Embedded newform 336.2.y.a.125.20

• Label: 336.2.y.a.125.20
• 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			125.20
Character	\(\chi\)	\(=\)	336.125
Dual form			336.2.y.a.293.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.621008 + 1.27057i) q^{2} +(0.254609 + 1.71324i) q^{3} +(-1.22870 - 1.57807i) q^{4} +(0.914067 - 0.914067i) q^{5} +(-2.33490 - 0.740435i) q^{6} +(-0.812405 + 2.51794i) q^{7} +(2.76808 - 0.581151i) q^{8} +(-2.87035 + 0.872410i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.621008	+	1.27057i	−0.439119	+	0.898429i
							\(3\)	0.254609	+	1.71324i	0.146999	+	0.989137i
\(5\)	0.914067	−	0.914067i	0.408783	−	0.408783i								−0.472531	−	0.881314i	\(-0.656660\pi\)
							0.881314	+	0.472531i	\(0.156660\pi\)
\(7\)	−0.812405	+	2.51794i	−0.307060	+	0.951690i
							\(11\)	0.967438	+	0.967438i	0.291693	+	0.291693i	0.837749	−	0.546056i	\(-0.183871\pi\)
−0.546056	+	0.837749i	\(0.683871\pi\)
\(13\)	−4.55813	+	4.55813i	−1.26420	+	1.26420i	−0.315158	+	0.949039i	\(0.602057\pi\)
							−0.949039	+	0.315158i	\(0.897943\pi\)
\(17\)	−5.60148			−1.35856			−0.679279	−	0.733880i	\(-0.737707\pi\)
							−0.679279	+	0.733880i	\(0.737707\pi\)
\(19\)	0.604835	−	0.604835i	0.138759	−	0.138759i	−0.634316	−	0.773074i	\(-0.718718\pi\)
							0.773074	+	0.634316i	\(0.218718\pi\)
\(23\)	6.34885			1.32383			0.661914	−	0.749580i	\(-0.269745\pi\)
							0.661914	+	0.749580i	\(0.269745\pi\)
\(29\)	2.82422	−	2.82422i	0.524445	−	0.524445i	−0.394466	−	0.918911i	\(-0.629070\pi\)
							0.918911	+	0.394466i	\(0.129070\pi\)
\(31\)		−	1.06892i		−	0.191984i	−0.995382	−	0.0959921i	\(-0.969398\pi\)
							0.995382	−	0.0959921i	\(-0.0306024\pi\)
\(37\)	4.37678	−	4.37678i	0.719538	−	0.719538i	−0.248972	−	0.968511i	\(-0.580093\pi\)
							0.968511	+	0.248972i	\(0.0800928\pi\)
\(41\)			0.292926i			0.0457474i	0.999738	+	0.0228737i	\(0.00728156\pi\)
							−0.999738	+	0.0228737i	\(0.992718\pi\)
\(43\)	−7.58906	+	7.58906i	−1.15732	+	1.15732i	−0.172271	+	0.985050i	\(0.555111\pi\)
							−0.985050	+	0.172271i	\(0.944889\pi\)
\(47\)	12.5000			1.82331			0.911657	−	0.410951i	\(-0.134803\pi\)
							0.911657	+	0.410951i	\(0.134803\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.y.a.125.20	yes	120
3.2	odd	2	inner	336.2.y.a.125.41	yes	120
7.6	odd	2	inner	336.2.y.a.125.19	✓	120
16.5	even	4	inner	336.2.y.a.293.42	yes	120
21.20	even	2	inner	336.2.y.a.125.42	yes	120
48.5	odd	4	inner	336.2.y.a.293.19	yes	120
112.69	odd	4	inner	336.2.y.a.293.41	yes	120
336.293	even	4	inner	336.2.y.a.293.20	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.y.a.125.19	✓	120	7.6	odd	2	inner
336.2.y.a.125.20	yes	120	1.1	even	1	trivial
336.2.y.a.125.41	yes	120	3.2	odd	2	inner
336.2.y.a.125.42	yes	120	21.20	even	2	inner
336.2.y.a.293.19	yes	120	48.5	odd	4	inner
336.2.y.a.293.20	yes	120	336.293	even	4	inner
336.2.y.a.293.41	yes	120	112.69	odd	4	inner
336.2.y.a.293.42	yes	120	16.5	even	4	inner