Embedded newform 25.6.d.a.6.8

• Label: 25.6.d.a.6.8
• Level: $25$
• Weight: $6$
• Character: 25.6
• Analytic conductor: $4.010$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• 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			6.8
Character	\(\chi\)	\(=\)	25.6
Dual form			25.6.d.a.21.8

$q$-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 \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.d.a.6.8	✓	44	1.1	even	1	trivial
25.6.d.a.21.8	yes	44	25.21	even	5	inner
625.6.a.c.1.8		22	25.14	even	10
625.6.a.d.1.15		22	25.11	even	5