Embedded newform 400.2.bm.a.67.17

• Label: 400.2.bm.a.67.17
• Level: $400$
• Weight: $2$
• Character: 400.67
• 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			67.17
Character	\(\chi\)	\(=\)	400.67
Dual form			400.2.bm.a.203.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.860012 - 1.12267i) q^{2} +(-1.12442 - 1.54763i) q^{3} +(-0.520759 + 1.93101i) q^{4} +(2.23605 + 0.00959339i) q^{5} +(-0.770459 + 2.59333i) q^{6} +(-1.00651 - 1.00651i) q^{7} +(2.61574 - 1.07606i) q^{8} +(-0.203793 + 0.627210i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.860012	−	1.12267i	−0.608120	−	0.793845i
							\(3\)	−1.12442	−	1.54763i	−0.649185	−	0.893526i	0.349879	−	0.936795i	\(-0.386223\pi\)
−0.999063	+	0.0432690i	\(0.986223\pi\)
\(5\)	2.23605	+	0.00959339i	0.999991	+	0.00429029i
							\(7\)	−1.00651	−	1.00651i	−0.380425	−	0.380425i	0.490830	−	0.871255i	\(-0.336694\pi\)
−0.871255	+	0.490830i	\(0.836694\pi\)
\(11\)	−0.625454	+	1.22752i	−0.188582	+	0.370112i	−0.965868	−	0.259035i	\(-0.916596\pi\)
							0.777286	+	0.629147i	\(0.216596\pi\)
\(13\)	1.56466	−	4.81552i	0.433958	−	1.33558i	−0.460194	−	0.887819i	\(-0.652220\pi\)
							0.894151	−	0.447765i	\(-0.147780\pi\)
\(17\)	−0.345943	−	2.18420i	−0.0839035	−	0.529746i	−0.993461	−	0.114168i	\(-0.963580\pi\)
							0.909558	−	0.415577i	\(-0.136420\pi\)
\(19\)	−0.133644	−	0.843793i	−0.0306600	−	0.193579i	0.967604	−	0.252471i	\(-0.0812433\pi\)
							−0.998264	+	0.0588919i	\(0.981243\pi\)
\(23\)	−3.78803	−	1.93010i	−0.789859	−	0.402453i	0.0120290	−	0.999928i	\(-0.496171\pi\)
							−0.801888	+	0.597474i	\(0.796171\pi\)
\(29\)	0.458110	−	2.89240i	0.0850690	−	0.537104i	−0.907943	−	0.419093i	\(-0.862348\pi\)
							0.993012	−	0.118011i	\(-0.0376519\pi\)
\(31\)	−2.14203	+	2.94825i	−0.384719	+	0.529521i	−0.956827	−	0.290658i	\(-0.906126\pi\)
							0.572108	+	0.820178i	\(0.306126\pi\)
\(37\)	−2.35601	+	7.25106i	−0.387326	+	1.19207i	0.547453	+	0.836837i	\(0.315598\pi\)
							−0.934779	+	0.355230i	\(0.884402\pi\)
\(41\)	−5.88085	−	1.91080i	−0.918434	−	0.298417i	−0.188610	−	0.982052i	\(-0.560398\pi\)
							−0.729824	+	0.683635i	\(0.760398\pi\)
\(43\)	2.93796			0.448034			0.224017	−	0.974585i	\(-0.428083\pi\)
							0.224017	+	0.974585i	\(0.428083\pi\)
\(47\)	1.11002	−	7.00837i	0.161912	−	1.02228i	−0.764187	−	0.644995i	\(-0.776859\pi\)
							0.926099	−	0.377280i	\(-0.123141\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.3.58	✓	464	25.3	odd	20
400.2.bd.a.267.58	yes	464	16.11	odd	4
400.2.bm.a.67.17	yes	464	1.1	even	1	trivial
400.2.bm.a.203.17	yes	464	400.203	even	20	inner