Embedded newform 33.5.g.a.19.3

• Label: 33.5.g.a.19.3
• 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			19.3
Character	\(\chi\)	\(=\)	33.19
Dual form			33.5.g.a.7.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.44632 - 3.36708i) q^{2} +(1.60570 + 4.94183i) q^{3} +(-0.408428 + 1.25701i) q^{4} +(-26.0050 - 18.8937i) q^{5} +(12.7115 - 17.4958i) q^{6} +(-15.8191 - 5.13994i) q^{7} +(-58.1002 + 18.8779i) q^{8} +(-21.8435 + 15.8702i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.44632	−	3.36708i	−0.611581	−	0.841769i	0.385125	−	0.922864i	\(-0.374158\pi\)
							−0.996706	+	0.0810953i	\(0.974158\pi\)
\(3\)	1.60570	+	4.94183i	0.178411	+	0.549093i
							\(5\)	−26.0050	−	18.8937i	−1.04020	−	0.755750i	−0.0698759	−	0.997556i	\(-0.522260\pi\)
−0.970325	+	0.241806i	\(0.922260\pi\)
\(7\)	−15.8191	−	5.13994i	−0.322839	−	0.104897i	0.143114	−	0.989706i	\(-0.454288\pi\)
							−0.465953	+	0.884810i	\(0.654288\pi\)
\(11\)	−113.903	−	40.8310i	−0.941345	−	0.337446i
							\(13\)	−8.90352	−	12.2546i	−0.0526835	−	0.0725127i	0.781862	−	0.623451i	\(-0.214270\pi\)
−0.834546	+	0.550938i	\(0.814270\pi\)
\(17\)	314.068	−	432.277i	1.08674	−	1.49577i	0.234861	−	0.972029i	\(-0.424536\pi\)
							0.851878	−	0.523740i	\(-0.175464\pi\)
\(19\)	243.564	−	79.1388i	0.674693	−	0.219221i	0.0484227	−	0.998827i	\(-0.484581\pi\)
							0.626271	+	0.779606i	\(0.284581\pi\)
\(23\)	−385.548			−0.728825			−0.364412	−	0.931238i	\(-0.618730\pi\)
							−0.364412	+	0.931238i	\(0.618730\pi\)
\(29\)	902.886	+	293.365i	1.07359	+	0.348829i	0.791883	−	0.610672i	\(-0.209101\pi\)
							0.281703	+	0.959502i	\(0.409101\pi\)
\(31\)	−849.184	+	616.968i	−0.883646	+	0.642006i	−0.934214	−	0.356714i	\(-0.883897\pi\)
							0.0505676	+	0.998721i	\(0.483897\pi\)
\(37\)	576.075	−	1772.98i	0.420800	−	1.29509i	−0.486159	−	0.873870i	\(-0.661602\pi\)
							0.906959	−	0.421219i	\(-0.138398\pi\)
\(41\)	−696.238	+	226.221i	−0.414181	+	0.134576i	−0.508693	−	0.860948i	\(-0.669871\pi\)
							0.0945117	+	0.995524i	\(0.469871\pi\)
\(43\)		−	1290.95i		−	0.698191i	−0.937087	−	0.349095i	\(-0.886489\pi\)
							0.937087	−	0.349095i	\(-0.113511\pi\)
\(47\)	−30.6010	−	94.1803i	−0.0138529	−	0.0426348i	0.943891	−	0.330257i	\(-0.107135\pi\)
							−0.957744	+	0.287622i	\(0.907135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.g.a.19.3	yes	32
3.2	odd	2		99.5.k.c.19.6		32
11.2	odd	10		363.5.c.e.241.9		32
11.7	odd	10	inner	33.5.g.a.7.3	✓	32
11.9	even	5		363.5.c.e.241.24		32
33.29	even	10		99.5.k.c.73.6		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.7.3	✓	32	11.7	odd	10	inner
33.5.g.a.19.3	yes	32	1.1	even	1	trivial
99.5.k.c.19.6		32	3.2	odd	2
99.5.k.c.73.6		32	33.29	even	10
363.5.c.e.241.9		32	11.2	odd	10
363.5.c.e.241.24		32	11.9	even	5