Embedded newform 25.8.e.a.4.14

• Label: 25.8.e.a.4.14
• Level: $25$
• Weight: $8$
• Character: 25.4
• Analytic conductor: $7.810$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	25.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.80962563710\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			4.14
Character	\(\chi\)	\(=\)	25.4
Dual form			25.8.e.a.19.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(17.1896 + 5.58523i) q^{2} +(-25.9999 + 35.7858i) q^{3} +(160.732 + 116.779i) q^{4} +(-164.394 + 226.052i) q^{5} +(-646.798 + 469.926i) q^{6} -372.101i q^{7} +(750.840 + 1033.44i) q^{8} +(71.1926 + 219.108i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	17.1896	+	5.58523i	1.51936	+	0.493669i	0.945593	−	0.325352i	\(-0.105483\pi\)
							0.573764	+	0.819021i	\(0.305483\pi\)
\(3\)	−25.9999	+	35.7858i	−0.555965	+	0.765220i	−0.990806	−	0.135287i	\(-0.956804\pi\)
							0.434842	+	0.900507i	\(0.356804\pi\)
\(5\)	−164.394	+	226.052i	−0.588153	+	0.808749i
							\(7\)		−	372.101i		−	0.410032i	−0.978759	−	0.205016i	\(-0.934275\pi\)
0.978759	−	0.205016i	\(-0.0657246\pi\)
\(11\)	−1419.64	+	4369.20i	−0.321591	+	0.989755i	0.651365	+	0.758764i	\(0.274197\pi\)
							−0.972956	+	0.230990i	\(0.925803\pi\)
\(13\)	12736.4	−	4138.29i	1.60784	−	0.522420i	0.638813	−	0.769362i	\(-0.279426\pi\)
							0.969030	+	0.246943i	\(0.0794259\pi\)
\(17\)	10014.6	+	13783.9i	0.494381	+	0.680458i	0.981189	−	0.193052i	\(-0.0618386\pi\)
							−0.486807	+	0.873509i	\(0.661839\pi\)
\(19\)	3945.75	−	2866.76i	0.131975	−	0.0958856i	−0.519839	−	0.854264i	\(-0.674008\pi\)
							0.651814	+	0.758379i	\(0.274008\pi\)
\(23\)	56065.1	+	18216.7i	0.960827	+	0.312192i	0.747108	−	0.664703i	\(-0.231442\pi\)
							0.213720	+	0.976895i	\(0.431442\pi\)
\(29\)	24432.2	+	17751.0i	0.186024	+	0.135155i	0.676900	−	0.736075i	\(-0.263323\pi\)
							−0.490876	+	0.871230i	\(0.663323\pi\)
\(31\)	164275.	−	119353.i	0.990390	−	0.719561i	0.0303838	−	0.999538i	\(-0.490327\pi\)
							0.960007	+	0.279978i	\(0.0903271\pi\)
\(37\)	−174283.	+	56628.1i	−0.565653	+	0.183792i	−0.577863	−	0.816134i	\(-0.696113\pi\)
							0.0122101	+	0.999925i	\(0.496113\pi\)
\(41\)	−109667.	−	337519.i	−0.248503	−	0.764812i	−0.995041	−	0.0994695i	\(-0.968285\pi\)
							0.746538	−	0.665343i	\(-0.231715\pi\)
\(43\)		−	86485.5i		−	0.165884i	−0.996554	−	0.0829419i	\(-0.973568\pi\)
							0.996554	−	0.0829419i	\(-0.0264316\pi\)
\(47\)	−622336.	+	856572.i	−0.874344	+	1.20343i	0.103611	+	0.994618i	\(0.466960\pi\)
							−0.977955	+	0.208814i	\(0.933040\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.4.14	✓	64
25.12	odd	20		625.8.a.g.1.9		64
25.13	odd	20		625.8.a.g.1.56		64
25.19	even	10	inner	25.8.e.a.19.14	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.4.14	✓	64	1.1	even	1	trivial
25.8.e.a.19.14	yes	64	25.19	even	10	inner
625.8.a.g.1.9		64	25.12	odd	20
625.8.a.g.1.56		64	25.13	odd	20