Embedded newform 52.3.j.a.35.11

• Label: 52.3.j.a.35.11
• Level: $52$
• Weight: $3$
• Character: 52.35
• Analytic conductor: $1.417$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	52.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.41689737467\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			35.11
Character	\(\chi\)	\(=\)	52.35
Dual form			52.3.j.a.3.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.88111 + 0.679287i) q^{2} +(-1.57700 - 0.910479i) q^{3} +(3.07714 + 2.55563i) q^{4} +4.64591 q^{5} +(-2.34802 - 2.78394i) q^{6} +(-2.72125 + 1.57112i) q^{7} +(4.05242 + 6.89767i) q^{8} +(-2.84206 - 4.92259i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} - q^{4} - 12 q^{5} - 6 q^{6} - 22 q^{8} + 22 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.88111	+	0.679287i	0.940554	+	0.339644i
							\(3\)	−1.57700	−	0.910479i	−0.525665	−	0.303493i	0.213584	−	0.976925i	\(-0.431486\pi\)
−0.739249	+	0.673432i	\(0.764820\pi\)
\(5\)	4.64591			0.929181			0.464591	−	0.885526i	\(-0.346202\pi\)
							0.464591	+	0.885526i	\(0.346202\pi\)
\(7\)	−2.72125	+	1.57112i	−0.388751	+	0.224445i	−0.681619	−	0.731708i	\(-0.738724\pi\)
							0.292868	+	0.956153i	\(0.405390\pi\)
\(11\)	−12.2659	−	7.08172i	−1.11508	−	0.643793i	−0.174941	−	0.984579i	\(-0.555973\pi\)
							−0.940141	+	0.340786i	\(0.889307\pi\)
\(13\)	−12.6512	+	2.99105i	−0.973172	+	0.230081i
							\(17\)	1.50517	+	2.60702i	0.0885392	+	0.153354i	0.906894	−	0.421359i	\(-0.138447\pi\)
−0.818355	+	0.574714i	\(0.805113\pi\)
\(19\)	18.7863	−	10.8463i	0.988752	−	0.570856i	0.0838510	−	0.996478i	\(-0.473278\pi\)
							0.904901	+	0.425622i	\(0.139945\pi\)
\(23\)	17.3703	+	10.0288i	0.755232	+	0.436033i	0.827581	−	0.561346i	\(-0.189716\pi\)
							−0.0723493	+	0.997379i	\(0.523050\pi\)
\(29\)	1.91404	−	3.31521i	0.0660014	−	0.114318i	−0.831136	−	0.556069i	\(-0.812309\pi\)
							0.897138	+	0.441751i	\(0.145642\pi\)
\(31\)			8.29993i			0.267740i	0.990999	+	0.133870i	\(0.0427404\pi\)
							−0.990999	+	0.133870i	\(0.957260\pi\)
\(37\)	26.8474	−	46.5010i	0.725604	−	1.25678i	−0.233121	−	0.972448i	\(-0.574894\pi\)
							0.958725	−	0.284335i	\(-0.0917729\pi\)
\(41\)	−26.0620	+	45.1407i	−0.635658	+	1.10099i	0.350717	+	0.936481i	\(0.385938\pi\)
							−0.986375	+	0.164511i	\(0.947396\pi\)
\(43\)	57.3526	−	33.1125i	1.33378	−	0.770059i	0.347904	−	0.937530i	\(-0.386894\pi\)
							0.985877	+	0.167471i	\(0.0535602\pi\)
\(47\)			31.6645i			0.673712i	0.941556	+	0.336856i	\(0.109364\pi\)
							−0.941556	+	0.336856i	\(0.890636\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.3.j.a.35.11	yes	24
4.3	odd	2	inner	52.3.j.a.35.4	yes	24
13.3	even	3	inner	52.3.j.a.3.4	✓	24
52.3	odd	6	inner	52.3.j.a.3.11	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.3.j.a.3.4	✓	24	13.3	even	3	inner
52.3.j.a.3.11	yes	24	52.3	odd	6	inner
52.3.j.a.35.4	yes	24	4.3	odd	2	inner
52.3.j.a.35.11	yes	24	1.1	even	1	trivial