Embedded newform 69.2.g.a.38.1

• Label: 69.2.g.a.38.1
• Level: $69$
• Weight: $2$
• Character: 69.38
• Analytic conductor: $0.551$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.550967773947\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			38.1
Character	\(\chi\)	\(=\)	69.38
Dual form			69.2.g.a.20.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79593 - 0.258215i) q^{2} +(1.73159 - 0.0398492i) q^{3} +(1.23969 + 0.364005i) q^{4} +(0.432952 + 0.499653i) q^{5} +(-3.12010 - 0.375557i) q^{6} +(1.19382 - 1.85763i) q^{7} +(1.16847 + 0.533622i) q^{8} +(2.99682 - 0.138005i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{17}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79593	−	0.258215i	−1.26991	−	0.182586i	−0.525781	−	0.850620i	\(-0.676227\pi\)
							−0.744130	+	0.668035i	\(0.767136\pi\)
\(3\)	1.73159	−	0.0398492i	0.999735	−	0.0230070i
							\(5\)	0.432952	+	0.499653i	0.193622	+	0.223452i	0.844257	−	0.535939i	\(-0.180042\pi\)
−0.650634	+	0.759391i	\(0.725497\pi\)
\(7\)	1.19382	−	1.85763i	0.451223	−	0.702117i	−0.538892	−	0.842375i	\(-0.681157\pi\)
							0.990115	+	0.140258i	\(0.0447932\pi\)
\(11\)	−0.201421	−	1.40092i	−0.0607308	−	0.422392i	−0.997393	−	0.0721585i	\(-0.977011\pi\)
							0.936662	−	0.350234i	\(-0.113898\pi\)
\(13\)	−3.07103	+	1.97363i	−0.851751	+	0.547387i	−0.892120	−	0.451798i	\(-0.850783\pi\)
							0.0403697	+	0.999185i	\(0.487146\pi\)
\(17\)	−4.29913	+	1.26234i	−1.04269	+	0.306162i	−0.757861	−	0.652416i	\(-0.773756\pi\)
							−0.284831	+	0.958578i	\(0.591937\pi\)
\(19\)	−2.06307	+	7.02618i	−0.473301	+	1.61192i	0.283983	+	0.958829i	\(0.408344\pi\)
							−0.757284	+	0.653086i	\(0.773474\pi\)
\(23\)	−2.51508	−	4.08343i	−0.524430	−	0.851454i
							\(29\)	1.18355	+	4.03079i	0.219779	+	0.748498i	0.993385	+	0.114831i	\(0.0366327\pi\)
−0.773606	+	0.633667i	\(0.781549\pi\)
\(31\)	−1.22499	+	2.68234i	−0.220014	+	0.481763i	−0.987165	−	0.159703i	\(-0.948946\pi\)
							0.767151	+	0.641466i	\(0.221674\pi\)
\(37\)	−4.70302	−	4.07519i	−0.773171	−	0.669957i	0.176115	−	0.984370i	\(-0.443647\pi\)
							−0.949286	+	0.314413i	\(0.898192\pi\)
\(41\)	−1.42850	+	1.23780i	−0.223094	+	0.193312i	−0.759235	−	0.650816i	\(-0.774427\pi\)
							0.536141	+	0.844128i	\(0.319881\pi\)
\(43\)	8.38440	−	3.82903i	1.27861	−	0.583921i	0.343790	−	0.939047i	\(-0.388289\pi\)
							0.934819	+	0.355126i	\(0.115562\pi\)
\(47\)		−	2.91904i		−	0.425786i	−0.977076	−	0.212893i	\(-0.931711\pi\)
							0.977076	−	0.212893i	\(-0.0682885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.38.1	yes	60
3.2	odd	2	inner	69.2.g.a.38.6	yes	60
23.20	odd	22	inner	69.2.g.a.20.6	yes	60
69.20	even	22	inner	69.2.g.a.20.1	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.20.1	✓	60	69.20	even	22	inner
69.2.g.a.20.6	yes	60	23.20	odd	22	inner
69.2.g.a.38.1	yes	60	1.1	even	1	trivial
69.2.g.a.38.6	yes	60	3.2	odd	2	inner