Embedded newform 363.3.g.h.94.5

• Label: 363.3.g.h.94.5
• Level: $363$
• Weight: $3$
• Character: 363.94
• Analytic conductor: $9.891$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.g (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89103359628\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			94.5
Character	\(\chi\)	\(=\)	363.94
Dual form			363.3.g.h.112.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.118031 - 0.0383504i) q^{2} +(-1.40126 - 1.01807i) q^{3} +(-3.22361 + 2.34209i) q^{4} +(1.40805 - 4.33353i) q^{5} +(-0.204435 - 0.0664249i) q^{6} +(2.97983 + 4.10138i) q^{7} +(-0.582452 + 0.801676i) q^{8} +(0.927051 + 2.85317i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 24 q^{4} - 16 q^{5} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.118031	−	0.0383504i	0.0590153	−	0.0191752i	−0.279361	−	0.960186i	\(-0.590122\pi\)
							0.338376	+	0.941011i	\(0.390122\pi\)
\(3\)	−1.40126	−	1.01807i	−0.467086	−	0.339358i
							\(5\)	1.40805	−	4.33353i	0.281610	−	0.866705i	−0.705785	−	0.708426i	\(-0.749405\pi\)
0.987394	−	0.158279i	\(-0.0505946\pi\)
\(7\)	2.97983	+	4.10138i	0.425690	+	0.585911i	0.966957	−	0.254939i	\(-0.0820554\pi\)
							−0.541268	+	0.840850i	\(0.682055\pi\)
\(11\)		0			0
							\(13\)	6.85393	−	2.22698i	0.527225	−	0.171306i	−0.0332967	−	0.999446i	\(-0.510601\pi\)
0.560522	+	0.828140i	\(0.310601\pi\)
\(17\)	−22.1790	−	7.20639i	−1.30465	−	0.423905i	−0.427450	−	0.904039i	\(-0.640588\pi\)
							−0.877195	+	0.480134i	\(0.840588\pi\)
\(19\)	15.3436	−	21.1186i	0.807557	−	1.11151i	−0.184139	−	0.982900i	\(-0.558950\pi\)
							0.991696	−	0.128606i	\(-0.0410504\pi\)
\(23\)	−41.1662			−1.78983			−0.894917	−	0.446233i	\(-0.852765\pi\)
							−0.894917	+	0.446233i	\(0.852765\pi\)
\(29\)	−29.1794	−	40.1620i	−1.00619	−	1.38490i	−0.921451	−	0.388495i	\(-0.872995\pi\)
							−0.0847366	−	0.996403i	\(-0.527005\pi\)
\(31\)	−2.92345	−	8.99745i	−0.0943048	−	0.290240i	0.892767	−	0.450519i	\(-0.148761\pi\)
							−0.987072	+	0.160278i	\(0.948761\pi\)
\(37\)	−40.0719	+	29.1139i	−1.08302	+	0.786863i	−0.978208	−	0.207629i	\(-0.933425\pi\)
							−0.104816	+	0.994492i	\(0.533425\pi\)
\(41\)	40.9082	−	56.3053i	0.997761	−	1.37330i	0.0710727	−	0.997471i	\(-0.477358\pi\)
							0.926689	−	0.375830i	\(-0.122642\pi\)
\(43\)		−	47.5666i		−	1.10620i	−0.833115	−	0.553100i	\(-0.813445\pi\)
							0.833115	−	0.553100i	\(-0.186555\pi\)
\(47\)	−2.31172	−	1.67956i	−0.0491855	−	0.0357354i	0.562921	−	0.826511i	\(-0.309678\pi\)
							−0.612106	+	0.790775i	\(0.709678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.g.h.94.5		32
11.2	odd	10	inner	363.3.g.h.112.5		32
11.3	even	5		363.3.c.d.241.4	✓	8
11.4	even	5	inner	363.3.g.h.118.5		32
11.5	even	5	inner	363.3.g.h.40.4		32
11.6	odd	10	inner	363.3.g.h.40.5		32
11.7	odd	10	inner	363.3.g.h.118.4		32
11.8	odd	10		363.3.c.d.241.5	yes	8
11.9	even	5	inner	363.3.g.h.112.4		32
11.10	odd	2	inner	363.3.g.h.94.4		32
33.8	even	10		1089.3.c.j.604.4		8
33.14	odd	10		1089.3.c.j.604.5		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.c.d.241.4	✓	8	11.3	even	5
363.3.c.d.241.5	yes	8	11.8	odd	10
363.3.g.h.40.4		32	11.5	even	5	inner
363.3.g.h.40.5		32	11.6	odd	10	inner
363.3.g.h.94.4		32	11.10	odd	2	inner
363.3.g.h.94.5		32	1.1	even	1	trivial
363.3.g.h.112.4		32	11.9	even	5	inner
363.3.g.h.112.5		32	11.2	odd	10	inner
363.3.g.h.118.4		32	11.7	odd	10	inner
363.3.g.h.118.5		32	11.4	even	5	inner
1089.3.c.j.604.4		8	33.8	even	10
1089.3.c.j.604.5		8	33.14	odd	10