Embedded newform 25.6.d.a.11.1

• Label: 25.6.d.a.11.1
• Level: $25$
• Weight: $6$
• Character: 25.11
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00959549532\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	25.11
Dual form			25.6.d.a.16.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.62958 + 6.26976i) q^{2} +(1.83172 - 5.63746i) q^{3} +(25.2713 - 77.7769i) q^{4} +(54.8570 + 10.7567i) q^{5} +(19.5385 + 60.1334i) q^{6} -120.405 q^{7} +(164.084 + 504.997i) q^{8} +(168.165 + 122.179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.62958	+	6.26976i	−1.52551	+	1.10835i	−0.566838	+	0.823829i	\(0.691833\pi\)
							−0.958671	+	0.284518i	\(0.908167\pi\)
\(3\)	1.83172	−	5.63746i	0.117505	−	0.361643i	−0.874956	−	0.484202i	\(-0.839110\pi\)
							0.992461	+	0.122559i	\(0.0391100\pi\)
\(5\)	54.8570	+	10.7567i	0.981312	+	0.192422i
							\(7\)	−120.405			−0.928748			−0.464374	−	0.885639i	\(-0.653721\pi\)
−0.464374	+	0.885639i	\(0.653721\pi\)
\(11\)	358.189	−	260.240i	0.892546	−	0.648472i	−0.0439948	−	0.999032i	\(-0.514008\pi\)
							0.936541	+	0.350559i	\(0.114008\pi\)
\(13\)	721.354	+	524.094i	1.18383	+	0.860104i	0.992599	−	0.121440i	\(-0.0387513\pi\)
							0.191233	+	0.981545i	\(0.438751\pi\)
\(17\)	148.458	+	456.908i	0.124590	+	0.383448i	0.993826	−	0.110949i	\(-0.0353890\pi\)
							−0.869236	+	0.494397i	\(0.835389\pi\)
\(19\)	36.3764	+	111.955i	0.0231172	+	0.0711476i	0.961950	−	0.273227i	\(-0.0880910\pi\)
							−0.938832	+	0.344375i	\(0.888091\pi\)
\(23\)	1726.95	−	1254.70i	0.680708	−	0.494563i	−0.192885	−	0.981221i	\(-0.561784\pi\)
							0.873592	+	0.486658i	\(0.161784\pi\)
\(29\)	198.050	−	609.537i	0.0437301	−	0.134588i	−0.926808	−	0.375536i	\(-0.877459\pi\)
							0.970538	+	0.240949i	\(0.0774586\pi\)
\(31\)	−501.577	−	1543.70i	−0.0937418	−	0.288508i	0.893182	−	0.449696i	\(-0.148468\pi\)
							−0.986924	+	0.161188i	\(0.948468\pi\)
\(37\)	−9784.27	−	7108.69i	−1.17496	−	0.853661i	−0.183368	−	0.983044i	\(-0.558700\pi\)
							−0.991595	+	0.129384i	\(0.958700\pi\)
\(41\)	10172.1	+	7390.45i	0.945040	+	0.686612i	0.949629	−	0.313378i	\(-0.101461\pi\)
							−0.00458868	+	0.999989i	\(0.501461\pi\)
\(43\)	−11083.8			−0.914154			−0.457077	−	0.889427i	\(-0.651104\pi\)
							−0.457077	+	0.889427i	\(0.651104\pi\)
\(47\)	−6173.90	+	19001.3i	−0.407676	+	1.25470i	0.510964	+	0.859602i	\(0.329288\pi\)
							−0.918640	+	0.395095i	\(0.870712\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.11.1	✓	44
25.4	even	10		625.6.a.c.1.1		22
25.16	even	5	inner	25.6.d.a.16.1	yes	44
25.21	even	5		625.6.a.d.1.22		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.1	✓	44	1.1	even	1	trivial
25.6.d.a.16.1	yes	44	25.16	even	5	inner
625.6.a.c.1.1		22	25.4	even	10
625.6.a.d.1.22		22	25.21	even	5