Embedded newform 287.3.bd.a.115.19

• Label: 287.3.bd.a.115.19
• Level: $287$
• Weight: $3$
• Character: 287.115
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $864$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.bd (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(864\)
Relative dimension:	\(54\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			115.19
Character	\(\chi\)	\(=\)	287.115
Dual form			287.3.bd.a.5.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32950 - 0.139736i) q^{2} +(-1.23979 + 0.332200i) q^{3} +(-2.16455 - 0.460089i) q^{4} +(-5.09510 + 5.65868i) q^{5} +(1.69472 - 0.268417i) q^{6} +(-6.54965 + 2.47023i) q^{7} +(7.89905 + 2.56656i) q^{8} +(-6.36751 + 3.67628i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 864 q - 10 q^{2} - 24 q^{3} - 214 q^{4} - 30 q^{5} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{9}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32950	−	0.139736i	−0.664749	−	0.0698680i	−0.233857	−	0.972271i	\(-0.575135\pi\)
							−0.430893	+	0.902403i	\(0.641801\pi\)
\(3\)	−1.23979	+	0.332200i	−0.413263	+	0.110733i	−0.459459	−	0.888199i	\(-0.651957\pi\)
							0.0461961	+	0.998932i	\(0.485290\pi\)
\(5\)	−5.09510	+	5.65868i	−1.01902	+	1.13174i	−0.0277866	+	0.999614i	\(0.508846\pi\)
							−0.991233	+	0.132122i	\(0.957821\pi\)
\(7\)	−6.54965	+	2.47023i	−0.935665	+	0.352890i
							\(11\)	2.03152	+	0.106468i	0.184684	+	0.00967887i	0.144454	−	0.989512i	\(-0.453858\pi\)
0.0402302	+	0.999190i	\(0.487191\pi\)
\(13\)	0.569066	+	3.59294i	0.0437743	+	0.276380i	0.999860	−	0.0167057i	\(-0.00531785\pi\)
							−0.956086	+	0.293086i	\(0.905318\pi\)
\(17\)	−29.7726	−	1.56031i	−1.75133	−	0.0917832i	−0.850504	−	0.525969i	\(-0.823703\pi\)
							−0.900823	+	0.434186i	\(0.857036\pi\)
\(19\)	31.3555	−	12.0362i	1.65029	−	0.633486i	0.657509	−	0.753447i	\(-0.271610\pi\)
							0.992779	+	0.119961i	\(0.0382770\pi\)
\(23\)	−0.0584870	+	0.556467i	−0.00254291	+	0.0241942i	−0.995720	−	0.0924236i	\(-0.970539\pi\)
							0.993177	+	0.116618i	\(0.0372053\pi\)
\(29\)	−6.66556	−	3.39627i	−0.229847	−	0.117113i	0.335275	−	0.942120i	\(-0.391171\pi\)
							−0.565122	+	0.825007i	\(0.691171\pi\)
\(31\)	−29.9248	+	26.9444i	−0.965317	+	0.869175i	−0.991451	−	0.130483i	\(-0.958347\pi\)
							0.0261334	+	0.999658i	\(0.491681\pi\)
\(37\)	2.58817	−	2.87446i	0.0699506	−	0.0776881i	−0.707154	−	0.707060i	\(-0.750021\pi\)
							0.777104	+	0.629372i	\(0.216688\pi\)
\(41\)	28.7857	+	29.1956i	0.702090	+	0.712088i
							\(43\)	34.4565	−	47.4254i	0.801315	−	1.10292i	−0.191291	−	0.981533i	\(-0.561267\pi\)
0.992606	−	0.121382i	\(-0.0387327\pi\)
\(47\)	−13.1562	−	16.2465i	−0.279919	−	0.345671i	0.617724	−	0.786395i	\(-0.288055\pi\)
							−0.897643	+	0.440724i	\(0.854722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.bd.a.115.19	yes	864
7.5	odd	6	inner	287.3.bd.a.33.36	yes	864
41.5	even	20	inner	287.3.bd.a.87.36	yes	864
287.5	odd	60	inner	287.3.bd.a.5.19	✓	864

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.bd.a.5.19	✓	864	287.5	odd	60	inner
287.3.bd.a.33.36	yes	864	7.5	odd	6	inner
287.3.bd.a.87.36	yes	864	41.5	even	20	inner
287.3.bd.a.115.19	yes	864	1.1	even	1	trivial