Embedded newform 11.10.c.a.3.7

• Label: 11.10.c.a.3.7
• Level: $11$
• Weight: $10$
• Character: 11.3
• 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			3.7
Character	\(\chi\)	\(=\)	11.3
Dual form			11.10.c.a.4.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(24.5942 - 17.8687i) q^{2} +(-45.7664 - 140.854i) q^{3} +(127.366 - 391.991i) q^{4} +(-601.873 - 437.286i) q^{5} +(-3642.47 - 2646.41i) q^{6} +(430.583 - 1325.20i) q^{7} +(937.875 + 2886.48i) q^{8} +(-1821.51 + 1323.41i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 21 q^{2} + 69 q^{3} - 1625 q^{4} + 225 q^{5} + 8173 q^{6} - 10675 q^{7} + 22863 q^{8} - 93587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	24.5942	−	17.8687i	1.08692	−	0.789693i	0.108043	−	0.994146i	\(-0.465542\pi\)
							0.978876	+	0.204453i	\(0.0655417\pi\)
\(3\)	−45.7664	−	140.854i	−0.326213	−	1.00398i	−0.970890	−	0.239525i	\(-0.923008\pi\)
							0.644678	−	0.764454i	\(-0.276992\pi\)
\(5\)	−601.873	−	437.286i	−0.430665	−	0.312897i	0.351250	−	0.936282i	\(-0.385757\pi\)
							−0.781915	+	0.623385i	\(0.785757\pi\)
\(7\)	430.583	−	1325.20i	0.0677821	−	0.208612i	−0.911428	−	0.411459i	\(-0.865019\pi\)
							0.979210	+	0.202847i	\(0.0650194\pi\)
\(11\)	2766.66	−	48479.8i	0.0569756	−	0.998376i
							\(13\)	72198.0	−	52454.9i	0.701100	−	0.509379i	−0.179190	−	0.983814i	\(-0.557348\pi\)
0.880290	+	0.474435i	\(0.157348\pi\)
\(17\)	237799.	+	172771.i	0.690541	+	0.501708i	0.876838	−	0.480786i	\(-0.159649\pi\)
							−0.186297	+	0.982494i	\(0.559649\pi\)
\(19\)	189386.	+	582870.i	0.333393	+	1.02608i	0.967508	+	0.252840i	\(0.0813645\pi\)
							−0.634115	+	0.773239i	\(0.718635\pi\)
\(23\)	385699.			0.287391			0.143696	−	0.989622i	\(-0.454101\pi\)
							0.143696	+	0.989622i	\(0.454101\pi\)
\(29\)	−1.33361e6	+	4.10443e6i	−0.350137	+	1.07761i	0.608639	+	0.793447i	\(0.291716\pi\)
							−0.958776	+	0.284164i	\(0.908284\pi\)
\(31\)	−4.15212e6	+	3.01669e6i	−0.807500	+	0.586683i	−0.913105	−	0.407725i	\(-0.866322\pi\)
							0.105605	+	0.994408i	\(0.466322\pi\)
\(37\)	5.92043e6	−	1.82212e7i	0.519332	−	1.59834i	−0.255926	−	0.966696i	\(-0.582380\pi\)
							0.775258	−	0.631644i	\(-0.217620\pi\)
\(41\)	7.78190e6	+	2.39502e7i	0.430089	+	1.32368i	0.898036	+	0.439921i	\(0.144994\pi\)
							−0.467947	+	0.883756i	\(0.655006\pi\)
\(43\)	−2.90204e7			−1.29448			−0.647239	−	0.762287i	\(-0.724076\pi\)
							−0.647239	+	0.762287i	\(0.724076\pi\)
\(47\)	8.55528e6	+	2.63304e7i	0.255737	+	0.787078i	0.993684	+	0.112219i	\(0.0357957\pi\)
							−0.737946	+	0.674859i	\(0.764204\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.10.c.a.3.7	✓	32
3.2	odd	2		99.10.f.a.91.2		32
11.2	odd	10		121.10.a.i.1.13		16
11.4	even	5	inner	11.10.c.a.4.7	yes	32
11.9	even	5		121.10.a.h.1.4		16
33.26	odd	10		99.10.f.a.37.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.c.a.3.7	✓	32	1.1	even	1	trivial
11.10.c.a.4.7	yes	32	11.4	even	5	inner
99.10.f.a.37.2		32	33.26	odd	10
99.10.f.a.91.2		32	3.2	odd	2
121.10.a.h.1.4		16	11.9	even	5
121.10.a.i.1.13		16	11.2	odd	10