Embedded newform 287.3.ba.a.15.13

• Label: 287.3.ba.a.15.13
• Level: $287$
• Weight: $3$
• Character: 287.15
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.ba (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(42\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			15.13
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96641 - 0.311449i) q^{2} +(3.24786 - 1.34531i) q^{3} +(-0.0344593 - 0.0111965i) q^{4} +(1.89942 + 3.72783i) q^{5} +(-6.80562 + 1.63388i) q^{6} +(2.57265 + 0.617638i) q^{7} +(7.15997 + 3.64819i) q^{8} +(2.37478 - 2.37478i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{37}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96641	−	0.311449i	−0.983205	−	0.155724i	−0.355922	−	0.934516i	\(-0.615833\pi\)
							−0.627283	+	0.778791i	\(0.715833\pi\)
\(3\)	3.24786	−	1.34531i	1.08262	−	0.448436i	0.231193	−	0.972908i	\(-0.425737\pi\)
							0.851428	+	0.524472i	\(0.175737\pi\)
\(5\)	1.89942	+	3.72783i	0.379885	+	0.745566i	0.999217	−	0.0395632i	\(-0.0125966\pi\)
							−0.619332	+	0.785129i	\(0.712597\pi\)
\(7\)	2.57265	+	0.617638i	0.367521	+	0.0882341i
							\(11\)	4.41743	−	3.77284i	0.401585	−	0.342986i	−0.425605	−	0.904909i	\(-0.639939\pi\)
0.827189	+	0.561924i	\(0.189939\pi\)
\(13\)	−7.71743	+	12.5937i	−0.593649	+	0.968747i	0.404872	+	0.914373i	\(0.367316\pi\)
							−0.998521	+	0.0543734i	\(0.982684\pi\)
\(17\)	−0.681313	+	8.65691i	−0.0400773	+	0.509230i	0.943816	+	0.330472i	\(0.107208\pi\)
							−0.983893	+	0.178758i	\(0.942792\pi\)
\(19\)	17.4305	+	28.4440i	0.917395	+	1.49705i	0.866991	+	0.498323i	\(0.166051\pi\)
							0.0504037	+	0.998729i	\(0.483949\pi\)
\(23\)	−5.73250	−	7.89011i	−0.249239	−	0.343048i	0.666006	−	0.745947i	\(-0.268003\pi\)
							−0.915245	+	0.402899i	\(0.868003\pi\)
\(29\)	0.0445607	+	0.566197i	0.00153657	+	0.0195240i	0.997638	−	0.0686933i	\(-0.0218830\pi\)
							−0.996101	+	0.0882173i	\(0.971883\pi\)
\(31\)	10.3269	−	3.35542i	0.333127	−	0.108239i	−0.137678	−	0.990477i	\(-0.543964\pi\)
							0.470804	+	0.882238i	\(0.343964\pi\)
\(37\)	11.5066	−	35.4135i	0.310988	−	0.957123i	−0.666386	−	0.745607i	\(-0.732160\pi\)
							0.977374	−	0.211516i	\(-0.0678402\pi\)
\(41\)	40.8321	+	3.70677i	0.995905	+	0.0904090i
							\(43\)	74.6803	+	11.8282i	1.73675	+	0.275074i	0.942909	−	0.333050i	\(-0.108078\pi\)
0.793842	+	0.608125i	\(0.208078\pi\)
\(47\)	−32.9746	+	7.91649i	−0.701587	+	0.168436i	−0.568518	−	0.822671i	\(-0.692483\pi\)
							−0.133069	+	0.991107i	\(0.542483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.13	✓	672
41.11	odd	40	inner	287.3.ba.a.134.13	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.13	✓	672	1.1	even	1	trivial
287.3.ba.a.134.13	yes	672	41.11	odd	40	inner