Embedded newform 25.7.f.a.2.9

• Label: 25.7.f.a.2.9
• Level: $25$
• Weight: $7$
• Character: 25.2
• Analytic conductor: $5.751$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.75135209050\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	25.2
Dual form			25.7.f.a.13.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.41905 + 0.858292i) q^{2} +(-10.2785 - 20.1728i) q^{3} +(-32.2382 - 10.4748i) q^{4} +(124.517 + 10.9798i) q^{5} +(-38.3857 - 118.139i) q^{6} +(-443.961 - 443.961i) q^{7} +(-478.580 - 243.849i) q^{8} +(127.203 - 175.080i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 40 q^{3} - 10 q^{4} + 60 q^{5} - 6 q^{6} - 560 q^{7} - 1870 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.41905	+	0.858292i	0.677381	+	0.107287i	0.485640	−	0.874159i	\(-0.338587\pi\)
							0.191741	+	0.981446i	\(0.438587\pi\)
\(3\)	−10.2785	−	20.1728i	−0.380687	−	0.747140i	0.618568	−	0.785731i	\(-0.287713\pi\)
							−0.999255	+	0.0385912i	\(0.987713\pi\)
\(5\)	124.517	+	10.9798i	0.996135	+	0.0878382i
							\(7\)	−443.961	−	443.961i	−1.29435	−	1.29435i	−0.932071	−	0.362276i	\(-0.882000\pi\)
−0.362276	−	0.932071i	\(-0.618000\pi\)
\(11\)	−74.0904	+	53.8298i	−0.0556652	+	0.0404432i	−0.615270	−	0.788317i	\(-0.710953\pi\)
							0.559605	+	0.828760i	\(0.310953\pi\)
\(13\)	2157.66	−	341.740i	0.982093	−	0.155548i	0.355315	−	0.934746i	\(-0.384373\pi\)
							0.626778	+	0.779198i	\(0.284373\pi\)
\(17\)	−3517.56	+	6903.60i	−0.715970	+	1.40517i	0.189982	+	0.981788i	\(0.439157\pi\)
							−0.905951	+	0.423382i	\(0.860843\pi\)
\(19\)	9096.04	−	2955.48i	1.32615	−	0.430891i	0.441546	−	0.897239i	\(-0.354430\pi\)
							0.884602	+	0.466347i	\(0.154430\pi\)
\(23\)	1012.02	−	6389.62i	0.0831771	−	0.525160i	−0.910556	−	0.413385i	\(-0.864346\pi\)
							0.993734	−	0.111775i	\(-0.0356536\pi\)
\(29\)	8513.80	+	2766.30i	0.349084	+	0.113424i	0.478310	−	0.878191i	\(-0.341249\pi\)
							−0.129227	+	0.991615i	\(0.541249\pi\)
\(31\)	−1191.44	−	3666.88i	−0.0399933	−	0.123087i	0.929067	−	0.369913i	\(-0.120612\pi\)
							−0.969060	+	0.246826i	\(0.920612\pi\)
\(37\)	−12750.9	−	80505.8i	−0.251730	−	1.58936i	−0.712391	−	0.701783i	\(-0.752388\pi\)
							0.460661	−	0.887576i	\(-0.347612\pi\)
\(41\)	11456.3	+	8323.51i	0.166224	+	0.120769i	0.667787	−	0.744352i	\(-0.267242\pi\)
							−0.501563	+	0.865121i	\(0.667242\pi\)
\(43\)	86357.7	−	86357.7i	1.08617	−	1.08617i	0.0902455	−	0.995920i	\(-0.471235\pi\)
							0.995920	−	0.0902455i	\(-0.0287652\pi\)
\(47\)	−50471.1	+	25716.3i	−0.486126	+	0.247694i	−0.679834	−	0.733366i	\(-0.737948\pi\)
							0.193709	+	0.981059i	\(0.437948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.7.f.a.2.9	✓	112
25.13	odd	20	inner	25.7.f.a.13.9	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.7.f.a.2.9	✓	112	1.1	even	1	trivial
25.7.f.a.13.9	yes	112	25.13	odd	20	inner