Embedded newform 287.2.be.a.12.1

• Label: 287.2.be.a.12.1
• 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			12.1
Character	\(\chi\)	\(=\)	287.12
Dual form			287.2.be.a.24.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.13508 - 1.72896i) q^{2} +(0.574751 - 0.0756675i) q^{3} +(1.15347 + 5.42664i) q^{4} +(-2.25766 - 0.118319i) q^{5} +(-1.35797 - 0.832164i) q^{6} +(2.57570 - 0.604798i) q^{7} +(4.42515 - 8.68484i) q^{8} +(-2.57316 + 0.689477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 832 q - 16 q^{2} - 48 q^{3} - 20 q^{4} - 48 q^{5} - 32 q^{7} - 48 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.13508	−	1.72896i	−1.50973	−	1.22256i	−0.904474	−	0.426528i	\(-0.859737\pi\)
							−0.605258	−	0.796029i	\(-0.706930\pi\)
\(3\)	0.574751	−	0.0756675i	0.331833	−	0.0436866i	0.0372310	−	0.999307i	\(-0.488146\pi\)
							0.294602	+	0.955620i	\(0.404813\pi\)
\(5\)	−2.25766	−	0.118319i	−1.00966	−	0.0529139i	−0.459673	−	0.888088i	\(-0.652033\pi\)
							−0.549985	+	0.835174i	\(0.685367\pi\)
\(7\)	2.57570	−	0.604798i	0.973522	−	0.228592i
							\(11\)	−1.04541	−	2.19175i	−0.315204	−	0.660838i	0.682329	−	0.731045i	\(-0.260967\pi\)
−0.997532	+	0.0702073i	\(0.977634\pi\)
\(13\)	−0.0817571	−	0.340543i	−0.0226753	−	0.0944496i	0.959885	−	0.280395i	\(-0.0904653\pi\)
							−0.982560	+	0.185945i	\(0.940465\pi\)
\(17\)	−5.86153	−	2.07568i	−1.42163	−	0.503425i	−0.491571	−	0.870837i	\(-0.663577\pi\)
							−0.930058	+	0.367412i	\(0.880244\pi\)
\(19\)	−4.18154	−	3.96814i	−0.959312	−	0.910353i	0.0367225	−	0.999326i	\(-0.488308\pi\)
							−0.996034	+	0.0889728i	\(0.971642\pi\)
\(23\)	4.81840	−	0.506434i	1.00471	−	0.105599i	0.412140	−	0.911121i	\(-0.364782\pi\)
							0.592566	+	0.805522i	\(0.298115\pi\)
\(29\)	−4.96754	+	5.81625i	−0.922450	+	1.08005i	0.0741165	+	0.997250i	\(0.476386\pi\)
							−0.996566	+	0.0828002i	\(0.973614\pi\)
\(31\)	3.08465	−	3.42585i	0.554020	−	0.615302i	−0.399463	−	0.916750i	\(-0.630803\pi\)
							0.953483	+	0.301448i	\(0.0974699\pi\)
\(37\)	−4.78760	−	5.31717i	−0.787077	−	0.874137i	0.207490	−	0.978237i	\(-0.433471\pi\)
							−0.994567	+	0.104100i	\(0.966804\pi\)
\(41\)	−3.41114	+	5.41886i	−0.532731	+	0.846284i
							\(43\)	−0.139649	+	0.881709i	−0.0212963	+	0.134459i	−0.996046	−	0.0888420i	\(-0.971683\pi\)
0.974749	+	0.223301i	\(0.0716834\pi\)
\(47\)	0.590228	−	0.320468i	0.0860936	−	0.0467450i	−0.433539	−	0.901135i	\(-0.642735\pi\)
							0.519633	+	0.854390i	\(0.326069\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.be.a.12.1	✓	832
7.3	odd	6	inner	287.2.be.a.94.26	yes	832
41.24	odd	40	inner	287.2.be.a.229.26	yes	832
287.24	even	120	inner	287.2.be.a.24.1	yes	832

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.be.a.12.1	✓	832	1.1	even	1	trivial
287.2.be.a.24.1	yes	832	287.24	even	120	inner
287.2.be.a.94.26	yes	832	7.3	odd	6	inner
287.2.be.a.229.26	yes	832	41.24	odd	40	inner