Embedded newform 400.2.bd.a.3.58

• Label: 400.2.bd.a.3.58
• Level: $400$
• Weight: $2$
• Character: 400.3
• 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.bd (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			3.58
Character	\(\chi\)	\(=\)	400.3
Dual form			400.2.bd.a.267.58

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41376 - 0.0358778i) q^{2} +(1.54763 + 1.12442i) q^{3} +(1.99743 - 0.101445i) q^{4} +(-0.00959339 - 2.23605i) q^{5} +(2.22832 + 1.53413i) q^{6} +(-1.00651 + 1.00651i) q^{7} +(2.82024 - 0.215082i) q^{8} +(0.203793 + 0.627210i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 6 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41376	−	0.0358778i	0.999678	−	0.0253694i
							\(3\)	1.54763	+	1.12442i	0.893526	+	0.649185i	0.936795	−	0.349879i	\(-0.113777\pi\)
−0.0432690	+	0.999063i	\(0.513777\pi\)
\(5\)	−0.00959339	−	2.23605i	−0.00429029	−	0.999991i
							\(7\)	−1.00651	+	1.00651i	−0.380425	+	0.380425i	−0.871255	−	0.490830i	\(-0.836694\pi\)
0.490830	+	0.871255i	\(0.336694\pi\)
\(11\)	1.22752	−	0.625454i	0.370112	−	0.188582i	−0.259035	−	0.965868i	\(-0.583404\pi\)
							0.629147	+	0.777286i	\(0.283404\pi\)
\(13\)	−4.81552	+	1.56466i	−1.33558	+	0.433958i	−0.887819	−	0.460194i	\(-0.847780\pi\)
							−0.447765	+	0.894151i	\(0.647780\pi\)
\(17\)	−0.345943	+	2.18420i	−0.0839035	+	0.529746i	0.909558	+	0.415577i	\(0.136420\pi\)
							−0.993461	+	0.114168i	\(0.963580\pi\)
\(19\)	0.843793	+	0.133644i	0.193579	+	0.0306600i	0.252471	−	0.967604i	\(-0.418757\pi\)
							−0.0588919	+	0.998264i	\(0.518757\pi\)
\(23\)	−3.78803	+	1.93010i	−0.789859	+	0.402453i	−0.801888	−	0.597474i	\(-0.796171\pi\)
							0.0120290	+	0.999928i	\(0.496171\pi\)
\(29\)	−2.89240	+	0.458110i	−0.537104	+	0.0850690i	−0.419093	−	0.907943i	\(-0.637652\pi\)
							−0.118011	+	0.993012i	\(0.537652\pi\)
\(31\)	2.14203	+	2.94825i	0.384719	+	0.529521i	0.956827	−	0.290658i	\(-0.0938741\pi\)
							−0.572108	+	0.820178i	\(0.693874\pi\)
\(37\)	−7.25106	+	2.35601i	−1.19207	+	0.387326i	−0.836837	−	0.547453i	\(-0.815598\pi\)
							−0.355230	+	0.934779i	\(0.615598\pi\)
\(41\)	5.88085	−	1.91080i	0.918434	−	0.298417i	0.188610	−	0.982052i	\(-0.439602\pi\)
							0.729824	+	0.683635i	\(0.239602\pi\)
\(43\)			2.93796i			0.448034i	0.974585	+	0.224017i	\(0.0719172\pi\)
							−0.974585	+	0.224017i	\(0.928083\pi\)
\(47\)	−1.11002	−	7.00837i	−0.161912	−	1.02228i	−0.926099	−	0.377280i	\(-0.876859\pi\)
							0.764187	−	0.644995i	\(-0.223141\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bd.a.3.58	✓	464
16.11	odd	4		400.2.bm.a.203.17	yes	464
25.17	odd	20		400.2.bm.a.67.17	yes	464
400.267	even	20	inner	400.2.bd.a.267.58	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.3.58	✓	464	1.1	even	1	trivial
400.2.bd.a.267.58	yes	464	400.267	even	20	inner
400.2.bm.a.67.17	yes	464	25.17	odd	20
400.2.bm.a.203.17	yes	464	16.11	odd	4