Embedded newform 448.2.bn.a.221.29

• Label: 448.2.bn.a.221.29
• 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			221.29
Character	\(\chi\)	\(=\)	448.221
Dual form			448.2.bn.a.373.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.215269 - 1.39773i) q^{2} +(-1.75966 + 2.00651i) q^{3} +(-1.90732 + 0.601777i) q^{4} +(-3.38412 - 0.221807i) q^{5} +(3.18336 + 2.02759i) q^{6} +(-1.23445 + 2.34011i) q^{7} +(1.25171 + 2.53638i) q^{8} +(-0.538089 - 4.08719i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.215269	−	1.39773i	−0.152218	−	0.988347i
							\(3\)	−1.75966	+	2.00651i	−1.01594	+	1.15846i	−0.0286387	+	0.999590i	\(0.509117\pi\)
−0.987300	+	0.158866i	\(0.949216\pi\)
\(5\)	−3.38412	−	0.221807i	−1.51342	−	0.0991951i	−0.714078	−	0.700066i	\(-0.753154\pi\)
							−0.799346	+	0.600871i	\(0.794821\pi\)
\(7\)	−1.23445	+	2.34011i	−0.466579	+	0.884479i
							\(11\)	1.96501	+	0.667030i	0.592472	+	0.201117i	0.601541	−	0.798842i	\(-0.294554\pi\)
−0.00906914	+	0.999959i	\(0.502887\pi\)
\(13\)	−2.67536	−	4.00395i	−0.742010	−	1.11050i	−0.989908	−	0.141710i	\(-0.954740\pi\)
							0.247898	−	0.968786i	\(-0.420260\pi\)
\(17\)	4.04519	−	1.08391i	0.981103	−	0.262886i	0.267594	−	0.963532i	\(-0.413771\pi\)
							0.713509	+	0.700646i	\(0.247105\pi\)
\(19\)	−2.43381	−	1.20022i	−0.558354	−	0.275350i	0.141138	−	0.989990i	\(-0.454924\pi\)
							−0.699492	+	0.714640i	\(0.746590\pi\)
\(23\)	1.42094	−	0.187070i	0.296286	−	0.0390068i	0.0190823	−	0.999818i	\(-0.493926\pi\)
							0.277204	+	0.960811i	\(0.410592\pi\)
\(29\)	−1.12925	+	5.67713i	−0.209697	+	1.05422i	0.722253	+	0.691629i	\(0.243106\pi\)
							−0.931950	+	0.362587i	\(0.881894\pi\)
\(31\)	2.38710	+	1.37819i	0.428735	+	0.247530i	0.698808	−	0.715310i	\(-0.253714\pi\)
							−0.270072	+	0.962840i	\(0.587048\pi\)
\(37\)	0.705237	−	10.7598i	0.115940	−	1.76891i	−0.404052	−	0.914736i	\(-0.632399\pi\)
							0.519992	−	0.854171i	\(-0.325935\pi\)
\(41\)	−2.94491	+	7.10964i	−0.459918	+	1.11034i	0.508513	+	0.861054i	\(0.330195\pi\)
							−0.968430	+	0.249285i	\(0.919805\pi\)
\(43\)	9.34352	−	1.85854i	1.42487	−	0.283425i	0.578355	−	0.815785i	\(-0.303695\pi\)
							0.846518	+	0.532360i	\(0.178695\pi\)
\(47\)	2.10454	−	7.85426i	0.306979	−	1.14566i	−0.624250	−	0.781225i	\(-0.714595\pi\)
							0.931228	−	0.364436i	\(-0.118738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bn.a.221.29	yes	992
7.2	even	3	inner	448.2.bn.a.93.13	yes	992
64.53	even	16	inner	448.2.bn.a.53.13	✓	992
448.373	even	48	inner	448.2.bn.a.373.29	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bn.a.53.13	✓	992	64.53	even	16	inner
448.2.bn.a.93.13	yes	992	7.2	even	3	inner
448.2.bn.a.221.29	yes	992	1.1	even	1	trivial
448.2.bn.a.373.29	yes	992	448.373	even	48	inner