Embedded newform 25.6.d.a.6.11

• Label: 25.6.d.a.6.11
• Level: $25$
• Weight: $6$
• Character: 25.6
• 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			6.11
Character	\(\chi\)	\(=\)	25.6
Dual form			25.6.d.a.21.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.27755 + 10.0873i) q^{2} +(20.2415 - 14.7063i) q^{3} +(-65.1220 + 47.3139i) q^{4} +(47.0312 + 30.2170i) q^{5} +(214.689 + 155.981i) q^{6} -69.7231 q^{7} +(-416.125 - 302.333i) q^{8} +(118.352 - 364.249i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.27755	+	10.0873i	0.579395	+	1.78319i	0.620702	+	0.784047i	\(0.286848\pi\)
							−0.0413071	+	0.999146i	\(0.513152\pi\)
\(3\)	20.2415	−	14.7063i	1.29849	−	0.943411i	0.298554	−	0.954393i	\(-0.403496\pi\)
							0.999940	+	0.0109824i	\(0.00349586\pi\)
\(5\)	47.0312	+	30.2170i	0.841320	+	0.540538i
							\(7\)	−69.7231			−0.537813			−0.268907	−	0.963166i	\(-0.586662\pi\)
−0.268907	+	0.963166i	\(0.586662\pi\)
\(11\)	−71.7303	−	220.763i	−0.178740	−	0.550104i	0.821045	−	0.570864i	\(-0.193392\pi\)
							−0.999784	+	0.0207594i	\(0.993392\pi\)
\(13\)	109.102	−	335.781i	0.179050	−	0.551059i	−0.820745	−	0.571294i	\(-0.806442\pi\)
							0.999795	+	0.0202355i	\(0.00644160\pi\)
\(17\)	−515.153	−	374.281i	−0.432329	−	0.314105i	0.350251	−	0.936656i	\(-0.386096\pi\)
							−0.782579	+	0.622551i	\(0.786096\pi\)
\(19\)	1577.48	+	1146.11i	1.00249	+	0.728351i	0.962621	−	0.270854i	\(-0.0873060\pi\)
							0.0398690	+	0.999205i	\(0.487306\pi\)
\(23\)	−458.775	−	1411.96i	−0.180834	−	0.556550i	0.819018	−	0.573768i	\(-0.194519\pi\)
							−0.999852	+	0.0172184i	\(0.994519\pi\)
\(29\)	588.169	−	427.330i	0.129870	−	0.0943558i	−0.520954	−	0.853585i	\(-0.674424\pi\)
							0.650824	+	0.759229i	\(0.274424\pi\)
\(31\)	−7225.68	−	5249.77i	−1.35044	−	0.981151i	−0.998990	−	0.0449432i	\(-0.985689\pi\)
							−0.351448	−	0.936207i	\(-0.614311\pi\)
\(37\)	−2132.85	+	6564.25i	−0.256128	+	0.788280i	0.737478	+	0.675372i	\(0.236017\pi\)
							−0.993605	+	0.112909i	\(0.963983\pi\)
\(41\)	−4471.72	+	13762.5i	−0.415446	+	1.27861i	0.496405	+	0.868091i	\(0.334653\pi\)
							−0.911851	+	0.410521i	\(0.865347\pi\)
\(43\)	3198.31			0.263785			0.131892	−	0.991264i	\(-0.457895\pi\)
							0.131892	+	0.991264i	\(0.457895\pi\)
\(47\)	13823.9	−	10043.7i	0.912822	−	0.663204i	−0.0289052	−	0.999582i	\(-0.509202\pi\)
							0.941727	+	0.336378i	\(0.109202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.6.11	✓	44
25.11	even	5		625.6.a.d.1.21		22
25.14	even	10		625.6.a.c.1.2		22
25.21	even	5	inner	25.6.d.a.21.11	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.6.11	✓	44	1.1	even	1	trivial
25.6.d.a.21.11	yes	44	25.21	even	5	inner
625.6.a.c.1.2		22	25.14	even	10
625.6.a.d.1.21		22	25.11	even	5