Embedded newform 25.9.f.a.2.11

• Label: 25.9.f.a.2.11
• Level: $25$
• Weight: $9$
• Character: 25.2
• Analytic conductor: $10.184$
• Analytic rank: $0$
• Dimension: $152$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1844652515\)
Analytic rank:	\(0\)
Dimension:	\(152\)
Relative dimension:	\(19\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			2.11
Character	\(\chi\)	\(=\)	25.2
Dual form			25.9.f.a.13.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.41297 + 0.857331i) q^{2} +(59.4837 + 116.743i) q^{3} +(-214.905 - 69.8269i) q^{4} +(608.428 - 142.970i) q^{5} +(221.896 + 682.926i) q^{6} +(2643.50 + 2643.50i) q^{7} +(-2353.49 - 1199.16i) q^{8} +(-6234.25 + 8580.71i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 8 q^{2} + 62 q^{3} - 10 q^{4} - 230 q^{5} - 6 q^{6} + 2342 q^{7} + 8210 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.41297	+	0.857331i	0.338311	+	0.0535832i	0.323279	−	0.946304i	\(-0.395215\pi\)
							0.0150322	+	0.999887i	\(0.495215\pi\)
\(3\)	59.4837	+	116.743i	0.734367	+	1.44128i	0.891181	+	0.453648i	\(0.149878\pi\)
							−0.156814	+	0.987628i	\(0.550122\pi\)
\(5\)	608.428	−	142.970i	0.973485	−	0.228752i
							\(7\)	2643.50	+	2643.50i	1.10100	+	1.10100i	0.994290	+	0.106711i	\(0.0340320\pi\)
0.106711	+	0.994290i	\(0.465968\pi\)
\(11\)	−7583.94	+	5510.05i	−0.517993	+	0.376344i	−0.815847	−	0.578267i	\(-0.803729\pi\)
							0.297854	+	0.954611i	\(0.403729\pi\)
\(13\)	−39121.7	+	6196.27i	−1.36976	+	0.216949i	−0.797597	−	0.603191i	\(-0.793896\pi\)
							−0.572164	+	0.820139i	\(0.693896\pi\)
\(17\)	−28373.0	+	55685.2i	−0.339712	+	0.666721i	−0.996151	−	0.0876575i	\(-0.972062\pi\)
							0.656439	+	0.754379i	\(0.272062\pi\)
\(19\)	150250.	−	48819.1i	1.15292	−	0.374607i	0.330679	−	0.943743i	\(-0.392722\pi\)
							0.822243	+	0.569137i	\(0.192722\pi\)
\(23\)	43771.7	−	276364.i	0.156416	−	0.987573i	−0.777188	−	0.629268i	\(-0.783355\pi\)
							0.933604	−	0.358305i	\(-0.116645\pi\)
\(29\)	947226.	+	307772.i	1.33925	+	0.435149i	0.889063	−	0.457786i	\(-0.151357\pi\)
							0.450187	+	0.892934i	\(0.351357\pi\)
\(31\)	56705.0	+	174520.i	0.0614009	+	0.188973i	0.977052	−	0.213002i	\(-0.0683241\pi\)
							−0.915651	+	0.401974i	\(0.868324\pi\)
\(37\)	−499407.	−	3.15313e6i	−0.266470	−	1.68242i	−0.650816	−	0.759235i	\(-0.725573\pi\)
							0.384346	−	0.923189i	\(-0.374427\pi\)
\(41\)	−189849.	−	137934.i	−0.0671853	−	0.0488129i	0.553686	−	0.832726i	\(-0.313221\pi\)
							−0.620871	+	0.783913i	\(0.713221\pi\)
\(43\)	−338441.	+	338441.i	−0.0989942	+	0.0989942i	−0.754869	−	0.655875i	\(-0.772300\pi\)
							0.655875	+	0.754869i	\(0.272300\pi\)
\(47\)	2.07213e6	−	1.05580e6i	0.424645	−	0.216367i	−0.228590	−	0.973523i	\(-0.573411\pi\)
							0.653235	+	0.757155i	\(0.273411\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.9.f.a.2.11	✓	152
25.13	odd	20	inner	25.9.f.a.13.11	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.9.f.a.2.11	✓	152	1.1	even	1	trivial
25.9.f.a.13.11	yes	152	25.13	odd	20	inner