Embedded newform 25.8.d.a.6.8

• Label: 25.8.d.a.6.8
• Level: $25$
• Weight: $8$
• Character: 25.6
• Analytic conductor: $7.810$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.80962563710\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(17\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			6.8
Character	\(\chi\)	\(=\)	25.6
Dual form			25.8.d.a.21.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0869750 - 0.267681i) q^{2} +(34.6603 - 25.1822i) q^{3} +(103.490 - 75.1899i) q^{4} +(-214.708 - 178.957i) q^{5} +(-9.75537 - 7.08769i) q^{6} -32.0119 q^{7} +(-58.2740 - 42.3386i) q^{8} +(-108.627 + 334.320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 11 q^{2} + 23 q^{3} - 1091 q^{4} + 475 q^{5} - 179 q^{6} + 1734 q^{7} + 220 q^{8} - 14554 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0869750	−	0.267681i	−0.00768757	−	0.0236599i	0.947139	−	0.320823i	\(-0.103959\pi\)
							−0.954827	+	0.297163i	\(0.903959\pi\)
\(3\)	34.6603	−	25.1822i	0.741152	−	0.538479i	−0.151919	−	0.988393i	\(-0.548545\pi\)
							0.893072	+	0.449914i	\(0.148545\pi\)
\(5\)	−214.708	−	178.957i	−0.768161	−	0.640256i
							\(7\)	−32.0119			−0.0352751			−0.0176375	−	0.999844i	\(-0.505614\pi\)
−0.0176375	+	0.999844i	\(0.505614\pi\)
\(11\)	−1425.13	−	4386.11i	−0.322836	−	0.993586i	−0.972408	−	0.233286i	\(-0.925052\pi\)
							0.649572	−	0.760300i	\(-0.274948\pi\)
\(13\)	2319.82	−	7139.66i	0.292855	−	0.901314i	−0.691079	−	0.722779i	\(-0.742864\pi\)
							0.983934	−	0.178534i	\(-0.0571356\pi\)
\(17\)	6384.27	+	4638.45i	0.315167	+	0.228982i	0.734110	−	0.679030i	\(-0.237600\pi\)
							−0.418944	+	0.908012i	\(0.637600\pi\)
\(19\)	24427.5	+	17747.6i	0.817037	+	0.593612i	0.915862	−	0.401493i	\(-0.131508\pi\)
							−0.0988252	+	0.995105i	\(0.531508\pi\)
\(23\)	−5556.95	−	17102.5i	−0.0952334	−	0.293098i	0.892081	−	0.451875i	\(-0.149245\pi\)
							−0.987314	+	0.158777i	\(0.949245\pi\)
\(29\)	−31246.1	+	22701.6i	−0.237905	+	0.172848i	−0.700349	−	0.713800i	\(-0.746972\pi\)
							0.462445	+	0.886648i	\(0.346972\pi\)
\(31\)	162380.	+	117976.i	0.978967	+	0.711261i	0.957477	−	0.288508i	\(-0.0931592\pi\)
							0.0214892	+	0.999769i	\(0.493159\pi\)
\(37\)	33231.0	−	102275.i	0.107854	−	0.331941i	−0.882535	−	0.470246i	\(-0.844165\pi\)
							0.990390	+	0.138304i	\(0.0441652\pi\)
\(41\)	−122269.	+	376305.i	−0.277059	+	0.852700i	0.711608	+	0.702577i	\(0.247967\pi\)
							−0.988667	+	0.150124i	\(0.952033\pi\)
\(43\)	361679.			0.693719			0.346859	−	0.937917i	\(-0.387248\pi\)
							0.346859	+	0.937917i	\(0.387248\pi\)
\(47\)	862323.	−	626515.i	1.21151	−	0.880215i	0.216144	−	0.976361i	\(-0.430652\pi\)
							0.995367	+	0.0961466i	\(0.0306518\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.d.a.6.8	✓	68
25.11	even	5		625.8.a.d.1.16		34
25.14	even	10		625.8.a.c.1.19		34
25.21	even	5	inner	25.8.d.a.21.8	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.d.a.6.8	✓	68	1.1	even	1	trivial
25.8.d.a.21.8	yes	68	25.21	even	5	inner
625.8.a.c.1.19		34	25.14	even	10
625.8.a.d.1.16		34	25.11	even	5