Embedded newform 400.2.bm.a.67.15

• Label: 400.2.bm.a.67.15
• Level: $400$
• Weight: $2$
• Character: 400.67
• 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			67.15
Character	\(\chi\)	\(=\)	400.67
Dual form			400.2.bm.a.203.15

$q$-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 \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(177\)	\(351\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{20}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.3.56	✓	464	25.3	odd	20
400.2.bd.a.267.56	yes	464	16.11	odd	4
400.2.bm.a.67.15	yes	464	1.1	even	1	trivial
400.2.bm.a.203.15	yes	464	400.203	even	20	inner