Embedded newform 912.2.bu.a.715.59

• Label: 912.2.bu.a.715.59
• Level: $912$
• Weight: $2$
• Character: 912.715
• Analytic conductor: $7.282$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.bu (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.28235666434\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(80\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			715.59
Character	\(\chi\)	\(=\)	912.715
Dual form			912.2.bu.a.787.59

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.950557 - 1.04711i) q^{2} +(-0.965926 - 0.258819i) q^{3} +(-0.192884 - 1.99068i) q^{4} +(0.840004 - 3.13494i) q^{5} +(-1.18918 + 0.765410i) q^{6} -2.76947 q^{7} +(-2.26781 - 1.69028i) q^{8} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(229\)	\(305\)	\(799\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.950557	−	1.04711i	0.672145	−	0.740419i
							\(3\)	−0.965926	−	0.258819i	−0.557678	−	0.149429i
\(5\)	0.840004	−	3.13494i	0.375661	−	1.40199i								−0.476715	−	0.879058i	\(-0.658173\pi\)
							0.852376	−	0.522929i	\(-0.175161\pi\)
\(7\)	−2.76947			−1.04676			−0.523381	−	0.852099i	\(-0.675329\pi\)
							−0.523381	+	0.852099i	\(0.675329\pi\)
\(11\)	4.35480	−	4.35480i	1.31302	−	1.31302i	0.393846	−	0.919177i	\(-0.371144\pi\)
							0.919177	−	0.393846i	\(-0.128856\pi\)
\(13\)	4.71111	−	1.26234i	1.30663	−	0.350109i	0.462674	−	0.886528i	\(-0.346890\pi\)
							0.843952	+	0.536419i	\(0.180223\pi\)
\(17\)	2.09735	+	3.63271i	0.508681	+	0.881061i	0.999949	+	0.0100532i	\(0.00320010\pi\)
							−0.491268	+	0.871008i	\(0.663467\pi\)
\(19\)	−3.61020	−	2.44263i	−0.828237	−	0.560378i
							\(23\)	−2.51098	+	4.34914i	−0.523575	+	0.906858i	0.476049	+	0.879419i	\(0.342069\pi\)
−0.999623	+	0.0274391i	\(0.991265\pi\)
\(29\)	−3.34439	+	0.896126i	−0.621037	+	0.166406i	−0.555599	−	0.831450i	\(-0.687511\pi\)
							−0.0654380	+	0.997857i	\(0.520844\pi\)
\(31\)	−7.69644			−1.38232			−0.691161	−	0.722701i	\(-0.742900\pi\)
							−0.691161	+	0.722701i	\(0.742900\pi\)
\(37\)	1.82499	−	1.82499i	0.300026	−	0.300026i	−0.540998	−	0.841024i	\(-0.681953\pi\)
							0.841024	+	0.540998i	\(0.181953\pi\)
\(41\)	2.09253	+	3.62437i	0.326798	+	0.566031i	0.981875	−	0.189532i	\(-0.0606969\pi\)
							−0.655076	+	0.755563i	\(0.727364\pi\)
\(43\)	5.23626	+	1.40305i	0.798522	+	0.213963i	0.634935	−	0.772566i	\(-0.281027\pi\)
							0.163587	+	0.986529i	\(0.447694\pi\)
\(47\)	2.37124	+	1.36904i	0.345881	+	0.199695i	0.662870	−	0.748735i	\(-0.269338\pi\)
							−0.316988	+	0.948429i	\(0.602672\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.bu.a.715.59	yes	320
16.3	odd	4	inner	912.2.bu.a.259.76	✓	320
19.8	odd	6	inner	912.2.bu.a.331.76	yes	320
304.179	even	12	inner	912.2.bu.a.787.59	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.bu.a.259.76	✓	320	16.3	odd	4	inner
912.2.bu.a.331.76	yes	320	19.8	odd	6	inner
912.2.bu.a.715.59	yes	320	1.1	even	1	trivial
912.2.bu.a.787.59	yes	320	304.179	even	12	inner