Embedded newform 25.8.e.a.9.4

• Label: 25.8.e.a.9.4
• 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			9.4
Character	\(\chi\)	\(=\)	25.9
Dual form			25.8.e.a.14.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.81540 + 10.7570i) q^{2} +(37.7432 - 12.2635i) q^{3} +(-15.0779 - 46.4050i) q^{4} +(-244.231 - 135.928i) q^{5} +(-163.060 + 501.847i) q^{6} +876.288i q^{7} +(-1001.62 - 325.445i) q^{8} +(-495.164 + 359.757i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.81540	+	10.7570i	−0.690791	+	0.950792i	−1.00000		0.000202341i	\(-0.999936\pi\)
							0.309209	+	0.950994i	\(0.399936\pi\)
\(3\)	37.7432	−	12.2635i	0.807076	−	0.262235i	0.123717	−	0.992318i	\(-0.460518\pi\)
							0.683359	+	0.730083i	\(0.260518\pi\)
\(5\)	−244.231	−	135.928i	−0.873786	−	0.486311i
							\(7\)			876.288i			0.965614i	0.875727	+	0.482807i	\(0.160383\pi\)
−0.875727	+	0.482807i	\(0.839617\pi\)
\(11\)	−1882.20	−	1367.49i	−0.426374	−	0.309779i	0.353824	−	0.935312i	\(-0.384881\pi\)
							−0.780197	+	0.625534i	\(0.784881\pi\)
\(13\)	−7814.94	−	10756.3i	−0.986561	−	1.35788i	−0.933219	−	0.359308i	\(-0.883013\pi\)
							−0.0533420	−	0.998576i	\(-0.516987\pi\)
\(17\)	583.589	+	189.619i	0.0288095	+	0.00936077i	0.323386	−	0.946267i	\(-0.395179\pi\)
							−0.294577	+	0.955628i	\(0.595179\pi\)
\(19\)	−4034.56	+	12417.1i	−0.134946	+	0.415320i	−0.995582	−	0.0939007i	\(-0.970066\pi\)
							0.860636	+	0.509221i	\(0.170066\pi\)
\(23\)	−44364.8	+	61062.9i	−0.760311	+	1.04648i	0.236878	+	0.971540i	\(0.423876\pi\)
							−0.997188	+	0.0749383i	\(0.976124\pi\)
\(29\)	−61427.1	−	189053.i	−0.467699	−	1.43943i	−0.855556	−	0.517711i	\(-0.826784\pi\)
							0.387856	−	0.921720i	\(-0.373216\pi\)
\(31\)	−9729.59	+	29944.6i	−0.0586582	+	0.180531i	−0.976092	−	0.217356i	\(-0.930257\pi\)
							0.917434	+	0.397888i	\(0.130257\pi\)
\(37\)	221004.	+	304185.i	0.717287	+	0.987261i	0.999610	+	0.0279417i	\(0.00889528\pi\)
							−0.282322	+	0.959320i	\(0.591105\pi\)
\(41\)	106355.	−	77271.5i	0.240999	−	0.175096i	−0.460729	−	0.887541i	\(-0.652412\pi\)
							0.701728	+	0.712445i	\(0.252412\pi\)
\(43\)			890431.i			1.70789i	0.520361	+	0.853946i	\(0.325797\pi\)
							−0.520361	+	0.853946i	\(0.674203\pi\)
\(47\)	−808424.	+	262673.i	−1.13579	+	0.369039i	−0.815772	−	0.578374i	\(-0.803688\pi\)
							−0.320014	+	0.947413i	\(0.603688\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.9.4	✓	64
25.8	odd	20		625.8.a.g.1.14		64
25.14	even	10	inner	25.8.e.a.14.4	yes	64
25.17	odd	20		625.8.a.g.1.51		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.9.4	✓	64	1.1	even	1	trivial
25.8.e.a.14.4	yes	64	25.14	even	10	inner
625.8.a.g.1.14		64	25.8	odd	20
625.8.a.g.1.51		64	25.17	odd	20