Embedded newform 336.2.bq.b.37.7

• Label: 336.2.bq.b.37.7
• Level: $336$
• Weight: $2$
• Character: 336.37
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.bq (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.7
Character	\(\chi\)	\(=\)	336.37
Dual form			336.2.bq.b.109.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02134 + 0.978190i) q^{2} +(-0.965926 - 0.258819i) q^{3} +(0.0862884 - 1.99814i) q^{4} +(-3.48258 + 0.933154i) q^{5} +(1.23972 - 0.680516i) q^{6} +(1.89801 - 1.84325i) q^{7} +(1.86643 + 2.12519i) q^{8} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02134	+	0.978190i	−0.722199	+	0.691685i
							\(3\)	−0.965926	−	0.258819i	−0.557678	−	0.149429i
\(5\)	−3.48258	+	0.933154i	−1.55746	+	0.417319i								−0.931857	−	0.362825i	\(-0.881812\pi\)
							−0.625599	+	0.780145i	\(0.715145\pi\)
\(7\)	1.89801	−	1.84325i	0.717381	−	0.696682i
							\(11\)	0.221968	−	0.828397i	0.0669260	−	0.249771i	−0.924356	−	0.381532i	\(-0.875397\pi\)
0.991281	+	0.131761i	\(0.0420632\pi\)
\(13\)	0.343455	−	0.343455i	0.0952572	−	0.0952572i	−0.657872	−	0.753129i	\(-0.728543\pi\)
							0.753129	+	0.657872i	\(0.228543\pi\)
\(17\)	1.72615	+	2.98978i	0.418653	+	0.725128i	0.995804	−	0.0915094i	\(-0.0291691\pi\)
							−0.577152	+	0.816637i	\(0.695836\pi\)
\(19\)	2.07058	+	7.72750i	0.475023	+	1.77281i	0.621307	+	0.783567i	\(0.286602\pi\)
							−0.146284	+	0.989243i	\(0.546731\pi\)
\(23\)	2.56988	+	1.48372i	0.535858	+	0.309378i	0.743399	−	0.668849i	\(-0.233213\pi\)
							−0.207541	+	0.978226i	\(0.566546\pi\)
\(29\)	4.67241	−	4.67241i	0.867644	−	0.867644i	−0.124567	−	0.992211i	\(-0.539754\pi\)
							0.992211	+	0.124567i	\(0.0397541\pi\)
\(31\)	3.93779	+	6.82046i	0.707249	+	1.22499i	0.965874	+	0.259013i	\(0.0833973\pi\)
							−0.258625	+	0.965978i	\(0.583269\pi\)
\(37\)	−3.23892	+	0.867866i	−0.532475	+	0.142676i	−0.515031	−	0.857172i	\(-0.672220\pi\)
							−0.0174440	+	0.999848i	\(0.505553\pi\)
\(41\)			4.63137i			0.723299i	0.932314	+	0.361649i	\(0.117786\pi\)
							−0.932314	+	0.361649i	\(0.882214\pi\)
\(43\)	4.39977	+	4.39977i	0.670958	+	0.670958i	0.957937	−	0.286979i	\(-0.0926509\pi\)
							−0.286979	+	0.957937i	\(0.592651\pi\)
\(47\)	4.92832	−	8.53611i	0.718870	−	1.24512i	−0.242578	−	0.970132i	\(-0.577993\pi\)
							0.961448	−	0.274987i	\(-0.0886736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bq.b.37.7	✓	120
7.4	even	3	inner	336.2.bq.b.277.13	yes	120
16.13	even	4	inner	336.2.bq.b.205.13	yes	120
112.109	even	12	inner	336.2.bq.b.109.7	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bq.b.37.7	✓	120	1.1	even	1	trivial
336.2.bq.b.109.7	yes	120	112.109	even	12	inner
336.2.bq.b.205.13	yes	120	16.13	even	4	inner
336.2.bq.b.277.13	yes	120	7.4	even	3	inner