Embedded newform 912.2.bu.a.715.10

• Label: 912.2.bu.a.715.10
• 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			715.10
Character	\(\chi\)	\(=\)	912.715
Dual form			912.2.bu.a.787.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28653 - 0.587229i) q^{2} +(-0.965926 - 0.258819i) q^{3} +(1.31032 + 1.51098i) q^{4} +(-0.216417 + 0.807678i) q^{5} +(1.09071 + 0.900198i) q^{6} +1.64911 q^{7} +(-0.798484 - 2.71338i) q^{8} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28653	−	0.587229i	−0.909715	−	0.415234i
							\(3\)	−0.965926	−	0.258819i	−0.557678	−	0.149429i
\(5\)	−0.216417	+	0.807678i	−0.0967844	+	0.361204i								−0.997284	−	0.0736525i	\(-0.976534\pi\)
							0.900500	+	0.434857i	\(0.143201\pi\)
\(7\)	1.64911			0.623304			0.311652	−	0.950196i	\(-0.399118\pi\)
							0.311652	+	0.950196i	\(0.399118\pi\)
\(11\)	−3.10431	+	3.10431i	−0.935985	+	0.935985i	−0.998071	−	0.0620857i	\(-0.980225\pi\)
							0.0620857	+	0.998071i	\(0.480225\pi\)
\(13\)	−4.07559	+	1.09205i	−1.13036	+	0.302880i	−0.775071	−	0.631874i	\(-0.782286\pi\)
							−0.355294	+	0.934755i	\(0.615619\pi\)
\(17\)	−1.17378	−	2.03305i	−0.284684	−	0.493086i	0.687849	−	0.725854i	\(-0.258555\pi\)
							−0.972532	+	0.232768i	\(0.925222\pi\)
\(19\)	−1.35914	−	4.14159i	−0.311809	−	0.950145i
							\(23\)	2.71701	−	4.70600i	0.566536	−	0.981269i	−0.430369	−	0.902653i	\(-0.641617\pi\)
0.996905	−	0.0786161i	\(-0.0250501\pi\)
\(29\)	−8.70881	+	2.33352i	−1.61719	+	0.433324i	−0.950173	−	0.311723i	\(-0.899094\pi\)
							−0.667013	+	0.745046i	\(0.732427\pi\)
\(31\)	5.51609			0.990720			0.495360	−	0.868688i	\(-0.335036\pi\)
							0.495360	+	0.868688i	\(0.335036\pi\)
\(37\)	3.82759	−	3.82759i	0.629251	−	0.629251i	−0.318629	−	0.947880i	\(-0.603222\pi\)
							0.947880	+	0.318629i	\(0.103222\pi\)
\(41\)	−5.68159	−	9.84081i	−0.887316	−	1.53688i	−0.843036	−	0.537857i	\(-0.819234\pi\)
							−0.0442795	−	0.999019i	\(-0.514099\pi\)
\(43\)	2.95474	+	0.791719i	0.450593	+	0.120736i	0.476977	−	0.878916i	\(-0.341733\pi\)
							−0.0263839	+	0.999652i	\(0.508399\pi\)
\(47\)	−3.19086	−	1.84225i	−0.465435	−	0.268719i	0.248892	−	0.968531i	\(-0.419934\pi\)
							−0.714327	+	0.699812i	\(0.753267\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.bu.a.715.10	yes	320
16.3	odd	4	inner	912.2.bu.a.259.37	✓	320
19.8	odd	6	inner	912.2.bu.a.331.37	yes	320
304.179	even	12	inner	912.2.bu.a.787.10	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.bu.a.259.37	✓	320	16.3	odd	4	inner
912.2.bu.a.331.37	yes	320	19.8	odd	6	inner
912.2.bu.a.715.10	yes	320	1.1	even	1	trivial
912.2.bu.a.787.10	yes	320	304.179	even	12	inner