Embedded newform 336.2.y.a.125.2

• Label: 336.2.y.a.125.2
• 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			125.2
Character	\(\chi\)	\(=\)	336.125
Dual form			336.2.y.a.293.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41405 - 0.0212996i) q^{2} +(1.55654 - 0.759733i) q^{3} +(1.99909 + 0.0602376i) q^{4} +(0.700186 - 0.700186i) q^{5} +(-2.21721 + 1.04115i) q^{6} +(-2.03996 - 1.68481i) q^{7} +(-2.82554 - 0.127759i) q^{8} +(1.84561 - 2.36510i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41405	−	0.0212996i	−0.999887	−	0.0150611i
							\(3\)	1.55654	−	0.759733i	0.898667	−	0.438632i
\(5\)	0.700186	−	0.700186i	0.313133	−	0.313133i								−0.532989	−	0.846122i	\(-0.678931\pi\)
							0.846122	+	0.532989i	\(0.178931\pi\)
\(7\)	−2.03996	−	1.68481i	−0.771031	−	0.636798i
							\(11\)	−2.30565	−	2.30565i	−0.695178	−	0.695178i	0.268188	−	0.963367i	\(-0.413575\pi\)
−0.963367	+	0.268188i	\(0.913575\pi\)
\(13\)	3.49099	−	3.49099i	0.968225	−	0.968225i	−0.0312853	−	0.999510i	\(-0.509960\pi\)
							0.999510	+	0.0312853i	\(0.00996004\pi\)
\(17\)	−3.77445			−0.915439			−0.457719	−	0.889097i	\(-0.651334\pi\)
							−0.457719	+	0.889097i	\(0.651334\pi\)
\(19\)	−4.72162	+	4.72162i	−1.08321	+	1.08321i	−0.0870070	+	0.996208i	\(0.527730\pi\)
							−0.996208	+	0.0870070i	\(0.972270\pi\)
\(23\)	8.15400			1.70023			0.850113	−	0.526601i	\(-0.176534\pi\)
							0.850113	+	0.526601i	\(0.176534\pi\)
\(29\)	4.19113	−	4.19113i	0.778273	−	0.778273i	−0.201264	−	0.979537i	\(-0.564505\pi\)
							0.979537	+	0.201264i	\(0.0645051\pi\)
\(31\)			2.83724i			0.509584i	0.966996	+	0.254792i	\(0.0820069\pi\)
							−0.966996	+	0.254792i	\(0.917993\pi\)
\(37\)	2.94154	−	2.94154i	0.483586	−	0.483586i	−0.422689	−	0.906275i	\(-0.638914\pi\)
							0.906275	+	0.422689i	\(0.138914\pi\)
\(41\)		−	7.00316i		−	1.09371i	−0.837228	−	0.546855i	\(-0.815825\pi\)
							0.837228	−	0.546855i	\(-0.184175\pi\)
\(43\)	1.95574	−	1.95574i	0.298248	−	0.298248i	−0.542079	−	0.840327i	\(-0.682363\pi\)
							0.840327	+	0.542079i	\(0.182363\pi\)
\(47\)	−2.61166			−0.380949			−0.190475	−	0.981692i	\(-0.561003\pi\)
							−0.190475	+	0.981692i	\(0.561003\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.y.a.125.2	yes	120
3.2	odd	2	inner	336.2.y.a.125.60	yes	120
7.6	odd	2	inner	336.2.y.a.125.1	✓	120
16.5	even	4	inner	336.2.y.a.293.59	yes	120
21.20	even	2	inner	336.2.y.a.125.59	yes	120
48.5	odd	4	inner	336.2.y.a.293.1	yes	120
112.69	odd	4	inner	336.2.y.a.293.60	yes	120
336.293	even	4	inner	336.2.y.a.293.2	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.y.a.125.1	✓	120	7.6	odd	2	inner
336.2.y.a.125.2	yes	120	1.1	even	1	trivial
336.2.y.a.125.59	yes	120	21.20	even	2	inner
336.2.y.a.125.60	yes	120	3.2	odd	2	inner
336.2.y.a.293.1	yes	120	48.5	odd	4	inner
336.2.y.a.293.2	yes	120	336.293	even	4	inner
336.2.y.a.293.59	yes	120	16.5	even	4	inner
336.2.y.a.293.60	yes	120	112.69	odd	4	inner