Embedded newform 25.6.d.a.11.5

• Label: 25.6.d.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	25.11
Dual form			25.6.d.a.16.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.39495 + 2.46657i) q^{2} +(-4.95437 + 15.2480i) q^{3} +(-4.44687 + 13.6860i) q^{4} +(23.5016 - 50.7215i) q^{5} +(-20.7904 - 63.9863i) q^{6} -192.901 q^{7} +(-60.1569 - 185.144i) q^{8} +(-11.3638 - 8.25627i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.39495	+	2.46657i	−0.600147	+	0.436033i	−0.845931	−	0.533292i	\(-0.820955\pi\)
							0.245784	+	0.969325i	\(0.420955\pi\)
\(3\)	−4.95437	+	15.2480i	−0.317823	+	0.978158i	0.656754	+	0.754105i	\(0.271929\pi\)
							−0.974577	+	0.224053i	\(0.928071\pi\)
\(5\)	23.5016	−	50.7215i	0.420410	−	0.907334i
							\(7\)	−192.901			−1.48796			−0.743978	−	0.668204i	\(-0.767063\pi\)
−0.743978	+	0.668204i	\(0.767063\pi\)
\(11\)	−140.546	+	102.113i	−0.350217	+	0.254447i	−0.748960	−	0.662615i	\(-0.769446\pi\)
							0.398743	+	0.917063i	\(0.369446\pi\)
\(13\)	−363.231	−	263.903i	−0.596108	−	0.433098i	0.248387	−	0.968661i	\(-0.420099\pi\)
							−0.844495	+	0.535563i	\(0.820099\pi\)
\(17\)	252.151	+	776.042i	0.211611	+	0.651273i	0.999377	+	0.0352974i	\(0.0112379\pi\)
							−0.787766	+	0.615975i	\(0.788762\pi\)
\(19\)	582.129	+	1791.61i	0.369943	+	1.13857i	0.946827	+	0.321743i	\(0.104269\pi\)
							−0.576884	+	0.816826i	\(0.695731\pi\)
\(23\)	−3521.25	+	2558.33i	−1.38796	+	1.00841i	−0.391873	+	0.920019i	\(0.628173\pi\)
							−0.996086	+	0.0883922i	\(0.971827\pi\)
\(29\)	−795.130	+	2447.16i	−0.175567	+	0.540340i	−0.999659	−	0.0261164i	\(-0.991686\pi\)
							0.824092	+	0.566456i	\(0.191686\pi\)
\(31\)	−2332.59	−	7178.99i	−0.435948	−	1.34171i	−0.892112	−	0.451815i	\(-0.850777\pi\)
							0.456164	−	0.889896i	\(-0.349223\pi\)
\(37\)	7078.55	+	5142.87i	0.850041	+	0.617591i	0.925157	−	0.379584i	\(-0.123933\pi\)
							−0.0751165	+	0.997175i	\(0.523933\pi\)
\(41\)	3457.12	+	2511.74i	0.321184	+	0.233354i	0.736681	−	0.676241i	\(-0.236392\pi\)
							−0.415496	+	0.909595i	\(0.636392\pi\)
\(43\)	2783.03			0.229534			0.114767	−	0.993392i	\(-0.463388\pi\)
							0.114767	+	0.993392i	\(0.463388\pi\)
\(47\)	−2994.89	+	9217.33i	−0.197759	+	0.608640i	0.802174	+	0.597090i	\(0.203676\pi\)
							−0.999933	+	0.0115498i	\(0.996324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.11.5	✓	44
25.4	even	10		625.6.a.c.1.9		22
25.16	even	5	inner	25.6.d.a.16.5	yes	44
25.21	even	5		625.6.a.d.1.14		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.11.5	✓	44	1.1	even	1	trivial
25.6.d.a.16.5	yes	44	25.16	even	5	inner
625.6.a.c.1.9		22	25.4	even	10
625.6.a.d.1.14		22	25.21	even	5