Embedded newform 400.2.bd.a.27.3

• Label: 400.2.bd.a.27.3
• Level: $400$
• Weight: $2$
• Character: 400.27
• 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			27.3
Character	\(\chi\)	\(=\)	400.27
Dual form			400.2.bd.a.163.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38880 + 0.266914i) q^{2} +(0.0662497 - 0.203896i) q^{3} +(1.85751 - 0.741379i) q^{4} +(-1.43828 - 1.71212i) q^{5} +(-0.0375847 + 0.300853i) q^{6} +(0.740084 + 0.740084i) q^{7} +(-2.38182 + 1.52542i) q^{8} +(2.38987 + 1.73634i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 464 q - 8 q^{2} - 6 q^{3} - 10 q^{4} - 8 q^{5} - 6 q^{6} - 16 q^{7} - 2 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38880	+	0.266914i	−0.982028	+	0.188737i
							\(3\)	0.0662497	−	0.203896i	0.0382493	−	0.117719i	−0.930109	−	0.367284i	\(-0.880288\pi\)
0.968358	+	0.249565i	\(0.0802876\pi\)
\(5\)	−1.43828	−	1.71212i	−0.643217	−	0.765684i
							\(7\)	0.740084	+	0.740084i	0.279725	+	0.279725i	0.832999	−	0.553274i	\(-0.186622\pi\)
−0.553274	+	0.832999i	\(0.686622\pi\)
\(11\)	−0.504658	+	3.18628i	−0.152160	+	0.960701i	0.786933	+	0.617038i	\(0.211668\pi\)
							−0.939093	+	0.343663i	\(0.888332\pi\)
\(13\)	0.673943	−	0.927603i	0.186918	−	0.257271i	−0.705266	−	0.708943i	\(-0.749172\pi\)
							0.892184	+	0.451672i	\(0.149172\pi\)
\(17\)	3.08791	−	6.06036i	0.748927	−	1.46985i	−0.129299	−	0.991606i	\(-0.541273\pi\)
							0.878226	−	0.478247i	\(-0.158727\pi\)
\(19\)	3.01685	+	1.53716i	0.692114	+	0.352649i	0.764407	−	0.644734i	\(-0.223032\pi\)
							−0.0722935	+	0.997383i	\(0.523032\pi\)
\(23\)	1.28772	−	8.13036i	0.268509	−	1.69530i	−0.372717	−	0.927945i	\(-0.621574\pi\)
							0.641226	−	0.767352i	\(-0.278426\pi\)
\(29\)	−2.10023	+	1.07012i	−0.390004	+	0.198717i	−0.637986	−	0.770048i	\(-0.720232\pi\)
							0.247983	+	0.968764i	\(0.420232\pi\)
\(31\)	8.74921	−	2.84279i	1.57140	−	0.510580i	0.611580	−	0.791182i	\(-0.290534\pi\)
							0.959824	+	0.280602i	\(0.0905341\pi\)
\(37\)	−0.685737	+	0.943836i	−0.112734	+	0.155166i	−0.861656	−	0.507493i	\(-0.830572\pi\)
							0.748921	+	0.662659i	\(0.230572\pi\)
\(41\)	5.17900	−	7.12828i	0.808824	−	1.11325i	−0.182680	−	0.983173i	\(-0.558477\pi\)
							0.991504	−	0.130078i	\(-0.0415229\pi\)
\(43\)			5.67991i			0.866179i	0.901351	+	0.433089i	\(0.142577\pi\)
							−0.901351	+	0.433089i	\(0.857423\pi\)
\(47\)	1.88101	+	3.69169i	0.274373	+	0.538488i	0.986540	−	0.163523i	\(-0.0522858\pi\)
							−0.712166	+	0.702011i	\(0.752286\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.bd.a.27.3	✓	464
16.3	odd	4		400.2.bm.a.227.10	yes	464
25.13	odd	20		400.2.bm.a.363.10	yes	464
400.163	even	20	inner	400.2.bd.a.163.3	yes	464

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.2.bd.a.27.3	✓	464	1.1	even	1	trivial
400.2.bd.a.163.3	yes	464	400.163	even	20	inner
400.2.bm.a.227.10	yes	464	16.3	odd	4
400.2.bm.a.363.10	yes	464	25.13	odd	20