Embedded newform 11.14.c.a.4.1

• Label: 11.14.c.a.4.1
• Level: $11$
• Weight: $14$
• Character: 11.4
• Analytic conductor: $11.795$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	11.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.7954021847\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	11.4
Dual form			11.14.c.a.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-133.057 - 96.6717i) q^{2} +(-300.655 + 925.320i) q^{3} +(5827.33 + 17934.7i) q^{4} +(6362.73 - 4622.79i) q^{5} +(129456. - 94055.7i) q^{6} +(-62415.4 - 192095. i) q^{7} +(542063. - 1.66830e6i) q^{8} +(524011. + 380716. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 59 q^{2} - 864 q^{3} - 44697 q^{4} - 52240 q^{5} - 130115 q^{6} + 390696 q^{7} + 876527 q^{8} - 3520244 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−133.057	−	96.6717i	−1.47009	−	1.06808i	−0.980588	−	0.196078i	\(-0.937179\pi\)
							−0.489500	−	0.872003i	\(-0.662821\pi\)
\(3\)	−300.655	+	925.320i	−0.238111	+	0.732831i	0.758582	+	0.651577i	\(0.225892\pi\)
							−0.996693	+	0.0812535i	\(0.974108\pi\)
\(5\)	6362.73	−	4622.79i	0.182112	−	0.132312i	−0.492994	−	0.870032i	\(-0.664098\pi\)
							0.675106	+	0.737720i	\(0.264098\pi\)
\(7\)	−62415.4	−	192095.i	−0.200518	−	0.617132i	−0.999868	−	0.0162657i	\(-0.994822\pi\)
							0.799349	−	0.600867i	\(-0.205178\pi\)
\(11\)	3.48525e6	+	4.73030e6i	0.593172	+	0.805075i
							\(13\)	−8.64075e6	−	6.27787e6i	−0.496500	−	0.360729i	0.311178	−	0.950352i	\(-0.399276\pi\)
−0.807679	+	0.589623i	\(0.799276\pi\)
\(17\)	−1.08903e8	+	7.91226e7i	−1.09426	+	0.795028i	−0.980114	−	0.198435i	\(-0.936414\pi\)
							−0.114148	+	0.993464i	\(0.536414\pi\)
\(19\)	8.65812e7	−	2.66470e8i	0.422207	−	1.29942i	−0.483437	−	0.875379i	\(-0.660612\pi\)
							0.905644	−	0.424040i	\(-0.139388\pi\)
\(23\)	−8.47834e8			−1.19421			−0.597104	−	0.802164i	\(-0.703682\pi\)
							−0.597104	+	0.802164i	\(0.703682\pi\)
\(29\)	−1.47122e9	−	4.52794e9i	−0.459293	−	1.41356i	−0.866021	−	0.500008i	\(-0.833330\pi\)
							0.406728	−	0.913549i	\(-0.366670\pi\)
\(31\)	−7.49358e8	−	5.44440e8i	−0.151649	−	0.110179i	0.509374	−	0.860545i	\(-0.329877\pi\)
							−0.661022	+	0.750366i	\(0.729877\pi\)
\(37\)	−9.23738e9	−	2.84297e10i	−0.591885	−	1.82163i	−0.569658	−	0.821882i	\(-0.692924\pi\)
							−0.0222273	−	0.999753i	\(-0.507076\pi\)
\(41\)	−4.47225e9	+	1.37642e10i	−0.147038	+	0.452538i	−0.997267	−	0.0738752i	\(-0.976463\pi\)
							0.850229	+	0.526413i	\(0.176463\pi\)
\(43\)	−2.03168e10			−0.490129			−0.245065	−	0.969507i	\(-0.578809\pi\)
							−0.245065	+	0.969507i	\(0.578809\pi\)
\(47\)	−1.97356e10	+	6.07399e10i	−0.267063	+	0.821936i	0.724148	+	0.689645i	\(0.242233\pi\)
							−0.991211	+	0.132291i	\(0.957767\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	11.14.c.a.4.1	yes	48
11.3	even	5	inner	11.14.c.a.3.1	✓	48
11.5	even	5		121.14.a.h.1.24		24
11.6	odd	10		121.14.a.i.1.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.14.c.a.3.1	✓	48	11.3	even	5	inner
11.14.c.a.4.1	yes	48	1.1	even	1	trivial
121.14.a.h.1.24		24	11.5	even	5
121.14.a.i.1.1		24	11.6	odd	10