Embedded newform 25.7.f.a.2.1

• Label: 25.7.f.a.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	25.2
Dual form			25.7.f.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.1993 - 2.24894i) q^{2} +(-7.91375 - 15.5316i) q^{3} +(135.693 + 44.0895i) q^{4} +(119.661 + 36.1404i) q^{5} +(77.4397 + 238.335i) q^{6} +(-114.344 - 114.344i) q^{7} +(-1007.79 - 513.497i) q^{8} +(249.892 - 343.947i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 40 q^{3} - 10 q^{4} + 60 q^{5} - 6 q^{6} - 560 q^{7} - 1870 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.1993	−	2.24894i	−1.77491	−	0.281118i	−0.818788	−	0.574096i	\(-0.805354\pi\)
							−0.956119	+	0.292979i	\(0.905354\pi\)
\(3\)	−7.91375	−	15.5316i	−0.293102	−	0.575245i	0.696756	−	0.717309i	\(-0.254626\pi\)
							−0.989857	+	0.142064i	\(0.954626\pi\)
\(5\)	119.661	+	36.1404i	0.957292	+	0.289123i
							\(7\)	−114.344	−	114.344i	−0.333366	−	0.333366i	0.520497	−	0.853863i	\(-0.325747\pi\)
−0.853863	+	0.520497i	\(0.825747\pi\)
\(11\)	314.184	−	228.268i	0.236051	−	0.171501i	−0.463471	−	0.886112i	\(-0.653396\pi\)
							0.699522	+	0.714611i	\(0.253396\pi\)
\(13\)	−2865.14	+	453.794i	−1.30411	+	0.206551i	−0.769556	−	0.638580i	\(-0.779522\pi\)
							−0.534559	+	0.845131i	\(0.679522\pi\)
\(17\)	2955.44	−	5800.38i	0.601555	−	1.18062i	−0.366626	−	0.930369i	\(-0.619487\pi\)
							0.968181	−	0.250250i	\(-0.0805130\pi\)
\(19\)	−5848.14	+	1900.18i	−0.852624	+	0.277034i	−0.702544	−	0.711640i	\(-0.747953\pi\)
							−0.150079	+	0.988674i	\(0.547953\pi\)
\(23\)	2705.49	−	17081.8i	0.222363	−	1.40394i	−0.583631	−	0.812019i	\(-0.698368\pi\)
							0.805993	−	0.591924i	\(-0.201632\pi\)
\(29\)	−25553.8	−	8302.93i	−1.04776	−	0.340437i	−0.265970	−	0.963981i	\(-0.585692\pi\)
							−0.781788	+	0.623544i	\(0.785692\pi\)
\(31\)	−18206.1	−	56032.6i	−0.611128	−	1.88086i	−0.447344	−	0.894362i	\(-0.647630\pi\)
							−0.163784	−	0.986496i	\(-0.552370\pi\)
\(37\)	5202.17	+	32845.2i	0.102702	+	0.648436i	0.984309	+	0.176452i	\(0.0564619\pi\)
							−0.881607	+	0.471984i	\(0.843538\pi\)
\(41\)	61282.7	+	44524.5i	0.889174	+	0.646023i	0.935662	−	0.352896i	\(-0.114803\pi\)
							−0.0464887	+	0.998919i	\(0.514803\pi\)
\(43\)	−44180.6	+	44180.6i	−0.555682	+	0.555682i	−0.928075	−	0.372393i	\(-0.878537\pi\)
							0.372393	+	0.928075i	\(0.378537\pi\)
\(47\)	59806.1	−	30472.7i	0.576039	−	0.293506i	−0.141598	−	0.989924i	\(-0.545224\pi\)
							0.717637	+	0.696418i	\(0.245224\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.7.f.a.2.1	✓	112
25.13	odd	20	inner	25.7.f.a.13.1	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.7.f.a.2.1	✓	112	1.1	even	1	trivial
25.7.f.a.13.1	yes	112	25.13	odd	20	inner