Embedded newform 33.5.g.a.7.1

• Label: 33.5.g.a.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	33.7
Dual form			33.5.g.a.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.37135 + 6.01665i) q^{2} +(-1.60570 + 4.94183i) q^{3} +(-12.1471 - 37.3849i) q^{4} +(-24.8968 + 18.0886i) q^{5} +(-22.7142 - 31.2634i) q^{6} +(40.6127 - 13.1959i) q^{7} +(164.863 + 53.5673i) q^{8} +(-21.8435 - 15.8702i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.37135	+	6.01665i	−1.09284	+	1.50416i	−0.248288	+	0.968686i	\(0.579868\pi\)
							−0.844550	+	0.535476i	\(0.820132\pi\)
\(3\)	−1.60570	+	4.94183i	−0.178411	+	0.549093i
							\(5\)	−24.8968	+	18.0886i	−0.995873	+	0.723544i	−0.961199	−	0.275855i	\(-0.911039\pi\)
−0.0346733	+	0.999399i	\(0.511039\pi\)
\(7\)	40.6127	−	13.1959i	0.828831	−	0.269304i	0.136278	−	0.990671i	\(-0.456486\pi\)
							0.692553	+	0.721367i	\(0.256486\pi\)
\(11\)	−55.8909	−	107.318i	−0.461908	−	0.886928i
							\(13\)	−92.9028	+	127.870i	−0.549721	+	0.756626i	−0.989974	−	0.141248i	\(-0.954889\pi\)
0.440253	+	0.897874i	\(0.354889\pi\)
\(17\)	−52.6147	−	72.4180i	−0.182058	−	0.250581i	0.708227	−	0.705984i	\(-0.249495\pi\)
							−0.890285	+	0.455403i	\(0.849495\pi\)
\(19\)	−624.099	−	202.782i	−1.72881	−	0.561723i	−0.735531	−	0.677492i	\(-0.763067\pi\)
							−0.993276	+	0.115768i	\(0.963067\pi\)
\(23\)	−292.602			−0.553123			−0.276561	−	0.960996i	\(-0.589195\pi\)
							−0.276561	+	0.960996i	\(0.589195\pi\)
\(29\)	−646.603	+	210.094i	−0.768850	+	0.249815i	−0.667073	−	0.744993i	\(-0.732453\pi\)
							−0.101777	+	0.994807i	\(0.532453\pi\)
\(31\)	1421.82	+	1033.01i	1.47952	+	1.07493i	0.977711	+	0.209955i	\(0.0673319\pi\)
							0.501809	+	0.864979i	\(0.332668\pi\)
\(37\)	56.5273	+	173.973i	0.0412909	+	0.127080i	0.969577	−	0.244786i	\(-0.0787178\pi\)
							−0.928286	+	0.371867i	\(0.878718\pi\)
\(41\)	−553.714	−	179.912i	−0.329395	−	0.107027i	0.139650	−	0.990201i	\(-0.455402\pi\)
							−0.469046	+	0.883174i	\(0.655402\pi\)
\(43\)		−	298.729i		−	0.161563i	−0.996732	−	0.0807813i	\(-0.974258\pi\)
							0.996732	−	0.0807813i	\(-0.0257415\pi\)
\(47\)	−662.592	+	2039.25i	−0.299951	+	0.923154i	0.681562	+	0.731760i	\(0.261301\pi\)
							−0.981513	+	0.191394i	\(0.938699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.g.a.7.1	✓	32
3.2	odd	2		99.5.k.c.73.8		32
11.5	even	5		363.5.c.e.241.2		32
11.6	odd	10		363.5.c.e.241.31		32
11.8	odd	10	inner	33.5.g.a.19.1	yes	32
33.8	even	10		99.5.k.c.19.8		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.7.1	✓	32	1.1	even	1	trivial
33.5.g.a.19.1	yes	32	11.8	odd	10	inner
99.5.k.c.19.8		32	33.8	even	10
99.5.k.c.73.8		32	3.2	odd	2
363.5.c.e.241.2		32	11.5	even	5
363.5.c.e.241.31		32	11.6	odd	10