LMFDB - Embedded newform 8496.2.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8496,2,Mod(1,8496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8496, 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("8496.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8496 = 2^{4} \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8496.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:	$67.8409015573$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	8496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32340 q^{5} -1.13919 q^{7} +O(q^{10})$	$q+3.32340 q^{5} -1.13919 q^{7} +0.398207 q^{11} +2.39821 q^{13} -7.10941 q^{17} -1.53740 q^{19} +3.25901 q^{23} +6.04502 q^{25} +3.93561 q^{29} -7.78600 q^{31} -3.78600 q^{35} +1.25901 q^{37} +7.50761 q^{41} +9.69182 q^{43} +8.71120 q^{47} -5.70224 q^{49} -10.7666 q^{53} +1.32340 q^{55} +1.00000 q^{59} +0.989588 q^{61} +7.97021 q^{65} -5.45219 q^{67} +15.0450 q^{71} +4.73202 q^{73} -0.453636 q^{77} -7.04502 q^{79} +13.0796 q^{83} -23.6274 q^{85} +16.4328 q^{89} -2.73202 q^{91} -5.10941 q^{95} +14.7666 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} - 9 q^{7}+O(q^{10}) $ 3 * q + 2 * q^5 - 9 * q^7	$ 3 q + 2 q^{5} - 9 q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} - 7 q^{19} + q^{23} - q^{25} + 11 q^{29} - 13 q^{31} - q^{35} - 5 q^{37} + q^{41} - 6 q^{43} + 11 q^{47} + 14 q^{49} - 2 q^{53} - 4 q^{55} + 3 q^{59} - q^{61} - 10 q^{67} + 26 q^{71} + 7 q^{73} + 17 q^{77} - 2 q^{79} - 3 q^{83} - 35 q^{85} + 23 q^{89} - q^{91} + 3 q^{95} + 14 q^{97}+O(q^{100}) $ 3 * q + 2 * q^5 - 9 * q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 - 7 * q^19 + q^23 - q^25 + 11 * q^29 - 13 * q^31 - q^35 - 5 * q^37 + q^41 - 6 * q^43 + 11 * q^47 + 14 * q^49 - 2 * q^53 - 4 * q^55 + 3 * q^59 - q^61 - 10 * q^67 + 26 * q^71 + 7 * q^73 + 17 * q^77 - 2 * q^79 - 3 * q^83 - 35 * q^85 + 23 * q^89 - q^91 + 3 * q^95 + 14 * 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
$5$	3.32340	1.48627	0.743136	−	0.669141i	$-0.233338\pi$
$5$	3.32340	1.48627	0.743136	+	0.669141i	$0.233338\pi$
$6$	0	0
$7$	−1.13919	−0.430575	−0.215287	−	0.976551i	$-0.569069\pi$
$7$	−1.13919	−0.430575	−0.215287	+	0.976551i	$0.569069\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.398207	0.120064	0.0600320	−	0.998196i	$-0.480880\pi$
$11$	0.398207	0.120064	0.0600320	+	0.998196i	$0.480880\pi$
$12$	0	0
$13$	2.39821	0.665143	0.332572	−	0.943078i	$-0.392084\pi$
$13$	2.39821	0.665143	0.332572	+	0.943078i	$0.392084\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.10941	−1.72428	−0.862142	−	0.506666i	$-0.830878\pi$
$17$	−7.10941	−1.72428	−0.862142	+	0.506666i	$0.830878\pi$
$18$	0	0
$19$	−1.53740	−0.352704	−0.176352	−	0.984327i	$-0.556430\pi$
$19$	−1.53740	−0.352704	−0.176352	+	0.984327i	$0.556430\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.25901	0.679551	0.339776	−	0.940507i	$-0.389649\pi$
$23$	3.25901	0.679551	0.339776	+	0.940507i	$0.389649\pi$
$24$	0	0
$25$	6.04502	1.20900
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.93561	0.730824	0.365412	−	0.930846i	$-0.380928\pi$
$29$	3.93561	0.730824	0.365412	+	0.930846i	$0.380928\pi$
$30$	0	0
$31$	−7.78600	−1.39841	−0.699204	−	0.714923i	$-0.746462\pi$
$31$	−7.78600	−1.39841	−0.699204	+	0.714923i	$0.746462\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.78600	−0.639951
$36$	0	0
$37$	1.25901	0.206981	0.103490	−	0.994630i	$-0.466999\pi$
$37$	1.25901	0.206981	0.103490	+	0.994630i	$0.466999\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.50761	1.17249	0.586246	−	0.810133i	$-0.300605\pi$
$41$	7.50761	1.17249	0.586246	+	0.810133i	$0.300605\pi$
$42$	0	0
$43$	9.69182	1.47799	0.738995	−	0.673711i	$-0.235301\pi$
$43$	9.69182	1.47799	0.738995	+	0.673711i	$0.235301\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.71120	1.27066	0.635330	−	0.772241i	$-0.280864\pi$
$47$	8.71120	1.27066	0.635330	+	0.772241i	$0.280864\pi$
$48$	0	0
$49$	−5.70224	−0.814605
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.7666	−1.47891	−0.739455	−	0.673206i	$-0.764917\pi$
$53$	−10.7666	−1.47891	−0.739455	+	0.673206i	$0.764917\pi$
$54$	0	0
$55$	1.32340	0.178448
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	0.989588	0.126704	0.0633519	−	0.997991i	$-0.479821\pi$
$61$	0.989588	0.126704	0.0633519	+	0.997991i	$0.479821\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.97021	0.988583
$66$	0	0
$67$	−5.45219	−0.666091	−0.333045	−	0.942911i	$-0.608076\pi$
$67$	−5.45219	−0.666091	−0.333045	+	0.942911i	$0.608076\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.0450	1.78551	0.892757	−	0.450538i	$-0.148768\pi$
$71$	15.0450	1.78551	0.892757	+	0.450538i	$0.148768\pi$
$72$	0	0
$73$	4.73202	0.553842	0.276921	−	0.960893i	$-0.410686\pi$
$73$	4.73202	0.553842	0.276921	+	0.960893i	$0.410686\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.453636	−0.0516966
$78$	0	0
$79$	−7.04502	−0.792626	−0.396313	−	0.918115i	$-0.629711\pi$
$79$	−7.04502	−0.792626	−0.396313	+	0.918115i	$0.629711\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.0796	1.43567	0.717837	−	0.696211i	$-0.245132\pi$
$83$	13.0796	1.43567	0.717837	+	0.696211i	$0.245132\pi$
$84$	0	0
$85$	−23.6274	−2.56275
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.4328	1.74187	0.870937	−	0.491394i	$-0.163513\pi$
$89$	16.4328	1.74187	0.870937	+	0.491394i	$0.163513\pi$
$90$	0	0
$91$	−2.73202	−0.286394
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.10941	−0.524214
$96$	0	0
$97$	14.7666	1.49932	0.749662	−	0.661821i	$-0.230216\pi$
$97$	14.7666	1.49932	0.749662	+	0.661821i	$0.230216\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.278388	0.0277007	0.0138503	−	0.999904i	$-0.495591\pi$
$101$	0.278388	0.0277007	0.0138503	+	0.999904i	$0.495591\pi$
$102$	0	0
$103$	15.0152	1.47949	0.739747	−	0.672885i	$-0.234945\pi$
$103$	15.0152	1.47949	0.739747	+	0.672885i	$0.234945\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.16898	0.113010	0.0565048	−	0.998402i	$-0.482004\pi$
$107$	1.16898	0.113010	0.0565048	+	0.998402i	$0.482004\pi$
$108$	0	0
$109$	−4.89059	−0.468434	−0.234217	−	0.972184i	$-0.575253\pi$
$109$	−4.89059	−0.468434	−0.234217	+	0.972184i	$0.575253\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.5630	1.46405	0.732024	−	0.681279i	$-0.238576\pi$
$113$	15.5630	1.46405	0.732024	+	0.681279i	$0.238576\pi$
$114$	0	0
$115$	10.8310	1.01000
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.09899	0.742434
$120$	0	0
$121$	−10.8414	−0.985585
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.47301	0.310636
$126$	0	0
$127$	−4.39821	−0.390278	−0.195139	−	0.980776i	$-0.562516\pi$
$127$	−4.39821	−0.390278	−0.195139	+	0.980776i	$0.562516\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.6620	1.54314	0.771570	−	0.636145i	$-0.219472\pi$
$131$	17.6620	1.54314	0.771570	+	0.636145i	$0.219472\pi$
$132$	0	0
$133$	1.75140	0.151866
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.7666	0.919855	0.459928	−	0.887956i	$-0.347875\pi$
$137$	10.7666	0.919855	0.459928	+	0.887956i	$0.347875\pi$
$138$	0	0
$139$	−9.82061	−0.832973	−0.416486	−	0.909142i	$-0.636739\pi$
$139$	−9.82061	−0.832973	−0.416486	+	0.909142i	$0.636739\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.954984	0.0798598
$144$	0	0
$145$	13.0796	1.08620
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.01938	0.411203	0.205602	−	0.978636i	$-0.434085\pi$
$149$	5.01938	0.411203	0.205602	+	0.978636i	$0.434085\pi$
$150$	0	0
$151$	−7.35801	−0.598786	−0.299393	−	0.954130i	$-0.596784\pi$
$151$	−7.35801	−0.598786	−0.299393	+	0.954130i	$0.596784\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−25.8760	−2.07841
$156$	0	0
$157$	−1.73057	−0.138115	−0.0690574	−	0.997613i	$-0.521999\pi$
$157$	−1.73057	−0.138115	−0.0690574	+	0.997613i	$0.521999\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.71265	−0.292598
$162$	0	0
$163$	8.58242	0.672227	0.336113	−	0.941822i	$-0.390887\pi$
$163$	8.58242	0.672227	0.336113	+	0.941822i	$0.390887\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.49239	0.502396	0.251198	−	0.967936i	$-0.419175\pi$
$167$	6.49239	0.502396	0.251198	+	0.967936i	$0.419175\pi$
$168$	0	0
$169$	−7.24860	−0.557585
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.5437	1.56191	0.780953	−	0.624590i	$-0.214734\pi$
$173$	20.5437	1.56191	0.780953	+	0.624590i	$0.214734\pi$
$174$	0	0
$175$	−6.88645	−0.520566
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23.9702	−1.79162	−0.895809	−	0.444439i	$-0.853403\pi$
$179$	−23.9702	−1.79162	−0.895809	+	0.444439i	$0.853403\pi$
$180$	0	0
$181$	14.7112	1.09347	0.546737	−	0.837304i	$-0.315870\pi$
$181$	14.7112	1.09347	0.546737	+	0.837304i	$0.315870\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.18421	0.307629
$186$	0	0
$187$	−2.83102	−0.207025
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.91623	0.211011	0.105506	−	0.994419i	$-0.466354\pi$
$191$	2.91623	0.211011	0.105506	+	0.994419i	$0.466354\pi$
$192$	0	0
$193$	−0.646809	−0.0465583	−0.0232791	−	0.999729i	$-0.507411\pi$
$193$	−0.646809	−0.0465583	−0.0232791	+	0.999729i	$0.507411\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.87122	0.418307	0.209153	−	0.977883i	$-0.432929\pi$
$197$	5.87122	0.418307	0.209153	+	0.977883i	$0.432929\pi$
$198$	0	0
$199$	−11.1004	−0.786890	−0.393445	−	0.919348i	$-0.628717\pi$
$199$	−11.1004	−0.786890	−0.393445	+	0.919348i	$0.628717\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.48342	−0.314675
$204$	0	0
$205$	24.9508	1.74264
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.612205	−0.0423471
$210$	0	0
$211$	13.4778	0.927852	0.463926	−	0.885874i	$-0.346440\pi$
$211$	13.4778	0.927852	0.463926	+	0.885874i	$0.346440\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	32.2099	2.19669
$216$	0	0
$217$	8.86977	0.602119
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−17.0498	−1.14690
$222$	0	0
$223$	−16.7368	−1.12078	−0.560391	−	0.828228i	$-0.689349\pi$
$223$	−16.7368	−1.12078	−0.560391	+	0.828228i	$0.689349\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.0346	−0.931509	−0.465755	−	0.884914i	$-0.654217\pi$
$227$	−14.0346	−0.931509	−0.465755	+	0.884914i	$0.654217\pi$
$228$	0	0
$229$	−12.2445	−0.809136	−0.404568	−	0.914508i	$-0.632578\pi$
$229$	−12.2445	−0.809136	−0.404568	+	0.914508i	$0.632578\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.87122	−0.122588	−0.0612938	−	0.998120i	$-0.519523\pi$
$233$	−1.87122	−0.122588	−0.0612938	+	0.998120i	$0.519523\pi$
$234$	0	0
$235$	28.9508	1.88854
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.9910	−0.905005	−0.452502	−	0.891763i	$-0.649469\pi$
$239$	−13.9910	−0.905005	−0.452502	+	0.891763i	$0.649469\pi$
$240$	0	0
$241$	27.5333	1.77357	0.886786	−	0.462179i	$-0.152932\pi$
$241$	27.5333	1.77357	0.886786	+	0.462179i	$0.152932\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−18.9508	−1.21072
$246$	0	0
$247$	−3.68701	−0.234599
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.61702	−0.165185	−0.0825925	−	0.996583i	$-0.526320\pi$
$251$	−2.61702	−0.165185	−0.0825925	+	0.996583i	$0.526320\pi$
$252$	0	0
$253$	1.29776	0.0815897
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.3892	−1.27185	−0.635923	−	0.771752i	$-0.719380\pi$
$257$	−20.3892	−1.27185	−0.635923	+	0.771752i	$0.719380\pi$
$258$	0	0
$259$	−1.43426	−0.0891206
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−23.8552	−1.47098	−0.735488	−	0.677538i	$-0.763047\pi$
$263$	−23.8552	−1.47098	−0.735488	+	0.677538i	$0.763047\pi$
$264$	0	0
$265$	−35.7819	−2.19806
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.2999	1.72547	0.862737	−	0.505653i	$-0.168748\pi$
$269$	28.2999	1.72547	0.862737	+	0.505653i	$0.168748\pi$
$270$	0	0
$271$	23.4134	1.42226	0.711132	−	0.703058i	$-0.248183\pi$
$271$	23.4134	1.42226	0.711132	+	0.703058i	$0.248183\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.40717	0.145158
$276$	0	0
$277$	15.6918	0.942830	0.471415	−	0.881911i	$-0.343743\pi$
$277$	15.6918	0.942830	0.471415	+	0.881911i	$0.343743\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.63158	0.335952	0.167976	−	0.985791i	$-0.446277\pi$
$281$	5.63158	0.335952	0.167976	+	0.985791i	$0.446277\pi$
$282$	0	0
$283$	−10.8310	−0.643837	−0.321919	−	0.946767i	$-0.604328\pi$
$283$	−10.8310	−0.643837	−0.321919	+	0.946767i	$0.604328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.55263	−0.504846
$288$	0	0
$289$	33.5437	1.97316
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−33.9959	−1.98606	−0.993029	−	0.117866i	$-0.962395\pi$
$293$	−33.9959	−1.98606	−0.993029	+	0.117866i	$0.962395\pi$
$294$	0	0
$295$	3.32340	0.193496
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.81579	0.451999
$300$	0	0
$301$	−11.0409	−0.636385
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.28880	0.188316
$306$	0	0
$307$	4.25756	0.242992	0.121496	−	0.992592i	$-0.461231\pi$
$307$	4.25756	0.242992	0.121496	+	0.992592i	$0.461231\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.5374	1.22127	0.610637	−	0.791911i	$-0.290913\pi$
$311$	21.5374	1.22127	0.610637	+	0.791911i	$0.290913\pi$
$312$	0	0
$313$	−3.96540	−0.224137	−0.112069	−	0.993700i	$-0.535748\pi$
$313$	−3.96540	−0.224137	−0.112069	+	0.993700i	$0.535748\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.58097	−0.425790	−0.212895	−	0.977075i	$-0.568289\pi$
$317$	−7.58097	−0.425790	−0.212895	+	0.977075i	$0.568289\pi$
$318$	0	0
$319$	1.56719	0.0877457
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.9300	0.608162
$324$	0	0
$325$	14.4972	0.804160
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.92375	−0.547114
$330$	0	0
$331$	0.667633	0.0366964	0.0183482	−	0.999832i	$-0.494159\pi$
$331$	0.667633	0.0366964	0.0183482	+	0.999832i	$0.494159\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−18.1198	−0.989991
$336$	0	0
$337$	7.47783	0.407343	0.203672	−	0.979039i	$-0.434713\pi$
$337$	7.47783	0.407343	0.203672	+	0.979039i	$0.434713\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.10044	−0.167898
$342$	0	0
$343$	14.4703	0.781323
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.0200	−1.55788	−0.778939	−	0.627100i	$-0.784242\pi$
$347$	−29.0200	−1.55788	−0.778939	+	0.627100i	$0.784242\pi$
$348$	0	0
$349$	−16.8746	−0.903276	−0.451638	−	0.892201i	$-0.649160\pi$
$349$	−16.8746	−0.903276	−0.451638	+	0.892201i	$0.649160\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.98062	0.158643	0.0793213	−	0.996849i	$-0.474725\pi$
$353$	2.98062	0.158643	0.0793213	+	0.996849i	$0.474725\pi$
$354$	0	0
$355$	50.0007	2.65376
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.85039	−0.0976600	−0.0488300	−	0.998807i	$-0.515549\pi$
$359$	−1.85039	−0.0976600	−0.0488300	+	0.998807i	$0.515549\pi$
$360$	0	0
$361$	−16.6364	−0.875600
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.7264	0.823159
$366$	0	0
$367$	−7.97021	−0.416042	−0.208021	−	0.978124i	$-0.566702\pi$
$367$	−7.97021	−0.416042	−0.208021	+	0.978124i	$0.566702\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.2653	0.636782
$372$	0	0
$373$	1.31859	0.0682739	0.0341369	−	0.999417i	$-0.489132\pi$
$373$	1.31859	0.0682739	0.0341369	+	0.999417i	$0.489132\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.43841	0.486103
$378$	0	0
$379$	−20.5783	−1.05703	−0.528517	−	0.848922i	$-0.677252\pi$
$379$	−20.5783	−1.05703	−0.528517	+	0.848922i	$0.677252\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.8802	1.11803	0.559013	−	0.829159i	$-0.311180\pi$
$383$	21.8802	1.11803	0.559013	+	0.829159i	$0.311180\pi$
$384$	0	0
$385$	−1.50761	−0.0768351
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.29921	0.319383	0.159691	−	0.987167i	$-0.448950\pi$
$389$	6.29921	0.319383	0.159691	+	0.987167i	$0.448950\pi$
$390$	0	0
$391$	−23.1697	−1.17174
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−23.4134	−1.17806
$396$	0	0
$397$	−12.1198	−0.608276	−0.304138	−	0.952628i	$-0.598368\pi$
$397$	−12.1198	−0.608276	−0.304138	+	0.952628i	$0.598368\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.3338	0.815672	0.407836	−	0.913055i	$-0.366284\pi$
$401$	16.3338	0.815672	0.407836	+	0.913055i	$0.366284\pi$
$402$	0	0
$403$	−18.6724	−0.930141
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.501348	0.0248509
$408$	0	0
$409$	37.0665	1.83282	0.916410	−	0.400240i	$-0.131073\pi$
$409$	37.0665	1.83282	0.916410	+	0.400240i	$0.131073\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.13919	−0.0560561
$414$	0	0
$415$	43.4689	2.13380
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.6468	0.715543	0.357772	−	0.933809i	$-0.383537\pi$
$419$	14.6468	0.715543	0.357772	+	0.933809i	$0.383537\pi$
$420$	0	0
$421$	−3.32340	−0.161973	−0.0809864	−	0.996715i	$-0.525807\pi$
$421$	−3.32340	−0.161973	−0.0809864	+	0.996715i	$0.525807\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−42.9765	−2.08467
$426$	0	0
$427$	−1.12733	−0.0545555
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.73202	0.131597	0.0657985	−	0.997833i	$-0.479041\pi$
$431$	2.73202	0.131597	0.0657985	+	0.997833i	$0.479041\pi$
$432$	0	0
$433$	−36.1038	−1.73504	−0.867519	−	0.497404i	$-0.834287\pi$
$433$	−36.1038	−1.73504	−0.867519	+	0.497404i	$0.834287\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.01041	−0.239681
$438$	0	0
$439$	37.9571	1.81159	0.905797	−	0.423712i	$-0.139273\pi$
$439$	37.9571	1.81159	0.905797	+	0.423712i	$0.139273\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.84625	0.0877179	0.0438589	−	0.999038i	$-0.486035\pi$
$443$	1.84625	0.0877179	0.0438589	+	0.999038i	$0.486035\pi$
$444$	0	0
$445$	54.6129	2.58890
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−27.9494	−1.31901	−0.659507	−	0.751699i	$-0.729235\pi$
$449$	−27.9494	−1.31901	−0.659507	+	0.751699i	$0.729235\pi$
$450$	0	0
$451$	2.98959	0.140774
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.07962	−0.425659
$456$	0	0
$457$	−23.2501	−1.08759	−0.543796	−	0.839218i	$-0.683013\pi$
$457$	−23.2501	−1.08759	−0.543796	+	0.839218i	$0.683013\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−20.4447	−0.952203	−0.476102	−	0.879390i	$-0.657951\pi$
$461$	−20.4447	−0.952203	−0.476102	+	0.879390i	$0.657951\pi$
$462$	0	0
$463$	12.6122	0.586139	0.293069	−	0.956091i	$-0.405323\pi$
$463$	12.6122	0.586139	0.293069	+	0.956091i	$0.405323\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.89204	−0.0875533	−0.0437766	−	0.999041i	$-0.513939\pi$
$467$	−1.89204	−0.0875533	−0.0437766	+	0.999041i	$0.513939\pi$
$468$	0	0
$469$	6.21110	0.286802
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.85936	0.177453
$474$	0	0
$475$	−9.29362	−0.426420
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.51658	−0.160677	−0.0803383	−	0.996768i	$-0.525600\pi$
$479$	−3.51658	−0.160677	−0.0803383	+	0.996768i	$0.525600\pi$
$480$	0	0
$481$	3.01938	0.137672
$482$	0	0
$483$	0	0
$484$	0	0
$485$	49.0755	2.22840
$486$	0	0
$487$	−21.2203	−0.961582	−0.480791	−	0.876835i	$-0.659650\pi$
$487$	−21.2203	−0.961582	−0.480791	+	0.876835i	$0.659650\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.4536	0.562025	0.281012	−	0.959704i	$-0.409330\pi$
$491$	12.4536	0.562025	0.281012	+	0.959704i	$0.409330\pi$
$492$	0	0
$493$	−27.9798	−1.26015
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.1392	−0.768798
$498$	0	0
$499$	−33.0755	−1.48066	−0.740331	−	0.672243i	$-0.765331\pi$
$499$	−33.0755	−1.48066	−0.740331	+	0.672243i	$0.765331\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.5824	0.561022	0.280511	−	0.959851i	$-0.409496\pi$
$503$	12.5824	0.561022	0.280511	+	0.959851i	$0.409496\pi$
$504$	0	0
$505$	0.925197	0.0411707
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−28.0465	−1.24314	−0.621569	−	0.783360i	$-0.713504\pi$
$509$	−28.0465	−1.24314	−0.621569	+	0.783360i	$0.713504\pi$
$510$	0	0
$511$	−5.39069	−0.238470
$512$	0	0
$513$	0	0
$514$	0	0
$515$	49.9017	2.19893
$516$	0	0
$517$	3.46886	0.152560
$518$	0	0
$519$	0	0
$520$	0	0
$521$	37.7956	1.65586	0.827928	−	0.560834i	$-0.189519\pi$
$521$	37.7956	1.65586	0.827928	+	0.560834i	$0.189519\pi$
$522$	0	0
$523$	17.1946	0.751868	0.375934	−	0.926646i	$-0.377322\pi$
$523$	17.1946	0.751868	0.375934	+	0.926646i	$0.377322\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	55.3539	2.41125
$528$	0	0
$529$	−12.3788	−0.538210
$530$	0	0
$531$	0	0
$532$	0	0
$533$	18.0048	0.779875
$534$	0	0
$535$	3.88500	0.167963
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.27067	−0.0978048
$540$	0	0
$541$	−3.94939	−0.169797	−0.0848987	−	0.996390i	$-0.527057\pi$
$541$	−3.94939	−0.169797	−0.0848987	+	0.996390i	$0.527057\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.2534	−0.696220
$546$	0	0
$547$	−17.1094	−0.731545	−0.365773	−	0.930704i	$-0.619195\pi$
$547$	−17.1094	−0.731545	−0.365773	+	0.930704i	$0.619195\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.05061	−0.257765
$552$	0	0
$553$	8.02564	0.341285
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.29362	−0.139555	−0.0697775	−	0.997563i	$-0.522229\pi$
$557$	−3.29362	−0.139555	−0.0697775	+	0.997563i	$0.522229\pi$
$558$	0	0
$559$	23.2430	0.983074
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.10459	0.130843	0.0654214	−	0.997858i	$-0.479161\pi$
$563$	3.10459	0.130843	0.0654214	+	0.997858i	$0.479161\pi$
$564$	0	0
$565$	51.7223	2.17597
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.4024	−0.603778	−0.301889	−	0.953343i	$-0.597617\pi$
$569$	−14.4024	−0.603778	−0.301889	+	0.953343i	$0.597617\pi$
$570$	0	0
$571$	34.1198	1.42787	0.713935	−	0.700212i	$-0.246911\pi$
$571$	34.1198	1.42787	0.713935	+	0.700212i	$0.246911\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	19.7008	0.821580
$576$	0	0
$577$	−43.6143	−1.81569	−0.907844	−	0.419308i	$-0.862273\pi$
$577$	−43.6143	−1.81569	−0.907844	+	0.419308i	$0.862273\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.9002	−0.618166
$582$	0	0
$583$	−4.28735	−0.177564
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10.6170	0.438211	0.219106	−	0.975701i	$-0.429686\pi$
$587$	10.6170	0.438211	0.219106	+	0.975701i	$0.429686\pi$
$588$	0	0
$589$	11.9702	0.493224
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.621168	0.0255083	0.0127541	−	0.999919i	$-0.495940\pi$
$593$	0.621168	0.0255083	0.0127541	+	0.999919i	$0.495940\pi$
$594$	0	0
$595$	26.9162	1.10346
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.01523	0.123199	0.0615995	−	0.998101i	$-0.480380\pi$
$599$	3.01523	0.123199	0.0615995	+	0.998101i	$0.480380\pi$
$600$	0	0
$601$	−6.24378	−0.254689	−0.127345	−	0.991859i	$-0.540645\pi$
$601$	−6.24378	−0.254689	−0.127345	+	0.991859i	$0.540645\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−36.0305	−1.46485
$606$	0	0
$607$	0.526989	0.0213898	0.0106949	−	0.999943i	$-0.496596\pi$
$607$	0.526989	0.0213898	0.0106949	+	0.999943i	$0.496596\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20.8913	0.845170
$612$	0	0
$613$	44.6564	1.80366	0.901828	−	0.432094i	$-0.142225\pi$
$613$	44.6564	1.80366	0.901828	+	0.432094i	$0.142225\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	44.5831	1.79485	0.897424	−	0.441170i	$-0.145436\pi$
$617$	44.5831	1.79485	0.897424	+	0.441170i	$0.145436\pi$
$618$	0	0
$619$	39.6531	1.59379	0.796896	−	0.604117i	$-0.206474\pi$
$619$	39.6531	1.59379	0.796896	+	0.604117i	$0.206474\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−18.7202	−0.750007
$624$	0	0
$625$	−18.6829	−0.747314
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.95084	−0.356893
$630$	0	0
$631$	6.30818	0.251125	0.125562	−	0.992086i	$-0.459927\pi$
$631$	6.30818	0.251125	0.125562	+	0.992086i	$0.459927\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−14.6170	−0.580059
$636$	0	0
$637$	−13.6751	−0.541829
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.5062	−0.651954	−0.325977	−	0.945378i	$-0.605693\pi$
$641$	−16.5062	−0.651954	−0.325977	+	0.945378i	$0.605693\pi$
$642$	0	0
$643$	22.1455	0.873332	0.436666	−	0.899624i	$-0.356159\pi$
$643$	22.1455	0.873332	0.436666	+	0.899624i	$0.356159\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.4376	−0.488974	−0.244487	−	0.969653i	$-0.578619\pi$
$647$	−12.4376	−0.488974	−0.244487	+	0.969653i	$0.578619\pi$
$648$	0	0
$649$	0.398207	0.0156310
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.3941	0.406751	0.203376	−	0.979101i	$-0.434809\pi$
$653$	10.3941	0.406751	0.203376	+	0.979101i	$0.434809\pi$
$654$	0	0
$655$	58.6981	2.29352
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−46.5139	−1.81192	−0.905962	−	0.423360i	$-0.860851\pi$
$659$	−46.5139	−1.81192	−0.905962	+	0.423360i	$0.860851\pi$
$660$	0	0
$661$	−31.3788	−1.22050	−0.610248	−	0.792211i	$-0.708930\pi$
$661$	−31.3788	−1.22050	−0.610248	+	0.792211i	$0.708930\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.82061	0.225713
$666$	0	0
$667$	12.8262	0.496633
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.394061	0.0152126
$672$	0	0
$673$	23.2936	0.897903	0.448951	−	0.893556i	$-0.351798\pi$
$673$	23.2936	0.897903	0.448951	+	0.893556i	$0.351798\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.775591	0.0298084	0.0149042	−	0.999889i	$-0.495256\pi$
$677$	0.775591	0.0298084	0.0149042	+	0.999889i	$0.495256\pi$
$678$	0	0
$679$	−16.8221	−0.645571
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13.5810	−0.519661	−0.259831	−	0.965654i	$-0.583667\pi$
$683$	−13.5810	−0.519661	−0.259831	+	0.965654i	$0.583667\pi$
$684$	0	0
$685$	35.7819	1.36715
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−25.8206	−0.983687
$690$	0	0
$691$	−18.6981	−0.711309	−0.355654	−	0.934618i	$-0.615742\pi$
$691$	−18.6981	−0.711309	−0.355654	+	0.934618i	$0.615742\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−32.6378	−1.23802
$696$	0	0
$697$	−53.3747	−2.02171
$698$	0	0
$699$	0	0
$700$	0	0
$701$	46.7625	1.76619	0.883097	−	0.469190i	$-0.155454\pi$
$701$	46.7625	1.76619	0.883097	+	0.469190i	$0.155454\pi$
$702$	0	0
$703$	−1.93561	−0.0730029
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.317138	−0.0119272
$708$	0	0
$709$	2.98477	0.112095	0.0560477	−	0.998428i	$-0.482150\pi$
$709$	2.98477	0.112095	0.0560477	+	0.998428i	$0.482150\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−25.3747	−0.950289
$714$	0	0
$715$	3.17380	0.118693
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33.0915	−1.23410	−0.617052	−	0.786922i	$-0.711673\pi$
$719$	−33.0915	−1.23410	−0.617052	+	0.786922i	$0.711673\pi$
$720$	0	0
$721$	−17.1053	−0.637033
$722$	0	0
$723$	0	0
$724$	0	0
$725$	23.7908	0.883569
$726$	0	0
$727$	−25.9100	−0.960948	−0.480474	−	0.877009i	$-0.659535\pi$
$727$	−25.9100	−0.960948	−0.480474	+	0.877009i	$0.659535\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−68.9031	−2.54847
$732$	0	0
$733$	−3.43426	−0.126847	−0.0634237	−	0.997987i	$-0.520202\pi$
$733$	−3.43426	−0.126847	−0.0634237	+	0.997987i	$0.520202\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.17110	−0.0799735
$738$	0	0
$739$	−19.0014	−0.698980	−0.349490	−	0.936940i	$-0.613645\pi$
$739$	−19.0014	−0.698980	−0.349490	+	0.936940i	$0.613645\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.1607	−0.482819	−0.241409	−	0.970423i	$-0.577610\pi$
$743$	−13.1607	−0.482819	−0.241409	+	0.970423i	$0.577610\pi$
$744$	0	0
$745$	16.6814	0.611160
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.33170	−0.0486591
$750$	0	0
$751$	14.6981	0.536341	0.268170	−	0.963371i	$-0.413581\pi$
$751$	14.6981	0.536341	0.268170	+	0.963371i	$0.413581\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−24.4536	−0.889959
$756$	0	0
$757$	3.27984	0.119208	0.0596039	−	0.998222i	$-0.481016\pi$
$757$	3.27984	0.119208	0.0596039	+	0.998222i	$0.481016\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15.5810	0.564810	0.282405	−	0.959295i	$-0.408868\pi$
$761$	15.5810	0.564810	0.282405	+	0.959295i	$0.408868\pi$
$762$	0	0
$763$	5.57133	0.201696
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.39821	0.0865943
$768$	0	0
$769$	11.0402	0.398120	0.199060	−	0.979987i	$-0.436211\pi$
$769$	11.0402	0.398120	0.199060	+	0.979987i	$0.436211\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−39.8600	−1.43367	−0.716833	−	0.697245i	$-0.754409\pi$
$773$	−39.8600	−1.43367	−0.716833	+	0.697245i	$0.754409\pi$
$774$	0	0
$775$	−47.0665	−1.69068
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.5422	−0.413543
$780$	0	0
$781$	5.99104	0.214376
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.75140	−0.205276
$786$	0	0
$787$	−24.0692	−0.857975	−0.428987	−	0.903311i	$-0.641129\pi$
$787$	−24.0692	−0.857975	−0.428987	+	0.903311i	$0.641129\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17.7293	−0.630382
$792$	0	0
$793$	2.37324	0.0842761
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.3088	−0.365158	−0.182579	−	0.983191i	$-0.558445\pi$
$797$	−10.3088	−0.365158	−0.182579	+	0.983191i	$0.558445\pi$
$798$	0	0
$799$	−61.9315	−2.19098
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.88433	0.0664965
$804$	0	0
$805$	−12.3386	−0.434880
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24.7119	−0.868823	−0.434412	−	0.900715i	$-0.643044\pi$
$809$	−24.7119	−0.868823	−0.434412	+	0.900715i	$0.643044\pi$
$810$	0	0
$811$	16.3088	0.572681	0.286341	−	0.958128i	$-0.407561\pi$
$811$	16.3088	0.572681	0.286341	+	0.958128i	$0.407561\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	28.5228	0.999112
$816$	0	0
$817$	−14.9002	−0.521293
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.5076	1.37883	0.689413	−	0.724369i	$-0.257869\pi$
$821$	39.5076	1.37883	0.689413	+	0.724369i	$0.257869\pi$
$822$	0	0
$823$	0.655771	0.0228588	0.0114294	−	0.999935i	$-0.496362\pi$
$823$	0.655771	0.0228588	0.0114294	+	0.999935i	$0.496362\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.5035	1.16503	0.582515	−	0.812820i	$-0.302069\pi$
$827$	33.5035	1.16503	0.582515	+	0.812820i	$0.302069\pi$
$828$	0	0
$829$	28.1752	0.978567	0.489283	−	0.872125i	$-0.337258\pi$
$829$	28.1752	0.978567	0.489283	+	0.872125i	$0.337258\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	40.5395	1.40461
$834$	0	0
$835$	21.5768	0.746697
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−21.9315	−0.757158	−0.378579	−	0.925569i	$-0.623587\pi$
$839$	−21.9315	−0.757158	−0.378579	+	0.925569i	$0.623587\pi$
$840$	0	0
$841$	−13.5110	−0.465896
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−24.0900	−0.828722
$846$	0	0
$847$	12.3505	0.424368
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.10314	0.140654
$852$	0	0
$853$	−22.9903	−0.787171	−0.393586	−	0.919288i	$-0.628766\pi$
$853$	−22.9903	−0.787171	−0.393586	+	0.919288i	$0.628766\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.5574	0.565592	0.282796	−	0.959180i	$-0.408738\pi$
$857$	16.5574	0.565592	0.282796	+	0.959180i	$0.408738\pi$
$858$	0	0
$859$	3.50280	0.119514	0.0597570	−	0.998213i	$-0.480967\pi$
$859$	3.50280	0.119514	0.0597570	+	0.998213i	$0.480967\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.1994	−0.653557	−0.326778	−	0.945101i	$-0.605963\pi$
$863$	−19.1994	−0.653557	−0.326778	+	0.945101i	$0.605963\pi$
$864$	0	0
$865$	68.2749	2.32142
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.80538	−0.0951659
$870$	0	0
$871$	−13.0755	−0.443046
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.95643	−0.133752
$876$	0	0
$877$	52.8864	1.78585	0.892924	−	0.450207i	$-0.148650\pi$
$877$	52.8864	1.78585	0.892924	+	0.450207i	$0.148650\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.86562	0.231309	0.115654	−	0.993290i	$-0.463104\pi$
$881$	6.86562	0.231309	0.115654	+	0.993290i	$0.463104\pi$
$882$	0	0
$883$	51.6233	1.73726	0.868631	−	0.495460i	$-0.165000\pi$
$883$	51.6233	1.73726	0.868631	+	0.495460i	$0.165000\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.7126	0.661886	0.330943	−	0.943651i	$-0.392633\pi$
$887$	19.7126	0.661886	0.330943	+	0.943651i	$0.392633\pi$
$888$	0	0
$889$	5.01041	0.168044
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.3926	−0.448167
$894$	0	0
$895$	−79.6627	−2.66283
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−30.6427	−1.02199
$900$	0	0
$901$	76.5443	2.55006
$902$	0	0
$903$	0	0
$904$	0	0
$905$	48.8913	1.62520
$906$	0	0
$907$	40.5366	1.34600	0.672998	−	0.739644i	$-0.265006\pi$
$907$	40.5366	1.34600	0.672998	+	0.739644i	$0.265006\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	54.5949	1.80881	0.904406	−	0.426674i	$-0.140315\pi$
$911$	54.5949	1.80881	0.904406	+	0.426674i	$0.140315\pi$
$912$	0	0
$913$	5.20840	0.172373
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20.1205	−0.664437
$918$	0	0
$919$	6.95565	0.229446	0.114723	−	0.993398i	$-0.463402\pi$
$919$	6.95565	0.229446	0.114723	+	0.993398i	$0.463402\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	36.0811	1.18762
$924$	0	0
$925$	7.61076	0.250240
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16.5478	−0.542916	−0.271458	−	0.962450i	$-0.587506\pi$
$929$	−16.5478	−0.542916	−0.271458	+	0.962450i	$0.587506\pi$
$930$	0	0
$931$	8.76663	0.287315
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.40862	−0.307695
$936$	0	0
$937$	−28.6766	−0.936824	−0.468412	−	0.883510i	$-0.655174\pi$
$937$	−28.6766	−0.936824	−0.468412	+	0.883510i	$0.655174\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24.1198	0.786284	0.393142	−	0.919478i	$-0.371388\pi$
$941$	24.1198	0.786284	0.393142	+	0.919478i	$0.371388\pi$
$942$	0	0
$943$	24.4674	0.796769
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.0617	1.07436	0.537180	−	0.843467i	$-0.319489\pi$
$947$	33.0617	1.07436	0.537180	+	0.843467i	$0.319489\pi$
$948$	0	0
$949$	11.3484	0.368384
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.79641	0.220157	0.110079	−	0.993923i	$-0.464890\pi$
$953$	6.79641	0.220157	0.110079	+	0.993923i	$0.464890\pi$
$954$	0	0
$955$	9.69182	0.313620
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.2653	−0.396067
$960$	0	0
$961$	29.6218	0.955543
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.14961	−0.0691983
$966$	0	0
$967$	43.7479	1.40684	0.703419	−	0.710775i	$-0.251656\pi$
$967$	43.7479	1.40684	0.703419	+	0.710775i	$0.251656\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.3434	0.973768	0.486884	−	0.873467i	$-0.338134\pi$
$971$	30.3434	0.973768	0.486884	+	0.873467i	$0.338134\pi$
$972$	0	0
$973$	11.1876	0.358657
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−47.2638	−1.51210	−0.756052	−	0.654512i	$-0.772874\pi$
$977$	−47.2638	−1.51210	−0.756052	+	0.654512i	$0.772874\pi$
$978$	0	0
$979$	6.54367	0.209137
$980$	0	0
$981$	0	0
$982$	0	0
$983$	30.9854	0.988282	0.494141	−	0.869382i	$-0.335483\pi$
$983$	30.9854	0.988282	0.494141	+	0.869382i	$0.335483\pi$
$984$	0	0
$985$	19.5124	0.621718
$986$	0	0
$987$	0	0
$988$	0	0
$989$	31.5858	1.00437
$990$	0	0
$991$	18.8131	0.597618	0.298809	−	0.954313i	$-0.403411\pi$
$991$	18.8131	0.597618	0.298809	+	0.954313i	$0.403411\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−36.8913	−1.16953
$996$	0	0
$997$	16.5020	0.522624	0.261312	−	0.965254i	$-0.415845\pi$
$997$	16.5020	0.522624	0.261312	+	0.965254i	$0.415845\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8496.2.a.bl.1.3		3
3.2	odd	2		2832.2.a.t.1.1		3
4.3	odd	2		531.2.a.d.1.3		3
12.11	even	2		177.2.a.d.1.1	✓	3
60.59	even	2		4425.2.a.w.1.3		3
84.83	odd	2		8673.2.a.s.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.d.1.1	✓	3	12.11	even	2
531.2.a.d.1.3		3	4.3	odd	2
2832.2.a.t.1.1		3	3.2	odd	2
4425.2.a.w.1.3		3	60.59	even	2
8496.2.a.bl.1.3		3	1.1	even	1	trivial
8673.2.a.s.1.1		3	84.83	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:	\( N \)	\(=\)	\( 8496 = 2^{4} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8496.a (trivial)

\(f(q)\)	\(=\)	\(q+3.32340 q^{5} -1.13919 q^{7} +O(q^{10})\)	\(q+3.32340 q^{5} -1.13919 q^{7} +0.398207 q^{11} +2.39821 q^{13} -7.10941 q^{17} -1.53740 q^{19} +3.25901 q^{23} +6.04502 q^{25} +3.93561 q^{29} -7.78600 q^{31} -3.78600 q^{35} +1.25901 q^{37} +7.50761 q^{41} +9.69182 q^{43} +8.71120 q^{47} -5.70224 q^{49} -10.7666 q^{53} +1.32340 q^{55} +1.00000 q^{59} +0.989588 q^{61} +7.97021 q^{65} -5.45219 q^{67} +15.0450 q^{71} +4.73202 q^{73} -0.453636 q^{77} -7.04502 q^{79} +13.0796 q^{83} -23.6274 q^{85} +16.4328 q^{89} -2.73202 q^{91} -5.10941 q^{95} +14.7666 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} - 9 q^{7}+O(q^{10}) \) 3 * q + 2 * q^5 - 9 * q^7	\( 3 q + 2 q^{5} - 9 q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} - 7 q^{19} + q^{23} - q^{25} + 11 q^{29} - 13 q^{31} - q^{35} - 5 q^{37} + q^{41} - 6 q^{43} + 11 q^{47} + 14 q^{49} - 2 q^{53} - 4 q^{55} + 3 q^{59} - q^{61} - 10 q^{67} + 26 q^{71} + 7 q^{73} + 17 q^{77} - 2 q^{79} - 3 q^{83} - 35 q^{85} + 23 q^{89} - q^{91} + 3 q^{95} + 14 q^{97}+O(q^{100}) \) 3 * q + 2 * q^5 - 9 * q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 - 7 * q^19 + q^23 - q^25 + 11 * q^29 - 13 * q^31 - q^35 - 5 * q^37 + q^41 - 6 * q^43 + 11 * q^47 + 14 * q^49 - 2 * q^53 - 4 * q^55 + 3 * q^59 - q^61 - 10 * q^67 + 26 * q^71 + 7 * q^73 + 17 * q^77 - 2 * q^79 - 3 * q^83 - 35 * q^85 + 23 * q^89 - q^91 + 3 * q^95 + 14 * q^97

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