Embedded newform 25.8.e.a.9.3

• Label: 25.8.e.a.9.3
• Level: $25$
• Weight: $8$
• Character: 25.9
• 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			9.3
Character	\(\chi\)	\(=\)	25.9
Dual form			25.8.e.a.14.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.3019 + 14.1794i) q^{2} +(-26.5031 + 8.61137i) q^{3} +(-55.3708 - 170.414i) q^{4} +(-226.305 + 164.046i) q^{5} +(150.928 - 464.510i) q^{6} -1206.49i q^{7} +(853.170 + 277.212i) q^{8} +(-1141.06 + 829.031i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.3019	+	14.1794i	−0.910569	+	1.25329i	0.0564035	+	0.998408i	\(0.482037\pi\)
							−0.966972	+	0.254882i	\(0.917963\pi\)
\(3\)	−26.5031	+	8.61137i	−0.566724	+	0.184140i	−0.578345	−	0.815793i	\(-0.696301\pi\)
							0.0116203	+	0.999932i	\(0.496301\pi\)
\(5\)	−226.305	+	164.046i	−0.809653	+	0.586909i
							\(7\)		−	1206.49i		−	1.32948i	−0.747076	−	0.664739i	\(-0.768543\pi\)
0.747076	−	0.664739i	\(-0.231457\pi\)
\(11\)	7016.60	+	5097.86i	1.58947	+	1.15482i	0.904691	+	0.426068i	\(0.140101\pi\)
							0.684779	+	0.728750i	\(0.259899\pi\)
\(13\)	1271.77	+	1750.44i	0.160549	+	0.220977i	0.881711	−	0.471790i	\(-0.156392\pi\)
							−0.721162	+	0.692766i	\(0.756392\pi\)
\(17\)	−23042.3	−	7486.91i	−1.13751	−	0.369599i	−0.321085	−	0.947050i	\(-0.604047\pi\)
							−0.816425	+	0.577451i	\(0.804047\pi\)
\(19\)	8457.65	−	26030.0i	0.282886	−	0.870634i	−0.704138	−	0.710063i	\(-0.748666\pi\)
							0.987024	−	0.160571i	\(-0.0513336\pi\)
\(23\)	25673.5	−	35336.5i	0.439985	−	0.605587i	−0.530224	−	0.847858i	\(-0.677892\pi\)
							0.970209	+	0.242271i	\(0.0778922\pi\)
\(29\)	−42594.7	−	131093.i	−0.324312	−	0.998128i	−0.971750	−	0.236011i	\(-0.924160\pi\)
							0.647439	−	0.762117i	\(-0.275840\pi\)
\(31\)	−35839.0	+	110301.i	−0.216068	+	0.664987i	0.783009	+	0.622011i	\(0.213684\pi\)
							−0.999076	+	0.0429765i	\(0.986316\pi\)
\(37\)	−125344.	−	172522.i	−0.406816	−	0.559934i	0.555622	−	0.831435i	\(-0.312480\pi\)
							−0.962438	+	0.271500i	\(0.912480\pi\)
\(41\)	115744.	−	84093.0i	0.262274	−	0.190553i	−0.448875	−	0.893595i	\(-0.648175\pi\)
							0.711149	+	0.703041i	\(0.248175\pi\)
\(43\)		−	581425.i		−	1.11520i	−0.830108	−	0.557602i	\(-0.811722\pi\)
							0.830108	−	0.557602i	\(-0.188278\pi\)
\(47\)	−126084.	+	40967.3i	−0.177141	+	0.0575566i	−0.396245	−	0.918145i	\(-0.629687\pi\)
							0.219104	+	0.975702i	\(0.429687\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.9.3	✓	64
25.8	odd	20		625.8.a.g.1.11		64
25.14	even	10	inner	25.8.e.a.14.3	yes	64
25.17	odd	20		625.8.a.g.1.54		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.9.3	✓	64	1.1	even	1	trivial
25.8.e.a.14.3	yes	64	25.14	even	10	inner
625.8.a.g.1.11		64	25.8	odd	20
625.8.a.g.1.54		64	25.17	odd	20