LMFDB - Embedded newform 6012.2.h.a.3005.48

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			3005.48
Character	$\chi$	$=$	6012.3005
Dual form			6012.2.h.a.3005.47

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04368 q^{5} +0.233711 q^{7} +O(q^{10})$	$q+3.04368 q^{5} +0.233711 q^{7} +2.11221i q^{11} +1.71375i q^{13} -5.18919 q^{17} -1.94672 q^{19} +2.99706 q^{23} +4.26397 q^{25} +2.73646i q^{29} -8.50175 q^{31} +0.711342 q^{35} +8.21183i q^{37} -3.33880 q^{41} -0.222843i q^{43} +8.47244i q^{47} -6.94538 q^{49} -9.15818 q^{53} +6.42890i q^{55} +10.8700 q^{59} +10.7341 q^{61} +5.21611i q^{65} -0.936932i q^{67} -9.56802 q^{71} +9.59038i q^{73} +0.493649i q^{77} -5.20452i q^{79} +3.37637 q^{83} -15.7942 q^{85} +16.1716i q^{89} +0.400524i q^{91} -5.92520 q^{95} -11.2646 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.04368			1.36117			0.680587	−	0.732668i	$-0.261725\pi$
$5$	3.04368			1.36117			0.680587	+	0.732668i	$0.261725\pi$
$6$		0			0
$7$	0.233711			0.0883346			0.0441673	−	0.999024i	$-0.485937\pi$
$7$	0.233711			0.0883346			0.0441673	+	0.999024i	$0.485937\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.11221i			0.636857i	0.947947	+	0.318428i	$0.103155\pi$
$11$			2.11221i			0.636857i	−0.947947	+	0.318428i	$0.896845\pi$
$12$		0			0
$13$			1.71375i			0.475310i	0.971350	+	0.237655i	$0.0763787\pi$
$13$			1.71375i			0.475310i	−0.971350	+	0.237655i	$0.923621\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.18919			−1.25856			−0.629281	−	0.777178i	$-0.716651\pi$
$17$	−5.18919			−1.25856			−0.629281	+	0.777178i	$0.716651\pi$
$18$		0			0
$19$	−1.94672			−0.446609			−0.223305	−	0.974749i	$-0.571684\pi$
$19$	−1.94672			−0.446609			−0.223305	+	0.974749i	$0.571684\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.99706			0.624931			0.312465	−	0.949929i	$-0.398845\pi$
$23$	2.99706			0.624931			0.312465	+	0.949929i	$0.398845\pi$
$24$		0			0
$25$	4.26397			0.852793
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.73646i			0.508149i	0.967185	+	0.254074i	$0.0817708\pi$
$29$			2.73646i			0.508149i	−0.967185	+	0.254074i	$0.918229\pi$
$30$		0			0
$31$	−8.50175			−1.52696			−0.763480	−	0.645832i	$-0.776511\pi$
$31$	−8.50175			−1.52696			−0.763480	+	0.645832i	$0.776511\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.711342			0.120239
$36$		0			0
$37$			8.21183i			1.35002i	0.737810	+	0.675008i	$0.235860\pi$
$37$			8.21183i			1.35002i	−0.737810	+	0.675008i	$0.764140\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.33880			−0.521433			−0.260717	−	0.965415i	$-0.583959\pi$
$41$	−3.33880			−0.521433			−0.260717	+	0.965415i	$0.583959\pi$
$42$		0			0
$43$		−	0.222843i		−	0.0339833i	−0.999856	−	0.0169916i	$-0.994591\pi$
$43$		−	0.222843i		−	0.0339833i	0.999856	−	0.0169916i	$-0.00540887\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.47244i			1.23583i	0.786244	+	0.617916i	$0.212023\pi$
$47$			8.47244i			1.23583i	−0.786244	+	0.617916i	$0.787977\pi$
$48$		0			0
$49$	−6.94538			−0.992197
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.15818			−1.25797			−0.628987	−	0.777416i	$-0.716530\pi$
$53$	−9.15818			−1.25797			−0.628987	+	0.777416i	$0.716530\pi$
$54$		0			0
$55$			6.42890i			0.866872i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10.8700			1.41516			0.707579	−	0.706634i	$-0.249787\pi$
$59$	10.8700			1.41516			0.707579	+	0.706634i	$0.249787\pi$
$60$		0			0
$61$	10.7341			1.37436			0.687179	−	0.726489i	$-0.258849\pi$
$61$	10.7341			1.37436			0.687179	+	0.726489i	$0.258849\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			5.21611i			0.646979i
$66$		0			0
$67$		−	0.936932i		−	0.114464i	−0.998361	−	0.0572322i	$-0.981772\pi$
$67$		−	0.936932i		−	0.114464i	0.998361	−	0.0572322i	$-0.0182275\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.56802			−1.13551			−0.567757	−	0.823196i	$-0.692189\pi$
$71$	−9.56802			−1.13551			−0.567757	+	0.823196i	$0.692189\pi$
$72$		0			0
$73$			9.59038i			1.12247i	0.827657	+	0.561234i	$0.189673\pi$
$73$			9.59038i			1.12247i	−0.827657	+	0.561234i	$0.810327\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.493649i			0.0562565i
$78$		0			0
$79$		−	5.20452i		−	0.585554i	−0.956181	−	0.292777i	$-0.905421\pi$
$79$		−	5.20452i		−	0.585554i	0.956181	−	0.292777i	$-0.0945794\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.37637			0.370605			0.185303	−	0.982682i	$-0.440673\pi$
$83$	3.37637			0.370605			0.185303	+	0.982682i	$0.440673\pi$
$84$		0			0
$85$	−15.7942			−1.71312
$86$		0			0
$87$		0			0
$88$		0			0
$89$			16.1716i			1.71419i	0.515161	+	0.857093i	$0.327732\pi$
$89$			16.1716i			1.71419i	−0.515161	+	0.857093i	$0.672268\pi$
$90$		0			0
$91$			0.400524i			0.0419863i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.92520			−0.607912
$96$		0			0
$97$	−11.2646			−1.14374			−0.571872	−	0.820343i	$-0.693782\pi$
$97$	−11.2646			−1.14374			−0.571872	+	0.820343i	$0.693782\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.22423			−0.121816			−0.0609080	−	0.998143i	$-0.519400\pi$
$101$	−1.22423			−0.121816			−0.0609080	+	0.998143i	$0.519400\pi$
$102$		0			0
$103$			0.764815i			0.0753594i	0.999290	+	0.0376797i	$0.0119967\pi$
$103$			0.764815i			0.0753594i	−0.999290	+	0.0376797i	$0.988003\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9.90034i		−	0.957102i	−0.878060	−	0.478551i	$-0.841162\pi$
$107$		−	9.90034i		−	0.957102i	0.878060	−	0.478551i	$-0.158838\pi$
$108$		0			0
$109$		−	0.0922689i		−	0.00883776i	−0.999990	−	0.00441888i	$-0.998593\pi$
$109$		−	0.0922689i		−	0.00883776i	0.999990	−	0.00441888i	$-0.00140658\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.87054			0.458182			0.229091	−	0.973405i	$-0.426425\pi$
$113$	4.87054			0.458182			0.229091	+	0.973405i	$0.426425\pi$
$114$		0			0
$115$	9.12209			0.850639
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.21277			−0.111175
$120$		0			0
$121$	6.53855			0.594414
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.24025			−0.200374
$126$		0			0
$127$	1.79810			0.159555			0.0797777	−	0.996813i	$-0.474579\pi$
$127$	1.79810			0.159555			0.0797777	+	0.996813i	$0.474579\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.0140			−0.962299			−0.481150	−	0.876639i	$-0.659781\pi$
$131$	−11.0140			−0.962299			−0.481150	+	0.876639i	$0.659781\pi$
$132$		0			0
$133$	−0.454972			−0.0394510
$134$		0			0
$135$		0			0
$136$		0			0
$137$			6.63132i			0.566552i	0.959038	+	0.283276i	$0.0914212\pi$
$137$			6.63132i			0.566552i	−0.959038	+	0.283276i	$0.908579\pi$
$138$		0			0
$139$			21.6598i			1.83716i	0.395236	+	0.918580i	$0.370663\pi$
$139$			21.6598i			1.83716i	−0.395236	+	0.918580i	$0.629337\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.61981			−0.302704
$144$		0			0
$145$			8.32891i			0.691678i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−17.6000			−1.44184			−0.720922	−	0.693016i	$-0.756282\pi$
$149$	−17.6000			−1.44184			−0.720922	+	0.693016i	$0.756282\pi$
$150$		0			0
$151$			23.5281i			1.91469i	0.288953	+	0.957343i	$0.406693\pi$
$151$			23.5281i			1.91469i	−0.288953	+	0.957343i	$0.593307\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−25.8766			−2.07846
$156$		0			0
$157$	−18.4566			−1.47300			−0.736499	−	0.676439i	$-0.763522\pi$
$157$	−18.4566			−1.47300			−0.736499	+	0.676439i	$0.763522\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.700448			0.0552030
$162$		0			0
$163$		−	16.5832i		−	1.29889i	−0.760407	−	0.649447i	$-0.775000\pi$
$163$		−	16.5832i		−	1.29889i	0.760407	−	0.649447i	$-0.225000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.8060	+	1.73367i	0.990960	+	0.134155i
$167$	12.8060	+	1.73367i	0.990960	+	0.134155i
$168$		0			0
$169$	10.0631			0.774081
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	25.6712i		−	1.95175i	−0.218332	−	0.975874i	$-0.570062\pi$
$173$		−	25.6712i		−	1.95175i	0.218332	−	0.975874i	$-0.429938\pi$
$174$		0			0
$175$	0.996537			0.0753311
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	17.9325i		−	1.34034i	−0.742208	−	0.670169i	$-0.766222\pi$
$179$		−	17.9325i		−	1.34034i	0.742208	−	0.670169i	$-0.233778\pi$
$180$		0			0
$181$	21.2042			1.57610			0.788049	−	0.615612i	$-0.211091\pi$
$181$	21.2042			1.57610			0.788049	+	0.615612i	$0.211091\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			24.9942i			1.83761i
$186$		0			0
$187$		−	10.9607i		−	0.801524i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	26.4287i		−	1.91231i	−0.292856	−	0.956157i	$-0.594606\pi$
$191$		−	26.4287i		−	1.91231i	0.292856	−	0.956157i	$-0.405394\pi$
$192$		0			0
$193$			13.7744i			0.991506i	0.868464	+	0.495753i	$0.165108\pi$
$193$			13.7744i			0.991506i	−0.868464	+	0.495753i	$0.834892\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.33812			0.665314			0.332657	−	0.943048i	$-0.392055\pi$
$197$	9.33812			0.665314			0.332657	+	0.943048i	$0.392055\pi$
$198$		0			0
$199$	16.7492			1.18732			0.593659	−	0.804717i	$-0.297683\pi$
$199$	16.7492			1.18732			0.593659	+	0.804717i	$0.297683\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			0.639543i			0.0448871i
$204$		0			0
$205$	−10.1622			−0.709761
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	4.11190i		−	0.284426i
$210$		0			0
$211$	13.9376			0.959504			0.479752	−	0.877404i	$-0.340727\pi$
$211$	13.9376			0.959504			0.479752	+	0.877404i	$0.340727\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	0.678263i		−	0.0462571i
$216$		0			0
$217$	−1.98696			−0.134883
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	8.89298i		−	0.598207i
$222$		0			0
$223$	12.9899			0.869865			0.434933	−	0.900463i	$-0.356772\pi$
$223$	12.9899			0.869865			0.434933	+	0.900463i	$0.356772\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.33538			0.619611			0.309805	−	0.950800i	$-0.399736\pi$
$227$	9.33538			0.619611			0.309805	+	0.950800i	$0.399736\pi$
$228$		0			0
$229$	26.0496			1.72141			0.860703	−	0.509107i	$-0.170024\pi$
$229$	26.0496			1.72141			0.860703	+	0.509107i	$0.170024\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1.10871i		−	0.0726338i	−0.999340	−	0.0363169i	$-0.988437\pi$
$233$		−	1.10871i		−	0.0726338i	0.999340	−	0.0363169i	$-0.0115626\pi$
$234$		0			0
$235$			25.7874i			1.68218i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	12.4479i		−	0.805184i	−0.915379	−	0.402592i	$-0.868109\pi$
$239$		−	12.4479i		−	0.805184i	0.915379	−	0.402592i	$-0.131891\pi$
$240$		0			0
$241$		−	11.4845i		−	0.739784i	−0.929075	−	0.369892i	$-0.879395\pi$
$241$		−	11.4845i		−	0.739784i	0.929075	−	0.369892i	$-0.120605\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−21.1395			−1.35055
$246$		0			0
$247$		−	3.33620i		−	0.212278i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			16.3240i			1.03036i	0.857082	+	0.515181i	$0.172275\pi$
$251$			16.3240i			1.03036i	−0.857082	+	0.515181i	$0.827725\pi$
$252$		0			0
$253$			6.33044i			0.397991i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−18.6077			−1.16072			−0.580359	−	0.814361i	$-0.697088\pi$
$257$	−18.6077			−1.16072			−0.580359	+	0.814361i	$0.697088\pi$
$258$		0			0
$259$			1.91920i			0.119253i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	16.3510i		−	1.00825i	−0.863631	−	0.504124i	$-0.831815\pi$
$263$		−	16.3510i		−	1.00825i	0.863631	−	0.504124i	$-0.168185\pi$
$264$		0			0
$265$	−27.8745			−1.71232
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.7864			−1.38931			−0.694657	−	0.719341i	$-0.744444\pi$
$269$	−22.7864			−1.38931			−0.694657	+	0.719341i	$0.744444\pi$
$270$		0			0
$271$			13.2759i			0.806453i	0.915100	+	0.403226i	$0.132111\pi$
$271$			13.2759i			0.806453i	−0.915100	+	0.403226i	$0.867889\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			9.00641i			0.543107i
$276$		0			0
$277$			24.9639i			1.49994i	0.661473	+	0.749969i	$0.269932\pi$
$277$			24.9639i			1.49994i	−0.661473	+	0.749969i	$0.730068\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			13.7440i			0.819898i	0.912108	+	0.409949i	$0.134454\pi$
$281$			13.7440i			0.819898i	−0.912108	+	0.409949i	$0.865546\pi$
$282$		0			0
$283$	4.64056			0.275853			0.137926	−	0.990442i	$-0.455956\pi$
$283$	4.64056			0.275853			0.137926	+	0.990442i	$0.455956\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.780316			−0.0460606
$288$		0			0
$289$	9.92764			0.583979
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8.10870i		−	0.473715i	−0.971544	−	0.236858i	$-0.923883\pi$
$293$		−	8.10870i		−	0.473715i	0.971544	−	0.236858i	$-0.0761175\pi$
$294$		0			0
$295$	33.0849			1.92627
$296$		0			0
$297$		0			0
$298$		0			0
$299$			5.13622i			0.297035i
$300$		0			0
$301$		−	0.0520810i		−	0.00300190i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	32.6710			1.87074
$306$		0			0
$307$		−	19.9446i		−	1.13830i	−0.822234	−	0.569150i	$-0.807272\pi$
$307$		−	19.9446i		−	1.13830i	0.822234	−	0.569150i	$-0.192728\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	3.07490i		−	0.174362i	−0.996193	−	0.0871808i	$-0.972214\pi$
$311$		−	3.07490i		−	0.174362i	0.996193	−	0.0871808i	$-0.0277858\pi$
$312$		0			0
$313$			8.69801i			0.491641i	0.969315	+	0.245820i	$0.0790573\pi$
$313$			8.69801i			0.491641i	−0.969315	+	0.245820i	$0.920943\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	0.795282i		−	0.0446675i	−0.999751	−	0.0223337i	$-0.992890\pi$
$317$		−	0.795282i		−	0.0446675i	0.999751	−	0.0223337i	$-0.00710964\pi$
$318$		0			0
$319$	−5.78000			−0.323618
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.1019			0.562085
$324$		0			0
$325$			7.30738i			0.405341i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			1.98011i			0.109167i
$330$		0			0
$331$			14.1043i			0.775240i	0.921819	+	0.387620i	$0.126703\pi$
$331$			14.1043i			0.775240i	−0.921819	+	0.387620i	$0.873297\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	2.85172i		−	0.155806i
$336$		0			0
$337$	−12.0058			−0.653998			−0.326999	−	0.945025i	$-0.606037\pi$
$337$	−12.0058			−0.653998			−0.326999	+	0.945025i	$0.606037\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	17.9575i		−	0.972454i
$342$		0			0
$343$	−3.25919			−0.175980
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−26.5883			−1.42734			−0.713668	−	0.700484i	$-0.752968\pi$
$347$	−26.5883			−1.42734			−0.713668	+	0.700484i	$0.752968\pi$
$348$		0			0
$349$			29.7666i			1.59337i	0.604393	+	0.796686i	$0.293416\pi$
$349$			29.7666i			1.59337i	−0.604393	+	0.796686i	$0.706584\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			31.1203i			1.65637i	0.560456	+	0.828184i	$0.310626\pi$
$353$			31.1203i			1.65637i	−0.560456	+	0.828184i	$0.689374\pi$
$354$		0			0
$355$	−29.1219			−1.54563
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	21.2836i		−	1.12330i	−0.827374	−	0.561652i	$-0.810166\pi$
$359$		−	21.2836i		−	1.12330i	0.827374	−	0.561652i	$-0.189834\pi$
$360$		0			0
$361$	−15.2103			−0.800540
$362$		0			0
$363$		0			0
$364$		0			0
$365$			29.1900i			1.52787i
$366$		0			0
$367$	7.18368			0.374985			0.187492	−	0.982266i	$-0.439964\pi$
$367$	7.18368			0.374985			0.187492	+	0.982266i	$0.439964\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.14037			−0.111123
$372$		0			0
$373$			12.3158i			0.637689i	0.947807	+	0.318844i	$0.103295\pi$
$373$			12.3158i			0.637689i	−0.947807	+	0.318844i	$0.896705\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.68962			−0.241528
$378$		0			0
$379$			12.8067i			0.657834i	0.944359	+	0.328917i	$0.106684\pi$
$379$			12.8067i			0.657834i	−0.944359	+	0.328917i	$0.893316\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	26.0935i		−	1.33332i	−0.745363	−	0.666659i	$-0.767724\pi$
$383$		−	26.0935i		−	1.33332i	0.745363	−	0.666659i	$-0.232276\pi$
$384$		0			0
$385$			1.50251i			0.0765748i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.1524			−0.768257			−0.384129	−	0.923280i	$-0.625498\pi$
$389$	−15.1524			−0.768257			−0.384129	+	0.923280i	$0.625498\pi$
$390$		0			0
$391$	−15.5523			−0.786514
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	15.8409i		−	0.797041i
$396$		0			0
$397$	21.1906			1.06353			0.531764	−	0.846893i	$-0.321530\pi$
$397$	21.1906			1.06353			0.531764	+	0.846893i	$0.321530\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0819			0.902965			0.451482	−	0.892280i	$-0.350895\pi$
$401$	18.0819			0.902965			0.451482	+	0.892280i	$0.350895\pi$
$402$		0			0
$403$		−	14.5699i		−	0.725778i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−17.3451			−0.859767
$408$		0			0
$409$	−2.21828			−0.109687			−0.0548434	−	0.998495i	$-0.517466\pi$
$409$	−2.21828			−0.109687			−0.0548434	+	0.998495i	$0.517466\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.54045			0.125007
$414$		0			0
$415$	10.2766			0.504458
$416$		0			0
$417$		0			0
$418$		0			0
$419$			1.24647i			0.0608939i	0.999536	+	0.0304469i	$0.00969306\pi$
$419$			1.24647i			0.0608939i	−0.999536	+	0.0304469i	$0.990307\pi$
$420$		0			0
$421$	4.24823			0.207046			0.103523	−	0.994627i	$-0.466988\pi$
$421$	4.24823			0.207046			0.103523	+	0.994627i	$0.466988\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−22.1265			−1.07329
$426$		0			0
$427$	2.50867			0.121403
$428$		0			0
$429$		0			0
$430$		0			0
$431$			22.3570i			1.07690i	0.842658	+	0.538448i	$0.180989\pi$
$431$			22.3570i			1.07690i	−0.842658	+	0.538448i	$0.819011\pi$
$432$		0			0
$433$	11.6183			0.558341			0.279171	−	0.960241i	$-0.409940\pi$
$433$	11.6183			0.558341			0.279171	+	0.960241i	$0.409940\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.83445			−0.279100
$438$		0			0
$439$		−	39.7712i		−	1.89817i	−0.315014	−	0.949087i	$-0.602009\pi$
$439$		−	39.7712i		−	1.89817i	0.315014	−	0.949087i	$-0.397991\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.7204			−0.794408			−0.397204	−	0.917730i	$-0.630019\pi$
$443$	−16.7204			−0.794408			−0.397204	+	0.917730i	$0.630019\pi$
$444$		0			0
$445$			49.2211i			2.33330i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			14.8333i			0.700028i	0.936744	+	0.350014i	$0.113823\pi$
$449$			14.8333i			0.700028i	−0.936744	+	0.350014i	$0.886177\pi$
$450$		0			0
$451$		−	7.05226i		−	0.332078i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			1.21906i			0.0571506i
$456$		0			0
$457$			29.3212i			1.37159i	0.727795	+	0.685794i	$0.240545\pi$
$457$			29.3212i			1.37159i	−0.727795	+	0.685794i	$0.759455\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	0.232415i		−	0.0108247i	−0.999985	−	0.00541233i	$-0.998277\pi$
$461$		−	0.232415i		−	0.0108247i	0.999985	−	0.00541233i	$-0.00172281\pi$
$462$		0			0
$463$			27.5746i			1.28150i	0.767749	+	0.640750i	$0.221377\pi$
$463$			27.5746i			1.28150i	−0.767749	+	0.640750i	$0.778623\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	11.6372i		−	0.538507i	−0.963069	−	0.269254i	$-0.913223\pi$
$467$		−	11.6372i		−	0.538507i	0.963069	−	0.269254i	$-0.0867770\pi$
$468$		0			0
$469$		−	0.218972i		−	0.0101112i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.470693			0.0216425
$474$		0			0
$475$	−8.30076			−0.380865
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.9459			0.500129			0.250065	−	0.968229i	$-0.419548\pi$
$479$	10.9459			0.500129			0.250065	+	0.968229i	$0.419548\pi$
$480$		0			0
$481$	−14.0730			−0.641676
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−34.2857			−1.55683
$486$		0			0
$487$		−	16.4976i		−	0.747579i	−0.927514	−	0.373789i	$-0.878058\pi$
$487$		−	16.4976i		−	0.747579i	0.927514	−	0.373789i	$-0.121942\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			24.7588i			1.11735i	0.829387	+	0.558674i	$0.188690\pi$
$491$			24.7588i			1.11735i	−0.829387	+	0.558674i	$0.811310\pi$
$492$		0			0
$493$		−	14.2000i		−	0.639537i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.23615			−0.100305
$498$		0			0
$499$		−	26.5947i		−	1.19054i	−0.803525	−	0.595272i	$-0.797045\pi$
$499$		−	26.5947i		−	1.19054i	0.803525	−	0.595272i	$-0.202955\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			39.4288i			1.75804i	0.476782	+	0.879022i	$0.341803\pi$
$503$			39.4288i			1.75804i	−0.476782	+	0.879022i	$0.658197\pi$
$504$		0			0
$505$	−3.72617			−0.165813
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	33.1756i		−	1.47048i	−0.677806	−	0.735241i	$-0.737069\pi$
$509$		−	33.1756i		−	1.47048i	0.677806	−	0.735241i	$-0.262931\pi$
$510$		0			0
$511$			2.24138i			0.0991528i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			2.32785i			0.102577i
$516$		0			0
$517$	−17.8956			−0.787048
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.16160			0.226134			0.113067	−	0.993587i	$-0.463933\pi$
$521$	5.16160			0.226134			0.113067	+	0.993587i	$0.463933\pi$
$522$		0			0
$523$	6.41931			0.280697			0.140348	−	0.990102i	$-0.455178\pi$
$523$	6.41931			0.280697			0.140348	+	0.990102i	$0.455178\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	44.1171			1.92177
$528$		0			0
$529$	−14.0176			−0.609462
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	5.72188i		−	0.247842i
$534$		0			0
$535$		−	30.1334i		−	1.30278i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	14.6701i		−	0.631887i
$540$		0			0
$541$			21.4097i			0.920476i	0.887796	+	0.460238i	$0.152236\pi$
$541$			21.4097i			0.920476i	−0.887796	+	0.460238i	$0.847764\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	0.280837i		−	0.0120297i
$546$		0			0
$547$		−	4.02545i		−	0.172116i	−0.996290	−	0.0860579i	$-0.972573\pi$
$547$		−	4.02545i		−	0.172116i	0.996290	−	0.0860579i	$-0.0274270\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	5.32714i		−	0.226944i
$552$		0			0
$553$		−	1.21636i		−	0.0517247i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			36.5466i			1.54853i	0.632861	+	0.774265i	$0.281880\pi$
$557$			36.5466i			1.54853i	−0.632861	+	0.774265i	$0.718120\pi$
$558$		0			0
$559$	0.381898			0.0161526
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	14.6535i		−	0.617571i	−0.951132	−	0.308786i	$-0.900077\pi$
$563$		−	14.6535i		−	0.617571i	0.951132	−	0.308786i	$-0.0999225\pi$
$564$		0			0
$565$	14.8243			0.623665
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−33.3881			−1.39970			−0.699850	−	0.714290i	$-0.746750\pi$
$569$	−33.3881			−1.39970			−0.699850	+	0.714290i	$0.746750\pi$
$570$		0			0
$571$		−	25.9499i		−	1.08597i	−0.839743	−	0.542984i	$-0.817294\pi$
$571$		−	25.9499i		−	1.08597i	0.839743	−	0.542984i	$-0.182706\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	12.7794			0.532936
$576$		0			0
$577$	22.5037			0.936840			0.468420	−	0.883506i	$-0.344824\pi$
$577$	22.5037			0.936840			0.468420	+	0.883506i	$0.344824\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.789097			0.0327373
$582$		0			0
$583$		−	19.3440i		−	0.801149i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.12151			−0.128838			−0.0644192	−	0.997923i	$-0.520519\pi$
$587$	−3.12151			−0.128838			−0.0644192	+	0.997923i	$0.520519\pi$
$588$		0			0
$589$	16.5506			0.681954
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.01684			0.206017			0.103009	−	0.994680i	$-0.467153\pi$
$593$	5.01684			0.206017			0.103009	+	0.994680i	$0.467153\pi$
$594$		0			0
$595$	−3.69128			−0.151328
$596$		0			0
$597$		0			0
$598$		0			0
$599$			41.9042i			1.71216i	0.516844	+	0.856080i	$0.327107\pi$
$599$			41.9042i			1.71216i	−0.516844	+	0.856080i	$0.672893\pi$
$600$		0			0
$601$	−16.0514			−0.654751			−0.327375	−	0.944894i	$-0.606164\pi$
$601$	−16.0514			−0.654751			−0.327375	+	0.944894i	$0.606164\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	19.9012			0.809100
$606$		0			0
$607$			6.91136i			0.280523i	0.990114	+	0.140262i	$0.0447944\pi$
$607$			6.91136i			0.280523i	−0.990114	+	0.140262i	$0.955206\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14.5197			−0.587403
$612$		0			0
$613$	22.9693			0.927723			0.463861	−	0.885908i	$-0.346464\pi$
$613$	22.9693			0.927723			0.463861	+	0.885908i	$0.346464\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			0.885982i			0.0356683i	0.999841	+	0.0178342i	$0.00567709\pi$
$617$			0.885982i			0.0356683i	−0.999841	+	0.0178342i	$0.994323\pi$
$618$		0			0
$619$		−	16.5451i		−	0.665005i	−0.943102	−	0.332503i	$-0.892107\pi$
$619$		−	16.5451i		−	0.665005i	0.943102	−	0.332503i	$-0.107893\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			3.77949i			0.151422i
$624$		0			0
$625$	−28.1384			−1.12554
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	42.6127i		−	1.69908i
$630$		0			0
$631$	−32.2675			−1.28455			−0.642274	−	0.766475i	$-0.722009\pi$
$631$	−32.2675			−1.28455			−0.642274	+	0.766475i	$0.722009\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5.47283			0.217183
$636$		0			0
$637$		−	11.9027i		−	0.471601i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	7.26760			0.287053			0.143526	−	0.989646i	$-0.454156\pi$
$641$	7.26760			0.287053			0.143526	+	0.989646i	$0.454156\pi$
$642$		0			0
$643$			11.1194i			0.438507i	0.975668	+	0.219254i	$0.0703622\pi$
$643$			11.1194i			0.438507i	−0.975668	+	0.219254i	$0.929638\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	38.0483			1.49583			0.747917	−	0.663792i	$-0.231054\pi$
$647$	38.0483			1.49583			0.747917	+	0.663792i	$0.231054\pi$
$648$		0			0
$649$			22.9598i			0.901253i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	49.0410i		−	1.91912i	−0.281498	−	0.959562i	$-0.590831\pi$
$653$		−	49.0410i		−	1.91912i	0.281498	−	0.959562i	$-0.409169\pi$
$654$		0			0
$655$	−33.5231			−1.30986
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.928331			0.0361626			0.0180813	−	0.999837i	$-0.494244\pi$
$659$	0.928331			0.0361626			0.0180813	+	0.999837i	$0.494244\pi$
$660$		0			0
$661$		−	6.60700i		−	0.256983i	−0.991711	−	0.128491i	$-0.958987\pi$
$661$		−	6.60700i		−	0.256983i	0.991711	−	0.128491i	$-0.0410134\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.38479			−0.0536997
$666$		0			0
$667$			8.20135i			0.317558i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			22.6727i			0.875268i
$672$		0			0
$673$			18.6494i			0.718881i	0.933168	+	0.359440i	$0.117032\pi$
$673$			18.6494i			0.718881i	−0.933168	+	0.359440i	$0.882968\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			40.2712i			1.54775i	0.633340	+	0.773874i	$0.281684\pi$
$677$			40.2712i			1.54775i	−0.633340	+	0.773874i	$0.718316\pi$
$678$		0			0
$679$	−2.63266			−0.101032
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.3335			0.586720			0.293360	−	0.956002i	$-0.405227\pi$
$683$	15.3335			0.586720			0.293360	+	0.956002i	$0.405227\pi$
$684$		0			0
$685$			20.1836i			0.771175i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	15.6949i		−	0.597927i
$690$		0			0
$691$			20.9822i			0.798201i	0.916907	+	0.399101i	$0.130678\pi$
$691$			20.9822i			0.798201i	−0.916907	+	0.399101i	$0.869322\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			65.9254i			2.50069i
$696$		0			0
$697$	17.3257			0.656256
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	8.00186i		−	0.302226i	−0.988516	−	0.151113i	$-0.951714\pi$
$701$		−	8.00186i		−	0.302226i	0.988516	−	0.151113i	$-0.0482858\pi$
$702$		0			0
$703$		−	15.9862i		−	0.602930i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.286118			−0.0107606
$708$		0			0
$709$			19.1316i			0.718502i	0.933241	+	0.359251i	$0.116968\pi$
$709$			19.1316i			0.718502i	−0.933241	+	0.359251i	$0.883032\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−25.4803			−0.954243
$714$		0			0
$715$	−11.0175			−0.412033
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.562268			−0.0209691			−0.0104845	−	0.999945i	$-0.503337\pi$
$719$	−0.562268			−0.0209691			−0.0104845	+	0.999945i	$0.503337\pi$
$720$		0			0
$721$			0.178746i			0.00665684i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			11.6682i			0.433346i
$726$		0			0
$727$			23.7745i			0.881747i	0.897569	+	0.440874i	$0.145331\pi$
$727$			23.7745i			0.881747i	−0.897569	+	0.440874i	$0.854669\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			1.15638i			0.0427701i
$732$		0			0
$733$	14.6661			0.541705			0.270853	−	0.962621i	$-0.412694\pi$
$733$	14.6661			0.541705			0.270853	+	0.962621i	$0.412694\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.97900			0.0728974
$738$		0			0
$739$			11.2062i			0.412225i	0.978528	+	0.206113i	$0.0660813\pi$
$739$			11.2062i			0.412225i	−0.978528	+	0.206113i	$0.933919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			18.3715i			0.673985i	0.941507	+	0.336992i	$0.109410\pi$
$743$			18.3715i			0.673985i	−0.941507	+	0.336992i	$0.890590\pi$
$744$		0			0
$745$	−53.5686			−1.96260
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	2.31382i		−	0.0845453i
$750$		0			0
$751$			28.1684i			1.02788i	0.857826	+	0.513940i	$0.171815\pi$
$751$			28.1684i			1.02788i	−0.857826	+	0.513940i	$0.828185\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			71.6118i			2.60622i
$756$		0			0
$757$	16.5297			0.600783			0.300392	−	0.953816i	$-0.402883\pi$
$757$	16.5297			0.600783			0.300392	+	0.953816i	$0.402883\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			21.7874i			0.789792i	0.918726	+	0.394896i	$0.129219\pi$
$761$			21.7874i			0.789792i	−0.918726	+	0.394896i	$0.870781\pi$
$762$		0			0
$763$		−	0.0215643i		−	0.000780680i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			18.6285i			0.672638i
$768$		0			0
$769$		−	47.7462i		−	1.72177i	−0.508799	−	0.860885i	$-0.669910\pi$
$769$		−	47.7462i		−	1.72177i	0.508799	−	0.860885i	$-0.330090\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.34511			0.0843478			0.0421739	−	0.999110i	$-0.486572\pi$
$773$	2.34511			0.0843478			0.0421739	+	0.999110i	$0.486572\pi$
$774$		0			0
$775$	−36.2512			−1.30218
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.49972			0.232877
$780$		0			0
$781$		−	20.2097i		−	0.723160i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−56.1759			−2.00500
$786$		0			0
$787$			28.3179i			1.00942i	0.863288	+	0.504711i	$0.168401\pi$
$787$			28.3179i			1.00942i	−0.863288	+	0.504711i	$0.831599\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.13830			0.0404733
$792$		0			0
$793$			18.3955i			0.653245i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.3671			−0.402642			−0.201321	−	0.979525i	$-0.564523\pi$
$797$	−11.3671			−0.402642			−0.201321	+	0.979525i	$0.564523\pi$
$798$		0			0
$799$		−	43.9651i		−	1.55537i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.2569			−0.714852
$804$		0			0
$805$	2.13194			0.0751409
$806$		0			0
$807$		0			0
$808$		0			0
$809$			27.1713i			0.955292i	0.878552	+	0.477646i	$0.158510\pi$
$809$			27.1713i			0.955292i	−0.878552	+	0.477646i	$0.841490\pi$
$810$		0			0
$811$		−	17.0530i		−	0.598812i	−0.954126	−	0.299406i	$-0.903211\pi$
$811$		−	17.0530i		−	0.598812i	0.954126	−	0.299406i	$-0.0967885\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	50.4738i		−	1.76802i
$816$		0			0
$817$			0.433814i			0.0151772i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.44058			0.189878			0.0949388	−	0.995483i	$-0.469734\pi$
$821$	5.44058			0.189878			0.0949388	+	0.995483i	$0.469734\pi$
$822$		0			0
$823$			23.9319i			0.834214i	0.908857	+	0.417107i	$0.136956\pi$
$823$			23.9319i			0.834214i	−0.908857	+	0.417107i	$0.863044\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	34.7390			1.20799			0.603997	−	0.796987i	$-0.293574\pi$
$827$	34.7390			1.20799			0.603997	+	0.796987i	$0.293574\pi$
$828$		0			0
$829$		−	52.6778i		−	1.82958i	−0.403934	−	0.914788i	$-0.632357\pi$
$829$		−	52.6778i		−	1.82958i	0.403934	−	0.914788i	$-0.367643\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	36.0409			1.24874
$834$		0			0
$835$	38.9774	+	5.27672i	1.34887	+	0.182608i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			13.3802i			0.461937i	0.972961	+	0.230969i	$0.0741895\pi$
$839$			13.3802i			0.461937i	−0.972961	+	0.230969i	$0.925811\pi$
$840$		0			0
$841$	21.5118			0.741785
$842$		0			0
$843$		0			0
$844$		0			0
$845$	30.6287			1.05366
$846$		0			0
$847$	1.52813			0.0525073
$848$		0			0
$849$		0			0
$850$		0			0
$851$			24.6114i			0.843667i
$852$		0			0
$853$	−27.8527			−0.953658			−0.476829	−	0.878996i	$-0.658214\pi$
$853$	−27.8527			−0.953658			−0.476829	+	0.878996i	$0.658214\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	44.3918i		−	1.51640i	−0.652024	−	0.758198i	$-0.726080\pi$
$857$		−	44.3918i		−	1.51640i	0.652024	−	0.758198i	$-0.273920\pi$
$858$		0			0
$859$	16.2278			0.553685			0.276842	−	0.960915i	$-0.410712\pi$
$859$	16.2278			0.553685			0.276842	+	0.960915i	$0.410712\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	41.3503i		−	1.40758i	−0.710407	−	0.703791i	$-0.751489\pi$
$863$		−	41.3503i		−	1.40758i	0.710407	−	0.703791i	$-0.248511\pi$
$864$		0			0
$865$		−	78.1350i		−	2.65667i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.9931			0.372914
$870$		0			0
$871$	1.60567			0.0544060
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.523572			−0.0177000
$876$		0			0
$877$	32.5337			1.09858			0.549292	−	0.835631i	$-0.314898\pi$
$877$	32.5337			1.09858			0.549292	+	0.835631i	$0.314898\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−22.7864			−0.767692			−0.383846	−	0.923397i	$-0.625401\pi$
$881$	−22.7864			−0.767692			−0.383846	+	0.923397i	$0.625401\pi$
$882$		0			0
$883$	−4.41110			−0.148445			−0.0742227	−	0.997242i	$-0.523648\pi$
$883$	−4.41110			−0.148445			−0.0742227	+	0.997242i	$0.523648\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	44.3070			1.48768			0.743842	−	0.668355i	$-0.233001\pi$
$887$	44.3070			1.48768			0.743842	+	0.668355i	$0.233001\pi$
$888$		0			0
$889$	0.420236			0.0140943
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	16.4935i		−	0.551934i
$894$		0			0
$895$		−	54.5807i		−	1.82443i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	23.2647i		−	0.775922i
$900$		0			0
$901$	47.5235			1.58324
$902$		0			0
$903$		0			0
$904$		0			0
$905$	64.5388			2.14534
$906$		0			0
$907$	−26.2595			−0.871931			−0.435966	−	0.899963i	$-0.643593\pi$
$907$	−26.2595			−0.871931			−0.435966	+	0.899963i	$0.643593\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	19.4779i		−	0.645332i	−0.946513	−	0.322666i	$-0.895421\pi$
$911$		−	19.4779i		−	0.645332i	0.946513	−	0.322666i	$-0.104579\pi$
$912$		0			0
$913$			7.13162i			0.236022i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.57410			−0.0850043
$918$		0			0
$919$	−16.9731			−0.559889			−0.279945	−	0.960016i	$-0.590316\pi$
$919$	−16.9731			−0.559889			−0.279945	+	0.960016i	$0.590316\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	16.3972i		−	0.539721i
$924$		0			0
$925$			35.0150i			1.15128i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	16.0513i		−	0.526625i	−0.964711	−	0.263312i	$-0.915185\pi$
$929$		−	16.0513i		−	0.526625i	0.964711	−	0.263312i	$-0.0848150\pi$
$930$		0			0
$931$	13.5207			0.443124
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	33.3607i		−	1.09101i
$936$		0			0
$937$			10.4512i			0.341426i	0.985321	+	0.170713i	$0.0546071\pi$
$937$			10.4512i			0.341426i	−0.985321	+	0.170713i	$0.945393\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	36.4202			1.18726			0.593632	−	0.804736i	$-0.297693\pi$
$941$	36.4202			1.18726			0.593632	+	0.804736i	$0.297693\pi$
$942$		0			0
$943$	−10.0066			−0.325860
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	24.1342i		−	0.784257i	−0.919910	−	0.392129i	$-0.871739\pi$
$947$		−	24.1342i		−	0.784257i	0.919910	−	0.392129i	$-0.128261\pi$
$948$		0			0
$949$	−16.4355			−0.533520
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.15154			−0.0696953			−0.0348476	−	0.999393i	$-0.511095\pi$
$953$	−2.15154			−0.0696953			−0.0348476	+	0.999393i	$0.511095\pi$
$954$		0			0
$955$		−	80.4404i		−	2.60299i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1.54981i			0.0500461i
$960$		0			0
$961$	41.2797			1.33160
$962$		0			0
$963$		0			0
$964$		0			0
$965$			41.9249i			1.34961i
$966$		0			0
$967$	−54.2527			−1.74465			−0.872324	−	0.488929i	$-0.837388\pi$
$967$	−54.2527			−1.74465			−0.872324	+	0.488929i	$0.837388\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	33.3533			1.07036			0.535180	−	0.844738i	$-0.320244\pi$
$971$	33.3533			1.07036			0.535180	+	0.844738i	$0.320244\pi$
$972$		0			0
$973$			5.06214i			0.162285i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.8130			0.473909			0.236955	−	0.971521i	$-0.423851\pi$
$977$	14.8130			0.473909			0.236955	+	0.971521i	$0.423851\pi$
$978$		0			0
$979$	−34.1579			−1.09169
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−30.3384			−0.967644			−0.483822	−	0.875166i	$-0.660752\pi$
$983$	−30.3384			−0.967644			−0.483822	+	0.875166i	$0.660752\pi$
$984$		0			0
$985$	28.4222			0.905607
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	0.667875i		−	0.0212372i
$990$		0			0
$991$		−	29.4997i		−	0.937088i	−0.883440	−	0.468544i	$-0.844779\pi$
$991$		−	29.4997i		−	0.937088i	0.883440	−	0.468544i	$-0.155221\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	50.9790			1.61614
$996$		0			0
$997$	−32.6928			−1.03539			−0.517696	−	0.855565i	$-0.673210\pi$
$997$	−32.6928			−1.03539			−0.517696	+	0.855565i	$0.673210\pi$
$998$		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.3005.48	yes	56
3.2	odd	2	inner	6012.2.h.a.3005.9	✓	56
167.166	odd	2	inner	6012.2.h.a.3005.10	yes	56
501.500	even	2	inner	6012.2.h.a.3005.47	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.h.a.3005.9	✓	56	3.2	odd	2	inner
6012.2.h.a.3005.10	yes	56	167.166	odd	2	inner
6012.2.h.a.3005.47	yes	56	501.500	even	2	inner
6012.2.h.a.3005.48	yes	56	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+3.04368 q^{5} +0.233711 q^{7} +O(q^{10})\)	\(q+3.04368 q^{5} +0.233711 q^{7} +2.11221i q^{11} +1.71375i q^{13} -5.18919 q^{17} -1.94672 q^{19} +2.99706 q^{23} +4.26397 q^{25} +2.73646i q^{29} -8.50175 q^{31} +0.711342 q^{35} +8.21183i q^{37} -3.33880 q^{41} -0.222843i q^{43} +8.47244i q^{47} -6.94538 q^{49} -9.15818 q^{53} +6.42890i q^{55} +10.8700 q^{59} +10.7341 q^{61} +5.21611i q^{65} -0.936932i q^{67} -9.56802 q^{71} +9.59038i q^{73} +0.493649i q^{77} -5.20452i q^{79} +3.37637 q^{83} -15.7942 q^{85} +16.1716i q^{89} +0.400524i q^{91} -5.92520 q^{95} -11.2646 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 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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