Embedded newform 336.2.y.a.125.9

• Label: 336.2.y.a.125.9
• 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			125.9
Character	\(\chi\)	\(=\)	336.125
Dual form			336.2.y.a.293.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22270 + 0.710642i) q^{2} +(-1.68333 - 0.407913i) q^{3} +(0.989977 - 1.73780i) q^{4} +(2.35722 - 2.35722i) q^{5} +(2.34809 - 0.697491i) q^{6} +(-0.422823 + 2.61175i) q^{7} +(0.0245110 + 2.82832i) q^{8} +(2.66721 + 1.37331i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22270	+	0.710642i	−0.864577	+	0.502500i
							\(3\)	−1.68333	−	0.407913i	−0.971872	−	0.235509i
\(5\)	2.35722	−	2.35722i	1.05418	−	1.05418i								0.0557374	−	0.998445i	\(-0.482249\pi\)
							0.998445	−	0.0557374i	\(-0.0177510\pi\)
\(7\)	−0.422823	+	2.61175i	−0.159812	+	0.987147i
							\(11\)	−3.31848	−	3.31848i	−1.00056	−	1.00056i	−1.00000		0.000559294i	\(-0.999822\pi\)
−0.000559294	−	1.00000i	\(-0.500178\pi\)
\(13\)	1.84991	−	1.84991i	0.513073	−	0.513073i	−0.402394	−	0.915467i	\(-0.631822\pi\)
							0.915467	+	0.402394i	\(0.131822\pi\)
\(17\)	−1.64810			−0.399723			−0.199861	−	0.979824i	\(-0.564049\pi\)
							−0.199861	+	0.979824i	\(0.564049\pi\)
\(19\)	3.26407	−	3.26407i	0.748828	−	0.748828i	−0.225431	−	0.974259i	\(-0.572379\pi\)
							0.974259	+	0.225431i	\(0.0723790\pi\)
\(23\)	2.91494			0.607806			0.303903	−	0.952703i	\(-0.401710\pi\)
							0.303903	+	0.952703i	\(0.401710\pi\)
\(29\)	−0.602748	+	0.602748i	−0.111927	+	0.111927i	−0.760852	−	0.648925i	\(-0.775219\pi\)
							0.648925	+	0.760852i	\(0.275219\pi\)
\(31\)		−	5.40799i		−	0.971303i	−0.874153	−	0.485651i	\(-0.838582\pi\)
							0.874153	−	0.485651i	\(-0.161418\pi\)
\(37\)	−0.00886229	+	0.00886229i	−0.00145695	+	0.00145695i	−0.707835	−	0.706378i	\(-0.750328\pi\)
							0.706378	+	0.707835i	\(0.250328\pi\)
\(41\)		−	11.8593i		−	1.85211i	−0.377389	−	0.926055i	\(-0.623178\pi\)
							0.377389	−	0.926055i	\(-0.376822\pi\)
\(43\)	7.23190	−	7.23190i	1.10285	−	1.10285i	0.108789	−	0.994065i	\(-0.465303\pi\)
							0.994065	−	0.108789i	\(-0.0346973\pi\)
\(47\)	−0.360279			−0.0525521			−0.0262761	−	0.999655i	\(-0.508365\pi\)
							−0.0262761	+	0.999655i	\(0.508365\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.y.a.125.9	✓	120
3.2	odd	2	inner	336.2.y.a.125.52	yes	120
7.6	odd	2	inner	336.2.y.a.125.10	yes	120
16.5	even	4	inner	336.2.y.a.293.51	yes	120
21.20	even	2	inner	336.2.y.a.125.51	yes	120
48.5	odd	4	inner	336.2.y.a.293.10	yes	120
112.69	odd	4	inner	336.2.y.a.293.52	yes	120
336.293	even	4	inner	336.2.y.a.293.9	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.y.a.125.9	✓	120	1.1	even	1	trivial
336.2.y.a.125.10	yes	120	7.6	odd	2	inner
336.2.y.a.125.51	yes	120	21.20	even	2	inner
336.2.y.a.125.52	yes	120	3.2	odd	2	inner
336.2.y.a.293.9	yes	120	336.293	even	4	inner
336.2.y.a.293.10	yes	120	48.5	odd	4	inner
336.2.y.a.293.51	yes	120	16.5	even	4	inner
336.2.y.a.293.52	yes	120	112.69	odd	4	inner