Embedded newform 25.8.e.a.4.6

• Label: 25.8.e.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	25.4
Dual form			25.8.e.a.19.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.12319 - 2.63939i) q^{2} +(22.8316 - 31.4250i) q^{3} +(-44.5343 - 32.3560i) q^{4} +(-192.399 + 202.750i) q^{5} +(-268.408 + 195.010i) q^{6} -1367.16i q^{7} +(918.973 + 1264.86i) q^{8} +(209.571 + 644.993i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.12319	−	2.63939i	−0.717996	−	0.233291i	−0.0728417	−	0.997344i	\(-0.523207\pi\)
							−0.645154	+	0.764053i	\(0.723207\pi\)
\(3\)	22.8316	−	31.4250i	0.488216	−	0.671972i	−0.491842	−	0.870685i	\(-0.663676\pi\)
							0.980058	+	0.198713i	\(0.0636761\pi\)
\(5\)	−192.399	+	202.750i	−0.688349	+	0.725380i
							\(7\)		−	1367.16i		−	1.50653i	−0.657718	−	0.753265i	\(-0.728478\pi\)
0.657718	−	0.753265i	\(-0.271522\pi\)
\(11\)	−1961.71	+	6037.53i	−0.444386	+	1.36768i	0.438769	+	0.898600i	\(0.355415\pi\)
							−0.883155	+	0.469080i	\(0.844585\pi\)
\(13\)	−8376.27	+	2721.62i	−1.05742	+	0.343578i	−0.785578	−	0.618763i	\(-0.787634\pi\)
							−0.271846	+	0.962341i	\(0.587634\pi\)
\(17\)	14440.5	+	19875.7i	0.712872	+	0.981184i	0.999731	+	0.0232111i	\(0.00738900\pi\)
							−0.286859	+	0.957973i	\(0.592611\pi\)
\(19\)	−4626.37	+	3361.25i	−0.154740	+	0.112425i	−0.662461	−	0.749096i	\(-0.730488\pi\)
							0.507721	+	0.861521i	\(0.330488\pi\)
\(23\)	−51422.5	−	16708.2i	−0.881264	−	0.286340i	−0.166782	−	0.985994i	\(-0.553338\pi\)
							−0.714482	+	0.699654i	\(0.753338\pi\)
\(29\)	−153902.	−	111817.i	−1.17180	−	0.851360i	−0.180573	−	0.983562i	\(-0.557795\pi\)
							−0.991223	+	0.132202i	\(0.957795\pi\)
\(31\)	−160670.	+	116734.i	−0.968655	+	0.703769i	−0.955145	−	0.296140i	\(-0.904301\pi\)
							−0.0135105	+	0.999909i	\(0.504301\pi\)
\(37\)	125663.	−	40830.4i	0.407851	−	0.132519i	−0.0979044	−	0.995196i	\(-0.531214\pi\)
							0.505755	+	0.862677i	\(0.331214\pi\)
\(41\)	−223935.	−	689201.i	−0.507433	−	1.56172i	−0.796642	−	0.604452i	\(-0.793392\pi\)
							0.289209	−	0.957266i	\(-0.406608\pi\)
\(43\)			510583.i			0.979324i	0.871912	+	0.489662i	\(0.162880\pi\)
							−0.871912	+	0.489662i	\(0.837120\pi\)
\(47\)	−277890.	+	382483.i	−0.390418	+	0.537365i	−0.958307	−	0.285741i	\(-0.907760\pi\)
							0.567889	+	0.823105i	\(0.307760\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.4.6	✓	64
25.12	odd	20		625.8.a.g.1.42		64
25.13	odd	20		625.8.a.g.1.23		64
25.19	even	10	inner	25.8.e.a.19.6	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.4.6	✓	64	1.1	even	1	trivial
25.8.e.a.19.6	yes	64	25.19	even	10	inner
625.8.a.g.1.23		64	25.13	odd	20
625.8.a.g.1.42		64	25.12	odd	20