LMFDB - Embedded newform 198.5.d.a.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [198,5,Mod(109,198)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(198, 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("198.109");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 198 = 2 \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	198.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:	$20.4672526906$
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:	$\Z[a_1, \ldots, a_{5}]$
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			109.1
Root			$12.2580 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	198.109
Dual form			198.5.d.a.109.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -4.25798 q^{5} +33.9411i q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} -4.25798 q^{5} +33.9411i q^{7} +22.6274i q^{8} +12.0434i q^{10} +(92.0319 + 78.5565i) q^{11} -326.819i q^{13} +96.0000 q^{14} +64.0000 q^{16} -280.285i q^{17} +12.5926i q^{19} +34.0638 q^{20} +(222.191 - 260.306i) q^{22} -184.641 q^{23} -606.870 q^{25} -924.383 q^{26} -271.529i q^{28} -1039.58i q^{29} +1492.51 q^{31} -181.019i q^{32} -792.766 q^{34} -144.520i q^{35} -1400.64 q^{37} +35.6172 q^{38} -96.3470i q^{40} -2563.10i q^{41} -870.960i q^{43} +(-736.255 - 628.452i) q^{44} +522.243i q^{46} -3021.08 q^{47} +1249.00 q^{49} +1716.49i q^{50} +2614.55i q^{52} +260.724 q^{53} +(-391.870 - 334.492i) q^{55} -768.000 q^{56} -2940.38 q^{58} -293.024 q^{59} -3593.92i q^{61} -4221.46i q^{62} -512.000 q^{64} +1391.59i q^{65} -2359.87 q^{67} +2242.28i q^{68} -408.766 q^{70} +2494.04 q^{71} +4481.31i q^{73} +3961.59i q^{74} -100.741i q^{76} +(-2666.30 + 3123.67i) q^{77} -9693.98i q^{79} -272.510 q^{80} -7249.53 q^{82} -6237.99i q^{83} +1193.45i q^{85} -2463.45 q^{86} +(-1777.53 + 2082.44i) q^{88} +9516.72 q^{89} +11092.6 q^{91} +1477.13 q^{92} +8544.92i q^{94} -53.6188i q^{95} +6401.66 q^{97} -3532.71i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} + 30 q^{5}+O(q^{10}) $ 4 * q - 32 * q^4 + 30 * q^5	$ 4 q - 32 q^{4} + 30 q^{5} + 180 q^{11} + 384 q^{14} + 256 q^{16} - 240 q^{20} - 240 q^{22} + 1566 q^{23} - 1722 q^{25} - 1440 q^{26} + 4418 q^{31} + 1344 q^{34} - 382 q^{37} + 2400 q^{38} - 1440 q^{44} - 5688 q^{47} + 4996 q^{49} + 8568 q^{53} - 862 q^{55} - 3072 q^{56} - 9504 q^{58} + 3390 q^{59} - 2048 q^{64} - 8734 q^{67} + 2880 q^{70} - 3522 q^{71} + 2880 q^{77} + 1920 q^{80} - 19968 q^{82} + 10464 q^{86} + 1920 q^{88} + 8766 q^{89} + 17280 q^{91} - 12528 q^{92} + 17282 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 + 30 * q^5 + 180 * q^11 + 384 * q^14 + 256 * q^16 - 240 * q^20 - 240 * q^22 + 1566 * q^23 - 1722 * q^25 - 1440 * q^26 + 4418 * q^31 + 1344 * q^34 - 382 * q^37 + 2400 * q^38 - 1440 * q^44 - 5688 * q^47 + 4996 * q^49 + 8568 * q^53 - 862 * q^55 - 3072 * q^56 - 9504 * q^58 + 3390 * q^59 - 2048 * q^64 - 8734 * q^67 + 2880 * q^70 - 3522 * q^71 + 2880 * q^77 + 1920 * q^80 - 19968 * q^82 + 10464 * q^86 + 1920 * q^88 + 8766 * q^89 + 17280 * q^91 - 12528 * q^92 + 17282 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/198\mathbb{Z}\right)^\times$.

$n$	$145$	$155$
$\chi(n)$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$	−8.00000			−0.500000
$5$	−4.25798			−0.170319			−0.0851595	−	0.996367i	$-0.527140\pi$
$5$	−4.25798			−0.170319			−0.0851595	+	0.996367i	$0.527140\pi$
$6$		0			0
$7$			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
$9$		0			0
$10$			12.0434i			0.120434i
$11$	92.0319	+	78.5565i	0.760594	+	0.649228i
$11$	92.0319	+	78.5565i	0.760594	+	0.649228i
$12$		0			0
$13$		−	326.819i		−	1.93384i	−0.255083	−	0.966919i	$-0.582103\pi$
$13$		−	326.819i		−	1.93384i	0.255083	−	0.966919i	$-0.417897\pi$
$14$	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	$-0.161152\pi$
$18$		0			0
$19$			12.5926i			0.0348825i	0.999848	+	0.0174412i	$0.00555200\pi$
$19$			12.5926i			0.0348825i	−0.999848	+	0.0174412i	$0.994448\pi$
$20$	34.0638			0.0851595
$21$		0			0
$22$	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	$0.555837\pi$
$24$		0			0
$25$	−606.870			−0.970991
$26$	−924.383			−1.36743
$27$		0			0
$28$		−	271.529i		−	0.346338i
$29$		−	1039.58i		−	1.23613i	−0.786128	−	0.618063i	$-0.787918\pi$
$29$		−	1039.58i		−	1.23613i	0.786128	−	0.618063i	$-0.212082\pi$
$30$		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
$33$		0			0
$34$	−792.766			−0.685783
$35$		−	144.520i		−	0.117976i
$36$		0			0
$37$	−1400.64			−1.02311			−0.511554	−	0.859251i	$-0.670930\pi$
$37$	−1400.64			−1.02311			−0.511554	+	0.859251i	$0.670930\pi$
$38$	35.6172			0.0246656
$39$		0			0
$40$		−	96.3470i		−	0.0602169i
$41$		−	2563.10i		−	1.52475i	−0.647138	−	0.762373i	$-0.724034\pi$
$41$		−	2563.10i		−	1.52475i	0.647138	−	0.762373i	$-0.275966\pi$
$42$		0			0
$43$		−	870.960i		−	0.471044i	−0.971869	−	0.235522i	$-0.924320\pi$
$43$		−	870.960i		−	0.471044i	0.971869	−	0.235522i	$-0.0756799\pi$
$44$	−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	$0.739679\pi$
$48$		0			0
$49$	1249.00			0.520200
$50$			1716.49i			0.686595i
$51$		0			0
$52$			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	$0.485222\pi$
$54$		0			0
$55$	−391.870	−	334.492i	−0.129544	−	0.110576i
$56$	−768.000			−0.244898
$57$		0			0
$58$	−2940.38			−0.874073
$59$	−293.024			−0.0841780			−0.0420890	−	0.999114i	$-0.513401\pi$
$59$	−293.024			−0.0841780			−0.0420890	+	0.999114i	$0.513401\pi$
$60$		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
$63$		0			0
$64$	−512.000			−0.125000
$65$			1391.59i			0.329370i
$66$		0			0
$67$	−2359.87			−0.525701			−0.262850	−	0.964837i	$-0.584663\pi$
$67$	−2359.87			−0.525701			−0.262850	+	0.964837i	$0.584663\pi$
$68$			2242.28i			0.484922i
$69$		0			0
$70$	−408.766			−0.0834216
$71$	2494.04			0.494751			0.247375	−	0.968920i	$-0.420432\pi$
$71$	2494.04			0.494751			0.247375	+	0.968920i	$0.420432\pi$
$72$		0			0
$73$			4481.31i			0.840929i	0.907309	+	0.420465i	$0.138133\pi$
$73$			4481.31i			0.840929i	−0.907309	+	0.420465i	$0.861867\pi$
$74$			3961.59i			0.723447i
$75$		0			0
$76$		−	100.741i		−	0.0174412i
$77$	−2666.30	+	3123.67i	−0.449704	+	0.526845i
$78$		0			0
$79$		−	9693.98i		−	1.55327i	−0.629949	−	0.776637i	$-0.716924\pi$
$79$		−	9693.98i		−	1.55327i	0.629949	−	0.776637i	$-0.283076\pi$
$80$	−272.510			−0.0425798
$81$		0			0
$82$	−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	$-0.149557\pi$
$84$		0			0
$85$			1193.45i			0.165183i
$86$	−2463.45			−0.333078
$87$		0			0
$88$	−1777.53	+	2082.44i	−0.229537	+	0.268911i
$89$	9516.72			1.20145			0.600727	−	0.799454i	$-0.294878\pi$
$89$	9516.72			1.20145			0.600727	+	0.799454i	$0.294878\pi$
$90$		0			0
$91$	11092.6			1.33952
$92$	1477.13			0.174519
$93$		0			0
$94$			8544.92i			0.967057i
$95$		−	53.6188i		−	0.00594115i
$96$		0			0
$97$	6401.66			0.680376			0.340188	−	0.940357i	$-0.389509\pi$
$97$	6401.66			0.680376			0.340188	+	0.940357i	$0.389509\pi$
$98$		−	3532.71i		−	0.367837i
$99$		0			0
$100$	4854.96			0.485496
$101$		−	15.9343i		−	0.00156203i	−1.00000		0.000781015i	$-0.999751\pi$
$101$		−	15.9343i		−	0.00156203i	1.00000		0.000781015i	$-0.000248605\pi$
$102$		0			0
$103$	12891.5			1.21514			0.607572	−	0.794265i	$-0.292144\pi$
$103$	12891.5			1.21514			0.607572	+	0.794265i	$0.292144\pi$
$104$	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	$0.843564\pi$
$108$		0			0
$109$		−	8470.43i		−	0.712939i	−0.934307	−	0.356470i	$-0.883980\pi$
$109$		−	8470.43i		−	0.712939i	0.934307	−	0.356470i	$-0.116020\pi$
$110$	−946.086	+	1108.37i	−0.0781889	+	0.0916012i
$111$		0			0
$112$			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	$0.487410\pi$
$114$		0			0
$115$	786.196			0.0594477
$116$			8316.66i			0.618063i
$117$		0			0
$118$			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
$123$		0			0
$124$	−11940.1			−0.776542
$125$	5245.27			0.335697
$126$		0			0
$127$			7859.09i			0.487264i	0.969868	+	0.243632i	$0.0783390\pi$
$127$			7859.09i			0.487264i	−0.969868	+	0.243632i	$0.921661\pi$
$128$			1448.15i			0.0883883i
$129$		0			0
$130$	3936.00			0.232899
$131$			24418.3i			1.42290i	0.702738	+	0.711448i	$0.251960\pi$
$131$			24418.3i			1.42290i	−0.702738	+	0.711448i	$0.748040\pi$
$132$		0			0
$133$	−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	$0.475419\pi$
$138$		0			0
$139$		−	14602.9i		−	0.755807i	−0.925845	−	0.377903i	$-0.876645\pi$
$139$		−	14602.9i		−	0.755807i	0.925845	−	0.377903i	$-0.123355\pi$
$140$			1156.16i			0.0589880i
$141$		0			0
$142$		−	7054.21i		−	0.349842i
$143$	25673.7	−	30077.7i	1.25550	−	1.47087i
$144$		0			0
$145$			4426.52i			0.210536i
$146$	12675.1			0.594627
$147$		0			0
$148$	11205.1			0.511554
$149$		−	9469.08i		−	0.426516i	−0.976996	−	0.213258i	$-0.931593\pi$
$149$		−	9469.08i		−	0.426516i	0.976996	−	0.213258i	$-0.0684075\pi$
$150$		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
$153$		0			0
$154$	8835.06	+	7541.43i	0.372536	+	0.317989i
$155$	−6355.09			−0.264520
$156$		0			0
$157$	−4810.17			−0.195147			−0.0975733	−	0.995228i	$-0.531108\pi$
$157$	−4810.17			−0.195147			−0.0975733	+	0.995228i	$0.531108\pi$
$158$	−27418.7			−1.09833
$159$		0			0
$160$			770.776i			0.0301084i
$161$		−	6266.92i		−	0.241770i
$162$		0			0
$163$	−9875.98			−0.371711			−0.185855	−	0.982577i	$-0.559506\pi$
$163$	−9875.98			−0.371711			−0.185855	+	0.982577i	$0.559506\pi$
$164$			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	$0.922555\pi$
$168$		0			0
$169$	−78249.5			−2.73973
$170$	3375.58			0.116802
$171$		0			0
$172$			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	$0.783373\pi$
$174$		0			0
$175$		−	20597.8i		−	0.672582i
$176$	5890.04	+	5027.62i	0.190149	+	0.162307i
$177$		0			0
$178$		−	26917.3i		−	0.849556i
$179$	−53941.3			−1.68351			−0.841755	−	0.539860i	$-0.818477\pi$
$179$	−53941.3			−1.68351			−0.841755	+	0.539860i	$0.818477\pi$
$180$		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
$183$		0			0
$184$		−	4177.94i		−	0.123403i
$185$	5963.87			0.174255
$186$		0			0
$187$	22018.2	−	25795.2i	0.629650	−	0.737658i
$188$	24168.7			0.683813
$189$		0			0
$190$	−151.657			−0.00420102
$191$	−46611.4			−1.27769			−0.638844	−	0.769336i	$-0.720587\pi$
$191$	−46611.4			−1.27769			−0.638844	+	0.769336i	$0.720587\pi$
$192$		0			0
$193$			57585.6i			1.54596i	0.634428	+	0.772982i	$0.281236\pi$
$193$			57585.6i			1.54596i	−0.634428	+	0.772982i	$0.718764\pi$
$194$		−	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	$-0.171261\pi$
$198$		0			0
$199$	18905.2			0.477393			0.238697	−	0.971094i	$-0.423280\pi$
$199$	18905.2			0.477393			0.238697	+	0.971094i	$0.423280\pi$
$200$		−	13731.9i		−	0.343297i
$201$		0			0
$202$	−45.0689			−0.00110452
$203$	35284.6			0.856235
$204$		0			0
$205$			10913.6i			0.259693i
$206$		−	36462.6i		−	0.859237i
$207$		0			0
$208$		−	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
$213$		0			0
$214$	30563.6			0.667386
$215$			3708.52i			0.0802277i
$216$		0			0
$217$			50657.6i			1.07578i
$218$	−23958.0			−0.504124
$219$		0			0
$220$	3134.96	+	2675.93i	0.0647718	+	0.0552879i
$221$	−91602.4			−1.87552
$222$		0			0
$223$	−19890.6			−0.399980			−0.199990	−	0.979798i	$-0.564091\pi$
$223$	−19890.6			−0.399980			−0.199990	+	0.979798i	$0.564091\pi$
$224$	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	$-0.266398\pi$
$228$		0			0
$229$	−56079.9			−1.06939			−0.534695	−	0.845045i	$-0.679574\pi$
$229$	−56079.9			−1.06939			−0.534695	+	0.845045i	$0.679574\pi$
$230$		−	2223.70i		−	0.0420359i
$231$		0			0
$232$	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	$-0.00983246\pi$
$234$		0			0
$235$	12863.7			0.232933
$236$	2344.19			0.0420890
$237$		0			0
$238$		−	26907.4i		−	0.475026i
$239$		−	8629.71i		−	0.151078i	−0.997143	−	0.0755388i	$-0.975932\pi$
$239$		−	8629.71i		−	0.151078i	0.997143	−	0.0755388i	$-0.0240677\pi$
$240$		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
$243$		0			0
$244$			28751.4i			0.482924i
$245$	−5318.21			−0.0886000
$246$		0			0
$247$	4115.49			0.0674570
$248$			33771.7i			0.549098i
$249$		0			0
$250$		−	14835.9i		−	0.237374i
$251$	81244.8			1.28958			0.644790	−	0.764360i	$-0.276945\pi$
$251$	81244.8			1.28958			0.644790	+	0.764360i	$0.276945\pi$
$252$		0			0
$253$	−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	$0.641916\pi$
$258$		0			0
$259$		−	47539.1i		−	0.708683i
$260$		−	11132.7i		−	0.164685i
$261$		0			0
$262$	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	$0.986188\pi$
$264$		0			0
$265$	−1110.16			−0.0158086
$266$			1208.89i			0.0170853i
$267$		0			0
$268$	18879.0			0.262850
$269$	−24832.0			−0.343169			−0.171584	−	0.985169i	$-0.554889\pi$
$269$	−24832.0			−0.343169			−0.171584	+	0.985169i	$0.554889\pi$
$270$		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
$273$		0			0
$274$		−	8190.88i		−	0.109101i
$275$	−55851.4	−	47673.6i	−0.738530	−	0.630394i
$276$		0			0
$277$			31675.1i			0.412818i	0.978466	+	0.206409i	$0.0661778\pi$
$277$			31675.1i			0.412818i	−0.978466	+	0.206409i	$0.933822\pi$
$278$	−41303.4			−0.534436
$279$		0			0
$280$	3270.13			0.0417108
$281$			130542.i			1.65325i	0.562753	+	0.826625i	$0.309742\pi$
$281$			130542.i			1.65325i	−0.562753	+	0.826625i	$0.690258\pi$
$282$		0			0
$283$			77156.4i			0.963384i	0.876341	+	0.481692i	$0.159978\pi$
$283$			77156.4i			0.963384i	−0.876341	+	0.481692i	$0.840022\pi$
$284$	−19952.3			−0.247375
$285$		0			0
$286$	−85072.7	−	72616.3i	−1.04006	−	0.887773i
$287$	86994.4			1.05615
$288$		0			0
$289$	4961.32			0.0594020
$290$	12520.1			0.148871
$291$		0			0
$292$		−	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	$0.859061\pi$
$294$		0			0
$295$	1247.69			0.0143371
$296$		−	31692.8i		−	0.361723i
$297$		0			0
$298$	−26782.6			−0.301592
$299$			60344.1i			0.674982i
$300$		0			0
$301$	29561.3			0.326281
$302$	43376.4			0.475598
$303$		0			0
$304$			805.924i			0.00872061i
$305$			15302.8i			0.164502i
$306$		0			0
$307$			14140.6i			0.150034i	0.997182	+	0.0750171i	$0.0239011\pi$
$307$			14140.6i			0.150034i	−0.997182	+	0.0750171i	$0.976099\pi$
$308$	21330.4	−	24989.3i	0.224852	−	0.263423i
$309$		0			0
$310$			17974.9i			0.187044i
$311$	115950.			1.19881			0.599406	−	0.800445i	$-0.295404\pi$
$311$	115950.			1.19881			0.599406	+	0.800445i	$0.295404\pi$
$312$		0			0
$313$	−167682.			−1.71159			−0.855793	−	0.517318i	$-0.826930\pi$
$313$	−167682.			−1.71159			−0.855793	+	0.517318i	$0.826930\pi$
$314$			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	$0.430442\pi$
$318$		0			0
$319$	81666.0	−	95674.7i	0.802527	−	0.940191i
$320$	2180.08			0.0212899
$321$		0			0
$322$	−17725.5			−0.170957
$323$	3529.51			0.0338305
$324$		0			0
$325$			198336.i			1.87774i
$326$			27933.5i			0.262839i
$327$		0			0
$328$	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
$333$		0			0
$334$	38006.3			0.340692
$335$	10048.3			0.0895368
$336$		0			0
$337$		−	41741.2i		−	0.367540i	−0.982969	−	0.183770i	$-0.941170\pi$
$337$		−	41741.2i		−	0.367540i	0.982969	−	0.183770i	$-0.0588302\pi$
$338$			221323.i			1.93728i
$339$		0			0
$340$		−	9547.57i		−	0.0825915i
$341$	137359.	+	117247.i	1.18127	+	1.00830i
$342$		0			0
$343$			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	$0.989277\pi$
$348$		0			0
$349$			16786.2i			0.137817i	0.997623	+	0.0689083i	$0.0219516\pi$
$349$			16786.2i			0.137817i	−0.997623	+	0.0689083i	$0.978048\pi$
$350$	−58259.5			−0.475588
$351$		0			0
$352$	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	$0.655137\pi$
$354$		0			0
$355$	−10619.6			−0.0842655
$356$	−76133.8			−0.600727
$357$		0			0
$358$			152569.i			1.19042i
$359$		−	35393.9i		−	0.274625i	−0.990528	−	0.137312i	$-0.956154\pi$
$359$		−	35393.9i		−	0.274625i	0.990528	−	0.137312i	$-0.0438464\pi$
$360$		0			0
$361$	130162.			0.998783
$362$		−	66127.5i		−	0.504621i
$363$		0			0
$364$	−88740.8			−0.669762
$365$		−	19081.3i		−	0.143226i
$366$		0			0
$367$	−91892.5			−0.682257			−0.341128	−	0.940017i	$-0.610809\pi$
$367$	−91892.5			−0.682257			−0.341128	+	0.940017i	$0.610809\pi$
$368$	−11817.0			−0.0872594
$369$		0			0
$370$		−	16868.4i		−	0.123217i
$371$			8849.26i			0.0642923i
$372$		0			0
$373$		−	35452.4i		−	0.254817i	−0.991850	−	0.127408i	$-0.959334\pi$
$373$		−	35452.4i		−	0.254817i	0.991850	−	0.127408i	$-0.0406659\pi$
$374$	−72959.7	−	62276.9i	−0.521603	−	0.445230i
$375$		0			0
$376$		−	68359.3i		−	0.483529i
$377$	−339755.			−2.39047
$378$		0			0
$379$	217027.			1.51090			0.755449	−	0.655207i	$-0.227419\pi$
$379$	217027.			1.51090			0.755449	+	0.655207i	$0.227419\pi$
$380$			428.951i			0.00297057i
$381$		0			0
$382$			131837.i			0.903462i
$383$	199025.			1.35678			0.678390	−	0.734702i	$-0.262678\pi$
$383$	199025.			1.35678			0.678390	+	0.734702i	$0.262678\pi$
$384$		0			0
$385$	11353.0	−	13300.5i	0.0765932	−	0.0897318i
$386$	162877.			1.09316
$387$		0			0
$388$	−51213.3			−0.340188
$389$	141391.			0.934380			0.467190	−	0.884157i	$-0.345266\pi$
$389$	141391.			0.934380			0.467190	+	0.884157i	$0.345266\pi$
$390$		0			0
$391$			51752.1i			0.338512i
$392$			28261.6i			0.183918i
$393$		0			0
$394$	−112501.			−0.724710
$395$			41276.7i			0.264552i
$396$		0			0
$397$	109000.			0.691585			0.345793	−	0.938311i	$-0.387610\pi$
$397$	109000.			0.691585			0.345793	+	0.938311i	$0.387610\pi$
$398$		−	53472.1i		−	0.337568i
$399$		0			0
$400$	−38839.7			−0.242748
$401$	262604.			1.63310			0.816549	−	0.577277i	$-0.195885\pi$
$401$	262604.			1.63310			0.816549	+	0.577277i	$0.195885\pi$
$402$		0			0
$403$		−	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
$408$		0			0
$409$			132695.i			0.793244i	0.917982	+	0.396622i	$0.129818\pi$
$409$			132695.i			0.793244i	−0.917982	+	0.396622i	$0.870182\pi$
$410$	30868.3			0.183631
$411$		0			0
$412$	−103132.			−0.607572
$413$		−	9945.55i		−	0.0583081i
$414$		0			0
$415$			26561.2i			0.154224i
$416$	−59160.5			−0.341858
$417$		0			0
$418$	3277.91	+	2797.96i	0.0187605	+	0.0160136i
$419$	−54888.5			−0.312646			−0.156323	−	0.987706i	$-0.549964\pi$
$419$	−54888.5			−0.312646			−0.156323	+	0.987706i	$0.549964\pi$
$420$		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
$423$		0			0
$424$			5899.51i			0.0328159i
$425$			170096.i			0.941710i
$426$		0			0
$427$	121982.			0.669020
$428$		−	86447.0i		−	0.471913i
$429$		0			0
$430$	10489.3			0.0567295
$431$		−	157298.i		−	0.846776i	−0.905949	−	0.423388i	$-0.860841\pi$
$431$		−	157298.i		−	0.846776i	0.905949	−	0.423388i	$-0.139159\pi$
$432$		0			0
$433$	−134223.			−0.715900			−0.357950	−	0.933741i	$-0.616524\pi$
$433$	−134223.			−0.715900			−0.357950	+	0.933741i	$0.616524\pi$
$434$	143281.			0.760694
$435$		0			0
$436$			67763.4i			0.356470i
$437$		−	2325.10i		−	0.0121753i
$438$		0			0
$439$		−	304612.i		−	1.58059i	−0.612729	−	0.790293i	$-0.709928\pi$
$439$		−	304612.i		−	1.58059i	0.612729	−	0.790293i	$-0.290072\pi$
$440$	7568.69	−	8867.00i	0.0390945	−	0.0458006i
$441$		0			0
$442$			259091.i			1.32619i
$443$	116115.			0.591671			0.295835	−	0.955239i	$-0.404402\pi$
$443$	116115.			0.591671			0.295835	+	0.955239i	$0.404402\pi$
$444$		0			0
$445$	−40522.0			−0.204631
$446$			56259.1i			0.282828i
$447$		0			0
$448$		−	17377.9i		−	0.0865845i
$449$	−156051.			−0.774059			−0.387030	−	0.922067i	$-0.626499\pi$
$449$	−156051.			−0.774059			−0.387030	+	0.922067i	$0.626499\pi$
$450$		0			0
$451$	201348.	−	235887.i	0.989906	−	1.15971i
$452$	−8078.94			−0.0395437
$453$		0			0
$454$	−216456.			−1.05017
$455$	−47232.0			−0.228146
$456$		0			0
$457$		−	154609.i		−	0.740290i	−0.928974	−	0.370145i	$-0.879308\pi$
$457$		−	154609.i		−	0.740290i	0.928974	−	0.370145i	$-0.120692\pi$
$458$			158618.i			0.756173i
$459$		0			0
$460$	−6289.57			−0.0297239
$461$		−	324927.i		−	1.52892i	−0.644674	−	0.764458i	$-0.723007\pi$
$461$		−	324927.i		−	1.52892i	0.644674	−	0.764458i	$-0.276993\pi$
$462$		0			0
$463$	−135116.			−0.630295			−0.315147	−	0.949043i	$-0.602054\pi$
$463$	−135116.			−0.630295			−0.315147	+	0.949043i	$0.602054\pi$
$464$		−	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	$0.666270\pi$
$468$		0			0
$469$		−	80096.6i		−	0.364140i
$470$		−	36384.1i		−	0.164708i
$471$		0			0
$472$		−	6630.37i		−	0.0297614i
$473$	68419.6	−	80156.1i	0.305814	−	0.358273i
$474$		0			0
$475$		−	7642.05i		−	0.0338706i
$476$	−76105.5			−0.335894
$477$		0			0
$478$	−24408.5			−0.106828
$479$			2332.66i			0.0101667i	0.999987	+	0.00508335i	$0.00161809\pi$
$479$			2332.66i			0.0101667i	−0.999987	+	0.00508335i	$0.998382\pi$
$480$		0			0
$481$			457754.i			1.97853i
$482$	−286792.			−1.23445
$483$		0			0
$484$	−18389.9	−	115675.i	−0.0785036	−	0.493799i
$485$	−27258.1			−0.115881
$486$		0			0
$487$	307695.			1.29737			0.648683	−	0.761059i	$-0.275320\pi$
$487$	307695.			1.29737			0.648683	+	0.761059i	$0.275320\pi$
$488$	81321.2			0.341479
$489$		0			0
$490$			15042.2i			0.0626496i
$491$		−	165203.i		−	0.685258i	−0.939471	−	0.342629i	$-0.888683\pi$
$491$		−	165203.i		−	0.685258i	0.939471	−	0.342629i	$-0.111317\pi$
$492$		0			0
$493$	−291379.			−1.19885
$494$		−	11640.4i		−	0.0476993i
$495$		0			0
$496$	95520.8			0.388271
$497$			84650.5i			0.342702i
$498$		0			0
$499$	271209.			1.08919			0.544594	−	0.838700i	$-0.316684\pi$
$499$	271209.			1.08919			0.544594	+	0.838700i	$0.316684\pi$
$500$	−41962.2			−0.167849
$501$		0			0
$502$		−	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	$0.840849\pi$
$504$		0			0
$505$			67.8477i			0.000266043i
$506$	−41025.6	+	48063.0i	−0.160234	+	0.187720i
$507$		0			0
$508$		−	62872.7i		−	0.243632i
$509$	231272.			0.892663			0.446331	−	0.894868i	$-0.352730\pi$
$509$	231272.			0.892663			0.446331	+	0.894868i	$0.352730\pi$
$510$		0			0
$511$	−152101.			−0.582491
$512$		−	11585.2i		−	0.0441942i
$513$		0			0
$514$			161116.i			0.609833i
$515$	−54891.6			−0.206962
$516$		0			0
$517$	−278036.	−	237326.i	−1.04021	−	0.887900i
$518$	−134461.			−0.501114
$519$		0			0
$520$	−31488.0			−0.116450
$521$	19525.5			0.0719327			0.0359664	−	0.999353i	$-0.488549\pi$
$521$	19525.5			0.0719327			0.0359664	+	0.999353i	$0.488549\pi$
$522$		0			0
$523$			493543.i			1.80435i	0.431366	+	0.902177i	$0.358032\pi$
$523$			493543.i			1.80435i	−0.431366	+	0.902177i	$0.641968\pi$
$524$		−	195347.i		−	0.711448i
$525$		0			0
$526$	16972.6			0.0613446
$527$		−	418329.i		−	1.50625i
$528$		0			0
$529$	−245749.			−0.878173
$530$			3139.99i			0.0111783i
$531$		0			0
$532$	3419.25			0.0120811
$533$	−837668.			−2.94861
$534$		0			0
$535$		−	46011.1i		−	0.160752i
$536$		−	53397.8i		−	0.185863i
$537$		0			0
$538$			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
$543$		0			0
$544$	−50737.0			−0.171446
$545$			36066.9i			0.121427i
$546$		0			0
$547$			271731.i			0.908163i	0.890960	+	0.454082i	$0.150033\pi$
$547$			271731.i			0.908163i	−0.890960	+	0.454082i	$0.849967\pi$
$548$	−23167.3			−0.0771462
$549$		0			0
$550$	−134841.	+	157972.i	−0.445756	+	0.522220i
$551$	13091.0			0.0431191
$552$		0			0
$553$	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	$-0.237223\pi$
$558$		0			0
$559$	−284646.			−0.910922
$560$		−	9249.31i		−	0.0294940i
$561$		0			0
$562$	369229.			1.16902
$563$		−	199781.i		−	0.630285i	−0.949044	−	0.315142i	$-0.897948\pi$
$563$		−	199781.i		−	0.630285i	0.949044	−	0.315142i	$-0.102052\pi$
$564$		0			0
$565$	−4299.99			−0.0134701
$566$	218231.			0.681215
$567$		0			0
$568$			56433.7i			0.174921i
$569$			393771.i			1.21624i	0.793845	+	0.608120i	$0.208076\pi$
$569$			393771.i			1.21624i	−0.793845	+	0.608120i	$0.791924\pi$
$570$		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
$573$		0			0
$574$		−	246057.i		−	0.746814i
$575$	112053.			0.338912
$576$		0			0
$577$	−260379.			−0.782086			−0.391043	−	0.920372i	$-0.627886\pi$
$577$	−260379.			−0.782086			−0.391043	+	0.920372i	$0.627886\pi$
$578$		−	14032.7i		−	0.0420036i
$579$		0			0
$580$		−	35412.1i		−	0.105268i
$581$	211724.			0.627218
$582$		0			0
$583$	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	$0.407419\pi$
$588$		0			0
$589$			18794.6i			0.0541754i
$590$		−	3528.99i		−	0.0101379i
$591$		0			0
$592$	−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	$-0.0470909\pi$
$594$		0			0
$595$	−40506.9			−0.114418
$596$			75752.6i			0.213258i
$597$		0			0
$598$	170679.			0.477284
$599$	−204660.			−0.570399			−0.285199	−	0.958468i	$-0.592060\pi$
$599$	−204660.			−0.570399			−0.285199	+	0.958468i	$0.592060\pi$
$600$		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
$603$		0			0
$604$		−	122687.i		−	0.336298i
$605$	−9787.99	−	61567.8i	−0.0267413	−	0.168207i
$606$		0			0
$607$		−	69648.7i		−	0.189032i	−0.995523	−	0.0945160i	$-0.969870\pi$
$607$		−	69648.7i		−	0.189032i	0.995523	−	0.0945160i	$-0.0301304\pi$
$608$	2279.50			0.00616640
$609$		0			0
$610$	43283.0			0.116321
$611$			987347.i			2.64477i
$612$		0			0
$613$			619709.i			1.64917i	0.565735	+	0.824587i	$0.308593\pi$
$613$			619709.i			1.64917i	−0.565735	+	0.824587i	$0.691407\pi$
$614$	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	$0.378138\pi$
$618$		0			0
$619$	−231315.			−0.603702			−0.301851	−	0.953355i	$-0.597605\pi$
$619$	−231315.			−0.603702			−0.301851	+	0.953355i	$0.597605\pi$
$620$	50840.7			0.132260
$621$		0			0
$622$		−	327957.i		−	0.847688i
$623$			323008.i			0.832219i
$624$		0			0
$625$	356959.			0.913816
$626$			474278.i			1.21027i
$627$		0			0
$628$	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
$633$		0			0
$634$		−	123234.i		−	0.306586i
$635$		−	33463.8i		−	0.0829904i
$636$		0			0
$637$		−	408197.i		−	1.00598i
$638$	−270609.	−	230986.i	−0.664815	−	0.567473i
$639$		0			0
$640$		−	6166.21i		−	0.0150542i
$641$	297618.			0.724341			0.362170	−	0.932112i	$-0.382036\pi$
$641$	297618.			0.724341			0.362170	+	0.932112i	$0.382036\pi$
$642$		0			0
$643$	−121820.			−0.294645			−0.147322	−	0.989089i	$-0.547065\pi$
$643$	−121820.			−0.294645			−0.147322	+	0.989089i	$0.547065\pi$
$644$			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	$0.351082\pi$
$648$		0			0
$649$	−26967.5	−	23018.9i	−0.0640253	−	0.0546507i
$650$	560980.			1.32776
$651$		0			0
$652$	79007.8			0.185855
$653$	558170.			1.30900			0.654501	−	0.756061i	$-0.272879\pi$
$653$	558170.			1.30900			0.654501	+	0.756061i	$0.272879\pi$
$654$		0			0
$655$		−	103973.i		−	0.242346i
$656$		−	164038.i		−	0.381186i
$657$		0			0
$658$	−290024.			−0.669857
$659$			357653.i			0.823551i	0.911285	+	0.411776i	$0.135091\pi$
$659$			357653.i			0.823551i	−0.911285	+	0.411776i	$0.864909\pi$
$660$		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
$663$		0			0
$664$	141150.			0.320143
$665$	1819.88			0.00411529
$666$		0			0
$667$			191949.i			0.431454i
$668$		−	107498.i		−	0.240906i
$669$		0			0
$670$		−	28420.8i		−	0.0633121i
$671$	282326.	−	330756.i	0.627056	−	0.734619i
$672$		0			0
$673$			465570.i			1.02791i	0.857817	+	0.513955i	$0.171820\pi$
$673$			465570.i			1.02791i	−0.857817	+	0.513955i	$0.828180\pi$
$674$	−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	$-0.129084\pi$
$678$		0			0
$679$			217280.i			0.471280i
$680$	−27004.6			−0.0584010
$681$		0			0
$682$	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	$0.756825\pi$
$684$		0			0
$685$	−12330.7			−0.0262789
$686$	350400.			0.744588
$687$		0			0
$688$		−	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
$693$		0			0
$694$	22940.3			0.0476300
$695$			62179.0i			0.128728i
$696$		0			0
$697$	−718397.			−1.47877
$698$	47478.5			0.0974510
$699$		0			0
$700$			164783.i			0.336291i
$701$			439863.i			0.895121i	0.894254	+	0.447560i	$0.147707\pi$
$701$			439863.i			0.895121i	−0.894254	+	0.447560i	$0.852293\pi$
$702$		0			0
$703$		−	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
$708$		0			0
$709$	511529.			1.01760			0.508801	−	0.860884i	$-0.330089\pi$
$709$	511529.			1.01760			0.508801	+	0.860884i	$0.330089\pi$
$710$			30036.6i			0.0595847i
$711$		0			0
$712$			215339.i			0.424778i
$713$	−275579.			−0.542084
$714$		0			0
$715$	−109318.	+	128070.i	−0.213836	+	0.250517i
$716$	431531.			0.841755
$717$		0			0
$718$	−100109.			−0.194189
$719$	−683729.			−1.32259			−0.661297	−	0.750124i	$-0.729994\pi$
$719$	−683729.			−1.32259			−0.661297	+	0.750124i	$0.729994\pi$
$720$		0			0
$721$			437551.i			0.841701i
$722$		−	368155.i		−	0.706246i
$723$		0			0
$724$	−187037.			−0.356821
$725$			630891.i			1.20027i
$726$		0			0
$727$	258952.			0.489949			0.244974	−	0.969530i	$-0.421220\pi$
$727$	258952.			0.489949			0.244974	+	0.969530i	$0.421220\pi$
$728$			250997.i			0.473593i
$729$		0			0
$730$	−53970.1			−0.101276
$731$	−244117.			−0.456839
$732$		0			0
$733$		−	474044.i		−	0.882288i	−0.897436	−	0.441144i	$-0.854573\pi$
$733$		−	474044.i		−	0.882288i	0.897436	−	0.441144i	$-0.145427\pi$
$734$			259911.i			0.482428i
$735$		0			0
$736$			33423.6i			0.0617017i
$737$	−217183.	−	185383.i	−0.399845	−	0.341299i
$738$		0			0
$739$		−	758640.i		−	1.38914i	−0.719424	−	0.694571i	$-0.755594\pi$
$739$		−	758640.i		−	1.38914i	0.719424	−	0.694571i	$-0.244406\pi$
$740$	−47711.0			−0.0871274
$741$		0			0
$742$	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	$-0.175993\pi$
$744$		0			0
$745$			40319.1i			0.0726438i
$746$	−100275.			−0.180183
$747$		0			0
$748$	−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
$753$		0			0
$754$			960972.i			1.69032i
$755$		−	65299.8i		−	0.114556i
$756$		0			0
$757$	801567.			1.39877			0.699387	−	0.714743i	$-0.253456\pi$
$757$	801567.			1.39877			0.699387	+	0.714743i	$0.253456\pi$
$758$		−	613845.i		−	1.06837i
$759$		0			0
$760$	1213.26			0.00210051
$761$			6823.03i			0.0117817i	0.999983	+	0.00589085i	$0.00187513\pi$
$761$			6823.03i			0.0117817i	−0.999983	+	0.00589085i	$0.998125\pi$
$762$		0			0
$763$	287496.			0.493836
$764$	372891.			0.638844
$765$		0			0
$766$		−	562927.i		−	0.959389i
$767$			95765.6i			0.162787i
$768$		0			0
$769$		−	375862.i		−	0.635588i	−0.948160	−	0.317794i	$-0.897058\pi$
$769$		−	375862.i		−	0.635588i	0.948160	−	0.317794i	$-0.102942\pi$
$770$	−37619.5	−	32111.2i	−0.0634500	−	0.0541596i
$771$		0			0
$772$		−	460685.i		−	0.772982i
$773$	857421.			1.43494			0.717472	−	0.696587i	$-0.245299\pi$
$773$	857421.			1.43494			0.717472	+	0.696587i	$0.245299\pi$
$774$		0			0
$775$	−905761.			−1.50803
$776$			144853.i			0.240549i
$777$		0			0
$778$		−	399915.i		−	0.660707i
$779$	32276.0			0.0531868
$780$		0			0
$781$	229531.	+	195923.i	0.376305	+	0.321206i
$782$	146377.			0.239364
$783$		0			0
$784$	79936.0			0.130050
$785$	20481.6			0.0332372
$786$		0			0
$787$			1.02294e6i			1.65158i	0.563976	+	0.825791i	$0.309271\pi$
$787$			1.02294e6i			1.65158i	−0.563976	+	0.825791i	$0.690729\pi$
$788$			318201.i			0.512448i
$789$		0			0
$790$	116748.			0.187067
$791$			34276.0i			0.0547820i
$792$		0			0
$793$	−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	$0.738619\pi$
$798$		0			0
$799$			846765.i			1.32638i
$800$			109855.i			0.171649i
$801$		0			0
$802$		−	742755.i		−	1.15477i
$803$	−352036.	+	412424.i	−0.545954	+	0.639606i
$804$		0			0
$805$			26684.4i			0.0411780i
$806$	−1.37965e6			−2.12373
$807$		0			0
$808$	360.551			0.000552261
$809$		−	886947.i		−	1.35519i	−0.735435	−	0.677596i	$-0.763022\pi$
$809$		−	886947.i		−	1.35519i	0.735435	−	0.677596i	$-0.236978\pi$
$810$		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
$813$		0			0
$814$	−311209.	+	364593.i	−0.469682	+	0.550250i
$815$	42051.7			0.0633094
$816$		0			0
$817$	10967.6			0.0164312
$818$	375317.			0.560908
$819$		0			0
$820$		−	87308.8i		−	0.129847i
$821$		−	102607.i		−	0.152227i	−0.997099	−	0.0761133i	$-0.975749\pi$
$821$		−	102607.i		−	0.152227i	0.997099	−	0.0761133i	$-0.0242511\pi$
$822$		0			0
$823$	338801.			0.500202			0.250101	−	0.968220i	$-0.419536\pi$
$823$	338801.			0.500202			0.250101	+	0.968220i	$0.419536\pi$
$824$			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	$-0.0712880\pi$
$828$		0			0
$829$	923889.			1.34435			0.672173	−	0.740394i	$-0.265361\pi$
$829$	923889.			1.34435			0.672173	+	0.740394i	$0.265361\pi$
$830$	75126.4			0.109053
$831$		0			0
$832$			167331.i			0.241730i
$833$		−	350076.i		−	0.504513i
$834$		0			0
$835$		−	57215.5i		−	0.0820617i
$836$	7913.83	−	9271.34i	0.0113233	−	0.0132657i
$837$		0			0
$838$			155248.i			0.221074i
$839$	44067.7			0.0626032			0.0313016	−	0.999510i	$-0.490035\pi$
$839$	44067.7			0.0626032			0.0313016	+	0.999510i	$0.490035\pi$
$840$		0			0
$841$	−373450.			−0.528009
$842$		−	57733.8i		−	0.0814340i
$843$		0			0
$844$			247794.i			0.347862i
$845$	333184.			0.466628
$846$		0			0
$847$	−490769.	+	78021.9i	−0.684085	+	0.108755i
$848$	16686.3			0.0232043
$849$		0			0
$850$	481105.			0.665890
$851$	258614.			0.357103
$852$		0			0
$853$		−	1.02952e6i		−	1.41494i	−0.706746	−	0.707468i	$-0.749837\pi$
$853$		−	1.02952e6i		−	1.41494i	0.706746	−	0.707468i	$-0.250163\pi$
$854$		−	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	$0.872784\pi$
$858$		0			0
$859$	−479749.			−0.650172			−0.325086	−	0.945685i	$-0.605393\pi$
$859$	−479749.			−0.650172			−0.325086	+	0.945685i	$0.605393\pi$
$860$		−	29668.2i		−	0.0401138i
$861$		0			0
$862$	−444906.			−0.598761
$863$	220319.			0.295822			0.147911	−	0.989001i	$-0.452745\pi$
$863$	220319.			0.295822			0.147911	+	0.989001i	$0.452745\pi$
$864$		0			0
$865$		−	160373.i		−	0.214338i
$866$			379641.i			0.506218i
$867$		0			0
$868$		−	405261.i		−	0.537892i
$869$	761526.	−	892156.i	1.00843	−	1.18141i
$870$		0			0
$871$			771250.i			1.01662i
$872$	191664.			0.252062
$873$		0			0
$874$	−6576.38			−0.00860922
$875$			178030.i			0.232530i
$876$		0			0
$877$		−	1.23303e6i		−	1.60315i	−0.597893	−	0.801576i	$-0.703995\pi$
$877$		−	1.23303e6i		−	1.60315i	0.597893	−	0.801576i	$-0.296005\pi$
$878$	−861573.			−1.11764
$879$		0			0
$880$	−25079.7	−	21407.5i	−0.0323859	−	0.0276440i
$881$	−227150.			−0.292659			−0.146329	−	0.989236i	$-0.546746\pi$
$881$	−227150.			−0.292659			−0.146329	+	0.989236i	$0.546746\pi$
$882$		0			0
$883$	−136322.			−0.174842			−0.0874210	−	0.996171i	$-0.527863\pi$
$883$	−136322.			−0.174842			−0.0874210	+	0.996171i	$0.527863\pi$
$884$	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	$0.817848\pi$
$888$		0			0
$889$	−266746.			−0.337516
$890$			114613.i			0.144696i
$891$		0			0
$892$	159125.			0.199990
$893$		−	38043.2i		−	0.0477061i
$894$		0			0
$895$	229681.			0.286734
$896$	−49152.0			−0.0612245
$897$		0			0
$898$			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
$903$		0			0
$904$			22850.7i			0.0279616i
$905$	−99549.8			−0.121547
$906$		0			0
$907$	−945937.			−1.14987			−0.574934	−	0.818200i	$-0.694972\pi$
$907$	−945937.			−1.14987			−0.574934	+	0.818200i	$0.694972\pi$
$908$			612231.i			0.742581i
$909$		0			0
$910$			133592.i			0.161324i
$911$	13013.5			0.0156803			0.00784017	−	0.999969i	$-0.497504\pi$
$911$	13013.5			0.0156803			0.00784017	+	0.999969i	$0.497504\pi$
$912$		0			0
$913$	490035.	−	574094.i	0.587875	−	0.688718i
$914$	−437300.			−0.523464
$915$		0			0
$916$	448639.			0.534695
$917$	−828786.			−0.985607
$918$		0			0
$919$		−	809716.i		−	0.958742i	−0.877612	−	0.479371i	$-0.840865\pi$
$919$		−	809716.i		−	0.958742i	0.877612	−	0.479371i	$-0.159135\pi$
$920$			17789.6i			0.0210179i
$921$		0			0
$922$	−919032.			−1.08111
$923$		−	815099.i		−	0.956768i
$924$		0			0
$925$	850003.			0.993429
$926$			382165.i			0.445686i
$927$		0			0
$928$	−188185.			−0.218518
$929$	−731171.			−0.847203			−0.423601	−	0.905849i	$-0.639234\pi$
$929$	−731171.			−0.847203			−0.423601	+	0.905849i	$0.639234\pi$
$930$		0			0
$931$			15728.1i			0.0181458i
$932$			26827.2i			0.0308847i
$933$		0			0
$934$			615517.i			0.705580i
$935$	−93753.0	+	109835.i	−0.107241	+	0.125637i
$936$		0			0
$937$		−	637069.i		−	0.725617i	−0.931864	−	0.362809i	$-0.881818\pi$
$937$		−	637069.i		−	0.725617i	0.931864	−	0.362809i	$-0.118182\pi$
$938$	−226547.			−0.257486
$939$		0			0
$940$	−102910.			−0.116466
$941$			77429.7i			0.0874437i	0.999044	+	0.0437218i	$0.0139215\pi$
$941$			77429.7i			0.0874437i	−0.999044	+	0.0437218i	$0.986078\pi$
$942$		0			0
$943$			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	$0.564713\pi$
$948$		0			0
$949$	1.46458e6			1.62622
$950$	−21615.0			−0.0239501
$951$		0			0
$952$			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	$-0.211182\pi$
$954$		0			0
$955$	198470.			0.217615
$956$			69037.7i			0.0755388i
$957$		0			0
$958$	6597.75			0.00718894
$959$			98290.6i			0.106875i
$960$		0			0
$961$	1.30407e6			1.41207
$962$	1.29472e6			1.39903
$963$		0			0
$964$			811170.i			0.872886i
$965$		−	245198.i		−	0.263307i
$966$		0			0
$967$			104836.i			0.112113i	0.998428	+	0.0560565i	$0.0178527\pi$
$967$			104836.i			0.112113i	−0.998428	+	0.0560565i	$0.982147\pi$
$968$	−327179.	+	52014.6i	−0.349168	+	0.0555104i
$969$		0			0
$970$			77097.6i			0.0819403i
$971$	1.63144e6			1.73034			0.865170	−	0.501478i	$-0.167210\pi$
$971$	1.63144e6			1.73034			0.865170	+	0.501478i	$0.167210\pi$
$972$		0			0
$973$	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	$0.500731\pi$
$978$		0			0
$979$	875842.	+	747600.i	0.913819	+	0.780017i
$980$	42545.7			0.0443000
$981$		0			0
$982$	−467264.			−0.484551
$983$	−147531.			−0.152678			−0.0763388	−	0.997082i	$-0.524323\pi$
$983$	−147531.			−0.152678			−0.0763388	+	0.997082i	$0.524323\pi$
$984$		0			0
$985$			169362.i			0.174559i
$986$			824145.i			0.847715i
$987$		0			0
$988$	−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
$993$		0			0
$994$	239428.			0.242327
$995$	−80498.1			−0.0813092
$996$		0			0
$997$			71451.7i			0.0718824i	0.999354	+	0.0359412i	$0.0114429\pi$
$997$			71451.7i			0.0718824i	−0.999354	+	0.0359412i	$0.988557\pi$
$998$		−	767094.i		−	0.770172i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	198.5.d.a.109.1		4
3.2	odd	2		22.5.b.a.21.4	yes	4
11.10	odd	2	inner	198.5.d.a.109.3		4
12.11	even	2		176.5.h.e.65.1		4
15.2	even	4		550.5.c.a.549.1		8
15.8	even	4		550.5.c.a.549.8		8
15.14	odd	2		550.5.d.a.351.1		4
24.5	odd	2		704.5.h.i.65.2		4
24.11	even	2		704.5.h.j.65.3		4
33.32	even	2		22.5.b.a.21.2	✓	4
132.131	odd	2		176.5.h.e.65.2		4
165.32	odd	4		550.5.c.a.549.5		8
165.98	odd	4		550.5.c.a.549.4		8
165.164	even	2		550.5.d.a.351.3		4
264.131	odd	2		704.5.h.j.65.4		4
264.197	even	2		704.5.h.i.65.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.5.b.a.21.2	✓	4	33.32	even	2
22.5.b.a.21.4	yes	4	3.2	odd	2
176.5.h.e.65.1		4	12.11	even	2
176.5.h.e.65.2		4	132.131	odd	2
198.5.d.a.109.1		4	1.1	even	1	trivial
198.5.d.a.109.3		4	11.10	odd	2	inner
550.5.c.a.549.1		8	15.2	even	4
550.5.c.a.549.4		8	165.98	odd	4
550.5.c.a.549.5		8	165.32	odd	4
550.5.c.a.549.8		8	15.8	even	4
550.5.d.a.351.1		4	15.14	odd	2
550.5.d.a.351.3		4	165.164	even	2
704.5.h.i.65.1		4	264.197	even	2
704.5.h.i.65.2		4	24.5	odd	2
704.5.h.j.65.3		4	24.11	even	2
704.5.h.j.65.4		4	264.131	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:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	198.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -4.25798 q^{5} +33.9411i q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -4.25798 q^{5} +33.9411i q^{7} +22.6274i q^{8} +12.0434i q^{10} +(92.0319 + 78.5565i) q^{11} -326.819i q^{13} +96.0000 q^{14} +64.0000 q^{16} -280.285i q^{17} +12.5926i q^{19} +34.0638 q^{20} +(222.191 - 260.306i) q^{22} -184.641 q^{23} -606.870 q^{25} -924.383 q^{26} -271.529i q^{28} -1039.58i q^{29} +1492.51 q^{31} -181.019i q^{32} -792.766 q^{34} -144.520i q^{35} -1400.64 q^{37} +35.6172 q^{38} -96.3470i q^{40} -2563.10i q^{41} -870.960i q^{43} +(-736.255 - 628.452i) q^{44} +522.243i q^{46} -3021.08 q^{47} +1249.00 q^{49} +1716.49i q^{50} +2614.55i q^{52} +260.724 q^{53} +(-391.870 - 334.492i) q^{55} -768.000 q^{56} -2940.38 q^{58} -293.024 q^{59} -3593.92i q^{61} -4221.46i q^{62} -512.000 q^{64} +1391.59i q^{65} -2359.87 q^{67} +2242.28i q^{68} -408.766 q^{70} +2494.04 q^{71} +4481.31i q^{73} +3961.59i q^{74} -100.741i q^{76} +(-2666.30 + 3123.67i) q^{77} -9693.98i q^{79} -272.510 q^{80} -7249.53 q^{82} -6237.99i q^{83} +1193.45i q^{85} -2463.45 q^{86} +(-1777.53 + 2082.44i) q^{88} +9516.72 q^{89} +11092.6 q^{91} +1477.13 q^{92} +8544.92i q^{94} -53.6188i q^{95} +6401.66 q^{97} -3532.71i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} + 30 q^{5}+O(q^{10}) \) 4 * q - 32 * q^4 + 30 * q^5	\( 4 q - 32 q^{4} + 30 q^{5} + 180 q^{11} + 384 q^{14} + 256 q^{16} - 240 q^{20} - 240 q^{22} + 1566 q^{23} - 1722 q^{25} - 1440 q^{26} + 4418 q^{31} + 1344 q^{34} - 382 q^{37} + 2400 q^{38} - 1440 q^{44} - 5688 q^{47} + 4996 q^{49} + 8568 q^{53} - 862 q^{55} - 3072 q^{56} - 9504 q^{58} + 3390 q^{59} - 2048 q^{64} - 8734 q^{67} + 2880 q^{70} - 3522 q^{71} + 2880 q^{77} + 1920 q^{80} - 19968 q^{82} + 10464 q^{86} + 1920 q^{88} + 8766 q^{89} + 17280 q^{91} - 12528 q^{92} + 17282 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 + 30 * q^5 + 180 * q^11 + 384 * q^14 + 256 * q^16 - 240 * q^20 - 240 * q^22 + 1566 * q^23 - 1722 * q^25 - 1440 * q^26 + 4418 * q^31 + 1344 * q^34 - 382 * q^37 + 2400 * q^38 - 1440 * q^44 - 5688 * q^47 + 4996 * q^49 + 8568 * q^53 - 862 * q^55 - 3072 * q^56 - 9504 * q^58 + 3390 * q^59 - 2048 * q^64 - 8734 * q^67 + 2880 * q^70 - 3522 * q^71 + 2880 * q^77 + 1920 * q^80 - 19968 * q^82 + 10464 * q^86 + 1920 * q^88 + 8766 * q^89 + 17280 * q^91 - 12528 * q^92 + 17282 * q^97

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