Embedded newform 25.6.e.a.4.3

• Label: 25.6.e.a.4.3
• Level: $25$
• Weight: $6$
• Character: 25.4
• 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			4.3
Character	\(\chi\)	\(=\)	25.4
Dual form			25.6.e.a.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.91215 - 2.24589i) q^{2} +(1.65280 - 2.27488i) q^{3} +(16.8453 + 12.2388i) q^{4} +(-34.0654 - 44.3232i) q^{5} +(-16.5335 + 12.0123i) q^{6} +197.082i q^{7} +(47.7523 + 65.7253i) q^{8} +(72.6478 + 223.587i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.91215	−	2.24589i	−1.22191	−	0.397022i	−0.374131	−	0.927376i	\(-0.622059\pi\)
							−0.847776	+	0.530354i	\(0.822059\pi\)
\(3\)	1.65280	−	2.27488i	0.106027	−	0.145934i	−0.752706	−	0.658357i	\(-0.771252\pi\)
							0.858733	+	0.512423i	\(0.171252\pi\)
\(5\)	−34.0654	−	44.3232i	−0.609381	−	0.792878i
							\(7\)			197.082i			1.52020i	0.649804	+	0.760102i	\(0.274851\pi\)
−0.649804	+	0.760102i	\(0.725149\pi\)
\(11\)	51.7706	−	159.333i	0.129003	−	0.397032i	−0.865606	−	0.500726i	\(-0.833066\pi\)
							0.994609	+	0.103694i	\(0.0330664\pi\)
\(13\)	−936.806	+	304.387i	−1.53742	+	0.499536i	−0.950662	−	0.310229i	\(-0.899594\pi\)
							−0.586753	+	0.809766i	\(0.699594\pi\)
\(17\)	901.830	+	1241.26i	0.756837	+	1.04170i	0.997471	+	0.0710804i	\(0.0226447\pi\)
							−0.240634	+	0.970616i	\(0.577355\pi\)
\(19\)	407.246	−	295.882i	0.258805	−	0.188033i	−0.450815	−	0.892617i	\(-0.648867\pi\)
							0.709620	+	0.704584i	\(0.248867\pi\)
\(23\)	−3416.07	−	1109.95i	−1.34650	−	0.437505i	−0.454987	−	0.890498i	\(-0.650356\pi\)
							−0.891514	+	0.452994i	\(0.850356\pi\)
\(29\)	−278.858	−	202.602i	−0.0615728	−	0.0447352i	0.556573	−	0.830799i	\(-0.312116\pi\)
							−0.618146	+	0.786063i	\(0.712116\pi\)
\(31\)	−3718.55	+	2701.69i	−0.694976	+	0.504930i	−0.878292	−	0.478124i	\(-0.841317\pi\)
							0.183316	+	0.983054i	\(0.441317\pi\)
\(37\)	−5493.33	+	1784.89i	−0.659677	+	0.214342i	−0.619676	−	0.784858i	\(-0.712736\pi\)
							−0.0400007	+	0.999200i	\(0.512736\pi\)
\(41\)	352.381	+	1084.52i	0.0327380	+	0.100757i	0.966090	−	0.258204i	\(-0.0831308\pi\)
							−0.933352	+	0.358962i	\(0.883131\pi\)
\(43\)			11234.2i			0.926558i	0.886212	+	0.463279i	\(0.153327\pi\)
							−0.886212	+	0.463279i	\(0.846673\pi\)
\(47\)	−4051.58	+	5576.52i	−0.267534	+	0.368230i	−0.921555	−	0.388247i	\(-0.873081\pi\)
							0.654021	+	0.756476i	\(0.273081\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.e.a.4.3	✓	48
25.12	odd	20		625.6.a.g.1.39		48
25.13	odd	20		625.6.a.g.1.10		48
25.19	even	10	inner	25.6.e.a.19.3	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.e.a.4.3	✓	48	1.1	even	1	trivial
25.6.e.a.19.3	yes	48	25.19	even	10	inner
625.6.a.g.1.10		48	25.13	odd	20
625.6.a.g.1.39		48	25.12	odd	20