LMFDB - Embedded newform 5824.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5824,2,Mod(1,5824)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5824, 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("5824.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5824 = 2^{6} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5824.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:	$46.5048741372$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	5824.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.14637 q^{3} +1.34292 q^{5} +1.00000 q^{7} -1.68585 q^{9} +O(q^{10})$	$q-1.14637 q^{3} +1.34292 q^{5} +1.00000 q^{7} -1.68585 q^{9} +1.14637 q^{11} -1.00000 q^{13} -1.53948 q^{15} +5.83221 q^{17} -3.34292 q^{19} -1.14637 q^{21} +3.17513 q^{23} -3.19656 q^{25} +5.37169 q^{27} -10.4893 q^{29} -1.63565 q^{31} -1.31415 q^{33} +1.34292 q^{35} -8.51806 q^{37} +1.14637 q^{39} -0.292731 q^{41} -8.15371 q^{43} -2.26396 q^{45} +10.6142 q^{47} +1.00000 q^{49} -6.68585 q^{51} +0.782020 q^{53} +1.53948 q^{55} +3.83221 q^{57} +12.6430 q^{59} +2.00000 q^{61} -1.68585 q^{63} -1.34292 q^{65} -6.10038 q^{67} -3.63986 q^{69} -1.53948 q^{71} -15.3001 q^{73} +3.66442 q^{75} +1.14637 q^{77} -0.882404 q^{79} -1.10038 q^{81} -12.1292 q^{83} +7.83221 q^{85} +12.0246 q^{87} +5.73604 q^{89} -1.00000 q^{91} +1.87506 q^{93} -4.48929 q^{95} -5.34292 q^{97} -1.93260 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 2 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - 2 * q^5 + 3 * q^7 + 7 * q^9	$ 3 q - 2 q^{3} - 2 q^{5} + 3 q^{7} + 7 q^{9} + 2 q^{11} - 3 q^{13} + 6 q^{15} + 4 q^{17} - 4 q^{19} - 2 q^{21} - 10 q^{23} - 5 q^{25} - 8 q^{27} - 24 q^{29} + 4 q^{31} - 16 q^{33} - 2 q^{35} + 2 q^{39} + 2 q^{41} + 10 q^{43} - 22 q^{45} + 8 q^{47} + 3 q^{49} - 8 q^{51} - 8 q^{53} - 6 q^{55} - 2 q^{57} - 4 q^{59} + 6 q^{61} + 7 q^{63} + 2 q^{65} - 12 q^{67} + 6 q^{69} + 6 q^{71} - 10 q^{73} - 16 q^{75} + 2 q^{77} + 14 q^{79} + 3 q^{81} - 12 q^{83} + 10 q^{85} + 26 q^{87} + 2 q^{89} - 3 q^{91} + 22 q^{93} - 6 q^{95} - 10 q^{97} + 14 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 2 * q^5 + 3 * q^7 + 7 * q^9 + 2 * q^11 - 3 * q^13 + 6 * q^15 + 4 * q^17 - 4 * q^19 - 2 * q^21 - 10 * q^23 - 5 * q^25 - 8 * q^27 - 24 * q^29 + 4 * q^31 - 16 * q^33 - 2 * q^35 + 2 * q^39 + 2 * q^41 + 10 * q^43 - 22 * q^45 + 8 * q^47 + 3 * q^49 - 8 * q^51 - 8 * q^53 - 6 * q^55 - 2 * q^57 - 4 * q^59 + 6 * q^61 + 7 * q^63 + 2 * q^65 - 12 * q^67 + 6 * q^69 + 6 * q^71 - 10 * q^73 - 16 * q^75 + 2 * q^77 + 14 * q^79 + 3 * q^81 - 12 * q^83 + 10 * q^85 + 26 * q^87 + 2 * q^89 - 3 * q^91 + 22 * q^93 - 6 * q^95 - 10 * q^97 + 14 * q^99

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
$3$	−1.14637	−0.661854	−0.330927	−	0.943656i	$-0.607361\pi$
$3$	−1.14637	−0.661854	−0.330927	+	0.943656i	$0.607361\pi$
$4$	0	0
$5$	1.34292	0.600573	0.300287	−	0.953849i	$-0.402918\pi$
$5$	1.34292	0.600573	0.300287	+	0.953849i	$0.402918\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.68585	−0.561949
$10$	0	0
$11$	1.14637	0.345642	0.172821	−	0.984953i	$-0.444712\pi$
$11$	1.14637	0.345642	0.172821	+	0.984953i	$0.444712\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.53948	−0.397492
$16$	0	0
$17$	5.83221	1.41452	0.707260	−	0.706954i	$-0.249931\pi$
$17$	5.83221	1.41452	0.707260	+	0.706954i	$0.249931\pi$
$18$	0	0
$19$	−3.34292	−0.766919	−0.383460	−	0.923558i	$-0.625267\pi$
$19$	−3.34292	−0.766919	−0.383460	+	0.923558i	$0.625267\pi$
$20$	0	0
$21$	−1.14637	−0.250157
$22$	0	0
$23$	3.17513	0.662061	0.331031	−	0.943620i	$-0.392604\pi$
$23$	3.17513	0.662061	0.331031	+	0.943620i	$0.392604\pi$
$24$	0	0
$25$	−3.19656	−0.639312
$26$	0	0
$27$	5.37169	1.03378
$28$	0	0
$29$	−10.4893	−1.94781	−0.973906	−	0.226952i	$-0.927124\pi$
$29$	−10.4893	−1.94781	−0.973906	+	0.226952i	$0.927124\pi$
$30$	0	0
$31$	−1.63565	−0.293772	−0.146886	−	0.989153i	$-0.546925\pi$
$31$	−1.63565	−0.293772	−0.146886	+	0.989153i	$0.546925\pi$
$32$	0	0
$33$	−1.31415	−0.228765
$34$	0	0
$35$	1.34292	0.226995
$36$	0	0
$37$	−8.51806	−1.40036	−0.700180	−	0.713966i	$-0.746897\pi$
$37$	−8.51806	−1.40036	−0.700180	+	0.713966i	$0.746897\pi$
$38$	0	0
$39$	1.14637	0.183565
$40$	0	0
$41$	−0.292731	−0.0457169	−0.0228584	−	0.999739i	$-0.507277\pi$
$41$	−0.292731	−0.0457169	−0.0228584	+	0.999739i	$0.507277\pi$
$42$	0	0
$43$	−8.15371	−1.24343	−0.621715	−	0.783244i	$-0.713564\pi$
$43$	−8.15371	−1.24343	−0.621715	+	0.783244i	$0.713564\pi$
$44$	0	0
$45$	−2.26396	−0.337491
$46$	0	0
$47$	10.6142	1.54824	0.774122	−	0.633036i	$-0.218191\pi$
$47$	10.6142	1.54824	0.774122	+	0.633036i	$0.218191\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.68585	−0.936206
$52$	0	0
$53$	0.782020	0.107419	0.0537093	−	0.998557i	$-0.482896\pi$
$53$	0.782020	0.107419	0.0537093	+	0.998557i	$0.482896\pi$
$54$	0	0
$55$	1.53948	0.207584
$56$	0	0
$57$	3.83221	0.507589
$58$	0	0
$59$	12.6430	1.64598	0.822989	−	0.568057i	$-0.192305\pi$
$59$	12.6430	1.64598	0.822989	+	0.568057i	$0.192305\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−1.68585	−0.212397
$64$	0	0
$65$	−1.34292	−0.166569
$66$	0	0
$67$	−6.10038	−0.745281	−0.372640	−	0.927976i	$-0.621547\pi$
$67$	−6.10038	−0.745281	−0.372640	+	0.927976i	$0.621547\pi$
$68$	0	0
$69$	−3.63986	−0.438188
$70$	0	0
$71$	−1.53948	−0.182703	−0.0913514	−	0.995819i	$-0.529119\pi$
$71$	−1.53948	−0.182703	−0.0913514	+	0.995819i	$0.529119\pi$
$72$	0	0
$73$	−15.3001	−1.79074	−0.895369	−	0.445324i	$-0.853088\pi$
$73$	−15.3001	−1.79074	−0.895369	+	0.445324i	$0.853088\pi$
$74$	0	0
$75$	3.66442	0.423131
$76$	0	0
$77$	1.14637	0.130640
$78$	0	0
$79$	−0.882404	−0.0992782	−0.0496391	−	0.998767i	$-0.515807\pi$
$79$	−0.882404	−0.0992782	−0.0496391	+	0.998767i	$0.515807\pi$
$80$	0	0
$81$	−1.10038	−0.122265
$82$	0	0
$83$	−12.1292	−1.33135	−0.665674	−	0.746243i	$-0.731856\pi$
$83$	−12.1292	−1.33135	−0.665674	+	0.746243i	$0.731856\pi$
$84$	0	0
$85$	7.83221	0.849523
$86$	0	0
$87$	12.0246	1.28917
$88$	0	0
$89$	5.73604	0.608019	0.304009	−	0.952669i	$-0.401675\pi$
$89$	5.73604	0.608019	0.304009	+	0.952669i	$0.401675\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	1.87506	0.194434
$94$	0	0
$95$	−4.48929	−0.460591
$96$	0	0
$97$	−5.34292	−0.542492	−0.271246	−	0.962510i	$-0.587436\pi$
$97$	−5.34292	−0.542492	−0.271246	+	0.962510i	$0.587436\pi$
$98$	0	0
$99$	−1.93260	−0.194233
$100$	0	0
$101$	−11.1464	−1.10910	−0.554552	−	0.832149i	$-0.687111\pi$
$101$	−11.1464	−1.10910	−0.554552	+	0.832149i	$0.687111\pi$
$102$	0	0
$103$	3.41454	0.336444	0.168222	−	0.985749i	$-0.446197\pi$
$103$	3.41454	0.336444	0.168222	+	0.985749i	$0.446197\pi$
$104$	0	0
$105$	−1.53948	−0.150238
$106$	0	0
$107$	4.97858	0.481297	0.240649	−	0.970612i	$-0.422640\pi$
$107$	4.97858	0.481297	0.240649	+	0.970612i	$0.422640\pi$
$108$	0	0
$109$	13.4966	1.29274	0.646372	−	0.763023i	$-0.276286\pi$
$109$	13.4966	1.29274	0.646372	+	0.763023i	$0.276286\pi$
$110$	0	0
$111$	9.76481	0.926835
$112$	0	0
$113$	16.4464	1.54715	0.773576	−	0.633704i	$-0.218466\pi$
$113$	16.4464	1.54715	0.773576	+	0.633704i	$0.218466\pi$
$114$	0	0
$115$	4.26396	0.397616
$116$	0	0
$117$	1.68585	0.155857
$118$	0	0
$119$	5.83221	0.534638
$120$	0	0
$121$	−9.68585	−0.880531
$122$	0	0
$123$	0.335577	0.0302579
$124$	0	0
$125$	−11.0073	−0.984527
$126$	0	0
$127$	−12.0575	−1.06993	−0.534967	−	0.844873i	$-0.679676\pi$
$127$	−12.0575	−1.06993	−0.534967	+	0.844873i	$0.679676\pi$
$128$	0	0
$129$	9.34713	0.822969
$130$	0	0
$131$	−3.66442	−0.320162	−0.160081	−	0.987104i	$-0.551176\pi$
$131$	−3.66442	−0.320162	−0.160081	+	0.987104i	$0.551176\pi$
$132$	0	0
$133$	−3.34292	−0.289868
$134$	0	0
$135$	7.21377	0.620862
$136$	0	0
$137$	−13.1035	−1.11951	−0.559755	−	0.828658i	$-0.689105\pi$
$137$	−13.1035	−1.11951	−0.559755	+	0.828658i	$0.689105\pi$
$138$	0	0
$139$	7.49663	0.635856	0.317928	−	0.948115i	$-0.397013\pi$
$139$	7.49663	0.635856	0.317928	+	0.948115i	$0.397013\pi$
$140$	0	0
$141$	−12.1678	−1.02471
$142$	0	0
$143$	−1.14637	−0.0958639
$144$	0	0
$145$	−14.0863	−1.16980
$146$	0	0
$147$	−1.14637	−0.0945506
$148$	0	0
$149$	−2.16779	−0.177592	−0.0887961	−	0.996050i	$-0.528302\pi$
$149$	−2.16779	−0.177592	−0.0887961	+	0.996050i	$0.528302\pi$
$150$	0	0
$151$	−14.9112	−1.21345	−0.606727	−	0.794910i	$-0.707518\pi$
$151$	−14.9112	−1.21345	−0.606727	+	0.794910i	$0.707518\pi$
$152$	0	0
$153$	−9.83221	−0.794887
$154$	0	0
$155$	−2.19656	−0.176432
$156$	0	0
$157$	−22.8683	−1.82509	−0.912546	−	0.408975i	$-0.865886\pi$
$157$	−22.8683	−1.82509	−0.912546	+	0.408975i	$0.865886\pi$
$158$	0	0
$159$	−0.896480	−0.0710955
$160$	0	0
$161$	3.17513	0.250236
$162$	0	0
$163$	7.07896	0.554467	0.277234	−	0.960803i	$-0.410582\pi$
$163$	7.07896	0.554467	0.277234	+	0.960803i	$0.410582\pi$
$164$	0	0
$165$	−1.76481	−0.137390
$166$	0	0
$167$	−2.61423	−0.202295	−0.101148	−	0.994871i	$-0.532251\pi$
$167$	−2.61423	−0.202295	−0.101148	+	0.994871i	$0.532251\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.63565	0.430969
$172$	0	0
$173$	−11.0031	−0.836553	−0.418276	−	0.908320i	$-0.637366\pi$
$173$	−11.0031	−0.836553	−0.418276	+	0.908320i	$0.637366\pi$
$174$	0	0
$175$	−3.19656	−0.241637
$176$	0	0
$177$	−14.4935	−1.08940
$178$	0	0
$179$	23.9614	1.79096	0.895478	−	0.445105i	$-0.146834\pi$
$179$	23.9614	1.79096	0.895478	+	0.445105i	$0.146834\pi$
$180$	0	0
$181$	−6.56090	−0.487668	−0.243834	−	0.969817i	$-0.578405\pi$
$181$	−6.56090	−0.487668	−0.243834	+	0.969817i	$0.578405\pi$
$182$	0	0
$183$	−2.29273	−0.169484
$184$	0	0
$185$	−11.4391	−0.841019
$186$	0	0
$187$	6.68585	0.488917
$188$	0	0
$189$	5.37169	0.390733
$190$	0	0
$191$	4.39312	0.317875	0.158937	−	0.987289i	$-0.449193\pi$
$191$	4.39312	0.317875	0.158937	+	0.987289i	$0.449193\pi$
$192$	0	0
$193$	−8.29273	−0.596924	−0.298462	−	0.954422i	$-0.596474\pi$
$193$	−8.29273	−0.596924	−0.298462	+	0.954422i	$0.596474\pi$
$194$	0	0
$195$	1.53948	0.110245
$196$	0	0
$197$	3.17092	0.225919	0.112959	−	0.993600i	$-0.463967\pi$
$197$	3.17092	0.225919	0.112959	+	0.993600i	$0.463967\pi$
$198$	0	0
$199$	13.5970	0.963867	0.481934	−	0.876208i	$-0.339935\pi$
$199$	13.5970	0.963867	0.481934	+	0.876208i	$0.339935\pi$
$200$	0	0
$201$	6.99327	0.493267
$202$	0	0
$203$	−10.4893	−0.736204
$204$	0	0
$205$	−0.393115	−0.0274564
$206$	0	0
$207$	−5.35279	−0.372045
$208$	0	0
$209$	−3.83221	−0.265080
$210$	0	0
$211$	9.27552	0.638553	0.319277	−	0.947662i	$-0.396560\pi$
$211$	9.27552	0.638553	0.319277	+	0.947662i	$0.396560\pi$
$212$	0	0
$213$	1.76481	0.120923
$214$	0	0
$215$	−10.9498	−0.746771
$216$	0	0
$217$	−1.63565	−0.111035
$218$	0	0
$219$	17.5395	1.18521
$220$	0	0
$221$	−5.83221	−0.392317
$222$	0	0
$223$	19.5928	1.31203	0.656016	−	0.754747i	$-0.272240\pi$
$223$	19.5928	1.31203	0.656016	+	0.754747i	$0.272240\pi$
$224$	0	0
$225$	5.38890	0.359260
$226$	0	0
$227$	−19.6644	−1.30517	−0.652587	−	0.757714i	$-0.726316\pi$
$227$	−19.6644	−1.30517	−0.652587	+	0.757714i	$0.726316\pi$
$228$	0	0
$229$	−7.76481	−0.513113	−0.256556	−	0.966529i	$-0.582588\pi$
$229$	−7.76481	−0.513113	−0.256556	+	0.966529i	$0.582588\pi$
$230$	0	0
$231$	−1.31415	−0.0864650
$232$	0	0
$233$	12.1966	0.799023	0.399512	−	0.916728i	$-0.369180\pi$
$233$	12.1966	0.799023	0.399512	+	0.916728i	$0.369180\pi$
$234$	0	0
$235$	14.2541	0.929835
$236$	0	0
$237$	1.01156	0.0657077
$238$	0	0
$239$	10.2927	0.665781	0.332891	−	0.942965i	$-0.391976\pi$
$239$	10.2927	0.665781	0.332891	+	0.942965i	$0.391976\pi$
$240$	0	0
$241$	−4.02877	−0.259516	−0.129758	−	0.991546i	$-0.541420\pi$
$241$	−4.02877	−0.259516	−0.129758	+	0.991546i	$0.541420\pi$
$242$	0	0
$243$	−14.8536	−0.952861
$244$	0	0
$245$	1.34292	0.0857962
$246$	0	0
$247$	3.34292	0.212705
$248$	0	0
$249$	13.9044	0.881158
$250$	0	0
$251$	2.91117	0.183752	0.0918758	−	0.995770i	$-0.470714\pi$
$251$	2.91117	0.183752	0.0918758	+	0.995770i	$0.470714\pi$
$252$	0	0
$253$	3.63986	0.228836
$254$	0	0
$255$	−8.97858	−0.562260
$256$	0	0
$257$	−19.5970	−1.22243	−0.611214	−	0.791465i	$-0.709319\pi$
$257$	−19.5970	−1.22243	−0.611214	+	0.791465i	$0.709319\pi$
$258$	0	0
$259$	−8.51806	−0.529286
$260$	0	0
$261$	17.6833	1.09457
$262$	0	0
$263$	−7.56825	−0.466678	−0.233339	−	0.972395i	$-0.574965\pi$
$263$	−7.56825	−0.466678	−0.233339	+	0.972395i	$0.574965\pi$
$264$	0	0
$265$	1.05019	0.0645128
$266$	0	0
$267$	−6.57560	−0.402420
$268$	0	0
$269$	9.47208	0.577523	0.288761	−	0.957401i	$-0.406757\pi$
$269$	9.47208	0.577523	0.288761	+	0.957401i	$0.406757\pi$
$270$	0	0
$271$	−29.3717	−1.78420	−0.892102	−	0.451835i	$-0.850770\pi$
$271$	−29.3717	−1.78420	−0.892102	+	0.451835i	$0.850770\pi$
$272$	0	0
$273$	1.14637	0.0693812
$274$	0	0
$275$	−3.66442	−0.220973
$276$	0	0
$277$	1.90383	0.114390	0.0571949	−	0.998363i	$-0.481784\pi$
$277$	1.90383	0.114390	0.0571949	+	0.998363i	$0.481784\pi$
$278$	0	0
$279$	2.75746	0.165085
$280$	0	0
$281$	−20.5756	−1.22744	−0.613719	−	0.789525i	$-0.710327\pi$
$281$	−20.5756	−1.22744	−0.613719	+	0.789525i	$0.710327\pi$
$282$	0	0
$283$	−26.9933	−1.60458	−0.802292	−	0.596932i	$-0.796386\pi$
$283$	−26.9933	−1.60458	−0.802292	+	0.596932i	$0.796386\pi$
$284$	0	0
$285$	5.14637	0.304844
$286$	0	0
$287$	−0.292731	−0.0172794
$288$	0	0
$289$	17.0147	1.00086
$290$	0	0
$291$	6.12494	0.359050
$292$	0	0
$293$	14.9070	0.870874	0.435437	−	0.900219i	$-0.356594\pi$
$293$	14.9070	0.870874	0.435437	+	0.900219i	$0.356594\pi$
$294$	0	0
$295$	16.9786	0.988531
$296$	0	0
$297$	6.15792	0.357319
$298$	0	0
$299$	−3.17513	−0.183623
$300$	0	0
$301$	−8.15371	−0.469972
$302$	0	0
$303$	12.7778	0.734066
$304$	0	0
$305$	2.68585	0.153791
$306$	0	0
$307$	26.0288	1.48554	0.742770	−	0.669546i	$-0.233512\pi$
$307$	26.0288	1.48554	0.742770	+	0.669546i	$0.233512\pi$
$308$	0	0
$309$	−3.91431	−0.222677
$310$	0	0
$311$	−19.4966	−1.10555	−0.552776	−	0.833330i	$-0.686432\pi$
$311$	−19.4966	−1.10555	−0.552776	+	0.833330i	$0.686432\pi$
$312$	0	0
$313$	−3.48194	−0.196811	−0.0984055	−	0.995146i	$-0.531374\pi$
$313$	−3.48194	−0.196811	−0.0984055	+	0.995146i	$0.531374\pi$
$314$	0	0
$315$	−2.26396	−0.127560
$316$	0	0
$317$	−5.02142	−0.282031	−0.141016	−	0.990007i	$-0.545037\pi$
$317$	−5.02142	−0.282031	−0.141016	+	0.990007i	$0.545037\pi$
$318$	0	0
$319$	−12.0246	−0.673246
$320$	0	0
$321$	−5.70727	−0.318549
$322$	0	0
$323$	−19.4966	−1.08482
$324$	0	0
$325$	3.19656	0.177313
$326$	0	0
$327$	−15.4721	−0.855608
$328$	0	0
$329$	10.6142	0.585182
$330$	0	0
$331$	−6.14950	−0.338007	−0.169004	−	0.985615i	$-0.554055\pi$
$331$	−6.14950	−0.338007	−0.169004	+	0.985615i	$0.554055\pi$
$332$	0	0
$333$	14.3601	0.786931
$334$	0	0
$335$	−8.19235	−0.447596
$336$	0	0
$337$	−25.6258	−1.39593	−0.697963	−	0.716134i	$-0.745910\pi$
$337$	−25.6258	−1.39593	−0.697963	+	0.716134i	$0.745910\pi$
$338$	0	0
$339$	−18.8536	−1.02399
$340$	0	0
$341$	−1.87506	−0.101540
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.88806	−0.263164
$346$	0	0
$347$	−16.7005	−0.896532	−0.448266	−	0.893900i	$-0.647958\pi$
$347$	−16.7005	−0.896532	−0.448266	+	0.893900i	$0.647958\pi$
$348$	0	0
$349$	23.5500	1.26060	0.630300	−	0.776351i	$-0.282932\pi$
$349$	23.5500	1.26060	0.630300	+	0.776351i	$0.282932\pi$
$350$	0	0
$351$	−5.37169	−0.286720
$352$	0	0
$353$	−7.64973	−0.407154	−0.203577	−	0.979059i	$-0.565257\pi$
$353$	−7.64973	−0.407154	−0.203577	+	0.979059i	$0.565257\pi$
$354$	0	0
$355$	−2.06740	−0.109726
$356$	0	0
$357$	−6.68585	−0.353853
$358$	0	0
$359$	−18.3748	−0.969786	−0.484893	−	0.874573i	$-0.661142\pi$
$359$	−18.3748	−0.969786	−0.484893	+	0.874573i	$0.661142\pi$
$360$	0	0
$361$	−7.82487	−0.411835
$362$	0	0
$363$	11.1035	0.582784
$364$	0	0
$365$	−20.5468	−1.07547
$366$	0	0
$367$	−5.33871	−0.278679	−0.139339	−	0.990245i	$-0.544498\pi$
$367$	−5.33871	−0.278679	−0.139339	+	0.990245i	$0.544498\pi$
$368$	0	0
$369$	0.493499	0.0256906
$370$	0	0
$371$	0.782020	0.0406004
$372$	0	0
$373$	−21.5212	−1.11433	−0.557163	−	0.830403i	$-0.688110\pi$
$373$	−21.5212	−1.11433	−0.557163	+	0.830403i	$0.688110\pi$
$374$	0	0
$375$	12.6184	0.651614
$376$	0	0
$377$	10.4893	0.540226
$378$	0	0
$379$	4.61002	0.236801	0.118400	−	0.992966i	$-0.462223\pi$
$379$	4.61002	0.236801	0.118400	+	0.992966i	$0.462223\pi$
$380$	0	0
$381$	13.8223	0.708140
$382$	0	0
$383$	8.33558	0.425928	0.212964	−	0.977060i	$-0.431688\pi$
$383$	8.33558	0.425928	0.212964	+	0.977060i	$0.431688\pi$
$384$	0	0
$385$	1.53948	0.0784592
$386$	0	0
$387$	13.7459	0.698744
$388$	0	0
$389$	6.44223	0.326634	0.163317	−	0.986574i	$-0.447781\pi$
$389$	6.44223	0.326634	0.163317	+	0.986574i	$0.447781\pi$
$390$	0	0
$391$	18.5181	0.936498
$392$	0	0
$393$	4.20077	0.211901
$394$	0	0
$395$	−1.18500	−0.0596238
$396$	0	0
$397$	−1.40046	−0.0702872	−0.0351436	−	0.999382i	$-0.511189\pi$
$397$	−1.40046	−0.0702872	−0.0351436	+	0.999382i	$0.511189\pi$
$398$	0	0
$399$	3.83221	0.191851
$400$	0	0
$401$	−6.97858	−0.348494	−0.174247	−	0.984702i	$-0.555749\pi$
$401$	−6.97858	−0.348494	−0.174247	+	0.984702i	$0.555749\pi$
$402$	0	0
$403$	1.63565	0.0814777
$404$	0	0
$405$	−1.47773	−0.0734291
$406$	0	0
$407$	−9.76481	−0.484024
$408$	0	0
$409$	−18.3790	−0.908785	−0.454392	−	0.890802i	$-0.650144\pi$
$409$	−18.3790	−0.908785	−0.454392	+	0.890802i	$0.650144\pi$
$410$	0	0
$411$	15.0214	0.740952
$412$	0	0
$413$	12.6430	0.622121
$414$	0	0
$415$	−16.2885	−0.799572
$416$	0	0
$417$	−8.59388	−0.420844
$418$	0	0
$419$	−30.0393	−1.46751	−0.733757	−	0.679412i	$-0.762235\pi$
$419$	−30.0393	−1.46751	−0.733757	+	0.679412i	$0.762235\pi$
$420$	0	0
$421$	8.31729	0.405360	0.202680	−	0.979245i	$-0.435035\pi$
$421$	8.31729	0.405360	0.202680	+	0.979245i	$0.435035\pi$
$422$	0	0
$423$	−17.8940	−0.870034
$424$	0	0
$425$	−18.6430	−0.904318
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	1.31415	0.0634479
$430$	0	0
$431$	−9.64973	−0.464811	−0.232406	−	0.972619i	$-0.574660\pi$
$431$	−9.64973	−0.464811	−0.232406	+	0.972619i	$0.574660\pi$
$432$	0	0
$433$	26.3074	1.26425	0.632127	−	0.774865i	$-0.282182\pi$
$433$	26.3074	1.26425	0.632127	+	0.774865i	$0.282182\pi$
$434$	0	0
$435$	16.1481	0.774240
$436$	0	0
$437$	−10.6142	−0.507748
$438$	0	0
$439$	−33.8139	−1.61385	−0.806925	−	0.590654i	$-0.798870\pi$
$439$	−33.8139	−1.61385	−0.806925	+	0.590654i	$0.798870\pi$
$440$	0	0
$441$	−1.68585	−0.0802784
$442$	0	0
$443$	26.4464	1.25651	0.628254	−	0.778008i	$-0.283770\pi$
$443$	26.4464	1.25651	0.628254	+	0.778008i	$0.283770\pi$
$444$	0	0
$445$	7.70306	0.365160
$446$	0	0
$447$	2.48508	0.117540
$448$	0	0
$449$	2.64300	0.124731	0.0623655	−	0.998053i	$-0.480136\pi$
$449$	2.64300	0.124731	0.0623655	+	0.998053i	$0.480136\pi$
$450$	0	0
$451$	−0.335577	−0.0158017
$452$	0	0
$453$	17.0937	0.803130
$454$	0	0
$455$	−1.34292	−0.0629572
$456$	0	0
$457$	−33.6890	−1.57590	−0.787952	−	0.615737i	$-0.788859\pi$
$457$	−33.6890	−1.57590	−0.787952	+	0.615737i	$0.788859\pi$
$458$	0	0
$459$	31.3288	1.46231
$460$	0	0
$461$	−33.0790	−1.54064	−0.770320	−	0.637657i	$-0.779904\pi$
$461$	−33.0790	−1.54064	−0.770320	+	0.637657i	$0.779904\pi$
$462$	0	0
$463$	2.51806	0.117024	0.0585120	−	0.998287i	$-0.481364\pi$
$463$	2.51806	0.117024	0.0585120	+	0.998287i	$0.481364\pi$
$464$	0	0
$465$	2.51806	0.116772
$466$	0	0
$467$	2.57560	0.119184	0.0595922	−	0.998223i	$-0.481020\pi$
$467$	2.57560	0.119184	0.0595922	+	0.998223i	$0.481020\pi$
$468$	0	0
$469$	−6.10038	−0.281690
$470$	0	0
$471$	26.2155	1.20794
$472$	0	0
$473$	−9.34713	−0.429782
$474$	0	0
$475$	10.6858	0.490300
$476$	0	0
$477$	−1.31836	−0.0603638
$478$	0	0
$479$	−0.513847	−0.0234783	−0.0117391	−	0.999931i	$-0.503737\pi$
$479$	−0.513847	−0.0234783	−0.0117391	+	0.999931i	$0.503737\pi$
$480$	0	0
$481$	8.51806	0.388390
$482$	0	0
$483$	−3.63986	−0.165620
$484$	0	0
$485$	−7.17513	−0.325806
$486$	0	0
$487$	−36.0575	−1.63392	−0.816962	−	0.576692i	$-0.804343\pi$
$487$	−36.0575	−1.63392	−0.816962	+	0.576692i	$0.804343\pi$
$488$	0	0
$489$	−8.11508	−0.366976
$490$	0	0
$491$	−9.22846	−0.416475	−0.208237	−	0.978078i	$-0.566773\pi$
$491$	−9.22846	−0.416475	−0.208237	+	0.978078i	$0.566773\pi$
$492$	0	0
$493$	−61.1758	−2.75522
$494$	0	0
$495$	−2.59533	−0.116651
$496$	0	0
$497$	−1.53948	−0.0690551
$498$	0	0
$499$	1.00314	0.0449065	0.0224533	−	0.999748i	$-0.492852\pi$
$499$	1.00314	0.0449065	0.0224533	+	0.999748i	$0.492852\pi$
$500$	0	0
$501$	2.99686	0.133890
$502$	0	0
$503$	30.3503	1.35325	0.676626	−	0.736327i	$-0.263441\pi$
$503$	30.3503	1.35325	0.676626	+	0.736327i	$0.263441\pi$
$504$	0	0
$505$	−14.9687	−0.666099
$506$	0	0
$507$	−1.14637	−0.0509119
$508$	0	0
$509$	−10.5995	−0.469816	−0.234908	−	0.972018i	$-0.575479\pi$
$509$	−10.5995	−0.469816	−0.234908	+	0.972018i	$0.575479\pi$
$510$	0	0
$511$	−15.3001	−0.676836
$512$	0	0
$513$	−17.9572	−0.792828
$514$	0	0
$515$	4.58546	0.202060
$516$	0	0
$517$	12.1678	0.535139
$518$	0	0
$519$	12.6136	0.553676
$520$	0	0
$521$	−16.2646	−0.712564	−0.356282	−	0.934378i	$-0.615956\pi$
$521$	−16.2646	−0.712564	−0.356282	+	0.934378i	$0.615956\pi$
$522$	0	0
$523$	7.22219	0.315804	0.157902	−	0.987455i	$-0.449527\pi$
$523$	7.22219	0.315804	0.157902	+	0.987455i	$0.449527\pi$
$524$	0	0
$525$	3.66442	0.159929
$526$	0	0
$527$	−9.53948	−0.415546
$528$	0	0
$529$	−12.9185	−0.561675
$530$	0	0
$531$	−21.3142	−0.924955
$532$	0	0
$533$	0.292731	0.0126796
$534$	0	0
$535$	6.68585	0.289054
$536$	0	0
$537$	−27.4685	−1.18535
$538$	0	0
$539$	1.14637	0.0493775
$540$	0	0
$541$	−17.3534	−0.746081	−0.373041	−	0.927815i	$-0.621685\pi$
$541$	−17.3534	−0.746081	−0.373041	+	0.927815i	$0.621685\pi$
$542$	0	0
$543$	7.52119	0.322765
$544$	0	0
$545$	18.1249	0.776387
$546$	0	0
$547$	−34.1109	−1.45848	−0.729238	−	0.684261i	$-0.760125\pi$
$547$	−34.1109	−1.45848	−0.729238	+	0.684261i	$0.760125\pi$
$548$	0	0
$549$	−3.37169	−0.143900
$550$	0	0
$551$	35.0649	1.49381
$552$	0	0
$553$	−0.882404	−0.0375236
$554$	0	0
$555$	13.1134	0.556632
$556$	0	0
$557$	13.2222	0.560242	0.280121	−	0.959965i	$-0.409625\pi$
$557$	13.2222	0.560242	0.280121	+	0.959965i	$0.409625\pi$
$558$	0	0
$559$	8.15371	0.344865
$560$	0	0
$561$	−7.66442	−0.323592
$562$	0	0
$563$	−41.3717	−1.74361	−0.871804	−	0.489854i	$-0.837050\pi$
$563$	−41.3717	−1.74361	−0.871804	+	0.489854i	$0.837050\pi$
$564$	0	0
$565$	22.0863	0.929178
$566$	0	0
$567$	−1.10038	−0.0462118
$568$	0	0
$569$	−2.68164	−0.112420	−0.0562100	−	0.998419i	$-0.517902\pi$
$569$	−2.68164	−0.112420	−0.0562100	+	0.998419i	$0.517902\pi$
$570$	0	0
$571$	−28.9315	−1.21075	−0.605373	−	0.795942i	$-0.706976\pi$
$571$	−28.9315	−1.21075	−0.605373	+	0.795942i	$0.706976\pi$
$572$	0	0
$573$	−5.03612	−0.210387
$574$	0	0
$575$	−10.1495	−0.423263
$576$	0	0
$577$	37.5296	1.56238	0.781189	−	0.624294i	$-0.214613\pi$
$577$	37.5296	1.56238	0.781189	+	0.624294i	$0.214613\pi$
$578$	0	0
$579$	9.50650	0.395077
$580$	0	0
$581$	−12.1292	−0.503202
$582$	0	0
$583$	0.896480	0.0371284
$584$	0	0
$585$	2.26396	0.0936033
$586$	0	0
$587$	23.0649	0.951990	0.475995	−	0.879448i	$-0.342088\pi$
$587$	23.0649	0.951990	0.475995	+	0.879448i	$0.342088\pi$
$588$	0	0
$589$	5.46787	0.225299
$590$	0	0
$591$	−3.63504	−0.149525
$592$	0	0
$593$	−11.0502	−0.453777	−0.226889	−	0.973921i	$-0.572855\pi$
$593$	−11.0502	−0.453777	−0.226889	+	0.973921i	$0.572855\pi$
$594$	0	0
$595$	7.83221	0.321089
$596$	0	0
$597$	−15.5872	−0.637940
$598$	0	0
$599$	44.6044	1.82248	0.911242	−	0.411870i	$-0.135124\pi$
$599$	44.6044	1.82248	0.911242	+	0.411870i	$0.135124\pi$
$600$	0	0
$601$	47.8715	1.95272	0.976359	−	0.216156i	$-0.0693520\pi$
$601$	47.8715	1.95272	0.976359	+	0.216156i	$0.0693520\pi$
$602$	0	0
$603$	10.2843	0.418809
$604$	0	0
$605$	−13.0073	−0.528824
$606$	0	0
$607$	−45.0691	−1.82930	−0.914649	−	0.404249i	$-0.867533\pi$
$607$	−45.0691	−1.82930	−0.914649	+	0.404249i	$0.867533\pi$
$608$	0	0
$609$	12.0246	0.487260
$610$	0	0
$611$	−10.6142	−0.429406
$612$	0	0
$613$	25.5212	1.03079	0.515396	−	0.856952i	$-0.327645\pi$
$613$	25.5212	1.03079	0.515396	+	0.856952i	$0.327645\pi$
$614$	0	0
$615$	0.450654	0.0181721
$616$	0	0
$617$	29.2432	1.17729	0.588643	−	0.808393i	$-0.299663\pi$
$617$	29.2432	1.17729	0.588643	+	0.808393i	$0.299663\pi$
$618$	0	0
$619$	4.78623	0.192375	0.0961874	−	0.995363i	$-0.469335\pi$
$619$	4.78623	0.192375	0.0961874	+	0.995363i	$0.469335\pi$
$620$	0	0
$621$	17.0558	0.684428
$622$	0	0
$623$	5.73604	0.229810
$624$	0	0
$625$	1.20077	0.0480307
$626$	0	0
$627$	4.39312	0.175444
$628$	0	0
$629$	−49.6791	−1.98084
$630$	0	0
$631$	28.3931	1.13031	0.565156	−	0.824984i	$-0.308816\pi$
$631$	28.3931	1.13031	0.565156	+	0.824984i	$0.308816\pi$
$632$	0	0
$633$	−10.6331	−0.422629
$634$	0	0
$635$	−16.1923	−0.642574
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	2.59533	0.102670
$640$	0	0
$641$	5.96137	0.235460	0.117730	−	0.993046i	$-0.462438\pi$
$641$	5.96137	0.235460	0.117730	+	0.993046i	$0.462438\pi$
$642$	0	0
$643$	31.1940	1.23017	0.615086	−	0.788460i	$-0.289121\pi$
$643$	31.1940	1.23017	0.615086	+	0.788460i	$0.289121\pi$
$644$	0	0
$645$	12.5525	0.494253
$646$	0	0
$647$	−14.9112	−0.586219	−0.293109	−	0.956079i	$-0.594690\pi$
$647$	−14.9112	−0.586219	−0.293109	+	0.956079i	$0.594690\pi$
$648$	0	0
$649$	14.4935	0.568920
$650$	0	0
$651$	1.87506	0.0734893
$652$	0	0
$653$	3.57246	0.139801	0.0699006	−	0.997554i	$-0.477732\pi$
$653$	3.57246	0.139801	0.0699006	+	0.997554i	$0.477732\pi$
$654$	0	0
$655$	−4.92104	−0.192281
$656$	0	0
$657$	25.7936	1.00630
$658$	0	0
$659$	−3.90383	−0.152071	−0.0760357	−	0.997105i	$-0.524226\pi$
$659$	−3.90383	−0.152071	−0.0760357	+	0.997105i	$0.524226\pi$
$660$	0	0
$661$	−13.7936	−0.536508	−0.268254	−	0.963348i	$-0.586447\pi$
$661$	−13.7936	−0.536508	−0.268254	+	0.963348i	$0.586447\pi$
$662$	0	0
$663$	6.68585	0.259657
$664$	0	0
$665$	−4.48929	−0.174087
$666$	0	0
$667$	−33.3049	−1.28957
$668$	0	0
$669$	−22.4605	−0.868374
$670$	0	0
$671$	2.29273	0.0885099
$672$	0	0
$673$	5.70306	0.219837	0.109918	−	0.993941i	$-0.464941\pi$
$673$	5.70306	0.219837	0.109918	+	0.993941i	$0.464941\pi$
$674$	0	0
$675$	−17.1709	−0.660909
$676$	0	0
$677$	−35.2614	−1.35521	−0.677604	−	0.735427i	$-0.736981\pi$
$677$	−35.2614	−1.35521	−0.677604	+	0.735427i	$0.736981\pi$
$678$	0	0
$679$	−5.34292	−0.205043
$680$	0	0
$681$	22.5426	0.863835
$682$	0	0
$683$	1.03612	0.0396459	0.0198229	−	0.999804i	$-0.493690\pi$
$683$	1.03612	0.0396459	0.0198229	+	0.999804i	$0.493690\pi$
$684$	0	0
$685$	−17.5970	−0.672348
$686$	0	0
$687$	8.90131	0.339606
$688$	0	0
$689$	−0.782020	−0.0297926
$690$	0	0
$691$	3.67850	0.139937	0.0699684	−	0.997549i	$-0.477710\pi$
$691$	3.67850	0.139937	0.0699684	+	0.997549i	$0.477710\pi$
$692$	0	0
$693$	−1.93260	−0.0734132
$694$	0	0
$695$	10.0674	0.381878
$696$	0	0
$697$	−1.70727	−0.0646674
$698$	0	0
$699$	−13.9817	−0.528837
$700$	0	0
$701$	0.0617493	0.00233224	0.00116612	−	0.999999i	$-0.499629\pi$
$701$	0.0617493	0.00233224	0.00116612	+	0.999999i	$0.499629\pi$
$702$	0	0
$703$	28.4752	1.07396
$704$	0	0
$705$	−16.3404	−0.615415
$706$	0	0
$707$	−11.1464	−0.419202
$708$	0	0
$709$	42.9834	1.61428	0.807138	−	0.590363i	$-0.201015\pi$
$709$	42.9834	1.61428	0.807138	+	0.590363i	$0.201015\pi$
$710$	0	0
$711$	1.48760	0.0557892
$712$	0	0
$713$	−5.19342	−0.194495
$714$	0	0
$715$	−1.53948	−0.0575733
$716$	0	0
$717$	−11.7992	−0.440650
$718$	0	0
$719$	−17.6546	−0.658404	−0.329202	−	0.944260i	$-0.606780\pi$
$719$	−17.6546	−0.658404	−0.329202	+	0.944260i	$0.606780\pi$
$720$	0	0
$721$	3.41454	0.127164
$722$	0	0
$723$	4.61844	0.171762
$724$	0	0
$725$	33.5296	1.24526
$726$	0	0
$727$	23.8077	0.882977	0.441488	−	0.897267i	$-0.354451\pi$
$727$	23.8077	0.882977	0.441488	+	0.897267i	$0.354451\pi$
$728$	0	0
$729$	20.3288	0.752920
$730$	0	0
$731$	−47.5542	−1.75885
$732$	0	0
$733$	−31.3492	−1.15791	−0.578954	−	0.815360i	$-0.696539\pi$
$733$	−31.3492	−1.15791	−0.578954	+	0.815360i	$0.696539\pi$
$734$	0	0
$735$	−1.53948	−0.0567846
$736$	0	0
$737$	−6.99327	−0.257600
$738$	0	0
$739$	−24.8108	−0.912680	−0.456340	−	0.889806i	$-0.650840\pi$
$739$	−24.8108	−0.912680	−0.456340	+	0.889806i	$0.650840\pi$
$740$	0	0
$741$	−3.83221	−0.140780
$742$	0	0
$743$	−21.3717	−0.784051	−0.392026	−	0.919954i	$-0.628226\pi$
$743$	−21.3717	−0.784051	−0.392026	+	0.919954i	$0.628226\pi$
$744$	0	0
$745$	−2.91117	−0.106657
$746$	0	0
$747$	20.4479	0.748149
$748$	0	0
$749$	4.97858	0.181913
$750$	0	0
$751$	−19.2243	−0.701503	−0.350751	−	0.936469i	$-0.614074\pi$
$751$	−19.2243	−0.701503	−0.350751	+	0.936469i	$0.614074\pi$
$752$	0	0
$753$	−3.33727	−0.121617
$754$	0	0
$755$	−20.0246	−0.728768
$756$	0	0
$757$	19.8610	0.721860	0.360930	−	0.932593i	$-0.382459\pi$
$757$	19.8610	0.721860	0.360930	+	0.932593i	$0.382459\pi$
$758$	0	0
$759$	−4.17262	−0.151456
$760$	0	0
$761$	6.12073	0.221876	0.110938	−	0.993827i	$-0.464614\pi$
$761$	6.12073	0.221876	0.110938	+	0.993827i	$0.464614\pi$
$762$	0	0
$763$	13.4966	0.488611
$764$	0	0
$765$	−13.2039	−0.477388
$766$	0	0
$767$	−12.6430	−0.456512
$768$	0	0
$769$	3.82800	0.138041	0.0690206	−	0.997615i	$-0.478013\pi$
$769$	3.82800	0.138041	0.0690206	+	0.997615i	$0.478013\pi$
$770$	0	0
$771$	22.4653	0.809070
$772$	0	0
$773$	6.53635	0.235096	0.117548	−	0.993067i	$-0.462497\pi$
$773$	6.53635	0.235096	0.117548	+	0.993067i	$0.462497\pi$
$774$	0	0
$775$	5.22846	0.187812
$776$	0	0
$777$	9.76481	0.350311
$778$	0	0
$779$	0.978577	0.0350612
$780$	0	0
$781$	−1.76481	−0.0631498
$782$	0	0
$783$	−56.3452	−2.01361
$784$	0	0
$785$	−30.7104	−1.09610
$786$	0	0
$787$	30.8066	1.09814	0.549068	−	0.835778i	$-0.314983\pi$
$787$	30.8066	1.09814	0.549068	+	0.835778i	$0.314983\pi$
$788$	0	0
$789$	8.67598	0.308873
$790$	0	0
$791$	16.4464	0.584768
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	−1.20390	−0.0426981
$796$	0	0
$797$	38.8156	1.37492	0.687460	−	0.726222i	$-0.258726\pi$
$797$	38.8156	1.37492	0.687460	+	0.726222i	$0.258726\pi$
$798$	0	0
$799$	61.9044	2.19002
$800$	0	0
$801$	−9.67008	−0.341675
$802$	0	0
$803$	−17.5395	−0.618955
$804$	0	0
$805$	4.26396	0.150285
$806$	0	0
$807$	−10.8585	−0.382236
$808$	0	0
$809$	1.04033	0.0365759	0.0182880	−	0.999833i	$-0.494178\pi$
$809$	1.04033	0.0365759	0.0182880	+	0.999833i	$0.494178\pi$
$810$	0	0
$811$	12.6712	0.444944	0.222472	−	0.974939i	$-0.428587\pi$
$811$	12.6712	0.444944	0.222472	+	0.974939i	$0.428587\pi$
$812$	0	0
$813$	33.6707	1.18088
$814$	0	0
$815$	9.50650	0.332998
$816$	0	0
$817$	27.2572	0.953610
$818$	0	0
$819$	1.68585	0.0589082
$820$	0	0
$821$	−27.0361	−0.943567	−0.471783	−	0.881714i	$-0.656390\pi$
$821$	−27.0361	−0.943567	−0.471783	+	0.881714i	$0.656390\pi$
$822$	0	0
$823$	31.6363	1.10277	0.551386	−	0.834251i	$-0.314099\pi$
$823$	31.6363	1.10277	0.551386	+	0.834251i	$0.314099\pi$
$824$	0	0
$825$	4.20077	0.146252
$826$	0	0
$827$	56.4800	1.96400	0.982002	−	0.188872i	$-0.0604832\pi$
$827$	56.4800	1.96400	0.982002	+	0.188872i	$0.0604832\pi$
$828$	0	0
$829$	42.6760	1.48220	0.741099	−	0.671396i	$-0.234305\pi$
$829$	42.6760	1.48220	0.741099	+	0.671396i	$0.234305\pi$
$830$	0	0
$831$	−2.18248	−0.0757094
$832$	0	0
$833$	5.83221	0.202074
$834$	0	0
$835$	−3.51071	−0.121493
$836$	0	0
$837$	−8.78623	−0.303697
$838$	0	0
$839$	40.1642	1.38662	0.693311	−	0.720639i	$-0.256151\pi$
$839$	40.1642	1.38662	0.693311	+	0.720639i	$0.256151\pi$
$840$	0	0
$841$	81.0252	2.79397
$842$	0	0
$843$	23.5872	0.812385
$844$	0	0
$845$	1.34292	0.0461980
$846$	0	0
$847$	−9.68585	−0.332810
$848$	0	0
$849$	30.9442	1.06200
$850$	0	0
$851$	−27.0460	−0.927124
$852$	0	0
$853$	−19.6932	−0.674282	−0.337141	−	0.941454i	$-0.609460\pi$
$853$	−19.6932	−0.674282	−0.337141	+	0.941454i	$0.609460\pi$
$854$	0	0
$855$	7.56825	0.258829
$856$	0	0
$857$	1.66442	0.0568556	0.0284278	−	0.999596i	$-0.490950\pi$
$857$	1.66442	0.0568556	0.0284278	+	0.999596i	$0.490950\pi$
$858$	0	0
$859$	−41.2944	−1.40895	−0.704474	−	0.709730i	$-0.748817\pi$
$859$	−41.2944	−1.40895	−0.704474	+	0.709730i	$0.748817\pi$
$860$	0	0
$861$	0.335577	0.0114364
$862$	0	0
$863$	31.3288	1.06645	0.533223	−	0.845975i	$-0.320981\pi$
$863$	31.3288	1.06645	0.533223	+	0.845975i	$0.320981\pi$
$864$	0	0
$865$	−14.7764	−0.502411
$866$	0	0
$867$	−19.5051	−0.662426
$868$	0	0
$869$	−1.01156	−0.0343147
$870$	0	0
$871$	6.10038	0.206704
$872$	0	0
$873$	9.00735	0.304852
$874$	0	0
$875$	−11.0073	−0.372116
$876$	0	0
$877$	53.8041	1.81683	0.908417	−	0.418065i	$-0.137292\pi$
$877$	53.8041	1.81683	0.908417	+	0.418065i	$0.137292\pi$
$878$	0	0
$879$	−17.0888	−0.576392
$880$	0	0
$881$	8.09196	0.272625	0.136313	−	0.990666i	$-0.456475\pi$
$881$	8.09196	0.272625	0.136313	+	0.990666i	$0.456475\pi$
$882$	0	0
$883$	−27.0705	−0.910996	−0.455498	−	0.890237i	$-0.650539\pi$
$883$	−27.0705	−0.910996	−0.455498	+	0.890237i	$0.650539\pi$
$884$	0	0
$885$	−19.4637	−0.654264
$886$	0	0
$887$	38.4935	1.29249	0.646243	−	0.763132i	$-0.276339\pi$
$887$	38.4935	1.29249	0.646243	+	0.763132i	$0.276339\pi$
$888$	0	0
$889$	−12.0575	−0.404397
$890$	0	0
$891$	−1.26144	−0.0422599
$892$	0	0
$893$	−35.4826	−1.18738
$894$	0	0
$895$	32.1783	1.07560
$896$	0	0
$897$	3.63986	0.121532
$898$	0	0
$899$	17.1568	0.572213
$900$	0	0
$901$	4.56090	0.151946
$902$	0	0
$903$	9.34713	0.311053
$904$	0	0
$905$	−8.81079	−0.292881
$906$	0	0
$907$	15.7031	0.521411	0.260706	−	0.965418i	$-0.416045\pi$
$907$	15.7031	0.521411	0.260706	+	0.965418i	$0.416045\pi$
$908$	0	0
$909$	18.7911	0.623260
$910$	0	0
$911$	−44.9399	−1.48893	−0.744463	−	0.667663i	$-0.767295\pi$
$911$	−44.9399	−1.48893	−0.744463	+	0.667663i	$0.767295\pi$
$912$	0	0
$913$	−13.9044	−0.460170
$914$	0	0
$915$	−3.07896	−0.101787
$916$	0	0
$917$	−3.66442	−0.121010
$918$	0	0
$919$	27.2432	0.898669	0.449334	−	0.893364i	$-0.351661\pi$
$919$	27.2432	0.898669	0.449334	+	0.893364i	$0.351661\pi$
$920$	0	0
$921$	−29.8385	−0.983211
$922$	0	0
$923$	1.53948	0.0506726
$924$	0	0
$925$	27.2285	0.895266
$926$	0	0
$927$	−5.75639	−0.189065
$928$	0	0
$929$	−17.1422	−0.562416	−0.281208	−	0.959647i	$-0.590735\pi$
$929$	−17.1422	−0.562416	−0.281208	+	0.959647i	$0.590735\pi$
$930$	0	0
$931$	−3.34292	−0.109560
$932$	0	0
$933$	22.3503	0.731715
$934$	0	0
$935$	8.97858	0.293631
$936$	0	0
$937$	−51.5197	−1.68308	−0.841538	−	0.540197i	$-0.818350\pi$
$937$	−51.5197	−1.68308	−0.841538	+	0.540197i	$0.818350\pi$
$938$	0	0
$939$	3.99158	0.130260
$940$	0	0
$941$	−32.7575	−1.06786	−0.533931	−	0.845528i	$-0.679286\pi$
$941$	−32.7575	−1.06786	−0.533931	+	0.845528i	$0.679286\pi$
$942$	0	0
$943$	−0.929460	−0.0302674
$944$	0	0
$945$	7.21377	0.234664
$946$	0	0
$947$	20.9295	0.680116	0.340058	−	0.940404i	$-0.389553\pi$
$947$	20.9295	0.680116	0.340058	+	0.940404i	$0.389553\pi$
$948$	0	0
$949$	15.3001	0.496662
$950$	0	0
$951$	5.75639	0.186664
$952$	0	0
$953$	46.4120	1.50343	0.751716	−	0.659487i	$-0.229226\pi$
$953$	46.4120	1.50343	0.751716	+	0.659487i	$0.229226\pi$
$954$	0	0
$955$	5.89962	0.190907
$956$	0	0
$957$	13.7845	0.445591
$958$	0	0
$959$	−13.1035	−0.423135
$960$	0	0
$961$	−28.3246	−0.913698
$962$	0	0
$963$	−8.39312	−0.270464
$964$	0	0
$965$	−11.1365	−0.358497
$966$	0	0
$967$	23.2186	0.746660	0.373330	−	0.927699i	$-0.378216\pi$
$967$	23.2186	0.746660	0.373330	+	0.927699i	$0.378216\pi$
$968$	0	0
$969$	22.3503	0.717994
$970$	0	0
$971$	−14.1004	−0.452503	−0.226251	−	0.974069i	$-0.572647\pi$
$971$	−14.1004	−0.452503	−0.226251	+	0.974069i	$0.572647\pi$
$972$	0	0
$973$	7.49663	0.240331
$974$	0	0
$975$	−3.66442	−0.117355
$976$	0	0
$977$	−2.95402	−0.0945074	−0.0472537	−	0.998883i	$-0.515047\pi$
$977$	−2.95402	−0.0945074	−0.0472537	+	0.998883i	$0.515047\pi$
$978$	0	0
$979$	6.57560	0.210157
$980$	0	0
$981$	−22.7533	−0.726455
$982$	0	0
$983$	35.0367	1.11750	0.558749	−	0.829337i	$-0.311281\pi$
$983$	35.0367	1.11750	0.558749	+	0.829337i	$0.311281\pi$
$984$	0	0
$985$	4.25831	0.135681
$986$	0	0
$987$	−12.1678	−0.387305
$988$	0	0
$989$	−25.8891	−0.823227
$990$	0	0
$991$	−47.9718	−1.52388	−0.761938	−	0.647650i	$-0.775752\pi$
$991$	−47.9718	−1.52388	−0.761938	+	0.647650i	$0.775752\pi$
$992$	0	0
$993$	7.04958	0.223712
$994$	0	0
$995$	18.2598	0.578873
$996$	0	0
$997$	38.4422	1.21748	0.608739	−	0.793371i	$-0.291676\pi$
$997$	38.4422	1.21748	0.608739	+	0.793371i	$0.291676\pi$
$998$	0	0
$999$	−45.7564	−1.44767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5824.2.a.bs.1.2		3
4.3	odd	2		5824.2.a.by.1.2		3
8.3	odd	2		91.2.a.d.1.3	✓	3
8.5	even	2		1456.2.a.t.1.2		3
24.11	even	2		819.2.a.i.1.1		3
40.19	odd	2		2275.2.a.m.1.1		3
56.3	even	6		637.2.e.i.79.1		6
56.11	odd	6		637.2.e.j.79.1		6
56.19	even	6		637.2.e.i.508.1		6
56.27	even	2		637.2.a.j.1.3		3
56.51	odd	6		637.2.e.j.508.1		6
104.51	odd	2		1183.2.a.i.1.1		3
104.83	even	4		1183.2.c.f.337.1		6
104.99	even	4		1183.2.c.f.337.6		6
168.83	odd	2		5733.2.a.x.1.1		3
728.363	even	2		8281.2.a.bg.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.d.1.3	✓	3	8.3	odd	2
637.2.a.j.1.3		3	56.27	even	2
637.2.e.i.79.1		6	56.3	even	6
637.2.e.i.508.1		6	56.19	even	6
637.2.e.j.79.1		6	56.11	odd	6
637.2.e.j.508.1		6	56.51	odd	6
819.2.a.i.1.1		3	24.11	even	2
1183.2.a.i.1.1		3	104.51	odd	2
1183.2.c.f.337.1		6	104.83	even	4
1183.2.c.f.337.6		6	104.99	even	4
1456.2.a.t.1.2		3	8.5	even	2
2275.2.a.m.1.1		3	40.19	odd	2
5733.2.a.x.1.1		3	168.83	odd	2
5824.2.a.bs.1.2		3	1.1	even	1	trivial
5824.2.a.by.1.2		3	4.3	odd	2
8281.2.a.bg.1.1		3	728.363	even	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:	\( N \)	\(=\)	\( 5824 = 2^{6} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5824.a (trivial)

\(f(q)\)	\(=\)	\(q-1.14637 q^{3} +1.34292 q^{5} +1.00000 q^{7} -1.68585 q^{9} +O(q^{10})\)	\(q-1.14637 q^{3} +1.34292 q^{5} +1.00000 q^{7} -1.68585 q^{9} +1.14637 q^{11} -1.00000 q^{13} -1.53948 q^{15} +5.83221 q^{17} -3.34292 q^{19} -1.14637 q^{21} +3.17513 q^{23} -3.19656 q^{25} +5.37169 q^{27} -10.4893 q^{29} -1.63565 q^{31} -1.31415 q^{33} +1.34292 q^{35} -8.51806 q^{37} +1.14637 q^{39} -0.292731 q^{41} -8.15371 q^{43} -2.26396 q^{45} +10.6142 q^{47} +1.00000 q^{49} -6.68585 q^{51} +0.782020 q^{53} +1.53948 q^{55} +3.83221 q^{57} +12.6430 q^{59} +2.00000 q^{61} -1.68585 q^{63} -1.34292 q^{65} -6.10038 q^{67} -3.63986 q^{69} -1.53948 q^{71} -15.3001 q^{73} +3.66442 q^{75} +1.14637 q^{77} -0.882404 q^{79} -1.10038 q^{81} -12.1292 q^{83} +7.83221 q^{85} +12.0246 q^{87} +5.73604 q^{89} -1.00000 q^{91} +1.87506 q^{93} -4.48929 q^{95} -5.34292 q^{97} -1.93260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 2 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - 2 * q^5 + 3 * q^7 + 7 * q^9	\( 3 q - 2 q^{3} - 2 q^{5} + 3 q^{7} + 7 q^{9} + 2 q^{11} - 3 q^{13} + 6 q^{15} + 4 q^{17} - 4 q^{19} - 2 q^{21} - 10 q^{23} - 5 q^{25} - 8 q^{27} - 24 q^{29} + 4 q^{31} - 16 q^{33} - 2 q^{35} + 2 q^{39} + 2 q^{41} + 10 q^{43} - 22 q^{45} + 8 q^{47} + 3 q^{49} - 8 q^{51} - 8 q^{53} - 6 q^{55} - 2 q^{57} - 4 q^{59} + 6 q^{61} + 7 q^{63} + 2 q^{65} - 12 q^{67} + 6 q^{69} + 6 q^{71} - 10 q^{73} - 16 q^{75} + 2 q^{77} + 14 q^{79} + 3 q^{81} - 12 q^{83} + 10 q^{85} + 26 q^{87} + 2 q^{89} - 3 q^{91} + 22 q^{93} - 6 q^{95} - 10 q^{97} + 14 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 2 * q^5 + 3 * q^7 + 7 * q^9 + 2 * q^11 - 3 * q^13 + 6 * q^15 + 4 * q^17 - 4 * q^19 - 2 * q^21 - 10 * q^23 - 5 * q^25 - 8 * q^27 - 24 * q^29 + 4 * q^31 - 16 * q^33 - 2 * q^35 + 2 * q^39 + 2 * q^41 + 10 * q^43 - 22 * q^45 + 8 * q^47 + 3 * q^49 - 8 * q^51 - 8 * q^53 - 6 * q^55 - 2 * q^57 - 4 * q^59 + 6 * q^61 + 7 * q^63 + 2 * q^65 - 12 * q^67 + 6 * q^69 + 6 * q^71 - 10 * q^73 - 16 * q^75 + 2 * q^77 + 14 * q^79 + 3 * q^81 - 12 * q^83 + 10 * q^85 + 26 * q^87 + 2 * q^89 - 3 * q^91 + 22 * q^93 - 6 * q^95 - 10 * q^97 + 14 * q^99

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