Embedded newform 33.5.h.a.14.2

• Label: 33.5.h.a.14.2
• Level: $33$
• Weight: $5$
• Character: 33.14
• Analytic conductor: $3.411$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.41120878177\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			14.2
Character	\(\chi\)	\(=\)	33.14
Dual form			33.5.h.a.26.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.94112 - 5.42449i) q^{2} +(-8.99437 + 0.318385i) q^{3} +(-8.94835 + 27.5402i) q^{4} +(2.90630 - 4.00017i) q^{5} +(37.1750 + 47.5350i) q^{6} +(-0.401365 + 1.23527i) q^{7} +(82.6281 - 26.8475i) q^{8} +(80.7973 - 5.72735i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 90 q^{4} - 83 q^{6} - 86 q^{7} + 232 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.94112	−	5.42449i	−0.985280	−	1.35612i	−0.933936	−	0.357440i	\(-0.883649\pi\)
							−0.0513440	−	0.998681i	\(-0.516351\pi\)
\(3\)	−8.99437	+	0.318385i	−0.999374	+	0.0353761i
							\(5\)	2.90630	−	4.00017i	0.116252	−	0.160007i	−0.746926	−	0.664908i	\(-0.768471\pi\)
0.863177	+	0.504901i	\(0.168471\pi\)
\(7\)	−0.401365	+	1.23527i	−0.00819111	+	0.0252097i	−0.955069	−	0.296385i	\(-0.904219\pi\)
							0.946877	+	0.321595i	\(0.104219\pi\)
\(11\)	56.0716	+	107.224i	0.463402	+	0.886148i
							\(13\)	−173.872	+	126.325i	−1.02883	+	0.747486i	−0.968073	−	0.250667i	\(-0.919350\pi\)
−0.0607527	+	0.998153i	\(0.519350\pi\)
\(17\)	−323.805	+	445.680i	−1.12043	+	1.54214i	−0.315358	+	0.948973i	\(0.602124\pi\)
							−0.805076	+	0.593172i	\(0.797876\pi\)
\(19\)	−64.0338	−	197.076i	−0.177379	−	0.545917i	0.822355	−	0.568975i	\(-0.192660\pi\)
							−0.999734	+	0.0230580i	\(0.992660\pi\)
\(23\)		−	602.543i		−	1.13902i	−0.821983	−	0.569511i	\(-0.807132\pi\)
							0.821983	−	0.569511i	\(-0.192868\pi\)
\(29\)	387.536	+	125.918i	0.460804	+	0.149724i	0.530214	−	0.847864i	\(-0.322112\pi\)
							−0.0694099	+	0.997588i	\(0.522112\pi\)
\(31\)	−646.607	+	469.787i	−0.672848	+	0.488853i	−0.870977	−	0.491323i	\(-0.836513\pi\)
							0.198129	+	0.980176i	\(0.436513\pi\)
\(37\)	−171.280	+	527.147i	−0.125114	+	0.385060i	−0.993922	−	0.110089i	\(-0.964886\pi\)
							0.868808	+	0.495149i	\(0.164886\pi\)
\(41\)	−844.917	+	274.530i	−0.502627	+	0.163314i	−0.549347	−	0.835594i	\(-0.685123\pi\)
							0.0467193	+	0.998908i	\(0.485123\pi\)
\(43\)	−2576.09			−1.39324			−0.696618	−	0.717442i	\(-0.745313\pi\)
							−0.696618	+	0.717442i	\(0.745313\pi\)
\(47\)	1407.45	−	457.307i	0.637141	−	0.207020i	0.0274055	−	0.999624i	\(-0.491275\pi\)
							0.609736	+	0.792605i	\(0.291275\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.h.a.14.2	✓	56
3.2	odd	2	inner	33.5.h.a.14.13	yes	56
11.4	even	5	inner	33.5.h.a.26.13	yes	56
33.26	odd	10	inner	33.5.h.a.26.2	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.h.a.14.2	✓	56	1.1	even	1	trivial
33.5.h.a.14.13	yes	56	3.2	odd	2	inner
33.5.h.a.26.2	yes	56	33.26	odd	10	inner
33.5.h.a.26.13	yes	56	11.4	even	5	inner