LMFDB - Embedded newform 819.2.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [819,2,Mod(1,819)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("819.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(819, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,0,3,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.53974792554$
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:	$\Z[a_1, a_2]$
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			1.1
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.34292 q^{2} +3.48929 q^{4} +1.34292 q^{5} -1.00000 q^{7} -3.48929 q^{8} -3.14637 q^{10} -1.14637 q^{11} +1.00000 q^{13} +2.34292 q^{14} +1.19656 q^{16} -5.83221 q^{17} -3.34292 q^{19} +4.68585 q^{20} +2.68585 q^{22} +3.17513 q^{23} -3.19656 q^{25} -2.34292 q^{26} -3.48929 q^{28} -10.4893 q^{29} +1.63565 q^{31} +4.17513 q^{32} +13.6644 q^{34} -1.34292 q^{35} +8.51806 q^{37} +7.83221 q^{38} -4.68585 q^{40} +0.292731 q^{41} -8.15371 q^{43} -4.00000 q^{44} -7.43910 q^{46} +10.6142 q^{47} +1.00000 q^{49} +7.48929 q^{50} +3.48929 q^{52} +0.782020 q^{53} -1.53948 q^{55} +3.48929 q^{56} +24.5756 q^{58} -12.6430 q^{59} -2.00000 q^{61} -3.83221 q^{62} -12.1751 q^{64} +1.34292 q^{65} -6.10038 q^{67} -20.3503 q^{68} +3.14637 q^{70} -1.53948 q^{71} -15.3001 q^{73} -19.9572 q^{74} -11.6644 q^{76} +1.14637 q^{77} +0.882404 q^{79} +1.60688 q^{80} -0.685846 q^{82} +12.1292 q^{83} -7.83221 q^{85} +19.1035 q^{86} +4.00000 q^{88} -5.73604 q^{89} -1.00000 q^{91} +11.0790 q^{92} -24.8683 q^{94} -4.48929 q^{95} -5.34292 q^{97} -2.34292 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{4} - 2 q^{5} - 3 q^{7} - 3 q^{8} - 8 q^{10} - 2 q^{11} + 3 q^{13} + q^{14} - q^{16} - 4 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} - 10 q^{23} - 5 q^{25} - q^{26} - 3 q^{28} - 24 q^{29}+ \cdots - q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^4 - 2 * q^5 - 3 * q^7 - 3 * q^8 - 8 * q^10 - 2 * q^11 + 3 * q^13 + q^14 - q^16 - 4 * q^17 - 4 * q^19 + 2 * q^20 - 4 * q^22 - 10 * q^23 - 5 * q^25 - q^26 - 3 * q^28 - 24 * q^29 - 4 * q^31 - 7 * q^32 + 14 * q^34 + 2 * q^35 + 10 * q^38 - 2 * q^40 - 2 * q^41 + 10 * q^43 - 12 * q^44 - 18 * q^46 + 8 * q^47 + 3 * q^49 + 15 * q^50 + 3 * q^52 - 8 * q^53 + 6 * q^55 + 3 * q^56 + 12 * q^58 + 4 * q^59 - 6 * q^61 + 2 * q^62 - 17 * q^64 - 2 * q^65 - 12 * q^67 - 22 * q^68 + 8 * q^70 + 6 * q^71 - 10 * q^73 - 30 * q^74 - 8 * q^76 + 2 * q^77 - 14 * q^79 + 14 * q^80 + 10 * q^82 + 12 * q^83 - 10 * q^85 + 26 * q^86 + 12 * q^88 - 2 * q^89 - 3 * q^91 + 12 * q^92 - 10 * q^94 - 6 * q^95 - 10 * q^97 - q^98

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 $

$2$	−2.34292	−1.65670	−0.828348	−	0.560213i	$-0.810719\pi$
$2$	−2.34292	−1.65670	−0.828348	+	0.560213i	$0.810719\pi$
$3$	0	0
$3$	0	0
$4$	3.48929	1.74464
$5$	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
$8$	−3.48929	−1.23365
$9$	0	0
$10$	−3.14637	−0.994968
$11$	−1.14637	−0.345642	−0.172821	−	0.984953i	$-0.555288\pi$
$11$	−1.14637	−0.345642	−0.172821	+	0.984953i	$0.555288\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.34292	0.626173
$15$	0	0
$16$	1.19656	0.299139
$17$	−5.83221	−1.41452	−0.707260	−	0.706954i	$-0.750069\pi$
$17$	−5.83221	−1.41452	−0.707260	+	0.706954i	$0.750069\pi$
$18$	0	0
$19$	−3.34292	−0.766919	−0.383460	−	0.923558i	$-0.625267\pi$
$19$	−3.34292	−0.766919	−0.383460	+	0.923558i	$0.625267\pi$
$20$	4.68585	1.04779
$21$	0	0
$22$	2.68585	0.572624
$23$	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
$26$	−2.34292	−0.459485
$27$	0	0
$28$	−3.48929	−0.659414
$29$	−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	$-0.453075\pi$
$31$	1.63565	0.293772	0.146886	+	0.989153i	$0.453075\pi$
$32$	4.17513	0.738067
$33$	0	0
$34$	13.6644	2.34343
$35$	−1.34292	−0.226995
$36$	0	0
$37$	8.51806	1.40036	0.700180	−	0.713966i	$-0.253103\pi$
$37$	8.51806	1.40036	0.700180	+	0.713966i	$0.253103\pi$
$38$	7.83221	1.27055
$39$	0	0
$40$	−4.68585	−0.740897
$41$	0.292731	0.0457169	0.0228584	−	0.999739i	$-0.492723\pi$
$41$	0.292731	0.0457169	0.0228584	+	0.999739i	$0.492723\pi$
$42$	0	0
$43$	−8.15371	−1.24343	−0.621715	−	0.783244i	$-0.713564\pi$
$43$	−8.15371	−1.24343	−0.621715	+	0.783244i	$0.713564\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−7.43910	−1.09683
$47$	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
$50$	7.48929	1.05915
$51$	0	0
$52$	3.48929	0.483877
$53$	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
$56$	3.48929	0.466276
$57$	0	0
$58$	24.5756	3.22693
$59$	−12.6430	−1.64598	−0.822989	−	0.568057i	$-0.807695\pi$
$59$	−12.6430	−1.64598	−0.822989	+	0.568057i	$0.807695\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−3.83221	−0.486691
$63$	0	0
$64$	−12.1751	−1.52189
$65$	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	$0.621547\pi$
$68$	−20.3503	−2.46783
$69$	0	0
$70$	3.14637	0.376063
$71$	−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	$0.853088\pi$
$74$	−19.9572	−2.31997
$75$	0	0
$76$	−11.6644	−1.33800
$77$	1.14637	0.130640
$78$	0	0
$79$	0.882404	0.0992782	0.0496391	−	0.998767i	$-0.484193\pi$
$79$	0.882404	0.0992782	0.0496391	+	0.998767i	$0.484193\pi$
$80$	1.60688	0.179655
$81$	0	0
$82$	−0.685846	−0.0757390
$83$	12.1292	1.33135	0.665674	−	0.746243i	$-0.268144\pi$
$83$	12.1292	1.33135	0.665674	+	0.746243i	$0.268144\pi$
$84$	0	0
$85$	−7.83221	−0.849523
$86$	19.1035	2.05999
$87$	0	0
$88$	4.00000	0.426401
$89$	−5.73604	−0.608019	−0.304009	−	0.952669i	$-0.598325\pi$
$89$	−5.73604	−0.608019	−0.304009	+	0.952669i	$0.598325\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	11.0790	1.15506
$93$	0	0
$94$	−24.8683	−2.56497
$95$	−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	$0.587436\pi$
$98$	−2.34292	−0.236671
$99$	0	0
$100$	−11.1537	−1.11537
$101$	−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	$-0.553803\pi$
$103$	−3.41454	−0.336444	−0.168222	+	0.985749i	$0.553803\pi$
$104$	−3.48929	−0.342153
$105$	0	0
$106$	−1.83221	−0.177960
$107$	−4.97858	−0.481297	−0.240649	−	0.970612i	$-0.577360\pi$
$107$	−4.97858	−0.481297	−0.240649	+	0.970612i	$0.577360\pi$
$108$	0	0
$109$	−13.4966	−1.29274	−0.646372	−	0.763023i	$-0.723714\pi$
$109$	−13.4966	−1.29274	−0.646372	+	0.763023i	$0.723714\pi$
$110$	3.60688	0.343903
$111$	0	0
$112$	−1.19656	−0.113064
$113$	−16.4464	−1.54715	−0.773576	−	0.633704i	$-0.781534\pi$
$113$	−16.4464	−1.54715	−0.773576	+	0.633704i	$0.781534\pi$
$114$	0	0
$115$	4.26396	0.397616
$116$	−36.6002	−3.39824
$117$	0	0
$118$	29.6216	2.72689
$119$	5.83221	0.534638
$120$	0	0
$121$	−9.68585	−0.880531
$122$	4.68585	0.424237
$123$	0	0
$124$	5.70727	0.512528
$125$	−11.0073	−0.984527
$126$	0	0
$127$	12.0575	1.06993	0.534967	−	0.844873i	$-0.320324\pi$
$127$	12.0575	1.06993	0.534967	+	0.844873i	$0.320324\pi$
$128$	20.1751	1.78325
$129$	0	0
$130$	−3.14637	−0.275955
$131$	3.66442	0.320162	0.160081	−	0.987104i	$-0.448824\pi$
$131$	3.66442	0.320162	0.160081	+	0.987104i	$0.448824\pi$
$132$	0	0
$133$	3.34292	0.289868
$134$	14.2927	1.23470
$135$	0	0
$136$	20.3503	1.74502
$137$	13.1035	1.11951	0.559755	−	0.828658i	$-0.310895\pi$
$137$	13.1035	1.11951	0.559755	+	0.828658i	$0.310895\pi$
$138$	0	0
$139$	7.49663	0.635856	0.317928	−	0.948115i	$-0.397013\pi$
$139$	7.49663	0.635856	0.317928	+	0.948115i	$0.397013\pi$
$140$	−4.68585	−0.396026
$141$	0	0
$142$	3.60688	0.302683
$143$	−1.14637	−0.0958639
$144$	0	0
$145$	−14.0863	−1.16980
$146$	35.8469	2.96671
$147$	0	0
$148$	29.7220	2.44313
$149$	−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	$-0.292482\pi$
$151$	14.9112	1.21345	0.606727	+	0.794910i	$0.292482\pi$
$152$	11.6644	0.946110
$153$	0	0
$154$	−2.68585	−0.216432
$155$	2.19656	0.176432
$156$	0	0
$157$	22.8683	1.82509	0.912546	−	0.408975i	$-0.134114\pi$
$157$	22.8683	1.82509	0.912546	+	0.408975i	$0.134114\pi$
$158$	−2.06740	−0.164474
$159$	0	0
$160$	5.60688	0.443263
$161$	−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	$0.410582\pi$
$164$	1.02142	0.0797597
$165$	0	0
$166$	−28.4177	−2.20564
$167$	−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
$170$	18.3503	1.40740
$171$	0	0
$172$	−28.4507	−2.16934
$173$	−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
$176$	−1.37169	−0.103395
$177$	0	0
$178$	13.4391	1.00730
$179$	−23.9614	−1.79096	−0.895478	−	0.445105i	$-0.853166\pi$
$179$	−23.9614	−1.79096	−0.895478	+	0.445105i	$0.853166\pi$
$180$	0	0
$181$	6.56090	0.487668	0.243834	−	0.969817i	$-0.421595\pi$
$181$	6.56090	0.487668	0.243834	+	0.969817i	$0.421595\pi$
$182$	2.34292	0.173669
$183$	0	0
$184$	−11.0790	−0.816752
$185$	11.4391	0.841019
$186$	0	0
$187$	6.68585	0.488917
$188$	37.0361	2.70114
$189$	0	0
$190$	10.5181	0.763060
$191$	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	$0.596474\pi$
$194$	12.5181	0.898744
$195$	0	0
$196$	3.48929	0.249235
$197$	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	$-0.660065\pi$
$199$	−13.5970	−0.963867	−0.481934	+	0.876208i	$0.660065\pi$
$200$	11.1537	0.788687
$201$	0	0
$202$	26.1151	1.83745
$203$	10.4893	0.736204
$204$	0	0
$205$	0.393115	0.0274564
$206$	8.00000	0.557386
$207$	0	0
$208$	1.19656	0.0829663
$209$	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	$0.396560\pi$
$212$	2.72869	0.187407
$213$	0	0
$214$	11.6644	0.797364
$215$	−10.9498	−0.746771
$216$	0	0
$217$	−1.63565	−0.111035
$218$	31.6216	2.14168
$219$	0	0
$220$	−5.37169	−0.362159
$221$	−5.83221	−0.392317
$222$	0	0
$223$	−19.5928	−1.31203	−0.656016	−	0.754747i	$-0.727760\pi$
$223$	−19.5928	−1.31203	−0.656016	+	0.754747i	$0.727760\pi$
$224$	−4.17513	−0.278963
$225$	0	0
$226$	38.5328	2.56316
$227$	19.6644	1.30517	0.652587	−	0.757714i	$-0.273684\pi$
$227$	19.6644	1.30517	0.652587	+	0.757714i	$0.273684\pi$
$228$	0	0
$229$	7.76481	0.513113	0.256556	−	0.966529i	$-0.417412\pi$
$229$	7.76481	0.513113	0.256556	+	0.966529i	$0.417412\pi$
$230$	−9.99013	−0.658730
$231$	0	0
$232$	36.6002	2.40292
$233$	−12.1966	−0.799023	−0.399512	−	0.916728i	$-0.630820\pi$
$233$	−12.1966	−0.799023	−0.399512	+	0.916728i	$0.630820\pi$
$234$	0	0
$235$	14.2541	0.929835
$236$	−44.1151	−2.87165
$237$	0	0
$238$	−13.6644	−0.885733
$239$	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	$0.541420\pi$
$242$	22.6932	1.45877
$243$	0	0
$244$	−6.97858	−0.446758
$245$	1.34292	0.0857962
$246$	0	0
$247$	−3.34292	−0.212705
$248$	−5.70727	−0.362412
$249$	0	0
$250$	25.7894	1.63106
$251$	−2.91117	−0.183752	−0.0918758	−	0.995770i	$-0.529286\pi$
$251$	−2.91117	−0.183752	−0.0918758	+	0.995770i	$0.529286\pi$
$252$	0	0
$253$	−3.63986	−0.228836
$254$	−28.2499	−1.77256
$255$	0	0
$256$	−22.9185	−1.43241
$257$	19.5970	1.22243	0.611214	−	0.791465i	$-0.290681\pi$
$257$	19.5970	1.22243	0.611214	+	0.791465i	$0.290681\pi$
$258$	0	0
$259$	−8.51806	−0.529286
$260$	4.68585	0.290604
$261$	0	0
$262$	−8.58546	−0.530412
$263$	−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
$266$	−7.83221	−0.480224
$267$	0	0
$268$	−21.2860	−1.30025
$269$	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	$-0.149230\pi$
$271$	29.3717	1.78420	0.892102	+	0.451835i	$0.149230\pi$
$272$	−6.97858	−0.423138
$273$	0	0
$274$	−30.7005	−1.85469
$275$	3.66442	0.220973
$276$	0	0
$277$	−1.90383	−0.114390	−0.0571949	−	0.998363i	$-0.518216\pi$
$277$	−1.90383	−0.114390	−0.0571949	+	0.998363i	$0.518216\pi$
$278$	−17.5640	−1.05342
$279$	0	0
$280$	4.68585	0.280033
$281$	20.5756	1.22744	0.613719	−	0.789525i	$-0.289673\pi$
$281$	20.5756	1.22744	0.613719	+	0.789525i	$0.289673\pi$
$282$	0	0
$283$	−26.9933	−1.60458	−0.802292	−	0.596932i	$-0.796386\pi$
$283$	−26.9933	−1.60458	−0.802292	+	0.596932i	$0.796386\pi$
$284$	−5.37169	−0.318751
$285$	0	0
$286$	2.68585	0.158817
$287$	−0.292731	−0.0172794
$288$	0	0
$289$	17.0147	1.00086
$290$	33.0031	1.93801
$291$	0	0
$292$	−53.3864	−3.12420
$293$	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
$296$	−29.7220	−1.72755
$297$	0	0
$298$	5.07896	0.294216
$299$	3.17513	0.183623
$300$	0	0
$301$	8.15371	0.469972
$302$	−34.9357	−2.01033
$303$	0	0
$304$	−4.00000	−0.229416
$305$	−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	$0.233512\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	−5.14637	−0.292294
$311$	−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	$0.531374\pi$
$314$	−53.5787	−3.02362
$315$	0	0
$316$	3.07896	0.173205
$317$	−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
$320$	−16.3503	−0.914008
$321$	0	0
$322$	7.43910	0.414565
$323$	19.4966	1.08482
$324$	0	0
$325$	−3.19656	−0.177313
$326$	−16.5855	−0.918584
$327$	0	0
$328$	−1.02142	−0.0563986
$329$	−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	$0.554055\pi$
$332$	42.3221	2.32273
$333$	0	0
$334$	6.12494	0.335142
$335$	−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	$0.745910\pi$
$338$	−2.34292	−0.127438
$339$	0	0
$340$	−27.3288	−1.48211
$341$	−1.87506	−0.101540
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	28.4507	1.53396
$345$	0	0
$346$	25.7795	1.38591
$347$	16.7005	0.896532	0.448266	−	0.893900i	$-0.352042\pi$
$347$	16.7005	0.896532	0.448266	+	0.893900i	$0.352042\pi$
$348$	0	0
$349$	−23.5500	−1.26060	−0.630300	−	0.776351i	$-0.717068\pi$
$349$	−23.5500	−1.26060	−0.630300	+	0.776351i	$0.717068\pi$
$350$	−7.48929	−0.400319
$351$	0	0
$352$	−4.78623	−0.255107
$353$	7.64973	0.407154	0.203577	−	0.979059i	$-0.434743\pi$
$353$	7.64973	0.407154	0.203577	+	0.979059i	$0.434743\pi$
$354$	0	0
$355$	−2.06740	−0.109726
$356$	−20.0147	−1.06078
$357$	0	0
$358$	56.1396	2.96707
$359$	−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
$362$	−15.3717	−0.807918
$363$	0	0
$364$	−3.48929	−0.182888
$365$	−20.5468	−1.07547
$366$	0	0
$367$	5.33871	0.278679	0.139339	−	0.990245i	$-0.455502\pi$
$367$	5.33871	0.278679	0.139339	+	0.990245i	$0.455502\pi$
$368$	3.79923	0.198049
$369$	0	0
$370$	−26.8009	−1.39331
$371$	−0.782020	−0.0406004
$372$	0	0
$373$	21.5212	1.11433	0.557163	−	0.830403i	$-0.311890\pi$
$373$	21.5212	1.11433	0.557163	+	0.830403i	$0.311890\pi$
$374$	−15.6644	−0.809988
$375$	0	0
$376$	−37.0361	−1.90999
$377$	−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	$0.462223\pi$
$380$	−15.6644	−0.803568
$381$	0	0
$382$	−10.2927	−0.526622
$383$	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
$386$	19.4292	0.988922
$387$	0	0
$388$	−18.6430	−0.946455
$389$	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
$392$	−3.48929	−0.176236
$393$	0	0
$394$	−7.42923	−0.374279
$395$	1.18500	0.0596238
$396$	0	0
$397$	1.40046	0.0702872	0.0351436	−	0.999382i	$-0.488811\pi$
$397$	1.40046	0.0702872	0.0351436	+	0.999382i	$0.488811\pi$
$398$	31.8568	1.59684
$399$	0	0
$400$	−3.82487	−0.191243
$401$	6.97858	0.348494	0.174247	−	0.984702i	$-0.444251\pi$
$401$	6.97858	0.348494	0.174247	+	0.984702i	$0.444251\pi$
$402$	0	0
$403$	1.63565	0.0814777
$404$	−38.8929	−1.93499
$405$	0	0
$406$	−24.5756	−1.21967
$407$	−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	$0.650144\pi$
$410$	−0.921039	−0.0454869
$411$	0	0
$412$	−11.9143	−0.586976
$413$	12.6430	0.622121
$414$	0	0
$415$	16.2885	0.799572
$416$	4.17513	0.204703
$417$	0	0
$418$	−8.97858	−0.439157
$419$	30.0393	1.46751	0.733757	−	0.679412i	$-0.237765\pi$
$419$	30.0393	1.46751	0.733757	+	0.679412i	$0.237765\pi$
$420$	0	0
$421$	−8.31729	−0.405360	−0.202680	−	0.979245i	$-0.564965\pi$
$421$	−8.31729	−0.405360	−0.202680	+	0.979245i	$0.564965\pi$
$422$	−21.7318	−1.05789
$423$	0	0
$424$	−2.72869	−0.132517
$425$	18.6430	0.904318
$426$	0	0
$427$	2.00000	0.0967868
$428$	−17.3717	−0.839692
$429$	0	0
$430$	25.6546	1.23717
$431$	−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	$0.282182\pi$
$434$	3.83221	0.183952
$435$	0	0
$436$	−47.0937	−2.25538
$437$	−10.6142	−0.507748
$438$	0	0
$439$	33.8139	1.61385	0.806925	−	0.590654i	$-0.201130\pi$
$439$	33.8139	1.61385	0.806925	+	0.590654i	$0.201130\pi$
$440$	5.37169	0.256085
$441$	0	0
$442$	13.6644	0.649950
$443$	−26.4464	−1.25651	−0.628254	−	0.778008i	$-0.716230\pi$
$443$	−26.4464	−1.25651	−0.628254	+	0.778008i	$0.716230\pi$
$444$	0	0
$445$	−7.70306	−0.365160
$446$	45.9044	2.17364
$447$	0	0
$448$	12.1751	0.575221
$449$	−2.64300	−0.124731	−0.0623655	−	0.998053i	$-0.519864\pi$
$449$	−2.64300	−0.124731	−0.0623655	+	0.998053i	$0.519864\pi$
$450$	0	0
$451$	−0.335577	−0.0158017
$452$	−57.3864	−2.69923
$453$	0	0
$454$	−46.0722	−2.16228
$455$	−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	$0.788859\pi$
$458$	−18.1923	−0.850073
$459$	0	0
$460$	14.8782	0.693699
$461$	−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	$-0.518636\pi$
$463$	−2.51806	−0.117024	−0.0585120	+	0.998287i	$0.518636\pi$
$464$	−12.5510	−0.582667
$465$	0	0
$466$	28.5756	1.32374
$467$	−2.57560	−0.119184	−0.0595922	−	0.998223i	$-0.518980\pi$
$467$	−2.57560	−0.119184	−0.0595922	+	0.998223i	$0.518980\pi$
$468$	0	0
$469$	6.10038	0.281690
$470$	−33.3963	−1.54045
$471$	0	0
$472$	44.1151	2.03056
$473$	9.34713	0.429782
$474$	0	0
$475$	10.6858	0.490300
$476$	20.3503	0.932753
$477$	0	0
$478$	−24.1151	−1.10300
$479$	−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
$482$	9.43910	0.429939
$483$	0	0
$484$	−33.7967	−1.53621
$485$	−7.17513	−0.325806
$486$	0	0
$487$	36.0575	1.63392	0.816962	−	0.576692i	$-0.195657\pi$
$487$	36.0575	1.63392	0.816962	+	0.576692i	$0.195657\pi$
$488$	6.97858	0.315905
$489$	0	0
$490$	−3.14637	−0.142138
$491$	9.22846	0.416475	0.208237	−	0.978078i	$-0.433227\pi$
$491$	9.22846	0.416475	0.208237	+	0.978078i	$0.433227\pi$
$492$	0	0
$493$	61.1758	2.75522
$494$	7.83221	0.352388
$495$	0	0
$496$	1.95715	0.0878788
$497$	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	$0.492852\pi$
$500$	−38.4078	−1.71765
$501$	0	0
$502$	6.82065	0.304421
$503$	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
$506$	8.52792	0.379112
$507$	0	0
$508$	42.0722	1.86665
$509$	−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
$512$	13.3461	0.589818
$513$	0	0
$514$	−45.9143	−2.02519
$515$	−4.58546	−0.202060
$516$	0	0
$517$	−12.1678	−0.535139
$518$	19.9572	0.876867
$519$	0	0
$520$	−4.68585	−0.205488
$521$	16.2646	0.712564	0.356282	−	0.934378i	$-0.384044\pi$
$521$	16.2646	0.712564	0.356282	+	0.934378i	$0.384044\pi$
$522$	0	0
$523$	7.22219	0.315804	0.157902	−	0.987455i	$-0.449527\pi$
$523$	7.22219	0.315804	0.157902	+	0.987455i	$0.449527\pi$
$524$	12.7862	0.558569
$525$	0	0
$526$	17.7318	0.773144
$527$	−9.53948	−0.415546
$528$	0	0
$529$	−12.9185	−0.561675
$530$	−2.46052	−0.106878
$531$	0	0
$532$	11.6644	0.505717
$533$	0.292731	0.0126796
$534$	0	0
$535$	−6.68585	−0.289054
$536$	21.2860	0.919415
$537$	0	0
$538$	−22.1923	−0.956780
$539$	−1.14637	−0.0493775
$540$	0	0
$541$	17.3534	0.746081	0.373041	−	0.927815i	$-0.378315\pi$
$541$	17.3534	0.746081	0.373041	+	0.927815i	$0.378315\pi$
$542$	−68.8156	−2.95588
$543$	0	0
$544$	−24.3503	−1.04401
$545$	−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	$0.760125\pi$
$548$	45.7220	1.95315
$549$	0	0
$550$	−8.58546	−0.366085
$551$	35.0649	1.49381
$552$	0	0
$553$	−0.882404	−0.0375236
$554$	4.46052	0.189509
$555$	0	0
$556$	26.1579	1.10934
$557$	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
$560$	−1.60688	−0.0679033
$561$	0	0
$562$	−48.2070	−2.03349
$563$	41.3717	1.74361	0.871804	−	0.489854i	$-0.162950\pi$
$563$	41.3717	1.74361	0.871804	+	0.489854i	$0.162950\pi$
$564$	0	0
$565$	−22.0863	−0.929178
$566$	63.2432	2.65831
$567$	0	0
$568$	5.37169	0.225391
$569$	2.68164	0.112420	0.0562100	−	0.998419i	$-0.482098\pi$
$569$	2.68164	0.112420	0.0562100	+	0.998419i	$0.482098\pi$
$570$	0	0
$571$	−28.9315	−1.21075	−0.605373	−	0.795942i	$-0.706976\pi$
$571$	−28.9315	−1.21075	−0.605373	+	0.795942i	$0.706976\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	0.685846	0.0286267
$575$	−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	$0.214613\pi$
$578$	−39.8641	−1.65813
$579$	0	0
$580$	−49.1512	−2.04089
$581$	−12.1292	−0.503202
$582$	0	0
$583$	−0.896480	−0.0371284
$584$	53.3864	2.20914
$585$	0	0
$586$	−34.9259	−1.44277
$587$	−23.0649	−0.951990	−0.475995	−	0.879448i	$-0.657912\pi$
$587$	−23.0649	−0.951990	−0.475995	+	0.879448i	$0.657912\pi$
$588$	0	0
$589$	−5.46787	−0.225299
$590$	39.7795	1.63770
$591$	0	0
$592$	10.1923	0.418903
$593$	11.0502	0.453777	0.226889	−	0.973921i	$-0.427145\pi$
$593$	11.0502	0.453777	0.226889	+	0.973921i	$0.427145\pi$
$594$	0	0
$595$	7.83221	0.321089
$596$	−7.56404	−0.309835
$597$	0	0
$598$	−7.43910	−0.304207
$599$	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	$0.0693520\pi$
$602$	−19.1035	−0.778601
$603$	0	0
$604$	52.0294	2.11705
$605$	−13.0073	−0.528824
$606$	0	0
$607$	45.0691	1.82930	0.914649	−	0.404249i	$-0.132467\pi$
$607$	45.0691	1.82930	0.914649	+	0.404249i	$0.132467\pi$
$608$	−13.9572	−0.566037
$609$	0	0
$610$	6.29273	0.254785
$611$	10.6142	0.429406
$612$	0	0
$613$	−25.5212	−1.03079	−0.515396	−	0.856952i	$-0.672355\pi$
$613$	−25.5212	−1.03079	−0.515396	+	0.856952i	$0.672355\pi$
$614$	−60.9834	−2.46109
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−29.2432	−1.17729	−0.588643	−	0.808393i	$-0.700337\pi$
$617$	−29.2432	−1.17729	−0.588643	+	0.808393i	$0.700337\pi$
$618$	0	0
$619$	4.78623	0.192375	0.0961874	−	0.995363i	$-0.469335\pi$
$619$	4.78623	0.192375	0.0961874	+	0.995363i	$0.469335\pi$
$620$	7.66442	0.307811
$621$	0	0
$622$	45.6791	1.83157
$623$	5.73604	0.229810
$624$	0	0
$625$	1.20077	0.0480307
$626$	8.15792	0.326056
$627$	0	0
$628$	79.7942	3.18413
$629$	−49.6791	−1.98084
$630$	0	0
$631$	−28.3931	−1.13031	−0.565156	−	0.824984i	$-0.691184\pi$
$631$	−28.3931	−1.13031	−0.565156	+	0.824984i	$0.691184\pi$
$632$	−3.07896	−0.122475
$633$	0	0
$634$	11.7648	0.467240
$635$	16.1923	0.642574
$636$	0	0
$637$	1.00000	0.0396214
$638$	−28.1726	−1.11536
$639$	0	0
$640$	27.0937	1.07097
$641$	−5.96137	−0.235460	−0.117730	−	0.993046i	$-0.537562\pi$
$641$	−5.96137	−0.235460	−0.117730	+	0.993046i	$0.537562\pi$
$642$	0	0
$643$	31.1940	1.23017	0.615086	−	0.788460i	$-0.289121\pi$
$643$	31.1940	1.23017	0.615086	+	0.788460i	$0.289121\pi$
$644$	−11.0790	−0.436572
$645$	0	0
$646$	−45.6791	−1.79722
$647$	−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
$650$	7.48929	0.293754
$651$	0	0
$652$	24.7005	0.967348
$653$	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
$656$	0.350269	0.0136757
$657$	0	0
$658$	24.8683	0.969468
$659$	3.90383	0.152071	0.0760357	−	0.997105i	$-0.475774\pi$
$659$	3.90383	0.152071	0.0760357	+	0.997105i	$0.475774\pi$
$660$	0	0
$661$	13.7936	0.536508	0.268254	−	0.963348i	$-0.413553\pi$
$661$	13.7936	0.536508	0.268254	+	0.963348i	$0.413553\pi$
$662$	14.4078	0.559975
$663$	0	0
$664$	−42.3221	−1.64242
$665$	4.48929	0.174087
$666$	0	0
$667$	−33.3049	−1.28957
$668$	−9.12181	−0.352933
$669$	0	0
$670$	19.1940	0.741530
$671$	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	$0.464941\pi$
$674$	60.0393	2.31263
$675$	0	0
$676$	3.48929	0.134203
$677$	−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
$680$	27.3288	1.04801
$681$	0	0
$682$	4.39312	0.168221
$683$	−1.03612	−0.0396459	−0.0198229	−	0.999804i	$-0.506310\pi$
$683$	−1.03612	−0.0396459	−0.0198229	+	0.999804i	$0.506310\pi$
$684$	0	0
$685$	17.5970	0.672348
$686$	2.34292	0.0894532
$687$	0	0
$688$	−9.75639	−0.371959
$689$	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	$0.477710\pi$
$692$	−38.3931	−1.45949
$693$	0	0
$694$	−39.1281	−1.48528
$695$	10.0674	0.381878
$696$	0	0
$697$	−1.70727	−0.0646674
$698$	55.1758	2.08843
$699$	0	0
$700$	11.1537	0.421571
$701$	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
$704$	13.9572	0.526030
$705$	0	0
$706$	−17.9227	−0.674531
$707$	11.1464	0.419202
$708$	0	0
$709$	−42.9834	−1.61428	−0.807138	−	0.590363i	$-0.798985\pi$
$709$	−42.9834	−1.61428	−0.807138	+	0.590363i	$0.798985\pi$
$710$	4.84377	0.181783
$711$	0	0
$712$	20.0147	0.750082
$713$	5.19342	0.194495
$714$	0	0
$715$	−1.53948	−0.0575733
$716$	−83.6081	−3.12458
$717$	0	0
$718$	43.0508	1.60664
$719$	−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
$722$	18.3331	0.682286
$723$	0	0
$724$	22.8929	0.850807
$725$	33.5296	1.24526
$726$	0	0
$727$	−23.8077	−0.882977	−0.441488	−	0.897267i	$-0.645549\pi$
$727$	−23.8077	−0.882977	−0.441488	+	0.897267i	$0.645549\pi$
$728$	3.48929	0.129322
$729$	0	0
$730$	48.1396	1.78173
$731$	47.5542	1.75885
$732$	0	0
$733$	31.3492	1.15791	0.578954	−	0.815360i	$-0.303461\pi$
$733$	31.3492	1.15791	0.578954	+	0.815360i	$0.303461\pi$
$734$	−12.5082	−0.461686
$735$	0	0
$736$	13.2566	0.488645
$737$	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	$0.650840\pi$
$740$	39.9143	1.46728
$741$	0	0
$742$	1.83221	0.0672626
$743$	−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
$746$	−50.4225	−1.84610
$747$	0	0
$748$	23.3288	0.852987
$749$	4.97858	0.181913
$750$	0	0
$751$	19.2243	0.701503	0.350751	−	0.936469i	$-0.385926\pi$
$751$	19.2243	0.701503	0.350751	+	0.936469i	$0.385926\pi$
$752$	12.7005	0.463141
$753$	0	0
$754$	24.5756	0.894990
$755$	20.0246	0.728768
$756$	0	0
$757$	−19.8610	−0.721860	−0.360930	−	0.932593i	$-0.617541\pi$
$757$	−19.8610	−0.721860	−0.360930	+	0.932593i	$0.617541\pi$
$758$	−10.8009	−0.392307
$759$	0	0
$760$	15.6644	0.568208
$761$	−6.12073	−0.221876	−0.110938	−	0.993827i	$-0.535386\pi$
$761$	−6.12073	−0.221876	−0.110938	+	0.993827i	$0.535386\pi$
$762$	0	0
$763$	13.4966	0.488611
$764$	15.3288	0.554578
$765$	0	0
$766$	−19.5296	−0.705634
$767$	−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	$0.478013\pi$
$770$	−3.60688	−0.129983
$771$	0	0
$772$	−28.9357	−1.04142
$773$	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
$776$	18.6430	0.669245
$777$	0	0
$778$	−15.0937	−0.541134
$779$	−0.978577	−0.0350612
$780$	0	0
$781$	1.76481	0.0631498
$782$	43.3864	1.55149
$783$	0	0
$784$	1.19656	0.0427342
$785$	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	$0.314983\pi$
$788$	11.0643	0.394148
$789$	0	0
$790$	−2.77636	−0.0987786
$791$	16.4464	0.584768
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	−3.28117	−0.116444
$795$	0	0
$796$	−47.4439	−1.68161
$797$	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
$800$	−13.3461	−0.471854
$801$	0	0
$802$	−16.3503	−0.577348
$803$	17.5395	0.618955
$804$	0	0
$805$	−4.26396	−0.150285
$806$	−3.83221	−0.134984
$807$	0	0
$808$	38.8929	1.36825
$809$	−1.04033	−0.0365759	−0.0182880	−	0.999833i	$-0.505822\pi$
$809$	−1.04033	−0.0365759	−0.0182880	+	0.999833i	$0.505822\pi$
$810$	0	0
$811$	12.6712	0.444944	0.222472	−	0.974939i	$-0.428587\pi$
$811$	12.6712	0.444944	0.222472	+	0.974939i	$0.428587\pi$
$812$	36.6002	1.28441
$813$	0	0
$814$	22.8782	0.801880
$815$	9.50650	0.332998
$816$	0	0
$817$	27.2572	0.953610
$818$	43.0607	1.50558
$819$	0	0
$820$	1.37169	0.0479016
$821$	−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	$-0.685901\pi$
$823$	−31.6363	−1.10277	−0.551386	+	0.834251i	$0.685901\pi$
$824$	11.9143	0.415055
$825$	0	0
$826$	−29.6216	−1.03067
$827$	−56.4800	−1.96400	−0.982002	−	0.188872i	$-0.939517\pi$
$827$	−56.4800	−1.96400	−0.982002	+	0.188872i	$0.939517\pi$
$828$	0	0
$829$	−42.6760	−1.48220	−0.741099	−	0.671396i	$-0.765695\pi$
$829$	−42.6760	−1.48220	−0.741099	+	0.671396i	$0.765695\pi$
$830$	−38.1627	−1.32465
$831$	0	0
$832$	−12.1751	−0.422097
$833$	−5.83221	−0.202074
$834$	0	0
$835$	−3.51071	−0.121493
$836$	13.3717	0.462470
$837$	0	0
$838$	−70.3797	−2.43122
$839$	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
$842$	19.4868	0.671558
$843$	0	0
$844$	32.3650	1.11405
$845$	1.34292	0.0461980
$846$	0	0
$847$	9.68585	0.332810
$848$	0.935731	0.0321331
$849$	0	0
$850$	−43.6791	−1.49818
$851$	27.0460	0.927124
$852$	0	0
$853$	19.6932	0.674282	0.337141	−	0.941454i	$-0.390540\pi$
$853$	19.6932	0.674282	0.337141	+	0.941454i	$0.390540\pi$
$854$	−4.68585	−0.160346
$855$	0	0
$856$	17.3717	0.593752
$857$	−1.66442	−0.0568556	−0.0284278	−	0.999596i	$-0.509050\pi$
$857$	−1.66442	−0.0568556	−0.0284278	+	0.999596i	$0.509050\pi$
$858$	0	0
$859$	−41.2944	−1.40895	−0.704474	−	0.709730i	$-0.748817\pi$
$859$	−41.2944	−1.40895	−0.704474	+	0.709730i	$0.748817\pi$
$860$	−38.2070	−1.30285
$861$	0	0
$862$	22.6086	0.770051
$863$	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
$866$	−61.6363	−2.09449
$867$	0	0
$868$	−5.70727	−0.193717
$869$	−1.01156	−0.0343147
$870$	0	0
$871$	−6.10038	−0.206704
$872$	47.0937	1.59479
$873$	0	0
$874$	24.8683	0.841184
$875$	11.0073	0.372116
$876$	0	0
$877$	−53.8041	−1.81683	−0.908417	−	0.418065i	$-0.862708\pi$
$877$	−53.8041	−1.81683	−0.908417	+	0.418065i	$0.862708\pi$
$878$	−79.2234	−2.67366
$879$	0	0
$880$	−1.84208	−0.0620964
$881$	−8.09196	−0.272625	−0.136313	−	0.990666i	$-0.543525\pi$
$881$	−8.09196	−0.272625	−0.136313	+	0.990666i	$0.543525\pi$
$882$	0	0
$883$	−27.0705	−0.910996	−0.455498	−	0.890237i	$-0.650539\pi$
$883$	−27.0705	−0.910996	−0.455498	+	0.890237i	$0.650539\pi$
$884$	−20.3503	−0.684454
$885$	0	0
$886$	61.9620	2.08165
$887$	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
$890$	18.0477	0.604959
$891$	0	0
$892$	−68.3650	−2.28903
$893$	−35.4826	−1.18738
$894$	0	0
$895$	−32.1783	−1.07560
$896$	−20.1751	−0.674004
$897$	0	0
$898$	6.19235	0.206641
$899$	−17.1568	−0.572213
$900$	0	0
$901$	−4.56090	−0.151946
$902$	0.786230	0.0261786
$903$	0	0
$904$	57.3864	1.90864
$905$	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	$0.416045\pi$
$908$	68.6148	2.27706
$909$	0	0
$910$	3.14637	0.104301
$911$	−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
$914$	78.9307	2.61080
$915$	0	0
$916$	27.0937	0.895200
$917$	−3.66442	−0.121010
$918$	0	0
$919$	−27.2432	−0.898669	−0.449334	−	0.893364i	$-0.648339\pi$
$919$	−27.2432	−0.898669	−0.449334	+	0.893364i	$0.648339\pi$
$920$	−14.8782	−0.490519
$921$	0	0
$922$	77.5015	2.55237
$923$	−1.53948	−0.0506726
$924$	0	0
$925$	−27.2285	−0.895266
$926$	5.89962	0.193873
$927$	0	0
$928$	−43.7942	−1.43761
$929$	17.1422	0.562416	0.281208	−	0.959647i	$-0.409265\pi$
$929$	17.1422	0.562416	0.281208	+	0.959647i	$0.409265\pi$
$930$	0	0
$931$	−3.34292	−0.109560
$932$	−42.5573	−1.39401
$933$	0	0
$934$	6.03442	0.197452
$935$	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	$0.818350\pi$
$938$	−14.2927	−0.466674
$939$	0	0
$940$	49.7367	1.62223
$941$	−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
$944$	−15.1281	−0.492377
$945$	0	0
$946$	−21.8996	−0.712018
$947$	−20.9295	−0.680116	−0.340058	−	0.940404i	$-0.610447\pi$
$947$	−20.9295	−0.680116	−0.340058	+	0.940404i	$0.610447\pi$
$948$	0	0
$949$	−15.3001	−0.496662
$950$	−25.0361	−0.812279
$951$	0	0
$952$	−20.3503	−0.659556
$953$	−46.4120	−1.50343	−0.751716	−	0.659487i	$-0.770774\pi$
$953$	−46.4120	−1.50343	−0.751716	+	0.659487i	$0.770774\pi$
$954$	0	0
$955$	5.89962	0.190907
$956$	35.9143	1.16155
$957$	0	0
$958$	1.20390	0.0388964
$959$	−13.1035	−0.423135
$960$	0	0
$961$	−28.3246	−0.913698
$962$	−19.9572	−0.643444
$963$	0	0
$964$	−14.0575	−0.452763
$965$	−11.1365	−0.358497
$966$	0	0
$967$	−23.2186	−0.746660	−0.373330	−	0.927699i	$-0.621784\pi$
$967$	−23.2186	−0.746660	−0.373330	+	0.927699i	$0.621784\pi$
$968$	33.7967	1.08627
$969$	0	0
$970$	16.8108	0.539762
$971$	14.1004	0.452503	0.226251	−	0.974069i	$-0.427353\pi$
$971$	14.1004	0.452503	0.226251	+	0.974069i	$0.427353\pi$
$972$	0	0
$973$	−7.49663	−0.240331
$974$	−84.4800	−2.70692
$975$	0	0
$976$	−2.39312	−0.0766018
$977$	2.95402	0.0945074	0.0472537	−	0.998883i	$-0.484953\pi$
$977$	2.95402	0.0945074	0.0472537	+	0.998883i	$0.484953\pi$
$978$	0	0
$979$	6.57560	0.210157
$980$	4.68585	0.149684
$981$	0	0
$982$	−21.6216	−0.689972
$983$	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
$986$	−143.330	−4.56456
$987$	0	0
$988$	−11.6644	−0.371095
$989$	−25.8891	−0.823227
$990$	0	0
$991$	47.9718	1.52388	0.761938	−	0.647650i	$-0.224248\pi$
$991$	47.9718	1.52388	0.761938	+	0.647650i	$0.224248\pi$
$992$	6.82908	0.216823
$993$	0	0
$994$	−3.60688	−0.114403
$995$	−18.2598	−0.578873
$996$	0	0
$997$	−38.4422	−1.21748	−0.608739	−	0.793371i	$-0.708324\pi$
$997$	−38.4422	−1.21748	−0.608739	+	0.793371i	$0.708324\pi$
$998$	−2.35027	−0.0743965
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.a.i.1.1		3
3.2	odd	2		91.2.a.d.1.3	✓	3
7.6	odd	2		5733.2.a.x.1.1		3
12.11	even	2		1456.2.a.t.1.2		3
15.14	odd	2		2275.2.a.m.1.1		3
21.2	odd	6		637.2.e.j.508.1		6
21.5	even	6		637.2.e.i.508.1		6
21.11	odd	6		637.2.e.j.79.1		6
21.17	even	6		637.2.e.i.79.1		6
21.20	even	2		637.2.a.j.1.3		3
24.5	odd	2		5824.2.a.by.1.2		3
24.11	even	2		5824.2.a.bs.1.2		3
39.5	even	4		1183.2.c.f.337.1		6
39.8	even	4		1183.2.c.f.337.6		6
39.38	odd	2		1183.2.a.i.1.1		3
273.272	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	3.2	odd	2
637.2.a.j.1.3		3	21.20	even	2
637.2.e.i.79.1		6	21.17	even	6
637.2.e.i.508.1		6	21.5	even	6
637.2.e.j.79.1		6	21.11	odd	6
637.2.e.j.508.1		6	21.2	odd	6
819.2.a.i.1.1		3	1.1	even	1	trivial
1183.2.a.i.1.1		3	39.38	odd	2
1183.2.c.f.337.1		6	39.5	even	4
1183.2.c.f.337.6		6	39.8	even	4
1456.2.a.t.1.2		3	12.11	even	2
2275.2.a.m.1.1		3	15.14	odd	2
5733.2.a.x.1.1		3	7.6	odd	2
5824.2.a.bs.1.2		3	24.11	even	2
5824.2.a.by.1.2		3	24.5	odd	2
8281.2.a.bg.1.1		3	273.272	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34292 q^{2} +3.48929 q^{4} +1.34292 q^{5} -1.00000 q^{7} -3.48929 q^{8} -3.14637 q^{10} -1.14637 q^{11} +1.00000 q^{13} +2.34292 q^{14} +1.19656 q^{16} -5.83221 q^{17} -3.34292 q^{19} +4.68585 q^{20} +2.68585 q^{22} +3.17513 q^{23} -3.19656 q^{25} -2.34292 q^{26} -3.48929 q^{28} -10.4893 q^{29} +1.63565 q^{31} +4.17513 q^{32} +13.6644 q^{34} -1.34292 q^{35} +8.51806 q^{37} +7.83221 q^{38} -4.68585 q^{40} +0.292731 q^{41} -8.15371 q^{43} -4.00000 q^{44} -7.43910 q^{46} +10.6142 q^{47} +1.00000 q^{49} +7.48929 q^{50} +3.48929 q^{52} +0.782020 q^{53} -1.53948 q^{55} +3.48929 q^{56} +24.5756 q^{58} -12.6430 q^{59} -2.00000 q^{61} -3.83221 q^{62} -12.1751 q^{64} +1.34292 q^{65} -6.10038 q^{67} -20.3503 q^{68} +3.14637 q^{70} -1.53948 q^{71} -15.3001 q^{73} -19.9572 q^{74} -11.6644 q^{76} +1.14637 q^{77} +0.882404 q^{79} +1.60688 q^{80} -0.685846 q^{82} +12.1292 q^{83} -7.83221 q^{85} +19.1035 q^{86} +4.00000 q^{88} -5.73604 q^{89} -1.00000 q^{91} +11.0790 q^{92} -24.8683 q^{94} -4.48929 q^{95} -5.34292 q^{97} -2.34292 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{4} - 2 q^{5} - 3 q^{7} - 3 q^{8} - 8 q^{10} - 2 q^{11} + 3 q^{13} + q^{14} - q^{16} - 4 q^{17} - 4 q^{19} + 2 q^{20} - 4 q^{22} - 10 q^{23} - 5 q^{25} - q^{26} - 3 q^{28} - 24 q^{29}+ \cdots - q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^4 - 2 * q^5 - 3 * q^7 - 3 * q^8 - 8 * q^10 - 2 * q^11 + 3 * q^13 + q^14 - q^16 - 4 * q^17 - 4 * q^19 + 2 * q^20 - 4 * q^22 - 10 * q^23 - 5 * q^25 - q^26 - 3 * q^28 - 24 * q^29 - 4 * q^31 - 7 * q^32 + 14 * q^34 + 2 * q^35 + 10 * q^38 - 2 * q^40 - 2 * q^41 + 10 * q^43 - 12 * q^44 - 18 * q^46 + 8 * q^47 + 3 * q^49 + 15 * q^50 + 3 * q^52 - 8 * q^53 + 6 * q^55 + 3 * q^56 + 12 * q^58 + 4 * q^59 - 6 * q^61 + 2 * q^62 - 17 * q^64 - 2 * q^65 - 12 * q^67 - 22 * q^68 + 8 * q^70 + 6 * q^71 - 10 * q^73 - 30 * q^74 - 8 * q^76 + 2 * q^77 - 14 * q^79 + 14 * q^80 + 10 * q^82 + 12 * q^83 - 10 * q^85 + 26 * q^86 + 12 * q^88 - 2 * q^89 - 3 * q^91 + 12 * q^92 - 10 * q^94 - 6 * q^95 - 10 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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