Embedded newform 55.4.g.b.31.3

• Label: 55.4.g.b.31.3
• Level: $55$
• Weight: $4$
• Character: 55.31
• Analytic conductor: $3.245$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	55.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.24510505032\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			31.3
Character	\(\chi\)	\(=\)	55.31
Dual form			55.4.g.b.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.513572 + 1.58061i) q^{2} +(-1.60233 + 1.16416i) q^{3} +(4.23755 + 3.07876i) q^{4} +(1.54508 + 4.75528i) q^{5} +(-1.01717 - 3.13054i) q^{6} +(-7.13418 - 5.18329i) q^{7} +(-17.7990 + 12.9317i) q^{8} +(-7.13127 + 21.9478i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{2} + 4 q^{3} - 36 q^{4} - 30 q^{5} + 30 q^{6} + 81 q^{7} - 10 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\)	\(12\)	\(46\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.513572	+	1.58061i	−0.181575	+	0.558831i	−0.999873	−	0.0159640i	\(-0.994918\pi\)
							0.818297	+	0.574795i	\(0.194918\pi\)
\(3\)	−1.60233	+	1.16416i	−0.308368	+	0.224042i	−0.731196	−	0.682168i	\(-0.761037\pi\)
							0.422828	+	0.906210i	\(0.361037\pi\)
\(5\)	1.54508	+	4.75528i	0.138197	+	0.425325i
							\(7\)	−7.13418	−	5.18329i	−0.385210	−	0.279871i	0.378280	−	0.925691i	\(-0.376516\pi\)
−0.763490	+	0.645820i	\(0.776516\pi\)
\(11\)	5.41658	+	36.0785i	0.148469	+	0.988917i
							\(13\)	3.19453	−	9.83177i	0.0681542	−	0.209757i	−0.911179	−	0.412011i	\(-0.864827\pi\)
0.979333	+	0.202254i	\(0.0648266\pi\)
\(17\)	−15.0140	−	46.2082i	−0.214201	−	0.659243i	−0.999209	−	0.0397580i	\(-0.987341\pi\)
							0.785008	−	0.619485i	\(-0.212659\pi\)
\(19\)	78.6063	−	57.1108i	0.949133	−	0.689585i	−0.00146880	−	0.999999i	\(-0.500468\pi\)
							0.950602	+	0.310414i	\(0.100468\pi\)
\(23\)	196.403			1.78056			0.890281	−	0.455411i	\(-0.150508\pi\)
							0.890281	+	0.455411i	\(0.150508\pi\)
\(29\)	136.759	+	99.3611i	0.875706	+	0.636238i	0.932112	−	0.362170i	\(-0.117964\pi\)
							−0.0564061	+	0.998408i	\(0.517964\pi\)
\(31\)	12.6165	−	38.8296i	0.0730965	−	0.224968i	−0.907833	−	0.419332i	\(-0.862264\pi\)
							0.980929	+	0.194364i	\(0.0622643\pi\)
\(37\)	−118.029	−	85.7530i	−0.524428	−	0.381019i	0.293841	−	0.955854i	\(-0.405066\pi\)
							−0.818269	+	0.574835i	\(0.805066\pi\)
\(41\)	127.285	−	92.4781i	0.484844	−	0.352260i	−0.318354	−	0.947972i	\(-0.603130\pi\)
							0.803198	+	0.595712i	\(0.203130\pi\)
\(43\)	266.051			0.943546			0.471773	−	0.881720i	\(-0.343614\pi\)
							0.471773	+	0.881720i	\(0.343614\pi\)
\(47\)	−390.791	+	283.926i	−1.21282	+	0.881167i	−0.995484	−	0.0949290i	\(-0.969738\pi\)
							−0.217339	+	0.976096i	\(0.569738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.4.g.b.31.3	yes	24
11.4	even	5		605.4.a.p.1.6		12
11.5	even	5	inner	55.4.g.b.16.3	✓	24
11.7	odd	10		605.4.a.t.1.7		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.g.b.16.3	✓	24	11.5	even	5	inner
55.4.g.b.31.3	yes	24	1.1	even	1	trivial
605.4.a.p.1.6		12	11.4	even	5
605.4.a.t.1.7		12	11.7	odd	10