Embedded newform 25.8.e.a.9.1

• Label: 25.8.e.a.9.1
• 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			9.1
Character	\(\chi\)	\(=\)	25.9
Dual form			25.8.e.a.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.2463 + 16.8556i) q^{2} +(48.7475 - 15.8390i) q^{3} +(-94.5855 - 291.104i) q^{4} +(150.427 + 235.578i) q^{5} +(-330.002 + 1015.64i) q^{6} +983.801i q^{7} +(3528.76 + 1146.56i) q^{8} +(356.126 - 258.741i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.2463	+	16.8556i	−1.08243	+	1.48984i	−0.225620	+	0.974215i	\(0.572441\pi\)
							−0.856813	+	0.515626i	\(0.827559\pi\)
\(3\)	48.7475	−	15.8390i	1.04239	−	0.338691i	0.262709	−	0.964875i	\(-0.415384\pi\)
							0.779676	+	0.626184i	\(0.215384\pi\)
\(5\)	150.427	+	235.578i	0.538183	+	0.842828i
							\(7\)			983.801i			1.08409i	0.840351	+	0.542043i	\(0.182349\pi\)
−0.840351	+	0.542043i	\(0.817651\pi\)
\(11\)	−3042.53	−	2210.52i	−0.689223	−	0.500750i	0.187181	−	0.982325i	\(-0.440065\pi\)
							−0.876405	+	0.481575i	\(0.840065\pi\)
\(13\)	6810.47	+	9373.80i	0.859756	+	1.18335i	0.981628	+	0.190807i	\(0.0611104\pi\)
							−0.121872	+	0.992546i	\(0.538890\pi\)
\(17\)	−26344.0	−	8559.68i	−1.30050	−	0.422558i	−0.424745	−	0.905313i	\(-0.639636\pi\)
							−0.875755	+	0.482755i	\(0.839636\pi\)
\(19\)	−8418.59	+	25909.8i	−0.281580	+	0.866614i	0.705823	+	0.708388i	\(0.250577\pi\)
							−0.987403	+	0.158226i	\(0.949423\pi\)
\(23\)	30112.4	−	41446.2i	0.516058	−	0.710292i	−0.468869	−	0.883268i	\(-0.655338\pi\)
							0.984926	+	0.172976i	\(0.0553382\pi\)
\(29\)	37761.4	+	116218.i	0.287512	+	0.884870i	0.985635	+	0.168892i	\(0.0540189\pi\)
							−0.698123	+	0.715978i	\(0.745981\pi\)
\(31\)	6233.40	−	19184.4i	0.0375802	−	0.115660i	−0.930507	−	0.366275i	\(-0.880633\pi\)
							0.968087	+	0.250615i	\(0.0806329\pi\)
\(37\)	13816.7	+	19017.1i	0.0448435	+	0.0617218i	0.830850	−	0.556497i	\(-0.187855\pi\)
							−0.786006	+	0.618218i	\(0.787855\pi\)
\(41\)	592664.	−	430596.i	1.34297	−	0.975722i	0.343637	−	0.939102i	\(-0.388341\pi\)
							0.999329	−	0.0366199i	\(-0.0116591\pi\)
\(43\)		−	98732.1i		−	0.189373i	−0.995507	−	0.0946867i	\(-0.969815\pi\)
							0.995507	−	0.0946867i	\(-0.0301849\pi\)
\(47\)	401164.	−	130346.i	0.563611	−	0.183128i	−0.0133345	−	0.999911i	\(-0.504245\pi\)
							0.576945	+	0.816783i	\(0.304245\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.e.a.9.1	✓	64
25.8	odd	20		625.8.a.g.1.3		64
25.14	even	10	inner	25.8.e.a.14.1	yes	64
25.17	odd	20		625.8.a.g.1.62		64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.e.a.9.1	✓	64	1.1	even	1	trivial
25.8.e.a.14.1	yes	64	25.14	even	10	inner
625.8.a.g.1.3		64	25.8	odd	20
625.8.a.g.1.62		64	25.17	odd	20