LMFDB - Embedded newform 6012.2.h.a.3005.14

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			3005.14
Character	$\chi$	$=$	6012.3005
Dual form			6012.2.h.a.3005.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21423 q^{5} -1.10661 q^{7} +O(q^{10})$	$q-2.21423 q^{5} -1.10661 q^{7} -1.52623i q^{11} +3.26430i q^{13} -5.68069 q^{17} +6.25231 q^{19} +4.86296 q^{23} -0.0972042 q^{25} -8.93164i q^{29} -1.89423 q^{31} +2.45029 q^{35} +4.41435i q^{37} -2.75307 q^{41} +0.818459i q^{43} +3.00307i q^{47} -5.77541 q^{49} -9.36175 q^{53} +3.37942i q^{55} -4.10719 q^{59} -12.1085 q^{61} -7.22790i q^{65} -9.17056i q^{67} +10.9769 q^{71} -5.25379i q^{73} +1.68895i q^{77} +11.5472i q^{79} +5.75501 q^{83} +12.5783 q^{85} -8.72974i q^{89} -3.61231i q^{91} -13.8440 q^{95} +16.1024 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.21423			−0.990232			−0.495116	−	0.868827i	$-0.664874\pi$
$5$	−2.21423			−0.990232			−0.495116	+	0.868827i	$0.664874\pi$
$6$		0			0
$7$	−1.10661			−0.418260			−0.209130	−	0.977888i	$-0.567063\pi$
$7$	−1.10661			−0.418260			−0.209130	+	0.977888i	$0.567063\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.52623i		−	0.460176i	−0.973170	−	0.230088i	$-0.926099\pi$
$11$		−	1.52623i		−	0.460176i	0.973170	−	0.230088i	$-0.0739015\pi$
$12$		0			0
$13$			3.26430i			0.905354i	0.891675	+	0.452677i	$0.149531\pi$
$13$			3.26430i			0.905354i	−0.891675	+	0.452677i	$0.850469\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.68069			−1.37777			−0.688885	−	0.724870i	$-0.741900\pi$
$17$	−5.68069			−1.37777			−0.688885	+	0.724870i	$0.741900\pi$
$18$		0			0
$19$	6.25231			1.43438			0.717189	−	0.696879i	$-0.245428\pi$
$19$	6.25231			1.43438			0.717189	+	0.696879i	$0.245428\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.86296			1.01400			0.506998	−	0.861947i	$-0.330755\pi$
$23$	4.86296			1.01400			0.506998	+	0.861947i	$0.330755\pi$
$24$		0			0
$25$	−0.0972042			−0.0194408
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.93164i		−	1.65856i	−0.558830	−	0.829282i	$-0.688750\pi$
$29$		−	8.93164i		−	1.65856i	0.558830	−	0.829282i	$-0.311250\pi$
$30$		0			0
$31$	−1.89423			−0.340214			−0.170107	−	0.985426i	$-0.554411\pi$
$31$	−1.89423			−0.340214			−0.170107	+	0.985426i	$0.554411\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.45029			0.414174
$36$		0			0
$37$			4.41435i			0.725714i	0.931845	+	0.362857i	$0.118199\pi$
$37$			4.41435i			0.725714i	−0.931845	+	0.362857i	$0.881801\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.75307			−0.429957			−0.214979	−	0.976619i	$-0.568968\pi$
$41$	−2.75307			−0.429957			−0.214979	+	0.976619i	$0.568968\pi$
$42$		0			0
$43$			0.818459i			0.124814i	0.998051	+	0.0624069i	$0.0198777\pi$
$43$			0.818459i			0.124814i	−0.998051	+	0.0624069i	$0.980122\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.00307i			0.438042i	0.975720	+	0.219021i	$0.0702864\pi$
$47$			3.00307i			0.438042i	−0.975720	+	0.219021i	$0.929714\pi$
$48$		0			0
$49$	−5.77541			−0.825059
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.36175			−1.28594			−0.642968	−	0.765893i	$-0.722297\pi$
$53$	−9.36175			−1.28594			−0.642968	+	0.765893i	$0.722297\pi$
$54$		0			0
$55$			3.37942i			0.455681i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.10719			−0.534710			−0.267355	−	0.963598i	$-0.586150\pi$
$59$	−4.10719			−0.534710			−0.267355	+	0.963598i	$0.586150\pi$
$60$		0			0
$61$	−12.1085			−1.55033			−0.775166	−	0.631757i	$-0.782334\pi$
$61$	−12.1085			−1.55033			−0.775166	+	0.631757i	$0.782334\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	7.22790i		−	0.896510i
$66$		0			0
$67$		−	9.17056i		−	1.12036i	−0.828370	−	0.560181i	$-0.810732\pi$
$67$		−	9.17056i		−	1.12036i	0.828370	−	0.560181i	$-0.189268\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.9769			1.30271			0.651357	−	0.758771i	$-0.274200\pi$
$71$	10.9769			1.30271			0.651357	+	0.758771i	$0.274200\pi$
$72$		0			0
$73$		−	5.25379i		−	0.614910i	−0.951563	−	0.307455i	$-0.900523\pi$
$73$		−	5.25379i		−	0.614910i	0.951563	−	0.307455i	$-0.0994773\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1.68895i			0.192473i
$78$		0			0
$79$			11.5472i			1.29916i	0.760294	+	0.649579i	$0.225055\pi$
$79$			11.5472i			1.29916i	−0.760294	+	0.649579i	$0.774945\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.75501			0.631694			0.315847	−	0.948810i	$-0.397711\pi$
$83$	5.75501			0.631694			0.315847	+	0.948810i	$0.397711\pi$
$84$		0			0
$85$	12.5783			1.36431
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	8.72974i		−	0.925350i	−0.886528	−	0.462675i	$-0.846890\pi$
$89$		−	8.72974i		−	0.925350i	0.886528	−	0.462675i	$-0.153110\pi$
$90$		0			0
$91$		−	3.61231i		−	0.378673i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13.8440			−1.42037
$96$		0			0
$97$	16.1024			1.63495			0.817473	−	0.575966i	$-0.195374\pi$
$97$	16.1024			1.63495			0.817473	+	0.575966i	$0.195374\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	13.4647			1.33978			0.669892	−	0.742458i	$-0.266340\pi$
$101$	13.4647			1.33978			0.669892	+	0.742458i	$0.266340\pi$
$102$		0			0
$103$			9.52408i			0.938435i	0.883083	+	0.469218i	$0.155464\pi$
$103$			9.52408i			0.938435i	−0.883083	+	0.469218i	$0.844536\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			15.5883i			1.50698i	0.657459	+	0.753490i	$0.271631\pi$
$107$			15.5883i			1.50698i	−0.657459	+	0.753490i	$0.728369\pi$
$108$		0			0
$109$			16.7828i			1.60750i	0.594969	+	0.803748i	$0.297164\pi$
$109$			16.7828i			1.60750i	−0.594969	+	0.803748i	$0.702836\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.82248			0.359589			0.179794	−	0.983704i	$-0.442457\pi$
$113$	3.82248			0.359589			0.179794	+	0.983704i	$0.442457\pi$
$114$		0			0
$115$	−10.7677			−1.00409
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.28632			0.576266
$120$		0			0
$121$	8.67061			0.788238
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.2864			1.00948
$126$		0			0
$127$	18.2667			1.62091			0.810454	−	0.585802i	$-0.199220\pi$
$127$	18.2667			1.62091			0.810454	+	0.585802i	$0.199220\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.659035			0.0575801			0.0287901	−	0.999585i	$-0.490835\pi$
$131$	0.659035			0.0575801			0.0287901	+	0.999585i	$0.490835\pi$
$132$		0			0
$133$	−6.91888			−0.599943
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	4.04200i		−	0.345331i	−0.984980	−	0.172666i	$-0.944762\pi$
$137$		−	4.04200i		−	0.345331i	0.984980	−	0.172666i	$-0.0552380\pi$
$138$		0			0
$139$		−	7.65339i		−	0.649152i	−0.945860	−	0.324576i	$-0.894778\pi$
$139$		−	7.65339i		−	0.649152i	0.945860	−	0.324576i	$-0.105222\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.98208			0.416623
$144$		0			0
$145$			19.7767i			1.64236i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.28994			0.597215			0.298608	−	0.954376i	$-0.403478\pi$
$149$	7.28994			0.597215			0.298608	+	0.954376i	$0.403478\pi$
$150$		0			0
$151$			3.47956i			0.283163i	0.989927	+	0.141581i	$0.0452187\pi$
$151$			3.47956i			0.283163i	−0.989927	+	0.141581i	$0.954781\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	4.19425			0.336890
$156$		0			0
$157$	3.65246			0.291498			0.145749	−	0.989322i	$-0.453441\pi$
$157$	3.65246			0.291498			0.145749	+	0.989322i	$0.453441\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.38140			−0.424114
$162$		0			0
$163$			0.688646i			0.0539389i	0.999636	+	0.0269695i	$0.00858569\pi$
$163$			0.688646i			0.0539389i	−0.999636	+	0.0269695i	$0.991414\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.71933	−	9.53799i	−0.674722	−	0.738072i
$167$	−8.71933	−	9.53799i	−0.674722	−	0.738072i
$168$		0			0
$169$	2.34434			0.180334
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	22.9194i		−	1.74253i	−0.490817	−	0.871263i	$-0.663301\pi$
$173$		−	22.9194i		−	1.74253i	0.490817	−	0.871263i	$-0.336699\pi$
$174$		0			0
$175$	0.107567			0.00813132
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	12.0216i		−	0.898538i	−0.893396	−	0.449269i	$-0.851684\pi$
$179$		−	12.0216i		−	0.898538i	0.893396	−	0.449269i	$-0.148316\pi$
$180$		0			0
$181$	3.52740			0.262190			0.131095	−	0.991370i	$-0.458151\pi$
$181$	3.52740			0.262190			0.131095	+	0.991370i	$0.458151\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	9.77436i		−	0.718625i
$186$		0			0
$187$			8.67006i			0.634018i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.9390i		−	0.936233i	−0.883667	−	0.468116i	$-0.844933\pi$
$191$		−	12.9390i		−	0.936233i	0.883667	−	0.468116i	$-0.155067\pi$
$192$		0			0
$193$			10.2433i			0.737327i	0.929563	+	0.368664i	$0.120185\pi$
$193$			10.2433i			0.737327i	−0.929563	+	0.368664i	$0.879815\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.8008			1.33950			0.669750	−	0.742587i	$-0.266401\pi$
$197$	18.8008			1.33950			0.669750	+	0.742587i	$0.266401\pi$
$198$		0			0
$199$	15.3128			1.08549			0.542747	−	0.839896i	$-0.317384\pi$
$199$	15.3128			1.08549			0.542747	+	0.839896i	$0.317384\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			9.88386i			0.693711i
$204$		0			0
$205$	6.09592			0.425757
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	9.54248i		−	0.660067i
$210$		0			0
$211$	5.41188			0.372569			0.186285	−	0.982496i	$-0.440355\pi$
$211$	5.41188			0.372569			0.186285	+	0.982496i	$0.440355\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.81225i		−	0.123595i
$216$		0			0
$217$	2.09618			0.142298
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	18.5435i		−	1.24737i
$222$		0			0
$223$	20.0075			1.33980			0.669902	−	0.742450i	$-0.266336\pi$
$223$	20.0075			1.33980			0.669902	+	0.742450i	$0.266336\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.2335			1.01108			0.505541	−	0.862802i	$-0.331293\pi$
$227$	15.2335			1.01108			0.505541	+	0.862802i	$0.331293\pi$
$228$		0			0
$229$	5.19317			0.343175			0.171587	−	0.985169i	$-0.445110\pi$
$229$	5.19317			0.343175			0.171587	+	0.985169i	$0.445110\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			17.8919i			1.17213i	0.810262	+	0.586067i	$0.199325\pi$
$233$			17.8919i			1.17213i	−0.810262	+	0.586067i	$0.800675\pi$
$234$		0			0
$235$		−	6.64947i		−	0.433763i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			9.92922i			0.642268i	0.947034	+	0.321134i	$0.104064\pi$
$239$			9.92922i			0.642268i	−0.947034	+	0.321134i	$0.895936\pi$
$240$		0			0
$241$			1.09323i			0.0704210i	0.999380	+	0.0352105i	$0.0112102\pi$
$241$			1.09323i			0.0704210i	−0.999380	+	0.0352105i	$0.988790\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.7881			0.816999
$246$		0			0
$247$			20.4094i			1.29862i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			12.1099i			0.764368i	0.924086	+	0.382184i	$0.124828\pi$
$251$			12.1099i			0.764368i	−0.924086	+	0.382184i	$0.875172\pi$
$252$		0			0
$253$		−	7.42200i		−	0.466617i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.89139			0.117982			0.0589908	−	0.998259i	$-0.481212\pi$
$257$	1.89139			0.117982			0.0589908	+	0.998259i	$0.481212\pi$
$258$		0			0
$259$		−	4.88497i		−	0.303537i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	0.0895652i		−	0.00552282i	−0.999996	−	0.00276141i	$-0.999121\pi$
$263$		−	0.0895652i		−	0.00552282i	0.999996	−	0.00276141i	$-0.000878986\pi$
$264$		0			0
$265$	20.7290			1.27337
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−25.1217			−1.53170			−0.765849	−	0.643020i	$-0.777681\pi$
$269$	−25.1217			−1.53170			−0.765849	+	0.643020i	$0.777681\pi$
$270$		0			0
$271$		−	32.9056i		−	1.99887i	−0.0335917	−	0.999436i	$-0.510695\pi$
$271$		−	32.9056i		−	1.99887i	0.0335917	−	0.999436i	$-0.489305\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			0.148356i			0.00894621i
$276$		0			0
$277$			3.38325i			0.203280i	0.994821	+	0.101640i	$0.0324090\pi$
$277$			3.38325i			0.203280i	−0.994821	+	0.101640i	$0.967591\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			16.9847i			1.01322i	0.862175	+	0.506611i	$0.169102\pi$
$281$			16.9847i			1.01322i	−0.862175	+	0.506611i	$0.830898\pi$
$282$		0			0
$283$	−15.2318			−0.905436			−0.452718	−	0.891654i	$-0.649546\pi$
$283$	−15.2318			−0.905436			−0.452718	+	0.891654i	$0.649546\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.04658			0.179834
$288$		0			0
$289$	15.2703			0.898252
$290$		0			0
$291$		0			0
$292$		0			0
$293$			5.64421i			0.329738i	0.986315	+	0.164869i	$0.0527202\pi$
$293$			5.64421i			0.329738i	−0.986315	+	0.164869i	$0.947280\pi$
$294$		0			0
$295$	9.09424			0.529487
$296$		0			0
$297$		0			0
$298$		0			0
$299$			15.8742i			0.918026i
$300$		0			0
$301$		−	0.905717i		−	0.0522046i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	26.8109			1.53519
$306$		0			0
$307$			23.8883i			1.36338i	0.731643	+	0.681688i	$0.238754\pi$
$307$			23.8883i			1.36338i	−0.731643	+	0.681688i	$0.761246\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	14.1909i		−	0.804690i	−0.915488	−	0.402345i	$-0.868195\pi$
$311$		−	14.1909i		−	0.804690i	0.915488	−	0.402345i	$-0.131805\pi$
$312$		0			0
$313$			9.69765i			0.548143i	0.961709	+	0.274072i	$0.0883706\pi$
$313$			9.69765i			0.548143i	−0.961709	+	0.274072i	$0.911629\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	8.30637i		−	0.466532i	−0.972413	−	0.233266i	$-0.925059\pi$
$317$		−	8.30637i		−	0.466532i	0.972413	−	0.233266i	$-0.0749413\pi$
$318$		0			0
$319$	−13.6318			−0.763232
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−35.5175			−1.97624
$324$		0			0
$325$		−	0.317304i		−	0.0176008i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	3.32323i		−	0.183216i
$330$		0			0
$331$		−	14.1080i		−	0.775447i	−0.921776	−	0.387723i	$-0.873262\pi$
$331$		−	14.1080i		−	0.775447i	0.921776	−	0.387723i	$-0.126738\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			20.3057i			1.10942i
$336$		0			0
$337$	29.0529			1.58261			0.791307	−	0.611419i	$-0.209401\pi$
$337$	29.0529			1.58261			0.791307	+	0.611419i	$0.209401\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			2.89103i			0.156558i
$342$		0			0
$343$	14.1374			0.763349
$344$		0			0
$345$		0			0
$346$		0			0
$347$	16.1997			0.869645			0.434822	−	0.900516i	$-0.356811\pi$
$347$	16.1997			0.869645			0.434822	+	0.900516i	$0.356811\pi$
$348$		0			0
$349$			16.0230i			0.857694i	0.903377	+	0.428847i	$0.141080\pi$
$349$			16.0230i			0.857694i	−0.903377	+	0.428847i	$0.858920\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			11.7462i			0.625185i	0.949887	+	0.312593i	$0.101197\pi$
$353$			11.7462i			0.625185i	−0.949887	+	0.312593i	$0.898803\pi$
$354$		0			0
$355$	−24.3053			−1.28999
$356$		0			0
$357$		0			0
$358$		0			0
$359$			9.78407i			0.516383i	0.966094	+	0.258192i	$0.0831266\pi$
$359$			9.78407i			0.516383i	−0.966094	+	0.258192i	$0.916873\pi$
$360$		0			0
$361$	20.0914			1.05744
$362$		0			0
$363$		0			0
$364$		0			0
$365$			11.6331i			0.608903i
$366$		0			0
$367$	8.46441			0.441838			0.220919	−	0.975292i	$-0.429094\pi$
$367$	8.46441			0.441838			0.220919	+	0.975292i	$0.429094\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.3598			0.537855
$372$		0			0
$373$			36.5181i			1.89084i	0.325859	+	0.945418i	$0.394346\pi$
$373$			36.5181i			1.89084i	−0.325859	+	0.945418i	$0.605654\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	29.1556			1.50159
$378$		0			0
$379$		−	35.7780i		−	1.83779i	−0.394497	−	0.918897i	$-0.629081\pi$
$379$		−	35.7780i		−	1.83779i	0.394497	−	0.918897i	$-0.370919\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	31.2989i		−	1.59930i	−0.600465	−	0.799651i	$-0.705018\pi$
$383$		−	31.2989i		−	1.59930i	0.600465	−	0.799651i	$-0.294982\pi$
$384$		0			0
$385$		−	3.73971i		−	0.190593i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	35.3084			1.79021			0.895104	−	0.445857i	$-0.147101\pi$
$389$	35.3084			1.79021			0.895104	+	0.445857i	$0.147101\pi$
$390$		0			0
$391$	−27.6250			−1.39705
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	25.5680i		−	1.28647i
$396$		0			0
$397$	16.7323			0.839773			0.419886	−	0.907577i	$-0.362070\pi$
$397$	16.7323			0.839773			0.419886	+	0.907577i	$0.362070\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.1969			−0.858772			−0.429386	−	0.903121i	$-0.641270\pi$
$401$	−17.1969			−0.858772			−0.429386	+	0.903121i	$0.641270\pi$
$402$		0			0
$403$		−	6.18333i		−	0.308014i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.73732			0.333957
$408$		0			0
$409$	−16.6768			−0.824614			−0.412307	−	0.911045i	$-0.635277\pi$
$409$	−16.6768			−0.824614			−0.412307	+	0.911045i	$0.635277\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	4.54506			0.223648
$414$		0			0
$415$	−12.7429			−0.625523
$416$		0			0
$417$		0			0
$418$		0			0
$419$			5.19272i			0.253681i	0.991923	+	0.126841i	$0.0404836\pi$
$419$			5.19272i			0.253681i	−0.991923	+	0.126841i	$0.959516\pi$
$420$		0			0
$421$	38.6414			1.88327			0.941633	−	0.336641i	$-0.109291\pi$
$421$	38.6414			1.88327			0.941633	+	0.336641i	$0.109291\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.552187			0.0267850
$426$		0			0
$427$	13.3994			0.648442
$428$		0			0
$429$		0			0
$430$		0			0
$431$			29.8086i			1.43583i	0.696130	+	0.717916i	$0.254904\pi$
$431$			29.8086i			1.43583i	−0.696130	+	0.717916i	$0.745096\pi$
$432$		0			0
$433$	−23.5768			−1.13303			−0.566515	−	0.824052i	$-0.691709\pi$
$433$	−23.5768			−1.13303			−0.566515	+	0.824052i	$0.691709\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	30.4047			1.45445
$438$		0			0
$439$		−	33.7776i		−	1.61212i	−0.591836	−	0.806058i	$-0.701597\pi$
$439$		−	33.7776i		−	1.61212i	0.591836	−	0.806058i	$-0.298403\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.82132			0.276579			0.138290	−	0.990392i	$-0.455839\pi$
$443$	5.82132			0.276579			0.138290	+	0.990392i	$0.455839\pi$
$444$		0			0
$445$			19.3296i			0.916311i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	31.1572i		−	1.47040i	−0.677850	−	0.735201i	$-0.737088\pi$
$449$		−	31.1572i		−	1.47040i	0.677850	−	0.735201i	$-0.262912\pi$
$450$		0			0
$451$			4.20183i			0.197856i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			7.99848i			0.374974i
$456$		0			0
$457$			9.38086i			0.438818i	0.975633	+	0.219409i	$0.0704129\pi$
$457$			9.38086i			0.438818i	−0.975633	+	0.219409i	$0.929587\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	42.4425i		−	1.97674i	−0.152053	−	0.988372i	$-0.548588\pi$
$461$		−	42.4425i		−	1.97674i	0.152053	−	0.988372i	$-0.451412\pi$
$462$		0			0
$463$			9.06979i			0.421509i	0.977539	+	0.210754i	$0.0675921\pi$
$463$			9.06979i			0.421509i	−0.977539	+	0.210754i	$0.932408\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	4.77793i		−	0.221096i	−0.993871	−	0.110548i	$-0.964739\pi$
$467$		−	4.77793i		−	0.221096i	0.993871	−	0.110548i	$-0.0352606\pi$
$468$		0			0
$469$			10.1482i			0.468603i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.24916			0.0574364
$474$		0			0
$475$	−0.607750			−0.0278855
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.0102115			−0.000466577			−0.000233289	−	1.00000i	$-0.500074\pi$
$479$	−0.0102115			−0.000466577			−0.000233289		1.00000i	$0.500074\pi$
$480$		0			0
$481$	−14.4098			−0.657028
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−35.6543			−1.61898
$486$		0			0
$487$			33.4280i			1.51477i	0.652971	+	0.757383i	$0.273522\pi$
$487$			33.4280i			1.51477i	−0.652971	+	0.757383i	$0.726478\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			22.1905i			1.00144i	0.865608	+	0.500722i	$0.166932\pi$
$491$			22.1905i			1.00144i	−0.865608	+	0.500722i	$0.833068\pi$
$492$		0			0
$493$			50.7379i			2.28512i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.1471			−0.544873
$498$		0			0
$499$			41.2840i			1.84813i	0.382238	+	0.924064i	$0.375153\pi$
$499$			41.2840i			1.84813i	−0.382238	+	0.924064i	$0.624847\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	32.2498i		−	1.43795i	−0.695037	−	0.718974i	$-0.744612\pi$
$503$		−	32.2498i		−	1.43795i	0.695037	−	0.718974i	$-0.255388\pi$
$504$		0			0
$505$	−29.8138			−1.32670
$506$		0			0
$507$		0			0
$508$		0			0
$509$			15.7838i			0.699603i	0.936824	+	0.349802i	$0.113751\pi$
$509$			15.7838i			0.699603i	−0.936824	+	0.349802i	$0.886249\pi$
$510$		0			0
$511$			5.81391i			0.257192i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	21.0885i		−	0.929269i
$516$		0			0
$517$	4.58338			0.201577
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−38.8058			−1.70011			−0.850057	−	0.526691i	$-0.823432\pi$
$521$	−38.8058			−1.70011			−0.850057	+	0.526691i	$0.823432\pi$
$522$		0			0
$523$	19.8033			0.865940			0.432970	−	0.901408i	$-0.357466\pi$
$523$	19.8033			0.865940			0.432970	+	0.901408i	$0.357466\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.7605			0.468736
$528$		0			0
$529$	0.648346			0.0281890
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	8.98685i		−	0.389264i
$534$		0			0
$535$		−	34.5161i		−	1.49226i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			8.81462i			0.379673i
$540$		0			0
$541$		−	29.3779i		−	1.26305i	−0.775354	−	0.631527i	$-0.782429\pi$
$541$		−	29.3779i		−	1.26305i	0.775354	−	0.631527i	$-0.217571\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	37.1608i		−	1.59179i
$546$		0			0
$547$		−	43.1267i		−	1.84397i	−0.387231	−	0.921983i	$-0.626568\pi$
$547$		−	43.1267i		−	1.84397i	0.387231	−	0.921983i	$-0.373432\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	55.8434i		−	2.37901i
$552$		0			0
$553$		−	12.7782i		−	0.543386i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			39.1934i			1.66068i	0.557259	+	0.830339i	$0.311853\pi$
$557$			39.1934i			1.66068i	−0.557259	+	0.830339i	$0.688147\pi$
$558$		0			0
$559$	−2.67170			−0.113001
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	32.6676i		−	1.37678i	−0.725342	−	0.688388i	$-0.758319\pi$
$563$		−	32.6676i		−	1.37678i	0.725342	−	0.688388i	$-0.241681\pi$
$564$		0			0
$565$	−8.46384			−0.356076
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−27.4263			−1.14977			−0.574885	−	0.818234i	$-0.694953\pi$
$569$	−27.4263			−1.14977			−0.574885	+	0.818234i	$0.694953\pi$
$570$		0			0
$571$			9.53874i			0.399184i	0.979879	+	0.199592i	$0.0639616\pi$
$571$			9.53874i			0.399184i	−0.979879	+	0.199592i	$0.936038\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.472700			−0.0197129
$576$		0			0
$577$	−8.45388			−0.351940			−0.175970	−	0.984396i	$-0.556306\pi$
$577$	−8.45388			−0.351940			−0.175970	+	0.984396i	$0.556306\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.36856			−0.264212
$582$		0			0
$583$			14.2882i			0.591757i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.7100			−0.689696			−0.344848	−	0.938659i	$-0.612070\pi$
$587$	−16.7100			−0.689696			−0.344848	+	0.938659i	$0.612070\pi$
$588$		0			0
$589$	−11.8433			−0.487995
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.86014			0.240647			0.120324	−	0.992735i	$-0.461607\pi$
$593$	5.86014			0.240647			0.120324	+	0.992735i	$0.461607\pi$
$594$		0			0
$595$	−13.9193			−0.570637
$596$		0			0
$597$		0			0
$598$		0			0
$599$			46.3364i			1.89326i	0.322329	+	0.946628i	$0.395534\pi$
$599$			46.3364i			1.89326i	−0.322329	+	0.946628i	$0.604466\pi$
$600$		0			0
$601$	14.8468			0.605615			0.302807	−	0.953052i	$-0.402076\pi$
$601$	14.8468			0.605615			0.302807	+	0.953052i	$0.402076\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−19.1987			−0.780538
$606$		0			0
$607$			16.2655i			0.660196i	0.943947	+	0.330098i	$0.107082\pi$
$607$			16.2655i			0.660196i	−0.943947	+	0.330098i	$0.892918\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.80291			−0.396583
$612$		0			0
$613$	0.116515			0.00470602			0.00235301	−	0.999997i	$-0.499251\pi$
$613$	0.116515			0.00470602			0.00235301	+	0.999997i	$0.499251\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.36390i			0.0549084i	0.999623	+	0.0274542i	$0.00874004\pi$
$617$			1.36390i			0.0549084i	−0.999623	+	0.0274542i	$0.991260\pi$
$618$		0			0
$619$			18.6289i			0.748759i	0.927276	+	0.374379i	$0.122144\pi$
$619$			18.6289i			0.748759i	−0.927276	+	0.374379i	$0.877856\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			9.66043i			0.387037i
$624$		0			0
$625$	−24.5045			−0.980181
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	25.0766i		−	0.999868i
$630$		0			0
$631$	8.81751			0.351019			0.175510	−	0.984478i	$-0.443843\pi$
$631$	8.81751			0.351019			0.175510	+	0.984478i	$0.443843\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−40.4466			−1.60508
$636$		0			0
$637$		−	18.8527i		−	0.746970i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14.4116			0.569224			0.284612	−	0.958643i	$-0.408135\pi$
$641$	14.4116			0.569224			0.284612	+	0.958643i	$0.408135\pi$
$642$		0			0
$643$			36.3292i			1.43268i	0.697749	+	0.716342i	$0.254185\pi$
$643$			36.3292i			1.43268i	−0.697749	+	0.716342i	$0.745815\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	13.6166			0.535322			0.267661	−	0.963513i	$-0.413749\pi$
$647$	13.6166			0.535322			0.267661	+	0.963513i	$0.413749\pi$
$648$		0			0
$649$			6.26853i			0.246061i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			22.5752i			0.883437i	0.897154	+	0.441718i	$0.145631\pi$
$653$			22.5752i			0.883437i	−0.897154	+	0.441718i	$0.854369\pi$
$654$		0			0
$655$	−1.45925			−0.0570177
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.04987			0.0798518			0.0399259	−	0.999203i	$-0.487288\pi$
$659$	2.04987			0.0798518			0.0399259	+	0.999203i	$0.487288\pi$
$660$		0			0
$661$			24.1084i			0.937708i	0.883276	+	0.468854i	$0.155333\pi$
$661$			24.1084i			0.937708i	−0.883276	+	0.468854i	$0.844667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	15.3200			0.594082
$666$		0			0
$667$		−	43.4342i		−	1.68178i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			18.4804i			0.713426i
$672$		0			0
$673$			5.80060i			0.223597i	0.993731	+	0.111798i	$0.0356611\pi$
$673$			5.80060i			0.223597i	−0.993731	+	0.111798i	$0.964339\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1.60083i		−	0.0615248i	−0.999527	−	0.0307624i	$-0.990206\pi$
$677$		−	1.60083i		−	0.0615248i	0.999527	−	0.0307624i	$-0.00979352\pi$
$678$		0			0
$679$	−17.8191			−0.683833
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26.7110			1.02207			0.511034	−	0.859560i	$-0.329263\pi$
$683$	26.7110			1.02207			0.511034	+	0.859560i	$0.329263\pi$
$684$		0			0
$685$			8.94990i			0.341958i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	30.5596i		−	1.16423i
$690$		0			0
$691$		−	14.0321i		−	0.533808i	−0.963723	−	0.266904i	$-0.913999\pi$
$691$		−	14.0321i		−	0.533808i	0.963723	−	0.266904i	$-0.0860006\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			16.9463i			0.642811i
$696$		0			0
$697$	15.6393			0.592382
$698$		0			0
$699$		0			0
$700$		0			0
$701$			50.1976i			1.89594i	0.318364	+	0.947969i	$0.396867\pi$
$701$			50.1976i			1.89594i	−0.318364	+	0.947969i	$0.603133\pi$
$702$		0			0
$703$			27.5999i			1.04095i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14.9002			−0.560378
$708$		0			0
$709$		−	23.4320i		−	0.880009i	−0.897996	−	0.440004i	$-0.854977\pi$
$709$		−	23.4320i		−	0.880009i	0.897996	−	0.440004i	$-0.145023\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.21155			−0.344975
$714$		0			0
$715$	−11.0315			−0.412553
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−36.5966			−1.36482			−0.682411	−	0.730969i	$-0.739069\pi$
$719$	−36.5966			−1.36482			−0.682411	+	0.730969i	$0.739069\pi$
$720$		0			0
$721$		−	10.5395i		−	0.392510i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			0.868193i			0.0322439i
$726$		0			0
$727$		−	24.8889i		−	0.923080i	−0.887119	−	0.461540i	$-0.847297\pi$
$727$		−	24.8889i		−	0.923080i	0.887119	−	0.461540i	$-0.152703\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	4.64942i		−	0.171965i
$732$		0			0
$733$	11.4373			0.422446			0.211223	−	0.977438i	$-0.432255\pi$
$733$	11.4373			0.422446			0.211223	+	0.977438i	$0.432255\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13.9964			−0.515564
$738$		0			0
$739$		−	36.0468i		−	1.32600i	−0.748618	−	0.663001i	$-0.769282\pi$
$739$		−	36.0468i		−	1.32600i	0.748618	−	0.663001i	$-0.230718\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			12.2477i			0.449324i	0.974437	+	0.224662i	$0.0721277\pi$
$743$			12.2477i			0.449324i	−0.974437	+	0.224662i	$0.927872\pi$
$744$		0			0
$745$	−16.1416			−0.591382
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	17.2502i		−	0.630310i
$750$		0			0
$751$			8.39744i			0.306427i	0.988193	+	0.153213i	$0.0489622\pi$
$751$			8.39744i			0.306427i	−0.988193	+	0.153213i	$0.951038\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	7.70453i		−	0.280397i
$756$		0			0
$757$	39.7794			1.44581			0.722904	−	0.690949i	$-0.242807\pi$
$757$	39.7794			1.44581			0.722904	+	0.690949i	$0.242807\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			16.9492i			0.614409i	0.951644	+	0.307205i	$0.0993936\pi$
$761$			16.9492i			0.614409i	−0.951644	+	0.307205i	$0.900606\pi$
$762$		0			0
$763$		−	18.5720i		−	0.672351i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	13.4071i		−	0.484102i
$768$		0			0
$769$		−	0.808888i		−	0.0291692i	−0.999894	−	0.0145846i	$-0.995357\pi$
$769$		−	0.808888i		−	0.0291692i	0.999894	−	0.0145846i	$-0.00464259\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21.1557			0.760919			0.380459	−	0.924798i	$-0.375766\pi$
$773$	21.1557			0.760919			0.380459	+	0.924798i	$0.375766\pi$
$774$		0			0
$775$	0.184127			0.00661404
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−17.2130			−0.616721
$780$		0			0
$781$		−	16.7533i		−	0.599479i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.08738			−0.288651
$786$		0			0
$787$		−	20.5075i		−	0.731015i	−0.930808	−	0.365508i	$-0.880895\pi$
$787$		−	20.5075i		−	0.731015i	0.930808	−	0.365508i	$-0.119105\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.23000			−0.150402
$792$		0			0
$793$		−	39.5257i		−	1.40360i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.49394			0.0883398			0.0441699	−	0.999024i	$-0.485936\pi$
$797$	2.49394			0.0883398			0.0441699	+	0.999024i	$0.485936\pi$
$798$		0			0
$799$		−	17.0595i		−	0.603522i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8.01851			−0.282967
$804$		0			0
$805$	11.9156			0.419971
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	45.2751i		−	1.59179i	−0.605437	−	0.795893i	$-0.707002\pi$
$809$		−	45.2751i		−	1.59179i	0.605437	−	0.795893i	$-0.292998\pi$
$810$		0			0
$811$			29.4613i			1.03453i	0.855826	+	0.517264i	$0.173049\pi$
$811$			29.4613i			1.03453i	−0.855826	+	0.517264i	$0.826951\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	1.52482i		−	0.0534120i
$816$		0			0
$817$			5.11726i			0.179030i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	32.1577			1.12231			0.561157	−	0.827710i	$-0.310357\pi$
$821$	32.1577			1.12231			0.561157	+	0.827710i	$0.310357\pi$
$822$		0			0
$823$		−	12.4505i		−	0.433997i	−0.976172	−	0.216998i	$-0.930373\pi$
$823$		−	12.4505i		−	0.433997i	0.976172	−	0.216998i	$-0.0696267\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	45.4844			1.58165			0.790824	−	0.612043i	$-0.209652\pi$
$827$	45.4844			1.58165			0.790824	+	0.612043i	$0.209652\pi$
$828$		0			0
$829$		−	26.3397i		−	0.914814i	−0.889257	−	0.457407i	$-0.848778\pi$
$829$		−	26.3397i		−	0.914814i	0.889257	−	0.457407i	$-0.151222\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	32.8083			1.13674
$834$		0			0
$835$	19.3066	+	21.1193i	0.668132	+	0.730862i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			7.89720i			0.272642i	0.990665	+	0.136321i	$0.0435278\pi$
$839$			7.89720i			0.272642i	−0.990665	+	0.136321i	$0.956472\pi$
$840$		0			0
$841$	−50.7742			−1.75083
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.19090			−0.178572
$846$		0			0
$847$	−9.59500			−0.329688
$848$		0			0
$849$		0			0
$850$		0			0
$851$			21.4668i			0.735872i
$852$		0			0
$853$	24.0348			0.822936			0.411468	−	0.911424i	$-0.365016\pi$
$853$	24.0348			0.822936			0.411468	+	0.911424i	$0.365016\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	21.7787i		−	0.743945i	−0.928244	−	0.371973i	$-0.878682\pi$
$857$		−	21.7787i		−	0.743945i	0.928244	−	0.371973i	$-0.121318\pi$
$858$		0			0
$859$	−9.15430			−0.312341			−0.156170	−	0.987730i	$-0.549915\pi$
$859$	−9.15430			−0.312341			−0.156170	+	0.987730i	$0.549915\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			0.493931i			0.0168136i	0.999965	+	0.00840681i	$0.00267600\pi$
$863$			0.493931i			0.0168136i	−0.999965	+	0.00840681i	$0.997324\pi$
$864$		0			0
$865$			50.7486i			1.72550i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	17.6237			0.597842
$870$		0			0
$871$	29.9355			1.01432
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.4896			−0.422226
$876$		0			0
$877$	2.67415			0.0902995			0.0451498	−	0.998980i	$-0.485623\pi$
$877$	2.67415			0.0902995			0.0451498	+	0.998980i	$0.485623\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.60269			0.323523			0.161761	−	0.986830i	$-0.448282\pi$
$881$	9.60269			0.323523			0.161761	+	0.986830i	$0.448282\pi$
$882$		0			0
$883$	−21.8319			−0.734702			−0.367351	−	0.930082i	$-0.619735\pi$
$883$	−21.8319			−0.734702			−0.367351	+	0.930082i	$0.619735\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15.7780			−0.529774			−0.264887	−	0.964279i	$-0.585335\pi$
$887$	−15.7780			−0.529774			−0.264887	+	0.964279i	$0.585335\pi$
$888$		0			0
$889$	−20.2142			−0.677961
$890$		0			0
$891$		0			0
$892$		0			0
$893$			18.7761i			0.628318i
$894$		0			0
$895$			26.6186i			0.889761i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			16.9186i			0.564266i
$900$		0			0
$901$	53.1812			1.77172
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.81047			−0.259629
$906$		0			0
$907$	35.1728			1.16789			0.583947	−	0.811792i	$-0.301508\pi$
$907$	35.1728			1.16789			0.583947	+	0.811792i	$0.301508\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			54.7431i			1.81372i	0.421432	+	0.906860i	$0.361528\pi$
$911$			54.7431i			1.81372i	−0.421432	+	0.906860i	$0.638472\pi$
$912$		0			0
$913$		−	8.78348i		−	0.290691i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.729295			−0.0240835
$918$		0			0
$919$	−47.4360			−1.56477			−0.782384	−	0.622796i	$-0.785997\pi$
$919$	−47.4360			−1.56477			−0.782384	+	0.622796i	$0.785997\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			35.8318i			1.17942i
$924$		0			0
$925$		−	0.429093i		−	0.0141085i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			20.8057i			0.682612i	0.939952	+	0.341306i	$0.110869\pi$
$929$			20.8057i			0.682612i	−0.939952	+	0.341306i	$0.889131\pi$
$930$		0			0
$931$	−36.1097			−1.18345
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	19.1975i		−	0.627825i
$936$		0			0
$937$			3.84765i			0.125697i	0.998023	+	0.0628486i	$0.0200185\pi$
$937$			3.84765i			0.125697i	−0.998023	+	0.0628486i	$0.979981\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	25.7928			0.840821			0.420411	−	0.907334i	$-0.361886\pi$
$941$	25.7928			0.840821			0.420411	+	0.907334i	$0.361886\pi$
$942$		0			0
$943$	−13.3881			−0.435975
$944$		0			0
$945$		0			0
$946$		0			0
$947$			13.5461i			0.440189i	0.975479	+	0.220094i	$0.0706365\pi$
$947$			13.5461i			0.440189i	−0.975479	+	0.220094i	$0.929363\pi$
$948$		0			0
$949$	17.1500			0.556711
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−43.3539			−1.40437			−0.702185	−	0.711994i	$-0.747792\pi$
$953$	−43.3539			−1.40437			−0.702185	+	0.711994i	$0.747792\pi$
$954$		0			0
$955$			28.6499i			0.927088i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			4.47292i			0.144438i
$960$		0			0
$961$	−27.4119			−0.884255
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	22.6809i		−	0.730125i
$966$		0			0
$967$	−36.1147			−1.16137			−0.580686	−	0.814128i	$-0.697215\pi$
$967$	−36.1147			−1.16137			−0.580686	+	0.814128i	$0.697215\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.5547			1.17310			0.586549	−	0.809914i	$-0.300486\pi$
$971$	36.5547			1.17310			0.586549	+	0.809914i	$0.300486\pi$
$972$		0			0
$973$			8.46933i			0.271514i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.0142			−0.672304			−0.336152	−	0.941808i	$-0.609126\pi$
$977$	−21.0142			−0.672304			−0.336152	+	0.941808i	$0.609126\pi$
$978$		0			0
$979$	−13.3236			−0.425824
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5.90153			−0.188230			−0.0941149	−	0.995561i	$-0.530002\pi$
$983$	−5.90153			−0.188230			−0.0941149	+	0.995561i	$0.530002\pi$
$984$		0			0
$985$	−41.6292			−1.32642
$986$		0			0
$987$		0			0
$988$		0			0
$989$			3.98013i			0.126561i
$990$		0			0
$991$		−	41.3636i		−	1.31396i	−0.753909	−	0.656979i	$-0.771834\pi$
$991$		−	41.3636i		−	1.31396i	0.753909	−	0.656979i	$-0.228166\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−33.9059			−1.07489
$996$		0			0
$997$	−25.2647			−0.800140			−0.400070	−	0.916485i	$-0.631014\pi$
$997$	−25.2647			−0.800140			−0.400070	+	0.916485i	$0.631014\pi$
$998$		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.3005.14	yes	56
3.2	odd	2	inner	6012.2.h.a.3005.44	yes	56
167.166	odd	2	inner	6012.2.h.a.3005.43	yes	56
501.500	even	2	inner	6012.2.h.a.3005.13	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.h.a.3005.13	✓	56	501.500	even	2	inner
6012.2.h.a.3005.14	yes	56	1.1	even	1	trivial
6012.2.h.a.3005.43	yes	56	167.166	odd	2	inner
6012.2.h.a.3005.44	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)

\(f(q)\)	\(=\)	\(q-2.21423 q^{5} -1.10661 q^{7} +O(q^{10})\)	\(q-2.21423 q^{5} -1.10661 q^{7} -1.52623i q^{11} +3.26430i q^{13} -5.68069 q^{17} +6.25231 q^{19} +4.86296 q^{23} -0.0972042 q^{25} -8.93164i q^{29} -1.89423 q^{31} +2.45029 q^{35} +4.41435i q^{37} -2.75307 q^{41} +0.818459i q^{43} +3.00307i q^{47} -5.77541 q^{49} -9.36175 q^{53} +3.37942i q^{55} -4.10719 q^{59} -12.1085 q^{61} -7.22790i q^{65} -9.17056i q^{67} +10.9769 q^{71} -5.25379i q^{73} +1.68895i q^{77} +11.5472i q^{79} +5.75501 q^{83} +12.5783 q^{85} -8.72974i q^{89} -3.61231i q^{91} -13.8440 q^{95} +16.1024 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 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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