Embedded newform 25.6.d.a.21.8

• Label: 25.6.d.a.21.8
• Level: $25$
• Weight: $6$
• Character: 25.21
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $44$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00959549532\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			21.8
Character	\(\chi\)	\(=\)	25.21
Dual form			25.6.d.a.6.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.31554 - 4.04881i) q^{2} +(13.2842 + 9.65154i) q^{3} +(11.2263 + 8.15638i) q^{4} +(-5.21916 + 55.6575i) q^{5} +(56.5532 - 41.0883i) q^{6} -22.2636 q^{7} +(158.004 - 114.797i) q^{8} +(8.22668 + 25.3191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 7 q^{2} - q^{3} - 147 q^{4} + 115 q^{5} + 133 q^{6} - 202 q^{7} - 500 q^{8} - 508 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.31554	−	4.04881i	0.232557	−	0.715736i	−0.764879	−	0.644174i	\(-0.777201\pi\)
							0.997436	−	0.0715625i	\(-0.0227985\pi\)
\(3\)	13.2842	+	9.65154i	0.852182	+	0.619146i	0.925747	−	0.378144i	\(-0.123438\pi\)
							−0.0735650	+	0.997290i	\(0.523438\pi\)
\(5\)	−5.21916	+	55.6575i	−0.0933631	+	0.995632i
							\(7\)	−22.2636			−0.171732			−0.0858660	−	0.996307i	\(-0.527366\pi\)
−0.0858660	+	0.996307i	\(0.527366\pi\)
\(11\)	94.4833	−	290.790i	0.235436	−	0.724598i	−0.761627	−	0.648016i	\(-0.775599\pi\)
							0.997063	−	0.0765825i	\(-0.0244009\pi\)
\(13\)	−123.394	−	379.767i	−0.202504	−	0.623244i	−0.999807	−	0.0196641i	\(-0.993740\pi\)
							0.797302	−	0.603580i	\(-0.206260\pi\)
\(17\)	−1240.73	+	901.440i	−1.04125	+	0.756510i	−0.970528	−	0.240987i	\(-0.922529\pi\)
							−0.0707180	+	0.997496i	\(0.522529\pi\)
\(19\)	−33.3183	+	24.2072i	−0.0211738	+	0.0153837i	−0.598322	−	0.801256i	\(-0.704166\pi\)
							0.577148	+	0.816640i	\(0.304166\pi\)
\(23\)	770.352	−	2370.90i	0.303648	−	0.934531i	−0.676531	−	0.736414i	\(-0.736517\pi\)
							0.980178	−	0.198117i	\(-0.0634825\pi\)
\(29\)	5283.57	+	3838.74i	1.16663	+	0.847604i	0.990601	−	0.136781i	\(-0.0436757\pi\)
							0.176026	+	0.984386i	\(0.443676\pi\)
\(31\)	−5368.36	+	3900.34i	−1.00332	+	0.728951i	−0.962797	−	0.270227i	\(-0.912901\pi\)
							−0.0405189	+	0.999179i	\(0.512901\pi\)
\(37\)	−3501.73	−	10777.2i	−0.420512	−	1.29420i	−0.907227	−	0.420642i	\(-0.861805\pi\)
							0.486715	−	0.873561i	\(-0.338195\pi\)
\(41\)	2279.55	+	7015.73i	0.211782	+	0.651798i	0.999366	+	0.0355914i	\(0.0113315\pi\)
							−0.787584	+	0.616207i	\(0.788669\pi\)
\(43\)	547.836			0.0451834			0.0225917	−	0.999745i	\(-0.492808\pi\)
							0.0225917	+	0.999745i	\(0.492808\pi\)
\(47\)	−18377.2	−	13351.8i	−1.21349	−	0.881650i	−0.217944	−	0.975961i	\(-0.569935\pi\)
							−0.995543	+	0.0943112i	\(0.969935\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.d.a.21.8	yes	44
25.6	even	5	inner	25.6.d.a.6.8	✓	44
25.9	even	10		625.6.a.c.1.8		22
25.16	even	5		625.6.a.d.1.15		22

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.6.8	✓	44	25.6	even	5	inner
25.6.d.a.21.8	yes	44	1.1	even	1	trivial
625.6.a.c.1.8		22	25.9	even	10
625.6.a.d.1.15		22	25.16	even	5