Embedded newform 33.5.g.a.7.4

• Label: 33.5.g.a.7.4
• Level: $33$
• Weight: $5$
• Character: 33.7
• 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			7.4
Character	\(\chi\)	\(=\)	33.7
Dual form			33.5.g.a.19.4

$q$-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 \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.7.4	✓	32	1.1	even	1	trivial
33.5.g.a.19.4	yes	32	11.8	odd	10	inner
99.5.k.c.19.5		32	33.8	even	10
99.5.k.c.73.5		32	3.2	odd	2
363.5.c.e.241.10		32	11.5	even	5
363.5.c.e.241.23		32	11.6	odd	10