Embedded newform 400.2.bm.a.147.39

• Label: 400.2.bm.a.147.39
• 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			147.39
Character	\(\chi\)	\(=\)	400.147
Dual form			400.2.bm.a.283.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.561429 + 1.29800i) q^{2} +(0.400074 - 0.550655i) q^{3} +(-1.36959 + 1.45747i) q^{4} +(-2.02565 - 0.946960i) q^{5} +(0.939363 + 0.210141i) q^{6} +(-3.10246 - 3.10246i) q^{7} +(-2.66072 - 0.959464i) q^{8} +(0.783889 + 2.41256i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.561429	+	1.29800i	0.396991	+	0.917823i
							\(3\)	0.400074	−	0.550655i	0.230983	−	0.317921i	−0.677755	−	0.735288i	\(-0.737047\pi\)
0.908738	+	0.417367i	\(0.137047\pi\)
\(5\)	−2.02565	−	0.946960i	−0.905899	−	0.423494i
							\(7\)	−3.10246	−	3.10246i	−1.17262	−	1.17262i	−0.981583	−	0.191035i	\(-0.938815\pi\)
−0.191035	−	0.981583i	\(-0.561185\pi\)
\(11\)	−4.42085	+	2.25254i	−1.33294	+	0.679166i	−0.967784	−	0.251781i	\(-0.918984\pi\)
							−0.365154	+	0.930947i	\(0.618984\pi\)
\(13\)	0.544917	+	1.67708i	0.151133	+	0.465139i	0.997749	−	0.0670655i	\(-0.0213636\pi\)
							−0.846616	+	0.532205i	\(0.821364\pi\)
\(17\)	−6.94276	−	1.09963i	−1.68387	−	0.266698i	−0.760142	−	0.649757i	\(-0.774871\pi\)
							−0.923724	+	0.383058i	\(0.874871\pi\)
\(19\)	0.811902	+	0.128593i	0.186263	+	0.0295012i	0.248869	−	0.968537i	\(-0.419941\pi\)
							−0.0626060	+	0.998038i	\(0.519941\pi\)
\(23\)	−1.41992	−	2.78674i	−0.296073	−	0.581076i	0.694270	−	0.719714i	\(-0.255727\pi\)
							−0.990343	+	0.138639i	\(0.955727\pi\)
\(29\)	8.71930	−	1.38100i	1.61913	−	0.256446i	0.719953	−	0.694023i	\(-0.244163\pi\)
							0.899181	+	0.437577i	\(0.144163\pi\)
\(31\)	−0.119066	−	0.163881i	−0.0213850	−	0.0294339i	0.798191	−	0.602405i	\(-0.205791\pi\)
							−0.819576	+	0.572971i	\(0.805791\pi\)
\(37\)	−0.280817	−	0.864265i	−0.0461660	−	0.142084i	0.925316	−	0.379196i	\(-0.123799\pi\)
							−0.971482	+	0.237112i	\(0.923799\pi\)
\(41\)	−3.50064	+	1.13743i	−0.546708	+	0.177636i	−0.569332	−	0.822108i	\(-0.692798\pi\)
							0.0226239	+	0.999744i	\(0.492798\pi\)
\(43\)	5.05926			0.771531			0.385765	−	0.922597i	\(-0.373937\pi\)
							0.385765	+	0.922597i	\(0.373937\pi\)
\(47\)	−6.06790	+	0.961061i	−0.885094	+	0.140185i	−0.582400	−	0.812903i	\(-0.697886\pi\)
							−0.302694	+	0.953088i	\(0.597886\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.147.39	yes	464
16.11	odd	4		400.2.bd.a.347.20	yes	464
25.8	odd	20		400.2.bd.a.83.20	✓	464
400.283	even	20	inner	400.2.bm.a.283.39	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.83.20	✓	464	25.8	odd	20
400.2.bd.a.347.20	yes	464	16.11	odd	4
400.2.bm.a.147.39	yes	464	1.1	even	1	trivial
400.2.bm.a.283.39	yes	464	400.283	even	20	inner