Embedded newform 525.2.bv.a.52.12

• Label: 525.2.bv.a.52.12
• 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			52.12
Character	\(\chi\)	\(=\)	525.52
Dual form			525.2.bv.a.313.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.496455 - 1.29331i) q^{2} +(0.0523360 - 0.998630i) q^{3} +(0.0601081 - 0.0541216i) q^{4} +(2.05147 - 0.889639i) q^{5} +(-1.31752 + 0.428088i) q^{6} +(2.12828 + 1.57176i) q^{7} +(-2.56850 - 1.30872i) q^{8} +(-0.994522 - 0.104528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q + 12 q^{5} - 8 q^{7} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.496455	−	1.29331i	−0.351047	−	0.914508i	−0.989459	−	0.144813i	\(-0.953742\pi\)
							0.638412	−	0.769695i	\(-0.279591\pi\)
\(3\)	0.0523360	−	0.998630i	0.0302162	−	0.576559i
							\(5\)	2.05147	−	0.889639i	0.917447	−	0.397859i
\(7\)	2.12828	+	1.57176i	0.804414	+	0.594069i
							\(11\)	−0.288382	−	2.74377i	−0.0869503	−	0.827277i	−0.947896	−	0.318580i	\(-0.896794\pi\)
0.860946	−	0.508697i	\(-0.169873\pi\)
\(13\)	0.0399667	+	0.252340i	0.0110848	+	0.0699865i	0.992611	−	0.121343i	\(-0.0387201\pi\)
							−0.981526	+	0.191329i	\(0.938720\pi\)
\(17\)	−0.783164	+	0.508592i	−0.189945	+	0.123352i	−0.636108	−	0.771600i	\(-0.719457\pi\)
							0.446163	+	0.894952i	\(0.352790\pi\)
\(19\)	2.44344	−	2.71371i	0.560563	−	0.622568i	−0.394527	−	0.918884i	\(-0.629091\pi\)
							0.955090	+	0.296316i	\(0.0957582\pi\)
\(23\)	3.48671	−	1.33842i	0.727029	−	0.279080i	0.0334294	−	0.999441i	\(-0.489357\pi\)
							0.693600	+	0.720361i	\(0.256024\pi\)
\(29\)	−3.04300	−	0.988731i	−0.565071	−	0.183603i	0.0125305	−	0.999921i	\(-0.496011\pi\)
							−0.577602	+	0.816319i	\(0.696011\pi\)
\(31\)	−0.418166	+	1.96732i	−0.0751048	+	0.353340i	−0.999613	−	0.0278221i	\(-0.991143\pi\)
							0.924508	+	0.381162i	\(0.124476\pi\)
\(37\)	−4.40627	+	3.56812i	−0.724386	+	0.586596i	−0.918988	−	0.394286i	\(-0.870992\pi\)
							0.194602	+	0.980882i	\(0.437658\pi\)
\(41\)	4.89727	−	6.74052i	0.764825	−	1.05269i	−0.231972	−	0.972722i	\(-0.574518\pi\)
							0.996797	−	0.0799694i	\(-0.0254823\pi\)
\(43\)	−5.17291	+	5.17291i	−0.788861	+	0.788861i	−0.981307	−	0.192446i	\(-0.938358\pi\)
							0.192446	+	0.981307i	\(0.438358\pi\)
\(47\)	−6.91306	+	10.6452i	−1.00837	+	1.55276i	−0.187006	+	0.982359i	\(0.559878\pi\)
							−0.821367	+	0.570400i	\(0.806788\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bv.a.52.12	✓	640
7.5	odd	6	inner	525.2.bv.a.502.12	yes	640
25.13	odd	20	inner	525.2.bv.a.388.12	yes	640
175.138	even	60	inner	525.2.bv.a.313.12	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bv.a.52.12	✓	640	1.1	even	1	trivial
525.2.bv.a.313.12	yes	640	175.138	even	60	inner
525.2.bv.a.388.12	yes	640	25.13	odd	20	inner
525.2.bv.a.502.12	yes	640	7.5	odd	6	inner