Embedded newform 287.2.be.a.12.6

• Label: 287.2.be.a.12.6
• 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			12.6
Character	\(\chi\)	\(=\)	287.12
Dual form			287.2.be.a.24.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25725 - 1.01810i) q^{2} +(0.937421 - 0.123414i) q^{3} +(0.128329 + 0.603740i) q^{4} +(-3.08596 - 0.161728i) q^{5} +(-1.30423 - 0.799230i) q^{6} +(0.923496 + 2.47935i) q^{7} +(-1.01559 + 1.99321i) q^{8} +(-2.03425 + 0.545076i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 832 q - 16 q^{2} - 48 q^{3} - 20 q^{4} - 48 q^{5} - 32 q^{7} - 48 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25725	−	1.01810i	−0.889013	−	0.719909i	0.0717073	−	0.997426i	\(-0.477155\pi\)
							−0.960721	+	0.277517i	\(0.910489\pi\)
\(3\)	0.937421	−	0.123414i	0.541220	−	0.0712530i	0.145040	−	0.989426i	\(-0.453669\pi\)
							0.396180	+	0.918173i	\(0.370336\pi\)
\(5\)	−3.08596	−	0.161728i	−1.38008	−	0.0723271i	−0.652210	−	0.758038i	\(-0.726158\pi\)
							−0.727874	+	0.685711i	\(0.759491\pi\)
\(7\)	0.923496	+	2.47935i	0.349049	+	0.937105i
							\(11\)	1.13452	+	2.37858i	0.342071	+	0.717168i	0.999334	−	0.0364942i	\(-0.0116190\pi\)
−0.657262	+	0.753662i	\(0.728286\pi\)
\(13\)	0.400038	+	1.66628i	0.110951	+	0.462143i	0.999991	+	0.00431335i	\(0.00137299\pi\)
							−0.889040	+	0.457830i	\(0.848627\pi\)
\(17\)	1.87273	+	0.663169i	0.454204	+	0.160842i	0.551076	−	0.834455i	\(-0.314217\pi\)
							−0.0968726	+	0.995297i	\(0.530884\pi\)
\(19\)	0.0996696	+	0.0945829i	0.0228658	+	0.0216988i	0.699702	−	0.714435i	\(-0.253316\pi\)
							−0.676836	+	0.736134i	\(0.736649\pi\)
\(23\)	−8.21751	+	0.863695i	−1.71347	+	0.180093i	−0.909818	−	0.415008i	\(-0.863779\pi\)
							−0.803651	+	0.595101i	\(0.797112\pi\)
\(29\)	2.88391	−	3.37663i	0.535529	−	0.627024i	−0.425004	−	0.905192i	\(-0.639727\pi\)
							0.960532	+	0.278168i	\(0.0897272\pi\)
\(31\)	−2.13573	+	2.37197i	−0.383589	+	0.426019i	−0.903758	−	0.428045i	\(-0.859203\pi\)
							0.520168	+	0.854064i	\(0.325869\pi\)
\(37\)	7.56343	+	8.40004i	1.24342	+	1.38096i	0.896488	+	0.443068i	\(0.146110\pi\)
							0.346933	+	0.937890i	\(0.387223\pi\)
\(41\)	−3.90428	+	5.07509i	−0.609746	+	0.792597i
							\(43\)	0.390477	−	2.46537i	0.0595471	−	0.375966i	−0.939861	−	0.341558i	\(-0.889045\pi\)
0.999408	−	0.0344079i	\(-0.0109545\pi\)
\(47\)	−3.31077	+	1.79760i	−0.482925	+	0.262207i	−0.700257	−	0.713891i	\(-0.746931\pi\)
							0.217332	+	0.976098i	\(0.430265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.be.a.12.6	✓	832
7.3	odd	6	inner	287.2.be.a.94.21	yes	832
41.24	odd	40	inner	287.2.be.a.229.21	yes	832
287.24	even	120	inner	287.2.be.a.24.6	yes	832

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.be.a.12.6	✓	832	1.1	even	1	trivial
287.2.be.a.24.6	yes	832	287.24	even	120	inner
287.2.be.a.94.21	yes	832	7.3	odd	6	inner
287.2.be.a.229.21	yes	832	41.24	odd	40	inner