Embedded newform 336.2.y.a.125.11

• Label: 336.2.y.a.125.11
• 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			125.11
Character	\(\chi\)	\(=\)	336.125
Dual form			336.2.y.a.293.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18025 + 0.779110i) q^{2} +(-0.970819 + 1.43440i) q^{3} +(0.785975 - 1.83909i) q^{4} +(-1.77648 + 1.77648i) q^{5} +(0.0282512 - 2.44933i) q^{6} +(-2.61801 - 0.382105i) q^{7} +(0.505204 + 2.78294i) q^{8} +(-1.11502 - 2.78509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18025	+	0.779110i	−0.834562	+	0.550914i
							\(3\)	−0.970819	+	1.43440i	−0.560503	+	0.828153i
\(5\)	−1.77648	+	1.77648i	−0.794466	+	0.794466i								−0.982217	−	0.187751i	\(-0.939880\pi\)
							0.187751	+	0.982217i	\(0.439880\pi\)
\(7\)	−2.61801	−	0.382105i	−0.989516	−	0.144422i
							\(11\)	0.0741148	+	0.0741148i	0.0223465	+	0.0223465i	0.718192	−	0.695845i	\(-0.244970\pi\)
−0.695845	+	0.718192i	\(0.744970\pi\)
\(13\)	2.31927	−	2.31927i	0.643251	−	0.643251i	−0.308102	−	0.951353i	\(-0.599694\pi\)
							0.951353	+	0.308102i	\(0.0996939\pi\)
\(17\)	−1.12306			−0.272382			−0.136191	−	0.990683i	\(-0.543486\pi\)
							−0.136191	+	0.990683i	\(0.543486\pi\)
\(19\)	0.438173	−	0.438173i	0.100524	−	0.100524i	−0.655056	−	0.755580i	\(-0.727355\pi\)
							0.755580	+	0.655056i	\(0.227355\pi\)
\(23\)	−5.33792			−1.11303			−0.556517	−	0.830836i	\(-0.687863\pi\)
							−0.556517	+	0.830836i	\(0.687863\pi\)
\(29\)	6.24342	−	6.24342i	1.15937	−	1.15937i	0.174764	−	0.984610i	\(-0.444084\pi\)
							0.984610	−	0.174764i	\(-0.0559162\pi\)
\(31\)		−	7.00163i		−	1.25753i	−0.777595	−	0.628765i	\(-0.783561\pi\)
							0.777595	−	0.628765i	\(-0.216439\pi\)
\(37\)	2.76432	−	2.76432i	0.454452	−	0.454452i	−0.442377	−	0.896829i	\(-0.645865\pi\)
							0.896829	+	0.442377i	\(0.145865\pi\)
\(41\)		−	5.12996i		−	0.801166i	−0.916261	−	0.400583i	\(-0.868808\pi\)
							0.916261	−	0.400583i	\(-0.131192\pi\)
\(43\)	−6.01001	+	6.01001i	−0.916519	+	0.916519i	−0.996774	−	0.0802558i	\(-0.974426\pi\)
							0.0802558	+	0.996774i	\(0.474426\pi\)
\(47\)	−9.67990			−1.41196			−0.705980	−	0.708232i	\(-0.749493\pi\)
							−0.705980	+	0.708232i	\(0.749493\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.y.a.125.11	✓	120
3.2	odd	2	inner	336.2.y.a.125.49	yes	120
7.6	odd	2	inner	336.2.y.a.125.12	yes	120
16.5	even	4	inner	336.2.y.a.293.50	yes	120
21.20	even	2	inner	336.2.y.a.125.50	yes	120
48.5	odd	4	inner	336.2.y.a.293.12	yes	120
112.69	odd	4	inner	336.2.y.a.293.49	yes	120
336.293	even	4	inner	336.2.y.a.293.11	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.y.a.125.11	✓	120	1.1	even	1	trivial
336.2.y.a.125.12	yes	120	7.6	odd	2	inner
336.2.y.a.125.49	yes	120	3.2	odd	2	inner
336.2.y.a.125.50	yes	120	21.20	even	2	inner
336.2.y.a.293.11	yes	120	336.293	even	4	inner
336.2.y.a.293.12	yes	120	48.5	odd	4	inner
336.2.y.a.293.49	yes	120	112.69	odd	4	inner
336.2.y.a.293.50	yes	120	16.5	even	4	inner