Embedded newform 121.4.g.a.4.13

• Label: 121.4.g.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	121.4
Dual form			121.4.g.a.91.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21959 - 0.0697386i) q^{2} +(1.08180 - 3.32944i) q^{3} +(-6.46532 - 0.741827i) q^{4} +(5.37679 + 10.1901i) q^{5} +(-1.55154 + 3.98510i) q^{6} +(-12.0041 - 5.07296i) q^{7} +(17.4628 + 3.02206i) q^{8} +(11.9286 + 8.66662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 37 q^{2} - 32 q^{3} + 73 q^{4} - 33 q^{5} + 73 q^{6} - 9 q^{7} - 91 q^{8} - 2748 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21959	−	0.0697386i	−0.431189	−	0.0246563i	−0.159853	−	0.987141i	\(-0.551102\pi\)
							−0.271337	+	0.962485i	\(0.587466\pi\)
\(3\)	1.08180	−	3.32944i	0.208193	−	0.640751i	−0.791374	−	0.611332i	\(-0.790634\pi\)
							0.999567	−	0.0294195i	\(-0.00936587\pi\)
\(5\)	5.37679	+	10.1901i	0.480914	+	0.911434i	0.998444	+	0.0557687i	\(0.0177609\pi\)
							−0.517529	+	0.855665i	\(0.673148\pi\)
\(7\)	−12.0041	−	5.07296i	−0.648158	−	0.273914i	0.0403026	−	0.999188i	\(-0.487168\pi\)
							−0.688461	+	0.725274i	\(0.741713\pi\)
\(11\)	35.9495	+	6.21536i	0.985381	+	0.170364i
							\(13\)	−21.3480	+	12.0558i	−0.455451	+	0.257205i	−0.702344	−	0.711838i	\(-0.747863\pi\)
0.246893	+	0.969043i	\(0.420590\pi\)
\(17\)	93.6897	−	33.4284i	1.33665	−	0.476916i	0.431541	−	0.902093i	\(-0.357970\pi\)
							0.905110	+	0.425177i	\(0.139788\pi\)
\(19\)	2.33399	−	81.7003i	0.0281818	−	0.986491i	−0.855330	−	0.518083i	\(-0.826646\pi\)
							0.883512	−	0.468408i	\(-0.155172\pi\)
\(23\)	91.9451	+	106.110i	0.833560	+	0.961979i	0.999709	−	0.0241212i	\(-0.00767875\pi\)
							−0.166149	+	0.986101i	\(0.553133\pi\)
\(29\)	146.515	+	214.271i	0.938175	+	1.37204i	0.928198	+	0.372088i	\(0.121358\pi\)
							0.00997782	+	0.999950i	\(0.496824\pi\)
\(31\)	−33.6713	+	166.174i	−0.195082	+	0.962767i	0.756176	+	0.654369i	\(0.227066\pi\)
							−0.951257	+	0.308398i	\(0.900207\pi\)
\(37\)	−140.221	+	128.694i	−0.623033	+	0.571815i	−0.924272	−	0.381734i	\(-0.875327\pi\)
							0.301239	+	0.953549i	\(0.402600\pi\)
\(41\)	8.21941	+	31.2699i	0.0313087	+	0.119111i	0.980966	−	0.194181i	\(-0.0622048\pi\)
							−0.949657	+	0.313291i	\(0.898568\pi\)
\(43\)	−7.87454	−	54.7686i	−0.0279269	−	0.194236i	0.971082	−	0.238746i	\(-0.0767364\pi\)
							−0.999009	+	0.0445103i	\(0.985827\pi\)
\(47\)	387.463	−	316.827i	1.20250	−	0.983276i	0.202504	−	0.979282i	\(-0.435092\pi\)
							0.999992	−	0.00399425i	\(-0.00127141\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	121.4.g.a.4.13	✓	1280
121.91	even	55	inner	121.4.g.a.91.13	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.4.g.a.4.13	✓	1280	1.1	even	1	trivial
121.4.g.a.91.13	yes	1280	121.91	even	55	inner