LMFDB - Embedded newform 550.5.d.a.351.1

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			351.1
Root			$12.2580 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	550.351
Dual form			550.5.d.a.351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -12.2580 q^{3} -8.00000 q^{4} +34.6708i q^{6} -33.9411i q^{7} +22.6274i q^{8} +69.2580 q^{9} +O(q^{10})$	\(q-2.82843i q^{2} -12.2580 q^{3} -8.00000 q^{4} +34.6708i q^{6} -33.9411i q^{7} +22.6274i q^{8} +69.2580 q^{9} +(-92.0319 - 78.5565i) q^{11} +98.0638 q^{12} +326.819i q^{13} -96.0000 q^{14} +64.0000 q^{16} -280.285i q^{17} -195.891i q^{18} +12.5926i q^{19} +416.050i q^{21} +(-222.191 + 260.306i) q^{22} -184.641 q^{23} -277.366i q^{24} +924.383 q^{26} +143.933 q^{27} +271.529i q^{28} +1039.58i q^{29} +1492.51 q^{31} -181.019i q^{32} +(1128.12 + 962.944i) q^{33} -792.766 q^{34} -554.064 q^{36} +1400.64 q^{37} +35.6172 q^{38} -4006.14i q^{39} +2563.10i q^{41} +1176.77 q^{42} +870.960i q^{43} +(736.255 + 628.452i) q^{44} +522.243i q^{46} -3021.08 q^{47} -784.510 q^{48} +1249.00 q^{49} +3435.73i q^{51} -2614.55i q^{52} +260.724 q^{53} -407.105i q^{54} +768.000 q^{56} -154.359i q^{57} +2940.38 q^{58} +293.024 q^{59} -3593.92i q^{61} -4221.46i q^{62} -2350.69i q^{63} -512.000 q^{64} +(2723.62 - 3190.82i) q^{66} +2359.87 q^{67} +2242.28i q^{68} +2263.32 q^{69} -2494.04 q^{71} +1567.13i q^{72} -4481.31i q^{73} -3961.59i q^{74} -100.741i q^{76} +(-2666.30 + 3123.67i) q^{77} -11331.1 q^{78} -9693.98i q^{79} -7374.23 q^{81} +7249.53 q^{82} -6237.99i q^{83} -3328.40i q^{84} +2463.45 q^{86} -12743.2i q^{87} +(1777.53 - 2082.44i) q^{88} -9516.72 q^{89} +11092.6 q^{91} +1477.13 q^{92} -18295.2 q^{93} +8544.92i q^{94} +2218.93i q^{96} -6401.66 q^{97} -3532.71i q^{98} +(-6373.94 - 5440.67i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−12.2580			−1.36200			−0.680999	−	0.732285i	$-0.738454\pi$
$3$	−12.2580			−1.36200			−0.680999	+	0.732285i	$0.738454\pi$
$4$	−8.00000			−0.500000
$5$		0			0
$5$		0			0
$6$			34.6708i			0.963078i
$7$		−	33.9411i		−	0.692676i	−0.938110	−	0.346338i	$-0.887425\pi$
$7$		−	33.9411i		−	0.692676i	0.938110	−	0.346338i	$-0.112575\pi$
$8$			22.6274i			0.353553i
$9$	69.2580			0.855037
$10$		0			0
$11$	−92.0319	−	78.5565i	−0.760594	−	0.649228i
$11$	−92.0319	−	78.5565i	−0.760594	−	0.649228i
$12$	98.0638			0.680999
$13$			326.819i			1.93384i	0.255083	+	0.966919i	$0.417897\pi$
$13$			326.819i			1.93384i	−0.255083	+	0.966919i	$0.582103\pi$
$14$	−96.0000			−0.489796
$15$		0			0
$16$	64.0000			0.250000
$17$		−	280.285i		−	0.969844i	−0.874557	−	0.484922i	$-0.838848\pi$
$17$		−	280.285i		−	0.969844i	0.874557	−	0.484922i	$-0.161152\pi$
$18$		−	195.891i		−	0.604602i
$19$			12.5926i			0.0348825i	0.999848	+	0.0174412i	$0.00555200\pi$
$19$			12.5926i			0.0348825i	−0.999848	+	0.0174412i	$0.994448\pi$
$20$		0			0
$21$			416.050i			0.943423i
$22$	−222.191	+	260.306i	−0.459073	+	0.537821i
$23$	−184.641			−0.349037			−0.174519	−	0.984654i	$-0.555837\pi$
$23$	−184.641			−0.349037			−0.174519	+	0.984654i	$0.555837\pi$
$24$		−	277.366i		−	0.481539i
$25$		0			0
$26$	924.383			1.36743
$27$	143.933			0.197440
$28$			271.529i			0.346338i
$29$			1039.58i			1.23613i	0.786128	+	0.618063i	$0.212082\pi$
$29$			1039.58i			1.23613i	−0.786128	+	0.618063i	$0.787918\pi$
$30$		0			0
$31$	1492.51			1.55308			0.776542	−	0.630066i	$-0.216972\pi$
$31$	1492.51			1.55308			0.776542	+	0.630066i	$0.216972\pi$
$32$		−	181.019i		−	0.176777i
$33$	1128.12	+	962.944i	1.03593	+	0.884246i
$34$	−792.766			−0.685783
$35$		0			0
$36$	−554.064			−0.427518
$37$	1400.64			1.02311			0.511554	−	0.859251i	$-0.329070\pi$
$37$	1400.64			1.02311			0.511554	+	0.859251i	$0.329070\pi$
$38$	35.6172			0.0246656
$39$		−	4006.14i		−	2.63388i
$40$		0			0
$41$			2563.10i			1.52475i	0.647138	+	0.762373i	$0.275966\pi$
$41$			2563.10i			1.52475i	−0.647138	+	0.762373i	$0.724034\pi$
$42$	1176.77			0.667101
$43$			870.960i			0.471044i	0.971869	+	0.235522i	$0.0756799\pi$
$43$			870.960i			0.471044i	−0.971869	+	0.235522i	$0.924320\pi$
$44$	736.255	+	628.452i	0.380297	+	0.324614i
$45$		0			0
$46$			522.243i			0.246807i
$47$	−3021.08			−1.36763			−0.683813	−	0.729658i	$-0.739679\pi$
$47$	−3021.08			−1.36763			−0.683813	+	0.729658i	$0.739679\pi$
$48$	−784.510			−0.340499
$49$	1249.00			0.520200
$50$		0			0
$51$			3435.73i			1.32093i
$52$		−	2614.55i		−	0.966919i
$53$	260.724			0.0928173			0.0464087	−	0.998923i	$-0.485222\pi$
$53$	260.724			0.0928173			0.0464087	+	0.998923i	$0.485222\pi$
$54$		−	407.105i		−	0.139611i
$55$		0			0
$56$	768.000			0.244898
$57$		−	154.359i		−	0.0475098i
$58$	2940.38			0.874073
$59$	293.024			0.0841780			0.0420890	−	0.999114i	$-0.486599\pi$
$59$	293.024			0.0841780			0.0420890	+	0.999114i	$0.486599\pi$
$60$		0			0
$61$		−	3593.92i		−	0.965849i	−0.875662	−	0.482924i	$-0.839575\pi$
$61$		−	3593.92i		−	0.965849i	0.875662	−	0.482924i	$-0.160425\pi$
$62$		−	4221.46i		−	1.09820i
$63$		−	2350.69i		−	0.592263i
$64$	−512.000			−0.125000
$65$		0			0
$66$	2723.62	−	3190.82i	0.625256	−	0.732511i
$67$	2359.87			0.525701			0.262850	−	0.964837i	$-0.415337\pi$
$67$	2359.87			0.525701			0.262850	+	0.964837i	$0.415337\pi$
$68$			2242.28i			0.484922i
$69$	2263.32			0.475388
$70$		0			0
$71$	−2494.04			−0.494751			−0.247375	−	0.968920i	$-0.579568\pi$
$71$	−2494.04			−0.494751			−0.247375	+	0.968920i	$0.579568\pi$
$72$			1567.13i			0.302301i
$73$		−	4481.31i		−	0.840929i	−0.907309	−	0.420465i	$-0.861867\pi$
$73$		−	4481.31i		−	0.840929i	0.907309	−	0.420465i	$-0.138133\pi$
$74$		−	3961.59i		−	0.723447i
$75$		0			0
$76$		−	100.741i		−	0.0174412i
$77$	−2666.30	+	3123.67i	−0.449704	+	0.526845i
$78$	−11331.1			−1.86244
$79$		−	9693.98i		−	1.55327i	−0.629949	−	0.776637i	$-0.716924\pi$
$79$		−	9693.98i		−	1.55327i	0.629949	−	0.776637i	$-0.283076\pi$
$80$		0			0
$81$	−7374.23			−1.12395
$82$	7249.53			1.07816
$83$		−	6237.99i		−	0.905500i	−0.891638	−	0.452750i	$-0.850443\pi$
$83$		−	6237.99i		−	0.905500i	0.891638	−	0.452750i	$-0.149557\pi$
$84$		−	3328.40i		−	0.471711i
$85$		0			0
$86$	2463.45			0.333078
$87$		−	12743.2i		−	1.68360i
$88$	1777.53	−	2082.44i	0.229537	−	0.268911i
$89$	−9516.72			−1.20145			−0.600727	−	0.799454i	$-0.705122\pi$
$89$	−9516.72			−1.20145			−0.600727	+	0.799454i	$0.705122\pi$
$90$		0			0
$91$	11092.6			1.33952
$92$	1477.13			0.174519
$93$	−18295.2			−2.11530
$94$			8544.92i			0.967057i
$95$		0			0
$96$			2218.93i			0.240769i
$97$	−6401.66			−0.680376			−0.340188	−	0.940357i	$-0.610491\pi$
$97$	−6401.66			−0.680376			−0.340188	+	0.940357i	$0.610491\pi$
$98$		−	3532.71i		−	0.367837i
$99$	−6373.94	−	5440.67i	−0.650336	−	0.555113i
$100$		0			0
$101$			15.9343i			0.00156203i	1.00000		0.000781015i	$0.000248605\pi$
$101$			15.9343i			0.00156203i	−1.00000		0.000781015i	$0.999751\pi$
$102$	9717.70			0.934035
$103$	−12891.5			−1.21514			−0.607572	−	0.794265i	$-0.707856\pi$
$103$	−12891.5			−1.21514			−0.607572	+	0.794265i	$0.707856\pi$
$104$	−7395.06			−0.683715
$105$		0			0
$106$		−	737.438i		−	0.0656318i
$107$			10805.9i			0.943827i	0.881645	+	0.471913i	$0.156436\pi$
$107$			10805.9i			0.943827i	−0.881645	+	0.471913i	$0.843564\pi$
$108$	−1151.47			−0.0987198
$109$		−	8470.43i		−	0.712939i	−0.934307	−	0.356470i	$-0.883980\pi$
$109$		−	8470.43i		−	0.712939i	0.934307	−	0.356470i	$-0.116020\pi$
$110$		0			0
$111$	−17169.0			−1.39347
$112$		−	2172.23i		−	0.173169i
$113$	1009.87			0.0790874			0.0395437	−	0.999218i	$-0.487410\pi$
$113$	1009.87			0.0790874			0.0395437	+	0.999218i	$0.487410\pi$
$114$	−436.594			−0.0335945
$115$		0			0
$116$		−	8316.66i		−	0.618063i
$117$			22634.8i			1.65350i
$118$		−	828.796i		−	0.0595228i
$119$	−9513.19			−0.671788
$120$		0			0
$121$	2298.74	+	14459.4i	0.157007	+	0.987597i
$122$	−10165.1			−0.682958
$123$		−	31418.4i		−	2.07670i
$124$	−11940.1			−0.776542
$125$		0			0
$126$	−6648.77			−0.418794
$127$		−	7859.09i		−	0.487264i	−0.969868	−	0.243632i	$-0.921661\pi$
$127$		−	7859.09i		−	0.487264i	0.969868	−	0.243632i	$-0.0783390\pi$
$128$			1448.15i			0.0883883i
$129$		−	10676.2i		−	0.641560i
$130$		0			0
$131$		−	24418.3i		−	1.42290i	−0.702738	−	0.711448i	$-0.748040\pi$
$131$		−	24418.3i		−	1.42290i	0.702738	−	0.711448i	$-0.251960\pi$
$132$	−9025.00	−	7703.55i	−0.517964	−	0.442123i
$133$	427.406			0.0241622
$134$		−	6674.72i		−	0.371726i
$135$		0			0
$136$	6342.13			0.342892
$137$	2895.91			0.154292			0.0771462	−	0.997020i	$-0.475419\pi$
$137$	2895.91			0.154292			0.0771462	+	0.997020i	$0.475419\pi$
$138$		−	6401.64i		−	0.336150i
$139$		−	14602.9i		−	0.755807i	−0.925845	−	0.377903i	$-0.876645\pi$
$139$		−	14602.9i		−	0.755807i	0.925845	−	0.377903i	$-0.123355\pi$
$140$		0			0
$141$	37032.4			1.86270
$142$			7054.21i			0.349842i
$143$	25673.7	−	30077.7i	1.25550	−	1.47087i
$144$	4432.51			0.213759
$145$		0			0
$146$	−12675.1			−0.594627
$147$	−15310.2			−0.708511
$148$	−11205.1			−0.511554
$149$			9469.08i			0.426516i	0.976996	+	0.213258i	$0.0684075\pi$
$149$			9469.08i			0.426516i	−0.976996	+	0.213258i	$0.931593\pi$
$150$		0			0
$151$			15335.9i			0.672597i	0.941755	+	0.336298i	$0.109175\pi$
$151$			15335.9i			0.672597i	−0.941755	+	0.336298i	$0.890825\pi$
$152$	−284.937			−0.0123328
$153$		−	19412.0i		−	0.829253i
$154$	8835.06	+	7541.43i	0.372536	+	0.317989i
$155$		0			0
$156$			32049.1i			1.31694i
$157$	4810.17			0.195147			0.0975733	−	0.995228i	$-0.468892\pi$
$157$	4810.17			0.195147			0.0975733	+	0.995228i	$0.468892\pi$
$158$	−27418.7			−1.09833
$159$	−3195.95			−0.126417
$160$		0			0
$161$			6266.92i			0.241770i
$162$			20857.5i			0.794752i
$163$	9875.98			0.371711			0.185855	−	0.982577i	$-0.440494\pi$
$163$	9875.98			0.371711			0.185855	+	0.982577i	$0.440494\pi$
$164$		−	20504.8i		−	0.762373i
$165$		0			0
$166$	−17643.7			−0.640285
$167$			13437.3i			0.481812i	0.970548	+	0.240906i	$0.0774445\pi$
$167$			13437.3i			0.481812i	−0.970548	+	0.240906i	$0.922555\pi$
$168$	−9414.13			−0.333550
$169$	−78249.5			−2.73973
$170$		0			0
$171$			872.136i			0.0298258i
$172$		−	6967.68i		−	0.235522i
$173$			37664.1i			1.25845i	0.777223	+	0.629225i	$0.216627\pi$
$173$			37664.1i			1.25845i	−0.777223	+	0.629225i	$0.783373\pi$
$174$	−36043.1			−1.19049
$175$		0			0
$176$	−5890.04	−	5027.62i	−0.190149	−	0.162307i
$177$	−3591.88			−0.114650
$178$			26917.3i			0.849556i
$179$	53941.3			1.68351			0.841755	−	0.539860i	$-0.181523\pi$
$179$	53941.3			1.68351			0.841755	+	0.539860i	$0.181523\pi$
$180$		0			0
$181$	23379.6			0.713642			0.356821	−	0.934173i	$-0.383861\pi$
$181$	23379.6			0.713642			0.356821	+	0.934173i	$0.383861\pi$
$182$		−	31374.6i		−	0.947186i
$183$			44054.2i			1.31548i
$184$		−	4177.94i		−	0.123403i
$185$		0			0
$186$			51746.6i			1.49574i
$187$	−22018.2	+	25795.2i	−0.629650	+	0.737658i
$188$	24168.7			0.683813
$189$		−	4885.26i		−	0.136762i
$190$		0			0
$191$	46611.4			1.27769			0.638844	−	0.769336i	$-0.279413\pi$
$191$	46611.4			1.27769			0.638844	+	0.769336i	$0.279413\pi$
$192$	6276.08			0.170250
$193$		−	57585.6i		−	1.54596i	−0.634428	−	0.772982i	$-0.718764\pi$
$193$		−	57585.6i		−	1.54596i	0.634428	−	0.772982i	$-0.281236\pi$
$194$			18106.6i			0.481099i
$195$		0			0
$196$	−9992.00			−0.260100
$197$		−	39775.2i		−	1.02490i	−0.858718	−	0.512448i	$-0.828739\pi$
$197$		−	39775.2i		−	1.02490i	0.858718	−	0.512448i	$-0.171261\pi$
$198$	−15388.5	+	18028.2i	−0.392524	+	0.459857i
$199$	18905.2			0.477393			0.238697	−	0.971094i	$-0.423280\pi$
$199$	18905.2			0.477393			0.238697	+	0.971094i	$0.423280\pi$
$200$		0			0
$201$	−28927.2			−0.716003
$202$	45.0689			0.00110452
$203$	35284.6			0.856235
$204$		−	27485.8i		−	0.660463i
$205$		0			0
$206$			36462.6i			0.859237i
$207$	−12787.8			−0.298440
$208$			20916.4i			0.483460i
$209$	989.228	−	1158.92i	0.0226466	−	0.0265314i
$210$		0			0
$211$		−	30974.3i		−	0.695723i	−0.937546	−	0.347862i	$-0.886908\pi$
$211$		−	30974.3i		−	0.695723i	0.937546	−	0.347862i	$-0.113092\pi$
$212$	−2085.79			−0.0464087
$213$	30571.9			0.673849
$214$	30563.6			0.667386
$215$		0			0
$216$			3256.84i			0.0698054i
$217$		−	50657.6i		−	1.07578i
$218$	−23958.0			−0.504124
$219$			54931.8i			1.14534i
$220$		0			0
$221$	91602.4			1.87552
$222$			48561.1i			0.985333i
$223$	19890.6			0.399980			0.199990	−	0.979798i	$-0.435909\pi$
$223$	19890.6			0.399980			0.199990	+	0.979798i	$0.435909\pi$
$224$	−6144.00			−0.122449
$225$		0			0
$226$		−	2856.34i		−	0.0559233i
$227$		−	76528.9i		−	1.48516i	−0.669757	−	0.742581i	$-0.733602\pi$
$227$		−	76528.9i		−	1.48516i	0.669757	−	0.742581i	$-0.266398\pi$
$228$			1234.87i			0.0237549i
$229$	−56079.9			−1.06939			−0.534695	−	0.845045i	$-0.679574\pi$
$229$	−56079.9			−1.06939			−0.534695	+	0.845045i	$0.679574\pi$
$230$		0			0
$231$	32683.4	−	38289.8i	0.612496	−	0.717562i
$232$	−23523.1			−0.437037
$233$		−	3353.39i		−	0.0617693i	−0.999523	−	0.0308847i	$-0.990168\pi$
$233$		−	3353.39i		−	0.0617693i	0.999523	−	0.0308847i	$-0.00983246\pi$
$234$	64020.9			1.16920
$235$		0			0
$236$	−2344.19			−0.0420890
$237$			118829.i			2.11555i
$238$			26907.4i			0.475026i
$239$			8629.71i			0.151078i	0.997143	+	0.0755388i	$0.0240677\pi$
$239$			8629.71i			0.151078i	−0.997143	+	0.0755388i	$0.975932\pi$
$240$		0			0
$241$		−	101396.i		−	1.74577i	−0.487924	−	0.872886i	$-0.662246\pi$
$241$		−	101396.i		−	1.74577i	0.487924	−	0.872886i	$-0.337754\pi$
$242$	40897.4	−	6501.83i	0.698337	−	0.111021i
$243$	78734.5			1.33338
$244$			28751.4i			0.482924i
$245$		0			0
$246$	−88864.6			−1.46845
$247$	−4115.49			−0.0674570
$248$			33771.7i			0.549098i
$249$			76465.1i			1.23329i
$250$		0			0
$251$	−81244.8			−1.28958			−0.644790	−	0.764360i	$-0.723055\pi$
$251$	−81244.8			−1.28958			−0.644790	+	0.764360i	$0.723055\pi$
$252$			18805.5i			0.296132i
$253$	16992.8	+	14504.7i	0.265476	+	0.226605i
$254$	−22228.9			−0.344548
$255$		0			0
$256$	4096.00			0.0625000
$257$	−56962.9			−0.862434			−0.431217	−	0.902248i	$-0.641916\pi$
$257$	−56962.9			−0.862434			−0.431217	+	0.902248i	$0.641916\pi$
$258$	−30196.9			−0.453651
$259$		−	47539.1i		−	0.708683i
$260$		0			0
$261$			71999.4i			1.05693i
$262$	−69065.5			−1.00614
$263$			6000.71i			0.0867543i	0.999059	+	0.0433772i	$0.0138117\pi$
$263$			6000.71i			0.0867543i	−0.999059	+	0.0433772i	$0.986188\pi$
$264$	−21788.9	+	25526.6i	−0.312628	+	0.366256i
$265$		0			0
$266$		−	1208.89i		−	0.0170853i
$267$	116656.			1.63638
$268$	−18879.0			−0.262850
$269$	24832.0			0.343169			0.171584	−	0.985169i	$-0.445111\pi$
$269$	24832.0			0.343169			0.171584	+	0.985169i	$0.445111\pi$
$270$		0			0
$271$			65976.9i			0.898367i	0.893440	+	0.449183i	$0.148285\pi$
$271$			65976.9i			0.898367i	−0.893440	+	0.449183i	$0.851715\pi$
$272$		−	17938.2i		−	0.242461i
$273$	−135973.			−1.82443
$274$		−	8190.88i		−	0.109101i
$275$		0			0
$276$	−18106.6			−0.237694
$277$		−	31675.1i		−	0.412818i	−0.978466	−	0.206409i	$-0.933822\pi$
$277$		−	31675.1i		−	0.412818i	0.978466	−	0.206409i	$-0.0661778\pi$
$278$	−41303.4			−0.534436
$279$	103368.			1.32794
$280$		0			0
$281$		−	130542.i		−	1.65325i	−0.562753	−	0.826625i	$-0.690258\pi$
$281$		−	130542.i		−	1.65325i	0.562753	−	0.826625i	$-0.309742\pi$
$282$		−	104743.i		−	1.31713i
$283$		−	77156.4i		−	0.963384i	−0.876341	−	0.481692i	$-0.840022\pi$
$283$		−	77156.4i		−	0.963384i	0.876341	−	0.481692i	$-0.159978\pi$
$284$	19952.3			0.247375
$285$		0			0
$286$	−85072.7	−	72616.3i	−1.04006	−	0.887773i
$287$	86994.4			1.05615
$288$		−	12537.0i		−	0.151151i
$289$	4961.32			0.0594020
$290$		0			0
$291$	78471.4			0.926671
$292$			35850.5i			0.420465i
$293$			73563.5i			0.856894i	0.903567	+	0.428447i	$0.140939\pi$
$293$			73563.5i			0.856894i	−0.903567	+	0.428447i	$0.859061\pi$
$294$			43303.8i			0.500993i
$295$		0			0
$296$			31692.8i			0.361723i
$297$	−13246.5	−	11306.9i	−0.150171	−	0.128183i
$298$	26782.6			0.301592
$299$		−	60344.1i		−	0.674982i
$300$		0			0
$301$	29561.3			0.326281
$302$	43376.4			0.475598
$303$		−	195.322i		−	0.00212748i
$304$			805.924i			0.00872061i
$305$		0			0
$306$	−54905.3			−0.586370
$307$		−	14140.6i		−	0.150034i	−0.997182	−	0.0750171i	$-0.976099\pi$
$307$		−	14140.6i		−	0.150034i	0.997182	−	0.0750171i	$-0.0239011\pi$
$308$	21330.4	−	24989.3i	0.224852	−	0.263423i
$309$	158023.			1.65502
$310$		0			0
$311$	−115950.			−1.19881			−0.599406	−	0.800445i	$-0.704596\pi$
$311$	−115950.			−1.19881			−0.599406	+	0.800445i	$0.704596\pi$
$312$	90648.5			0.931218
$313$	167682.			1.71159			0.855793	−	0.517318i	$-0.173070\pi$
$313$	167682.			1.71159			0.855793	+	0.517318i	$0.173070\pi$
$314$		−	13605.2i		−	0.137989i
$315$		0			0
$316$			77551.9i			0.776637i
$317$	43569.8			0.433577			0.216789	−	0.976219i	$-0.430442\pi$
$317$	43569.8			0.433577			0.216789	+	0.976219i	$0.430442\pi$
$318$			9039.50i			0.0893903i
$319$	81666.0	−	95674.7i	0.802527	−	0.940191i
$320$		0			0
$321$		−	132458.i		−	1.28549i
$322$	17725.5			0.170957
$323$	3529.51			0.0338305
$324$	58993.8			0.561974
$325$		0			0
$326$		−	27933.5i		−	0.262839i
$327$			103830.i			0.971021i
$328$	−57996.3			−0.539079
$329$			102539.i			0.947321i
$330$		0			0
$331$	−13167.8			−0.120187			−0.0600934	−	0.998193i	$-0.519140\pi$
$331$	−13167.8			−0.120187			−0.0600934	+	0.998193i	$0.519140\pi$
$332$			49903.9i			0.452750i
$333$	97005.2			0.874795
$334$	38006.3			0.340692
$335$		0			0
$336$			26627.2i			0.235856i
$337$			41741.2i			0.367540i	0.982969	+	0.183770i	$0.0588302\pi$
$337$			41741.2i			0.367540i	−0.982969	+	0.183770i	$0.941170\pi$
$338$			221323.i			1.93728i
$339$	−12378.9			−0.107717
$340$		0			0
$341$	−137359.	−	117247.i	−1.18127	−	1.00830i
$342$	2466.77			0.0210900
$343$		−	123885.i		−	1.05301i
$344$	−19707.6			−0.166539
$345$		0			0
$346$	106530.			0.889858
$347$			8110.63i			0.0673590i	0.999433	+	0.0336795i	$0.0107225\pi$
$347$			8110.63i			0.0673590i	−0.999433	+	0.0336795i	$0.989277\pi$
$348$			101945.i			0.841800i
$349$			16786.2i			0.137817i	0.997623	+	0.0689083i	$0.0219516\pi$
$349$			16786.2i			0.137817i	−0.997623	+	0.0689083i	$0.978048\pi$
$350$		0			0
$351$			47040.1i			0.381816i
$352$	−14220.3	+	16659.6i	−0.114768	+	0.134455i
$353$	−116711.			−0.936618			−0.468309	−	0.883565i	$-0.655137\pi$
$353$	−116711.			−0.936618			−0.468309	+	0.883565i	$0.655137\pi$
$354$			10159.4i			0.0810700i
$355$		0			0
$356$	76133.8			0.600727
$357$	116612.			0.914973
$358$		−	152569.i		−	1.19042i
$359$			35393.9i			0.274625i	0.990528	+	0.137312i	$0.0438464\pi$
$359$			35393.9i			0.274625i	−0.990528	+	0.137312i	$0.956154\pi$
$360$		0			0
$361$	130162.			0.998783
$362$		−	66127.5i		−	0.504621i
$363$	−28177.9	−	177243.i	−0.213843	−	1.34511i
$364$	−88740.8			−0.669762
$365$		0			0
$366$	124604.			0.930187
$367$	91892.5			0.682257			0.341128	−	0.940017i	$-0.389191\pi$
$367$	91892.5			0.682257			0.341128	+	0.940017i	$0.389191\pi$
$368$	−11817.0			−0.0872594
$369$			177515.i			1.30371i
$370$		0			0
$371$		−	8849.26i		−	0.0642923i
$372$	146362.			1.05765
$373$			35452.4i			0.254817i	0.991850	+	0.127408i	$0.0406659\pi$
$373$			35452.4i			0.254817i	−0.991850	+	0.127408i	$0.959334\pi$
$374$	72959.7	+	62276.9i	0.521603	+	0.445230i
$375$		0			0
$376$		−	68359.3i		−	0.483529i
$377$	−339755.			−2.39047
$378$	−13817.6			−0.0967051
$379$	217027.			1.51090			0.755449	−	0.655207i	$-0.227419\pi$
$379$	217027.			1.51090			0.755449	+	0.655207i	$0.227419\pi$
$380$		0			0
$381$			96336.5i			0.663653i
$382$		−	131837.i		−	0.903462i
$383$	199025.			1.35678			0.678390	−	0.734702i	$-0.262678\pi$
$383$	199025.			1.35678			0.678390	+	0.734702i	$0.262678\pi$
$384$		−	17751.4i		−	0.120385i
$385$		0			0
$386$	−162877.			−1.09316
$387$			60320.9i			0.402760i
$388$	51213.3			0.340188
$389$	−141391.			−0.934380			−0.467190	−	0.884157i	$-0.654734\pi$
$389$	−141391.			−0.934380			−0.467190	+	0.884157i	$0.654734\pi$
$390$		0			0
$391$			51752.1i			0.338512i
$392$			28261.6i			0.183918i
$393$			299319.i			1.93798i
$394$	−112501.			−0.724710
$395$		0			0
$396$	50991.5	+	43525.3i	0.325168	+	0.277557i
$397$	−109000.			−0.691585			−0.345793	−	0.938311i	$-0.612390\pi$
$397$	−109000.			−0.691585			−0.345793	+	0.938311i	$0.612390\pi$
$398$		−	53472.1i		−	0.337568i
$399$	−5239.13			−0.0329089
$400$		0			0
$401$	−262604.			−1.63310			−0.816549	−	0.577277i	$-0.804115\pi$
$401$	−262604.			−1.63310			−0.816549	+	0.577277i	$0.804115\pi$
$402$			81818.5i			0.506290i
$403$			487781.i			3.00341i
$404$		−	127.474i		−	0.000781015i
$405$		0			0
$406$		−	99799.9i		−	0.605450i
$407$	−128903.	−	110029.i	−0.778170	−	0.664230i
$408$	−77741.6			−0.467018
$409$			132695.i			0.793244i	0.917982	+	0.396622i	$0.129818\pi$
$409$			132695.i			0.793244i	−0.917982	+	0.396622i	$0.870182\pi$
$410$		0			0
$411$	−35498.0			−0.210146
$412$	103132.			0.607572
$413$		−	9945.55i		−	0.0583081i
$414$			36169.5i			0.211029i
$415$		0			0
$416$	59160.5			0.341858
$417$			179003.i			1.02941i
$418$	−3277.91	−	2797.96i	−0.0187605	−	0.0160136i
$419$	54888.5			0.312646			0.156323	−	0.987706i	$-0.450036\pi$
$419$	54888.5			0.312646			0.156323	+	0.987706i	$0.450036\pi$
$420$		0			0
$421$	20412.0			0.115165			0.0575826	−	0.998341i	$-0.481661\pi$
$421$	20412.0			0.115165			0.0575826	+	0.998341i	$0.481661\pi$
$422$	−87608.5			−0.491951
$423$	−209234.			−1.16937
$424$			5899.51i			0.0328159i
$425$		0			0
$426$		−	86470.3i		−	0.476483i
$427$	−121982.			−0.669020
$428$		−	86447.0i		−	0.471913i
$429$	−314708.	+	368692.i	−1.70999	+	2.00332i
$430$		0			0
$431$			157298.i			0.846776i	0.905949	+	0.423388i	$0.139159\pi$
$431$			157298.i			0.846776i	−0.905949	+	0.423388i	$0.860841\pi$
$432$	9211.74			0.0493599
$433$	134223.			0.715900			0.357950	−	0.933741i	$-0.383476\pi$
$433$	134223.			0.715900			0.357950	+	0.933741i	$0.383476\pi$
$434$	−143281.			−0.760694
$435$		0			0
$436$			67763.4i			0.356470i
$437$		−	2325.10i		−	0.0121753i
$438$	155371.			0.809880
$439$		−	304612.i		−	1.58059i	−0.612729	−	0.790293i	$-0.709928\pi$
$439$		−	304612.i		−	1.58059i	0.612729	−	0.790293i	$-0.290072\pi$
$440$		0			0
$441$	86503.2			0.444790
$442$		−	259091.i		−	1.32619i
$443$	116115.			0.591671			0.295835	−	0.955239i	$-0.404402\pi$
$443$	116115.			0.591671			0.295835	+	0.955239i	$0.404402\pi$
$444$	137352.			0.696735
$445$		0			0
$446$		−	56259.1i		−	0.282828i
$447$		−	116072.i		−	0.580913i
$448$			17377.9i			0.0865845i
$449$	156051.			0.774059			0.387030	−	0.922067i	$-0.373501\pi$
$449$	156051.			0.774059			0.387030	+	0.922067i	$0.373501\pi$
$450$		0			0
$451$	201348.	−	235887.i	0.989906	−	1.15971i
$452$	−8078.94			−0.0395437
$453$		−	187987.i		−	0.916075i
$454$	−216456.			−1.05017
$455$		0			0
$456$	3492.75			0.0167973
$457$			154609.i			0.740290i	0.928974	+	0.370145i	$0.120692\pi$
$457$			154609.i			0.740290i	−0.928974	+	0.370145i	$0.879308\pi$
$458$			158618.i			0.756173i
$459$		−	40342.4i		−	0.191486i
$460$		0			0
$461$			324927.i			1.52892i	0.644674	+	0.764458i	$0.276993\pi$
$461$			324927.i			1.52892i	−0.644674	+	0.764458i	$0.723007\pi$
$462$	−108300.	−	92442.6i	−0.507393	−	0.433100i
$463$	135116.			0.630295			0.315147	−	0.949043i	$-0.397946\pi$
$463$	135116.			0.630295			0.315147	+	0.949043i	$0.397946\pi$
$464$			66533.3i			0.309032i
$465$		0			0
$466$	−9484.83			−0.0436775
$467$	−217618.			−0.997840			−0.498920	−	0.866648i	$-0.666270\pi$
$467$	−217618.			−0.997840			−0.498920	+	0.866648i	$0.666270\pi$
$468$		−	181078.i		−	0.826751i
$469$		−	80096.6i		−	0.364140i
$470$		0			0
$471$	−58962.9			−0.265789
$472$			6630.37i			0.0297614i
$473$	68419.6	−	80156.1i	0.305814	−	0.358273i
$474$	336098.			1.49592
$475$		0			0
$476$	76105.5			0.335894
$477$	18057.2			0.0793622
$478$	24408.5			0.106828
$479$		−	2332.66i		−	0.0101667i	−0.999987	−	0.00508335i	$-0.998382\pi$
$479$		−	2332.66i		−	0.0101667i	0.999987	−	0.00508335i	$-0.00161809\pi$
$480$		0			0
$481$			457754.i			1.97853i
$482$	−286792.			−1.23445
$483$		−	76819.7i		−	0.329290i
$484$	−18389.9	−	115675.i	−0.0785036	−	0.493799i
$485$		0			0
$486$		−	222695.i		−	0.942839i
$487$	−307695.			−1.29737			−0.648683	−	0.761059i	$-0.724680\pi$
$487$	−307695.			−1.29737			−0.648683	+	0.761059i	$0.724680\pi$
$488$	81321.2			0.341479
$489$	−121060.			−0.506269
$490$		0			0
$491$			165203.i			0.685258i	0.939471	+	0.342629i	$0.111317\pi$
$491$			165203.i			0.685258i	−0.939471	+	0.342629i	$0.888683\pi$
$492$			251347.i			1.03835i
$493$	291379.			1.19885
$494$			11640.4i			0.0476993i
$495$		0			0
$496$	95520.8			0.388271
$497$			84650.5i			0.342702i
$498$	216276.			0.872067
$499$	271209.			1.08919			0.544594	−	0.838700i	$-0.316684\pi$
$499$	271209.			1.08919			0.544594	+	0.838700i	$0.316684\pi$
$500$		0			0
$501$		−	164713.i		−	0.656226i
$502$			229795.i			0.911870i
$503$			242593.i			0.958831i	0.877588	+	0.479415i	$0.159151\pi$
$503$			242593.i			0.958831i	−0.877588	+	0.479415i	$0.840849\pi$
$504$	53190.1			0.209397
$505$		0			0
$506$	41025.6	−	48063.0i	0.160234	−	0.187720i
$507$	959180.			3.73151
$508$			62872.7i			0.243632i
$509$	−231272.			−0.892663			−0.446331	−	0.894868i	$-0.647270\pi$
$509$	−231272.			−0.892663			−0.446331	+	0.894868i	$0.647270\pi$
$510$		0			0
$511$	−152101.			−0.582491
$512$		−	11585.2i		−	0.0441942i
$513$			1812.49i			0.00688718i
$514$			161116.i			0.609833i
$515$		0			0
$516$			85409.6i			0.320780i
$517$	278036.	+	237326.i	1.04021	+	0.887900i
$518$	−134461.			−0.501114
$519$		−	461686.i		−	1.71400i
$520$		0			0
$521$	−19525.5			−0.0719327			−0.0359664	−	0.999353i	$-0.511451\pi$
$521$	−19525.5			−0.0719327			−0.0359664	+	0.999353i	$0.511451\pi$
$522$	203645.			0.747365
$523$		−	493543.i		−	1.80435i	−0.431366	−	0.902177i	$-0.641968\pi$
$523$		−	493543.i		−	1.80435i	0.431366	−	0.902177i	$-0.358032\pi$
$524$			195347.i			0.711448i
$525$		0			0
$526$	16972.6			0.0613446
$527$		−	418329.i		−	1.50625i
$528$	72200.0	+	61628.4i	0.258982	+	0.221062i
$529$	−245749.			−0.878173
$530$		0			0
$531$	20294.2			0.0719753
$532$	−3419.25			−0.0120811
$533$	−837668.			−2.94861
$534$		−	329952.i		−	1.15709i
$535$		0			0
$536$			53397.8i			0.185863i
$537$	−661211.			−2.29293
$538$		−	70235.6i		−	0.242657i
$539$	−114948.	−	98117.1i	−0.395661	−	0.337728i
$540$		0			0
$541$		−	211511.i		−	0.722669i	−0.932436	−	0.361334i	$-0.882321\pi$
$541$		−	211511.i		−	0.722669i	0.932436	−	0.361334i	$-0.117679\pi$
$542$	186611.			0.635241
$543$	−286587.			−0.971978
$544$	−50737.0			−0.171446
$545$		0			0
$546$			384589.i			1.29007i
$547$		−	271731.i		−	0.908163i	−0.890960	−	0.454082i	$-0.849967\pi$
$547$		−	271731.i		−	0.908163i	0.890960	−	0.454082i	$-0.150033\pi$
$548$	−23167.3			−0.0771462
$549$		−	248908.i		−	0.825836i
$550$		0			0
$551$	−13091.0			−0.0431191
$552$			51213.1i			0.168075i
$553$	−329025.			−1.07592
$554$	−89590.8			−0.291907
$555$		0			0
$556$			116824.i			0.377903i
$557$		−	420797.i		−	1.35632i	−0.734914	−	0.678160i	$-0.762777\pi$
$557$		−	420797.i		−	1.35632i	0.734914	−	0.678160i	$-0.237223\pi$
$558$		−	292370.i		−	0.938998i
$559$	−284646.			−0.910922
$560$		0			0
$561$	269899.	−	316196.i	0.857581	−	1.00469i
$562$	−369229.			−1.16902
$563$		−	199781.i		−	0.630285i	−0.949044	−	0.315142i	$-0.897948\pi$
$563$		−	199781.i		−	0.630285i	0.949044	−	0.315142i	$-0.102052\pi$
$564$	−296259.			−0.931351
$565$		0			0
$566$	−218231.			−0.681215
$567$			250290.i			0.778532i
$568$		−	56433.7i		−	0.174921i
$569$		−	393771.i		−	1.21624i	−0.793845	−	0.608120i	$-0.791924\pi$
$569$		−	393771.i		−	1.21624i	0.793845	−	0.608120i	$-0.208076\pi$
$570$		0			0
$571$		−	152076.i		−	0.466431i	−0.972425	−	0.233215i	$-0.925075\pi$
$571$		−	152076.i		−	0.466431i	0.972425	−	0.233215i	$-0.0749247\pi$
$572$	−205390.	+	240622.i	−0.627751	+	0.735433i
$573$	−571361.			−1.74021
$574$		−	246057.i		−	0.746814i
$575$		0			0
$576$	−35460.1			−0.106880
$577$	260379.			0.782086			0.391043	−	0.920372i	$-0.372114\pi$
$577$	260379.			0.782086			0.391043	+	0.920372i	$0.372114\pi$
$578$		−	14032.7i		−	0.0420036i
$579$			705883.i			2.10560i
$580$		0			0
$581$	−211724.			−0.627218
$582$		−	221951.i		−	0.655255i
$583$	−23994.9	−	20481.6i	−0.0705963	−	0.0602596i
$584$	101401.			0.297313
$585$		0			0
$586$	208069.			0.605916
$587$	197623.			0.573538			0.286769	−	0.958000i	$-0.407419\pi$
$587$	197623.			0.573538			0.286769	+	0.958000i	$0.407419\pi$
$588$	122482.			0.354255
$589$			18794.6i			0.0541754i
$590$		0			0
$591$			487563.i			1.39590i
$592$	89640.7			0.255777
$593$		−	103667.i		−	0.294803i	−0.989077	−	0.147401i	$-0.952909\pi$
$593$		−	103667.i		−	0.294803i	0.989077	−	0.147401i	$-0.0470909\pi$
$594$	−31980.8	+	37466.7i	−0.0906392	+	0.106187i
$595$		0			0
$596$		−	75752.6i		−	0.213258i
$597$	−231740.			−0.650208
$598$	−170679.			−0.477284
$599$	204660.			0.570399			0.285199	−	0.958468i	$-0.407940\pi$
$599$	204660.			0.570399			0.285199	+	0.958468i	$0.407940\pi$
$600$		0			0
$601$			140115.i			0.387913i	0.981010	+	0.193957i	$0.0621321\pi$
$601$			140115.i			0.387913i	−0.981010	+	0.193957i	$0.937868\pi$
$602$		−	83612.1i		−	0.230715i
$603$	163440.			0.449493
$604$		−	122687.i		−	0.336298i
$605$		0			0
$606$	−552.453			−0.00150436
$607$			69648.7i			0.189032i	0.995523	+	0.0945160i	$0.0301304\pi$
$607$			69648.7i			0.189032i	−0.995523	+	0.0945160i	$0.969870\pi$
$608$	2279.50			0.00616640
$609$	−432518.			−1.16619
$610$		0			0
$611$		−	987347.i		−	2.64477i
$612$			155296.i			0.414626i
$613$		−	619709.i		−	1.64917i	−0.565735	−	0.824587i	$-0.691407\pi$
$613$		−	619709.i		−	1.64917i	0.565735	−	0.824587i	$-0.308593\pi$
$614$	−39995.6			−0.106090
$615$		0			0
$616$	−70680.5	−	60331.4i	−0.186268	−	0.158994i
$617$	284417.			0.747112			0.373556	−	0.927608i	$-0.378138\pi$
$617$	284417.			0.747112			0.373556	+	0.927608i	$0.378138\pi$
$618$		−	446957.i		−	1.17028i
$619$	−231315.			−0.603702			−0.301851	−	0.953355i	$-0.597605\pi$
$619$	−231315.			−0.603702			−0.301851	+	0.953355i	$0.597605\pi$
$620$		0			0
$621$	−26576.0			−0.0689138
$622$			327957.i			0.847688i
$623$			323008.i			0.832219i
$624$		−	256393.i		−	0.658471i
$625$		0			0
$626$		−	474278.i		−	1.21027i
$627$	−12125.9	+	14206.0i	−0.0308447	+	0.0361357i
$628$	−38481.3			−0.0975733
$629$		−	392577.i		−	0.992256i
$630$		0			0
$631$	311488.			0.782316			0.391158	−	0.920324i	$-0.372075\pi$
$631$	311488.			0.782316			0.391158	+	0.920324i	$0.372075\pi$
$632$	219350.			0.549165
$633$			379682.i			0.947573i
$634$		−	123234.i		−	0.306586i
$635$		0			0
$636$	25567.6			0.0632085
$637$			408197.i			1.00598i
$638$	−270609.	−	230986.i	−0.664815	−	0.567473i
$639$	−172732.			−0.423030
$640$		0			0
$641$	−297618.			−0.724341			−0.362170	−	0.932112i	$-0.617964\pi$
$641$	−297618.			−0.724341			−0.362170	+	0.932112i	$0.617964\pi$
$642$	−374648.			−0.908978
$643$	121820.			0.294645			0.147322	−	0.989089i	$-0.452935\pi$
$643$	121820.			0.294645			0.147322	+	0.989089i	$0.452935\pi$
$644$		−	50135.3i		−	0.120885i
$645$		0			0
$646$		−	9982.95i		−	0.0239218i
$647$	377550.			0.901917			0.450958	−	0.892545i	$-0.351082\pi$
$647$	377550.			0.901917			0.450958	+	0.892545i	$0.351082\pi$
$648$		−	166860.i		−	0.397376i
$649$	−26967.5	−	23018.9i	−0.0640253	−	0.0546507i
$650$		0			0
$651$			620959.i			1.46521i
$652$	−79007.8			−0.185855
$653$	558170.			1.30900			0.654501	−	0.756061i	$-0.272879\pi$
$653$	558170.			1.30900			0.654501	+	0.756061i	$0.272879\pi$
$654$	293677.			0.686616
$655$		0			0
$656$			164038.i			0.381186i
$657$		−	310367.i		−	0.719025i
$658$	290024.			0.669857
$659$		−	357653.i		−	0.823551i	−0.911285	−	0.411776i	$-0.864909\pi$
$659$		−	357653.i		−	0.823551i	0.911285	−	0.411776i	$-0.135091\pi$
$660$		0			0
$661$	290959.			0.665931			0.332965	−	0.942939i	$-0.391951\pi$
$661$	290959.			0.665931			0.332965	+	0.942939i	$0.391951\pi$
$662$			37244.1i			0.0849849i
$663$	−1.12286e6			−2.55446
$664$	141150.			0.320143
$665$		0			0
$666$		−	274372.i		−	0.618574i
$667$		−	191949.i		−	0.431454i
$668$		−	107498.i		−	0.240906i
$669$	−243818.			−0.544771
$670$		0			0
$671$	−282326.	+	330756.i	−0.627056	+	0.734619i
$672$	75313.0			0.166775
$673$		−	465570.i		−	1.02791i	−0.857817	−	0.513955i	$-0.828180\pi$
$673$		−	465570.i		−	1.02791i	0.857817	−	0.513955i	$-0.171820\pi$
$674$	118062.			0.259890
$675$		0			0
$676$	625996.			1.36987
$677$		−	361627.i		−	0.789012i	−0.918893	−	0.394506i	$-0.870916\pi$
$677$		−	361627.i		−	0.789012i	0.918893	−	0.394506i	$-0.129084\pi$
$678$			35012.9i			0.0761673i
$679$			217280.i			0.471280i
$680$		0			0
$681$			938089.i			2.02279i
$682$	−331624.	+	388509.i	−0.712979	+	0.835281i
$683$	−673707.			−1.44421			−0.722103	−	0.691785i	$-0.756825\pi$
$683$	−673707.			−1.44421			−0.722103	+	0.691785i	$0.756825\pi$
$684$		−	6977.08i		−	0.0149129i
$685$		0			0
$686$	−350400.			−0.744588
$687$	687426.			1.45651
$688$			55741.4i			0.117761i
$689$			85209.4i			0.179494i
$690$		0			0
$691$	66171.2			0.138584			0.0692920	−	0.997596i	$-0.477926\pi$
$691$	66171.2			0.138584			0.0692920	+	0.997596i	$0.477926\pi$
$692$		−	301313.i		−	0.629225i
$693$	−184662.	+	216339.i	−0.384514	+	0.450472i
$694$	22940.3			0.0476300
$695$		0			0
$696$	288345.			0.595243
$697$	718397.			1.47877
$698$	47478.5			0.0974510
$699$			41105.8i			0.0841297i
$700$		0			0
$701$		−	439863.i		−	0.895121i	−0.894254	−	0.447560i	$-0.852293\pi$
$701$		−	439863.i		−	0.895121i	0.894254	−	0.447560i	$-0.147707\pi$
$702$	133050.			0.269985
$703$			17637.6i			0.0356885i
$704$	47120.3	+	40220.9i	0.0950743	+	0.0811534i
$705$		0			0
$706$			330109.i			0.662289i
$707$	540.827			0.00108198
$708$	28735.0			0.0573251
$709$	511529.			1.01760			0.508801	−	0.860884i	$-0.330089\pi$
$709$	511529.			1.01760			0.508801	+	0.860884i	$0.330089\pi$
$710$		0			0
$711$		−	671386.i		−	1.32811i
$712$		−	215339.i		−	0.424778i
$713$	−275579.			−0.542084
$714$		−	329830.i		−	0.646984i
$715$		0			0
$716$	−431531.			−0.841755
$717$		−	105783.i		−	0.205767i
$718$	100109.			0.194189
$719$	683729.			1.32259			0.661297	−	0.750124i	$-0.270006\pi$
$719$	683729.			1.32259			0.661297	+	0.750124i	$0.270006\pi$
$720$		0			0
$721$			437551.i			0.841701i
$722$		−	368155.i		−	0.706246i
$723$			1.24291e6i			2.37774i
$724$	−187037.			−0.356821
$725$		0			0
$726$	−501319.	+	79699.2i	−0.951133	+	0.151210i
$727$	−258952.			−0.489949			−0.244974	−	0.969530i	$-0.578780\pi$
$727$	−258952.			−0.489949			−0.244974	+	0.969530i	$0.578780\pi$
$728$			250997.i			0.473593i
$729$	−367813.			−0.692105
$730$		0			0
$731$	244117.			0.456839
$732$		−	352434.i		−	0.657742i
$733$			474044.i			0.882288i	0.897436	+	0.441144i	$0.145427\pi$
$733$			474044.i			0.882288i	−0.897436	+	0.441144i	$0.854573\pi$
$734$		−	259911.i		−	0.482428i
$735$		0			0
$736$			33423.6i			0.0617017i
$737$	−217183.	−	185383.i	−0.399845	−	0.341299i
$738$	502088.			0.921864
$739$		−	758640.i		−	1.38914i	−0.719424	−	0.694571i	$-0.755594\pi$
$739$		−	758640.i		−	1.38914i	0.719424	−	0.694571i	$-0.244406\pi$
$740$		0			0
$741$	50447.5			0.0918763
$742$	−25029.5			−0.0454615
$743$		−	579823.i		−	1.05031i	−0.851006	−	0.525156i	$-0.824007\pi$
$743$		−	579823.i		−	1.05031i	0.851006	−	0.525156i	$-0.175993\pi$
$744$		−	413973.i		−	0.747870i
$745$		0			0
$746$	100275.			0.180183
$747$		−	432030.i		−	0.774236i
$748$	176146.	−	206361.i	0.314825	−	0.368829i
$749$	366763.			0.653766
$750$		0			0
$751$	−844322.			−1.49702			−0.748511	−	0.663122i	$-0.769231\pi$
$751$	−844322.			−1.49702			−0.748511	+	0.663122i	$0.769231\pi$
$752$	−193349.			−0.341906
$753$	995897.			1.75640
$754$			960972.i			1.69032i
$755$		0			0
$756$			39082.1i			0.0683808i
$757$	−801567.			−1.39877			−0.699387	−	0.714743i	$-0.746544\pi$
$757$	−801567.			−1.39877			−0.699387	+	0.714743i	$0.746544\pi$
$758$		−	613845.i		−	1.06837i
$759$	−208298.	−	177799.i	−0.361577	−	0.308635i
$760$		0			0
$761$		−	6823.03i		−	0.0117817i	−0.999983	−	0.00589085i	$-0.998125\pi$
$761$		−	6823.03i		−	0.0117817i	0.999983	−	0.00589085i	$-0.00187513\pi$
$762$	272481.			0.469273
$763$	−287496.			−0.493836
$764$	−372891.			−0.638844
$765$		0			0
$766$		−	562927.i		−	0.959389i
$767$			95765.6i			0.162787i
$768$	−50208.7			−0.0851248
$769$		−	375862.i		−	0.635588i	−0.948160	−	0.317794i	$-0.897058\pi$
$769$		−	375862.i		−	0.635588i	0.948160	−	0.317794i	$-0.102942\pi$
$770$		0			0
$771$	698250.			1.17463
$772$			460685.i			0.772982i
$773$	857421.			1.43494			0.717472	−	0.696587i	$-0.245299\pi$
$773$	857421.			1.43494			0.717472	+	0.696587i	$0.245299\pi$
$774$	170613.			0.284794
$775$		0			0
$776$		−	144853.i		−	0.240549i
$777$			582734.i			0.965224i
$778$			399915.i			0.660707i
$779$	−32276.0			−0.0531868
$780$		0			0
$781$	229531.	+	195923.i	0.376305	+	0.321206i
$782$	146377.			0.239364
$783$			149631.i			0.244060i
$784$	79936.0			0.130050
$785$		0			0
$786$	846603.			1.37036
$787$		−	1.02294e6i		−	1.65158i	−0.563976	−	0.825791i	$-0.690729\pi$
$787$		−	1.02294e6i		−	1.65158i	0.563976	−	0.825791i	$-0.309271\pi$
$788$			318201.i			0.512448i
$789$		−	73556.6i		−	0.118159i
$790$		0			0
$791$		−	34276.0i		−	0.0547820i
$792$	123108.	−	144226.i	0.196262	−	0.229929i
$793$	1.17456e6			1.86780
$794$			308299.i			0.489024i
$795$		0			0
$796$	−151242.			−0.238697
$797$	−865636.			−1.36276			−0.681379	−	0.731931i	$-0.738619\pi$
$797$	−865636.			−1.36276			−0.681379	+	0.731931i	$0.738619\pi$
$798$			14818.5i			0.0232701i
$799$			846765.i			1.32638i
$800$		0			0
$801$	−659109.			−1.02729
$802$			742755.i			1.15477i
$803$	−352036.	+	412424.i	−0.545954	+	0.639606i
$804$	231418.			0.358001
$805$		0			0
$806$	1.37965e6			2.12373
$807$	−304390.			−0.467395
$808$	−360.551			−0.000552261
$809$			886947.i			1.35519i	0.735435	+	0.677596i	$0.236978\pi$
$809$			886947.i			1.35519i	−0.735435	+	0.677596i	$0.763022\pi$
$810$		0			0
$811$		−	608105.i		−	0.924563i	−0.886733	−	0.462282i	$-0.847031\pi$
$811$		−	608105.i		−	0.924563i	0.886733	−	0.462282i	$-0.152969\pi$
$812$	−282277.			−0.428118
$813$		−	808744.i		−	1.22357i
$814$	−311209.	+	364593.i	−0.469682	+	0.550250i
$815$		0			0
$816$			219887.i			0.330231i
$817$	−10967.6			−0.0164312
$818$	375317.			0.560908
$819$	768251.			1.14534
$820$		0			0
$821$			102607.i			0.152227i	0.997099	+	0.0761133i	$0.0242511\pi$
$821$			102607.i			0.152227i	−0.997099	+	0.0761133i	$0.975749\pi$
$822$			100404.i			0.148596i
$823$	−338801.			−0.500202			−0.250101	−	0.968220i	$-0.580464\pi$
$823$	−338801.			−0.500202			−0.250101	+	0.968220i	$0.580464\pi$
$824$		−	291701.i		−	0.429618i
$825$		0			0
$826$	−28130.3			−0.0412300
$827$		−	303788.i		−	0.444181i	−0.975026	−	0.222090i	$-0.928712\pi$
$827$		−	303788.i		−	0.444181i	0.975026	−	0.222090i	$-0.0712880\pi$
$828$	102303.			0.149220
$829$	923889.			1.34435			0.672173	−	0.740394i	$-0.265361\pi$
$829$	923889.			1.34435			0.672173	+	0.740394i	$0.265361\pi$
$830$		0			0
$831$			388273.i			0.562258i
$832$		−	167331.i		−	0.241730i
$833$		−	350076.i		−	0.504513i
$834$	506296.			0.727901
$835$		0			0
$836$	−7913.83	+	9271.34i	−0.0113233	+	0.0132657i
$837$	214823.			0.306640
$838$		−	155248.i		−	0.221074i
$839$	−44067.7			−0.0626032			−0.0313016	−	0.999510i	$-0.509965\pi$
$839$	−44067.7			−0.0626032			−0.0313016	+	0.999510i	$0.509965\pi$
$840$		0			0
$841$	−373450.			−0.528009
$842$		−	57733.8i		−	0.0814340i
$843$			1.60018e6i			2.25172i
$844$			247794.i			0.347862i
$845$		0			0
$846$			591804.i			0.826869i
$847$	490769.	−	78021.9i	0.684085	−	0.108755i
$848$	16686.3			0.0232043
$849$			945782.i			1.31213i
$850$		0			0
$851$	−258614.			−0.357103
$852$	−244575.			−0.336925
$853$			1.02952e6i			1.41494i	0.706746	+	0.707468i	$0.250163\pi$
$853$			1.02952e6i			1.41494i	−0.706746	+	0.707468i	$0.749837\pi$
$854$			345017.i			0.473069i
$855$		0			0
$856$	−244509.			−0.333693
$857$			571557.i			0.778212i	0.921193	+	0.389106i	$0.127216\pi$
$857$			571557.i			0.778212i	−0.921193	+	0.389106i	$0.872784\pi$
$858$	1.04282e6	+	890129.i	1.41656	+	1.20914i
$859$	−479749.			−0.650172			−0.325086	−	0.945685i	$-0.605393\pi$
$859$	−479749.			−0.650172			−0.325086	+	0.945685i	$0.605393\pi$
$860$		0			0
$861$	−1.06637e6			−1.43848
$862$	444906.			0.598761
$863$	220319.			0.295822			0.147911	−	0.989001i	$-0.452745\pi$
$863$	220319.			0.295822			0.147911	+	0.989001i	$0.452745\pi$
$864$		−	26054.7i		−	0.0349027i
$865$		0			0
$866$		−	379641.i		−	0.506218i
$867$	−60815.7			−0.0809054
$868$			405261.i			0.537892i
$869$	−761526.	+	892156.i	−1.00843	+	1.18141i
$870$		0			0
$871$			771250.i			1.01662i
$872$	191664.			0.252062
$873$	−443366.			−0.581747
$874$	−6576.38			−0.00860922
$875$		0			0
$876$		−	439454.i		−	0.572672i
$877$			1.23303e6i			1.60315i	0.597893	+	0.801576i	$0.296005\pi$
$877$			1.23303e6i			1.60315i	−0.597893	+	0.801576i	$0.703995\pi$
$878$	−861573.			−1.11764
$879$		−	901740.i		−	1.16709i
$880$		0			0
$881$	227150.			0.292659			0.146329	−	0.989236i	$-0.453254\pi$
$881$	227150.			0.292659			0.146329	+	0.989236i	$0.453254\pi$
$882$		−	244668.i		−	0.314514i
$883$	136322.			0.174842			0.0874210	−	0.996171i	$-0.472137\pi$
$883$	136322.			0.174842			0.0874210	+	0.996171i	$0.472137\pi$
$884$	−732819.			−0.937761
$885$		0			0
$886$		−	328422.i		−	0.418374i
$887$			852106.i			1.08304i	0.840686	+	0.541522i	$0.182152\pi$
$887$			852106.i			1.08304i	−0.840686	+	0.541522i	$0.817848\pi$
$888$		−	388489.i		−	0.492666i
$889$	−266746.			−0.337516
$890$		0			0
$891$	678664.	+	579294.i	0.854869	+	0.729699i
$892$	−159125.			−0.199990
$893$		−	38043.2i		−	0.0477061i
$894$	−328300.			−0.410768
$895$		0			0
$896$	49152.0			0.0612245
$897$			739696.i			0.919324i
$898$		−	441379.i		−	0.547342i
$899$			1.55159e6i			1.91981i
$900$		0			0
$901$		−	73077.0i		−	0.0900183i
$902$	−667188.	−	569498.i	−0.820040	−	0.699970i
$903$	−362362.			−0.444393
$904$			22850.7i			0.0279616i
$905$		0			0
$906$	−531707.			−0.647763
$907$	945937.			1.14987			0.574934	−	0.818200i	$-0.305028\pi$
$907$	945937.			1.14987			0.574934	+	0.818200i	$0.305028\pi$
$908$			612231.i			0.742581i
$909$			1103.57i			0.00133559i
$910$		0			0
$911$	−13013.5			−0.0156803			−0.00784017	−	0.999969i	$-0.502496\pi$
$911$	−13013.5			−0.0156803			−0.00784017	+	0.999969i	$0.502496\pi$
$912$		−	9879.00i		−	0.0118775i
$913$	−490035.	+	574094.i	−0.587875	+	0.688718i
$914$	437300.			0.523464
$915$		0			0
$916$	448639.			0.534695
$917$	−828786.			−0.985607
$918$	−114106.			−0.135401
$919$		−	809716.i		−	0.958742i	−0.877612	−	0.479371i	$-0.840865\pi$
$919$		−	809716.i		−	0.958742i	0.877612	−	0.479371i	$-0.159135\pi$
$920$		0			0
$921$			173335.i			0.204346i
$922$	919032.			1.08111
$923$		−	815099.i		−	0.956768i
$924$	−261467.	+	306319.i	−0.306248	+	0.358781i
$925$		0			0
$926$		−	382165.i		−	0.445686i
$927$	−892837.			−1.03899
$928$	188185.			0.218518
$929$	731171.			0.847203			0.423601	−	0.905849i	$-0.360766\pi$
$929$	731171.			0.847203			0.423601	+	0.905849i	$0.360766\pi$
$930$		0			0
$931$			15728.1i			0.0181458i
$932$			26827.2i			0.0308847i
$933$	1.42131e6			1.63278
$934$			615517.i			0.705580i
$935$		0			0
$936$	−512167.			−0.584602
$937$			637069.i			0.725617i	0.931864	+	0.362809i	$0.118182\pi$
$937$			637069.i			0.725617i	−0.931864	+	0.362809i	$0.881818\pi$
$938$	−226547.			−0.257486
$939$	−2.05545e6			−2.33118
$940$		0			0
$941$		−	77429.7i		−	0.0874437i	−0.999044	−	0.0437218i	$-0.986078\pi$
$941$		−	77429.7i		−	0.0874437i	0.999044	−	0.0437218i	$-0.0139215\pi$
$942$			166772.i			0.187941i
$943$		−	473252.i		−	0.532193i
$944$	18753.5			0.0210445
$945$		0			0
$946$	−226716.	−	193520.i	−0.253337	−	0.216243i
$947$	−362138.			−0.403807			−0.201904	−	0.979405i	$-0.564713\pi$
$947$	−362138.			−0.403807			−0.201904	+	0.979405i	$0.564713\pi$
$948$		−	950629.i		−	1.05778i
$949$	1.46458e6			1.62622
$950$		0			0
$951$	−534077.			−0.590531
$952$		−	215259.i		−	0.237513i
$953$		−	1.11862e6i		−	1.23168i	−0.787873	−	0.615838i	$-0.788818\pi$
$953$		−	1.11862e6i		−	1.23168i	0.787873	−	0.615838i	$-0.211182\pi$
$954$		−	51073.5i		−	0.0561176i
$955$		0			0
$956$		−	69037.7i		−	0.0755388i
$957$	−1.00106e6	+	1.17278e6i	−1.09304	+	1.28054i
$958$	−6597.75			−0.00718894
$959$		−	98290.6i		−	0.106875i
$960$		0			0
$961$	1.30407e6			1.41207
$962$	1.29472e6			1.39903
$963$			748393.i			0.807006i
$964$			811170.i			0.872886i
$965$		0			0
$966$	−217279.			−0.232843
$967$		−	104836.i		−	0.112113i	−0.998428	−	0.0560565i	$-0.982147\pi$
$967$		−	104836.i		−	0.112113i	0.998428	−	0.0560565i	$-0.0178527\pi$
$968$	−327179.	+	52014.6i	−0.349168	+	0.0555104i
$969$	−43264.6			−0.0460771
$970$		0			0
$971$	−1.63144e6			−1.73034			−0.865170	−	0.501478i	$-0.832790\pi$
$971$	−1.63144e6			−1.73034			−0.865170	+	0.501478i	$0.832790\pi$
$972$	−629876.			−0.666688
$973$	−495640.			−0.523529
$974$			870293.i			0.917376i
$975$		0			0
$976$		−	230011.i		−	0.241462i
$977$	−4386.95			−0.00459593			−0.00229796	−	0.999997i	$-0.500731\pi$
$977$	−4386.95			−0.00459593			−0.00229796	+	0.999997i	$0.500731\pi$
$978$			342408.i			0.357986i
$979$	875842.	+	747600.i	0.913819	+	0.780017i
$980$		0			0
$981$		−	586645.i		−	0.609589i
$982$	467264.			0.484551
$983$	−147531.			−0.152678			−0.0763388	−	0.997082i	$-0.524323\pi$
$983$	−147531.			−0.152678			−0.0763388	+	0.997082i	$0.524323\pi$
$984$	710917.			0.734224
$985$		0			0
$986$		−	824145.i		−	0.847715i
$987$		−	1.25692e6i		−	1.29025i
$988$	32923.9			0.0337285
$989$		−	160815.i		−	0.164412i
$990$		0			0
$991$	373627.			0.380444			0.190222	−	0.981741i	$-0.439079\pi$
$991$	373627.			0.380444			0.190222	+	0.981741i	$0.439079\pi$
$992$		−	270174.i		−	0.274549i
$993$	161410.			0.163694
$994$	239428.			0.242327
$995$		0			0
$996$		−	611721.i		−	0.616644i
$997$		−	71451.7i		−	0.0718824i	−0.999354	−	0.0359412i	$-0.988557\pi$
$997$		−	71451.7i		−	0.0718824i	0.999354	−	0.0359412i	$-0.0114429\pi$
$998$		−	767094.i		−	0.770172i
$999$	201598.			0.202002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.5.d.a.351.1		4
5.2	odd	4		550.5.c.a.549.8		8
5.3	odd	4		550.5.c.a.549.1		8
5.4	even	2		22.5.b.a.21.4	yes	4
11.10	odd	2	inner	550.5.d.a.351.3		4
15.14	odd	2		198.5.d.a.109.1		4
20.19	odd	2		176.5.h.e.65.1		4
40.19	odd	2		704.5.h.j.65.3		4
40.29	even	2		704.5.h.i.65.2		4
55.32	even	4		550.5.c.a.549.4		8
55.43	even	4		550.5.c.a.549.5		8
55.54	odd	2		22.5.b.a.21.2	✓	4
165.164	even	2		198.5.d.a.109.3		4
220.219	even	2		176.5.h.e.65.2		4
440.109	odd	2		704.5.h.i.65.1		4
440.219	even	2		704.5.h.j.65.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.5.b.a.21.2	✓	4	55.54	odd	2
22.5.b.a.21.4	yes	4	5.4	even	2
176.5.h.e.65.1		4	20.19	odd	2
176.5.h.e.65.2		4	220.219	even	2
198.5.d.a.109.1		4	15.14	odd	2
198.5.d.a.109.3		4	165.164	even	2
550.5.c.a.549.1		8	5.3	odd	4
550.5.c.a.549.4		8	55.32	even	4
550.5.c.a.549.5		8	55.43	even	4
550.5.c.a.549.8		8	5.2	odd	4
550.5.d.a.351.1		4	1.1	even	1	trivial
550.5.d.a.351.3		4	11.10	odd	2	inner
704.5.h.i.65.1		4	440.109	odd	2
704.5.h.i.65.2		4	40.29	even	2
704.5.h.j.65.3		4	40.19	odd	2
704.5.h.j.65.4		4	440.219	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:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	550.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -12.2580 q^{3} -8.00000 q^{4} +34.6708i q^{6} -33.9411i q^{7} +22.6274i q^{8} +69.2580 q^{9} +O(q^{10})\)	\(q-2.82843i q^{2} -12.2580 q^{3} -8.00000 q^{4} +34.6708i q^{6} -33.9411i q^{7} +22.6274i q^{8} +69.2580 q^{9} +(-92.0319 - 78.5565i) q^{11} +98.0638 q^{12} +326.819i q^{13} -96.0000 q^{14} +64.0000 q^{16} -280.285i q^{17} -195.891i q^{18} +12.5926i q^{19} +416.050i q^{21} +(-222.191 + 260.306i) q^{22} -184.641 q^{23} -277.366i q^{24} +924.383 q^{26} +143.933 q^{27} +271.529i q^{28} +1039.58i q^{29} +1492.51 q^{31} -181.019i q^{32} +(1128.12 + 962.944i) q^{33} -792.766 q^{34} -554.064 q^{36} +1400.64 q^{37} +35.6172 q^{38} -4006.14i q^{39} +2563.10i q^{41} +1176.77 q^{42} +870.960i q^{43} +(736.255 + 628.452i) q^{44} +522.243i q^{46} -3021.08 q^{47} -784.510 q^{48} +1249.00 q^{49} +3435.73i q^{51} -2614.55i q^{52} +260.724 q^{53} -407.105i q^{54} +768.000 q^{56} -154.359i q^{57} +2940.38 q^{58} +293.024 q^{59} -3593.92i q^{61} -4221.46i q^{62} -2350.69i q^{63} -512.000 q^{64} +(2723.62 - 3190.82i) q^{66} +2359.87 q^{67} +2242.28i q^{68} +2263.32 q^{69} -2494.04 q^{71} +1567.13i q^{72} -4481.31i q^{73} -3961.59i q^{74} -100.741i q^{76} +(-2666.30 + 3123.67i) q^{77} -11331.1 q^{78} -9693.98i q^{79} -7374.23 q^{81} +7249.53 q^{82} -6237.99i q^{83} -3328.40i q^{84} +2463.45 q^{86} -12743.2i q^{87} +(1777.53 - 2082.44i) q^{88} -9516.72 q^{89} +11092.6 q^{91} +1477.13 q^{92} -18295.2 q^{93} +8544.92i q^{94} +2218.93i q^{96} -6401.66 q^{97} -3532.71i q^{98} +(-6373.94 - 5440.67i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 32 q^{4} + 230 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more