Embedded newform 448.2.bm.a.187.46

• Label: 448.2.bm.a.187.46
• Level: $448$
• Weight: $2$
• Character: 448.187
• 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			187.46
Character	\(\chi\)	\(=\)	448.187
Dual form			448.2.bm.a.115.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.907566 - 1.08458i) q^{2} +(-0.165563 + 2.52601i) q^{3} +(-0.352649 - 1.96866i) q^{4} +(-2.63170 - 2.30794i) q^{5} +(2.58941 + 2.47208i) q^{6} +(2.24854 - 1.39430i) q^{7} +(-2.45524 - 1.40421i) q^{8} +(-3.37896 - 0.444848i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 24 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.907566	−	1.08458i	0.641746	−	0.766917i
							\(3\)	−0.165563	+	2.52601i	−0.0955879	+	1.45839i	0.633379	+	0.773841i	\(0.281667\pi\)
−0.728967	+	0.684549i	\(0.759999\pi\)
\(5\)	−2.63170	−	2.30794i	−1.17693	−	1.03214i	−0.998841	−	0.0481289i	\(-0.984674\pi\)
							−0.178092	−	0.984014i	\(-0.556992\pi\)
\(7\)	2.24854	−	1.39430i	0.849869	−	0.526994i
							\(11\)	3.51312	+	1.73248i	1.05924	+	0.522361i	0.886559	−	0.462616i	\(-0.153089\pi\)
0.172685	+	0.984977i	\(0.444756\pi\)
\(13\)	−1.13875	−	5.72487i	−0.315832	−	1.58779i	−0.733822	−	0.679341i	\(-0.762266\pi\)
							0.417991	−	0.908451i	\(-0.362734\pi\)
\(17\)	1.44497	−	5.39272i	0.350458	−	1.30793i	−0.535647	−	0.844442i	\(-0.679932\pi\)
							0.886105	−	0.463484i	\(-0.153401\pi\)
\(19\)	−3.47146	−	1.17840i	−0.796408	−	0.270344i	−0.106567	−	0.994305i	\(-0.533986\pi\)
							−0.689840	+	0.723962i	\(0.742319\pi\)
\(23\)	5.58439	+	0.735199i	1.16443	+	0.153300i	0.687863	−	0.725841i	\(-0.258549\pi\)
							0.476563	+	0.879140i	\(0.341882\pi\)
\(29\)	1.50020	−	2.24521i	0.278581	−	0.416925i	−0.665623	−	0.746288i	\(-0.731834\pi\)
							0.944203	+	0.329363i	\(0.106834\pi\)
\(31\)	−1.48512	+	0.857432i	−0.266735	+	0.153999i	−0.627403	−	0.778695i	\(-0.715882\pi\)
							0.360668	+	0.932694i	\(0.382549\pi\)
\(37\)	2.49605	+	2.18898i	0.410348	+	0.359866i	0.839494	−	0.543370i	\(-0.182852\pi\)
							−0.429145	+	0.903236i	\(0.641185\pi\)
\(41\)	2.30821	+	5.57251i	0.360482	+	0.870280i	0.995230	+	0.0975606i	\(0.0311040\pi\)
							−0.634748	+	0.772719i	\(0.718896\pi\)
\(43\)	−4.72036	−	7.06452i	−0.719848	−	1.07733i	−0.993314	−	0.115447i	\(-0.963170\pi\)
							0.273465	−	0.961882i	\(-0.411830\pi\)
\(47\)	−1.28596	−	4.79927i	−0.187577	−	0.700045i	−0.994064	−	0.108794i	\(-0.965301\pi\)
							0.806488	−	0.591251i	\(-0.201366\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bm.a.187.46	yes	992
7.3	odd	6	inner	448.2.bm.a.59.37	✓	992
64.51	odd	16	inner	448.2.bm.a.243.37	yes	992
448.115	even	48	inner	448.2.bm.a.115.46	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bm.a.59.37	✓	992	7.3	odd	6	inner
448.2.bm.a.115.46	yes	992	448.115	even	48	inner
448.2.bm.a.187.46	yes	992	1.1	even	1	trivial
448.2.bm.a.243.37	yes	992	64.51	odd	16	inner