LMFDB - Embedded newform 6012.2.a.j.1.6

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:	$10$
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			1.6
Root			$2.09707$ 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)

$f(q)$	$=$	$q-0.399927 q^{5} +2.09707 q^{7} +O(q^{10})$	$q-0.399927 q^{5} +2.09707 q^{7} +4.31478 q^{11} +4.27739 q^{13} -5.86662 q^{17} -0.918430 q^{19} -7.26465 q^{23} -4.84006 q^{25} -8.94985 q^{29} -3.68539 q^{31} -0.838677 q^{35} -8.80497 q^{37} +1.00134 q^{41} +7.32968 q^{43} -3.15565 q^{47} -2.60228 q^{49} -3.31402 q^{53} -1.72560 q^{55} -0.721570 q^{59} -12.2242 q^{61} -1.71064 q^{65} +11.0918 q^{67} -13.2885 q^{71} -12.1358 q^{73} +9.04841 q^{77} +0.0176137 q^{79} -2.24460 q^{83} +2.34622 q^{85} -5.28794 q^{89} +8.97000 q^{91} +0.367305 q^{95} +7.03334 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.399927	−0.178853	−0.0894264	−	0.995993i	$-0.528503\pi$
$5$	−0.399927	−0.178853	−0.0894264	+	0.995993i	$0.528503\pi$
$6$	0	0
$7$	2.09707	0.792619	0.396310	−	0.918117i	$-0.370291\pi$
$7$	2.09707	0.792619	0.396310	+	0.918117i	$0.370291\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.31478	1.30096	0.650478	−	0.759525i	$-0.274569\pi$
$11$	4.31478	1.30096	0.650478	+	0.759525i	$0.274569\pi$
$12$	0	0
$13$	4.27739	1.18633	0.593167	−	0.805079i	$-0.297877\pi$
$13$	4.27739	1.18633	0.593167	+	0.805079i	$0.297877\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.86662	−1.42287	−0.711433	−	0.702754i	$-0.751953\pi$
$17$	−5.86662	−1.42287	−0.711433	+	0.702754i	$0.751953\pi$
$18$	0	0
$19$	−0.918430	−0.210702	−0.105351	−	0.994435i	$-0.533597\pi$
$19$	−0.918430	−0.210702	−0.105351	+	0.994435i	$0.533597\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.26465	−1.51478	−0.757392	−	0.652961i	$-0.773527\pi$
$23$	−7.26465	−1.51478	−0.757392	+	0.652961i	$0.773527\pi$
$24$	0	0
$25$	−4.84006	−0.968012
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.94985	−1.66195	−0.830973	−	0.556313i	$-0.812215\pi$
$29$	−8.94985	−1.66195	−0.830973	+	0.556313i	$0.812215\pi$
$30$	0	0
$31$	−3.68539	−0.661915	−0.330957	−	0.943646i	$-0.607372\pi$
$31$	−3.68539	−0.661915	−0.330957	+	0.943646i	$0.607372\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.838677	−0.141762
$36$	0	0
$37$	−8.80497	−1.44753	−0.723764	−	0.690047i	$-0.757590\pi$
$37$	−8.80497	−1.44753	−0.723764	+	0.690047i	$0.757590\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.00134	0.156383	0.0781917	−	0.996938i	$-0.475085\pi$
$41$	1.00134	0.156383	0.0781917	+	0.996938i	$0.475085\pi$
$42$	0	0
$43$	7.32968	1.11777	0.558883	−	0.829247i	$-0.311230\pi$
$43$	7.32968	1.11777	0.558883	+	0.829247i	$0.311230\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.15565	−0.460299	−0.230150	−	0.973155i	$-0.573922\pi$
$47$	−3.15565	−0.460299	−0.230150	+	0.973155i	$0.573922\pi$
$48$	0	0
$49$	−2.60228	−0.371755
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.31402	−0.455216	−0.227608	−	0.973753i	$-0.573091\pi$
$53$	−3.31402	−0.455216	−0.227608	+	0.973753i	$0.573091\pi$
$54$	0	0
$55$	−1.72560	−0.232680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.721570	−0.0939404	−0.0469702	−	0.998896i	$-0.514957\pi$
$59$	−0.721570	−0.0939404	−0.0469702	+	0.998896i	$0.514957\pi$
$60$	0	0
$61$	−12.2242	−1.56515	−0.782574	−	0.622558i	$-0.786094\pi$
$61$	−12.2242	−1.56515	−0.782574	+	0.622558i	$0.786094\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.71064	−0.212179
$66$	0	0
$67$	11.0918	1.35508	0.677538	−	0.735488i	$-0.263047\pi$
$67$	11.0918	1.35508	0.677538	+	0.735488i	$0.263047\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.2885	−1.57705	−0.788526	−	0.615001i	$-0.789155\pi$
$71$	−13.2885	−1.57705	−0.788526	+	0.615001i	$0.789155\pi$
$72$	0	0
$73$	−12.1358	−1.42039	−0.710195	−	0.704005i	$-0.751393\pi$
$73$	−12.1358	−1.42039	−0.710195	+	0.704005i	$0.751393\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.04841	1.03116
$78$	0	0
$79$	0.0176137	0.00198169	0.000990847	−	1.00000i	$-0.499685\pi$
$79$	0.0176137	0.00198169	0.000990847		1.00000i	$0.499685\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.24460	−0.246377	−0.123189	−	0.992383i	$-0.539312\pi$
$83$	−2.24460	−0.246377	−0.123189	+	0.992383i	$0.539312\pi$
$84$	0	0
$85$	2.34622	0.254484
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.28794	−0.560521	−0.280260	−	0.959924i	$-0.590421\pi$
$89$	−5.28794	−0.560521	−0.280260	+	0.959924i	$0.590421\pi$
$90$	0	0
$91$	8.97000	0.940312
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.367305	0.0376847
$96$	0	0
$97$	7.03334	0.714127	0.357064	−	0.934080i	$-0.383778\pi$
$97$	7.03334	0.714127	0.357064	+	0.934080i	$0.383778\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	19.4931	1.93963	0.969816	−	0.243839i	$-0.0784068\pi$
$101$	19.4931	1.93963	0.969816	+	0.243839i	$0.0784068\pi$
$102$	0	0
$103$	−12.5038	−1.23203	−0.616016	−	0.787734i	$-0.711254\pi$
$103$	−12.5038	−1.23203	−0.616016	+	0.787734i	$0.711254\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.31701	−0.320667	−0.160334	−	0.987063i	$-0.551257\pi$
$107$	−3.31701	−0.320667	−0.160334	+	0.987063i	$0.551257\pi$
$108$	0	0
$109$	17.1877	1.64628	0.823142	−	0.567835i	$-0.192219\pi$
$109$	17.1877	1.64628	0.823142	+	0.567835i	$0.192219\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.0349	1.03807	0.519037	−	0.854752i	$-0.326291\pi$
$113$	11.0349	1.03807	0.519037	+	0.854752i	$0.326291\pi$
$114$	0	0
$115$	2.90533	0.270923
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.3027	−1.12779
$120$	0	0
$121$	7.61733	0.692485
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.93531	0.351985
$126$	0	0
$127$	10.7388	0.952916	0.476458	−	0.879197i	$-0.341920\pi$
$127$	10.7388	0.952916	0.476458	+	0.879197i	$0.341920\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.3253	−1.16424	−0.582118	−	0.813105i	$-0.697776\pi$
$131$	−13.3253	−1.16424	−0.582118	+	0.813105i	$0.697776\pi$
$132$	0	0
$133$	−1.92602	−0.167007
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.24766	0.192031	0.0960153	−	0.995380i	$-0.469390\pi$
$137$	2.24766	0.192031	0.0960153	+	0.995380i	$0.469390\pi$
$138$	0	0
$139$	20.1855	1.71211	0.856056	−	0.516882i	$-0.172908\pi$
$139$	20.1855	1.71211	0.856056	+	0.516882i	$0.172908\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	18.4560	1.54337
$144$	0	0
$145$	3.57929	0.297244
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.7380	−1.28931	−0.644655	−	0.764474i	$-0.722999\pi$
$149$	−15.7380	−1.28931	−0.644655	+	0.764474i	$0.722999\pi$
$150$	0	0
$151$	−6.33525	−0.515555	−0.257778	−	0.966204i	$-0.582990\pi$
$151$	−6.33525	−0.515555	−0.257778	+	0.966204i	$0.582990\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.47389	0.118385
$156$	0	0
$157$	13.0814	1.04401	0.522006	−	0.852942i	$-0.325184\pi$
$157$	13.0814	1.04401	0.522006	+	0.852942i	$0.325184\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−15.2345	−1.20065
$162$	0	0
$163$	6.75257	0.528902	0.264451	−	0.964399i	$-0.414809\pi$
$163$	6.75257	0.528902	0.264451	+	0.964399i	$0.414809\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	5.29607	0.407390
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.04073	0.155154	0.0775768	−	0.996986i	$-0.475282\pi$
$173$	2.04073	0.155154	0.0775768	+	0.996986i	$0.475282\pi$
$174$	0	0
$175$	−10.1500	−0.767265
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.94349	0.444237	0.222119	−	0.975020i	$-0.428703\pi$
$179$	5.94349	0.444237	0.222119	+	0.975020i	$0.428703\pi$
$180$	0	0
$181$	−20.8975	−1.55330	−0.776649	−	0.629933i	$-0.783082\pi$
$181$	−20.8975	−1.55330	−0.776649	+	0.629933i	$0.783082\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.52135	0.258895
$186$	0	0
$187$	−25.3132	−1.85108
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.3045	−1.75861	−0.879306	−	0.476257i	$-0.841993\pi$
$191$	−24.3045	−1.75861	−0.879306	+	0.476257i	$0.841993\pi$
$192$	0	0
$193$	−9.16298	−0.659566	−0.329783	−	0.944057i	$-0.606976\pi$
$193$	−9.16298	−0.659566	−0.329783	+	0.944057i	$0.606976\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.8782	−1.63000	−0.815001	−	0.579459i	$-0.803264\pi$
$197$	−22.8782	−1.63000	−0.815001	+	0.579459i	$0.803264\pi$
$198$	0	0
$199$	18.7146	1.32664	0.663320	−	0.748336i	$-0.269147\pi$
$199$	18.7146	1.32664	0.663320	+	0.748336i	$0.269147\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−18.7685	−1.31729
$204$	0	0
$205$	−0.400464	−0.0279696
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.96283	−0.274114
$210$	0	0
$211$	13.9444	0.959974	0.479987	−	0.877276i	$-0.340641\pi$
$211$	13.9444	0.959974	0.479987	+	0.877276i	$0.340641\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.93134	−0.199915
$216$	0	0
$217$	−7.72853	−0.524647
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−25.0938	−1.68799
$222$	0	0
$223$	−16.4828	−1.10377	−0.551884	−	0.833921i	$-0.686091\pi$
$223$	−16.4828	−1.10377	−0.551884	+	0.833921i	$0.686091\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.88068	−0.124825	−0.0624124	−	0.998050i	$-0.519879\pi$
$227$	−1.88068	−0.124825	−0.0624124	+	0.998050i	$0.519879\pi$
$228$	0	0
$229$	8.52426	0.563299	0.281649	−	0.959517i	$-0.409118\pi$
$229$	8.52426	0.563299	0.281649	+	0.959517i	$0.409118\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.748022	0.0490045	0.0245023	−	0.999700i	$-0.492200\pi$
$233$	0.748022	0.0490045	0.0245023	+	0.999700i	$0.492200\pi$
$234$	0	0
$235$	1.26203	0.0823259
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.8828	1.60954	0.804769	−	0.593588i	$-0.202289\pi$
$239$	24.8828	1.60954	0.804769	+	0.593588i	$0.202289\pi$
$240$	0	0
$241$	1.35372	0.0872010	0.0436005	−	0.999049i	$-0.486117\pi$
$241$	1.35372	0.0872010	0.0436005	+	0.999049i	$0.486117\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.04072	0.0664894
$246$	0	0
$247$	−3.92849	−0.249964
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.76789	−0.111588	−0.0557941	−	0.998442i	$-0.517769\pi$
$251$	−1.76789	−0.111588	−0.0557941	+	0.998442i	$0.517769\pi$
$252$	0	0
$253$	−31.3454	−1.97067
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.1592	−0.758469	−0.379234	−	0.925301i	$-0.623813\pi$
$257$	−12.1592	−0.758469	−0.379234	+	0.925301i	$0.623813\pi$
$258$	0	0
$259$	−18.4647	−1.14734
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.09839	0.252717	0.126359	−	0.991985i	$-0.459671\pi$
$263$	4.09839	0.252717	0.126359	+	0.991985i	$0.459671\pi$
$264$	0	0
$265$	1.32537	0.0814167
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.8602	0.967012	0.483506	−	0.875341i	$-0.339363\pi$
$269$	15.8602	0.967012	0.483506	+	0.875341i	$0.339363\pi$
$270$	0	0
$271$	−8.22895	−0.499873	−0.249937	−	0.968262i	$-0.580410\pi$
$271$	−8.22895	−0.499873	−0.249937	+	0.968262i	$0.580410\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.8838	−1.25934
$276$	0	0
$277$	−22.5561	−1.35527	−0.677634	−	0.735400i	$-0.736995\pi$
$277$	−22.5561	−1.35527	−0.677634	+	0.735400i	$0.736995\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−31.0830	−1.85426	−0.927129	−	0.374742i	$-0.877731\pi$
$281$	−31.0830	−1.85426	−0.927129	+	0.374742i	$0.877731\pi$
$282$	0	0
$283$	8.27233	0.491739	0.245869	−	0.969303i	$-0.420927\pi$
$283$	8.27233	0.491739	0.245869	+	0.969303i	$0.420927\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.09989	0.123953
$288$	0	0
$289$	17.4173	1.02455
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.61606	0.211252	0.105626	−	0.994406i	$-0.466315\pi$
$293$	3.61606	0.211252	0.105626	+	0.994406i	$0.466315\pi$
$294$	0	0
$295$	0.288575	0.0168015
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−31.0737	−1.79704
$300$	0	0
$301$	15.3709	0.885962
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.88879	0.279931
$306$	0	0
$307$	−0.541176	−0.0308866	−0.0154433	−	0.999881i	$-0.504916\pi$
$307$	−0.541176	−0.0308866	−0.0154433	+	0.999881i	$0.504916\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.79572	0.555465	0.277732	−	0.960659i	$-0.410417\pi$
$311$	9.79572	0.555465	0.277732	+	0.960659i	$0.410417\pi$
$312$	0	0
$313$	3.71536	0.210005	0.105002	−	0.994472i	$-0.466515\pi$
$313$	3.71536	0.210005	0.105002	+	0.994472i	$0.466515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.5198	1.54567	0.772833	−	0.634610i	$-0.218839\pi$
$317$	27.5198	1.54567	0.772833	+	0.634610i	$0.218839\pi$
$318$	0	0
$319$	−38.6166	−2.16212
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.38809	0.299801
$324$	0	0
$325$	−20.7028	−1.14839
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.61764	−0.364842
$330$	0	0
$331$	20.5753	1.13092	0.565462	−	0.824775i	$-0.308698\pi$
$331$	20.5753	1.13092	0.565462	+	0.824775i	$0.308698\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.43590	−0.242359
$336$	0	0
$337$	12.3238	0.671320	0.335660	−	0.941983i	$-0.391041\pi$
$337$	12.3238	0.671320	0.335660	+	0.941983i	$0.391041\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15.9016	−0.861122
$342$	0	0
$343$	−20.1367	−1.08728
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.9634	−1.34010	−0.670052	−	0.742314i	$-0.733728\pi$
$347$	−24.9634	−1.34010	−0.670052	+	0.742314i	$0.733728\pi$
$348$	0	0
$349$	32.0104	1.71348	0.856738	−	0.515752i	$-0.172487\pi$
$349$	32.0104	1.71348	0.856738	+	0.515752i	$0.172487\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.1498	0.699894	0.349947	−	0.936770i	$-0.386200\pi$
$353$	13.1498	0.699894	0.349947	+	0.936770i	$0.386200\pi$
$354$	0	0
$355$	5.31442	0.282060
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.246074	0.0129873	0.00649365	−	0.999979i	$-0.497933\pi$
$359$	0.246074	0.0129873	0.00649365	+	0.999979i	$0.497933\pi$
$360$	0	0
$361$	−18.1565	−0.955605
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.85344	0.254041
$366$	0	0
$367$	−18.8123	−0.981994	−0.490997	−	0.871161i	$-0.663367\pi$
$367$	−18.8123	−0.981994	−0.490997	+	0.871161i	$0.663367\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.94975	−0.360813
$372$	0	0
$373$	13.5033	0.699176	0.349588	−	0.936904i	$-0.386322\pi$
$373$	13.5033	0.699176	0.349588	+	0.936904i	$0.386322\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−38.2820	−1.97162
$378$	0	0
$379$	−5.85536	−0.300770	−0.150385	−	0.988628i	$-0.548051\pi$
$379$	−5.85536	−0.300770	−0.150385	+	0.988628i	$0.548051\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.10081	−0.260639	−0.130319	−	0.991472i	$-0.541600\pi$
$383$	−5.10081	−0.260639	−0.130319	+	0.991472i	$0.541600\pi$
$384$	0	0
$385$	−3.61871	−0.184426
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.9505	0.960827	0.480413	−	0.877042i	$-0.340487\pi$
$389$	18.9505	0.960827	0.480413	+	0.877042i	$0.340487\pi$
$390$	0	0
$391$	42.6190	2.15533
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.00704419	−0.000354432 0
$396$	0	0
$397$	21.3751	1.07279	0.536394	−	0.843968i	$-0.319786\pi$
$397$	21.3751	1.07279	0.536394	+	0.843968i	$0.319786\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.34328	−0.216893	−0.108447	−	0.994102i	$-0.534588\pi$
$401$	−4.34328	−0.216893	−0.108447	+	0.994102i	$0.534588\pi$
$402$	0	0
$403$	−15.7638	−0.785253
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−37.9915	−1.88317
$408$	0	0
$409$	21.7057	1.07328	0.536638	−	0.843812i	$-0.319694\pi$
$409$	21.7057	1.07328	0.536638	+	0.843812i	$0.319694\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.51318	−0.0744590
$414$	0	0
$415$	0.897677	0.0440653
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.0956	−1.07944	−0.539721	−	0.841844i	$-0.681470\pi$
$419$	−22.0956	−1.07944	−0.539721	+	0.841844i	$0.681470\pi$
$420$	0	0
$421$	−1.30389	−0.0635475	−0.0317738	−	0.999495i	$-0.510116\pi$
$421$	−1.30389	−0.0635475	−0.0317738	+	0.999495i	$0.510116\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	28.3948	1.37735
$426$	0	0
$427$	−25.6350	−1.24057
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.2043	0.876873	0.438436	−	0.898762i	$-0.355532\pi$
$431$	18.2043	0.876873	0.438436	+	0.898762i	$0.355532\pi$
$432$	0	0
$433$	−35.6941	−1.71535	−0.857676	−	0.514191i	$-0.828092\pi$
$433$	−35.6941	−1.71535	−0.857676	+	0.514191i	$0.828092\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.67207	0.319168
$438$	0	0
$439$	1.00502	0.0479668	0.0239834	−	0.999712i	$-0.492365\pi$
$439$	1.00502	0.0479668	0.0239834	+	0.999712i	$0.492365\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.8148	1.08396	0.541981	−	0.840391i	$-0.317675\pi$
$443$	22.8148	1.08396	0.541981	+	0.840391i	$0.317675\pi$
$444$	0	0
$445$	2.11479	0.100251
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−21.4156	−1.01066	−0.505332	−	0.862925i	$-0.668630\pi$
$449$	−21.4156	−1.01066	−0.505332	+	0.862925i	$0.668630\pi$
$450$	0	0
$451$	4.32057	0.203448
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.58735	−0.168177
$456$	0	0
$457$	−8.42517	−0.394113	−0.197056	−	0.980392i	$-0.563138\pi$
$457$	−8.42517	−0.394113	−0.197056	+	0.980392i	$0.563138\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−36.1271	−1.68261	−0.841304	−	0.540562i	$-0.818212\pi$
$461$	−36.1271	−1.68261	−0.841304	+	0.540562i	$0.818212\pi$
$462$	0	0
$463$	23.8681	1.10924	0.554622	−	0.832103i	$-0.312863\pi$
$463$	23.8681	1.10924	0.554622	+	0.832103i	$0.312863\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.1611	0.609022	0.304511	−	0.952509i	$-0.401507\pi$
$467$	13.1611	0.609022	0.304511	+	0.952509i	$0.401507\pi$
$468$	0	0
$469$	23.2603	1.07406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	31.6259	1.45416
$474$	0	0
$475$	4.44526	0.203962
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.38701	−0.428904	−0.214452	−	0.976735i	$-0.568796\pi$
$479$	−9.38701	−0.428904	−0.214452	+	0.976735i	$0.568796\pi$
$480$	0	0
$481$	−37.6623	−1.71725
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.81282	−0.127724
$486$	0	0
$487$	13.0360	0.590719	0.295359	−	0.955386i	$-0.404561\pi$
$487$	13.0360	0.590719	0.295359	+	0.955386i	$0.404561\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.21421	0.280443	0.140222	−	0.990120i	$-0.455218\pi$
$491$	6.21421	0.280443	0.140222	+	0.990120i	$0.455218\pi$
$492$	0	0
$493$	52.5054	2.36472
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.8669	−1.25000
$498$	0	0
$499$	23.9470	1.07201	0.536007	−	0.844214i	$-0.319932\pi$
$499$	23.9470	1.07201	0.536007	+	0.844214i	$0.319932\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−27.6753	−1.23398	−0.616989	−	0.786971i	$-0.711648\pi$
$503$	−27.6753	−1.23398	−0.616989	+	0.786971i	$0.711648\pi$
$504$	0	0
$505$	−7.79580	−0.346909
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.75522	0.122123	0.0610614	−	0.998134i	$-0.480551\pi$
$509$	2.75522	0.122123	0.0610614	+	0.998134i	$0.480551\pi$
$510$	0	0
$511$	−25.4497	−1.12583
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.00059	0.220352
$516$	0	0
$517$	−13.6160	−0.598829
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.4376	0.807765	0.403882	−	0.914811i	$-0.367661\pi$
$521$	18.4376	0.807765	0.403882	+	0.914811i	$0.367661\pi$
$522$	0	0
$523$	10.6145	0.464140	0.232070	−	0.972699i	$-0.425450\pi$
$523$	10.6145	0.464140	0.232070	+	0.972699i	$0.425450\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.6208	0.941816
$528$	0	0
$529$	29.7751	1.29457
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.28313	0.185523
$534$	0	0
$535$	1.32656	0.0573522
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−11.2283	−0.483636
$540$	0	0
$541$	15.4622	0.664774	0.332387	−	0.943143i	$-0.392146\pi$
$541$	15.4622	0.664774	0.332387	+	0.943143i	$0.392146\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.87384	−0.294443
$546$	0	0
$547$	14.1452	0.604805	0.302403	−	0.953180i	$-0.402211\pi$
$547$	14.1452	0.604805	0.302403	+	0.953180i	$0.402211\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.21981	0.350176
$552$	0	0
$553$	0.0369372	0.00157073
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.5888	0.533405	0.266702	−	0.963779i	$-0.414066\pi$
$557$	12.5888	0.533405	0.266702	+	0.963779i	$0.414066\pi$
$558$	0	0
$559$	31.3519	1.32604
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−42.1884	−1.77803	−0.889015	−	0.457878i	$-0.848610\pi$
$563$	−42.1884	−1.77803	−0.889015	+	0.457878i	$0.848610\pi$
$564$	0	0
$565$	−4.41315	−0.185663
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.84546	0.119288	0.0596439	−	0.998220i	$-0.481003\pi$
$569$	2.84546	0.119288	0.0596439	+	0.998220i	$0.481003\pi$
$570$	0	0
$571$	−47.2379	−1.97684	−0.988422	−	0.151730i	$-0.951515\pi$
$571$	−47.2379	−1.97684	−0.988422	+	0.151730i	$0.951515\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	35.1613	1.46633
$576$	0	0
$577$	14.1649	0.589692	0.294846	−	0.955545i	$-0.404732\pi$
$577$	14.1649	0.589692	0.294846	+	0.955545i	$0.404732\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.70710	−0.195283
$582$	0	0
$583$	−14.2993	−0.592216
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.41772	−0.182339	−0.0911695	−	0.995835i	$-0.529061\pi$
$587$	−4.41772	−0.182339	−0.0911695	+	0.995835i	$0.529061\pi$
$588$	0	0
$589$	3.38477	0.139467
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12.7417	−0.523240	−0.261620	−	0.965171i	$-0.584257\pi$
$593$	−12.7417	−0.523240	−0.261620	+	0.965171i	$0.584257\pi$
$594$	0	0
$595$	4.92020	0.201709
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.54331	0.349070	0.174535	−	0.984651i	$-0.444158\pi$
$599$	8.54331	0.349070	0.174535	+	0.984651i	$0.444158\pi$
$600$	0	0
$601$	8.34074	0.340226	0.170113	−	0.985425i	$-0.445587\pi$
$601$	8.34074	0.340226	0.170113	+	0.985425i	$0.445587\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.04638	−0.123853
$606$	0	0
$607$	−36.9494	−1.49973	−0.749865	−	0.661591i	$-0.769882\pi$
$607$	−36.9494	−1.49973	−0.749865	+	0.661591i	$0.769882\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.4980	−0.546069
$612$	0	0
$613$	32.3308	1.30583	0.652914	−	0.757432i	$-0.273546\pi$
$613$	32.3308	1.30583	0.652914	+	0.757432i	$0.273546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.9853	1.24742	0.623710	−	0.781656i	$-0.285625\pi$
$617$	30.9853	1.24742	0.623710	+	0.781656i	$0.285625\pi$
$618$	0	0
$619$	10.2392	0.411549	0.205775	−	0.978599i	$-0.434029\pi$
$619$	10.2392	0.411549	0.205775	+	0.978599i	$0.434029\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.0892	−0.444280
$624$	0	0
$625$	22.6265	0.905058
$626$	0	0
$627$	0	0
$628$	0	0
$629$	51.6555	2.05964
$630$	0	0
$631$	−24.2880	−0.966890	−0.483445	−	0.875375i	$-0.660615\pi$
$631$	−24.2880	−0.966890	−0.483445	+	0.875375i	$0.660615\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.29474	−0.170432
$636$	0	0
$637$	−11.1310	−0.441025
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.43569	0.0962039	0.0481019	−	0.998842i	$-0.484683\pi$
$641$	2.43569	0.0962039	0.0481019	+	0.998842i	$0.484683\pi$
$642$	0	0
$643$	−37.6634	−1.48530	−0.742650	−	0.669680i	$-0.766431\pi$
$643$	−37.6634	−1.48530	−0.742650	+	0.669680i	$0.766431\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.5245	−0.885530	−0.442765	−	0.896638i	$-0.646002\pi$
$647$	−22.5245	−0.885530	−0.442765	+	0.896638i	$0.646002\pi$
$648$	0	0
$649$	−3.11341	−0.122212
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.6457	−1.19926	−0.599630	−	0.800277i	$-0.704686\pi$
$653$	−30.6457	−1.19926	−0.599630	+	0.800277i	$0.704686\pi$
$654$	0	0
$655$	5.32914	0.208227
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−43.9293	−1.71124	−0.855622	−	0.517601i	$-0.826825\pi$
$659$	−43.9293	−1.71124	−0.855622	+	0.517601i	$0.826825\pi$
$660$	0	0
$661$	33.4286	1.30022	0.650111	−	0.759839i	$-0.274722\pi$
$661$	33.4286	1.30022	0.650111	+	0.759839i	$0.274722\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.770266	0.0298696
$666$	0	0
$667$	65.0175	2.51749
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−52.7447	−2.03619
$672$	0	0
$673$	−8.42437	−0.324736	−0.162368	−	0.986730i	$-0.551913\pi$
$673$	−8.42437	−0.324736	−0.162368	+	0.986730i	$0.551913\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.48309	−0.287599	−0.143799	−	0.989607i	$-0.545932\pi$
$677$	−7.48309	−0.287599	−0.143799	+	0.989607i	$0.545932\pi$
$678$	0	0
$679$	14.7494	0.566031
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−48.0138	−1.83720	−0.918598	−	0.395193i	$-0.870678\pi$
$683$	−48.0138	−1.83720	−0.918598	+	0.395193i	$0.870678\pi$
$684$	0	0
$685$	−0.898901	−0.0343452
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.1754	−0.540039
$690$	0	0
$691$	22.5102	0.856330	0.428165	−	0.903701i	$-0.359160\pi$
$691$	22.5102	0.856330	0.428165	+	0.903701i	$0.359160\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.07274	−0.306216
$696$	0	0
$697$	−5.87450	−0.222513
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.4577	1.18814	0.594070	−	0.804414i	$-0.297520\pi$
$701$	31.4577	1.18814	0.594070	+	0.804414i	$0.297520\pi$
$702$	0	0
$703$	8.08675	0.304998
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40.8784	1.53739
$708$	0	0
$709$	−15.4045	−0.578529	−0.289265	−	0.957249i	$-0.593411\pi$
$709$	−15.4045	−0.578529	−0.289265	+	0.957249i	$0.593411\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	26.7730	1.00266
$714$	0	0
$715$	−7.38106	−0.276036
$716$	0	0
$717$	0	0
$718$	0	0
$719$	39.3745	1.46842	0.734211	−	0.678922i	$-0.237552\pi$
$719$	39.3745	1.46842	0.734211	+	0.678922i	$0.237552\pi$
$720$	0	0
$721$	−26.2213	−0.976532
$722$	0	0
$723$	0	0
$724$	0	0
$725$	43.3178	1.60878
$726$	0	0
$727$	−30.5155	−1.13176	−0.565879	−	0.824488i	$-0.691463\pi$
$727$	−30.5155	−1.13176	−0.565879	+	0.824488i	$0.691463\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−43.0005	−1.59043
$732$	0	0
$733$	−43.0844	−1.59136	−0.795679	−	0.605719i	$-0.792886\pi$
$733$	−43.0844	−1.59136	−0.795679	+	0.605719i	$0.792886\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	47.8585	1.76289
$738$	0	0
$739$	−15.8759	−0.584005	−0.292003	−	0.956418i	$-0.594322\pi$
$739$	−15.8759	−0.584005	−0.292003	+	0.956418i	$0.594322\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.5110	−0.605731	−0.302865	−	0.953033i	$-0.597943\pi$
$743$	−16.5110	−0.605731	−0.302865	+	0.953033i	$0.597943\pi$
$744$	0	0
$745$	6.29407	0.230597
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.95601	−0.254167
$750$	0	0
$751$	−37.5264	−1.36936	−0.684680	−	0.728844i	$-0.740058\pi$
$751$	−37.5264	−1.36936	−0.684680	+	0.728844i	$0.740058\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.53364	0.0922085
$756$	0	0
$757$	−25.4575	−0.925268	−0.462634	−	0.886549i	$-0.653096\pi$
$757$	−25.4575	−0.925268	−0.462634	+	0.886549i	$0.653096\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31.4592	1.14039	0.570197	−	0.821508i	$-0.306867\pi$
$761$	31.4592	1.14039	0.570197	+	0.821508i	$0.306867\pi$
$762$	0	0
$763$	36.0439	1.30488
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.08644	−0.111445
$768$	0	0
$769$	−14.9585	−0.539418	−0.269709	−	0.962942i	$-0.586927\pi$
$769$	−14.9585	−0.539418	−0.269709	+	0.962942i	$0.586927\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28.9178	1.04010	0.520051	−	0.854135i	$-0.325913\pi$
$773$	28.9178	1.04010	0.520051	+	0.854135i	$0.325913\pi$
$774$	0	0
$775$	17.8375	0.640741
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.919663	−0.0329504
$780$	0	0
$781$	−57.3369	−2.05167
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.23162	−0.186725
$786$	0	0
$787$	7.47819	0.266569	0.133284	−	0.991078i	$-0.457448\pi$
$787$	7.47819	0.266569	0.133284	+	0.991078i	$0.457448\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	23.1410	0.822798
$792$	0	0
$793$	−52.2876	−1.85679
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.4432	0.936665	0.468333	−	0.883552i	$-0.344855\pi$
$797$	26.4432	0.936665	0.468333	+	0.883552i	$0.344855\pi$
$798$	0	0
$799$	18.5130	0.654944
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−52.3634	−1.84786
$804$	0	0
$805$	6.09269	0.214739
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.6375	1.14747	0.573737	−	0.819040i	$-0.305493\pi$
$809$	32.6375	1.14747	0.573737	+	0.819040i	$0.305493\pi$
$810$	0	0
$811$	−27.7632	−0.974897	−0.487448	−	0.873152i	$-0.662072\pi$
$811$	−27.7632	−0.974897	−0.487448	+	0.873152i	$0.662072\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.70054	−0.0945957
$816$	0	0
$817$	−6.73180	−0.235516
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.6848	−0.931307	−0.465654	−	0.884967i	$-0.654181\pi$
$821$	−26.6848	−0.931307	−0.465654	+	0.884967i	$0.654181\pi$
$822$	0	0
$823$	−28.4082	−0.990247	−0.495124	−	0.868823i	$-0.664877\pi$
$823$	−28.4082	−0.990247	−0.495124	+	0.868823i	$0.664877\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.3462	1.54207	0.771035	−	0.636793i	$-0.219740\pi$
$827$	44.3462	1.54207	0.771035	+	0.636793i	$0.219740\pi$
$828$	0	0
$829$	19.5676	0.679611	0.339806	−	0.940496i	$-0.389639\pi$
$829$	19.5676	0.679611	0.339806	+	0.940496i	$0.389639\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15.2666	0.528957
$834$	0	0
$835$	−0.399927	−0.0138401
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.17993	−0.109783	−0.0548917	−	0.998492i	$-0.517481\pi$
$839$	−3.17993	−0.109783	−0.0548917	+	0.998492i	$0.517481\pi$
$840$	0	0
$841$	51.0998	1.76206
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.11804	−0.0728629
$846$	0	0
$847$	15.9741	0.548877
$848$	0	0
$849$	0	0
$850$	0	0
$851$	63.9650	2.19269
$852$	0	0
$853$	−7.27317	−0.249029	−0.124514	−	0.992218i	$-0.539737\pi$
$853$	−7.27317	−0.249029	−0.124514	+	0.992218i	$0.539737\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29.0285	0.991594	0.495797	−	0.868438i	$-0.334876\pi$
$857$	29.0285	0.991594	0.495797	+	0.868438i	$0.334876\pi$
$858$	0	0
$859$	−51.0952	−1.74335	−0.871673	−	0.490089i	$-0.836964\pi$
$859$	−51.0952	−1.74335	−0.871673	+	0.490089i	$0.836964\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.3073	0.725307	0.362654	−	0.931924i	$-0.381871\pi$
$863$	21.3073	0.725307	0.362654	+	0.931924i	$0.381871\pi$
$864$	0	0
$865$	−0.816142	−0.0277497
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.0759992	0.00257810
$870$	0	0
$871$	47.4438	1.60757
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.25263	0.278990
$876$	0	0
$877$	−42.3051	−1.42854	−0.714270	−	0.699870i	$-0.753241\pi$
$877$	−42.3051	−1.42854	−0.714270	+	0.699870i	$0.753241\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.8082	1.40855	0.704277	−	0.709926i	$-0.251271\pi$
$881$	41.8082	1.40855	0.704277	+	0.709926i	$0.251271\pi$
$882$	0	0
$883$	−20.7589	−0.698592	−0.349296	−	0.937012i	$-0.613579\pi$
$883$	−20.7589	−0.698592	−0.349296	+	0.937012i	$0.613579\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.86158	−0.0960826	−0.0480413	−	0.998845i	$-0.515298\pi$
$887$	−2.86158	−0.0960826	−0.0480413	+	0.998845i	$0.515298\pi$
$888$	0	0
$889$	22.5201	0.755300
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.89825	0.0969862
$894$	0	0
$895$	−2.37696	−0.0794531
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.9837	1.10007
$900$	0	0
$901$	19.4421	0.647712
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.35748	0.277812
$906$	0	0
$907$	12.0443	0.399924	0.199962	−	0.979804i	$-0.435918\pi$
$907$	12.0443	0.399924	0.199962	+	0.979804i	$0.435918\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−20.6182	−0.683111	−0.341556	−	0.939862i	$-0.610954\pi$
$911$	−20.6182	−0.683111	−0.341556	+	0.939862i	$0.610954\pi$
$912$	0	0
$913$	−9.68497	−0.320526
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−27.9441	−0.922796
$918$	0	0
$919$	32.6622	1.07743	0.538714	−	0.842489i	$-0.318910\pi$
$919$	32.6622	1.07743	0.538714	+	0.842489i	$0.318910\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−56.8400	−1.87091
$924$	0	0
$925$	42.6166	1.40122
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18.4228	0.604433	0.302217	−	0.953239i	$-0.402273\pi$
$929$	18.4228	0.604433	0.302217	+	0.953239i	$0.402273\pi$
$930$	0	0
$931$	2.39001	0.0783296
$932$	0	0
$933$	0	0
$934$	0	0
$935$	10.1234	0.331072
$936$	0	0
$937$	−17.9437	−0.586196	−0.293098	−	0.956082i	$-0.594686\pi$
$937$	−17.9437	−0.586196	−0.293098	+	0.956082i	$0.594686\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−41.1099	−1.34014	−0.670072	−	0.742296i	$-0.733737\pi$
$941$	−41.1099	−1.34014	−0.670072	+	0.742296i	$0.733737\pi$
$942$	0	0
$943$	−7.27440	−0.236887
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.10189	0.133294	0.0666468	−	0.997777i	$-0.478770\pi$
$947$	4.10189	0.133294	0.0666468	+	0.997777i	$0.478770\pi$
$948$	0	0
$949$	−51.9096	−1.68506
$950$	0	0
$951$	0	0
$952$	0	0
$953$	23.7257	0.768550	0.384275	−	0.923219i	$-0.374451\pi$
$953$	23.7257	0.768550	0.384275	+	0.923219i	$0.374451\pi$
$954$	0	0
$955$	9.72003	0.314533
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.71351	0.152207
$960$	0	0
$961$	−17.4179	−0.561869
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.66453	0.117965
$966$	0	0
$967$	27.9993	0.900398	0.450199	−	0.892928i	$-0.351353\pi$
$967$	27.9993	0.900398	0.450199	+	0.892928i	$0.351353\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	48.8585	1.56794	0.783971	−	0.620797i	$-0.213191\pi$
$971$	48.8585	1.56794	0.783971	+	0.620797i	$0.213191\pi$
$972$	0	0
$973$	42.3305	1.35705
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.76631	0.0885021	0.0442510	−	0.999020i	$-0.485910\pi$
$977$	2.76631	0.0885021	0.0442510	+	0.999020i	$0.485910\pi$
$978$	0	0
$979$	−22.8163	−0.729212
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.4903	−0.876805	−0.438403	−	0.898779i	$-0.644456\pi$
$983$	−27.4903	−0.876805	−0.438403	+	0.898779i	$0.644456\pi$
$984$	0	0
$985$	9.14960	0.291531
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−53.2475	−1.69317
$990$	0	0
$991$	−43.9219	−1.39523	−0.697613	−	0.716475i	$-0.745754\pi$
$991$	−43.9219	−1.39523	−0.697613	+	0.716475i	$0.745754\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7.48446	−0.237273
$996$	0	0
$997$	−24.2800	−0.768956	−0.384478	−	0.923134i	$-0.625619\pi$
$997$	−24.2800	−0.768956	−0.384478	+	0.923134i	$0.625619\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.j.1.6	✓	10
3.2	odd	2		6012.2.a.k.1.5	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.a.j.1.6	✓	10	1.1	even	1	trivial
6012.2.a.k.1.5	yes	10	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)

\(f(q)\)	\(=\)	\(q-0.399927 q^{5} +2.09707 q^{7} +O(q^{10})\)	\(q-0.399927 q^{5} +2.09707 q^{7} +4.31478 q^{11} +4.27739 q^{13} -5.86662 q^{17} -0.918430 q^{19} -7.26465 q^{23} -4.84006 q^{25} -8.94985 q^{29} -3.68539 q^{31} -0.838677 q^{35} -8.80497 q^{37} +1.00134 q^{41} +7.32968 q^{43} -3.15565 q^{47} -2.60228 q^{49} -3.31402 q^{53} -1.72560 q^{55} -0.721570 q^{59} -12.2242 q^{61} -1.71064 q^{65} +11.0918 q^{67} -13.2885 q^{71} -12.1358 q^{73} +9.04841 q^{77} +0.0176137 q^{79} -2.24460 q^{83} +2.34622 q^{85} -5.28794 q^{89} +8.97000 q^{91} +0.367305 q^{95} +7.03334 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{5} + 4 q^{7}+O(q^{10}) \) 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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