LMFDB - Embedded newform 3060.2.be.b.361.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(361,3060)] 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("3060.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.be (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 6 x^{10} - 16 x^{9} + 9 x^{8} - 72 x^{7} + 114 x^{6} - 144 x^{5} + 391 x^{4} - 484 x^{3} + \cdots + 121 $ x^12 + 6x^10 - 16x^9 + 9x^8 - 72x^7 + 114x^6 - 144x^5 + 391x^4 - 484x^3 + 504x^2 - 396x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 340)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			361.1
Root			$-1.47591 + 1.40653i$ of defining polynomial
Character	$\chi$	$=$	3060.361
Dual form			3060.2.be.b.1441.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{5} +(-3.70031 - 3.70031i) q^{7} +(3.73762 + 3.73762i) q^{11} +2.63690 q^{13} +(-4.12309 - 0.0118173i) q^{17} +4.31329i q^{19} +(-2.94577 - 2.94577i) q^{23} -1.00000i q^{25} +(-1.80095 + 1.80095i) q^{29} +(2.12819 - 2.12819i) q^{31} +5.23303 q^{35} +(6.69720 - 6.69720i) q^{37} +(-0.956555 - 0.956555i) q^{41} +0.613266i q^{43} +5.18546 q^{47} +20.3846i q^{49} +1.77451i q^{53} -5.28579 q^{55} +7.97014i q^{59} +(-2.57942 - 2.57942i) q^{61} +(-1.86457 + 1.86457i) q^{65} +11.7984 q^{67} +(5.08713 - 5.08713i) q^{71} +(-4.58989 + 4.58989i) q^{73} -27.6607i q^{77} +(1.26649 + 1.26649i) q^{79} -12.8929i q^{83} +(2.92382 - 2.90711i) q^{85} +3.34014 q^{89} +(-9.75736 - 9.75736i) q^{91} +(-3.04995 - 3.04995i) q^{95} +(-3.27294 + 3.27294i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{11} + 16 q^{13} + 4 q^{17} - 12 q^{23} - 4 q^{29} - 8 q^{31} + 8 q^{35} - 20 q^{37} - 8 q^{41} + 8 q^{47} + 16 q^{55} - 8 q^{61} - 4 q^{65} + 32 q^{67} - 28 q^{71} - 24 q^{79} - 4 q^{85} + 56 q^{89}+ \cdots - 44 q^{97}+O(q^{100}) $ 12 * q + 12 * q^11 + 16 * q^13 + 4 * q^17 - 12 * q^23 - 4 * q^29 - 8 * q^31 + 8 * q^35 - 20 * q^37 - 8 * q^41 + 8 * q^47 + 16 * q^55 - 8 * q^61 - 4 * q^65 + 32 * q^67 - 28 * q^71 - 24 * q^79 - 4 * q^85 + 56 * q^89 - 4 * q^91 - 8 * q^95 - 44 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$5$	−0.707107	+	0.707107i	−0.316228	+	0.316228i
$6$		0			0
$7$	−3.70031	−	3.70031i	−1.39859	−	1.39859i	−0.804121	−	0.594466i	$-0.797363\pi$
$7$	−3.70031	−	3.70031i	−1.39859	−	1.39859i	−0.594466	−	0.804121i	$-0.702637\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.73762	+	3.73762i	1.12693	+	1.12693i	0.990673	+	0.136261i	$0.0435084\pi$
$11$	3.73762	+	3.73762i	1.12693	+	1.12693i	0.136261	+	0.990673i	$0.456492\pi$
$12$		0			0
$13$	2.63690			0.731345			0.365672	−	0.930744i	$-0.380839\pi$
$13$	2.63690			0.731345			0.365672	+	0.930744i	$0.380839\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.12309	−	0.0118173i	−0.999996	−	0.00286611i
$17$	−4.12309	−	0.0118173i	−0.999996	−	0.00286611i
$18$		0			0
$19$			4.31329i			0.989536i	0.869025	+	0.494768i	$0.164747\pi$
$19$			4.31329i			0.989536i	−0.869025	+	0.494768i	$0.835253\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.94577	−	2.94577i	−0.614236	−	0.614236i	0.329811	−	0.944047i	$-0.393015\pi$
$23$	−2.94577	−	2.94577i	−0.614236	−	0.614236i	−0.944047	+	0.329811i	$0.893015\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.80095	+	1.80095i	−0.334428	+	0.334428i	−0.854265	−	0.519838i	$-0.825992\pi$
$29$	−1.80095	+	1.80095i	−0.334428	+	0.334428i	0.519838	+	0.854265i	$0.325992\pi$
$30$		0			0
$31$	2.12819	−	2.12819i	0.382235	−	0.382235i	−0.489672	−	0.871907i	$-0.662883\pi$
$31$	2.12819	−	2.12819i	0.382235	−	0.382235i	0.871907	+	0.489672i	$0.162883\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.23303			0.884544
$36$		0			0
$37$	6.69720	−	6.69720i	1.10101	−	1.10101i	0.106724	−	0.994289i	$-0.465964\pi$
$37$	6.69720	−	6.69720i	1.10101	−	1.10101i	0.994289	−	0.106724i	$-0.0340363\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.956555	−	0.956555i	−0.149389	−	0.149389i	0.628456	−	0.777845i	$-0.283687\pi$
$41$	−0.956555	−	0.956555i	−0.149389	−	0.149389i	−0.777845	+	0.628456i	$0.783687\pi$
$42$		0			0
$43$			0.613266i			0.0935222i	0.998906	+	0.0467611i	$0.0148900\pi$
$43$			0.613266i			0.0935222i	−0.998906	+	0.0467611i	$0.985110\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.18546			0.756378			0.378189	−	0.925728i	$-0.376547\pi$
$47$	5.18546			0.756378			0.378189	+	0.925728i	$0.376547\pi$
$48$		0			0
$49$			20.3846i			2.91209i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1.77451i			0.243748i	0.992546	+	0.121874i	$0.0388904\pi$
$53$			1.77451i			0.243748i	−0.992546	+	0.121874i	$0.961110\pi$
$54$		0			0
$55$	−5.28579			−0.712735
$56$		0			0
$57$		0			0
$58$		0			0
$59$			7.97014i			1.03762i	0.854888	+	0.518812i	$0.173626\pi$
$59$			7.97014i			1.03762i	−0.854888	+	0.518812i	$0.826374\pi$
$60$		0			0
$61$	−2.57942	−	2.57942i	−0.330261	−	0.330261i	0.522425	−	0.852685i	$-0.325028\pi$
$61$	−2.57942	−	2.57942i	−0.330261	−	0.330261i	−0.852685	+	0.522425i	$0.825028\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.86457	+	1.86457i	−0.231272	+	0.231272i
$66$		0			0
$67$	11.7984			1.44141			0.720703	−	0.693244i	$-0.243819\pi$
$67$	11.7984			1.44141			0.720703	+	0.693244i	$0.243819\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.08713	−	5.08713i	0.603731	−	0.603731i	−0.337570	−	0.941301i	$-0.609605\pi$
$71$	5.08713	−	5.08713i	0.603731	−	0.603731i	0.941301	+	0.337570i	$0.109605\pi$
$72$		0			0
$73$	−4.58989	+	4.58989i	−0.537206	+	0.537206i	−0.922707	−	0.385501i	$-0.874028\pi$
$73$	−4.58989	+	4.58989i	−0.537206	+	0.537206i	0.385501	+	0.922707i	$0.374028\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	27.6607i		−	3.15223i
$78$		0			0
$79$	1.26649	+	1.26649i	0.142491	+	0.142491i	0.774754	−	0.632263i	$-0.217874\pi$
$79$	1.26649	+	1.26649i	0.142491	+	0.142491i	−0.632263	+	0.774754i	$0.717874\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	12.8929i		−	1.41518i	−0.706625	−	0.707588i	$-0.749783\pi$
$83$		−	12.8929i		−	1.41518i	0.706625	−	0.707588i	$-0.250217\pi$
$84$		0			0
$85$	2.92382	−	2.90711i	0.317133	−	0.315320i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.34014			0.354055			0.177027	−	0.984206i	$-0.443352\pi$
$89$	3.34014			0.354055			0.177027	+	0.984206i	$0.443352\pi$
$90$		0			0
$91$	−9.75736	−	9.75736i	−1.02285	−	1.02285i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.04995	−	3.04995i	−0.312919	−	0.312919i
$96$		0			0
$97$	−3.27294	+	3.27294i	−0.332317	+	0.332317i	−0.853466	−	0.521149i	$-0.825504\pi$
$97$	−3.27294	+	3.27294i	−0.332317	+	0.332317i	0.521149	+	0.853466i	$0.325504\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	17.3002			1.72144			0.860719	−	0.509081i	$-0.170015\pi$
$101$	17.3002			1.72144			0.860719	+	0.509081i	$0.170015\pi$
$102$		0			0
$103$	18.0638			1.77988			0.889940	−	0.456077i	$-0.150746\pi$
$103$	18.0638			1.77988			0.889940	+	0.456077i	$0.150746\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.72098	−	8.72098i	0.843089	−	0.843089i	−0.146170	−	0.989259i	$-0.546695\pi$
$107$	8.72098	−	8.72098i	0.843089	−	0.843089i	0.989259	+	0.146170i	$0.0466947\pi$
$108$		0			0
$109$	4.14463	+	4.14463i	0.396983	+	0.396983i	0.877168	−	0.480184i	$-0.159430\pi$
$109$	4.14463	+	4.14463i	0.396983	+	0.396983i	−0.480184	+	0.877168i	$0.659430\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.11083	+	3.11083i	0.292642	+	0.292642i	0.838123	−	0.545481i	$-0.183653\pi$
$113$	3.11083	+	3.11083i	0.292642	+	0.292642i	−0.545481	+	0.838123i	$0.683653\pi$
$114$		0			0
$115$	4.16595			0.388477
$116$		0			0
$117$		0			0
$118$		0			0
$119$	15.2130	+	15.3004i	1.39457	+	1.40259i
$120$		0			0
$121$			16.9395i			1.53996i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	+	0.707107i	0.0632456	+	0.0632456i
$126$		0			0
$127$		−	4.13619i		−	0.367027i	−0.983017	−	0.183514i	$-0.941253\pi$
$127$		−	4.13619i		−	0.367027i	0.983017	−	0.183514i	$-0.0587472\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	15.6272	−	15.6272i	1.36536	−	1.36536i	0.498419	−	0.866936i	$-0.333914\pi$
$131$	15.6272	−	15.6272i	1.36536	−	1.36536i	0.866936	−	0.498419i	$-0.166086\pi$
$132$		0			0
$133$	15.9605	−	15.9605i	1.38395	−	1.38395i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.63690			0.225286			0.112643	−	0.993636i	$-0.464068\pi$
$137$	2.63690			0.225286			0.112643	+	0.993636i	$0.464068\pi$
$138$		0			0
$139$	9.17535	−	9.17535i	0.778243	−	0.778243i	−0.201289	−	0.979532i	$-0.564513\pi$
$139$	9.17535	−	9.17535i	0.778243	−	0.778243i	0.979532	+	0.201289i	$0.0645130\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.85572	+	9.85572i	0.824177	+	0.824177i
$144$		0			0
$145$		−	2.54692i		−	0.211511i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.47255			−0.448329			−0.224164	−	0.974551i	$-0.571965\pi$
$149$	−5.47255			−0.448329			−0.224164	+	0.974551i	$0.571965\pi$
$150$		0			0
$151$		−	22.6375i		−	1.84221i	−0.389314	−	0.921105i	$-0.627288\pi$
$151$		−	22.6375i		−	1.84221i	0.389314	−	0.921105i	$-0.372712\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			3.00972i			0.241747i
$156$		0			0
$157$	11.0324			0.880480			0.440240	−	0.897880i	$-0.354893\pi$
$157$	11.0324			0.880480			0.440240	+	0.897880i	$0.354893\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			21.8006i			1.71813i
$162$		0			0
$163$	6.98640	+	6.98640i	0.547217	+	0.547217i	0.925635	−	0.378418i	$-0.123532\pi$
$163$	6.98640	+	6.98640i	0.547217	+	0.547217i	−0.378418	+	0.925635i	$0.623532\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.4318	+	11.4318i	−0.884617	+	0.884617i	−0.994000	−	0.109383i	$-0.965113\pi$
$167$	−11.4318	+	11.4318i	−0.884617	+	0.884617i	0.109383	+	0.994000i	$0.465113\pi$
$168$		0			0
$169$	−6.04675			−0.465135
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.25225	+	4.25225i	−0.323292	+	0.323292i	−0.850029	−	0.526736i	$-0.823416\pi$
$173$	−4.25225	+	4.25225i	−0.323292	+	0.323292i	0.526736	+	0.850029i	$0.323416\pi$
$174$		0			0
$175$	−3.70031	+	3.70031i	−0.279717	+	0.279717i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	12.0604i		−	0.901438i	−0.892666	−	0.450719i	$-0.851168\pi$
$179$		−	12.0604i		−	0.901438i	0.892666	−	0.450719i	$-0.148832\pi$
$180$		0			0
$181$	−2.45754	−	2.45754i	−0.182668	−	0.182668i	0.609850	−	0.792517i	$-0.291230\pi$
$181$	−2.45754	−	2.45754i	−0.182668	−	0.182668i	−0.792517	+	0.609850i	$0.791230\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			9.47127i			0.696342i
$186$		0			0
$187$	−15.3664	−	15.4547i	−1.12370	−	1.13016i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.77866			0.345772			0.172886	−	0.984942i	$-0.444691\pi$
$191$	4.77866			0.345772			0.172886	+	0.984942i	$0.444691\pi$
$192$		0			0
$193$	−16.0784	−	16.0784i	−1.15735	−	1.15735i	−0.985043	−	0.172307i	$-0.944878\pi$
$193$	−16.0784	−	16.0784i	−1.15735	−	1.15735i	−0.172307	−	0.985043i	$-0.555122\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.96349	+	8.96349i	0.638623	+	0.638623i	0.950216	−	0.311593i	$-0.100863\pi$
$197$	8.96349	+	8.96349i	0.638623	+	0.638623i	−0.311593	+	0.950216i	$0.600863\pi$
$198$		0			0
$199$	4.36005	−	4.36005i	0.309076	−	0.309076i	−0.535475	−	0.844551i	$-0.679867\pi$
$199$	4.36005	−	4.36005i	0.309076	−	0.309076i	0.844551	+	0.535475i	$0.179867\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	13.3281			0.935452
$204$		0			0
$205$	1.35277			0.0944818
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.1214	+	16.1214i	−1.11514	+	1.11514i
$210$		0			0
$211$	13.9501	+	13.9501i	0.960367	+	0.960367i	0.999244	−	0.0388769i	$-0.0123780\pi$
$211$	13.9501	+	13.9501i	0.960367	+	0.960367i	−0.0388769	+	0.999244i	$0.512378\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.433645	−	0.433645i	−0.0295743	−	0.0295743i
$216$		0			0
$217$	−15.7500			−1.06918
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−10.8722	−	0.0311610i	−0.731342	−	0.00209611i
$222$		0			0
$223$			23.7682i			1.59164i	0.605536	+	0.795818i	$0.292959\pi$
$223$			23.7682i			1.59164i	−0.605536	+	0.795818i	$0.707041\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.12811	+	9.12811i	0.605854	+	0.605854i	0.941860	−	0.336006i	$-0.109076\pi$
$227$	9.12811	+	9.12811i	0.605854	+	0.605854i	−0.336006	+	0.941860i	$0.609076\pi$
$228$		0			0
$229$			23.9466i			1.58244i	0.611534	+	0.791218i	$0.290553\pi$
$229$			23.9466i			1.58244i	−0.611534	+	0.791218i	$0.709447\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.84567	−	6.84567i	0.448475	−	0.448475i	−0.446372	−	0.894847i	$-0.647284\pi$
$233$	6.84567	−	6.84567i	0.448475	−	0.448475i	0.894847	+	0.446372i	$0.147284\pi$
$234$		0			0
$235$	−3.66668	+	3.66668i	−0.239188	+	0.239188i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	18.7624			1.21364			0.606820	−	0.794839i	$-0.292445\pi$
$239$	18.7624			1.21364			0.606820	+	0.794839i	$0.292445\pi$
$240$		0			0
$241$	−0.400091	+	0.400091i	−0.0257721	+	0.0257721i	−0.719875	−	0.694103i	$-0.755801\pi$
$241$	−0.400091	+	0.400091i	−0.0257721	+	0.0257721i	0.694103	+	0.719875i	$0.255801\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.4141	−	14.4141i	−0.920884	−	0.920884i
$246$		0			0
$247$			11.3737i			0.723692i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−22.5898			−1.42585			−0.712927	−	0.701238i	$-0.752631\pi$
$251$	−22.5898			−1.42585			−0.712927	+	0.701238i	$0.752631\pi$
$252$		0			0
$253$		−	22.0203i		−	1.38441i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			19.4988i			1.21630i	0.793823	+	0.608149i	$0.208088\pi$
$257$			19.4988i			1.21630i	−0.793823	+	0.608149i	$0.791912\pi$
$258$		0			0
$259$	−49.5635			−3.07972
$260$		0			0
$261$		0			0
$262$		0			0
$263$			3.17067i			0.195512i	0.995210	+	0.0977560i	$0.0311665\pi$
$263$			3.17067i			0.195512i	−0.995210	+	0.0977560i	$0.968834\pi$
$264$		0			0
$265$	−1.25477	−	1.25477i	−0.0770799	−	0.0770799i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	11.2514	−	11.2514i	0.686009	−	0.686009i	−0.275338	−	0.961347i	$-0.588790\pi$
$269$	11.2514	−	11.2514i	0.686009	−	0.686009i	0.961347	+	0.275338i	$0.0887899\pi$
$270$		0			0
$271$	−12.7342			−0.773546			−0.386773	−	0.922175i	$-0.626410\pi$
$271$	−12.7342			−0.773546			−0.386773	+	0.922175i	$0.626410\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.73762	−	3.73762i	0.225387	−	0.225387i
$276$		0			0
$277$	−8.48144	+	8.48144i	−0.509600	+	0.509600i	−0.914404	−	0.404804i	$-0.867340\pi$
$277$	−8.48144	+	8.48144i	−0.509600	+	0.509600i	0.404804	+	0.914404i	$0.367340\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3.10194i		−	0.185046i	−0.995711	−	0.0925232i	$-0.970507\pi$
$281$		−	3.10194i		−	0.185046i	0.995711	−	0.0925232i	$-0.0294932\pi$
$282$		0			0
$283$	18.3585	+	18.3585i	1.09130	+	1.09130i	0.995390	+	0.0959087i	$0.0305757\pi$
$283$	18.3585	+	18.3585i	1.09130	+	1.09130i	0.0959087	+	0.995390i	$0.469424\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			7.07911i			0.417866i
$288$		0			0
$289$	16.9997	+	0.0974473i	0.999984	+	0.00573220i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.39037			−0.256488			−0.128244	−	0.991743i	$-0.540934\pi$
$293$	−4.39037			−0.256488			−0.128244	+	0.991743i	$0.540934\pi$
$294$		0			0
$295$	−5.63574	−	5.63574i	−0.328126	−	0.328126i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.76771	−	7.76771i	−0.449218	−	0.449218i
$300$		0			0
$301$	2.26928	−	2.26928i	0.130799	−	0.130799i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.64785			0.208875
$306$		0			0
$307$	14.1398			0.806998			0.403499	−	0.914980i	$-0.367794\pi$
$307$	14.1398			0.806998			0.403499	+	0.914980i	$0.367794\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.00643	−	7.00643i	0.397298	−	0.397298i	−0.479981	−	0.877279i	$-0.659356\pi$
$311$	7.00643	−	7.00643i	0.397298	−	0.397298i	0.877279	+	0.479981i	$0.159356\pi$
$312$		0			0
$313$	−0.805586	−	0.805586i	−0.0455344	−	0.0455344i	0.683973	−	0.729507i	$-0.260251\pi$
$313$	−0.805586	−	0.805586i	−0.0455344	−	0.0455344i	−0.729507	+	0.683973i	$0.760251\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.19894	−	1.19894i	−0.0673391	−	0.0673391i	0.672635	−	0.739974i	$-0.265162\pi$
$317$	−1.19894	−	1.19894i	−0.0673391	−	0.0673391i	−0.739974	+	0.672635i	$0.765162\pi$
$318$		0			0
$319$	−13.4625			−0.753755
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.0509713	−	17.7841i	0.00283612	−	0.989532i
$324$		0			0
$325$		−	2.63690i		−	0.146269i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−19.1878	−	19.1878i	−1.05786	−	1.05786i
$330$		0			0
$331$			14.1128i			0.775711i	0.921720	+	0.387856i	$0.126784\pi$
$331$			14.1128i			0.775711i	−0.921720	+	0.387856i	$0.873216\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.34274	+	8.34274i	−0.455813	+	0.455813i
$336$		0			0
$337$	2.18076	−	2.18076i	0.118794	−	0.118794i	−0.645211	−	0.764005i	$-0.723230\pi$
$337$	2.18076	−	2.18076i	0.118794	−	0.118794i	0.764005	+	0.645211i	$0.223230\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	15.9087			0.861507
$342$		0			0
$343$	49.5273	−	49.5273i	2.67422	−	2.67422i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.47271	+	4.47271i	0.240107	+	0.240107i	0.816895	−	0.576787i	$-0.195694\pi$
$347$	4.47271	+	4.47271i	0.240107	+	0.240107i	−0.576787	+	0.816895i	$0.695694\pi$
$348$		0			0
$349$			7.89867i			0.422806i	0.977399	+	0.211403i	$0.0678032\pi$
$349$			7.89867i			0.422806i	−0.977399	+	0.211403i	$0.932197\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−29.9338			−1.59321			−0.796607	−	0.604498i	$-0.793374\pi$
$353$	−29.9338			−1.59321			−0.796607	+	0.604498i	$0.793374\pi$
$354$		0			0
$355$			7.19428i			0.381833i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	5.08403i		−	0.268325i	−0.990959	−	0.134163i	$-0.957166\pi$
$359$		−	5.08403i		−	0.268325i	0.990959	−	0.134163i	$-0.0428344\pi$
$360$		0			0
$361$	0.395551			0.0208185
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	6.49108i		−	0.339759i
$366$		0			0
$367$	−6.47084	−	6.47084i	−0.337775	−	0.337775i	0.517754	−	0.855529i	$-0.326768\pi$
$367$	−6.47084	−	6.47084i	−0.337775	−	0.337775i	−0.855529	+	0.517754i	$0.826768\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.56625	−	6.56625i	0.340903	−	0.340903i
$372$		0			0
$373$	−9.45731			−0.489681			−0.244840	−	0.969563i	$-0.578736\pi$
$373$	−9.45731			−0.489681			−0.244840	+	0.969563i	$0.578736\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.74892	+	4.74892i	−0.244582	+	0.244582i
$378$		0			0
$379$	0.326606	−	0.326606i	0.0167766	−	0.0167766i	−0.698669	−	0.715445i	$-0.746224\pi$
$379$	0.326606	−	0.326606i	0.0167766	−	0.0167766i	0.715445	+	0.698669i	$0.246224\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			24.2666i			1.23996i	0.784616	+	0.619982i	$0.212860\pi$
$383$			24.2666i			1.23996i	−0.784616	+	0.619982i	$0.787140\pi$
$384$		0			0
$385$	19.5591	+	19.5591i	0.996822	+	0.996822i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	10.9870i		−	0.557065i	−0.960427	−	0.278533i	$-0.910152\pi$
$389$		−	10.9870i		−	0.557065i	0.960427	−	0.278533i	$-0.0898481\pi$
$390$		0			0
$391$	12.1109	+	12.1805i	0.612473	+	0.615994i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.79108			−0.0901191
$396$		0			0
$397$	−12.2054	−	12.2054i	−0.612571	−	0.612571i	0.331044	−	0.943615i	$-0.392599\pi$
$397$	−12.2054	−	12.2054i	−0.612571	−	0.612571i	−0.943615	+	0.331044i	$0.892599\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.9192	+	10.9192i	0.545278	+	0.545278i	0.925071	−	0.379793i	$-0.124005\pi$
$401$	10.9192	+	10.9192i	0.545278	+	0.545278i	−0.379793	+	0.925071i	$0.624005\pi$
$402$		0			0
$403$	5.61184	−	5.61184i	0.279546	−	0.279546i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	50.0631			2.48154
$408$		0			0
$409$	−30.1548			−1.49106			−0.745530	−	0.666472i	$-0.767803\pi$
$409$	−30.1548			−1.49106			−0.745530	+	0.666472i	$0.767803\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	29.4920	−	29.4920i	1.45121	−	1.45121i
$414$		0			0
$415$	9.11663	+	9.11663i	0.447518	+	0.447518i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.23437	+	1.23437i	0.0603028	+	0.0603028i	0.736615	−	0.676312i	$-0.236423\pi$
$419$	1.23437	+	1.23437i	0.0603028	+	0.0603028i	−0.676312	+	0.736615i	$0.736423\pi$
$420$		0			0
$421$	27.3370			1.33233			0.666163	−	0.745806i	$-0.267936\pi$
$421$	27.3370			1.33233			0.666163	+	0.745806i	$0.267936\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.0118173	+	4.12309i	−0.000573222	+	0.199999i
$426$		0			0
$427$			19.0893i			0.923797i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−26.1296	−	26.1296i	−1.25862	−	1.25862i	−0.951753	−	0.306866i	$-0.900720\pi$
$431$	−26.1296	−	26.1296i	−1.25862	−	1.25862i	−0.306866	−	0.951753i	$-0.599280\pi$
$432$		0			0
$433$		−	30.8419i		−	1.48217i	−0.671413	−	0.741084i	$-0.734312\pi$
$433$		−	30.8419i		−	1.48217i	0.671413	−	0.741084i	$-0.265688\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.7060	−	12.7060i	0.607809	−	0.607809i
$438$		0			0
$439$	3.80871	−	3.80871i	0.181780	−	0.181780i	−0.610351	−	0.792131i	$-0.708972\pi$
$439$	3.80871	−	3.80871i	0.181780	−	0.181780i	0.792131	+	0.610351i	$0.208972\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−26.4761			−1.25792			−0.628958	−	0.777440i	$-0.716518\pi$
$443$	−26.4761			−1.25792			−0.628958	+	0.777440i	$0.716518\pi$
$444$		0			0
$445$	−2.36184	+	2.36184i	−0.111962	+	0.111962i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−8.03081	−	8.03081i	−0.378998	−	0.378998i	0.491743	−	0.870740i	$-0.336360\pi$
$449$	−8.03081	−	8.03081i	−0.378998	−	0.378998i	−0.870740	+	0.491743i	$0.836360\pi$
$450$		0			0
$451$		−	7.15047i		−	0.336703i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	13.7990			0.646907
$456$		0			0
$457$		−	3.71394i		−	0.173731i	−0.996220	−	0.0868653i	$-0.972315\pi$
$457$		−	3.71394i		−	0.173731i	0.996220	−	0.0868653i	$-0.0276850\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	13.3511i		−	0.621821i	−0.950439	−	0.310910i	$-0.899366\pi$
$461$		−	13.3511i		−	0.621821i	0.950439	−	0.310910i	$-0.100634\pi$
$462$		0			0
$463$	−30.5506			−1.41981			−0.709904	−	0.704298i	$-0.751262\pi$
$463$	−30.5506			−1.41981			−0.709904	+	0.704298i	$0.751262\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	12.1558i		−	0.562504i	−0.959634	−	0.281252i	$-0.909250\pi$
$467$		−	12.1558i		−	0.562504i	0.959634	−	0.281252i	$-0.0907496\pi$
$468$		0			0
$469$	−43.6579	−	43.6579i	−2.01593	−	2.01593i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.29215	+	2.29215i	−0.105393	+	0.105393i
$474$		0			0
$475$	4.31329			0.197907
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.10864	−	5.10864i	0.233420	−	0.233420i	−0.580699	−	0.814119i	$-0.697221\pi$
$479$	5.10864	−	5.10864i	0.233420	−	0.233420i	0.814119	+	0.580699i	$0.197221\pi$
$480$		0			0
$481$	17.6599	−	17.6599i	0.805220	−	0.805220i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	4.62864i		−	0.210176i
$486$		0			0
$487$	0.819615	+	0.819615i	0.0371403	+	0.0371403i	0.725433	−	0.688293i	$-0.241640\pi$
$487$	0.819615	+	0.819615i	0.0371403	+	0.0371403i	−0.688293	+	0.725433i	$0.741640\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			19.9684i			0.901163i	0.892735	+	0.450581i	$0.148783\pi$
$491$			19.9684i			0.901163i	−0.892735	+	0.450581i	$0.851217\pi$
$492$		0			0
$493$	7.44675	−	7.40418i	0.335385	−	0.333468i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−37.6479			−1.68874
$498$		0			0
$499$	0.471180	+	0.471180i	0.0210929	+	0.0210929i	0.717575	−	0.696482i	$-0.245252\pi$
$499$	0.471180	+	0.471180i	0.0210929	+	0.0210929i	−0.696482	+	0.717575i	$0.745252\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.01218	+	4.01218i	0.178894	+	0.178894i	0.790874	−	0.611979i	$-0.209626\pi$
$503$	4.01218	+	4.01218i	0.178894	+	0.178894i	−0.611979	+	0.790874i	$0.709626\pi$
$504$		0			0
$505$	−12.2331	+	12.2331i	−0.544366	+	0.544366i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.8648			0.614545			0.307273	−	0.951622i	$-0.400584\pi$
$509$	13.8648			0.614545			0.307273	+	0.951622i	$0.400584\pi$
$510$		0			0
$511$	33.9680			1.50266
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−12.7730	+	12.7730i	−0.562848	+	0.562848i
$516$		0			0
$517$	19.3813	+	19.3813i	0.852387	+	0.852387i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.6179	−	21.6179i	−0.947097	−	0.947097i	0.0515719	−	0.998669i	$-0.483577\pi$
$521$	−21.6179	−	21.6179i	−0.947097	−	0.947097i	−0.998669	+	0.0515719i	$0.983577\pi$
$522$		0			0
$523$	−10.0946			−0.441405			−0.220703	−	0.975341i	$-0.570835\pi$
$523$	−10.0946			−0.441405			−0.220703	+	0.975341i	$0.570835\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.79989	+	8.74959i	−0.383329	+	0.381138i
$528$		0			0
$529$		−	5.64483i		−	0.245428i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.52234	−	2.52234i	−0.109255	−	0.109255i
$534$		0			0
$535$			12.3333i			0.533217i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−76.1899	+	76.1899i	−3.28173	+	3.28173i
$540$		0			0
$541$	−18.3093	+	18.3093i	−0.787179	+	0.787179i	−0.981031	−	0.193852i	$-0.937902\pi$
$541$	−18.3093	+	18.3093i	−0.787179	+	0.787179i	0.193852	+	0.981031i	$0.437902\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.86139			−0.251074
$546$		0			0
$547$	−8.69458	+	8.69458i	−0.371753	+	0.371753i	−0.868116	−	0.496362i	$-0.834669\pi$
$547$	−8.69458	+	8.69458i	−0.371753	+	0.371753i	0.496362	+	0.868116i	$0.334669\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.76800	−	7.76800i	−0.330928	−	0.330928i
$552$		0			0
$553$		−	9.37279i		−	0.398572i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−28.6723			−1.21489			−0.607443	−	0.794363i	$-0.707805\pi$
$557$	−28.6723			−1.21489			−0.607443	+	0.794363i	$0.707805\pi$
$558$		0			0
$559$			1.61712i			0.0683970i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			29.7196i			1.25253i	0.779609	+	0.626266i	$0.215418\pi$
$563$			29.7196i			1.25253i	−0.779609	+	0.626266i	$0.784582\pi$
$564$		0			0
$565$	−4.39938			−0.185083
$566$		0			0
$567$		0			0
$568$		0			0
$569$			28.0263i			1.17492i	0.809251	+	0.587462i	$0.199873\pi$
$569$			28.0263i			1.17492i	−0.809251	+	0.587462i	$0.800127\pi$
$570$		0			0
$571$	−29.9668	−	29.9668i	−1.25407	−	1.25407i	−0.953879	−	0.300192i	$-0.902949\pi$
$571$	−29.9668	−	29.9668i	−1.25407	−	1.25407i	−0.300192	−	0.953879i	$-0.597051\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.94577	+	2.94577i	−0.122847	+	0.122847i
$576$		0			0
$577$	8.16367			0.339858			0.169929	−	0.985456i	$-0.445646\pi$
$577$	8.16367			0.339858			0.169929	+	0.985456i	$0.445646\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−47.7076	+	47.7076i	−1.97925	+	1.97925i
$582$		0			0
$583$	−6.63245	+	6.63245i	−0.274688	+	0.274688i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	13.4440i		−	0.554894i	−0.960741	−	0.277447i	$-0.910512\pi$
$587$		−	13.4440i		−	0.554894i	0.960741	−	0.277447i	$-0.0894884\pi$
$588$		0			0
$589$	9.17952	+	9.17952i	0.378235	+	0.378235i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	14.4936i		−	0.595182i	−0.954693	−	0.297591i	$-0.903817\pi$
$593$		−	14.4936i		−	0.595182i	0.954693	−	0.297591i	$-0.0961832\pi$
$594$		0			0
$595$	−21.5763	−	0.0618402i	−0.884540	−	0.00253520i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	24.1901			0.988382			0.494191	−	0.869353i	$-0.335464\pi$
$599$	24.1901			0.988382			0.494191	+	0.869353i	$0.335464\pi$
$600$		0			0
$601$	5.85477	+	5.85477i	0.238821	+	0.238821i	0.816362	−	0.577541i	$-0.195988\pi$
$601$	5.85477	+	5.85477i	0.238821	+	0.238821i	−0.577541	+	0.816362i	$0.695988\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.9781	−	11.9781i	−0.486978	−	0.486978i
$606$		0			0
$607$	−21.2283	+	21.2283i	−0.861631	+	0.861631i	−0.991528	−	0.129897i	$-0.958535\pi$
$607$	−21.2283	+	21.2283i	−0.861631	+	0.861631i	0.129897	+	0.991528i	$0.458535\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.6736			0.553173
$612$		0			0
$613$	1.77264			0.0715963			0.0357982	−	0.999359i	$-0.488603\pi$
$613$	1.77264			0.0715963			0.0357982	+	0.999359i	$0.488603\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.6189	+	19.6189i	−0.789827	+	0.789827i	−0.981466	−	0.191638i	$-0.938620\pi$
$617$	−19.6189	+	19.6189i	−0.789827	+	0.789827i	0.191638	+	0.981466i	$0.438620\pi$
$618$		0			0
$619$	33.5392	+	33.5392i	1.34806	+	1.34806i	0.887772	+	0.460284i	$0.152252\pi$
$619$	33.5392	+	33.5392i	1.34806	+	1.34806i	0.460284	+	0.887772i	$0.347748\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.3596	−	12.3596i	−0.495176	−	0.495176i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27.6923	+	27.5340i	−1.10416	+	1.09785i
$630$		0			0
$631$		−	4.14957i		−	0.165192i	−0.996583	−	0.0825960i	$-0.973679\pi$
$631$		−	4.14957i		−	0.165192i	0.996583	−	0.0825960i	$-0.0263211\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.92473	+	2.92473i	0.116064	+	0.116064i
$636$		0			0
$637$			53.7522i			2.12974i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.7918	−	23.7918i	0.939721	−	0.939721i	−0.0585627	−	0.998284i	$-0.518652\pi$
$641$	23.7918	−	23.7918i	0.939721	−	0.939721i	0.998284	+	0.0585627i	$0.0186518\pi$
$642$		0			0
$643$	9.76174	−	9.76174i	0.384966	−	0.384966i	−0.487922	−	0.872887i	$-0.662245\pi$
$643$	9.76174	−	9.76174i	0.384966	−	0.384966i	0.872887	+	0.487922i	$0.162245\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.0014			0.943594			0.471797	−	0.881707i	$-0.343605\pi$
$647$	24.0014			0.943594			0.471797	+	0.881707i	$0.343605\pi$
$648$		0			0
$649$	−29.7893	+	29.7893i	−1.16933	+	1.16933i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.75190	−	7.75190i	−0.303355	−	0.303355i	0.538970	−	0.842325i	$-0.318814\pi$
$653$	−7.75190	−	7.75190i	−0.303355	−	0.303355i	−0.842325	+	0.538970i	$0.818814\pi$
$654$		0			0
$655$			22.1002i			0.863526i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13.8658			0.540136			0.270068	−	0.962841i	$-0.412954\pi$
$659$	13.8658			0.540136			0.270068	+	0.962841i	$0.412954\pi$
$660$		0			0
$661$			2.31432i			0.0900166i	0.998987	+	0.0450083i	$0.0143314\pi$
$661$			2.31432i			0.0900166i	−0.998987	+	0.0450083i	$0.985669\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			22.5716i			0.875288i
$666$		0			0
$667$	10.6104			0.410835
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	19.2818i		−	0.744364i
$672$		0			0
$673$	−1.18174	−	1.18174i	−0.0455527	−	0.0455527i	0.683964	−	0.729516i	$-0.260255\pi$
$673$	−1.18174	−	1.18174i	−0.0455527	−	0.0455527i	−0.729516	+	0.683964i	$0.760255\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.62427	+	6.62427i	−0.254592	+	0.254592i	−0.822850	−	0.568259i	$-0.807617\pi$
$677$	−6.62427	+	6.62427i	−0.254592	+	0.254592i	0.568259	+	0.822850i	$0.307617\pi$
$678$		0			0
$679$	24.2218			0.929548
$680$		0			0
$681$		0			0
$682$		0			0
$683$	24.4455	−	24.4455i	0.935382	−	0.935382i	−0.0626534	−	0.998035i	$-0.519956\pi$
$683$	24.4455	−	24.4455i	0.935382	−	0.935382i	0.998035	+	0.0626534i	$0.0199563\pi$
$684$		0			0
$685$	−1.86457	+	1.86457i	−0.0712416	+	0.0712416i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			4.67922i			0.178264i
$690$		0			0
$691$	−13.5700	−	13.5700i	−0.516226	−	0.516226i	0.400201	−	0.916427i	$-0.368940\pi$
$691$	−13.5700	−	13.5700i	−0.516226	−	0.516226i	−0.916427	+	0.400201i	$0.868940\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			12.9759i			0.492204i
$696$		0			0
$697$	3.93266	+	3.95527i	0.148960	+	0.149816i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−31.8819			−1.20416			−0.602081	−	0.798435i	$-0.705661\pi$
$701$	−31.8819			−1.20416			−0.602081	+	0.798435i	$0.705661\pi$
$702$		0			0
$703$	28.8870	+	28.8870i	1.08949	+	1.08949i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−64.0163	−	64.0163i	−2.40758	−	2.40758i
$708$		0			0
$709$	−27.5837	+	27.5837i	−1.03593	+	1.03593i	−0.0365978	+	0.999330i	$0.511652\pi$
$709$	−27.5837	+	27.5837i	−1.03593	+	1.03593i	−0.999330	+	0.0365978i	$0.988348\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.5384			−0.469565
$714$		0			0
$715$	−13.9381			−0.521255
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.12908	−	9.12908i	0.340457	−	0.340457i	−0.516082	−	0.856539i	$-0.672610\pi$
$719$	9.12908	−	9.12908i	0.340457	−	0.340457i	0.856539	+	0.516082i	$0.172610\pi$
$720$		0			0
$721$	−66.8418	−	66.8418i	−2.48932	−	2.48932i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.80095	+	1.80095i	0.0668855	+	0.0668855i
$726$		0			0
$727$	−10.4543			−0.387728			−0.193864	−	0.981028i	$-0.562102\pi$
$727$	−10.4543			−0.387728			−0.193864	+	0.981028i	$0.562102\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.00724714	−	2.52855i	0.000268045	−	0.0935218i
$732$		0			0
$733$			32.9360i			1.21652i	0.793739	+	0.608259i	$0.208132\pi$
$733$			32.9360i			1.21652i	−0.793739	+	0.608259i	$0.791868\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	44.0980	+	44.0980i	1.62437	+	1.62437i
$738$		0			0
$739$			18.8281i			0.692601i	0.938124	+	0.346301i	$0.112562\pi$
$739$			18.8281i			0.692601i	−0.938124	+	0.346301i	$0.887438\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.5719	−	15.5719i	0.571278	−	0.571278i	−0.361207	−	0.932485i	$-0.617635\pi$
$743$	15.5719	−	15.5719i	0.571278	−	0.571278i	0.932485	+	0.361207i	$0.117635\pi$
$744$		0			0
$745$	3.86968	−	3.86968i	0.141774	−	0.141774i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−64.5407			−2.35827
$750$		0			0
$751$	−11.0755	+	11.0755i	−0.404149	+	0.404149i	−0.879692	−	0.475543i	$-0.842252\pi$
$751$	−11.0755	+	11.0755i	−0.404149	+	0.404149i	0.475543	+	0.879692i	$0.342252\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0071	+	16.0071i	0.582558	+	0.582558i
$756$		0			0
$757$			0.298820i			0.0108608i	0.999985	+	0.00543041i	$0.00172856\pi$
$757$			0.298820i			0.0108608i	−0.999985	+	0.00543041i	$0.998271\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	49.5107			1.79476			0.897381	−	0.441257i	$-0.145467\pi$
$761$	49.5107			1.79476			0.897381	+	0.441257i	$0.145467\pi$
$762$		0			0
$763$		−	30.6728i		−	1.11043i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			21.0165i			0.758861i
$768$		0			0
$769$	−28.9566			−1.04420			−0.522100	−	0.852884i	$-0.674851\pi$
$769$	−28.9566			−1.04420			−0.522100	+	0.852884i	$0.674851\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	2.56888i		−	0.0923962i	−0.998932	−	0.0461981i	$-0.985289\pi$
$773$		−	2.56888i		−	0.0923962i	0.998932	−	0.0461981i	$-0.0147105\pi$
$774$		0			0
$775$	−2.12819	−	2.12819i	−0.0764470	−	0.0764470i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.12590	−	4.12590i	0.147826	−	0.147826i
$780$		0			0
$781$	38.0275			1.36073
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.80107	+	7.80107i	−0.278432	+	0.278432i
$786$		0			0
$787$	28.8848	−	28.8848i	1.02963	−	1.02963i	0.0300851	−	0.999547i	$-0.490422\pi$
$787$	28.8848	−	28.8848i	1.02963	−	1.02963i	0.999547	−	0.0300851i	$-0.00957783\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	23.0221i		−	0.818571i
$792$		0			0
$793$	−6.80167	−	6.80167i	−0.241535	−	0.241535i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			21.4521i			0.759873i	0.925013	+	0.379937i	$0.124054\pi$
$797$			21.4521i			0.759873i	−0.925013	+	0.379937i	$0.875946\pi$
$798$		0			0
$799$	−21.3801	−	0.0612781i	−0.756375	−	0.00216786i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−34.3105			−1.21079
$804$		0			0
$805$	−15.4153	−	15.4153i	−0.543319	−	0.543319i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15.3542	−	15.3542i	−0.539825	−	0.539825i	0.383653	−	0.923477i	$-0.374666\pi$
$809$	−15.3542	−	15.3542i	−0.539825	−	0.539825i	−0.923477	+	0.383653i	$0.874666\pi$
$810$		0			0
$811$	30.4708	−	30.4708i	1.06998	−	1.06998i	0.0726162	−	0.997360i	$-0.476865\pi$
$811$	30.4708	−	30.4708i	1.06998	−	1.06998i	0.997360	−	0.0726162i	$-0.0231348\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.88026			−0.346090
$816$		0			0
$817$	−2.64519			−0.0925436
$818$		0			0
$819$		0			0
$820$		0			0
$821$	20.8172	−	20.8172i	0.726526	−	0.726526i	−0.243400	−	0.969926i	$-0.578263\pi$
$821$	20.8172	−	20.8172i	0.726526	−	0.726526i	0.969926	+	0.243400i	$0.0782628\pi$
$822$		0			0
$823$	−19.2852	−	19.2852i	−0.672241	−	0.672241i	0.285991	−	0.958232i	$-0.407677\pi$
$823$	−19.2852	−	19.2852i	−0.672241	−	0.672241i	−0.958232	+	0.285991i	$0.907677\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.15465	+	2.15465i	0.0749244	+	0.0749244i	0.743576	−	0.668652i	$-0.233128\pi$
$827$	2.15465	+	2.15465i	0.0749244	+	0.0749244i	−0.668652	+	0.743576i	$0.733128\pi$
$828$		0			0
$829$	36.6800			1.27395			0.636975	−	0.770885i	$-0.280186\pi$
$829$	36.6800			1.27395			0.636975	+	0.770885i	$0.280186\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.240891	−	84.0476i	0.00834637	−	2.91208i
$834$		0			0
$835$		−	16.1670i		−	0.559481i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.8209	−	11.8209i	−0.408101	−	0.408101i	0.472975	−	0.881076i	$-0.343180\pi$
$839$	−11.8209	−	11.8209i	−0.408101	−	0.408101i	−0.881076	+	0.472975i	$0.843180\pi$
$840$		0			0
$841$			22.5132i			0.776316i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.27570	−	4.27570i	0.147089	−	0.147089i
$846$		0			0
$847$	62.6816	−	62.6816i	2.15377	−	2.15377i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−39.4569			−1.35256
$852$		0			0
$853$	−0.894962	+	0.894962i	−0.0306429	+	0.0306429i	−0.722262	−	0.691619i	$-0.756898\pi$
$853$	−0.894962	+	0.894962i	−0.0306429	+	0.0306429i	0.691619	+	0.722262i	$0.256898\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.83375	−	8.83375i	−0.301755	−	0.301755i	0.539945	−	0.841700i	$-0.318445\pi$
$857$	−8.83375	−	8.83375i	−0.301755	−	0.301755i	−0.841700	+	0.539945i	$0.818445\pi$
$858$		0			0
$859$			30.5126i			1.04108i	0.853839	+	0.520538i	$0.174269\pi$
$859$			30.5126i			1.04108i	−0.853839	+	0.520538i	$0.825731\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.4910			0.833685			0.416842	−	0.908979i	$-0.363137\pi$
$863$	24.4910			0.833685			0.416842	+	0.908979i	$0.363137\pi$
$864$		0			0
$865$		−	6.01359i		−	0.204468i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			9.46728i			0.321155i
$870$		0			0
$871$	31.1113			1.05417
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	5.23303i		−	0.176909i
$876$		0			0
$877$	−17.6487	−	17.6487i	−0.595953	−	0.595953i	0.343280	−	0.939233i	$-0.388462\pi$
$877$	−17.6487	−	17.6487i	−0.595953	−	0.595953i	−0.939233	+	0.343280i	$0.888462\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.3106	+	24.3106i	−0.819045	+	0.819045i	−0.985970	−	0.166925i	$-0.946616\pi$
$881$	−24.3106	+	24.3106i	−0.819045	+	0.819045i	0.166925	+	0.985970i	$0.446616\pi$
$882$		0			0
$883$	−32.9362			−1.10839			−0.554196	−	0.832386i	$-0.686974\pi$
$883$	−32.9362			−1.10839			−0.554196	+	0.832386i	$0.686974\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.9126	−	20.9126i	0.702175	−	0.702175i	−0.262702	−	0.964877i	$-0.584614\pi$
$887$	20.9126	−	20.9126i	0.702175	−	0.702175i	0.964877	+	0.262702i	$0.0846136\pi$
$888$		0			0
$889$	−15.3052	+	15.3052i	−0.513320	+	0.513320i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			22.3664i			0.748463i
$894$		0			0
$895$	8.52801	+	8.52801i	0.285060	+	0.285060i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			7.66553i			0.255660i
$900$		0			0
$901$	0.0209699	−	7.31648i	0.000698609	−	0.243747i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.47549			0.115529
$906$		0			0
$907$	1.70827	+	1.70827i	0.0567222	+	0.0567222i	0.734899	−	0.678177i	$-0.237230\pi$
$907$	1.70827	+	1.70827i	0.0567222	+	0.0567222i	−0.678177	+	0.734899i	$0.737230\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	38.2691	+	38.2691i	1.26791	+	1.26791i	0.947164	+	0.320749i	$0.103935\pi$
$911$	38.2691	+	38.2691i	1.26791	+	1.26791i	0.320749	+	0.947164i	$0.396065\pi$
$912$		0			0
$913$	48.1886	−	48.1886i	1.59481	−	1.59481i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−115.651			−3.81914
$918$		0			0
$919$	−9.64435			−0.318138			−0.159069	−	0.987267i	$-0.550849\pi$
$919$	−9.64435			−0.318138			−0.159069	+	0.987267i	$0.550849\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.4142	−	13.4142i	0.441535	−	0.441535i
$924$		0			0
$925$	−6.69720	−	6.69720i	−0.220203	−	0.220203i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20.8704	+	20.8704i	0.684737	+	0.684737i	0.961064	−	0.276327i	$-0.0891171\pi$
$929$	20.8704	+	20.8704i	0.684737	+	0.684737i	−0.276327	+	0.961064i	$0.589117\pi$
$930$		0			0
$931$	−87.9248			−2.88162
$932$		0			0
$933$		0			0
$934$		0			0
$935$	21.7938	+	0.0624636i	0.712732	+	0.00204278i
$936$		0			0
$937$		−	8.00891i		−	0.261640i	−0.991406	−	0.130820i	$-0.958239\pi$
$937$		−	8.00891i		−	0.261640i	0.991406	−	0.130820i	$-0.0417609\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	33.4390	+	33.4390i	1.09008	+	1.09008i	0.995519	+	0.0945620i	$0.0301451\pi$
$941$	33.4390	+	33.4390i	1.09008	+	1.09008i	0.0945620	+	0.995519i	$0.469855\pi$
$942$		0			0
$943$			5.63559i			0.183520i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.6772	−	19.6772i	0.639424	−	0.639424i	−0.310990	−	0.950413i	$-0.600660\pi$
$947$	19.6772	−	19.6772i	0.639424	−	0.639424i	0.950413	+	0.310990i	$0.100660\pi$
$948$		0			0
$949$	−12.1031	+	12.1031i	−0.392883	+	0.392883i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	32.2796			1.04564			0.522819	−	0.852443i	$-0.324880\pi$
$953$	32.2796			1.04564			0.522819	+	0.852443i	$0.324880\pi$
$954$		0			0
$955$	−3.37903	+	3.37903i	−0.109343	+	0.109343i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9.75736	−	9.75736i	−0.315082	−	0.315082i
$960$		0			0
$961$			21.9416i			0.707793i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	22.7383			0.731973
$966$		0			0
$967$		−	60.1211i		−	1.93337i	−0.255978	−	0.966683i	$-0.582397\pi$
$967$		−	60.1211i		−	1.93337i	0.255978	−	0.966683i	$-0.417603\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			56.9650i			1.82809i	0.405608	+	0.914047i	$0.367060\pi$
$971$			56.9650i			1.82809i	−0.405608	+	0.914047i	$0.632940\pi$
$972$		0			0
$973$	−67.9033			−2.17688
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	24.2881i		−	0.777045i	−0.921439	−	0.388522i	$-0.872986\pi$
$977$		−	24.2881i		−	0.777045i	0.921439	−	0.388522i	$-0.127014\pi$
$978$		0			0
$979$	12.4842	+	12.4842i	0.398996	+	0.398996i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.24397	+	4.24397i	−0.135362	+	0.135362i	−0.771541	−	0.636179i	$-0.780514\pi$
$983$	−4.24397	+	4.24397i	−0.135362	+	0.135362i	0.636179	+	0.771541i	$0.280514\pi$
$984$		0			0
$985$	−12.6763			−0.403900
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.80654	−	1.80654i	0.0574447	−	0.0574447i
$990$		0			0
$991$	27.5531	−	27.5531i	0.875254	−	0.875254i	−0.117785	−	0.993039i	$-0.537579\pi$
$991$	27.5531	−	27.5531i	0.875254	−	0.875254i	0.993039	+	0.117785i	$0.0375794\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			6.16604i			0.195477i
$996$		0			0
$997$	−19.1160	−	19.1160i	−0.605410	−	0.605410i	0.336333	−	0.941743i	$-0.390813\pi$
$997$	−19.1160	−	19.1160i	−0.605410	−	0.605410i	−0.941743	+	0.336333i	$0.890813\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.be.b.361.1		12
3.2	odd	2		340.2.o.a.21.5	✓	12
12.11	even	2		1360.2.bt.c.1041.2		12
15.2	even	4		1700.2.m.c.1449.2		12
15.8	even	4		1700.2.m.f.1449.5		12
15.14	odd	2		1700.2.o.d.701.2		12
17.13	even	4	inner	3060.2.be.b.1441.1		12
51.2	odd	8		5780.2.c.h.5201.2		12
51.8	odd	8		5780.2.a.n.1.1		6
51.26	odd	8		5780.2.a.m.1.6		6
51.32	odd	8		5780.2.c.h.5201.11		12
51.47	odd	4		340.2.o.a.81.5	yes	12
204.47	even	4		1360.2.bt.c.81.2		12
255.47	even	4		1700.2.m.f.149.5		12
255.98	even	4		1700.2.m.c.149.2		12
255.149	odd	4		1700.2.o.d.1101.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.o.a.21.5	✓	12	3.2	odd	2
340.2.o.a.81.5	yes	12	51.47	odd	4
1360.2.bt.c.81.2		12	204.47	even	4
1360.2.bt.c.1041.2		12	12.11	even	2
1700.2.m.c.149.2		12	255.98	even	4
1700.2.m.c.1449.2		12	15.2	even	4
1700.2.m.f.149.5		12	255.47	even	4
1700.2.m.f.1449.5		12	15.8	even	4
1700.2.o.d.701.2		12	15.14	odd	2
1700.2.o.d.1101.2		12	255.149	odd	4
3060.2.be.b.361.1		12	1.1	even	1	trivial
3060.2.be.b.1441.1		12	17.13	even	4	inner
5780.2.a.m.1.6		6	51.26	odd	8
5780.2.a.n.1.1		6	51.8	odd	8
5780.2.c.h.5201.2		12	51.2	odd	8
5780.2.c.h.5201.11		12	51.32	odd	8

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:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.be (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{5} +(-3.70031 - 3.70031i) q^{7} +(3.73762 + 3.73762i) q^{11} +2.63690 q^{13} +(-4.12309 - 0.0118173i) q^{17} +4.31329i q^{19} +(-2.94577 - 2.94577i) q^{23} -1.00000i q^{25} +(-1.80095 + 1.80095i) q^{29} +(2.12819 - 2.12819i) q^{31} +5.23303 q^{35} +(6.69720 - 6.69720i) q^{37} +(-0.956555 - 0.956555i) q^{41} +0.613266i q^{43} +5.18546 q^{47} +20.3846i q^{49} +1.77451i q^{53} -5.28579 q^{55} +7.97014i q^{59} +(-2.57942 - 2.57942i) q^{61} +(-1.86457 + 1.86457i) q^{65} +11.7984 q^{67} +(5.08713 - 5.08713i) q^{71} +(-4.58989 + 4.58989i) q^{73} -27.6607i q^{77} +(1.26649 + 1.26649i) q^{79} -12.8929i q^{83} +(2.92382 - 2.90711i) q^{85} +3.34014 q^{89} +(-9.75736 - 9.75736i) q^{91} +(-3.04995 - 3.04995i) q^{95} +(-3.27294 + 3.27294i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{11} + 16 q^{13} + 4 q^{17} - 12 q^{23} - 4 q^{29} - 8 q^{31} + 8 q^{35} - 20 q^{37} - 8 q^{41} + 8 q^{47} + 16 q^{55} - 8 q^{61} - 4 q^{65} + 32 q^{67} - 28 q^{71} - 24 q^{79} - 4 q^{85} + 56 q^{89}+ \cdots - 44 q^{97}+O(q^{100}) \) 12 * q + 12 * q^11 + 16 * q^13 + 4 * q^17 - 12 * q^23 - 4 * q^29 - 8 * q^31 + 8 * q^35 - 20 * q^37 - 8 * q^41 + 8 * q^47 + 16 * q^55 - 8 * q^61 - 4 * q^65 + 32 * q^67 - 28 * q^71 - 24 * q^79 - 4 * q^85 + 56 * q^89 - 4 * q^91 - 8 * q^95 - 44 * q^97

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