Embedded newform 25.6.d.a.11.2

• Label: 25.6.d.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	25.11
Dual form			25.6.d.a.16.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.12278 + 4.44846i) q^{2} +(-7.99178 + 24.5962i) q^{3} +(7.81108 - 24.0400i) q^{4} +(-12.4773 + 54.4914i) q^{5} +(-60.4831 - 186.148i) q^{6} +130.915 q^{7} +(-15.7226 - 48.3892i) q^{8} +(-344.512 - 250.302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.12278	+	4.44846i	−1.08236	+	0.786384i	−0.978094	−	0.208165i	\(-0.933251\pi\)
							−0.104271	+	0.994549i	\(0.533251\pi\)
\(3\)	−7.99178	+	24.5962i	−0.512673	+	1.57784i	0.274804	+	0.961500i	\(0.411387\pi\)
							−0.787477	+	0.616344i	\(0.788613\pi\)
\(5\)	−12.4773	+	54.4914i	−0.223201	+	0.974772i
							\(7\)	130.915			1.00982			0.504912	−	0.863171i	\(-0.331525\pi\)
0.504912	+	0.863171i	\(0.331525\pi\)
\(11\)	−118.894	+	86.3814i	−0.296263	+	0.215248i	−0.725980	−	0.687716i	\(-0.758613\pi\)
							0.429717	+	0.902964i	\(0.358613\pi\)
\(13\)	623.412	+	452.936i	1.02310	+	0.743324i	0.966916	−	0.255097i	\(-0.0821073\pi\)
							0.0561819	+	0.998421i	\(0.482107\pi\)
\(17\)	184.142	+	566.731i	0.154536	+	0.475614i	0.998114	−	0.0613936i	\(-0.0195545\pi\)
							−0.843577	+	0.537008i	\(0.819554\pi\)
\(19\)	−520.137	−	1600.82i	−0.330548	−	1.01732i	−0.968874	−	0.247555i	\(-0.920373\pi\)
							0.638326	−	0.769766i	\(-0.279627\pi\)
\(23\)	−3119.97	+	2266.79i	−1.22979	+	0.893494i	−0.996874	−	0.0790035i	\(-0.974826\pi\)
							−0.232914	+	0.972497i	\(0.574826\pi\)
\(29\)	2018.70	−	6212.92i	0.445735	−	1.37183i	−0.435940	−	0.899976i	\(-0.643584\pi\)
							0.881675	−	0.471857i	\(-0.156416\pi\)
\(31\)	876.762	+	2698.40i	0.163862	+	0.504314i	0.998951	−	0.0458006i	\(-0.0145839\pi\)
							−0.835089	+	0.550115i	\(0.814584\pi\)
\(37\)	−6406.86	−	4654.86i	−0.769381	−	0.558988i	0.132393	−	0.991197i	\(-0.457734\pi\)
							−0.901773	+	0.432210i	\(0.857734\pi\)
\(41\)	14617.4	+	10620.2i	1.35804	+	0.986672i	0.998567	+	0.0535151i	\(0.0170425\pi\)
							0.359470	+	0.933157i	\(0.382957\pi\)
\(43\)	11961.8			0.986563			0.493282	−	0.869870i	\(-0.335797\pi\)
							0.493282	+	0.869870i	\(0.335797\pi\)
\(47\)	−5523.66	+	17000.1i	−0.364739	+	1.12255i	0.585406	+	0.810741i	\(0.300935\pi\)
							−0.950144	+	0.311810i	\(0.899065\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.11.2	✓	44
25.4	even	10		625.6.a.c.1.5		22
25.16	even	5	inner	25.6.d.a.16.2	yes	44
25.21	even	5		625.6.a.d.1.18		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.2	✓	44	1.1	even	1	trivial
25.6.d.a.16.2	yes	44	25.16	even	5	inner
625.6.a.c.1.5		22	25.4	even	10
625.6.a.d.1.18		22	25.21	even	5