Embedded newform 525.2.bh.a.13.1

• Label: 525.2.bh.a.13.1
• Level: $525$
• Weight: $2$
• Character: 525.13
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bh (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(40\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	525.13
Dual form			525.2.bh.a.202.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.65979 + 0.421270i) q^{2} +(-0.891007 - 0.453990i) q^{3} +(4.99492 - 1.62295i) q^{4} +(0.301881 + 2.21560i) q^{5} +(2.56115 + 0.832167i) q^{6} +(2.34376 - 1.22751i) q^{7} +(-7.80289 + 3.97577i) q^{8} +(0.587785 + 0.809017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(176\)	\(451\)
\(\chi(n)\)	\(e\left(\frac{19}{20}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.65979	+	0.421270i	−1.88076	+	0.297883i	−0.988214	−	0.153078i	\(-0.951082\pi\)
							−0.892544	+	0.450960i	\(0.851082\pi\)
\(3\)	−0.891007	−	0.453990i	−0.514423	−	0.262112i
							\(5\)	0.301881	+	2.21560i	0.135005	+	0.990845i
\(7\)	2.34376	−	1.22751i	0.885858	−	0.463956i
							\(11\)	2.88419	+	2.09549i	0.869616	+	0.631813i	0.930484	−	0.366333i	\(-0.119387\pi\)
−0.0608676	+	0.998146i	\(0.519387\pi\)
\(13\)	−0.934003	+	5.89706i	−0.259046	+	1.63555i	0.424346	+	0.905500i	\(0.360504\pi\)
							−0.683392	+	0.730051i	\(0.739496\pi\)
\(17\)	0.635671	−	0.323890i	0.154173	−	0.0785549i	−0.375202	−	0.926943i	\(-0.622427\pi\)
							0.529375	+	0.848388i	\(0.322427\pi\)
\(19\)	1.08852	−	3.35012i	0.249724	−	0.768571i	−0.745100	−	0.666953i	\(-0.767598\pi\)
							0.994824	−	0.101618i	\(-0.0324018\pi\)
\(23\)	−0.967431	−	6.10812i	−0.201723	−	1.27363i	−0.855842	−	0.517238i	\(-0.826960\pi\)
							0.654118	−	0.756392i	\(-0.273040\pi\)
\(29\)	−0.902636	+	0.293284i	−0.167615	+	0.0544615i	−0.391623	−	0.920126i	\(-0.628086\pi\)
							0.224007	+	0.974587i	\(0.428086\pi\)
\(31\)	0.610193	+	0.198264i	0.109594	+	0.0356092i	0.363301	−	0.931672i	\(-0.381650\pi\)
							−0.253707	+	0.967281i	\(0.581650\pi\)
\(37\)	−1.17169	+	7.39773i	−0.192624	+	1.21618i	0.681991	+	0.731361i	\(0.261115\pi\)
							−0.874615	+	0.484819i	\(0.838885\pi\)
\(41\)	5.83596	+	8.03251i	0.911424	+	1.25447i	0.966678	+	0.255996i	\(0.0824033\pi\)
							−0.0552535	+	0.998472i	\(0.517597\pi\)
\(43\)	−0.144018	−	0.144018i	−0.0219626	−	0.0219626i	0.696040	−	0.718003i	\(-0.254944\pi\)
							−0.718003	+	0.696040i	\(0.754944\pi\)
\(47\)	−0.0925464	+	0.181633i	−0.0134993	+	0.0264938i	−0.897658	−	0.440693i	\(-0.854733\pi\)
							0.884158	+	0.467187i	\(0.154733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.bh.a.13.1	✓	320
7.6	odd	2	inner	525.2.bh.a.13.2	yes	320
25.2	odd	20	inner	525.2.bh.a.202.2	yes	320
175.27	even	20	inner	525.2.bh.a.202.1	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.bh.a.13.1	✓	320	1.1	even	1	trivial
525.2.bh.a.13.2	yes	320	7.6	odd	2	inner
525.2.bh.a.202.1	yes	320	175.27	even	20	inner
525.2.bh.a.202.2	yes	320	25.2	odd	20	inner