LMFDB - Embedded newform 6012.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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, 0, 0]))

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

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

chi := DirichletCharacter("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.a (trivial)

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:	yes
Analytic conductor:	$48.0060616952$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67513 q^{5} +3.35026 q^{7} +O(q^{10})$	$q-1.67513 q^{5} +3.35026 q^{7} -1.19394 q^{11} -2.96239 q^{13} +0.324869 q^{17} -1.61213 q^{19} +5.92478 q^{23} -2.19394 q^{25} -6.70052 q^{29} +8.31265 q^{31} -5.61213 q^{35} -0.574515 q^{37} -6.06300 q^{41} -1.36248 q^{43} +5.19394 q^{47} +4.22425 q^{49} -11.2120 q^{53} +2.00000 q^{55} -4.64974 q^{59} -0.806063 q^{61} +4.96239 q^{65} +5.28726 q^{67} -4.00000 q^{71} -3.61213 q^{73} -4.00000 q^{77} +12.1744 q^{79} -11.6629 q^{83} -0.544198 q^{85} -10.7005 q^{89} -9.92478 q^{91} +2.70052 q^{95} +16.0508 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q+O(q^{10}) $ 3 * q	$ 3 q - 4 q^{11} + 2 q^{13} + 6 q^{17} - 4 q^{19} - 4 q^{23} - 7 q^{25} + 4 q^{31} - 16 q^{35} + 10 q^{37} - 14 q^{41} - 20 q^{43} + 16 q^{47} + 11 q^{49} - 6 q^{53} + 6 q^{55} - 24 q^{59} - 2 q^{61} + 4 q^{65} + 10 q^{67} - 12 q^{71} - 10 q^{73} - 12 q^{77} - 2 q^{79} - 4 q^{83} + 8 q^{85} - 12 q^{89} - 8 q^{91} - 12 q^{95} + 18 q^{97}+O(q^{100}) $ 3 * q - 4 * q^11 + 2 * q^13 + 6 * q^17 - 4 * q^19 - 4 * q^23 - 7 * q^25 + 4 * q^31 - 16 * q^35 + 10 * q^37 - 14 * q^41 - 20 * q^43 + 16 * q^47 + 11 * q^49 - 6 * q^53 + 6 * q^55 - 24 * q^59 - 2 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 - 10 * q^73 - 12 * q^77 - 2 * q^79 - 4 * q^83 + 8 * q^85 - 12 * q^89 - 8 * q^91 - 12 * q^95 + 18 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−1.67513	−0.749141	−0.374571	−	0.927198i	$-0.622210\pi$
$5$	−1.67513	−0.749141	−0.374571	+	0.927198i	$0.622210\pi$
$6$	0	0
$7$	3.35026	1.26628	0.633140	−	0.774037i	$-0.281766\pi$
$7$	3.35026	1.26628	0.633140	+	0.774037i	$0.281766\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.19394	−0.359985	−0.179993	−	0.983668i	$-0.557607\pi$
$11$	−1.19394	−0.359985	−0.179993	+	0.983668i	$0.557607\pi$
$12$	0	0
$13$	−2.96239	−0.821619	−0.410809	−	0.911721i	$-0.634754\pi$
$13$	−2.96239	−0.821619	−0.410809	+	0.911721i	$0.634754\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.324869	0.0787923	0.0393962	−	0.999224i	$-0.487457\pi$
$17$	0.324869	0.0787923	0.0393962	+	0.999224i	$0.487457\pi$
$18$	0	0
$19$	−1.61213	−0.369847	−0.184924	−	0.982753i	$-0.559204\pi$
$19$	−1.61213	−0.369847	−0.184924	+	0.982753i	$0.559204\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.92478	1.23540	0.617701	−	0.786413i	$-0.288064\pi$
$23$	5.92478	1.23540	0.617701	+	0.786413i	$0.288064\pi$
$24$	0	0
$25$	−2.19394	−0.438787
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.70052	−1.24426	−0.622128	−	0.782916i	$-0.713732\pi$
$29$	−6.70052	−1.24426	−0.622128	+	0.782916i	$0.713732\pi$
$30$	0	0
$31$	8.31265	1.49300	0.746498	−	0.665388i	$-0.231734\pi$
$31$	8.31265	1.49300	0.746498	+	0.665388i	$0.231734\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.61213	−0.948623
$36$	0	0
$37$	−0.574515	−0.0944498	−0.0472249	−	0.998884i	$-0.515038\pi$
$37$	−0.574515	−0.0944498	−0.0472249	+	0.998884i	$0.515038\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.06300	−0.946882	−0.473441	−	0.880825i	$-0.656988\pi$
$41$	−6.06300	−0.946882	−0.473441	+	0.880825i	$0.656988\pi$
$42$	0	0
$43$	−1.36248	−0.207776	−0.103888	−	0.994589i	$-0.533128\pi$
$43$	−1.36248	−0.207776	−0.103888	+	0.994589i	$0.533128\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.19394	0.757614	0.378807	−	0.925476i	$-0.376335\pi$
$47$	5.19394	0.757614	0.378807	+	0.925476i	$0.376335\pi$
$48$	0	0
$49$	4.22425	0.603465
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.2120	−1.54009	−0.770046	−	0.637989i	$-0.779767\pi$
$53$	−11.2120	−1.54009	−0.770046	+	0.637989i	$0.779767\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.64974	−0.605344	−0.302672	−	0.953095i	$-0.597879\pi$
$59$	−4.64974	−0.605344	−0.302672	+	0.953095i	$0.597879\pi$
$60$	0	0
$61$	−0.806063	−0.103206	−0.0516029	−	0.998668i	$-0.516433\pi$
$61$	−0.806063	−0.103206	−0.0516029	+	0.998668i	$0.516433\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.96239	0.615509
$66$	0	0
$67$	5.28726	0.645941	0.322971	−	0.946409i	$-0.395318\pi$
$67$	5.28726	0.645941	0.322971	+	0.946409i	$0.395318\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−3.61213	−0.422767	−0.211384	−	0.977403i	$-0.567797\pi$
$73$	−3.61213	−0.422767	−0.211384	+	0.977403i	$0.567797\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	12.1744	1.36973	0.684865	−	0.728670i	$-0.259861\pi$
$79$	12.1744	1.36973	0.684865	+	0.728670i	$0.259861\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.6629	−1.28017	−0.640085	−	0.768304i	$-0.721101\pi$
$83$	−11.6629	−1.28017	−0.640085	+	0.768304i	$0.721101\pi$
$84$	0	0
$85$	−0.544198	−0.0590266
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.7005	−1.13425	−0.567127	−	0.823631i	$-0.691945\pi$
$89$	−10.7005	−1.13425	−0.567127	+	0.823631i	$0.691945\pi$
$90$	0	0
$91$	−9.92478	−1.04040
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.70052	0.277068
$96$	0	0
$97$	16.0508	1.62971	0.814855	−	0.579665i	$-0.196816\pi$
$97$	16.0508	1.62971	0.814855	+	0.579665i	$0.196816\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.98778	−0.197792	−0.0988958	−	0.995098i	$-0.531531\pi$
$101$	−1.98778	−0.197792	−0.0988958	+	0.995098i	$0.531531\pi$
$102$	0	0
$103$	7.02539	0.692233	0.346116	−	0.938192i	$-0.387500\pi$
$103$	7.02539	0.692233	0.346116	+	0.938192i	$0.387500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.3430	−0.999892	−0.499946	−	0.866056i	$-0.666647\pi$
$107$	−10.3430	−0.999892	−0.499946	+	0.866056i	$0.666647\pi$
$108$	0	0
$109$	1.35026	0.129332	0.0646658	−	0.997907i	$-0.479402\pi$
$109$	1.35026	0.129332	0.0646658	+	0.997907i	$0.479402\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.4133	−1.07367	−0.536835	−	0.843687i	$-0.680380\pi$
$113$	−11.4133	−1.07367	−0.536835	+	0.843687i	$0.680380\pi$
$114$	0	0
$115$	−9.92478	−0.925490
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.08840	0.0997732
$120$	0	0
$121$	−9.57452	−0.870410
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0508	1.07785
$126$	0	0
$127$	−0.962389	−0.0853982	−0.0426991	−	0.999088i	$-0.513596\pi$
$127$	−0.962389	−0.0853982	−0.0426991	+	0.999088i	$0.513596\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.27504	0.810364	0.405182	−	0.914236i	$-0.367208\pi$
$131$	9.27504	0.810364	0.405182	+	0.914236i	$0.367208\pi$
$132$	0	0
$133$	−5.40105	−0.468330
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.64974	−0.397254	−0.198627	−	0.980075i	$-0.563648\pi$
$137$	−4.64974	−0.397254	−0.198627	+	0.980075i	$0.563648\pi$
$138$	0	0
$139$	−17.6751	−1.49919	−0.749593	−	0.661900i	$-0.769751\pi$
$139$	−17.6751	−1.49919	−0.749593	+	0.661900i	$0.769751\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.53690	0.295771
$144$	0	0
$145$	11.2243	0.932124
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.47390	−0.448439	−0.224220	−	0.974539i	$-0.571983\pi$
$149$	−5.47390	−0.448439	−0.224220	+	0.974539i	$0.571983\pi$
$150$	0	0
$151$	−9.33804	−0.759919	−0.379960	−	0.925003i	$-0.624062\pi$
$151$	−9.33804	−0.759919	−0.379960	+	0.925003i	$0.624062\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−13.9248	−1.11847
$156$	0	0
$157$	−20.0508	−1.60023	−0.800113	−	0.599849i	$-0.795227\pi$
$157$	−20.0508	−1.60023	−0.800113	+	0.599849i	$0.795227\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	19.8496	1.56436
$162$	0	0
$163$	18.3004	1.43340	0.716700	−	0.697381i	$-0.245652\pi$
$163$	18.3004	1.43340	0.716700	+	0.697381i	$0.245652\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−4.22425	−0.324943
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.6121	−1.33903	−0.669513	−	0.742801i	$-0.733497\pi$
$173$	−17.6121	−1.33903	−0.669513	+	0.742801i	$0.733497\pi$
$174$	0	0
$175$	−7.35026	−0.555628
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.3733	1.22380	0.611898	−	0.790936i	$-0.290406\pi$
$179$	16.3733	1.22380	0.611898	+	0.790936i	$0.290406\pi$
$180$	0	0
$181$	1.43136	0.106392	0.0531962	−	0.998584i	$-0.483059\pi$
$181$	1.43136	0.106392	0.0531962	+	0.998584i	$0.483059\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.962389	0.0707562
$186$	0	0
$187$	−0.387873	−0.0283641
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.50659	0.109013	0.0545064	−	0.998513i	$-0.482641\pi$
$191$	1.50659	0.109013	0.0545064	+	0.998513i	$0.482641\pi$
$192$	0	0
$193$	−0.911603	−0.0656186	−0.0328093	−	0.999462i	$-0.510445\pi$
$193$	−0.911603	−0.0656186	−0.0328093	+	0.999462i	$0.510445\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.4641	0.959274	0.479637	−	0.877467i	$-0.340768\pi$
$197$	13.4641	0.959274	0.479637	+	0.877467i	$0.340768\pi$
$198$	0	0
$199$	−18.3634	−1.30175	−0.650875	−	0.759185i	$-0.725598\pi$
$199$	−18.3634	−1.30175	−0.650875	+	0.759185i	$0.725598\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−22.4485	−1.57558
$204$	0	0
$205$	10.1563	0.709349
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.92478	0.133140
$210$	0	0
$211$	−7.01317	−0.482807	−0.241403	−	0.970425i	$-0.577608\pi$
$211$	−7.01317	−0.482807	−0.241403	+	0.970425i	$0.577608\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.28233	0.155654
$216$	0	0
$217$	27.8496	1.89055
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.962389	−0.0647373
$222$	0	0
$223$	21.2750	1.42468	0.712341	−	0.701834i	$-0.247635\pi$
$223$	21.2750	1.42468	0.712341	+	0.701834i	$0.247635\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.03761	−0.201613	−0.100807	−	0.994906i	$-0.532142\pi$
$227$	−3.03761	−0.201613	−0.100807	+	0.994906i	$0.532142\pi$
$228$	0	0
$229$	−16.8061	−1.11058	−0.555288	−	0.831658i	$-0.687392\pi$
$229$	−16.8061	−1.11058	−0.555288	+	0.831658i	$0.687392\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.43866	−0.552835	−0.276417	−	0.961038i	$-0.589147\pi$
$233$	−8.43866	−0.552835	−0.276417	+	0.961038i	$0.589147\pi$
$234$	0	0
$235$	−8.70052	−0.567560
$236$	0	0
$237$	0	0
$238$	0	0
$239$	27.2809	1.76466	0.882328	−	0.470635i	$-0.155975\pi$
$239$	27.2809	1.76466	0.882328	+	0.470635i	$0.155975\pi$
$240$	0	0
$241$	−22.9380	−1.47756	−0.738782	−	0.673945i	$-0.764599\pi$
$241$	−22.9380	−1.47756	−0.738782	+	0.673945i	$0.764599\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−7.07618	−0.452080
$246$	0	0
$247$	4.77575	0.303873
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.11283	−0.322719	−0.161360	−	0.986896i	$-0.551588\pi$
$251$	−5.11283	−0.322719	−0.161360	+	0.986896i	$0.551588\pi$
$252$	0	0
$253$	−7.07381	−0.444727
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.5623	1.03313	0.516564	−	0.856249i	$-0.327211\pi$
$257$	16.5623	1.03313	0.516564	+	0.856249i	$0.327211\pi$
$258$	0	0
$259$	−1.92478	−0.119600
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−26.8872	−1.65793	−0.828967	−	0.559298i	$-0.811071\pi$
$263$	−26.8872	−1.65793	−0.828967	+	0.559298i	$0.811071\pi$
$264$	0	0
$265$	18.7816	1.15375
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.1890	−1.10900	−0.554502	−	0.832183i	$-0.687091\pi$
$269$	−18.1890	−1.10900	−0.554502	+	0.832183i	$0.687091\pi$
$270$	0	0
$271$	18.5623	1.12758	0.563790	−	0.825918i	$-0.309343\pi$
$271$	18.5623	1.12758	0.563790	+	0.825918i	$0.309343\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.61942	0.157957
$276$	0	0
$277$	1.09825	0.0659872	0.0329936	−	0.999456i	$-0.489496\pi$
$277$	1.09825	0.0659872	0.0329936	+	0.999456i	$0.489496\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.1114	−0.841817	−0.420908	−	0.907103i	$-0.638289\pi$
$281$	−14.1114	−0.841817	−0.420908	+	0.907103i	$0.638289\pi$
$282$	0	0
$283$	18.2374	1.08410	0.542051	−	0.840345i	$-0.317648\pi$
$283$	18.2374	1.08410	0.542051	+	0.840345i	$0.317648\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.3127	−1.19902
$288$	0	0
$289$	−16.8945	−0.993792
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.92478	0.346129	0.173065	−	0.984910i	$-0.444633\pi$
$293$	5.92478	0.346129	0.173065	+	0.984910i	$0.444633\pi$
$294$	0	0
$295$	7.78892	0.453488
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−17.5515	−1.01503
$300$	0	0
$301$	−4.56467	−0.263103
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.35026	0.0773158
$306$	0	0
$307$	−10.0484	−0.573493	−0.286747	−	0.958006i	$-0.592574\pi$
$307$	−10.0484	−0.573493	−0.286747	+	0.958006i	$0.592574\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.8872	−1.52463	−0.762316	−	0.647205i	$-0.775938\pi$
$311$	−26.8872	−1.52463	−0.762316	+	0.647205i	$0.775938\pi$
$312$	0	0
$313$	8.44851	0.477538	0.238769	−	0.971076i	$-0.423256\pi$
$313$	8.44851	0.477538	0.238769	+	0.971076i	$0.423256\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.83638	−0.271638	−0.135819	−	0.990734i	$-0.543367\pi$
$317$	−4.83638	−0.271638	−0.135819	+	0.990734i	$0.543367\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.523730	−0.0291411
$324$	0	0
$325$	6.49929	0.360516
$326$	0	0
$327$	0	0
$328$	0	0
$329$	17.4010	0.959351
$330$	0	0
$331$	0.787965	0.0433105	0.0216552	−	0.999765i	$-0.493106\pi$
$331$	0.787965	0.0433105	0.0216552	+	0.999765i	$0.493106\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.85685	−0.483901
$336$	0	0
$337$	−12.4241	−0.676782	−0.338391	−	0.941006i	$-0.609883\pi$
$337$	−12.4241	−0.676782	−0.338391	+	0.941006i	$0.609883\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.92478	−0.537457
$342$	0	0
$343$	−9.29948	−0.502125
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.27504	0.283179	0.141589	−	0.989925i	$-0.454779\pi$
$347$	5.27504	0.283179	0.141589	+	0.989925i	$0.454779\pi$
$348$	0	0
$349$	−16.5745	−0.887213	−0.443607	−	0.896222i	$-0.646301\pi$
$349$	−16.5745	−0.887213	−0.443607	+	0.896222i	$0.646301\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.58769	−0.297403	−0.148701	−	0.988882i	$-0.547509\pi$
$353$	−5.58769	−0.297403	−0.148701	+	0.988882i	$0.547509\pi$
$354$	0	0
$355$	6.70052	0.355627
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.5501	−0.556812	−0.278406	−	0.960464i	$-0.589806\pi$
$359$	−10.5501	−0.556812	−0.278406	+	0.960464i	$0.589806\pi$
$360$	0	0
$361$	−16.4010	−0.863213
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.05079	0.316713
$366$	0	0
$367$	−22.6107	−1.18027	−0.590135	−	0.807305i	$-0.700925\pi$
$367$	−22.6107	−1.18027	−0.590135	+	0.807305i	$0.700925\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−37.5633	−1.95019
$372$	0	0
$373$	1.66291	0.0861023	0.0430512	−	0.999073i	$-0.486292\pi$
$373$	1.66291	0.0861023	0.0430512	+	0.999073i	$0.486292\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.8496	1.02230
$378$	0	0
$379$	17.8618	0.917498	0.458749	−	0.888566i	$-0.348298\pi$
$379$	17.8618	0.917498	0.458749	+	0.888566i	$0.348298\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.67021	0.238636	0.119318	−	0.992856i	$-0.461929\pi$
$383$	4.67021	0.238636	0.119318	+	0.992856i	$0.461929\pi$
$384$	0	0
$385$	6.70052	0.341490
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.78797	0.141355	0.0706777	−	0.997499i	$-0.477484\pi$
$389$	2.78797	0.141355	0.0706777	+	0.997499i	$0.477484\pi$
$390$	0	0
$391$	1.92478	0.0973402
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−20.3938	−1.02612
$396$	0	0
$397$	12.0303	0.603784	0.301892	−	0.953342i	$-0.402382\pi$
$397$	12.0303	0.603784	0.301892	+	0.953342i	$0.402382\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−39.3235	−1.96372	−0.981860	−	0.189608i	$-0.939278\pi$
$401$	−39.3235	−1.96372	−0.981860	+	0.189608i	$0.939278\pi$
$402$	0	0
$403$	−24.6253	−1.22667
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.685935	0.0340005
$408$	0	0
$409$	−0.956509	−0.0472963	−0.0236482	−	0.999720i	$-0.507528\pi$
$409$	−0.956509	−0.0472963	−0.0236482	+	0.999720i	$0.507528\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−15.5778	−0.766535
$414$	0	0
$415$	19.5369	0.959029
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.49341	0.317224	0.158612	−	0.987341i	$-0.449298\pi$
$419$	6.49341	0.317224	0.158612	+	0.987341i	$0.449298\pi$
$420$	0	0
$421$	−29.8251	−1.45359	−0.726794	−	0.686856i	$-0.758990\pi$
$421$	−29.8251	−1.45359	−0.726794	+	0.686856i	$0.758990\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.712742	−0.0345731
$426$	0	0
$427$	−2.70052	−0.130687
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.2882	1.16992	0.584961	−	0.811061i	$-0.301110\pi$
$431$	24.2882	1.16992	0.584961	+	0.811061i	$0.301110\pi$
$432$	0	0
$433$	−18.2677	−0.877892	−0.438946	−	0.898513i	$-0.644648\pi$
$433$	−18.2677	−0.877892	−0.438946	+	0.898513i	$0.644648\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.55149	−0.456910
$438$	0	0
$439$	−20.5623	−0.981385	−0.490692	−	0.871333i	$-0.663256\pi$
$439$	−20.5623	−0.981385	−0.490692	+	0.871333i	$0.663256\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−40.0771	−1.90412	−0.952061	−	0.305908i	$-0.901040\pi$
$443$	−40.0771	−1.90412	−0.952061	+	0.305908i	$0.901040\pi$
$444$	0	0
$445$	17.9248	0.849716
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.4993	0.967421	0.483711	−	0.875228i	$-0.339289\pi$
$449$	20.4993	0.967421	0.483711	+	0.875228i	$0.339289\pi$
$450$	0	0
$451$	7.23884	0.340864
$452$	0	0
$453$	0	0
$454$	0	0
$455$	16.6253	0.779406
$456$	0	0
$457$	−21.4763	−1.00462	−0.502309	−	0.864688i	$-0.667516\pi$
$457$	−21.4763	−1.00462	−0.502309	+	0.864688i	$0.667516\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.48612	0.0692155	0.0346077	−	0.999401i	$-0.488982\pi$
$461$	1.48612	0.0692155	0.0346077	+	0.999401i	$0.488982\pi$
$462$	0	0
$463$	−24.6982	−1.14782	−0.573910	−	0.818918i	$-0.694574\pi$
$463$	−24.6982	−1.14782	−0.573910	+	0.818918i	$0.694574\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.14903	0.238269	0.119134	−	0.992878i	$-0.461988\pi$
$467$	5.14903	0.238269	0.119134	+	0.992878i	$0.461988\pi$
$468$	0	0
$469$	17.7137	0.817943
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.62672	0.0747964
$474$	0	0
$475$	3.53690	0.162284
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.8773	−0.953909	−0.476954	−	0.878928i	$-0.658259\pi$
$479$	−20.8773	−0.953909	−0.476954	+	0.878928i	$0.658259\pi$
$480$	0	0
$481$	1.70194	0.0776017
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−26.8872	−1.22088
$486$	0	0
$487$	11.8373	0.536401	0.268200	−	0.963363i	$-0.413571\pi$
$487$	11.8373	0.536401	0.268200	+	0.963363i	$0.413571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.3576	−0.557689	−0.278844	−	0.960336i	$-0.589951\pi$
$491$	−12.3576	−0.557689	−0.278844	+	0.960336i	$0.589951\pi$
$492$	0	0
$493$	−2.17679	−0.0980378
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.4010	−0.601119
$498$	0	0
$499$	23.5853	1.05582	0.527912	−	0.849299i	$-0.322975\pi$
$499$	23.5853	1.05582	0.527912	+	0.849299i	$0.322975\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.0449056	−0.00200224	−0.00100112	−	0.999999i	$-0.500319\pi$
$503$	−0.0449056	−0.00200224	−0.00100112	+	0.999999i	$0.500319\pi$
$504$	0	0
$505$	3.32979	0.148174
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.4387	−1.08322	−0.541612	−	0.840628i	$-0.682186\pi$
$509$	−24.4387	−1.08322	−0.541612	+	0.840628i	$0.682186\pi$
$510$	0	0
$511$	−12.1016	−0.535342
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.7685	−0.518580
$516$	0	0
$517$	−6.20123	−0.272730
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.2252	0.798461	0.399230	−	0.916851i	$-0.369277\pi$
$521$	18.2252	0.798461	0.399230	+	0.916851i	$0.369277\pi$
$522$	0	0
$523$	29.3112	1.28169	0.640845	−	0.767670i	$-0.278584\pi$
$523$	29.3112	1.28169	0.640845	+	0.767670i	$0.278584\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.70052	0.117637
$528$	0	0
$529$	12.1030	0.526217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.9610	0.777976
$534$	0	0
$535$	17.3258	0.749061
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.04349	−0.217239
$540$	0	0
$541$	24.9887	1.07435	0.537175	−	0.843471i	$-0.319492\pi$
$541$	24.9887	1.07435	0.537175	+	0.843471i	$0.319492\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.26187	−0.0968877
$546$	0	0
$547$	−1.10062	−0.0470589	−0.0235295	−	0.999723i	$-0.507490\pi$
$547$	−1.10062	−0.0470589	−0.0235295	+	0.999723i	$0.507490\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.8021	0.460185
$552$	0	0
$553$	40.7875	1.73446
$554$	0	0
$555$	0	0
$556$	0	0
$557$	44.4993	1.88550	0.942748	−	0.333507i	$-0.108232\pi$
$557$	44.4993	1.88550	0.942748	+	0.333507i	$0.108232\pi$
$558$	0	0
$559$	4.03620	0.170713
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−37.4372	−1.57779	−0.788896	−	0.614527i	$-0.789347\pi$
$563$	−37.4372	−1.57779	−0.788896	+	0.614527i	$0.789347\pi$
$564$	0	0
$565$	19.1187	0.804330
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.1744	1.09729	0.548644	−	0.836056i	$-0.315144\pi$
$569$	26.1744	1.09729	0.548644	+	0.836056i	$0.315144\pi$
$570$	0	0
$571$	1.92715	0.0806486	0.0403243	−	0.999187i	$-0.487161\pi$
$571$	1.92715	0.0806486	0.0403243	+	0.999187i	$0.487161\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.9986	−0.542078
$576$	0	0
$577$	−16.4934	−0.686630	−0.343315	−	0.939220i	$-0.611550\pi$
$577$	−16.4934	−0.686630	−0.343315	+	0.939220i	$0.611550\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−39.0738	−1.62105
$582$	0	0
$583$	13.3865	0.554410
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.2130	−0.916828	−0.458414	−	0.888739i	$-0.651582\pi$
$587$	−22.2130	−0.916828	−0.458414	+	0.888739i	$0.651582\pi$
$588$	0	0
$589$	−13.4010	−0.552181
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.90080	0.160187	0.0800933	−	0.996787i	$-0.474478\pi$
$593$	3.90080	0.160187	0.0800933	+	0.996787i	$0.474478\pi$
$594$	0	0
$595$	−1.82321	−0.0747442
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.4010	−1.20129	−0.600647	−	0.799514i	$-0.705090\pi$
$599$	−29.4010	−1.20129	−0.600647	+	0.799514i	$0.705090\pi$
$600$	0	0
$601$	13.8496	0.564935	0.282468	−	0.959277i	$-0.408847\pi$
$601$	13.8496	0.564935	0.282468	+	0.959277i	$0.408847\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16.0386	0.652060
$606$	0	0
$607$	19.2022	0.779393	0.389696	−	0.920943i	$-0.372580\pi$
$607$	19.2022	0.779393	0.389696	+	0.920943i	$0.372580\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.3865	−0.622469
$612$	0	0
$613$	12.1319	0.490002	0.245001	−	0.969523i	$-0.421212\pi$
$613$	12.1319	0.490002	0.245001	+	0.969523i	$0.421212\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.0620	−0.445341	−0.222671	−	0.974894i	$-0.571477\pi$
$617$	−11.0620	−0.445341	−0.222671	+	0.974894i	$0.571477\pi$
$618$	0	0
$619$	−10.3150	−0.414596	−0.207298	−	0.978278i	$-0.566467\pi$
$619$	−10.3150	−0.414596	−0.207298	+	0.978278i	$0.566467\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−35.8496	−1.43628
$624$	0	0
$625$	−9.21696	−0.368678
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.186642	−0.00744192
$630$	0	0
$631$	−26.4847	−1.05434	−0.527170	−	0.849760i	$-0.676747\pi$
$631$	−26.4847	−1.05434	−0.527170	+	0.849760i	$0.676747\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.61213	0.0639753
$636$	0	0
$637$	−12.5139	−0.495818
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.3136	1.31581	0.657904	−	0.753102i	$-0.271443\pi$
$641$	33.3136	1.31581	0.657904	+	0.753102i	$0.271443\pi$
$642$	0	0
$643$	23.9224	0.943408	0.471704	−	0.881757i	$-0.343639\pi$
$643$	23.9224	0.943408	0.471704	+	0.881757i	$0.343639\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.8021	1.36821	0.684106	−	0.729383i	$-0.260193\pi$
$647$	34.8021	1.36821	0.684106	+	0.729383i	$0.260193\pi$
$648$	0	0
$649$	5.55149	0.217915
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.1622	0.945540	0.472770	−	0.881186i	$-0.343254\pi$
$653$	24.1622	0.945540	0.472770	+	0.881186i	$0.343254\pi$
$654$	0	0
$655$	−15.5369	−0.607077
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.1721	0.396247	0.198123	−	0.980177i	$-0.436515\pi$
$659$	10.1721	0.396247	0.198123	+	0.980177i	$0.436515\pi$
$660$	0	0
$661$	−40.2736	−1.56646	−0.783231	−	0.621731i	$-0.786430\pi$
$661$	−40.2736	−1.56646	−0.783231	+	0.621731i	$0.786430\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.04746	0.350845
$666$	0	0
$667$	−39.6991	−1.53716
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.962389	0.0371526
$672$	0	0
$673$	−15.1490	−0.583952	−0.291976	−	0.956426i	$-0.594313\pi$
$673$	−15.1490	−0.583952	−0.291976	+	0.956426i	$0.594313\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.6531	0.832195	0.416097	−	0.909320i	$-0.363398\pi$
$677$	21.6531	0.832195	0.416097	+	0.909320i	$0.363398\pi$
$678$	0	0
$679$	53.7743	2.06367
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.63656	0.368733	0.184366	−	0.982858i	$-0.440977\pi$
$683$	9.63656	0.368733	0.184366	+	0.982858i	$0.440977\pi$
$684$	0	0
$685$	7.78892	0.297599
$686$	0	0
$687$	0	0
$688$	0	0
$689$	33.2144	1.26537
$690$	0	0
$691$	−10.4001	−0.395638	−0.197819	−	0.980239i	$-0.563386\pi$
$691$	−10.4001	−0.395638	−0.197819	+	0.980239i	$0.563386\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	29.6082	1.12310
$696$	0	0
$697$	−1.96968	−0.0746071
$698$	0	0
$699$	0	0
$700$	0	0
$701$	40.6615	1.53576	0.767882	−	0.640592i	$-0.221311\pi$
$701$	40.6615	1.53576	0.767882	+	0.640592i	$0.221311\pi$
$702$	0	0
$703$	0.926192	0.0349320
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.65959	−0.250460
$708$	0	0
$709$	−28.7367	−1.07923	−0.539615	−	0.841912i	$-0.681430\pi$
$709$	−28.7367	−1.07923	−0.539615	+	0.841912i	$0.681430\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	49.2506	1.84445
$714$	0	0
$715$	−5.92478	−0.221574
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−11.8496	−0.441914	−0.220957	−	0.975284i	$-0.570918\pi$
$719$	−11.8496	−0.441914	−0.220957	+	0.975284i	$0.570918\pi$
$720$	0	0
$721$	23.5369	0.876560
$722$	0	0
$723$	0	0
$724$	0	0
$725$	14.7005	0.545964
$726$	0	0
$727$	−3.13681	−0.116338	−0.0581690	−	0.998307i	$-0.518526\pi$
$727$	−3.13681	−0.116338	−0.0581690	+	0.998307i	$0.518526\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.442628	−0.0163712
$732$	0	0
$733$	15.1939	0.561201	0.280600	−	0.959825i	$-0.409466\pi$
$733$	15.1939	0.561201	0.280600	+	0.959825i	$0.409466\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.31265	−0.232529
$738$	0	0
$739$	19.9271	0.733032	0.366516	−	0.930412i	$-0.380551\pi$
$739$	19.9271	0.733032	0.366516	+	0.930412i	$0.380551\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.1490	−0.482391	−0.241196	−	0.970477i	$-0.577540\pi$
$743$	−13.1490	−0.482391	−0.241196	+	0.970477i	$0.577540\pi$
$744$	0	0
$745$	9.16950	0.335944
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−34.6516	−1.26614
$750$	0	0
$751$	24.2252	0.883990	0.441995	−	0.897017i	$-0.354271\pi$
$751$	24.2252	0.883990	0.441995	+	0.897017i	$0.354271\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.6424	0.569287
$756$	0	0
$757$	16.3272	0.593424	0.296712	−	0.954967i	$-0.404110\pi$
$757$	16.3272	0.593424	0.296712	+	0.954967i	$0.404110\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	47.2652	1.71336	0.856681	−	0.515847i	$-0.172523\pi$
$761$	47.2652	1.71336	0.856681	+	0.515847i	$0.172523\pi$
$762$	0	0
$763$	4.52373	0.163770
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.7743	0.497362
$768$	0	0
$769$	10.1504	0.366034	0.183017	−	0.983110i	$-0.441414\pi$
$769$	10.1504	0.366034	0.183017	+	0.983110i	$0.441414\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.0616	0.901403	0.450701	−	0.892675i	$-0.351174\pi$
$773$	25.0616	0.901403	0.450701	+	0.892675i	$0.351174\pi$
$774$	0	0
$775$	−18.2374	−0.655108
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.77433	0.350202
$780$	0	0
$781$	4.77575	0.170890
$782$	0	0
$783$	0	0
$784$	0	0
$785$	33.5877	1.19880
$786$	0	0
$787$	25.5491	0.910728	0.455364	−	0.890305i	$-0.349509\pi$
$787$	25.5491	0.910728	0.455364	+	0.890305i	$0.349509\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−38.2374	−1.35957
$792$	0	0
$793$	2.38787	0.0847959
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.2384	1.17736	0.588682	−	0.808365i	$-0.299647\pi$
$797$	33.2384	1.17736	0.588682	+	0.808365i	$0.299647\pi$
$798$	0	0
$799$	1.68735	0.0596941
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.31265	0.152190
$804$	0	0
$805$	−33.2506	−1.17193
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.0997	0.355085	0.177543	−	0.984113i	$-0.443185\pi$
$809$	10.0997	0.355085	0.177543	+	0.984113i	$0.443185\pi$
$810$	0	0
$811$	7.13681	0.250607	0.125304	−	0.992118i	$-0.460009\pi$
$811$	7.13681	0.250607	0.125304	+	0.992118i	$0.460009\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−30.6556	−1.07382
$816$	0	0
$817$	2.19649	0.0768455
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.9878	0.557977	0.278989	−	0.960294i	$-0.410001\pi$
$821$	15.9878	0.557977	0.278989	+	0.960294i	$0.410001\pi$
$822$	0	0
$823$	−37.7377	−1.31545	−0.657726	−	0.753257i	$-0.728482\pi$
$823$	−37.7377	−1.31545	−0.657726	+	0.753257i	$0.728482\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.6155	0.786416	0.393208	−	0.919449i	$-0.371365\pi$
$827$	22.6155	0.786416	0.393208	+	0.919449i	$0.371365\pi$
$828$	0	0
$829$	−21.3503	−0.741525	−0.370763	−	0.928728i	$-0.620904\pi$
$829$	−21.3503	−0.741525	−0.370763	+	0.928728i	$0.620904\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.37233	0.0475484
$834$	0	0
$835$	−1.67513	−0.0579703
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27.1187	−0.936242	−0.468121	−	0.883664i	$-0.655069\pi$
$839$	−27.1187	−0.936242	−0.468121	+	0.883664i	$0.655069\pi$
$840$	0	0
$841$	15.8970	0.548173
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.07618	0.243428
$846$	0	0
$847$	−32.0771	−1.10218
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.40388	−0.116683
$852$	0	0
$853$	33.1305	1.13437	0.567183	−	0.823592i	$-0.308033\pi$
$853$	33.1305	1.13437	0.567183	+	0.823592i	$0.308033\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.9868	0.853534	0.426767	−	0.904362i	$-0.359652\pi$
$857$	24.9868	0.853534	0.426767	+	0.904362i	$0.359652\pi$
$858$	0	0
$859$	−23.2896	−0.794632	−0.397316	−	0.917682i	$-0.630058\pi$
$859$	−23.2896	−0.794632	−0.397316	+	0.917682i	$0.630058\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.0362	−1.09052	−0.545262	−	0.838265i	$-0.683570\pi$
$863$	−32.0362	−1.09052	−0.545262	+	0.838265i	$0.683570\pi$
$864$	0	0
$865$	29.5026	1.00312
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.5355	−0.493083
$870$	0	0
$871$	−15.6629	−0.530718
$872$	0	0
$873$	0	0
$874$	0	0
$875$	40.3733	1.36487
$876$	0	0
$877$	−20.1721	−0.681162	−0.340581	−	0.940215i	$-0.610624\pi$
$877$	−20.1721	−0.681162	−0.340581	+	0.940215i	$0.610624\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.3757	0.551710	0.275855	−	0.961199i	$-0.411039\pi$
$881$	16.3757	0.551710	0.275855	+	0.961199i	$0.411039\pi$
$882$	0	0
$883$	5.98541	0.201425	0.100713	−	0.994916i	$-0.467888\pi$
$883$	5.98541	0.201425	0.100713	+	0.994916i	$0.467888\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	55.6093	1.86718	0.933589	−	0.358346i	$-0.116659\pi$
$887$	55.6093	1.86718	0.933589	+	0.358346i	$0.116659\pi$
$888$	0	0
$889$	−3.22425	−0.108138
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.37328	−0.280201
$894$	0	0
$895$	−27.4274	−0.916797
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−55.6991	−1.85767
$900$	0	0
$901$	−3.64244	−0.121347
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.39772	−0.0797030
$906$	0	0
$907$	9.11283	0.302587	0.151293	−	0.988489i	$-0.451656\pi$
$907$	9.11283	0.302587	0.151293	+	0.988489i	$0.451656\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.6991	−1.31529	−0.657645	−	0.753328i	$-0.728447\pi$
$911$	−39.6991	−1.31529	−0.657645	+	0.753328i	$0.728447\pi$
$912$	0	0
$913$	13.9248	0.460843
$914$	0	0
$915$	0	0
$916$	0	0
$917$	31.0738	1.02615
$918$	0	0
$919$	5.79877	0.191284	0.0956419	−	0.995416i	$-0.469510\pi$
$919$	5.79877	0.191284	0.0956419	+	0.995416i	$0.469510\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.8496	0.390033
$924$	0	0
$925$	1.26045	0.0414434
$926$	0	0
$927$	0	0
$928$	0	0
$929$	43.1636	1.41615	0.708076	−	0.706136i	$-0.249563\pi$
$929$	43.1636	1.41615	0.708076	+	0.706136i	$0.249563\pi$
$930$	0	0
$931$	−6.81003	−0.223190
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.649738	0.0212487
$936$	0	0
$937$	48.1886	1.57425	0.787126	−	0.616793i	$-0.211568\pi$
$937$	48.1886	1.57425	0.787126	+	0.616793i	$0.211568\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.31028	0.270907	0.135454	−	0.990784i	$-0.456751\pi$
$941$	8.31028	0.270907	0.135454	+	0.990784i	$0.456751\pi$
$942$	0	0
$943$	−35.9219	−1.16978
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.66879	0.184211	0.0921055	−	0.995749i	$-0.470640\pi$
$947$	5.66879	0.184211	0.0921055	+	0.995749i	$0.470640\pi$
$948$	0	0
$949$	10.7005	0.347354
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−25.6869	−0.832080	−0.416040	−	0.909346i	$-0.636582\pi$
$953$	−25.6869	−0.832080	−0.416040	+	0.909346i	$0.636582\pi$
$954$	0	0
$955$	−2.52373	−0.0816660
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15.5778	−0.503035
$960$	0	0
$961$	38.1002	1.22904
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.52705	0.0491576
$966$	0	0
$967$	−41.6893	−1.34064	−0.670318	−	0.742074i	$-0.733842\pi$
$967$	−41.6893	−1.34064	−0.670318	+	0.742074i	$0.733842\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	48.4993	1.55642	0.778208	−	0.628006i	$-0.216129\pi$
$971$	48.4993	1.55642	0.778208	+	0.628006i	$0.216129\pi$
$972$	0	0
$973$	−59.2163	−1.89839
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.7851	−0.984904	−0.492452	−	0.870340i	$-0.663899\pi$
$977$	−30.7851	−0.984904	−0.492452	+	0.870340i	$0.663899\pi$
$978$	0	0
$979$	12.7757	0.408315
$980$	0	0
$981$	0	0
$982$	0	0
$983$	31.2652	0.997205	0.498602	−	0.866831i	$-0.333847\pi$
$983$	31.2652	0.997205	0.498602	+	0.866831i	$0.333847\pi$
$984$	0	0
$985$	−22.5540	−0.718632
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.07239	−0.256687
$990$	0	0
$991$	7.39868	0.235027	0.117513	−	0.993071i	$-0.462508\pi$
$991$	7.39868	0.235027	0.117513	+	0.993071i	$0.462508\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	30.7612	0.975194
$996$	0	0
$997$	50.2189	1.59045	0.795224	−	0.606316i	$-0.207353\pi$
$997$	50.2189	1.59045	0.795224	+	0.606316i	$0.207353\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.b.1.1	✓	3
3.2	odd	2		6012.2.a.c.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.a.b.1.1	✓	3	1.1	even	1	trivial
6012.2.a.c.1.3	yes	3	3.2	odd	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67513 q^{5} +3.35026 q^{7} +O(q^{10})\)	\(q-1.67513 q^{5} +3.35026 q^{7} -1.19394 q^{11} -2.96239 q^{13} +0.324869 q^{17} -1.61213 q^{19} +5.92478 q^{23} -2.19394 q^{25} -6.70052 q^{29} +8.31265 q^{31} -5.61213 q^{35} -0.574515 q^{37} -6.06300 q^{41} -1.36248 q^{43} +5.19394 q^{47} +4.22425 q^{49} -11.2120 q^{53} +2.00000 q^{55} -4.64974 q^{59} -0.806063 q^{61} +4.96239 q^{65} +5.28726 q^{67} -4.00000 q^{71} -3.61213 q^{73} -4.00000 q^{77} +12.1744 q^{79} -11.6629 q^{83} -0.544198 q^{85} -10.7005 q^{89} -9.92478 q^{91} +2.70052 q^{95} +16.0508 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q+O(q^{10}) \) 3 * q	\( 3 q - 4 q^{11} + 2 q^{13} + 6 q^{17} - 4 q^{19} - 4 q^{23} - 7 q^{25} + 4 q^{31} - 16 q^{35} + 10 q^{37} - 14 q^{41} - 20 q^{43} + 16 q^{47} + 11 q^{49} - 6 q^{53} + 6 q^{55} - 24 q^{59} - 2 q^{61} + 4 q^{65} + 10 q^{67} - 12 q^{71} - 10 q^{73} - 12 q^{77} - 2 q^{79} - 4 q^{83} + 8 q^{85} - 12 q^{89} - 8 q^{91} - 12 q^{95} + 18 q^{97}+O(q^{100}) \) 3 * q - 4 * q^11 + 2 * q^13 + 6 * q^17 - 4 * q^19 - 4 * q^23 - 7 * q^25 + 4 * q^31 - 16 * q^35 + 10 * q^37 - 14 * q^41 - 20 * q^43 + 16 * q^47 + 11 * q^49 - 6 * q^53 + 6 * q^55 - 24 * q^59 - 2 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 - 10 * q^73 - 12 * q^77 - 2 * q^79 - 4 * q^83 + 8 * q^85 - 12 * q^89 - 8 * q^91 - 12 * q^95 + 18 * q^97

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