Embedded newform 52.3.j.a.3.2

• Label: 52.3.j.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	52.3
Dual form			52.3.j.a.35.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85348 - 0.751419i) q^{2} +(4.85432 - 2.80264i) q^{3} +(2.87074 + 2.78547i) q^{4} -3.52764 q^{5} +(-11.1033 + 1.54700i) q^{6} +(3.09209 + 1.78522i) q^{7} +(-3.22779 - 7.31993i) q^{8} +(11.2096 - 19.4156i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} - q^{4} - 12 q^{5} - 6 q^{6} - 22 q^{8} + 22 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85348	−	0.751419i	−0.926738	−	0.375709i
							\(3\)	4.85432	−	2.80264i	1.61811	−	0.934215i	0.630698	−	0.776028i	\(-0.282769\pi\)
0.987409	−	0.158186i	\(-0.0505647\pi\)
\(5\)	−3.52764			−0.705527			−0.352764	−	0.935712i	\(-0.614758\pi\)
							−0.352764	+	0.935712i	\(0.614758\pi\)
\(7\)	3.09209	+	1.78522i	0.441727	+	0.255031i	0.704330	−	0.709873i	\(-0.251248\pi\)
							−0.262603	+	0.964904i	\(0.584581\pi\)
\(11\)	−2.86800	+	1.65584i	−0.260727	+	0.150531i	−0.624666	−	0.780892i	\(-0.714765\pi\)
							0.363939	+	0.931423i	\(0.381432\pi\)
\(13\)	−11.3419	+	6.35302i	−0.872455	+	0.488694i
							\(17\)	−4.90584	+	8.49717i	−0.288579	+	0.499833i	−0.973471	−	0.228811i	\(-0.926516\pi\)
0.684892	+	0.728645i	\(0.259849\pi\)
\(19\)	25.5521	+	14.7525i	1.34485	+	0.776447i	0.987514	−	0.157530i	\(-0.0503532\pi\)
							0.357332	+	0.933977i	\(0.383687\pi\)
\(23\)	−10.6382	+	6.14197i	−0.462531	+	0.267042i	−0.713108	−	0.701054i	\(-0.752713\pi\)
							0.250577	+	0.968097i	\(0.419380\pi\)
\(29\)	18.9463	+	32.8160i	0.653321	+	1.13159i	0.982312	+	0.187252i	\(0.0599582\pi\)
							−0.328991	+	0.944333i	\(0.606708\pi\)
\(31\)			3.81678i			0.123122i	0.998103	+	0.0615610i	\(0.0196079\pi\)
							−0.998103	+	0.0615610i	\(0.980392\pi\)
\(37\)	−10.0042	−	17.3277i	−0.270382	−	0.468316i	0.698577	−	0.715535i	\(-0.253817\pi\)
							−0.968960	+	0.247218i	\(0.920483\pi\)
\(41\)	−22.2921	−	38.6110i	−0.543709	−	0.941731i	−0.998687	−	0.0512284i	\(-0.983686\pi\)
							0.454978	−	0.890502i	\(-0.349647\pi\)
\(43\)	10.3722	+	5.98839i	0.241214	+	0.139265i	0.615734	−	0.787954i	\(-0.288859\pi\)
							−0.374521	+	0.927219i	\(0.622193\pi\)
\(47\)		−	52.1794i		−	1.11020i	−0.831784	−	0.555100i	\(-0.812680\pi\)
							0.831784	−	0.555100i	\(-0.187320\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.3.j.a.3.2	✓	24
4.3	odd	2	inner	52.3.j.a.3.7	yes	24
13.9	even	3	inner	52.3.j.a.35.7	yes	24
52.35	odd	6	inner	52.3.j.a.35.2	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.3.j.a.3.2	✓	24	1.1	even	1	trivial
52.3.j.a.3.7	yes	24	4.3	odd	2	inner
52.3.j.a.35.2	yes	24	52.35	odd	6	inner
52.3.j.a.35.7	yes	24	13.9	even	3	inner