Embedded newform 575.2.g.b.116.6

• Label: 575.2.g.b.116.6
• Level: $575$
• Weight: $2$
• Character: 575.116
• Analytic conductor: $4.591$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	575.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.59139811622\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(27\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			116.6
Character	\(\chi\)	\(=\)	575.116
Dual form			575.2.g.b.461.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62357 - 1.17959i) q^{2} +(-1.06461 - 3.27654i) q^{3} +(0.626505 + 1.92818i) q^{4} +(1.50900 + 1.65013i) q^{5} +(-2.13651 + 6.57550i) q^{6} +4.84819 q^{7} +(0.0169997 - 0.0523197i) q^{8} +(-7.17528 + 5.21315i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 4 q^{2} + q^{3} - 20 q^{4} + 3 q^{5} + 8 q^{6} + 24 q^{7} + 13 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\)	\(51\)	\(277\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62357	−	1.17959i	−1.14804	−	0.834097i	−0.159818	−	0.987147i	\(-0.551091\pi\)
							−0.988219	+	0.153049i	\(0.951091\pi\)
\(3\)	−1.06461	−	3.27654i	−0.614655	−	1.89171i	−0.406645	−	0.913586i	\(-0.633301\pi\)
							−0.208010	−	0.978127i	\(-0.566699\pi\)
\(5\)	1.50900	+	1.65013i	0.674846	+	0.737959i
							\(7\)	4.84819			1.83244			0.916221	−	0.400673i	\(-0.131224\pi\)
0.916221	+	0.400673i	\(0.131224\pi\)
\(11\)	2.24190	+	1.62883i	0.675957	+	0.491112i	0.872014	−	0.489481i	\(-0.162814\pi\)
							−0.196057	+	0.980593i	\(0.562814\pi\)
\(13\)	0.559162	−	0.406255i	0.155084	−	0.112675i	−0.507537	−	0.861630i	\(-0.669444\pi\)
							0.662621	+	0.748955i	\(0.269444\pi\)
\(17\)	1.43564	−	4.41845i	0.348194	−	1.07163i	−0.611658	−	0.791123i	\(-0.709497\pi\)
							0.959852	−	0.280508i	\(-0.0905030\pi\)
\(19\)	0.270031	−	0.831069i	0.0619493	−	0.190660i	−0.915292	−	0.402791i	\(-0.868040\pi\)
							0.977241	+	0.212131i	\(0.0680402\pi\)
\(23\)	−0.809017	−	0.587785i	−0.168692	−	0.122562i
							\(29\)	−0.922231	−	2.83834i	−0.171254	−	0.527066i	0.828189	−	0.560450i	\(-0.189372\pi\)
−0.999443	+	0.0333837i	\(0.989372\pi\)
\(31\)	1.44216	−	4.43852i	0.259020	−	0.797182i	−0.733991	−	0.679159i	\(-0.762345\pi\)
							0.993011	−	0.118022i	\(-0.0376555\pi\)
\(37\)	1.63902	−	1.19082i	0.269453	−	0.195769i	−0.444851	−	0.895605i	\(-0.646743\pi\)
							0.714304	+	0.699835i	\(0.246743\pi\)
\(41\)	−1.02897	+	0.747589i	−0.160698	+	0.116754i	−0.665228	−	0.746640i	\(-0.731666\pi\)
							0.504530	+	0.863394i	\(0.331666\pi\)
\(43\)	4.35214			0.663696			0.331848	−	0.943333i	\(-0.392328\pi\)
							0.331848	+	0.943333i	\(0.392328\pi\)
\(47\)	−0.286775	−	0.882602i	−0.0418304	−	0.128741i	0.927960	−	0.372679i	\(-0.121561\pi\)
							−0.969791	+	0.243938i	\(0.921561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.2.g.b.116.6	✓	108
25.11	even	5	inner	575.2.g.b.461.6	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
575.2.g.b.116.6	✓	108	1.1	even	1	trivial
575.2.g.b.461.6	yes	108	25.11	even	5	inner