
$652$	−173824.			−0.408898
$653$	−380939.			−0.893366			−0.446683	−	0.894692i	$-0.647395\pi$
$653$	−380939.			−0.893366			−0.446683	+	0.894692i	$0.647395\pi$
$654$	327539.			0.765787
$655$		0			0
$656$		−	61874.0i		−	0.143781i
$657$			58946.8i			0.136562i
$658$	−17000.1			−0.0392645
$659$		−	149577.i		−	0.344424i	−0.985060	−	0.172212i	$-0.944909\pi$
$659$		−	149577.i		−	0.344424i	0.985060	−	0.172212i	$-0.0550914\pi$
$660$		0			0
$661$	−637662.			−1.45945			−0.729723	−	0.683743i	$-0.760351\pi$
$661$	−637662.			−1.45945			−0.729723	+	0.683743i	$0.760351\pi$
$662$		−	170337.i		−	0.388681i
$663$	421292.			0.958420
$664$	−229086.			−0.519591
$665$		0			0
$666$		−	156500.i		−	0.352831i
$667$		−	619777.i		−	1.39311i
$668$		−	211765.i		−	0.474571i
$669$	−135857.			−0.303549
$670$		0			0
$671$	289958.	+	4869.84i	0.644006	+	0.0108161i
$672$	−69169.0			−0.153170
$673$			165463.i			0.365318i	0.983176	+	0.182659i	$0.0584704\pi$
$673$			165463.i			0.365318i	−0.983176	+	0.182659i	$0.941530\pi$
$674$	242226.			0.533214
$675$		0			0
$676$	−186716.			−0.408590
$677$			420292.i			0.917009i	0.888692	+	0.458504i	$0.151615\pi$
$677$			420292.i			0.917009i	−0.888692	+	0.458504i	$0.848385\pi$
$678$		−	542176.i		−	1.17945i
$679$		−	76005.7i		−	0.164857i
$680$		0			0
$681$			434742.i			0.937428i
$682$	245176.	+	4117.72i	0.527119	+	0.00885294i
$683$	−265281.			−0.568677			−0.284338	−	0.958724i	$-0.591774\pi$
$683$	−265281.			−0.568677			−0.284338	+	0.958724i	$0.591774\pi$
$684$			150646.i			0.321991i
$685$		0			0
$686$	−350400.			−0.744588
$687$	834001.			1.76707
$688$			174128.i			0.367868i
$689$			290723.i			0.612408i
$690$		0			0
$691$	−72314.2			−0.151449			−0.0757247	−	0.997129i	$-0.524127\pi$
$691$	−72314.2			−0.151449			−0.0757247	+	0.997129i	$0.524127\pi$
$692$			49095.1i			0.102524i
$693$	187830.	+	3154.60i	0.391110	+	0.00656868i
$694$	281428.			0.584316
$695$		0			0
$696$	−163161.			−0.336820
$697$	−500669.			−1.03059
$698$	134393.			0.275846
$699$		−	582257.i		−	1.19168i
$700$		0			0
$701$			142549.i			0.290088i	0.989425	+	0.145044i	$0.0463324\pi$
$701$			142549.i			0.290088i	−0.989425	+	0.145044i	$0.953668\pi$
$702$	81126.4			0.164622
$703$			497972.i			1.00761i
$704$	−1040.33	+	61943.3i	−0.00209907	+	0.124982i
$705$		0			0
$706$		−	473711.i		−	0.950394i
$707$	528803.			1.05793
$708$	179049.			0.357195
$709$	−588664.			−1.17105			−0.585525	−	0.810654i	$-0.699112\pi$
$709$	−588664.			−1.17105			−0.585525	+	0.810654i	$0.699112\pi$
$710$		0			0
$711$		−	67708.7i		−	0.133938i
$712$		−	116163.i		−	0.229143i
$713$	693302.			1.36378
$714$			559699.i			1.09789i
$715$		0			0
$716$	65314.5			0.127404
$717$			190387.i			0.370340i
$718$	14322.9			0.0277832
$719$	−133520.			−0.258280			−0.129140	−	0.991626i	$-0.541222\pi$
$719$	−133520.			−0.258280			−0.129140	+	0.991626i	$0.541222\pi$
$720$		0			0
$721$			309525.i			0.595422i
$722$		−	110740.i		−	0.212437i
$723$			674263.i			1.28989i
$724$	81044.9			0.154614
$725$		0			0
$726$	15655.3	−	465941.i	0.0297023	−	0.884011i
$727$	−224313.			−0.424410			−0.212205	−	0.977225i	$-0.568064\pi$
$727$	−224313.			−0.424410			−0.212205	+	0.977225i	$0.568064\pi$
$728$			55495.9i			0.104712i
$729$	−11922.8			−0.0224348
$730$		0			0
$731$	1.40900e6			2.63680
$732$		−	215855.i		−	0.402846i
$733$		−	297252.i		−	0.553243i	−0.960979	−	0.276622i	$-0.910785\pi$
$733$		−	297252.i		−	0.553243i	0.960979	−	0.276622i	$-0.0892149\pi$
$734$		−	746765.i		−	1.38609i
$735$		0			0
$736$			175162.i			0.323358i
$737$	4078.30	−	242829.i	0.00750834	−	0.447059i
$738$	125080.			0.229655
$739$		−	268988.i		−	0.492543i	−0.969201	−	0.246272i	$-0.920794\pi$
$739$		−	268988.i		−	0.492543i	0.969201	−	0.246272i	$-0.0792056\pi$
$740$		0			0
$741$	334896.			0.609922
$742$	−386235.			−0.701525
$743$			133653.i			0.242104i	0.992646	+	0.121052i	$0.0386267\pi$
$743$			133653.i			0.242104i	−0.992646	+	0.121052i	$0.961373\pi$
$744$		−	182517.i		−	0.329729i
$745$		0			0
$746$	765117.			1.37483
$747$		−	463104.i		−	0.829921i
$748$	501230.	+	8418.14i	0.895848	+	0.0150457i
$749$	380309.			0.677911
$750$		0			0
$751$	201267.			0.356856			0.178428	−	0.983953i	$-0.442899\pi$
$751$	201267.			0.356856			0.178428	+	0.983953i	$0.442899\pi$
$752$	11333.4			0.0200413
$753$	−215998.			−0.380942
$754$			130908.i			0.230262i
$755$		0			0
$756$			107779.i			0.188578i
$757$	−835429.			−1.45787			−0.728934	−	0.684584i	$-0.759984\pi$
$757$	−835429.			−1.45787			−0.728934	+	0.684584i	$0.759984\pi$
$758$		−	109884.i		−	0.191248i
$759$	22134.9	−	1.31795e6i	0.0384233	−	2.28778i
$760$		0			0
$761$		−	345963.i		−	0.597393i	−0.954348	−	0.298696i	$-0.903448\pi$
$761$		−	345963.i		−	0.597393i	0.954348	−	0.298696i	$-0.0965518\pi$
$762$	−885633.			−1.52526
$763$	349128.			0.599702
$764$	−110829.			−0.189875
$765$		0			0
$766$			687506.i			1.17171i
$767$		−	143655.i		−	0.244191i
$768$	46112.7			0.0781804
$769$			1.05669e6i			1.78688i	0.449183	+	0.893440i	$0.351715\pi$
$769$			1.05669e6i			1.78688i	−0.449183	+	0.893440i	$0.648285\pi$
$770$		0			0
$771$	−344776.			−0.580001
$772$			446022.i			0.748380i
$773$	224559.			0.375813			0.187907	−	0.982187i	$-0.439830\pi$
$773$	224559.			0.375813			0.187907	+	0.982187i	$0.439830\pi$
$774$	−352005.			−0.587580
$775$		0			0
$776$			50670.4i			0.0841456i
$777$		−	462212.i		−	0.765595i
$778$		−	403174.i		−	0.666091i
$779$	−397996.			−0.655849
$780$		0			0
$781$	8645.83	−	514787.i	0.0141744	−	0.843967i
$782$	1.41737e6			2.31776
$783$			254237.i			0.414682i
$784$	79936.0			0.130050
$785$		0			0
$786$	200949.			0.325268
$787$		−	227376.i		−	0.367109i	−0.983010	−	0.183554i	$-0.941240\pi$
$787$		−	227376.i		−	0.367109i	0.983010	−	0.183554i	$-0.0587603\pi$
$788$		−	308623.i		−	0.497023i
$789$			831008.i			1.33491i
$790$		0			0
$791$		−	577911.i		−	0.923651i
$792$	−125220.	−	2103.07i	−0.199629	−	0.00335277i
$793$	−173185.			−0.275400
$794$			208908.i			0.331370i
$795$		0			0
$796$	349930.			0.552275
$797$	972019.			1.53023			0.765117	−	0.643891i	$-0.222681\pi$
$797$	972019.			1.53023			0.765117	+	0.643891i	$0.222681\pi$
$798$			444920.i			0.698677i
$799$		−	91707.4i		−	0.143652i
$800$		0			0
$801$	234827.			0.366001
$802$			638590.i			0.992826i
$803$	155908.	+	2618.47i	0.241790	+	0.00406085i
$804$	−180770.			−0.279649
$805$		0			0
$806$	−146438.			−0.225415
$807$	−982732.			−1.50900
$808$	−352535.			−0.539983
$809$			56475.6i			0.0862906i	0.999069	+	0.0431453i	$0.0137378\pi$
$809$			56475.6i			0.0862906i	−0.999069	+	0.0431453i	$0.986262\pi$
$810$		0			0
$811$			224590.i			0.341467i	0.985317	+	0.170733i	$0.0546137\pi$
$811$			224590.i			0.341467i	−0.985317	+	0.170733i	$0.945386\pi$
$812$	−173915.			−0.263770
$813$		−	50872.3i		−	0.0769663i
$814$	−413927.	−	6951.89i	−0.624705	−	0.0104919i
$815$		0			0
$816$		−	373133.i		−	0.560381i
$817$	1.12006e6			1.67801
$818$	−277109.			−0.414137
$819$	−112187.			−0.167253
$820$		0			0
$821$			719097.i			1.06684i	0.845849	+	0.533422i	$0.179094\pi$
$821$			719097.i			1.06684i	−0.845849	+	0.533422i	$0.820906\pi$
$822$			452387.i			0.669525i
$823$	−52447.6			−0.0774330			−0.0387165	−	0.999250i	$-0.512327\pi$
$823$	−52447.6			−0.0774330			−0.0387165	+	0.999250i	$0.512327\pi$
$824$		−	206350.i		−	0.303913i
$825$		0			0
$826$	190850.			0.279726
$827$			121409.i			0.177517i	0.996053	+	0.0887586i	$0.0282900\pi$
$827$			121409.i			0.177517i	−0.996053	+	0.0887586i	$0.971710\pi$
$828$	−354095.			−0.516486
$829$	555136.			0.807774			0.403887	−	0.914809i	$-0.367659\pi$
$829$	555136.			0.807774			0.403887	+	0.914809i	$0.367659\pi$
$830$		0			0
$831$		−	52248.7i		−	0.0756612i
$832$		−	36997.2i		−	0.0534469i
$833$		−	646823.i		−	0.932171i
$834$	−884344.			−1.27142
$835$		0			0
$836$	398442.	+	6691.82i	0.570102	+	0.00957484i
$837$	−284398.			−0.405952
$838$		−	182639.i		−	0.260079i
$839$	−728123.			−1.03438			−0.517191	−	0.855870i	$-0.673022\pi$
$839$	−728123.			−1.03438			−0.517191	+	0.855870i	$0.673022\pi$
$840$		0			0
$841$	297036.			0.419969
$842$			574940.i			0.810958i
$843$		−	659955.i		−	0.928665i
$844$			323687.i			0.454402i
$845$		0			0
$846$			22910.9i			0.0320111i
$847$	16687.2	−	496652.i	0.0232604	−	0.692285i
$848$	257490.			0.358070
$849$			1.01689e6i			1.41078i
$850$		0			0
$851$	−1.17049e6			−1.61625
$852$	−383225.			−0.527928
$853$			560810.i			0.770757i	0.922759	+	0.385379i	$0.125929\pi$
$853$			560810.i			0.770757i	−0.922759	+	0.385379i	$0.874071\pi$
$854$		−	230082.i		−	0.315476i
$855$		0			0
$856$	−253539.			−0.346017
$857$			1.02077e6i			1.38984i	0.719086	+	0.694921i	$0.244561\pi$
$857$			1.02077e6i			1.38984i	−0.719086	+	0.694921i	$0.755439\pi$
$858$	−4675.28	+	278374.i	−0.00635087	+	0.378141i
$859$	350246.			0.474665			0.237332	−	0.971428i	$-0.423727\pi$
$859$	350246.			0.474665			0.237332	+	0.971428i	$0.423727\pi$
$860$		0			0
$861$	369415.			0.498320
$862$	−31433.6			−0.0423038
$863$	516997.			0.694170			0.347085	−	0.937834i	$-0.387171\pi$
$863$	516997.			0.694170			0.347085	+	0.937834i	$0.387171\pi$
$864$		−	71852.6i		−	0.0962532i
$865$		0			0
$866$			816034.i			1.08811i
$867$	−2.07903e6			−2.76581
$868$		−	194547.i		−	0.258217i
$869$	−179082.	−	3007.68i	−0.237145	−	0.00398284i
$870$		0			0
$871$			145036.i			0.191178i
$872$	−232752.			−0.306098
$873$	−102432.			−0.134402
$874$	1.12670e6			1.47498
$875$		0			0
$876$		−	116063.i		−	0.151247i
$877$			657406.i			0.854741i	0.904077	+	0.427371i	$0.140560\pi$
$877$			657406.i			0.854741i	−0.904077	+	0.427371i	$0.859440\pi$
$878$	−303963.			−0.394304
$879$			874603.i			1.13197i
$880$		0			0
$881$	869395.			1.12012			0.560061	−	0.828452i	$-0.310778\pi$
$881$	869395.			1.12012			0.560061	+	0.828452i	$0.310778\pi$
$882$			161593.i			0.207724i
$883$	−955758.			−1.22582			−0.612910	−	0.790153i	$-0.710001\pi$
$883$	−955758.			−1.22582			−0.612910	+	0.790153i	$0.710001\pi$
$884$	−299373.			−0.383096
$885$		0			0
$886$			369993.i			0.471331i
$887$		−	436265.i		−	0.554503i	−0.960797	−	0.277251i	$-0.910577\pi$
$887$		−	436265.i		−	0.554503i	0.960797	−	0.277251i	$-0.0894235\pi$
$888$			308141.i			0.390772i
$889$	−944006.			−1.19446
$890$		0			0
$891$	−16608.3	+	988887.i	−0.0209204	+	1.24564i
$892$	96540.7			0.121334
$893$		−	72900.7i		−	0.0914174i
$894$	746284.			0.933747
$895$		0			0
$896$	49152.0			0.0612245
$897$			787180.i			0.978338i
$898$		−	449432.i		−	0.557328i
$899$		−	458912.i		−	0.567819i
$900$		0			0
$901$		−	2.08355e6i		−	2.56657i
$902$	5556.18	−	330824.i	0.00682909	−	0.406616i
$903$	−1.03962e6			−1.27497
$904$			385274.i			0.471447i
$905$		0			0
$906$	208763.			0.254330
$907$	707203.			0.859666			0.429833	−	0.902909i	$-0.358573\pi$
$907$	707203.			0.859666			0.429833	+	0.902909i	$0.358573\pi$
$908$		−	308931.i		−	0.374705i
$909$		−	712661.i		−	0.862493i
$910$		0			0
$911$	132785.			0.159998			0.0799989	−	0.996795i	$-0.474508\pi$
$911$	132785.			0.159998			0.0799989	+	0.996795i	$0.474508\pi$
$912$		−	296614.i		−	0.356616i
$913$	−1.22486e6	−	20571.5i	−1.46942	−	0.0246788i
$914$	−824660.			−0.987149
$915$		0			0
$916$	−592647.			−0.706326
$917$	214194.			0.254723
$918$	−581414.			−0.689923
$919$		−	964589.i		−	1.14212i	−0.820909	−	0.571059i	$-0.806533\pi$
$919$		−	964589.i		−	1.14212i	0.820909	−	0.571059i	$-0.193467\pi$
$920$		0			0
$921$		−	680963.i		−	0.802795i
$922$	177432.			0.208723
$923$			307470.i			0.360911i
$924$	−369829.	−	6211.26i	−0.433168	−	0.00727505i
$925$		0			0
$926$			483331.i			0.563667i
$927$	417143.			0.485429
$928$	115943.			0.134633
$929$	634849.			0.735596			0.367798	−	0.929906i	$-0.380112\pi$
$929$	634849.			0.735596			0.367798	+	0.929906i	$0.380112\pi$
$930$		0			0
$931$		−	514178.i		−	0.593217i
$932$			413756.i			0.476335i
$933$	361451.			0.415228
$934$		−	267453.i		−	0.306587i
$935$		0			0
$936$	74791.1			0.0853686
$937$		−	3331.90i		−	0.00379501i	−0.999998	−	0.00189750i	$-0.999396\pi$
$937$		−	3331.90i		−	0.00379501i	0.999998	−	0.00189750i	$-0.000603995\pi$
$938$	−192685.			−0.218998
$939$	−1.25499e6			−1.42334
$940$		0			0
$941$			1.13619e6i			1.28313i	0.767069	+	0.641564i	$0.221714\pi$
$941$			1.13619e6i			1.28313i	−0.767069	+	0.641564i	$0.778286\pi$
$942$		−	73720.4i		−	0.0830780i
$943$		−	935496.i		−	1.05201i
$944$	−127234.			−0.142777
$945$		0			0
$946$	−15636.4	+	931018.i	−0.0174725	+	1.04034i
$947$	−156679.			−0.174707			−0.0873536	−	0.996177i	$-0.527841\pi$
$947$	−156679.			−0.174707			−0.0873536	+	0.996177i	$0.527841\pi$
$948$			133315.i			0.148341i
$949$	−93120.3			−0.103398
$950$		0			0
$951$	−1.56786e6			−1.73359
$952$		−	397726.i		−	0.438844i
$953$		−	343607.i		−	0.378334i	−0.981945	−	0.189167i	$-0.939421\pi$
$953$		−	343607.i		−	0.378334i	0.981945	−	0.189167i	$-0.0605788\pi$
$954$			520523.i			0.571931i
$955$		0			0
$956$		−	135291.i		−	0.148031i
$957$	−872380.	−	14651.6i	−0.952537	−	0.0159978i
$958$	−437786.			−0.477014
$959$			482205.i			0.524317i
$960$		0			0
$961$	−410168.			−0.444135
$962$	247229.			0.267146
$963$		−	512537.i		−	0.552679i
$964$		−	479136.i		−	0.515590i
$965$		0			0
$966$	−1.04579e6			−1.12071
$967$		−	395609.i		−	0.423071i	−0.977370	−	0.211536i	$-0.932154\pi$
$967$		−	395609.i		−	0.423071i	0.977370	−	0.211536i	$-0.0678464\pi$
$968$	−11124.8	+	331101.i	−0.0118725	+	0.353354i
$969$	−2.40013e6			−2.55615
$970$		0			0
$971$	1.28188e6			1.35960			0.679799	−	0.733398i	$-0.262067\pi$
$971$	1.28188e6			1.35960			0.679799	+	0.733398i	$0.262067\pi$
$972$	478948.			0.506939
$973$	−942632.			−0.995672
$974$		−	580761.i		−	0.612181i
$975$		0			0
$976$			153388.i			0.161024i
$977$	1.02959e6			1.07863			0.539316	−	0.842103i	$-0.318683\pi$
$977$	1.02959e6			1.07863			0.539316	+	0.842103i	$0.318683\pi$
$978$			691872.i			0.723349i
$979$	10431.2	−	621092.i	0.0108835	−	0.648024i
$980$		0			0
$981$		−	470515.i		−	0.488918i
$982$	−511376.			−0.530295
$983$	−1.35529e6			−1.40257			−0.701284	−	0.712882i	$-0.747390\pi$
$983$	−1.35529e6			−1.40257			−0.701284	+	0.712882i	$0.747390\pi$
$984$	−246277.			−0.254351
$985$		0			0
$986$		−	938187.i		−	0.965019i
$987$			67665.6i			0.0694598i
$988$	−237980.			−0.243796
$989$			2.63271e6i			2.69160i
$990$		0			0
$991$	−47967.0			−0.0488422			−0.0244211	−	0.999702i	$-0.507774\pi$
$991$	−47967.0			−0.0488422			−0.0244211	+	0.999702i	$0.507774\pi$
$992$			129698.i			0.131798i
$993$	−677991.			−0.687584
$994$	−408484.			−0.413430
$995$		0			0
$996$			911828.i			0.919167i
$997$			1.71845e6i			1.72881i	0.502798	+	0.864404i	$0.332304\pi$
$997$			1.71845e6i			1.72881i	−0.502798	+	0.864404i	$0.667696\pi$
$998$			561169.i			0.563421i
$999$	480145.			0.481106

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.5.d.a.351.4		4
5.2	odd	4		550.5.c.a.549.2		8
5.3	odd	4		550.5.c.a.549.7		8
5.4	even	2		22.5.b.a.21.1	✓	4
11.10	odd	2	inner	550.5.d.a.351.2		4
15.14	odd	2		198.5.d.a.109.4		4
20.19	odd	2		176.5.h.e.65.4		4
40.19	odd	2		704.5.h.j.65.2		4
40.29	even	2		704.5.h.i.65.3		4
55.32	even	4		550.5.c.a.549.6		8
55.43	even	4		550.5.c.a.549.3		8
55.54	odd	2		22.5.b.a.21.3	yes	4
165.164	even	2		198.5.d.a.109.2		4
220.219	even	2		176.5.h.e.65.3		4
440.109	odd	2		704.5.h.i.65.4		4
440.219	even	2		704.5.h.j.65.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.5.b.a.21.1	✓	4	5.4	even	2
22.5.b.a.21.3	yes	4	55.54	odd	2
176.5.h.e.65.3		4	220.219	even	2
176.5.h.e.65.4		4	20.19	odd	2
198.5.d.a.109.2		4	165.164	even	2
198.5.d.a.109.4		4	15.14	odd	2
550.5.c.a.549.2		8	5.2	odd	4
550.5.c.a.549.3		8	55.43	even	4
550.5.c.a.549.6		8	55.32	even	4
550.5.c.a.549.7		8	5.3	odd	4
550.5.d.a.351.2		4	11.10	odd	2	inner
550.5.d.a.351.4		4	1.1	even	1	trivial
704.5.h.i.65.3		4	40.29	even	2
704.5.h.i.65.4		4	440.109	odd	2
704.5.h.j.65.1		4	440.219	even	2
704.5.h.j.65.2		4	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	550.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} +11.2580 q^{3} -8.00000 q^{4} +31.8424i q^{6} +33.9411i q^{7} -22.6274i q^{8} +45.7420 q^{9} +O(q^{10})\)	\(q+2.82843i q^{2} +11.2580 q^{3} -8.00000 q^{4} +31.8424i q^{6} +33.9411i q^{7} -22.6274i q^{8} +45.7420 q^{9} +(2.03190 - 120.983i) q^{11} -90.0638 q^{12} +72.2602i q^{13} -96.0000 q^{14} +64.0000 q^{16} -517.873i q^{17} +129.378i q^{18} -411.672i q^{19} +382.108i q^{21} +(342.191 + 5.74709i) q^{22} +967.641 q^{23} -254.739i q^{24} -204.383 q^{26} -396.933 q^{27} -271.529i q^{28} -640.503i q^{29} +716.487 q^{31} +181.019i q^{32} +(22.8751 - 1362.02i) q^{33} +1464.77 q^{34} -365.936 q^{36} -1209.64 q^{37} +1164.38 q^{38} +813.504i q^{39} -966.781i q^{41} -1080.77 q^{42} +2720.75i q^{43} +(-16.2552 + 967.864i) q^{44} +2736.90i q^{46} +177.085 q^{47} +720.510 q^{48} +1249.00 q^{49} -5830.20i q^{51} -578.082i q^{52} +4023.28 q^{53} -1122.70i q^{54} +768.000 q^{56} -4634.59i q^{57} +1811.62 q^{58} -1988.02 q^{59} +2396.69i q^{61} +2026.53i q^{62} +1552.54i q^{63} -512.000 q^{64} +(3852.38 + 64.7006i) q^{66} +2007.13 q^{67} +4142.98i q^{68} +10893.7 q^{69} +4255.04 q^{71} -1035.02i q^{72} +1288.68i q^{73} -3421.37i q^{74} +3293.37i q^{76} +(4106.30 + 68.9651i) q^{77} -2300.94 q^{78} -1480.23i q^{79} -8173.77 q^{81} +2734.47 q^{82} -10124.2i q^{83} -3056.87i q^{84} -7695.45 q^{86} -7210.77i q^{87} +(-2737.53 - 45.9767i) q^{88} +5133.72 q^{89} -2452.59 q^{91} -7741.13 q^{92} +8066.19 q^{93} +500.871i q^{94} +2037.91i q^{96} -2239.34 q^{97} +3532.71i q^{98} +(92.9434 - 5534.00i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 32 q^{4} + 230 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 32 * q^4 + 230 * q^9	\( 4 q - 2 q^{3} - 32 q^{4} + 230 q^{9} - 180 q^{11} + 16 q^{12} - 384 q^{14} + 256 q^{16} + 240 q^{22} + 1566 q^{23} + 1440 q^{26} - 506 q^{27} + 4418 q^{31} + 2302 q^{33} + 1344 q^{34} - 1840 q^{36} + 382 q^{37} + 2400 q^{38} + 192 q^{42} + 1440 q^{44} - 5688 q^{47} - 128 q^{48} + 4996 q^{49} + 8568 q^{53} + 3072 q^{56} + 9504 q^{58} - 3390 q^{59} - 2048 q^{64} + 13152 q^{66} + 8734 q^{67} + 26314 q^{69} + 3522 q^{71} + 2880 q^{77} - 27264 q^{78} - 31096 q^{81} + 19968 q^{82} - 10464 q^{86} - 1920 q^{88} - 8766 q^{89} + 17280 q^{91} - 12528 q^{92} - 20458 q^{93} - 17282 q^{97} - 12562 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 32 * q^4 + 230 * q^9 - 180 * q^11 + 16 * q^12 - 384 * q^14 + 256 * q^16 + 240 * q^22 + 1566 * q^23 + 1440 * q^26 - 506 * q^27 + 4418 * q^31 + 2302 * q^33 + 1344 * q^34 - 1840 * q^36 + 382 * q^37 + 2400 * q^38 + 192 * q^42 + 1440 * q^44 - 5688 * q^47 - 128 * q^48 + 4996 * q^49 + 8568 * q^53 + 3072 * q^56 + 9504 * q^58 - 3390 * q^59 - 2048 * q^64 + 13152 * q^66 + 8734 * q^67 + 26314 * q^69 + 3522 * q^71 + 2880 * q^77 - 27264 * q^78 - 31096 * q^81 + 19968 * q^82 - 10464 * q^86 - 1920 * q^88 - 8766 * q^89 + 17280 * q^91 - 12528 * q^92 - 20458 * q^93 - 17282 * q^97 - 12562 * q^99

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