Embedded newform 25.8.e.a.4.3

• Label: 25.8.e.a.4.3
• 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			4.3
Character	\(\chi\)	\(=\)	25.4
Dual form			25.8.e.a.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.8860 - 5.48660i) q^{2} +(-41.4495 + 57.0503i) q^{3} +(151.480 + 110.057i) q^{4} +(37.1286 + 277.032i) q^{5} +(1012.93 - 735.935i) q^{6} +1074.80i q^{7} +(-618.235 - 850.927i) q^{8} +(-860.857 - 2649.45i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.8860	−	5.48660i	−1.49253	−	0.484951i	−0.554698	−	0.832052i	\(-0.687166\pi\)
							−0.937828	+	0.347101i	\(0.887166\pi\)
\(3\)	−41.4495	+	57.0503i	−0.886328	+	1.21993i	0.0882999	+	0.996094i	\(0.471857\pi\)
							−0.974628	+	0.223832i	\(0.928143\pi\)
\(5\)	37.1286	+	277.032i	0.132835	+	0.991138i
							\(7\)			1074.80i			1.18437i	0.805804	+	0.592183i	\(0.201734\pi\)
−0.805804	+	0.592183i	\(0.798266\pi\)
\(11\)	−1778.23	+	5472.83i	−0.402822	+	1.23976i	0.519878	+	0.854240i	\(0.325977\pi\)
							−0.922700	+	0.385518i	\(0.874023\pi\)
\(13\)	−9029.04	+	2933.71i	−1.13983	+	0.370353i	−0.817303	−	0.576208i	\(-0.804532\pi\)
							−0.322526	+	0.946561i	\(0.604532\pi\)
\(17\)	7062.48	+	9720.66i	0.348647	+	0.479871i	0.946942	−	0.321405i	\(-0.104155\pi\)
							−0.598295	+	0.801276i	\(0.704155\pi\)
\(19\)	40134.3	−	29159.3i	1.34239	−	0.975302i	0.343035	−	0.939323i	\(-0.388545\pi\)
							0.999353	−	0.0359788i	\(-0.0114549\pi\)
\(23\)	45277.4	+	14711.5i	0.775951	+	0.252122i	0.670111	−	0.742261i	\(-0.266247\pi\)
							0.105841	+	0.994383i	\(0.466247\pi\)
\(29\)	−24145.3	−	17542.6i	−0.183840	−	0.133568i	0.492059	−	0.870562i	\(-0.336244\pi\)
							−0.675899	+	0.736994i	\(0.736244\pi\)
\(31\)	−33332.6	+	24217.5i	−0.200957	+	0.146004i	−0.683713	−	0.729751i	\(-0.739636\pi\)
							0.482756	+	0.875755i	\(0.339636\pi\)
\(37\)	169619.	−	55112.7i	0.550515	−	0.178873i	−0.0205338	−	0.999789i	\(-0.506537\pi\)
							0.571049	+	0.820916i	\(0.306537\pi\)
\(41\)	166713.	+	513091.i	0.377770	+	1.16266i	0.941591	+	0.336759i	\(0.109331\pi\)
							−0.563821	+	0.825897i	\(0.690669\pi\)
\(43\)		−	390136.i		−	0.748302i	−0.927368	−	0.374151i	\(-0.877934\pi\)
							0.927368	−	0.374151i	\(-0.122066\pi\)
\(47\)	437821.	−	602608.i	0.615111	−	0.846628i	−0.381874	−	0.924214i	\(-0.624721\pi\)
							0.996986	+	0.0775863i	\(0.0247213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.4.3	✓	64
25.12	odd	20		625.8.a.g.1.55		64
25.13	odd	20		625.8.a.g.1.10		64
25.19	even	10	inner	25.8.e.a.19.3	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.4.3	✓	64	1.1	even	1	trivial
25.8.e.a.19.3	yes	64	25.19	even	10	inner
625.8.a.g.1.10		64	25.13	odd	20
625.8.a.g.1.55		64	25.12	odd	20