LMFDB - Embedded newform 6012.2.h.a.3005.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(3005,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6012.3005");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.h (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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3005.12
Character	$\chi$	$=$	6012.3005
Dual form			6012.2.h.a.3005.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76473 q^{5} +0.618008 q^{7} +O(q^{10})$	$q-2.76473 q^{5} +0.618008 q^{7} +5.51365i q^{11} +2.58896i q^{13} +5.12683 q^{17} -4.09998 q^{19} +5.83852 q^{23} +2.64374 q^{25} +0.795676i q^{29} +4.82779 q^{31} -1.70862 q^{35} -1.19611i q^{37} +8.20721 q^{41} -1.84848i q^{43} +1.33447i q^{47} -6.61807 q^{49} +4.14070 q^{53} -15.2438i q^{55} +4.91196 q^{59} -14.6371 q^{61} -7.15778i q^{65} +1.39489i q^{67} -2.37783 q^{71} -5.69062i q^{73} +3.40748i q^{77} +7.74996i q^{79} -0.0266917 q^{83} -14.1743 q^{85} +8.65105i q^{89} +1.60000i q^{91} +11.3353 q^{95} -5.26085 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * q	$ 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) $ 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6012\mathbb{Z}\right)^\times$.

$n$	$3007$	$3341$	$4681$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	−2.76473			−1.23643			−0.618213	−	0.786011i	$-0.712143\pi$
$5$	−2.76473			−1.23643			−0.618213	+	0.786011i	$0.712143\pi$
$6$		0			0
$7$	0.618008			0.233585			0.116792	−	0.993156i	$-0.462739\pi$
$7$	0.618008			0.233585			0.116792	+	0.993156i	$0.462739\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.51365i			1.66243i	0.555951	+	0.831215i	$0.312354\pi$
$11$			5.51365i			1.66243i	−0.555951	+	0.831215i	$0.687646\pi$
$12$		0			0
$13$			2.58896i			0.718049i	0.933328	+	0.359024i	$0.116890\pi$
$13$			2.58896i			0.718049i	−0.933328	+	0.359024i	$0.883110\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.12683			1.24344			0.621720	−	0.783240i	$-0.286434\pi$
$17$	5.12683			1.24344			0.621720	+	0.783240i	$0.286434\pi$
$18$		0			0
$19$	−4.09998			−0.940600			−0.470300	−	0.882507i	$-0.655854\pi$
$19$	−4.09998			−0.940600			−0.470300	+	0.882507i	$0.655854\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.83852			1.21741			0.608707	−	0.793395i	$-0.291688\pi$
$23$	5.83852			1.21741			0.608707	+	0.793395i	$0.291688\pi$
$24$		0			0
$25$	2.64374			0.528747
$26$		0			0
$27$		0			0
$28$		0			0
$29$			0.795676i			0.147753i	0.997267	+	0.0738767i	$0.0235371\pi$
$29$			0.795676i			0.147753i	−0.997267	+	0.0738767i	$0.976463\pi$
$30$		0			0
$31$	4.82779			0.867097			0.433549	−	0.901130i	$-0.357261\pi$
$31$	4.82779			0.867097			0.433549	+	0.901130i	$0.357261\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.70862			−0.288810
$36$		0			0
$37$		−	1.19611i		−	0.196639i	−0.995155	−	0.0983193i	$-0.968653\pi$
$37$		−	1.19611i		−	0.196639i	0.995155	−	0.0983193i	$-0.0313467\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.20721			1.28175			0.640875	−	0.767645i	$-0.278572\pi$
$41$	8.20721			1.28175			0.640875	+	0.767645i	$0.278572\pi$
$42$		0			0
$43$		−	1.84848i		−	0.281890i	−0.990017	−	0.140945i	$-0.954986\pi$
$43$		−	1.84848i		−	0.281890i	0.990017	−	0.140945i	$-0.0450141\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.33447i			0.194653i	0.995253	+	0.0973263i	$0.0310290\pi$
$47$			1.33447i			0.194653i	−0.995253	+	0.0973263i	$0.968971\pi$
$48$		0			0
$49$	−6.61807			−0.945438
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.14070			0.568769			0.284385	−	0.958710i	$-0.408211\pi$
$53$	4.14070			0.568769			0.284385	+	0.958710i	$0.408211\pi$
$54$		0			0
$55$		−	15.2438i		−	2.05547i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.91196			0.639483			0.319741	−	0.947505i	$-0.396404\pi$
$59$	4.91196			0.639483			0.319741	+	0.947505i	$0.396404\pi$
$60$		0			0
$61$	−14.6371			−1.87409			−0.937047	−	0.349202i	$-0.886453\pi$
$61$	−14.6371			−1.87409			−0.937047	+	0.349202i	$0.886453\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	7.15778i		−	0.887813i
$66$		0			0
$67$			1.39489i			0.170412i	0.996363	+	0.0852062i	$0.0271549\pi$
$67$			1.39489i			0.170412i	−0.996363	+	0.0852062i	$0.972845\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.37783			−0.282196			−0.141098	−	0.989996i	$-0.545063\pi$
$71$	−2.37783			−0.282196			−0.141098	+	0.989996i	$0.545063\pi$
$72$		0			0
$73$		−	5.69062i		−	0.666037i	−0.942920	−	0.333018i	$-0.891933\pi$
$73$		−	5.69062i		−	0.666037i	0.942920	−	0.333018i	$-0.108067\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			3.40748i			0.388318i
$78$		0			0
$79$			7.74996i			0.871939i	0.899961	+	0.435970i	$0.143594\pi$
$79$			7.74996i			0.871939i	−0.899961	+	0.435970i	$0.856406\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.0266917			−0.00292979			−0.00146490	−	0.999999i	$-0.500466\pi$
$83$	−0.0266917			−0.00292979			−0.00146490	+	0.999999i	$0.500466\pi$
$84$		0			0
$85$	−14.1743			−1.53742
$86$		0			0
$87$		0			0
$88$		0			0
$89$			8.65105i			0.917009i	0.888692	+	0.458505i	$0.151615\pi$
$89$			8.65105i			0.917009i	−0.888692	+	0.458505i	$0.848385\pi$
$90$		0			0
$91$			1.60000i			0.167725i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	11.3353			1.16298
$96$		0			0
$97$	−5.26085			−0.534158			−0.267079	−	0.963675i	$-0.586059\pi$
$97$	−5.26085			−0.534158			−0.267079	+	0.963675i	$0.586059\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.49685			−0.347949			−0.173975	−	0.984750i	$-0.555661\pi$
$101$	−3.49685			−0.347949			−0.173975	+	0.984750i	$0.555661\pi$
$102$		0			0
$103$		−	0.704239i		−	0.0693907i	−0.999398	−	0.0346954i	$-0.988954\pi$
$103$		−	0.704239i		−	0.0693907i	0.999398	−	0.0346954i	$-0.0110461\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			4.27785i			0.413555i	0.978388	+	0.206778i	$0.0662977\pi$
$107$			4.27785i			0.413555i	−0.978388	+	0.206778i	$0.933702\pi$
$108$		0			0
$109$			11.3752i			1.08955i	0.838583	+	0.544773i	$0.183384\pi$
$109$			11.3752i			1.08955i	−0.838583	+	0.544773i	$0.816616\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.2226			1.14980			0.574902	−	0.818222i	$-0.305040\pi$
$113$	12.2226			1.14980			0.574902	+	0.818222i	$0.305040\pi$
$114$		0			0
$115$	−16.1419			−1.50524
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.16842			0.290449
$120$		0			0
$121$	−19.4004			−1.76367
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.51444			0.582669
$126$		0			0
$127$	6.02121			0.534296			0.267148	−	0.963656i	$-0.413919\pi$
$127$	6.02121			0.534296			0.267148	+	0.963656i	$0.413919\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−19.3564			−1.69118			−0.845588	−	0.533836i	$-0.820750\pi$
$131$	−19.3564			−1.69118			−0.845588	+	0.533836i	$0.820750\pi$
$132$		0			0
$133$	−2.53382			−0.219710
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	21.6540i		−	1.85003i	−0.379931	−	0.925015i	$-0.624052\pi$
$137$		−	21.6540i		−	1.85003i	0.379931	−	0.925015i	$-0.375948\pi$
$138$		0			0
$139$		−	7.89202i		−	0.669393i	−0.942326	−	0.334696i	$-0.891366\pi$
$139$		−	7.89202i		−	0.669393i	0.942326	−	0.334696i	$-0.108634\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.2746			−1.19371
$144$		0			0
$145$		−	2.19983i		−	0.182686i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.6114			−1.03316			−0.516582	−	0.856238i	$-0.672796\pi$
$149$	−12.6114			−1.03316			−0.516582	+	0.856238i	$0.672796\pi$
$150$		0			0
$151$			13.8446i			1.12666i	0.826233	+	0.563328i	$0.190479\pi$
$151$			13.8446i			1.12666i	−0.826233	+	0.563328i	$0.809521\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−13.3475			−1.07210
$156$		0			0
$157$	6.78446			0.541459			0.270730	−	0.962655i	$-0.412735\pi$
$157$	6.78446			0.541459			0.270730	+	0.962655i	$0.412735\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.60825			0.284370
$162$		0			0
$163$			10.6704i			0.835771i	0.908500	+	0.417886i	$0.137229\pi$
$163$			10.6704i			0.835771i	−0.908500	+	0.417886i	$0.862771\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.30520	+	12.1846i	−0.333147	+	0.942875i
$167$	−4.30520	+	12.1846i	−0.333147	+	0.942875i
$168$		0			0
$169$	6.29728			0.484406
$170$		0			0
$171$		0			0
$172$		0			0
$173$			15.1637i			1.15288i	0.817140	+	0.576439i	$0.195558\pi$
$173$			15.1637i			1.15288i	−0.817140	+	0.576439i	$0.804442\pi$
$174$		0			0
$175$	1.63385			0.123507
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	0.202535i		−	0.0151382i	−0.999971	−	0.00756908i	$-0.997591\pi$
$179$		−	0.202535i		−	0.0151382i	0.999971	−	0.00756908i	$-0.00240934\pi$
$180$		0			0
$181$	10.3582			0.769918			0.384959	−	0.922934i	$-0.374216\pi$
$181$	10.3582			0.769918			0.384959	+	0.922934i	$0.374216\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			3.30691i			0.243129i
$186$		0			0
$187$			28.2676i			2.06713i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			13.7071i			0.991813i	0.868376	+	0.495906i	$0.165164\pi$
$191$			13.7071i			0.991813i	−0.868376	+	0.495906i	$0.834836\pi$
$192$		0			0
$193$		−	9.11688i		−	0.656247i	−0.944635	−	0.328124i	$-0.893584\pi$
$193$		−	9.11688i		−	0.656247i	0.944635	−	0.328124i	$-0.106416\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.2003			−0.869234			−0.434617	−	0.900615i	$-0.643116\pi$
$197$	−12.2003			−0.869234			−0.434617	+	0.900615i	$0.643116\pi$
$198$		0			0
$199$	2.79531			0.198154			0.0990770	−	0.995080i	$-0.468411\pi$
$199$	2.79531			0.198154			0.0990770	+	0.995080i	$0.468411\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			0.491734i			0.0345129i
$204$		0			0
$205$	−22.6907			−1.58479
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	22.6059i		−	1.56368i
$210$		0			0
$211$	−10.0550			−0.692216			−0.346108	−	0.938195i	$-0.612497\pi$
$211$	−10.0550			−0.692216			−0.346108	+	0.938195i	$0.612497\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			5.11055i			0.348536i
$216$		0			0
$217$	2.98361			0.202541
$218$		0			0
$219$		0			0
$220$		0			0
$221$			13.2732i			0.892850i
$222$		0			0
$223$	15.9618			1.06888			0.534442	−	0.845205i	$-0.320522\pi$
$223$	15.9618			1.06888			0.534442	+	0.845205i	$0.320522\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.7386			1.17736			0.588678	−	0.808368i	$-0.299649\pi$
$227$	17.7386			1.17736			0.588678	+	0.808368i	$0.299649\pi$
$228$		0			0
$229$	−3.06625			−0.202624			−0.101312	−	0.994855i	$-0.532304\pi$
$229$	−3.06625			−0.202624			−0.101312	+	0.994855i	$0.532304\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			27.3615i			1.79251i	0.443535	+	0.896257i	$0.353724\pi$
$233$			27.3615i			1.79251i	−0.443535	+	0.896257i	$0.646276\pi$
$234$		0			0
$235$		−	3.68945i		−	0.240673i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	11.6002i		−	0.750354i	−0.926953	−	0.375177i	$-0.877582\pi$
$239$		−	11.6002i		−	0.750354i	0.926953	−	0.375177i	$-0.122418\pi$
$240$		0			0
$241$			21.6048i			1.39169i	0.718192	+	0.695845i	$0.244970\pi$
$241$			21.6048i			1.39169i	−0.718192	+	0.695845i	$0.755030\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	18.2972			1.16896
$246$		0			0
$247$		−	10.6147i		−	0.675396i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			8.54576i			0.539404i	0.962944	+	0.269702i	$0.0869251\pi$
$251$			8.54576i			0.539404i	−0.962944	+	0.269702i	$0.913075\pi$
$252$		0			0
$253$			32.1916i			2.02387i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.15043			0.383653			0.191827	−	0.981429i	$-0.438559\pi$
$257$	6.15043			0.383653			0.191827	+	0.981429i	$0.438559\pi$
$258$		0			0
$259$		−	0.739203i		−	0.0459318i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			22.6243i			1.39507i	0.716549	+	0.697537i	$0.245721\pi$
$263$			22.6243i			1.39507i	−0.716549	+	0.697537i	$0.754279\pi$
$264$		0			0
$265$	−11.4479			−0.703241
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.2318			−1.11161			−0.555807	−	0.831312i	$-0.687591\pi$
$269$	−18.2318			−1.11161			−0.555807	+	0.831312i	$0.687591\pi$
$270$		0			0
$271$			3.06817i			0.186378i	0.995648	+	0.0931891i	$0.0297061\pi$
$271$			3.06817i			0.186378i	−0.995648	+	0.0931891i	$0.970294\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			14.5766i			0.879005i
$276$		0			0
$277$			1.76542i			0.106074i	0.998593	+	0.0530368i	$0.0168901\pi$
$277$			1.76542i			0.106074i	−0.998593	+	0.0530368i	$0.983110\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			18.2786i			1.09041i	0.838303	+	0.545205i	$0.183548\pi$
$281$			18.2786i			1.09041i	−0.838303	+	0.545205i	$0.816452\pi$
$282$		0			0
$283$	−27.0531			−1.60814			−0.804070	−	0.594535i	$-0.797336\pi$
$283$	−27.0531			−1.60814			−0.804070	+	0.594535i	$0.797336\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.07212			0.299398
$288$		0			0
$289$	9.28439			0.546141
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	17.2388i		−	1.00710i	−0.863965	−	0.503552i	$-0.832026\pi$
$293$		−	17.2388i		−	1.00710i	0.863965	−	0.503552i	$-0.167974\pi$
$294$		0			0
$295$	−13.5802			−0.790673
$296$		0			0
$297$		0			0
$298$		0			0
$299$			15.1157i			0.874163i
$300$		0			0
$301$		−	1.14237i		−	0.0658453i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	40.4678			2.31718
$306$		0			0
$307$		−	16.6785i		−	0.951891i	−0.879475	−	0.475945i	$-0.842106\pi$
$307$		−	16.6785i		−	0.951891i	0.879475	−	0.475945i	$-0.157894\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8.38908i		−	0.475701i	−0.971302	−	0.237851i	$-0.923557\pi$
$311$		−	8.38908i		−	0.475701i	0.971302	−	0.237851i	$-0.0764429\pi$
$312$		0			0
$313$		−	25.7289i		−	1.45429i	−0.686486	−	0.727143i	$-0.740848\pi$
$313$		−	25.7289i		−	1.45429i	0.686486	−	0.727143i	$-0.259152\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			8.35143i			0.469063i	0.972109	+	0.234532i	$0.0753556\pi$
$317$			8.35143i			0.469063i	−0.972109	+	0.234532i	$0.924644\pi$
$318$		0			0
$319$	−4.38708			−0.245630
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−21.0199			−1.16958
$324$		0			0
$325$			6.84453i			0.379666i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			0.824713i			0.0454679i
$330$		0			0
$331$			28.1622i			1.54794i	0.633225	+	0.773968i	$0.281731\pi$
$331$			28.1622i			1.54794i	−0.633225	+	0.773968i	$0.718269\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	3.85648i		−	0.210702i
$336$		0			0
$337$	−15.7525			−0.858092			−0.429046	−	0.903283i	$-0.641150\pi$
$337$	−15.7525			−0.858092			−0.429046	+	0.903283i	$0.641150\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			26.6188i			1.44149i
$342$		0			0
$343$	−8.41607			−0.454425
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.6851			−0.680972			−0.340486	−	0.940250i	$-0.610592\pi$
$347$	−12.6851			−0.680972			−0.340486	+	0.940250i	$0.610592\pi$
$348$		0			0
$349$		−	7.51912i		−	0.402489i	−0.979541	−	0.201245i	$-0.935501\pi$
$349$		−	7.51912i		−	0.402489i	0.979541	−	0.201245i	$-0.0644986\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			29.8355i			1.58798i	0.607928	+	0.793992i	$0.292001\pi$
$353$			29.8355i			1.58798i	−0.607928	+	0.793992i	$0.707999\pi$
$354$		0			0
$355$	6.57405			0.348914
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	7.29188i		−	0.384851i	−0.981312	−	0.192425i	$-0.938365\pi$
$359$		−	7.29188i		−	0.384851i	0.981312	−	0.192425i	$-0.0616353\pi$
$360$		0			0
$361$	−2.19016			−0.115272
$362$		0			0
$363$		0			0
$364$		0			0
$365$			15.7330i			0.823504i
$366$		0			0
$367$	1.45949			0.0761846			0.0380923	−	0.999274i	$-0.487872\pi$
$367$	1.45949			0.0761846			0.0380923	+	0.999274i	$0.487872\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.55899			0.132856
$372$		0			0
$373$		−	11.4525i		−	0.592986i	−0.955035	−	0.296493i	$-0.904183\pi$
$373$		−	11.4525i		−	0.592986i	0.955035	−	0.296493i	$-0.0958172\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.05997			−0.106094
$378$		0			0
$379$			28.7140i			1.47494i	0.675379	+	0.737471i	$0.263980\pi$
$379$			28.7140i			1.47494i	−0.675379	+	0.737471i	$0.736020\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			8.96528i			0.458104i	0.973414	+	0.229052i	$0.0735626\pi$
$383$			8.96528i			0.458104i	−0.973414	+	0.229052i	$0.926437\pi$
$384$		0			0
$385$		−	9.42077i		−	0.480127i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−23.0490			−1.16863			−0.584315	−	0.811527i	$-0.698637\pi$
$389$	−23.0490			−1.16863			−0.584315	+	0.811527i	$0.698637\pi$
$390$		0			0
$391$	29.9331			1.51378
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	21.4266i		−	1.07809i
$396$		0			0
$397$	6.18808			0.310571			0.155285	−	0.987870i	$-0.450370\pi$
$397$	6.18808			0.310571			0.155285	+	0.987870i	$0.450370\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	21.7599			1.08664			0.543319	−	0.839527i	$-0.317168\pi$
$401$	21.7599			1.08664			0.543319	+	0.839527i	$0.317168\pi$
$402$		0			0
$403$			12.4990i			0.622618i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.59492			0.326898
$408$		0			0
$409$	−31.0725			−1.53644			−0.768218	−	0.640189i	$-0.778856\pi$
$409$	−31.0725			−1.53644			−0.768218	+	0.640189i	$0.778856\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.03563			0.149374
$414$		0			0
$415$	0.0737954			0.00362247
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	29.5123i		−	1.44177i	−0.693054	−	0.720885i	$-0.743735\pi$
$419$		−	29.5123i		−	1.44177i	0.693054	−	0.720885i	$-0.256265\pi$
$420$		0			0
$421$	−22.4021			−1.09181			−0.545907	−	0.837846i	$-0.683815\pi$
$421$	−22.4021			−1.09181			−0.545907	+	0.837846i	$0.683815\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13.5540			0.657465
$426$		0			0
$427$	−9.04587			−0.437760
$428$		0			0
$429$		0			0
$430$		0			0
$431$			6.69219i			0.322351i	0.986926	+	0.161176i	$0.0515286\pi$
$431$			6.69219i			0.322351i	−0.986926	+	0.161176i	$0.948471\pi$
$432$		0			0
$433$	−21.8508			−1.05008			−0.525041	−	0.851077i	$-0.675950\pi$
$433$	−21.8508			−1.05008			−0.525041	+	0.851077i	$0.675950\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−23.9378			−1.14510
$438$		0			0
$439$			12.3049i			0.587283i	0.955916	+	0.293641i	$0.0948672\pi$
$439$			12.3049i			0.587283i	−0.955916	+	0.293641i	$0.905133\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.3204			−0.870429			−0.435215	−	0.900327i	$-0.643328\pi$
$443$	−18.3204			−0.870429			−0.435215	+	0.900327i	$0.643328\pi$
$444$		0			0
$445$		−	23.9178i		−	1.13381i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	15.1799i		−	0.716382i	−0.933648	−	0.358191i	$-0.883394\pi$
$449$		−	15.1799i		−	0.716382i	0.933648	−	0.358191i	$-0.116606\pi$
$450$		0			0
$451$			45.2517i			2.13082i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	4.42356i		−	0.207380i
$456$		0			0
$457$		−	4.65645i		−	0.217819i	−0.994052	−	0.108910i	$-0.965264\pi$
$457$		−	4.65645i		−	0.217819i	0.994052	−	0.108910i	$-0.0347359\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			32.4248i			1.51017i	0.655626	+	0.755086i	$0.272405\pi$
$461$			32.4248i			1.51017i	−0.655626	+	0.755086i	$0.727595\pi$
$462$		0			0
$463$		−	23.2195i		−	1.07910i	−0.841953	−	0.539551i	$-0.818594\pi$
$463$		−	23.2195i		−	1.07910i	0.841953	−	0.539551i	$-0.181406\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	25.2354i		−	1.16776i	−0.811841	−	0.583878i	$-0.801535\pi$
$467$		−	25.2354i		−	1.16776i	0.811841	−	0.583878i	$-0.198465\pi$
$468$		0			0
$469$			0.862050i			0.0398058i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.1919			0.468623
$474$		0			0
$475$	−10.8393			−0.497340
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−39.7086			−1.81433			−0.907166	−	0.420773i	$-0.861759\pi$
$479$	−39.7086			−1.81433			−0.907166	+	0.420773i	$0.861759\pi$
$480$		0			0
$481$	3.09667			0.141196
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.5448			0.660447
$486$		0			0
$487$		−	28.8653i		−	1.30801i	−0.756490	−	0.654005i	$-0.773087\pi$
$487$		−	28.8653i		−	1.30801i	0.756490	−	0.654005i	$-0.226913\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	14.1667i		−	0.639336i	−0.947530	−	0.319668i	$-0.896429\pi$
$491$		−	14.1667i		−	0.639336i	0.947530	−	0.319668i	$-0.103571\pi$
$492$		0			0
$493$			4.07930i			0.183722i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.46951			−0.0659167
$498$		0			0
$499$		−	0.585757i		−	0.0262221i	−0.999914	−	0.0131110i	$-0.995827\pi$
$499$		−	0.585757i		−	0.0262221i	0.999914	−	0.0131110i	$-0.00417349\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.21259i		−	0.410769i	−0.978681	−	0.205385i	$-0.934155\pi$
$503$		−	9.21259i		−	0.410769i	0.978681	−	0.205385i	$-0.0658445\pi$
$504$		0			0
$505$	9.66784			0.430213
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	18.8442i		−	0.835253i	−0.908619	−	0.417626i	$-0.862862\pi$
$509$		−	18.8442i		−	0.835253i	0.908619	−	0.417626i	$-0.137138\pi$
$510$		0			0
$511$		−	3.51685i		−	0.155576i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			1.94703i			0.0857965i
$516$		0			0
$517$	−7.35781			−0.323596
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18.7521			−0.821546			−0.410773	−	0.911738i	$-0.634741\pi$
$521$	−18.7521			−0.821546			−0.410773	+	0.911738i	$0.634741\pi$
$522$		0			0
$523$	4.82149			0.210829			0.105415	−	0.994428i	$-0.466383\pi$
$523$	4.82149			0.210829			0.105415	+	0.994428i	$0.466383\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	24.7513			1.07818
$528$		0			0
$529$	11.0883			0.482099
$530$		0			0
$531$		0			0
$532$		0			0
$533$			21.2481i			0.920359i
$534$		0			0
$535$		−	11.8271i		−	0.511330i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	36.4897i		−	1.57172i
$540$		0			0
$541$		−	0.602251i		−	0.0258928i	−0.999916	−	0.0129464i	$-0.995879\pi$
$541$		−	0.602251i		−	0.0258928i	0.999916	−	0.0129464i	$-0.00412108\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	31.4494i		−	1.34714i
$546$		0			0
$547$			8.33181i			0.356242i	0.984009	+	0.178121i	$0.0570019\pi$
$547$			8.33181i			0.356242i	−0.984009	+	0.178121i	$0.942998\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	3.26226i		−	0.138977i
$552$		0			0
$553$			4.78954i			0.203672i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			25.3948i			1.07601i	0.842941	+	0.538005i	$0.180822\pi$
$557$			25.3948i			1.07601i	−0.842941	+	0.538005i	$0.819178\pi$
$558$		0			0
$559$	4.78564			0.202411
$560$		0			0
$561$		0			0
$562$		0			0
$563$			0.239030i			0.0100739i	0.999987	+	0.00503696i	$0.00160332\pi$
$563$			0.239030i			0.0100739i	−0.999987	+	0.00503696i	$0.998397\pi$
$564$		0			0
$565$	−33.7922			−1.42165
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.16148			0.132536			0.0662680	−	0.997802i	$-0.478891\pi$
$569$	3.16148			0.132536			0.0662680	+	0.997802i	$0.478891\pi$
$570$		0			0
$571$		−	46.4685i		−	1.94465i	−0.233640	−	0.972323i	$-0.575064\pi$
$571$		−	46.4685i		−	1.94465i	0.233640	−	0.972323i	$-0.424936\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	15.4355			0.643705
$576$		0			0
$577$	9.69066			0.403428			0.201714	−	0.979445i	$-0.435349\pi$
$577$	9.69066			0.403428			0.201714	+	0.979445i	$0.435349\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.0164957			−0.000684356
$582$		0			0
$583$			22.8304i			0.945539i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	42.3954			1.74985			0.874924	−	0.484261i	$-0.160911\pi$
$587$	42.3954			1.74985			0.874924	+	0.484261i	$0.160911\pi$
$588$		0			0
$589$	−19.7939			−0.815592
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12.7221			0.522435			0.261218	−	0.965280i	$-0.415876\pi$
$593$	12.7221			0.522435			0.261218	+	0.965280i	$0.415876\pi$
$594$		0			0
$595$	−8.75983			−0.359118
$596$		0			0
$597$		0			0
$598$		0			0
$599$			30.2589i			1.23635i	0.786042	+	0.618173i	$0.212127\pi$
$599$			30.2589i			1.23635i	−0.786042	+	0.618173i	$0.787873\pi$
$600$		0			0
$601$	−13.6377			−0.556293			−0.278146	−	0.960539i	$-0.589720\pi$
$601$	−13.6377			−0.556293			−0.278146	+	0.960539i	$0.589720\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	53.6369			2.18065
$606$		0			0
$607$			21.7351i			0.882201i	0.897458	+	0.441101i	$0.145412\pi$
$607$			21.7351i			0.882201i	−0.897458	+	0.441101i	$0.854588\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.45489			−0.139770
$612$		0			0
$613$	−20.4741			−0.826943			−0.413471	−	0.910517i	$-0.635684\pi$
$613$	−20.4741			−0.826943			−0.413471	+	0.910517i	$0.635684\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	19.2770i		−	0.776061i	−0.921646	−	0.388031i	$-0.873155\pi$
$617$		−	19.2770i		−	0.776061i	0.921646	−	0.388031i	$-0.126845\pi$
$618$		0			0
$619$		−	25.9858i		−	1.04446i	−0.852805	−	0.522229i	$-0.825101\pi$
$619$		−	25.9858i		−	1.04446i	0.852805	−	0.522229i	$-0.174899\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			5.34641i			0.214200i
$624$		0			0
$625$	−31.2293			−1.24917
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	6.13223i		−	0.244508i
$630$		0			0
$631$	−13.7377			−0.546891			−0.273445	−	0.961888i	$-0.588163\pi$
$631$	−13.7377			−0.546891			−0.273445	+	0.961888i	$0.588163\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−16.6470			−0.660617
$636$		0			0
$637$		−	17.1339i		−	0.678870i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.9736			0.709915			0.354957	−	0.934882i	$-0.384495\pi$
$641$	17.9736			0.709915			0.354957	+	0.934882i	$0.384495\pi$
$642$		0			0
$643$		−	39.1122i		−	1.54243i	−0.636573	−	0.771216i	$-0.719649\pi$
$643$		−	39.1122i		−	1.54243i	0.636573	−	0.771216i	$-0.280351\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20.8207			−0.818547			−0.409274	−	0.912412i	$-0.634218\pi$
$647$	−20.8207			−0.818547			−0.409274	+	0.912412i	$0.634218\pi$
$648$		0			0
$649$			27.0829i			1.06310i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			22.1623i			0.867278i	0.901087	+	0.433639i	$0.142771\pi$
$653$			22.1623i			0.867278i	−0.901087	+	0.433639i	$0.857229\pi$
$654$		0			0
$655$	53.5152			2.09101
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.16750			−0.240252			−0.120126	−	0.992759i	$-0.538330\pi$
$659$	−6.16750			−0.240252			−0.120126	+	0.992759i	$0.538330\pi$
$660$		0			0
$661$			35.0646i			1.36385i	0.731420	+	0.681927i	$0.238858\pi$
$661$			35.0646i			1.36385i	−0.731420	+	0.681927i	$0.761142\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	7.00533			0.271655
$666$		0			0
$667$			4.64557i			0.179877i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	80.7042i		−	3.11555i
$672$		0			0
$673$		−	19.9031i		−	0.767206i	−0.923498	−	0.383603i	$-0.874683\pi$
$673$		−	19.9031i		−	0.767206i	0.923498	−	0.383603i	$-0.125317\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			15.4258i			0.592861i	0.955054	+	0.296431i	$0.0957963\pi$
$677$			15.4258i			0.592861i	−0.955054	+	0.296431i	$0.904204\pi$
$678$		0			0
$679$	−3.25124			−0.124771
$680$		0			0
$681$		0			0
$682$		0			0
$683$	35.6351			1.36354			0.681770	−	0.731567i	$-0.261211\pi$
$683$	35.6351			1.36354			0.681770	+	0.731567i	$0.261211\pi$
$684$		0			0
$685$			59.8676i			2.28742i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			10.7201i			0.408404i
$690$		0			0
$691$		−	19.3096i		−	0.734573i	−0.930108	−	0.367287i	$-0.880287\pi$
$691$		−	19.3096i		−	0.734573i	0.930108	−	0.367287i	$-0.119713\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			21.8193i			0.827654i
$696$		0			0
$697$	42.0770			1.59378
$698$		0			0
$699$		0			0
$700$		0			0
$701$			5.57426i			0.210537i	0.994444	+	0.105269i	$0.0335702\pi$
$701$			5.57426i			0.210537i	−0.994444	+	0.105269i	$0.966430\pi$
$702$		0			0
$703$			4.90401i			0.184958i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.16108			−0.0812757
$708$		0			0
$709$		−	7.79896i		−	0.292896i	−0.989218	−	0.146448i	$-0.953216\pi$
$709$		−	7.79896i		−	0.292896i	0.989218	−	0.146448i	$-0.0467841\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	28.1871			1.05562
$714$		0			0
$715$	39.4655			1.47593
$716$		0			0
$717$		0			0
$718$		0			0
$719$	0.766739			0.0285946			0.0142973	−	0.999898i	$-0.495449\pi$
$719$	0.766739			0.0285946			0.0142973	+	0.999898i	$0.495449\pi$
$720$		0			0
$721$		−	0.435225i		−	0.0162086i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			2.10356i			0.0781242i
$726$		0			0
$727$		−	19.4239i		−	0.720391i	−0.932877	−	0.360196i	$-0.882710\pi$
$727$		−	19.4239i		−	0.720391i	0.932877	−	0.360196i	$-0.117290\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	9.47684i		−	0.350514i
$732$		0			0
$733$	−3.26000			−0.120411			−0.0602053	−	0.998186i	$-0.519176\pi$
$733$	−3.26000			−0.120411			−0.0602053	+	0.998186i	$0.519176\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7.69092			−0.283299
$738$		0			0
$739$		−	29.4351i		−	1.08279i	−0.840769	−	0.541394i	$-0.817897\pi$
$739$		−	29.4351i		−	1.08279i	0.840769	−	0.541394i	$-0.182103\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.35428i			0.0496838i	0.999691	+	0.0248419i	$0.00790824\pi$
$743$			1.35428i			0.0496838i	−0.999691	+	0.0248419i	$0.992092\pi$
$744$		0			0
$745$	34.8670			1.27743
$746$		0			0
$747$		0			0
$748$		0			0
$749$			2.64374i			0.0966003i
$750$		0			0
$751$			9.26564i			0.338108i	0.985607	+	0.169054i	$0.0540712\pi$
$751$			9.26564i			0.338108i	−0.985607	+	0.169054i	$0.945929\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	38.2766i		−	1.39303i
$756$		0			0
$757$	−4.77283			−0.173471			−0.0867357	−	0.996231i	$-0.527644\pi$
$757$	−4.77283			−0.173471			−0.0867357	+	0.996231i	$0.527644\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			40.4481i			1.46624i	0.680098	+	0.733121i	$0.261937\pi$
$761$			40.4481i			1.46624i	−0.680098	+	0.733121i	$0.738063\pi$
$762$		0			0
$763$			7.02996i			0.254502i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			12.7169i			0.459180i
$768$		0			0
$769$			36.9631i			1.33292i	0.745540	+	0.666461i	$0.232192\pi$
$769$			36.9631i			1.33292i	−0.745540	+	0.666461i	$0.767808\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−42.7701			−1.53833			−0.769167	−	0.639048i	$-0.779328\pi$
$773$	−42.7701			−1.53833			−0.769167	+	0.639048i	$0.779328\pi$
$774$		0			0
$775$	12.7634			0.458475
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−33.6494			−1.20561
$780$		0			0
$781$		−	13.1105i		−	0.469131i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−18.7572			−0.669474
$786$		0			0
$787$		−	10.8862i		−	0.388052i	−0.980996	−	0.194026i	$-0.937845\pi$
$787$		−	10.8862i		−	0.388052i	0.980996	−	0.194026i	$-0.0621546\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.55365			0.268577
$792$		0			0
$793$		−	37.8950i		−	1.34569i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	38.3384			1.35802			0.679008	−	0.734130i	$-0.262410\pi$
$797$	38.3384			1.35802			0.679008	+	0.734130i	$0.262410\pi$
$798$		0			0
$799$			6.84161i			0.242039i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	31.3761			1.10724
$804$		0			0
$805$	−9.97583			−0.351602
$806$		0			0
$807$		0			0
$808$		0			0
$809$			9.55069i			0.335784i	0.985805	+	0.167892i	$0.0536960\pi$
$809$			9.55069i			0.335784i	−0.985805	+	0.167892i	$0.946304\pi$
$810$		0			0
$811$		−	41.3699i		−	1.45269i	−0.687328	−	0.726347i	$-0.741216\pi$
$811$		−	41.3699i		−	1.45269i	0.687328	−	0.726347i	$-0.258784\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	29.5008i		−	1.03337i
$816$		0			0
$817$			7.57873i			0.265146i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	16.3279			0.569848			0.284924	−	0.958550i	$-0.408032\pi$
$821$	16.3279			0.569848			0.284924	+	0.958550i	$0.408032\pi$
$822$		0			0
$823$		−	10.1789i		−	0.354816i	−0.984137	−	0.177408i	$-0.943229\pi$
$823$		−	10.1789i		−	0.354816i	0.984137	−	0.177408i	$-0.0567712\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	42.5412			1.47930			0.739652	−	0.672990i	$-0.234990\pi$
$827$	42.5412			1.47930			0.739652	+	0.672990i	$0.234990\pi$
$828$		0			0
$829$			38.9049i			1.35122i	0.737257	+	0.675612i	$0.236120\pi$
$829$			38.9049i			1.35122i	−0.737257	+	0.675612i	$0.763880\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−33.9297			−1.17559
$834$		0			0
$835$	11.9027	−	33.6872i	0.411911	−	1.16579i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			14.3161i			0.494246i	0.968984	+	0.247123i	$0.0794852\pi$
$839$			14.3161i			0.494246i	−0.968984	+	0.247123i	$0.920515\pi$
$840$		0			0
$841$	28.3669			0.978169
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17.4103			−0.598932
$846$		0			0
$847$	−11.9896			−0.411967
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	6.98349i		−	0.239391i
$852$		0			0
$853$	−0.567890			−0.0194442			−0.00972210	−	0.999953i	$-0.503095\pi$
$853$	−0.567890			−0.0194442			−0.00972210	+	0.999953i	$0.503095\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			7.62281i			0.260390i	0.991488	+	0.130195i	$0.0415604\pi$
$857$			7.62281i			0.260390i	−0.991488	+	0.130195i	$0.958440\pi$
$858$		0			0
$859$	37.9124			1.29355			0.646777	−	0.762679i	$-0.276117\pi$
$859$	37.9124			1.29355			0.646777	+	0.762679i	$0.276117\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			49.3748i			1.68074i	0.542016	+	0.840368i	$0.317661\pi$
$863$			49.3748i			1.68074i	−0.542016	+	0.840368i	$0.682339\pi$
$864$		0			0
$865$		−	41.9237i		−	1.42545i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−42.7306			−1.44954
$870$		0			0
$871$	−3.61130			−0.122364
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.02597			0.136103
$876$		0			0
$877$	−47.5617			−1.60604			−0.803022	−	0.595949i	$-0.796776\pi$
$877$	−47.5617			−1.60604			−0.803022	+	0.595949i	$0.796776\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.77578			−0.160900			−0.0804500	−	0.996759i	$-0.525636\pi$
$881$	−4.77578			−0.160900			−0.0804500	+	0.996759i	$0.525636\pi$
$882$		0			0
$883$	51.4846			1.73259			0.866297	−	0.499529i	$-0.166494\pi$
$883$	51.4846			1.73259			0.866297	+	0.499529i	$0.166494\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	54.1904			1.81953			0.909767	−	0.415118i	$-0.136260\pi$
$887$	54.1904			1.81953			0.909767	+	0.415118i	$0.136260\pi$
$888$		0			0
$889$	3.72115			0.124803
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	5.47131i		−	0.183090i
$894$		0			0
$895$			0.559954i			0.0187172i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			3.84136i			0.128117i
$900$		0			0
$901$	21.2287			0.707230
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−28.6376			−0.951946
$906$		0			0
$907$	11.5142			0.382324			0.191162	−	0.981558i	$-0.438774\pi$
$907$	11.5142			0.382324			0.191162	+	0.981558i	$0.438774\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			22.5980i			0.748705i	0.927286	+	0.374353i	$0.122135\pi$
$911$			22.5980i			0.748705i	−0.927286	+	0.374353i	$0.877865\pi$
$912$		0			0
$913$		−	0.147169i		−	0.00487058i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−11.9624			−0.395033
$918$		0			0
$919$	0.315062			0.0103929			0.00519646	−	0.999986i	$-0.498346\pi$
$919$	0.315062			0.0103929			0.00519646	+	0.999986i	$0.498346\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	6.15610i		−	0.202630i
$924$		0			0
$925$		−	3.16219i		−	0.103972i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	45.5324i		−	1.49387i	−0.664897	−	0.746935i	$-0.731524\pi$
$929$		−	45.5324i		−	1.49387i	0.664897	−	0.746935i	$-0.268476\pi$
$930$		0			0
$931$	27.1339			0.889279
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	78.1522i		−	2.55585i
$936$		0			0
$937$			48.3931i			1.58093i	0.612504	+	0.790467i	$0.290162\pi$
$937$			48.3931i			1.58093i	−0.612504	+	0.790467i	$0.709838\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−31.6769			−1.03264			−0.516319	−	0.856397i	$-0.672698\pi$
$941$	−31.6769			−1.03264			−0.516319	+	0.856397i	$0.672698\pi$
$942$		0			0
$943$	47.9179			1.56042
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	30.4342i		−	0.988979i	−0.869184	−	0.494489i	$-0.835355\pi$
$947$		−	30.4342i		−	0.988979i	0.869184	−	0.494489i	$-0.164645\pi$
$948$		0			0
$949$	14.7328			0.478247
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.2749			0.332835			0.166418	−	0.986055i	$-0.446780\pi$
$953$	10.2749			0.332835			0.166418	+	0.986055i	$0.446780\pi$
$954$		0			0
$955$		−	37.8965i		−	1.22630i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	13.3824i		−	0.432139i
$960$		0			0
$961$	−7.69242			−0.248142
$962$		0			0
$963$		0			0
$964$		0			0
$965$			25.2057i			0.811401i
$966$		0			0
$967$	16.4511			0.529032			0.264516	−	0.964381i	$-0.414788\pi$
$967$	16.4511			0.529032			0.264516	+	0.964381i	$0.414788\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.5143			0.401603			0.200802	−	0.979632i	$-0.435645\pi$
$971$	12.5143			0.401603			0.200802	+	0.979632i	$0.435645\pi$
$972$		0			0
$973$		−	4.87733i		−	0.156360i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−28.8557			−0.923174			−0.461587	−	0.887095i	$-0.652720\pi$
$977$	−28.8557			−0.923174			−0.461587	+	0.887095i	$0.652720\pi$
$978$		0			0
$979$	−47.6989			−1.52446
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.58421			−0.146214			−0.0731069	−	0.997324i	$-0.523291\pi$
$983$	−4.58421			−0.146214			−0.0731069	+	0.997324i	$0.523291\pi$
$984$		0			0
$985$	33.7305			1.07474
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	10.7924i		−	0.343178i
$990$		0			0
$991$			0.284587i			0.00904022i	0.999990	+	0.00452011i	$0.00143880\pi$
$991$			0.284587i			0.00904022i	−0.999990	+	0.00452011i	$0.998561\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.72827			−0.245003
$996$		0			0
$997$	50.4186			1.59677			0.798386	−	0.602145i	$-0.205687\pi$
$997$	50.4186			1.59677			0.798386	+	0.602145i	$0.205687\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.h.a.3005.12	yes	56
3.2	odd	2	inner	6012.2.h.a.3005.46	yes	56
167.166	odd	2	inner	6012.2.h.a.3005.45	yes	56
501.500	even	2	inner	6012.2.h.a.3005.11	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.h.a.3005.11	✓	56	501.500	even	2	inner
6012.2.h.a.3005.12	yes	56	1.1	even	1	trivial
6012.2.h.a.3005.45	yes	56	167.166	odd	2	inner
6012.2.h.a.3005.46	yes	56	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.76473 q^{5} +0.618008 q^{7} +O(q^{10})\)	\(q-2.76473 q^{5} +0.618008 q^{7} +5.51365i q^{11} +2.58896i q^{13} +5.12683 q^{17} -4.09998 q^{19} +5.83852 q^{23} +2.64374 q^{25} +0.795676i q^{29} +4.82779 q^{31} -1.70862 q^{35} -1.19611i q^{37} +8.20721 q^{41} -1.84848i q^{43} +1.33447i q^{47} -6.61807 q^{49} +4.14070 q^{53} -15.2438i q^{55} +4.91196 q^{59} -14.6371 q^{61} -7.15778i q^{65} +1.39489i q^{67} -2.37783 q^{71} -5.69062i q^{73} +3.40748i q^{77} +7.74996i q^{79} -0.0266917 q^{83} -14.1743 q^{85} +8.65105i q^{89} +1.60000i q^{91} +11.3353 q^{95} -5.26085 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) \) 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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