Embedded newform 448.2.bn.a.221.23

• Label: 448.2.bn.a.221.23
• 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			221.23
Character	\(\chi\)	\(=\)	448.221
Dual form			448.2.bn.a.373.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.592063 + 1.28431i) q^{2} +(1.56634 - 1.78607i) q^{3} +(-1.29892 - 1.52079i) q^{4} +(0.825894 + 0.0541320i) q^{5} +(1.36650 + 3.06914i) q^{6} +(-1.37747 - 2.25889i) q^{7} +(2.72221 - 0.767820i) q^{8} +(-0.345042 - 2.62085i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.592063	+	1.28431i	−0.418652	+	0.908147i
							\(3\)	1.56634	−	1.78607i	0.904327	−	1.03119i	−0.0950348	−	0.995474i	\(-0.530296\pi\)
0.999362	−	0.0357136i	\(-0.0113704\pi\)
\(5\)	0.825894	+	0.0541320i	0.369351	+	0.0242086i	0.248949	−	0.968517i	\(-0.419915\pi\)
							0.120402	+	0.992725i	\(0.461582\pi\)
\(7\)	−1.37747	−	2.25889i	−0.520635	−	0.853780i
							\(11\)	1.55630	+	0.528293i	0.469243	+	0.159286i	0.546038	−	0.837760i	\(-0.316135\pi\)
−0.0767952	+	0.997047i	\(0.524469\pi\)
\(13\)	−2.80968	−	4.20499i	−0.779266	−	1.16625i	−0.982343	−	0.187091i	\(-0.940094\pi\)
							0.203077	−	0.979163i	\(-0.434906\pi\)
\(17\)	0.864344	−	0.231600i	0.209634	−	0.0561713i	−0.152474	−	0.988308i	\(-0.548724\pi\)
							0.362108	+	0.932136i	\(0.382057\pi\)
\(19\)	−0.533748	−	0.263216i	−0.122450	−	0.0603858i	0.380032	−	0.924973i	\(-0.375913\pi\)
							−0.502482	+	0.864588i	\(0.667580\pi\)
\(23\)	1.72542	−	0.227156i	0.359774	−	0.0473652i	0.0515275	−	0.998672i	\(-0.483591\pi\)
							0.308247	+	0.951306i	\(0.400258\pi\)
\(29\)	1.23352	−	6.20131i	0.229058	−	1.15155i	−0.679461	−	0.733711i	\(-0.737786\pi\)
							0.908520	−	0.417842i	\(-0.137214\pi\)
\(31\)	2.79441	+	1.61336i	0.501891	+	0.289767i	0.729494	−	0.683987i	\(-0.239755\pi\)
							−0.227603	+	0.973754i	\(0.573089\pi\)
\(37\)	0.368329	−	5.61962i	0.0605530	−	0.923860i	−0.854225	−	0.519903i	\(-0.825968\pi\)
							0.914778	−	0.403957i	\(-0.132365\pi\)
\(41\)	−2.37476	+	5.73317i	−0.370875	+	0.895370i	0.622728	+	0.782438i	\(0.286024\pi\)
							−0.993603	+	0.112932i	\(0.963976\pi\)
\(43\)	10.7574	−	2.13979i	1.64049	−	0.326314i	0.713289	−	0.700870i	\(-0.247205\pi\)
							0.927205	+	0.374555i	\(0.122205\pi\)
\(47\)	0.934756	−	3.48856i	0.136348	−	0.508858i	−0.863641	−	0.504108i	\(-0.831821\pi\)
							0.999989	−	0.00475018i	\(-0.00151203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bn.a.221.23	yes	992
7.2	even	3	inner	448.2.bn.a.93.62	yes	992
64.53	even	16	inner	448.2.bn.a.53.62	✓	992
448.373	even	48	inner	448.2.bn.a.373.23	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bn.a.53.62	✓	992	64.53	even	16	inner
448.2.bn.a.93.62	yes	992	7.2	even	3	inner
448.2.bn.a.221.23	yes	992	1.1	even	1	trivial
448.2.bn.a.373.23	yes	992	448.373	even	48	inner