Embedded newform 400.2.bd.a.267.58

• Label: 400.2.bd.a.267.58
• Level: $400$
• Weight: $2$
• Character: 400.267
• 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			267.58
Character	\(\chi\)	\(=\)	400.267
Dual form			400.2.bd.a.3.58

$q$-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 \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.3.58	✓	464	400.3	even	20	inner
400.2.bd.a.267.58	yes	464	1.1	even	1	trivial
400.2.bm.a.67.17	yes	464	16.3	odd	4
400.2.bm.a.203.17	yes	464	25.3	odd	20