LMFDB - Embedded newform 1305.2.d.b.811.5

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 = [6,0,0,-6,-6] 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:	$6$
Coefficient field:	6.0.16516096.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 13x^{2} + 1 $ x^6 + 9x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			811.5
Root			$1.30397i$ of defining polynomial
Character	$\chi$	$=$	1305.811
Dual form			1305.2.d.b.811.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.30397i q^{2} +0.299664 q^{4} -1.00000 q^{5} -1.29966 q^{7} +2.99869i q^{8} -1.30397i q^{10} -0.537080i q^{11} -5.71155 q^{13} -1.69472i q^{14} -3.31087 q^{16} -6.91060i q^{17} -4.30266i q^{19} -0.299664 q^{20} +0.700336 q^{22} -5.01121 q^{23} +1.00000 q^{25} -7.44768i q^{26} -0.389463 q^{28} +(-0.299664 - 5.37682i) q^{29} +8.44438i q^{31} +1.68011i q^{32} +9.01121 q^{34} +1.29966 q^{35} +7.98476i q^{37} +5.61054 q^{38} -2.99869i q^{40} -0.698024i q^{41} -10.2166i q^{43} -0.160944i q^{44} -6.53446i q^{46} -0.537080i q^{47} -5.31087 q^{49} +1.30397i q^{50} -1.71155 q^{52} -7.11222 q^{53} +0.537080i q^{55} -3.89729i q^{56} +(7.01121 - 0.390753i) q^{58} +7.71155 q^{59} -5.91390i q^{61} -11.0112 q^{62} -8.81255 q^{64} +5.71155 q^{65} -9.01121 q^{67} -2.07086i q^{68} +1.69472i q^{70} +4.28845 q^{71} -6.21258i q^{73} -10.4119 q^{74} -1.28935i q^{76} +0.698024i q^{77} -10.2166i q^{79} +3.31087 q^{80} +0.910201 q^{82} +2.70034 q^{83} +6.91060i q^{85} +13.3221 q^{86} +1.61054 q^{88} +9.67948i q^{89} +7.42309 q^{91} -1.50168 q^{92} +0.700336 q^{94} +4.30266i q^{95} +2.07086i q^{97} -6.92522i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 6 q^{5} - 4 q^{13} + 26 q^{16} + 6 q^{20} + 12 q^{22} + 8 q^{23} + 6 q^{25} - 56 q^{28} + 6 q^{29} + 16 q^{34} - 20 q^{38} + 14 q^{49} + 20 q^{52} - 28 q^{53} + 4 q^{58} + 16 q^{59} - 28 q^{62}+ \cdots + 12 q^{94}+O(q^{100}) $ 6 * q - 6 * q^4 - 6 * q^5 - 4 * q^13 + 26 * q^16 + 6 * q^20 + 12 * q^22 + 8 * q^23 + 6 * q^25 - 56 * q^28 + 6 * q^29 + 16 * q^34 - 20 * q^38 + 14 * q^49 + 20 * q^52 - 28 * q^53 + 4 * q^58 + 16 * q^59 - 28 * q^62 - 46 * q^64 + 4 * q^65 - 16 * q^67 + 56 * q^71 - 40 * q^74 - 26 * q^80 - 56 * q^82 + 24 * q^83 - 4 * q^86 - 44 * q^88 - 16 * q^91 - 48 * q^92 + 12 * 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 $

$2$			1.30397i			0.922046i	0.887388	+	0.461023i	$0.152517\pi$
$2$			1.30397i			0.922046i	−0.887388	+	0.461023i	$0.847483\pi$
$3$		0			0
$3$		0			0
$4$	0.299664			0.149832
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−1.29966			−0.491227			−0.245613	−	0.969368i	$-0.578989\pi$
$7$	−1.29966			−0.491227			−0.245613	+	0.969368i	$0.578989\pi$
$8$			2.99869i			1.06020i
$9$		0			0
$10$		−	1.30397i		−	0.412351i
$11$		−	0.537080i		−	0.161936i	−0.996717	−	0.0809679i	$-0.974199\pi$
$11$		−	0.537080i		−	0.161936i	0.996717	−	0.0809679i	$-0.0258011\pi$
$12$		0			0
$13$	−5.71155			−1.58410			−0.792049	−	0.610458i	$-0.790985\pi$
$13$	−5.71155			−1.58410			−0.792049	+	0.610458i	$0.790985\pi$
$14$		−	1.69472i		−	0.452934i
$15$		0			0
$16$	−3.31087			−0.827718
$17$		−	6.91060i		−	1.67607i	−0.545619	−	0.838033i	$-0.683705\pi$
$17$		−	6.91060i		−	1.67607i	0.545619	−	0.838033i	$-0.316295\pi$
$18$		0			0
$19$		−	4.30266i		−	0.987098i	−0.869718	−	0.493549i	$-0.835699\pi$
$19$		−	4.30266i		−	0.987098i	0.869718	−	0.493549i	$-0.164301\pi$
$20$	−0.299664			−0.0670069
$21$		0			0
$22$	0.700336			0.149312
$23$	−5.01121			−1.04491			−0.522455	−	0.852667i	$-0.674984\pi$
$23$	−5.01121			−1.04491			−0.522455	+	0.852667i	$0.674984\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	7.44768i		−	1.46061i
$27$		0			0
$28$	−0.389463			−0.0736015
$29$	−0.299664	−	5.37682i	−0.0556462	−	0.998451i
$29$	−0.299664	−	5.37682i	−0.0556462	−	0.998451i
$30$		0			0
$31$			8.44438i			1.51666i	0.651874	+	0.758328i	$0.273983\pi$
$31$			8.44438i			1.51666i	−0.651874	+	0.758328i	$0.726017\pi$
$32$			1.68011i			0.297004i
$33$		0			0
$34$	9.01121			1.54541
$35$	1.29966			0.219683
$36$		0			0
$37$			7.98476i			1.31269i	0.754462	+	0.656343i	$0.227898\pi$
$37$			7.98476i			1.31269i	−0.754462	+	0.656343i	$0.772102\pi$
$38$	5.61054			0.910149
$39$		0			0
$40$		−	2.99869i		−	0.474135i
$41$		−	0.698024i		−	0.109013i	−0.998513	−	0.0545065i	$-0.982641\pi$
$41$		−	0.698024i		−	0.109013i	0.998513	−	0.0545065i	$-0.0173586\pi$
$42$		0			0
$43$		−	10.2166i		−	1.55801i	−0.627017	−	0.779006i	$-0.715724\pi$
$43$		−	10.2166i		−	1.55801i	0.627017	−	0.779006i	$-0.284276\pi$
$44$		−	0.160944i		−	0.0242632i
$45$		0			0
$46$		−	6.53446i		−	0.963454i
$47$		−	0.537080i		−	0.0783412i	−0.999233	−	0.0391706i	$-0.987528\pi$
$47$		−	0.537080i		−	0.0783412i	0.999233	−	0.0391706i	$-0.0124716\pi$
$48$		0			0
$49$	−5.31087			−0.758696
$50$			1.30397i			0.184409i
$51$		0			0
$52$	−1.71155			−0.237349
$53$	−7.11222			−0.976938			−0.488469	−	0.872581i	$-0.662444\pi$
$53$	−7.11222			−0.976938			−0.488469	+	0.872581i	$0.662444\pi$
$54$		0			0
$55$			0.537080i			0.0724199i
$56$		−	3.89729i		−	0.520797i
$57$		0			0
$58$	7.01121	−	0.390753i	0.920617	−	0.0513084i
$59$	7.71155			1.00396			0.501979	−	0.864880i	$-0.332606\pi$
$59$	7.71155			1.00396			0.501979	+	0.864880i	$0.332606\pi$
$60$		0			0
$61$		−	5.91390i		−	0.757197i	−0.925561	−	0.378599i	$-0.876406\pi$
$61$		−	5.91390i		−	0.757197i	0.925561	−	0.378599i	$-0.123594\pi$
$62$	−11.0112			−1.39842
$63$		0			0
$64$	−8.81255			−1.10157
$65$	5.71155			0.708430
$66$		0			0
$67$	−9.01121			−1.10089			−0.550447	−	0.834870i	$-0.685543\pi$
$67$	−9.01121			−1.10089			−0.550447	+	0.834870i	$0.685543\pi$
$68$		−	2.07086i		−	0.251128i
$69$		0			0
$70$			1.69472i			0.202558i
$71$	4.28845			0.508946			0.254473	−	0.967080i	$-0.418098\pi$
$71$	4.28845			0.508946			0.254473	+	0.967080i	$0.418098\pi$
$72$		0			0
$73$		−	6.21258i		−	0.727127i	−0.931569	−	0.363563i	$-0.881560\pi$
$73$		−	6.21258i		−	0.727127i	0.931569	−	0.363563i	$-0.118440\pi$
$74$	−10.4119			−1.21036
$75$		0			0
$76$		−	1.28935i		−	0.147899i
$77$			0.698024i			0.0795472i
$78$		0			0
$79$		−	10.2166i		−	1.14945i	−0.818346	−	0.574726i	$-0.805108\pi$
$79$		−	10.2166i		−	1.14945i	0.818346	−	0.574726i	$-0.194892\pi$
$80$	3.31087			0.370167
$81$		0			0
$82$	0.910201			0.100515
$83$	2.70034			0.296400			0.148200	−	0.988957i	$-0.452652\pi$
$83$	2.70034			0.296400			0.148200	+	0.988957i	$0.452652\pi$
$84$		0			0
$85$			6.91060i			0.749560i
$86$	13.3221			1.43656
$87$		0			0
$88$	1.61054			0.171684
$89$			9.67948i			1.02602i	0.858382	+	0.513011i	$0.171470\pi$
$89$			9.67948i			1.02602i	−0.858382	+	0.513011i	$0.828530\pi$
$90$		0			0
$91$	7.42309			0.778151
$92$	−1.50168			−0.156561
$93$		0			0
$94$	0.700336			0.0722341
$95$			4.30266i			0.441444i
$96$		0			0
$97$			2.07086i			0.210264i	0.994458	+	0.105132i	$0.0335265\pi$
$97$			2.07086i			0.210264i	−0.994458	+	0.105132i	$0.966474\pi$
$98$		−	6.92522i		−	0.699552i
$99$		0			0
$100$	0.299664			0.0299664
$101$		−	3.06756i		−	0.305233i	−0.988285	−	0.152617i	$-0.951230\pi$
$101$		−	3.06756i		−	0.305233i	0.988285	−	0.152617i	$-0.0487700\pi$
$102$		0			0
$103$	−0.187447			−0.0184697			−0.00923487	−	0.999957i	$-0.502940\pi$
$103$	−0.187447			−0.0184697			−0.00923487	+	0.999957i	$0.502940\pi$
$104$		−	17.1272i		−	1.67946i
$105$		0			0
$106$		−	9.27411i		−	0.900781i
$107$	−12.7228			−1.22996			−0.614978	−	0.788545i	$-0.710835\pi$
$107$	−12.7228			−1.22996			−0.614978	+	0.788545i	$0.710835\pi$
$108$		0			0
$109$	−11.1122			−1.06436			−0.532179	−	0.846632i	$-0.678627\pi$
$109$	−11.1122			−1.06436			−0.532179	+	0.846632i	$0.678627\pi$
$110$	−0.700336			−0.0667744
$111$		0			0
$112$	4.30302			0.406597
$113$		−	9.60202i		−	0.903282i	−0.892200	−	0.451641i	$-0.850839\pi$
$113$		−	9.60202i		−	0.903282i	0.892200	−	0.451641i	$-0.149161\pi$
$114$		0			0
$115$	5.01121			0.467298
$116$	−0.0897986	−	1.61124i	−0.00833759	−	0.149600i
$117$		0			0
$118$			10.0556i			0.925695i
$119$			8.98146i			0.823329i
$120$		0			0
$121$	10.7115			0.973777
$122$	7.71155			0.698170
$123$		0			0
$124$			2.53048i			0.227244i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			6.07484i			0.539055i	0.962993	+	0.269528i	$0.0868676\pi$
$127$			6.07484i			0.539055i	−0.962993	+	0.269528i	$0.913132\pi$
$128$		−	8.13109i		−	0.718693i
$129$		0			0
$130$			7.44768i			0.653205i
$131$			18.1239i			1.58349i	0.610852	+	0.791744i	$0.290827\pi$
$131$			18.1239i			1.58349i	−0.610852	+	0.791744i	$0.709173\pi$
$132$		0			0
$133$			5.59201i			0.484889i
$134$		−	11.7503i		−	1.01507i
$135$		0			0
$136$	20.7228			1.77696
$137$			13.2006i			1.12781i	0.825841	+	0.563903i	$0.190701\pi$
$137$			13.2006i			1.12781i	−0.825841	+	0.563903i	$0.809299\pi$
$138$		0			0
$139$	14.5993			1.23830			0.619149	−	0.785273i	$-0.287478\pi$
$139$	14.5993			1.23830			0.619149	+	0.785273i	$0.287478\pi$
$140$	0.389463			0.0329156
$141$		0			0
$142$			5.59201i			0.469271i
$143$			3.06756i			0.256522i
$144$		0			0
$145$	0.299664	+	5.37682i	0.0248857	+	0.446521i
$146$	8.10101			0.670444
$147$		0			0
$148$			2.39275i			0.196682i
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	−21.7340			−1.76868			−0.884342	−	0.466839i	$-0.845393\pi$
$151$	−21.7340			−1.76868			−0.884342	+	0.466839i	$0.845393\pi$
$152$	12.9024			1.04652
$153$		0			0
$154$	−0.910201			−0.0733461
$155$		−	8.44438i		−	0.678269i
$156$		0			0
$157$			5.29334i			0.422454i	0.977437	+	0.211227i	$0.0677460\pi$
$157$			5.29334i			0.422454i	−0.977437	+	0.211227i	$0.932254\pi$
$158$	13.3221			1.05985
$159$		0			0
$160$		−	1.68011i		−	0.132824i
$161$	6.51289			0.513288
$162$		0			0
$163$		−	6.07484i		−	0.475819i	−0.971287	−	0.237909i	$-0.923538\pi$
$163$		−	6.07484i		−	0.475819i	0.971287	−	0.237909i	$-0.0764621\pi$
$164$		−	0.209173i		−	0.0163336i
$165$		0			0
$166$			3.52116i			0.273295i
$167$	−5.01121			−0.387779			−0.193890	−	0.981023i	$-0.562110\pi$
$167$	−5.01121			−0.387779			−0.193890	+	0.981023i	$0.562110\pi$
$168$		0			0
$169$	19.6217			1.50937
$170$	−9.01121			−0.691128
$171$		0			0
$172$		−	3.06154i		−	0.233440i
$173$	8.31087			0.631864			0.315932	−	0.948782i	$-0.397683\pi$
$173$	8.31087			0.631864			0.315932	+	0.948782i	$0.397683\pi$
$174$		0			0
$175$	−1.29966			−0.0982454
$176$			1.77820i			0.134037i
$177$		0			0
$178$	−12.6217			−0.946040
$179$	−6.59933			−0.493257			−0.246628	−	0.969110i	$-0.579323\pi$
$179$	−6.59933			−0.493257			−0.246628	+	0.969110i	$0.579323\pi$
$180$		0			0
$181$	−3.48711			−0.259195			−0.129597	−	0.991567i	$-0.541369\pi$
$181$	−3.48711			−0.259195			−0.129597	+	0.991567i	$0.541369\pi$
$182$			9.67948i			0.717491i
$183$		0			0
$184$		−	15.0271i		−	1.10781i
$185$		−	7.98476i		−	0.587051i
$186$		0			0
$187$	−3.71155			−0.271415
$188$		−	0.160944i		−	0.0117380i
$189$		0			0
$190$	−5.61054			−0.407031
$191$			2.68540i			0.194309i	0.995269	+	0.0971544i	$0.0309741\pi$
$191$			2.68540i			0.194309i	−0.995269	+	0.0971544i	$0.969026\pi$
$192$		0			0
$193$			4.76228i			0.342796i	0.985202	+	0.171398i	$0.0548285\pi$
$193$			4.76228i			0.342796i	−0.985202	+	0.171398i	$0.945172\pi$
$194$	−2.70034			−0.193873
$195$		0			0
$196$	−1.59148			−0.113677
$197$	−16.0224			−1.14155			−0.570775	−	0.821106i	$-0.693357\pi$
$197$	−16.0224			−1.14155			−0.570775	+	0.821106i	$0.693357\pi$
$198$		0			0
$199$	−1.68913			−0.119739			−0.0598695	−	0.998206i	$-0.519068\pi$
$199$	−1.68913			−0.119739			−0.0598695	+	0.998206i	$0.519068\pi$
$200$			2.99869i			0.212039i
$201$		0			0
$202$	4.00000			0.281439
$203$	0.389463	+	6.98806i	0.0273349	+	0.490466i
$204$		0			0
$205$			0.698024i			0.0487521i
$206$		−	0.244426i		−	0.0170299i
$207$		0			0
$208$	18.9102			1.31119
$209$	−2.31087			−0.159846
$210$		0			0
$211$		−	6.99408i		−	0.481492i	−0.970588	−	0.240746i	$-0.922608\pi$
$211$		−	6.99408i		−	0.481492i	0.970588	−	0.240746i	$-0.0773921\pi$
$212$	−2.13128			−0.146377
$213$		0			0
$214$		−	16.5901i		−	1.13407i
$215$			10.2166i			0.696764i
$216$		0			0
$217$		−	10.9749i		−	0.745022i
$218$		−	14.4900i		−	0.981386i
$219$		0			0
$220$			0.160944i			0.0108508i
$221$			39.4702i			2.65505i
$222$		0			0
$223$	2.98879			0.200144			0.100072	−	0.994980i	$-0.468093\pi$
$223$	2.98879			0.200144			0.100072	+	0.994980i	$0.468093\pi$
$224$		−	2.18357i		−	0.145896i
$225$		0			0
$226$	12.5207			0.832867
$227$	23.8574			1.58347			0.791735	−	0.610864i	$-0.209178\pi$
$227$	23.8574			1.58347			0.791735	+	0.610864i	$0.209178\pi$
$228$		0			0
$229$			0.321887i			0.0212709i	0.999943	+	0.0106355i	$0.00338543\pi$
$229$			0.321887i			0.0212709i	−0.999943	+	0.0106355i	$0.996615\pi$
$230$			6.53446i			0.430870i
$231$		0			0
$232$	16.1234	−	0.898600i	1.05855	−	0.0589960i
$233$	16.0224			1.04966			0.524832	−	0.851206i	$-0.324128\pi$
$233$	16.0224			1.04966			0.524832	+	0.851206i	$0.324128\pi$
$234$		0			0
$235$			0.537080i			0.0350352i
$236$	2.31087			0.150425
$237$		0			0
$238$	−11.7115			−0.759147
$239$	−19.7115			−1.27503			−0.637517	−	0.770436i	$-0.720038\pi$
$239$	−19.7115			−1.27503			−0.637517	+	0.770436i	$0.720038\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$			13.9675i			0.897867i
$243$		0			0
$244$		−	1.77218i		−	0.113452i
$245$	5.31087			0.339299
$246$		0			0
$247$			24.5748i			1.56366i
$248$	−25.3221			−1.60795
$249$		0			0
$250$		−	1.30397i		−	0.0824703i
$251$		−	24.7900i		−	1.56473i	−0.622818	−	0.782366i	$-0.714012\pi$
$251$		−	24.7900i		−	1.56473i	0.622818	−	0.782366i	$-0.285988\pi$
$252$		0			0
$253$			2.69142i			0.169208i
$254$	−7.92141			−0.497034
$255$		0			0
$256$	−7.02242			−0.438901
$257$	−16.8878			−1.05343			−0.526715	−	0.850042i	$-0.676577\pi$
$257$	−16.8878			−1.05343			−0.526715	+	0.850042i	$0.676577\pi$
$258$		0			0
$259$		−	10.3775i		−	0.644827i
$260$	1.71155			0.106146
$261$		0			0
$262$	−23.6330			−1.46005
$263$			7.74635i			0.477661i	0.971061	+	0.238830i	$0.0767640\pi$
$263$			7.74635i			0.477661i	−0.971061	+	0.238830i	$0.923236\pi$
$264$		0			0
$265$	7.11222			0.436900
$266$	−7.29181			−0.447090
$267$		0			0
$268$	−2.70034			−0.164949
$269$			11.4517i			0.698220i	0.937082	+	0.349110i	$0.113516\pi$
$269$			11.4517i			0.698220i	−0.937082	+	0.349110i	$0.886484\pi$
$270$		0			0
$271$			25.8642i			1.57114i	0.618774	+	0.785569i	$0.287630\pi$
$271$			25.8642i			1.57114i	−0.618774	+	0.785569i	$0.712370\pi$
$272$			22.8801i			1.38731i
$273$		0			0
$274$	−17.2132			−1.03989
$275$		−	0.537080i		−	0.0323871i
$276$		0			0
$277$	−8.88778			−0.534015			−0.267008	−	0.963694i	$-0.586035\pi$
$277$	−8.88778			−0.534015			−0.267008	+	0.963694i	$0.586035\pi$
$278$			19.0371i			1.14177i
$279$		0			0
$280$			3.89729i			0.232908i
$281$	22.5353			1.34434			0.672172	−	0.740395i	$-0.265362\pi$
$281$	22.5353			1.34434			0.672172	+	0.740395i	$0.265362\pi$
$282$		0			0
$283$	−14.4119			−0.856697			−0.428349	−	0.903614i	$-0.640904\pi$
$283$	−14.4119			−0.856697			−0.428349	+	0.903614i	$0.640904\pi$
$284$	1.28510			0.0762564
$285$		0			0
$286$	−4.00000			−0.236525
$287$			0.907196i			0.0535501i
$288$		0			0
$289$	−30.7564			−1.80920
$290$	−7.01121	+	0.390753i	−0.411712	+	0.0229458i
$291$		0			0
$292$		−	1.86169i		−	0.108947i
$293$		−	0.996698i		−	0.0582277i	−0.999576	−	0.0291139i	$-0.990731\pi$
$293$		−	0.996698i		−	0.0582277i	0.999576	−	0.0291139i	$-0.00926854\pi$
$294$		0			0
$295$	−7.71155			−0.448984
$296$	−23.9438			−1.39171
$297$		0			0
$298$		−	7.82382i		−	0.453221i
$299$	28.6217			1.65524
$300$		0			0
$301$			13.2781i			0.765337i
$302$		−	28.3404i		−	1.63081i
$303$		0			0
$304$			14.2456i			0.817039i
$305$			5.91390i			0.338629i
$306$		0			0
$307$			27.1053i			1.54698i	0.633807	+	0.773491i	$0.281491\pi$
$307$			27.1053i			1.54698i	−0.633807	+	0.773491i	$0.718509\pi$
$308$			0.209173i			0.0119187i
$309$		0			0
$310$	11.0112			0.625395
$311$		−	18.8761i		−	1.07037i	−0.844736	−	0.535184i	$-0.820242\pi$
$311$		−	18.8761i		−	1.07037i	0.844736	−	0.535184i	$-0.179758\pi$
$312$		0			0
$313$	6.28845			0.355444			0.177722	−	0.984081i	$-0.443127\pi$
$313$	6.28845			0.355444			0.177722	+	0.984081i	$0.443127\pi$
$314$	−6.90235			−0.389522
$315$		0			0
$316$		−	3.06154i		−	0.172225i
$317$			13.5225i			0.759501i	0.925089	+	0.379750i	$0.123990\pi$
$317$			13.5225i			0.759501i	−0.925089	+	0.379750i	$0.876010\pi$
$318$		0			0
$319$	−2.88778	+	0.160944i	−0.161685	+	0.00901111i
$320$	8.81255			0.492637
$321$		0			0
$322$			8.49261i			0.473274i
$323$	−29.7340			−1.65444
$324$		0			0
$325$	−5.71155			−0.316820
$326$	7.92141			0.438726
$327$		0			0
$328$	2.09316			0.115575
$329$			0.698024i			0.0384833i
$330$		0			0
$331$		−	20.8695i		−	1.14709i	−0.819173	−	0.573547i	$-0.805567\pi$
$331$		−	20.8695i		−	1.14709i	0.819173	−	0.573547i	$-0.194433\pi$
$332$	0.809194			0.0444103
$333$		0			0
$334$		−	6.53446i		−	0.357550i
$335$	9.01121			0.492335
$336$		0			0
$337$			31.1093i			1.69463i	0.531089	+	0.847316i	$0.321783\pi$
$337$			31.1093i			1.69463i	−0.531089	+	0.847316i	$0.678217\pi$
$338$			25.5862i			1.39170i
$339$		0			0
$340$			2.07086i			0.112308i
$341$	4.53531			0.245601
$342$		0			0
$343$	16.0000			0.863919
$344$	30.6363			1.65180
$345$		0			0
$346$			10.8371i			0.582607i
$347$	−32.7676			−1.75906			−0.879528	−	0.475847i	$-0.842142\pi$
$347$	−32.7676			−1.75906			−0.879528	+	0.475847i	$0.842142\pi$
$348$		0			0
$349$	24.0224			1.28589			0.642945	−	0.765912i	$-0.277712\pi$
$349$	24.0224			1.28589			0.642945	+	0.765912i	$0.277712\pi$
$350$		−	1.69472i		−	0.0905867i
$351$		0			0
$352$	0.902351			0.0480955
$353$	−28.3557			−1.50922			−0.754611	−	0.656172i	$-0.772174\pi$
$353$	−28.3557			−1.50922			−0.754611	+	0.656172i	$0.772174\pi$
$354$		0			0
$355$	−4.28845			−0.227608
$356$			2.90059i			0.153731i
$357$		0			0
$358$		−	8.60532i		−	0.454805i
$359$		−	30.7039i		−	1.62049i	−0.586090	−	0.810246i	$-0.699334\pi$
$359$		−	30.7039i		−	1.62049i	0.586090	−	0.810246i	$-0.300666\pi$
$360$		0			0
$361$	0.487111			0.0256374
$362$		−	4.54709i		−	0.238990i
$363$		0			0
$364$	2.22443			0.116592
$365$			6.21258i			0.325181i
$366$		0			0
$367$			1.93313i			0.100908i	0.998726	+	0.0504542i	$0.0160669\pi$
$367$			1.93313i			0.100908i	−0.998726	+	0.0504542i	$0.983933\pi$
$368$	16.5915			0.864891
$369$		0			0
$370$	10.4119			0.541288
$371$	9.24349			0.479898
$372$		0			0
$373$	0.801344			0.0414920			0.0207460	−	0.999785i	$-0.493396\pi$
$373$	0.801344			0.0414920			0.0207460	+	0.999785i	$0.493396\pi$
$374$		−	4.83974i		−	0.250257i
$375$		0			0
$376$	1.61054			0.0830571
$377$	1.71155	+	30.7100i	0.0881491	+	1.58164i
$378$		0			0
$379$		−	25.8642i		−	1.32855i	−0.747486	−	0.664277i	$-0.768739\pi$
$379$		−	25.8642i		−	1.32855i	0.747486	−	0.664277i	$-0.231261\pi$
$380$			1.28935i			0.0661424i
$381$		0			0
$382$	−3.50168			−0.179162
$383$	20.7676			1.06117			0.530587	−	0.847630i	$-0.321971\pi$
$383$	20.7676			1.06117			0.530587	+	0.847630i	$0.321971\pi$
$384$		0			0
$385$		−	0.698024i		−	0.0355746i
$386$	−6.20987			−0.316074
$387$		0			0
$388$			0.620562i			0.0315043i
$389$			2.52446i			0.127995i	0.997950	+	0.0639975i	$0.0203850\pi$
$389$			2.52446i			0.127995i	−0.997950	+	0.0639975i	$0.979615\pi$
$390$		0			0
$391$			34.6305i			1.75134i
$392$		−	15.9257i		−	0.804368i
$393$		0			0
$394$		−	20.8927i		−	1.05256i
$395$			10.2166i			0.514051i
$396$		0			0
$397$	13.2211			0.663547			0.331773	−	0.943359i	$-0.392353\pi$
$397$	13.2211			0.663547			0.331773	+	0.943359i	$0.392353\pi$
$398$		−	2.20257i		−	0.110405i
$399$		0			0
$400$	−3.31087			−0.165544
$401$	−25.1795			−1.25740			−0.628701	−	0.777647i	$-0.716413\pi$
$401$	−25.1795			−1.25740			−0.628701	+	0.777647i	$0.716413\pi$
$402$		0			0
$403$		−	48.2304i		−	2.40253i
$404$		−	0.919237i		−	0.0457337i
$405$		0			0
$406$	−9.11222	+	0.507847i	−0.452232	+	0.0252040i
$407$	4.28845			0.212571
$408$		0			0
$409$			28.1855i			1.39368i	0.717225	+	0.696842i	$0.245412\pi$
$409$			28.1855i			1.39368i	−0.717225	+	0.696842i	$0.754588\pi$
$410$	−0.910201			−0.0449516
$411$		0			0
$412$	−0.0561712			−0.00276736
$413$	−10.0224			−0.493171
$414$		0			0
$415$	−2.70034			−0.132554
$416$		−	9.59600i		−	0.470483i
$417$		0			0
$418$		−	3.01331i		−	0.147386i
$419$	20.9102			1.02153			0.510765	−	0.859720i	$-0.329362\pi$
$419$	20.9102			1.02153			0.510765	+	0.859720i	$0.329362\pi$
$420$		0			0
$421$		−	4.67278i		−	0.227737i	−0.993496	−	0.113869i	$-0.963676\pi$
$421$		−	4.67278i		−	0.227737i	0.993496	−	0.113869i	$-0.0363243\pi$
$422$	9.12007			0.443958
$423$		0			0
$424$		−	21.3273i		−	1.03575i
$425$		−	6.91060i		−	0.335213i
$426$		0			0
$427$			7.68608i			0.371956i
$428$	−3.81255			−0.184287
$429$		0			0
$430$	−13.3221			−0.642448
$431$	−12.2469			−0.589910			−0.294955	−	0.955511i	$-0.595305\pi$
$431$	−12.2469			−0.589910			−0.294955	+	0.955511i	$0.595305\pi$
$432$		0			0
$433$		−	11.8046i		−	0.567292i	−0.958929	−	0.283646i	$-0.908456\pi$
$433$		−	11.8046i		−	0.567292i	0.958929	−	0.283646i	$-0.0915441\pi$
$434$	14.3109			0.686944
$435$		0			0
$436$	−3.32993			−0.159475
$437$			21.5615i			1.03143i
$438$		0			0
$439$	−29.5319			−1.40948			−0.704741	−	0.709464i	$-0.748937\pi$
$439$	−29.5319			−1.40948			−0.704741	+	0.709464i	$0.748937\pi$
$440$	−1.61054			−0.0767794
$441$		0			0
$442$	−51.4679			−2.44808
$443$			3.28275i			0.155968i	0.996955	+	0.0779841i	$0.0248483\pi$
$443$			3.28275i			0.155968i	−0.996955	+	0.0779841i	$0.975152\pi$
$444$		0			0
$445$		−	9.67948i		−	0.458851i
$446$			3.89729i			0.184542i
$447$		0			0
$448$	11.4534			0.541120
$449$			28.9378i			1.36566i	0.730578	+	0.682829i	$0.239251\pi$
$449$			28.9378i			1.36566i	−0.730578	+	0.682829i	$0.760749\pi$
$450$		0			0
$451$	−0.374895			−0.0176531
$452$		−	2.87738i		−	0.135341i
$453$		0			0
$454$			31.1093i			1.46003i
$455$	−7.42309			−0.348000
$456$		0			0
$457$	−14.5353			−0.679933			−0.339966	−	0.940438i	$-0.610416\pi$
$457$	−14.5353			−0.679933			−0.339966	+	0.940438i	$0.610416\pi$
$458$	−0.419731			−0.0196127
$459$		0			0
$460$	1.50168			0.0700162
$461$			22.4265i			1.04451i	0.852790	+	0.522254i	$0.174909\pi$
$461$			22.4265i			1.04451i	−0.852790	+	0.522254i	$0.825091\pi$
$462$		0			0
$463$	−10.0370			−0.466458			−0.233229	−	0.972422i	$-0.574929\pi$
$463$	−10.0370			−0.466458			−0.233229	+	0.972422i	$0.574929\pi$
$464$	0.992150	+	17.8020i	0.0460594	+	0.826436i
$465$		0			0
$466$			20.8927i			0.967838i
$467$		−	4.35691i		−	0.201614i	−0.994906	−	0.100807i	$-0.967858\pi$
$467$		−	4.35691i		−	0.201614i	0.994906	−	0.100807i	$-0.0321424\pi$
$468$		0			0
$469$	11.7115			0.540789
$470$	−0.700336			−0.0323041
$471$		0			0
$472$			23.1245i			1.06439i
$473$	−5.48711			−0.252298
$474$		0			0
$475$		−	4.30266i		−	0.197420i
$476$			2.69142i			0.123361i
$477$		0			0
$478$		−	25.7032i		−	1.17564i
$479$		−	28.0125i		−	1.27992i	−0.768406	−	0.639962i	$-0.778950\pi$
$479$		−	28.0125i		−	1.27992i	0.768406	−	0.639962i	$-0.221050\pi$
$480$		0			0
$481$		−	45.6053i		−	2.07942i
$482$			2.60794i			0.118788i
$483$		0			0
$484$	3.20987			0.145903
$485$		−	2.07086i		−	0.0940328i
$486$		0			0
$487$	19.5241			0.884721			0.442361	−	0.896837i	$-0.354141\pi$
$487$	19.5241			0.884721			0.442361	+	0.896837i	$0.354141\pi$
$488$	17.7340			0.802779
$489$		0			0
$490$			6.92522i			0.312849i
$491$		−	36.4087i		−	1.64310i	−0.570137	−	0.821550i	$-0.693110\pi$
$491$		−	36.4087i		−	1.64310i	0.570137	−	0.821550i	$-0.306890\pi$
$492$		0			0
$493$	−37.1571	+	2.07086i	−1.67347	+	0.0932668i
$494$	−32.0448			−1.44177
$495$		0			0
$496$		−	27.9583i		−	1.25536i
$497$	−5.57355			−0.250008
$498$		0			0
$499$	31.4679			1.40870			0.704349	−	0.709854i	$-0.251239\pi$
$499$	31.4679			1.40870			0.704349	+	0.709854i	$0.251239\pi$
$500$	−0.299664			−0.0134014
$501$		0			0
$502$	32.3254			1.44276
$503$		−	8.82051i		−	0.393287i	−0.980475	−	0.196644i	$-0.936996\pi$
$503$		−	8.82051i		−	0.393287i	0.980475	−	0.196644i	$-0.0630042\pi$
$504$		0			0
$505$			3.06756i			0.136504i
$506$	−3.50953			−0.156018
$507$		0			0
$508$			1.82041i			0.0807678i
$509$	−14.9102			−0.660883			−0.330442	−	0.943826i	$-0.607198\pi$
$509$	−14.9102			−0.660883			−0.330442	+	0.943826i	$0.607198\pi$
$510$		0			0
$511$			8.07426i			0.357184i
$512$		−	25.4192i		−	1.12338i
$513$		0			0
$514$		−	22.0212i		−	0.971311i
$515$	0.187447			0.00825991
$516$		0			0
$517$	−0.288455			−0.0126862
$518$	13.5319			0.594560
$519$		0			0
$520$			17.1272i			0.751076i
$521$	−0.512889			−0.0224701			−0.0112350	−	0.999937i	$-0.503576\pi$
$521$	−0.512889			−0.0224701			−0.0112350	+	0.999937i	$0.503576\pi$
$522$		0			0
$523$	32.8092			1.43465			0.717323	−	0.696741i	$-0.245367\pi$
$523$	32.8092			1.43465			0.717323	+	0.696741i	$0.245367\pi$
$524$			5.43107i			0.237257i
$525$		0			0
$526$	−10.1010			−0.440425
$527$	58.3557			2.54201
$528$		0			0
$529$	2.11222			0.0918355
$530$			9.27411i			0.402842i
$531$		0			0
$532$			1.67573i			0.0726519i
$533$			3.98679i			0.172687i
$534$		0			0
$535$	12.7228			0.550053
$536$		−	27.0218i		−	1.16717i
$537$		0			0
$538$	−14.9326			−0.643791
$539$			2.85236i			0.122860i
$540$		0			0
$541$		−	25.8160i		−	1.10991i	−0.831879	−	0.554957i	$-0.812735\pi$
$541$		−	25.8160i		−	1.10991i	0.831879	−	0.554957i	$-0.187265\pi$
$542$	−33.7261			−1.44866
$543$		0			0
$544$	11.6105			0.497798
$545$	11.1122			0.475995
$546$		0			0
$547$	39.3221			1.68129			0.840645	−	0.541586i	$-0.182176\pi$
$547$	39.3221			1.68129			0.840645	+	0.541586i	$0.182176\pi$
$548$			3.95576i			0.168982i
$549$		0			0
$550$	0.700336			0.0298624
$551$	−23.1346	+	1.28935i	−0.985569	+	0.0549283i
$552$		0			0
$553$			13.2781i			0.564642i
$554$		−	11.5894i		−	0.492386i
$555$		0			0
$556$	4.37489			0.185537
$557$	30.2469			1.28160			0.640800	−	0.767708i	$-0.278603\pi$
$557$	30.2469			1.28160			0.640800	+	0.767708i	$0.278603\pi$
$558$		0			0
$559$			58.3524i			2.46804i
$560$	−4.30302			−0.181836
$561$		0			0
$562$			29.3853i			1.23955i
$563$		−	18.0696i		−	0.761543i	−0.924669	−	0.380772i	$-0.875658\pi$
$563$		−	18.0696i		−	0.761543i	0.924669	−	0.380772i	$-0.124342\pi$
$564$		0			0
$565$			9.60202i			0.403960i
$566$		−	18.7927i		−	0.789914i
$567$		0			0
$568$			12.8598i			0.539583i
$569$		−	0.0542492i		−	0.00227425i	−0.999999	−	0.00113712i	$-0.999638\pi$
$569$		−	0.0542492i		−	0.00227425i	0.999999	−	0.00113712i	$-0.000361957\pi$
$570$		0			0
$571$	−24.8238			−1.03884			−0.519421	−	0.854518i	$-0.673852\pi$
$571$	−24.8238			−1.03884			−0.519421	+	0.854518i	$0.673852\pi$
$572$			0.919237i			0.0384352i
$573$		0			0
$574$	−1.18296			−0.0493756
$575$	−5.01121			−0.208982
$576$		0			0
$577$		−	33.8007i		−	1.40714i	−0.710625	−	0.703571i	$-0.751588\pi$
$577$		−	33.8007i		−	1.40714i	0.710625	−	0.703571i	$-0.248412\pi$
$578$		−	40.1054i		−	1.66816i
$579$		0			0
$580$	0.0897986	+	1.61124i	0.00372868	+	0.0669031i
$581$	−3.50953			−0.145600
$582$		0			0
$583$			3.81983i			0.158201i
$584$	18.6296			0.770898
$585$		0			0
$586$	1.29966			0.0536886
$587$	16.2323			0.669978			0.334989	−	0.942222i	$-0.391267\pi$
$587$	16.2323			0.669978			0.334989	+	0.942222i	$0.391267\pi$
$588$		0			0
$589$	36.3333			1.49709
$590$		−	10.0556i		−	0.413983i
$591$		0			0
$592$		−	26.4365i		−	1.08653i
$593$	−9.08980			−0.373273			−0.186637	−	0.982429i	$-0.559759\pi$
$593$	−9.08980			−0.373273			−0.186637	+	0.982429i	$0.559759\pi$
$594$		0			0
$595$		−	8.98146i		−	0.368204i
$596$	−1.79798			−0.0736483
$597$		0			0
$598$			37.3219i			1.52621i
$599$		−	13.2841i		−	0.542774i	−0.962470	−	0.271387i	$-0.912518\pi$
$599$		−	13.2841i		−	0.542774i	0.962470	−	0.271387i	$-0.0874824\pi$
$600$		0			0
$601$			18.6609i			0.761196i	0.924741	+	0.380598i	$0.124282\pi$
$601$			18.6609i			0.761196i	−0.924741	+	0.380598i	$0.875718\pi$
$602$	−17.3142			−0.705675
$603$		0			0
$604$	−6.51289			−0.265006
$605$	−10.7115			−0.435486
$606$		0			0
$607$		−	35.3888i		−	1.43639i	−0.695844	−	0.718193i	$-0.744970\pi$
$607$		−	35.3888i		−	1.43639i	0.695844	−	0.718193i	$-0.255030\pi$
$608$	7.22892			0.293172
$609$		0			0
$610$	−7.71155			−0.312231
$611$			3.06756i			0.124100i
$612$		0			0
$613$	−7.77557			−0.314052			−0.157026	−	0.987594i	$-0.550191\pi$
$613$	−7.77557			−0.314052			−0.157026	+	0.987594i	$0.550191\pi$
$614$	−35.3445			−1.42639
$615$		0			0
$616$	−2.09316			−0.0843357
$617$		−	1.31859i		−	0.0530843i	−0.999648	−	0.0265421i	$-0.991550\pi$
$617$		−	1.31859i		−	0.0530843i	0.999648	−	0.0265421i	$-0.00844961\pi$
$618$		0			0
$619$		−	49.5198i		−	1.99037i	−0.0980212	−	0.995184i	$-0.531251\pi$
$619$		−	49.5198i		−	1.99037i	0.0980212	−	0.995184i	$-0.468749\pi$
$620$		−	2.53048i		−	0.101626i
$621$		0			0
$622$	24.6139			0.986927
$623$		−	12.5801i		−	0.504010i
$624$		0			0
$625$	1.00000			0.0400000
$626$			8.19995i			0.327736i
$627$		0			0
$628$			1.58622i			0.0632972i
$629$	55.1795			2.20015
$630$		0			0
$631$	10.2244			0.407028			0.203514	−	0.979072i	$-0.434764\pi$
$631$	10.2244			0.407028			0.203514	+	0.979072i	$0.434764\pi$
$632$	30.6363			1.21865
$633$		0			0
$634$	−17.6330			−0.700294
$635$		−	6.07484i		−	0.241073i
$636$		0			0
$637$	30.3333			1.20185
$638$	−0.209866	−	3.76558i	−0.00830866	−	0.149081i
$639$		0			0
$640$			8.13109i			0.321409i
$641$			31.8848i			1.25937i	0.776849	+	0.629687i	$0.216817\pi$
$641$			31.8848i			1.25937i	−0.776849	+	0.629687i	$0.783183\pi$
$642$		0			0
$643$	−29.8350			−1.17658			−0.588288	−	0.808651i	$-0.700198\pi$
$643$	−29.8350			−1.17658			−0.588288	+	0.808651i	$0.700198\pi$
$644$	1.95168			0.0769069
$645$		0			0
$646$		−	38.7722i		−	1.52547i
$647$	23.9438			0.941329			0.470665	−	0.882312i	$-0.344014\pi$
$647$	23.9438			0.941329			0.470665	+	0.882312i	$0.344014\pi$
$648$		0			0
$649$		−	4.14172i		−	0.162577i
$650$		−	7.44768i		−	0.292122i
$651$		0			0
$652$		−	1.82041i		−	0.0712929i
$653$		−	34.1769i		−	1.33744i	−0.743513	−	0.668722i	$-0.766842\pi$
$653$		−	34.1769i		−	1.33744i	0.743513	−	0.668722i	$-0.233158\pi$
$654$		0			0
$655$		−	18.1239i		−	0.708158i
$656$			2.31107i			0.0902320i
$657$		0			0
$658$	−0.910201			−0.0354833
$659$		−	9.67346i		−	0.376825i	−0.982090	−	0.188412i	$-0.939666\pi$
$659$		−	9.67346i		−	0.376825i	0.982090	−	0.188412i	$-0.0603341\pi$
$660$		0			0
$661$	27.3591			1.06414			0.532072	−	0.846699i	$-0.321413\pi$
$661$	27.3591			1.06414			0.532072	+	0.846699i	$0.321413\pi$
$662$	27.2132			1.05767
$663$		0			0
$664$			8.09747i			0.314243i
$665$		−	5.59201i		−	0.216849i
$666$		0			0
$667$	1.50168	+	26.9444i	0.0581453	+	1.04329i
$668$	−1.50168			−0.0581017
$669$		0			0
$670$			11.7503i			0.453955i
$671$	−3.17624			−0.122617
$672$		0			0
$673$	32.0673			1.23610			0.618051	−	0.786138i	$-0.287923\pi$
$673$	32.0673			1.23610			0.618051	+	0.786138i	$0.287923\pi$
$674$	−40.5656			−1.56253
$675$		0			0
$676$	5.87993			0.226151
$677$		−	21.7053i		−	0.834202i	−0.908860	−	0.417101i	$-0.863046\pi$
$677$		−	21.7053i		−	0.834202i	0.908860	−	0.417101i	$-0.136954\pi$
$678$		0			0
$679$		−	2.69142i		−	0.103287i
$680$	−20.7228			−0.794681
$681$		0			0
$682$			5.91390i			0.226455i
$683$	8.43430			0.322729			0.161365	−	0.986895i	$-0.448410\pi$
$683$	8.43430			0.322729			0.161365	+	0.986895i	$0.448410\pi$
$684$		0			0
$685$		−	13.2006i		−	0.504370i
$686$			20.8635i			0.796572i
$687$		0			0
$688$			33.8257i			1.28959i
$689$	40.6217			1.54757
$690$		0			0
$691$	−12.7373			−0.484551			−0.242275	−	0.970208i	$-0.577894\pi$
$691$	−12.7373			−0.484551			−0.242275	+	0.970208i	$0.577894\pi$
$692$	2.49047			0.0946735
$693$		0			0
$694$		−	42.7279i		−	1.62193i
$695$	−14.5993			−0.553784
$696$		0			0
$697$	−4.82376			−0.182713
$698$			31.3245i			1.18565i
$699$		0			0
$700$	−0.389463			−0.0147203
$701$	31.5319			1.19095			0.595473	−	0.803376i	$-0.296965\pi$
$701$	31.5319			1.19095			0.595473	+	0.803376i	$0.296965\pi$
$702$		0			0
$703$	34.3557			1.29575
$704$			4.73305i			0.178383i
$705$		0			0
$706$		−	36.9750i		−	1.39157i
$707$			3.98679i			0.149939i
$708$		0			0
$709$	−5.71155			−0.214502			−0.107251	−	0.994232i	$-0.534205\pi$
$709$	−5.71155			−0.214502			−0.107251	+	0.994232i	$0.534205\pi$
$710$		−	5.59201i		−	0.209865i
$711$		0			0
$712$	−29.0258			−1.08779
$713$		−	42.3165i		−	1.58477i
$714$		0			0
$715$		−	3.06756i		−	0.114720i
$716$	−1.97758			−0.0739057
$717$		0			0
$718$	40.0370			1.49417
$719$	−3.33665			−0.124436			−0.0622180	−	0.998063i	$-0.519817\pi$
$719$	−3.33665			−0.124436			−0.0622180	+	0.998063i	$0.519817\pi$
$720$		0			0
$721$	0.243619			0.00907283
$722$			0.635178i			0.0236389i
$723$		0			0
$724$	−1.04496			−0.0388357
$725$	−0.299664	−	5.37682i	−0.0111292	−	0.199690i
$726$		0			0
$727$		−	38.3358i		−	1.42179i	−0.703296	−	0.710897i	$-0.748289\pi$
$727$		−	38.3358i		−	1.42179i	0.703296	−	0.710897i	$-0.251711\pi$
$728$			22.2596i			0.824994i
$729$		0			0
$730$	−8.10101			−0.299832
$731$	−70.6026			−2.61133
$732$		0			0
$733$		−	34.7200i		−	1.28241i	−0.767369	−	0.641205i	$-0.778435\pi$
$733$		−	34.7200i		−	1.28241i	0.767369	−	0.641205i	$-0.221565\pi$
$734$	−2.52074			−0.0930422
$735$		0			0
$736$		−	8.41936i		−	0.310342i
$737$			4.83974i			0.178274i
$738$		0			0
$739$		−	4.30266i		−	0.158276i	−0.996864	−	0.0791380i	$-0.974783\pi$
$739$		−	4.30266i		−	0.158276i	0.996864	−	0.0791380i	$-0.0252168\pi$
$740$		−	2.39275i		−	0.0879591i
$741$		0			0
$742$			12.0532i			0.442488i
$743$			23.7159i			0.870051i	0.900418	+	0.435025i	$0.143261\pi$
$743$			23.7159i			0.870051i	−0.900418	+	0.435025i	$0.856739\pi$
$744$		0			0
$745$	6.00000			0.219823
$746$			1.04493i			0.0382575i
$747$		0			0
$748$	−1.11222			−0.0406667
$749$	16.5353			0.604187
$750$		0			0
$751$			45.1689i			1.64824i	0.566418	+	0.824118i	$0.308329\pi$
$751$			45.1689i			1.64824i	−0.566418	+	0.824118i	$0.691671\pi$
$752$			1.77820i			0.0648444i
$753$		0			0
$754$	−40.0448	+	2.23180i	−1.45835	+	0.0812774i
$755$	21.7340			0.790980
$756$		0			0
$757$			3.63387i			0.132075i	0.997817	+	0.0660376i	$0.0210357\pi$
$757$			3.63387i			0.132075i	−0.997817	+	0.0660376i	$0.978964\pi$
$758$	33.7261			1.22499
$759$		0			0
$760$	−12.9024			−0.468017
$761$	−30.4197			−1.10271			−0.551357	−	0.834269i	$-0.685890\pi$
$761$	−30.4197			−1.10271			−0.551357	+	0.834269i	$0.685890\pi$
$762$		0			0
$763$	14.4421			0.522841
$764$			0.804718i			0.0291137i
$765$		0			0
$766$			27.0803i			0.978451i
$767$	−44.0448			−1.59037
$768$		0			0
$769$		−	20.7086i		−	0.746771i	−0.927676	−	0.373385i	$-0.878197\pi$
$769$		−	20.7086i		−	0.746771i	0.927676	−	0.373385i	$-0.121803\pi$
$770$	0.910201			0.0328014
$771$		0			0
$772$			1.42708i			0.0513619i
$773$		−	32.0828i		−	1.15394i	−0.816766	−	0.576969i	$-0.804235\pi$
$773$		−	32.0828i		−	1.15394i	0.816766	−	0.576969i	$-0.195765\pi$
$774$		0			0
$775$			8.44438i			0.303331i
$776$	−6.20987			−0.222921
$777$		0			0
$778$	−3.29181			−0.118017
$779$	−3.00336			−0.107607
$780$		0			0
$781$		−	2.30324i		−	0.0824165i
$782$	−45.1571			−1.61481
$783$		0			0
$784$	17.5836			0.627987
$785$		−	5.29334i		−	0.188927i
$786$		0			0
$787$	−6.45348			−0.230042			−0.115021	−	0.993363i	$-0.536694\pi$
$787$	−6.45348			−0.230042			−0.115021	+	0.993363i	$0.536694\pi$
$788$	−4.80134			−0.171041
$789$		0			0
$790$	−13.3221			−0.473978
$791$			12.4794i			0.443716i
$792$		0			0
$793$			33.7775i			1.19947i
$794$			17.2399i			0.611820i
$795$		0			0
$796$	−0.506171			−0.0179407
$797$			36.9225i			1.30786i	0.756554	+	0.653932i	$0.226882\pi$
$797$			36.9225i			1.30786i	−0.756554	+	0.653932i	$0.773118\pi$
$798$		0			0
$799$	−3.71155			−0.131305
$800$			1.68011i			0.0594007i
$801$		0			0
$802$		−	32.8333i		−	1.15938i
$803$	−3.33665			−0.117748
$804$		0			0
$805$	−6.51289			−0.229549
$806$	62.8910			2.21524
$807$		0			0
$808$	9.19866			0.323608
$809$		−	22.2596i		−	0.782604i	−0.920262	−	0.391302i	$-0.872025\pi$
$809$		−	22.2596i		−	0.782604i	0.920262	−	0.391302i	$-0.127975\pi$
$810$		0			0
$811$	−16.2469			−0.570504			−0.285252	−	0.958453i	$-0.592077\pi$
$811$	−16.2469			−0.570504			−0.285252	+	0.958453i	$0.592077\pi$
$812$	0.116708	+	2.09407i	0.00409565	+	0.0734875i
$813$		0			0
$814$			5.59201i			0.196000i
$815$			6.07484i			0.212793i
$816$		0			0
$817$	−43.9584			−1.53791
$818$	−36.7530			−1.28504
$819$		0			0
$820$			0.209173i			0.00730463i
$821$	0.352476			0.0123015			0.00615075	−	0.999981i	$-0.498042\pi$
$821$	0.352476			0.0123015			0.00615075	+	0.999981i	$0.498042\pi$
$822$		0			0
$823$			15.0021i			0.522939i	0.965212	+	0.261469i	$0.0842070\pi$
$823$			15.0021i			0.522939i	−0.965212	+	0.261469i	$0.915793\pi$
$824$		−	0.562097i		−	0.0195816i
$825$		0			0
$826$		−	13.0689i		−	0.454726i
$827$		−	27.4152i		−	0.953319i	−0.879088	−	0.476659i	$-0.841848\pi$
$827$		−	27.4152i		−	0.953319i	0.879088	−	0.476659i	$-0.158152\pi$
$828$		0			0
$829$			50.9219i			1.76859i	0.466929	+	0.884295i	$0.345360\pi$
$829$			50.9219i			1.76859i	−0.466929	+	0.884295i	$0.654640\pi$
$830$		−	3.52116i		−	0.122221i
$831$		0			0
$832$	50.3333			1.74499
$833$			36.7013i			1.27163i
$834$		0			0
$835$	5.01121			0.173420
$836$	−0.692486			−0.0239501
$837$		0			0
$838$			27.2663i			0.941897i
$839$		−	37.5371i		−	1.29592i	−0.761673	−	0.647962i	$-0.775622\pi$
$839$		−	37.5371i		−	1.29592i	0.761673	−	0.647962i	$-0.224378\pi$
$840$		0			0
$841$	−28.8204	+	3.22248i	−0.993807	+	0.111120i
$842$	6.09316			0.209984
$843$		0			0
$844$		−	2.09588i		−	0.0721430i
$845$	−19.6217			−0.675009
$846$		0			0
$847$	−13.9214			−0.478345
$848$	23.5476			0.808630
$849$		0			0
$850$	9.01121			0.309082
$851$		−	40.0133i		−	1.37164i
$852$		0			0
$853$		−	0.942449i		−	0.0322688i	−0.999870	−	0.0161344i	$-0.994864\pi$
$853$		−	0.942449i		−	0.0322688i	0.999870	−	0.0161344i	$-0.00513597\pi$
$854$	−10.0224			−0.342960
$855$		0			0
$856$		−	38.1516i		−	1.30400i
$857$	2.66335			0.0909783			0.0454891	−	0.998965i	$-0.485515\pi$
$857$	2.66335			0.0909783			0.0454891	+	0.998965i	$0.485515\pi$
$858$		0			0
$859$		−	22.9636i		−	0.783508i	−0.920070	−	0.391754i	$-0.871868\pi$
$859$		−	22.9636i		−	0.783508i	0.920070	−	0.391754i	$-0.128132\pi$
$860$			3.06154i			0.104398i
$861$		0			0
$862$		−	15.9695i		−	0.543924i
$863$	13.7485			0.468005			0.234003	−	0.972236i	$-0.424818\pi$
$863$	13.7485			0.468005			0.234003	+	0.972236i	$0.424818\pi$
$864$		0			0
$865$	−8.31087			−0.282578
$866$	15.3928			0.523069
$867$		0			0
$868$		−	3.28877i		−	0.111628i
$869$	−5.48711			−0.186138
$870$		0			0
$871$	51.4679			1.74392
$872$		−	33.3221i		−	1.12843i
$873$		0			0
$874$	−28.1156			−0.951024
$875$	1.29966			0.0439367
$876$		0			0
$877$	−42.1313			−1.42267			−0.711336	−	0.702852i	$-0.751910\pi$
$877$	−42.1313			−1.42267			−0.711336	+	0.702852i	$0.751910\pi$
$878$		−	38.5088i		−	1.29961i
$879$		0			0
$880$		−	1.77820i		−	0.0599432i
$881$		−	13.1232i		−	0.442131i	−0.975259	−	0.221065i	$-0.929047\pi$
$881$		−	13.1232i		−	0.442131i	0.975259	−	0.221065i	$-0.0709535\pi$
$882$		0			0
$883$	−49.5465			−1.66737			−0.833687	−	0.552238i	$-0.813774\pi$
$883$	−49.5465			−1.66737			−0.833687	+	0.552238i	$0.813774\pi$
$884$			11.8278i			0.397812i
$885$		0			0
$886$	−4.28060			−0.143810
$887$		−	3.43767i		−	0.115426i	−0.998333	−	0.0577129i	$-0.981619\pi$
$887$		−	3.43767i		−	0.115426i	0.998333	−	0.0577129i	$-0.0183808\pi$
$888$		0			0
$889$		−	7.89526i		−	0.264798i
$890$	12.6217			0.423082
$891$		0			0
$892$	0.895633			0.0299880
$893$	−2.31087			−0.0773304
$894$		0			0
$895$	6.59933			0.220591
$896$			10.5677i			0.353041i
$897$		0			0
$898$	−37.7340			−1.25920
$899$	45.4039	−	2.53048i	1.51431	−	0.0843961i
$900$		0			0
$901$			49.1497i			1.63741i
$902$		−	0.488851i		−	0.0162770i
$903$		0			0
$904$	28.7935			0.957657
$905$	3.48711			0.115916
$906$		0			0
$907$			3.44971i			0.114546i	0.998359	+	0.0572729i	$0.0182405\pi$
$907$			3.44971i			0.114546i	−0.998359	+	0.0572729i	$0.981759\pi$
$908$	7.14920			0.237255
$909$		0			0
$910$		−	9.67948i		−	0.320872i
$911$			30.8709i			1.02280i	0.859343	+	0.511399i	$0.170873\pi$
$911$			30.8709i			1.02280i	−0.859343	+	0.511399i	$0.829127\pi$
$912$		0			0
$913$		−	1.45030i		−	0.0479978i
$914$		−	18.9536i		−	0.626929i
$915$		0			0
$916$			0.0964580i			0.00318706i
$917$		−	23.5549i		−	0.777852i
$918$		0			0
$919$	−49.5768			−1.63539			−0.817694	−	0.575654i	$-0.804748\pi$
$919$	−49.5768			−1.63539			−0.817694	+	0.575654i	$0.804748\pi$
$920$			15.0271i			0.495428i
$921$		0			0
$922$	−29.2435			−0.963083
$923$	−24.4937			−0.806220
$924$		0			0
$925$			7.98476i			0.262537i
$926$		−	13.0879i		−	0.430096i
$927$		0			0
$928$	9.03363	−	0.503467i	0.296543	−	0.0165271i
$929$	−58.0897			−1.90586			−0.952930	−	0.303190i	$-0.901948\pi$
$929$	−58.0897			−1.90586			−0.952930	+	0.303190i	$0.901948\pi$
$930$		0			0
$931$			22.8509i			0.748908i
$932$	4.80134			0.157273
$933$		0			0
$934$	5.68128			0.185897
$935$	3.71155			0.121381
$936$		0			0
$937$	54.9550			1.79530			0.897651	−	0.440706i	$-0.145272\pi$
$937$	54.9550			1.79530			0.897651	+	0.440706i	$0.145272\pi$
$938$			15.2715i			0.498632i
$939$		0			0
$940$			0.160944i			0.00524940i
$941$	−43.6924			−1.42433			−0.712165	−	0.702012i	$-0.752285\pi$
$941$	−43.6924			−1.42433			−0.712165	+	0.702012i	$0.752285\pi$
$942$		0			0
$943$			3.49794i			0.113909i
$944$	−25.5319			−0.830994
$945$		0			0
$946$		−	7.15502i		−	0.232630i
$947$		−	13.9279i		−	0.452596i	−0.974058	−	0.226298i	$-0.927338\pi$
$947$		−	13.9279i		−	0.452596i	0.974058	−	0.226298i	$-0.0726623\pi$
$948$		0			0
$949$			35.4834i			1.15184i
$950$	5.61054			0.182030
$951$		0			0
$952$	−26.9326			−0.872891
$953$	12.3525			0.400136			0.200068	−	0.979782i	$-0.435884\pi$
$953$	12.3525			0.400136			0.200068	+	0.979782i	$0.435884\pi$
$954$		0			0
$955$		−	2.68540i		−	0.0868975i
$956$	−5.90684			−0.191041
$957$		0			0
$958$	36.5275			1.18015
$959$		−	17.1564i		−	0.554009i
$960$		0			0
$961$	−40.3075			−1.30024
$962$	59.4679			1.91732
$963$		0			0
$964$	0.599328			0.0193031
$965$		−	4.76228i		−	0.153303i
$966$		0			0
$967$		−	26.5080i		−	0.852439i	−0.904620	−	0.426219i	$-0.859845\pi$
$967$		−	26.5080i		−	0.852439i	0.904620	−	0.426219i	$-0.140155\pi$
$968$			32.1206i			1.03240i
$969$		0			0
$970$	2.70034			0.0867026
$971$			22.2656i			0.714536i	0.934002	+	0.357268i	$0.116292\pi$
$971$			22.2656i			0.714536i	−0.934002	+	0.357268i	$0.883708\pi$
$972$		0			0
$973$	−18.9742			−0.608286
$974$			25.4588i			0.815753i
$975$		0			0
$976$			19.5802i			0.626746i
$977$	18.0864			0.578636			0.289318	−	0.957233i	$-0.406571\pi$
$977$	18.0864			0.578636			0.289318	+	0.957233i	$0.406571\pi$
$978$		0			0
$979$	5.19866			0.166150
$980$	1.59148			0.0508379
$981$		0			0
$982$	47.4758			1.51501
$983$		−	21.8894i		−	0.698165i	−0.937092	−	0.349082i	$-0.886493\pi$
$983$		−	21.8894i		−	0.698165i	0.937092	−	0.349082i	$-0.113507\pi$
$984$		0			0
$985$	16.0224			0.510517
$986$	−2.70034	−	48.4517i	−0.0859962	−	1.54302i
$987$		0			0
$988$			7.36420i			0.234286i
$989$			51.1973i			1.62798i
$990$		0			0
$991$	−14.8878			−0.472926			−0.236463	−	0.971640i	$-0.575988\pi$
$991$	−14.8878			−0.472926			−0.236463	+	0.971640i	$0.575988\pi$
$992$	−14.1874			−0.450452
$993$		0			0
$994$		−	7.26774i		−	0.230519i
$995$	1.68913			0.0535489
$996$		0			0
$997$		−	38.4735i		−	1.21847i	−0.792991	−	0.609234i	$-0.791477\pi$
$997$		−	38.4735i		−	1.21847i	0.792991	−	0.609234i	$-0.208523\pi$
$998$			41.0332i			1.29888i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.d.b.811.5		6
3.2	odd	2		145.2.c.b.86.2	✓	6
12.11	even	2		2320.2.g.i.1681.4		6
15.2	even	4		725.2.d.c.724.9		12
15.8	even	4		725.2.d.c.724.4		12
15.14	odd	2		725.2.c.e.376.5		6
29.28	even	2	inner	1305.2.d.b.811.2		6
87.17	even	4		4205.2.a.m.1.2		6
87.41	even	4		4205.2.a.m.1.5		6
87.86	odd	2		145.2.c.b.86.5	yes	6
348.347	even	2		2320.2.g.i.1681.3		6
435.173	even	4		725.2.d.c.724.10		12
435.347	even	4		725.2.d.c.724.3		12
435.434	odd	2		725.2.c.e.376.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.c.b.86.2	✓	6	3.2	odd	2
145.2.c.b.86.5	yes	6	87.86	odd	2
725.2.c.e.376.2		6	435.434	odd	2
725.2.c.e.376.5		6	15.14	odd	2
725.2.d.c.724.3		12	435.347	even	4
725.2.d.c.724.4		12	15.8	even	4
725.2.d.c.724.9		12	15.2	even	4
725.2.d.c.724.10		12	435.173	even	4
1305.2.d.b.811.2		6	29.28	even	2	inner
1305.2.d.b.811.5		6	1.1	even	1	trivial
2320.2.g.i.1681.3		6	348.347	even	2
2320.2.g.i.1681.4		6	12.11	even	2
4205.2.a.m.1.2		6	87.17	even	4
4205.2.a.m.1.5		6	87.41	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)

\(f(q)\)	\(=\)	\(q+1.30397i q^{2} +0.299664 q^{4} -1.00000 q^{5} -1.29966 q^{7} +2.99869i q^{8} -1.30397i q^{10} -0.537080i q^{11} -5.71155 q^{13} -1.69472i q^{14} -3.31087 q^{16} -6.91060i q^{17} -4.30266i q^{19} -0.299664 q^{20} +0.700336 q^{22} -5.01121 q^{23} +1.00000 q^{25} -7.44768i q^{26} -0.389463 q^{28} +(-0.299664 - 5.37682i) q^{29} +8.44438i q^{31} +1.68011i q^{32} +9.01121 q^{34} +1.29966 q^{35} +7.98476i q^{37} +5.61054 q^{38} -2.99869i q^{40} -0.698024i q^{41} -10.2166i q^{43} -0.160944i q^{44} -6.53446i q^{46} -0.537080i q^{47} -5.31087 q^{49} +1.30397i q^{50} -1.71155 q^{52} -7.11222 q^{53} +0.537080i q^{55} -3.89729i q^{56} +(7.01121 - 0.390753i) q^{58} +7.71155 q^{59} -5.91390i q^{61} -11.0112 q^{62} -8.81255 q^{64} +5.71155 q^{65} -9.01121 q^{67} -2.07086i q^{68} +1.69472i q^{70} +4.28845 q^{71} -6.21258i q^{73} -10.4119 q^{74} -1.28935i q^{76} +0.698024i q^{77} -10.2166i q^{79} +3.31087 q^{80} +0.910201 q^{82} +2.70034 q^{83} +6.91060i q^{85} +13.3221 q^{86} +1.61054 q^{88} +9.67948i q^{89} +7.42309 q^{91} -1.50168 q^{92} +0.700336 q^{94} +4.30266i q^{95} +2.07086i q^{97} -6.92522i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 6 q^{5} - 4 q^{13} + 26 q^{16} + 6 q^{20} + 12 q^{22} + 8 q^{23} + 6 q^{25} - 56 q^{28} + 6 q^{29} + 16 q^{34} - 20 q^{38} + 14 q^{49} + 20 q^{52} - 28 q^{53} + 4 q^{58} + 16 q^{59} - 28 q^{62}+ \cdots + 12 q^{94}+O(q^{100}) \) 6 * q - 6 * q^4 - 6 * q^5 - 4 * q^13 + 26 * q^16 + 6 * q^20 + 12 * q^22 + 8 * q^23 + 6 * q^25 - 56 * q^28 + 6 * q^29 + 16 * q^34 - 20 * q^38 + 14 * q^49 + 20 * q^52 - 28 * q^53 + 4 * q^58 + 16 * q^59 - 28 * q^62 - 46 * q^64 + 4 * q^65 - 16 * q^67 + 56 * q^71 - 40 * q^74 - 26 * q^80 - 56 * q^82 + 24 * q^83 - 4 * q^86 - 44 * q^88 - 16 * q^91 - 48 * q^92 + 12 * q^94

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