Embedded newform 400.2.bm.a.203.25

• Label: 400.2.bm.a.203.25
• Level: $400$
• Weight: $2$
• Character: 400.203
• 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			203.25
Character	\(\chi\)	\(=\)	400.203
Dual form			400.2.bm.a.67.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.337370 + 1.37338i) q^{2} +(-1.50268 + 2.06827i) q^{3} +(-1.77236 - 0.926677i) q^{4} +(0.470031 - 2.18611i) q^{5} +(-2.33356 - 2.76153i) q^{6} +(2.52238 - 2.52238i) q^{7} +(1.87063 - 2.12150i) q^{8} +(-1.09262 - 3.36273i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{7}{20}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.337370	+	1.37338i	−0.238557	+	0.971129i
							\(3\)	−1.50268	+	2.06827i	−0.867575	+	1.19411i	0.112135	+	0.993693i	\(0.464231\pi\)
−0.979710	+	0.200421i	\(0.935769\pi\)
\(5\)	0.470031	−	2.18611i	0.210204	−	0.977657i
							\(7\)	2.52238	−	2.52238i	0.953370	−	0.953370i	−0.0455906	−	0.998960i	\(-0.514517\pi\)
0.998960	+	0.0455906i	\(0.0145170\pi\)
\(11\)	2.45771	+	4.82352i	0.741027	+	1.45435i	0.885403	+	0.464824i	\(0.153883\pi\)
							−0.144376	+	0.989523i	\(0.546117\pi\)
\(13\)	−0.583088	−	1.79456i	−0.161719	−	0.497721i	0.837060	−	0.547111i	\(-0.184272\pi\)
							−0.998780	+	0.0493896i	\(0.984272\pi\)
\(17\)	0.781964	−	4.93713i	0.189654	−	1.19743i	−0.690710	−	0.723131i	\(-0.742702\pi\)
							0.880365	−	0.474298i	\(-0.157298\pi\)
\(19\)	0.781517	−	4.93430i	0.179292	−	1.13201i	−0.719779	−	0.694204i	\(-0.755757\pi\)
							0.899071	−	0.437803i	\(-0.144243\pi\)
\(23\)	3.24767	−	1.65477i	0.677187	−	0.345044i	−0.0813393	−	0.996686i	\(-0.525920\pi\)
							0.758526	+	0.651642i	\(0.225920\pi\)
\(29\)	0.977218	+	6.16991i	0.181465	+	1.14572i	0.895318	+	0.445428i	\(0.146949\pi\)
							−0.713853	+	0.700296i	\(0.753051\pi\)
\(31\)	1.65481	+	2.27766i	0.297213	+	0.409079i	0.931340	−	0.364150i	\(-0.118640\pi\)
							−0.634127	+	0.773229i	\(0.718640\pi\)
\(37\)	2.09728	+	6.45477i	0.344791	+	1.06116i	0.961695	+	0.274120i	\(0.0883866\pi\)
							−0.616904	+	0.787038i	\(0.711613\pi\)
\(41\)	1.70913	−	0.555331i	0.266922	−	0.0867281i	−0.172498	−	0.985010i	\(-0.555184\pi\)
							0.439420	+	0.898282i	\(0.355184\pi\)
\(43\)	6.56785			1.00159			0.500794	−	0.865566i	\(-0.333041\pi\)
							0.500794	+	0.865566i	\(0.333041\pi\)
\(47\)	−0.206806	−	1.30572i	−0.0301658	−	0.190460i	0.968004	−	0.250935i	\(-0.0807382\pi\)
							−0.998170	+	0.0604759i	\(0.980738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.203.25	yes	464
16.3	odd	4		400.2.bd.a.3.53	✓	464
25.17	odd	20		400.2.bd.a.267.53	yes	464
400.67	even	20	inner	400.2.bm.a.67.25	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.3.53	✓	464	16.3	odd	4
400.2.bd.a.267.53	yes	464	25.17	odd	20
400.2.bm.a.67.25	yes	464	400.67	even	20	inner
400.2.bm.a.203.25	yes	464	1.1	even	1	trivial