Embedded newform 400.2.bm.a.123.39

• Label: 400.2.bm.a.123.39
• Level: $400$
• Weight: $2$
• Character: 400.123
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bm (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.19401608085\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(58\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			123.39
Character	\(\chi\)	\(=\)	400.123
Dual form			400.2.bm.a.387.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.608428 - 1.27664i) q^{2} +(3.11215 + 1.01120i) q^{3} +(-1.25963 - 1.55349i) q^{4} +(0.0755856 - 2.23479i) q^{5} +(3.18446 - 3.35786i) q^{6} +(-1.39631 + 1.39631i) q^{7} +(-2.74965 + 0.662909i) q^{8} +(6.23591 + 4.53066i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.608428	−	1.27664i	0.430224	−	0.902722i
							\(3\)	3.11215	+	1.01120i	1.79680	+	0.583816i	0.999797	−	0.0201570i	\(-0.00641660\pi\)
0.797005	+	0.603973i	\(0.206417\pi\)
\(5\)	0.0755856	−	2.23479i	0.0338029	−	0.999429i
							\(7\)	−1.39631	+	1.39631i	−0.527756	+	0.527756i	−0.919903	−	0.392147i	\(-0.871733\pi\)
0.392147	+	0.919903i	\(0.371733\pi\)
\(11\)	0.459641	−	2.90206i	0.138587	−	0.875005i	−0.816212	−	0.577753i	\(-0.803930\pi\)
							0.954799	−	0.297252i	\(-0.0960702\pi\)
\(13\)	1.35193	+	0.982238i	0.374959	+	0.272424i	0.759264	−	0.650782i	\(-0.225559\pi\)
							−0.384305	+	0.923206i	\(0.625559\pi\)
\(17\)	−3.30404	−	1.68349i	−0.801347	−	0.408306i	0.00482500	−	0.999988i	\(-0.498464\pi\)
							−0.806172	+	0.591682i	\(0.798464\pi\)
\(19\)	5.31606	+	2.70867i	1.21959	+	0.621411i	0.940807	−	0.338944i	\(-0.110070\pi\)
							0.278781	+	0.960355i	\(0.410070\pi\)
\(23\)	−6.20311	−	0.982475i	−1.29344	−	0.204860i	−0.528482	−	0.848944i	\(-0.677239\pi\)
							−0.764955	+	0.644084i	\(0.777239\pi\)
\(29\)	−6.23497	+	3.17688i	−1.15780	+	0.589931i	−0.924015	−	0.382356i	\(-0.875113\pi\)
							−0.233790	+	0.972287i	\(0.575113\pi\)
\(31\)	3.15378	−	1.02473i	0.566436	−	0.184046i	−0.0117791	−	0.999931i	\(-0.503750\pi\)
							0.578215	+	0.815884i	\(0.303750\pi\)
\(37\)	3.86968	+	2.81148i	0.636171	+	0.462205i	0.858533	−	0.512759i	\(-0.171377\pi\)
							−0.222362	+	0.974964i	\(0.571377\pi\)
\(41\)	−4.18238	+	5.75655i	−0.653178	+	0.899023i	−0.999232	−	0.0391920i	\(-0.987522\pi\)
							0.346053	+	0.938215i	\(0.387522\pi\)
\(43\)	4.66764			0.711809			0.355904	−	0.934522i	\(-0.384173\pi\)
							0.355904	+	0.934522i	\(0.384173\pi\)
\(47\)	−2.15928	+	1.10021i	−0.314964	+	0.160482i	−0.604325	−	0.796738i	\(-0.706557\pi\)
							0.289361	+	0.957220i	\(0.406557\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.123.39	yes	464
16.3	odd	4		400.2.bd.a.323.27	yes	464
25.12	odd	20		400.2.bd.a.187.27	✓	464
400.387	even	20	inner	400.2.bm.a.387.39	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.187.27	✓	464	25.12	odd	20
400.2.bd.a.323.27	yes	464	16.3	odd	4
400.2.bm.a.123.39	yes	464	1.1	even	1	trivial
400.2.bm.a.387.39	yes	464	400.387	even	20	inner