LMFDB - Embedded newform 5976.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5976,2,Mod(1,5976)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5976, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5976.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5976 = 2^{3} \cdot 3^{2} \cdot 83 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5976.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:	$47.7186002479$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 18x^{6} + 33x^{5} + 87x^{4} - 127x^{3} - 126x^{2} + 100x - 16 $ x^8 - 2x^7 - 18x^6 + 33x^5 + 87x^4 - 127x^3 - 126x^2 + 100*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 664)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.297635$ of defining polynomial
Character	$\chi$	$=$	5976.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.84665 q^{5} +1.95246 q^{7} +O(q^{10})$	$q-3.84665 q^{5} +1.95246 q^{7} -3.36179 q^{11} +4.30849 q^{13} +3.55797 q^{17} -6.21341 q^{19} +6.13909 q^{23} +9.79670 q^{25} -9.67464 q^{29} -4.49202 q^{31} -7.51043 q^{35} +5.91141 q^{37} +7.02818 q^{41} -7.98694 q^{43} -6.67793 q^{47} -3.18789 q^{49} -0.940210 q^{53} +12.9316 q^{55} +4.84423 q^{59} +9.67464 q^{61} -16.5732 q^{65} +12.8131 q^{67} +8.78876 q^{71} -5.27320 q^{73} -6.56376 q^{77} +13.0101 q^{79} +1.00000 q^{83} -13.6863 q^{85} -1.89620 q^{89} +8.41216 q^{91} +23.9008 q^{95} +5.58949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 7 q^{5} + q^{7}+O(q^{10}) $ 8 * q - 7 * q^5 + q^7	$ 8 q - 7 q^{5} + q^{7} - 10 q^{11} + 7 q^{13} - 15 q^{17} + 7 q^{19} - 4 q^{23} + 27 q^{25} - 16 q^{29} + 3 q^{31} - 2 q^{35} + 8 q^{37} - 24 q^{41} - q^{43} + 2 q^{47} + 3 q^{49} - 7 q^{53} - 30 q^{55} + 2 q^{59} + 16 q^{61} - 2 q^{65} - 25 q^{67} - 8 q^{71} + 14 q^{73} + 7 q^{77} - 4 q^{79} + 8 q^{83} - 21 q^{85} - 20 q^{89} - 45 q^{91} - 8 q^{95} + 2 q^{97}+O(q^{100}) $ 8 * q - 7 * q^5 + q^7 - 10 * q^11 + 7 * q^13 - 15 * q^17 + 7 * q^19 - 4 * q^23 + 27 * q^25 - 16 * q^29 + 3 * q^31 - 2 * q^35 + 8 * q^37 - 24 * q^41 - q^43 + 2 * q^47 + 3 * q^49 - 7 * q^53 - 30 * q^55 + 2 * q^59 + 16 * q^61 - 2 * q^65 - 25 * q^67 - 8 * q^71 + 14 * q^73 + 7 * q^77 - 4 * q^79 + 8 * q^83 - 21 * q^85 - 20 * q^89 - 45 * q^91 - 8 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.84665	−1.72027	−0.860136	−	0.510064i	$-0.829622\pi$
$5$	−3.84665	−1.72027	−0.860136	+	0.510064i	$0.829622\pi$
$6$	0	0
$7$	1.95246	0.737961	0.368981	−	0.929437i	$-0.379707\pi$
$7$	1.95246	0.737961	0.368981	+	0.929437i	$0.379707\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.36179	−1.01362	−0.506808	−	0.862059i	$-0.669175\pi$
$11$	−3.36179	−1.01362	−0.506808	+	0.862059i	$0.669175\pi$
$12$	0	0
$13$	4.30849	1.19496	0.597480	−	0.801884i	$-0.296169\pi$
$13$	4.30849	1.19496	0.597480	+	0.801884i	$0.296169\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.55797	0.862934	0.431467	−	0.902129i	$-0.357996\pi$
$17$	3.55797	0.862934	0.431467	+	0.902129i	$0.357996\pi$
$18$	0	0
$19$	−6.21341	−1.42545	−0.712727	−	0.701441i	$-0.752540\pi$
$19$	−6.21341	−1.42545	−0.712727	+	0.701441i	$0.752540\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.13909	1.28009	0.640044	−	0.768338i	$-0.278916\pi$
$23$	6.13909	1.28009	0.640044	+	0.768338i	$0.278916\pi$
$24$	0	0
$25$	9.79670	1.95934
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.67464	−1.79653	−0.898267	−	0.439449i	$-0.855174\pi$
$29$	−9.67464	−1.79653	−0.898267	+	0.439449i	$0.855174\pi$
$30$	0	0
$31$	−4.49202	−0.806790	−0.403395	−	0.915026i	$-0.632170\pi$
$31$	−4.49202	−0.806790	−0.403395	+	0.915026i	$0.632170\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.51043	−1.26949
$36$	0	0
$37$	5.91141	0.971830	0.485915	−	0.874006i	$-0.338486\pi$
$37$	5.91141	0.971830	0.485915	+	0.874006i	$0.338486\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.02818	1.09762	0.548809	−	0.835948i	$-0.315081\pi$
$41$	7.02818	1.09762	0.548809	+	0.835948i	$0.315081\pi$
$42$	0	0
$43$	−7.98694	−1.21800	−0.608998	−	0.793172i	$-0.708428\pi$
$43$	−7.98694	−1.21800	−0.608998	+	0.793172i	$0.708428\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.67793	−0.974076	−0.487038	−	0.873381i	$-0.661923\pi$
$47$	−6.67793	−0.974076	−0.487038	+	0.873381i	$0.661923\pi$
$48$	0	0
$49$	−3.18789	−0.455413
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.940210	−0.129148	−0.0645739	−	0.997913i	$-0.520569\pi$
$53$	−0.940210	−0.129148	−0.0645739	+	0.997913i	$0.520569\pi$
$54$	0	0
$55$	12.9316	1.74370
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.84423	0.630666	0.315333	−	0.948981i	$-0.397884\pi$
$59$	4.84423	0.630666	0.315333	+	0.948981i	$0.397884\pi$
$60$	0	0
$61$	9.67464	1.23871	0.619355	−	0.785111i	$-0.287394\pi$
$61$	9.67464	1.23871	0.619355	+	0.785111i	$0.287394\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−16.5732	−2.05566
$66$	0	0
$67$	12.8131	1.56537	0.782687	−	0.622415i	$-0.213848\pi$
$67$	12.8131	1.56537	0.782687	+	0.622415i	$0.213848\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.78876	1.04303	0.521517	−	0.853241i	$-0.325366\pi$
$71$	8.78876	1.04303	0.521517	+	0.853241i	$0.325366\pi$
$72$	0	0
$73$	−5.27320	−0.617181	−0.308591	−	0.951195i	$-0.599857\pi$
$73$	−5.27320	−0.617181	−0.308591	+	0.951195i	$0.599857\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.56376	−0.748010
$78$	0	0
$79$	13.0101	1.46375	0.731876	−	0.681438i	$-0.238645\pi$
$79$	13.0101	1.46375	0.731876	+	0.681438i	$0.238645\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.00000	0.109764
$83$	1.00000	0.109764
$84$	0	0
$85$	−13.6863	−1.48448
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.89620	−0.200997	−0.100498	−	0.994937i	$-0.532044\pi$
$89$	−1.89620	−0.200997	−0.100498	+	0.994937i	$0.532044\pi$
$90$	0	0
$91$	8.41216	0.881834
$92$	0	0
$93$	0	0
$94$	0	0
$95$	23.9008	2.45217
$96$	0	0
$97$	5.58949	0.567527	0.283764	−	0.958894i	$-0.408417\pi$
$97$	5.58949	0.567527	0.283764	+	0.958894i	$0.408417\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.92663	0.987736	0.493868	−	0.869537i	$-0.335583\pi$
$101$	9.92663	0.987736	0.493868	+	0.869537i	$0.335583\pi$
$102$	0	0
$103$	−17.1172	−1.68660	−0.843302	−	0.537439i	$-0.819392\pi$
$103$	−17.1172	−1.68660	−0.843302	+	0.537439i	$0.819392\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.96471	−0.479957	−0.239978	−	0.970778i	$-0.577140\pi$
$107$	−4.96471	−0.479957	−0.239978	+	0.970778i	$0.577140\pi$
$108$	0	0
$109$	−20.2864	−1.94309	−0.971543	−	0.236862i	$-0.923881\pi$
$109$	−20.2864	−1.94309	−0.971543	+	0.236862i	$0.923881\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.69082	0.441275	0.220637	−	0.975356i	$-0.429186\pi$
$113$	4.69082	0.441275	0.220637	+	0.975356i	$0.429186\pi$
$114$	0	0
$115$	−23.6149	−2.20210
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.94680	0.636812
$120$	0	0
$121$	0.301612	0.0274193
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−18.4512	−1.65033
$126$	0	0
$127$	5.30316	0.470579	0.235290	−	0.971925i	$-0.424396\pi$
$127$	5.30316	0.470579	0.235290	+	0.971925i	$0.424396\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.57603	−0.312439	−0.156219	−	0.987722i	$-0.549931\pi$
$131$	−3.57603	−0.312439	−0.156219	+	0.987722i	$0.549931\pi$
$132$	0	0
$133$	−12.1314	−1.05193
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.89497	−0.332770	−0.166385	−	0.986061i	$-0.553209\pi$
$137$	−3.89497	−0.332770	−0.166385	+	0.986061i	$0.553209\pi$
$138$	0	0
$139$	−11.8799	−1.00764	−0.503819	−	0.863809i	$-0.668072\pi$
$139$	−11.8799	−1.00764	−0.503819	+	0.863809i	$0.668072\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−14.4842	−1.21123
$144$	0	0
$145$	37.2149	3.09053
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.607903	0.0498013	0.0249007	−	0.999690i	$-0.492073\pi$
$149$	0.607903	0.0498013	0.0249007	+	0.999690i	$0.492073\pi$
$150$	0	0
$151$	−18.9340	−1.54083	−0.770415	−	0.637543i	$-0.779951\pi$
$151$	−18.9340	−1.54083	−0.770415	+	0.637543i	$0.779951\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.2792	1.38790
$156$	0	0
$157$	−17.6682	−1.41008	−0.705040	−	0.709168i	$-0.749071\pi$
$157$	−17.6682	−1.41008	−0.705040	+	0.709168i	$0.749071\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	11.9863	0.944656
$162$	0	0
$163$	−17.8366	−1.39707	−0.698534	−	0.715577i	$-0.746164\pi$
$163$	−17.8366	−1.39707	−0.698534	+	0.715577i	$0.746164\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.87561	−0.145139	−0.0725693	−	0.997363i	$-0.523120\pi$
$167$	−1.87561	−0.145139	−0.0725693	+	0.997363i	$0.523120\pi$
$168$	0	0
$169$	5.56306	0.427928
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.83264	−0.215361	−0.107681	−	0.994186i	$-0.534342\pi$
$173$	−2.83264	−0.215361	−0.107681	+	0.994186i	$0.534342\pi$
$174$	0	0
$175$	19.1277	1.44592
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.63502	−0.720155	−0.360078	−	0.932922i	$-0.617250\pi$
$179$	−9.63502	−0.720155	−0.360078	+	0.932922i	$0.617250\pi$
$180$	0	0
$181$	−14.0440	−1.04388	−0.521941	−	0.852982i	$-0.674792\pi$
$181$	−14.0440	−1.04388	−0.521941	+	0.852982i	$0.674792\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−22.7391	−1.67181
$186$	0	0
$187$	−11.9611	−0.874685
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.7727	−0.851846	−0.425923	−	0.904759i	$-0.640050\pi$
$191$	−11.7727	−0.851846	−0.425923	+	0.904759i	$0.640050\pi$
$192$	0	0
$193$	−21.4497	−1.54399	−0.771993	−	0.635631i	$-0.780740\pi$
$193$	−21.4497	−1.54399	−0.771993	+	0.635631i	$0.780740\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.87874	−0.703831	−0.351916	−	0.936032i	$-0.614470\pi$
$197$	−9.87874	−0.703831	−0.351916	+	0.936032i	$0.614470\pi$
$198$	0	0
$199$	−15.5924	−1.10531	−0.552656	−	0.833409i	$-0.686386\pi$
$199$	−15.5924	−1.10531	−0.552656	+	0.833409i	$0.686386\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−18.8894	−1.32577
$204$	0	0
$205$	−27.0349	−1.88820
$206$	0	0
$207$	0	0
$208$	0	0
$209$	20.8882	1.44486
$210$	0	0
$211$	10.5994	0.729695	0.364848	−	0.931067i	$-0.381121\pi$
$211$	10.5994	0.729695	0.364848	+	0.931067i	$0.381121\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	30.7229	2.09529
$216$	0	0
$217$	−8.77049	−0.595380
$218$	0	0
$219$	0	0
$220$	0	0
$221$	15.3295	1.03117
$222$	0	0
$223$	21.2459	1.42273	0.711365	−	0.702822i	$-0.248077\pi$
$223$	21.2459	1.42273	0.711365	+	0.702822i	$0.248077\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.67282	−0.111029	−0.0555144	−	0.998458i	$-0.517680\pi$
$227$	−1.67282	−0.111029	−0.0555144	+	0.998458i	$0.517680\pi$
$228$	0	0
$229$	13.6746	0.903641	0.451821	−	0.892109i	$-0.350775\pi$
$229$	13.6746	0.903641	0.451821	+	0.892109i	$0.350775\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.0677	−0.987120	−0.493560	−	0.869712i	$-0.664305\pi$
$233$	−15.0677	−0.987120	−0.493560	+	0.869712i	$0.664305\pi$
$234$	0	0
$235$	25.6876	1.67568
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.13774	−0.202963	−0.101482	−	0.994837i	$-0.532358\pi$
$239$	−3.13774	−0.202963	−0.101482	+	0.994837i	$0.532358\pi$
$240$	0	0
$241$	21.0347	1.35497	0.677483	−	0.735538i	$-0.263071\pi$
$241$	21.0347	1.35497	0.677483	+	0.735538i	$0.263071\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.2627	0.783435
$246$	0	0
$247$	−26.7704	−1.70336
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.7003	−1.49595	−0.747976	−	0.663726i	$-0.768974\pi$
$251$	−23.7003	−1.49595	−0.747976	+	0.663726i	$0.768974\pi$
$252$	0	0
$253$	−20.6383	−1.29752
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.78673	0.548101	0.274050	−	0.961715i	$-0.411636\pi$
$257$	8.78673	0.548101	0.274050	+	0.961715i	$0.411636\pi$
$258$	0	0
$259$	11.5418	0.717173
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.33838	−0.575829	−0.287915	−	0.957656i	$-0.592962\pi$
$263$	−9.33838	−0.575829	−0.287915	+	0.957656i	$0.592962\pi$
$264$	0	0
$265$	3.61666	0.222169
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.1829	−1.59640	−0.798198	−	0.602395i	$-0.794213\pi$
$269$	−26.1829	−1.59640	−0.798198	+	0.602395i	$0.794213\pi$
$270$	0	0
$271$	17.8483	1.08421	0.542103	−	0.840312i	$-0.317628\pi$
$271$	17.8483	1.08421	0.542103	+	0.840312i	$0.317628\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−32.9344	−1.98602
$276$	0	0
$277$	−14.1906	−0.852631	−0.426315	−	0.904575i	$-0.640189\pi$
$277$	−14.1906	−0.852631	−0.426315	+	0.904575i	$0.640189\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.1483	−0.605397	−0.302699	−	0.953086i	$-0.597888\pi$
$281$	−10.1483	−0.605397	−0.302699	+	0.953086i	$0.597888\pi$
$282$	0	0
$283$	−10.5364	−0.626327	−0.313163	−	0.949699i	$-0.601389\pi$
$283$	−10.5364	−0.626327	−0.313163	+	0.949699i	$0.601389\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.7223	0.809999
$288$	0	0
$289$	−4.34085	−0.255344
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.49114	−0.203954	−0.101977	−	0.994787i	$-0.532517\pi$
$293$	−3.49114	−0.203954	−0.101977	+	0.994787i	$0.532517\pi$
$294$	0	0
$295$	−18.6341	−1.08492
$296$	0	0
$297$	0	0
$298$	0	0
$299$	26.4502	1.52965
$300$	0	0
$301$	−15.5942	−0.898834
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−37.2149	−2.13092
$306$	0	0
$307$	−4.48031	−0.255705	−0.127852	−	0.991793i	$-0.540808\pi$
$307$	−4.48031	−0.255705	−0.127852	+	0.991793i	$0.540808\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.9209	−1.52654	−0.763271	−	0.646079i	$-0.776408\pi$
$311$	−26.9209	−1.52654	−0.763271	+	0.646079i	$0.776408\pi$
$312$	0	0
$313$	15.7811	0.892000	0.446000	−	0.895033i	$-0.352848\pi$
$313$	15.7811	0.892000	0.446000	+	0.895033i	$0.352848\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.6425	0.766242	0.383121	−	0.923698i	$-0.374849\pi$
$317$	13.6425	0.766242	0.383121	+	0.923698i	$0.374849\pi$
$318$	0	0
$319$	32.5241	1.82100
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−22.1071	−1.23007
$324$	0	0
$325$	42.2089	2.34133
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−13.0384	−0.718831
$330$	0	0
$331$	−4.95696	−0.272459	−0.136229	−	0.990677i	$-0.543498\pi$
$331$	−4.95696	−0.272459	−0.136229	+	0.990677i	$0.543498\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−49.2876	−2.69287
$336$	0	0
$337$	−7.29444	−0.397354	−0.198677	−	0.980065i	$-0.563664\pi$
$337$	−7.29444	−0.397354	−0.198677	+	0.980065i	$0.563664\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15.1012	0.817776
$342$	0	0
$343$	−19.8915	−1.07404
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.57397	−0.0844949	−0.0422474	−	0.999107i	$-0.513452\pi$
$347$	−1.57397	−0.0844949	−0.0422474	+	0.999107i	$0.513452\pi$
$348$	0	0
$349$	26.8257	1.43595	0.717973	−	0.696071i	$-0.245070\pi$
$349$	26.8257	1.43595	0.717973	+	0.696071i	$0.245070\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.4802	0.717480	0.358740	−	0.933438i	$-0.383206\pi$
$353$	13.4802	0.717480	0.358740	+	0.933438i	$0.383206\pi$
$354$	0	0
$355$	−33.8072	−1.79430
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.5146	−0.924387	−0.462194	−	0.886779i	$-0.652938\pi$
$359$	−17.5146	−0.924387	−0.462194	+	0.886779i	$0.652938\pi$
$360$	0	0
$361$	19.6065	1.03192
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.2841	1.06172
$366$	0	0
$367$	−35.0719	−1.83074	−0.915368	−	0.402618i	$-0.868100\pi$
$367$	−35.0719	−1.83074	−0.915368	+	0.402618i	$0.868100\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.83572	−0.0953061
$372$	0	0
$373$	−10.1613	−0.526134	−0.263067	−	0.964778i	$-0.584734\pi$
$373$	−10.1613	−0.526134	−0.263067	+	0.964778i	$0.584734\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−41.6831	−2.14679
$378$	0	0
$379$	−3.67281	−0.188659	−0.0943297	−	0.995541i	$-0.530071\pi$
$379$	−3.67281	−0.188659	−0.0943297	+	0.995541i	$0.530071\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.03677	0.104074	0.0520369	−	0.998645i	$-0.483429\pi$
$383$	2.03677	0.104074	0.0520369	+	0.998645i	$0.483429\pi$
$384$	0	0
$385$	25.2485	1.28678
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.2283	1.83685	0.918424	−	0.395598i	$-0.129463\pi$
$389$	36.2283	1.83685	0.918424	+	0.395598i	$0.129463\pi$
$390$	0	0
$391$	21.8427	1.10463
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−50.0453	−2.51805
$396$	0	0
$397$	9.63505	0.483569	0.241785	−	0.970330i	$-0.422267\pi$
$397$	9.63505	0.483569	0.241785	+	0.970330i	$0.422267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.2376	1.01062	0.505310	−	0.862938i	$-0.331378\pi$
$401$	20.2376	1.01062	0.505310	+	0.862938i	$0.331378\pi$
$402$	0	0
$403$	−19.3538	−0.964082
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−19.8729	−0.985064
$408$	0	0
$409$	−17.7170	−0.876049	−0.438025	−	0.898963i	$-0.644322\pi$
$409$	−17.7170	−0.876049	−0.438025	+	0.898963i	$0.644322\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.45818	0.465407
$414$	0	0
$415$	−3.84665	−0.188824
$416$	0	0
$417$	0	0
$418$	0	0
$419$	25.9831	1.26936	0.634679	−	0.772776i	$-0.281132\pi$
$419$	25.9831	1.26936	0.634679	+	0.772776i	$0.281132\pi$
$420$	0	0
$421$	32.7959	1.59837	0.799186	−	0.601084i	$-0.205264\pi$
$421$	32.7959	1.59837	0.799186	+	0.601084i	$0.205264\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	34.8563	1.69078
$426$	0	0
$427$	18.8894	0.914120
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.6017	−1.08869	−0.544343	−	0.838862i	$-0.683221\pi$
$431$	−22.6017	−1.08869	−0.544343	+	0.838862i	$0.683221\pi$
$432$	0	0
$433$	−13.9384	−0.669837	−0.334919	−	0.942247i	$-0.608709\pi$
$433$	−13.9384	−0.669837	−0.334919	+	0.942247i	$0.608709\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−38.1447	−1.82471
$438$	0	0
$439$	−0.493599	−0.0235582	−0.0117791	−	0.999931i	$-0.503749\pi$
$439$	−0.493599	−0.0235582	−0.0117791	+	0.999931i	$0.503749\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.85429	0.135611	0.0678056	−	0.997699i	$-0.478400\pi$
$443$	2.85429	0.135611	0.0678056	+	0.997699i	$0.478400\pi$
$444$	0	0
$445$	7.29401	0.345769
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.1490	−0.762118	−0.381059	−	0.924551i	$-0.624440\pi$
$449$	−16.1490	−0.762118	−0.381059	+	0.924551i	$0.624440\pi$
$450$	0	0
$451$	−23.6273	−1.11256
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−32.3586	−1.51699
$456$	0	0
$457$	16.1732	0.756552	0.378276	−	0.925693i	$-0.376517\pi$
$457$	16.1732	0.756552	0.378276	+	0.925693i	$0.376517\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16.0789	−0.748868	−0.374434	−	0.927253i	$-0.622163\pi$
$461$	−16.0789	−0.748868	−0.374434	+	0.927253i	$0.622163\pi$
$462$	0	0
$463$	−6.53936	−0.303910	−0.151955	−	0.988387i	$-0.548557\pi$
$463$	−6.53936	−0.303910	−0.151955	+	0.988387i	$0.548557\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.9651	1.52544	0.762720	−	0.646728i	$-0.223863\pi$
$467$	32.9651	1.52544	0.762720	+	0.646728i	$0.223863\pi$
$468$	0	0
$469$	25.0172	1.15519
$470$	0	0
$471$	0	0
$472$	0	0
$473$	26.8504	1.23458
$474$	0	0
$475$	−60.8709	−2.79295
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.1201	−1.19346	−0.596729	−	0.802443i	$-0.703533\pi$
$479$	−26.1201	−1.19346	−0.596729	+	0.802443i	$0.703533\pi$
$480$	0	0
$481$	25.4693	1.16130
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−21.5008	−0.976301
$486$	0	0
$487$	−19.4977	−0.883524	−0.441762	−	0.897132i	$-0.645646\pi$
$487$	−19.4977	−0.883524	−0.441762	+	0.897132i	$0.645646\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10.3626	−0.467658	−0.233829	−	0.972278i	$-0.575126\pi$
$491$	−10.3626	−0.467658	−0.233829	+	0.972278i	$0.575126\pi$
$492$	0	0
$493$	−34.4221	−1.55029
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.1597	0.769718
$498$	0	0
$499$	28.4833	1.27509	0.637543	−	0.770415i	$-0.279951\pi$
$499$	28.4833	1.27509	0.637543	+	0.770415i	$0.279951\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−27.6186	−1.23145	−0.615726	−	0.787960i	$-0.711137\pi$
$503$	−27.6186	−1.23145	−0.615726	+	0.787960i	$0.711137\pi$
$504$	0	0
$505$	−38.1842	−1.69918
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.4135	0.638868	0.319434	−	0.947608i	$-0.396507\pi$
$509$	14.4135	0.638868	0.319434	+	0.947608i	$0.396507\pi$
$510$	0	0
$511$	−10.2957	−0.455456
$512$	0	0
$513$	0	0
$514$	0	0
$515$	65.8437	2.90142
$516$	0	0
$517$	22.4498	0.987340
$518$	0	0
$519$	0	0
$520$	0	0
$521$	34.2567	1.50081	0.750407	−	0.660976i	$-0.229858\pi$
$521$	34.2567	1.50081	0.750407	+	0.660976i	$0.229858\pi$
$522$	0	0
$523$	−38.1449	−1.66796	−0.833979	−	0.551796i	$-0.813943\pi$
$523$	−38.1449	−1.66796	−0.833979	+	0.551796i	$0.813943\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.9825	−0.696207
$528$	0	0
$529$	14.6884	0.638627
$530$	0	0
$531$	0	0
$532$	0	0
$533$	30.2808	1.31161
$534$	0	0
$535$	19.0975	0.825657
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.7170	0.461615
$540$	0	0
$541$	−10.0253	−0.431020	−0.215510	−	0.976502i	$-0.569141\pi$
$541$	−10.0253	−0.431020	−0.215510	+	0.976502i	$0.569141\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	78.0347	3.34264
$546$	0	0
$547$	16.9306	0.723902	0.361951	−	0.932197i	$-0.382111\pi$
$547$	16.9306	0.723902	0.361951	+	0.932197i	$0.382111\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	60.1125	2.56088
$552$	0	0
$553$	25.4017	1.08019
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.9200	−1.26775	−0.633875	−	0.773435i	$-0.718537\pi$
$557$	−29.9200	−1.26775	−0.633875	+	0.773435i	$0.718537\pi$
$558$	0	0
$559$	−34.4116	−1.45546
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.9551	−1.22031	−0.610156	−	0.792281i	$-0.708893\pi$
$563$	−28.9551	−1.22031	−0.610156	+	0.792281i	$0.708893\pi$
$564$	0	0
$565$	−18.0439	−0.759113
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.68034	−0.0704436	−0.0352218	−	0.999380i	$-0.511214\pi$
$569$	−1.68034	−0.0704436	−0.0352218	+	0.999380i	$0.511214\pi$
$570$	0	0
$571$	34.5579	1.44620	0.723101	−	0.690742i	$-0.242716\pi$
$571$	34.5579	1.44620	0.723101	+	0.690742i	$0.242716\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	60.1428	2.50813
$576$	0	0
$577$	21.8924	0.911391	0.455696	−	0.890136i	$-0.349391\pi$
$577$	21.8924	0.911391	0.455696	+	0.890136i	$0.349391\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.95246	0.0810018
$582$	0	0
$583$	3.16079	0.130906
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.6431	−0.852033	−0.426017	−	0.904715i	$-0.640083\pi$
$587$	−20.6431	−0.852033	−0.426017	+	0.904715i	$0.640083\pi$
$588$	0	0
$589$	27.9108	1.15004
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.3731	0.508101	0.254051	−	0.967191i	$-0.418237\pi$
$593$	12.3731	0.508101	0.254051	+	0.967191i	$0.418237\pi$
$594$	0	0
$595$	−26.7219	−1.09549
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.1498	−0.945874	−0.472937	−	0.881096i	$-0.656806\pi$
$599$	−23.1498	−0.945874	−0.472937	+	0.881096i	$0.656806\pi$
$600$	0	0
$601$	3.86867	0.157806	0.0789031	−	0.996882i	$-0.474858\pi$
$601$	3.86867	0.157806	0.0789031	+	0.996882i	$0.474858\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.16019	−0.0471686
$606$	0	0
$607$	0.269972	0.0109578	0.00547892	−	0.999985i	$-0.498256\pi$
$607$	0.269972	0.0109578	0.00547892	+	0.999985i	$0.498256\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−28.7718	−1.16398
$612$	0	0
$613$	−5.98773	−0.241842	−0.120921	−	0.992662i	$-0.538585\pi$
$613$	−5.98773	−0.241842	−0.120921	+	0.992662i	$0.538585\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.35394	−0.255800	−0.127900	−	0.991787i	$-0.540824\pi$
$617$	−6.35394	−0.255800	−0.127900	+	0.991787i	$0.540824\pi$
$618$	0	0
$619$	40.5269	1.62891	0.814457	−	0.580224i	$-0.197035\pi$
$619$	40.5269	1.62891	0.814457	+	0.580224i	$0.197035\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.70225	−0.148328
$624$	0	0
$625$	21.9918	0.879671
$626$	0	0
$627$	0	0
$628$	0	0
$629$	21.0326	0.838626
$630$	0	0
$631$	5.95010	0.236870	0.118435	−	0.992962i	$-0.462212\pi$
$631$	5.95010	0.236870	0.118435	+	0.992962i	$0.462212\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−20.3994	−0.809525
$636$	0	0
$637$	−13.7350	−0.544200
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20.4627	−0.808229	−0.404114	−	0.914708i	$-0.632420\pi$
$641$	−20.4627	−0.808229	−0.404114	+	0.914708i	$0.632420\pi$
$642$	0	0
$643$	30.7374	1.21217	0.606083	−	0.795401i	$-0.292740\pi$
$643$	30.7374	1.21217	0.606083	+	0.795401i	$0.292740\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.2440	0.835188	0.417594	−	0.908634i	$-0.362873\pi$
$647$	21.2440	0.835188	0.417594	+	0.908634i	$0.362873\pi$
$648$	0	0
$649$	−16.2853	−0.639253
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.26029	−0.244984	−0.122492	−	0.992470i	$-0.539089\pi$
$653$	−6.26029	−0.244984	−0.122492	+	0.992470i	$0.539089\pi$
$654$	0	0
$655$	13.7557	0.537480
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.2457	0.438071	0.219035	−	0.975717i	$-0.429709\pi$
$659$	11.2457	0.438071	0.219035	+	0.975717i	$0.429709\pi$
$660$	0	0
$661$	−37.9620	−1.47655	−0.738276	−	0.674499i	$-0.764360\pi$
$661$	−37.9620	−1.47655	−0.738276	+	0.674499i	$0.764360\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	46.6654	1.80961
$666$	0	0
$667$	−59.3935	−2.29972
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−32.5241	−1.25558
$672$	0	0
$673$	19.4720	0.750590	0.375295	−	0.926905i	$-0.377541\pi$
$673$	19.4720	0.750590	0.375295	+	0.926905i	$0.377541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−45.4654	−1.74738	−0.873689	−	0.486485i	$-0.838279\pi$
$677$	−45.4654	−1.74738	−0.873689	+	0.486485i	$0.838279\pi$
$678$	0	0
$679$	10.9133	0.418813
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13.8810	0.531143	0.265572	−	0.964091i	$-0.414439\pi$
$683$	13.8810	0.531143	0.265572	+	0.964091i	$0.414439\pi$
$684$	0	0
$685$	14.9826	0.572455
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.05088	−0.154326
$690$	0	0
$691$	36.3175	1.38158	0.690791	−	0.723055i	$-0.257263\pi$
$691$	36.3175	1.38158	0.690791	+	0.723055i	$0.257263\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	45.6977	1.73341
$696$	0	0
$697$	25.0061	0.947172
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13.8289	−0.522310	−0.261155	−	0.965297i	$-0.584103\pi$
$701$	−13.8289	−0.522310	−0.261155	+	0.965297i	$0.584103\pi$
$702$	0	0
$703$	−36.7300	−1.38530
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.3814	0.728911
$708$	0	0
$709$	47.6187	1.78836	0.894179	−	0.447709i	$-0.147760\pi$
$709$	47.6187	1.78836	0.894179	+	0.447709i	$0.147760\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−27.5769	−1.03276
$714$	0	0
$715$	55.7157	2.08365
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.8666	0.442551	0.221276	−	0.975211i	$-0.428978\pi$
$719$	11.8666	0.442551	0.221276	+	0.975211i	$0.428978\pi$
$720$	0	0
$721$	−33.4206	−1.24465
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−94.7795	−3.52002
$726$	0	0
$727$	−40.3613	−1.49692	−0.748459	−	0.663181i	$-0.769206\pi$
$727$	−40.3613	−1.49692	−0.748459	+	0.663181i	$0.769206\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−28.4173	−1.05105
$732$	0	0
$733$	−20.1559	−0.744474	−0.372237	−	0.928138i	$-0.621409\pi$
$733$	−20.1559	−0.744474	−0.372237	+	0.928138i	$0.621409\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.0751	−1.58669
$738$	0	0
$739$	−14.2713	−0.524979	−0.262490	−	0.964935i	$-0.584544\pi$
$739$	−14.2713	−0.524979	−0.262490	+	0.964935i	$0.584544\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.60118	0.352233	0.176117	−	0.984369i	$-0.443646\pi$
$743$	9.60118	0.352233	0.176117	+	0.984369i	$0.443646\pi$
$744$	0	0
$745$	−2.33839	−0.0856719
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.69341	−0.354190
$750$	0	0
$751$	23.5839	0.860588	0.430294	−	0.902689i	$-0.358410\pi$
$751$	23.5839	0.860588	0.430294	+	0.902689i	$0.358410\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	72.8326	2.65065
$756$	0	0
$757$	34.7513	1.26306	0.631529	−	0.775352i	$-0.282428\pi$
$757$	34.7513	1.26306	0.631529	+	0.775352i	$0.282428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.3425	1.06366	0.531832	−	0.846850i	$-0.321504\pi$
$761$	29.3425	1.06366	0.531832	+	0.846850i	$0.321504\pi$
$762$	0	0
$763$	−39.6085	−1.43392
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.8713	0.753620
$768$	0	0
$769$	−23.0896	−0.832634	−0.416317	−	0.909220i	$-0.636679\pi$
$769$	−23.0896	−0.832634	−0.416317	+	0.909220i	$0.636679\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.8924	0.535642	0.267821	−	0.963469i	$-0.413696\pi$
$773$	14.8924	0.535642	0.267821	+	0.963469i	$0.413696\pi$
$774$	0	0
$775$	−44.0069	−1.58078
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−43.6690	−1.56460
$780$	0	0
$781$	−29.5459	−1.05724
$782$	0	0
$783$	0	0
$784$	0	0
$785$	67.9635	2.42572
$786$	0	0
$787$	−27.6419	−0.985329	−0.492664	−	0.870219i	$-0.663977\pi$
$787$	−27.6419	−0.985329	−0.492664	+	0.870219i	$0.663977\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.15864	0.325644
$792$	0	0
$793$	41.6831	1.48021
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.0693	0.746313	0.373156	−	0.927768i	$-0.378275\pi$
$797$	21.0693	0.746313	0.373156	+	0.927768i	$0.378275\pi$
$798$	0	0
$799$	−23.7599	−0.840564
$800$	0	0
$801$	0	0
$802$	0	0
$803$	17.7274	0.625586
$804$	0	0
$805$	−46.1072	−1.62507
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.7319	0.728894	0.364447	−	0.931224i	$-0.381258\pi$
$809$	20.7319	0.728894	0.364447	+	0.931224i	$0.381258\pi$
$810$	0	0
$811$	6.37186	0.223746	0.111873	−	0.993723i	$-0.464315\pi$
$811$	6.37186	0.223746	0.111873	+	0.993723i	$0.464315\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	68.6110	2.40334
$816$	0	0
$817$	49.6261	1.73620
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.34818	0.326254	0.163127	−	0.986605i	$-0.447842\pi$
$821$	9.34818	0.326254	0.163127	+	0.986605i	$0.447842\pi$
$822$	0	0
$823$	−41.2177	−1.43676	−0.718380	−	0.695651i	$-0.755116\pi$
$823$	−41.2177	−1.43676	−0.718380	+	0.695651i	$0.755116\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−51.7970	−1.80116	−0.900579	−	0.434692i	$-0.856857\pi$
$827$	−51.7970	−1.80116	−0.900579	+	0.434692i	$0.856857\pi$
$828$	0	0
$829$	−25.3785	−0.881432	−0.440716	−	0.897647i	$-0.645275\pi$
$829$	−25.3785	−0.881432	−0.440716	+	0.897647i	$0.645275\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11.3424	−0.392992
$834$	0	0
$835$	7.21479	0.249678
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27.7612	−0.958423	−0.479211	−	0.877699i	$-0.659077\pi$
$839$	−27.7612	−0.958423	−0.479211	+	0.877699i	$0.659077\pi$
$840$	0	0
$841$	64.5986	2.22754
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−21.3991	−0.736153
$846$	0	0
$847$	0.588886	0.0202343
$848$	0	0
$849$	0	0
$850$	0	0
$851$	36.2907	1.24403
$852$	0	0
$853$	30.2122	1.03445	0.517223	−	0.855851i	$-0.326966\pi$
$853$	30.2122	1.03445	0.517223	+	0.855851i	$0.326966\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.9691	1.67275	0.836376	−	0.548156i	$-0.184670\pi$
$857$	48.9691	1.67275	0.836376	+	0.548156i	$0.184670\pi$
$858$	0	0
$859$	21.9356	0.748434	0.374217	−	0.927341i	$-0.377911\pi$
$859$	21.9356	0.748434	0.374217	+	0.927341i	$0.377911\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.2136	−0.892322	−0.446161	−	0.894953i	$-0.647209\pi$
$863$	−26.2136	−0.892322	−0.446161	+	0.894953i	$0.647209\pi$
$864$	0	0
$865$	10.8962	0.370480
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−43.7372	−1.48368
$870$	0	0
$871$	55.2053	1.87056
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−36.0253	−1.21788
$876$	0	0
$877$	−31.1584	−1.05214	−0.526072	−	0.850440i	$-0.676336\pi$
$877$	−31.1584	−1.05214	−0.526072	+	0.850440i	$0.676336\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.0918	−0.542148	−0.271074	−	0.962559i	$-0.587379\pi$
$881$	−16.0918	−0.542148	−0.271074	+	0.962559i	$0.587379\pi$
$882$	0	0
$883$	−27.5401	−0.926800	−0.463400	−	0.886149i	$-0.653371\pi$
$883$	−27.5401	−0.926800	−0.463400	+	0.886149i	$0.653371\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23.8811	−0.801848	−0.400924	−	0.916111i	$-0.631311\pi$
$887$	−23.8811	−0.801848	−0.400924	+	0.916111i	$0.631311\pi$
$888$	0	0
$889$	10.3542	0.347269
$890$	0	0
$891$	0	0
$892$	0	0
$893$	41.4927	1.38850
$894$	0	0
$895$	37.0625	1.23886
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43.4586	1.44943
$900$	0	0
$901$	−3.34524	−0.111446
$902$	0	0
$903$	0	0
$904$	0	0
$905$	54.0223	1.79576
$906$	0	0
$907$	18.5303	0.615289	0.307645	−	0.951501i	$-0.400459\pi$
$907$	18.5303	0.615289	0.307645	+	0.951501i	$0.400459\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44.0016	−1.45784	−0.728920	−	0.684599i	$-0.759977\pi$
$911$	−44.0016	−1.45784	−0.728920	+	0.684599i	$0.759977\pi$
$912$	0	0
$913$	−3.36179	−0.111259
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.98206	−0.230568
$918$	0	0
$919$	21.0668	0.694931	0.347465	−	0.937693i	$-0.387042\pi$
$919$	21.0668	0.694931	0.347465	+	0.937693i	$0.387042\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	37.8662	1.24638
$924$	0	0
$925$	57.9123	1.90415
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.43686	−0.112760	−0.0563798	−	0.998409i	$-0.517956\pi$
$929$	−3.43686	−0.112760	−0.0563798	+	0.998409i	$0.517956\pi$
$930$	0	0
$931$	19.8077	0.649171
$932$	0	0
$933$	0	0
$934$	0	0
$935$	46.0103	1.50470
$936$	0	0
$937$	21.9722	0.717800	0.358900	−	0.933376i	$-0.383152\pi$
$937$	21.9722	0.717800	0.358900	+	0.933376i	$0.383152\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−43.3096	−1.41185	−0.705926	−	0.708286i	$-0.749469\pi$
$941$	−43.3096	−1.41185	−0.705926	+	0.708286i	$0.749469\pi$
$942$	0	0
$943$	43.1466	1.40505
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.9590	1.26600	0.632999	−	0.774153i	$-0.281824\pi$
$947$	38.9590	1.26600	0.632999	+	0.774153i	$0.281824\pi$
$948$	0	0
$949$	−22.7195	−0.737507
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.1725	0.523877	0.261939	−	0.965085i	$-0.415638\pi$
$953$	16.1725	0.523877	0.261939	+	0.965085i	$0.415638\pi$
$954$	0	0
$955$	45.2856	1.46541
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.60478	−0.245571
$960$	0	0
$961$	−10.8218	−0.349089
$962$	0	0
$963$	0	0
$964$	0	0
$965$	82.5096	2.65608
$966$	0	0
$967$	−7.34049	−0.236054	−0.118027	−	0.993010i	$-0.537657\pi$
$967$	−7.34049	−0.236054	−0.118027	+	0.993010i	$0.537657\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.6682	0.952097	0.476049	−	0.879419i	$-0.342069\pi$
$971$	29.6682	0.952097	0.476049	+	0.879419i	$0.342069\pi$
$972$	0	0
$973$	−23.1950	−0.743598
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.1712	0.485369	0.242684	−	0.970105i	$-0.421972\pi$
$977$	15.1712	0.485369	0.242684	+	0.970105i	$0.421972\pi$
$978$	0	0
$979$	6.37462	0.203734
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.3016	0.743206	0.371603	−	0.928392i	$-0.378808\pi$
$983$	23.3016	0.743206	0.371603	+	0.928392i	$0.378808\pi$
$984$	0	0
$985$	38.0000	1.21078
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−49.0325	−1.55914
$990$	0	0
$991$	−17.2039	−0.546501	−0.273251	−	0.961943i	$-0.588099\pi$
$991$	−17.2039	−0.546501	−0.273251	+	0.961943i	$0.588099\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	59.9783	1.90144
$996$	0	0
$997$	22.5624	0.714559	0.357279	−	0.933998i	$-0.383704\pi$
$997$	22.5624	0.714559	0.357279	+	0.933998i	$0.383704\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5976.2.a.s.1.2		8
3.2	odd	2		664.2.a.f.1.4	✓	8
12.11	even	2		1328.2.a.n.1.5		8
24.5	odd	2		5312.2.a.br.1.5		8
24.11	even	2		5312.2.a.bs.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
664.2.a.f.1.4	✓	8	3.2	odd	2
1328.2.a.n.1.5		8	12.11	even	2
5312.2.a.br.1.5		8	24.5	odd	2
5312.2.a.bs.1.4		8	24.11	even	2
5976.2.a.s.1.2		8	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 5976 = 2^{3} \cdot 3^{2} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5976.a (trivial)

\(f(q)\)	\(=\)	\(q-3.84665 q^{5} +1.95246 q^{7} +O(q^{10})\)	\(q-3.84665 q^{5} +1.95246 q^{7} -3.36179 q^{11} +4.30849 q^{13} +3.55797 q^{17} -6.21341 q^{19} +6.13909 q^{23} +9.79670 q^{25} -9.67464 q^{29} -4.49202 q^{31} -7.51043 q^{35} +5.91141 q^{37} +7.02818 q^{41} -7.98694 q^{43} -6.67793 q^{47} -3.18789 q^{49} -0.940210 q^{53} +12.9316 q^{55} +4.84423 q^{59} +9.67464 q^{61} -16.5732 q^{65} +12.8131 q^{67} +8.78876 q^{71} -5.27320 q^{73} -6.56376 q^{77} +13.0101 q^{79} +1.00000 q^{83} -13.6863 q^{85} -1.89620 q^{89} +8.41216 q^{91} +23.9008 q^{95} +5.58949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 7 q^{5} + q^{7}+O(q^{10}) \) 8 * q - 7 * q^5 + q^7	\( 8 q - 7 q^{5} + q^{7} - 10 q^{11} + 7 q^{13} - 15 q^{17} + 7 q^{19} - 4 q^{23} + 27 q^{25} - 16 q^{29} + 3 q^{31} - 2 q^{35} + 8 q^{37} - 24 q^{41} - q^{43} + 2 q^{47} + 3 q^{49} - 7 q^{53} - 30 q^{55} + 2 q^{59} + 16 q^{61} - 2 q^{65} - 25 q^{67} - 8 q^{71} + 14 q^{73} + 7 q^{77} - 4 q^{79} + 8 q^{83} - 21 q^{85} - 20 q^{89} - 45 q^{91} - 8 q^{95} + 2 q^{97}+O(q^{100}) \) 8 * q - 7 * q^5 + q^7 - 10 * q^11 + 7 * q^13 - 15 * q^17 + 7 * q^19 - 4 * q^23 + 27 * q^25 - 16 * q^29 + 3 * q^31 - 2 * q^35 + 8 * q^37 - 24 * q^41 - q^43 + 2 * q^47 + 3 * q^49 - 7 * q^53 - 30 * q^55 + 2 * q^59 + 16 * q^61 - 2 * q^65 - 25 * q^67 - 8 * q^71 + 14 * q^73 + 7 * q^77 - 4 * q^79 + 8 * q^83 - 21 * q^85 - 20 * q^89 - 45 * q^91 - 8 * q^95 + 2 * q^97

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