Embedded newform 336.2.y.a.293.7

• Label: 336.2.y.a.293.7
• Level: $336$
• Weight: $2$
• Character: 336.293
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $120$
• 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			293.7
Character	\(\chi\)	\(=\)	336.293
Dual form			336.2.y.a.125.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28455 + 0.591550i) q^{2} +(-1.44974 - 0.947768i) q^{3} +(1.30014 - 1.51975i) q^{4} +(0.639017 + 0.639017i) q^{5} +(2.42291 + 0.359865i) q^{6} +(-1.10656 - 2.40323i) q^{7} +(-0.771085 + 2.72129i) q^{8} +(1.20347 + 2.74803i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 8 q^{4}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28455	+	0.591550i	−0.908314	+	0.418289i
							\(3\)	−1.44974	−	0.947768i	−0.837006	−	0.547194i
\(5\)	0.639017	+	0.639017i	0.285777	+	0.285777i								0.835408	−	0.549631i	\(-0.185232\pi\)
							−0.549631	+	0.835408i	\(0.685232\pi\)
\(7\)	−1.10656	−	2.40323i	−0.418240	−	0.908336i
							\(11\)	2.11808	−	2.11808i	0.638625	−	0.638625i	−0.311591	−	0.950216i	\(-0.600862\pi\)
0.950216	+	0.311591i	\(0.100862\pi\)
\(13\)	0.505248	+	0.505248i	0.140131	+	0.140131i	0.773692	−	0.633562i	\(-0.218408\pi\)
							−0.633562	+	0.773692i	\(0.718408\pi\)
\(17\)	−5.88112			−1.42638			−0.713190	−	0.700971i	\(-0.752750\pi\)
							−0.713190	+	0.700971i	\(0.752750\pi\)
\(19\)	−1.69500	−	1.69500i	−0.388860	−	0.388860i	0.485421	−	0.874281i	\(-0.338666\pi\)
							−0.874281	+	0.485421i	\(0.838666\pi\)
\(23\)	−2.20670			−0.460128			−0.230064	−	0.973176i	\(-0.573894\pi\)
							−0.230064	+	0.973176i	\(0.573894\pi\)
\(29\)	−2.64164	−	2.64164i	−0.490541	−	0.490541i	0.417936	−	0.908477i	\(-0.362754\pi\)
							−0.908477	+	0.417936i	\(0.862754\pi\)
\(31\)		−	8.14559i		−	1.46299i	−0.681847	−	0.731495i	\(-0.738823\pi\)
							0.681847	−	0.731495i	\(-0.261177\pi\)
\(37\)	−1.18455	−	1.18455i	−0.194739	−	0.194739i	0.603001	−	0.797740i	\(-0.293971\pi\)
							−0.797740	+	0.603001i	\(0.793971\pi\)
\(41\)		−	8.70605i		−	1.35966i	−0.733371	−	0.679829i	\(-0.762054\pi\)
							0.733371	−	0.679829i	\(-0.237946\pi\)
\(43\)	2.47677	+	2.47677i	0.377704	+	0.377704i	0.870273	−	0.492569i	\(-0.163942\pi\)
							−0.492569	+	0.870273i	\(0.663942\pi\)
\(47\)	−10.2932			−1.50141			−0.750707	−	0.660635i	\(-0.770287\pi\)
							−0.750707	+	0.660635i	\(0.770287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.y.a.293.7	yes	120
3.2	odd	2	inner	336.2.y.a.293.53	yes	120
7.6	odd	2	inner	336.2.y.a.293.8	yes	120
16.13	even	4	inner	336.2.y.a.125.54	yes	120
21.20	even	2	inner	336.2.y.a.293.54	yes	120
48.29	odd	4	inner	336.2.y.a.125.8	yes	120
112.13	odd	4	inner	336.2.y.a.125.53	yes	120
336.125	even	4	inner	336.2.y.a.125.7	✓	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.y.a.125.7	✓	120	336.125	even	4	inner
336.2.y.a.125.8	yes	120	48.29	odd	4	inner
336.2.y.a.125.53	yes	120	112.13	odd	4	inner
336.2.y.a.125.54	yes	120	16.13	even	4	inner
336.2.y.a.293.7	yes	120	1.1	even	1	trivial
336.2.y.a.293.8	yes	120	7.6	odd	2	inner
336.2.y.a.293.53	yes	120	3.2	odd	2	inner
336.2.y.a.293.54	yes	120	21.20	even	2	inner