Embedded newform 336.2.y.a.125.3

• Label: 336.2.y.a.125.3
• 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			125.3
Character	\(\chi\)	\(=\)	336.125
Dual form			336.2.y.a.293.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37838 + 0.316322i) q^{2} +(-0.367327 - 1.69265i) q^{3} +(1.79988 - 0.872025i) q^{4} +(-0.461355 + 0.461355i) q^{5} +(1.04174 + 2.21693i) q^{6} +(-2.58125 + 0.580663i) q^{7} +(-2.20509 + 1.77133i) q^{8} +(-2.73014 + 1.24351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37838	+	0.316322i	−0.974664	+	0.223673i
							\(3\)	−0.367327	−	1.69265i	−0.212076	−	0.977253i
\(5\)	−0.461355	+	0.461355i	−0.206324	+	0.206324i								−0.802703	−	0.596379i	\(-0.796606\pi\)
							0.596379	+	0.802703i	\(0.296606\pi\)
\(7\)	−2.58125	+	0.580663i	−0.975619	+	0.219470i
							\(11\)	2.02654	+	2.02654i	0.611026	+	0.611026i	0.943213	−	0.332188i	\(-0.107787\pi\)
−0.332188	+	0.943213i	\(0.607787\pi\)
\(13\)	−3.73127	+	3.73127i	−1.03487	+	1.03487i	−0.0354973	+	0.999370i	\(0.511302\pi\)
							−0.999370	+	0.0354973i	\(0.988698\pi\)
\(17\)	1.00011			0.242562			0.121281	−	0.992618i	\(-0.461300\pi\)
							0.121281	+	0.992618i	\(0.461300\pi\)
\(19\)	0.833607	−	0.833607i	0.191243	−	0.191243i	−0.604990	−	0.796233i	\(-0.706823\pi\)
							0.796233	+	0.604990i	\(0.206823\pi\)
\(23\)	3.08625			0.643527			0.321763	−	0.946820i	\(-0.395724\pi\)
							0.321763	+	0.946820i	\(0.395724\pi\)
\(29\)	−3.19477	+	3.19477i	−0.593255	+	0.593255i	−0.938509	−	0.345254i	\(-0.887793\pi\)
							0.345254	+	0.938509i	\(0.387793\pi\)
\(31\)		−	2.56995i		−	0.461576i	−0.973004	−	0.230788i	\(-0.925870\pi\)
							0.973004	−	0.230788i	\(-0.0741305\pi\)
\(37\)	−7.39802	+	7.39802i	−1.21623	+	1.21623i	−0.247285	+	0.968943i	\(0.579538\pi\)
							−0.968943	+	0.247285i	\(0.920462\pi\)
\(41\)			5.51962i			0.862019i	0.902347	+	0.431010i	\(0.141842\pi\)
							−0.902347	+	0.431010i	\(0.858158\pi\)
\(43\)	0.484910	−	0.484910i	0.0739481	−	0.0739481i	−0.669165	−	0.743114i	\(-0.733348\pi\)
							0.743114	+	0.669165i	\(0.233348\pi\)
\(47\)	−5.66080			−0.825713			−0.412856	−	0.910796i	\(-0.635469\pi\)
							−0.412856	+	0.910796i	\(0.635469\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.y.a.125.3	✓	120
3.2	odd	2	inner	336.2.y.a.125.58	yes	120
7.6	odd	2	inner	336.2.y.a.125.4	yes	120
16.5	even	4	inner	336.2.y.a.293.57	yes	120
21.20	even	2	inner	336.2.y.a.125.57	yes	120
48.5	odd	4	inner	336.2.y.a.293.4	yes	120
112.69	odd	4	inner	336.2.y.a.293.58	yes	120
336.293	even	4	inner	336.2.y.a.293.3	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.y.a.125.3	✓	120	1.1	even	1	trivial
336.2.y.a.125.4	yes	120	7.6	odd	2	inner
336.2.y.a.125.57	yes	120	21.20	even	2	inner
336.2.y.a.125.58	yes	120	3.2	odd	2	inner
336.2.y.a.293.3	yes	120	336.293	even	4	inner
336.2.y.a.293.4	yes	120	48.5	odd	4	inner
336.2.y.a.293.57	yes	120	16.5	even	4	inner
336.2.y.a.293.58	yes	120	112.69	odd	4	inner