Embedded newform 400.2.bm.a.123.33

• Label: 400.2.bm.a.123.33
• Level: $400$
• Weight: $2$
• Character: 400.123
• 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			123.33
Character	\(\chi\)	\(=\)	400.123
Dual form			400.2.bm.a.387.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.325391 - 1.37627i) q^{2} +(0.591076 + 0.192052i) q^{3} +(-1.78824 - 0.895652i) q^{4} +(-2.13420 + 0.667231i) q^{5} +(0.456646 - 0.750988i) q^{6} +(2.61466 - 2.61466i) q^{7} +(-1.81454 + 2.16967i) q^{8} +(-2.11456 - 1.53632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.325391	−	1.37627i	0.230086	−	0.973170i
							\(3\)	0.591076	+	0.192052i	0.341258	+	0.110881i	0.474632	−	0.880184i	\(-0.342581\pi\)
−0.133374	+	0.991066i	\(0.542581\pi\)
\(5\)	−2.13420	+	0.667231i	−0.954443	+	0.298395i
							\(7\)	2.61466	−	2.61466i	0.988248	−	0.988248i	−0.0116838	−	0.999932i	\(-0.503719\pi\)
0.999932	+	0.0116838i	\(0.00371916\pi\)
\(11\)	0.494093	−	3.11958i	0.148975	−	0.940588i	−0.794047	−	0.607856i	\(-0.792030\pi\)
							0.943022	−	0.332732i	\(-0.107970\pi\)
\(13\)	−1.00932	−	0.733313i	−0.279935	−	0.203384i	0.438954	−	0.898509i	\(-0.355349\pi\)
							−0.718889	+	0.695125i	\(0.755349\pi\)
\(17\)	−5.14892	−	2.62351i	−1.24880	−	0.636294i	−0.300531	−	0.953772i	\(-0.597164\pi\)
							−0.948266	+	0.317478i	\(0.897164\pi\)
\(19\)	2.65815	+	1.35440i	0.609822	+	0.310720i	0.731491	−	0.681851i	\(-0.238825\pi\)
							−0.121669	+	0.992571i	\(0.538825\pi\)
\(23\)	−5.17050	−	0.818926i	−1.07812	−	0.170758i	−0.407981	−	0.912990i	\(-0.633767\pi\)
							−0.670142	+	0.742233i	\(0.733767\pi\)
\(29\)	2.82388	−	1.43884i	0.524382	−	0.267186i	−0.171699	−	0.985149i	\(-0.554926\pi\)
							0.696081	+	0.717964i	\(0.254926\pi\)
\(31\)	−4.97309	+	1.61586i	−0.893194	+	0.290216i	−0.719425	−	0.694570i	\(-0.755595\pi\)
							−0.173769	+	0.984786i	\(0.555595\pi\)
\(37\)	5.09962	+	3.70509i	0.838372	+	0.609113i	0.921916	−	0.387391i	\(-0.126624\pi\)
							−0.0835431	+	0.996504i	\(0.526624\pi\)
\(41\)	2.88956	−	3.97713i	0.451273	−	0.621124i	−0.521398	−	0.853314i	\(-0.674589\pi\)
							0.972670	+	0.232190i	\(0.0745891\pi\)
\(43\)	9.48598			1.44660			0.723299	−	0.690535i	\(-0.242625\pi\)
							0.723299	+	0.690535i	\(0.242625\pi\)
\(47\)	4.36870	−	2.22596i	0.637240	−	0.324690i	−0.105351	−	0.994435i	\(-0.533597\pi\)
							0.742591	+	0.669745i	\(0.233597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.123.33	yes	464
16.3	odd	4		400.2.bd.a.323.32	yes	464
25.12	odd	20		400.2.bd.a.187.32	✓	464
400.387	even	20	inner	400.2.bm.a.387.33	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.187.32	✓	464	25.12	odd	20
400.2.bd.a.323.32	yes	464	16.3	odd	4
400.2.bm.a.123.33	yes	464	1.1	even	1	trivial
400.2.bm.a.387.33	yes	464	400.387	even	20	inner