Embedded newform 336.2.y.a.125.15

• Label: 336.2.y.a.125.15
• 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			125.15
Character	\(\chi\)	\(=\)	336.125
Dual form			336.2.y.a.293.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.957005 + 1.04122i) q^{2} +(-1.52023 - 0.830004i) q^{3} +(-0.168282 - 1.99291i) q^{4} +(-1.24321 + 1.24321i) q^{5} +(2.31908 - 0.788576i) q^{6} +(1.10620 - 2.40340i) q^{7} +(2.23610 + 1.73200i) q^{8} +(1.62219 + 2.52359i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.957005	+	1.04122i	−0.676705	+	0.736254i
							\(3\)	−1.52023	−	0.830004i	−0.877704	−	0.479203i
\(5\)	−1.24321	+	1.24321i	−0.555980	+	0.555980i								−0.928160	−	0.372180i	\(-0.878610\pi\)
							0.372180	+	0.928160i	\(0.378610\pi\)
\(7\)	1.10620	−	2.40340i	0.418104	−	0.908399i
							\(11\)	0.0609709	+	0.0609709i	0.0183834	+	0.0183834i	0.716239	−	0.697855i	\(-0.245862\pi\)
−0.697855	+	0.716239i	\(0.745862\pi\)
\(13\)	−1.21487	+	1.21487i	−0.336943	+	0.336943i	−0.855216	−	0.518272i	\(-0.826575\pi\)
							0.518272	+	0.855216i	\(0.326575\pi\)
\(17\)	−5.01681			−1.21675			−0.608377	−	0.793648i	\(-0.708179\pi\)
							−0.608377	+	0.793648i	\(0.708179\pi\)
\(19\)	−3.53049	+	3.53049i	−0.809950	+	0.809950i	−0.984626	−	0.174676i	\(-0.944112\pi\)
							0.174676	+	0.984626i	\(0.444112\pi\)
\(23\)	0.280155			0.0584164			0.0292082	−	0.999573i	\(-0.490701\pi\)
							0.0292082	+	0.999573i	\(0.490701\pi\)
\(29\)	−6.89820	+	6.89820i	−1.28096	+	1.28096i	−0.340844	+	0.940120i	\(0.610713\pi\)
							−0.940120	+	0.340844i	\(0.889287\pi\)
\(31\)			5.91625i			1.06259i	0.847187	+	0.531295i	\(0.178294\pi\)
							−0.847187	+	0.531295i	\(0.821706\pi\)
\(37\)	0.0123848	−	0.0123848i	0.00203605	−	0.00203605i	−0.706088	−	0.708124i	\(-0.749542\pi\)
							0.708124	+	0.706088i	\(0.249542\pi\)
\(41\)		−	3.16937i		−	0.494973i	−0.968891	−	0.247486i	\(-0.920395\pi\)
							0.968891	−	0.247486i	\(-0.0796046\pi\)
\(43\)	−3.09016	+	3.09016i	−0.471245	+	0.471245i	−0.902317	−	0.431073i	\(-0.858135\pi\)
							0.431073	+	0.902317i	\(0.358135\pi\)
\(47\)	4.07120			0.593846			0.296923	−	0.954901i	\(-0.404040\pi\)
							0.296923	+	0.954901i	\(0.404040\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.y.a.125.15	✓	120
3.2	odd	2	inner	336.2.y.a.125.46	yes	120
7.6	odd	2	inner	336.2.y.a.125.16	yes	120
16.5	even	4	inner	336.2.y.a.293.45	yes	120
21.20	even	2	inner	336.2.y.a.125.45	yes	120
48.5	odd	4	inner	336.2.y.a.293.16	yes	120
112.69	odd	4	inner	336.2.y.a.293.46	yes	120
336.293	even	4	inner	336.2.y.a.293.15	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.y.a.125.15	✓	120	1.1	even	1	trivial
336.2.y.a.125.16	yes	120	7.6	odd	2	inner
336.2.y.a.125.45	yes	120	21.20	even	2	inner
336.2.y.a.125.46	yes	120	3.2	odd	2	inner
336.2.y.a.293.15	yes	120	336.293	even	4	inner
336.2.y.a.293.16	yes	120	48.5	odd	4	inner
336.2.y.a.293.45	yes	120	16.5	even	4	inner
336.2.y.a.293.46	yes	120	112.69	odd	4	inner