Embedded newform 320.2.bd.a.43.14

• Label: 320.2.bd.a.43.14
• Level: $320$
• Weight: $2$
• Character: 320.43
• Analytic conductor: $2.555$
• Analytic rank: $0$
• Dimension: $368$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	320.bd (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.55521286468\)
Analytic rank:	\(0\)
Dimension:	\(368\)
Relative dimension:	\(46\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			43.14
Character	\(\chi\)	\(=\)	320.43
Dual form			320.2.bd.a.67.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.850415 + 1.12995i) q^{2} +(-2.07904 - 1.38917i) q^{3} +(-0.553590 - 1.92186i) q^{4} +(-2.12829 - 0.685857i) q^{5} +(3.33774 - 1.16785i) q^{6} +(-1.73532 - 4.18944i) q^{7} +(2.64239 + 1.00885i) q^{8} +(1.24456 + 3.00463i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.850415	+	1.12995i	−0.601334	+	0.798998i
							\(3\)	−2.07904	−	1.38917i	−1.20033	−	0.802037i	−0.215664	−	0.976468i	\(-0.569192\pi\)
−0.984669	+	0.174431i	\(0.944192\pi\)
\(5\)	−2.12829	−	0.685857i	−0.951798	−	0.306725i
							\(7\)	−1.73532	−	4.18944i	−0.655891	−	1.58346i	−0.804093	−	0.594504i	\(-0.797349\pi\)
0.148202	−	0.988957i	\(-0.452651\pi\)
\(11\)	−0.915073	+	4.60038i	−0.275905	+	1.38707i	0.555553	+	0.831481i	\(0.312506\pi\)
							−0.831458	+	0.555587i	\(0.812494\pi\)
\(13\)	0.701281	+	3.52558i	0.194500	+	0.977819i	0.947490	+	0.319787i	\(0.103611\pi\)
							−0.752989	+	0.658033i	\(0.771389\pi\)
\(17\)		−	0.591087i		−	0.143360i	−0.997428	−	0.0716798i	\(-0.977164\pi\)
							0.997428	−	0.0716798i	\(-0.0228360\pi\)
\(19\)	3.16889	−	4.74257i	0.726993	−	1.08802i	−0.265307	−	0.964164i	\(-0.585473\pi\)
							0.992300	−	0.123857i	\(-0.0395265\pi\)
\(23\)	−3.61628	−	1.49791i	−0.754046	−	0.312336i	−0.0276550	−	0.999618i	\(-0.508804\pi\)
							−0.726391	+	0.687281i	\(0.758804\pi\)
\(29\)	1.31990	+	6.63559i	0.245100	+	1.23220i	0.885676	+	0.464304i	\(0.153696\pi\)
							−0.640576	+	0.767895i	\(0.721304\pi\)
\(31\)	−0.228216			−0.0409888			−0.0204944	−	0.999790i	\(-0.506524\pi\)
							−0.0204944	+	0.999790i	\(0.506524\pi\)
\(37\)	−6.35158	−	1.26341i	−1.04419	−	0.207703i	−0.356945	−	0.934125i	\(-0.616182\pi\)
							−0.687249	+	0.726422i	\(0.741182\pi\)
\(41\)	1.59517	+	3.85109i	0.249124	+	0.601440i	0.998130	−	0.0611239i	\(-0.0194685\pi\)
							−0.749006	+	0.662564i	\(0.769468\pi\)
\(43\)	0.512197	+	0.766558i	0.0781094	+	0.116899i	0.868467	−	0.495746i	\(-0.165105\pi\)
							−0.790358	+	0.612645i	\(0.790105\pi\)
\(47\)	5.89819			0.860339			0.430170	−	0.902748i	\(-0.358454\pi\)
							0.430170	+	0.902748i	\(0.358454\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.2.bd.a.43.14	✓	368
5.2	odd	4		320.2.bj.a.107.11	yes	368
64.3	odd	16		320.2.bj.a.3.11	yes	368
320.67	even	16	inner	320.2.bd.a.67.14	yes	368

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.14	✓	368	1.1	even	1	trivial
320.2.bd.a.67.14	yes	368	320.67	even	16	inner
320.2.bj.a.3.11	yes	368	64.3	odd	16
320.2.bj.a.107.11	yes	368	5.2	odd	4