Embedded newform 52.3.j.a.35.12

• Label: 52.3.j.a.35.12
• 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			35.12
Character	\(\chi\)	\(=\)	52.35
Dual form			52.3.j.a.3.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.93496 - 0.505890i) q^{2} +(2.13241 + 1.23115i) q^{3} +(3.48815 - 1.95776i) q^{4} -6.97792 q^{5} +(4.74896 + 1.30346i) q^{6} +(-6.03665 + 3.48526i) q^{7} +(5.75902 - 5.55280i) q^{8} +(-1.46855 - 2.54360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} - q^{4} - 12 q^{5} - 6 q^{6} - 22 q^{8} + 22 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.93496	−	0.505890i	0.967481	−	0.252945i
							\(3\)	2.13241	+	1.23115i	0.710804	+	0.410383i	0.811359	−	0.584549i	\(-0.198728\pi\)
−0.100555	+	0.994932i	\(0.532062\pi\)
\(5\)	−6.97792			−1.39558			−0.697792	−	0.716300i	\(-0.745834\pi\)
							−0.697792	+	0.716300i	\(0.745834\pi\)
\(7\)	−6.03665	+	3.48526i	−0.862379	+	0.497895i	−0.864808	−	0.502102i	\(-0.832560\pi\)
							0.00242936	+	0.999997i	\(0.499227\pi\)
\(11\)	10.5088	+	6.06725i	0.955344	+	0.551568i	0.894737	−	0.446594i	\(-0.147363\pi\)
							0.0606068	+	0.998162i	\(0.480696\pi\)
\(13\)	−2.28998	−	12.7967i	−0.176152	−	0.984363i
							\(17\)	9.02161	+	15.6259i	0.530683	+	0.919169i	0.999359	+	0.0357994i	\(0.0113977\pi\)
−0.468676	+	0.883370i	\(0.655269\pi\)
\(19\)	−18.5938	+	10.7351i	−0.978619	+	0.565006i	−0.901853	−	0.432043i	\(-0.857793\pi\)
							−0.0767660	+	0.997049i	\(0.524459\pi\)
\(23\)	16.3602	+	9.44559i	0.711315	+	0.410678i	0.811548	−	0.584286i	\(-0.198625\pi\)
							−0.100233	+	0.994964i	\(0.531959\pi\)
\(29\)	−0.863823	+	1.49619i	−0.0297870	+	0.0515926i	−0.880535	−	0.473982i	\(-0.842816\pi\)
							0.850748	+	0.525574i	\(0.176150\pi\)
\(31\)		−	3.09722i		−	0.0999103i	−0.998751	−	0.0499551i	\(-0.984092\pi\)
							0.998751	−	0.0499551i	\(-0.0159078\pi\)
\(37\)	−1.81099	+	3.13672i	−0.0489456	+	0.0847763i	−0.889460	−	0.457013i	\(-0.848919\pi\)
							0.840515	+	0.541789i	\(0.182253\pi\)
\(41\)	24.1701	−	41.8639i	0.589516	−	1.02107i	−0.404780	−	0.914414i	\(-0.632652\pi\)
							0.994296	−	0.106657i	\(-0.0340147\pi\)
\(43\)	34.7006	−	20.0344i	0.806991	−	0.465917i	−0.0389188	−	0.999242i	\(-0.512391\pi\)
							0.845910	+	0.533326i	\(0.179058\pi\)
\(47\)			65.4862i			1.39332i	0.717400	+	0.696662i	\(0.245332\pi\)
							−0.717400	+	0.696662i	\(0.754668\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.3.j.a.35.12	yes	24
4.3	odd	2	inner	52.3.j.a.35.5	yes	24
13.3	even	3	inner	52.3.j.a.3.5	✓	24
52.3	odd	6	inner	52.3.j.a.3.12	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.3.j.a.3.5	✓	24	13.3	even	3	inner
52.3.j.a.3.12	yes	24	52.3	odd	6	inner
52.3.j.a.35.5	yes	24	4.3	odd	2	inner
52.3.j.a.35.12	yes	24	1.1	even	1	trivial