Embedded newform 25.7.f.a.3.2

• Label: 25.7.f.a.3.2
• Level: $25$
• Weight: $7$
• Character: 25.3
• 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			3.2
Character	\(\chi\)	\(=\)	25.3
Dual form			25.7.f.a.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.01647 - 11.8080i) q^{2} +(-4.30463 - 27.1784i) q^{3} +(-65.6123 + 90.3075i) q^{4} +(-122.996 - 22.2949i) q^{5} +(-295.023 + 214.347i) q^{6} +(251.807 - 251.807i) q^{7} +(623.392 + 98.7355i) q^{8} +(-26.8129 + 8.71203i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 40 q^{3} - 10 q^{4} + 60 q^{5} - 6 q^{6} - 560 q^{7} - 1870 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.01647	−	11.8080i	−0.752058	−	1.47600i	−0.875279	−	0.483618i	\(-0.839322\pi\)
							0.123221	−	0.992379i	\(-0.460678\pi\)
\(3\)	−4.30463	−	27.1784i	−0.159431	−	1.00661i	−0.929547	−	0.368703i	\(-0.879802\pi\)
							0.770117	−	0.637903i	\(-0.220198\pi\)
\(5\)	−122.996	−	22.2949i	−0.983965	−	0.178359i
							\(7\)	251.807	−	251.807i	0.734130	−	0.734130i	−0.237305	−	0.971435i	\(-0.576264\pi\)
0.971435	+	0.237305i	\(0.0762641\pi\)
\(11\)	−349.328	+	1075.12i	−0.262455	+	0.807753i	0.729814	+	0.683646i	\(0.239607\pi\)
							−0.992269	+	0.124107i	\(0.960393\pi\)
\(13\)	643.141	−	1262.24i	0.292736	−	0.574527i	−0.697061	−	0.717012i	\(-0.745509\pi\)
							0.989797	+	0.142485i	\(0.0455093\pi\)
\(17\)	−428.136	+	2703.15i	−0.0871436	+	0.550203i	0.905031	+	0.425345i	\(0.139847\pi\)
							−0.992175	+	0.124857i	\(0.960153\pi\)
\(19\)	−6080.10	−	8368.54i	−0.886442	−	1.22008i	−0.974595	−	0.223975i	\(-0.928097\pi\)
							0.0881532	−	0.996107i	\(-0.471903\pi\)
\(23\)	−21068.7	+	10735.0i	−1.73162	+	0.882306i	−0.758614	+	0.651541i	\(0.774123\pi\)
							−0.973009	+	0.230766i	\(0.925877\pi\)
\(29\)	−14172.3	+	19506.5i	−0.581094	+	0.799807i	−0.993815	−	0.111050i	\(-0.964579\pi\)
							0.412721	+	0.910858i	\(0.364579\pi\)
\(31\)	14825.1	−	10771.1i	0.497637	−	0.361555i	−0.310477	−	0.950581i	\(-0.600489\pi\)
							0.808114	+	0.589026i	\(0.200489\pi\)
\(37\)	1583.30	+	806.730i	0.0312577	+	0.0159266i	0.469550	−	0.882906i	\(-0.344416\pi\)
							−0.438292	+	0.898833i	\(0.644416\pi\)
\(41\)	−9070.20	−	27915.2i	−0.131603	−	0.405032i	0.863443	−	0.504446i	\(-0.168303\pi\)
							−0.995046	+	0.0994139i	\(0.968303\pi\)
\(43\)	−84932.0	−	84932.0i	−1.06823	−	1.06823i	−0.997495	−	0.0707382i	\(-0.977465\pi\)
							−0.0707382	−	0.997495i	\(-0.522535\pi\)
\(47\)	166652.	−	26395.1i	1.60516	−	0.254232i	0.711401	−	0.702787i	\(-0.248061\pi\)
							0.893755	+	0.448555i	\(0.148061\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.7.f.a.3.2	✓	112
25.17	odd	20	inner	25.7.f.a.17.2	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.7.f.a.3.2	✓	112	1.1	even	1	trivial
25.7.f.a.17.2	yes	112	25.17	odd	20	inner