Embedded newform 448.2.bm.a.227.39

• Label: 448.2.bm.a.227.39
• Level: $448$
• Weight: $2$
• Character: 448.227
• 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.bm (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			227.39
Character	\(\chi\)	\(=\)	448.227
Dual form			448.2.bm.a.75.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.480630 - 1.33004i) q^{2} +(-0.903902 + 1.03070i) q^{3} +(-1.53799 - 1.27851i) q^{4} +(-0.0681498 + 1.03976i) q^{5} +(0.936428 + 1.69761i) q^{6} +(-1.01073 - 2.44508i) q^{7} +(-2.43967 + 1.43109i) q^{8} +(0.146269 + 1.11103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 24 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.480630	−	1.33004i	0.339857	−	0.940477i
							\(3\)	−0.903902	+	1.03070i	−0.521868	+	0.595076i	−0.951393	−	0.307978i	\(-0.900348\pi\)
0.429525	+	0.903055i	\(0.358681\pi\)
\(5\)	−0.0681498	+	1.03976i	−0.0304775	+	0.464997i	0.955156	+	0.296101i	\(0.0956866\pi\)
							−0.985634	+	0.168895i	\(0.945980\pi\)
\(7\)	−1.01073	−	2.44508i	−0.382019	−	0.924154i
							\(11\)	−0.907687	+	2.67396i	−0.273678	+	0.806229i	0.720001	+	0.693973i	\(0.244141\pi\)
−0.993679	+	0.112256i	\(0.964192\pi\)
\(13\)	−4.34734	+	2.90480i	−1.20573	+	0.805646i	−0.985479	−	0.169796i	\(-0.945689\pi\)
							−0.220255	+	0.975442i	\(0.570689\pi\)
\(17\)	−0.941509	−	3.51376i	−0.228349	−	0.852211i	−0.981035	−	0.193831i	\(-0.937909\pi\)
							0.752686	−	0.658380i	\(-0.228758\pi\)
\(19\)	−3.98125	−	1.96334i	−0.913362	−	0.450420i	−0.0760243	−	0.997106i	\(-0.524223\pi\)
							−0.837338	+	0.546685i	\(0.815889\pi\)
\(23\)	0.977158	+	7.42225i	0.203752	+	1.54765i	0.721683	+	0.692224i	\(0.243369\pi\)
							−0.517931	+	0.855422i	\(0.673298\pi\)
\(29\)	−1.21709	+	6.11875i	−0.226009	+	1.13622i	0.686489	+	0.727140i	\(0.259151\pi\)
							−0.912498	+	0.409082i	\(0.865849\pi\)
\(31\)	−3.68668	−	2.12851i	−0.662147	−	0.382291i	0.130947	−	0.991389i	\(-0.458198\pi\)
							−0.793095	+	0.609098i	\(0.791531\pi\)
\(37\)	−0.680934	+	10.3890i	−0.111945	+	1.70795i	0.460216	+	0.887807i	\(0.347772\pi\)
							−0.572161	+	0.820141i	\(0.693895\pi\)
\(41\)	3.12552	+	1.29463i	0.488125	+	0.202188i	0.613151	−	0.789965i	\(-0.289902\pi\)
							−0.125027	+	0.992153i	\(0.539902\pi\)
\(43\)	−1.82692	−	9.18457i	−0.278603	−	1.40063i	−0.825958	−	0.563732i	\(-0.809365\pi\)
							0.547354	−	0.836901i	\(-0.315635\pi\)
\(47\)	0.886224	−	3.30743i	0.129269	−	0.482439i	−0.870687	−	0.491838i	\(-0.836325\pi\)
							0.999956	+	0.00939925i	\(0.00299192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bm.a.227.39	yes	992
7.5	odd	6	inner	448.2.bm.a.355.3	yes	992
64.11	odd	16	inner	448.2.bm.a.395.3	yes	992
448.75	even	48	inner	448.2.bm.a.75.39	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bm.a.75.39	✓	992	448.75	even	48	inner
448.2.bm.a.227.39	yes	992	1.1	even	1	trivial
448.2.bm.a.355.3	yes	992	7.5	odd	6	inner
448.2.bm.a.395.3	yes	992	64.11	odd	16	inner