Embedded newform 287.3.x.a.31.15

• Label: 287.3.x.a.31.15
• Level: $287$
• Weight: $3$
• Character: 287.31
• 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.x (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			31.15
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.216725 + 2.06200i) q^{2} +(0.981587 - 1.70016i) q^{3} +(-0.292279 - 0.0621259i) q^{4} +(5.16597 + 4.65146i) q^{5} +(3.29299 + 2.39250i) q^{6} +(-6.36962 + 2.90310i) q^{7} +(-2.37136 + 7.29830i) q^{8} +(2.57297 + 4.45652i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.216725	+	2.06200i	−0.108362	+	1.03100i	0.796309	+	0.604890i	\(0.206783\pi\)
							−0.904672	+	0.426110i	\(0.859884\pi\)
\(3\)	0.981587	−	1.70016i	0.327196	−	0.566720i	−0.654759	−	0.755838i	\(-0.727230\pi\)
							0.981954	+	0.189118i	\(0.0605630\pi\)
\(5\)	5.16597	+	4.65146i	1.03319	+	0.930292i	0.997611	−	0.0690746i	\(-0.0220047\pi\)
							0.0355826	+	0.999367i	\(0.488671\pi\)
\(7\)	−6.36962	+	2.90310i	−0.909945	+	0.414729i
							\(11\)	5.42056	−	4.88069i	0.492778	−	0.443699i	−0.384894	−	0.922961i	\(-0.625762\pi\)
0.877672	+	0.479261i	\(0.159095\pi\)
\(13\)	−15.4911	−	11.2550i	−1.19162	−	0.865766i	−0.198189	−	0.980164i	\(-0.563506\pi\)
							−0.993435	+	0.114398i	\(0.963506\pi\)
\(17\)	16.1485	+	17.9348i	0.949914	+	1.05499i	0.998422	+	0.0561633i	\(0.0178867\pi\)
							−0.0485079	+	0.998823i	\(0.515447\pi\)
\(19\)	4.11035	+	1.83004i	0.216334	+	0.0963181i	0.512043	−	0.858960i	\(-0.328889\pi\)
							−0.295708	+	0.955278i	\(0.595556\pi\)
\(23\)	0.113331	−	1.07828i	0.00492745	−	0.0468816i	−0.991782	−	0.127940i	\(-0.959163\pi\)
							0.996709	+	0.0810586i	\(0.0258301\pi\)
\(29\)	−4.72678	+	1.53582i	−0.162992	+	0.0529595i	−0.389377	−	0.921079i	\(-0.627310\pi\)
							0.226384	+	0.974038i	\(0.427310\pi\)
\(31\)	−32.0799	+	28.8849i	−1.03484	+	0.931770i	−0.997716	−	0.0675412i	\(-0.978485\pi\)
							−0.0371185	+	0.999311i	\(0.511818\pi\)
\(37\)	37.2383	−	41.3573i	1.00644	−	1.11776i	0.0134087	−	0.999910i	\(-0.495732\pi\)
							0.993031	−	0.117854i	\(-0.0376016\pi\)
\(41\)	28.3286	+	29.6393i	0.690943	+	0.722910i
							\(43\)	−8.28771	−	6.02137i	−0.192737	−	0.140032i	0.487232	−	0.873273i	\(-0.338007\pi\)
−0.679969	+	0.733241i	\(0.738007\pi\)
\(47\)	9.43054	−	89.7256i	0.200650	−	1.90906i	−0.178870	−	0.983873i	\(-0.557244\pi\)
							0.379520	−	0.925183i	\(-0.376089\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.15	✓	432
7.5	odd	6	inner	287.3.x.a.236.40	yes	432
41.4	even	10	inner	287.3.x.a.45.40	yes	432
287.250	odd	30	inner	287.3.x.a.250.15	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.15	✓	432	1.1	even	1	trivial
287.3.x.a.45.40	yes	432	41.4	even	10	inner
287.3.x.a.236.40	yes	432	7.5	odd	6	inner
287.3.x.a.250.15	yes	432	287.250	odd	30	inner