Embedded newform 52.3.j.a.3.10

• Label: 52.3.j.a.3.10
• Level: $52$
• Weight: $3$
• Character: 52.3
• 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			3.10
Character	\(\chi\)	\(=\)	52.3
Dual form			52.3.j.a.35.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39069 + 1.43736i) q^{2} +(-3.19741 + 1.84603i) q^{3} +(-0.131987 + 3.99782i) q^{4} +2.13128 q^{5} +(-7.10000 - 2.02858i) q^{6} +(1.80105 + 1.03984i) q^{7} +(-5.92985 + 5.37000i) q^{8} +(2.31563 - 4.01080i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} - q^{4} - 12 q^{5} - 6 q^{6} - 22 q^{8} + 22 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39069	+	1.43736i	0.695343	+	0.718678i
							\(3\)	−3.19741	+	1.84603i	−1.06580	+	0.615342i	−0.927032	−	0.374982i	\(-0.877649\pi\)
−0.138772	+	0.990324i	\(0.544316\pi\)
\(5\)	2.13128			0.426257			0.213128	−	0.977024i	\(-0.431635\pi\)
							0.213128	+	0.977024i	\(0.431635\pi\)
\(7\)	1.80105	+	1.03984i	0.257294	+	0.148548i	0.623099	−	0.782143i	\(-0.285873\pi\)
							−0.365806	+	0.930691i	\(0.619207\pi\)
\(11\)	14.3664	−	8.29445i	1.30604	−	0.754041i	0.324604	−	0.945850i	\(-0.394769\pi\)
							0.981432	+	0.191809i	\(0.0614355\pi\)
\(13\)	6.93730	−	10.9943i	0.533639	−	0.845713i
							\(17\)	1.97862	−	3.42707i	0.116389	−	0.201592i	−0.801945	−	0.597398i	\(-0.796201\pi\)
0.918334	+	0.395806i	\(0.129535\pi\)
\(19\)	11.8435	+	6.83787i	0.623345	+	0.359888i	0.778170	−	0.628054i	\(-0.216148\pi\)
							−0.154825	+	0.987942i	\(0.549482\pi\)
\(23\)	−32.3654	+	18.6862i	−1.40719	+	0.812442i	−0.995116	−	0.0987077i	\(-0.968529\pi\)
							−0.412075	+	0.911150i	\(0.635196\pi\)
\(29\)	11.0969	+	19.2204i	0.382652	+	0.662773i	0.991440	−	0.130560i	\(-0.0416775\pi\)
							−0.608788	+	0.793333i	\(0.708344\pi\)
\(31\)		−	47.2231i		−	1.52333i	−0.647974	−	0.761663i	\(-0.724383\pi\)
							0.647974	−	0.761663i	\(-0.275617\pi\)
\(37\)	−11.7978	−	20.4344i	−0.318860	−	0.552281i	0.661391	−	0.750042i	\(-0.269967\pi\)
							−0.980250	+	0.197760i	\(0.936633\pi\)
\(41\)	31.2023	+	54.0439i	0.761031	+	1.31814i	0.942320	+	0.334714i	\(0.108640\pi\)
							−0.181289	+	0.983430i	\(0.558027\pi\)
\(43\)	12.3875	+	7.15195i	0.288082	+	0.166324i	0.637077	−	0.770800i	\(-0.280143\pi\)
							−0.348994	+	0.937125i	\(0.613477\pi\)
\(47\)		−	19.4471i		−	0.413768i	−0.978366	−	0.206884i	\(-0.933668\pi\)
							0.978366	−	0.206884i	\(-0.0663322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.3.j.a.3.10	yes	24
4.3	odd	2	inner	52.3.j.a.3.8	✓	24
13.9	even	3	inner	52.3.j.a.35.8	yes	24
52.35	odd	6	inner	52.3.j.a.35.10	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.3.j.a.3.8	✓	24	4.3	odd	2	inner
52.3.j.a.3.10	yes	24	1.1	even	1	trivial
52.3.j.a.35.8	yes	24	13.9	even	3	inner
52.3.j.a.35.10	yes	24	52.35	odd	6	inner