Embedded newform 287.2.bc.a.2.2

• Label: 287.2.bc.a.2.2
• Level: $287$
• Weight: $2$
• Character: 287.2
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $416$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.bc (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(416\)
Relative dimension:	\(26\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			2.2
Character	\(\chi\)	\(=\)	287.2
Dual form			287.2.bc.a.144.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.519333 + 2.44327i) q^{2} +(2.09987 + 0.562657i) q^{3} +(-3.87276 - 1.72426i) q^{4} +(2.28837 - 0.240517i) q^{5} +(-2.46525 + 4.83833i) q^{6} +(-0.880269 + 2.49502i) q^{7} +(3.28769 - 4.52512i) q^{8} +(1.49478 + 0.863010i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.519333	+	2.44327i	−0.367224	+	1.72765i	0.275276	+	0.961365i	\(0.411231\pi\)
							−0.642499	+	0.766286i	\(0.722102\pi\)
\(3\)	2.09987	+	0.562657i	1.21236	+	0.324850i	0.807687	−	0.589612i	\(-0.200719\pi\)
							0.404671	+	0.914462i	\(0.367386\pi\)
\(5\)	2.28837	−	0.240517i	1.02339	−	0.107562i	0.422063	−	0.906567i	\(-0.361306\pi\)
							0.601325	+	0.799004i	\(0.294640\pi\)
\(7\)	−0.880269	+	2.49502i	−0.332710	+	0.943029i
							\(11\)	−1.16914	−	1.44377i	−0.352509	−	0.435312i	0.569790	−	0.821791i	\(-0.307025\pi\)
−0.922298	+	0.386479i	\(0.873691\pi\)
\(13\)	2.99249	+	1.52475i	0.829968	+	0.422890i	0.816728	−	0.577023i	\(-0.195786\pi\)
							0.0132396	+	0.999912i	\(0.495786\pi\)
\(17\)	−3.63856	+	2.94645i	−0.882480	+	0.714618i	−0.959289	−	0.282428i	\(-0.908860\pi\)
							0.0768086	+	0.997046i	\(0.475527\pi\)
\(19\)	−0.0257997	+	0.492288i	−0.00591886	+	0.112939i	0.994072	+	0.108721i	\(0.0346757\pi\)
							−0.999991	+	0.00421705i	\(0.998658\pi\)
\(23\)	4.94672	+	1.05146i	1.03146	+	0.219244i	0.692396	−	0.721518i	\(-0.256555\pi\)
							0.339067	+	0.940762i	\(0.389889\pi\)
\(29\)	0.598959	−	3.78168i	0.111224	−	0.702240i	−0.867558	−	0.497336i	\(-0.834312\pi\)
							0.978782	−	0.204904i	\(-0.0656883\pi\)
\(31\)	0.702666	−	6.68542i	0.126202	−	1.20074i	−0.729762	−	0.683701i	\(-0.760369\pi\)
							0.855964	−	0.517035i	\(-0.172964\pi\)
\(37\)	−0.974382	−	9.27062i	−0.160187	−	1.52408i	−0.719130	−	0.694875i	\(-0.755460\pi\)
							0.558943	−	0.829206i	\(-0.311207\pi\)
\(41\)	3.85426	−	5.11319i	0.601934	−	0.798546i
							\(43\)	10.9729	−	3.56531i	1.67335	−	0.543705i	0.689747	−	0.724050i	\(-0.257722\pi\)
0.983603	+	0.180345i	\(0.0577216\pi\)
\(47\)	5.08558	+	7.83111i	0.741809	+	1.14229i	0.985042	+	0.172317i	\(0.0551254\pi\)
							−0.243233	+	0.969968i	\(0.578208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.bc.a.2.2	✓	416
7.4	even	3	inner	287.2.bc.a.207.2	yes	416
41.21	even	20	inner	287.2.bc.a.226.2	yes	416
287.144	even	60	inner	287.2.bc.a.144.2	yes	416

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.bc.a.2.2	✓	416	1.1	even	1	trivial
287.2.bc.a.144.2	yes	416	287.144	even	60	inner
287.2.bc.a.207.2	yes	416	7.4	even	3	inner
287.2.bc.a.226.2	yes	416	41.21	even	20	inner