Embedded newform 287.3.t.a.244.10

• Label: 287.3.t.a.244.10
• Level: $287$
• Weight: $3$
• Character: 287.244
• 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			244.10
Character	\(\chi\)	\(=\)	287.244
Dual form			287.3.t.a.20.10

$q$-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 \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.t.a.20.9	✓	432	41.20	even	20	inner
287.3.t.a.20.10	yes	432	287.20	odd	20	inner
287.3.t.a.244.9	yes	432	7.6	odd	2	inner
287.3.t.a.244.10	yes	432	1.1	even	1	trivial