Embedded newform 25.8.e.a.4.4

• Label: 25.8.e.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	25.4
Dual form			25.8.e.a.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.0786 - 4.57442i) q^{2} +(21.9300 - 30.1841i) q^{3} +(73.7277 + 53.5663i) q^{4} +(222.231 + 169.524i) q^{5} +(-446.819 + 324.633i) q^{6} +198.096i q^{7} +(320.787 + 441.525i) q^{8} +(245.667 + 756.085i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.0786	−	4.57442i	−1.24438	−	0.404325i	−0.388479	−	0.921457i	\(-0.627000\pi\)
							−0.855905	+	0.517132i	\(0.827000\pi\)
\(3\)	21.9300	−	30.1841i	0.468937	−	0.645437i	−0.507395	−	0.861714i	\(-0.669391\pi\)
							0.976332	+	0.216277i	\(0.0693914\pi\)
\(5\)	222.231	+	169.524i	0.795078	+	0.606507i
							\(7\)			198.096i			0.218289i	0.994026	+	0.109144i	\(0.0348111\pi\)
−0.994026	+	0.109144i	\(0.965189\pi\)
\(11\)	2581.07	−	7943.71i	0.584689	−	1.79949i	−0.0158270	−	0.999875i	\(-0.505038\pi\)
							0.600516	−	0.799613i	\(-0.294962\pi\)
\(13\)	4726.07	−	1535.59i	0.596621	−	0.193854i	0.00488795	−	0.999988i	\(-0.498444\pi\)
							0.591733	+	0.806134i	\(0.298444\pi\)
\(17\)	2045.78	+	2815.78i	0.100992	+	0.139004i	0.856522	−	0.516110i	\(-0.172621\pi\)
							−0.755530	+	0.655114i	\(0.772621\pi\)
\(19\)	14920.5	−	10840.3i	0.499050	−	0.362581i	−0.309604	−	0.950866i	\(-0.600196\pi\)
							0.808654	+	0.588284i	\(0.200196\pi\)
\(23\)	−63325.4	−	20575.7i	−1.08525	−	0.352620i	−0.288842	−	0.957377i	\(-0.593270\pi\)
							−0.796410	+	0.604757i	\(0.793270\pi\)
\(29\)	27704.4	+	20128.4i	0.210938	+	0.153256i	0.688238	−	0.725485i	\(-0.258384\pi\)
							−0.477300	+	0.878740i	\(0.658384\pi\)
\(31\)	252751.	−	183635.i	1.52380	−	1.10710i	0.564235	−	0.825614i	\(-0.309171\pi\)
							0.959564	−	0.281491i	\(-0.0908288\pi\)
\(37\)	202542.	−	65810.0i	0.657370	−	0.213592i	0.0387089	−	0.999251i	\(-0.487675\pi\)
							0.618661	+	0.785658i	\(0.287675\pi\)
\(41\)	−36298.6	−	111716.i	−0.0822520	−	0.253145i	0.901470	−	0.432841i	\(-0.142489\pi\)
							−0.983722	+	0.179695i	\(0.942489\pi\)
\(43\)		−	129563.i		−	0.248508i	−0.992250	−	0.124254i	\(-0.960346\pi\)
							0.992250	−	0.124254i	\(-0.0396537\pi\)
\(47\)	−475659.	+	654688.i	−0.668272	+	0.919797i	−0.999720	−	0.0236766i	\(-0.992463\pi\)
							0.331448	+	0.943473i	\(0.392463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.4.4	✓	64
25.12	odd	20		625.8.a.g.1.52		64
25.13	odd	20		625.8.a.g.1.13		64
25.19	even	10	inner	25.8.e.a.19.4	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.4.4	✓	64	1.1	even	1	trivial
25.8.e.a.19.4	yes	64	25.19	even	10	inner
625.8.a.g.1.13		64	25.13	odd	20
625.8.a.g.1.52		64	25.12	odd	20