Embedded newform 448.2.bn.a.221.60

• Label: 448.2.bn.a.221.60
• Level: $448$
• Weight: $2$
• Character: 448.221
• Analytic conductor: $3.577$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.bn (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.57729801055\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(62\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			221.60
Character	\(\chi\)	\(=\)	448.221
Dual form			448.2.bn.a.373.60

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40966 - 0.113413i) q^{2} +(1.57593 - 1.79700i) q^{3} +(1.97428 - 0.319747i) q^{4} +(-4.01986 - 0.263476i) q^{5} +(2.01772 - 2.71189i) q^{6} +(0.0172974 - 2.64569i) q^{7} +(2.74679 - 0.674643i) q^{8} +(-0.354087 - 2.68956i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40966	−	0.113413i	0.996779	−	0.0801951i
							\(3\)	1.57593	−	1.79700i	0.909863	−	1.03750i	−0.0892762	−	0.996007i	\(-0.528455\pi\)
0.999139	−	0.0414926i	\(-0.0132113\pi\)
\(5\)	−4.01986	−	0.263476i	−1.79774	−	0.117830i	−0.870468	−	0.492225i	\(-0.836184\pi\)
							−0.927269	+	0.374396i	\(0.877850\pi\)
\(7\)	0.0172974	−	2.64569i	0.00653779	−	0.999979i
							\(11\)	1.76522	+	0.599212i	0.532234	+	0.180669i	0.574602	−	0.818433i	\(-0.305157\pi\)
−0.0423679	+	0.999102i	\(0.513490\pi\)
\(13\)	0.0225970	+	0.0338187i	0.00626727	+	0.00937963i	0.834590	−	0.550872i	\(-0.185705\pi\)
							−0.828323	+	0.560251i	\(0.810705\pi\)
\(17\)	−6.25204	+	1.67523i	−1.51634	+	0.406303i	−0.918536	−	0.395337i	\(-0.870628\pi\)
							−0.597807	+	0.801640i	\(0.703961\pi\)
\(19\)	5.65792	+	2.79018i	1.29802	+	0.640111i	0.954268	−	0.298951i	\(-0.0966368\pi\)
							0.343748	+	0.939062i	\(0.388303\pi\)
\(23\)	2.19513	−	0.288994i	0.457715	−	0.0602594i	0.101857	−	0.994799i	\(-0.467522\pi\)
							0.355859	+	0.934540i	\(0.384188\pi\)
\(29\)	1.07541	−	5.40643i	0.199698	−	1.00395i	−0.742743	−	0.669576i	\(-0.766476\pi\)
							0.942441	−	0.334372i	\(-0.108524\pi\)
\(31\)	0.0165809	+	0.00957299i	0.00297802	+	0.00171936i	0.501488	−	0.865164i	\(-0.332786\pi\)
							−0.498510	+	0.866884i	\(0.666119\pi\)
\(37\)	−0.763384	+	11.6470i	−0.125500	+	1.91475i	0.218591	+	0.975817i	\(0.429854\pi\)
							−0.344090	+	0.938937i	\(0.611813\pi\)
\(41\)	−0.510861	+	1.23333i	−0.0797831	+	0.192613i	−0.958738	−	0.284292i	\(-0.908242\pi\)
							0.878955	+	0.476905i	\(0.158242\pi\)
\(43\)	0.363216	−	0.0722482i	0.0553899	−	0.0110177i	−0.167318	−	0.985903i	\(-0.553511\pi\)
							0.222707	+	0.974885i	\(0.428511\pi\)
\(47\)	0.928221	−	3.46417i	0.135395	−	0.505301i	−0.864601	−	0.502459i	\(-0.832429\pi\)
							0.999996	−	0.00284186i	\(-0.000904593\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bn.a.221.60	yes	992
7.2	even	3	inner	448.2.bn.a.93.18	yes	992
64.53	even	16	inner	448.2.bn.a.53.18	✓	992
448.373	even	48	inner	448.2.bn.a.373.60	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bn.a.53.18	✓	992	64.53	even	16	inner
448.2.bn.a.93.18	yes	992	7.2	even	3	inner
448.2.bn.a.221.60	yes	992	1.1	even	1	trivial
448.2.bn.a.373.60	yes	992	448.373	even	48	inner