Embedded newform 25.7.f.a.2.10

• Label: 25.7.f.a.2.10
• 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			2.10
Character	\(\chi\)	\(=\)	25.2
Dual form			25.7.f.a.13.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.09592 + 1.12388i) q^{2} +(11.5411 + 22.6507i) q^{3} +(-11.7786 - 3.82710i) q^{4} +(2.31555 + 124.979i) q^{5} +(56.4381 + 173.699i) q^{6} +(273.850 + 273.850i) q^{7} +(-488.964 - 249.139i) q^{8} +(48.6381 - 66.9445i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 40 q^{3} - 10 q^{4} + 60 q^{5} - 6 q^{6} - 560 q^{7} - 1870 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.09592	+	1.12388i	0.886990	+	0.140485i	0.583273	−	0.812277i	\(-0.301772\pi\)
							0.303718	+	0.952762i	\(0.401772\pi\)
\(3\)	11.5411	+	22.6507i	0.427449	+	0.838915i	0.999821	+	0.0189214i	\(0.00602321\pi\)
							−0.572372	+	0.819994i	\(0.693977\pi\)
\(5\)	2.31555	+	124.979i	0.0185244	+	0.999828i
							\(7\)	273.850	+	273.850i	0.798397	+	0.798397i	0.982843	−	0.184445i	\(-0.0590489\pi\)
−0.184445	+	0.982843i	\(0.559049\pi\)
\(11\)	−612.645	+	445.113i	−0.460289	+	0.334420i	−0.793645	−	0.608381i	\(-0.791819\pi\)
							0.333355	+	0.942801i	\(0.391819\pi\)
\(13\)	3253.33	−	515.277i	1.48081	−	0.234537i	0.636867	−	0.770974i	\(-0.280230\pi\)
							0.843940	+	0.536437i	\(0.180230\pi\)
\(17\)	1026.41	−	2014.45i	0.208918	−	0.410024i	−0.762641	−	0.646822i	\(-0.776098\pi\)
							0.971559	+	0.236798i	\(0.0760978\pi\)
\(19\)	24.7824	−	8.05230i	0.00361313	−	0.00117398i	−0.307210	−	0.951642i	\(-0.599395\pi\)
							0.310823	+	0.950468i	\(0.399395\pi\)
\(23\)	3491.75	−	22046.0i	0.286985	−	1.81195i	−0.249871	−	0.968279i	\(-0.580388\pi\)
							0.536856	−	0.843674i	\(-0.319612\pi\)
\(29\)	−8094.08	−	2629.93i	−0.331874	−	0.107833i	0.138340	−	0.990385i	\(-0.455823\pi\)
							−0.470214	+	0.882552i	\(0.655823\pi\)
\(31\)	−6623.35	−	20384.6i	−0.222327	−	0.684253i	−0.998552	−	0.0537964i	\(-0.982868\pi\)
							0.776225	−	0.630456i	\(-0.217132\pi\)
\(37\)	11699.5	+	73867.8i	0.230974	+	1.45831i	0.781716	+	0.623635i	\(0.214345\pi\)
							−0.550742	+	0.834676i	\(0.685655\pi\)
\(41\)	74028.4	+	53784.8i	1.07411	+	0.780383i	0.976646	−	0.214856i	\(-0.0689282\pi\)
							0.0974598	+	0.995239i	\(0.468928\pi\)
\(43\)	−55767.5	+	55767.5i	−0.701417	+	0.701417i	−0.964715	−	0.263298i	\(-0.915190\pi\)
							0.263298	+	0.964715i	\(0.415190\pi\)
\(47\)	−55530.8	+	28294.3i	−0.534860	+	0.272525i	−0.700489	−	0.713663i	\(-0.747035\pi\)
							0.165629	+	0.986188i	\(0.447035\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.7.f.a.2.10	✓	112
25.13	odd	20	inner	25.7.f.a.13.10	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.7.f.a.2.10	✓	112	1.1	even	1	trivial
25.7.f.a.13.10	yes	112	25.13	odd	20	inner