Embedded newform 287.3.ba.a.15.2

• Label: 287.3.ba.a.15.2
• 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			15.2
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.66968 - 0.581219i) q^{2} +(5.26184 - 2.17952i) q^{3} +(9.32448 + 3.02971i) q^{4} +(3.63247 + 7.12913i) q^{5} +(-20.5760 + 4.93986i) q^{6} +(2.57265 + 0.617638i) q^{7} +(-19.2150 - 9.79055i) q^{8} +(16.5726 - 16.5726i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.66968	−	0.581219i	−1.83484	−	0.290610i	−0.859469	−	0.511189i	\(-0.829205\pi\)
							−0.975369	+	0.220579i	\(0.929205\pi\)
\(3\)	5.26184	−	2.17952i	1.75395	−	0.726508i	0.756585	−	0.653895i	\(-0.226866\pi\)
							0.997360	−	0.0726129i	\(-0.0231338\pi\)
\(5\)	3.63247	+	7.12913i	0.726494	+	1.42583i	0.897708	+	0.440592i	\(0.145231\pi\)
							−0.171213	+	0.985234i	\(0.554769\pi\)
\(7\)	2.57265	+	0.617638i	0.367521	+	0.0882341i
							\(11\)	−7.62085	+	6.50882i	−0.692805	+	0.591711i	−0.924548	−	0.381066i	\(-0.875557\pi\)
0.231743	+	0.972777i	\(0.425557\pi\)
\(13\)	7.89267	−	12.8797i	0.607129	−	0.990744i	−0.390411	−	0.920641i	\(-0.627667\pi\)
							0.997540	−	0.0701032i	\(-0.0223329\pi\)
\(17\)	0.738362	−	9.38178i	0.0434331	−	0.551870i	−0.936252	−	0.351329i	\(-0.885730\pi\)
							0.979685	−	0.200541i	\(-0.0642700\pi\)
\(19\)	−3.96822	−	6.47555i	−0.208854	−	0.340818i	0.731011	−	0.682366i	\(-0.239049\pi\)
							−0.939864	+	0.341548i	\(0.889049\pi\)
\(23\)	6.43998	+	8.86387i	0.279999	+	0.385386i	0.925734	−	0.378176i	\(-0.123449\pi\)
							−0.645734	+	0.763562i	\(0.723449\pi\)
\(29\)	1.43364	+	18.2161i	0.0494359	+	0.628142i	0.970870	+	0.239607i	\(0.0770187\pi\)
							−0.921434	+	0.388535i	\(0.872981\pi\)
\(31\)	0.457028	−	0.148497i	0.0147428	−	0.00479024i	−0.301636	−	0.953423i	\(-0.597533\pi\)
							0.316379	+	0.948633i	\(0.397533\pi\)
\(37\)	−11.6366	+	35.8138i	−0.314503	+	0.967940i	0.661456	+	0.749984i	\(0.269939\pi\)
							−0.975959	+	0.217956i	\(0.930061\pi\)
\(41\)	28.5376	+	29.4382i	0.696038	+	0.718005i
							\(43\)	25.5467	+	4.04619i	0.594108	+	0.0940975i	0.446246	−	0.894910i	\(-0.352761\pi\)
0.147862	+	0.989008i	\(0.452761\pi\)
\(47\)	79.9399	−	19.1919i	1.70085	−	0.408338i	0.737235	−	0.675636i	\(-0.236131\pi\)
							0.963614	+	0.267299i	\(0.0861310\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.2	✓	672
41.11	odd	40	inner	287.3.ba.a.134.2	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.2	✓	672	1.1	even	1	trivial
287.3.ba.a.134.2	yes	672	41.11	odd	40	inner