Embedded newform 448.2.bm.a.227.24

• Label: 448.2.bm.a.227.24
• 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			227.24
Character	\(\chi\)	\(=\)	448.227
Dual form			448.2.bm.a.75.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.425829 + 1.34858i) q^{2} +(-1.61499 + 1.84154i) q^{3} +(-1.63734 - 1.14853i) q^{4} +(-0.00363568 + 0.0554698i) q^{5} +(-1.79576 - 2.96213i) q^{6} +(2.12402 - 1.57751i) q^{7} +(2.24611 - 1.71900i) q^{8} +(-0.391511 - 2.97382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 24 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.425829	+	1.34858i	−0.301107	+	0.953590i
							\(3\)	−1.61499	+	1.84154i	−0.932416	+	1.06322i	0.0654459	+	0.997856i	\(0.479153\pi\)
−0.997862	+	0.0653603i	\(0.979180\pi\)
\(5\)	−0.00363568	+	0.0554698i	−0.00162593	+	0.0248069i	−0.998595	−	0.0529946i	\(-0.983123\pi\)
							0.996969	+	0.0778015i	\(0.0247900\pi\)
\(7\)	2.12402	−	1.57751i	0.802803	−	0.596244i
							\(11\)	1.96520	−	5.78928i	0.592529	−	1.74553i	−0.0710963	−	0.997469i	\(-0.522650\pi\)
0.663625	−	0.748065i	\(-0.269017\pi\)
\(13\)	2.24517	−	1.50018i	0.622699	−	0.416074i	−0.203800	−	0.979013i	\(-0.565329\pi\)
							0.826499	+	0.562938i	\(0.190329\pi\)
\(17\)	−0.0852833	−	0.318281i	−0.0206842	−	0.0771946i	0.954812	−	0.297210i	\(-0.0960560\pi\)
							−0.975496	+	0.220015i	\(0.929389\pi\)
\(19\)	−6.14743	−	3.03158i	−1.41032	−	0.695492i	−0.431861	−	0.901940i	\(-0.642143\pi\)
							−0.978458	+	0.206448i	\(0.933809\pi\)
\(23\)	0.893615	+	6.78768i	0.186332	+	1.41533i	0.787553	+	0.616247i	\(0.211348\pi\)
							−0.601222	+	0.799082i	\(0.705319\pi\)
\(29\)	1.81107	−	9.10489i	0.336308	−	1.69074i	−0.329132	−	0.944284i	\(-0.606756\pi\)
							0.665440	−	0.746451i	\(-0.268244\pi\)
\(31\)	−4.72146	−	2.72594i	−0.848000	−	0.489593i	0.0119755	−	0.999928i	\(-0.496188\pi\)
							−0.859976	+	0.510335i	\(0.829521\pi\)
\(37\)	0.0177698	−	0.271115i	0.00292134	−	0.0445711i	−0.996149	−	0.0876723i	\(-0.972057\pi\)
							0.999071	+	0.0431013i	\(0.0137238\pi\)
\(41\)	9.05062	+	3.74889i	1.41347	+	0.585478i	0.953211	−	0.302307i	\(-0.0977567\pi\)
							0.460259	+	0.887785i	\(0.347757\pi\)
\(43\)	−0.291491	−	1.46543i	−0.0444520	−	0.223475i	0.952174	−	0.305558i	\(-0.0988429\pi\)
							−0.996626	+	0.0820824i	\(0.973843\pi\)
\(47\)	0.342298	−	1.27747i	0.0499293	−	0.186339i	−0.936457	−	0.350782i	\(-0.885916\pi\)
							0.986387	+	0.164443i	\(0.0525826\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bm.a.227.24	yes	992
7.5	odd	6	inner	448.2.bm.a.355.59	yes	992
64.11	odd	16	inner	448.2.bm.a.395.59	yes	992
448.75	even	48	inner	448.2.bm.a.75.24	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bm.a.75.24	✓	992	448.75	even	48	inner
448.2.bm.a.227.24	yes	992	1.1	even	1	trivial
448.2.bm.a.355.59	yes	992	7.5	odd	6	inner
448.2.bm.a.395.59	yes	992	64.11	odd	16	inner