Embedded newform 287.3.y.a.10.15

• Label: 287.3.y.a.10.15
• Level: $287$
• Weight: $3$
• Character: 287.10
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.y (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			10.15
Character	\(\chi\)	\(=\)	287.10
Dual form			287.3.y.a.201.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.197672 + 1.88072i) q^{2} +(-2.25015 - 1.29912i) q^{3} +(0.414549 + 0.0881151i) q^{4} +(-2.05729 - 1.85239i) q^{5} +(2.88808 - 3.97510i) q^{6} +(6.85731 - 1.40618i) q^{7} +(-2.58517 + 7.95634i) q^{8} +(-1.12456 - 1.94779i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} - 24 q^{3} + 101 q^{4} - 9 q^{5} - 6 q^{7} - 16 q^{8} + 576 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.197672	+	1.88072i	−0.0988359	+	0.940361i	0.826940	+	0.562290i	\(0.190079\pi\)
							−0.925776	+	0.378072i	\(0.876587\pi\)
\(3\)	−2.25015	−	1.29912i	−0.750049	−	0.433041i	0.0756625	−	0.997133i	\(-0.475893\pi\)
							−0.825712	+	0.564092i	\(0.809226\pi\)
\(5\)	−2.05729	−	1.85239i	−0.411459	−	0.370479i	0.437250	−	0.899340i	\(-0.355953\pi\)
							−0.848708	+	0.528861i	\(0.822619\pi\)
\(7\)	6.85731	−	1.40618i	0.979615	−	0.200883i
							\(11\)	−10.2753	−	11.4119i	−0.934121	−	1.03745i	−0.999217	−	0.0395679i	\(-0.987402\pi\)
0.0650955	−	0.997879i	\(-0.479265\pi\)
\(13\)	−13.3668	+	18.3978i	−1.02822	+	1.41522i	−0.121938	+	0.992538i	\(0.538911\pi\)
							−0.906279	+	0.422681i	\(0.861089\pi\)
\(17\)	−20.6682	+	18.6097i	−1.21578	+	1.09469i	−0.222987	+	0.974821i	\(0.571581\pi\)
							−0.992790	+	0.119869i	\(0.961753\pi\)
\(19\)	12.4393	−	27.9390i	0.654698	−	1.47048i	−0.214861	−	0.976645i	\(-0.568930\pi\)
							0.869558	−	0.493830i	\(-0.164404\pi\)
\(23\)	2.23196	−	21.2357i	0.0970418	−	0.923291i	−0.832363	−	0.554230i	\(-0.813013\pi\)
							0.929405	−	0.369061i	\(-0.120321\pi\)
\(29\)	−13.5244	−	41.6238i	−0.466358	−	1.43530i	−0.857266	−	0.514873i	\(-0.827839\pi\)
							0.390908	−	0.920430i	\(-0.372161\pi\)
\(31\)	−14.0794	+	12.6771i	−0.454174	+	0.408940i	−0.864212	−	0.503129i	\(-0.832182\pi\)
							0.410038	+	0.912069i	\(0.365516\pi\)
\(37\)	2.46331	−	2.73579i	0.0665760	−	0.0739402i	−0.708937	−	0.705272i	\(-0.750825\pi\)
							0.775513	+	0.631332i	\(0.217491\pi\)
\(41\)	−40.2409	−	7.85326i	−0.981484	−	0.191543i
							\(43\)	−31.2322	−	22.6915i	−0.726330	−	0.527709i	0.162071	−	0.986779i	\(-0.448183\pi\)
−0.888400	+	0.459070i	\(0.848183\pi\)
\(47\)	3.70375	+	0.389280i	0.0788031	+	0.00828254i	0.143848	−	0.989600i	\(-0.454052\pi\)
							−0.0650448	+	0.997882i	\(0.520719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.y.a.10.15	✓	432
7.5	odd	6	inner	287.3.y.a.215.40	yes	432
41.37	even	5	inner	287.3.y.a.283.40	yes	432
287.201	odd	30	inner	287.3.y.a.201.15	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.y.a.10.15	✓	432	1.1	even	1	trivial
287.3.y.a.201.15	yes	432	287.201	odd	30	inner
287.3.y.a.215.40	yes	432	7.5	odd	6	inner
287.3.y.a.283.40	yes	432	41.37	even	5	inner