LMFDB - Embedded newform 2268.4.f.a.1133.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,4,Mod(1133,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2268.1133");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2268.f (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:	$133.816331893$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1133.9
Character	$\chi$	$=$	2268.1133
Dual form			2268.4.f.a.1133.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.0714 q^{5} +(14.8834 + 11.0221i) q^{7} +O(q^{10})$	$q-12.0714 q^{5} +(14.8834 + 11.0221i) q^{7} +0.00255549i q^{11} -7.00719i q^{13} -28.3143 q^{17} +49.1973i q^{19} -51.3482i q^{23} +20.7189 q^{25} +113.099i q^{29} +32.8248i q^{31} +(-179.663 - 133.052i) q^{35} -101.961 q^{37} +22.4898 q^{41} +454.406 q^{43} -462.520 q^{47} +(100.029 + 328.090i) q^{49} -567.586i q^{53} -0.0308484i q^{55} +291.308 q^{59} +683.586i q^{61} +84.5867i q^{65} -539.560 q^{67} +307.517i q^{71} -495.192i q^{73} +(-0.0281668 + 0.0380343i) q^{77} +649.730 q^{79} -1131.16 q^{83} +341.794 q^{85} +130.217 q^{89} +(77.2337 - 104.291i) q^{91} -593.880i q^{95} +1455.39i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 12 q^{7}+O(q^{10}) $ 48 * q - 12 * q^7	$ 48 q - 12 q^{7} + 1200 q^{25} + 336 q^{37} - 168 q^{43} - 636 q^{49} + 1176 q^{67} - 408 q^{79} + 720 q^{85} - 1080 q^{91}+O(q^{100}) $ 48 * q - 12 * q^7 + 1200 * q^25 + 336 * q^37 - 168 * q^43 - 636 * q^49 + 1176 * q^67 - 408 * q^79 + 720 * q^85 - 1080 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−12.0714			−1.07970			−0.539850	−	0.841761i	$-0.681519\pi$
$5$	−12.0714			−1.07970			−0.539850	+	0.841761i	$0.681519\pi$
$6$		0			0
$7$	14.8834	+	11.0221i	0.803626	+	0.595135i
$7$	14.8834	+	11.0221i	0.803626	+	0.595135i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			0.00255549i			7.00463e-5i	1.00000		3.50232e-5i	$1.11482e-5\pi$
$11$			0.00255549i			7.00463e-5i	−1.00000		3.50232e-5i	$0.999989\pi$
$12$		0			0
$13$		−	7.00719i		−	0.149496i	−0.997202	−	0.0747479i	$-0.976185\pi$
$13$		−	7.00719i		−	0.149496i	0.997202	−	0.0747479i	$-0.0238152\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−28.3143			−0.403955			−0.201977	−	0.979390i	$-0.564737\pi$
$17$	−28.3143			−0.403955			−0.201977	+	0.979390i	$0.564737\pi$
$18$		0			0
$19$			49.1973i			0.594033i	0.954872	+	0.297016i	$0.0959916\pi$
$19$			49.1973i			0.594033i	−0.954872	+	0.297016i	$0.904008\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	51.3482i		−	0.465515i	−0.972535	−	0.232758i	$-0.925225\pi$
$23$		−	51.3482i		−	0.465515i	0.972535	−	0.232758i	$-0.0747749\pi$
$24$		0			0
$25$	20.7189			0.165751
$26$		0			0
$27$		0			0
$28$		0			0
$29$			113.099i			0.724207i	0.932138	+	0.362104i	$0.117941\pi$
$29$			113.099i			0.724207i	−0.932138	+	0.362104i	$0.882059\pi$
$30$		0			0
$31$			32.8248i			0.190178i	0.995469	+	0.0950889i	$0.0303135\pi$
$31$			32.8248i			0.190178i	−0.995469	+	0.0950889i	$0.969686\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−179.663	−	133.052i	−0.867674	−	0.642567i
$36$		0			0
$37$	−101.961			−0.453034			−0.226517	−	0.974007i	$-0.572734\pi$
$37$	−101.961			−0.453034			−0.226517	+	0.974007i	$0.572734\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	22.4898			0.0856661			0.0428331	−	0.999082i	$-0.486362\pi$
$41$	22.4898			0.0856661			0.0428331	+	0.999082i	$0.486362\pi$
$42$		0			0
$43$	454.406			1.61154			0.805771	−	0.592227i	$-0.201751\pi$
$43$	454.406			1.61154			0.805771	+	0.592227i	$0.201751\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−462.520			−1.43543			−0.717717	−	0.696334i	$-0.754813\pi$
$47$	−462.520			−1.43543			−0.717717	+	0.696334i	$0.754813\pi$
$48$		0			0
$49$	100.029	+	328.090i	0.291628	+	0.956532i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	567.586i		−	1.47102i	−0.677515	−	0.735509i	$-0.736943\pi$
$53$		−	567.586i		−	1.47102i	0.677515	−	0.735509i	$-0.263057\pi$
$54$		0			0
$55$		−	0.0308484i		−	7.56290e-5i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	291.308			0.642798			0.321399	−	0.946944i	$-0.395847\pi$
$59$	291.308			0.642798			0.321399	+	0.946944i	$0.395847\pi$
$60$		0			0
$61$			683.586i			1.43482i	0.696650	+	0.717412i	$0.254673\pi$
$61$			683.586i			1.43482i	−0.696650	+	0.717412i	$0.745327\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			84.5867i			0.161411i
$66$		0			0
$67$	−539.560			−0.983848			−0.491924	−	0.870638i	$-0.663706\pi$
$67$	−539.560			−0.983848			−0.491924	+	0.870638i	$0.663706\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			307.517i			0.514022i	0.966408	+	0.257011i	$0.0827377\pi$
$71$			307.517i			0.514022i	−0.966408	+	0.257011i	$0.917262\pi$
$72$		0			0
$73$		−	495.192i		−	0.793943i	−0.917831	−	0.396971i	$-0.870061\pi$
$73$		−	495.192i		−	0.793943i	0.917831	−	0.396971i	$-0.129939\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.0281668	+	0.0380343i	−4.16870e−5	+	5.62910e-5i
$78$		0			0
$79$	649.730			0.925321			0.462661	−	0.886536i	$-0.346895\pi$
$79$	649.730			0.925321			0.462661	+	0.886536i	$0.346895\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1131.16			−1.49592			−0.747959	−	0.663745i	$-0.768966\pi$
$83$	−1131.16			−1.49592			−0.747959	+	0.663745i	$0.768966\pi$
$84$		0			0
$85$	341.794			0.436150
$86$		0			0
$87$		0			0
$88$		0			0
$89$	130.217			0.155089			0.0775447	−	0.996989i	$-0.475292\pi$
$89$	130.217			0.155089			0.0775447	+	0.996989i	$0.475292\pi$
$90$		0			0
$91$	77.2337	−	104.291i	0.0889702	−	0.120139i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	593.880i		−	0.641377i
$96$		0			0
$97$			1455.39i			1.52343i	0.647912	+	0.761715i	$0.275642\pi$
$97$			1455.39i			1.52343i	−0.647912	+	0.761715i	$0.724358\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−395.201			−0.389346			−0.194673	−	0.980868i	$-0.562365\pi$
$101$	−395.201			−0.389346			−0.194673	+	0.980868i	$0.562365\pi$
$102$		0			0
$103$		−	629.612i		−	0.602306i	−0.953576	−	0.301153i	$-0.902629\pi$
$103$		−	629.612i		−	0.602306i	0.953576	−	0.301153i	$-0.0973715\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			515.616i			0.465855i	0.972494	+	0.232927i	$0.0748305\pi$
$107$			515.616i			0.465855i	−0.972494	+	0.232927i	$0.925170\pi$
$108$		0			0
$109$	794.011			0.697729			0.348864	−	0.937173i	$-0.386567\pi$
$109$	794.011			0.697729			0.348864	+	0.937173i	$0.386567\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2058.04i		−	1.71331i	−0.515891	−	0.856654i	$-0.672539\pi$
$113$		−	2058.04i		−	1.71331i	0.515891	−	0.856654i	$-0.327461\pi$
$114$		0			0
$115$			619.846i			0.502617i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−421.412	−	312.082i	−0.324628	−	0.240408i
$120$		0			0
$121$	1331.00			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1258.82			0.900738
$126$		0			0
$127$	−2071.63			−1.44746			−0.723729	−	0.690084i	$-0.757573\pi$
$127$	−2071.63			−1.44746			−0.723729	+	0.690084i	$0.757573\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	934.421			0.623211			0.311606	−	0.950211i	$-0.399133\pi$
$131$	934.421			0.623211			0.311606	+	0.950211i	$0.399133\pi$
$132$		0			0
$133$	−542.255	+	732.220i	−0.353530	+	0.477380i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			2679.59i			1.67104i	0.549459	+	0.835521i	$0.314834\pi$
$137$			2679.59i			1.67104i	−0.549459	+	0.835521i	$0.685166\pi$
$138$		0			0
$139$		−	917.672i		−	0.559970i	−0.960004	−	0.279985i	$-0.909670\pi$
$139$		−	917.672i		−	0.559970i	0.960004	−	0.279985i	$-0.0903296\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.0179068			1.04716e−5
$144$		0			0
$145$		−	1365.27i		−	0.781926i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	1371.14i		−	0.753882i	−0.926237	−	0.376941i	$-0.876976\pi$
$149$		−	1371.14i		−	0.753882i	0.926237	−	0.376941i	$-0.123024\pi$
$150$		0			0
$151$	−254.371			−0.137089			−0.0685445	−	0.997648i	$-0.521836\pi$
$151$	−254.371			−0.137089			−0.0685445	+	0.997648i	$0.521836\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	396.242i		−	0.205335i
$156$		0			0
$157$		−	590.915i		−	0.300383i	−0.988657	−	0.150191i	$-0.952011\pi$
$157$		−	590.915i		−	0.300383i	0.988657	−	0.150191i	$-0.0479890\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	565.963	−	764.234i	0.277044	−	0.374100i
$162$		0			0
$163$	−2197.17			−1.05580			−0.527901	−	0.849306i	$-0.677021\pi$
$163$	−2197.17			−1.05580			−0.527901	+	0.849306i	$0.677021\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−421.806			−0.195451			−0.0977256	−	0.995213i	$-0.531157\pi$
$167$	−421.806			−0.195451			−0.0977256	+	0.995213i	$0.531157\pi$
$168$		0			0
$169$	2147.90			0.977651
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1744.86			−0.766815			−0.383407	−	0.923579i	$-0.625249\pi$
$173$	−1744.86			−0.766815			−0.383407	+	0.923579i	$0.625249\pi$
$174$		0			0
$175$	308.367	+	228.365i	0.133202	+	0.0986444i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	4311.61i		−	1.80036i	−0.435518	−	0.900180i	$-0.643435\pi$
$179$		−	4311.61i		−	1.80036i	0.435518	−	0.900180i	$-0.356565\pi$
$180$		0			0
$181$		−	1307.99i		−	0.537141i	−0.963260	−	0.268570i	$-0.913449\pi$
$181$		−	1307.99i		−	0.537141i	0.963260	−	0.268570i	$-0.0865512\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1230.81			0.489141
$186$		0			0
$187$		−	0.0723570i		−	2.82955e-5i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	4036.82i		−	1.52929i	−0.644453	−	0.764644i	$-0.722915\pi$
$191$		−	4036.82i		−	1.52929i	0.644453	−	0.764644i	$-0.277085\pi$
$192$		0			0
$193$	−4419.28			−1.64822			−0.824112	−	0.566428i	$-0.808325\pi$
$193$	−4419.28			−1.64822			−0.824112	+	0.566428i	$0.808325\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	151.673i		−	0.0548540i	−0.999624	−	0.0274270i	$-0.991269\pi$
$197$		−	151.673i		−	0.0548540i	0.999624	−	0.0274270i	$-0.00873138\pi$
$198$		0			0
$199$		−	1838.65i		−	0.654966i	−0.944857	−	0.327483i	$-0.893800\pi$
$199$		−	1838.65i		−	0.654966i	0.944857	−	0.327483i	$-0.106200\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1246.59	+	1683.30i	−0.431001	+	0.581992i
$204$		0			0
$205$	−271.483			−0.0924937
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.125723			−4.16098e−5
$210$		0			0
$211$	−2458.54			−0.802145			−0.401072	−	0.916046i	$-0.631362\pi$
$211$	−2458.54			−0.802145			−0.401072	+	0.916046i	$0.631362\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5485.32			−1.73998
$216$		0			0
$217$	−361.797	+	488.543i	−0.113181	+	0.152832i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			198.404i			0.0603895i
$222$		0			0
$223$		−	5185.98i		−	1.55731i	−0.627455	−	0.778653i	$-0.715903\pi$
$223$		−	5185.98i		−	1.55731i	0.627455	−	0.778653i	$-0.284097\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−995.062			−0.290945			−0.145473	−	0.989362i	$-0.546470\pi$
$227$	−995.062			−0.290945			−0.145473	+	0.989362i	$0.546470\pi$
$228$		0			0
$229$		−	117.622i		−	0.0339419i	−0.999856	−	0.0169710i	$-0.994598\pi$
$229$		−	117.622i		−	0.0339419i	0.999856	−	0.0169710i	$-0.00540228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1484.08i		−	0.417276i	−0.977993	−	0.208638i	$-0.933097\pi$
$233$		−	1484.08i		−	0.417276i	0.977993	−	0.208638i	$-0.0669030\pi$
$234$		0			0
$235$	5583.26			1.54984
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2762.14i		−	0.747563i	−0.927517	−	0.373782i	$-0.878061\pi$
$239$		−	2762.14i		−	0.747563i	0.927517	−	0.373782i	$-0.121939\pi$
$240$		0			0
$241$		−	3883.25i		−	1.03793i	−0.854794	−	0.518967i	$-0.826317\pi$
$241$		−	3883.25i		−	1.03793i	0.854794	−	0.518967i	$-0.173683\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1207.49	−	3960.51i	−0.314871	−	1.03277i
$246$		0			0
$247$	344.735			0.0888054
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5549.57			−1.39556			−0.697780	−	0.716312i	$-0.745829\pi$
$251$	−5549.57			−1.39556			−0.697780	+	0.716312i	$0.745829\pi$
$252$		0			0
$253$	0.131220			3.26076e−5
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1619.17			−0.393000			−0.196500	−	0.980504i	$-0.562958\pi$
$257$	−1619.17			−0.393000			−0.196500	+	0.980504i	$0.562958\pi$
$258$		0			0
$259$	−1517.52	−	1123.82i	−0.364070	−	0.269616i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	686.831i		−	0.161033i	−0.996753	−	0.0805167i	$-0.974343\pi$
$263$		−	686.831i		−	0.161033i	0.996753	−	0.0805167i	$-0.0256570\pi$
$264$		0			0
$265$			6851.56i			1.58826i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1805.03			−0.409124			−0.204562	−	0.978854i	$-0.565577\pi$
$269$	−1805.03			−0.409124			−0.204562	+	0.978854i	$0.565577\pi$
$270$		0			0
$271$			6726.19i			1.50770i	0.657046	+	0.753851i	$0.271806\pi$
$271$			6726.19i			1.50770i	−0.657046	+	0.753851i	$0.728194\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			0.0529470i			1.16103e-5i
$276$		0			0
$277$	672.248			0.145818			0.0729088	−	0.997339i	$-0.476772\pi$
$277$	672.248			0.145818			0.0729088	+	0.997339i	$0.476772\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2317.62i		−	0.492020i	−0.969267	−	0.246010i	$-0.920880\pi$
$281$		−	2317.62i		−	0.492020i	0.969267	−	0.246010i	$-0.0791196\pi$
$282$		0			0
$283$		−	9254.78i		−	1.94396i	−0.235072	−	0.971978i	$-0.575533\pi$
$283$		−	9254.78i		−	1.94396i	0.235072	−	0.971978i	$-0.424467\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	334.723	+	247.883i	0.0688435	+	0.0509829i
$288$		0			0
$289$	−4111.30			−0.836821
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8785.44			−1.75171			−0.875855	−	0.482575i	$-0.839702\pi$
$293$	−8785.44			−1.75171			−0.875855	+	0.482575i	$0.839702\pi$
$294$		0			0
$295$	−3516.50			−0.694029
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−359.807			−0.0695926
$300$		0			0
$301$	6763.09	+	5008.49i	1.29508	+	0.959085i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	8251.85i		−	1.54918i
$306$		0			0
$307$		−	599.516i		−	0.111453i	−0.998446	−	0.0557267i	$-0.982252\pi$
$307$		−	599.516i		−	0.111453i	0.998446	−	0.0557267i	$-0.0177476\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8988.58			−1.63889			−0.819447	−	0.573155i	$-0.805719\pi$
$311$	−8988.58			−1.63889			−0.819447	+	0.573155i	$0.805719\pi$
$312$		0			0
$313$			8201.48i			1.48107i	0.672017	+	0.740535i	$0.265428\pi$
$313$			8201.48i			1.48107i	−0.672017	+	0.740535i	$0.734572\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6806.29i		−	1.20593i	−0.797768	−	0.602964i	$-0.793986\pi$
$317$		−	6806.29i		−	1.20593i	0.797768	−	0.602964i	$-0.206014\pi$
$318$		0			0
$319$	−0.289024			−5.07281e−5
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1392.99i		−	0.239962i
$324$		0			0
$325$		−	145.181i		−	0.0247791i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6883.85	−	5097.92i	−1.15355	−	0.854278i
$330$		0			0
$331$	−7750.82			−1.28708			−0.643540	−	0.765412i	$-0.722535\pi$
$331$	−7750.82			−1.28708			−0.643540	+	0.765412i	$0.722535\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6513.25			1.06226
$336$		0			0
$337$	3088.47			0.499228			0.249614	−	0.968345i	$-0.419696\pi$
$337$	3088.47			0.499228			0.249614	+	0.968345i	$0.419696\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.0838835			−1.33213e−5
$342$		0			0
$343$	−2127.47	+	5985.61i	−0.334906	+	0.942252i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	11677.4i		−	1.80656i	−0.429050	−	0.903281i	$-0.641152\pi$
$347$		−	11677.4i		−	1.80656i	0.429050	−	0.903281i	$-0.358848\pi$
$348$		0			0
$349$		−	7717.09i		−	1.18363i	−0.806074	−	0.591814i	$-0.798412\pi$
$349$		−	7717.09i		−	1.18363i	0.806074	−	0.591814i	$-0.201588\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5520.23			−0.832329			−0.416165	−	0.909289i	$-0.636626\pi$
$353$	−5520.23			−0.832329			−0.416165	+	0.909289i	$0.636626\pi$
$354$		0			0
$355$		−	3712.16i		−	0.554989i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6290.95i			0.924857i	0.886657	+	0.462429i	$0.153022\pi$
$359$			6290.95i			0.924857i	−0.886657	+	0.462429i	$0.846978\pi$
$360$		0			0
$361$	4438.63			0.647125
$362$		0			0
$363$		0			0
$364$		0			0
$365$			5977.66i			0.857220i
$366$		0			0
$367$		−	3859.50i		−	0.548949i	−0.961594	−	0.274474i	$-0.911496\pi$
$367$		−	3859.50i		−	0.548949i	0.961594	−	0.274474i	$-0.0885039\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6255.96	−	8447.58i	0.875454	−	1.18215i
$372$		0			0
$373$	8616.57			1.19611			0.598055	−	0.801455i	$-0.295940\pi$
$373$	8616.57			1.19611			0.598055	+	0.801455i	$0.295940\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	792.509			0.108266
$378$		0			0
$379$	11100.5			1.50447			0.752237	−	0.658893i	$-0.228975\pi$
$379$	11100.5			1.50447			0.752237	+	0.658893i	$0.228975\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6588.58			0.879009			0.439505	−	0.898240i	$-0.355154\pi$
$383$	6588.58			0.879009			0.439505	+	0.898240i	$0.355154\pi$
$384$		0			0
$385$	0.340013	−	0.459127i	4.50095e−5	−	6.07774e-5i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	7982.68i		−	1.04046i	−0.854027	−	0.520228i	$-0.825847\pi$
$389$		−	7982.68i		−	1.04046i	0.854027	−	0.520228i	$-0.174153\pi$
$390$		0			0
$391$			1453.89i			0.188047i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7843.16			−0.999069
$396$		0			0
$397$			9040.06i			1.14284i	0.820658	+	0.571420i	$0.193607\pi$
$397$			9040.06i			1.14284i	−0.820658	+	0.571420i	$0.806393\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			965.529i			0.120240i	0.998191	+	0.0601200i	$0.0191483\pi$
$401$			965.529i			0.120240i	−0.998191	+	0.0601200i	$0.980852\pi$
$402$		0			0
$403$	230.010			0.0284308
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	0.260560i		−	3.17334e-5i
$408$		0			0
$409$			3312.57i			0.400479i	0.979747	+	0.200240i	$0.0641721\pi$
$409$			3312.57i			0.400479i	−0.979747	+	0.200240i	$0.935828\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4335.64	+	3210.82i	0.516569	+	0.382552i
$414$		0			0
$415$	13654.7			1.61514
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8344.63			−0.972940			−0.486470	−	0.873697i	$-0.661716\pi$
$419$	−8344.63			−0.972940			−0.486470	+	0.873697i	$0.661716\pi$
$420$		0			0
$421$	−6701.17			−0.775760			−0.387880	−	0.921710i	$-0.626793\pi$
$421$	−6701.17			−0.775760			−0.387880	+	0.921710i	$0.626793\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−586.641			−0.0669560
$426$		0			0
$427$	−7534.52	+	10174.1i	−0.853914	+	1.15306i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			9168.55i			1.02467i	0.858785	+	0.512336i	$0.171220\pi$
$431$			9168.55i			1.02467i	−0.858785	+	0.512336i	$0.828780\pi$
$432$		0			0
$433$			1346.90i			0.149487i	0.997203	+	0.0747436i	$0.0238138\pi$
$433$			1346.90i			0.149487i	−0.997203	+	0.0747436i	$0.976186\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2526.19			0.276531
$438$		0			0
$439$			5648.88i			0.614137i	0.951687	+	0.307069i	$0.0993481\pi$
$439$			5648.88i			0.614137i	−0.951687	+	0.307069i	$0.900652\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	8689.91i		−	0.931986i	−0.884788	−	0.465993i	$-0.845697\pi$
$443$		−	8689.91i		−	0.931986i	0.884788	−	0.465993i	$-0.154303\pi$
$444$		0			0
$445$	−1571.90			−0.167450
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	14492.3i		−	1.52324i	−0.648026	−	0.761618i	$-0.724405\pi$
$449$		−	14492.3i		−	1.52324i	0.648026	−	0.761618i	$-0.275595\pi$
$450$		0			0
$451$			0.0574724i			6.00060e-6i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−932.319	+	1258.93i	−0.0960611	+	0.129714i
$456$		0			0
$457$	8263.35			0.845828			0.422914	−	0.906170i	$-0.361007\pi$
$457$	8263.35			0.845828			0.422914	+	0.906170i	$0.361007\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9896.66			−0.999855			−0.499928	−	0.866067i	$-0.666640\pi$
$461$	−9896.66			−0.999855			−0.499928	+	0.866067i	$0.666640\pi$
$462$		0			0
$463$	−3975.57			−0.399051			−0.199525	−	0.979893i	$-0.563940\pi$
$463$	−3975.57			−0.399051			−0.199525	+	0.979893i	$0.563940\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17044.5			−1.68892			−0.844460	−	0.535619i	$-0.820079\pi$
$467$	−17044.5			−1.68892			−0.844460	+	0.535619i	$0.820079\pi$
$468$		0			0
$469$	−8030.47	−	5947.07i	−0.790645	−	0.585522i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.16123i			0.000112883i
$474$		0			0
$475$			1019.31i			0.0984616i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10729.9			1.02351			0.511753	−	0.859132i	$-0.328996\pi$
$479$	10729.9			1.02351			0.511753	+	0.859132i	$0.328996\pi$
$480$		0			0
$481$			714.459i			0.0677267i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	17568.6i		−	1.64485i
$486$		0			0
$487$	−12461.1			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$487$	−12461.1			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			4679.09i			0.430070i	0.976606	+	0.215035i	$0.0689866\pi$
$491$			4679.09i			0.430070i	−0.976606	+	0.215035i	$0.931013\pi$
$492$		0			0
$493$		−	3202.33i		−	0.292547i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3389.47	+	4576.89i	−0.305913	+	0.413081i
$498$		0			0
$499$	−5371.08			−0.481848			−0.240924	−	0.970544i	$-0.577451\pi$
$499$	−5371.08			−0.481848			−0.240924	+	0.970544i	$0.577451\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	22263.9			1.97355			0.986777	−	0.162086i	$-0.0518222\pi$
$503$	22263.9			1.97355			0.986777	+	0.162086i	$0.0518222\pi$
$504$		0			0
$505$	4770.63			0.420376
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14499.5			1.26263			0.631316	−	0.775526i	$-0.282515\pi$
$509$	14499.5			1.26263			0.631316	+	0.775526i	$0.282515\pi$
$510$		0			0
$511$	5458.03	−	7370.12i	0.472503	−	0.638033i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			7600.30i			0.650309i
$516$		0			0
$517$		−	1.18197i		−	0.000100547i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7009.79			−0.589452			−0.294726	−	0.955582i	$-0.595228\pi$
$521$	−7009.79			−0.589452			−0.294726	+	0.955582i	$0.595228\pi$
$522$		0			0
$523$			3631.76i			0.303644i	0.988408	+	0.151822i	$0.0485141\pi$
$523$			3631.76i			0.303644i	−0.988408	+	0.151822i	$0.951486\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	929.412i		−	0.0768232i
$528$		0			0
$529$	9530.36			0.783296
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	157.590i		−	0.0128067i
$534$		0			0
$535$		−	6224.21i		−	0.502983i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.838432	+	0.255622i	−6.70015e−5	+	2.04275e-5i
$540$		0			0
$541$	−12760.4			−1.01407			−0.507037	−	0.861925i	$-0.669259\pi$
$541$	−12760.4			−1.01407			−0.507037	+	0.861925i	$0.669259\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9584.83			−0.753338
$546$		0			0
$547$	7839.24			0.612763			0.306382	−	0.951909i	$-0.400882\pi$
$547$	7839.24			0.612763			0.306382	+	0.951909i	$0.400882\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5564.18			−0.430203
$552$		0			0
$553$	9670.17	+	7161.36i	0.743612	+	0.550691i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	9443.19i		−	0.718350i	−0.933270	−	0.359175i	$-0.883058\pi$
$557$		−	9443.19i		−	0.718350i	0.933270	−	0.359175i	$-0.116942\pi$
$558$		0			0
$559$		−	3184.11i		−	0.240919i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16044.3			1.20104			0.600519	−	0.799610i	$-0.294961\pi$
$563$	16044.3			1.20104			0.600519	+	0.799610i	$0.294961\pi$
$564$		0			0
$565$			24843.4i			1.84986i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			22186.3i			1.63462i	0.576198	+	0.817310i	$0.304536\pi$
$569$			22186.3i			1.63462i	−0.576198	+	0.817310i	$0.695464\pi$
$570$		0			0
$571$	6422.21			0.470685			0.235343	−	0.971912i	$-0.424379\pi$
$571$	6422.21			0.470685			0.235343	+	0.971912i	$0.424379\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1063.88i		−	0.0771597i
$576$		0			0
$577$		−	21441.1i		−	1.54697i	−0.633813	−	0.773486i	$-0.718511\pi$
$577$		−	21441.1i		−	1.54697i	0.633813	−	0.773486i	$-0.281489\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16835.5	−	12467.7i	−1.20216	−	0.890273i
$582$		0			0
$583$	1.45046			0.000103039
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−685.524			−0.0482021			−0.0241010	−	0.999710i	$-0.507672\pi$
$587$	−685.524			−0.0482021			−0.0241010	+	0.999710i	$0.507672\pi$
$588$		0			0
$589$	−1614.89			−0.112972
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11503.8			−0.796635			−0.398317	−	0.917248i	$-0.630406\pi$
$593$	−11503.8			−0.796635			−0.398317	+	0.917248i	$0.630406\pi$
$594$		0			0
$595$	5087.04	+	3767.27i	0.350501	+	0.259568i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	5917.21i		−	0.403624i	−0.979424	−	0.201812i	$-0.935317\pi$
$599$		−	5917.21i		−	0.403624i	0.979424	−	0.201812i	$-0.0646829\pi$
$600$		0			0
$601$		−	11741.3i		−	0.796904i	−0.917189	−	0.398452i	$-0.869548\pi$
$601$		−	11741.3i		−	0.796904i	0.917189	−	0.398452i	$-0.130452\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16067.0			−1.07970
$606$		0			0
$607$		−	27733.6i		−	1.85448i	−0.374464	−	0.927241i	$-0.622173\pi$
$607$		−	27733.6i		−	1.85448i	0.374464	−	0.927241i	$-0.377827\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			3240.96i			0.214592i
$612$		0			0
$613$	−19338.2			−1.27416			−0.637081	−	0.770797i	$-0.719858\pi$
$613$	−19338.2			−1.27416			−0.637081	+	0.770797i	$0.719858\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	11463.1i		−	0.747953i	−0.927438	−	0.373976i	$-0.877994\pi$
$617$		−	11463.1i		−	0.747953i	0.927438	−	0.373976i	$-0.122006\pi$
$618$		0			0
$619$		−	3960.15i		−	0.257143i	−0.991700	−	0.128572i	$-0.958961\pi$
$619$		−	3960.15i		−	0.257143i	0.991700	−	0.128572i	$-0.0410393\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1938.06	+	1435.26i	0.124634	+	0.0922991i
$624$		0			0
$625$	−17785.6			−1.13828
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2886.95			0.183005
$630$		0			0
$631$	−5409.20			−0.341263			−0.170631	−	0.985335i	$-0.554581\pi$
$631$	−5409.20			−0.341263			−0.170631	+	0.985335i	$0.554581\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	25007.5			1.56282
$636$		0			0
$637$	2298.99	−	700.919i	0.142997	−	0.0435972i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	24028.8i		−	1.48063i	−0.672262	−	0.740313i	$-0.734677\pi$
$641$		−	24028.8i		−	1.48063i	0.672262	−	0.740313i	$-0.265323\pi$
$642$		0			0
$643$			4135.99i			0.253667i	0.991924	+	0.126833i	$0.0404813\pi$
$643$			4135.99i			0.253667i	−0.991924	+	0.126833i	$0.959519\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1393.75			0.0846895			0.0423447	−	0.999103i	$-0.486517\pi$
$647$	1393.75			0.0846895			0.0423447	+	0.999103i	$0.486517\pi$
$648$		0			0
$649$			0.744436i			4.50257e-5i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	9059.90i		−	0.542942i	−0.962447	−	0.271471i	$-0.912490\pi$
$653$		−	9059.90i		−	0.542942i	0.962447	−	0.271471i	$-0.0875101\pi$
$654$		0			0
$655$	−11279.8			−0.672881
$656$		0			0
$657$		0			0
$658$		0			0
$659$			16186.7i			0.956819i	0.878137	+	0.478409i	$0.158786\pi$
$659$			16186.7i			0.956819i	−0.878137	+	0.478409i	$0.841214\pi$
$660$		0			0
$661$		−	28661.5i		−	1.68654i	−0.537492	−	0.843269i	$-0.680628\pi$
$661$		−	28661.5i		−	1.68654i	0.537492	−	0.843269i	$-0.319372\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6545.78	−	8838.93i	0.381706	−	0.515427i
$666$		0			0
$667$	5807.45			0.337130
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.74690			−0.000100504
$672$		0			0
$673$	−7341.76			−0.420511			−0.210255	−	0.977646i	$-0.567430\pi$
$673$	−7341.76			−0.420511			−0.210255	+	0.977646i	$0.567430\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−32956.8			−1.87095			−0.935474	−	0.353396i	$-0.885027\pi$
$677$	−32956.8			−1.87095			−0.935474	+	0.353396i	$0.885027\pi$
$678$		0			0
$679$	−16041.4	+	21661.1i	−0.906647	+	1.22427i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			22213.0i			1.24444i	0.782841	+	0.622222i	$0.213770\pi$
$683$			22213.0i			1.24444i	−0.782841	+	0.622222i	$0.786230\pi$
$684$		0			0
$685$		−	32346.4i		−	1.80422i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3977.18			−0.219911
$690$		0			0
$691$			25486.7i			1.40313i	0.712608	+	0.701563i	$0.247514\pi$
$691$			25486.7i			1.40313i	−0.712608	+	0.701563i	$0.752486\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			11077.6i			0.604600i
$696$		0			0
$697$	−636.782			−0.0346052
$698$		0			0
$699$		0			0
$700$		0			0
$701$			22316.4i			1.20239i	0.799101	+	0.601197i	$0.205309\pi$
$701$			22316.4i			1.20239i	−0.799101	+	0.601197i	$0.794691\pi$
$702$		0			0
$703$		−	5016.19i		−	0.269117i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5881.91	−	4355.92i	−0.312888	−	0.231713i
$708$		0			0
$709$	1240.50			0.0657095			0.0328547	−	0.999460i	$-0.489540\pi$
$709$	1240.50			0.0657095			0.0328547	+	0.999460i	$0.489540\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1685.50			0.0885306
$714$		0			0
$715$	−0.216161			−1.13062e−5
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1718.52			0.0891374			0.0445687	−	0.999006i	$-0.485809\pi$
$719$	1718.52			0.0891374			0.0445687	+	0.999006i	$0.485809\pi$
$720$		0			0
$721$	6939.62	−	9370.74i	0.358453	−	0.484028i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			2343.29i			0.120038i
$726$		0			0
$727$			2545.82i			0.129875i	0.997889	+	0.0649376i	$0.0206848\pi$
$727$			2545.82i			0.129875i	−0.997889	+	0.0649376i	$0.979315\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12866.2			−0.650990
$732$		0			0
$733$		−	22084.9i		−	1.11285i	−0.830896	−	0.556427i	$-0.812172\pi$
$733$		−	22084.9i		−	1.11285i	0.830896	−	0.556427i	$-0.187828\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.37884i		−	6.89149e-5i
$738$		0			0
$739$	−5401.50			−0.268873			−0.134437	−	0.990922i	$-0.542922\pi$
$739$	−5401.50			−0.268873			−0.134437	+	0.990922i	$0.542922\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			31143.5i			1.53774i	0.639403	+	0.768872i	$0.279182\pi$
$743$			31143.5i			1.53774i	−0.639403	+	0.768872i	$0.720818\pi$
$744$		0			0
$745$			16551.6i			0.813966i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5683.15	+	7674.09i	−0.277247	+	0.374373i
$750$		0			0
$751$	5289.23			0.257000			0.128500	−	0.991710i	$-0.458984\pi$
$751$	5289.23			0.257000			0.128500	+	0.991710i	$0.458984\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3070.62			0.148015
$756$		0			0
$757$	−20794.2			−0.998384			−0.499192	−	0.866491i	$-0.666370\pi$
$757$	−20794.2			−0.998384			−0.499192	+	0.866491i	$0.666370\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18253.9			0.869517			0.434759	−	0.900547i	$-0.356834\pi$
$761$	18253.9			0.869517			0.434759	+	0.900547i	$0.356834\pi$
$762$		0			0
$763$	11817.5	+	8751.63i	0.560713	+	0.415243i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	2041.25i		−	0.0960956i
$768$		0			0
$769$			32445.2i			1.52146i	0.649068	+	0.760730i	$0.275159\pi$
$769$			32445.2i			1.52146i	−0.649068	+	0.760730i	$0.724841\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	20600.1			0.958519			0.479259	−	0.877673i	$-0.340905\pi$
$773$	20600.1			0.958519			0.479259	+	0.877673i	$0.340905\pi$
$774$		0			0
$775$			680.094i			0.0315222i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1106.43i			0.0508885i
$780$		0			0
$781$	−0.785857			−3.60054e−5
$782$		0			0
$783$		0			0
$784$		0			0
$785$			7133.17i			0.324323i
$786$		0			0
$787$		−	14148.7i		−	0.640848i	−0.947274	−	0.320424i	$-0.896175\pi$
$787$		−	14148.7i		−	0.640848i	0.947274	−	0.320424i	$-0.103825\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	22683.8	−	30630.5i	1.01965	−	1.37686i
$792$		0			0
$793$	4790.02			0.214500
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−19961.5			−0.887166			−0.443583	−	0.896233i	$-0.646293\pi$
$797$	−19961.5			−0.887166			−0.443583	+	0.896233i	$0.646293\pi$
$798$		0			0
$799$	13095.9			0.579851
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.26546			5.56128e−5
$804$		0			0
$805$	−6831.97	+	9225.38i	−0.299125	+	0.403916i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	13027.7i		−	0.566169i	−0.959095	−	0.283085i	$-0.908642\pi$
$809$		−	13027.7i		−	0.566169i	0.959095	−	0.283085i	$-0.0913578\pi$
$810$		0			0
$811$			2467.58i			0.106841i	0.998572	+	0.0534207i	$0.0170124\pi$
$811$			2467.58i			0.106841i	−0.998572	+	0.0534207i	$0.982988\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	26522.9			1.13995
$816$		0			0
$817$			22355.5i			0.957309i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			19979.6i			0.849322i	0.905353	+	0.424661i	$0.139607\pi$
$821$			19979.6i			0.849322i	−0.905353	+	0.424661i	$0.860393\pi$
$822$		0			0
$823$	40963.6			1.73499			0.867497	−	0.497442i	$-0.165727\pi$
$823$	40963.6			1.73499			0.867497	+	0.497442i	$0.165727\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			38066.2i			1.60059i	0.599604	+	0.800297i	$0.295325\pi$
$827$			38066.2i			1.60059i	−0.599604	+	0.800297i	$0.704675\pi$
$828$		0			0
$829$		−	5157.09i		−	0.216059i	−0.994148	−	0.108030i	$-0.965546\pi$
$829$		−	5157.09i		−	0.216059i	0.994148	−	0.108030i	$-0.0344542\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2832.24	−	9289.65i	−0.117805	−	0.386395i
$834$		0			0
$835$	5091.80			0.211029
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24384.4			1.00339			0.501695	−	0.865045i	$-0.332710\pi$
$839$	24384.4			1.00339			0.501695	+	0.865045i	$0.332710\pi$
$840$		0			0
$841$	11597.5			0.475524
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−25928.2			−1.05557
$846$		0			0
$847$	19809.7	+	14670.4i	0.803626	+	0.595135i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			5235.51i			0.210894i
$852$		0			0
$853$		−	40661.2i		−	1.63214i	−0.577956	−	0.816068i	$-0.696150\pi$
$853$		−	40661.2i		−	1.63214i	0.577956	−	0.816068i	$-0.303850\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10849.9			0.432469			0.216234	−	0.976342i	$-0.430622\pi$
$857$	10849.9			0.432469			0.216234	+	0.976342i	$0.430622\pi$
$858$		0			0
$859$		−	33997.1i		−	1.35037i	−0.737649	−	0.675184i	$-0.764064\pi$
$859$		−	33997.1i		−	1.35037i	0.737649	−	0.675184i	$-0.235936\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			21617.2i			0.852675i	0.904564	+	0.426337i	$0.140196\pi$
$863$			21617.2i			0.852675i	−0.904564	+	0.426337i	$0.859804\pi$
$864$		0			0
$865$	21062.9			0.827930
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.66038i			6.48154e-5i
$870$		0			0
$871$			3780.80i			0.147081i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	18735.5	+	13874.8i	0.723856	+	0.536061i
$876$		0			0
$877$	−14904.8			−0.573889			−0.286945	−	0.957947i	$-0.592640\pi$
$877$	−14904.8			−0.573889			−0.286945	+	0.957947i	$0.592640\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18956.5			0.724928			0.362464	−	0.931998i	$-0.381936\pi$
$881$	18956.5			0.724928			0.362464	+	0.931998i	$0.381936\pi$
$882$		0			0
$883$	−35931.1			−1.36940			−0.684700	−	0.728825i	$-0.740067\pi$
$883$	−35931.1			−1.36940			−0.684700	+	0.728825i	$0.740067\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	38644.7			1.46287			0.731434	−	0.681912i	$-0.238851\pi$
$887$	38644.7			1.46287			0.731434	+	0.681912i	$0.238851\pi$
$888$		0			0
$889$	−30832.8	−	22833.6i	−1.16321	−	0.861433i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	22754.7i		−	0.852695i
$894$		0			0
$895$			52047.1i			1.94385i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3712.46			−0.137728
$900$		0			0
$901$			16070.8i			0.594224i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			15789.3i			0.579951i
$906$		0			0
$907$	572.311			0.0209518			0.0104759	−	0.999945i	$-0.496665\pi$
$907$	572.311			0.0209518			0.0104759	+	0.999945i	$0.496665\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			28445.1i			1.03450i	0.855835	+	0.517248i	$0.173044\pi$
$911$			28445.1i			1.03450i	−0.855835	+	0.517248i	$0.826956\pi$
$912$		0			0
$913$		−	2.89068i		−	0.000104784i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	13907.3	+	10299.2i	0.500829	+	0.370895i
$918$		0			0
$919$	20019.4			0.718585			0.359292	−	0.933225i	$-0.383018\pi$
$919$	20019.4			0.718585			0.359292	+	0.933225i	$0.383018\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2154.83			0.0768441
$924$		0			0
$925$	−2112.52			−0.0750909
$926$		0			0
$927$		0			0
$928$		0			0
$929$	31582.5			1.11538			0.557690	−	0.830049i	$-0.311688\pi$
$929$	31582.5			1.11538			0.557690	+	0.830049i	$0.311688\pi$
$930$		0			0
$931$	−16141.1	+	4921.13i	−0.568211	+	0.173237i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			0.873451i			3.05507e-5i
$936$		0			0
$937$			39841.1i			1.38906i	0.719463	+	0.694531i	$0.244388\pi$
$937$			39841.1i			1.38906i	−0.719463	+	0.694531i	$0.755612\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	10887.2			0.377165			0.188583	−	0.982057i	$-0.439611\pi$
$941$	10887.2			0.377165			0.188583	+	0.982057i	$0.439611\pi$
$942$		0			0
$943$		−	1154.81i		−	0.0398789i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			22840.6i			0.783758i	0.920017	+	0.391879i	$0.128175\pi$
$947$			22840.6i			0.783758i	−0.920017	+	0.391879i	$0.871825\pi$
$948$		0			0
$949$	−3469.90			−0.118691
$950$		0			0
$951$		0			0
$952$		0			0
$953$			51243.3i			1.74180i	0.491462	+	0.870899i	$0.336463\pi$
$953$			51243.3i			1.74180i	−0.491462	+	0.870899i	$0.663537\pi$
$954$		0			0
$955$			48730.1i			1.65117i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−29534.6	+	39881.3i	−0.994496	+	1.34289i
$960$		0			0
$961$	28713.5			0.963832
$962$		0			0
$963$		0			0
$964$		0			0
$965$	53347.0			1.77959
$966$		0			0
$967$	14979.1			0.498135			0.249067	−	0.968486i	$-0.419876\pi$
$967$	14979.1			0.498135			0.249067	+	0.968486i	$0.419876\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12705.2			−0.419908			−0.209954	−	0.977711i	$-0.567331\pi$
$971$	−12705.2			−0.419908			−0.209954	+	0.977711i	$0.567331\pi$
$972$		0			0
$973$	10114.6	−	13658.0i	0.333258	−	0.450007i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			20325.4i			0.665575i	0.943002	+	0.332788i	$0.107989\pi$
$977$			20325.4i			0.665575i	−0.943002	+	0.332788i	$0.892011\pi$
$978$		0			0
$979$			0.332768i			1.08634e-5i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	40514.2			1.31455			0.657274	−	0.753652i	$-0.271709\pi$
$983$	40514.2			1.31455			0.657274	+	0.753652i	$0.271709\pi$
$984$		0			0
$985$			1830.90i			0.0592258i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	23333.0i		−	0.750197i
$990$		0			0
$991$	−5083.59			−0.162952			−0.0814761	−	0.996675i	$-0.525963\pi$
$991$	−5083.59			−0.162952			−0.0814761	+	0.996675i	$0.525963\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			22195.1i			0.707167i
$996$		0			0
$997$			18471.0i			0.586742i	0.955999	+	0.293371i	$0.0947771\pi$
$997$			18471.0i			0.586742i	−0.955999	+	0.293371i	$0.905223\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.4.f.a.1133.9		48
3.2	odd	2	inner	2268.4.f.a.1133.40		48
7.6	odd	2	inner	2268.4.f.a.1133.39		48
9.2	odd	6		756.4.x.a.125.5		48
9.4	even	3		756.4.x.a.629.20		48
9.5	odd	6		252.4.x.a.209.9	yes	48
9.7	even	3		252.4.x.a.41.16	yes	48
21.20	even	2	inner	2268.4.f.a.1133.10		48
63.13	odd	6		756.4.x.a.629.5		48
63.20	even	6		756.4.x.a.125.20		48
63.34	odd	6		252.4.x.a.41.9	✓	48
63.41	even	6		252.4.x.a.209.16	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.9	✓	48	63.34	odd	6
252.4.x.a.41.16	yes	48	9.7	even	3
252.4.x.a.209.9	yes	48	9.5	odd	6
252.4.x.a.209.16	yes	48	63.41	even	6
756.4.x.a.125.5		48	9.2	odd	6
756.4.x.a.125.20		48	63.20	even	6
756.4.x.a.629.5		48	63.13	odd	6
756.4.x.a.629.20		48	9.4	even	3
2268.4.f.a.1133.9		48	1.1	even	1	trivial
2268.4.f.a.1133.10		48	21.20	even	2	inner
2268.4.f.a.1133.39		48	7.6	odd	2	inner
2268.4.f.a.1133.40		48	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2268.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-12.0714 q^{5} +(14.8834 + 11.0221i) q^{7} +O(q^{10})\)	\(q-12.0714 q^{5} +(14.8834 + 11.0221i) q^{7} +0.00255549i q^{11} -7.00719i q^{13} -28.3143 q^{17} +49.1973i q^{19} -51.3482i q^{23} +20.7189 q^{25} +113.099i q^{29} +32.8248i q^{31} +(-179.663 - 133.052i) q^{35} -101.961 q^{37} +22.4898 q^{41} +454.406 q^{43} -462.520 q^{47} +(100.029 + 328.090i) q^{49} -567.586i q^{53} -0.0308484i q^{55} +291.308 q^{59} +683.586i q^{61} +84.5867i q^{65} -539.560 q^{67} +307.517i q^{71} -495.192i q^{73} +(-0.0281668 + 0.0380343i) q^{77} +649.730 q^{79} -1131.16 q^{83} +341.794 q^{85} +130.217 q^{89} +(77.2337 - 104.291i) q^{91} -593.880i q^{95} +1455.39i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{7}+O(q^{10}) \) 48 * q - 12 * q^7	\( 48 q - 12 q^{7} + 1200 q^{25} + 336 q^{37} - 168 q^{43} - 636 q^{49} + 1176 q^{67} - 408 q^{79} + 720 q^{85} - 1080 q^{91}+O(q^{100}) \) 48 * q - 12 * q^7 + 1200 * q^25 + 336 * q^37 - 168 * q^43 - 636 * q^49 + 1176 * q^67 - 408 * q^79 + 720 * q^85 - 1080 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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