Embedded newform 25.6.d.a.11.8

• Label: 25.6.d.a.11.8
• 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			11.8
Character	\(\chi\)	\(=\)	25.11
Dual form			25.6.d.a.16.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.28450 - 1.65978i) q^{2} +(7.91379 - 24.3561i) q^{3} +(-7.42450 + 22.8503i) q^{4} +(-0.762635 - 55.8965i) q^{5} +(-22.3469 - 68.7767i) q^{6} -43.3673 q^{7} +(48.8885 + 150.463i) q^{8} +(-334.002 - 242.667i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.28450	−	1.65978i	0.403846	−	0.293411i	−0.367260	−	0.930118i	\(-0.619704\pi\)
							0.771106	+	0.636707i	\(0.219704\pi\)
\(3\)	7.91379	−	24.3561i	0.507670	−	1.56245i	−0.288566	−	0.957460i	\(-0.593179\pi\)
							0.796236	−	0.604986i	\(-0.206821\pi\)
\(5\)	−0.762635	−	55.8965i	−0.0136424	−	0.999907i
							\(7\)	−43.3673			−0.334516			−0.167258	−	0.985913i	\(-0.553491\pi\)
−0.167258	+	0.985913i	\(0.553491\pi\)
\(11\)	545.305	−	396.187i	1.35881	−	0.987230i	0.360286	−	0.932842i	\(-0.382679\pi\)
							0.998520	−	0.0543886i	\(-0.0173210\pi\)
\(13\)	261.277	+	189.829i	0.428788	+	0.311533i	0.781164	−	0.624326i	\(-0.214626\pi\)
							−0.352376	+	0.935858i	\(0.614626\pi\)
\(17\)	648.971	+	1997.33i	0.544632	+	1.67621i	0.721863	+	0.692036i	\(0.243286\pi\)
							−0.177231	+	0.984169i	\(0.556714\pi\)
\(19\)	317.136	+	976.045i	0.201540	+	0.620277i	0.999838	+	0.0180137i	\(0.00573425\pi\)
							−0.798298	+	0.602263i	\(0.794266\pi\)
\(23\)	1206.36	−	876.474i	0.475509	−	0.345477i	−0.324076	−	0.946031i	\(-0.605053\pi\)
							0.799584	+	0.600554i	\(0.205053\pi\)
\(29\)	654.646	−	2014.79i	0.144548	−	0.444873i	−0.852405	−	0.522883i	\(-0.824857\pi\)
							0.996953	+	0.0780099i	\(0.0248566\pi\)
\(31\)	319.838	+	984.361i	0.0597759	+	0.183971i	0.976486	−	0.215583i	\(-0.0691650\pi\)
							−0.916710	+	0.399554i	\(0.869165\pi\)
\(37\)	4669.93	+	3392.91i	0.560798	+	0.407444i	0.831751	−	0.555149i	\(-0.187339\pi\)
							−0.270953	+	0.962593i	\(0.587339\pi\)
\(41\)	−14349.1	−	10425.2i	−1.33311	−	0.968559i	−0.999667	−	0.0257879i	\(-0.991791\pi\)
							−0.333440	−	0.942771i	\(-0.608209\pi\)
\(43\)	−4140.62			−0.341503			−0.170751	−	0.985314i	\(-0.554619\pi\)
							−0.170751	+	0.985314i	\(0.554619\pi\)
\(47\)	−4084.71	+	12571.4i	−0.269722	+	0.830119i	0.720846	+	0.693095i	\(0.243754\pi\)
							−0.990568	+	0.137023i	\(0.956246\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.11.8	✓	44
25.4	even	10		625.6.a.c.1.15		22
25.16	even	5	inner	25.6.d.a.16.8	yes	44
25.21	even	5		625.6.a.d.1.8		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.8	✓	44	1.1	even	1	trivial
25.6.d.a.16.8	yes	44	25.16	even	5	inner
625.6.a.c.1.15		22	25.4	even	10
625.6.a.d.1.8		22	25.21	even	5