Embedded newform 287.3.ba.a.15.11

• Label: 287.3.ba.a.15.11
• 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			15.11
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.42937 - 0.384775i) q^{2} +(1.48197 - 0.613852i) q^{3} +(1.94957 + 0.633455i) q^{4} +(3.86932 + 7.59398i) q^{5} +(-3.83645 + 0.921050i) q^{6} +(-2.57265 - 0.617638i) q^{7} +(4.27377 + 2.17760i) q^{8} +(-4.54454 + 4.54454i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.42937	−	0.384775i	−1.21469	−	0.192387i	−0.483980	−	0.875079i	\(-0.660809\pi\)
							−0.730706	+	0.682692i	\(0.760809\pi\)
\(3\)	1.48197	−	0.613852i	0.493990	−	0.204617i	−0.121759	−	0.992560i	\(-0.538854\pi\)
							0.615749	+	0.787942i	\(0.288854\pi\)
\(5\)	3.86932	+	7.59398i	0.773865	+	1.51880i	0.852988	+	0.521930i	\(0.174788\pi\)
							−0.0791233	+	0.996865i	\(0.525212\pi\)
\(7\)	−2.57265	−	0.617638i	−0.367521	−	0.0882341i
							\(11\)	−5.51663	+	4.71165i	−0.501512	+	0.428332i	−0.863876	−	0.503705i	\(-0.831970\pi\)
0.362364	+	0.932037i	\(0.381970\pi\)
\(13\)	−5.95975	+	9.72542i	−0.458442	+	0.748109i	−0.995431	−	0.0954797i	\(-0.969562\pi\)
							0.536989	+	0.843589i	\(0.319562\pi\)
\(17\)	1.04351	−	13.2591i	0.0613830	−	0.779944i	−0.886787	−	0.462179i	\(-0.847068\pi\)
							0.948169	−	0.317765i	\(-0.102932\pi\)
\(19\)	−18.7130	−	30.5369i	−0.984895	−	1.60720i	−0.777438	−	0.628959i	\(-0.783481\pi\)
							−0.207457	−	0.978244i	\(-0.566519\pi\)
\(23\)	−0.762145	−	1.04900i	−0.0331367	−	0.0456088i	0.792127	−	0.610356i	\(-0.208974\pi\)
							−0.825264	+	0.564747i	\(0.808974\pi\)
\(29\)	−1.73548	−	22.0514i	−0.0598441	−	0.760392i	−0.951494	−	0.307667i	\(-0.900452\pi\)
							0.891650	−	0.452725i	\(-0.149548\pi\)
\(31\)	−6.11031	+	1.98536i	−0.197107	+	0.0640439i	−0.405907	−	0.913915i	\(-0.633044\pi\)
							0.208800	+	0.977958i	\(0.433044\pi\)
\(37\)	−9.70179	+	29.8590i	−0.262210	+	0.807001i	0.730112	+	0.683327i	\(0.239468\pi\)
							−0.992323	+	0.123674i	\(0.960532\pi\)
\(41\)	17.2546	+	37.1924i	0.420845	+	0.907133i
							\(43\)	10.2067	+	1.61659i	0.237366	+	0.0375951i	0.273984	−	0.961734i	\(-0.411658\pi\)
−0.0366179	+	0.999329i	\(0.511658\pi\)
\(47\)	−79.6660	+	19.1261i	−1.69502	+	0.406939i	−0.961953	−	0.273216i	\(-0.911913\pi\)
							−0.733069	+	0.680155i	\(0.761913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.11	✓	672
41.11	odd	40	inner	287.3.ba.a.134.11	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.11	✓	672	1.1	even	1	trivial
287.3.ba.a.134.11	yes	672	41.11	odd	40	inner