Embedded newform 25.8.e.a.4.7

• Label: 25.8.e.a.4.7
• 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			4.7
Character	\(\chi\)	\(=\)	25.4
Dual form			25.8.e.a.19.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.59051 - 1.81647i) q^{2} +(42.7242 - 58.8048i) q^{3} +(-75.5999 - 54.9266i) q^{4} +(12.2594 - 279.240i) q^{5} +(-345.667 + 251.142i) q^{6} +531.360i q^{7} +(765.126 + 1053.11i) q^{8} +(-956.829 - 2944.82i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.59051	−	1.81647i	−0.494136	−	0.160554i	0.0513396	−	0.998681i	\(-0.483651\pi\)
							−0.545475	+	0.838127i	\(0.683651\pi\)
\(3\)	42.7242	−	58.8048i	0.913586	−	1.25744i	−0.0523407	−	0.998629i	\(-0.516668\pi\)
							0.965927	−	0.258814i	\(-0.0833318\pi\)
\(5\)	12.2594	−	279.240i	0.0438607	−	0.999038i
							\(7\)			531.360i			0.585526i	0.956185	+	0.292763i	\(0.0945746\pi\)
−0.956185	+	0.292763i	\(0.905425\pi\)
\(11\)	128.961	−	396.901i	0.0292135	−	0.0899100i	−0.935387	−	0.353627i	\(-0.884948\pi\)
							0.964600	+	0.263717i	\(0.0849484\pi\)
\(13\)	−6932.26	+	2252.43i	−0.875131	+	0.284347i	−0.711934	−	0.702246i	\(-0.752181\pi\)
							−0.163197	+	0.986594i	\(0.552181\pi\)
\(17\)	−18397.8	−	25322.3i	−0.908226	−	1.25007i	−0.967769	−	0.251840i	\(-0.918964\pi\)
							0.0595434	−	0.998226i	\(-0.481036\pi\)
\(19\)	12231.1	−	8886.44i	0.409100	−	0.297228i	−0.364138	−	0.931345i	\(-0.618636\pi\)
							0.773237	+	0.634117i	\(0.218636\pi\)
\(23\)	104969.	+	34106.5i	1.79893	+	0.584507i	0.999860	−	0.0167336i	\(-0.00532672\pi\)
							0.799068	+	0.601241i	\(0.205327\pi\)
\(29\)	−102720.	−	74630.8i	−0.782103	−	0.568231i	0.123506	−	0.992344i	\(-0.460586\pi\)
							−0.905609	+	0.424113i	\(0.860586\pi\)
\(31\)	59764.7	−	43421.6i	0.360312	−	0.261782i	−0.392870	−	0.919594i	\(-0.628518\pi\)
							0.753182	+	0.657812i	\(0.228518\pi\)
\(37\)	−26242.7	+	8526.78i	−0.0851732	+	0.0276745i	−0.351294	−	0.936265i	\(-0.614258\pi\)
							0.266120	+	0.963940i	\(0.414258\pi\)
\(41\)	−1471.08	−	4527.52i	−0.00333344	−	0.0102593i	0.949376	−	0.314142i	\(-0.101717\pi\)
							−0.952709	+	0.303883i	\(0.901717\pi\)
\(43\)		−	997421.i		−	1.91311i	−0.291560	−	0.956553i	\(-0.594174\pi\)
							0.291560	−	0.956553i	\(-0.405826\pi\)
\(47\)	−281125.	+	386935.i	−0.394963	+	0.543620i	−0.959471	−	0.281808i	\(-0.909066\pi\)
							0.564508	+	0.825428i	\(0.309066\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.4.7	✓	64
25.12	odd	20		625.8.a.g.1.40		64
25.13	odd	20		625.8.a.g.1.25		64
25.19	even	10	inner	25.8.e.a.19.7	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.4.7	✓	64	1.1	even	1	trivial
25.8.e.a.19.7	yes	64	25.19	even	10	inner
625.8.a.g.1.25		64	25.13	odd	20
625.8.a.g.1.40		64	25.12	odd	20