LMFDB - Embedded newform 6012.2.a.c.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:	$0$
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			$0.311108$ 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)

$f(q)$	$=$	$q-2.21432 q^{5} -4.42864 q^{7} +O(q^{10})$	$q-2.21432 q^{5} -4.42864 q^{7} -0.903212 q^{11} +0.622216 q^{13} -4.21432 q^{17} -5.80642 q^{19} +1.24443 q^{23} -0.0967881 q^{25} -8.85728 q^{29} -3.05086 q^{31} +9.80642 q^{35} -1.18421 q^{37} -2.02074 q^{41} -8.83654 q^{43} -3.09679 q^{47} +12.6128 q^{49} +4.34767 q^{53} +2.00000 q^{55} +12.4286 q^{59} -2.90321 q^{61} -1.37778 q^{65} +5.59210 q^{67} +4.00000 q^{71} -7.80642 q^{73} +4.00000 q^{77} +1.72546 q^{79} -7.47949 q^{83} +9.33185 q^{85} -4.85728 q^{89} -2.75557 q^{91} +12.8573 q^{95} -7.28592 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.21432	−0.990274	−0.495137	−	0.868815i	$-0.664882\pi$
$5$	−2.21432	−0.990274	−0.495137	+	0.868815i	$0.664882\pi$
$6$	0	0
$7$	−4.42864	−1.67387	−0.836934	−	0.547304i	$-0.815654\pi$
$7$	−4.42864	−1.67387	−0.836934	+	0.547304i	$0.815654\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.903212	−0.272329	−0.136164	−	0.990686i	$-0.543478\pi$
$11$	−0.903212	−0.272329	−0.136164	+	0.990686i	$0.543478\pi$
$12$	0	0
$13$	0.622216	0.172572	0.0862858	−	0.996270i	$-0.472500\pi$
$13$	0.622216	0.172572	0.0862858	+	0.996270i	$0.472500\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.21432	−1.02212	−0.511061	−	0.859544i	$-0.670748\pi$
$17$	−4.21432	−1.02212	−0.511061	+	0.859544i	$0.670748\pi$
$18$	0	0
$19$	−5.80642	−1.33208	−0.666042	−	0.745914i	$-0.732013\pi$
$19$	−5.80642	−1.33208	−0.666042	+	0.745914i	$0.732013\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.24443	0.259482	0.129741	−	0.991548i	$-0.458585\pi$
$23$	1.24443	0.259482	0.129741	+	0.991548i	$0.458585\pi$
$24$	0	0
$25$	−0.0967881	−0.0193576
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.85728	−1.64476	−0.822378	−	0.568942i	$-0.807353\pi$
$29$	−8.85728	−1.64476	−0.822378	+	0.568942i	$0.807353\pi$
$30$	0	0
$31$	−3.05086	−0.547950	−0.273975	−	0.961737i	$-0.588339\pi$
$31$	−3.05086	−0.547950	−0.273975	+	0.961737i	$0.588339\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.80642	1.65759
$36$	0	0
$37$	−1.18421	−0.194683	−0.0973413	−	0.995251i	$-0.531034\pi$
$37$	−1.18421	−0.194683	−0.0973413	+	0.995251i	$0.531034\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.02074	−0.315587	−0.157794	−	0.987472i	$-0.550438\pi$
$41$	−2.02074	−0.315587	−0.157794	+	0.987472i	$0.550438\pi$
$42$	0	0
$43$	−8.83654	−1.34756	−0.673780	−	0.738932i	$-0.735330\pi$
$43$	−8.83654	−1.34756	−0.673780	+	0.738932i	$0.735330\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.09679	−0.451713	−0.225856	−	0.974161i	$-0.572518\pi$
$47$	−3.09679	−0.451713	−0.225856	+	0.974161i	$0.572518\pi$
$48$	0	0
$49$	12.6128	1.80184
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.34767	0.597199	0.298599	−	0.954379i	$-0.403481\pi$
$53$	4.34767	0.597199	0.298599	+	0.954379i	$0.403481\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.4286	1.61807	0.809036	−	0.587760i	$-0.199990\pi$
$59$	12.4286	1.61807	0.809036	+	0.587760i	$0.199990\pi$
$60$	0	0
$61$	−2.90321	−0.371718	−0.185859	−	0.982576i	$-0.559507\pi$
$61$	−2.90321	−0.371718	−0.185859	+	0.982576i	$0.559507\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.37778	−0.170893
$66$	0	0
$67$	5.59210	0.683184	0.341592	−	0.939848i	$-0.389034\pi$
$67$	5.59210	0.683184	0.341592	+	0.939848i	$0.389034\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−7.80642	−0.913673	−0.456836	−	0.889551i	$-0.651018\pi$
$73$	−7.80642	−0.913673	−0.456836	+	0.889551i	$0.651018\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	1.72546	0.194129	0.0970646	−	0.995278i	$-0.469055\pi$
$79$	1.72546	0.194129	0.0970646	+	0.995278i	$0.469055\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.47949	−0.820981	−0.410491	−	0.911865i	$-0.634643\pi$
$83$	−7.47949	−0.820981	−0.410491	+	0.911865i	$0.634643\pi$
$84$	0	0
$85$	9.33185	1.01218
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.85728	−0.514871	−0.257435	−	0.966296i	$-0.582877\pi$
$89$	−4.85728	−0.514871	−0.257435	+	0.966296i	$0.582877\pi$
$90$	0	0
$91$	−2.75557	−0.288862
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.8573	1.31913
$96$	0	0
$97$	−7.28592	−0.739773	−0.369886	−	0.929077i	$-0.620603\pi$
$97$	−7.28592	−0.739773	−0.369886	+	0.929077i	$0.620603\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.2652	−1.31993	−0.659967	−	0.751294i	$-0.729430\pi$
$101$	−13.2652	−1.31993	−0.659967	+	0.751294i	$0.729430\pi$
$102$	0	0
$103$	−4.64296	−0.457484	−0.228742	−	0.973487i	$-0.573461\pi$
$103$	−4.64296	−0.457484	−0.228742	+	0.973487i	$0.573461\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.46520	0.915036	0.457518	−	0.889200i	$-0.348739\pi$
$107$	9.46520	0.915036	0.457518	+	0.889200i	$0.348739\pi$
$108$	0	0
$109$	−6.42864	−0.615752	−0.307876	−	0.951426i	$-0.599618\pi$
$109$	−6.42864	−0.615752	−0.307876	+	0.951426i	$0.599618\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.44938	−0.418563	−0.209281	−	0.977855i	$-0.567112\pi$
$113$	−4.44938	−0.418563	−0.209281	+	0.977855i	$0.567112\pi$
$114$	0	0
$115$	−2.75557	−0.256958
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18.6637	1.71090
$120$	0	0
$121$	−10.1842	−0.925837
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.2859	1.00944
$126$	0	0
$127$	2.62222	0.232684	0.116342	−	0.993209i	$-0.462883\pi$
$127$	2.62222	0.232684	0.116342	+	0.993209i	$0.462883\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.67307	0.495658	0.247829	−	0.968804i	$-0.420283\pi$
$131$	5.67307	0.495658	0.247829	+	0.968804i	$0.420283\pi$
$132$	0	0
$133$	25.7146	2.22973
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.4286	1.06185	0.530925	−	0.847419i	$-0.321845\pi$
$137$	12.4286	1.06185	0.530925	+	0.847419i	$0.321845\pi$
$138$	0	0
$139$	−13.7857	−1.16929	−0.584643	−	0.811291i	$-0.698765\pi$
$139$	−13.7857	−1.16929	−0.584643	+	0.811291i	$0.698765\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.561993	−0.0469962
$144$	0	0
$145$	19.6128	1.62876
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.5827	0.866972	0.433486	−	0.901160i	$-0.357283\pi$
$149$	10.5827	0.866972	0.433486	+	0.901160i	$0.357283\pi$
$150$	0	0
$151$	13.6938	1.11439	0.557193	−	0.830383i	$-0.311878\pi$
$151$	13.6938	1.11439	0.557193	+	0.830383i	$0.311878\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.75557	0.542620
$156$	0	0
$157$	3.28592	0.262245	0.131122	−	0.991366i	$-0.458142\pi$
$157$	3.28592	0.262245	0.131122	+	0.991366i	$0.458142\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.51114	−0.434338
$162$	0	0
$163$	−8.31603	−0.651362	−0.325681	−	0.945480i	$-0.605594\pi$
$163$	−8.31603	−0.651362	−0.325681	+	0.945480i	$0.605594\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.6128	−0.970219
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.8064	1.65791	0.828956	−	0.559314i	$-0.188935\pi$
$173$	21.8064	1.65791	0.828956	+	0.559314i	$0.188935\pi$
$174$	0	0
$175$	0.428639	0.0324021
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−25.9813	−1.94193	−0.970965	−	0.239220i	$-0.923108\pi$
$179$	−25.9813	−1.94193	−0.970965	+	0.239220i	$0.923108\pi$
$180$	0	0
$181$	−19.1985	−1.42701	−0.713507	−	0.700649i	$-0.752894\pi$
$181$	−19.1985	−1.42701	−0.713507	+	0.700649i	$0.752894\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.62222	0.192789
$186$	0	0
$187$	3.80642	0.278353
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.9541	0.864966	0.432483	−	0.901642i	$-0.357638\pi$
$191$	11.9541	0.864966	0.432483	+	0.901642i	$0.357638\pi$
$192$	0	0
$193$	−20.6637	−1.48741	−0.743703	−	0.668510i	$-0.766932\pi$
$193$	−20.6637	−1.48741	−0.743703	+	0.668510i	$0.766932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.7353	1.83356	0.916782	−	0.399388i	$-0.130777\pi$
$197$	25.7353	1.83356	0.916782	+	0.399388i	$0.130777\pi$
$198$	0	0
$199$	16.3368	1.15808	0.579042	−	0.815298i	$-0.303427\pi$
$199$	16.3368	1.15808	0.579042	+	0.815298i	$0.303427\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	39.2257	2.75310
$204$	0	0
$205$	4.47457	0.312518
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.24443	0.362765
$210$	0	0
$211$	19.9081	1.37053	0.685266	−	0.728293i	$-0.259686\pi$
$211$	19.9081	1.37053	0.685266	+	0.728293i	$0.259686\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	19.5669	1.33445
$216$	0	0
$217$	13.5111	0.917196
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.62222	−0.176389
$222$	0	0
$223$	6.32693	0.423683	0.211841	−	0.977304i	$-0.432054\pi$
$223$	6.32693	0.423683	0.211841	+	0.977304i	$0.432054\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.62222	0.439532	0.219766	−	0.975553i	$-0.429471\pi$
$227$	6.62222	0.439532	0.219766	+	0.975553i	$0.429471\pi$
$228$	0	0
$229$	−18.9032	−1.24916	−0.624580	−	0.780961i	$-0.714730\pi$
$229$	−18.9032	−1.24916	−0.624580	+	0.780961i	$0.714730\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.0923	−1.25078	−0.625390	−	0.780312i	$-0.715060\pi$
$233$	−19.0923	−1.25078	−0.625390	+	0.780312i	$0.715060\pi$
$234$	0	0
$235$	6.85728	0.447320
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.68736	0.497254	0.248627	−	0.968599i	$-0.420021\pi$
$239$	7.68736	0.497254	0.248627	+	0.968599i	$0.420021\pi$
$240$	0	0
$241$	11.1526	0.718400	0.359200	−	0.933261i	$-0.383050\pi$
$241$	11.1526	0.718400	0.359200	+	0.933261i	$0.383050\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−27.9289	−1.78431
$246$	0	0
$247$	−3.61285	−0.229880
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.8666	1.00149	0.500747	−	0.865594i	$-0.333059\pi$
$251$	15.8666	1.00149	0.500747	+	0.865594i	$0.333059\pi$
$252$	0	0
$253$	−1.12399	−0.0706643
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.91903	−0.119706	−0.0598530	−	0.998207i	$-0.519063\pi$
$257$	−1.91903	−0.119706	−0.0598530	+	0.998207i	$0.519063\pi$
$258$	0	0
$259$	5.24443	0.325873
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.1334	0.994825	0.497413	−	0.867514i	$-0.334284\pi$
$263$	16.1334	0.994825	0.497413	+	0.867514i	$0.334284\pi$
$264$	0	0
$265$	−9.62714	−0.591390
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.06223	−0.369621	−0.184810	−	0.982774i	$-0.559167\pi$
$269$	−6.06223	−0.369621	−0.184810	+	0.982774i	$0.559167\pi$
$270$	0	0
$271$	3.91903	0.238064	0.119032	−	0.992890i	$-0.462021\pi$
$271$	3.91903	0.238064	0.119032	+	0.992890i	$0.462021\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.0874201	0.00527163
$276$	0	0
$277$	25.6543	1.54142	0.770710	−	0.637186i	$-0.219902\pi$
$277$	25.6543	1.54142	0.770710	+	0.637186i	$0.219902\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.7462	0.700720	0.350360	−	0.936615i	$-0.386059\pi$
$281$	11.7462	0.700720	0.350360	+	0.936615i	$0.386059\pi$
$282$	0	0
$283$	−0.295286	−0.0175530	−0.00877648	−	0.999961i	$-0.502794\pi$
$283$	−0.295286	−0.0175530	−0.00877648	+	0.999961i	$0.502794\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.94914	0.528251
$288$	0	0
$289$	0.760491	0.0447348
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.24443	0.0727005	0.0363502	−	0.999339i	$-0.488427\pi$
$293$	1.24443	0.0727005	0.0363502	+	0.999339i	$0.488427\pi$
$294$	0	0
$295$	−27.5210	−1.60233
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.774305	0.0447792
$300$	0	0
$301$	39.1338	2.25564
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.42864	0.368103
$306$	0	0
$307$	−15.7669	−0.899867	−0.449934	−	0.893062i	$-0.648552\pi$
$307$	−15.7669	−0.899867	−0.449934	+	0.893062i	$0.648552\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.1334	0.914839	0.457419	−	0.889251i	$-0.348774\pi$
$311$	16.1334	0.914839	0.457419	+	0.889251i	$0.348774\pi$
$312$	0	0
$313$	25.2257	1.42584	0.712920	−	0.701245i	$-0.247372\pi$
$313$	25.2257	1.42584	0.712920	+	0.701245i	$0.247372\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.4193	0.978364	0.489182	−	0.872182i	$-0.337295\pi$
$317$	17.4193	0.978364	0.489182	+	0.872182i	$0.337295\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.4701	1.36155
$324$	0	0
$325$	−0.0602231	−0.00334057
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13.7146	0.756108
$330$	0	0
$331$	7.65233	0.420610	0.210305	−	0.977636i	$-0.432554\pi$
$331$	7.65233	0.420610	0.210305	+	0.977636i	$0.432554\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−12.3827	−0.676540
$336$	0	0
$337$	1.30465	0.0710690	0.0355345	−	0.999368i	$-0.488687\pi$
$337$	1.30465	0.0710690	0.0355345	+	0.999368i	$0.488687\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.75557	0.149222
$342$	0	0
$343$	−24.8573	−1.34217
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.67307	0.519278	0.259639	−	0.965706i	$-0.416396\pi$
$347$	9.67307	0.519278	0.259639	+	0.965706i	$0.416396\pi$
$348$	0	0
$349$	−17.1842	−0.919850	−0.459925	−	0.887958i	$-0.652124\pi$
$349$	−17.1842	−0.919850	−0.459925	+	0.887958i	$0.652124\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.7239	−1.10302	−0.551512	−	0.834167i	$-0.685949\pi$
$353$	−20.7239	−1.10302	−0.551512	+	0.834167i	$0.685949\pi$
$354$	0	0
$355$	−8.85728	−0.470096
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.3461	−1.02105	−0.510525	−	0.859863i	$-0.670549\pi$
$359$	−19.3461	−1.02105	−0.510525	+	0.859863i	$0.670549\pi$
$360$	0	0
$361$	14.7146	0.774450
$362$	0	0
$363$	0	0
$364$	0	0
$365$	17.2859	0.904786
$366$	0	0
$367$	−13.6860	−0.714402	−0.357201	−	0.934027i	$-0.616269\pi$
$367$	−13.6860	−0.714402	−0.357201	+	0.934027i	$0.616269\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−19.2543	−0.999632
$372$	0	0
$373$	−17.4795	−0.905054	−0.452527	−	0.891751i	$-0.649477\pi$
$373$	−17.4795	−0.905054	−0.452527	+	0.891751i	$0.649477\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.51114	−0.283838
$378$	0	0
$379$	18.7763	0.964474	0.482237	−	0.876041i	$-0.339824\pi$
$379$	18.7763	0.964474	0.482237	+	0.876041i	$0.339824\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.3733	1.09213	0.546063	−	0.837744i	$-0.316126\pi$
$383$	21.3733	1.09213	0.546063	+	0.837744i	$0.316126\pi$
$384$	0	0
$385$	−8.85728	−0.451409
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.65233	−0.489392	−0.244696	−	0.969600i	$-0.578688\pi$
$389$	−9.65233	−0.489392	−0.244696	+	0.969600i	$0.578688\pi$
$390$	0	0
$391$	−5.24443	−0.265222
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.82071	−0.192241
$396$	0	0
$397$	22.5161	1.13005	0.565024	−	0.825074i	$-0.308867\pi$
$397$	22.5161	1.13005	0.565024	+	0.825074i	$0.308867\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0939	1.50282	0.751408	−	0.659838i	$-0.229375\pi$
$401$	30.0939	1.50282	0.751408	+	0.659838i	$0.229375\pi$
$402$	0	0
$403$	−1.89829	−0.0945605
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.06959	0.0530177
$408$	0	0
$409$	−17.3921	−0.859983	−0.429991	−	0.902833i	$-0.641483\pi$
$409$	−17.3921	−0.859983	−0.429991	+	0.902833i	$0.641483\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−55.0420	−2.70844
$414$	0	0
$415$	16.5620	0.812996
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−19.9541	−0.974820	−0.487410	−	0.873173i	$-0.662058\pi$
$419$	−19.9541	−0.974820	−0.487410	+	0.873173i	$0.662058\pi$
$420$	0	0
$421$	15.0192	0.731992	0.365996	−	0.930616i	$-0.380728\pi$
$421$	15.0192	0.731992	0.365996	+	0.930616i	$0.380728\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.407896	0.0197859
$426$	0	0
$427$	12.8573	0.622207
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.5812	0.846857	0.423428	−	0.905930i	$-0.360827\pi$
$431$	17.5812	0.846857	0.423428	+	0.905930i	$0.360827\pi$
$432$	0	0
$433$	−10.2208	−0.491179	−0.245590	−	0.969374i	$-0.578982\pi$
$433$	−10.2208	−0.491179	−0.245590	+	0.969374i	$0.578982\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.22570	−0.345652
$438$	0	0
$439$	−5.91903	−0.282500	−0.141250	−	0.989974i	$-0.545112\pi$
$439$	−5.91903	−0.282500	−0.141250	+	0.989974i	$0.545112\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−37.1022	−1.76278	−0.881389	−	0.472391i	$-0.843391\pi$
$443$	−37.1022	−1.76278	−0.881389	+	0.472391i	$0.843391\pi$
$444$	0	0
$445$	10.7556	0.509863
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.9398	−0.657859	−0.328929	−	0.944355i	$-0.606688\pi$
$449$	−13.9398	−0.657859	−0.328929	+	0.944355i	$0.606688\pi$
$450$	0	0
$451$	1.82516	0.0859434
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.10171	0.286053
$456$	0	0
$457$	2.47013	0.115548	0.0577738	−	0.998330i	$-0.481600\pi$
$457$	2.47013	0.115548	0.0577738	+	0.998330i	$0.481600\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.8479	−1.01756	−0.508779	−	0.860897i	$-0.669903\pi$
$461$	−21.8479	−1.01756	−0.508779	+	0.860897i	$0.669903\pi$
$462$	0	0
$463$	−38.1956	−1.77510	−0.887550	−	0.460712i	$-0.847594\pi$
$463$	−38.1956	−1.77510	−0.887550	+	0.460712i	$0.847594\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.36842	−0.294695	−0.147348	−	0.989085i	$-0.547074\pi$
$467$	−6.36842	−0.294695	−0.147348	+	0.989085i	$0.547074\pi$
$468$	0	0
$469$	−24.7654	−1.14356
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.98126	0.366979
$474$	0	0
$475$	0.561993	0.0257860
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−34.1847	−1.56194	−0.780969	−	0.624570i	$-0.785274\pi$
$479$	−34.1847	−1.56194	−0.780969	+	0.624570i	$0.785274\pi$
$480$	0	0
$481$	−0.736833	−0.0335967
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.1334	0.732578
$486$	0	0
$487$	−17.7540	−0.804512	−0.402256	−	0.915527i	$-0.631774\pi$
$487$	−17.7540	−0.804512	−0.402256	+	0.915527i	$0.631774\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.32248	−0.104812	−0.0524061	−	0.998626i	$-0.516689\pi$
$491$	−2.32248	−0.104812	−0.0524061	+	0.998626i	$0.516689\pi$
$492$	0	0
$493$	37.3274	1.68114
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.7146	−0.794607
$498$	0	0
$499$	26.3289	1.17865	0.589323	−	0.807898i	$-0.299395\pi$
$499$	26.3289	1.17865	0.589323	+	0.807898i	$0.299395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.27163	−0.145875	−0.0729373	−	0.997337i	$-0.523237\pi$
$503$	−3.27163	−0.145875	−0.0729373	+	0.997337i	$0.523237\pi$
$504$	0	0
$505$	29.3733	1.30710
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.09234	−0.137066	−0.0685328	−	0.997649i	$-0.521832\pi$
$509$	−3.09234	−0.137066	−0.0685328	+	0.997649i	$0.521832\pi$
$510$	0	0
$511$	34.5718	1.52937
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.2810	0.453035
$516$	0	0
$517$	2.79706	0.123014
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.5605	0.681716	0.340858	−	0.940115i	$-0.389282\pi$
$521$	15.5605	0.681716	0.340858	+	0.940115i	$0.389282\pi$
$522$	0	0
$523$	4.82870	0.211144	0.105572	−	0.994412i	$-0.466333\pi$
$523$	4.82870	0.211144	0.105572	+	0.994412i	$0.466333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.8573	0.560072
$528$	0	0
$529$	−21.4514	−0.932669
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.25734	−0.0544614
$534$	0	0
$535$	−20.9590	−0.906136
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−11.3921	−0.490691
$540$	0	0
$541$	−32.4385	−1.39464	−0.697320	−	0.716760i	$-0.745624\pi$
$541$	−32.4385	−1.39464	−0.697320	+	0.716760i	$0.745624\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14.2351	0.609763
$546$	0	0
$547$	3.39853	0.145311	0.0726553	−	0.997357i	$-0.476853\pi$
$547$	3.39853	0.145311	0.0726553	+	0.997357i	$0.476853\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	51.4291	2.19095
$552$	0	0
$553$	−7.64143	−0.324947
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−37.9398	−1.60756	−0.803780	−	0.594927i	$-0.797181\pi$
$557$	−37.9398	−1.60756	−0.803780	+	0.594927i	$0.797181\pi$
$558$	0	0
$559$	−5.49823	−0.232550
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.21279	−0.135403	−0.0677014	−	0.997706i	$-0.521567\pi$
$563$	−3.21279	−0.135403	−0.0677014	+	0.997706i	$0.521567\pi$
$564$	0	0
$565$	9.85236	0.414492
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15.7255	−0.659245	−0.329623	−	0.944113i	$-0.606922\pi$
$569$	−15.7255	−0.659245	−0.329623	+	0.944113i	$0.606922\pi$
$570$	0	0
$571$	−34.2973	−1.43530	−0.717649	−	0.696405i	$-0.754782\pi$
$571$	−34.2973	−1.43530	−0.717649	+	0.696405i	$0.754782\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.120446	−0.00502295
$576$	0	0
$577$	−29.9541	−1.24700	−0.623502	−	0.781822i	$-0.714291\pi$
$577$	−29.9541	−1.24700	−0.623502	+	0.781822i	$0.714291\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	33.1240	1.37421
$582$	0	0
$583$	−3.92687	−0.162634
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.8256	−1.10721	−0.553606	−	0.832779i	$-0.686749\pi$
$587$	−26.8256	−1.10721	−0.553606	+	0.832779i	$0.686749\pi$
$588$	0	0
$589$	17.7146	0.729916
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.5190	−0.883678	−0.441839	−	0.897094i	$-0.645674\pi$
$593$	−21.5190	−0.883678	−0.441839	+	0.897094i	$0.645674\pi$
$594$	0	0
$595$	−41.3274	−1.69426
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.71456	−0.0700549	−0.0350275	−	0.999386i	$-0.511152\pi$
$599$	−1.71456	−0.0700549	−0.0350275	+	0.999386i	$0.511152\pi$
$600$	0	0
$601$	−0.488863	−0.0199411	−0.00997056	−	0.999950i	$-0.503174\pi$
$601$	−0.488863	−0.0199411	−0.00997056	+	0.999950i	$0.503174\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22.5511	0.916832
$606$	0	0
$607$	−31.9704	−1.29764	−0.648819	−	0.760943i	$-0.724737\pi$
$607$	−31.9704	−1.29764	−0.648819	+	0.760943i	$0.724737\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.92687	−0.0779528
$612$	0	0
$613$	−24.0558	−0.971604	−0.485802	−	0.874069i	$-0.661472\pi$
$613$	−24.0558	−0.971604	−0.485802	+	0.874069i	$0.661472\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.1526	1.81777	0.908887	−	0.417043i	$-0.136933\pi$
$617$	45.1526	1.81777	0.908887	+	0.417043i	$0.136933\pi$
$618$	0	0
$619$	30.1037	1.20997	0.604985	−	0.796237i	$-0.293179\pi$
$619$	30.1037	1.20997	0.604985	+	0.796237i	$0.293179\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	21.5111	0.861826
$624$	0	0
$625$	−24.5067	−0.980268
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.99063	0.198990
$630$	0	0
$631$	−33.7275	−1.34267	−0.671335	−	0.741154i	$-0.734279\pi$
$631$	−33.7275	−1.34267	−0.671335	+	0.741154i	$0.734279\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.80642	−0.230421
$636$	0	0
$637$	7.84791	0.310946
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.2242	0.798806	0.399403	−	0.916775i	$-0.369217\pi$
$641$	20.2242	0.798806	0.399403	+	0.916775i	$0.369217\pi$
$642$	0	0
$643$	45.8084	1.80651	0.903254	−	0.429106i	$-0.141171\pi$
$643$	45.8084	1.80651	0.903254	+	0.429106i	$0.141171\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.4291	1.07835	0.539175	−	0.842194i	$-0.318736\pi$
$647$	27.4291	1.07835	0.539175	+	0.842194i	$0.318736\pi$
$648$	0	0
$649$	−11.2257	−0.440647
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.53972	0.0602538	0.0301269	−	0.999546i	$-0.490409\pi$
$653$	1.53972	0.0602538	0.0301269	+	0.999546i	$0.490409\pi$
$654$	0	0
$655$	−12.5620	−0.490838
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−28.7783	−1.12104	−0.560522	−	0.828139i	$-0.689400\pi$
$659$	−28.7783	−1.12104	−0.560522	+	0.828139i	$0.689400\pi$
$660$	0	0
$661$	−12.2065	−0.474777	−0.237389	−	0.971415i	$-0.576291\pi$
$661$	−12.2065	−0.474777	−0.237389	+	0.971415i	$0.576291\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−56.9403	−2.20805
$666$	0	0
$667$	−11.0223	−0.426784
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.62222	0.101230
$672$	0	0
$673$	−16.3684	−0.630956	−0.315478	−	0.948933i	$-0.602165\pi$
$673$	−16.3684	−0.630956	−0.315478	+	0.948933i	$0.602165\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.7975	1.60641	0.803205	−	0.595703i	$-0.203126\pi$
$677$	41.7975	1.60641	0.803205	+	0.595703i	$0.203126\pi$
$678$	0	0
$679$	32.2667	1.23828
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−44.3368	−1.69650	−0.848250	−	0.529596i	$-0.822343\pi$
$683$	−44.3368	−1.69650	−0.848250	+	0.529596i	$0.822343\pi$
$684$	0	0
$685$	−27.5210	−1.05152
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.70519	0.103060
$690$	0	0
$691$	−21.4588	−0.816329	−0.408165	−	0.912908i	$-0.633831\pi$
$691$	−21.4588	−0.816329	−0.408165	+	0.912908i	$0.633831\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	30.5259	1.15791
$696$	0	0
$697$	8.51606	0.322569
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.40006	−0.317266	−0.158633	−	0.987338i	$-0.550709\pi$
$701$	−8.40006	−0.317266	−0.158633	+	0.987338i	$0.550709\pi$
$702$	0	0
$703$	6.87601	0.259334
$704$	0	0
$705$	0	0
$706$	0	0
$707$	58.7467	2.20940
$708$	0	0
$709$	−3.64449	−0.136872	−0.0684359	−	0.997656i	$-0.521801\pi$
$709$	−3.64449	−0.136872	−0.0684359	+	0.997656i	$0.521801\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.79658	−0.142183
$714$	0	0
$715$	1.24443	0.0465391
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.48886	−0.0928189	−0.0464095	−	0.998923i	$-0.514778\pi$
$719$	−2.48886	−0.0928189	−0.0464095	+	0.998923i	$0.514778\pi$
$720$	0	0
$721$	20.5620	0.765769
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.857279	0.0318385
$726$	0	0
$727$	10.8968	0.404138	0.202069	−	0.979371i	$-0.435233\pi$
$727$	10.8968	0.404138	0.202069	+	0.979371i	$0.435233\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	37.2400	1.37737
$732$	0	0
$733$	13.0968	0.483741	0.241870	−	0.970309i	$-0.422239\pi$
$733$	13.0968	0.483741	0.241870	+	0.970309i	$0.422239\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.05086	−0.186051
$738$	0	0
$739$	−16.2973	−0.599506	−0.299753	−	0.954017i	$-0.596904\pi$
$739$	−16.2973	−0.599506	−0.299753	+	0.954017i	$0.596904\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.3684	0.527126	0.263563	−	0.964642i	$-0.415102\pi$
$743$	14.3684	0.527126	0.263563	+	0.964642i	$0.415102\pi$
$744$	0	0
$745$	−23.4336	−0.858539
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−41.9180	−1.53165
$750$	0	0
$751$	−9.56046	−0.348866	−0.174433	−	0.984669i	$-0.555809\pi$
$751$	−9.56046	−0.348866	−0.174433	+	0.984669i	$0.555809\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−30.3225	−1.10355
$756$	0	0
$757$	−8.83854	−0.321242	−0.160621	−	0.987016i	$-0.551350\pi$
$757$	−8.83854	−0.321242	−0.160621	+	0.987016i	$0.551350\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.9911	0.434677	0.217339	−	0.976096i	$-0.430262\pi$
$761$	11.9911	0.434677	0.217339	+	0.976096i	$0.430262\pi$
$762$	0	0
$763$	28.4701	1.03069
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.73329	0.279233
$768$	0	0
$769$	24.4889	0.883091	0.441545	−	0.897239i	$-0.354430\pi$
$769$	24.4889	0.883091	0.441545	+	0.897239i	$0.354430\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.85881	−0.138792	−0.0693959	−	0.997589i	$-0.522107\pi$
$773$	−3.85881	−0.138792	−0.0693959	+	0.997589i	$0.522107\pi$
$774$	0	0
$775$	0.295286	0.0106070
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.7333	0.420389
$780$	0	0
$781$	−3.61285	−0.129278
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.27607	−0.259694
$786$	0	0
$787$	37.8272	1.34839	0.674196	−	0.738552i	$-0.264490\pi$
$787$	37.8272	1.34839	0.674196	+	0.738552i	$0.264490\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	19.7047	0.700619
$792$	0	0
$793$	−1.80642	−0.0641480
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.4686	0.972988	0.486494	−	0.873684i	$-0.338276\pi$
$797$	27.4686	0.972988	0.486494	+	0.873684i	$0.338276\pi$
$798$	0	0
$799$	13.0509	0.461706
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.05086	0.248819
$804$	0	0
$805$	12.2034	0.430114
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−47.7748	−1.67967	−0.839836	−	0.542840i	$-0.817349\pi$
$809$	−47.7748	−1.67967	−0.839836	+	0.542840i	$0.817349\pi$
$810$	0	0
$811$	−6.89676	−0.242178	−0.121089	−	0.992642i	$-0.538639\pi$
$811$	−6.89676	−0.242178	−0.121089	+	0.992642i	$0.538639\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	18.4143	0.645027
$816$	0	0
$817$	51.3087	1.79506
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.734825	−0.0256456	−0.0128228	−	0.999918i	$-0.504082\pi$
$821$	−0.734825	−0.0256456	−0.0128228	+	0.999918i	$0.504082\pi$
$822$	0	0
$823$	29.5288	1.02931	0.514655	−	0.857397i	$-0.327920\pi$
$823$	29.5288	1.02931	0.514655	+	0.857397i	$0.327920\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.4197	1.54463	0.772313	−	0.635242i	$-0.219100\pi$
$827$	44.4197	1.54463	0.772313	+	0.635242i	$0.219100\pi$
$828$	0	0
$829$	−13.5714	−0.471353	−0.235676	−	0.971832i	$-0.575731\pi$
$829$	−13.5714	−0.471353	−0.235676	+	0.971832i	$0.575731\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−53.1546	−1.84170
$834$	0	0
$835$	2.21432	0.0766297
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.8524	0.616332	0.308166	−	0.951333i	$-0.400285\pi$
$839$	17.8524	0.616332	0.308166	+	0.951333i	$0.400285\pi$
$840$	0	0
$841$	49.4514	1.70522
$842$	0	0
$843$	0	0
$844$	0	0
$845$	27.9289	0.960783
$846$	0	0
$847$	45.1022	1.54973
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.47367	−0.0505166
$852$	0	0
$853$	−16.1762	−0.553863	−0.276932	−	0.960890i	$-0.589318\pi$
$853$	−16.1762	−0.553863	−0.276932	+	0.960890i	$0.589318\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.9081	−1.77315	−0.886574	−	0.462587i	$-0.846921\pi$
$857$	−51.9081	−1.77315	−0.886574	+	0.462587i	$0.846921\pi$
$858$	0	0
$859$	5.46076	0.186319	0.0931593	−	0.995651i	$-0.470303\pi$
$859$	5.46076	0.186319	0.0931593	+	0.995651i	$0.470303\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.5018	0.765969	0.382985	−	0.923755i	$-0.374896\pi$
$863$	22.5018	0.765969	0.382985	+	0.923755i	$0.374896\pi$
$864$	0	0
$865$	−48.2864	−1.64179
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.55845	−0.0528669
$870$	0	0
$871$	3.47949	0.117898
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−49.9813	−1.68968
$876$	0	0
$877$	−38.7783	−1.30945	−0.654725	−	0.755867i	$-0.727216\pi$
$877$	−38.7783	−1.30945	−0.654725	+	0.755867i	$0.727216\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.07160	0.103485	0.0517424	−	0.998660i	$-0.483523\pi$
$881$	3.07160	0.103485	0.0517424	+	0.998660i	$0.483523\pi$
$882$	0	0
$883$	19.7877	0.665909	0.332954	−	0.942943i	$-0.391954\pi$
$883$	19.7877	0.665909	0.332954	+	0.942943i	$0.391954\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33.5655	−1.12702	−0.563510	−	0.826109i	$-0.690549\pi$
$887$	−33.5655	−1.12702	−0.563510	+	0.826109i	$0.690549\pi$
$888$	0	0
$889$	−11.6128	−0.389482
$890$	0	0
$891$	0	0
$892$	0	0
$893$	17.9813	0.601720
$894$	0	0
$895$	57.5308	1.92304
$896$	0	0
$897$	0	0
$898$	0	0
$899$	27.0223	0.901243
$900$	0	0
$901$	−18.3225	−0.610410
$902$	0	0
$903$	0	0
$904$	0	0
$905$	42.5116	1.41313
$906$	0	0
$907$	19.8666	0.659661	0.329831	−	0.944040i	$-0.393008\pi$
$907$	19.8666	0.659661	0.329831	+	0.944040i	$0.393008\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11.0223	0.365184	0.182592	−	0.983189i	$-0.441551\pi$
$911$	11.0223	0.365184	0.182592	+	0.983189i	$0.441551\pi$
$912$	0	0
$913$	6.75557	0.223577
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−25.1240	−0.829667
$918$	0	0
$919$	14.7971	0.488110	0.244055	−	0.969761i	$-0.421522\pi$
$919$	14.7971	0.488110	0.244055	+	0.969761i	$0.421522\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.48886	0.0819219
$924$	0	0
$925$	0.114617	0.00376859
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30.5807	−1.00332	−0.501661	−	0.865065i	$-0.667277\pi$
$929$	−30.5807	−1.00332	−0.501661	+	0.865065i	$0.667277\pi$
$930$	0	0
$931$	−73.2355	−2.40020
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.42864	−0.275646
$936$	0	0
$937$	−31.3560	−1.02436	−0.512178	−	0.858880i	$-0.671161\pi$
$937$	−31.3560	−1.02436	−0.512178	+	0.858880i	$0.671161\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.0020	−0.847641	−0.423821	−	0.905746i	$-0.639311\pi$
$941$	−26.0020	−0.847641	−0.423821	+	0.905746i	$0.639311\pi$
$942$	0	0
$943$	−2.51468	−0.0818891
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.4938	1.08840	0.544201	−	0.838955i	$-0.316833\pi$
$947$	33.4938	1.08840	0.544201	+	0.838955i	$0.316833\pi$
$948$	0	0
$949$	−4.85728	−0.157674
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−18.2429	−0.590945	−0.295473	−	0.955351i	$-0.595477\pi$
$953$	−18.2429	−0.590945	−0.295473	+	0.955351i	$0.595477\pi$
$954$	0	0
$955$	−26.4701	−0.856553
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−55.0420	−1.77740
$960$	0	0
$961$	−21.6923	−0.699751
$962$	0	0
$963$	0	0
$964$	0	0
$965$	45.7560	1.47294
$966$	0	0
$967$	31.2958	1.00640	0.503202	−	0.864169i	$-0.332155\pi$
$967$	31.2958	1.00640	0.503202	+	0.864169i	$0.332155\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−41.9398	−1.34591	−0.672956	−	0.739683i	$-0.734976\pi$
$971$	−41.9398	−1.34591	−0.672956	+	0.739683i	$0.734976\pi$
$972$	0	0
$973$	61.0518	1.95723
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.4114	0.365084	0.182542	−	0.983198i	$-0.441567\pi$
$977$	11.4114	0.365084	0.182542	+	0.983198i	$0.441567\pi$
$978$	0	0
$979$	4.38715	0.140214
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.9911	0.892778	0.446389	−	0.894839i	$-0.352710\pi$
$983$	27.9911	0.892778	0.446389	+	0.894839i	$0.352710\pi$
$984$	0	0
$985$	−56.9862	−1.81573
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.9965	−0.349667
$990$	0	0
$991$	5.33830	0.169577	0.0847884	−	0.996399i	$-0.472979\pi$
$991$	5.33830	0.169577	0.0847884	+	0.996399i	$0.472979\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−36.1748	−1.14682
$996$	0	0
$997$	−18.8399	−0.596666	−0.298333	−	0.954462i	$-0.596431\pi$
$997$	−18.8399	−0.596666	−0.298333	+	0.954462i	$0.596431\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.21432 q^{5} -4.42864 q^{7} +O(q^{10})\)	\(q-2.21432 q^{5} -4.42864 q^{7} -0.903212 q^{11} +0.622216 q^{13} -4.21432 q^{17} -5.80642 q^{19} +1.24443 q^{23} -0.0967881 q^{25} -8.85728 q^{29} -3.05086 q^{31} +9.80642 q^{35} -1.18421 q^{37} -2.02074 q^{41} -8.83654 q^{43} -3.09679 q^{47} +12.6128 q^{49} +4.34767 q^{53} +2.00000 q^{55} +12.4286 q^{59} -2.90321 q^{61} -1.37778 q^{65} +5.59210 q^{67} +4.00000 q^{71} -7.80642 q^{73} +4.00000 q^{77} +1.72546 q^{79} -7.47949 q^{83} +9.33185 q^{85} -4.85728 q^{89} -2.75557 q^{91} +12.8573 q^{95} -7.28592 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q+O(q^{10}) \) 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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