Embedded newform 400.2.bm.a.203.15

• Label: 400.2.bm.a.203.15
• 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			203.15
Character	\(\chi\)	\(=\)	400.203
Dual form			400.2.bm.a.67.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.977689 + 1.02182i) q^{2} +(1.05492 - 1.45197i) q^{3} +(-0.0882476 - 1.99805i) q^{4} +(1.30519 + 1.81562i) q^{5} +(0.452275 + 2.49751i) q^{6} +(0.298708 - 0.298708i) q^{7} +(2.12794 + 1.86330i) q^{8} +(-0.0683109 - 0.210239i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 10 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} + 108 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.977689	+	1.02182i	−0.691331	+	0.722539i
							\(3\)	1.05492	−	1.45197i	0.609056	−	0.838294i	−0.387443	−	0.921894i	\(-0.626642\pi\)
0.996499	+	0.0835996i	\(0.0266417\pi\)
\(5\)	1.30519	+	1.81562i	0.583698	+	0.811971i
							\(7\)	0.298708	−	0.298708i	0.112901	−	0.112901i	−0.648399	−	0.761300i	\(-0.724561\pi\)
0.761300	+	0.648399i	\(0.224561\pi\)
\(11\)	1.69140	+	3.31956i	0.509977	+	1.00089i	0.992179	+	0.124826i	\(0.0398373\pi\)
							−0.482202	+	0.876060i	\(0.660163\pi\)
\(13\)	−0.516740	−	1.59036i	−0.143318	−	0.441087i	0.853473	−	0.521137i	\(-0.174492\pi\)
							−0.996791	+	0.0800502i	\(0.974492\pi\)
\(17\)	−0.281871	+	1.77966i	−0.0683636	+	0.431631i	0.929640	+	0.368470i	\(0.120118\pi\)
							−0.998003	+	0.0631612i	\(0.979882\pi\)
\(19\)	0.465649	−	2.93999i	0.106827	−	0.674481i	−0.874917	−	0.484274i	\(-0.839084\pi\)
							0.981744	−	0.190207i	\(-0.0609161\pi\)
\(23\)	5.68038	−	2.89430i	1.18444	−	0.603502i	0.253025	−	0.967460i	\(-0.418575\pi\)
							0.931416	+	0.363957i	\(0.118575\pi\)
\(29\)	−0.231330	−	1.46056i	−0.0429568	−	0.271219i	0.956856	−	0.290562i	\(-0.0938424\pi\)
							−0.999813	+	0.0193436i	\(0.993842\pi\)
\(31\)	1.00053	+	1.37711i	0.179700	+	0.247336i	0.889359	−	0.457209i	\(-0.151151\pi\)
							−0.709659	+	0.704545i	\(0.751151\pi\)
\(37\)	−1.51796	−	4.67180i	−0.249551	−	0.768040i	−0.994854	−	0.101314i	\(-0.967695\pi\)
							0.745303	−	0.666726i	\(-0.232305\pi\)
\(41\)	−3.76182	+	1.22229i	−0.587498	+	0.190890i	−0.587657	−	0.809110i	\(-0.699949\pi\)
							0.000158983		1.00000i	\(0.499949\pi\)
\(43\)	6.34865			0.968161			0.484080	−	0.875024i	\(-0.339154\pi\)
							0.484080	+	0.875024i	\(0.339154\pi\)
\(47\)	−1.88925	−	11.9283i	−0.275576	−	1.73992i	−0.605442	−	0.795890i	\(-0.707003\pi\)
							0.329866	−	0.944028i	\(-0.392997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bm.a.203.15	yes	464
16.3	odd	4		400.2.bd.a.3.56	✓	464
25.17	odd	20		400.2.bd.a.267.56	yes	464
400.67	even	20	inner	400.2.bm.a.67.15	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.3.56	✓	464	16.3	odd	4
400.2.bd.a.267.56	yes	464	25.17	odd	20
400.2.bm.a.67.15	yes	464	400.67	even	20	inner
400.2.bm.a.203.15	yes	464	1.1	even	1	trivial