LMFDB - Embedded newform 6012.2.a.k.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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 26x^{8} + 82x^{7} + 211x^{6} - 340x^{5} - 593x^{4} + 192x^{3} + 423x^{2} + 126x + 9 $ x^10 - 4x^9 - 26x^8 + 82x^7 + 211x^6 - 340x^5 - 593x^4 + 192x^3 + 423x^2 + 126*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.291281$ 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)

$f(q)$	$=$	$q-3.41200 q^{5} -0.291281 q^{7} +O(q^{10})$	$q-3.41200 q^{5} -0.291281 q^{7} +6.12719 q^{11} +2.59878 q^{13} +3.77398 q^{17} -1.94877 q^{19} +4.34133 q^{23} +6.64172 q^{25} -1.08847 q^{29} +4.46417 q^{31} +0.993850 q^{35} -6.62870 q^{37} -4.23469 q^{41} -4.37204 q^{43} -1.99179 q^{47} -6.91516 q^{49} +12.6312 q^{53} -20.9060 q^{55} -13.9769 q^{59} +11.3261 q^{61} -8.86703 q^{65} -5.07954 q^{67} +9.79943 q^{71} -4.08179 q^{73} -1.78474 q^{77} -1.63850 q^{79} +10.9884 q^{83} -12.8768 q^{85} -6.60084 q^{89} -0.756976 q^{91} +6.64919 q^{95} +0.569922 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{5} + 4 q^{7}+O(q^{10}) $ 10 * q + 6 * q^5 + 4 * q^7	$ 10 q + 6 q^{5} + 4 q^{7} + 8 q^{11} - 2 q^{13} + 6 q^{17} + 20 q^{23} + 24 q^{25} + 8 q^{29} - 4 q^{31} - 4 q^{37} - 14 q^{41} + 20 q^{43} + 48 q^{47} - 2 q^{49} + 22 q^{53} - 6 q^{55} + 2 q^{59} - 8 q^{61} + 28 q^{65} - 6 q^{67} + 20 q^{71} + 20 q^{73} + 24 q^{77} - 4 q^{79} + 46 q^{83} - 18 q^{85} - 8 q^{89} + 28 q^{91} + 36 q^{95} - 34 q^{97}+O(q^{100}) $ 10 * q + 6 * q^5 + 4 * q^7 + 8 * q^11 - 2 * q^13 + 6 * q^17 + 20 * q^23 + 24 * q^25 + 8 * q^29 - 4 * q^31 - 4 * q^37 - 14 * q^41 + 20 * q^43 + 48 * q^47 - 2 * q^49 + 22 * q^53 - 6 * q^55 + 2 * q^59 - 8 * q^61 + 28 * q^65 - 6 * q^67 + 20 * q^71 + 20 * q^73 + 24 * q^77 - 4 * q^79 + 46 * q^83 - 18 * q^85 - 8 * q^89 + 28 * q^91 + 36 * q^95 - 34 * 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
$5$	−3.41200	−1.52589	−0.762946	−	0.646463i	$-0.776248\pi$
$5$	−3.41200	−1.52589	−0.762946	+	0.646463i	$0.776248\pi$
$6$	0	0
$7$	−0.291281	−0.110094	−0.0550470	−	0.998484i	$-0.517531\pi$
$7$	−0.291281	−0.110094	−0.0550470	+	0.998484i	$0.517531\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.12719	1.84742	0.923709	−	0.383095i	$-0.125142\pi$
$11$	6.12719	1.84742	0.923709	+	0.383095i	$0.125142\pi$
$12$	0	0
$13$	2.59878	0.720772	0.360386	−	0.932803i	$-0.382645\pi$
$13$	2.59878	0.720772	0.360386	+	0.932803i	$0.382645\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.77398	0.915324	0.457662	−	0.889126i	$-0.348687\pi$
$17$	3.77398	0.915324	0.457662	+	0.889126i	$0.348687\pi$
$18$	0	0
$19$	−1.94877	−0.447078	−0.223539	−	0.974695i	$-0.571761\pi$
$19$	−1.94877	−0.447078	−0.223539	+	0.974695i	$0.571761\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.34133	0.905230	0.452615	−	0.891706i	$-0.350491\pi$
$23$	4.34133	0.905230	0.452615	+	0.891706i	$0.350491\pi$
$24$	0	0
$25$	6.64172	1.32834
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.08847	−0.202123	−0.101062	−	0.994880i	$-0.532224\pi$
$29$	−1.08847	−0.202123	−0.101062	+	0.994880i	$0.532224\pi$
$30$	0	0
$31$	4.46417	0.801789	0.400894	−	0.916124i	$-0.368699\pi$
$31$	4.46417	0.801789	0.400894	+	0.916124i	$0.368699\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.993850	0.167991
$36$	0	0
$37$	−6.62870	−1.08975	−0.544875	−	0.838517i	$-0.683423\pi$
$37$	−6.62870	−1.08975	−0.544875	+	0.838517i	$0.683423\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.23469	−0.661347	−0.330674	−	0.943745i	$-0.607276\pi$
$41$	−4.23469	−0.661347	−0.330674	+	0.943745i	$0.607276\pi$
$42$	0	0
$43$	−4.37204	−0.666730	−0.333365	−	0.942798i	$-0.608184\pi$
$43$	−4.37204	−0.666730	−0.333365	+	0.942798i	$0.608184\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.99179	−0.290532	−0.145266	−	0.989393i	$-0.546404\pi$
$47$	−1.99179	−0.290532	−0.145266	+	0.989393i	$0.546404\pi$
$48$	0	0
$49$	−6.91516	−0.987879
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.6312	1.73503	0.867515	−	0.497411i	$-0.165716\pi$
$53$	12.6312	1.73503	0.867515	+	0.497411i	$0.165716\pi$
$54$	0	0
$55$	−20.9060	−2.81896
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.9769	−1.81964	−0.909822	−	0.414999i	$-0.863782\pi$
$59$	−13.9769	−1.81964	−0.909822	+	0.414999i	$0.863782\pi$
$60$	0	0
$61$	11.3261	1.45016	0.725082	−	0.688663i	$-0.241802\pi$
$61$	11.3261	1.45016	0.725082	+	0.688663i	$0.241802\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.86703	−1.09982
$66$	0	0
$67$	−5.07954	−0.620565	−0.310282	−	0.950644i	$-0.600424\pi$
$67$	−5.07954	−0.620565	−0.310282	+	0.950644i	$0.600424\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.79943	1.16298	0.581489	−	0.813554i	$-0.302470\pi$
$71$	9.79943	1.16298	0.581489	+	0.813554i	$0.302470\pi$
$72$	0	0
$73$	−4.08179	−0.477738	−0.238869	−	0.971052i	$-0.576777\pi$
$73$	−4.08179	−0.477738	−0.238869	+	0.971052i	$0.576777\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.78474	−0.203390
$78$	0	0
$79$	−1.63850	−0.184345	−0.0921726	−	0.995743i	$-0.529381\pi$
$79$	−1.63850	−0.184345	−0.0921726	+	0.995743i	$0.529381\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.9884	1.20613	0.603067	−	0.797690i	$-0.293945\pi$
$83$	10.9884	1.20613	0.603067	+	0.797690i	$0.293945\pi$
$84$	0	0
$85$	−12.8768	−1.39669
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.60084	−0.699688	−0.349844	−	0.936808i	$-0.613765\pi$
$89$	−6.60084	−0.699688	−0.349844	+	0.936808i	$0.613765\pi$
$90$	0	0
$91$	−0.756976	−0.0793526
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.64919	0.682192
$96$	0	0
$97$	0.569922	0.0578668	0.0289334	−	0.999581i	$-0.490789\pi$
$97$	0.569922	0.0578668	0.0289334	+	0.999581i	$0.490789\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.0234	1.09687	0.548433	−	0.836194i	$-0.315225\pi$
$101$	11.0234	1.09687	0.548433	+	0.836194i	$0.315225\pi$
$102$	0	0
$103$	−16.5918	−1.63484	−0.817419	−	0.576043i	$-0.804596\pi$
$103$	−16.5918	−1.63484	−0.817419	+	0.576043i	$0.804596\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.9330	1.05693	0.528464	−	0.848956i	$-0.322768\pi$
$107$	10.9330	1.05693	0.528464	+	0.848956i	$0.322768\pi$
$108$	0	0
$109$	5.78467	0.554071	0.277035	−	0.960860i	$-0.410648\pi$
$109$	5.78467	0.554071	0.277035	+	0.960860i	$0.410648\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.52855	0.143794	0.0718968	−	0.997412i	$-0.477095\pi$
$113$	1.52855	0.143794	0.0718968	+	0.997412i	$0.477095\pi$
$114$	0	0
$115$	−14.8126	−1.38128
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.09929	−0.100772
$120$	0	0
$121$	26.5425	2.41295
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.60155	−0.501018
$126$	0	0
$127$	−5.07418	−0.450261	−0.225130	−	0.974329i	$-0.572281\pi$
$127$	−5.07418	−0.450261	−0.225130	+	0.974329i	$0.572281\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	19.7354	1.72429	0.862147	−	0.506659i	$-0.169120\pi$
$131$	19.7354	1.72429	0.862147	+	0.506659i	$0.169120\pi$
$132$	0	0
$133$	0.567639	0.0492206
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.8992	1.52923	0.764616	−	0.644486i	$-0.222929\pi$
$137$	17.8992	1.52923	0.764616	+	0.644486i	$0.222929\pi$
$138$	0	0
$139$	−18.2401	−1.54711	−0.773553	−	0.633731i	$-0.781523\pi$
$139$	−18.2401	−1.54711	−0.773553	+	0.633731i	$0.781523\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.9232	1.33157
$144$	0	0
$145$	3.71385	0.308418
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.61393	0.214142	0.107071	−	0.994251i	$-0.465853\pi$
$149$	2.61393	0.214142	0.107071	+	0.994251i	$0.465853\pi$
$150$	0	0
$151$	13.9441	1.13475	0.567376	−	0.823459i	$-0.307959\pi$
$151$	13.9441	1.13475	0.567376	+	0.823459i	$0.307959\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−15.2317	−1.22344
$156$	0	0
$157$	3.56066	0.284171	0.142086	−	0.989854i	$-0.454619\pi$
$157$	3.56066	0.284171	0.142086	+	0.989854i	$0.454619\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.26455	−0.0996603
$162$	0	0
$163$	4.31659	0.338102	0.169051	−	0.985607i	$-0.445930\pi$
$163$	4.31659	0.338102	0.169051	+	0.985607i	$0.445930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−6.24634	−0.480488
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.0442	−1.06776	−0.533882	−	0.845559i	$-0.679267\pi$
$173$	−14.0442	−1.06776	−0.533882	+	0.845559i	$0.679267\pi$
$174$	0	0
$175$	−1.93461	−0.146243
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.8702	0.961964	0.480982	−	0.876730i	$-0.340280\pi$
$179$	12.8702	0.961964	0.480982	+	0.876730i	$0.340280\pi$
$180$	0	0
$181$	5.87456	0.436653	0.218326	−	0.975876i	$-0.429940\pi$
$181$	5.87456	0.436653	0.218326	+	0.975876i	$0.429940\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	22.6171	1.66284
$186$	0	0
$187$	23.1239	1.69099
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.8130	−1.50597	−0.752987	−	0.658035i	$-0.771388\pi$
$191$	−20.8130	−1.50597	−0.752987	+	0.658035i	$0.771388\pi$
$192$	0	0
$193$	11.1647	0.803651	0.401825	−	0.915716i	$-0.368376\pi$
$193$	11.1647	0.803651	0.401825	+	0.915716i	$0.368376\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.0757	−0.931603	−0.465801	−	0.884889i	$-0.654234\pi$
$197$	−13.0757	−0.931603	−0.465801	+	0.884889i	$0.654234\pi$
$198$	0	0
$199$	2.39537	0.169803	0.0849017	−	0.996389i	$-0.472942\pi$
$199$	2.39537	0.169803	0.0849017	+	0.996389i	$0.472942\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.317050	0.0222526
$204$	0	0
$205$	14.4487	1.00914
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−11.9405	−0.825940
$210$	0	0
$211$	−5.62852	−0.387483	−0.193742	−	0.981053i	$-0.562062\pi$
$211$	−5.62852	−0.387483	−0.193742	+	0.981053i	$0.562062\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.9174	1.01736
$216$	0	0
$217$	−1.30033	−0.0882721
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.80774	0.659740
$222$	0	0
$223$	18.8661	1.26337	0.631685	−	0.775225i	$-0.282364\pi$
$223$	18.8661	1.26337	0.631685	+	0.775225i	$0.282364\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.7474	1.44343	0.721714	−	0.692191i	$-0.243354\pi$
$227$	21.7474	1.44343	0.721714	+	0.692191i	$0.243354\pi$
$228$	0	0
$229$	24.3968	1.61218	0.806092	−	0.591790i	$-0.201579\pi$
$229$	24.3968	1.61218	0.806092	+	0.591790i	$0.201579\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.50508	−0.360650	−0.180325	−	0.983607i	$-0.557715\pi$
$233$	−5.50508	−0.360650	−0.180325	+	0.983607i	$0.557715\pi$
$234$	0	0
$235$	6.79597	0.443320
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.833851	0.0539373	0.0269687	−	0.999636i	$-0.491415\pi$
$239$	0.833851	0.0539373	0.0269687	+	0.999636i	$0.491415\pi$
$240$	0	0
$241$	−18.4934	−1.19126	−0.595631	−	0.803258i	$-0.703098\pi$
$241$	−18.4934	−1.19126	−0.595631	+	0.803258i	$0.703098\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	23.5945	1.50740
$246$	0	0
$247$	−5.06442	−0.322241
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.16073	−0.451981	−0.225990	−	0.974130i	$-0.572562\pi$
$251$	−7.16073	−0.451981	−0.225990	+	0.974130i	$0.572562\pi$
$252$	0	0
$253$	26.6002	1.67234
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.1460	0.882403	0.441202	−	0.897408i	$-0.354552\pi$
$257$	14.1460	0.882403	0.441202	+	0.897408i	$0.354552\pi$
$258$	0	0
$259$	1.93081	0.119975
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.30539	0.450470	0.225235	−	0.974304i	$-0.427685\pi$
$263$	7.30539	0.450470	0.225235	+	0.974304i	$0.427685\pi$
$264$	0	0
$265$	−43.0976	−2.64747
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.7796	−0.718215	−0.359108	−	0.933296i	$-0.616919\pi$
$269$	−11.7796	−0.718215	−0.359108	+	0.933296i	$0.616919\pi$
$270$	0	0
$271$	20.8381	1.26583	0.632913	−	0.774223i	$-0.281859\pi$
$271$	20.8381	1.26583	0.632913	+	0.774223i	$0.281859\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	40.6951	2.45401
$276$	0	0
$277$	28.7622	1.72815	0.864077	−	0.503359i	$-0.167903\pi$
$277$	28.7622	1.72815	0.864077	+	0.503359i	$0.167903\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.5293	1.04571	0.522854	−	0.852422i	$-0.324867\pi$
$281$	17.5293	1.04571	0.522854	+	0.852422i	$0.324867\pi$
$282$	0	0
$283$	−27.0623	−1.60869	−0.804345	−	0.594163i	$-0.797483\pi$
$283$	−27.0623	−1.60869	−0.804345	+	0.594163i	$0.797483\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.23349	0.0728103
$288$	0	0
$289$	−2.75708	−0.162181
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.3794	0.898475	0.449238	−	0.893412i	$-0.351696\pi$
$293$	15.3794	0.898475	0.449238	+	0.893412i	$0.351696\pi$
$294$	0	0
$295$	47.6893	2.77658
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.2822	0.652464
$300$	0	0
$301$	1.27349	0.0734029
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−38.6448	−2.21279
$306$	0	0
$307$	−13.7824	−0.786604	−0.393302	−	0.919409i	$-0.628667\pi$
$307$	−13.7824	−0.786604	−0.393302	+	0.919409i	$0.628667\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.85650	0.445501	0.222751	−	0.974875i	$-0.428496\pi$
$311$	7.85650	0.445501	0.222751	+	0.974875i	$0.428496\pi$
$312$	0	0
$313$	−4.93813	−0.279120	−0.139560	−	0.990214i	$-0.544569\pi$
$313$	−4.93813	−0.279120	−0.139560	+	0.990214i	$0.544569\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.7567	0.997315	0.498658	−	0.866799i	$-0.333826\pi$
$317$	17.7567	0.997315	0.498658	+	0.866799i	$0.333826\pi$
$318$	0	0
$319$	−6.66925	−0.373406
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.35461	−0.409221
$324$	0	0
$325$	17.2604	0.957434
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.580170	0.0319858
$330$	0	0
$331$	32.2486	1.77254	0.886272	−	0.463165i	$-0.153286\pi$
$331$	32.2486	1.77254	0.886272	+	0.463165i	$0.153286\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	17.3314	0.946915
$336$	0	0
$337$	13.4246	0.731285	0.365643	−	0.930755i	$-0.380849\pi$
$337$	13.4246	0.731285	0.365643	+	0.930755i	$0.380849\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	27.3528	1.48124
$342$	0	0
$343$	4.05322	0.218853
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.4797	−0.562580	−0.281290	−	0.959623i	$-0.590762\pi$
$347$	−10.4797	−0.562580	−0.281290	+	0.959623i	$0.590762\pi$
$348$	0	0
$349$	28.9229	1.54821	0.774104	−	0.633059i	$-0.218201\pi$
$349$	28.9229	1.54821	0.774104	+	0.633059i	$0.218201\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.5377	−1.62536	−0.812679	−	0.582711i	$-0.801992\pi$
$353$	−30.5377	−1.62536	−0.812679	+	0.582711i	$0.801992\pi$
$354$	0	0
$355$	−33.4356	−1.77458
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0301	1.05715	0.528573	−	0.848888i	$-0.322727\pi$
$359$	20.0301	1.05715	0.528573	+	0.848888i	$0.322727\pi$
$360$	0	0
$361$	−15.2023	−0.800121
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.9271	0.728976
$366$	0	0
$367$	−15.1487	−0.790757	−0.395378	−	0.918518i	$-0.629387\pi$
$367$	−15.1487	−0.790757	−0.395378	+	0.918518i	$0.629387\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.67923	−0.191016
$372$	0	0
$373$	23.0327	1.19259	0.596293	−	0.802767i	$-0.296640\pi$
$373$	23.0327	1.19259	0.596293	+	0.802767i	$0.296640\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.82869	−0.145685
$378$	0	0
$379$	−5.64472	−0.289950	−0.144975	−	0.989435i	$-0.546310\pi$
$379$	−5.64472	−0.289950	−0.144975	+	0.989435i	$0.546310\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.1795	−1.43991	−0.719953	−	0.694022i	$-0.755837\pi$
$383$	−28.1795	−1.43991	−0.719953	+	0.694022i	$0.755837\pi$
$384$	0	0
$385$	6.08951	0.310350
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.3995	0.983593	0.491796	−	0.870710i	$-0.336340\pi$
$389$	19.3995	0.983593	0.491796	+	0.870710i	$0.336340\pi$
$390$	0	0
$391$	16.3841	0.828579
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.59054	0.281291
$396$	0	0
$397$	−14.3396	−0.719684	−0.359842	−	0.933013i	$-0.617169\pi$
$397$	−14.3396	−0.719684	−0.359842	+	0.933013i	$0.617169\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.5009	1.52314	0.761572	−	0.648080i	$-0.224428\pi$
$401$	30.5009	1.52314	0.761572	+	0.648080i	$0.224428\pi$
$402$	0	0
$403$	11.6014	0.577907
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−40.6153	−2.01323
$408$	0	0
$409$	14.7855	0.731094	0.365547	−	0.930793i	$-0.380882\pi$
$409$	14.7855	0.731094	0.365547	+	0.930793i	$0.380882\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.07122	0.200332
$414$	0	0
$415$	−37.4924	−1.84043
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.93938	0.143598	0.0717992	−	0.997419i	$-0.477126\pi$
$419$	2.93938	0.143598	0.0717992	+	0.997419i	$0.477126\pi$
$420$	0	0
$421$	−21.1339	−1.03000	−0.515002	−	0.857189i	$-0.672209\pi$
$421$	−21.1339	−1.03000	−0.515002	+	0.857189i	$0.672209\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	25.0657	1.21587
$426$	0	0
$427$	−3.29909	−0.159654
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.3640	−0.643721	−0.321860	−	0.946787i	$-0.604308\pi$
$431$	−13.3640	−0.643721	−0.321860	+	0.946787i	$0.604308\pi$
$432$	0	0
$433$	−37.7325	−1.81331	−0.906654	−	0.421876i	$-0.861372\pi$
$433$	−37.7325	−1.81331	−0.906654	+	0.421876i	$0.861372\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.46024	−0.404708
$438$	0	0
$439$	8.22737	0.392671	0.196335	−	0.980537i	$-0.437096\pi$
$439$	8.22737	0.392671	0.196335	+	0.980537i	$0.437096\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.6602	−0.601504	−0.300752	−	0.953702i	$-0.597238\pi$
$443$	−12.6602	−0.601504	−0.300752	+	0.953702i	$0.597238\pi$
$444$	0	0
$445$	22.5221	1.06765
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.43988	−0.351109	−0.175555	−	0.984470i	$-0.556172\pi$
$449$	−7.43988	−0.351109	−0.175555	+	0.984470i	$0.556172\pi$
$450$	0	0
$451$	−25.9468	−1.22179
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.58280	0.121083
$456$	0	0
$457$	3.22759	0.150980	0.0754902	−	0.997147i	$-0.475948\pi$
$457$	3.22759	0.150980	0.0754902	+	0.997147i	$0.475948\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.58839	0.306852	0.153426	−	0.988160i	$-0.450969\pi$
$461$	6.58839	0.306852	0.153426	+	0.988160i	$0.450969\pi$
$462$	0	0
$463$	37.9414	1.76329	0.881644	−	0.471914i	$-0.156437\pi$
$463$	37.9414	1.76329	0.881644	+	0.471914i	$0.156437\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.95363	0.321775	0.160888	−	0.986973i	$-0.448564\pi$
$467$	6.95363	0.321775	0.160888	+	0.986973i	$0.448564\pi$
$468$	0	0
$469$	1.47958	0.0683204
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−26.7883	−1.23173
$474$	0	0
$475$	−12.9432	−0.593873
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15.1113	0.690454	0.345227	−	0.938519i	$-0.387802\pi$
$479$	15.1113	0.690454	0.345227	+	0.938519i	$0.387802\pi$
$480$	0	0
$481$	−17.2265	−0.785462
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.94457	−0.0882985
$486$	0	0
$487$	−21.8089	−0.988258	−0.494129	−	0.869389i	$-0.664513\pi$
$487$	−21.8089	−0.988258	−0.494129	+	0.869389i	$0.664513\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.33861	−0.376316	−0.188158	−	0.982139i	$-0.560252\pi$
$491$	−8.33861	−0.376316	−0.188158	+	0.982139i	$0.560252\pi$
$492$	0	0
$493$	−4.10786	−0.185008
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.85439	−0.128037
$498$	0	0
$499$	−42.4209	−1.89902	−0.949510	−	0.313738i	$-0.898419\pi$
$499$	−42.4209	−1.89902	−0.949510	+	0.313738i	$0.898419\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.32276	0.0589788	0.0294894	−	0.999565i	$-0.490612\pi$
$503$	1.32276	0.0589788	0.0294894	+	0.999565i	$0.490612\pi$
$504$	0	0
$505$	−37.6117	−1.67370
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.9046	−1.10388	−0.551938	−	0.833885i	$-0.686111\pi$
$509$	−24.9046	−1.10388	−0.551938	+	0.833885i	$0.686111\pi$
$510$	0	0
$511$	1.18895	0.0525960
$512$	0	0
$513$	0	0
$514$	0	0
$515$	56.6112	2.49459
$516$	0	0
$517$	−12.2041	−0.536734
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.5005	−0.591468	−0.295734	−	0.955270i	$-0.595564\pi$
$521$	−13.5005	−0.591468	−0.295734	+	0.955270i	$0.595564\pi$
$522$	0	0
$523$	0.156569	0.00684628	0.00342314	−	0.999994i	$-0.498910\pi$
$523$	0.156569	0.00684628	0.00342314	+	0.999994i	$0.498910\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.8477	0.733897
$528$	0	0
$529$	−4.15287	−0.180559
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.0050	−0.476681
$534$	0	0
$535$	−37.3032	−1.61276
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−42.3705	−1.82503
$540$	0	0
$541$	−37.3245	−1.60470	−0.802352	−	0.596851i	$-0.796418\pi$
$541$	−37.3245	−1.60470	−0.802352	+	0.596851i	$0.796418\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−19.7373	−0.845452
$546$	0	0
$547$	7.48123	0.319874	0.159937	−	0.987127i	$-0.448871\pi$
$547$	7.48123	0.319874	0.159937	+	0.987127i	$0.448871\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.12117	0.0903649
$552$	0	0
$553$	0.477263	0.0202953
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−27.8003	−1.17794	−0.588969	−	0.808156i	$-0.700466\pi$
$557$	−27.8003	−1.17794	−0.588969	+	0.808156i	$0.700466\pi$
$558$	0	0
$559$	−11.3620	−0.480560
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.2028	1.02003	0.510014	−	0.860166i	$-0.329640\pi$
$563$	24.2028	1.02003	0.510014	+	0.860166i	$0.329640\pi$
$564$	0	0
$565$	−5.21540	−0.219413
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.5231	−0.734606	−0.367303	−	0.930101i	$-0.619719\pi$
$569$	−17.5231	−0.734606	−0.367303	+	0.930101i	$0.619719\pi$
$570$	0	0
$571$	−8.32586	−0.348426	−0.174213	−	0.984708i	$-0.555738\pi$
$571$	−8.32586	−0.348426	−0.174213	+	0.984708i	$0.555738\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	28.8339	1.20246
$576$	0	0
$577$	47.7900	1.98952	0.994762	−	0.102220i	$-0.0325945\pi$
$577$	47.7900	1.98952	0.994762	+	0.102220i	$0.0325945\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.20072	−0.132788
$582$	0	0
$583$	77.3938	3.20532
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.6967	1.72101	0.860504	−	0.509443i	$-0.170149\pi$
$587$	41.6967	1.72101	0.860504	+	0.509443i	$0.170149\pi$
$588$	0	0
$589$	−8.69963	−0.358462
$590$	0	0
$591$	0	0
$592$	0	0
$593$	36.3286	1.49184	0.745919	−	0.666037i	$-0.232011\pi$
$593$	36.3286	1.49184	0.745919	+	0.666037i	$0.232011\pi$
$594$	0	0
$595$	3.75077	0.153767
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.7675	0.971112	0.485556	−	0.874206i	$-0.338617\pi$
$599$	23.7675	0.971112	0.485556	+	0.874206i	$0.338617\pi$
$600$	0	0
$601$	−42.4398	−1.73115	−0.865577	−	0.500776i	$-0.833048\pi$
$601$	−42.4398	−1.73115	−0.865577	+	0.500776i	$0.833048\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−90.5629	−3.68191
$606$	0	0
$607$	29.7149	1.20609	0.603046	−	0.797706i	$-0.293954\pi$
$607$	29.7149	1.20609	0.603046	+	0.797706i	$0.293954\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.17622	−0.209407
$612$	0	0
$613$	19.5598	0.790012	0.395006	−	0.918679i	$-0.370743\pi$
$613$	19.5598	0.790012	0.395006	+	0.918679i	$0.370743\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.38720	−0.0558464	−0.0279232	−	0.999610i	$-0.508889\pi$
$617$	−1.38720	−0.0558464	−0.0279232	+	0.999610i	$0.508889\pi$
$618$	0	0
$619$	9.22037	0.370598	0.185299	−	0.982682i	$-0.440675\pi$
$619$	9.22037	0.370598	0.185299	+	0.982682i	$0.440675\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.92270	0.0770314
$624$	0	0
$625$	−14.0961	−0.563845
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−25.0166	−0.997476
$630$	0	0
$631$	31.2883	1.24557	0.622785	−	0.782393i	$-0.286001\pi$
$631$	31.2883	1.24557	0.622785	+	0.782393i	$0.286001\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	17.3131	0.687049
$636$	0	0
$637$	−17.9710	−0.712036
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.5813	−0.654922	−0.327461	−	0.944865i	$-0.606193\pi$
$641$	−16.5813	−0.654922	−0.327461	+	0.944865i	$0.606193\pi$
$642$	0	0
$643$	6.23113	0.245732	0.122866	−	0.992423i	$-0.460792\pi$
$643$	6.23113	0.245732	0.122866	+	0.992423i	$0.460792\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.7254	0.854115	0.427058	−	0.904224i	$-0.359550\pi$
$647$	21.7254	0.854115	0.427058	+	0.904224i	$0.359550\pi$
$648$	0	0
$649$	−85.6395	−3.36164
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.9754	0.859964	0.429982	−	0.902837i	$-0.358520\pi$
$653$	21.9754	0.859964	0.429982	+	0.902837i	$0.358520\pi$
$654$	0	0
$655$	−67.3373	−2.63108
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.6318	−0.959520	−0.479760	−	0.877400i	$-0.659276\pi$
$659$	−24.6318	−0.959520	−0.479760	+	0.877400i	$0.659276\pi$
$660$	0	0
$661$	−11.2582	−0.437893	−0.218947	−	0.975737i	$-0.570262\pi$
$661$	−11.2582	−0.437893	−0.218947	+	0.975737i	$0.570262\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.93678	−0.0751052
$666$	0	0
$667$	−4.72540	−0.182968
$668$	0	0
$669$	0	0
$670$	0	0
$671$	69.3975	2.67906
$672$	0	0
$673$	4.17977	0.161118	0.0805592	−	0.996750i	$-0.474329\pi$
$673$	4.17977	0.161118	0.0805592	+	0.996750i	$0.474329\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.31505	0.281140	0.140570	−	0.990071i	$-0.455106\pi$
$677$	7.31505	0.281140	0.140570	+	0.990071i	$0.455106\pi$
$678$	0	0
$679$	−0.166008	−0.00637078
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.7789	0.871611	0.435806	−	0.900041i	$-0.356464\pi$
$683$	22.7789	0.871611	0.435806	+	0.900041i	$0.356464\pi$
$684$	0	0
$685$	−61.0720	−2.33344
$686$	0	0
$687$	0	0
$688$	0	0
$689$	32.8257	1.25056
$690$	0	0
$691$	44.2391	1.68293	0.841467	−	0.540308i	$-0.181692\pi$
$691$	44.2391	1.68293	0.841467	+	0.540308i	$0.181692\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	62.2352	2.36072
$696$	0	0
$697$	−15.9816	−0.605347
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20.0507	0.757305	0.378652	−	0.925539i	$-0.376388\pi$
$701$	20.0507	0.757305	0.378652	+	0.925539i	$0.376388\pi$
$702$	0	0
$703$	12.9178	0.487203
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.21090	−0.120758
$708$	0	0
$709$	−0.221177	−0.00830648	−0.00415324	−	0.999991i	$-0.501322\pi$
$709$	−0.221177	−0.00830648	−0.00415324	+	0.999991i	$0.501322\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.3804	0.725803
$714$	0	0
$715$	−54.3300	−2.03183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.9058	−0.406718	−0.203359	−	0.979104i	$-0.565186\pi$
$719$	−10.9058	−0.406718	−0.203359	+	0.979104i	$0.565186\pi$
$720$	0	0
$721$	4.83288	0.179986
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.22930	−0.268490
$726$	0	0
$727$	−20.4906	−0.759952	−0.379976	−	0.924996i	$-0.624068\pi$
$727$	−20.4906	−0.759952	−0.379976	+	0.924996i	$0.624068\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16.5000	−0.610274
$732$	0	0
$733$	10.9090	0.402932	0.201466	−	0.979495i	$-0.435429\pi$
$733$	10.9090	0.402932	0.201466	+	0.979495i	$0.435429\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.1233	−1.14644
$738$	0	0
$739$	41.8962	1.54118	0.770588	−	0.637333i	$-0.219963\pi$
$739$	41.8962	1.54118	0.770588	+	0.637333i	$0.219963\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.1544	0.592647	0.296324	−	0.955088i	$-0.404239\pi$
$743$	16.1544	0.592647	0.296324	+	0.955088i	$0.404239\pi$
$744$	0	0
$745$	−8.91874	−0.326757
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.18456	−0.116361
$750$	0	0
$751$	45.1657	1.64812	0.824061	−	0.566502i	$-0.191704\pi$
$751$	45.1657	1.64812	0.824061	+	0.566502i	$0.191704\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−47.5771	−1.73151
$756$	0	0
$757$	−48.0409	−1.74608	−0.873039	−	0.487651i	$-0.837854\pi$
$757$	−48.0409	−1.74608	−0.873039	+	0.487651i	$0.837854\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.6614	−0.785227	−0.392613	−	0.919704i	$-0.628429\pi$
$761$	−21.6614	−0.785227	−0.392613	+	0.919704i	$0.628429\pi$
$762$	0	0
$763$	−1.68496	−0.0609998
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−36.3230	−1.31155
$768$	0	0
$769$	−8.30670	−0.299547	−0.149774	−	0.988720i	$-0.547854\pi$
$769$	−8.30670	−0.299547	−0.149774	+	0.988720i	$0.547854\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−22.6148	−0.813399	−0.406699	−	0.913562i	$-0.633320\pi$
$773$	−22.6148	−0.813399	−0.406699	+	0.913562i	$0.633320\pi$
$774$	0	0
$775$	29.6498	1.06505
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.25242	0.295674
$780$	0	0
$781$	60.0430	2.14851
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.1489	−0.433615
$786$	0	0
$787$	12.3569	0.440475	0.220237	−	0.975446i	$-0.429317\pi$
$787$	12.3569	0.440475	0.220237	+	0.975446i	$0.429317\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.445237	−0.0158308
$792$	0	0
$793$	29.4342	1.04524
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.85494	0.136549	0.0682746	−	0.997667i	$-0.478251\pi$
$797$	3.85494	0.136549	0.0682746	+	0.997667i	$0.478251\pi$
$798$	0	0
$799$	−7.51697	−0.265931
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−25.0099	−0.882581
$804$	0	0
$805$	4.31463	0.152071
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.6187	1.07650	0.538248	−	0.842786i	$-0.319086\pi$
$809$	30.6187	1.07650	0.538248	+	0.842786i	$0.319086\pi$
$810$	0	0
$811$	31.8189	1.11731	0.558656	−	0.829399i	$-0.311317\pi$
$811$	31.8189	1.11731	0.558656	+	0.829399i	$0.311317\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14.7282	−0.515907
$816$	0	0
$817$	8.52009	0.298080
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.7882	−0.795312	−0.397656	−	0.917534i	$-0.630176\pi$
$821$	−22.7882	−0.795312	−0.397656	+	0.917534i	$0.630176\pi$
$822$	0	0
$823$	15.4610	0.538936	0.269468	−	0.963009i	$-0.413152\pi$
$823$	15.4610	0.538936	0.269468	+	0.963009i	$0.413152\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−54.4784	−1.89440	−0.947200	−	0.320643i	$-0.896101\pi$
$827$	−54.4784	−1.89440	−0.947200	+	0.320643i	$0.896101\pi$
$828$	0	0
$829$	15.4131	0.535320	0.267660	−	0.963513i	$-0.413750\pi$
$829$	15.4131	0.535320	0.267660	+	0.963513i	$0.413750\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−26.0977	−0.904230
$834$	0	0
$835$	3.41200	0.118077
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−0.749809	−0.0258863	−0.0129431	−	0.999916i	$-0.504120\pi$
$839$	−0.749809	−0.0258863	−0.0129431	+	0.999916i	$0.504120\pi$
$840$	0	0
$841$	−27.8152	−0.959146
$842$	0	0
$843$	0	0
$844$	0	0
$845$	21.3125	0.733172
$846$	0	0
$847$	−7.73133	−0.265652
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−28.7773	−0.986475
$852$	0	0
$853$	3.21107	0.109945	0.0549724	−	0.998488i	$-0.482493\pi$
$853$	3.21107	0.109945	0.0549724	+	0.998488i	$0.482493\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.7551	1.22137	0.610686	−	0.791873i	$-0.290894\pi$
$857$	35.7551	1.22137	0.610686	+	0.791873i	$0.290894\pi$
$858$	0	0
$859$	26.9461	0.919388	0.459694	−	0.888077i	$-0.347959\pi$
$859$	26.9461	0.919388	0.459694	+	0.888077i	$0.347959\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.0039	−0.714982	−0.357491	−	0.933917i	$-0.616368\pi$
$863$	−21.0039	−0.714982	−0.357491	+	0.933917i	$0.616368\pi$
$864$	0	0
$865$	47.9189	1.62929
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10.0394	−0.340563
$870$	0	0
$871$	−13.2006	−0.447286
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.63163	0.0551591
$876$	0	0
$877$	46.2387	1.56137	0.780686	−	0.624924i	$-0.214870\pi$
$877$	46.2387	1.56137	0.780686	+	0.624924i	$0.214870\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.2918	−1.02056	−0.510278	−	0.860010i	$-0.670457\pi$
$881$	−30.2918	−1.02056	−0.510278	+	0.860010i	$0.670457\pi$
$882$	0	0
$883$	4.42858	0.149034	0.0745168	−	0.997220i	$-0.476259\pi$
$883$	4.42858	0.149034	0.0745168	+	0.997220i	$0.476259\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.5581	1.22750	0.613750	−	0.789500i	$-0.289660\pi$
$887$	36.5581	1.22750	0.613750	+	0.789500i	$0.289660\pi$
$888$	0	0
$889$	1.47801	0.0495710
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.88153	0.129890
$894$	0	0
$895$	−43.9131	−1.46785
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.85911	−0.162060
$900$	0	0
$901$	47.6699	1.58811
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20.0440	−0.666285
$906$	0	0
$907$	−2.29168	−0.0760941	−0.0380471	−	0.999276i	$-0.512114\pi$
$907$	−2.29168	−0.0760941	−0.0380471	+	0.999276i	$0.512114\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.8010	0.689169	0.344585	−	0.938755i	$-0.388020\pi$
$911$	20.8010	0.689169	0.344585	+	0.938755i	$0.388020\pi$
$912$	0	0
$913$	67.3281	2.22824
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.74856	−0.189834
$918$	0	0
$919$	−24.4125	−0.805294	−0.402647	−	0.915355i	$-0.631910\pi$
$919$	−24.4125	−0.805294	−0.402647	+	0.915355i	$0.631910\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	25.4666	0.838242
$924$	0	0
$925$	−44.0260	−1.44756
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27.6851	0.908320	0.454160	−	0.890920i	$-0.349939\pi$
$929$	27.6851	0.908320	0.454160	+	0.890920i	$0.349939\pi$
$930$	0	0
$931$	13.4760	0.441659
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−78.8987	−2.58026
$936$	0	0
$937$	−31.1986	−1.01921	−0.509606	−	0.860408i	$-0.670209\pi$
$937$	−31.1986	−1.01921	−0.509606	+	0.860408i	$0.670209\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28.8519	−0.940545	−0.470273	−	0.882521i	$-0.655844\pi$
$941$	−28.8519	−0.940545	−0.470273	+	0.882521i	$0.655844\pi$
$942$	0	0
$943$	−18.3842	−0.598671
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.12473	0.231523	0.115761	−	0.993277i	$-0.463069\pi$
$947$	7.12473	0.231523	0.115761	+	0.993277i	$0.463069\pi$
$948$	0	0
$949$	−10.6077	−0.344340
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29.6450	−0.960297	−0.480148	−	0.877187i	$-0.659417\pi$
$953$	−29.6450	−0.960297	−0.480148	+	0.877187i	$0.659417\pi$
$954$	0	0
$955$	71.0138	2.29795
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.21370	−0.168359
$960$	0	0
$961$	−11.0712	−0.357135
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−38.0938	−1.22628
$966$	0	0
$967$	36.0455	1.15914	0.579572	−	0.814921i	$-0.303220\pi$
$967$	36.0455	1.15914	0.579572	+	0.814921i	$0.303220\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.800880	0.0257015	0.0128507	−	0.999917i	$-0.495909\pi$
$971$	0.800880	0.0257015	0.0128507	+	0.999917i	$0.495909\pi$
$972$	0	0
$973$	5.31300	0.170327
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.95875	0.318608	0.159304	−	0.987230i	$-0.449075\pi$
$977$	9.95875	0.318608	0.159304	+	0.987230i	$0.449075\pi$
$978$	0	0
$979$	−40.4446	−1.29262
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−34.5905	−1.10327	−0.551633	−	0.834087i	$-0.685995\pi$
$983$	−34.5905	−1.10327	−0.551633	+	0.834087i	$0.685995\pi$
$984$	0	0
$985$	44.6141	1.42152
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18.9805	−0.603543
$990$	0	0
$991$	−0.751978	−0.0238874	−0.0119437	−	0.999929i	$-0.503802\pi$
$991$	−0.751978	−0.0238874	−0.0119437	+	0.999929i	$0.503802\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.17300	−0.259102
$996$	0	0
$997$	4.17186	0.132124	0.0660620	−	0.997816i	$-0.478956\pi$
$997$	4.17186	0.132124	0.0660620	+	0.997816i	$0.478956\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.k.1.1	yes	10
3.2	odd	2		6012.2.a.j.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.a.j.1.10	✓	10	3.2	odd	2
6012.2.a.k.1.1	yes	10	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)

\(f(q)\)	\(=\)	\(q-3.41200 q^{5} -0.291281 q^{7} +O(q^{10})\)	\(q-3.41200 q^{5} -0.291281 q^{7} +6.12719 q^{11} +2.59878 q^{13} +3.77398 q^{17} -1.94877 q^{19} +4.34133 q^{23} +6.64172 q^{25} -1.08847 q^{29} +4.46417 q^{31} +0.993850 q^{35} -6.62870 q^{37} -4.23469 q^{41} -4.37204 q^{43} -1.99179 q^{47} -6.91516 q^{49} +12.6312 q^{53} -20.9060 q^{55} -13.9769 q^{59} +11.3261 q^{61} -8.86703 q^{65} -5.07954 q^{67} +9.79943 q^{71} -4.08179 q^{73} -1.78474 q^{77} -1.63850 q^{79} +10.9884 q^{83} -12.8768 q^{85} -6.60084 q^{89} -0.756976 q^{91} +6.64919 q^{95} +0.569922 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{5} + 4 q^{7}+O(q^{10}) \) 10 * q + 6 * q^5 + 4 * q^7	\( 10 q + 6 q^{5} + 4 q^{7} + 8 q^{11} - 2 q^{13} + 6 q^{17} + 20 q^{23} + 24 q^{25} + 8 q^{29} - 4 q^{31} - 4 q^{37} - 14 q^{41} + 20 q^{43} + 48 q^{47} - 2 q^{49} + 22 q^{53} - 6 q^{55} + 2 q^{59} - 8 q^{61} + 28 q^{65} - 6 q^{67} + 20 q^{71} + 20 q^{73} + 24 q^{77} - 4 q^{79} + 46 q^{83} - 18 q^{85} - 8 q^{89} + 28 q^{91} + 36 q^{95} - 34 q^{97}+O(q^{100}) \) 10 * q + 6 * q^5 + 4 * q^7 + 8 * q^11 - 2 * q^13 + 6 * q^17 + 20 * q^23 + 24 * q^25 + 8 * q^29 - 4 * q^31 - 4 * q^37 - 14 * q^41 + 20 * q^43 + 48 * q^47 - 2 * q^49 + 22 * q^53 - 6 * q^55 + 2 * q^59 - 8 * q^61 + 28 * q^65 - 6 * q^67 + 20 * q^71 + 20 * q^73 + 24 * q^77 - 4 * q^79 + 46 * q^83 - 18 * q^85 - 8 * q^89 + 28 * q^91 + 36 * q^95 - 34 * q^97

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