Embedded newform 55.4.g.b.31.6

• Label: 55.4.g.b.31.6
• 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			31.6
Character	\(\chi\)	\(=\)	55.31
Dual form			55.4.g.b.16.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.50387 - 4.62843i) q^{2} +(4.25544 - 3.09176i) q^{3} +(-12.6886 - 9.21884i) q^{4} +(1.54508 + 4.75528i) q^{5} +(-7.91037 - 24.3456i) q^{6} +(4.59294 + 3.33697i) q^{7} +(-30.2534 + 21.9804i) q^{8} +(0.206327 - 0.635008i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{2} + 4 q^{3} - 36 q^{4} - 30 q^{5} + 30 q^{6} + 81 q^{7} - 10 q^{8} - 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.50387	−	4.62843i	0.531698	−	1.63640i	−0.218979	−	0.975730i	\(-0.570273\pi\)
							0.750677	−	0.660669i	\(-0.229727\pi\)
\(3\)	4.25544	−	3.09176i	0.818959	−	0.595009i	−0.0974551	−	0.995240i	\(-0.531070\pi\)
							0.916414	+	0.400231i	\(0.131070\pi\)
\(5\)	1.54508	+	4.75528i	0.138197	+	0.425325i
							\(7\)	4.59294	+	3.33697i	0.247996	+	0.180179i	0.704838	−	0.709368i	\(-0.251020\pi\)
−0.456842	+	0.889548i	\(0.651020\pi\)
\(11\)	33.8148	−	13.6951i	0.926869	−	0.375385i
							\(13\)	−24.4801	+	75.3419i	−0.522273	+	1.60739i	0.247373	+	0.968920i	\(0.420433\pi\)
−0.769646	+	0.638471i	\(0.779567\pi\)
\(17\)	−28.7566	−	88.5037i	−0.410264	−	1.26266i	−0.916419	−	0.400221i	\(-0.868933\pi\)
							0.506154	−	0.862443i	\(-0.331067\pi\)
\(19\)	6.75924	−	4.91087i	0.0816145	−	0.0592964i	−0.546230	−	0.837635i	\(-0.683937\pi\)
							0.627844	+	0.778339i	\(0.283937\pi\)
\(23\)	25.5926			0.232018			0.116009	−	0.993248i	\(-0.462990\pi\)
							0.116009	+	0.993248i	\(0.462990\pi\)
\(29\)	97.5933	+	70.9057i	0.624918	+	0.454029i	0.854636	−	0.519228i	\(-0.173780\pi\)
							−0.229718	+	0.973257i	\(0.573780\pi\)
\(31\)	50.9402	−	156.778i	0.295133	−	0.908327i	−0.688043	−	0.725670i	\(-0.741530\pi\)
							0.983177	−	0.182657i	\(-0.0584699\pi\)
\(37\)	89.4056	+	64.9570i	0.397248	+	0.288618i	0.768419	−	0.639947i	\(-0.221044\pi\)
							−0.371171	+	0.928565i	\(0.621044\pi\)
\(41\)	−389.354	+	282.882i	−1.48310	+	1.07753i	−0.506553	+	0.862209i	\(0.669081\pi\)
							−0.976543	+	0.215324i	\(0.930919\pi\)
\(43\)	−341.001			−1.20935			−0.604677	−	0.796471i	\(-0.706698\pi\)
							−0.604677	+	0.796471i	\(0.706698\pi\)
\(47\)	65.5726	−	47.6413i	0.203505	−	0.147855i	−0.481365	−	0.876520i	\(-0.659859\pi\)
							0.684870	+	0.728665i	\(0.259859\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	55.4.g.b.31.6	yes	24
11.4	even	5		605.4.a.p.1.12		12
11.5	even	5	inner	55.4.g.b.16.6	✓	24
11.7	odd	10		605.4.a.t.1.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.g.b.16.6	✓	24	11.5	even	5	inner
55.4.g.b.31.6	yes	24	1.1	even	1	trivial
605.4.a.p.1.12		12	11.4	even	5
605.4.a.t.1.1		12	11.7	odd	10