Embedded newform 525.2.bv.a.52.14

• Label: 525.2.bv.a.52.14
• Level: $525$
• Weight: $2$
• Character: 525.52
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bv (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(40\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			52.14
Character	\(\chi\)	\(=\)	525.52
Dual form			525.2.bv.a.313.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.385788 - 1.00501i) q^{2} +(0.0523360 - 0.998630i) q^{3} +(0.625072 - 0.562818i) q^{4} +(-2.23091 + 0.151798i) q^{5} +(-1.02383 + 0.332661i) q^{6} +(-1.22058 + 2.34738i) q^{7} +(-2.72515 - 1.38853i) q^{8} +(-0.994522 - 0.104528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 12 q^{5} - 8 q^{7} + 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.385788	−	1.00501i	−0.272793	−	0.710651i	−0.999714	−	0.0239292i	\(-0.992382\pi\)
							0.726920	−	0.686722i	\(-0.240951\pi\)
\(3\)	0.0523360	−	0.998630i	0.0302162	−	0.576559i
							\(5\)	−2.23091	+	0.151798i	−0.997693	+	0.0678861i
\(7\)	−1.22058	+	2.34738i	−0.461335	+	0.887226i
							\(11\)	0.449287	+	4.27468i	0.135465	+	1.28886i	0.825216	+	0.564817i	\(0.191053\pi\)
−0.689751	+	0.724047i	\(0.742280\pi\)
\(13\)	−0.889274	−	5.61465i	−0.246640	−	1.55722i	−0.731014	−	0.682363i	\(-0.760952\pi\)
							0.484373	−	0.874861i	\(-0.339048\pi\)
\(17\)	−5.95982	+	3.87035i	−1.44547	+	0.938698i	−0.446432	+	0.894818i	\(0.647306\pi\)
							−0.999037	+	0.0438808i	\(0.986028\pi\)
\(19\)	−3.11072	+	3.45481i	−0.713649	+	0.792587i	−0.985486	−	0.169756i	\(-0.945702\pi\)
							0.271837	+	0.962343i	\(0.412369\pi\)
\(23\)	−1.00823	+	0.387022i	−0.210230	+	0.0806997i	−0.461204	−	0.887294i	\(-0.652582\pi\)
							0.250974	+	0.967994i	\(0.419249\pi\)
\(29\)	−2.22440	−	0.722751i	−0.413061	−	0.134212i	0.0951124	−	0.995467i	\(-0.469679\pi\)
							−0.508173	+	0.861255i	\(0.669679\pi\)
\(31\)	0.746330	−	3.51121i	0.134045	−	0.630631i	−0.858909	−	0.512128i	\(-0.828857\pi\)
							0.992954	−	0.118503i	\(-0.0378095\pi\)
\(37\)	4.87730	−	3.94956i	0.801823	−	0.649304i	−0.138204	−	0.990404i	\(-0.544133\pi\)
							0.940027	+	0.341100i	\(0.110800\pi\)
\(41\)	4.43598	−	6.10560i	0.692784	−	0.953535i	−0.307214	−	0.951640i	\(-0.599397\pi\)
							0.999998	−	0.00189478i	\(-0.000603128\pi\)
\(43\)	−7.09017	+	7.09017i	−1.08124	+	1.08124i	−0.0848474	+	0.996394i	\(0.527040\pi\)
							−0.996394	+	0.0848474i	\(0.972960\pi\)
\(47\)	0.748888	−	1.15319i	0.109237	−	0.168210i	−0.779824	−	0.625998i	\(-0.784692\pi\)
							0.889061	+	0.457789i	\(0.151358\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bv.a.52.14	✓	640
7.5	odd	6	inner	525.2.bv.a.502.14	yes	640
25.13	odd	20	inner	525.2.bv.a.388.14	yes	640
175.138	even	60	inner	525.2.bv.a.313.14	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bv.a.52.14	✓	640	1.1	even	1	trivial
525.2.bv.a.313.14	yes	640	175.138	even	60	inner
525.2.bv.a.388.14	yes	640	25.13	odd	20	inner
525.2.bv.a.502.14	yes	640	7.5	odd	6	inner