Embedded newform 448.2.bn.a.221.52

• Label: 448.2.bn.a.221.52
• 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			221.52
Character	\(\chi\)	\(=\)	448.221
Dual form			448.2.bn.a.373.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.15676 + 0.813569i) q^{2} +(-1.25280 + 1.42855i) q^{3} +(0.676210 + 1.88222i) q^{4} +(2.17561 + 0.142597i) q^{5} +(-2.61142 + 0.633251i) q^{6} +(-2.44948 + 1.00003i) q^{7} +(-0.749097 + 2.72743i) q^{8} +(-0.0796537 - 0.605030i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.15676	+	0.813569i	0.817956	+	0.575280i
							\(3\)	−1.25280	+	1.42855i	−0.723305	+	0.824771i	−0.990303	−	0.138926i	\(-0.955635\pi\)
0.266998	+	0.963697i	\(0.413968\pi\)
\(5\)	2.17561	+	0.142597i	0.972962	+	0.0637713i	0.543585	−	0.839354i	\(-0.317067\pi\)
							0.429377	+	0.903125i	\(0.358733\pi\)
\(7\)	−2.44948	+	1.00003i	−0.925816	+	0.377975i
							\(11\)	−1.10179	−	0.374006i	−0.332201	−	0.112767i	0.150355	−	0.988632i	\(-0.451958\pi\)
−0.482556	+	0.875865i	\(0.660292\pi\)
\(13\)	−0.641365	−	0.959870i	−0.177883	−	0.266220i	0.731804	−	0.681515i	\(-0.238678\pi\)
							−0.909687	+	0.415295i	\(0.863678\pi\)
\(17\)	3.80832	−	1.02044i	0.923654	−	0.247492i	0.234507	−	0.972114i	\(-0.424652\pi\)
							0.689146	+	0.724622i	\(0.257986\pi\)
\(19\)	2.86604	+	1.41338i	0.657515	+	0.324251i	0.740261	−	0.672320i	\(-0.234702\pi\)
							−0.0827452	+	0.996571i	\(0.526369\pi\)
\(23\)	−1.20469	+	0.158600i	−0.251195	+	0.0330705i	−0.255072	−	0.966922i	\(-0.582099\pi\)
							0.00387699	+	0.999992i	\(0.498766\pi\)
\(29\)	−0.134821	+	0.677790i	−0.0250356	+	0.125862i	−0.991282	−	0.131757i	\(-0.957938\pi\)
							0.966246	+	0.257619i	\(0.0829381\pi\)
\(31\)	6.02043	+	3.47590i	1.08130	+	0.624290i	0.931247	−	0.364388i	\(-0.118722\pi\)
							0.150054	+	0.988678i	\(0.452055\pi\)
\(37\)	−0.332866	+	5.07856i	−0.0547229	+	0.834910i	0.878979	+	0.476860i	\(0.158225\pi\)
							−0.933702	+	0.358050i	\(0.883442\pi\)
\(41\)	1.97362	−	4.76474i	0.308228	−	0.744127i	−0.691535	−	0.722343i	\(-0.743065\pi\)
							0.999763	−	0.0217845i	\(-0.00693478\pi\)
\(43\)	−3.43533	+	0.683331i	−0.523884	+	0.104207i	−0.449946	−	0.893056i	\(-0.648557\pi\)
							−0.0739379	+	0.997263i	\(0.523557\pi\)
\(47\)	2.79194	−	10.4197i	0.407246	−	1.51986i	−0.392630	−	0.919697i	\(-0.628435\pi\)
							0.799876	−	0.600166i	\(-0.204899\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bn.a.221.52	yes	992
7.2	even	3	inner	448.2.bn.a.93.33	yes	992
64.53	even	16	inner	448.2.bn.a.53.33	✓	992
448.373	even	48	inner	448.2.bn.a.373.52	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bn.a.53.33	✓	992	64.53	even	16	inner
448.2.bn.a.93.33	yes	992	7.2	even	3	inner
448.2.bn.a.221.52	yes	992	1.1	even	1	trivial
448.2.bn.a.373.52	yes	992	448.373	even	48	inner