Embedded newform 287.3.ba.a.15.6

• Label: 287.3.ba.a.15.6
• 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			15.6
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.42144 - 0.541903i) q^{2} +(2.05380 - 0.850711i) q^{3} +(7.60836 + 2.47211i) q^{4} +(0.819120 + 1.60761i) q^{5} +(-7.48794 + 1.79770i) q^{6} +(-2.57265 - 0.617638i) q^{7} +(-12.3458 - 6.29050i) q^{8} +(-2.86959 + 2.86959i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.42144	−	0.541903i	−1.71072	−	0.270951i	−0.777142	−	0.629326i	\(-0.783331\pi\)
							−0.933578	+	0.358374i	\(0.883331\pi\)
\(3\)	2.05380	−	0.850711i	0.684599	−	0.283570i	−0.0131489	−	0.999914i	\(-0.504186\pi\)
							0.697748	+	0.716343i	\(0.254186\pi\)
\(5\)	0.819120	+	1.60761i	0.163824	+	0.321523i	0.958296	−	0.285778i	\(-0.0922521\pi\)
							−0.794472	+	0.607301i	\(0.792252\pi\)
\(7\)	−2.57265	−	0.617638i	−0.367521	−	0.0882341i
							\(11\)	14.3939	−	12.2936i	1.30854	−	1.11760i	0.323320	−	0.946290i	\(-0.395201\pi\)
0.985218	−	0.171307i	\(-0.0547991\pi\)
\(13\)	3.91076	−	6.38178i	0.300828	−	0.490906i	−0.666133	−	0.745833i	\(-0.732052\pi\)
							0.966960	+	0.254927i	\(0.0820515\pi\)
\(17\)	−1.11925	+	14.2214i	−0.0658383	+	0.836555i	0.872005	+	0.489497i	\(0.162819\pi\)
							−0.937843	+	0.347059i	\(0.887181\pi\)
\(19\)	−6.13285	−	10.0079i	−0.322781	−	0.526731i	0.649555	−	0.760314i	\(-0.274955\pi\)
							−0.972337	+	0.233583i	\(0.924955\pi\)
\(23\)	18.4699	+	25.4217i	0.803041	+	1.10529i	0.992360	+	0.123375i	\(0.0393719\pi\)
							−0.189319	+	0.981916i	\(0.560628\pi\)
\(29\)	1.08593	+	13.7980i	0.0374458	+	0.475794i	0.986839	+	0.161704i	\(0.0516991\pi\)
							−0.949393	+	0.314089i	\(0.898301\pi\)
\(31\)	35.8747	−	11.6564i	1.15725	−	0.376013i	0.333380	−	0.942793i	\(-0.391811\pi\)
							0.823870	+	0.566779i	\(0.191811\pi\)
\(37\)	17.8898	−	55.0591i	0.483508	−	1.48808i	−0.350623	−	0.936517i	\(-0.614030\pi\)
							0.834131	−	0.551567i	\(-0.185970\pi\)
\(41\)	−14.3161	−	38.4194i	−0.349172	−	0.937059i
							\(43\)	29.3969	+	4.65602i	0.683649	+	0.108279i	0.488592	−	0.872512i	\(-0.337511\pi\)
0.195057	+	0.980792i	\(0.437511\pi\)
\(47\)	49.2744	−	11.8297i	1.04839	−	0.251697i	0.327577	−	0.944824i	\(-0.393768\pi\)
							0.720815	+	0.693128i	\(0.243768\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.6	✓	672
41.11	odd	40	inner	287.3.ba.a.134.6	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.6	✓	672	1.1	even	1	trivial
287.3.ba.a.134.6	yes	672	41.11	odd	40	inner