Embedded newform 448.2.bn.a.205.54

• Label: 448.2.bn.a.205.54
• Level: $448$
• Weight: $2$
• Character: 448.205
• 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			205.54
Character	\(\chi\)	\(=\)	448.205
Dual form			448.2.bn.a.389.54

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.24130 - 0.677621i) q^{2} +(0.156866 + 0.0773576i) q^{3} +(1.08166 - 1.68226i) q^{4} +(-0.864953 + 2.54807i) q^{5} +(0.247137 - 0.0102713i) q^{6} +(1.29694 + 2.30607i) q^{7} +(0.202732 - 2.82115i) q^{8} +(-1.80766 - 2.35579i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.24130	−	0.677621i	0.877733	−	0.479150i
							\(3\)	0.156866	+	0.0773576i	0.0905665	+	0.0446624i	0.487008	−	0.873398i	\(-0.338088\pi\)
−0.396441	+	0.918060i	\(0.629755\pi\)
\(5\)	−0.864953	+	2.54807i	−0.386819	+	1.13953i	0.562926	+	0.826507i	\(0.309676\pi\)
							−0.949745	+	0.313025i	\(0.898658\pi\)
\(7\)	1.29694	+	2.30607i	0.490197	+	0.871612i
							\(11\)	4.13225	+	0.270842i	1.24592	+	0.0816619i	0.674052	−	0.738684i	\(-0.264552\pi\)
0.571868	+	0.820346i	\(0.306219\pi\)
\(13\)	5.46319	+	1.08670i	1.51522	+	0.301395i	0.881505	−	0.472174i	\(-0.156531\pi\)
							0.633711	+	0.773570i	\(0.281531\pi\)
\(17\)	−0.118798	+	0.443362i	−0.0288129	+	0.107531i	−0.978835	−	0.204652i	\(-0.934394\pi\)
							0.950022	+	0.312183i	\(0.101060\pi\)
\(19\)	−1.68913	−	1.48133i	−0.387513	−	0.339840i	0.443318	−	0.896364i	\(-0.353801\pi\)
							−0.830831	+	0.556524i	\(0.812135\pi\)
\(23\)	0.429264	−	0.329386i	0.0895078	−	0.0686818i	−0.563038	−	0.826431i	\(-0.690368\pi\)
							0.652546	+	0.757749i	\(0.273701\pi\)
\(29\)	1.67445	+	2.50600i	0.310938	+	0.465352i	0.953719	−	0.300698i	\(-0.0972196\pi\)
							−0.642781	+	0.766050i	\(0.722220\pi\)
\(31\)	−6.75348	+	3.89913i	−1.21296	+	0.700304i	−0.963403	−	0.268056i	\(-0.913619\pi\)
							−0.249558	+	0.968360i	\(0.580285\pi\)
\(37\)	−10.4757	−	3.55601i	−1.72219	−	0.584604i	−0.727367	−	0.686249i	\(-0.759256\pi\)
							−0.994821	+	0.101645i	\(0.967589\pi\)
\(41\)	1.37184	+	0.568236i	0.214246	+	0.0887436i	0.487225	−	0.873276i	\(-0.338009\pi\)
							−0.272979	+	0.962020i	\(0.588009\pi\)
\(43\)	−9.76435	−	6.52433i	−1.48905	−	0.994951i	−0.991863	−	0.127306i	\(-0.959367\pi\)
							−0.497185	−	0.867644i	\(-0.665633\pi\)
\(47\)	−2.69579	+	0.722336i	−0.393222	+	0.105363i	−0.450012	−	0.893023i	\(-0.648580\pi\)
							0.0567898	+	0.998386i	\(0.481914\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bn.a.205.54	✓	992
7.4	even	3	inner	448.2.bn.a.333.30	yes	992
64.5	even	16	inner	448.2.bn.a.261.30	yes	992
448.389	even	48	inner	448.2.bn.a.389.54	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bn.a.205.54	✓	992	1.1	even	1	trivial
448.2.bn.a.261.30	yes	992	64.5	even	16	inner
448.2.bn.a.333.30	yes	992	7.4	even	3	inner
448.2.bn.a.389.54	yes	992	448.389	even	48	inner