LMFDB - Embedded newform 6012.2.a.g.1.2

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 11x^{5} - 7x^{4} + 21x^{3} + 17x^{2} - 4x - 4 $ x^7 - 11x^5 - 7x^4 + 21x^3 + 17x^2 - 4*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 668)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.47217$ 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)

$f(q)$	$=$	$q-1.35854 q^{5} -3.73540 q^{7} +O(q^{10})$	$q-1.35854 q^{5} -3.73540 q^{7} -3.46328 q^{11} +5.65282 q^{13} +5.24666 q^{17} -5.91175 q^{19} +3.74509 q^{23} -3.15438 q^{25} +2.15273 q^{29} +10.5485 q^{31} +5.07467 q^{35} -6.12685 q^{37} +9.96026 q^{41} -6.73611 q^{43} -3.69405 q^{47} +6.95320 q^{49} +10.3613 q^{53} +4.70499 q^{55} +2.38711 q^{59} -7.77996 q^{61} -7.67956 q^{65} -10.1595 q^{67} +0.525143 q^{71} +4.96794 q^{73} +12.9367 q^{77} -3.47337 q^{79} -16.2088 q^{83} -7.12778 q^{85} -1.12957 q^{89} -21.1155 q^{91} +8.03133 q^{95} +3.37062 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{5} - 12 q^{7}+O(q^{10}) $ 7 * q + 2 * q^5 - 12 * q^7	$ 7 q + 2 q^{5} - 12 q^{7} + 7 q^{11} - 9 q^{13} + q^{17} - 11 q^{19} + 19 q^{23} + 3 q^{25} + 5 q^{29} - 13 q^{31} + 7 q^{35} - 26 q^{37} + 2 q^{41} - 24 q^{43} + 11 q^{47} + 19 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} - 5 q^{61} - 13 q^{65} - 42 q^{67} - 9 q^{71} + 27 q^{73} - 12 q^{77} - 8 q^{79} - 16 q^{83} - 27 q^{85} - 9 q^{89} - 2 q^{91} - 10 q^{95} - 8 q^{97}+O(q^{100}) $ 7 * q + 2 * q^5 - 12 * q^7 + 7 * q^11 - 9 * q^13 + q^17 - 11 * q^19 + 19 * q^23 + 3 * q^25 + 5 * q^29 - 13 * q^31 + 7 * q^35 - 26 * q^37 + 2 * q^41 - 24 * q^43 + 11 * q^47 + 19 * q^49 - 4 * q^53 - 4 * q^55 + 4 * q^59 - 5 * q^61 - 13 * q^65 - 42 * q^67 - 9 * q^71 + 27 * q^73 - 12 * q^77 - 8 * q^79 - 16 * q^83 - 27 * q^85 - 9 * q^89 - 2 * q^91 - 10 * q^95 - 8 * 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
$5$	−1.35854	−0.607556	−0.303778	−	0.952743i	$-0.598248\pi$
$5$	−1.35854	−0.607556	−0.303778	+	0.952743i	$0.598248\pi$
$6$	0	0
$7$	−3.73540	−1.41185	−0.705924	−	0.708288i	$-0.749468\pi$
$7$	−3.73540	−1.41185	−0.705924	+	0.708288i	$0.749468\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.46328	−1.04422	−0.522109	−	0.852879i	$-0.674855\pi$
$11$	−3.46328	−1.04422	−0.522109	+	0.852879i	$0.674855\pi$
$12$	0	0
$13$	5.65282	1.56781	0.783904	−	0.620881i	$-0.213225\pi$
$13$	5.65282	1.56781	0.783904	+	0.620881i	$0.213225\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.24666	1.27250	0.636252	−	0.771482i	$-0.280484\pi$
$17$	5.24666	1.27250	0.636252	+	0.771482i	$0.280484\pi$
$18$	0	0
$19$	−5.91175	−1.35625	−0.678124	−	0.734947i	$-0.737207\pi$
$19$	−5.91175	−1.35625	−0.678124	+	0.734947i	$0.737207\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.74509	0.780905	0.390452	−	0.920623i	$-0.372319\pi$
$23$	3.74509	0.780905	0.390452	+	0.920623i	$0.372319\pi$
$24$	0	0
$25$	−3.15438	−0.630876
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.15273	0.399752	0.199876	−	0.979821i	$-0.435946\pi$
$29$	2.15273	0.399752	0.199876	+	0.979821i	$0.435946\pi$
$30$	0	0
$31$	10.5485	1.89456	0.947282	−	0.320401i	$-0.103817\pi$
$31$	10.5485	1.89456	0.947282	+	0.320401i	$0.103817\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.07467	0.857776
$36$	0	0
$37$	−6.12685	−1.00725	−0.503624	−	0.863923i	$-0.668000\pi$
$37$	−6.12685	−1.00725	−0.503624	+	0.863923i	$0.668000\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.96026	1.55553	0.777765	−	0.628555i	$-0.216353\pi$
$41$	9.96026	1.55553	0.777765	+	0.628555i	$0.216353\pi$
$42$	0	0
$43$	−6.73611	−1.02725	−0.513624	−	0.858016i	$-0.671697\pi$
$43$	−6.73611	−1.02725	−0.513624	+	0.858016i	$0.671697\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.69405	−0.538832	−0.269416	−	0.963024i	$-0.586831\pi$
$47$	−3.69405	−0.538832	−0.269416	+	0.963024i	$0.586831\pi$
$48$	0	0
$49$	6.95320	0.993314
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.3613	1.42323	0.711616	−	0.702569i	$-0.247964\pi$
$53$	10.3613	1.42323	0.711616	+	0.702569i	$0.247964\pi$
$54$	0	0
$55$	4.70499	0.634421
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.38711	0.310776	0.155388	−	0.987854i	$-0.450337\pi$
$59$	2.38711	0.310776	0.155388	+	0.987854i	$0.450337\pi$
$60$	0	0
$61$	−7.77996	−0.996121	−0.498061	−	0.867142i	$-0.665954\pi$
$61$	−7.77996	−0.996121	−0.498061	+	0.867142i	$0.665954\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.67956	−0.952532
$66$	0	0
$67$	−10.1595	−1.24118	−0.620590	−	0.784135i	$-0.713107\pi$
$67$	−10.1595	−1.24118	−0.620590	+	0.784135i	$0.713107\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.525143	0.0623229	0.0311615	−	0.999514i	$-0.490079\pi$
$71$	0.525143	0.0623229	0.0311615	+	0.999514i	$0.490079\pi$
$72$	0	0
$73$	4.96794	0.581454	0.290727	−	0.956806i	$-0.406103\pi$
$73$	4.96794	0.581454	0.290727	+	0.956806i	$0.406103\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.9367	1.47428
$78$	0	0
$79$	−3.47337	−0.390785	−0.195393	−	0.980725i	$-0.562598\pi$
$79$	−3.47337	−0.390785	−0.195393	+	0.980725i	$0.562598\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.2088	−1.77915	−0.889575	−	0.456789i	$-0.849000\pi$
$83$	−16.2088	−1.77915	−0.889575	+	0.456789i	$0.849000\pi$
$84$	0	0
$85$	−7.12778	−0.773117
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.12957	−0.119735	−0.0598673	−	0.998206i	$-0.519068\pi$
$89$	−1.12957	−0.119735	−0.0598673	+	0.998206i	$0.519068\pi$
$90$	0	0
$91$	−21.1155	−2.21351
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.03133	0.823997
$96$	0	0
$97$	3.37062	0.342235	0.171117	−	0.985251i	$-0.445262\pi$
$97$	3.37062	0.342235	0.171117	+	0.985251i	$0.445262\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.81750	−0.280351	−0.140176	−	0.990127i	$-0.544767\pi$
$101$	−2.81750	−0.280351	−0.140176	+	0.990127i	$0.544767\pi$
$102$	0	0
$103$	−0.231833	−0.0228432	−0.0114216	−	0.999935i	$-0.503636\pi$
$103$	−0.231833	−0.0228432	−0.0114216	+	0.999935i	$0.503636\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.7696	1.81452	0.907262	−	0.420566i	$-0.138168\pi$
$107$	18.7696	1.81452	0.907262	+	0.420566i	$0.138168\pi$
$108$	0	0
$109$	1.02824	0.0984873	0.0492437	−	0.998787i	$-0.484319\pi$
$109$	1.02824	0.0984873	0.0492437	+	0.998787i	$0.484319\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.36583	0.128487	0.0642433	−	0.997934i	$-0.479537\pi$
$113$	1.36583	0.128487	0.0642433	+	0.997934i	$0.479537\pi$
$114$	0	0
$115$	−5.08784	−0.474443
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−19.5984	−1.79658
$120$	0	0
$121$	0.994304	0.0903912
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.0780	0.990848
$126$	0	0
$127$	12.9283	1.14720	0.573601	−	0.819135i	$-0.305546\pi$
$127$	12.9283	1.14720	0.573601	+	0.819135i	$0.305546\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.5957	−1.62471	−0.812356	−	0.583162i	$-0.801815\pi$
$131$	−18.5957	−1.62471	−0.812356	+	0.583162i	$0.801815\pi$
$132$	0	0
$133$	22.0827	1.91482
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.14323	−0.439416	−0.219708	−	0.975566i	$-0.570510\pi$
$137$	−5.14323	−0.439416	−0.219708	+	0.975566i	$0.570510\pi$
$138$	0	0
$139$	−2.75646	−0.233800	−0.116900	−	0.993144i	$-0.537296\pi$
$139$	−2.75646	−0.233800	−0.116900	+	0.993144i	$0.537296\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−19.5773	−1.63713
$144$	0	0
$145$	−2.92456	−0.242872
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.0712	−1.23468	−0.617339	−	0.786697i	$-0.711789\pi$
$149$	−15.0712	−1.23468	−0.617339	+	0.786697i	$0.711789\pi$
$150$	0	0
$151$	−11.7438	−0.955694	−0.477847	−	0.878443i	$-0.658583\pi$
$151$	−11.7438	−0.955694	−0.477847	+	0.878443i	$0.658583\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−14.3305	−1.15105
$156$	0	0
$157$	−19.5884	−1.56332	−0.781662	−	0.623702i	$-0.785628\pi$
$157$	−19.5884	−1.56332	−0.781662	+	0.623702i	$0.785628\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.9894	−1.10252
$162$	0	0
$163$	−17.7961	−1.39390	−0.696950	−	0.717119i	$-0.745460\pi$
$163$	−17.7961	−1.39390	−0.696950	+	0.717119i	$0.745460\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	18.9543	1.45803
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.2960	−1.08690	−0.543451	−	0.839441i	$-0.682883\pi$
$173$	−14.2960	−1.08690	−0.543451	+	0.839441i	$0.682883\pi$
$174$	0	0
$175$	11.7829	0.890701
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.8108	1.63022	0.815108	−	0.579310i	$-0.196678\pi$
$179$	21.8108	1.63022	0.815108	+	0.579310i	$0.196678\pi$
$180$	0	0
$181$	−10.6609	−0.792418	−0.396209	−	0.918160i	$-0.629674\pi$
$181$	−10.6609	−0.792418	−0.396209	+	0.918160i	$0.629674\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.32355	0.611959
$186$	0	0
$187$	−18.1707	−1.32877
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.42495	−0.609608	−0.304804	−	0.952415i	$-0.598591\pi$
$191$	−8.42495	−0.609608	−0.304804	+	0.952415i	$0.598591\pi$
$192$	0	0
$193$	19.9264	1.43433	0.717167	−	0.696902i	$-0.245439\pi$
$193$	19.9264	1.43433	0.717167	+	0.696902i	$0.245439\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.67355	0.190483	0.0952414	−	0.995454i	$-0.469638\pi$
$197$	2.67355	0.190483	0.0952414	+	0.995454i	$0.469638\pi$
$198$	0	0
$199$	−22.9740	−1.62858	−0.814292	−	0.580456i	$-0.802875\pi$
$199$	−22.9740	−1.62858	−0.814292	+	0.580456i	$0.802875\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.04130	−0.564389
$204$	0	0
$205$	−13.5314	−0.945072
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.4740	1.41622
$210$	0	0
$211$	−14.8171	−1.02005	−0.510025	−	0.860160i	$-0.670364\pi$
$211$	−14.8171	−1.02005	−0.510025	+	0.860160i	$0.670364\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.15125	0.624110
$216$	0	0
$217$	−39.4028	−2.67484
$218$	0	0
$219$	0	0
$220$	0	0
$221$	29.6584	1.99504
$222$	0	0
$223$	−4.33025	−0.289975	−0.144988	−	0.989433i	$-0.546314\pi$
$223$	−4.33025	−0.289975	−0.144988	+	0.989433i	$0.546314\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.81874	−0.187086	−0.0935432	−	0.995615i	$-0.529819\pi$
$227$	−2.81874	−0.187086	−0.0935432	+	0.995615i	$0.529819\pi$
$228$	0	0
$229$	−4.12400	−0.272521	−0.136261	−	0.990673i	$-0.543508\pi$
$229$	−4.12400	−0.272521	−0.136261	+	0.990673i	$0.543508\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.19333	−0.209202	−0.104601	−	0.994514i	$-0.533357\pi$
$233$	−3.19333	−0.209202	−0.104601	+	0.994514i	$0.533357\pi$
$234$	0	0
$235$	5.01850	0.327371
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−18.5384	−1.19915	−0.599574	−	0.800319i	$-0.704663\pi$
$239$	−18.5384	−1.19915	−0.599574	+	0.800319i	$0.704663\pi$
$240$	0	0
$241$	−18.5958	−1.19786	−0.598931	−	0.800800i	$-0.704408\pi$
$241$	−18.5958	−1.19786	−0.598931	+	0.800800i	$0.704408\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.44617	−0.603494
$246$	0	0
$247$	−33.4180	−2.12634
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.5175	−1.54753	−0.773767	−	0.633471i	$-0.781630\pi$
$251$	−24.5175	−1.54753	−0.773767	+	0.633471i	$0.781630\pi$
$252$	0	0
$253$	−12.9703	−0.815435
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.01478	−0.188057	−0.0940284	−	0.995570i	$-0.529974\pi$
$257$	−3.01478	−0.188057	−0.0940284	+	0.995570i	$0.529974\pi$
$258$	0	0
$259$	22.8862	1.42208
$260$	0	0
$261$	0	0
$262$	0	0
$263$	29.3036	1.80694	0.903469	−	0.428653i	$-0.141012\pi$
$263$	29.3036	1.80694	0.903469	+	0.428653i	$0.141012\pi$
$264$	0	0
$265$	−14.0762	−0.864693
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.97312	0.120303	0.0601515	−	0.998189i	$-0.480842\pi$
$269$	1.97312	0.120303	0.0601515	+	0.998189i	$0.480842\pi$
$270$	0	0
$271$	18.6146	1.13076	0.565380	−	0.824831i	$-0.308730\pi$
$271$	18.6146	1.13076	0.565380	+	0.824831i	$0.308730\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.9245	0.658772
$276$	0	0
$277$	−17.4286	−1.04718	−0.523592	−	0.851969i	$-0.675408\pi$
$277$	−17.4286	−1.04718	−0.523592	+	0.851969i	$0.675408\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.9852	−1.49049	−0.745246	−	0.666790i	$-0.767668\pi$
$281$	−24.9852	−1.49049	−0.745246	+	0.666790i	$0.767668\pi$
$282$	0	0
$283$	1.77892	0.105746	0.0528728	−	0.998601i	$-0.483162\pi$
$283$	1.77892	0.105746	0.0528728	+	0.998601i	$0.483162\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−37.2055	−2.19617
$288$	0	0
$289$	10.5275	0.619264
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.2815	−0.775915	−0.387957	−	0.921677i	$-0.626819\pi$
$293$	−13.2815	−0.775915	−0.387957	+	0.921677i	$0.626819\pi$
$294$	0	0
$295$	−3.24298	−0.188814
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.1703	1.22431
$300$	0	0
$301$	25.1621	1.45032
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.5694	0.605199
$306$	0	0
$307$	0.916704	0.0523191	0.0261595	−	0.999658i	$-0.491672\pi$
$307$	0.916704	0.0523191	0.0261595	+	0.999658i	$0.491672\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.5632	1.10933	0.554664	−	0.832075i	$-0.312847\pi$
$311$	19.5632	1.10933	0.554664	+	0.832075i	$0.312847\pi$
$312$	0	0
$313$	−19.0566	−1.07715	−0.538573	−	0.842579i	$-0.681036\pi$
$313$	−19.0566	−1.07715	−0.538573	+	0.842579i	$0.681036\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.33422	−0.355765	−0.177883	−	0.984052i	$-0.556925\pi$
$317$	−6.33422	−0.355765	−0.177883	+	0.984052i	$0.556925\pi$
$318$	0	0
$319$	−7.45551	−0.417428
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−31.0170	−1.72583
$324$	0	0
$325$	−17.8311	−0.989093
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13.7987	0.760749
$330$	0	0
$331$	5.49519	0.302043	0.151022	−	0.988530i	$-0.451744\pi$
$331$	5.49519	0.302043	0.151022	+	0.988530i	$0.451744\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.8021	0.754087
$336$	0	0
$337$	9.69085	0.527894	0.263947	−	0.964537i	$-0.414976\pi$
$337$	9.69085	0.527894	0.263947	+	0.964537i	$0.414976\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−36.5324	−1.97834
$342$	0	0
$343$	0.174827	0.00943975
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.49206	0.294829	0.147415	−	0.989075i	$-0.452905\pi$
$347$	5.49206	0.294829	0.147415	+	0.989075i	$0.452905\pi$
$348$	0	0
$349$	25.6773	1.37447	0.687236	−	0.726434i	$-0.258824\pi$
$349$	25.6773	1.37447	0.687236	+	0.726434i	$0.258824\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.17231	−0.328519	−0.164260	−	0.986417i	$-0.552523\pi$
$353$	−6.17231	−0.328519	−0.164260	+	0.986417i	$0.552523\pi$
$354$	0	0
$355$	−0.713425	−0.0378647
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.966490	0.0510094	0.0255047	−	0.999675i	$-0.491881\pi$
$359$	0.966490	0.0510094	0.0255047	+	0.999675i	$0.491881\pi$
$360$	0	0
$361$	15.9488	0.839409
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.74913	−0.353266
$366$	0	0
$367$	−4.35207	−0.227176	−0.113588	−	0.993528i	$-0.536234\pi$
$367$	−4.35207	−0.227176	−0.113588	+	0.993528i	$0.536234\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−38.7035	−2.00939
$372$	0	0
$373$	18.9655	0.981998	0.490999	−	0.871160i	$-0.336632\pi$
$373$	18.9655	0.981998	0.490999	+	0.871160i	$0.336632\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.1690	0.626735
$378$	0	0
$379$	14.2692	0.732962	0.366481	−	0.930426i	$-0.380562\pi$
$379$	14.2692	0.732962	0.366481	+	0.930426i	$0.380562\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.89756	0.250254	0.125127	−	0.992141i	$-0.460066\pi$
$383$	4.89756	0.250254	0.125127	+	0.992141i	$0.460066\pi$
$384$	0	0
$385$	−17.5750	−0.895706
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−0.524174	−0.0265767	−0.0132883	−	0.999912i	$-0.504230\pi$
$389$	−0.524174	−0.0265767	−0.0132883	+	0.999912i	$0.504230\pi$
$390$	0	0
$391$	19.6492	0.993703
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.71871	0.237424
$396$	0	0
$397$	−20.8607	−1.04697	−0.523484	−	0.852035i	$-0.675368\pi$
$397$	−20.8607	−1.04697	−0.523484	+	0.852035i	$0.675368\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.1324	−0.855553	−0.427776	−	0.903885i	$-0.640703\pi$
$401$	−17.1324	−0.855553	−0.427776	+	0.903885i	$0.640703\pi$
$402$	0	0
$403$	59.6287	2.97031
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21.2190	1.05179
$408$	0	0
$409$	7.68942	0.380217	0.190109	−	0.981763i	$-0.439116\pi$
$409$	7.68942	0.380217	0.190109	+	0.981763i	$0.439116\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.91682	−0.438768
$414$	0	0
$415$	22.0203	1.08093
$416$	0	0
$417$	0	0
$418$	0	0
$419$	32.1640	1.57131	0.785656	−	0.618663i	$-0.212325\pi$
$419$	32.1640	1.57131	0.785656	+	0.618663i	$0.212325\pi$
$420$	0	0
$421$	−4.11781	−0.200690	−0.100345	−	0.994953i	$-0.531995\pi$
$421$	−4.11781	−0.200690	−0.100345	+	0.994953i	$0.531995\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−16.5500	−0.802791
$426$	0	0
$427$	29.0612	1.40637
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.0788	0.918994	0.459497	−	0.888179i	$-0.348030\pi$
$431$	19.0788	0.918994	0.459497	+	0.888179i	$0.348030\pi$
$432$	0	0
$433$	20.0805	0.965006	0.482503	−	0.875894i	$-0.339728\pi$
$433$	20.0805	0.965006	0.482503	+	0.875894i	$0.339728\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−22.1400	−1.05910
$438$	0	0
$439$	−17.6120	−0.840577	−0.420288	−	0.907391i	$-0.638071\pi$
$439$	−17.6120	−0.840577	−0.420288	+	0.907391i	$0.638071\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.6825	0.792610	0.396305	−	0.918119i	$-0.370292\pi$
$443$	16.6825	0.792610	0.396305	+	0.918119i	$0.370292\pi$
$444$	0	0
$445$	1.53457	0.0727454
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.7438	−0.884572	−0.442286	−	0.896874i	$-0.645832\pi$
$449$	−18.7438	−0.884572	−0.442286	+	0.896874i	$0.645832\pi$
$450$	0	0
$451$	−34.4952	−1.62431
$452$	0	0
$453$	0	0
$454$	0	0
$455$	28.6862	1.34483
$456$	0	0
$457$	0.235876	0.0110338	0.00551691	−	0.999985i	$-0.498244\pi$
$457$	0.235876	0.0110338	0.00551691	+	0.999985i	$0.498244\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.84630	0.225714	0.112857	−	0.993611i	$-0.464000\pi$
$461$	4.84630	0.225714	0.112857	+	0.993611i	$0.464000\pi$
$462$	0	0
$463$	18.1375	0.842922	0.421461	−	0.906847i	$-0.361517\pi$
$463$	18.1375	0.842922	0.421461	+	0.906847i	$0.361517\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.1549	−0.655009	−0.327504	−	0.944850i	$-0.606208\pi$
$467$	−14.1549	−0.655009	−0.327504	+	0.944850i	$0.606208\pi$
$468$	0	0
$469$	37.9498	1.75236
$470$	0	0
$471$	0	0
$472$	0	0
$473$	23.3290	1.07267
$474$	0	0
$475$	18.6479	0.855624
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.5238	−1.21190	−0.605952	−	0.795501i	$-0.707208\pi$
$479$	−26.5238	−1.21190	−0.605952	+	0.795501i	$0.707208\pi$
$480$	0	0
$481$	−34.6339	−1.57917
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.57911	−0.207927
$486$	0	0
$487$	7.12246	0.322750	0.161375	−	0.986893i	$-0.448407\pi$
$487$	7.12246	0.322750	0.161375	+	0.986893i	$0.448407\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.06416	0.183413	0.0917066	−	0.995786i	$-0.470768\pi$
$491$	4.06416	0.183413	0.0917066	+	0.995786i	$0.470768\pi$
$492$	0	0
$493$	11.2947	0.508686
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.96162	−0.0879905
$498$	0	0
$499$	−17.1820	−0.769172	−0.384586	−	0.923089i	$-0.625656\pi$
$499$	−17.1820	−0.769172	−0.384586	+	0.923089i	$0.625656\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−34.1952	−1.52469	−0.762345	−	0.647171i	$-0.775952\pi$
$503$	−34.1952	−1.52469	−0.762345	+	0.647171i	$0.775952\pi$
$504$	0	0
$505$	3.82767	0.170329
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.31236	0.102493	0.0512467	−	0.998686i	$-0.483681\pi$
$509$	2.31236	0.102493	0.0512467	+	0.998686i	$0.483681\pi$
$510$	0	0
$511$	−18.5572	−0.820924
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.314953	0.0138785
$516$	0	0
$517$	12.7935	0.562658
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.99538	−0.437905	−0.218953	−	0.975735i	$-0.570264\pi$
$521$	−9.99538	−0.437905	−0.218953	+	0.975735i	$0.570264\pi$
$522$	0	0
$523$	31.1314	1.36128	0.680642	−	0.732617i	$-0.261701\pi$
$523$	31.1314	1.36128	0.680642	+	0.732617i	$0.261701\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	55.3444	2.41084
$528$	0	0
$529$	−8.97433	−0.390188
$530$	0	0
$531$	0	0
$532$	0	0
$533$	56.3035	2.43878
$534$	0	0
$535$	−25.4992	−1.10242
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−24.0809	−1.03724
$540$	0	0
$541$	−9.66706	−0.415620	−0.207810	−	0.978169i	$-0.566633\pi$
$541$	−9.66706	−0.415620	−0.207810	+	0.978169i	$0.566633\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.39690	−0.0598365
$546$	0	0
$547$	41.2694	1.76455	0.882276	−	0.470733i	$-0.156010\pi$
$547$	41.2694	1.76455	0.882276	+	0.470733i	$0.156010\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.7264	−0.542163
$552$	0	0
$553$	12.9744	0.551729
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.04280	−0.340785	−0.170392	−	0.985376i	$-0.554503\pi$
$557$	−8.04280	−0.340785	−0.170392	+	0.985376i	$0.554503\pi$
$558$	0	0
$559$	−38.0780	−1.61053
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−34.5717	−1.45703	−0.728513	−	0.685032i	$-0.759788\pi$
$563$	−34.5717	−1.45703	−0.728513	+	0.685032i	$0.759788\pi$
$564$	0	0
$565$	−1.85553	−0.0780628
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.7585	−0.744476	−0.372238	−	0.928137i	$-0.621410\pi$
$569$	−17.7585	−0.744476	−0.372238	+	0.928137i	$0.621410\pi$
$570$	0	0
$571$	−7.35563	−0.307824	−0.153912	−	0.988085i	$-0.549187\pi$
$571$	−7.35563	−0.307824	−0.153912	+	0.988085i	$0.549187\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.8134	−0.492654
$576$	0	0
$577$	−32.3499	−1.34675	−0.673373	−	0.739303i	$-0.735155\pi$
$577$	−32.3499	−1.34675	−0.673373	+	0.739303i	$0.735155\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	60.5464	2.51189
$582$	0	0
$583$	−35.8840	−1.48616
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.2634	−1.24910	−0.624552	−	0.780984i	$-0.714718\pi$
$587$	−30.2634	−1.24910	−0.624552	+	0.780984i	$0.714718\pi$
$588$	0	0
$589$	−62.3600	−2.56950
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−35.3338	−1.45098	−0.725492	−	0.688231i	$-0.758388\pi$
$593$	−35.3338	−1.45098	−0.725492	+	0.688231i	$0.758388\pi$
$594$	0	0
$595$	26.6251	1.09152
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.7744	−0.971394	−0.485697	−	0.874127i	$-0.661434\pi$
$599$	−23.7744	−0.971394	−0.485697	+	0.874127i	$0.661434\pi$
$600$	0	0
$601$	−3.75741	−0.153268	−0.0766340	−	0.997059i	$-0.524417\pi$
$601$	−3.75741	−0.153268	−0.0766340	+	0.997059i	$0.524417\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.35080	−0.0549177
$606$	0	0
$607$	9.76801	0.396471	0.198236	−	0.980154i	$-0.436479\pi$
$607$	9.76801	0.396471	0.198236	+	0.980154i	$0.436479\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.8818	−0.844786
$612$	0	0
$613$	−46.3303	−1.87126	−0.935632	−	0.352977i	$-0.885170\pi$
$613$	−46.3303	−1.87126	−0.935632	+	0.352977i	$0.885170\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.47963	−0.341377	−0.170688	−	0.985325i	$-0.554599\pi$
$617$	−8.47963	−0.341377	−0.170688	+	0.985325i	$0.554599\pi$
$618$	0	0
$619$	4.67093	0.187740	0.0938702	−	0.995584i	$-0.470076\pi$
$619$	4.67093	0.187740	0.0938702	+	0.995584i	$0.470076\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.21941	0.169047
$624$	0	0
$625$	0.722001	0.0288800
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−32.1455	−1.28173
$630$	0	0
$631$	−8.82339	−0.351253	−0.175627	−	0.984457i	$-0.556195\pi$
$631$	−8.82339	−0.351253	−0.175627	+	0.984457i	$0.556195\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−17.5636	−0.696989
$636$	0	0
$637$	39.3051	1.55733
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.8502	0.705040	0.352520	−	0.935804i	$-0.385325\pi$
$641$	17.8502	0.705040	0.352520	+	0.935804i	$0.385325\pi$
$642$	0	0
$643$	−12.2272	−0.482193	−0.241096	−	0.970501i	$-0.577507\pi$
$643$	−12.2272	−0.482193	−0.241096	+	0.970501i	$0.577507\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.9410	−0.823274	−0.411637	−	0.911348i	$-0.635043\pi$
$647$	−20.9410	−0.823274	−0.411637	+	0.911348i	$0.635043\pi$
$648$	0	0
$649$	−8.26724	−0.324517
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.83643	0.345796	0.172898	−	0.984940i	$-0.444687\pi$
$653$	8.83643	0.345796	0.172898	+	0.984940i	$0.444687\pi$
$654$	0	0
$655$	25.2629	0.987103
$656$	0	0
$657$	0	0
$658$	0	0
$659$	17.9043	0.697454	0.348727	−	0.937224i	$-0.386614\pi$
$659$	17.9043	0.697454	0.348727	+	0.937224i	$0.386614\pi$
$660$	0	0
$661$	−11.0640	−0.430341	−0.215171	−	0.976576i	$-0.569031\pi$
$661$	−11.0640	−0.430341	−0.215171	+	0.976576i	$0.569031\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−30.0002	−1.16336
$666$	0	0
$667$	8.06216	0.312168
$668$	0	0
$669$	0	0
$670$	0	0
$671$	26.9442	1.04017
$672$	0	0
$673$	38.1470	1.47046	0.735229	−	0.677819i	$-0.237075\pi$
$673$	38.1470	1.47046	0.735229	+	0.677819i	$0.237075\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.6262	0.638997	0.319499	−	0.947587i	$-0.396485\pi$
$677$	16.6262	0.638997	0.319499	+	0.947587i	$0.396485\pi$
$678$	0	0
$679$	−12.5906	−0.483183
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.2239	1.30954	0.654771	−	0.755827i	$-0.272765\pi$
$683$	34.2239	1.30954	0.654771	+	0.755827i	$0.272765\pi$
$684$	0	0
$685$	6.98726	0.266970
$686$	0	0
$687$	0	0
$688$	0	0
$689$	58.5704	2.23136
$690$	0	0
$691$	−36.1393	−1.37481	−0.687403	−	0.726277i	$-0.741249\pi$
$691$	−36.1393	−1.37481	−0.687403	+	0.726277i	$0.741249\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.74475	0.142046
$696$	0	0
$697$	52.2581	1.97942
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.6295	0.741395	0.370697	−	0.928754i	$-0.379119\pi$
$701$	19.6295	0.741395	0.370697	+	0.928754i	$0.379119\pi$
$702$	0	0
$703$	36.2204	1.36608
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.5245	0.395813
$708$	0	0
$709$	20.7410	0.778946	0.389473	−	0.921038i	$-0.372657\pi$
$709$	20.7410	0.778946	0.389473	+	0.921038i	$0.372657\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39.5050	1.47947
$714$	0	0
$715$	26.5964	0.994651
$716$	0	0
$717$	0	0
$718$	0	0
$719$	43.2159	1.61168	0.805841	−	0.592132i	$-0.201713\pi$
$719$	43.2159	1.61168	0.805841	+	0.592132i	$0.201713\pi$
$720$	0	0
$721$	0.865987	0.0322510
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.79053	−0.252194
$726$	0	0
$727$	−33.6198	−1.24689	−0.623445	−	0.781867i	$-0.714267\pi$
$727$	−33.6198	−1.24689	−0.623445	+	0.781867i	$0.714267\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−35.3421	−1.30718
$732$	0	0
$733$	−10.8105	−0.399296	−0.199648	−	0.979868i	$-0.563980\pi$
$733$	−10.8105	−0.399296	−0.199648	+	0.979868i	$0.563980\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35.1852	1.29606
$738$	0	0
$739$	19.6414	0.722521	0.361260	−	0.932465i	$-0.382347\pi$
$739$	19.6414	0.722521	0.361260	+	0.932465i	$0.382347\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−46.2216	−1.69571	−0.847853	−	0.530231i	$-0.822105\pi$
$743$	−46.2216	−1.69571	−0.847853	+	0.530231i	$0.822105\pi$
$744$	0	0
$745$	20.4747	0.750136
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−70.1119	−2.56183
$750$	0	0
$751$	−40.5429	−1.47943	−0.739715	−	0.672920i	$-0.765040\pi$
$751$	−40.5429	−1.47943	−0.739715	+	0.672920i	$0.765040\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.9543	0.580637
$756$	0	0
$757$	−33.4877	−1.21713	−0.608566	−	0.793504i	$-0.708255\pi$
$757$	−33.4877	−1.21713	−0.608566	+	0.793504i	$0.708255\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16.6129	0.602216	0.301108	−	0.953590i	$-0.402644\pi$
$761$	16.6129	0.602216	0.301108	+	0.953590i	$0.402644\pi$
$762$	0	0
$763$	−3.84088	−0.139049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.4939	0.487237
$768$	0	0
$769$	21.1342	0.762119	0.381059	−	0.924551i	$-0.375559\pi$
$769$	21.1342	0.762119	0.381059	+	0.924551i	$0.375559\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19.6281	0.705975	0.352988	−	0.935628i	$-0.385166\pi$
$773$	19.6281	0.705975	0.352988	+	0.935628i	$0.385166\pi$
$774$	0	0
$775$	−33.2739	−1.19523
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−58.8825	−2.10969
$780$	0	0
$781$	−1.81872	−0.0650787
$782$	0	0
$783$	0	0
$784$	0	0
$785$	26.6116	0.949807
$786$	0	0
$787$	−7.14146	−0.254566	−0.127283	−	0.991866i	$-0.540626\pi$
$787$	−7.14146	−0.254566	−0.127283	+	0.991866i	$0.540626\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.10193	−0.181404
$792$	0	0
$793$	−43.9787	−1.56173
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.3622	1.14633	0.573164	−	0.819441i	$-0.305716\pi$
$797$	32.3622	1.14633	0.573164	+	0.819441i	$0.305716\pi$
$798$	0	0
$799$	−19.3814	−0.685666
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17.2054	−0.607165
$804$	0	0
$805$	19.0051	0.669842
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−17.6100	−0.619134	−0.309567	−	0.950878i	$-0.600184\pi$
$809$	−17.6100	−0.619134	−0.309567	+	0.950878i	$0.600184\pi$
$810$	0	0
$811$	20.6634	0.725588	0.362794	−	0.931869i	$-0.381823\pi$
$811$	20.6634	0.725588	0.362794	+	0.931869i	$0.381823\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	24.1767	0.846873
$816$	0	0
$817$	39.8222	1.39320
$818$	0	0
$819$	0	0
$820$	0	0
$821$	14.5424	0.507534	0.253767	−	0.967265i	$-0.418330\pi$
$821$	14.5424	0.507534	0.253767	+	0.967265i	$0.418330\pi$
$822$	0	0
$823$	−18.4446	−0.642939	−0.321469	−	0.946920i	$-0.604177\pi$
$823$	−18.4446	−0.642939	−0.321469	+	0.946920i	$0.604177\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.6524	−1.27453	−0.637265	−	0.770645i	$-0.719934\pi$
$827$	−36.6524	−1.27453	−0.637265	+	0.770645i	$0.719934\pi$
$828$	0	0
$829$	52.5477	1.82506	0.912528	−	0.409015i	$-0.134128\pi$
$829$	52.5477	1.82506	0.912528	+	0.409015i	$0.134128\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	36.4811	1.26399
$834$	0	0
$835$	1.35854	0.0470141
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−44.7378	−1.54452	−0.772260	−	0.635307i	$-0.780874\pi$
$839$	−44.7378	−1.54452	−0.772260	+	0.635307i	$0.780874\pi$
$840$	0	0
$841$	−24.3658	−0.840198
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−25.7501	−0.885832
$846$	0	0
$847$	−3.71412	−0.127619
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−22.9456	−0.786564
$852$	0	0
$853$	−23.9929	−0.821501	−0.410751	−	0.911748i	$-0.634733\pi$
$853$	−23.9929	−0.821501	−0.410751	+	0.911748i	$0.634733\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	47.8544	1.63467	0.817337	−	0.576160i	$-0.195449\pi$
$857$	47.8544	1.63467	0.817337	+	0.576160i	$0.195449\pi$
$858$	0	0
$859$	4.54528	0.155083	0.0775415	−	0.996989i	$-0.475293\pi$
$859$	4.54528	0.155083	0.0775415	+	0.996989i	$0.475293\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.00469	−0.102281	−0.0511405	−	0.998691i	$-0.516286\pi$
$863$	−3.00469	−0.102281	−0.0511405	+	0.998691i	$0.516286\pi$
$864$	0	0
$865$	19.4216	0.660354
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.0293	0.408065
$870$	0	0
$871$	−57.4298	−1.94593
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−41.3808	−1.39893
$876$	0	0
$877$	−39.3309	−1.32811	−0.664055	−	0.747683i	$-0.731166\pi$
$877$	−39.3309	−1.32811	−0.664055	+	0.747683i	$0.731166\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	24.6172	0.829373	0.414687	−	0.909964i	$-0.363891\pi$
$881$	24.6172	0.829373	0.414687	+	0.909964i	$0.363891\pi$
$882$	0	0
$883$	24.2688	0.816711	0.408355	−	0.912823i	$-0.366102\pi$
$883$	24.2688	0.816711	0.408355	+	0.912823i	$0.366102\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.7424	−0.495001	−0.247501	−	0.968888i	$-0.579609\pi$
$887$	−14.7424	−0.495001	−0.247501	+	0.968888i	$0.579609\pi$
$888$	0	0
$889$	−48.2923	−1.61967
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21.8383	0.730790
$894$	0	0
$895$	−29.6308	−0.990447
$896$	0	0
$897$	0	0
$898$	0	0
$899$	22.7080	0.757356
$900$	0	0
$901$	54.3622	1.81107
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.4832	0.481438
$906$	0	0
$907$	4.04344	0.134260	0.0671300	−	0.997744i	$-0.478616\pi$
$907$	4.04344	0.134260	0.0671300	+	0.997744i	$0.478616\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.1517	−0.866445	−0.433223	−	0.901287i	$-0.642624\pi$
$911$	−26.1517	−0.866445	−0.433223	+	0.901287i	$0.642624\pi$
$912$	0	0
$913$	56.1357	1.85782
$914$	0	0
$915$	0	0
$916$	0	0
$917$	69.4623	2.29385
$918$	0	0
$919$	−11.7942	−0.389053	−0.194527	−	0.980897i	$-0.562317\pi$
$919$	−11.7942	−0.389053	−0.194527	+	0.980897i	$0.562317\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.96853	0.0977105
$924$	0	0
$925$	19.3264	0.635448
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.4933	1.16450	0.582248	−	0.813011i	$-0.302173\pi$
$929$	35.4933	1.16450	0.582248	+	0.813011i	$0.302173\pi$
$930$	0	0
$931$	−41.1056	−1.34718
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.6855	0.807302
$936$	0	0
$937$	19.6733	0.642700	0.321350	−	0.946961i	$-0.395863\pi$
$937$	19.6733	0.642700	0.321350	+	0.946961i	$0.395863\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.50813	−0.114362	−0.0571809	−	0.998364i	$-0.518211\pi$
$941$	−3.50813	−0.114362	−0.0571809	+	0.998364i	$0.518211\pi$
$942$	0	0
$943$	37.3020	1.21472
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.4558	0.957183	0.478592	−	0.878038i	$-0.341147\pi$
$947$	29.4558	0.957183	0.478592	+	0.878038i	$0.341147\pi$
$948$	0	0
$949$	28.0829	0.911609
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.45573	0.209121	0.104561	−	0.994519i	$-0.466656\pi$
$953$	6.45573	0.209121	0.104561	+	0.994519i	$0.466656\pi$
$954$	0	0
$955$	11.4456	0.370371
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19.2120	0.620388
$960$	0	0
$961$	80.2706	2.58937
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−27.0707	−0.871438
$966$	0	0
$967$	−46.5083	−1.49560	−0.747802	−	0.663921i	$-0.768891\pi$
$967$	−46.5083	−1.49560	−0.747802	+	0.663921i	$0.768891\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.5794	0.724607	0.362304	−	0.932060i	$-0.381990\pi$
$971$	22.5794	0.724607	0.362304	+	0.932060i	$0.381990\pi$
$972$	0	0
$973$	10.2965	0.330090
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.6174	−1.04352	−0.521762	−	0.853091i	$-0.674725\pi$
$977$	−32.6174	−1.04352	−0.521762	+	0.853091i	$0.674725\pi$
$978$	0	0
$979$	3.91203	0.125029
$980$	0	0
$981$	0	0
$982$	0	0
$983$	38.6062	1.23135	0.615674	−	0.788001i	$-0.288884\pi$
$983$	38.6062	1.23135	0.615674	+	0.788001i	$0.288884\pi$
$984$	0	0
$985$	−3.63212	−0.115729
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25.2273	−0.802182
$990$	0	0
$991$	4.63294	0.147170	0.0735851	−	0.997289i	$-0.476556\pi$
$991$	4.63294	0.147170	0.0735851	+	0.997289i	$0.476556\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	31.2110	0.989456
$996$	0	0
$997$	−34.0242	−1.07756	−0.538779	−	0.842447i	$-0.681114\pi$
$997$	−34.0242	−1.07756	−0.538779	+	0.842447i	$0.681114\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.g.1.2		7
3.2	odd	2		668.2.a.c.1.2	✓	7
12.11	even	2		2672.2.a.k.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.c.1.2	✓	7	3.2	odd	2
2672.2.a.k.1.6		7	12.11	even	2
6012.2.a.g.1.2		7	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)

\(f(q)\)	\(=\)	\(q-1.35854 q^{5} -3.73540 q^{7} +O(q^{10})\)	\(q-1.35854 q^{5} -3.73540 q^{7} -3.46328 q^{11} +5.65282 q^{13} +5.24666 q^{17} -5.91175 q^{19} +3.74509 q^{23} -3.15438 q^{25} +2.15273 q^{29} +10.5485 q^{31} +5.07467 q^{35} -6.12685 q^{37} +9.96026 q^{41} -6.73611 q^{43} -3.69405 q^{47} +6.95320 q^{49} +10.3613 q^{53} +4.70499 q^{55} +2.38711 q^{59} -7.77996 q^{61} -7.67956 q^{65} -10.1595 q^{67} +0.525143 q^{71} +4.96794 q^{73} +12.9367 q^{77} -3.47337 q^{79} -16.2088 q^{83} -7.12778 q^{85} -1.12957 q^{89} -21.1155 q^{91} +8.03133 q^{95} +3.37062 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{5} - 12 q^{7}+O(q^{10}) \) 7 * q + 2 * q^5 - 12 * q^7	\( 7 q + 2 q^{5} - 12 q^{7} + 7 q^{11} - 9 q^{13} + q^{17} - 11 q^{19} + 19 q^{23} + 3 q^{25} + 5 q^{29} - 13 q^{31} + 7 q^{35} - 26 q^{37} + 2 q^{41} - 24 q^{43} + 11 q^{47} + 19 q^{49} - 4 q^{53} - 4 q^{55} + 4 q^{59} - 5 q^{61} - 13 q^{65} - 42 q^{67} - 9 q^{71} + 27 q^{73} - 12 q^{77} - 8 q^{79} - 16 q^{83} - 27 q^{85} - 9 q^{89} - 2 q^{91} - 10 q^{95} - 8 q^{97}+O(q^{100}) \) 7 * q + 2 * q^5 - 12 * q^7 + 7 * q^11 - 9 * q^13 + q^17 - 11 * q^19 + 19 * q^23 + 3 * q^25 + 5 * q^29 - 13 * q^31 + 7 * q^35 - 26 * q^37 + 2 * q^41 - 24 * q^43 + 11 * q^47 + 19 * q^49 - 4 * q^53 - 4 * q^55 + 4 * q^59 - 5 * q^61 - 13 * q^65 - 42 * q^67 - 9 * q^71 + 27 * q^73 - 12 * q^77 - 8 * q^79 - 16 * q^83 - 27 * q^85 - 9 * q^89 - 2 * q^91 - 10 * q^95 - 8 * q^97

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