Embedded newform 400.2.bm.a.147.12

• Label: 400.2.bm.a.147.12
• Level: $400$
• Weight: $2$
• Character: 400.147
• 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			147.12
Character	\(\chi\)	\(=\)	400.147
Dual form			400.2.bm.a.283.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13487 + 0.843846i) q^{2} +(-0.244928 + 0.337115i) q^{3} +(0.575847 - 1.91531i) q^{4} +(2.16963 - 0.541009i) q^{5} +(-0.00651207 - 0.589263i) q^{6} +(-1.90565 - 1.90565i) q^{7} +(0.962715 + 2.65955i) q^{8} +(0.873394 + 2.68803i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{17}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13487	+	0.843846i	−0.802472	+	0.596689i
							\(3\)	−0.244928	+	0.337115i	−0.141409	+	0.194633i	−0.873847	−	0.486200i	\(-0.838382\pi\)
0.732438	+	0.680834i	\(0.238382\pi\)
\(5\)	2.16963	−	0.541009i	0.970290	−	0.241947i
							\(7\)	−1.90565	−	1.90565i	−0.720268	−	0.720268i	0.248391	−	0.968660i	\(-0.420098\pi\)
−0.968660	+	0.248391i	\(0.920098\pi\)
\(11\)	1.58358	−	0.806873i	0.477466	−	0.243281i	−0.198658	−	0.980069i	\(-0.563658\pi\)
							0.676124	+	0.736788i	\(0.263658\pi\)
\(13\)	−1.93548	−	5.95679i	−0.536805	−	1.65212i	−0.739717	−	0.672919i	\(-0.765040\pi\)
							0.202912	−	0.979197i	\(-0.434960\pi\)
\(17\)	4.90475	+	0.776836i	1.18958	+	0.188410i	0.719677	−	0.694309i	\(-0.244290\pi\)
							0.469900	+	0.882720i	\(0.344290\pi\)
\(19\)	1.47454	+	0.233545i	0.338284	+	0.0535788i	0.323265	−	0.946308i	\(-0.395219\pi\)
							0.0150182	+	0.999887i	\(0.495219\pi\)
\(23\)	−3.65143	−	7.16634i	−0.761377	−	1.49429i	−0.866147	−	0.499790i	\(-0.833411\pi\)
							0.104770	−	0.994496i	\(-0.466589\pi\)
\(29\)	4.21069	−	0.666908i	0.781906	−	0.123842i	0.247295	−	0.968940i	\(-0.420458\pi\)
							0.534611	+	0.845098i	\(0.320458\pi\)
\(31\)	3.43449	+	4.72717i	0.616852	+	0.849025i	0.997119	−	0.0758523i	\(-0.0241678\pi\)
							−0.380267	+	0.924877i	\(0.624168\pi\)
\(37\)	2.25019	+	6.92538i	0.369929	+	1.13853i	0.946836	+	0.321716i	\(0.104259\pi\)
							−0.576907	+	0.816810i	\(0.695741\pi\)
\(41\)	1.69093	−	0.549415i	0.264078	−	0.0858042i	−0.173985	−	0.984748i	\(-0.555665\pi\)
							0.438063	+	0.898944i	\(0.355665\pi\)
\(43\)	7.79633			1.18893			0.594465	−	0.804122i	\(-0.297364\pi\)
							0.594465	+	0.804122i	\(0.297364\pi\)
\(47\)	−1.98824	+	0.314906i	−0.290014	+	0.0459337i	−0.299746	−	0.954019i	\(-0.596902\pi\)
							0.00973226	+	0.999953i	\(0.496902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.147.12	yes	464
16.11	odd	4		400.2.bd.a.347.6	yes	464
25.8	odd	20		400.2.bd.a.83.6	✓	464
400.283	even	20	inner	400.2.bm.a.283.12	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.83.6	✓	464	25.8	odd	20
400.2.bd.a.347.6	yes	464	16.11	odd	4
400.2.bm.a.147.12	yes	464	1.1	even	1	trivial
400.2.bm.a.283.12	yes	464	400.283	even	20	inner