Embedded newform 33.5.g.a.19.4

• Label: 33.5.g.a.19.4
• Level: $33$
• Weight: $5$
• Character: 33.19
• Analytic conductor: $3.411$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	33.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.41120878177\)
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			19.4
Character	\(\chi\)	\(=\)	33.19
Dual form			33.5.g.a.7.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95593 - 2.69210i) q^{2} +(-1.60570 - 4.94183i) q^{3} +(1.52251 - 4.68579i) q^{4} +(6.63359 + 4.81958i) q^{5} +(-10.1633 + 13.9886i) q^{6} +(-71.1959 - 23.1329i) q^{7} +(-66.2286 + 21.5190i) q^{8} +(-21.8435 + 15.8702i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95593	−	2.69210i	−0.488982	−	0.673026i	0.491218	−	0.871037i	\(-0.336552\pi\)
							−0.980200	+	0.198011i	\(0.936552\pi\)
\(3\)	−1.60570	−	4.94183i	−0.178411	−	0.549093i
							\(5\)	6.63359	+	4.81958i	0.265344	+	0.192783i	0.712499	−	0.701673i	\(-0.247563\pi\)
−0.447156	+	0.894456i	\(0.647563\pi\)
\(7\)	−71.1959	−	23.1329i	−1.45298	−	0.472101i	−0.527060	−	0.849828i	\(-0.676706\pi\)
							−0.925917	+	0.377727i	\(0.876706\pi\)
\(11\)	118.191	+	25.9219i	0.976783	+	0.214231i
							\(13\)	−135.531	−	186.543i	−0.801961	−	1.10380i	−0.992514	−	0.122127i	\(-0.961028\pi\)
0.190554	−	0.981677i	\(-0.438972\pi\)
\(17\)	106.423	−	146.479i	0.368246	−	0.506847i	−0.584177	−	0.811626i	\(-0.698583\pi\)
							0.952423	+	0.304779i	\(0.0985828\pi\)
\(19\)	436.761	−	141.912i	1.20986	−	0.393109i	0.366484	−	0.930424i	\(-0.380562\pi\)
							0.843381	+	0.537316i	\(0.180562\pi\)
\(23\)	331.404			0.626474			0.313237	−	0.949675i	\(-0.398587\pi\)
							0.313237	+	0.949675i	\(0.398587\pi\)
\(29\)	−1156.66	−	375.821i	−1.37534	−	0.446874i	−0.474203	−	0.880415i	\(-0.657264\pi\)
							−0.901134	+	0.433541i	\(0.857264\pi\)
\(31\)	675.256	−	490.602i	0.702659	−	0.510512i	−0.178138	−	0.984006i	\(-0.557007\pi\)
							0.880797	+	0.473494i	\(0.157007\pi\)
\(37\)	−378.977	+	1166.37i	−0.276828	+	0.851988i	0.711903	+	0.702278i	\(0.247834\pi\)
							−0.988730	+	0.149709i	\(0.952166\pi\)
\(41\)	2038.87	−	662.470i	1.21289	−	0.394093i	0.368404	−	0.929666i	\(-0.379904\pi\)
							0.844489	+	0.535573i	\(0.179904\pi\)
\(43\)		−	630.662i		−	0.341083i	−0.985351	−	0.170541i	\(-0.945448\pi\)
							0.985351	−	0.170541i	\(-0.0545516\pi\)
\(47\)	372.078	+	1145.14i	0.168437	+	0.518397i	0.999273	−	0.0381209i	\(-0.0121372\pi\)
							−0.830836	+	0.556518i	\(0.812137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.g.a.19.4	yes	32
3.2	odd	2		99.5.k.c.19.5		32
11.2	odd	10		363.5.c.e.241.10		32
11.7	odd	10	inner	33.5.g.a.7.4	✓	32
11.9	even	5		363.5.c.e.241.23		32
33.29	even	10		99.5.k.c.73.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.7.4	✓	32	11.7	odd	10	inner
33.5.g.a.19.4	yes	32	1.1	even	1	trivial
99.5.k.c.19.5		32	3.2	odd	2
99.5.k.c.73.5		32	33.29	even	10
363.5.c.e.241.10		32	11.2	odd	10
363.5.c.e.241.23		32	11.9	even	5