Embedded newform 25.6.d.a.11.4

• Label: 25.6.d.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	25.11
Dual form			25.6.d.a.16.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.49484 + 3.26569i) q^{2} +(6.79188 - 20.9033i) q^{3} +(-0.349697 + 1.07626i) q^{4} +(-34.6743 + 43.8485i) q^{5} +(37.7352 + 116.137i) q^{6} -153.453 q^{7} +(-56.8829 - 175.068i) q^{8} +(-194.225 - 141.113i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.49484	+	3.26569i	−0.794583	+	0.577299i	−0.909320	−	0.416097i	\(-0.863398\pi\)
							0.114737	+	0.993396i	\(0.463398\pi\)
\(3\)	6.79188	−	20.9033i	0.435699	−	1.34094i	−0.456669	−	0.889637i	\(-0.650958\pi\)
							0.892368	−	0.451308i	\(-0.149042\pi\)
\(5\)	−34.6743	+	43.8485i	−0.620273	+	0.784386i
							\(7\)	−153.453			−1.18367			−0.591835	−	0.806059i	\(-0.701596\pi\)
−0.591835	+	0.806059i	\(0.701596\pi\)
\(11\)	−431.726	+	313.667i	−1.07579	+	0.781605i	−0.976943	−	0.213499i	\(-0.931514\pi\)
							−0.0988430	+	0.995103i	\(0.531514\pi\)
\(13\)	−177.821	−	129.194i	−0.291826	−	0.212024i	0.432233	−	0.901762i	\(-0.357726\pi\)
							−0.724059	+	0.689738i	\(0.757726\pi\)
\(17\)	273.555	+	841.916i	0.229574	+	0.706556i	0.997795	+	0.0663712i	\(0.0211421\pi\)
							−0.768221	+	0.640185i	\(0.778858\pi\)
\(19\)	−488.104	−	1502.23i	−0.310190	−	0.954668i	−0.977689	−	0.210057i	\(-0.932635\pi\)
							0.667499	−	0.744611i	\(-0.267365\pi\)
\(23\)	2762.52	−	2007.09i	1.08889	−	0.791128i	0.109682	−	0.993967i	\(-0.465017\pi\)
							0.979212	+	0.202839i	\(0.0650168\pi\)
\(29\)	−2598.97	+	7998.82i	−0.573861	+	1.76616i	0.0661625	+	0.997809i	\(0.478924\pi\)
							−0.640024	+	0.768355i	\(0.721076\pi\)
\(31\)	−182.652	−	562.145i	−0.0341366	−	0.105062i	0.932536	−	0.361076i	\(-0.117591\pi\)
							−0.966673	+	0.256014i	\(0.917591\pi\)
\(37\)	−5748.99	−	4176.88i	−0.690378	−	0.501589i	0.186406	−	0.982473i	\(-0.440316\pi\)
							−0.876784	+	0.480884i	\(0.840316\pi\)
\(41\)	−7612.11	−	5530.52i	−0.707205	−	0.513814i	0.175066	−	0.984557i	\(-0.443986\pi\)
							−0.882271	+	0.470742i	\(0.843986\pi\)
\(43\)	−12992.7			−1.07159			−0.535793	−	0.844350i	\(-0.679987\pi\)
							−0.535793	+	0.844350i	\(0.679987\pi\)
\(47\)	−1444.03	+	4444.25i	−0.0953521	+	0.293464i	−0.987345	−	0.158585i	\(-0.949307\pi\)
							0.891993	+	0.452049i	\(0.149307\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.11.4	✓	44
25.4	even	10		625.6.a.c.1.7		22
25.16	even	5	inner	25.6.d.a.16.4	yes	44
25.21	even	5		625.6.a.d.1.16		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.4	✓	44	1.1	even	1	trivial
25.6.d.a.16.4	yes	44	25.16	even	5	inner
625.6.a.c.1.7		22	25.4	even	10
625.6.a.d.1.16		22	25.21	even	5