Embedded newform 121.4.g.a.4.7

• Label: 121.4.g.a.4.7
• Level: $121$
• Weight: $4$
• Character: 121.4
• Analytic conductor: $7.139$
• Analytic rank: $0$
• Dimension: $1280$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	121.g (of order \(55\), degree \(40\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.13923111069\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(32\) over \(\Q(\zeta_{55})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label			4.7
Character	\(\chi\)	\(=\)	121.4
Dual form			121.4.g.a.91.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.63025 - 0.207585i) q^{2} +(-1.34595 + 4.14240i) q^{3} +(5.18774 + 0.595238i) q^{4} +(-8.43183 - 15.9801i) q^{5} +(5.74602 - 14.7585i) q^{6} +(11.1935 + 4.73039i) q^{7} +(9.95414 + 1.72263i) q^{8} +(6.49556 + 4.71930i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 37 q^{2} - 32 q^{3} + 73 q^{4} - 33 q^{5} + 73 q^{6} - 9 q^{7} - 91 q^{8} - 2748 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{55}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.63025	−	0.207585i	−1.28349	−	0.0733924i	−0.598011	−	0.801488i	\(-0.704042\pi\)
							−0.685475	+	0.728096i	\(0.740406\pi\)
\(3\)	−1.34595	+	4.14240i	−0.259028	+	0.797205i	0.733982	+	0.679169i	\(0.237660\pi\)
							−0.993009	+	0.118036i	\(0.962340\pi\)
\(5\)	−8.43183	−	15.9801i	−0.754166	−	1.42930i	−0.898134	−	0.439721i	\(-0.855077\pi\)
							0.143968	−	0.989582i	\(-0.454014\pi\)
\(7\)	11.1935	+	4.73039i	0.604390	+	0.255417i	0.669920	−	0.742433i	\(-0.266328\pi\)
							−0.0655301	+	0.997851i	\(0.520874\pi\)
\(11\)	23.4307	−	27.9643i	0.642239	−	0.766504i
							\(13\)	−55.7872	+	31.5045i	−1.19020	+	0.672136i	−0.954069	−	0.299586i	\(-0.903152\pi\)
−0.236129	+	0.971722i	\(0.575879\pi\)
\(17\)	−72.3898	+	25.8286i	−1.03277	+	0.368492i	−0.797424	−	0.603419i	\(-0.793804\pi\)
							−0.235347	+	0.971911i	\(0.575623\pi\)
\(19\)	0.259590	−	9.08683i	0.00313442	−	0.109719i	−0.996518	−	0.0833725i	\(-0.973431\pi\)
							0.999653	−	0.0263465i	\(-0.00838732\pi\)
\(23\)	13.1250	+	15.1471i	0.118989	+	0.137321i	0.812119	−	0.583492i	\(-0.198314\pi\)
							−0.693129	+	0.720813i	\(0.743769\pi\)
\(29\)	97.9186	+	143.201i	0.627001	+	0.916960i	0.999941	−	0.0108960i	\(-0.00346836\pi\)
							−0.372940	+	0.927856i	\(0.621650\pi\)
\(31\)	−32.5145	+	160.466i	−0.188380	+	0.929692i	0.768642	+	0.639679i	\(0.220933\pi\)
							−0.957023	+	0.290014i	\(0.906340\pi\)
\(37\)	−109.442	+	100.445i	−0.486277	+	0.446301i	−0.881430	−	0.472315i	\(-0.843419\pi\)
							0.395154	+	0.918615i	\(0.370691\pi\)
\(41\)	29.2065	+	111.113i	0.111251	+	0.423243i	0.999336	−	0.0364364i	\(-0.0116006\pi\)
							−0.888085	+	0.459680i	\(0.847964\pi\)
\(43\)	23.2604	+	161.780i	0.0824925	+	0.573748i	0.988585	+	0.150667i	\(0.0481422\pi\)
							−0.906092	+	0.423081i	\(0.860949\pi\)
\(47\)	−113.491	+	92.8015i	−0.352222	+	0.288010i	−0.792526	−	0.609838i	\(-0.791234\pi\)
							0.440304	+	0.897849i	\(0.354871\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.4.g.a.4.7	✓	1280
121.91	even	55	inner	121.4.g.a.91.7	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.4.g.a.4.7	✓	1280	1.1	even	1	trivial
121.4.g.a.91.7	yes	1280	121.91	even	55	inner