LMFDB - Embedded newform 6012.2.a.k.1.3

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			1.3
Root			$4.25727$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.39178 q^{5} +4.25727 q^{7} +O(q^{10})$	$q-1.39178 q^{5} +4.25727 q^{7} +3.66003 q^{11} -2.28747 q^{13} +5.79296 q^{17} +4.93268 q^{19} +1.43986 q^{23} -3.06294 q^{25} +3.08505 q^{29} -7.83454 q^{31} -5.92519 q^{35} +1.91243 q^{37} +1.76508 q^{41} +0.276646 q^{43} +9.79548 q^{47} +11.1243 q^{49} +10.1870 q^{53} -5.09396 q^{55} +13.3421 q^{59} -5.62064 q^{61} +3.18367 q^{65} -9.11949 q^{67} -13.3824 q^{71} +3.42992 q^{73} +15.5817 q^{77} +5.57520 q^{79} -7.04794 q^{83} -8.06255 q^{85} -3.10114 q^{89} -9.73839 q^{91} -6.86522 q^{95} -12.0143 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.39178	−0.622424	−0.311212	−	0.950340i	$-0.600735\pi$
$5$	−1.39178	−0.622424	−0.311212	+	0.950340i	$0.600735\pi$
$6$	0	0
$7$	4.25727	1.60910	0.804548	−	0.593888i	$-0.202408\pi$
$7$	4.25727	1.60910	0.804548	+	0.593888i	$0.202408\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.66003	1.10354	0.551770	−	0.833997i	$-0.313953\pi$
$11$	3.66003	1.10354	0.551770	+	0.833997i	$0.313953\pi$
$12$	0	0
$13$	−2.28747	−0.634431	−0.317216	−	0.948353i	$-0.602748\pi$
$13$	−2.28747	−0.634431	−0.317216	+	0.948353i	$0.602748\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.79296	1.40500	0.702500	−	0.711684i	$-0.252067\pi$
$17$	5.79296	1.40500	0.702500	+	0.711684i	$0.252067\pi$
$18$	0	0
$19$	4.93268	1.13163	0.565817	−	0.824531i	$-0.308561\pi$
$19$	4.93268	1.13163	0.565817	+	0.824531i	$0.308561\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.43986	0.300232	0.150116	−	0.988668i	$-0.452035\pi$
$23$	1.43986	0.300232	0.150116	+	0.988668i	$0.452035\pi$
$24$	0	0
$25$	−3.06294	−0.612588
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.08505	0.572880	0.286440	−	0.958098i	$-0.407528\pi$
$29$	3.08505	0.572880	0.286440	+	0.958098i	$0.407528\pi$
$30$	0	0
$31$	−7.83454	−1.40712	−0.703562	−	0.710634i	$-0.748408\pi$
$31$	−7.83454	−1.40712	−0.703562	+	0.710634i	$0.748408\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.92519	−1.00154
$36$	0	0
$37$	1.91243	0.314402	0.157201	−	0.987567i	$-0.449753\pi$
$37$	1.91243	0.314402	0.157201	+	0.987567i	$0.449753\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.76508	0.275658	0.137829	−	0.990456i	$-0.455987\pi$
$41$	1.76508	0.275658	0.137829	+	0.990456i	$0.455987\pi$
$42$	0	0
$43$	0.276646	0.0421882	0.0210941	−	0.999777i	$-0.493285\pi$
$43$	0.276646	0.0421882	0.0210941	+	0.999777i	$0.493285\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.79548	1.42882	0.714409	−	0.699728i	$-0.246696\pi$
$47$	9.79548	1.42882	0.714409	+	0.699728i	$0.246696\pi$
$48$	0	0
$49$	11.1243	1.58919
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.1870	1.39929	0.699643	−	0.714493i	$-0.253343\pi$
$53$	10.1870	1.39929	0.699643	+	0.714493i	$0.253343\pi$
$54$	0	0
$55$	−5.09396	−0.686870
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.3421	1.73700	0.868499	−	0.495690i	$-0.165085\pi$
$59$	13.3421	1.73700	0.868499	+	0.495690i	$0.165085\pi$
$60$	0	0
$61$	−5.62064	−0.719649	−0.359825	−	0.933020i	$-0.617163\pi$
$61$	−5.62064	−0.719649	−0.359825	+	0.933020i	$0.617163\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.18367	0.394885
$66$	0	0
$67$	−9.11949	−1.11412	−0.557061	−	0.830471i	$-0.688071\pi$
$67$	−9.11949	−1.11412	−0.557061	+	0.830471i	$0.688071\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.3824	−1.58820	−0.794098	−	0.607790i	$-0.792056\pi$
$71$	−13.3824	−1.58820	−0.794098	+	0.607790i	$0.792056\pi$
$72$	0	0
$73$	3.42992	0.401442	0.200721	−	0.979648i	$-0.435671\pi$
$73$	3.42992	0.401442	0.200721	+	0.979648i	$0.435671\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.5817	1.77570
$78$	0	0
$79$	5.57520	0.627259	0.313630	−	0.949545i	$-0.398455\pi$
$79$	5.57520	0.627259	0.313630	+	0.949545i	$0.398455\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.04794	−0.773612	−0.386806	−	0.922161i	$-0.626422\pi$
$83$	−7.04794	−0.773612	−0.386806	+	0.922161i	$0.626422\pi$
$84$	0	0
$85$	−8.06255	−0.874507
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.10114	−0.328720	−0.164360	−	0.986400i	$-0.552556\pi$
$89$	−3.10114	−0.328720	−0.164360	+	0.986400i	$0.552556\pi$
$90$	0	0
$91$	−9.73839	−1.02086
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.86522	−0.704356
$96$	0	0
$97$	−12.0143	−1.21986	−0.609932	−	0.792454i	$-0.708803\pi$
$97$	−12.0143	−1.21986	−0.609932	+	0.792454i	$0.708803\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.22400	0.818318	0.409159	−	0.912463i	$-0.365822\pi$
$101$	8.22400	0.818318	0.409159	+	0.912463i	$0.365822\pi$
$102$	0	0
$103$	−7.28200	−0.717516	−0.358758	−	0.933431i	$-0.616800\pi$
$103$	−7.28200	−0.717516	−0.358758	+	0.933431i	$0.616800\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.1613	−1.17568	−0.587838	−	0.808978i	$-0.700021\pi$
$107$	−12.1613	−1.17568	−0.587838	+	0.808978i	$0.700021\pi$
$108$	0	0
$109$	−12.0963	−1.15862	−0.579308	−	0.815108i	$-0.696677\pi$
$109$	−12.0963	−1.15862	−0.579308	+	0.815108i	$0.696677\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.87674	0.176549	0.0882746	−	0.996096i	$-0.471865\pi$
$113$	1.87674	0.176549	0.0882746	+	0.996096i	$0.471865\pi$
$114$	0	0
$115$	−2.00398	−0.186872
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.6622	2.26078
$120$	0	0
$121$	2.39579	0.217799
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.2219	1.00371
$126$	0	0
$127$	−10.5165	−0.933190	−0.466595	−	0.884471i	$-0.654519\pi$
$127$	−10.5165	−0.933190	−0.466595	+	0.884471i	$0.654519\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.5723	−1.36056	−0.680281	−	0.732952i	$-0.738142\pi$
$131$	−15.5723	−1.36056	−0.680281	+	0.732952i	$0.738142\pi$
$132$	0	0
$133$	20.9997	1.82091
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.64636	−0.567837	−0.283919	−	0.958848i	$-0.591635\pi$
$137$	−6.64636	−0.567837	−0.283919	+	0.958848i	$0.591635\pi$
$138$	0	0
$139$	4.35341	0.369251	0.184626	−	0.982809i	$-0.440893\pi$
$139$	4.35341	0.369251	0.184626	+	0.982809i	$0.440893\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.37221	−0.700120
$144$	0	0
$145$	−4.29373	−0.356575
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.06650	0.333141	0.166570	−	0.986030i	$-0.446731\pi$
$149$	4.06650	0.333141	0.166570	+	0.986030i	$0.446731\pi$
$150$	0	0
$151$	23.3590	1.90093	0.950463	−	0.310837i	$-0.100609\pi$
$151$	23.3590	1.90093	0.950463	+	0.310837i	$0.100609\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.9040	0.875828
$156$	0	0
$157$	24.2451	1.93497	0.967484	−	0.252932i	$-0.0813949\pi$
$157$	24.2451	1.93497	0.967484	+	0.252932i	$0.0813949\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.12988	0.483102
$162$	0	0
$163$	−6.96711	−0.545706	−0.272853	−	0.962056i	$-0.587967\pi$
$163$	−6.96711	−0.545706	−0.272853	+	0.962056i	$0.587967\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−7.76746	−0.597497
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.07920	0.538222	0.269111	−	0.963109i	$-0.413270\pi$
$173$	7.07920	0.538222	0.269111	+	0.963109i	$0.413270\pi$
$174$	0	0
$175$	−13.0397	−0.985712
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.59761	−0.119411	−0.0597056	−	0.998216i	$-0.519016\pi$
$179$	−1.59761	−0.119411	−0.0597056	+	0.998216i	$0.519016\pi$
$180$	0	0
$181$	21.3709	1.58848	0.794242	−	0.607601i	$-0.207868\pi$
$181$	21.3709	1.58848	0.794242	+	0.607601i	$0.207868\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.66169	−0.195691
$186$	0	0
$187$	21.2024	1.55047
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.52311	−0.110208	−0.0551042	−	0.998481i	$-0.517549\pi$
$191$	−1.52311	−0.110208	−0.0551042	+	0.998481i	$0.517549\pi$
$192$	0	0
$193$	7.42971	0.534802	0.267401	−	0.963585i	$-0.413835\pi$
$193$	7.42971	0.534802	0.267401	+	0.963585i	$0.413835\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.32951	0.593453	0.296727	−	0.954962i	$-0.404105\pi$
$197$	8.32951	0.593453	0.296727	+	0.954962i	$0.404105\pi$
$198$	0	0
$199$	13.5576	0.961071	0.480536	−	0.876975i	$-0.340442\pi$
$199$	13.5576	0.961071	0.480536	+	0.876975i	$0.340442\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	13.1339	0.921819
$204$	0	0
$205$	−2.45660	−0.171577
$206$	0	0
$207$	0	0
$208$	0	0
$209$	18.0537	1.24880
$210$	0	0
$211$	−18.1444	−1.24911	−0.624554	−	0.780981i	$-0.714719\pi$
$211$	−18.1444	−1.24911	−0.624554	+	0.780981i	$0.714719\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.385032	−0.0262590
$216$	0	0
$217$	−33.3537	−2.26420
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−13.2513	−0.891376
$222$	0	0
$223$	1.45475	0.0974170	0.0487085	−	0.998813i	$-0.484489\pi$
$223$	1.45475	0.0974170	0.0487085	+	0.998813i	$0.484489\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.2575	1.21179	0.605895	−	0.795544i	$-0.292815\pi$
$227$	18.2575	1.21179	0.605895	+	0.795544i	$0.292815\pi$
$228$	0	0
$229$	22.4108	1.48095	0.740473	−	0.672086i	$-0.234601\pi$
$229$	22.4108	1.48095	0.740473	+	0.672086i	$0.234601\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.06077	−0.266030	−0.133015	−	0.991114i	$-0.542466\pi$
$233$	−4.06077	−0.266030	−0.133015	+	0.991114i	$0.542466\pi$
$234$	0	0
$235$	−13.6332	−0.889331
$236$	0	0
$237$	0	0
$238$	0	0
$239$	29.2144	1.88972	0.944860	−	0.327473i	$-0.106197\pi$
$239$	29.2144	1.88972	0.944860	+	0.327473i	$0.106197\pi$
$240$	0	0
$241$	24.9206	1.60528	0.802638	−	0.596466i	$-0.203429\pi$
$241$	24.9206	1.60528	0.802638	+	0.596466i	$0.203429\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−15.4826	−0.989150
$246$	0	0
$247$	−11.2834	−0.717943
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.87443	−0.118313	−0.0591565	−	0.998249i	$-0.518841\pi$
$251$	−1.87443	−0.118313	−0.0591565	+	0.998249i	$0.518841\pi$
$252$	0	0
$253$	5.26993	0.331318
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.2965	−1.51557	−0.757787	−	0.652502i	$-0.773719\pi$
$257$	−24.2965	−1.51557	−0.757787	+	0.652502i	$0.773719\pi$
$258$	0	0
$259$	8.14173	0.505902
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.3727	0.701270	0.350635	−	0.936512i	$-0.385966\pi$
$263$	11.3727	0.701270	0.350635	+	0.936512i	$0.385966\pi$
$264$	0	0
$265$	−14.1780	−0.870949
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.0789	−1.16326	−0.581631	−	0.813453i	$-0.697585\pi$
$269$	−19.0789	−1.16326	−0.581631	+	0.813453i	$0.697585\pi$
$270$	0	0
$271$	−1.76028	−0.106929	−0.0534647	−	0.998570i	$-0.517026\pi$
$271$	−1.76028	−0.106929	−0.0534647	+	0.998570i	$0.517026\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.2104	−0.676015
$276$	0	0
$277$	15.2189	0.914414	0.457207	−	0.889360i	$-0.348850\pi$
$277$	15.2189	0.914414	0.457207	+	0.889360i	$0.348850\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.14037	−0.425958	−0.212979	−	0.977057i	$-0.568317\pi$
$281$	−7.14037	−0.425958	−0.212979	+	0.977057i	$0.568317\pi$
$282$	0	0
$283$	0.290145	0.0172473	0.00862366	−	0.999963i	$-0.497255\pi$
$283$	0.290145	0.0172473	0.00862366	+	0.999963i	$0.497255\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.51440	0.443561
$288$	0	0
$289$	16.5584	0.974026
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.0213	−1.52018	−0.760090	−	0.649818i	$-0.774845\pi$
$293$	−26.0213	−1.52018	−0.760090	+	0.649818i	$0.774845\pi$
$294$	0	0
$295$	−18.5694	−1.08115
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.29365	−0.190477
$300$	0	0
$301$	1.17776	0.0678848
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.82271	0.447927
$306$	0	0
$307$	−21.6191	−1.23386	−0.616932	−	0.787016i	$-0.711625\pi$
$307$	−21.6191	−1.23386	−0.616932	+	0.787016i	$0.711625\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.1636	−1.59701	−0.798505	−	0.601988i	$-0.794375\pi$
$311$	−28.1636	−1.59701	−0.798505	+	0.601988i	$0.794375\pi$
$312$	0	0
$313$	12.3640	0.698857	0.349429	−	0.936963i	$-0.386376\pi$
$313$	12.3640	0.698857	0.349429	+	0.936963i	$0.386376\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.9600	0.671740	0.335870	−	0.941908i	$-0.390970\pi$
$317$	11.9600	0.671740	0.335870	+	0.941908i	$0.390970\pi$
$318$	0	0
$319$	11.2914	0.632196
$320$	0	0
$321$	0	0
$322$	0	0
$323$	28.5748	1.58995
$324$	0	0
$325$	7.00639	0.388645
$326$	0	0
$327$	0	0
$328$	0	0
$329$	41.7020	2.29911
$330$	0	0
$331$	−2.96837	−0.163156	−0.0815781	−	0.996667i	$-0.525996\pi$
$331$	−2.96837	−0.163156	−0.0815781	+	0.996667i	$0.525996\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.6923	0.693457
$336$	0	0
$337$	6.68862	0.364353	0.182176	−	0.983266i	$-0.441686\pi$
$337$	6.68862	0.364353	0.182176	+	0.983266i	$0.441686\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−28.6746	−1.55282
$342$	0	0
$343$	17.5583	0.948061
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.24229	0.388786	0.194393	−	0.980924i	$-0.437726\pi$
$347$	7.24229	0.388786	0.194393	+	0.980924i	$0.437726\pi$
$348$	0	0
$349$	−33.4467	−1.79036	−0.895180	−	0.445705i	$-0.852953\pi$
$349$	−33.4467	−1.79036	−0.895180	+	0.445705i	$0.852953\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.6909	1.04804	0.524020	−	0.851706i	$-0.324432\pi$
$353$	19.6909	1.04804	0.524020	+	0.851706i	$0.324432\pi$
$354$	0	0
$355$	18.6254	0.988532
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.2198	0.803270	0.401635	−	0.915800i	$-0.368442\pi$
$359$	15.2198	0.803270	0.401635	+	0.915800i	$0.368442\pi$
$360$	0	0
$361$	5.33129	0.280594
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.77371	−0.249868
$366$	0	0
$367$	5.88580	0.307236	0.153618	−	0.988130i	$-0.450907\pi$
$367$	5.88580	0.307236	0.153618	+	0.988130i	$0.450907\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	43.3686	2.25158
$372$	0	0
$373$	16.6428	0.861731	0.430865	−	0.902416i	$-0.358208\pi$
$373$	16.6428	0.861731	0.430865	+	0.902416i	$0.358208\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.05698	−0.363453
$378$	0	0
$379$	25.5510	1.31247	0.656234	−	0.754557i	$-0.272148\pi$
$379$	25.5510	1.31247	0.656234	+	0.754557i	$0.272148\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.5475	0.743343	0.371671	−	0.928364i	$-0.378785\pi$
$383$	14.5475	0.743343	0.371671	+	0.928364i	$0.378785\pi$
$384$	0	0
$385$	−21.6864	−1.10524
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.3033	0.826608	0.413304	−	0.910593i	$-0.364375\pi$
$389$	16.3033	0.826608	0.413304	+	0.910593i	$0.364375\pi$
$390$	0	0
$391$	8.34107	0.421826
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.75947	−0.390421
$396$	0	0
$397$	−3.26247	−0.163739	−0.0818693	−	0.996643i	$-0.526089\pi$
$397$	−3.26247	−0.163739	−0.0818693	+	0.996643i	$0.526089\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.46004	0.272661	0.136331	−	0.990663i	$-0.456469\pi$
$401$	5.46004	0.272661	0.136331	+	0.990663i	$0.456469\pi$
$402$	0	0
$403$	17.9213	0.892723
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.99955	0.346955
$408$	0	0
$409$	−38.8473	−1.92088	−0.960439	−	0.278491i	$-0.910166\pi$
$409$	−38.8473	−1.92088	−0.960439	+	0.278491i	$0.910166\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	56.8011	2.79500
$414$	0	0
$415$	9.80921	0.481515
$416$	0	0
$417$	0	0
$418$	0	0
$419$	34.1733	1.66948	0.834738	−	0.550648i	$-0.185619\pi$
$419$	34.1733	1.66948	0.834738	+	0.550648i	$0.185619\pi$
$420$	0	0
$421$	7.67271	0.373945	0.186972	−	0.982365i	$-0.440132\pi$
$421$	7.67271	0.373945	0.186972	+	0.982365i	$0.440132\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−17.7435	−0.860686
$426$	0	0
$427$	−23.9286	−1.15798
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.5706	1.18352	0.591762	−	0.806112i	$-0.298432\pi$
$431$	24.5706	1.18352	0.591762	+	0.806112i	$0.298432\pi$
$432$	0	0
$433$	−22.1231	−1.06317	−0.531583	−	0.847006i	$-0.678403\pi$
$433$	−22.1231	−1.06317	−0.531583	+	0.847006i	$0.678403\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.10237	0.339753
$438$	0	0
$439$	−27.3940	−1.30744	−0.653722	−	0.756735i	$-0.726793\pi$
$439$	−27.3940	−1.30744	−0.653722	+	0.756735i	$0.726793\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.2766	1.72355	0.861776	−	0.507290i	$-0.169353\pi$
$443$	36.2766	1.72355	0.861776	+	0.507290i	$0.169353\pi$
$444$	0	0
$445$	4.31612	0.204604
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.8698	−0.843329	−0.421664	−	0.906752i	$-0.638554\pi$
$449$	−17.8698	−0.843329	−0.421664	+	0.906752i	$0.638554\pi$
$450$	0	0
$451$	6.46022	0.304200
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13.5537	0.635408
$456$	0	0
$457$	−6.92837	−0.324095	−0.162048	−	0.986783i	$-0.551810\pi$
$457$	−6.92837	−0.324095	−0.162048	+	0.986783i	$0.551810\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.6138	1.42583	0.712913	−	0.701253i	$-0.247375\pi$
$461$	30.6138	1.42583	0.712913	+	0.701253i	$0.247375\pi$
$462$	0	0
$463$	−31.1919	−1.44961	−0.724806	−	0.688953i	$-0.758071\pi$
$463$	−31.1919	−1.44961	−0.724806	+	0.688953i	$0.758071\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.7822	−1.51698	−0.758489	−	0.651686i	$-0.774062\pi$
$467$	−32.7822	−1.51698	−0.758489	+	0.651686i	$0.774062\pi$
$468$	0	0
$469$	−38.8241	−1.79273
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.01253	0.0465563
$474$	0	0
$475$	−15.1085	−0.693225
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.7758	−1.40618	−0.703090	−	0.711101i	$-0.748197\pi$
$479$	−30.7758	−1.40618	−0.703090	+	0.711101i	$0.748197\pi$
$480$	0	0
$481$	−4.37464	−0.199466
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.7213	0.759273
$486$	0	0
$487$	32.6527	1.47964	0.739818	−	0.672807i	$-0.234912\pi$
$487$	32.6527	1.47964	0.739818	+	0.672807i	$0.234912\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	33.9388	1.53164	0.765818	−	0.643057i	$-0.222334\pi$
$491$	33.9388	1.53164	0.765818	+	0.643057i	$0.222334\pi$
$492$	0	0
$493$	17.8716	0.804897
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−56.9723	−2.55556
$498$	0	0
$499$	4.56450	0.204335	0.102168	−	0.994767i	$-0.467422\pi$
$499$	4.56450	0.204335	0.102168	+	0.994767i	$0.467422\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.2468	1.21487	0.607437	−	0.794368i	$-0.292198\pi$
$503$	27.2468	1.21487	0.607437	+	0.794368i	$0.292198\pi$
$504$	0	0
$505$	−11.4460	−0.509341
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.1193	−1.06907	−0.534536	−	0.845146i	$-0.679513\pi$
$509$	−24.1193	−1.06907	−0.534536	+	0.845146i	$0.679513\pi$
$510$	0	0
$511$	14.6021	0.645959
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.1350	0.446600
$516$	0	0
$517$	35.8517	1.57676
$518$	0	0
$519$	0	0
$520$	0	0
$521$	23.4790	1.02863	0.514316	−	0.857601i	$-0.328046\pi$
$521$	23.4790	1.02863	0.514316	+	0.857601i	$0.328046\pi$
$522$	0	0
$523$	−16.9668	−0.741907	−0.370954	−	0.928651i	$-0.620969\pi$
$523$	−16.9668	−0.741907	−0.370954	+	0.928651i	$0.620969\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−45.3852	−1.97701
$528$	0	0
$529$	−20.9268	−0.909861
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.03756	−0.174886
$534$	0	0
$535$	16.9259	0.731770
$536$	0	0
$537$	0	0
$538$	0	0
$539$	40.7153	1.75373
$540$	0	0
$541$	−6.27042	−0.269587	−0.134793	−	0.990874i	$-0.543037\pi$
$541$	−6.27042	−0.269587	−0.134793	+	0.990874i	$0.543037\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.8354	0.721151
$546$	0	0
$547$	10.8787	0.465140	0.232570	−	0.972580i	$-0.425287\pi$
$547$	10.8787	0.465140	0.232570	+	0.972580i	$0.425287\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15.2176	0.648290
$552$	0	0
$553$	23.7351	1.00932
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.9464	0.718042	0.359021	−	0.933329i	$-0.383111\pi$
$557$	16.9464	0.718042	0.359021	+	0.933329i	$0.383111\pi$
$558$	0	0
$559$	−0.632822	−0.0267655
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.5295	1.28667	0.643333	−	0.765586i	$-0.277551\pi$
$563$	30.5295	1.28667	0.643333	+	0.765586i	$0.277551\pi$
$564$	0	0
$565$	−2.61202	−0.109889
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.95359	0.333432	0.166716	−	0.986005i	$-0.446684\pi$
$569$	7.95359	0.333432	0.166716	+	0.986005i	$0.446684\pi$
$570$	0	0
$571$	0.378342	0.0158331	0.00791655	−	0.999969i	$-0.497480\pi$
$571$	0.378342	0.0158331	0.00791655	+	0.999969i	$0.497480\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.41021	−0.183918
$576$	0	0
$577$	−17.3452	−0.722092	−0.361046	−	0.932548i	$-0.617580\pi$
$577$	−17.3452	−0.722092	−0.361046	+	0.932548i	$0.617580\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−30.0050	−1.24482
$582$	0	0
$583$	37.2845	1.54417
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.1126	1.61435	0.807176	−	0.590311i	$-0.200995\pi$
$587$	39.1126	1.61435	0.807176	+	0.590311i	$0.200995\pi$
$588$	0	0
$589$	−38.6452	−1.59235
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.4775	−0.881977	−0.440989	−	0.897513i	$-0.645372\pi$
$593$	−21.4775	−0.881977	−0.440989	+	0.897513i	$0.645372\pi$
$594$	0	0
$595$	−34.3244	−1.40716
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.2438	−0.745422	−0.372711	−	0.927947i	$-0.621572\pi$
$599$	−18.2438	−0.745422	−0.372711	+	0.927947i	$0.621572\pi$
$600$	0	0
$601$	0.437641	0.0178518	0.00892588	−	0.999960i	$-0.497159\pi$
$601$	0.437641	0.0178518	0.00892588	+	0.999960i	$0.497159\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.33442	−0.135563
$606$	0	0
$607$	−29.2308	−1.18644	−0.593222	−	0.805039i	$-0.702144\pi$
$607$	−29.2308	−1.18644	−0.593222	+	0.805039i	$0.702144\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22.4069	−0.906487
$612$	0	0
$613$	3.15674	0.127500	0.0637499	−	0.997966i	$-0.479694\pi$
$613$	3.15674	0.127500	0.0637499	+	0.997966i	$0.479694\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−47.9082	−1.92871	−0.964356	−	0.264607i	$-0.914758\pi$
$617$	−47.9082	−1.92871	−0.964356	+	0.264607i	$0.914758\pi$
$618$	0	0
$619$	0.744274	0.0299149	0.0149574	−	0.999888i	$-0.495239\pi$
$619$	0.744274	0.0299149	0.0149574	+	0.999888i	$0.495239\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.2024	−0.528943
$624$	0	0
$625$	−0.303709	−0.0121483
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.0786	0.441735
$630$	0	0
$631$	12.2918	0.489328	0.244664	−	0.969608i	$-0.421322\pi$
$631$	12.2918	0.489328	0.244664	+	0.969608i	$0.421322\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.6367	0.580840
$636$	0	0
$637$	−25.4466	−1.00823
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.04354	−0.199208	−0.0996039	−	0.995027i	$-0.531758\pi$
$641$	−5.04354	−0.199208	−0.0996039	+	0.995027i	$0.531758\pi$
$642$	0	0
$643$	−13.1671	−0.519259	−0.259629	−	0.965708i	$-0.583600\pi$
$643$	−13.1671	−0.519259	−0.259629	+	0.965708i	$0.583600\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.5460	−1.12226	−0.561130	−	0.827727i	$-0.689633\pi$
$647$	−28.5460	−1.12226	−0.561130	+	0.827727i	$0.689633\pi$
$648$	0	0
$649$	48.8326	1.91685
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.2922	−0.637561	−0.318781	−	0.947829i	$-0.603273\pi$
$653$	−16.2922	−0.637561	−0.318781	+	0.947829i	$0.603273\pi$
$654$	0	0
$655$	21.6733	0.846847
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.1719	−1.48696	−0.743482	−	0.668756i	$-0.766827\pi$
$659$	−38.1719	−1.48696	−0.743482	+	0.668756i	$0.766827\pi$
$660$	0	0
$661$	−20.3397	−0.791123	−0.395561	−	0.918440i	$-0.629450\pi$
$661$	−20.3397	−0.791123	−0.395561	+	0.918440i	$0.629450\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−29.2271	−1.13338
$666$	0	0
$667$	4.44205	0.171997
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−20.5717	−0.794161
$672$	0	0
$673$	9.21558	0.355235	0.177617	−	0.984100i	$-0.443161\pi$
$673$	9.21558	0.355235	0.177617	+	0.984100i	$0.443161\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.6465	−0.409178	−0.204589	−	0.978848i	$-0.565586\pi$
$677$	−10.6465	−0.409178	−0.204589	+	0.978848i	$0.565586\pi$
$678$	0	0
$679$	−51.1479	−1.96288
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11.3921	0.435908	0.217954	−	0.975959i	$-0.430062\pi$
$683$	11.3921	0.435908	0.217954	+	0.975959i	$0.430062\pi$
$684$	0	0
$685$	9.25030	0.353436
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−23.3024	−0.887750
$690$	0	0
$691$	−8.59096	−0.326815	−0.163408	−	0.986559i	$-0.552249\pi$
$691$	−8.59096	−0.326815	−0.163408	+	0.986559i	$0.552249\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.05900	−0.229831
$696$	0	0
$697$	10.2250	0.387300
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.1410	1.13841	0.569205	−	0.822195i	$-0.307251\pi$
$701$	30.1410	1.13841	0.569205	+	0.822195i	$0.307251\pi$
$702$	0	0
$703$	9.43340	0.355787
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35.0117	1.31675
$708$	0	0
$709$	51.0845	1.91852	0.959259	−	0.282528i	$-0.0911731\pi$
$709$	51.0845	1.91852	0.959259	+	0.282528i	$0.0911731\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11.2807	−0.422464
$714$	0	0
$715$	11.6523	0.435772
$716$	0	0
$717$	0	0
$718$	0	0
$719$	17.5285	0.653701	0.326850	−	0.945076i	$-0.394013\pi$
$719$	17.5285	0.653701	0.326850	+	0.945076i	$0.394013\pi$
$720$	0	0
$721$	−31.0014	−1.15455
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.44933	−0.350939
$726$	0	0
$727$	35.9025	1.33155	0.665775	−	0.746153i	$-0.268101\pi$
$727$	35.9025	1.33155	0.665775	+	0.746153i	$0.268101\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.60260	0.0592744
$732$	0	0
$733$	9.82592	0.362929	0.181464	−	0.983398i	$-0.441916\pi$
$733$	9.82592	0.362929	0.181464	+	0.983398i	$0.441916\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−33.3776	−1.22948
$738$	0	0
$739$	−40.0365	−1.47277	−0.736383	−	0.676565i	$-0.763468\pi$
$739$	−40.0365	−1.47277	−0.736383	+	0.676565i	$0.763468\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.3902	−1.55515	−0.777573	−	0.628793i	$-0.783549\pi$
$743$	−42.3902	−1.55515	−0.777573	+	0.628793i	$0.783549\pi$
$744$	0	0
$745$	−5.65969	−0.207355
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−51.7739	−1.89178
$750$	0	0
$751$	3.34009	0.121882	0.0609408	−	0.998141i	$-0.480590\pi$
$751$	3.34009	0.121882	0.0609408	+	0.998141i	$0.480590\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−32.5106	−1.18318
$756$	0	0
$757$	−8.37333	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$757$	−8.37333	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.09937	−0.0398522	−0.0199261	−	0.999801i	$-0.506343\pi$
$761$	−1.09937	−0.0398522	−0.0199261	+	0.999801i	$0.506343\pi$
$762$	0	0
$763$	−51.4972	−1.86433
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−30.5198	−1.10201
$768$	0	0
$769$	−11.5728	−0.417325	−0.208662	−	0.977988i	$-0.566911\pi$
$769$	−11.5728	−0.417325	−0.208662	+	0.977988i	$0.566911\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19.7006	−0.708583	−0.354292	−	0.935135i	$-0.615278\pi$
$773$	−19.7006	−0.708583	−0.354292	+	0.935135i	$0.615278\pi$
$774$	0	0
$775$	23.9967	0.861987
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.70654	0.311944
$780$	0	0
$781$	−48.9798	−1.75264
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−33.7439	−1.20437
$786$	0	0
$787$	−35.6505	−1.27080	−0.635401	−	0.772182i	$-0.719165\pi$
$787$	−35.6505	−1.27080	−0.635401	+	0.772182i	$0.719165\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7.98980	0.284085
$792$	0	0
$793$	12.8571	0.456568
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.60959	0.0924365	0.0462183	−	0.998931i	$-0.485283\pi$
$797$	2.60959	0.0924365	0.0462183	+	0.998931i	$0.485283\pi$
$798$	0	0
$799$	56.7449	2.00749
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.5536	0.443007
$804$	0	0
$805$	−8.53146	−0.300695
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−45.7455	−1.60833	−0.804164	−	0.594408i	$-0.797386\pi$
$809$	−45.7455	−1.60833	−0.804164	+	0.594408i	$0.797386\pi$
$810$	0	0
$811$	−0.145852	−0.00512157	−0.00256079	−	0.999997i	$-0.500815\pi$
$811$	−0.145852	−0.00512157	−0.00256079	+	0.999997i	$0.500815\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.69671	0.339661
$816$	0	0
$817$	1.36461	0.0477416
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.13087	0.109268	0.0546341	−	0.998506i	$-0.482601\pi$
$821$	3.13087	0.109268	0.0546341	+	0.998506i	$0.482601\pi$
$822$	0	0
$823$	45.2203	1.57628	0.788141	−	0.615495i	$-0.211044\pi$
$823$	45.2203	1.57628	0.788141	+	0.615495i	$0.211044\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.7575	0.895676	0.447838	−	0.894115i	$-0.352194\pi$
$827$	25.7575	0.895676	0.447838	+	0.894115i	$0.352194\pi$
$828$	0	0
$829$	17.1370	0.595192	0.297596	−	0.954692i	$-0.403815\pi$
$829$	17.1370	0.595192	0.297596	+	0.954692i	$0.403815\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	64.4428	2.23281
$834$	0	0
$835$	1.39178	0.0481646
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.3875	0.600283	0.300141	−	0.953895i	$-0.402966\pi$
$839$	17.3875	0.600283	0.300141	+	0.953895i	$0.402966\pi$
$840$	0	0
$841$	−19.4824	−0.671808
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.8106	0.371897
$846$	0	0
$847$	10.1995	0.350459
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.75364	0.0943935
$852$	0	0
$853$	−5.57073	−0.190738	−0.0953691	−	0.995442i	$-0.530403\pi$
$853$	−5.57073	−0.190738	−0.0953691	+	0.995442i	$0.530403\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.6239	−0.943615	−0.471807	−	0.881702i	$-0.656398\pi$
$857$	−27.6239	−0.943615	−0.471807	+	0.881702i	$0.656398\pi$
$858$	0	0
$859$	2.62310	0.0894991	0.0447495	−	0.998998i	$-0.485751\pi$
$859$	2.62310	0.0894991	0.0447495	+	0.998998i	$0.485751\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−52.9595	−1.80276	−0.901381	−	0.433027i	$-0.857445\pi$
$863$	−52.9595	−1.80276	−0.901381	+	0.433027i	$0.857445\pi$
$864$	0	0
$865$	−9.85272	−0.335002
$866$	0	0
$867$	0	0
$868$	0	0
$869$	20.4054	0.692205
$870$	0	0
$871$	20.8606	0.706834
$872$	0	0
$873$	0	0
$874$	0	0
$875$	47.7745	1.61507
$876$	0	0
$877$	2.82548	0.0954097	0.0477048	−	0.998861i	$-0.484809\pi$
$877$	2.82548	0.0954097	0.0477048	+	0.998861i	$0.484809\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42.6592	−1.43722	−0.718612	−	0.695411i	$-0.755222\pi$
$881$	−42.6592	−1.43722	−0.718612	+	0.695411i	$0.755222\pi$
$882$	0	0
$883$	17.2707	0.581205	0.290603	−	0.956844i	$-0.406144\pi$
$883$	17.2707	0.581205	0.290603	+	0.956844i	$0.406144\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.7354	0.696226	0.348113	−	0.937453i	$-0.386823\pi$
$887$	20.7354	0.696226	0.348113	+	0.937453i	$0.386823\pi$
$888$	0	0
$889$	−44.7716	−1.50159
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48.3179	1.61690
$894$	0	0
$895$	2.22353	0.0743244
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−24.1700	−0.806113
$900$	0	0
$901$	59.0126	1.96600
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−29.7436	−0.988712
$906$	0	0
$907$	−28.4892	−0.945968	−0.472984	−	0.881071i	$-0.656823\pi$
$907$	−28.4892	−0.945968	−0.472984	+	0.881071i	$0.656823\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−45.6076	−1.51105	−0.755524	−	0.655121i	$-0.772618\pi$
$911$	−45.6076	−1.51105	−0.755524	+	0.655121i	$0.772618\pi$
$912$	0	0
$913$	−25.7957	−0.853712
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−66.2956	−2.18927
$918$	0	0
$919$	−56.3742	−1.85961	−0.929806	−	0.368049i	$-0.880026\pi$
$919$	−56.3742	−1.85961	−0.929806	+	0.368049i	$0.880026\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	30.6118	1.00760
$924$	0	0
$925$	−5.85766	−0.192599
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.46953	0.212258	0.106129	−	0.994352i	$-0.466154\pi$
$929$	6.46953	0.212258	0.106129	+	0.994352i	$0.466154\pi$
$930$	0	0
$931$	54.8727	1.79838
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−29.5091	−0.965052
$936$	0	0
$937$	−5.48633	−0.179231	−0.0896154	−	0.995976i	$-0.528564\pi$
$937$	−5.48633	−0.179231	−0.0896154	+	0.995976i	$0.528564\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.44965	0.275451	0.137725	−	0.990470i	$-0.456021\pi$
$941$	8.44965	0.275451	0.137725	+	0.990470i	$0.456021\pi$
$942$	0	0
$943$	2.54147	0.0827615
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.8452	1.39228	0.696141	−	0.717905i	$-0.254899\pi$
$947$	42.8452	1.39228	0.696141	+	0.717905i	$0.254899\pi$
$948$	0	0
$949$	−7.84586	−0.254688
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−52.3756	−1.69661	−0.848307	−	0.529505i	$-0.822378\pi$
$953$	−52.3756	−1.69661	−0.848307	+	0.529505i	$0.822378\pi$
$954$	0	0
$955$	2.11984	0.0685964
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−28.2953	−0.913704
$960$	0	0
$961$	30.3799	0.979998
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.3406	−0.332874
$966$	0	0
$967$	22.9969	0.739529	0.369765	−	0.929125i	$-0.379438\pi$
$967$	22.9969	0.739529	0.369765	+	0.929125i	$0.379438\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.61085	−0.115878	−0.0579389	−	0.998320i	$-0.518453\pi$
$971$	−3.61085	−0.115878	−0.0579389	+	0.998320i	$0.518453\pi$
$972$	0	0
$973$	18.5336	0.594160
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.75567	0.184140	0.0920701	−	0.995753i	$-0.470652\pi$
$977$	5.75567	0.184140	0.0920701	+	0.995753i	$0.470652\pi$
$978$	0	0
$979$	−11.3503	−0.362756
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.6439	0.722228	0.361114	−	0.932522i	$-0.382396\pi$
$983$	22.6439	0.722228	0.361114	+	0.932522i	$0.382396\pi$
$984$	0	0
$985$	−11.5929	−0.369380
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.398333	0.0126662
$990$	0	0
$991$	2.47503	0.0786218	0.0393109	−	0.999227i	$-0.487484\pi$
$991$	2.47503	0.0786218	0.0393109	+	0.999227i	$0.487484\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−18.8692	−0.598194
$996$	0	0
$997$	1.54549	0.0489461	0.0244731	−	0.999700i	$-0.492209\pi$
$997$	1.54549	0.0489461	0.0244731	+	0.999700i	$0.492209\pi$
$998$	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.1.3	yes	10
3.2	odd	2		6012.2.a.j.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.a.j.1.8	✓	10	3.2	odd	2
6012.2.a.k.1.3	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)

\(f(q)\)	\(=\)	\(q-1.39178 q^{5} +4.25727 q^{7} +O(q^{10})\)	\(q-1.39178 q^{5} +4.25727 q^{7} +3.66003 q^{11} -2.28747 q^{13} +5.79296 q^{17} +4.93268 q^{19} +1.43986 q^{23} -3.06294 q^{25} +3.08505 q^{29} -7.83454 q^{31} -5.92519 q^{35} +1.91243 q^{37} +1.76508 q^{41} +0.276646 q^{43} +9.79548 q^{47} +11.1243 q^{49} +10.1870 q^{53} -5.09396 q^{55} +13.3421 q^{59} -5.62064 q^{61} +3.18367 q^{65} -9.11949 q^{67} -13.3824 q^{71} +3.42992 q^{73} +15.5817 q^{77} +5.57520 q^{79} -7.04794 q^{83} -8.06255 q^{85} -3.10114 q^{89} -9.73839 q^{91} -6.86522 q^{95} -12.0143 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{5} + 4 q^{7}+O(q^{10}) \) 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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