Embedded newform 287.3.t.a.20.10

• Label: 287.3.t.a.20.10
• Level: $287$
• Weight: $3$
• Character: 287.20
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.t (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			20.10
Character	\(\chi\)	\(=\)	287.20
Dual form			287.3.t.a.244.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.90133 - 0.942701i) q^{2} +(1.41180 + 1.41180i) q^{3} +(4.29299 + 3.11904i) q^{4} +(4.40256 + 3.19865i) q^{5} +(-2.76519 - 5.42699i) q^{6} +(-6.97987 + 0.530441i) q^{7} +(-2.34258 - 3.22429i) q^{8} -5.01366i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 20 q^{2} + 196 q^{4} - 8 q^{7} - 20 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.90133	−	0.942701i	−1.45067	−	0.471350i	−0.525462	−	0.850817i	\(-0.676107\pi\)
							−0.925205	+	0.379467i	\(0.876107\pi\)
\(3\)	1.41180	+	1.41180i	0.470599	+	0.470599i	0.902108	−	0.431510i	\(-0.142019\pi\)
							−0.431510	+	0.902108i	\(0.642019\pi\)
\(5\)	4.40256	+	3.19865i	0.880511	+	0.639729i	0.933387	−	0.358872i	\(-0.116839\pi\)
							−0.0528753	+	0.998601i	\(0.516839\pi\)
\(7\)	−6.97987	+	0.530441i	−0.997125	+	0.0757772i
							\(11\)	2.94967	−	18.6235i	0.268152	−	1.69304i	−0.374774	−	0.927116i	\(-0.622280\pi\)
0.642926	−	0.765929i	\(-0.277720\pi\)
\(13\)	14.0289	−	7.14808i	1.07915	−	0.549852i	0.178293	−	0.983977i	\(-0.442942\pi\)
							0.900853	+	0.434125i	\(0.142942\pi\)
\(17\)	−3.64499	+	23.0136i	−0.214411	+	1.35374i	0.612083	+	0.790793i	\(0.290332\pi\)
							−0.826494	+	0.562945i	\(0.809668\pi\)
\(19\)	4.83632	+	2.46423i	0.254543	+	0.129696i	0.576607	−	0.817022i	\(-0.304376\pi\)
							−0.322064	+	0.946718i	\(0.604376\pi\)
\(23\)	1.44876	−	4.45883i	0.0629896	−	0.193862i	−0.914609	−	0.404339i	\(-0.867502\pi\)
							0.977599	+	0.210477i	\(0.0675016\pi\)
\(29\)	29.0443	−	4.60017i	1.00153	−	0.158626i	0.365926	−	0.930644i	\(-0.380752\pi\)
							0.635601	+	0.772017i	\(0.280752\pi\)
\(31\)	25.3730	+	34.9230i	0.818484	+	1.12655i	0.989959	+	0.141358i	\(0.0451470\pi\)
							−0.171474	+	0.985189i	\(0.554853\pi\)
\(37\)	44.6934	+	32.4716i	1.20793	+	0.877612i	0.995041	−	0.0994646i	\(-0.0317130\pi\)
							0.212888	+	0.977077i	\(0.431713\pi\)
\(41\)	−3.32195	−	40.8652i	−0.0810232	−	0.996712i
							\(43\)	10.8486	+	3.52492i	0.252293	+	0.0819749i	0.432433	−	0.901666i	\(-0.357655\pi\)
−0.180140	+	0.983641i	\(0.557655\pi\)
\(47\)	−23.4114	−	45.9475i	−0.498115	−	0.977606i	−0.994017	−	0.109227i	\(-0.965162\pi\)
							0.495902	−	0.868379i	\(-0.334838\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.t.a.20.10	yes	432
7.6	odd	2	inner	287.3.t.a.20.9	✓	432
41.39	even	20	inner	287.3.t.a.244.9	yes	432
287.244	odd	20	inner	287.3.t.a.244.10	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.t.a.20.9	✓	432	7.6	odd	2	inner
287.3.t.a.20.10	yes	432	1.1	even	1	trivial
287.3.t.a.244.9	yes	432	41.39	even	20	inner
287.3.t.a.244.10	yes	432	287.244	odd	20	inner