Embedded newform 25.6.e.a.9.1

• Label: 25.6.e.a.9.1
• Level: $25$
• Weight: $6$
• Character: 25.9
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00959549532\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			9.1
Character	\(\chi\)	\(=\)	25.9
Dual form			25.6.e.a.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.42181 + 8.83886i) q^{2} +(13.2720 - 4.31235i) q^{3} +(-26.9973 - 83.0892i) q^{4} +(-22.7744 - 51.0522i) q^{5} +(-47.1143 + 145.003i) q^{6} -134.700i q^{7} +(575.283 + 186.921i) q^{8} +(-39.0403 + 28.3645i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.42181	+	8.83886i	−1.13523	+	1.56251i	−0.357487	+	0.933918i	\(0.616367\pi\)
							−0.777739	+	0.628587i	\(0.783633\pi\)
\(3\)	13.2720	−	4.31235i	0.851402	−	0.276637i	0.149369	−	0.988782i	\(-0.452276\pi\)
							0.702033	+	0.712144i	\(0.252276\pi\)
\(5\)	−22.7744	−	51.0522i	−0.407401	−	0.913249i
							\(7\)		−	134.700i		−	1.03901i	−0.854466	−	0.519507i	\(-0.826116\pi\)
0.854466	−	0.519507i	\(-0.173884\pi\)
\(11\)	−15.5182	−	11.2746i	−0.0386686	−	0.0280944i	0.568283	−	0.822833i	\(-0.307608\pi\)
							−0.606952	+	0.794739i	\(0.707608\pi\)
\(13\)	−356.764	−	491.044i	−0.585495	−	0.805865i	0.408789	−	0.912629i	\(-0.365951\pi\)
							−0.994284	+	0.106764i	\(0.965951\pi\)
\(17\)	440.708	+	143.195i	0.369852	+	0.120172i	0.488045	−	0.872818i	\(-0.337710\pi\)
							−0.118193	+	0.992991i	\(0.537710\pi\)
\(19\)	384.174	−	1182.37i	0.244143	−	0.751395i	−0.751633	−	0.659581i	\(-0.770734\pi\)
							0.995776	−	0.0918134i	\(-0.0292663\pi\)
\(23\)	2518.98	−	3467.08i	0.992898	−	1.36661i	0.0633154	−	0.997994i	\(-0.479833\pi\)
							0.929583	−	0.368613i	\(-0.120167\pi\)
\(29\)	783.871	+	2412.51i	0.173081	+	0.532689i	0.999541	−	0.0303071i	\(-0.00964854\pi\)
							−0.826459	+	0.562996i	\(0.809649\pi\)
\(31\)	−208.877	+	642.858i	−0.0390380	+	0.120146i	−0.968676	−	0.248327i	\(-0.920119\pi\)
							0.929638	+	0.368473i	\(0.120119\pi\)
\(37\)	654.184	+	900.407i	0.0785589	+	0.108127i	0.846488	−	0.532408i	\(-0.178713\pi\)
							−0.767929	+	0.640535i	\(0.778713\pi\)
\(41\)	−7612.59	+	5530.87i	−0.707249	+	0.513847i	−0.882285	−	0.470715i	\(-0.843996\pi\)
							0.175036	+	0.984562i	\(0.443996\pi\)
\(43\)		−	4827.27i		−	0.398135i	−0.979986	−	0.199068i	\(-0.936209\pi\)
							0.979986	−	0.199068i	\(-0.0637913\pi\)
\(47\)	14856.2	−	4827.07i	0.980987	−	0.318742i	0.225744	−	0.974187i	\(-0.427519\pi\)
							0.755243	+	0.655445i	\(0.227519\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.9.1	✓	48
25.8	odd	20		625.6.a.g.1.1		48
25.14	even	10	inner	25.6.e.a.14.1	yes	48
25.17	odd	20		625.6.a.g.1.48		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.9.1	✓	48	1.1	even	1	trivial
25.6.e.a.14.1	yes	48	25.14	even	10	inner
625.6.a.g.1.1		48	25.8	odd	20
625.6.a.g.1.48		48	25.17	odd	20