Embedded newform 11.10.c.a.3.8

• Label: 11.10.c.a.3.8
• Level: $11$
• Weight: $10$
• Character: 11.3
• Analytic conductor: $5.665$
• Analytic rank: $0$
• Dimension: $32$
• 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			3.8
Character	\(\chi\)	\(=\)	11.3
Dual form			11.10.c.a.4.8

$q$-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 \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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	0.669300	−	0.742992i	\(-0.266594\pi\)
							0.913453	−	0.406945i	\(-0.133406\pi\)
\(3\)	63.0538	+	194.060i	0.449434	+	1.38321i	0.877548	+	0.479490i	\(0.159178\pi\)
							−0.428114	+	0.903725i	\(0.640822\pi\)
\(5\)	376.543	+	273.575i	0.269432	+	0.195754i	0.714295	−	0.699845i	\(-0.246747\pi\)
							−0.444863	+	0.895599i	\(0.646747\pi\)
\(7\)	−442.773	+	1362.72i	−0.0697012	+	0.214518i	−0.979839	−	0.199786i	\(-0.935975\pi\)
							0.910138	+	0.414305i	\(0.135975\pi\)
\(11\)	−48486.9	−	2640.41i	−0.998521	−	0.0543755i
							\(13\)	−64523.9	+	46879.4i	−0.626579	+	0.455236i	−0.855213	−	0.518276i	\(-0.826574\pi\)
0.228634	+	0.973512i	\(0.426574\pi\)
\(17\)	291931.	+	212100.i	0.847736	+	0.615916i	0.924521	−	0.381132i	\(-0.124466\pi\)
							−0.0767850	+	0.997048i	\(0.524466\pi\)
\(19\)	85194.7	+	262202.i	0.149976	+	0.461578i	0.997617	−	0.0689914i	\(-0.0219781\pi\)
							−0.847641	+	0.530569i	\(0.821978\pi\)
\(23\)	−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	−0.908280	−	0.418363i	\(-0.862604\pi\)
							0.980722	+	0.195410i	\(0.0626039\pi\)
\(31\)	1.84539e6	−	1.34075e6i	0.358889	−	0.260748i	−0.393699	−	0.919239i	\(-0.628805\pi\)
							0.752589	+	0.658491i	\(0.228805\pi\)
\(37\)	2.38092e6	−	7.32771e6i	0.208851	−	0.642777i	−0.790682	−	0.612227i	\(-0.790274\pi\)
							0.999533	−	0.0305505i	\(-0.00972605\pi\)
\(41\)	−5.47307e6	−	1.68444e7i	−0.302485	−	0.930953i	−0.980604	−	0.196001i	\(-0.937204\pi\)
							0.678119	−	0.734953i	\(-0.262796\pi\)
\(43\)	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	0.957232	+	0.289320i	\(0.0934292\pi\)
							−0.604359	+	0.796712i	\(0.706571\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.3.8	✓	32	1.1	even	1	trivial
11.10.c.a.4.8	yes	32	11.4	even	5	inner
99.10.f.a.37.1		32	33.26	odd	10
99.10.f.a.91.1		32	3.2	odd	2
121.10.a.h.1.1		16	11.9	even	5
121.10.a.i.1.16		16	11.2	odd	10