Embedded newform 400.2.bd.a.323.55

• Label: 400.2.bd.a.323.55
• Level: $400$
• Weight: $2$
• Character: 400.323
• 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			323.55
Character	\(\chi\)	\(=\)	400.323
Dual form			400.2.bd.a.187.55

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37516 - 0.330066i) q^{2} +(0.784993 - 2.41596i) q^{3} +(1.78211 - 0.907785i) q^{4} +(0.998071 - 2.00096i) q^{5} +(0.282062 - 3.58143i) q^{6} +(-3.39821 + 3.39821i) q^{7} +(2.15106 - 1.83656i) q^{8} +(-2.79360 - 2.02967i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 6 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{11}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37516	−	0.330066i	0.972383	−	0.233392i
							\(3\)	0.784993	−	2.41596i	0.453216	−	1.39486i	−0.420001	−	0.907524i	\(-0.637970\pi\)
0.873217	−	0.487332i	\(-0.162030\pi\)
\(5\)	0.998071	−	2.00096i	0.446351	−	0.894858i
							\(7\)	−3.39821	+	3.39821i	−1.28440	+	1.28440i	−0.346268	+	0.938136i	\(0.612551\pi\)
−0.938136	+	0.346268i	\(0.887449\pi\)
\(11\)	3.25510	+	0.515557i	0.981449	+	0.155446i	0.626485	−	0.779434i	\(-0.284493\pi\)
							0.354964	+	0.934880i	\(0.384493\pi\)
\(13\)	−2.10969	+	2.90374i	−0.585122	+	0.805352i	−0.994245	−	0.107128i	\(-0.965835\pi\)
							0.409123	+	0.912479i	\(0.365835\pi\)
\(17\)	2.18846	+	1.11508i	0.530780	+	0.270446i	0.698774	−	0.715342i	\(-0.253729\pi\)
							−0.167995	+	0.985788i	\(0.553729\pi\)
\(19\)	−2.29828	+	4.51062i	−0.527261	+	1.03481i	0.461756	+	0.887007i	\(0.347219\pi\)
							−0.989017	+	0.147801i	\(0.952781\pi\)
\(23\)	0.415386	+	0.0657906i	0.0866139	+	0.0137183i	0.199591	−	0.979879i	\(-0.436039\pi\)
							−0.112977	+	0.993598i	\(0.536039\pi\)
\(29\)	−1.34958	−	2.64870i	−0.250611	−	0.491851i	0.731090	−	0.682281i	\(-0.239012\pi\)
							−0.981701	+	0.190430i	\(0.939012\pi\)
\(31\)	−3.43253	+	1.11530i	−0.616500	+	0.200313i	−0.600586	−	0.799560i	\(-0.705066\pi\)
							−0.0159144	+	0.999873i	\(0.505066\pi\)
\(37\)	−2.48816	+	3.42467i	−0.409052	+	0.563012i	−0.962987	−	0.269548i	\(-0.913126\pi\)
							0.553935	+	0.832560i	\(0.313126\pi\)
\(41\)	4.66765	−	6.42447i	0.728965	−	1.00333i	−0.270213	−	0.962801i	\(-0.587094\pi\)
							0.999178	−	0.0405337i	\(-0.0129058\pi\)
\(43\)			6.41683i			0.978558i	0.872127	+	0.489279i	\(0.162740\pi\)
							−0.872127	+	0.489279i	\(0.837260\pi\)
\(47\)	−5.76213	+	2.93595i	−0.840492	+	0.428252i	−0.820568	−	0.571549i	\(-0.806343\pi\)
							−0.0199246	+	0.999801i	\(0.506343\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bd.a.323.55	yes	464
16.11	odd	4		400.2.bm.a.123.11	yes	464
25.12	odd	20		400.2.bm.a.387.11	yes	464
400.187	even	20	inner	400.2.bd.a.187.55	✓	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.187.55	✓	464	400.187	even	20	inner
400.2.bd.a.323.55	yes	464	1.1	even	1	trivial
400.2.bm.a.123.11	yes	464	16.11	odd	4
400.2.bm.a.387.11	yes	464	25.12	odd	20