Embedded newform 448.2.bn.a.261.27

• Label: 448.2.bn.a.261.27
• Level: $448$
• Weight: $2$
• Character: 448.261
• 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			261.27
Character	\(\chi\)	\(=\)	448.261
Dual form			448.2.bn.a.333.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.355304 + 1.36885i) q^{2} +(-0.0920779 + 1.40484i) q^{3} +(-1.74752 - 0.972717i) q^{4} +(1.91346 - 2.18188i) q^{5} +(-1.89030 - 0.625185i) q^{6} +(0.753625 + 2.53615i) q^{7} +(1.95241 - 2.04649i) q^{8} +(1.00925 + 0.132870i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.355304	+	1.36885i	−0.251238	+	0.967925i
							\(3\)	−0.0920779	+	1.40484i	−0.0531612	+	0.811083i	0.885153	+	0.465300i	\(0.154053\pi\)
−0.938314	+	0.345783i	\(0.887613\pi\)
\(5\)	1.91346	−	2.18188i	0.855725	−	0.975767i	−0.144180	−	0.989551i	\(-0.546054\pi\)
							0.999905	+	0.0137842i	\(0.00438778\pi\)
\(7\)	0.753625	+	2.53615i	0.284843	+	0.958574i
							\(11\)	1.44162	−	2.92331i	0.434663	−	0.881410i	−0.563796	−	0.825914i	\(-0.690659\pi\)
0.998459	−	0.0554958i	\(-0.0176739\pi\)
\(13\)	1.92976	−	0.383853i	0.535219	−	0.106462i	0.0799215	−	0.996801i	\(-0.474533\pi\)
							0.455297	+	0.890340i	\(0.349533\pi\)
\(17\)	−3.77806	−	1.01233i	−0.916313	−	0.245525i	−0.230304	−	0.973119i	\(-0.573972\pi\)
							−0.686009	+	0.727593i	\(0.740639\pi\)
\(19\)	5.88170	+	1.99657i	1.34935	+	0.458044i	0.900226	−	0.435423i	\(-0.143401\pi\)
							0.449129	+	0.893467i	\(0.351734\pi\)
\(23\)	−0.434223	+	3.29825i	−0.0905418	+	0.687733i	0.884632	+	0.466291i	\(0.154410\pi\)
							−0.975173	+	0.221443i	\(0.928923\pi\)
\(29\)	1.79099	−	2.68040i	0.332578	−	0.497738i	−0.627062	−	0.778969i	\(-0.715743\pi\)
							0.959640	+	0.281231i	\(0.0907428\pi\)
\(31\)	−2.95022	+	1.70331i	−0.529875	+	0.305923i	−0.740965	−	0.671543i	\(-0.765632\pi\)
							0.211091	+	0.977466i	\(0.432298\pi\)
\(37\)	−3.75196	−	3.29038i	−0.616818	−	0.540935i	0.292910	−	0.956140i	\(-0.405376\pi\)
							−0.909728	+	0.415205i	\(0.863710\pi\)
\(41\)	1.87936	−	0.778454i	0.293506	−	0.121574i	−0.231072	−	0.972937i	\(-0.574223\pi\)
							0.524578	+	0.851363i	\(0.324223\pi\)
\(43\)	−5.22158	+	3.48895i	−0.796284	+	0.532060i	−0.885878	−	0.463919i	\(-0.846443\pi\)
							0.0895943	+	0.995978i	\(0.471443\pi\)
\(47\)	−2.47085	−	9.22133i	−0.360410	−	1.34507i	−0.873538	−	0.486757i	\(-0.838180\pi\)
							0.513128	−	0.858312i	\(-0.328487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bn.a.261.27	yes	992
7.4	even	3	inner	448.2.bn.a.389.17	yes	992
64.13	even	16	inner	448.2.bn.a.205.17	✓	992
448.333	even	48	inner	448.2.bn.a.333.27	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bn.a.205.17	✓	992	64.13	even	16	inner
448.2.bn.a.261.27	yes	992	1.1	even	1	trivial
448.2.bn.a.333.27	yes	992	448.333	even	48	inner
448.2.bn.a.389.17	yes	992	7.4	even	3	inner