Embedded newform 25.6.d.a.11.9

• Label: 25.6.d.a.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	25.11
Dual form			25.6.d.a.16.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.56327 - 4.04195i) q^{2} +(-7.98941 + 24.5889i) q^{3} +(4.72404 - 14.5391i) q^{4} +(52.9093 + 18.0446i) q^{5} +(54.9398 + 169.087i) q^{6} -5.99482 q^{7} +(35.5141 + 109.301i) q^{8} +(-344.191 - 250.069i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.56327	−	4.04195i	0.983456	−	0.714523i	0.0249777	−	0.999688i	\(-0.492049\pi\)
							0.958478	+	0.285165i	\(0.0920485\pi\)
\(3\)	−7.98941	+	24.5889i	−0.512521	+	1.57738i	0.275227	+	0.961379i	\(0.411247\pi\)
							−0.787748	+	0.615998i	\(0.788753\pi\)
\(5\)	52.9093	+	18.0446i	0.946470	+	0.322791i
							\(7\)	−5.99482			−0.0462414			−0.0231207	−	0.999733i	\(-0.507360\pi\)
−0.0231207	+	0.999733i	\(0.507360\pi\)
\(11\)	460.750	−	334.754i	1.14811	−	0.834151i	0.159881	−	0.987136i	\(-0.448889\pi\)
							0.988228	+	0.152986i	\(0.0488888\pi\)
\(13\)	−694.810	−	504.809i	−1.14027	−	0.828454i	−0.153113	−	0.988209i	\(-0.548930\pi\)
							−0.987157	+	0.159754i	\(0.948930\pi\)
\(17\)	282.676	+	869.988i	0.237229	+	0.730115i	0.996818	+	0.0797117i	\(0.0254000\pi\)
							−0.759589	+	0.650403i	\(0.774600\pi\)
\(19\)	−482.215	−	1484.11i	−0.306448	−	0.943150i	−0.979133	−	0.203221i	\(-0.934859\pi\)
							0.672685	−	0.739929i	\(-0.265141\pi\)
\(23\)	2350.86	−	1708.00i	0.926632	−	0.673237i	−0.0185341	−	0.999828i	\(-0.505900\pi\)
							0.945166	+	0.326591i	\(0.105900\pi\)
\(29\)	−308.658	+	949.952i	−0.0681527	+	0.209752i	−0.979333	−	0.202256i	\(-0.935173\pi\)
							0.911180	+	0.412009i	\(0.135173\pi\)
\(31\)	−1593.48	−	4904.24i	−0.297813	−	0.916573i	−0.982262	−	0.187513i	\(-0.939957\pi\)
							0.684449	−	0.729060i	\(-0.260043\pi\)
\(37\)	−477.227	−	346.726i	−0.0573087	−	0.0416372i	0.558762	−	0.829328i	\(-0.311276\pi\)
							−0.616071	+	0.787691i	\(0.711276\pi\)
\(41\)	−4422.90	−	3213.42i	−0.410911	−	0.298544i	0.363060	−	0.931766i	\(-0.381732\pi\)
							−0.773970	+	0.633222i	\(0.781732\pi\)
\(43\)	9774.65			0.806176			0.403088	−	0.915161i	\(-0.367937\pi\)
							0.403088	+	0.915161i	\(0.367937\pi\)
\(47\)	−6146.94	+	18918.3i	−0.405895	+	1.24922i	0.514250	+	0.857640i	\(0.328070\pi\)
							−0.920145	+	0.391577i	\(0.871930\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.11.9	✓	44
25.4	even	10		625.6.a.c.1.18		22
25.16	even	5	inner	25.6.d.a.16.9	yes	44
25.21	even	5		625.6.a.d.1.5		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.9	✓	44	1.1	even	1	trivial
25.6.d.a.16.9	yes	44	25.16	even	5	inner
625.6.a.c.1.18		22	25.4	even	10
625.6.a.d.1.5		22	25.21	even	5