Embedded newform 25.7.f.a.2.7

• Label: 25.7.f.a.2.7
• Level: $25$
• Weight: $7$
• Character: 25.2
• Analytic conductor: $5.751$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.75135209050\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			2.7
Character	\(\chi\)	\(=\)	25.2
Dual form			25.7.f.a.13.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86717 - 0.295730i) q^{2} +(-17.0200 - 33.4037i) q^{3} +(-57.4688 - 18.6727i) q^{4} +(17.1804 + 123.814i) q^{5} +(21.9008 + 67.4036i) q^{6} +(280.929 + 280.929i) q^{7} +(209.583 + 106.788i) q^{8} +(-397.628 + 547.288i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 40 q^{3} - 10 q^{4} + 60 q^{5} - 6 q^{6} - 560 q^{7} - 1870 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86717	−	0.295730i	−0.233396	−	0.0369663i	0.0386407	−	0.999253i	\(-0.487697\pi\)
							−0.272037	+	0.962287i	\(0.587697\pi\)
\(3\)	−17.0200	−	33.4037i	−0.630371	−	1.23717i	−0.956468	−	0.291836i	\(-0.905734\pi\)
							0.326097	−	0.945336i	\(-0.394266\pi\)
\(5\)	17.1804	+	123.814i	0.137443	+	0.990510i
							\(7\)	280.929	+	280.929i	0.819034	+	0.819034i	0.985968	−	0.166934i	\(-0.0533868\pi\)
−0.166934	+	0.985968i	\(0.553387\pi\)
\(11\)	−87.4690	+	63.5500i	−0.0657168	+	0.0477460i	−0.620159	−	0.784476i	\(-0.712932\pi\)
							0.554442	+	0.832222i	\(0.312932\pi\)
\(13\)	−3824.07	+	605.674i	−1.74059	+	0.275682i	−0.944264	−	0.329188i	\(-0.893225\pi\)
							−0.796324	+	0.604870i	\(0.793225\pi\)
\(17\)	−1604.54	+	3149.09i	−0.326591	+	0.640971i	−0.994669	−	0.103120i	\(-0.967117\pi\)
							0.668078	+	0.744091i	\(0.267117\pi\)
\(19\)	−2537.98	+	824.640i	−0.370022	+	0.120227i	−0.488125	−	0.872774i	\(-0.662319\pi\)
							0.118103	+	0.993001i	\(0.462319\pi\)
\(23\)	−765.472	+	4833.00i	−0.0629138	+	0.397222i	0.936056	+	0.351850i	\(0.114447\pi\)
							−0.998970	+	0.0453720i	\(0.985553\pi\)
\(29\)	24734.1	+	8036.60i	1.01415	+	0.329517i	0.768506	−	0.639843i	\(-0.221000\pi\)
							0.245644	+	0.969360i	\(0.421000\pi\)
\(31\)	7121.35	+	21917.3i	0.239044	+	0.735700i	0.996559	+	0.0828835i	\(0.0264129\pi\)
							−0.757516	+	0.652817i	\(0.773587\pi\)
\(37\)	5145.68	+	32488.5i	0.101587	+	0.641394i	0.984968	+	0.172738i	\(0.0552615\pi\)
							−0.883381	+	0.468656i	\(0.844738\pi\)
\(41\)	−99083.3	−	71988.2i	−1.43764	−	1.04450i	−0.988530	−	0.151025i	\(-0.951743\pi\)
							−0.449105	−	0.893479i	\(-0.648257\pi\)
\(43\)	31494.9	−	31494.9i	0.396127	−	0.396127i	−0.480737	−	0.876865i	\(-0.659631\pi\)
							0.876865	+	0.480737i	\(0.159631\pi\)
\(47\)	−152024.	+	77460.1i	−1.46426	+	0.746078i	−0.990876	−	0.134778i	\(-0.956968\pi\)
							−0.473385	+	0.880856i	\(0.656968\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.7.f.a.2.7	✓	112
25.13	odd	20	inner	25.7.f.a.13.7	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.7.f.a.2.7	✓	112	1.1	even	1	trivial
25.7.f.a.13.7	yes	112	25.13	odd	20	inner