LMFDB - Embedded newform 4140.2.f.d.829.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.12
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.d.829.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.274117 + 2.21920i) q^{5} +0.615120i q^{7} +O(q^{10})$	$q+(-0.274117 + 2.21920i) q^{5} +0.615120i q^{7} -5.98437 q^{11} -1.31179i q^{13} -5.22391i q^{17} +6.47909 q^{19} -1.00000i q^{23} +(-4.84972 - 1.21664i) q^{25} -3.81196 q^{29} +2.89560 q^{31} +(-1.36508 - 0.168615i) q^{35} +3.05956i q^{37} +8.67483 q^{41} +2.78753i q^{43} -6.52870i q^{47} +6.62163 q^{49} -8.11595i q^{53} +(1.64042 - 13.2805i) q^{55} -8.93735 q^{59} -1.29231 q^{61} +(2.91112 + 0.359583i) q^{65} -11.1665i q^{67} +15.0033 q^{71} +16.1444i q^{73} -3.68110i q^{77} +1.59081 q^{79} -7.29587i q^{83} +(11.5929 + 1.43196i) q^{85} +12.8138 q^{89} +0.806908 q^{91} +(-1.77603 + 14.3784i) q^{95} -17.2881i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100}) $ 24 * q + 8 * q^19 - 8 * q^25 - 12 * q^31 - 28 * q^49 - 16 * q^55 - 16 * q^61 + 8 * q^79 + 12 * q^85 - 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	−0.274117	+	2.21920i	−0.122589	+	0.992458i
$5$	−0.274117	+	2.21920i	−0.122589	+	0.992458i
$6$		0			0
$7$			0.615120i			0.232494i	0.993220	+	0.116247i	$0.0370864\pi$
$7$			0.615120i			0.232494i	−0.993220	+	0.116247i	$0.962914\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.98437			−1.80435			−0.902177	−	0.431366i	$-0.858032\pi$
$11$	−5.98437			−1.80435			−0.902177	+	0.431366i	$0.858032\pi$
$12$		0			0
$13$		−	1.31179i		−	0.363825i	−0.983315	−	0.181912i	$-0.941771\pi$
$13$		−	1.31179i		−	0.363825i	0.983315	−	0.181912i	$-0.0582287\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.22391i		−	1.26698i	−0.773749	−	0.633492i	$-0.781621\pi$
$17$		−	5.22391i		−	1.26698i	0.773749	−	0.633492i	$-0.218379\pi$
$18$		0			0
$19$	6.47909			1.48641			0.743203	−	0.669066i	$-0.233306\pi$
$19$	6.47909			1.48641			0.743203	+	0.669066i	$0.233306\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.84972	−	1.21664i	−0.969944	−	0.243328i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.81196			−0.707863			−0.353932	−	0.935271i	$-0.615155\pi$
$29$	−3.81196			−0.707863			−0.353932	+	0.935271i	$0.615155\pi$
$30$		0			0
$31$	2.89560			0.520065			0.260033	−	0.965600i	$-0.416267\pi$
$31$	2.89560			0.520065			0.260033	+	0.965600i	$0.416267\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.36508	−	0.168615i	−0.230740	−	0.0285011i
$36$		0			0
$37$			3.05956i			0.502988i	0.967859	+	0.251494i	$0.0809219\pi$
$37$			3.05956i			0.502988i	−0.967859	+	0.251494i	$0.919078\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.67483			1.35478			0.677390	−	0.735624i	$-0.263111\pi$
$41$	8.67483			1.35478			0.677390	+	0.735624i	$0.263111\pi$
$42$		0			0
$43$			2.78753i			0.425094i	0.977151	+	0.212547i	$0.0681759\pi$
$43$			2.78753i			0.425094i	−0.977151	+	0.212547i	$0.931824\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.52870i		−	0.952309i	−0.879362	−	0.476154i	$-0.842030\pi$
$47$		−	6.52870i		−	0.952309i	0.879362	−	0.476154i	$-0.157970\pi$
$48$		0			0
$49$	6.62163			0.945947
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.11595i		−	1.11481i	−0.830240	−	0.557405i	$-0.811797\pi$
$53$		−	8.11595i		−	1.11481i	0.830240	−	0.557405i	$-0.188203\pi$
$54$		0			0
$55$	1.64042	−	13.2805i	0.221194	−	1.79075i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.93735			−1.16354			−0.581772	−	0.813352i	$-0.697640\pi$
$59$	−8.93735			−1.16354			−0.581772	+	0.813352i	$0.697640\pi$
$60$		0			0
$61$	−1.29231			−0.165463			−0.0827315	−	0.996572i	$-0.526364\pi$
$61$	−1.29231			−0.165463			−0.0827315	+	0.996572i	$0.526364\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.91112	+	0.359583i	0.361080	+	0.0446008i
$66$		0			0
$67$		−	11.1665i		−	1.36420i	−0.731258	−	0.682100i	$-0.761067\pi$
$67$		−	11.1665i		−	1.36420i	0.731258	−	0.682100i	$-0.238933\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	15.0033			1.78056			0.890280	−	0.455414i	$-0.150509\pi$
$71$	15.0033			1.78056			0.890280	+	0.455414i	$0.150509\pi$
$72$		0			0
$73$			16.1444i			1.88956i	0.327699	+	0.944782i	$0.393727\pi$
$73$			16.1444i			1.88956i	−0.327699	+	0.944782i	$0.606273\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.68110i		−	0.419501i
$78$		0			0
$79$	1.59081			0.178980			0.0894900	−	0.995988i	$-0.471476\pi$
$79$	1.59081			0.178980			0.0894900	+	0.995988i	$0.471476\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	7.29587i		−	0.800826i	−0.916335	−	0.400413i	$-0.868867\pi$
$83$		−	7.29587i		−	0.800826i	0.916335	−	0.400413i	$-0.131133\pi$
$84$		0			0
$85$	11.5929	+	1.43196i	1.25743	+	0.155318i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.8138			1.35826			0.679132	−	0.734016i	$-0.262356\pi$
$89$	12.8138			1.35826			0.679132	+	0.734016i	$0.262356\pi$
$90$		0			0
$91$	0.806908			0.0845869
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.77603	+	14.3784i	−0.182217	+	1.47519i
$96$		0			0
$97$		−	17.2881i		−	1.75534i	−0.479264	−	0.877671i	$-0.659096\pi$
$97$		−	17.2881i		−	1.75534i	0.479264	−	0.877671i	$-0.340904\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.0366			−1.19769			−0.598843	−	0.800866i	$-0.704373\pi$
$101$	−12.0366			−1.19769			−0.598843	+	0.800866i	$0.704373\pi$
$102$		0			0
$103$			16.2834i			1.60445i	0.597023	+	0.802224i	$0.296350\pi$
$103$			16.2834i			1.60445i	−0.597023	+	0.802224i	$0.703650\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			9.27352i			0.896505i	0.893907	+	0.448252i	$0.147953\pi$
$107$			9.27352i			0.896505i	−0.893907	+	0.448252i	$0.852047\pi$
$108$		0			0
$109$	13.2547			1.26957			0.634786	−	0.772688i	$-0.281088\pi$
$109$	13.2547			1.26957			0.634786	+	0.772688i	$0.281088\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			10.7077i			1.00729i	0.863909	+	0.503647i	$0.168009\pi$
$113$			10.7077i			1.00729i	−0.863909	+	0.503647i	$0.831991\pi$
$114$		0			0
$115$	2.21920	+	0.274117i	0.206942	+	0.0255615i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.21333			0.294566
$120$		0			0
$121$	24.8126			2.25569
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.02937	−	10.4290i	0.360397	−	0.932799i
$126$		0			0
$127$			15.4392i			1.37001i	0.728538	+	0.685006i	$0.240200\pi$
$127$			15.4392i			1.37001i	−0.728538	+	0.685006i	$0.759800\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.639958			−0.0559134			−0.0279567	−	0.999609i	$-0.508900\pi$
$131$	−0.639958			−0.0559134			−0.0279567	+	0.999609i	$0.508900\pi$
$132$		0			0
$133$			3.98542i			0.345580i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			16.2886i			1.39163i	0.718220	+	0.695816i	$0.244957\pi$
$137$			16.2886i			1.39163i	−0.718220	+	0.695816i	$0.755043\pi$
$138$		0			0
$139$	10.4692			0.887988			0.443994	−	0.896030i	$-0.353561\pi$
$139$	10.4692			0.887988			0.443994	+	0.896030i	$0.353561\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			7.85022i			0.656469i
$144$		0			0
$145$	1.04492	−	8.45951i	0.0867761	−	0.702524i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.0451			1.15062			0.575308	−	0.817937i	$-0.304882\pi$
$149$	14.0451			1.15062			0.575308	+	0.817937i	$0.304882\pi$
$150$		0			0
$151$	−1.97765			−0.160939			−0.0804693	−	0.996757i	$-0.525642\pi$
$151$	−1.97765			−0.160939			−0.0804693	+	0.996757i	$0.525642\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.793733	+	6.42592i	−0.0637542	+	0.516142i
$156$		0			0
$157$		−	22.4924i		−	1.79509i	−0.440921	−	0.897546i	$-0.645348\pi$
$157$		−	22.4924i		−	1.79509i	0.440921	−	0.897546i	$-0.354652\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.615120			0.0484783
$162$		0			0
$163$		−	22.7909i		−	1.78512i	−0.450924	−	0.892562i	$-0.648906\pi$
$163$		−	22.7909i		−	1.78512i	0.450924	−	0.892562i	$-0.351094\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			16.4369i			1.27193i	0.771718	+	0.635964i	$0.219397\pi$
$167$			16.4369i			1.27193i	−0.771718	+	0.635964i	$0.780603\pi$
$168$		0			0
$169$	11.2792			0.867632
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	14.9290i		−	1.13503i	−0.823363	−	0.567514i	$-0.807905\pi$
$173$		−	14.9290i		−	1.13503i	0.823363	−	0.567514i	$-0.192095\pi$
$174$		0			0
$175$	0.748381	−	2.98316i	0.0565723	−	0.225506i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−14.8548			−1.11030			−0.555151	−	0.831750i	$-0.687340\pi$
$179$	−14.8548			−1.11030			−0.555151	+	0.831750i	$0.687340\pi$
$180$		0			0
$181$	21.4238			1.59242			0.796211	−	0.605019i	$-0.206835\pi$
$181$	21.4238			1.59242			0.796211	+	0.605019i	$0.206835\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.78978	−	0.838677i	−0.499194	−	0.0616607i
$186$		0			0
$187$			31.2618i			2.28609i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.03651			−0.581502			−0.290751	−	0.956799i	$-0.593905\pi$
$191$	−8.03651			−0.581502			−0.290751	+	0.956799i	$0.593905\pi$
$192$		0			0
$193$			9.96609i			0.717375i	0.933458	+	0.358687i	$0.116776\pi$
$193$			9.96609i			0.717375i	−0.933458	+	0.358687i	$0.883224\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	16.6755i		−	1.18808i	−0.804436	−	0.594039i	$-0.797533\pi$
$197$		−	16.6755i		−	1.18808i	0.804436	−	0.594039i	$-0.202467\pi$
$198$		0			0
$199$	−23.5993			−1.67291			−0.836454	−	0.548037i	$-0.815375\pi$
$199$	−23.5993			−1.67291			−0.836454	+	0.548037i	$0.815375\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	2.34481i		−	0.164574i
$204$		0			0
$205$	−2.37792	+	19.2512i	−0.166081	+	1.34456i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−38.7733			−2.68200
$210$		0			0
$211$	−8.18416			−0.563420			−0.281710	−	0.959500i	$-0.590902\pi$
$211$	−8.18416			−0.563420			−0.281710	+	0.959500i	$0.590902\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.18609	−	0.764109i	−0.421888	−	0.0521118i
$216$		0			0
$217$			1.78114i			0.120912i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.85266			−0.460960
$222$		0			0
$223$		−	19.8895i		−	1.33190i	−0.745996	−	0.665950i	$-0.768026\pi$
$223$		−	19.8895i		−	1.33190i	0.745996	−	0.665950i	$-0.231974\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			24.3204i			1.61420i	0.590414	+	0.807101i	$0.298965\pi$
$227$			24.3204i			1.61420i	−0.590414	+	0.807101i	$0.701035\pi$
$228$		0			0
$229$	13.3872			0.884650			0.442325	−	0.896855i	$-0.354154\pi$
$229$	13.3872			0.884650			0.442325	+	0.896855i	$0.354154\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.06679i		−	0.135400i	−0.997706	−	0.0677001i	$-0.978434\pi$
$233$		−	2.06679i		−	0.135400i	0.997706	−	0.0677001i	$-0.0215661\pi$
$234$		0			0
$235$	14.4885	+	1.78963i	0.945126	+	0.116742i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	20.8529			1.34886			0.674430	−	0.738338i	$-0.264389\pi$
$239$	20.8529			1.34886			0.674430	+	0.738338i	$0.264389\pi$
$240$		0			0
$241$	15.7172			1.01243			0.506216	−	0.862407i	$-0.331044\pi$
$241$	15.7172			1.01243			0.506216	+	0.862407i	$0.331044\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.81510	+	14.6947i	−0.115963	+	0.938812i
$246$		0			0
$247$		−	8.49920i		−	0.540791i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.55695			−0.224513			−0.112256	−	0.993679i	$-0.535808\pi$
$251$	−3.55695			−0.224513			−0.112256	+	0.993679i	$0.535808\pi$
$252$		0			0
$253$			5.98437i			0.376234i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			7.19995i			0.449121i	0.974460	+	0.224560i	$0.0720946\pi$
$257$			7.19995i			0.449121i	−0.974460	+	0.224560i	$0.927905\pi$
$258$		0			0
$259$	−1.88200			−0.116942
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.1503i		−	1.36585i	−0.730490	−	0.682924i	$-0.760708\pi$
$263$		−	22.1503i		−	1.36585i	0.730490	−	0.682924i	$-0.239292\pi$
$264$		0			0
$265$	18.0109	+	2.22472i	1.10640	+	0.136663i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.18838			−0.0724570			−0.0362285	−	0.999344i	$-0.511534\pi$
$269$	−1.18838			−0.0724570			−0.0362285	+	0.999344i	$0.511534\pi$
$270$		0			0
$271$	−10.4443			−0.634443			−0.317222	−	0.948351i	$-0.602750\pi$
$271$	−10.4443			−0.634443			−0.317222	+	0.948351i	$0.602750\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	29.0225	+	7.28083i	1.75012	+	0.439051i
$276$		0			0
$277$		−	17.9541i		−	1.07876i	−0.842064	−	0.539378i	$-0.818659\pi$
$277$		−	17.9541i		−	1.07876i	0.842064	−	0.539378i	$-0.181341\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	21.5059			1.28293			0.641466	−	0.767151i	$-0.278326\pi$
$281$	21.5059			1.28293			0.641466	+	0.767151i	$0.278326\pi$
$282$		0			0
$283$		−	11.0567i		−	0.657255i	−0.944460	−	0.328628i	$-0.893414\pi$
$283$		−	11.0567i		−	0.657255i	0.944460	−	0.328628i	$-0.106586\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			5.33606i			0.314978i
$288$		0			0
$289$	−10.2892			−0.605249
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	20.6040i		−	1.20370i	−0.798611	−	0.601848i	$-0.794431\pi$
$293$		−	20.6040i		−	1.20370i	0.798611	−	0.601848i	$-0.205569\pi$
$294$		0			0
$295$	2.44988	−	19.8338i	0.142637	−	1.15477i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.31179			−0.0758627
$300$		0			0
$301$	−1.71466			−0.0988316
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.354243	−	2.86789i	0.0202839	−	0.164215i
$306$		0			0
$307$			18.0065i			1.02768i	0.857885	+	0.513842i	$0.171778\pi$
$307$			18.0065i			1.02768i	−0.857885	+	0.513842i	$0.828222\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.2370			0.750603			0.375301	−	0.926903i	$-0.377539\pi$
$311$	13.2370			0.750603			0.375301	+	0.926903i	$0.377539\pi$
$312$		0			0
$313$		−	26.6691i		−	1.50743i	−0.657203	−	0.753714i	$-0.728260\pi$
$313$		−	26.6691i		−	1.50743i	0.657203	−	0.753714i	$-0.271740\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	10.2856i		−	0.577695i	−0.957375	−	0.288847i	$-0.906728\pi$
$317$		−	10.2856i		−	0.577695i	0.957375	−	0.288847i	$-0.0932720\pi$
$318$		0			0
$319$	22.8122			1.27724
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	33.8462i		−	1.88325i
$324$		0			0
$325$	−1.59598	+	6.36181i	−0.0885289	+	0.352889i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.01594			0.221406
$330$		0			0
$331$	25.6054			1.40740			0.703700	−	0.710498i	$-0.251530\pi$
$331$	25.6054			1.40740			0.703700	+	0.710498i	$0.251530\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	24.7806	+	3.06092i	1.35391	+	0.167236i
$336$		0			0
$337$			4.79102i			0.260984i	0.991449	+	0.130492i	$0.0416556\pi$
$337$			4.79102i			0.260984i	−0.991449	+	0.130492i	$0.958344\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−17.3283			−0.938382
$342$		0			0
$343$			8.37894i			0.452420i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			30.1722i			1.61973i	0.586616	+	0.809865i	$0.300460\pi$
$347$			30.1722i			1.61973i	−0.586616	+	0.809865i	$0.699540\pi$
$348$		0			0
$349$	−1.56320			−0.0836761			−0.0418380	−	0.999124i	$-0.513321\pi$
$349$	−1.56320			−0.0836761			−0.0418380	+	0.999124i	$0.513321\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			17.8462i			0.949858i	0.880024	+	0.474929i	$0.157526\pi$
$353$			17.8462i			0.949858i	−0.880024	+	0.474929i	$0.842474\pi$
$354$		0			0
$355$	−4.11265	+	33.2953i	−0.218277	+	1.76713i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	21.6438			1.14232			0.571159	−	0.820840i	$-0.306494\pi$
$359$	21.6438			1.14232			0.571159	+	0.820840i	$0.306494\pi$
$360$		0			0
$361$	22.9787			1.20940
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−35.8278	−	4.42547i	−1.87531	−	0.231640i
$366$		0			0
$367$		−	3.44815i		−	0.179992i	−0.995942	−	0.0899960i	$-0.971315\pi$
$367$		−	3.44815i		−	0.179992i	0.995942	−	0.0899960i	$-0.0286854\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.99228			0.259186
$372$		0			0
$373$		−	29.8283i		−	1.54445i	−0.635349	−	0.772225i	$-0.719144\pi$
$373$		−	29.8283i		−	1.54445i	0.635349	−	0.772225i	$-0.280856\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			5.00048i			0.257538i
$378$		0			0
$379$	16.2121			0.832759			0.416380	−	0.909191i	$-0.363299\pi$
$379$	16.2121			0.832759			0.416380	+	0.909191i	$0.363299\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	16.9807i		−	0.867675i	−0.900991	−	0.433837i	$-0.857159\pi$
$383$		−	16.9807i		−	0.867675i	0.900991	−	0.433837i	$-0.142841\pi$
$384$		0			0
$385$	8.16912	+	1.00905i	0.416337	+	0.0514261i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.3955			1.13550			0.567749	−	0.823202i	$-0.307815\pi$
$389$	22.3955			1.13550			0.567749	+	0.823202i	$0.307815\pi$
$390$		0			0
$391$	−5.22391			−0.264185
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.436068	+	3.53033i	−0.0219410	+	0.177630i
$396$		0			0
$397$		−	5.16344i		−	0.259145i	−0.991570	−	0.129573i	$-0.958639\pi$
$397$		−	5.16344i		−	0.259145i	0.991570	−	0.129573i	$-0.0413606\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.3874			−0.518722			−0.259361	−	0.965780i	$-0.583512\pi$
$401$	−10.3874			−0.518722			−0.259361	+	0.965780i	$0.583512\pi$
$402$		0			0
$403$		−	3.79841i		−	0.189212i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	18.3095i		−	0.907569i
$408$		0			0
$409$	−20.3243			−1.00497			−0.502486	−	0.864586i	$-0.667581\pi$
$409$	−20.3243			−1.00497			−0.502486	+	0.864586i	$0.667581\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	5.49754i		−	0.270516i
$414$		0			0
$415$	16.1910	+	1.99992i	0.794786	+	0.0981723i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.91981			−0.386908			−0.193454	−	0.981109i	$-0.561969\pi$
$419$	−7.91981			−0.386908			−0.193454	+	0.981109i	$0.561969\pi$
$420$		0			0
$421$	7.58961			0.369895			0.184948	−	0.982748i	$-0.440788\pi$
$421$	7.58961			0.369895			0.184948	+	0.982748i	$0.440788\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.35563	+	25.3345i	−0.308293	+	1.22890i
$426$		0			0
$427$		−	0.794924i		−	0.0384691i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−9.65376			−0.465005			−0.232503	−	0.972596i	$-0.574691\pi$
$431$	−9.65376			−0.465005			−0.232503	+	0.972596i	$0.574691\pi$
$432$		0			0
$433$		−	14.0796i		−	0.676622i	−0.941034	−	0.338311i	$-0.890144\pi$
$433$		−	14.0796i		−	0.676622i	0.941034	−	0.338311i	$-0.109856\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	6.47909i		−	0.309937i
$438$		0			0
$439$	−30.3743			−1.44969			−0.724843	−	0.688914i	$-0.758088\pi$
$439$	−30.3743			−1.44969			−0.724843	+	0.688914i	$0.758088\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	29.5307i		−	1.40305i	−0.712646	−	0.701524i	$-0.752503\pi$
$443$		−	29.5307i		−	1.40305i	0.712646	−	0.701524i	$-0.247497\pi$
$444$		0			0
$445$	−3.51249	+	28.4365i	−0.166508	+	1.34802i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.10463			−0.288095			−0.144048	−	0.989571i	$-0.546012\pi$
$449$	−6.10463			−0.288095			−0.144048	+	0.989571i	$0.546012\pi$
$450$		0			0
$451$	−51.9134			−2.44450
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.221187	+	1.79069i	−0.0103694	+	0.0839489i
$456$		0			0
$457$			20.0009i			0.935602i	0.883834	+	0.467801i	$0.154954\pi$
$457$			20.0009i			0.935602i	−0.883834	+	0.467801i	$0.845046\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.0209			0.653017			0.326509	−	0.945194i	$-0.394128\pi$
$461$	14.0209			0.653017			0.326509	+	0.945194i	$0.394128\pi$
$462$		0			0
$463$		−	7.32433i		−	0.340390i	−0.985410	−	0.170195i	$-0.945560\pi$
$463$		−	7.32433i		−	0.340390i	0.985410	−	0.170195i	$-0.0544398\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	19.8644i		−	0.919213i	−0.888123	−	0.459606i	$-0.847991\pi$
$467$		−	19.8644i		−	0.919213i	0.888123	−	0.459606i	$-0.152009\pi$
$468$		0			0
$469$	6.86872			0.317168
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	16.6816i		−	0.767020i
$474$		0			0
$475$	−31.4218	−	7.88274i	−1.44173	−	0.361685i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.6308			0.485732			0.242866	−	0.970060i	$-0.421912\pi$
$479$	10.6308			0.485732			0.242866	+	0.970060i	$0.421912\pi$
$480$		0			0
$481$	4.01349			0.183000
$482$		0			0
$483$		0			0
$484$		0			0
$485$	38.3658	+	4.73897i	1.74210	+	0.215185i
$486$		0			0
$487$			30.9923i			1.40439i	0.711982	+	0.702197i	$0.247798\pi$
$487$			30.9923i			1.40439i	−0.711982	+	0.702197i	$0.752202\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.57209			−0.386853			−0.193426	−	0.981115i	$-0.561960\pi$
$491$	−8.57209			−0.386853			−0.193426	+	0.981115i	$0.561960\pi$
$492$		0			0
$493$			19.9133i			0.896851i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			9.22881i			0.413969i
$498$		0			0
$499$	3.35700			0.150280			0.0751400	−	0.997173i	$-0.476060\pi$
$499$	3.35700			0.150280			0.0751400	+	0.997173i	$0.476060\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	30.4182i		−	1.35628i	−0.734934	−	0.678139i	$-0.762787\pi$
$503$		−	30.4182i		−	1.35628i	0.734934	−	0.678139i	$-0.237213\pi$
$504$		0			0
$505$	3.29944	−	26.7116i	0.146823	−	1.18865i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−29.0135			−1.28600			−0.643000	−	0.765866i	$-0.722311\pi$
$509$	−29.0135			−1.28600			−0.643000	+	0.765866i	$0.722311\pi$
$510$		0			0
$511$	−9.93078			−0.439312
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−36.1361	−	4.46355i	−1.59235	−	0.196687i
$516$		0			0
$517$			39.0701i			1.71830i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.5383			−0.549313			−0.274656	−	0.961542i	$-0.588564\pi$
$521$	−12.5383			−0.549313			−0.274656	+	0.961542i	$0.588564\pi$
$522$		0			0
$523$		−	36.5882i		−	1.59989i	−0.600073	−	0.799945i	$-0.704862\pi$
$523$		−	36.5882i		−	1.59989i	0.600073	−	0.799945i	$-0.295138\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	15.1264i		−	0.658914i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	11.3795i		−	0.492903i
$534$		0			0
$535$	−20.5798	−	2.54203i	−0.889743	−	0.109901i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−39.6262			−1.70682
$540$		0			0
$541$	−31.7280			−1.36409			−0.682047	−	0.731308i	$-0.738910\pi$
$541$	−31.7280			−1.36409			−0.682047	+	0.731308i	$0.738910\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.63335	+	29.4149i	−0.155635	+	1.26000i
$546$		0			0
$547$			0.532055i			0.0227490i	0.999935	+	0.0113745i	$0.00362070\pi$
$547$			0.532055i			0.0227490i	−0.999935	+	0.0113745i	$0.996379\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−24.6980			−1.05217
$552$		0			0
$553$			0.978539i			0.0416117i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	11.0877i		−	0.469801i	−0.972019	−	0.234901i	$-0.924524\pi$
$557$		−	11.0877i		−	0.469801i	0.972019	−	0.234901i	$-0.0754765\pi$
$558$		0			0
$559$	3.65665			0.154660
$560$		0			0
$561$		0			0
$562$		0			0
$563$			5.41332i			0.228144i	0.993472	+	0.114072i	$0.0363895\pi$
$563$			5.41332i			0.228144i	−0.993472	+	0.114072i	$0.963610\pi$
$564$		0			0
$565$	−23.7625	−	2.93516i	−0.999697	−	0.123483i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.7784			0.912998			0.456499	−	0.889724i	$-0.349103\pi$
$569$	21.7784			0.912998			0.456499	+	0.889724i	$0.349103\pi$
$570$		0			0
$571$	−1.80463			−0.0755216			−0.0377608	−	0.999287i	$-0.512022\pi$
$571$	−1.80463			−0.0755216			−0.0377608	+	0.999287i	$0.512022\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.21664	+	4.84972i	−0.0507375	+	0.202247i
$576$		0			0
$577$		−	18.9643i		−	0.789495i	−0.918790	−	0.394748i	$-0.870832\pi$
$577$		−	18.9643i		−	0.789495i	0.918790	−	0.394748i	$-0.129168\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.48784			0.186187
$582$		0			0
$583$			48.5688i			2.01151i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			11.2880i			0.465904i	0.972488	+	0.232952i	$0.0748385\pi$
$587$			11.2880i			0.465904i	−0.972488	+	0.232952i	$0.925161\pi$
$588$		0			0
$589$	18.7609			0.773028
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	14.4723i		−	0.594306i	−0.954830	−	0.297153i	$-0.903963\pi$
$593$		−	14.4723i		−	0.594306i	0.954830	−	0.297153i	$-0.0960371\pi$
$594$		0			0
$595$	−0.880829	+	7.13104i	−0.0361105	+	0.292344i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.97328			0.284920			0.142460	−	0.989801i	$-0.454499\pi$
$599$	6.97328			0.284920			0.142460	+	0.989801i	$0.454499\pi$
$600$		0			0
$601$	43.3811			1.76955			0.884776	−	0.466017i	$-0.154311\pi$
$601$	43.3811			1.76955			0.884776	+	0.466017i	$0.154311\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.80157	+	55.0643i	−0.276523	+	2.23868i
$606$		0			0
$607$		−	43.8861i		−	1.78128i	−0.454709	−	0.890640i	$-0.650257\pi$
$607$		−	43.8861i		−	1.78128i	0.454709	−	0.890640i	$-0.349743\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.56427			−0.346473
$612$		0			0
$613$		−	0.202301i		−	0.00817086i	−0.999992	−	0.00408543i	$-0.998700\pi$
$613$		−	0.202301i		−	0.00817086i	0.999992	−	0.00408543i	$-0.00130044\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	27.2791i		−	1.09822i	−0.835752	−	0.549108i	$-0.814968\pi$
$617$		−	27.2791i		−	1.09822i	0.835752	−	0.549108i	$-0.185032\pi$
$618$		0			0
$619$	−40.5971			−1.63173			−0.815867	−	0.578240i	$-0.803740\pi$
$619$	−40.5971			−1.63173			−0.815867	+	0.578240i	$0.803740\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			7.88205i			0.315788i
$624$		0			0
$625$	22.0396	+	11.8007i	0.881583	+	0.472030i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15.9829			0.637278
$630$		0			0
$631$	−12.7891			−0.509127			−0.254564	−	0.967056i	$-0.581932\pi$
$631$	−12.7891			−0.509127			−0.254564	+	0.967056i	$0.581932\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−34.2628	−	4.23216i	−1.35968	−	0.167948i
$636$		0			0
$637$		−	8.68617i		−	0.344159i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.6207			0.498487			0.249243	−	0.968441i	$-0.419818\pi$
$641$	12.6207			0.498487			0.249243	+	0.968441i	$0.419818\pi$
$642$		0			0
$643$			21.3299i			0.841168i	0.907254	+	0.420584i	$0.138175\pi$
$643$			21.3299i			0.841168i	−0.907254	+	0.420584i	$0.861825\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			3.55402i			0.139723i	0.997557	+	0.0698614i	$0.0222557\pi$
$647$			3.55402i			0.139723i	−0.997557	+	0.0698614i	$0.977744\pi$
$648$		0			0
$649$	53.4844			2.09945
$650$		0			0
$651$		0			0
$652$		0			0
$653$			11.3608i			0.444583i	0.974980	+	0.222291i	$0.0713536\pi$
$653$			11.3608i			0.444583i	−0.974980	+	0.222291i	$0.928646\pi$
$654$		0			0
$655$	0.175423	−	1.42020i	0.00685436	−	0.0554917i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	33.3654			1.29973			0.649865	−	0.760050i	$-0.274825\pi$
$659$	33.3654			1.29973			0.649865	+	0.760050i	$0.274825\pi$
$660$		0			0
$661$	16.8236			0.654364			0.327182	−	0.944961i	$-0.393901\pi$
$661$	16.8236			0.654364			0.327182	+	0.944961i	$0.393901\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.84446	−	1.09247i	−0.342973	−	0.0423642i
$666$		0			0
$667$			3.81196i			0.147600i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7.73364			0.298554
$672$		0			0
$673$		−	3.68855i		−	0.142183i	−0.997470	−	0.0710916i	$-0.977352\pi$
$673$		−	3.68855i		−	0.142183i	0.997470	−	0.0710916i	$-0.0226483\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			7.30588i			0.280788i	0.990096	+	0.140394i	$0.0448369\pi$
$677$			7.30588i			0.280788i	−0.990096	+	0.140394i	$0.955163\pi$
$678$		0			0
$679$	10.6343			0.408106
$680$		0			0
$681$		0			0
$682$		0			0
$683$			23.3440i			0.893232i	0.894726	+	0.446616i	$0.147371\pi$
$683$			23.3440i			0.893232i	−0.894726	+	0.446616i	$0.852629\pi$
$684$		0			0
$685$	−36.1478	−	4.46499i	−1.38114	−	0.170598i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10.6464			−0.405596
$690$		0			0
$691$	−30.1048			−1.14524			−0.572621	−	0.819820i	$-0.694073\pi$
$691$	−30.1048			−1.14524			−0.572621	+	0.819820i	$0.694073\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.86979	+	23.2333i	−0.108857	+	0.881290i
$696$		0			0
$697$		−	45.3165i		−	1.71649i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14.8106			0.559389			0.279695	−	0.960089i	$-0.409767\pi$
$701$	14.8106			0.559389			0.279695	+	0.960089i	$0.409767\pi$
$702$		0			0
$703$			19.8232i			0.747645i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	7.40395i		−	0.278454i
$708$		0			0
$709$	−8.49889			−0.319183			−0.159591	−	0.987183i	$-0.551018\pi$
$709$	−8.49889			−0.319183			−0.159591	+	0.987183i	$0.551018\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	2.89560i		−	0.108441i
$714$		0			0
$715$	−17.4212	−	2.15188i	−0.651517	−	0.0804757i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−48.3151			−1.80185			−0.900924	−	0.433977i	$-0.857110\pi$
$719$	−48.3151			−1.80185			−0.900924	+	0.433977i	$0.857110\pi$
$720$		0			0
$721$	−10.0162			−0.373024
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.4869	+	4.63779i	0.686587	+	0.172243i
$726$		0			0
$727$		−	7.49278i		−	0.277892i	−0.990300	−	0.138946i	$-0.955629\pi$
$727$		−	7.49278i		−	0.277892i	0.990300	−	0.138946i	$-0.0443714\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	14.5618			0.538587
$732$		0			0
$733$		−	38.3132i		−	1.41513i	−0.706647	−	0.707566i	$-0.749793\pi$
$733$		−	38.3132i		−	1.41513i	0.706647	−	0.707566i	$-0.250207\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			66.8242i			2.46150i
$738$		0			0
$739$	−43.0010			−1.58182			−0.790909	−	0.611933i	$-0.790392\pi$
$739$	−43.0010			−1.58182			−0.790909	+	0.611933i	$0.790392\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			5.91637i			0.217051i	0.994094	+	0.108525i	$0.0346129\pi$
$743$			5.91637i			0.217051i	−0.994094	+	0.108525i	$0.965387\pi$
$744$		0			0
$745$	−3.84999	+	31.1688i	−0.141053	+	1.14194i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.70433			−0.208432
$750$		0			0
$751$	−17.4950			−0.638403			−0.319201	−	0.947687i	$-0.603415\pi$
$751$	−17.4950			−0.638403			−0.319201	+	0.947687i	$0.603415\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.542106	−	4.38880i	0.0197293	−	0.159725i
$756$		0			0
$757$			23.6493i			0.859547i	0.902937	+	0.429773i	$0.141407\pi$
$757$			23.6493i			0.859547i	−0.902937	+	0.429773i	$0.858593\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9.31465			0.337656			0.168828	−	0.985646i	$-0.446002\pi$
$761$	9.31465			0.337656			0.168828	+	0.985646i	$0.446002\pi$
$762$		0			0
$763$			8.15325i			0.295168i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			11.7239i			0.423326i
$768$		0			0
$769$	31.9156			1.15091			0.575453	−	0.817835i	$-0.304826\pi$
$769$	31.9156			1.15091			0.575453	+	0.817835i	$0.304826\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			4.65567i			0.167453i	0.996489	+	0.0837263i	$0.0266822\pi$
$773$			4.65567i			0.167453i	−0.996489	+	0.0837263i	$0.973318\pi$
$774$		0			0
$775$	−14.0428	−	3.52291i	−0.504434	−	0.126547i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	56.2050			2.01375
$780$		0			0
$781$	−89.7850			−3.21276
$782$		0			0
$783$		0			0
$784$		0			0
$785$	49.9153	+	6.16556i	1.78155	+	0.220058i
$786$		0			0
$787$			3.10123i			0.110547i	0.998471	+	0.0552734i	$0.0176030\pi$
$787$			3.10123i			0.110547i	−0.998471	+	0.0552734i	$0.982397\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.58652			−0.234190
$792$		0			0
$793$			1.69523i			0.0601995i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			22.5887i			0.800134i	0.916486	+	0.400067i	$0.131013\pi$
$797$			22.5887i			0.800134i	−0.916486	+	0.400067i	$0.868987\pi$
$798$		0			0
$799$	−34.1053			−1.20656
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	96.6143i		−	3.40944i
$804$		0			0
$805$	−0.168615	+	1.36508i	−0.00594289	+	0.0481126i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.9807			0.667327			0.333663	−	0.942692i	$-0.391715\pi$
$809$	18.9807			0.667327			0.333663	+	0.942692i	$0.391715\pi$
$810$		0			0
$811$	7.34794			0.258021			0.129010	−	0.991643i	$-0.458820\pi$
$811$	7.34794			0.258021			0.129010	+	0.991643i	$0.458820\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	50.5777	+	6.24738i	1.77166	+	0.218836i
$816$		0			0
$817$			18.0607i			0.631862i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	35.9606			1.25503			0.627517	−	0.778603i	$-0.284071\pi$
$821$	35.9606			1.25503			0.627517	+	0.778603i	$0.284071\pi$
$822$		0			0
$823$			12.3877i			0.431810i	0.976414	+	0.215905i	$0.0692701\pi$
$823$			12.3877i			0.431810i	−0.976414	+	0.215905i	$0.930730\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			40.5239i			1.40915i	0.709628	+	0.704577i	$0.248863\pi$
$827$			40.5239i			1.40915i	−0.709628	+	0.704577i	$0.751137\pi$
$828$		0			0
$829$	−13.2131			−0.458910			−0.229455	−	0.973319i	$-0.573694\pi$
$829$	−13.2131			−0.458910			−0.229455	+	0.973319i	$0.573694\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	34.5908i		−	1.19850i
$834$		0			0
$835$	−36.4769	−	4.50564i	−1.26234	−	0.155924i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.2715			−0.768898			−0.384449	−	0.923146i	$-0.625609\pi$
$839$	−22.2715			−0.768898			−0.384449	+	0.923146i	$0.625609\pi$
$840$		0			0
$841$	−14.4690			−0.498930
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.09182	+	25.0309i	−0.106362	+	0.861088i
$846$		0			0
$847$			15.2628i			0.524435i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.05956			0.104880
$852$		0			0
$853$			20.5864i			0.704864i	0.935837	+	0.352432i	$0.114645\pi$
$853$			20.5864i			0.704864i	−0.935837	+	0.352432i	$0.885355\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	12.6200i		−	0.431090i	−0.976494	−	0.215545i	$-0.930847\pi$
$857$		−	12.6200i		−	0.431090i	0.976494	−	0.215545i	$-0.0691528\pi$
$858$		0			0
$859$	−45.2157			−1.54274			−0.771370	−	0.636387i	$-0.780428\pi$
$859$	−45.2157			−1.54274			−0.771370	+	0.636387i	$0.780428\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			25.1398i			0.855768i	0.903834	+	0.427884i	$0.140741\pi$
$863$			25.1398i			0.855768i	−0.903834	+	0.427884i	$0.859259\pi$
$864$		0			0
$865$	33.1304	+	4.09228i	1.12647	+	0.139142i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.51999			−0.322943
$870$		0			0
$871$	−14.6480			−0.496330
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.41509	+	2.47854i	0.216870	+	0.0837901i
$876$		0			0
$877$		−	36.8075i		−	1.24290i	−0.783454	−	0.621450i	$-0.786544\pi$
$877$		−	36.8075i		−	1.24290i	0.783454	−	0.621450i	$-0.213456\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13.6144			0.458681			0.229341	−	0.973346i	$-0.426343\pi$
$881$	13.6144			0.458681			0.229341	+	0.973346i	$0.426343\pi$
$882$		0			0
$883$		−	21.4076i		−	0.720424i	−0.932871	−	0.360212i	$-0.882704\pi$
$883$		−	21.4076i		−	0.720424i	0.932871	−	0.360212i	$-0.117296\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			40.9342i			1.37444i	0.726451	+	0.687218i	$0.241168\pi$
$887$			40.9342i			1.37444i	−0.726451	+	0.687218i	$0.758832\pi$
$888$		0			0
$889$	−9.49699			−0.318519
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	42.3001i		−	1.41552i
$894$		0			0
$895$	4.07196	−	32.9659i	0.136111	−	1.10193i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.0379			−0.368135
$900$		0			0
$901$	−42.3970			−1.41245
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.87264	+	47.5439i	−0.195213	+	1.58041i
$906$		0			0
$907$		−	1.66355i		−	0.0552372i	−0.999619	−	0.0276186i	$-0.991208\pi$
$907$		−	1.66355i		−	0.0552372i	0.999619	−	0.0276186i	$-0.00879240\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−13.3624			−0.442718			−0.221359	−	0.975192i	$-0.571049\pi$
$911$	−13.3624			−0.442718			−0.221359	+	0.975192i	$0.571049\pi$
$912$		0			0
$913$			43.6612i			1.44497i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	0.393651i		−	0.0129995i
$918$		0			0
$919$	−33.9281			−1.11919			−0.559593	−	0.828768i	$-0.689043\pi$
$919$	−33.9281			−1.11919			−0.559593	+	0.828768i	$0.689043\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	19.6811i		−	0.647811i
$924$		0			0
$925$	3.72239	−	14.8380i	0.122391	−	0.487870i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	41.0428			1.34657			0.673285	−	0.739383i	$-0.264883\pi$
$929$	41.0428			1.34657			0.673285	+	0.739383i	$0.264883\pi$
$930$		0			0
$931$	42.9021			1.40606
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−69.3763	−	8.56939i	−2.26885	−	0.280249i
$936$		0			0
$937$			26.6877i			0.871848i	0.899983	+	0.435924i	$0.143578\pi$
$937$			26.6877i			0.871848i	−0.899983	+	0.435924i	$0.856422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−24.2512			−0.790567			−0.395284	−	0.918559i	$-0.629354\pi$
$941$	−24.2512			−0.790567			−0.395284	+	0.918559i	$0.629354\pi$
$942$		0			0
$943$		−	8.67483i		−	0.282491i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			37.2400i			1.21014i	0.796173	+	0.605069i	$0.206854\pi$
$947$			37.2400i			1.21014i	−0.796173	+	0.605069i	$0.793146\pi$
$948$		0			0
$949$	21.1781			0.687470
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	36.8798i		−	1.19465i	−0.801998	−	0.597327i	$-0.796230\pi$
$953$		−	36.8798i		−	1.19465i	0.801998	−	0.597327i	$-0.203770\pi$
$954$		0			0
$955$	2.20294	−	17.8347i	0.0712856	−	0.577116i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.0195			−0.323545
$960$		0			0
$961$	−22.6155			−0.729532
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.1168	−	2.73187i	−0.711964	−	0.0879421i
$966$		0			0
$967$			30.2608i			0.973121i	0.873647	+	0.486560i	$0.161749\pi$
$967$			30.2608i			0.973121i	−0.873647	+	0.486560i	$0.838251\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.07623			0.0666294			0.0333147	−	0.999445i	$-0.489394\pi$
$971$	2.07623			0.0666294			0.0333147	+	0.999445i	$0.489394\pi$
$972$		0			0
$973$			6.43983i			0.206451i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	37.1834i		−	1.18960i	−0.803873	−	0.594801i	$-0.797231\pi$
$977$		−	37.1834i		−	1.18960i	0.803873	−	0.594801i	$-0.202769\pi$
$978$		0			0
$979$	−76.6827			−2.45079
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	11.1581i		−	0.355888i	−0.984041	−	0.177944i	$-0.943055\pi$
$983$		−	11.1581i		−	0.355888i	0.984041	−	0.177944i	$-0.0569445\pi$
$984$		0			0
$985$	37.0062	+	4.57103i	1.17912	+	0.145645i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.78753			0.0886382
$990$		0			0
$991$	−19.8963			−0.632027			−0.316014	−	0.948755i	$-0.602345\pi$
$991$	−19.8963			−0.632027			−0.316014	+	0.948755i	$0.602345\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.46896	−	52.3716i	0.205080	−	1.66029i
$996$		0			0
$997$		−	38.5276i		−	1.22018i	−0.792331	−	0.610091i	$-0.791133\pi$
$997$		−	38.5276i		−	1.22018i	0.792331	−	0.610091i	$-0.208867\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.d.829.12	yes	24
3.2	odd	2	inner	4140.2.f.d.829.13	yes	24
5.4	even	2	inner	4140.2.f.d.829.11	✓	24
15.14	odd	2	inner	4140.2.f.d.829.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.f.d.829.11	✓	24	5.4	even	2	inner
4140.2.f.d.829.12	yes	24	1.1	even	1	trivial
4140.2.f.d.829.13	yes	24	3.2	odd	2	inner
4140.2.f.d.829.14	yes	24	15.14	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.274117 + 2.21920i) q^{5} +0.615120i q^{7} +O(q^{10})\)	\(q+(-0.274117 + 2.21920i) q^{5} +0.615120i q^{7} -5.98437 q^{11} -1.31179i q^{13} -5.22391i q^{17} +6.47909 q^{19} -1.00000i q^{23} +(-4.84972 - 1.21664i) q^{25} -3.81196 q^{29} +2.89560 q^{31} +(-1.36508 - 0.168615i) q^{35} +3.05956i q^{37} +8.67483 q^{41} +2.78753i q^{43} -6.52870i q^{47} +6.62163 q^{49} -8.11595i q^{53} +(1.64042 - 13.2805i) q^{55} -8.93735 q^{59} -1.29231 q^{61} +(2.91112 + 0.359583i) q^{65} -11.1665i q^{67} +15.0033 q^{71} +16.1444i q^{73} -3.68110i q^{77} +1.59081 q^{79} -7.29587i q^{83} +(11.5929 + 1.43196i) q^{85} +12.8138 q^{89} +0.806908 q^{91} +(-1.77603 + 14.3784i) q^{95} -17.2881i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100}) \) 24 * q + 8 * q^19 - 8 * q^25 - 12 * q^31 - 28 * q^49 - 16 * q^55 - 16 * q^61 + 8 * q^79 + 12 * q^85 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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