LMFDB - Embedded newform 1305.2.d.c.811.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1305,2,Mod(811,1305)] 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("1305.811"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-14,-10] 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:	no
Analytic conductor:	$10.4204774638$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.281900339052544.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 17x^{8} + 104x^{6} + 273x^{4} + 281x^{2} + 64 $ x^10 + 17x^8 + 104x^6 + 273x^4 + 281x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			811.3
Root			$-1.78296i$ of defining polynomial
Character	$\chi$	$=$	1305.811
Dual form			1305.2.d.c.811.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.78296i q^{2} -1.17894 q^{4} -1.00000 q^{5} +2.60402 q^{7} -1.46391i q^{8} +1.78296i q^{10} +6.61083i q^{11} +4.14692 q^{13} -4.64285i q^{14} -4.96798 q^{16} +3.78904i q^{17} +7.10882i q^{19} +1.17894 q^{20} +11.7868 q^{22} +2.46391 q^{23} +1.00000 q^{25} -7.39379i q^{26} -3.06999 q^{28} +(-4.42581 - 3.06793i) q^{29} +6.39305i q^{31} +5.92988i q^{32} +6.75570 q^{34} -2.60402 q^{35} +4.64491i q^{37} +12.6747 q^{38} +1.46391i q^{40} -0.721109i q^{41} -3.57274i q^{43} -7.79380i q^{44} -4.39305i q^{46} +1.85308i q^{47} -0.219100 q^{49} -1.78296i q^{50} -4.88898 q^{52} -0.644913 q^{53} -6.61083i q^{55} -3.81205i q^{56} +(-5.46999 + 7.89104i) q^{58} +13.2217 q^{59} -5.92380i q^{61} +11.3986 q^{62} +0.636777 q^{64} -4.14692 q^{65} -5.45564 q^{67} -4.46706i q^{68} +4.64285i q^{70} +3.13184 q^{71} -15.7086i q^{73} +8.28169 q^{74} -8.38090i q^{76} +17.2147i q^{77} +1.18502i q^{79} +4.96798 q^{80} -1.28571 q^{82} +10.5236 q^{83} -3.78904i q^{85} -6.37004 q^{86} +9.67767 q^{88} -4.34682i q^{89} +10.7987 q^{91} -2.90481 q^{92} +3.30396 q^{94} -7.10882i q^{95} -6.44090i q^{97} +0.390646i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 14 q^{4} - 10 q^{5} + 8 q^{7} - 4 q^{13} - 2 q^{16} + 14 q^{20} - 26 q^{22} + 10 q^{23} + 10 q^{25} + 34 q^{28} - 16 q^{29} - 6 q^{34} - 8 q^{35} + 36 q^{38} + 14 q^{49} - 14 q^{52} + 38 q^{53} - 6 q^{58}+ \cdots + 14 q^{94}+O(q^{100}) $ 10 * q - 14 * q^4 - 10 * q^5 + 8 * q^7 - 4 * q^13 - 2 * q^16 + 14 * q^20 - 26 * q^22 + 10 * q^23 + 10 * q^25 + 34 * q^28 - 16 * q^29 - 6 * q^34 - 8 * q^35 + 36 * q^38 + 14 * q^49 - 14 * q^52 + 38 * q^53 - 6 * q^58 + 12 * q^59 + 28 * q^62 + 28 * q^64 + 4 * q^65 + 20 * q^67 - 32 * q^71 + 60 * q^74 + 2 * q^80 + 12 * q^82 - 2 * q^83 + 92 * q^88 - 24 * q^91 - 4 * q^92 + 14 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$.

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		−	1.78296i		−	1.26074i	−0.776294	−	0.630371i	$-0.782903\pi$
$2$		−	1.78296i		−	1.26074i	0.776294	−	0.630371i	$-0.217097\pi$
$3$		0			0
$3$		0			0
$4$	−1.17894			−0.589471
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	2.60402			0.984226			0.492113	−	0.870531i	$-0.336225\pi$
$7$	2.60402			0.984226			0.492113	+	0.870531i	$0.336225\pi$
$8$		−	1.46391i		−	0.517571i
$9$		0			0
$10$			1.78296i			0.563821i
$11$			6.61083i			1.99324i	0.0821418	+	0.996621i	$0.473824\pi$
$11$			6.61083i			1.99324i	−0.0821418	+	0.996621i	$0.526176\pi$
$12$		0			0
$13$	4.14692			1.15015			0.575075	−	0.818101i	$-0.304973\pi$
$13$	4.14692			1.15015			0.575075	+	0.818101i	$0.304973\pi$
$14$		−	4.64285i		−	1.24085i
$15$		0			0
$16$	−4.96798			−1.24199
$17$			3.78904i			0.918976i	0.888184	+	0.459488i	$0.151967\pi$
$17$			3.78904i			0.918976i	−0.888184	+	0.459488i	$0.848033\pi$
$18$		0			0
$19$			7.10882i			1.63088i	0.578844	+	0.815438i	$0.303504\pi$
$19$			7.10882i			1.63088i	−0.578844	+	0.815438i	$0.696496\pi$
$20$	1.17894			0.263620
$21$		0			0
$22$	11.7868			2.51296
$23$	2.46391			0.513761			0.256881	−	0.966443i	$-0.417305\pi$
$23$	2.46391			0.513761			0.256881	+	0.966443i	$0.417305\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	7.39379i		−	1.45004i
$27$		0			0
$28$	−3.06999			−0.580173
$29$	−4.42581	−	3.06793i	−0.821853	−	0.569700i
$29$	−4.42581	−	3.06793i	−0.821853	−	0.569700i
$30$		0			0
$31$			6.39305i			1.14823i	0.818776	+	0.574113i	$0.194653\pi$
$31$			6.39305i			1.14823i	−0.818776	+	0.574113i	$0.805347\pi$
$32$			5.92988i			1.04826i
$33$		0			0
$34$	6.75570			1.15859
$35$	−2.60402			−0.440159
$36$		0			0
$37$			4.64491i			0.763619i	0.924241	+	0.381809i	$0.124699\pi$
$37$			4.64491i			0.763619i	−0.924241	+	0.381809i	$0.875301\pi$
$38$	12.6747			2.05611
$39$		0			0
$40$			1.46391i			0.231465i
$41$		−	0.721109i		−	0.112618i	−0.998413	−	0.0563092i	$-0.982067\pi$
$41$		−	0.721109i		−	0.112618i	0.998413	−	0.0563092i	$-0.0179333\pi$
$42$		0			0
$43$		−	3.57274i		−	0.544837i	−0.962179	−	0.272419i	$-0.912176\pi$
$43$		−	3.57274i		−	0.544837i	0.962179	−	0.272419i	$-0.0878235\pi$
$44$		−	7.79380i		−	1.17496i
$45$		0			0
$46$		−	4.39305i		−	0.647720i
$47$			1.85308i			0.270299i	0.990825	+	0.135150i	$0.0431515\pi$
$47$			1.85308i			0.270299i	−0.990825	+	0.135150i	$0.956849\pi$
$48$		0			0
$49$	−0.219100			−0.0313000
$50$		−	1.78296i		−	0.252148i
$51$		0			0
$52$	−4.88898			−0.677980
$53$	−0.644913			−0.0885856			−0.0442928	−	0.999019i	$-0.514103\pi$
$53$	−0.644913			−0.0885856			−0.0442928	+	0.999019i	$0.514103\pi$
$54$		0			0
$55$		−	6.61083i		−	0.891405i
$56$		−	3.81205i		−	0.509406i
$57$		0			0
$58$	−5.46999	+	7.89104i	−0.718245	+	1.03614i
$59$	13.2217			1.72131			0.860657	−	0.509185i	$-0.170053\pi$
$59$	13.2217			1.72131			0.860657	+	0.509185i	$0.170053\pi$
$60$		0			0
$61$		−	5.92380i		−	0.758465i	−0.925301	−	0.379233i	$-0.876188\pi$
$61$		−	5.92380i		−	0.758465i	0.925301	−	0.379233i	$-0.123812\pi$
$62$	11.3986			1.44762
$63$		0			0
$64$	0.636777			0.0795971
$65$	−4.14692			−0.514362
$66$		0			0
$67$	−5.45564			−0.666513			−0.333256	−	0.942836i	$-0.608147\pi$
$67$	−5.45564			−0.666513			−0.333256	+	0.942836i	$0.608147\pi$
$68$		−	4.46706i		−	0.541710i
$69$		0			0
$70$			4.64285i			0.554927i
$71$	3.13184			0.371681			0.185840	−	0.982580i	$-0.440499\pi$
$71$	3.13184			0.371681			0.185840	+	0.982580i	$0.440499\pi$
$72$		0			0
$73$		−	15.7086i		−	1.83855i	−0.393614	−	0.919276i	$-0.628775\pi$
$73$		−	15.7086i		−	1.83855i	0.393614	−	0.919276i	$-0.371225\pi$
$74$	8.28169			0.962727
$75$		0			0
$76$		−	8.38090i		−	0.961355i
$77$			17.2147i			1.96180i
$78$		0			0
$79$			1.18502i			0.133325i	0.997776	+	0.0666626i	$0.0212351\pi$
$79$			1.18502i			0.133325i	−0.997776	+	0.0666626i	$0.978765\pi$
$80$	4.96798			0.555437
$81$		0			0
$82$	−1.28571			−0.141983
$83$	10.5236			1.15511			0.577556	−	0.816351i	$-0.304007\pi$
$83$	10.5236			1.15511			0.577556	+	0.816351i	$0.304007\pi$
$84$		0			0
$85$		−	3.78904i		−	0.410979i
$86$	−6.37004			−0.686899
$87$		0			0
$88$	9.67767			1.03164
$89$		−	4.34682i		−	0.460762i	−0.973101	−	0.230381i	$-0.926003\pi$
$89$		−	4.34682i		−	0.460762i	0.973101	−	0.230381i	$-0.0739972\pi$
$90$		0			0
$91$	10.7987			1.13201
$92$	−2.90481			−0.302847
$93$		0			0
$94$	3.30396			0.340778
$95$		−	7.10882i		−	0.729350i
$96$		0			0
$97$		−	6.44090i		−	0.653974i	−0.945029	−	0.326987i	$-0.893967\pi$
$97$		−	6.44090i		−	0.653974i	0.945029	−	0.326987i	$-0.106033\pi$
$98$			0.390646i			0.0394612i
$99$		0			0
$100$	−1.17894			−0.117894
$101$		−	17.6420i		−	1.75544i	−0.479171	−	0.877721i	$-0.659063\pi$
$101$		−	17.6420i		−	1.75544i	0.479171	−	0.877721i	$-0.340937\pi$
$102$		0			0
$103$	1.97699			0.194798			0.0973992	−	0.995245i	$-0.468948\pi$
$103$	1.97699			0.194798			0.0973992	+	0.995245i	$0.468948\pi$
$104$		−	6.07073i		−	0.595284i
$105$		0			0
$106$			1.14985i			0.111684i
$107$	−7.03263			−0.679870			−0.339935	−	0.940449i	$-0.610405\pi$
$107$	−7.03263			−0.679870			−0.339935	+	0.940449i	$0.610405\pi$
$108$		0			0
$109$	18.1357			1.73709			0.868544	−	0.495613i	$-0.165056\pi$
$109$	18.1357			1.73709			0.868544	+	0.495613i	$0.165056\pi$
$110$	−11.7868			−1.12383
$111$		0			0
$112$	−12.9367			−1.22240
$113$			4.99707i			0.470085i	0.971985	+	0.235042i	$0.0755229\pi$
$113$			4.99707i			0.470085i	−0.971985	+	0.235042i	$0.924477\pi$
$114$		0			0
$115$	−2.46391			−0.229761
$116$	5.21778	+	3.61691i	0.484459	+	0.335822i
$117$		0			0
$118$		−	23.5737i		−	2.17013i
$119$			9.86671i			0.904480i
$120$		0			0
$121$	−32.7031			−2.97301
$122$	−10.5619			−0.956229
$123$		0			0
$124$		−	7.53704i		−	0.676847i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			4.93316i			0.437747i	0.975753	+	0.218874i	$0.0702382\pi$
$127$			4.93316i			0.437747i	−0.975753	+	0.218874i	$0.929762\pi$
$128$			10.7244i			0.947913i
$129$		0			0
$130$			7.39379i			0.648479i
$131$			8.09374i			0.707153i	0.935406	+	0.353577i	$0.115035\pi$
$131$			8.09374i			0.707153i	−0.935406	+	0.353577i	$0.884965\pi$
$132$		0			0
$133$			18.5115i			1.60515i
$134$			9.72719i			0.840301i
$135$		0			0
$136$	5.54681			0.475635
$137$		−	9.08581i		−	0.776253i	−0.921606	−	0.388127i	$-0.873122\pi$
$137$		−	9.08581i		−	0.776253i	0.921606	−	0.388127i	$-0.126878\pi$
$138$		0			0
$139$	−13.9957			−1.18710			−0.593552	−	0.804796i	$-0.702275\pi$
$139$	−13.9957			−1.18710			−0.593552	+	0.804796i	$0.702275\pi$
$140$	3.06999			0.259461
$141$		0			0
$142$		−	5.58394i		−	0.468593i
$143$			27.4146i			2.29253i
$144$		0			0
$145$	4.42581	+	3.06793i	0.367544	+	0.254778i
$146$	−28.0078			−2.31794
$147$		0			0
$148$		−	5.47609i		−	0.450132i
$149$	6.57405			0.538568			0.269284	−	0.963061i	$-0.413213\pi$
$149$	6.57405			0.538568			0.269284	+	0.963061i	$0.413213\pi$
$150$		0			0
$151$	−10.8570			−0.883527			−0.441764	−	0.897131i	$-0.645647\pi$
$151$	−10.8570			−0.883527			−0.441764	+	0.897131i	$0.645647\pi$
$152$	10.4067			0.844094
$153$		0			0
$154$	30.6931			2.47332
$155$		−	6.39305i		−	0.513502i
$156$		0			0
$157$			13.0637i			1.04259i	0.853375	+	0.521297i	$0.174552\pi$
$157$			13.0637i			1.04259i	−0.853375	+	0.521297i	$0.825448\pi$
$158$	2.11284			0.168089
$159$		0			0
$160$		−	5.92988i		−	0.468798i
$161$	6.41606			0.505657
$162$		0			0
$163$		−	23.8666i		−	1.86937i	−0.355471	−	0.934687i	$-0.615679\pi$
$163$		−	23.8666i		−	1.86937i	0.355471	−	0.934687i	$-0.384321\pi$
$164$			0.850147i			0.0663853i
$165$		0			0
$166$		−	18.7631i		−	1.45630i
$167$	−18.5182			−1.43298			−0.716492	−	0.697595i	$-0.754253\pi$
$167$	−18.5182			−1.43298			−0.716492	+	0.697595i	$0.754253\pi$
$168$		0			0
$169$	4.19696			0.322843
$170$	−6.75570			−0.518138
$171$		0			0
$172$			4.21205i			0.321166i
$173$	−4.07086			−0.309502			−0.154751	−	0.987954i	$-0.549457\pi$
$173$	−4.07086			−0.309502			−0.154751	+	0.987954i	$0.549457\pi$
$174$		0			0
$175$	2.60402			0.196845
$176$		−	32.8425i		−	2.47560i
$177$		0			0
$178$	−7.75020			−0.580902
$179$	17.2677			1.29065			0.645324	−	0.763909i	$-0.276722\pi$
$179$	17.2677			1.29065			0.645324	+	0.763909i	$0.276722\pi$
$180$		0			0
$181$	−8.86803			−0.659156			−0.329578	−	0.944128i	$-0.606906\pi$
$181$	−8.86803			−0.659156			−0.329578	+	0.944128i	$0.606906\pi$
$182$		−	19.2536i		−	1.42717i
$183$		0			0
$184$		−	3.60695i		−	0.265908i
$185$		−	4.64491i		−	0.341501i
$186$		0			0
$187$	−25.0487			−1.83174
$188$		−	2.18467i		−	0.159334i
$189$		0			0
$190$	−12.6747			−0.919522
$191$		−	4.68446i		−	0.338956i	−0.985534	−	0.169478i	$-0.945792\pi$
$191$		−	4.68446i		−	0.338956i	0.985534	−	0.169478i	$-0.0542081\pi$
$192$		0			0
$193$			19.5479i			1.40709i	0.710651	+	0.703544i	$0.248400\pi$
$193$			19.5479i			1.40709i	−0.710651	+	0.703544i	$0.751600\pi$
$194$	−11.4839			−0.824493
$195$		0			0
$196$	0.258306			0.0184504
$197$	25.5128			1.81771			0.908856	−	0.417109i	$-0.136957\pi$
$197$	25.5128			1.81771			0.908856	+	0.417109i	$0.136957\pi$
$198$		0			0
$199$	−1.36177			−0.0965335			−0.0482667	−	0.998834i	$-0.515370\pi$
$199$	−1.36177			−0.0965335			−0.0482667	+	0.998834i	$0.515370\pi$
$200$		−	1.46391i		−	0.103514i
$201$		0			0
$202$	−31.4549			−2.21316
$203$	−11.5249	−	7.98893i	−0.808889	−	0.560713i
$204$		0			0
$205$			0.721109i			0.0503645i
$206$		−	3.52489i		−	0.245591i
$207$		0			0
$208$	−20.6018			−1.42848
$209$	−46.9953			−3.25073
$210$		0			0
$211$			0.904811i			0.0622897i	0.999515	+	0.0311449i	$0.00991532\pi$
$211$			0.904811i			0.0622897i	−0.999515	+	0.0311449i	$0.990085\pi$
$212$	0.760316			0.0522187
$213$		0			0
$214$			12.5389i			0.857141i
$215$			3.57274i			0.243659i
$216$		0			0
$217$			16.6476i			1.13011i
$218$		−	32.3352i		−	2.19002i
$219$		0			0
$220$			7.79380i			0.525458i
$221$			15.7128i			1.05696i
$222$		0			0
$223$	26.9575			1.80521			0.902605	−	0.430471i	$-0.141652\pi$
$223$	26.9575			1.80521			0.902605	+	0.430471i	$0.141652\pi$
$224$			15.4415i			1.03173i
$225$		0			0
$226$	8.90957			0.592656
$227$	3.33208			0.221158			0.110579	−	0.993867i	$-0.464730\pi$
$227$	3.33208			0.221158			0.110579	+	0.993867i	$0.464730\pi$
$228$		0			0
$229$		−	3.07218i		−	0.203015i	−0.994835	−	0.101508i	$-0.967633\pi$
$229$		−	3.07218i		−	0.203015i	0.994835	−	0.101508i	$-0.0323666\pi$
$230$			4.39305i			0.289669i
$231$		0			0
$232$	−4.49117	+	6.47900i	−0.294860	+	0.425367i
$233$	−18.6744			−1.22340			−0.611700	−	0.791090i	$-0.709514\pi$
$233$	−18.6744			−1.22340			−0.611700	+	0.791090i	$0.709514\pi$
$234$		0			0
$235$		−	1.85308i		−	0.120881i
$236$	−15.5876			−1.01467
$237$		0			0
$238$	17.5919			1.14032
$239$	−13.5155			−0.874246			−0.437123	−	0.899402i	$-0.644003\pi$
$239$	−13.5155			−0.874246			−0.437123	+	0.899402i	$0.644003\pi$
$240$		0			0
$241$	9.28996			0.598419			0.299209	−	0.954187i	$-0.403277\pi$
$241$	9.28996			0.598419			0.299209	+	0.954187i	$0.403277\pi$
$242$			58.3083i			3.74820i
$243$		0			0
$244$			6.98383i			0.447094i
$245$	0.219100			0.0139978
$246$		0			0
$247$			29.4797i			1.87575i
$248$	9.35886			0.594288
$249$		0			0
$250$			1.78296i			0.112764i
$251$			7.79989i			0.492325i	0.969229	+	0.246162i	$0.0791697\pi$
$251$			7.79989i			0.492325i	−0.969229	+	0.246162i	$0.920830\pi$
$252$		0			0
$253$			16.2885i			1.02405i
$254$	8.79562			0.551886
$255$		0			0
$256$	20.3947			1.27467
$257$	−3.42838			−0.213856			−0.106928	−	0.994267i	$-0.534102\pi$
$257$	−3.42838			−0.213856			−0.106928	+	0.994267i	$0.534102\pi$
$258$		0			0
$259$			12.0954i			0.751573i
$260$	4.88898			0.303202
$261$		0			0
$262$	14.4308			0.891538
$263$		−	15.1455i		−	0.933910i	−0.884281	−	0.466955i	$-0.845351\pi$
$263$		−	15.1455i		−	0.933910i	0.884281	−	0.466955i	$-0.154649\pi$
$264$		0			0
$265$	0.644913			0.0396167
$266$	33.0052			2.02368
$267$		0			0
$268$	6.43189			0.392890
$269$		−	25.6963i		−	1.56673i	−0.621561	−	0.783366i	$-0.713501\pi$
$269$		−	25.6963i		−	1.56673i	0.621561	−	0.783366i	$-0.286499\pi$
$270$		0			0
$271$			5.24468i			0.318592i	0.987231	+	0.159296i	$0.0509224\pi$
$271$			5.24468i			0.318592i	−0.987231	+	0.159296i	$0.949078\pi$
$272$		−	18.8239i		−	1.14136i
$273$		0			0
$274$	−16.1996			−0.978656
$275$			6.61083i			0.398648i
$276$		0			0
$277$	15.4448			0.927987			0.463994	−	0.885839i	$-0.346416\pi$
$277$	15.4448			0.927987			0.463994	+	0.885839i	$0.346416\pi$
$278$			24.9538i			1.49663i
$279$		0			0
$280$			3.81205i			0.227813i
$281$	22.1658			1.32230			0.661149	−	0.750254i	$-0.270069\pi$
$281$	22.1658			1.32230			0.661149	+	0.750254i	$0.270069\pi$
$282$		0			0
$283$	−13.1225			−0.780049			−0.390025	−	0.920804i	$-0.627534\pi$
$283$	−13.1225			−0.780049			−0.390025	+	0.920804i	$0.627534\pi$
$284$	−3.69226			−0.219095
$285$		0			0
$286$	48.8791			2.89028
$287$		−	1.87778i		−	0.110842i
$288$		0			0
$289$	2.64320			0.155482
$290$	5.46999	−	7.89104i	0.321209	−	0.463378i
$291$		0			0
$292$			18.5195i			1.08377i
$293$			17.1631i			1.00268i	0.865251	+	0.501339i	$0.167159\pi$
$293$			17.1631i			1.00268i	−0.865251	+	0.501339i	$0.832841\pi$
$294$		0			0
$295$	−13.2217			−0.769795
$296$	6.79974			0.395227
$297$		0			0
$298$		−	11.7213i		−	0.678995i
$299$	10.2176			0.590902
$300$		0			0
$301$		−	9.30346i		−	0.536243i
$302$			19.3575i			1.11390i
$303$		0			0
$304$		−	35.3165i		−	2.02554i
$305$			5.92380i			0.339196i
$306$		0			0
$307$			8.04069i			0.458906i	0.973320	+	0.229453i	$0.0736938\pi$
$307$			8.04069i			0.458906i	−0.973320	+	0.229453i	$0.926306\pi$
$308$		−	20.2952i		−	1.15642i
$309$		0			0
$310$	−11.3986			−0.647394
$311$			3.67741i			0.208527i	0.994550	+	0.104263i	$0.0332485\pi$
$311$			3.67741i			0.208527i	−0.994550	+	0.104263i	$0.966751\pi$
$312$		0			0
$313$	17.6267			0.996318			0.498159	−	0.867086i	$-0.334010\pi$
$313$	17.6267			0.996318			0.498159	+	0.867086i	$0.334010\pi$
$314$	23.2920			1.31444
$315$		0			0
$316$		−	1.39707i		−	0.0785914i
$317$			17.4787i			0.981699i	0.871244	+	0.490850i	$0.163314\pi$
$317$			17.4787i			0.981699i	−0.871244	+	0.490850i	$0.836686\pi$
$318$		0			0
$319$	20.2816	−	29.2583i	1.13555	−	1.63815i
$320$	−0.636777			−0.0355969
$321$		0			0
$322$		−	11.4396i		−	0.637503i
$323$	−26.9356			−1.49874
$324$		0			0
$325$	4.14692			0.230030
$326$	−42.5531			−2.35680
$327$		0			0
$328$	−1.05564			−0.0582880
$329$			4.82544i			0.266035i
$330$		0			0
$331$		−	22.9479i		−	1.26133i	−0.776054	−	0.630666i	$-0.782782\pi$
$331$		−	22.9479i		−	1.26133i	0.776054	−	0.630666i	$-0.217218\pi$
$332$	−12.4067			−0.680905
$333$		0			0
$334$			33.0172i			1.80662i
$335$	5.45564			0.298074
$336$		0			0
$337$			25.2051i			1.37301i	0.727125	+	0.686505i	$0.240856\pi$
$337$			25.2051i			1.37301i	−0.727125	+	0.686505i	$0.759144\pi$
$338$		−	7.48302i		−	0.407022i
$339$		0			0
$340$			4.46706i			0.242260i
$341$	−42.2634			−2.28869
$342$		0			0
$343$	−18.7987			−1.01503
$344$	−5.23017			−0.281992
$345$		0			0
$346$			7.25817i			0.390202i
$347$	21.1875			1.13740			0.568701	−	0.822544i	$-0.307446\pi$
$347$	21.1875			1.13740			0.568701	+	0.822544i	$0.307446\pi$
$348$		0			0
$349$	30.9222			1.65523			0.827614	−	0.561297i	$-0.189698\pi$
$349$	30.9222			1.65523			0.827614	+	0.561297i	$0.189698\pi$
$350$		−	4.64285i		−	0.248171i
$351$		0			0
$352$	−39.2015			−2.08944
$353$	−27.0800			−1.44132			−0.720660	−	0.693288i	$-0.756161\pi$
$353$	−27.0800			−1.44132			−0.720660	+	0.693288i	$0.756161\pi$
$354$		0			0
$355$	−3.13184			−0.166221
$356$			5.12465i			0.271606i
$357$		0			0
$358$		−	30.7876i		−	1.62717i
$359$			4.10164i			0.216476i	0.994125	+	0.108238i	$0.0345209\pi$
$359$			4.10164i			0.216476i	−0.994125	+	0.108238i	$0.965479\pi$
$360$		0			0
$361$	−31.5354			−1.65976
$362$			15.8113i			0.831025i
$363$		0			0
$364$	−12.7310			−0.667285
$365$			15.7086i			0.822225i
$366$		0			0
$367$		−	3.44492i		−	0.179823i	−0.995950	−	0.0899116i	$-0.971342\pi$
$367$		−	3.44492i		−	0.179823i	0.995950	−	0.0899116i	$-0.0286585\pi$
$368$	−12.2407			−0.638089
$369$		0			0
$370$	−8.28169			−0.430545
$371$	−1.67936			−0.0871882
$372$		0			0
$373$	−1.89363			−0.0980485			−0.0490243	−	0.998798i	$-0.515611\pi$
$373$	−1.89363			−0.0980485			−0.0490243	+	0.998798i	$0.515611\pi$
$374$			44.6608i			2.30935i
$375$		0			0
$376$	2.71274			0.139899
$377$	−18.3535	−	12.7225i	−0.945254	−	0.655240i
$378$		0			0
$379$		−	19.0161i		−	0.976791i	−0.872622	−	0.488395i	$-0.837582\pi$
$379$		−	19.0161i		−	0.976791i	0.872622	−	0.488395i	$-0.162418\pi$
$380$			8.38090i			0.429931i
$381$		0			0
$382$	−8.35220			−0.427336
$383$	33.6337			1.71860			0.859301	−	0.511471i	$-0.170899\pi$
$383$	33.6337			1.71860			0.859301	+	0.511471i	$0.170899\pi$
$384$		0			0
$385$		−	17.2147i		−	0.877343i
$386$	34.8531			1.77398
$387$		0			0
$388$			7.59345i			0.385499i
$389$		−	23.2259i		−	1.17760i	−0.808278	−	0.588800i	$-0.799600\pi$
$389$		−	23.2259i		−	1.17760i	0.808278	−	0.588800i	$-0.200400\pi$
$390$		0			0
$391$			9.33585i			0.472134i
$392$			0.320743i			0.0162000i
$393$		0			0
$394$		−	45.4883i		−	2.29167i
$395$		−	1.18502i		−	0.0596248i
$396$		0			0
$397$	4.93046			0.247453			0.123726	−	0.992316i	$-0.460515\pi$
$397$	4.93046			0.247453			0.123726	+	0.992316i	$0.460515\pi$
$398$			2.42798i			0.121704i
$399$		0			0
$400$	−4.96798			−0.248399
$401$	−21.8391			−1.09059			−0.545296	−	0.838243i	$-0.683583\pi$
$401$	−21.8391			−1.09059			−0.545296	+	0.838243i	$0.683583\pi$
$402$		0			0
$403$			26.5115i			1.32063i
$404$			20.7989i			1.03478i
$405$		0			0
$406$	−14.2439	+	20.5484i	−0.706915	+	1.01980i
$407$	−30.7067			−1.52208
$408$		0			0
$409$		−	14.0163i		−	0.693060i	−0.938039	−	0.346530i	$-0.887360\pi$
$409$		−	14.0163i		−	0.693060i	0.938039	−	0.346530i	$-0.112640\pi$
$410$	1.28571			0.0634966
$411$		0			0
$412$	−2.33076			−0.114828
$413$	34.4294			1.69416
$414$		0			0
$415$	−10.5236			−0.516582
$416$			24.5908i			1.20566i
$417$		0			0
$418$			83.7906i			4.09833i
$419$	−12.0925			−0.590756			−0.295378	−	0.955381i	$-0.595446\pi$
$419$	−12.0925			−0.590756			−0.295378	+	0.955381i	$0.595446\pi$
$420$		0			0
$421$			32.3671i			1.57748i	0.614728	+	0.788739i	$0.289266\pi$
$421$			32.3671i			1.57748i	−0.614728	+	0.788739i	$0.710734\pi$
$422$	1.61324			0.0785313
$423$		0			0
$424$			0.944095i			0.0458493i
$425$			3.78904i			0.183795i
$426$		0			0
$427$		−	15.4257i		−	0.746501i
$428$	8.29107			0.400764
$429$		0			0
$430$	6.37004			0.307191
$431$	19.3797			0.933485			0.466743	−	0.884393i	$-0.345427\pi$
$431$	19.3797			0.933485			0.466743	+	0.884393i	$0.345427\pi$
$432$		0			0
$433$		−	20.3511i		−	0.978010i	−0.872281	−	0.489005i	$-0.837360\pi$
$433$		−	20.3511i		−	0.978010i	0.872281	−	0.489005i	$-0.162640\pi$
$434$	29.6820			1.42478
$435$		0			0
$436$	−21.3810			−1.02396
$437$			17.5155i			0.837881i
$438$		0			0
$439$	−8.42565			−0.402134			−0.201067	−	0.979577i	$-0.564441\pi$
$439$	−8.42565			−0.402134			−0.201067	+	0.979577i	$0.564441\pi$
$440$	−9.67767			−0.461365
$441$		0			0
$442$	28.0154			1.33255
$443$			7.52948i			0.357736i	0.983873	+	0.178868i	$0.0572436\pi$
$443$			7.52948i			0.357736i	−0.983873	+	0.178868i	$0.942756\pi$
$444$		0			0
$445$			4.34682i			0.206059i
$446$		−	48.0642i		−	2.27590i
$447$		0			0
$448$	1.65818			0.0783415
$449$		−	31.1439i		−	1.46977i	−0.678192	−	0.734885i	$-0.737236\pi$
$449$		−	31.1439i		−	1.46977i	0.678192	−	0.734885i	$-0.262764\pi$
$450$		0			0
$451$	4.76713			0.224476
$452$		−	5.89126i		−	0.277102i
$453$		0			0
$454$		−	5.94095i		−	0.278823i
$455$	−10.7987			−0.506249
$456$		0			0
$457$	−5.96504			−0.279033			−0.139516	−	0.990220i	$-0.544555\pi$
$457$	−5.96504			−0.279033			−0.139516	+	0.990220i	$0.544555\pi$
$458$	−5.47757			−0.255950
$459$		0			0
$460$	2.90481			0.135437
$461$			22.3287i			1.03995i	0.854182	+	0.519975i	$0.174059\pi$
$461$			22.3287i			1.03995i	−0.854182	+	0.519975i	$0.825941\pi$
$462$		0			0
$463$	12.1195			0.563242			0.281621	−	0.959526i	$-0.409128\pi$
$463$	12.1195			0.563242			0.281621	+	0.959526i	$0.409128\pi$
$464$	21.9873	+	15.2414i	1.02074	+	0.707564i
$465$		0			0
$466$			33.2957i			1.54239i
$467$		−	35.8252i		−	1.65779i	−0.559402	−	0.828896i	$-0.688969\pi$
$467$		−	35.8252i		−	1.65779i	0.559402	−	0.828896i	$-0.311031\pi$
$468$		0			0
$469$	−14.2066			−0.655999
$470$	−3.30396			−0.152400
$471$		0			0
$472$		−	19.3553i		−	0.890902i
$473$	23.6188			1.08599
$474$		0			0
$475$			7.10882i			0.326175i
$476$		−	11.6323i		−	0.533165i
$477$		0			0
$478$			24.0976i			1.10220i
$479$			26.7495i			1.22222i	0.791547	+	0.611108i	$0.209276\pi$
$479$			26.7495i			1.22222i	−0.791547	+	0.611108i	$0.790724\pi$
$480$		0			0
$481$			19.2621i			0.878276i
$482$		−	16.5636i		−	0.754452i
$483$		0			0
$484$	38.5551			1.75251
$485$			6.44090i			0.292466i
$486$		0			0
$487$	−0.382377			−0.0173272			−0.00866358	−	0.999962i	$-0.502758\pi$
$487$	−0.382377			−0.0173272			−0.00866358	+	0.999962i	$0.502758\pi$
$488$	−8.67192			−0.392560
$489$		0			0
$490$		−	0.390646i		−	0.0176476i
$491$			25.6067i			1.15561i	0.816174	+	0.577807i	$0.196091\pi$
$491$			25.6067i			1.15561i	−0.816174	+	0.577807i	$0.803909\pi$
$492$		0			0
$493$	11.6245	−	16.7696i	0.523541	−	0.755263i
$494$	52.5612			2.36484
$495$		0			0
$496$		−	31.7606i		−	1.42609i
$497$	8.15535			0.365818
$498$		0			0
$499$	−20.2051			−0.904505			−0.452252	−	0.891890i	$-0.649379\pi$
$499$	−20.2051			−0.904505			−0.452252	+	0.891890i	$0.649379\pi$
$500$	1.17894			0.0527239
$501$		0			0
$502$	13.9069			0.620695
$503$			6.63918i			0.296027i	0.988985	+	0.148013i	$0.0472878\pi$
$503$			6.63918i			0.296027i	−0.988985	+	0.148013i	$0.952712\pi$
$504$		0			0
$505$			17.6420i			0.785058i
$506$	29.0417			1.29106
$507$		0			0
$508$		−	5.81591i		−	0.258039i
$509$	6.71577			0.297671			0.148836	−	0.988862i	$-0.452447\pi$
$509$	6.71577			0.297671			0.148836	+	0.988862i	$0.452447\pi$
$510$		0			0
$511$		−	40.9054i		−	1.80955i
$512$		−	14.9142i		−	0.659119i
$513$		0			0
$514$			6.11266i			0.269618i
$515$	−1.97699			−0.0871165
$516$		0			0
$517$	−12.2504			−0.538771
$518$	21.5657			0.947540
$519$		0			0
$520$			6.07073i			0.266219i
$521$	−43.7273			−1.91573			−0.957864	−	0.287223i	$-0.907268\pi$
$521$	−43.7273			−1.91573			−0.957864	+	0.287223i	$0.907268\pi$
$522$		0			0
$523$	24.1412			1.05562			0.527811	−	0.849362i	$-0.323013\pi$
$523$	24.1412			1.05562			0.527811	+	0.849362i	$0.323013\pi$
$524$		−	9.54205i		−	0.416847i
$525$		0			0
$526$	−27.0038			−1.17742
$527$	−24.2235			−1.05519
$528$		0			0
$529$	−16.9291			−0.736050
$530$		−	1.14985i		−	0.0499464i
$531$		0			0
$532$		−	21.8240i		−	0.946190i
$533$		−	2.99038i		−	0.129528i
$534$		0			0
$535$	7.03263			0.304047
$536$			7.98658i			0.344968i
$537$		0			0
$538$	−45.8155			−1.97524
$539$		−	1.44843i		−	0.0623884i
$540$		0			0
$541$			12.6252i			0.542800i	0.962467	+	0.271400i	$0.0874867\pi$
$541$			12.6252i			0.542800i	−0.962467	+	0.271400i	$0.912513\pi$
$542$	9.35105			0.401662
$543$		0			0
$544$	−22.4685			−0.963331
$545$	−18.1357			−0.776849
$546$		0			0
$547$	12.2476			0.523670			0.261835	−	0.965113i	$-0.415672\pi$
$547$	12.2476			0.523670			0.261835	+	0.965113i	$0.415672\pi$
$548$			10.7117i			0.457579i
$549$		0			0
$550$	11.7868			0.502593
$551$	21.8094	−	31.4623i	0.929110	−	1.34034i
$552$		0			0
$553$			3.08581i			0.131222i
$554$		−	27.5374i		−	1.16995i
$555$		0			0
$556$	16.5002			0.699764
$557$	−21.9729			−0.931024			−0.465512	−	0.885042i	$-0.654130\pi$
$557$	−21.9729			−0.931024			−0.465512	+	0.885042i	$0.654130\pi$
$558$		0			0
$559$		−	14.8159i		−	0.626644i
$560$	12.9367			0.546675
$561$		0			0
$562$		−	39.5206i		−	1.66708i
$563$		−	32.5545i		−	1.37201i	−0.727598	−	0.686004i	$-0.759363\pi$
$563$		−	32.5545i		−	1.37201i	0.727598	−	0.686004i	$-0.240637\pi$
$564$		0			0
$565$		−	4.99707i		−	0.210228i
$566$			23.3968i			0.983441i
$567$		0			0
$568$		−	4.58473i		−	0.192371i
$569$		−	33.2040i		−	1.39199i	−0.718049	−	0.695993i	$-0.754964\pi$
$569$		−	33.2040i		−	1.39199i	0.718049	−	0.695993i	$-0.245036\pi$
$570$		0			0
$571$	17.3794			0.727305			0.363653	−	0.931535i	$-0.381530\pi$
$571$	17.3794			0.727305			0.363653	+	0.931535i	$0.381530\pi$
$572$		−	32.3203i		−	1.35138i
$573$		0			0
$574$	−3.34800			−0.139743
$575$	2.46391			0.102752
$576$		0			0
$577$		−	38.4323i		−	1.59996i	−0.600027	−	0.799980i	$-0.704844\pi$
$577$		−	38.4323i		−	1.59996i	0.600027	−	0.799980i	$-0.295156\pi$
$578$		−	4.71272i		−	0.196023i
$579$		0			0
$580$	−5.21778	−	3.61691i	−0.216657	−	0.150184i
$581$	27.4035			1.13689
$582$		0			0
$583$		−	4.26341i		−	0.176573i
$584$	−22.9960			−0.951580
$585$		0			0
$586$	30.6011			1.26412
$587$	−16.8853			−0.696931			−0.348466	−	0.937322i	$-0.613297\pi$
$587$	−16.8853			−0.696931			−0.348466	+	0.937322i	$0.613297\pi$
$588$		0			0
$589$	−45.4471			−1.87261
$590$			23.5737i			0.970513i
$591$		0			0
$592$		−	23.0758i		−	0.948411i
$593$	−12.1712			−0.499811			−0.249906	−	0.968270i	$-0.580400\pi$
$593$	−12.1712			−0.499811			−0.249906	+	0.968270i	$0.580400\pi$
$594$		0			0
$595$		−	9.86671i		−	0.404496i
$596$	−7.75043			−0.317470
$597$		0			0
$598$		−	18.2176i		−	0.744975i
$599$		−	18.4881i		−	0.755403i	−0.925927	−	0.377701i	$-0.876715\pi$
$599$		−	18.4881i		−	0.755403i	0.925927	−	0.377701i	$-0.123285\pi$
$600$		0			0
$601$		−	25.7139i		−	1.04889i	−0.851443	−	0.524446i	$-0.824272\pi$
$601$		−	25.7139i		−	1.04889i	0.851443	−	0.524446i	$-0.175728\pi$
$602$	−16.5877			−0.676064
$603$		0			0
$604$	12.7997			0.520814
$605$	32.7031			1.32957
$606$		0			0
$607$		−	24.3863i		−	0.989810i	−0.868947	−	0.494905i	$-0.835203\pi$
$607$		−	24.3863i		−	0.989810i	0.868947	−	0.494905i	$-0.164797\pi$
$608$	−42.1545			−1.70959
$609$		0			0
$610$	10.5619			0.427639
$611$			7.68457i			0.310884i
$612$		0			0
$613$	−13.4750			−0.544248			−0.272124	−	0.962262i	$-0.587726\pi$
$613$	−13.4750			−0.544248			−0.272124	+	0.962262i	$0.587726\pi$
$614$	14.3362			0.578562
$615$		0			0
$616$	25.2008			1.01537
$617$		−	1.36560i		−	0.0549770i	−0.999622	−	0.0274885i	$-0.991249\pi$
$617$		−	1.36560i		−	0.0549770i	0.999622	−	0.0274885i	$-0.00875096\pi$
$618$		0			0
$619$		−	22.6187i		−	0.909124i	−0.890715	−	0.454562i	$-0.849796\pi$
$619$		−	22.6187i		−	0.909124i	0.890715	−	0.454562i	$-0.150204\pi$
$620$			7.53704i			0.302695i
$621$		0			0
$622$	6.55668			0.262899
$623$		−	11.3192i		−	0.453494i
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	31.4276i		−	1.25610i
$627$		0			0
$628$		−	15.4013i		−	0.614580i
$629$	−17.5997			−0.701748
$630$		0			0
$631$	10.6701			0.424772			0.212386	−	0.977186i	$-0.431877\pi$
$631$	10.6701			0.424772			0.212386	+	0.977186i	$0.431877\pi$
$632$	1.73476			0.0690052
$633$		0			0
$634$	31.1637			1.23767
$635$		−	4.93316i		−	0.195766i
$636$		0			0
$637$	−0.908590			−0.0359997
$638$	−52.1664	−	36.1612i	−2.06529	−	1.43164i
$639$		0			0
$640$		−	10.7244i		−	0.423920i
$641$			11.7428i			0.463811i	0.972738	+	0.231905i	$0.0744960\pi$
$641$			11.7428i			0.463811i	−0.972738	+	0.231905i	$0.925504\pi$
$642$		0			0
$643$	37.2947			1.47076			0.735381	−	0.677654i	$-0.237003\pi$
$643$	37.2947			1.47076			0.735381	+	0.677654i	$0.237003\pi$
$644$	−7.56417			−0.298070
$645$		0			0
$646$			48.0251i			1.88952i
$647$	−14.6160			−0.574616			−0.287308	−	0.957838i	$-0.592760\pi$
$647$	−14.6160			−0.574616			−0.287308	+	0.957838i	$0.592760\pi$
$648$		0			0
$649$			87.4062i			3.43100i
$650$		−	7.39379i		−	0.290008i
$651$		0			0
$652$			28.1373i			1.10194i
$653$			20.8225i			0.814849i	0.913239	+	0.407425i	$0.133573\pi$
$653$			20.8225i			0.814849i	−0.913239	+	0.407425i	$0.866427\pi$
$654$		0			0
$655$		−	8.09374i		−	0.316249i
$656$			3.58246i			0.139871i
$657$		0			0
$658$	8.60357			0.335402
$659$			5.84686i			0.227761i	0.993494	+	0.113881i	$0.0363282\pi$
$659$			5.84686i			0.227761i	−0.993494	+	0.113881i	$0.963672\pi$
$660$		0			0
$661$	−16.9767			−0.660316			−0.330158	−	0.943926i	$-0.607102\pi$
$661$	−16.9767			−0.660316			−0.330158	+	0.943926i	$0.607102\pi$
$662$	−40.9152			−1.59022
$663$		0			0
$664$		−	15.4056i		−	0.597852i
$665$		−	18.5115i		−	0.717845i
$666$		0			0
$667$	−10.9048	−	7.55910i	−0.422236	−	0.292690i
$668$	21.8319			0.844703
$669$		0			0
$670$		−	9.72719i		−	0.375794i
$671$	39.1613			1.51180
$672$		0			0
$673$	18.2039			0.701710			0.350855	−	0.936430i	$-0.385891\pi$
$673$	18.2039			0.701710			0.350855	+	0.936430i	$0.385891\pi$
$674$	44.9397			1.73101
$675$		0			0
$676$	−4.94798			−0.190307
$677$		−	16.5888i		−	0.637558i	−0.947829	−	0.318779i	$-0.896727\pi$
$677$		−	16.5888i		−	0.637558i	0.947829	−	0.318779i	$-0.103273\pi$
$678$		0			0
$679$		−	16.7722i		−	0.643658i
$680$	−5.54681			−0.212711
$681$		0			0
$682$			75.3539i			2.88545i
$683$	36.6485			1.40232			0.701158	−	0.713006i	$-0.252667\pi$
$683$	36.6485			1.40232			0.701158	+	0.713006i	$0.252667\pi$
$684$		0			0
$685$			9.08581i			0.347151i
$686$			33.5172i			1.27969i
$687$		0			0
$688$			17.7493i			0.676685i
$689$	−2.67440			−0.101887
$690$		0			0
$691$	20.4746			0.778892			0.389446	−	0.921049i	$-0.372667\pi$
$691$	20.4746			0.778892			0.389446	+	0.921049i	$0.372667\pi$
$692$	4.79931			0.182442
$693$		0			0
$694$		−	37.7764i		−	1.43397i
$695$	13.9957			0.530889
$696$		0			0
$697$	2.73231			0.103494
$698$		−	55.1331i		−	2.08682i
$699$		0			0
$700$	−3.06999			−0.116035
$701$	31.8417			1.20265			0.601323	−	0.799006i	$-0.294640\pi$
$701$	31.8417			1.20265			0.601323	+	0.799006i	$0.294640\pi$
$702$		0			0
$703$	−33.0199			−1.24537
$704$			4.20962i			0.158656i
$705$		0			0
$706$			48.2824i			1.81713i
$707$		−	45.9400i		−	1.72775i
$708$		0			0
$709$	−14.0789			−0.528744			−0.264372	−	0.964421i	$-0.585165\pi$
$709$	−14.0789			−0.528744			−0.264372	+	0.964421i	$0.585165\pi$
$710$			5.58394i			0.209561i
$711$		0			0
$712$	−6.36336			−0.238477
$713$			15.7519i			0.589914i
$714$		0			0
$715$		−	27.4146i		−	1.02525i
$716$	−20.3576			−0.760800
$717$		0			0
$718$	7.31306			0.272921
$719$	−5.94838			−0.221837			−0.110919	−	0.993830i	$-0.535379\pi$
$719$	−5.94838			−0.221837			−0.110919	+	0.993830i	$0.535379\pi$
$720$		0			0
$721$	5.14811			0.191726
$722$			56.2263i			2.09253i
$723$		0			0
$724$	10.4549			0.388553
$725$	−4.42581	−	3.06793i	−0.164371	−	0.113940i
$726$		0			0
$727$		−	12.3620i		−	0.458481i	−0.973370	−	0.229241i	$-0.926376\pi$
$727$		−	12.3620i		−	0.458481i	0.973370	−	0.229241i	$-0.0736243\pi$
$728$		−	15.8083i		−	0.585893i
$729$		0			0
$730$	28.0078			1.03661
$731$	13.5372			0.500692
$732$		0			0
$733$		−	49.3006i		−	1.82096i	−0.413556	−	0.910479i	$-0.635713\pi$
$733$		−	49.3006i		−	1.82096i	0.413556	−	0.910479i	$-0.364287\pi$
$734$	−6.14215			−0.226711
$735$		0			0
$736$			14.6107i			0.538558i
$737$		−	36.0663i		−	1.32852i
$738$		0			0
$739$		−	8.72784i		−	0.321059i	−0.987031	−	0.160529i	$-0.948680\pi$
$739$		−	8.72784i		−	0.321059i	0.987031	−	0.160529i	$-0.0513202\pi$
$740$			5.47609i			0.201305i
$741$		0			0
$742$			2.99424i			0.109922i
$743$		−	41.6374i		−	1.52753i	−0.645496	−	0.763764i	$-0.723349\pi$
$743$		−	41.6374i		−	1.52753i	0.645496	−	0.763764i	$-0.276651\pi$
$744$		0			0
$745$	−6.57405			−0.240855
$746$			3.37627i			0.123614i
$747$		0			0
$748$	29.5310			1.07976
$749$	−18.3131			−0.669145
$750$		0			0
$751$			14.9756i			0.546468i	0.961948	+	0.273234i	$0.0880933\pi$
$751$			14.9756i			0.546468i	−0.961948	+	0.273234i	$0.911907\pi$
$752$		−	9.20605i		−	0.335710i
$753$		0			0
$754$	−22.6836	+	32.7235i	−0.826089	+	1.19172i
$755$	10.8570			0.395125
$756$		0			0
$757$			7.16735i			0.260502i	0.991481	+	0.130251i	$0.0415783\pi$
$757$			7.16735i			0.260502i	−0.991481	+	0.130251i	$0.958422\pi$
$758$	−33.9049			−1.23148
$759$		0			0
$760$	−10.4067			−0.377490
$761$	8.16312			0.295913			0.147957	−	0.988994i	$-0.452730\pi$
$761$	8.16312			0.295913			0.147957	+	0.988994i	$0.452730\pi$
$762$		0			0
$763$	47.2257			1.70969
$764$			5.52271i			0.199805i
$765$		0			0
$766$		−	59.9675i		−	2.16671i
$767$	54.8292			1.97977
$768$		0			0
$769$			9.40091i			0.339005i	0.985530	+	0.169503i	$0.0542161\pi$
$769$			9.40091i			0.339005i	−0.985530	+	0.169503i	$0.945784\pi$
$770$	−30.6931			−1.10610
$771$		0			0
$772$		−	23.0459i		−	0.829439i
$773$			14.0490i			0.505307i	0.967557	+	0.252653i	$0.0813032\pi$
$773$			14.0490i			0.505307i	−0.967557	+	0.252653i	$0.918697\pi$
$774$		0			0
$775$			6.39305i			0.229645i
$776$	−9.42890			−0.338478
$777$		0			0
$778$	−41.4109			−1.48465
$779$	5.12624			0.183667
$780$		0			0
$781$			20.7040i			0.740849i
$782$	16.6454			0.595240
$783$		0			0
$784$	1.08848			0.0388744
$785$		−	13.0637i		−	0.466263i
$786$		0			0
$787$	28.6760			1.02219			0.511094	−	0.859525i	$-0.329240\pi$
$787$	28.6760			1.02219			0.511094	+	0.859525i	$0.329240\pi$
$788$	−30.0781			−1.07149
$789$		0			0
$790$	−2.11284			−0.0751716
$791$			13.0124i			0.462669i
$792$		0			0
$793$		−	24.5656i		−	0.872348i
$794$		−	8.79081i		−	0.311974i
$795$		0			0
$796$	1.60545			0.0569037
$797$			17.0067i			0.602409i	0.953560	+	0.301204i	$0.0973886\pi$
$797$			17.0067i			0.602409i	−0.953560	+	0.301204i	$0.902611\pi$
$798$		0			0
$799$	−7.02138			−0.248399
$800$			5.92988i			0.209653i
$801$		0			0
$802$			38.9382i			1.37496i
$803$	103.847			3.66468
$804$		0			0
$805$	−6.41606			−0.226137
$806$	47.2689			1.66498
$807$		0			0
$808$	−25.8263			−0.908566
$809$			37.8813i			1.33184i	0.746025	+	0.665918i	$0.231960\pi$
$809$			37.8813i			1.33184i	−0.746025	+	0.665918i	$0.768040\pi$
$810$		0			0
$811$	−53.0728			−1.86364			−0.931819	−	0.362923i	$-0.881779\pi$
$811$	−53.0728			−1.86364			−0.931819	+	0.362923i	$0.881779\pi$
$812$	13.5872	+	9.41849i	0.476817	+	0.330524i
$813$		0			0
$814$			54.7489i			1.91895i
$815$			23.8666i			0.836010i
$816$		0			0
$817$	25.3979			0.888562
$818$	−24.9904			−0.873770
$819$		0			0
$820$		−	0.850147i		−	0.0296884i
$821$	−23.5402			−0.821558			−0.410779	−	0.911735i	$-0.634743\pi$
$821$	−23.5402			−0.821558			−0.410779	+	0.911735i	$0.634743\pi$
$822$		0			0
$823$			5.33654i			0.186020i	0.995665	+	0.0930100i	$0.0296488\pi$
$823$			5.33654i			0.186020i	−0.995665	+	0.0930100i	$0.970351\pi$
$824$		−	2.89414i		−	0.100822i
$825$		0			0
$826$		−	61.3863i		−	2.13590i
$827$		−	16.3727i		−	0.569334i	−0.958626	−	0.284667i	$-0.908117\pi$
$827$		−	16.3727i		−	0.569334i	0.958626	−	0.284667i	$-0.0918830\pi$
$828$		0			0
$829$		−	4.76206i		−	0.165393i	−0.996575	−	0.0826966i	$-0.973647\pi$
$829$		−	4.76206i		−	0.165393i	0.996575	−	0.0826966i	$-0.0263532\pi$
$830$			18.7631i			0.651276i
$831$		0			0
$832$	2.64066			0.0915485
$833$		−	0.830177i		−	0.0287639i
$834$		0			0
$835$	18.5182			0.640850
$836$	55.4047			1.91621
$837$		0			0
$838$			21.5604i			0.744791i
$839$			1.38600i			0.0478502i	0.999714	+	0.0239251i	$0.00761632\pi$
$839$			1.38600i			0.0478502i	−0.999714	+	0.0239251i	$0.992384\pi$
$840$		0			0
$841$	10.1756	+	27.1561i	0.350884	+	0.936419i
$842$	57.7093			1.98879
$843$		0			0
$844$		−	1.06672i		−	0.0367180i
$845$	−4.19696			−0.144380
$846$		0			0
$847$	−85.1595			−2.92611
$848$	3.20391			0.110023
$849$		0			0
$850$	6.75570			0.231718
$851$			11.4447i			0.392318i
$852$		0			0
$853$		−	36.3453i		−	1.24444i	−0.782843	−	0.622219i	$-0.786231\pi$
$853$		−	36.3453i		−	1.24444i	0.782843	−	0.622219i	$-0.213769\pi$
$854$	−27.5034			−0.941145
$855$		0			0
$856$			10.2951i			0.351881i
$857$	−42.3964			−1.44823			−0.724116	−	0.689678i	$-0.757752\pi$
$857$	−42.3964			−1.44823			−0.724116	+	0.689678i	$0.757752\pi$
$858$		0			0
$859$		−	14.1971i		−	0.484398i	−0.970227	−	0.242199i	$-0.922131\pi$
$859$		−	14.1971i		−	0.484398i	0.970227	−	0.242199i	$-0.0778687\pi$
$860$		−	4.21205i		−	0.143630i
$861$		0			0
$862$		−	34.5531i		−	1.17688i
$863$	23.0565			0.784853			0.392426	−	0.919783i	$-0.371636\pi$
$863$	23.0565			0.784853			0.392426	+	0.919783i	$0.371636\pi$
$864$		0			0
$865$	4.07086			0.138413
$866$	−36.2851			−1.23302
$867$		0			0
$868$		−	19.6266i		−	0.666170i
$869$	−7.83397			−0.265749
$870$		0			0
$871$	−22.6241			−0.766589
$872$		−	26.5491i		−	0.899066i
$873$		0			0
$874$	31.2294			1.05635
$875$	−2.60402			−0.0880318
$876$		0			0
$877$	−31.7600			−1.07246			−0.536229	−	0.844073i	$-0.680152\pi$
$877$	−31.7600			−1.07246			−0.536229	+	0.844073i	$0.680152\pi$
$878$			15.0226i			0.506988i
$879$		0			0
$880$			32.8425i			1.10712i
$881$			12.5647i			0.423316i	0.977344	+	0.211658i	$0.0678864\pi$
$881$			12.5647i			0.423316i	−0.977344	+	0.211658i	$0.932114\pi$
$882$		0			0
$883$	39.7039			1.33614			0.668071	−	0.744098i	$-0.267120\pi$
$883$	39.7039			1.33614			0.668071	+	0.744098i	$0.267120\pi$
$884$		−	18.5245i		−	0.623048i
$885$		0			0
$886$	13.4248			0.451014
$887$			19.0472i			0.639543i	0.947495	+	0.319771i	$0.103606\pi$
$887$			19.0472i			0.639543i	−0.947495	+	0.319771i	$0.896394\pi$
$888$		0			0
$889$			12.8460i			0.430842i
$890$	7.75020			0.259787
$891$		0			0
$892$	−31.7814			−1.06412
$893$	−13.1732			−0.440824
$894$		0			0
$895$	−17.2677			−0.577195
$896$			27.9266i			0.932961i
$897$		0			0
$898$	−55.5282			−1.85300
$899$	19.6134	−	28.2945i	0.654144	−	0.943673i
$900$		0			0
$901$		−	2.44360i		−	0.0814081i
$902$		−	8.49960i		−	0.283006i
$903$		0			0
$904$	7.31527			0.243302
$905$	8.86803			0.294783
$906$		0			0
$907$			9.30880i			0.309094i	0.987985	+	0.154547i	$0.0493917\pi$
$907$			9.30880i			0.309094i	−0.987985	+	0.154547i	$0.950608\pi$
$908$	−3.92833			−0.130366
$909$		0			0
$910$			19.2536i			0.638249i
$911$		−	15.8924i		−	0.526539i	−0.964722	−	0.263270i	$-0.915199\pi$
$911$		−	15.8924i		−	0.526539i	0.964722	−	0.263270i	$-0.0848009\pi$
$912$		0			0
$913$			69.5696i			2.30242i
$914$			10.6354i			0.351789i
$915$		0			0
$916$			3.62192i			0.119672i
$917$			21.0762i			0.695998i
$918$		0			0
$919$	26.1892			0.863904			0.431952	−	0.901897i	$-0.357825\pi$
$919$	26.1892			0.863904			0.431952	+	0.901897i	$0.357825\pi$
$920$			3.60695i			0.118918i
$921$		0			0
$922$	39.8111			1.31111
$923$	12.9875			0.427488
$924$		0			0
$925$			4.64491i			0.152724i
$926$		−	21.6086i		−	0.710103i
$927$		0			0
$928$	18.1924	−	26.2445i	0.597196	−	0.861519i
$929$	−44.7041			−1.46669			−0.733347	−	0.679854i	$-0.762043\pi$
$929$	−44.7041			−1.46669			−0.733347	+	0.679854i	$0.762043\pi$
$930$		0			0
$931$		−	1.55754i		−	0.0510464i
$932$	22.0160			0.721160
$933$		0			0
$934$	−63.8749			−2.09005
$935$	25.0487			0.819180
$936$		0			0
$937$	28.9245			0.944923			0.472462	−	0.881351i	$-0.343366\pi$
$937$	28.9245			0.944923			0.472462	+	0.881351i	$0.343366\pi$
$938$			25.3298i			0.827046i
$939$		0			0
$940$			2.18467i			0.0712562i
$941$	−1.16465			−0.0379664			−0.0189832	−	0.999820i	$-0.506043\pi$
$941$	−1.16465			−0.0379664			−0.0189832	+	0.999820i	$0.506043\pi$
$942$		0			0
$943$		−	1.77675i		−	0.0578589i
$944$	−65.6850			−2.13786
$945$		0			0
$946$		−	42.1113i		−	1.36916i
$947$			12.4430i			0.404343i	0.979350	+	0.202171i	$0.0647998\pi$
$947$			12.4430i			0.404343i	−0.979350	+	0.202171i	$0.935200\pi$
$948$		0			0
$949$		−	65.1423i		−	2.11461i
$950$	12.6747			0.411223
$951$		0			0
$952$	14.4440			0.468132
$953$	−15.1160			−0.489655			−0.244827	−	0.969567i	$-0.578731\pi$
$953$	−15.1160			−0.489655			−0.244827	+	0.969567i	$0.578731\pi$
$954$		0			0
$955$			4.68446i			0.151586i
$956$	15.9340			0.515343
$957$		0			0
$958$	47.6933			1.54090
$959$		−	23.6596i		−	0.764008i
$960$		0			0
$961$	−9.87112			−0.318423
$962$	34.3435			1.10728
$963$		0			0
$964$	−10.9523			−0.352751
$965$		−	19.5479i		−	0.629269i
$966$		0			0
$967$			51.6458i			1.66082i	0.557156	+	0.830408i	$0.311893\pi$
$967$			51.6458i			1.66082i	−0.557156	+	0.830408i	$0.688107\pi$
$968$			47.8745i			1.53874i
$969$		0			0
$970$	11.4839			0.368725
$971$		−	7.30486i		−	0.234424i	−0.993107	−	0.117212i	$-0.962604\pi$
$971$		−	7.30486i		−	0.234424i	0.993107	−	0.117212i	$-0.0373957\pi$
$972$		0			0
$973$	−36.4452			−1.16838
$974$			0.681763i			0.0218451i
$975$		0			0
$976$			29.4293i			0.942010i
$977$	13.5367			0.433077			0.216539	−	0.976274i	$-0.430523\pi$
$977$	13.5367			0.433077			0.216539	+	0.976274i	$0.430523\pi$
$978$		0			0
$979$	28.7361			0.918409
$980$	−0.258306			−0.00825129
$981$		0			0
$982$	45.6557			1.45693
$983$			17.1481i			0.546940i	0.961881	+	0.273470i	$0.0881714\pi$
$983$			17.1481i			0.546940i	−0.961881	+	0.273470i	$0.911829\pi$
$984$		0			0
$985$	−25.5128			−0.812906
$986$	−29.8995	−	20.7260i	−0.952193	−	0.660050i
$987$		0			0
$988$		−	34.7549i		−	1.10570i
$989$		−	8.80290i		−	0.279916i
$990$		0			0
$991$	62.2864			1.97859			0.989296	−	0.145923i	$-0.0466153\pi$
$991$	62.2864			1.97859			0.989296	+	0.145923i	$0.0466153\pi$
$992$	−37.9100			−1.20365
$993$		0			0
$994$		−	14.5407i		−	0.461202i
$995$	1.36177			0.0431711
$996$		0			0
$997$		−	17.0038i		−	0.538516i	−0.963068	−	0.269258i	$-0.913222\pi$
$997$		−	17.0038i		−	0.538516i	0.963068	−	0.269258i	$-0.0867785\pi$
$998$			36.0249i			1.14035i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.d.c.811.3		10
3.2	odd	2		435.2.d.b.376.8	yes	10
15.2	even	4		2175.2.f.c.724.4		10
15.8	even	4		2175.2.f.f.724.7		10
15.14	odd	2		2175.2.d.g.376.3		10
29.28	even	2	inner	1305.2.d.c.811.8		10
87.86	odd	2		435.2.d.b.376.3	✓	10
435.173	even	4		2175.2.f.c.724.3		10
435.347	even	4		2175.2.f.f.724.8		10
435.434	odd	2		2175.2.d.g.376.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.d.b.376.3	✓	10	87.86	odd	2
435.2.d.b.376.8	yes	10	3.2	odd	2
1305.2.d.c.811.3		10	1.1	even	1	trivial
1305.2.d.c.811.8		10	29.28	even	2	inner
2175.2.d.g.376.3		10	15.14	odd	2
2175.2.d.g.376.8		10	435.434	odd	2
2175.2.f.c.724.3		10	435.173	even	4
2175.2.f.c.724.4		10	15.2	even	4
2175.2.f.f.724.7		10	15.8	even	4
2175.2.f.f.724.8		10	435.347	even	4

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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.78296i q^{2} -1.17894 q^{4} -1.00000 q^{5} +2.60402 q^{7} -1.46391i q^{8} +1.78296i q^{10} +6.61083i q^{11} +4.14692 q^{13} -4.64285i q^{14} -4.96798 q^{16} +3.78904i q^{17} +7.10882i q^{19} +1.17894 q^{20} +11.7868 q^{22} +2.46391 q^{23} +1.00000 q^{25} -7.39379i q^{26} -3.06999 q^{28} +(-4.42581 - 3.06793i) q^{29} +6.39305i q^{31} +5.92988i q^{32} +6.75570 q^{34} -2.60402 q^{35} +4.64491i q^{37} +12.6747 q^{38} +1.46391i q^{40} -0.721109i q^{41} -3.57274i q^{43} -7.79380i q^{44} -4.39305i q^{46} +1.85308i q^{47} -0.219100 q^{49} -1.78296i q^{50} -4.88898 q^{52} -0.644913 q^{53} -6.61083i q^{55} -3.81205i q^{56} +(-5.46999 + 7.89104i) q^{58} +13.2217 q^{59} -5.92380i q^{61} +11.3986 q^{62} +0.636777 q^{64} -4.14692 q^{65} -5.45564 q^{67} -4.46706i q^{68} +4.64285i q^{70} +3.13184 q^{71} -15.7086i q^{73} +8.28169 q^{74} -8.38090i q^{76} +17.2147i q^{77} +1.18502i q^{79} +4.96798 q^{80} -1.28571 q^{82} +10.5236 q^{83} -3.78904i q^{85} -6.37004 q^{86} +9.67767 q^{88} -4.34682i q^{89} +10.7987 q^{91} -2.90481 q^{92} +3.30396 q^{94} -7.10882i q^{95} -6.44090i q^{97} +0.390646i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 14 q^{4} - 10 q^{5} + 8 q^{7} - 4 q^{13} - 2 q^{16} + 14 q^{20} - 26 q^{22} + 10 q^{23} + 10 q^{25} + 34 q^{28} - 16 q^{29} - 6 q^{34} - 8 q^{35} + 36 q^{38} + 14 q^{49} - 14 q^{52} + 38 q^{53} - 6 q^{58}+ \cdots + 14 q^{94}+O(q^{100}) \) 10 * q - 14 * q^4 - 10 * q^5 + 8 * q^7 - 4 * q^13 - 2 * q^16 + 14 * q^20 - 26 * q^22 + 10 * q^23 + 10 * q^25 + 34 * q^28 - 16 * q^29 - 6 * q^34 - 8 * q^35 + 36 * q^38 + 14 * q^49 - 14 * q^52 + 38 * q^53 - 6 * q^58 + 12 * q^59 + 28 * q^62 + 28 * q^64 + 4 * q^65 + 20 * q^67 - 32 * q^71 + 60 * q^74 + 2 * q^80 + 12 * q^82 - 2 * q^83 + 92 * q^88 - 24 * q^91 - 4 * q^92 + 14 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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