LMFDB - Embedded newform 8712.2.a.ca.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8712,2,Mod(1,8712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, 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("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8712.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:	$69.5656702409$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.46224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 15x^{2} + 16x + 16 $ x^4 - 2x^3 - 15x^2 + 16*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2904)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.649694$ of defining polynomial
Character	$\chi$	$=$	8712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.649694 q^{5} +4.15674 q^{7} +O(q^{10})$	$q+0.649694 q^{5} +4.15674 q^{7} -6.42469 q^{13} -6.42469 q^{17} -2.85735 q^{19} +5.55000 q^{23} -4.57790 q^{25} +8.58940 q^{29} -1.64969 q^{31} +2.70061 q^{35} -2.55000 q^{37} -5.88879 q^{41} +7.01410 q^{43} +7.62881 q^{47} +10.2785 q^{49} -4.27851 q^{53} -1.55000 q^{59} -8.24264 q^{61} -4.17409 q^{65} +3.19969 q^{67} -7.77759 q^{71} -8.07085 q^{73} -12.0708 q^{79} +4.31349 q^{83} -4.17409 q^{85} -9.49908 q^{89} -26.7058 q^{91} -1.85641 q^{95} +7.59878 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q - 2 * q^5 - 4 * q^7	$ 4 q - 2 q^{5} - 4 q^{7} - 12 q^{13} - 12 q^{17} + 4 q^{23} + 14 q^{25} + 16 q^{29} - 2 q^{31} + 20 q^{35} + 8 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 18 q^{49} + 6 q^{53} + 12 q^{59} - 8 q^{61} - 12 q^{65} - 10 q^{67} + 24 q^{71} - 16 q^{73} - 32 q^{79} - 24 q^{83} - 12 q^{85} - 6 q^{89} - 24 q^{91} + 48 q^{95} + 12 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 - 4 * q^7 - 12 * q^13 - 12 * q^17 + 4 * q^23 + 14 * q^25 + 16 * q^29 - 2 * q^31 + 20 * q^35 + 8 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 18 * q^49 + 6 * q^53 + 12 * q^59 - 8 * q^61 - 12 * q^65 - 10 * q^67 + 24 * q^71 - 16 * q^73 - 32 * q^79 - 24 * q^83 - 12 * q^85 - 6 * q^89 - 24 * q^91 + 48 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.649694	0.290552	0.145276	−	0.989391i	$-0.453593\pi$
$5$	0.649694	0.290552	0.145276	+	0.989391i	$0.453593\pi$
$6$	0	0
$7$	4.15674	1.57110	0.785551	−	0.618798i	$-0.212380\pi$
$7$	4.15674	1.57110	0.785551	+	0.618798i	$0.212380\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.42469	−1.78189	−0.890944	−	0.454112i	$-0.849956\pi$
$13$	−6.42469	−1.78189	−0.890944	+	0.454112i	$0.849956\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.42469	−1.55822	−0.779108	−	0.626889i	$-0.784328\pi$
$17$	−6.42469	−1.55822	−0.779108	+	0.626889i	$0.784328\pi$
$18$	0	0
$19$	−2.85735	−0.655522	−0.327761	−	0.944761i	$-0.606294\pi$
$19$	−2.85735	−0.655522	−0.327761	+	0.944761i	$0.606294\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.55000	1.15725	0.578627	−	0.815592i	$-0.303589\pi$
$23$	5.55000	1.15725	0.578627	+	0.815592i	$0.303589\pi$
$24$	0	0
$25$	−4.57790	−0.915579
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.58940	1.59501	0.797506	−	0.603311i	$-0.206152\pi$
$29$	8.58940	1.59501	0.797506	+	0.603311i	$0.206152\pi$
$30$	0	0
$31$	−1.64969	−0.296294	−0.148147	−	0.988965i	$-0.547331\pi$
$31$	−1.64969	−0.296294	−0.148147	+	0.988965i	$0.547331\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.70061	0.456487
$36$	0	0
$37$	−2.55000	−0.419217	−0.209608	−	0.977785i	$-0.567219\pi$
$37$	−2.55000	−0.419217	−0.209608	+	0.977785i	$0.567219\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.88879	−0.919675	−0.459838	−	0.888003i	$-0.652092\pi$
$41$	−5.88879	−0.919675	−0.459838	+	0.888003i	$0.652092\pi$
$42$	0	0
$43$	7.01410	1.06964	0.534820	−	0.844966i	$-0.320379\pi$
$43$	7.01410	1.06964	0.534820	+	0.844966i	$0.320379\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.62881	1.11278	0.556389	−	0.830922i	$-0.312187\pi$
$47$	7.62881	1.11278	0.556389	+	0.830922i	$0.312187\pi$
$48$	0	0
$49$	10.2785	1.46836
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.27851	−0.587698	−0.293849	−	0.955852i	$-0.594936\pi$
$53$	−4.27851	−0.587698	−0.293849	+	0.955852i	$0.594936\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.55000	−0.201792	−0.100896	−	0.994897i	$-0.532171\pi$
$59$	−1.55000	−0.201792	−0.100896	+	0.994897i	$0.532171\pi$
$60$	0	0
$61$	−8.24264	−1.05536	−0.527681	−	0.849443i	$-0.676938\pi$
$61$	−8.24264	−1.05536	−0.527681	+	0.849443i	$0.676938\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.17409	−0.517732
$66$	0	0
$67$	3.19969	0.390904	0.195452	−	0.980713i	$-0.437383\pi$
$67$	3.19969	0.390904	0.195452	+	0.980713i	$0.437383\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.77759	−0.923030	−0.461515	−	0.887132i	$-0.652694\pi$
$71$	−7.77759	−0.923030	−0.461515	+	0.887132i	$0.652694\pi$
$72$	0	0
$73$	−8.07085	−0.944621	−0.472311	−	0.881432i	$-0.656580\pi$
$73$	−8.07085	−0.944621	−0.472311	+	0.881432i	$0.656580\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.0708	−1.35808	−0.679038	−	0.734103i	$-0.737603\pi$
$79$	−12.0708	−1.35808	−0.679038	+	0.734103i	$0.737603\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.31349	0.473467	0.236733	−	0.971575i	$-0.423923\pi$
$83$	4.31349	0.473467	0.236733	+	0.971575i	$0.423923\pi$
$84$	0	0
$85$	−4.17409	−0.452743
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.49908	−1.00690	−0.503450	−	0.864024i	$-0.667936\pi$
$89$	−9.49908	−1.00690	−0.503450	+	0.864024i	$0.667936\pi$
$90$	0	0
$91$	−26.7058	−2.79953
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.85641	−0.190463
$96$	0	0
$97$	7.59878	0.771539	0.385769	−	0.922595i	$-0.373936\pi$
$97$	7.59878	0.771539	0.385769	+	0.922595i	$0.373936\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.31349	−0.429208	−0.214604	−	0.976701i	$-0.568846\pi$
$101$	−4.31349	−0.429208	−0.214604	+	0.976701i	$0.568846\pi$
$102$	0	0
$103$	−10.4991	−1.03450	−0.517252	−	0.855833i	$-0.673045\pi$
$103$	−10.4991	−1.03450	−0.517252	+	0.855833i	$0.673045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.5411	1.98578	0.992890	−	0.119032i	$-0.0379791\pi$
$107$	20.5411	1.98578	0.992890	+	0.119032i	$0.0379791\pi$
$108$	0	0
$109$	−17.9596	−1.72022	−0.860111	−	0.510107i	$-0.829606\pi$
$109$	−17.9596	−1.72022	−0.860111	+	0.510107i	$0.829606\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.12789	0.482392	0.241196	−	0.970476i	$-0.422460\pi$
$113$	5.12789	0.482392	0.241196	+	0.970476i	$0.422460\pi$
$114$	0	0
$115$	3.60580	0.336243
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−26.7058	−2.44812
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.22270	−0.556576
$126$	0	0
$127$	−12.0708	−1.07111	−0.535557	−	0.844499i	$-0.679898\pi$
$127$	−12.0708	−1.07111	−0.535557	+	0.844499i	$0.679898\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.01410	0.612824	0.306412	−	0.951899i	$-0.400871\pi$
$131$	7.01410	0.612824	0.306412	+	0.951899i	$0.400871\pi$
$132$	0	0
$133$	−11.8773	−1.02989
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.2994	−1.13624	−0.568122	−	0.822945i	$-0.692330\pi$
$137$	−13.2994	−1.13624	−0.568122	+	0.822945i	$0.692330\pi$
$138$	0	0
$139$	−10.6847	−0.906262	−0.453131	−	0.891444i	$-0.649693\pi$
$139$	−10.6847	−0.906262	−0.453131	+	0.891444i	$0.649693\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	5.58049	0.463434
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−23.8311	−1.95232	−0.976160	−	0.217054i	$-0.930355\pi$
$149$	−23.8311	−1.95232	−0.976160	+	0.217054i	$0.930355\pi$
$150$	0	0
$151$	7.24169	0.589320	0.294660	−	0.955602i	$-0.404794\pi$
$151$	7.24169	0.589320	0.294660	+	0.955602i	$0.404794\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.07180	−0.0860888
$156$	0	0
$157$	0.0208794	0.00166636	0.000833178	−	1.00000i	$-0.499735\pi$
$157$	0.0208794	0.00166636	0.000833178		1.00000i	$0.499735\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	23.0699	1.81816
$162$	0	0
$163$	−14.4991	−1.13566	−0.567828	−	0.823147i	$-0.692216\pi$
$163$	−14.4991	−1.13566	−0.567828	+	0.823147i	$0.692216\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.9423	1.38842	0.694208	−	0.719774i	$-0.255755\pi$
$167$	17.9423	1.38842	0.694208	+	0.719774i	$0.255755\pi$
$168$	0	0
$169$	28.2767	2.17513
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.07180	0.0814872	0.0407436	−	0.999170i	$-0.487027\pi$
$173$	1.07180	0.0814872	0.0407436	+	0.999170i	$0.487027\pi$
$174$	0	0
$175$	−19.0291	−1.43847
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.32758	−0.697176	−0.348588	−	0.937276i	$-0.613339\pi$
$179$	−9.32758	−0.697176	−0.348588	+	0.937276i	$0.613339\pi$
$180$	0	0
$181$	7.30641	0.543081	0.271541	−	0.962427i	$-0.412467\pi$
$181$	7.30641	0.543081	0.271541	+	0.962427i	$0.412467\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.65672	−0.121804
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.0282	1.59390	0.796952	−	0.604043i	$-0.206444\pi$
$191$	22.0282	1.59390	0.796952	+	0.604043i	$0.206444\pi$
$192$	0	0
$193$	2.04554	0.147241	0.0736205	−	0.997286i	$-0.476545\pi$
$193$	2.04554	0.147241	0.0736205	+	0.997286i	$0.476545\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.0029	−1.71013	−0.855067	−	0.518517i	$-0.826484\pi$
$197$	−24.0029	−1.71013	−0.855067	+	0.518517i	$0.826484\pi$
$198$	0	0
$199$	−20.3555	−1.44296	−0.721481	−	0.692434i	$-0.756538\pi$
$199$	−20.3555	−1.44296	−0.721481	+	0.692434i	$0.756538\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	35.7039	2.50593
$204$	0	0
$205$	−3.82591	−0.267213
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−22.7555	−1.56655	−0.783277	−	0.621673i	$-0.786453\pi$
$211$	−22.7555	−1.56655	−0.783277	+	0.621673i	$0.786453\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.55702	0.310786
$216$	0	0
$217$	−6.85735	−0.465508
$218$	0	0
$219$	0	0
$220$	0	0
$221$	41.2767	2.77657
$222$	0	0
$223$	2.15091	0.144035	0.0720177	−	0.997403i	$-0.477056\pi$
$223$	2.15091	0.144035	0.0720177	+	0.997403i	$0.477056\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.2276	−0.811574	−0.405787	−	0.913968i	$-0.633002\pi$
$227$	−12.2276	−0.811574	−0.405787	+	0.913968i	$0.633002\pi$
$228$	0	0
$229$	−3.32729	−0.219874	−0.109937	−	0.993939i	$-0.535065\pi$
$229$	−3.32729	−0.219874	−0.109937	+	0.993939i	$0.535065\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.6594	1.09139	0.545695	−	0.837984i	$-0.316266\pi$
$233$	16.6594	1.09139	0.545695	+	0.837984i	$0.316266\pi$
$234$	0	0
$235$	4.95640	0.323320
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.61472	0.169132	0.0845660	−	0.996418i	$-0.473050\pi$
$239$	2.61472	0.169132	0.0845660	+	0.996418i	$0.473050\pi$
$240$	0	0
$241$	−22.9282	−1.47694	−0.738468	−	0.674289i	$-0.764450\pi$
$241$	−22.9282	−1.47694	−0.738468	+	0.674289i	$0.764450\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.67789	0.426635
$246$	0	0
$247$	18.3576	1.16807
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.37821	−0.339469	−0.169735	−	0.985490i	$-0.554291\pi$
$251$	−5.37821	−0.339469	−0.169735	+	0.985490i	$0.554291\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.9285	1.18073	0.590364	−	0.807137i	$-0.298984\pi$
$257$	18.9285	1.18073	0.590364	+	0.807137i	$0.298984\pi$
$258$	0	0
$259$	−10.5997	−0.658632
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−21.7147	−1.33899	−0.669493	−	0.742818i	$-0.733489\pi$
$263$	−21.7147	−1.33899	−0.669493	+	0.742818i	$0.733489\pi$
$264$	0	0
$265$	−2.77972	−0.170757
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.04907	−0.185905	−0.0929526	−	0.995671i	$-0.529631\pi$
$269$	−3.04907	−0.185905	−0.0929526	+	0.995671i	$0.529631\pi$
$270$	0	0
$271$	2.51288	0.152647	0.0763234	−	0.997083i	$-0.475682\pi$
$271$	2.51288	0.152647	0.0763234	+	0.997083i	$0.475682\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.0756	−0.785635	−0.392818	−	0.919616i	$-0.628500\pi$
$277$	−13.0756	−0.785635	−0.392818	+	0.919616i	$0.628500\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.6270	1.23050	0.615251	−	0.788331i	$-0.289055\pi$
$281$	20.6270	1.23050	0.615251	+	0.788331i	$0.289055\pi$
$282$	0	0
$283$	−16.7997	−0.998636	−0.499318	−	0.866419i	$-0.666416\pi$
$283$	−16.7997	−0.998636	−0.499318	+	0.866419i	$0.666416\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−24.4782	−1.44490
$288$	0	0
$289$	24.2767	1.42804
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.20412	0.187187	0.0935934	−	0.995611i	$-0.470165\pi$
$293$	3.20412	0.187187	0.0935934	+	0.995611i	$0.470165\pi$
$294$	0	0
$295$	−1.00702	−0.0586311
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−35.6570	−2.06210
$300$	0	0
$301$	29.1558	1.68051
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.35519	−0.306637
$306$	0	0
$307$	−6.30034	−0.359579	−0.179790	−	0.983705i	$-0.557542\pi$
$307$	−6.30034	−0.359579	−0.179790	+	0.983705i	$0.557542\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	35.2257	1.99747	0.998734	−	0.0502944i	$-0.0160160\pi$
$311$	35.2257	1.99747	0.998734	+	0.0502944i	$0.0160160\pi$
$312$	0	0
$313$	33.9773	1.92051	0.960255	−	0.279126i	$-0.0900446\pi$
$313$	33.9773	1.92051	0.960255	+	0.279126i	$0.0900446\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.2994	0.746968	0.373484	−	0.927637i	$-0.378163\pi$
$317$	13.2994	0.746968	0.373484	+	0.927637i	$0.378163\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	18.3576	1.02145
$324$	0	0
$325$	29.4116	1.63146
$326$	0	0
$327$	0	0
$328$	0	0
$329$	31.7110	1.74829
$330$	0	0
$331$	4.94908	0.272026	0.136013	−	0.990707i	$-0.456571\pi$
$331$	4.94908	0.272026	0.136013	+	0.990707i	$0.456571\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.07882	0.113578
$336$	0	0
$337$	0.205071	0.0111709	0.00558546	−	0.999984i	$-0.498222\pi$
$337$	0.205071	0.0111709	0.00558546	+	0.999984i	$0.498222\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	13.6279	0.735838
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.14170	−0.437069	−0.218535	−	0.975829i	$-0.570128\pi$
$347$	−8.14170	−0.437069	−0.218535	+	0.975829i	$0.570128\pi$
$348$	0	0
$349$	5.48852	0.293794	0.146897	−	0.989152i	$-0.453071\pi$
$349$	5.48852	0.293794	0.146897	+	0.989152i	$0.453071\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.5991	−0.670581	−0.335290	−	0.942115i	$-0.608834\pi$
$353$	−12.5991	−0.670581	−0.335290	+	0.942115i	$0.608834\pi$
$354$	0	0
$355$	−5.05305	−0.268188
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.4852	−1.02839	−0.514195	−	0.857673i	$-0.671909\pi$
$359$	−19.4852	−1.02839	−0.514195	+	0.857673i	$0.671909\pi$
$360$	0	0
$361$	−10.8355	−0.570291
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.24358	−0.274462
$366$	0	0
$367$	18.7058	0.976434	0.488217	−	0.872722i	$-0.337647\pi$
$367$	18.7058	0.976434	0.488217	+	0.872722i	$0.337647\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17.7847	−0.923334
$372$	0	0
$373$	−6.68557	−0.346166	−0.173083	−	0.984907i	$-0.555373\pi$
$373$	−6.68557	−0.346166	−0.173083	+	0.984907i	$0.555373\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−55.1843	−2.84214
$378$	0	0
$379$	36.4046	1.86998	0.934988	−	0.354679i	$-0.115410\pi$
$379$	36.4046	1.86998	0.934988	+	0.354679i	$0.115410\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.4782	−1.25078	−0.625389	−	0.780313i	$-0.715060\pi$
$383$	−24.4782	−1.25078	−0.625389	+	0.780313i	$0.715060\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−23.1049	−1.17146	−0.585732	−	0.810505i	$-0.699193\pi$
$389$	−23.1049	−1.17146	−0.585732	+	0.810505i	$0.699193\pi$
$390$	0	0
$391$	−35.6570	−1.80325
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.84236	−0.394592
$396$	0	0
$397$	22.1052	1.10943	0.554713	−	0.832042i	$-0.312828\pi$
$397$	22.1052	1.10943	0.554713	+	0.832042i	$0.312828\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.5273	1.17490	0.587448	−	0.809262i	$-0.300133\pi$
$401$	23.5273	1.17490	0.587448	+	0.809262i	$0.300133\pi$
$402$	0	0
$403$	10.5988	0.527963
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	15.4468	0.763793	0.381897	−	0.924205i	$-0.375271\pi$
$409$	15.4468	0.763793	0.381897	+	0.924205i	$0.375271\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.44293	−0.317036
$414$	0	0
$415$	2.80245	0.137567
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.8076	−1.21193	−0.605966	−	0.795491i	$-0.707213\pi$
$419$	−24.8076	−1.21193	−0.605966	+	0.795491i	$0.707213\pi$
$420$	0	0
$421$	5.97210	0.291062	0.145531	−	0.989354i	$-0.453511\pi$
$421$	5.97210	0.291062	0.145531	+	0.989354i	$0.453511\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	29.4116	1.42667
$426$	0	0
$427$	−34.2625	−1.65808
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.6847	0.514662	0.257331	−	0.966323i	$-0.417157\pi$
$431$	10.6847	0.514662	0.257331	+	0.966323i	$0.417157\pi$
$432$	0	0
$433$	−11.9982	−0.576595	−0.288297	−	0.957541i	$-0.593089\pi$
$433$	−11.9982	−0.576595	−0.288297	+	0.957541i	$0.593089\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.8583	−0.758605
$438$	0	0
$439$	19.6260	0.936699	0.468349	−	0.883543i	$-0.344849\pi$
$439$	19.6260	0.936699	0.468349	+	0.883543i	$0.344849\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.51996	0.404795	0.202398	−	0.979303i	$-0.435127\pi$
$443$	8.51996	0.404795	0.202398	+	0.979303i	$0.435127\pi$
$444$	0	0
$445$	−6.17150	−0.292557
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.35733	0.394407	0.197203	−	0.980363i	$-0.436814\pi$
$449$	8.35733	0.394407	0.197203	+	0.980363i	$0.436814\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.3506	−0.813409
$456$	0	0
$457$	−27.5017	−1.28647	−0.643237	−	0.765667i	$-0.722409\pi$
$457$	−27.5017	−1.28647	−0.643237	+	0.765667i	$0.722409\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.7400	−0.546787	−0.273394	−	0.961902i	$-0.588146\pi$
$461$	−11.7400	−0.546787	−0.273394	+	0.961902i	$0.588146\pi$
$462$	0	0
$463$	1.00702	0.0468003	0.0234002	−	0.999726i	$-0.492551\pi$
$463$	1.00702	0.0468003	0.0234002	+	0.999726i	$0.492551\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−26.2187	−1.21326	−0.606629	−	0.794985i	$-0.707479\pi$
$467$	−26.2187	−1.21326	−0.606629	+	0.794985i	$0.707479\pi$
$468$	0	0
$469$	13.3003	0.614150
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	13.0807	0.600183
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.2577	−0.560068	−0.280034	−	0.959990i	$-0.590346\pi$
$479$	−12.2577	−0.560068	−0.280034	+	0.959990i	$0.590346\pi$
$480$	0	0
$481$	16.3829	0.746997
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.93688	0.224172
$486$	0	0
$487$	0.551840	0.0250062	0.0125031	−	0.999922i	$-0.496020\pi$
$487$	0.551840	0.0250062	0.0125031	+	0.999922i	$0.496020\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14.0441	−0.633803	−0.316901	−	0.948458i	$-0.602642\pi$
$491$	−14.0441	−0.633803	−0.316901	+	0.948458i	$0.602642\pi$
$492$	0	0
$493$	−55.1843	−2.48537
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−32.3294	−1.45017
$498$	0	0
$499$	−19.9003	−0.890860	−0.445430	−	0.895317i	$-0.646949\pi$
$499$	−19.9003	−0.890860	−0.445430	+	0.895317i	$0.646949\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.4994	0.512732	0.256366	−	0.966580i	$-0.417475\pi$
$503$	11.4994	0.512732	0.256366	+	0.966580i	$0.417475\pi$
$504$	0	0
$505$	−2.80245	−0.124707
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15.8564	−0.702823	−0.351411	−	0.936221i	$-0.614298\pi$
$509$	−15.8564	−0.702823	−0.351411	+	0.936221i	$0.614298\pi$
$510$	0	0
$511$	−33.5484	−1.48410
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.82119	−0.300578
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	34.6270	1.51704	0.758518	−	0.651653i	$-0.225924\pi$
$521$	34.6270	1.51704	0.758518	+	0.651653i	$0.225924\pi$
$522$	0	0
$523$	−35.7978	−1.56533	−0.782664	−	0.622444i	$-0.786140\pi$
$523$	−35.7978	−1.56533	−0.782664	+	0.622444i	$0.786140\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.5988	0.461690
$528$	0	0
$529$	7.80245	0.339237
$530$	0	0
$531$	0	0
$532$	0	0
$533$	37.8337	1.63876
$534$	0	0
$535$	13.3454	0.576973
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−42.1540	−1.81234	−0.906170	−	0.422914i	$-0.861007\pi$
$541$	−42.1540	−1.81234	−0.906170	+	0.422914i	$0.861007\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.6683	−0.499814
$546$	0	0
$547$	23.0699	0.986398	0.493199	−	0.869917i	$-0.335827\pi$
$547$	23.0699	0.986398	0.493199	+	0.869917i	$0.335827\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.5430	−1.04557
$552$	0	0
$553$	−50.1754	−2.13368
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−26.0282	−1.10285	−0.551425	−	0.834224i	$-0.685916\pi$
$557$	−26.0282	−1.10285	−0.551425	+	0.834224i	$0.685916\pi$
$558$	0	0
$559$	−45.0634	−1.90598
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18.7006	−0.788137	−0.394068	−	0.919081i	$-0.628933\pi$
$563$	−18.7006	−0.788137	−0.394068	+	0.919081i	$0.628933\pi$
$564$	0	0
$565$	3.33156	0.140160
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.73065	−0.240241	−0.120121	−	0.992759i	$-0.538328\pi$
$569$	−5.73065	−0.240241	−0.120121	+	0.992759i	$0.538328\pi$
$570$	0	0
$571$	−8.86902	−0.371157	−0.185579	−	0.982629i	$-0.559416\pi$
$571$	−8.86902	−0.371157	−0.185579	+	0.982629i	$0.559416\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−25.4073	−1.05956
$576$	0	0
$577$	−15.7988	−0.657711	−0.328855	−	0.944380i	$-0.606663\pi$
$577$	−15.7988	−0.657711	−0.328855	+	0.944380i	$0.606663\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	17.9300	0.743864
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.2858	−1.53895	−0.769475	−	0.638677i	$-0.779482\pi$
$587$	−37.2858	−1.53895	−0.769475	+	0.638677i	$0.779482\pi$
$588$	0	0
$589$	4.71376	0.194227
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−35.9170	−1.47493	−0.737467	−	0.675383i	$-0.763978\pi$
$593$	−35.9170	−1.47493	−0.737467	+	0.675383i	$0.763978\pi$
$594$	0	0
$595$	−17.3506	−0.711305
$596$	0	0
$597$	0	0
$598$	0	0
$599$	46.9846	1.91974	0.959869	−	0.280448i	$-0.0904831\pi$
$599$	46.9846	1.91974	0.959869	+	0.280448i	$0.0904831\pi$
$600$	0	0
$601$	34.8790	1.42274	0.711372	−	0.702816i	$-0.248074\pi$
$601$	34.8790	1.42274	0.711372	+	0.702816i	$0.248074\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.34537	0.338728	0.169364	−	0.985554i	$-0.445829\pi$
$607$	8.34537	0.338728	0.169364	+	0.985554i	$0.445829\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−49.0128	−1.98284
$612$	0	0
$613$	−10.8250	−0.437216	−0.218608	−	0.975813i	$-0.570152\pi$
$613$	−10.8250	−0.437216	−0.218608	+	0.975813i	$0.570152\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.19784	0.289774	0.144887	−	0.989448i	$-0.453718\pi$
$617$	7.19784	0.289774	0.144887	+	0.989448i	$0.453718\pi$
$618$	0	0
$619$	21.8058	0.876448	0.438224	−	0.898866i	$-0.355608\pi$
$619$	21.8058	0.876448	0.438224	+	0.898866i	$0.355608\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−39.4852	−1.58194
$624$	0	0
$625$	18.8466	0.753865
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.3829	0.653230
$630$	0	0
$631$	42.7058	1.70009	0.850045	−	0.526710i	$-0.176574\pi$
$631$	42.7058	1.70009	0.850045	+	0.526710i	$0.176574\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7.84236	−0.311215
$636$	0	0
$637$	−66.0363	−2.61645
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.6709	−0.539967	−0.269983	−	0.962865i	$-0.587018\pi$
$641$	−13.6709	−0.539967	−0.269983	+	0.962865i	$0.587018\pi$
$642$	0	0
$643$	−37.1631	−1.46557	−0.732785	−	0.680460i	$-0.761780\pi$
$643$	−37.1631	−1.46557	−0.732785	+	0.680460i	$0.761780\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	35.1103	1.38033	0.690165	−	0.723652i	$-0.257538\pi$
$647$	35.1103	1.38033	0.690165	+	0.723652i	$0.257538\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.48707	0.0973264	0.0486632	−	0.998815i	$-0.484504\pi$
$653$	2.48707	0.0973264	0.0486632	+	0.998815i	$0.484504\pi$
$654$	0	0
$655$	4.55702	0.178057
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25.2699	0.984375	0.492187	−	0.870489i	$-0.336197\pi$
$659$	25.2699	0.984375	0.492187	+	0.870489i	$0.336197\pi$
$660$	0	0
$661$	−25.7058	−0.999839	−0.499920	−	0.866072i	$-0.666637\pi$
$661$	−25.7058	−0.999839	−0.499920	+	0.866072i	$0.666637\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.71660	−0.299237
$666$	0	0
$667$	47.6712	1.84583
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	41.4984	1.59964	0.799822	−	0.600237i	$-0.204927\pi$
$673$	41.4984	1.59964	0.799822	+	0.600237i	$0.204927\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.72408	−0.296861	−0.148430	−	0.988923i	$-0.547422\pi$
$677$	−7.72408	−0.296861	−0.148430	+	0.988923i	$0.547422\pi$
$678$	0	0
$679$	31.5862	1.21217
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−32.9334	−1.26016	−0.630080	−	0.776530i	$-0.716978\pi$
$683$	−32.9334	−1.26016	−0.630080	+	0.776530i	$0.716978\pi$
$684$	0	0
$685$	−8.64054	−0.330138
$686$	0	0
$687$	0	0
$688$	0	0
$689$	27.4881	1.04721
$690$	0	0
$691$	7.24145	0.275478	0.137739	−	0.990469i	$-0.456017\pi$
$691$	7.24145	0.275478	0.137739	+	0.990469i	$0.456017\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.94177	−0.263316
$696$	0	0
$697$	37.8337	1.43305
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.8081	1.01253	0.506265	−	0.862378i	$-0.331026\pi$
$701$	26.8081	1.01253	0.506265	+	0.862378i	$0.331026\pi$
$702$	0	0
$703$	7.28624	0.274806
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.9300	−0.674329
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.15579	−0.342887
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.25245	0.0467085	0.0233543	−	0.999727i	$-0.492565\pi$
$719$	1.25245	0.0467085	0.0233543	+	0.999727i	$0.492565\pi$
$720$	0	0
$721$	−43.6420	−1.62531
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−39.3214	−1.46036
$726$	0	0
$727$	20.9058	0.775352	0.387676	−	0.921796i	$-0.373278\pi$
$727$	20.9058	0.775352	0.387676	+	0.921796i	$0.373278\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−45.0634	−1.66673
$732$	0	0
$733$	37.1117	1.37075	0.685377	−	0.728189i	$-0.259638\pi$
$733$	37.1117	1.37075	0.685377	+	0.728189i	$0.259638\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−16.1272	−0.593250	−0.296625	−	0.954994i	$-0.595861\pi$
$739$	−16.1272	−0.593250	−0.296625	+	0.954994i	$0.595861\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.3558	1.22370	0.611852	−	0.790972i	$-0.290425\pi$
$743$	33.3558	1.22370	0.611852	+	0.790972i	$0.290425\pi$
$744$	0	0
$745$	−15.4829	−0.567250
$746$	0	0
$747$	0	0
$748$	0	0
$749$	85.3840	3.11986
$750$	0	0
$751$	−3.00029	−0.109482	−0.0547411	−	0.998501i	$-0.517433\pi$
$751$	−3.00029	−0.109482	−0.0547411	+	0.998501i	$0.517433\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.70488	0.171228
$756$	0	0
$757$	21.4923	0.781153	0.390576	−	0.920571i	$-0.372276\pi$
$757$	21.4923	0.781153	0.390576	+	0.920571i	$0.372276\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.6575	−1.25633	−0.628166	−	0.778079i	$-0.716194\pi$
$761$	−34.6575	−1.25633	−0.628166	+	0.778079i	$0.716194\pi$
$762$	0	0
$763$	−74.6536	−2.70264
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.95824	0.359571
$768$	0	0
$769$	27.7096	0.999234	0.499617	−	0.866246i	$-0.333474\pi$
$769$	27.7096	0.999234	0.499617	+	0.866246i	$0.333474\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−26.4134	−0.950025	−0.475012	−	0.879979i	$-0.657556\pi$
$773$	−26.4134	−0.950025	−0.475012	+	0.879979i	$0.657556\pi$
$774$	0	0
$775$	7.55213	0.271281
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.8264	0.602867
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.0135652	0.000484163 0
$786$	0	0
$787$	−14.7865	−0.527082	−0.263541	−	0.964648i	$-0.584890\pi$
$787$	−14.7865	−0.527082	−0.263541	+	0.964648i	$0.584890\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	21.3153	0.757886
$792$	0	0
$793$	52.9564	1.88054
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.2294	0.539454	0.269727	−	0.962937i	$-0.413066\pi$
$797$	15.2294	0.539454	0.269727	+	0.962937i	$0.413066\pi$
$798$	0	0
$799$	−49.0128	−1.73395
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	14.9884	0.528271
$806$	0	0
$807$	0	0
$808$	0	0
$809$	51.1422	1.79807	0.899033	−	0.437882i	$-0.144271\pi$
$809$	51.1422	1.79807	0.899033	+	0.437882i	$0.144271\pi$
$810$	0	0
$811$	0.368393	0.0129360	0.00646801	−	0.999979i	$-0.497941\pi$
$811$	0.368393	0.0129360	0.00646801	+	0.999979i	$0.497941\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.41997	−0.329967
$816$	0	0
$817$	−20.0418	−0.701172
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.71471	−0.199445	−0.0997223	−	0.995015i	$-0.531795\pi$
$821$	−5.71471	−0.199445	−0.0997223	+	0.995015i	$0.531795\pi$
$822$	0	0
$823$	1.05610	0.0368132	0.0184066	−	0.999831i	$-0.494141\pi$
$823$	1.05610	0.0368132	0.0184066	+	0.999831i	$0.494141\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.7565	0.408812	0.204406	−	0.978886i	$-0.434474\pi$
$827$	11.7565	0.408812	0.204406	+	0.978886i	$0.434474\pi$
$828$	0	0
$829$	−22.2070	−0.771281	−0.385641	−	0.922649i	$-0.626020\pi$
$829$	−22.2070	−0.771281	−0.385641	+	0.922649i	$0.626020\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−66.0363	−2.28802
$834$	0	0
$835$	11.6570	0.403407
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.42942	−0.118397	−0.0591983	−	0.998246i	$-0.518854\pi$
$839$	−3.42942	−0.118397	−0.0591983	+	0.998246i	$0.518854\pi$
$840$	0	0
$841$	44.7779	1.54406
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.3712	0.631988
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.1525	−0.485140
$852$	0	0
$853$	1.43969	0.0492939	0.0246469	−	0.999696i	$-0.492154\pi$
$853$	1.43969	0.0492939	0.0246469	+	0.999696i	$0.492154\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.2140	−1.51032	−0.755161	−	0.655539i	$-0.772441\pi$
$857$	−44.2140	−1.51032	−0.755161	+	0.655539i	$0.772441\pi$
$858$	0	0
$859$	−29.0561	−0.991381	−0.495691	−	0.868499i	$-0.665085\pi$
$859$	−29.0561	−0.991381	−0.495691	+	0.868499i	$0.665085\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.6076	0.361088	0.180544	−	0.983567i	$-0.442214\pi$
$863$	10.6076	0.361088	0.180544	+	0.983567i	$0.442214\pi$
$864$	0	0
$865$	0.696340	0.0236763
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−20.5570	−0.696548
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−25.8662	−0.874437
$876$	0	0
$877$	−6.31916	−0.213383	−0.106691	−	0.994292i	$-0.534026\pi$
$877$	−6.31916	−0.213383	−0.106691	+	0.994292i	$0.534026\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.8825	−1.04046	−0.520228	−	0.854027i	$-0.674153\pi$
$881$	−30.8825	−1.04046	−0.520228	+	0.854027i	$0.674153\pi$
$882$	0	0
$883$	43.8036	1.47411	0.737055	−	0.675833i	$-0.236216\pi$
$883$	43.8036	1.47411	0.737055	+	0.675833i	$0.236216\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.71286	0.292549	0.146275	−	0.989244i	$-0.453272\pi$
$887$	8.71286	0.292549	0.146275	+	0.989244i	$0.453272\pi$
$888$	0	0
$889$	−50.1754	−1.68283
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−21.7982	−0.729450
$894$	0	0
$895$	−6.06008	−0.202566
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.1699	−0.472592
$900$	0	0
$901$	27.4881	0.915761
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.74693	0.157793
$906$	0	0
$907$	5.79847	0.192535	0.0962675	−	0.995356i	$-0.469310\pi$
$907$	5.79847	0.192535	0.0962675	+	0.995356i	$0.469310\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0.348170	0.0115354	0.00576769	−	0.999983i	$-0.498164\pi$
$911$	0.348170	0.0115354	0.00576769	+	0.999983i	$0.498164\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	29.1558	0.962809
$918$	0	0
$919$	6.27025	0.206836	0.103418	−	0.994638i	$-0.467022\pi$
$919$	6.27025	0.206836	0.103418	+	0.994638i	$0.467022\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	49.9686	1.64474
$924$	0	0
$925$	11.6736	0.383826
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−14.7567	−0.484152	−0.242076	−	0.970257i	$-0.577828\pi$
$929$	−14.7567	−0.484152	−0.242076	+	0.970257i	$0.577828\pi$
$930$	0	0
$931$	−29.3693	−0.962541
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.5725	0.900755	0.450377	−	0.892838i	$-0.351289\pi$
$937$	27.5725	0.900755	0.450377	+	0.892838i	$0.351289\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41.5645	1.35497	0.677483	−	0.735539i	$-0.263071\pi$
$941$	41.5645	1.35497	0.677483	+	0.735539i	$0.263071\pi$
$942$	0	0
$943$	−32.6828	−1.06430
$944$	0	0
$945$	0	0
$946$	0	0
$947$	56.5346	1.83713	0.918564	−	0.395273i	$-0.129350\pi$
$947$	56.5346	1.83713	0.918564	+	0.395273i	$0.129350\pi$
$948$	0	0
$949$	51.8527	1.68321
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.2688	0.332641	0.166320	−	0.986072i	$-0.446811\pi$
$953$	10.2688	0.332641	0.166320	+	0.986072i	$0.446811\pi$
$954$	0	0
$955$	14.3116	0.463112
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−55.2821	−1.78515
$960$	0	0
$961$	−28.2785	−0.912210
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.32897	0.0427812
$966$	0	0
$967$	−35.5702	−1.14386	−0.571931	−	0.820302i	$-0.693805\pi$
$967$	−35.5702	−1.14386	−0.571931	+	0.820302i	$0.693805\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11.5218	−0.369752	−0.184876	−	0.982762i	$-0.559188\pi$
$971$	−11.5218	−0.369752	−0.184876	+	0.982762i	$0.559188\pi$
$972$	0	0
$973$	−44.4134	−1.42383
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.04389	−0.0973828	−0.0486914	−	0.998814i	$-0.515505\pi$
$977$	−3.04389	−0.0973828	−0.0486914	+	0.998814i	$0.515505\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.1300	−0.897209	−0.448604	−	0.893730i	$-0.648079\pi$
$983$	−28.1300	−0.897209	−0.448604	+	0.893730i	$0.648079\pi$
$984$	0	0
$985$	−15.5945	−0.496883
$986$	0	0
$987$	0	0
$988$	0	0
$989$	38.9282	1.23784
$990$	0	0
$991$	−13.3552	−0.424242	−0.212121	−	0.977243i	$-0.568037\pi$
$991$	−13.3552	−0.424242	−0.212121	+	0.977243i	$0.568037\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−13.2248	−0.419256
$996$	0	0
$997$	47.0577	1.49033	0.745167	−	0.666878i	$-0.232370\pi$
$997$	47.0577	1.49033	0.745167	+	0.666878i	$0.232370\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.ca.1.3		4
3.2	odd	2		2904.2.a.bc.1.2	✓	4
11.10	odd	2		8712.2.a.cb.1.3		4
12.11	even	2		5808.2.a.cn.1.2		4
33.32	even	2		2904.2.a.bd.1.2	yes	4
132.131	odd	2		5808.2.a.cm.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2904.2.a.bc.1.2	✓	4	3.2	odd	2
2904.2.a.bd.1.2	yes	4	33.32	even	2
5808.2.a.cm.1.2		4	132.131	odd	2
5808.2.a.cn.1.2		4	12.11	even	2
8712.2.a.ca.1.3		4	1.1	even	1	trivial
8712.2.a.cb.1.3		4	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q+0.649694 q^{5} +4.15674 q^{7} +O(q^{10})\)	\(q+0.649694 q^{5} +4.15674 q^{7} -6.42469 q^{13} -6.42469 q^{17} -2.85735 q^{19} +5.55000 q^{23} -4.57790 q^{25} +8.58940 q^{29} -1.64969 q^{31} +2.70061 q^{35} -2.55000 q^{37} -5.88879 q^{41} +7.01410 q^{43} +7.62881 q^{47} +10.2785 q^{49} -4.27851 q^{53} -1.55000 q^{59} -8.24264 q^{61} -4.17409 q^{65} +3.19969 q^{67} -7.77759 q^{71} -8.07085 q^{73} -12.0708 q^{79} +4.31349 q^{83} -4.17409 q^{85} -9.49908 q^{89} -26.7058 q^{91} -1.85641 q^{95} +7.59878 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q - 2 * q^5 - 4 * q^7	\( 4 q - 2 q^{5} - 4 q^{7} - 12 q^{13} - 12 q^{17} + 4 q^{23} + 14 q^{25} + 16 q^{29} - 2 q^{31} + 20 q^{35} + 8 q^{37} + 4 q^{41} - 4 q^{43} + 12 q^{47} + 18 q^{49} + 6 q^{53} + 12 q^{59} - 8 q^{61} - 12 q^{65} - 10 q^{67} + 24 q^{71} - 16 q^{73} - 32 q^{79} - 24 q^{83} - 12 q^{85} - 6 q^{89} - 24 q^{91} + 48 q^{95} + 12 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 - 4 * q^7 - 12 * q^13 - 12 * q^17 + 4 * q^23 + 14 * q^25 + 16 * q^29 - 2 * q^31 + 20 * q^35 + 8 * q^37 + 4 * q^41 - 4 * q^43 + 12 * q^47 + 18 * q^49 + 6 * q^53 + 12 * q^59 - 8 * q^61 - 12 * q^65 - 10 * q^67 + 24 * q^71 - 16 * q^73 - 32 * q^79 - 24 * q^83 - 12 * q^85 - 6 * q^89 - 24 * q^91 + 48 * q^95 + 12 * q^97

Introduction

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