Embedded newform 25.8.e.a.4.13

• Label: 25.8.e.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	25.4
Dual form			25.8.e.a.19.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.2396 + 3.65197i) q^{2} +(-50.2306 + 69.1364i) q^{3} +(9.43754 + 6.85677i) q^{4} +(-36.2236 - 277.151i) q^{5} +(-817.055 + 593.625i) q^{6} +865.676i q^{7} +(-808.113 - 1112.27i) q^{8} +(-1580.92 - 4865.56i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.2396	+	3.65197i	0.993450	+	0.322791i	0.760245	−	0.649636i	\(-0.225079\pi\)
							0.233204	+	0.972428i	\(0.425079\pi\)
\(3\)	−50.2306	+	69.1364i	−1.07410	+	1.47837i	−0.208240	+	0.978078i	\(0.566773\pi\)
							−0.865858	+	0.500290i	\(0.833227\pi\)
\(5\)	−36.2236	−	277.151i	−0.129598	−	0.991567i
							\(7\)			865.676i			0.953921i	0.878925	+	0.476960i	\(0.158262\pi\)
−0.878925	+	0.476960i	\(0.841738\pi\)
\(11\)	−221.975	+	683.168i	−0.0502840	+	0.154758i	−0.973045	−	0.230613i	\(-0.925927\pi\)
							0.922761	+	0.385372i	\(0.125927\pi\)
\(13\)	−11070.8	+	3597.12i	−1.39758	+	0.454102i	−0.908408	−	0.418084i	\(-0.862702\pi\)
							−0.489174	+	0.872186i	\(0.662702\pi\)
\(17\)	13588.8	+	18703.3i	0.670824	+	0.923310i	0.999779	−	0.0210359i	\(-0.00669642\pi\)
							−0.328955	+	0.944346i	\(0.606696\pi\)
\(19\)	−41032.5	+	29811.9i	−1.37243	+	0.997129i	−0.374888	+	0.927070i	\(0.622319\pi\)
							−0.997543	+	0.0700591i	\(0.977681\pi\)
\(23\)	47536.8	+	15445.6i	0.814671	+	0.264703i	0.686575	−	0.727059i	\(-0.259113\pi\)
							0.128096	+	0.991762i	\(0.459113\pi\)
\(29\)	−30293.1	−	22009.2i	−0.230649	−	0.167576i	0.466458	−	0.884543i	\(-0.345530\pi\)
							−0.697107	+	0.716967i	\(0.745530\pi\)
\(31\)	48351.5	−	35129.4i	0.291504	−	0.211790i	−0.432416	−	0.901674i	\(-0.642339\pi\)
							0.723919	+	0.689885i	\(0.242339\pi\)
\(37\)	−22157.3	+	7199.34i	−0.0719135	+	0.0233661i	−0.344753	−	0.938693i	\(-0.612037\pi\)
							0.272839	+	0.962060i	\(0.412037\pi\)
\(41\)	−79085.0	−	243398.i	−0.179205	−	0.551537i	0.820595	−	0.571510i	\(-0.193642\pi\)
							−0.999801	+	0.0199729i	\(0.993642\pi\)
\(43\)		−	162575.i		−	0.311828i	−0.987771	−	0.155914i	\(-0.950168\pi\)
							0.987771	−	0.155914i	\(-0.0498323\pi\)
\(47\)	−174864.	+	240680.i	−0.245673	+	0.338141i	−0.913990	−	0.405736i	\(-0.867015\pi\)
							0.668317	+	0.743877i	\(0.267015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.4.13	✓	64
25.12	odd	20		625.8.a.g.1.16		64
25.13	odd	20		625.8.a.g.1.49		64
25.19	even	10	inner	25.8.e.a.19.13	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.4.13	✓	64	1.1	even	1	trivial
25.8.e.a.19.13	yes	64	25.19	even	10	inner
625.8.a.g.1.16		64	25.12	odd	20
625.8.a.g.1.49		64	25.13	odd	20