LMFDB - Embedded newform 8281.2.a.by.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8281,2,Mod(1,8281)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8281, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8281.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-4,0,4,6,4,0,-12,4,-12,-4,-2,0,0,-20,8,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7674048.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*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			1.6
Root			$2.38595$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.38595 q^{2} +2.82577 q^{3} -0.0791355 q^{4} -0.518957 q^{5} +3.91639 q^{6} -2.88158 q^{8} +4.98500 q^{9} -0.719250 q^{10} +1.62416 q^{11} -0.223619 q^{12} -1.46646 q^{15} -3.83547 q^{16} -1.94825 q^{17} +6.90897 q^{18} -2.49115 q^{19} +0.0410679 q^{20} +2.25101 q^{22} -9.14058 q^{23} -8.14270 q^{24} -4.73068 q^{25} +5.60916 q^{27} -5.22996 q^{29} -2.03244 q^{30} -5.79391 q^{31} +0.447392 q^{32} +4.58951 q^{33} -2.70019 q^{34} -0.394491 q^{36} -10.2293 q^{37} -3.45262 q^{38} +1.49542 q^{40} +4.20903 q^{41} -0.997311 q^{43} -0.128529 q^{44} -2.58700 q^{45} -12.6684 q^{46} +4.51725 q^{47} -10.8382 q^{48} -6.55650 q^{50} -5.50532 q^{51} -8.89651 q^{53} +7.77403 q^{54} -0.842869 q^{55} -7.03944 q^{57} -7.24847 q^{58} -6.20526 q^{59} +0.116049 q^{60} +13.4707 q^{61} -8.03008 q^{62} +8.29100 q^{64} +6.36084 q^{66} +8.37266 q^{67} +0.154176 q^{68} -25.8292 q^{69} -5.19809 q^{71} -14.3647 q^{72} +11.8395 q^{73} -14.1773 q^{74} -13.3678 q^{75} +0.197139 q^{76} +0.982310 q^{79} +1.99044 q^{80} +0.895217 q^{81} +5.83352 q^{82} +8.91851 q^{83} +1.01106 q^{85} -1.38223 q^{86} -14.7787 q^{87} -4.68015 q^{88} +12.0190 q^{89} -3.58546 q^{90} +0.723345 q^{92} -16.3723 q^{93} +6.26070 q^{94} +1.29280 q^{95} +1.26423 q^{96} -4.42228 q^{97} +8.09643 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 12 q^{8} + 4 q^{9} - 12 q^{10} - 4 q^{11} - 2 q^{12} - 20 q^{15} + 8 q^{16} + 4 q^{17} + 16 q^{18} + 2 q^{19} + 26 q^{20} - 6 q^{22} - 12 q^{23} + 2 q^{24}+ \cdots - 16 q^{99}+O(q^{100}) $ 6 * q - 4 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^6 - 12 * q^8 + 4 * q^9 - 12 * q^10 - 4 * q^11 - 2 * q^12 - 20 * q^15 + 8 * q^16 + 4 * q^17 + 16 * q^18 + 2 * q^19 + 26 * q^20 - 6 * q^22 - 12 * q^23 + 2 * q^24 + 10 * q^25 - 6 * q^27 - 8 * q^29 + 8 * q^30 - 14 * q^31 - 8 * q^32 + 16 * q^33 - 2 * q^34 - 10 * q^36 - 12 * q^37 + 2 * q^38 - 46 * q^40 + 28 * q^41 + 2 * q^43 + 20 * q^44 + 16 * q^45 - 20 * q^46 + 14 * q^47 - 2 * q^48 - 32 * q^50 - 26 * q^51 - 22 * q^53 + 14 * q^54 - 6 * q^55 - 4 * q^58 - 2 * q^59 + 14 * q^61 + 4 * q^62 + 26 * q^64 + 26 * q^66 - 24 * q^67 - 8 * q^68 - 4 * q^69 - 4 * q^71 - 8 * q^72 + 36 * q^73 - 6 * q^74 - 46 * q^75 - 26 * q^76 - 28 * q^79 + 36 * q^80 - 2 * q^81 - 14 * q^82 + 26 * q^83 + 20 * q^85 + 24 * q^86 - 2 * q^87 - 14 * q^88 + 42 * q^89 + 12 * q^90 + 12 * q^92 + 4 * q^94 - 22 * q^95 - 42 * q^96 + 24 * q^97 - 16 * q^99

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 $

$2$	1.38595	0.980016	0.490008	−	0.871718i	$-0.336994\pi$
$2$	1.38595	0.980016	0.490008	+	0.871718i	$0.336994\pi$
$3$	2.82577	1.63146	0.815731	−	0.578432i	$-0.196335\pi$
$3$	2.82577	1.63146	0.815731	+	0.578432i	$0.196335\pi$
$4$	−0.0791355	−0.0395678
$5$	−0.518957	−0.232085	−0.116042	−	0.993244i	$-0.537021\pi$
$5$	−0.518957	−0.232085	−0.116042	+	0.993244i	$0.537021\pi$
$6$	3.91639	1.59886
$7$	0	0
$7$	0	0
$8$	−2.88158	−1.01879
$9$	4.98500	1.66167
$10$	−0.719250	−0.227447
$11$	1.62416	0.489702	0.244851	−	0.969561i	$-0.421261\pi$
$11$	1.62416	0.489702	0.244851	+	0.969561i	$0.421261\pi$
$12$	−0.223619	−0.0645533
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.46646	−0.378637
$16$	−3.83547	−0.958867
$17$	−1.94825	−0.472521	−0.236260	−	0.971690i	$-0.575922\pi$
$17$	−1.94825	−0.472521	−0.236260	+	0.971690i	$0.575922\pi$
$18$	6.90897	1.62846
$19$	−2.49115	−0.571510	−0.285755	−	0.958303i	$-0.592244\pi$
$19$	−2.49115	−0.571510	−0.285755	+	0.958303i	$0.592244\pi$
$20$	0.0410679	0.00918307
$21$	0	0
$22$	2.25101	0.479916
$23$	−9.14058	−1.90594	−0.952971	−	0.303060i	$-0.901992\pi$
$23$	−9.14058	−1.90594	−0.952971	+	0.303060i	$0.901992\pi$
$24$	−8.14270	−1.66212
$25$	−4.73068	−0.946137
$26$	0	0
$27$	5.60916	1.07948
$28$	0	0
$29$	−5.22996	−0.971179	−0.485589	−	0.874187i	$-0.661395\pi$
$29$	−5.22996	−0.971179	−0.485589	+	0.874187i	$0.661395\pi$
$30$	−2.03244	−0.371071
$31$	−5.79391	−1.04062	−0.520308	−	0.853978i	$-0.674183\pi$
$31$	−5.79391	−1.04062	−0.520308	+	0.853978i	$0.674183\pi$
$32$	0.447392	0.0790885
$33$	4.58951	0.798931
$34$	−2.70019	−0.463078
$35$	0	0
$36$	−0.394491	−0.0657484
$37$	−10.2293	−1.68168	−0.840840	−	0.541284i	$-0.817938\pi$
$37$	−10.2293	−1.68168	−0.840840	+	0.541284i	$0.817938\pi$
$38$	−3.45262	−0.560089
$39$	0	0
$40$	1.49542	0.236446
$41$	4.20903	0.657340	0.328670	−	0.944445i	$-0.393400\pi$
$41$	4.20903	0.657340	0.328670	+	0.944445i	$0.393400\pi$
$42$	0	0
$43$	−0.997311	−0.152088	−0.0760442	−	0.997104i	$-0.524229\pi$
$43$	−0.997311	−0.152088	−0.0760442	+	0.997104i	$0.524229\pi$
$44$	−0.128529	−0.0193764
$45$	−2.58700	−0.385647
$46$	−12.6684	−1.86786
$47$	4.51725	0.658909	0.329455	−	0.944171i	$-0.393135\pi$
$47$	4.51725	0.658909	0.329455	+	0.944171i	$0.393135\pi$
$48$	−10.8382	−1.56435
$49$	0	0
$50$	−6.55650	−0.927230
$51$	−5.50532	−0.770899
$52$	0	0
$53$	−8.89651	−1.22203	−0.611015	−	0.791619i	$-0.709238\pi$
$53$	−8.89651	−1.22203	−0.611015	+	0.791619i	$0.709238\pi$
$54$	7.77403	1.05791
$55$	−0.842869	−0.113652
$56$	0	0
$57$	−7.03944	−0.932397
$58$	−7.24847	−0.951771
$59$	−6.20526	−0.807856	−0.403928	−	0.914791i	$-0.632355\pi$
$59$	−6.20526	−0.807856	−0.403928	+	0.914791i	$0.632355\pi$
$60$	0.116049	0.0149818
$61$	13.4707	1.72475	0.862375	−	0.506270i	$-0.168976\pi$
$61$	13.4707	1.72475	0.862375	+	0.506270i	$0.168976\pi$
$62$	−8.03008	−1.01982
$63$	0	0
$64$	8.29100	1.03637
$65$	0	0
$66$	6.36084	0.782965
$67$	8.37266	1.02288	0.511442	−	0.859318i	$-0.329112\pi$
$67$	8.37266	1.02288	0.511442	+	0.859318i	$0.329112\pi$
$68$	0.154176	0.0186966
$69$	−25.8292	−3.10947
$70$	0	0
$71$	−5.19809	−0.616900	−0.308450	−	0.951241i	$-0.599810\pi$
$71$	−5.19809	−0.616900	−0.308450	+	0.951241i	$0.599810\pi$
$72$	−14.3647	−1.69289
$73$	11.8395	1.38571	0.692856	−	0.721076i	$-0.256352\pi$
$73$	11.8395	1.38571	0.692856	+	0.721076i	$0.256352\pi$
$74$	−14.1773	−1.64807
$75$	−13.3678	−1.54359
$76$	0.197139	0.0226134
$77$	0	0
$78$	0	0
$79$	0.982310	0.110518	0.0552592	−	0.998472i	$-0.482401\pi$
$79$	0.982310	0.110518	0.0552592	+	0.998472i	$0.482401\pi$
$80$	1.99044	0.222538
$81$	0.895217	0.0994686
$82$	5.83352	0.644204
$83$	8.91851	0.978934	0.489467	−	0.872022i	$-0.337191\pi$
$83$	8.91851	0.978934	0.489467	+	0.872022i	$0.337191\pi$
$84$	0	0
$85$	1.01106	0.109665
$86$	−1.38223	−0.149049
$87$	−14.7787	−1.58444
$88$	−4.68015	−0.498906
$89$	12.0190	1.27401	0.637005	−	0.770860i	$-0.280173\pi$
$89$	12.0190	1.27401	0.637005	+	0.770860i	$0.280173\pi$
$90$	−3.58546	−0.377941
$91$	0	0
$92$	0.723345	0.0754139
$93$	−16.3723	−1.69773
$94$	6.26070	0.645742
$95$	1.29280	0.132639
$96$	1.26423	0.129030
$97$	−4.42228	−0.449015	−0.224507	−	0.974472i	$-0.572077\pi$
$97$	−4.42228	−0.449015	−0.224507	+	0.974472i	$0.572077\pi$
$98$	0	0
$99$	8.09643	0.813722
$100$	0.374365	0.0374365
$101$	−18.3026	−1.82118	−0.910591	−	0.413309i	$-0.864373\pi$
$101$	−18.3026	−1.82118	−0.910591	+	0.413309i	$0.864373\pi$
$102$	−7.63012	−0.755494
$103$	−5.02046	−0.494680	−0.247340	−	0.968929i	$-0.579556\pi$
$103$	−5.02046	−0.494680	−0.247340	+	0.968929i	$0.579556\pi$
$104$	0	0
$105$	0	0
$106$	−12.3301	−1.19761
$107$	6.14456	0.594017	0.297008	−	0.954875i	$-0.404011\pi$
$107$	6.14456	0.594017	0.297008	+	0.954875i	$0.404011\pi$
$108$	−0.443884	−0.0427127
$109$	11.8962	1.13945	0.569727	−	0.821834i	$-0.307049\pi$
$109$	11.8962	1.13945	0.569727	+	0.821834i	$0.307049\pi$
$110$	−1.16818	−0.111381
$111$	−28.9056	−2.74359
$112$	0	0
$113$	3.55612	0.334532	0.167266	−	0.985912i	$-0.446506\pi$
$113$	3.55612	0.334532	0.167266	+	0.985912i	$0.446506\pi$
$114$	−9.75633	−0.913764
$115$	4.74357	0.442340
$116$	0.413875	0.0384274
$117$	0	0
$118$	−8.60020	−0.791713
$119$	0	0
$120$	4.22571	0.385753
$121$	−8.36211	−0.760191
$122$	18.6698	1.69028
$123$	11.8938	1.07243
$124$	0.458504	0.0411749
$125$	5.04981	0.451668
$126$	0	0
$127$	−1.42350	−0.126315	−0.0631575	−	0.998004i	$-0.520117\pi$
$127$	−1.42350	−0.126315	−0.0631575	+	0.998004i	$0.520117\pi$
$128$	10.5961	0.936576
$129$	−2.81818	−0.248127
$130$	0	0
$131$	−8.67374	−0.757828	−0.378914	−	0.925432i	$-0.623702\pi$
$131$	−8.67374	−0.757828	−0.378914	+	0.925432i	$0.623702\pi$
$132$	−0.363193	−0.0316119
$133$	0	0
$134$	11.6041	1.00244
$135$	−2.91091	−0.250531
$136$	5.61405	0.481401
$137$	−8.51784	−0.727728	−0.363864	−	0.931452i	$-0.618543\pi$
$137$	−8.51784	−0.727728	−0.363864	+	0.931452i	$0.618543\pi$
$138$	−35.7981	−3.04733
$139$	5.03844	0.427355	0.213677	−	0.976904i	$-0.431456\pi$
$139$	5.03844	0.427355	0.213677	+	0.976904i	$0.431456\pi$
$140$	0	0
$141$	12.7647	1.07499
$142$	−7.20431	−0.604572
$143$	0	0
$144$	−19.1198	−1.59332
$145$	2.71412	0.225396
$146$	16.4090	1.35802
$147$	0	0
$148$	0.809498	0.0665403
$149$	3.36490	0.275663	0.137832	−	0.990456i	$-0.455987\pi$
$149$	3.36490	0.275663	0.137832	+	0.990456i	$0.455987\pi$
$150$	−18.5272	−1.51274
$151$	12.6566	1.02998	0.514991	−	0.857196i	$-0.327795\pi$
$151$	12.6566	1.02998	0.514991	+	0.857196i	$0.327795\pi$
$152$	7.17847	0.582251
$153$	−9.71204	−0.785172
$154$	0	0
$155$	3.00679	0.241511
$156$	0	0
$157$	−10.3691	−0.827547	−0.413773	−	0.910380i	$-0.635789\pi$
$157$	−10.3691	−0.827547	−0.413773	+	0.910380i	$0.635789\pi$
$158$	1.36143	0.108310
$159$	−25.1395	−1.99369
$160$	−0.232177	−0.0183552
$161$	0	0
$162$	1.24073	0.0974808
$163$	−15.7534	−1.23390	−0.616950	−	0.787002i	$-0.711632\pi$
$163$	−15.7534	−1.23390	−0.616950	+	0.787002i	$0.711632\pi$
$164$	−0.333084	−0.0260095
$165$	−2.38176	−0.185420
$166$	12.3606	0.959371
$167$	16.3986	1.26896	0.634481	−	0.772939i	$-0.281214\pi$
$167$	16.3986	1.26896	0.634481	+	0.772939i	$0.281214\pi$
$168$	0	0
$169$	0	0
$170$	1.40128	0.107473
$171$	−12.4184	−0.949659
$172$	0.0789227	0.00601780
$173$	0.301355	0.0229116	0.0114558	−	0.999934i	$-0.496353\pi$
$173$	0.301355	0.0229116	0.0114558	+	0.999934i	$0.496353\pi$
$174$	−20.4825	−1.55278
$175$	0	0
$176$	−6.22941	−0.469559
$177$	−17.5347	−1.31799
$178$	16.6577	1.24855
$179$	−9.81582	−0.733669	−0.366834	−	0.930286i	$-0.619558\pi$
$179$	−9.81582	−0.733669	−0.366834	+	0.930286i	$0.619558\pi$
$180$	0.204724	0.0152592
$181$	12.4320	0.924062	0.462031	−	0.886864i	$-0.347121\pi$
$181$	12.4320	0.924062	0.462031	+	0.886864i	$0.347121\pi$
$182$	0	0
$183$	38.0652	2.81386
$184$	26.3393	1.94176
$185$	5.30854	0.390292
$186$	−22.6912	−1.66380
$187$	−3.16427	−0.231395
$188$	−0.357475	−0.0260716
$189$	0	0
$190$	1.79176	0.129988
$191$	−12.2469	−0.886156	−0.443078	−	0.896483i	$-0.646114\pi$
$191$	−12.2469	−0.886156	−0.443078	+	0.896483i	$0.646114\pi$
$192$	23.4285	1.69081
$193$	−11.6338	−0.837422	−0.418711	−	0.908119i	$-0.637518\pi$
$193$	−11.6338	−0.837422	−0.418711	+	0.908119i	$0.637518\pi$
$194$	−6.12908	−0.440042
$195$	0	0
$196$	0	0
$197$	1.80114	0.128326	0.0641631	−	0.997939i	$-0.479562\pi$
$197$	1.80114	0.128326	0.0641631	+	0.997939i	$0.479562\pi$
$198$	11.2213	0.797461
$199$	−6.59313	−0.467375	−0.233687	−	0.972312i	$-0.575079\pi$
$199$	−6.59313	−0.467375	−0.233687	+	0.972312i	$0.575079\pi$
$200$	13.6319	0.963918
$201$	23.6592	1.66879
$202$	−25.3666	−1.78479
$203$	0	0
$204$	0.435667	0.0305028
$205$	−2.18431	−0.152559
$206$	−6.95811	−0.484795
$207$	−45.5658	−3.16704
$208$	0	0
$209$	−4.04603	−0.279870
$210$	0	0
$211$	10.7199	0.737990	0.368995	−	0.929431i	$-0.379702\pi$
$211$	10.7199	0.737990	0.368995	+	0.929431i	$0.379702\pi$
$212$	0.704030	0.0483530
$213$	−14.6886	−1.00645
$214$	8.51607	0.582146
$215$	0.517562	0.0352974
$216$	−16.1633	−1.09977
$217$	0	0
$218$	16.4876	1.11668
$219$	33.4558	2.26074
$220$	0.0667009	0.00449697
$221$	0	0
$222$	−40.0617	−2.68877
$223$	−12.8878	−0.863034	−0.431517	−	0.902105i	$-0.642021\pi$
$223$	−12.8878	−0.863034	−0.431517	+	0.902105i	$0.642021\pi$
$224$	0	0
$225$	−23.5825	−1.57216
$226$	4.92862	0.327847
$227$	0.699155	0.0464045	0.0232023	−	0.999731i	$-0.492614\pi$
$227$	0.699155	0.0464045	0.0232023	+	0.999731i	$0.492614\pi$
$228$	0.557070	0.0368929
$229$	−18.2868	−1.20843	−0.604214	−	0.796822i	$-0.706513\pi$
$229$	−18.2868	−1.20843	−0.604214	+	0.796822i	$0.706513\pi$
$230$	6.57436	0.433501
$231$	0	0
$232$	15.0706	0.989431
$233$	26.7796	1.75439	0.877194	−	0.480137i	$-0.159413\pi$
$233$	26.7796	1.75439	0.877194	+	0.480137i	$0.159413\pi$
$234$	0	0
$235$	−2.34426	−0.152923
$236$	0.491057	0.0319651
$237$	2.77579	0.180307
$238$	0	0
$239$	−16.6177	−1.07491	−0.537454	−	0.843293i	$-0.680614\pi$
$239$	−16.6177	−1.07491	−0.537454	+	0.843293i	$0.680614\pi$
$240$	5.62454	0.363063
$241$	17.4129	1.12166	0.560830	−	0.827931i	$-0.310482\pi$
$241$	17.4129	1.12166	0.560830	+	0.827931i	$0.310482\pi$
$242$	−11.5895	−0.745000
$243$	−14.2978	−0.917204
$244$	−1.06601	−0.0682445
$245$	0	0
$246$	16.4842	1.05099
$247$	0	0
$248$	16.6956	1.06017
$249$	25.2017	1.59709
$250$	6.99879	0.442642
$251$	6.44982	0.407109	0.203554	−	0.979064i	$-0.434751\pi$
$251$	6.44982	0.407109	0.203554	+	0.979064i	$0.434751\pi$
$252$	0	0
$253$	−14.8458	−0.933345
$254$	−1.97290	−0.123791
$255$	2.85703	0.178914
$256$	−1.89624	−0.118515
$257$	3.67156	0.229025	0.114513	−	0.993422i	$-0.463469\pi$
$257$	3.67156	0.229025	0.114513	+	0.993422i	$0.463469\pi$
$258$	−3.90586	−0.243168
$259$	0	0
$260$	0	0
$261$	−26.0713	−1.61377
$262$	−12.0214	−0.742684
$263$	18.3193	1.12961	0.564807	−	0.825223i	$-0.308950\pi$
$263$	18.3193	1.12961	0.564807	+	0.825223i	$0.308950\pi$
$264$	−13.2250	−0.813945
$265$	4.61690	0.283614
$266$	0	0
$267$	33.9629	2.07850
$268$	−0.662575	−0.0404732
$269$	−27.5429	−1.67932	−0.839661	−	0.543111i	$-0.817246\pi$
$269$	−27.5429	−1.67932	−0.839661	+	0.543111i	$0.817246\pi$
$270$	−4.03439	−0.245525
$271$	−6.51923	−0.396015	−0.198007	−	0.980201i	$-0.563447\pi$
$271$	−6.51923	−0.396015	−0.198007	+	0.980201i	$0.563447\pi$
$272$	7.47246	0.453084
$273$	0	0
$274$	−11.8053	−0.713186
$275$	−7.68338	−0.463325
$276$	2.04401	0.123035
$277$	5.44186	0.326970	0.163485	−	0.986546i	$-0.447727\pi$
$277$	5.44186	0.326970	0.163485	+	0.986546i	$0.447727\pi$
$278$	6.98304	0.418815
$279$	−28.8826	−1.72916
$280$	0	0
$281$	3.54237	0.211320	0.105660	−	0.994402i	$-0.466304\pi$
$281$	3.54237	0.211320	0.105660	+	0.994402i	$0.466304\pi$
$282$	17.6913	1.05350
$283$	14.1391	0.840484	0.420242	−	0.907412i	$-0.361945\pi$
$283$	14.1391	0.840484	0.420242	+	0.907412i	$0.361945\pi$
$284$	0.411354	0.0244094
$285$	3.65317	0.216395
$286$	0	0
$287$	0	0
$288$	2.23025	0.131419
$289$	−13.2043	−0.776724
$290$	3.76165	0.220891
$291$	−12.4964	−0.732551
$292$	−0.936928	−0.0548295
$293$	−8.34931	−0.487772	−0.243886	−	0.969804i	$-0.578422\pi$
$293$	−8.34931	−0.487772	−0.243886	+	0.969804i	$0.578422\pi$
$294$	0	0
$295$	3.22027	0.187491
$296$	29.4765	1.71328
$297$	9.11017	0.528626
$298$	4.66359	0.270155
$299$	0	0
$300$	1.05787	0.0610762
$301$	0	0
$302$	17.5415	1.00940
$303$	−51.7192	−2.97119
$304$	9.55474	0.548002
$305$	−6.99073	−0.400288
$306$	−13.4604	−0.769481
$307$	8.33362	0.475625	0.237813	−	0.971311i	$-0.423570\pi$
$307$	8.33362	0.475625	0.237813	+	0.971311i	$0.423570\pi$
$308$	0	0
$309$	−14.1867	−0.807052
$310$	4.16727	0.236685
$311$	14.6227	0.829176	0.414588	−	0.910009i	$-0.363926\pi$
$311$	14.6227	0.829176	0.414588	+	0.910009i	$0.363926\pi$
$312$	0	0
$313$	−17.1328	−0.968404	−0.484202	−	0.874956i	$-0.660890\pi$
$313$	−17.1328	−0.968404	−0.484202	+	0.874956i	$0.660890\pi$
$314$	−14.3711	−0.811009
$315$	0	0
$316$	−0.0777356	−0.00437297
$317$	−14.0000	−0.786320	−0.393160	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786320	−0.393160	+	0.919470i	$0.628618\pi$
$318$	−34.8422	−1.95385
$319$	−8.49428	−0.475589
$320$	−4.30267	−0.240527
$321$	17.3631	0.969116
$322$	0	0
$323$	4.85340	0.270050
$324$	−0.0708435	−0.00393575
$325$	0	0
$326$	−21.8334	−1.20924
$327$	33.6161	1.85897
$328$	−12.1287	−0.669694
$329$	0	0
$330$	−3.30100	−0.181714
$331$	6.91996	0.380355	0.190178	−	0.981750i	$-0.439094\pi$
$331$	6.91996	0.380355	0.190178	+	0.981750i	$0.439094\pi$
$332$	−0.705771	−0.0387342
$333$	−50.9928	−2.79439
$334$	22.7277	1.24360
$335$	−4.34505	−0.237395
$336$	0	0
$337$	−11.1559	−0.607703	−0.303852	−	0.952719i	$-0.598273\pi$
$337$	−11.1559	−0.607703	−0.303852	+	0.952719i	$0.598273\pi$
$338$	0	0
$339$	10.0488	0.545776
$340$	−0.0800108	−0.00433919
$341$	−9.41023	−0.509593
$342$	−17.2113	−0.930682
$343$	0	0
$344$	2.87383	0.154947
$345$	13.4043	0.721661
$346$	0.417663	0.0224537
$347$	−4.92511	−0.264393	−0.132197	−	0.991223i	$-0.542203\pi$
$347$	−4.92511	−0.264393	−0.132197	+	0.991223i	$0.542203\pi$
$348$	1.16952	0.0626928
$349$	−1.52335	−0.0815430	−0.0407715	−	0.999168i	$-0.512982\pi$
$349$	−1.52335	−0.0815430	−0.0407715	+	0.999168i	$0.512982\pi$
$350$	0	0
$351$	0	0
$352$	0.726636	0.0387298
$353$	17.9280	0.954212	0.477106	−	0.878846i	$-0.341686\pi$
$353$	17.9280	0.954212	0.477106	+	0.878846i	$0.341686\pi$
$354$	−24.3022	−1.29165
$355$	2.69759	0.143173
$356$	−0.951129	−0.0504097
$357$	0	0
$358$	−13.6043	−0.719007
$359$	−20.0014	−1.05563	−0.527816	−	0.849359i	$-0.676989\pi$
$359$	−20.0014	−1.05563	−0.527816	+	0.849359i	$0.676989\pi$
$360$	7.45466	0.392895
$361$	−12.7941	−0.673376
$362$	17.2301	0.905596
$363$	−23.6294	−1.24022
$364$	0	0
$365$	−6.14421	−0.321602
$366$	52.7566	2.75763
$367$	−27.4157	−1.43109	−0.715544	−	0.698568i	$-0.753821\pi$
$367$	−27.4157	−1.43109	−0.715544	+	0.698568i	$0.753821\pi$
$368$	35.0584	1.82754
$369$	20.9820	1.09228
$370$	7.35739	0.382492
$371$	0	0
$372$	1.29563	0.0671752
$373$	−15.8929	−0.822901	−0.411451	−	0.911432i	$-0.634978\pi$
$373$	−15.8929	−0.822901	−0.411451	+	0.911432i	$0.634978\pi$
$374$	−4.38553	−0.226771
$375$	14.2696	0.736880
$376$	−13.0168	−0.671292
$377$	0	0
$378$	0	0
$379$	−8.77900	−0.450947	−0.225474	−	0.974249i	$-0.572393\pi$
$379$	−8.77900	−0.450947	−0.225474	+	0.974249i	$0.572393\pi$
$380$	−0.102307	−0.00524822
$381$	−4.02249	−0.206078
$382$	−16.9736	−0.868447
$383$	−7.96237	−0.406858	−0.203429	−	0.979090i	$-0.565209\pi$
$383$	−7.96237	−0.406858	−0.203429	+	0.979090i	$0.565209\pi$
$384$	29.9423	1.52799
$385$	0	0
$386$	−16.1240	−0.820688
$387$	−4.97159	−0.252720
$388$	0.349960	0.0177665
$389$	32.0434	1.62467	0.812333	−	0.583194i	$-0.198197\pi$
$389$	32.0434	1.62467	0.812333	+	0.583194i	$0.198197\pi$
$390$	0	0
$391$	17.8082	0.900598
$392$	0	0
$393$	−24.5100	−1.23637
$394$	2.49630	0.125762
$395$	−0.509777	−0.0256496
$396$	−0.640716	−0.0321972
$397$	6.43457	0.322942	0.161471	−	0.986877i	$-0.448376\pi$
$397$	6.43457	0.322942	0.161471	+	0.986877i	$0.448376\pi$
$398$	−9.13777	−0.458035
$399$	0	0
$400$	18.1444	0.907219
$401$	−0.533577	−0.0266456	−0.0133228	−	0.999911i	$-0.504241\pi$
$401$	−0.533577	−0.0266456	−0.0133228	+	0.999911i	$0.504241\pi$
$402$	32.7906	1.63545
$403$	0	0
$404$	1.44839	0.0720601
$405$	−0.464579	−0.0230851
$406$	0	0
$407$	−16.6139	−0.823522
$408$	15.8640	0.785387
$409$	39.7528	1.96565	0.982824	−	0.184547i	$-0.0590817\pi$
$409$	39.7528	1.96565	0.982824	+	0.184547i	$0.0590817\pi$
$410$	−3.02735	−0.149510
$411$	−24.0695	−1.18726
$412$	0.397296	0.0195734
$413$	0	0
$414$	−63.1520	−3.10375
$415$	−4.62832	−0.227195
$416$	0	0
$417$	14.2375	0.697213
$418$	−5.60761	−0.274277
$419$	−23.8176	−1.16357	−0.581783	−	0.813344i	$-0.697645\pi$
$419$	−23.8176	−1.16357	−0.581783	+	0.813344i	$0.697645\pi$
$420$	0	0
$421$	23.2419	1.13274	0.566370	−	0.824151i	$-0.308347\pi$
$421$	23.2419	1.13274	0.566370	+	0.824151i	$0.308347\pi$
$422$	14.8573	0.723243
$423$	22.5185	1.09489
$424$	25.6360	1.24500
$425$	9.21657	0.447069
$426$	−20.3578	−0.986336
$427$	0	0
$428$	−0.486253	−0.0235039
$429$	0	0
$430$	0.717316	0.0345920
$431$	2.70689	0.130386	0.0651932	−	0.997873i	$-0.479234\pi$
$431$	2.70689	0.130386	0.0651932	+	0.997873i	$0.479234\pi$
$432$	−21.5137	−1.03508
$433$	5.81890	0.279638	0.139819	−	0.990177i	$-0.455348\pi$
$433$	5.81890	0.279638	0.139819	+	0.990177i	$0.455348\pi$
$434$	0	0
$435$	7.66950	0.367724
$436$	−0.941416	−0.0450856
$437$	22.7706	1.08927
$438$	46.3682	2.21556
$439$	−38.1702	−1.82176	−0.910882	−	0.412668i	$-0.864597\pi$
$439$	−38.1702	−1.82176	−0.910882	+	0.412668i	$0.864597\pi$
$440$	2.42880	0.115788
$441$	0	0
$442$	0	0
$443$	−31.6740	−1.50488	−0.752440	−	0.658661i	$-0.771123\pi$
$443$	−31.6740	−1.50488	−0.752440	+	0.658661i	$0.771123\pi$
$444$	2.28746	0.108558
$445$	−6.23734	−0.295678
$446$	−17.8619	−0.845787
$447$	9.50845	0.449734
$448$	0	0
$449$	−31.4049	−1.48209	−0.741045	−	0.671455i	$-0.765669\pi$
$449$	−31.4049	−1.48209	−0.741045	+	0.671455i	$0.765669\pi$
$450$	−32.6842	−1.54075
$451$	6.83614	0.321901
$452$	−0.281416	−0.0132367
$453$	35.7648	1.68037
$454$	0.968995	0.0454772
$455$	0	0
$456$	20.2847	0.949920
$457$	−31.8281	−1.48886	−0.744429	−	0.667702i	$-0.767278\pi$
$457$	−31.8281	−1.48886	−0.744429	+	0.667702i	$0.767278\pi$
$458$	−25.3447	−1.18428
$459$	−10.9281	−0.510078
$460$	−0.375385	−0.0175024
$461$	−1.16631	−0.0543203	−0.0271601	−	0.999631i	$-0.508646\pi$
$461$	−1.16631	−0.0543203	−0.0271601	+	0.999631i	$0.508646\pi$
$462$	0	0
$463$	20.3441	0.945469	0.472734	−	0.881205i	$-0.343267\pi$
$463$	20.3441	0.945469	0.472734	+	0.881205i	$0.343267\pi$
$464$	20.0593	0.931231
$465$	8.49651	0.394016
$466$	37.1152	1.71933
$467$	1.56939	0.0726229	0.0363114	−	0.999341i	$-0.488439\pi$
$467$	1.56939	0.0726229	0.0363114	+	0.999341i	$0.488439\pi$
$468$	0	0
$469$	0	0
$470$	−3.24903	−0.149867
$471$	−29.3008	−1.35011
$472$	17.8810	0.823039
$473$	−1.61979	−0.0744781
$474$	3.84711	0.176703
$475$	11.7849	0.540727
$476$	0	0
$477$	−44.3491	−2.03060
$478$	−23.0313	−1.05343
$479$	−7.71918	−0.352699	−0.176349	−	0.984328i	$-0.556429\pi$
$479$	−7.71918	−0.352699	−0.176349	+	0.984328i	$0.556429\pi$
$480$	−0.656080	−0.0299458
$481$	0	0
$482$	24.1334	1.09925
$483$	0	0
$484$	0.661740	0.0300791
$485$	2.29498	0.104209
$486$	−19.8161	−0.898875
$487$	−0.0761801	−0.00345205	−0.00172602	−	0.999999i	$-0.500549\pi$
$487$	−0.0761801	−0.00345205	−0.00172602	+	0.999999i	$0.500549\pi$
$488$	−38.8170	−1.75716
$489$	−44.5155	−2.01306
$490$	0	0
$491$	1.78715	0.0806529	0.0403264	−	0.999187i	$-0.487160\pi$
$491$	1.78715	0.0806529	0.0403264	+	0.999187i	$0.487160\pi$
$492$	−0.941220	−0.0424335
$493$	10.1893	0.458902
$494$	0	0
$495$	−4.20170	−0.188852
$496$	22.2223	0.997813
$497$	0	0
$498$	34.9284	1.56518
$499$	8.33493	0.373123	0.186561	−	0.982443i	$-0.440266\pi$
$499$	8.33493	0.373123	0.186561	+	0.982443i	$0.440266\pi$
$500$	−0.399619	−0.0178715
$501$	46.3387	2.07026
$502$	8.93914	0.398973
$503$	−1.44048	−0.0642277	−0.0321138	−	0.999484i	$-0.510224\pi$
$503$	−1.44048	−0.0642277	−0.0321138	+	0.999484i	$0.510224\pi$
$504$	0	0
$505$	9.49829	0.422668
$506$	−20.5755	−0.914693
$507$	0	0
$508$	0.112649	0.00499801
$509$	14.8256	0.657135	0.328568	−	0.944480i	$-0.393434\pi$
$509$	14.8256	0.657135	0.328568	+	0.944480i	$0.393434\pi$
$510$	3.95970	0.175339
$511$	0	0
$512$	−23.8204	−1.05272
$513$	−13.9733	−0.616935
$514$	5.08860	0.224449
$515$	2.60540	0.114808
$516$	0.223018	0.00981781
$517$	7.33674	0.322670
$518$	0	0
$519$	0.851561	0.0373794
$520$	0	0
$521$	0.334388	0.0146498	0.00732489	−	0.999973i	$-0.497668\pi$
$521$	0.334388	0.0146498	0.00732489	+	0.999973i	$0.497668\pi$
$522$	−36.1336	−1.58153
$523$	32.5065	1.42141	0.710705	−	0.703490i	$-0.248376\pi$
$523$	32.5065	1.42141	0.710705	+	0.703490i	$0.248376\pi$
$524$	0.686401	0.0299856
$525$	0	0
$526$	25.3896	1.10704
$527$	11.2880	0.491713
$528$	−17.6029	−0.766068
$529$	60.5502	2.63262
$530$	6.39881	0.277947
$531$	−30.9332	−1.34239
$532$	0	0
$533$	0	0
$534$	47.0710	2.03696
$535$	−3.18876	−0.137862
$536$	−24.1265	−1.04211
$537$	−27.7373	−1.19695
$538$	−38.1732	−1.64576
$539$	0	0
$540$	0.230357	0.00991297
$541$	10.6015	0.455796	0.227898	−	0.973685i	$-0.426815\pi$
$541$	10.6015	0.455796	0.227898	+	0.973685i	$0.426815\pi$
$542$	−9.03534	−0.388101
$543$	35.1300	1.50757
$544$	−0.871633	−0.0373709
$545$	−6.17364	−0.264450
$546$	0	0
$547$	10.2327	0.437519	0.218760	−	0.975779i	$-0.429799\pi$
$547$	10.2327	0.437519	0.218760	+	0.975779i	$0.429799\pi$
$548$	0.674064	0.0287946
$549$	67.1515	2.86596
$550$	−10.6488	−0.454067
$551$	13.0286	0.555038
$552$	74.4290	3.16791
$553$	0	0
$554$	7.54216	0.320436
$555$	15.0007	0.636746
$556$	−0.398720	−0.0169095
$557$	31.9930	1.35559	0.677793	−	0.735253i	$-0.262937\pi$
$557$	31.9930	1.35559	0.677793	+	0.735253i	$0.262937\pi$
$558$	−40.0300	−1.69460
$559$	0	0
$560$	0	0
$561$	−8.94152	−0.377511
$562$	4.90956	0.207097
$563$	−10.7913	−0.454800	−0.227400	−	0.973801i	$-0.573022\pi$
$563$	−10.7913	−0.454800	−0.227400	+	0.973801i	$0.573022\pi$
$564$	−1.01014	−0.0425348
$565$	−1.84547	−0.0776397
$566$	19.5962	0.823688
$567$	0	0
$568$	14.9787	0.628494
$569$	−24.6014	−1.03134	−0.515672	−	0.856786i	$-0.672458\pi$
$569$	−24.6014	−1.03134	−0.515672	+	0.856786i	$0.672458\pi$
$570$	5.06312	0.212071
$571$	16.5724	0.693534	0.346767	−	0.937951i	$-0.387279\pi$
$571$	16.5724	0.693534	0.346767	+	0.937951i	$0.387279\pi$
$572$	0	0
$573$	−34.6070	−1.44573
$574$	0	0
$575$	43.2412	1.80328
$576$	41.3306	1.72211
$577$	−14.6611	−0.610348	−0.305174	−	0.952297i	$-0.598715\pi$
$577$	−14.6611	−0.610348	−0.305174	+	0.952297i	$0.598715\pi$
$578$	−18.3005	−0.761202
$579$	−32.8746	−1.36622
$580$	−0.214784	−0.00891840
$581$	0	0
$582$	−17.3194	−0.717912
$583$	−14.4493	−0.598431
$584$	−34.1166	−1.41175
$585$	0	0
$586$	−11.5717	−0.478024
$587$	35.3336	1.45837	0.729186	−	0.684315i	$-0.239899\pi$
$587$	35.3336	1.45837	0.729186	+	0.684315i	$0.239899\pi$
$588$	0	0
$589$	14.4335	0.594723
$590$	4.46313	0.183744
$591$	5.08963	0.209359
$592$	39.2340	1.61251
$593$	−16.4294	−0.674675	−0.337338	−	0.941384i	$-0.609526\pi$
$593$	−16.4294	−0.674675	−0.337338	+	0.941384i	$0.609526\pi$
$594$	12.6263	0.518062
$595$	0	0
$596$	−0.266283	−0.0109074
$597$	−18.6307	−0.762504
$598$	0	0
$599$	12.0819	0.493653	0.246826	−	0.969060i	$-0.420612\pi$
$599$	12.0819	0.493653	0.246826	+	0.969060i	$0.420612\pi$
$600$	38.5206	1.57259
$601$	−7.81486	−0.318775	−0.159387	−	0.987216i	$-0.550952\pi$
$601$	−7.81486	−0.318775	−0.159387	+	0.987216i	$0.550952\pi$
$602$	0	0
$603$	41.7377	1.69969
$604$	−1.00159	−0.0407541
$605$	4.33957	0.176429
$606$	−71.6803	−2.91181
$607$	−35.5649	−1.44354	−0.721768	−	0.692135i	$-0.756670\pi$
$607$	−35.5649	−1.44354	−0.721768	+	0.692135i	$0.756670\pi$
$608$	−1.11452	−0.0451999
$609$	0	0
$610$	−9.68882	−0.392289
$611$	0	0
$612$	0.768568	0.0310675
$613$	11.9368	0.482122	0.241061	−	0.970510i	$-0.422505\pi$
$613$	11.9368	0.482122	0.241061	+	0.970510i	$0.422505\pi$
$614$	11.5500	0.466120
$615$	−6.17236	−0.248893
$616$	0	0
$617$	−23.5702	−0.948901	−0.474450	−	0.880282i	$-0.657353\pi$
$617$	−23.5702	−0.948901	−0.474450	+	0.880282i	$0.657353\pi$
$618$	−19.6621	−0.790924
$619$	−28.5571	−1.14781	−0.573904	−	0.818923i	$-0.694572\pi$
$619$	−28.5571	−1.14781	−0.573904	+	0.818923i	$0.694572\pi$
$620$	−0.237944	−0.00955606
$621$	−51.2710	−2.05743
$622$	20.2663	0.812607
$623$	0	0
$624$	0	0
$625$	21.0328	0.841311
$626$	−23.7453	−0.949052
$627$	−11.4332	−0.456597
$628$	0.820567	0.0327442
$629$	19.9292	0.794628
$630$	0	0
$631$	44.9925	1.79112	0.895561	−	0.444938i	$-0.146774\pi$
$631$	44.9925	1.79112	0.895561	+	0.444938i	$0.146774\pi$
$632$	−2.83061	−0.112596
$633$	30.2921	1.20400
$634$	−19.4034	−0.770607
$635$	0.738735	0.0293158
$636$	1.98943	0.0788860
$637$	0	0
$638$	−11.7727	−0.466085
$639$	−25.9125	−1.02508
$640$	−5.49894	−0.217365
$641$	−2.53300	−0.100048	−0.0500238	−	0.998748i	$-0.515930\pi$
$641$	−2.53300	−0.100048	−0.0500238	+	0.998748i	$0.515930\pi$
$642$	24.0645	0.949749
$643$	−18.3771	−0.724721	−0.362360	−	0.932038i	$-0.618029\pi$
$643$	−18.3771	−0.724721	−0.362360	+	0.932038i	$0.618029\pi$
$644$	0	0
$645$	1.46251	0.0575863
$646$	6.72658	0.264654
$647$	−20.9287	−0.822791	−0.411396	−	0.911457i	$-0.634959\pi$
$647$	−20.9287	−0.822791	−0.411396	+	0.911457i	$0.634959\pi$
$648$	−2.57964	−0.101338
$649$	−10.0783	−0.395609
$650$	0	0
$651$	0	0
$652$	1.24665	0.0488227
$653$	−48.1160	−1.88292	−0.941461	−	0.337121i	$-0.890547\pi$
$653$	−48.1160	−1.88292	−0.941461	+	0.337121i	$0.890547\pi$
$654$	46.5903	1.82183
$655$	4.50130	0.175880
$656$	−16.1436	−0.630302
$657$	59.0200	2.30259
$658$	0	0
$659$	−2.21638	−0.0863379	−0.0431690	−	0.999068i	$-0.513745\pi$
$659$	−2.21638	−0.0863379	−0.0431690	+	0.999068i	$0.513745\pi$
$660$	0.188482	0.00733664
$661$	0.637434	0.0247933	0.0123966	−	0.999923i	$-0.496054\pi$
$661$	0.637434	0.0247933	0.0123966	+	0.999923i	$0.496054\pi$
$662$	9.59073	0.372754
$663$	0	0
$664$	−25.6994	−0.997331
$665$	0	0
$666$	−70.6736	−2.73855
$667$	47.8048	1.85101
$668$	−1.29771	−0.0502100
$669$	−36.4181	−1.40801
$670$	−6.02203	−0.232651
$671$	21.8786	0.844614
$672$	0	0
$673$	15.4069	0.593891	0.296945	−	0.954895i	$-0.404032\pi$
$673$	15.4069	0.593891	0.296945	+	0.954895i	$0.404032\pi$
$674$	−15.4616	−0.595559
$675$	−26.5352	−1.02134
$676$	0	0
$677$	11.6812	0.448945	0.224473	−	0.974480i	$-0.427934\pi$
$677$	11.6812	0.448945	0.224473	+	0.974480i	$0.427934\pi$
$678$	13.9272	0.534869
$679$	0	0
$680$	−2.91345	−0.111726
$681$	1.97565	0.0757072
$682$	−13.0421	−0.499409
$683$	−22.9114	−0.876680	−0.438340	−	0.898809i	$-0.644433\pi$
$683$	−22.9114	−0.876680	−0.438340	+	0.898809i	$0.644433\pi$
$684$	0.982737	0.0375759
$685$	4.42039	0.168895
$686$	0	0
$687$	−51.6745	−1.97150
$688$	3.82515	0.145833
$689$	0	0
$690$	18.5777	0.707239
$691$	−47.2325	−1.79681	−0.898405	−	0.439167i	$-0.855274\pi$
$691$	−47.2325	−1.79681	−0.898405	+	0.439167i	$0.855274\pi$
$692$	−0.0238479	−0.000906560 0
$693$	0	0
$694$	−6.82596	−0.259110
$695$	−2.61473	−0.0991825
$696$	42.5860	1.61422
$697$	−8.20026	−0.310607
$698$	−2.11129	−0.0799135
$699$	75.6730	2.86222
$700$	0	0
$701$	−12.2098	−0.461158	−0.230579	−	0.973054i	$-0.574062\pi$
$701$	−12.2098	−0.461158	−0.230579	+	0.973054i	$0.574062\pi$
$702$	0	0
$703$	25.4827	0.961097
$704$	13.4659	0.507515
$705$	−6.62435	−0.249488
$706$	24.8474	0.935144
$707$	0	0
$708$	1.38762	0.0521498
$709$	17.8875	0.671778	0.335889	−	0.941902i	$-0.390963\pi$
$709$	17.8875	0.671778	0.335889	+	0.941902i	$0.390963\pi$
$710$	3.73873	0.140312
$711$	4.89681	0.183645
$712$	−34.6337	−1.29795
$713$	52.9597	1.98336
$714$	0	0
$715$	0	0
$716$	0.776780	0.0290296
$717$	−46.9578	−1.75367
$718$	−27.7210	−1.03454
$719$	−9.12634	−0.340355	−0.170178	−	0.985413i	$-0.554434\pi$
$719$	−9.12634	−0.340355	−0.170178	+	0.985413i	$0.554434\pi$
$720$	9.92235	0.369784
$721$	0	0
$722$	−17.7321	−0.659920
$723$	49.2048	1.82995
$724$	−0.983812	−0.0365631
$725$	24.7413	0.918868
$726$	−32.7493	−1.21544
$727$	33.6859	1.24934	0.624670	−	0.780889i	$-0.285233\pi$
$727$	33.6859	1.24934	0.624670	+	0.780889i	$0.285233\pi$
$728$	0	0
$729$	−43.0880	−1.59585
$730$	−8.51558	−0.315176
$731$	1.94301	0.0718650
$732$	−3.01231	−0.111338
$733$	−46.4344	−1.71509	−0.857547	−	0.514406i	$-0.828013\pi$
$733$	−46.4344	−1.71509	−0.857547	+	0.514406i	$0.828013\pi$
$734$	−37.9968	−1.40249
$735$	0	0
$736$	−4.08942	−0.150738
$737$	13.5985	0.500908
$738$	29.0801	1.07045
$739$	−1.85025	−0.0680627	−0.0340314	−	0.999421i	$-0.510835\pi$
$739$	−1.85025	−0.0680627	−0.0340314	+	0.999421i	$0.510835\pi$
$740$	−0.420095	−0.0154430
$741$	0	0
$742$	0	0
$743$	−33.1509	−1.21619	−0.608094	−	0.793865i	$-0.708066\pi$
$743$	−33.1509	−1.21619	−0.608094	+	0.793865i	$0.708066\pi$
$744$	47.1781	1.72963
$745$	−1.74624	−0.0639773
$746$	−22.0268	−0.806457
$747$	44.4588	1.62666
$748$	0.250407	0.00915577
$749$	0	0
$750$	19.7770	0.722154
$751$	−20.7743	−0.758064	−0.379032	−	0.925384i	$-0.623743\pi$
$751$	−20.7743	−0.758064	−0.379032	+	0.925384i	$0.623743\pi$
$752$	−17.3258	−0.631806
$753$	18.2257	0.664182
$754$	0	0
$755$	−6.56824	−0.239043
$756$	0	0
$757$	43.6150	1.58521	0.792607	−	0.609733i	$-0.208723\pi$
$757$	43.6150	1.58521	0.792607	+	0.609733i	$0.208723\pi$
$758$	−12.1673	−0.441936
$759$	−41.9508	−1.52272
$760$	−3.72532	−0.135131
$761$	12.3902	0.449145	0.224573	−	0.974457i	$-0.427901\pi$
$761$	12.3902	0.449145	0.224573	+	0.974457i	$0.427901\pi$
$762$	−5.57497	−0.201960
$763$	0	0
$764$	0.969167	0.0350632
$765$	5.04013	0.182226
$766$	−11.0355	−0.398728
$767$	0	0
$768$	−5.35835	−0.193353
$769$	−5.55359	−0.200268	−0.100134	−	0.994974i	$-0.531927\pi$
$769$	−5.55359	−0.200268	−0.100134	+	0.994974i	$0.531927\pi$
$770$	0	0
$771$	10.3750	0.373646
$772$	0.920651	0.0331349
$773$	−43.8042	−1.57553	−0.787764	−	0.615977i	$-0.788761\pi$
$773$	−43.8042	−1.57553	−0.787764	+	0.615977i	$0.788761\pi$
$774$	−6.89039	−0.247670
$775$	27.4092	0.984566
$776$	12.7432	0.457454
$777$	0	0
$778$	44.4107	1.59220
$779$	−10.4853	−0.375677
$780$	0	0
$781$	−8.44253	−0.302098
$782$	24.6813	0.882600
$783$	−29.3357	−1.04837
$784$	0	0
$785$	5.38113	0.192061
$786$	−33.9697	−1.21166
$787$	24.8009	0.884057	0.442029	−	0.897001i	$-0.354259\pi$
$787$	24.8009	0.884057	0.442029	+	0.897001i	$0.354259\pi$
$788$	−0.142535	−0.00507758
$789$	51.7661	1.84292
$790$	−0.706526	−0.0251371
$791$	0	0
$792$	−23.3305	−0.829015
$793$	0	0
$794$	8.91801	0.316489
$795$	13.0463	0.462706
$796$	0.521751	0.0184930
$797$	16.4715	0.583451	0.291725	−	0.956502i	$-0.405771\pi$
$797$	16.4715	0.583451	0.291725	+	0.956502i	$0.405771\pi$
$798$	0	0
$799$	−8.80076	−0.311348
$800$	−2.11647	−0.0748285
$801$	59.9146	2.11698
$802$	−0.739513	−0.0261131
$803$	19.2293	0.678587
$804$	−1.87229	−0.0660305
$805$	0	0
$806$	0	0
$807$	−77.8301	−2.73975
$808$	52.7406	1.85541
$809$	1.38194	0.0485863	0.0242932	−	0.999705i	$-0.492266\pi$
$809$	1.38194	0.0485863	0.0242932	+	0.999705i	$0.492266\pi$
$810$	−0.643885	−0.0226238
$811$	6.83571	0.240034	0.120017	−	0.992772i	$-0.461705\pi$
$811$	6.83571	0.240034	0.120017	+	0.992772i	$0.461705\pi$
$812$	0	0
$813$	−18.4219	−0.646083
$814$	−23.0261	−0.807066
$815$	8.17533	0.286369
$816$	21.1155	0.739190
$817$	2.48446	0.0869201
$818$	55.0954	1.92637
$819$	0	0
$820$	0.172856	0.00603640
$821$	10.5425	0.367936	0.183968	−	0.982932i	$-0.441106\pi$
$821$	10.5425	0.367936	0.183968	+	0.982932i	$0.441106\pi$
$822$	−33.3592	−1.16353
$823$	14.8330	0.517047	0.258524	−	0.966005i	$-0.416764\pi$
$823$	14.8330	0.517047	0.258524	+	0.966005i	$0.416764\pi$
$824$	14.4669	0.503977
$825$	−21.7115	−0.755898
$826$	0	0
$827$	55.6758	1.93604	0.968018	−	0.250879i	$-0.0807197\pi$
$827$	55.6758	1.93604	0.968018	+	0.250879i	$0.0807197\pi$
$828$	3.60587	0.125313
$829$	0.0464848	0.00161448	0.000807242	−	1.00000i	$-0.499743\pi$
$829$	0.0464848	0.00161448	0.000807242		1.00000i	$0.499743\pi$
$830$	−6.41464	−0.222655
$831$	15.3775	0.533438
$832$	0	0
$833$	0	0
$834$	19.7325	0.683280
$835$	−8.51016	−0.294506
$836$	0.320185	0.0110738
$837$	−32.4990	−1.12333
$838$	−33.0100	−1.14031
$839$	25.5475	0.881999	0.440999	−	0.897507i	$-0.354624\pi$
$839$	25.5475	0.881999	0.440999	+	0.897507i	$0.354624\pi$
$840$	0	0
$841$	−1.64755	−0.0568120
$842$	32.2121	1.11010
$843$	10.0099	0.344761
$844$	−0.848327	−0.0292006
$845$	0	0
$846$	31.2096	1.07301
$847$	0	0
$848$	34.1223	1.17176
$849$	39.9540	1.37122
$850$	12.7737	0.438135
$851$	93.5013	3.20518
$852$	1.16239	0.0398229
$853$	22.6671	0.776105	0.388053	−	0.921637i	$-0.373148\pi$
$853$	22.6671	0.776105	0.388053	+	0.921637i	$0.373148\pi$
$854$	0	0
$855$	6.44462	0.220401
$856$	−17.7061	−0.605181
$857$	37.0535	1.26572	0.632862	−	0.774264i	$-0.281880\pi$
$857$	37.0535	1.26572	0.632862	+	0.774264i	$0.281880\pi$
$858$	0	0
$859$	4.24339	0.144782	0.0723912	−	0.997376i	$-0.476937\pi$
$859$	4.24339	0.144782	0.0723912	+	0.997376i	$0.476937\pi$
$860$	−0.0409575	−0.00139664
$861$	0	0
$862$	3.75162	0.127781
$863$	−7.50051	−0.255320	−0.127660	−	0.991818i	$-0.540747\pi$
$863$	−7.50051	−0.255320	−0.127660	+	0.991818i	$0.540747\pi$
$864$	2.50949	0.0853746
$865$	−0.156390	−0.00531743
$866$	8.06472	0.274050
$867$	−37.3124	−1.26720
$868$	0	0
$869$	1.59543	0.0541212
$870$	10.6296	0.360376
$871$	0	0
$872$	−34.2800	−1.16087
$873$	−22.0451	−0.746113
$874$	31.5590	1.06750
$875$	0	0
$876$	−2.64755	−0.0894523
$877$	−28.2897	−0.955274	−0.477637	−	0.878557i	$-0.658507\pi$
$877$	−28.2897	−0.955274	−0.477637	+	0.878557i	$0.658507\pi$
$878$	−52.9021	−1.78536
$879$	−23.5933	−0.795781
$880$	3.23280	0.108978
$881$	39.5721	1.33322	0.666609	−	0.745408i	$-0.267745\pi$
$881$	39.5721	1.33322	0.666609	+	0.745408i	$0.267745\pi$
$882$	0	0
$883$	28.3609	0.954419	0.477209	−	0.878790i	$-0.341648\pi$
$883$	28.3609	0.954419	0.477209	+	0.878790i	$0.341648\pi$
$884$	0	0
$885$	9.09974	0.305884
$886$	−43.8987	−1.47481
$887$	43.7186	1.46793	0.733963	−	0.679189i	$-0.237669\pi$
$887$	43.7186	1.46793	0.733963	+	0.679189i	$0.237669\pi$
$888$	83.2938	2.79516
$889$	0	0
$890$	−8.64465	−0.289769
$891$	1.45398	0.0487100
$892$	1.01989	0.0341483
$893$	−11.2532	−0.376573
$894$	13.1783	0.440747
$895$	5.09399	0.170273
$896$	0	0
$897$	0	0
$898$	−43.5257	−1.45247
$899$	30.3019	1.01062
$900$	1.86621	0.0622070
$901$	17.3326	0.577434
$902$	9.47456	0.315468
$903$	0	0
$904$	−10.2473	−0.340819
$905$	−6.45167	−0.214461
$906$	49.5683	1.64679
$907$	−36.6985	−1.21855	−0.609277	−	0.792957i	$-0.708540\pi$
$907$	−36.6985	−1.21855	−0.609277	+	0.792957i	$0.708540\pi$
$908$	−0.0553280	−0.00183612
$909$	−91.2387	−3.02620
$910$	0	0
$911$	−35.5211	−1.17686	−0.588432	−	0.808546i	$-0.700255\pi$
$911$	−35.5211	−1.17686	−0.588432	+	0.808546i	$0.700255\pi$
$912$	26.9995	0.894044
$913$	14.4851	0.479386
$914$	−44.1123	−1.45911
$915$	−19.7542	−0.653054
$916$	1.44714	0.0478148
$917$	0	0
$918$	−15.1458	−0.499885
$919$	21.9334	0.723516	0.361758	−	0.932272i	$-0.382177\pi$
$919$	21.9334	0.723516	0.361758	+	0.932272i	$0.382177\pi$
$920$	−13.6690	−0.450653
$921$	23.5489	0.775964
$922$	−1.61644	−0.0532348
$923$	0	0
$924$	0	0
$925$	48.3914	1.59110
$926$	28.1959	0.926575
$927$	−25.0270	−0.821993
$928$	−2.33984	−0.0768090
$929$	15.5172	0.509102	0.254551	−	0.967059i	$-0.418072\pi$
$929$	15.5172	0.509102	0.254551	+	0.967059i	$0.418072\pi$
$930$	11.7758	0.386142
$931$	0	0
$932$	−2.11922	−0.0694172
$933$	41.3204	1.35277
$934$	2.17510	0.0711716
$935$	1.64212	0.0537031
$936$	0	0
$937$	−40.8110	−1.33324	−0.666618	−	0.745399i	$-0.732259\pi$
$937$	−40.8110	−1.33324	−0.666618	+	0.745399i	$0.732259\pi$
$938$	0	0
$939$	−48.4135	−1.57991
$940$	0.185514	0.00605081
$941$	51.5936	1.68190	0.840952	−	0.541109i	$-0.181996\pi$
$941$	51.5936	1.68190	0.840952	+	0.541109i	$0.181996\pi$
$942$	−40.6096	−1.32313
$943$	−38.4730	−1.25285
$944$	23.8001	0.774627
$945$	0	0
$946$	−2.24495	−0.0729898
$947$	−4.98209	−0.161896	−0.0809482	−	0.996718i	$-0.525795\pi$
$947$	−4.98209	−0.161896	−0.0809482	+	0.996718i	$0.525795\pi$
$948$	−0.219663	−0.00713433
$949$	0	0
$950$	16.3333	0.529921
$951$	−39.5609	−1.28285
$952$	0	0
$953$	30.2325	0.979328	0.489664	−	0.871911i	$-0.337119\pi$
$953$	30.2325	0.979328	0.489664	+	0.871911i	$0.337119\pi$
$954$	−61.4657	−1.99003
$955$	6.35562	0.205663
$956$	1.31505	0.0425317
$957$	−24.0029	−0.775904
$958$	−10.6984	−0.345650
$959$	0	0
$960$	−12.1584	−0.392410
$961$	2.56939	0.0828835
$962$	0	0
$963$	30.6306	0.987058
$964$	−1.37798	−0.0443816
$965$	6.03746	0.194353
$966$	0	0
$967$	−29.9990	−0.964703	−0.482352	−	0.875978i	$-0.660217\pi$
$967$	−29.9990	−0.964703	−0.482352	+	0.875978i	$0.660217\pi$
$968$	24.0961	0.774478
$969$	13.7146	0.440577
$970$	3.18073	0.102127
$971$	−44.1240	−1.41601	−0.708003	−	0.706209i	$-0.750404\pi$
$971$	−44.1240	−1.41601	−0.708003	+	0.706209i	$0.750404\pi$
$972$	1.13146	0.0362917
$973$	0	0
$974$	−0.105582	−0.00338306
$975$	0	0
$976$	−51.6665	−1.65380
$977$	15.0024	0.479970	0.239985	−	0.970777i	$-0.422858\pi$
$977$	15.0024	0.479970	0.239985	+	0.970777i	$0.422858\pi$
$978$	−61.6964	−1.97283
$979$	19.5207	0.623886
$980$	0	0
$981$	59.3028	1.89339
$982$	2.47690	0.0790411
$983$	6.01856	0.191962	0.0959812	−	0.995383i	$-0.469401\pi$
$983$	6.01856	0.191962	0.0959812	+	0.995383i	$0.469401\pi$
$984$	−34.2729	−1.09258
$985$	−0.934717	−0.0297825
$986$	14.1219	0.449732
$987$	0	0
$988$	0	0
$989$	9.11600	0.289872
$990$	−5.82336	−0.185078
$991$	33.8400	1.07496	0.537482	−	0.843275i	$-0.319376\pi$
$991$	33.8400	1.07496	0.537482	+	0.843275i	$0.319376\pi$
$992$	−2.59215	−0.0823008
$993$	19.5542	0.620535
$994$	0	0
$995$	3.42155	0.108471
$996$	−1.99435	−0.0631934
$997$	−26.3215	−0.833610	−0.416805	−	0.908996i	$-0.636850\pi$
$997$	−26.3215	−0.833610	−0.416805	+	0.908996i	$0.636850\pi$
$998$	11.5518	0.365666
$999$	−57.3775	−1.81534

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.by.1.6		6
7.6	odd	2		1183.2.a.m.1.6		6
13.6	odd	12		637.2.q.h.491.2		12
13.11	odd	12		637.2.q.h.589.2		12
13.12	even	2		8281.2.a.ch.1.1		6
91.6	even	12		91.2.q.a.36.2	✓	12
91.11	odd	12		637.2.u.i.30.5		12
91.19	even	12		637.2.u.h.361.5		12
91.24	even	12		637.2.u.h.30.5		12
91.32	odd	12		637.2.k.g.569.2		12
91.34	even	4		1183.2.c.i.337.10		12
91.37	odd	12		637.2.k.g.459.5		12
91.45	even	12		637.2.k.h.569.2		12
91.58	odd	12		637.2.u.i.361.5		12
91.76	even	12		91.2.q.a.43.2	yes	12
91.83	even	4		1183.2.c.i.337.3		12
91.89	even	12		637.2.k.h.459.5		12
91.90	odd	2		1183.2.a.p.1.1		6
273.167	odd	12		819.2.ct.a.316.5		12
273.188	odd	12		819.2.ct.a.127.5		12
364.167	odd	12		1456.2.cc.c.225.1		12
364.279	odd	12		1456.2.cc.c.673.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.q.a.36.2	✓	12	91.6	even	12
91.2.q.a.43.2	yes	12	91.76	even	12
637.2.k.g.459.5		12	91.37	odd	12
637.2.k.g.569.2		12	91.32	odd	12
637.2.k.h.459.5		12	91.89	even	12
637.2.k.h.569.2		12	91.45	even	12
637.2.q.h.491.2		12	13.6	odd	12
637.2.q.h.589.2		12	13.11	odd	12
637.2.u.h.30.5		12	91.24	even	12
637.2.u.h.361.5		12	91.19	even	12
637.2.u.i.30.5		12	91.11	odd	12
637.2.u.i.361.5		12	91.58	odd	12
819.2.ct.a.127.5		12	273.188	odd	12
819.2.ct.a.316.5		12	273.167	odd	12
1183.2.a.m.1.6		6	7.6	odd	2
1183.2.a.p.1.1		6	91.90	odd	2
1183.2.c.i.337.3		12	91.83	even	4
1183.2.c.i.337.10		12	91.34	even	4
1456.2.cc.c.225.1		12	364.167	odd	12
1456.2.cc.c.673.1		12	364.279	odd	12
8281.2.a.by.1.6		6	1.1	even	1	trivial
8281.2.a.ch.1.1		6	13.12	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+1.38595 q^{2} +2.82577 q^{3} -0.0791355 q^{4} -0.518957 q^{5} +3.91639 q^{6} -2.88158 q^{8} +4.98500 q^{9} -0.719250 q^{10} +1.62416 q^{11} -0.223619 q^{12} -1.46646 q^{15} -3.83547 q^{16} -1.94825 q^{17} +6.90897 q^{18} -2.49115 q^{19} +0.0410679 q^{20} +2.25101 q^{22} -9.14058 q^{23} -8.14270 q^{24} -4.73068 q^{25} +5.60916 q^{27} -5.22996 q^{29} -2.03244 q^{30} -5.79391 q^{31} +0.447392 q^{32} +4.58951 q^{33} -2.70019 q^{34} -0.394491 q^{36} -10.2293 q^{37} -3.45262 q^{38} +1.49542 q^{40} +4.20903 q^{41} -0.997311 q^{43} -0.128529 q^{44} -2.58700 q^{45} -12.6684 q^{46} +4.51725 q^{47} -10.8382 q^{48} -6.55650 q^{50} -5.50532 q^{51} -8.89651 q^{53} +7.77403 q^{54} -0.842869 q^{55} -7.03944 q^{57} -7.24847 q^{58} -6.20526 q^{59} +0.116049 q^{60} +13.4707 q^{61} -8.03008 q^{62} +8.29100 q^{64} +6.36084 q^{66} +8.37266 q^{67} +0.154176 q^{68} -25.8292 q^{69} -5.19809 q^{71} -14.3647 q^{72} +11.8395 q^{73} -14.1773 q^{74} -13.3678 q^{75} +0.197139 q^{76} +0.982310 q^{79} +1.99044 q^{80} +0.895217 q^{81} +5.83352 q^{82} +8.91851 q^{83} +1.01106 q^{85} -1.38223 q^{86} -14.7787 q^{87} -4.68015 q^{88} +12.0190 q^{89} -3.58546 q^{90} +0.723345 q^{92} -16.3723 q^{93} +6.26070 q^{94} +1.29280 q^{95} +1.26423 q^{96} -4.42228 q^{97} +8.09643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 12 q^{8} + 4 q^{9} - 12 q^{10} - 4 q^{11} - 2 q^{12} - 20 q^{15} + 8 q^{16} + 4 q^{17} + 16 q^{18} + 2 q^{19} + 26 q^{20} - 6 q^{22} - 12 q^{23} + 2 q^{24}+ \cdots - 16 q^{99}+O(q^{100}) \) 6 * q - 4 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^6 - 12 * q^8 + 4 * q^9 - 12 * q^10 - 4 * q^11 - 2 * q^12 - 20 * q^15 + 8 * q^16 + 4 * q^17 + 16 * q^18 + 2 * q^19 + 26 * q^20 - 6 * q^22 - 12 * q^23 + 2 * q^24 + 10 * q^25 - 6 * q^27 - 8 * q^29 + 8 * q^30 - 14 * q^31 - 8 * q^32 + 16 * q^33 - 2 * q^34 - 10 * q^36 - 12 * q^37 + 2 * q^38 - 46 * q^40 + 28 * q^41 + 2 * q^43 + 20 * q^44 + 16 * q^45 - 20 * q^46 + 14 * q^47 - 2 * q^48 - 32 * q^50 - 26 * q^51 - 22 * q^53 + 14 * q^54 - 6 * q^55 - 4 * q^58 - 2 * q^59 + 14 * q^61 + 4 * q^62 + 26 * q^64 + 26 * q^66 - 24 * q^67 - 8 * q^68 - 4 * q^69 - 4 * q^71 - 8 * q^72 + 36 * q^73 - 6 * q^74 - 46 * q^75 - 26 * q^76 - 28 * q^79 + 36 * q^80 - 2 * q^81 - 14 * q^82 + 26 * q^83 + 20 * q^85 + 24 * q^86 - 2 * q^87 - 14 * q^88 + 42 * q^89 + 12 * q^90 + 12 * q^92 + 4 * q^94 - 22 * q^95 - 42 * q^96 + 24 * q^97 - 16 * q^99

Introduction

L-functions

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