Embedded newform 287.3.ba.a.15.5

• Label: 287.3.ba.a.15.5
• 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			15.5
Character	\(\chi\)	\(=\)	287.15
Dual form			287.3.ba.a.134.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.43813 - 0.544547i) q^{2} +(3.38252 - 1.40108i) q^{3} +(7.72000 + 2.50838i) q^{4} +(-2.27644 - 4.46777i) q^{5} +(-12.3925 + 2.97517i) q^{6} +(-2.57265 - 0.617638i) q^{7} +(-12.7701 - 6.50670i) q^{8} +(3.11441 - 3.11441i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q - 8 q^{2} + 16 q^{3} - 24 q^{6} + 48 q^{8} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.43813	−	0.544547i	−1.71907	−	0.272273i	−0.782471	−	0.622687i	\(-0.786041\pi\)
							−0.936595	+	0.350413i	\(0.886041\pi\)
\(3\)	3.38252	−	1.40108i	1.12751	−	0.467028i	0.260573	−	0.965454i	\(-0.416089\pi\)
							0.866932	+	0.498426i	\(0.166089\pi\)
\(5\)	−2.27644	−	4.46777i	−0.455289	−	0.893555i	−0.998542	−	0.0539856i	\(-0.982808\pi\)
							0.543253	−	0.839569i	\(-0.317192\pi\)
\(7\)	−2.57265	−	0.617638i	−0.367521	−	0.0882341i
							\(11\)	−14.2293	+	12.1529i	−1.29357	+	1.10481i	−0.305273	+	0.952265i	\(0.598748\pi\)
−0.988296	+	0.152548i	\(0.951252\pi\)
\(13\)	−4.34228	+	7.08595i	−0.334021	+	0.545073i	−0.974921	−	0.222551i	\(-0.928561\pi\)
							0.640900	+	0.767625i	\(0.278561\pi\)
\(17\)	−2.30566	+	29.2962i	−0.135627	+	1.72330i	0.433585	+	0.901113i	\(0.357248\pi\)
							−0.569212	+	0.822191i	\(0.692752\pi\)
\(19\)	8.19693	+	13.3762i	0.431418	+	0.704009i	0.992296	−	0.123889i	\(-0.0395366\pi\)
							−0.560879	+	0.827898i	\(0.689537\pi\)
\(23\)	−23.7800	−	32.7303i	−1.03391	−	1.42306i	−0.901969	−	0.431801i	\(-0.857878\pi\)
							−0.131944	−	0.991257i	\(-0.542122\pi\)
\(29\)	−2.97892	−	37.8508i	−0.102721	−	1.30520i	−0.804991	−	0.593288i	\(-0.797830\pi\)
							0.702269	−	0.711912i	\(-0.252170\pi\)
\(31\)	26.6870	−	8.67112i	0.860870	−	0.279714i	0.154878	−	0.987934i	\(-0.450502\pi\)
							0.705992	+	0.708220i	\(0.250502\pi\)
\(37\)	−16.5136	+	50.8236i	−0.446313	+	1.37361i	0.434724	+	0.900564i	\(0.356846\pi\)
							−0.881037	+	0.473047i	\(0.843154\pi\)
\(41\)	−15.1253	+	38.1081i	−0.368910	+	0.929465i
							\(43\)	25.4550	+	4.03168i	0.591977	+	0.0937600i	0.445234	−	0.895415i	\(-0.353121\pi\)
0.146744	+	0.989175i	\(0.453121\pi\)
\(47\)	−31.0907	+	7.46422i	−0.661504	+	0.158813i	−0.550277	−	0.834982i	\(-0.685478\pi\)
							−0.111227	+	0.993795i	\(0.535478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.ba.a.15.5	✓	672
41.11	odd	40	inner	287.3.ba.a.134.5	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.ba.a.15.5	✓	672	1.1	even	1	trivial
287.3.ba.a.134.5	yes	672	41.11	odd	40	inner