Embedded newform 336.2.y.a.125.5

• Label: 336.2.y.a.125.5
• Level: $336$
• Weight: $2$
• Character: 336.125
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.y (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(60\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			125.5
Character	\(\chi\)	\(=\)	336.125
Dual form			336.2.y.a.293.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37255 - 0.340756i) q^{2} +(-1.24999 - 1.19897i) q^{3} +(1.76777 + 0.935408i) q^{4} +(2.22287 - 2.22287i) q^{5} +(1.30712 + 2.07158i) q^{6} +(1.16923 - 2.37337i) q^{7} +(-2.10760 - 1.88627i) q^{8} +(0.124963 + 2.99740i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 8 q^{4}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37255	−	0.340756i	−0.970537	−	0.240951i
							\(3\)	−1.24999	−	1.19897i	−0.721684	−	0.692223i
\(5\)	2.22287	−	2.22287i	0.994099	−	0.994099i								−0.00588320	−	0.999983i	\(-0.501873\pi\)
							0.999983	+	0.00588320i	\(0.00187269\pi\)
\(7\)	1.16923	−	2.37337i	0.441928	−	0.897050i
							\(11\)	2.89998	+	2.89998i	0.874378	+	0.874378i	0.992946	−	0.118568i	\(-0.0378304\pi\)
−0.118568	+	0.992946i	\(0.537830\pi\)
\(13\)	2.85803	−	2.85803i	0.792676	−	0.792676i	−0.189252	−	0.981928i	\(-0.560606\pi\)
							0.981928	+	0.189252i	\(0.0606064\pi\)
\(17\)	3.18041			0.771364			0.385682	−	0.922632i	\(-0.373966\pi\)
							0.385682	+	0.922632i	\(0.373966\pi\)
\(19\)	−1.73347	+	1.73347i	−0.397684	+	0.397684i	−0.877415	−	0.479731i	\(-0.840734\pi\)
							0.479731	+	0.877415i	\(0.340734\pi\)
\(23\)	−7.79850			−1.62610			−0.813050	−	0.582194i	\(-0.802194\pi\)
							−0.813050	+	0.582194i	\(0.802194\pi\)
\(29\)	1.10738	−	1.10738i	0.205635	−	0.205635i	−0.596774	−	0.802409i	\(-0.703551\pi\)
							0.802409	+	0.596774i	\(0.203551\pi\)
\(31\)		−	9.18848i		−	1.65030i	−0.564914	−	0.825150i	\(-0.691091\pi\)
							0.564914	−	0.825150i	\(-0.308909\pi\)
\(37\)	1.36244	−	1.36244i	0.223984	−	0.223984i	−0.586190	−	0.810174i	\(-0.699373\pi\)
							0.810174	+	0.586190i	\(0.199373\pi\)
\(41\)			4.51251i			0.704735i	0.935862	+	0.352368i	\(0.114623\pi\)
							−0.935862	+	0.352368i	\(0.885377\pi\)
\(43\)	−3.61139	+	3.61139i	−0.550732	+	0.550732i	−0.926652	−	0.375920i	\(-0.877327\pi\)
							0.375920	+	0.926652i	\(0.377327\pi\)
\(47\)	−9.27141			−1.35237			−0.676187	−	0.736730i	\(-0.736369\pi\)
							−0.676187	+	0.736730i	\(0.736369\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.y.a.125.5	✓	120
3.2	odd	2	inner	336.2.y.a.125.56	yes	120
7.6	odd	2	inner	336.2.y.a.125.6	yes	120
16.5	even	4	inner	336.2.y.a.293.55	yes	120
21.20	even	2	inner	336.2.y.a.125.55	yes	120
48.5	odd	4	inner	336.2.y.a.293.6	yes	120
112.69	odd	4	inner	336.2.y.a.293.56	yes	120
336.293	even	4	inner	336.2.y.a.293.5	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.y.a.125.5	✓	120	1.1	even	1	trivial
336.2.y.a.125.6	yes	120	7.6	odd	2	inner
336.2.y.a.125.55	yes	120	21.20	even	2	inner
336.2.y.a.125.56	yes	120	3.2	odd	2	inner
336.2.y.a.293.5	yes	120	336.293	even	4	inner
336.2.y.a.293.6	yes	120	48.5	odd	4	inner
336.2.y.a.293.55	yes	120	16.5	even	4	inner
336.2.y.a.293.56	yes	120	112.69	odd	4	inner