Embedded newform 61.8.k.a.4.14

• Label: 61.8.k.a.4.14
• Level: $61$
• Weight: $8$
• Character: 61.4
• Analytic conductor: $19.055$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	61.k (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0554865545\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			4.14
Character	\(\chi\)	\(=\)	61.4
Dual form			61.8.k.a.46.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.93518 - 0.834021i) q^{2} +(-69.1989 - 50.2759i) q^{3} +(-62.9314 - 13.3765i) q^{4} +(-215.468 + 239.302i) q^{5} +(507.174 + 456.662i) q^{6} +(51.7877 - 116.317i) q^{7} +(1459.53 + 474.229i) q^{8} +(1585.00 + 4878.11i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 7 q^{2} - 60 q^{3} - 2237 q^{4} - 1098 q^{5} + 2039 q^{6} + 5925 q^{7} - 10 q^{8} - 44400 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/61\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{30}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.93518	−	0.834021i	−0.701377	−	0.0737177i	−0.252874	−	0.967499i	\(-0.581376\pi\)
							−0.448503	+	0.893781i	\(0.648043\pi\)
\(3\)	−69.1989	−	50.2759i	−1.47970	−	1.07507i	−0.977652	−	0.210229i	\(-0.932579\pi\)
							−0.502051	−	0.864838i	\(-0.667421\pi\)
\(5\)	−215.468	+	239.302i	−0.770883	+	0.856152i	−0.992908	−	0.118889i	\(-0.962067\pi\)
							0.222024	+	0.975041i	\(0.428734\pi\)
\(7\)	51.7877	−	116.317i	0.0570668	−	0.128174i	−0.882761	−	0.469822i	\(-0.844318\pi\)
							0.939828	+	0.341647i	\(0.110985\pi\)
\(11\)			4543.70i			1.02928i	0.857405	+	0.514642i	\(0.172075\pi\)
							−0.857405	+	0.514642i	\(0.827925\pi\)
\(13\)	−5698.15	−	9869.49i	−0.719337	−	1.24593i	−0.961263	−	0.275633i	\(-0.911112\pi\)
							0.241926	−	0.970295i	\(-0.422221\pi\)
\(17\)	−5012.59	+	23582.4i	−0.247452	+	1.16417i	0.662371	+	0.749176i	\(0.269550\pi\)
							−0.909824	+	0.414995i	\(0.863783\pi\)
\(19\)	−38516.1	+	17148.5i	−1.28826	+	0.573571i	−0.932556	−	0.361026i	\(-0.882427\pi\)
							−0.355707	+	0.934598i	\(0.615760\pi\)
\(23\)	−100022.	+	32499.1i	−1.71415	+	0.556960i	−0.991015	−	0.133751i	\(-0.957298\pi\)
							−0.723131	+	0.690711i	\(0.757298\pi\)
\(29\)	205138.	+	118437.i	1.56190	+	0.901765i	0.997065	+	0.0765625i	\(0.0243945\pi\)
							0.564837	+	0.825202i	\(0.308939\pi\)
\(31\)	40781.6	−	4286.32i	0.245866	−	0.0258416i	0.0192062	−	0.999816i	\(-0.493886\pi\)
							0.226660	+	0.973974i	\(0.427219\pi\)
\(37\)	−191462.	−	263525.i	−0.621408	−	0.855295i	0.376046	−	0.926601i	\(-0.377283\pi\)
							−0.997455	+	0.0713057i	\(0.977283\pi\)
\(41\)	263060.	−	191125.i	0.596090	−	0.433085i	−0.248399	−	0.968658i	\(-0.579904\pi\)
							0.844489	+	0.535573i	\(0.179904\pi\)
\(43\)	−25498.7	−	119962.i	−0.0489079	−	0.230094i	0.946904	−	0.321515i	\(-0.104192\pi\)
							−0.995812	+	0.0914217i	\(0.970859\pi\)
\(47\)	−273719.	+	474095.i	−0.384559	+	0.666075i	−0.991708	−	0.128513i	\(-0.958980\pi\)
							0.607149	+	0.794588i	\(0.292313\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.8.k.a.4.14	✓	280
61.46	even	30	inner	61.8.k.a.46.14	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.8.k.a.4.14	✓	280	1.1	even	1	trivial
61.8.k.a.46.14	yes	280	61.46	even	30	inner