Embedded newform 287.2.be.a.12.16

• Label: 287.2.be.a.12.16
• Level: $287$
• Weight: $2$
• Character: 287.12
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $832$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.be (of order \(120\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(832\)
Relative dimension:	\(26\) over \(\Q(\zeta_{120})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{120}]$

Embedding invariants

Embedding label			12.16
Character	\(\chi\)	\(=\)	287.12
Dual form			287.2.be.a.24.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.507849 + 0.411248i) q^{2} +(-3.12411 + 0.411296i) q^{3} +(-0.327038 - 1.53859i) q^{4} +(-1.16320 - 0.0609607i) q^{5} +(-1.75572 - 1.07591i) q^{6} +(1.58731 + 2.11671i) q^{7} +(1.06000 - 2.08038i) q^{8} +(6.69310 - 1.79341i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 832 q - 16 q^{2} - 48 q^{3} - 20 q^{4} - 48 q^{5} - 32 q^{7} - 48 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{27}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.507849	+	0.411248i	0.359104	+	0.290796i	0.791845	−	0.610721i	\(-0.209120\pi\)
							−0.432742	+	0.901518i	\(0.642454\pi\)
\(3\)	−3.12411	+	0.411296i	−1.80370	+	0.237462i	−0.956074	−	0.293126i	\(-0.905305\pi\)
							−0.847630	+	0.530588i	\(0.821971\pi\)
\(5\)	−1.16320	−	0.0609607i	−0.520199	−	0.0272625i	−0.209570	−	0.977794i	\(-0.567206\pi\)
							−0.310629	+	0.950531i	\(0.600540\pi\)
\(7\)	1.58731	+	2.11671i	0.599948	+	0.800039i
							\(11\)	1.30926	+	2.74493i	0.394758	+	0.827628i	0.999513	+	0.0312157i	\(0.00993788\pi\)
−0.604755	+	0.796412i	\(0.706729\pi\)
\(13\)	0.451853	+	1.88210i	0.125321	+	0.522001i	0.999347	+	0.0361221i	\(0.0115005\pi\)
							−0.874026	+	0.485879i	\(0.838499\pi\)
\(17\)	−0.268621	−	0.0951238i	−0.0651502	−	0.0230709i	0.301032	−	0.953614i	\(-0.402669\pi\)
							−0.366183	+	0.930543i	\(0.619335\pi\)
\(19\)	5.74096	+	5.44796i	1.31707	+	1.24985i	0.947153	+	0.320782i	\(0.103946\pi\)
							0.369912	+	0.929067i	\(0.379388\pi\)
\(23\)	4.96174	−	0.521500i	1.03459	−	0.108740i	0.428022	−	0.903768i	\(-0.359211\pi\)
							0.606572	+	0.795028i	\(0.292544\pi\)
\(29\)	4.01366	−	4.69940i	0.745318	−	0.872656i	−0.250105	−	0.968219i	\(-0.580465\pi\)
							0.995423	+	0.0955630i	\(0.0304651\pi\)
\(31\)	5.61422	−	6.23522i	1.00834	−	1.11988i	0.0155690	−	0.999879i	\(-0.495044\pi\)
							0.992774	−	0.120000i	\(-0.0382893\pi\)
\(37\)	4.07856	+	4.52970i	0.670511	+	0.744678i	0.978395	−	0.206746i	\(-0.0662874\pi\)
							−0.307884	+	0.951424i	\(0.599621\pi\)
\(41\)	−6.06471	+	2.05409i	−0.947149	+	0.320795i
							\(43\)	−1.79260	+	11.3180i	−0.273369	+	1.72598i	0.343706	+	0.939077i	\(0.388318\pi\)
−0.617074	+	0.786905i	\(0.711682\pi\)
\(47\)	2.87225	−	1.55950i	0.418960	−	0.227477i	−0.253925	−	0.967224i	\(-0.581722\pi\)
							0.672885	+	0.739747i	\(0.265055\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.be.a.12.16	✓	832
7.3	odd	6	inner	287.2.be.a.94.11	yes	832
41.24	odd	40	inner	287.2.be.a.229.11	yes	832
287.24	even	120	inner	287.2.be.a.24.16	yes	832

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.be.a.12.16	✓	832	1.1	even	1	trivial
287.2.be.a.24.16	yes	832	287.24	even	120	inner
287.2.be.a.94.11	yes	832	7.3	odd	6	inner
287.2.be.a.229.11	yes	832	41.24	odd	40	inner