Embedded newform 11.10.c.a.4.8

• Label: 11.10.c.a.4.8
• Level: $11$
• Weight: $10$
• Character: 11.4
• Analytic conductor: $5.665$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.66539419780\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			4.8
Character	\(\chi\)	\(=\)	11.4
Dual form			11.10.c.a.3.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(35.8136 + 26.0201i) q^{2} +(63.0538 - 194.060i) q^{3} +(447.352 + 1376.81i) q^{4} +(376.543 - 273.575i) q^{5} +(7307.64 - 5309.31i) q^{6} +(-442.773 - 1362.72i) q^{7} +(-12799.4 + 39392.6i) q^{8} +(-17759.5 - 12903.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	35.8136	+	26.0201i	1.58275	+	1.14994i	0.913453	+	0.406945i	\(0.133406\pi\)
							0.669300	+	0.742992i	\(0.266594\pi\)
\(3\)	63.0538	−	194.060i	0.449434	−	1.38321i	−0.428114	−	0.903725i	\(-0.640822\pi\)
							0.877548	−	0.479490i	\(-0.159178\pi\)
\(5\)	376.543	−	273.575i	0.269432	−	0.195754i	−0.444863	−	0.895599i	\(-0.646747\pi\)
							0.714295	+	0.699845i	\(0.246747\pi\)
\(7\)	−442.773	−	1362.72i	−0.0697012	−	0.214518i	0.910138	−	0.414305i	\(-0.135975\pi\)
							−0.979839	+	0.199786i	\(0.935975\pi\)
\(11\)	−48486.9	+	2640.41i	−0.998521	+	0.0543755i
							\(13\)	−64523.9	−	46879.4i	−0.626579	−	0.455236i	0.228634	−	0.973512i	\(-0.426574\pi\)
−0.855213	+	0.518276i	\(0.826574\pi\)
\(17\)	291931.	−	212100.i	0.847736	−	0.615916i	−0.0767850	−	0.997048i	\(-0.524466\pi\)
							0.924521	+	0.381132i	\(0.124466\pi\)
\(19\)	85194.7	−	262202.i	0.149976	−	0.461578i	−0.847641	−	0.530569i	\(-0.821978\pi\)
							0.997617	+	0.0689914i	\(0.0219781\pi\)
\(23\)	−157157.			−0.117101			−0.0585504	−	0.998284i	\(-0.518648\pi\)
							−0.0585504	+	0.998284i	\(0.518648\pi\)
\(29\)	275918.	+	849189.i	0.0724419	+	0.222953i	0.980722	−	0.195410i	\(-0.0626039\pi\)
							−0.908280	+	0.418363i	\(0.862604\pi\)
\(31\)	1.84539e6	+	1.34075e6i	0.358889	+	0.260748i	0.752589	−	0.658491i	\(-0.228805\pi\)
							−0.393699	+	0.919239i	\(0.628805\pi\)
\(37\)	2.38092e6	+	7.32771e6i	0.208851	+	0.642777i	0.999533	+	0.0305505i	\(0.00972605\pi\)
							−0.790682	+	0.612227i	\(0.790274\pi\)
\(41\)	−5.47307e6	+	1.68444e7i	−0.302485	+	0.930953i	0.678119	+	0.734953i	\(0.262796\pi\)
							−0.980604	+	0.196001i	\(0.937204\pi\)
\(43\)	5.17055e6			0.230637			0.115318	−	0.993329i	\(-0.463211\pi\)
							0.115318	+	0.993329i	\(0.463211\pi\)
\(47\)	1.18048e7	−	3.63315e7i	0.352873	−	1.08603i	−0.604359	−	0.796712i	\(-0.706571\pi\)
							0.957232	−	0.289320i	\(-0.0934292\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.4.8	yes	32
3.2	odd	2		99.10.f.a.37.1		32
11.3	even	5	inner	11.10.c.a.3.8	✓	32
11.5	even	5		121.10.a.h.1.1		16
11.6	odd	10		121.10.a.i.1.16		16
33.14	odd	10		99.10.f.a.91.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.3.8	✓	32	11.3	even	5	inner
11.10.c.a.4.8	yes	32	1.1	even	1	trivial
99.10.f.a.37.1		32	3.2	odd	2
99.10.f.a.91.1		32	33.14	odd	10
121.10.a.h.1.1		16	11.5	even	5
121.10.a.i.1.16		16	11.6	odd	10