Embedded newform 320.2.bj.a.107.11

• Label: 320.2.bj.a.107.11
• Level: $320$
• Weight: $2$
• Character: 320.107
• 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.bj (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			107.11
Character	\(\chi\)	\(=\)	320.107
Dual form			320.2.bj.a.3.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12995 - 0.850415i) q^{2} +(-1.38917 + 2.07904i) q^{3} +(0.553590 + 1.92186i) q^{4} +(-0.180810 + 2.22875i) q^{5} +(3.33774 - 1.16785i) q^{6} +(4.18944 - 1.73532i) q^{7} +(1.00885 - 2.64239i) q^{8} +(-1.24456 - 3.00463i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 368 q - 8 q^{2} - 8 q^{3} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12995	−	0.850415i	−0.798998	−	0.601334i
							\(3\)	−1.38917	+	2.07904i	−0.802037	+	1.20033i	0.174431	+	0.984669i	\(0.444192\pi\)
−0.976468	+	0.215664i	\(0.930808\pi\)
\(5\)	−0.180810	+	2.22875i	−0.0808607	+	0.996725i
							\(7\)	4.18944	−	1.73532i	1.58346	−	0.655891i	0.594504	−	0.804093i	\(-0.297349\pi\)
0.988957	+	0.148202i	\(0.0473485\pi\)
\(11\)	−0.915073	+	4.60038i	−0.275905	+	1.38707i	0.555553	+	0.831481i	\(0.312506\pi\)
							−0.831458	+	0.555587i	\(0.812494\pi\)
\(13\)	3.52558	−	0.701281i	0.977819	−	0.194500i	0.319787	−	0.947490i	\(-0.396389\pi\)
							0.658033	+	0.752989i	\(0.271389\pi\)
\(17\)	0.591087			0.143360			0.0716798	−	0.997428i	\(-0.477164\pi\)
							0.0716798	+	0.997428i	\(0.477164\pi\)
\(19\)	−3.16889	+	4.74257i	−0.726993	+	1.08802i	0.265307	+	0.964164i	\(0.414527\pi\)
							−0.992300	+	0.123857i	\(0.960473\pi\)
\(23\)	−1.49791	+	3.61628i	−0.312336	+	0.754046i	0.687281	+	0.726391i	\(0.258804\pi\)
							−0.999618	+	0.0276550i	\(0.991196\pi\)
\(29\)	−1.31990	−	6.63559i	−0.245100	−	1.23220i	−0.885676	−	0.464304i	\(-0.846304\pi\)
							0.640576	−	0.767895i	\(-0.278696\pi\)
\(31\)	−0.228216			−0.0409888			−0.0204944	−	0.999790i	\(-0.506524\pi\)
							−0.0204944	+	0.999790i	\(0.506524\pi\)
\(37\)	1.26341	−	6.35158i	0.207703	−	1.04419i	−0.726422	−	0.687249i	\(-0.758818\pi\)
							0.934125	−	0.356945i	\(-0.116182\pi\)
\(41\)	1.59517	+	3.85109i	0.249124	+	0.601440i	0.998130	−	0.0611239i	\(-0.0194685\pi\)
							−0.749006	+	0.662564i	\(0.769468\pi\)
\(43\)	0.766558	−	0.512197i	0.116899	−	0.0781094i	−0.495746	−	0.868467i	\(-0.665105\pi\)
							0.612645	+	0.790358i	\(0.290105\pi\)
\(47\)			5.89819i			0.860339i	0.902748	+	0.430170i	\(0.141546\pi\)
							−0.902748	+	0.430170i	\(0.858454\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.bd.a.43.14	✓	368	5.3	odd	4
320.2.bd.a.67.14	yes	368	64.3	odd	16
320.2.bj.a.3.11	yes	368	320.3	even	16	inner
320.2.bj.a.107.11	yes	368	1.1	even	1	trivial