Embedded newform 33.6.f.a.2.13

• Label: 33.6.f.a.2.13
• Level: $33$
• Weight: $6$
• Character: 33.2
• Analytic conductor: $5.293$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.29266605383\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			2.13
Character	\(\chi\)	\(=\)	33.2
Dual form			33.6.f.a.17.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36557 - 4.20281i) q^{2} +(-11.6039 - 10.4091i) q^{3} +(10.0898 + 7.33064i) q^{4} +(-98.1154 + 31.8796i) q^{5} +(-59.5934 + 34.5547i) q^{6} +(-5.71539 + 7.86655i) q^{7} +(158.991 - 115.514i) q^{8} +(26.3023 + 241.572i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 18 q^{3} - 262 q^{4} + 15 q^{6} - 10 q^{7} + 292 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36557	−	4.20281i	0.241402	−	0.742958i	−0.754806	−	0.655948i	\(-0.772269\pi\)
							0.996207	−	0.0870098i	\(-0.0277311\pi\)
\(3\)	−11.6039	−	10.4091i	−0.744392	−	0.667742i
							\(5\)	−98.1154	+	31.8796i	−1.75514	+	0.570280i	−0.996679	−	0.0814355i	\(-0.974050\pi\)
−0.758463	+	0.651716i	\(0.774050\pi\)
\(7\)	−5.71539	+	7.86655i	−0.0440860	+	0.0606791i	−0.830491	−	0.557032i	\(-0.811940\pi\)
							0.786405	+	0.617711i	\(0.211940\pi\)
\(11\)	−294.033	+	273.122i	−0.732681	+	0.680573i
							\(13\)	−752.218	−	244.410i	−1.23448	−	0.401108i	−0.382147	−	0.924102i	\(-0.624815\pi\)
−0.852336	+	0.522994i	\(0.824815\pi\)
\(17\)	−218.298	−	671.853i	−0.183201	−	0.563835i	0.816712	−	0.577046i	\(-0.195795\pi\)
							−0.999913	+	0.0132112i	\(0.995795\pi\)
\(19\)	−1020.61	−	1404.76i	−0.648601	−	0.892723i	0.350436	−	0.936587i	\(-0.386033\pi\)
							−0.999038	+	0.0438636i	\(0.986033\pi\)
\(23\)			1918.02i			0.756020i	0.925801	+	0.378010i	\(0.123392\pi\)
							−0.925801	+	0.378010i	\(0.876608\pi\)
\(29\)	−4400.01	−	3196.79i	−0.971535	−	0.705861i	−0.0157339	−	0.999876i	\(-0.505008\pi\)
							−0.955801	+	0.294015i	\(0.905008\pi\)
\(31\)	−805.352	+	2478.62i	−0.150516	+	0.463239i	−0.997679	−	0.0680931i	\(-0.978309\pi\)
							0.847163	+	0.531333i	\(0.178309\pi\)
\(37\)	533.898	+	387.900i	0.0641142	+	0.0465817i	0.619381	−	0.785091i	\(-0.287384\pi\)
							−0.555267	+	0.831672i	\(0.687384\pi\)
\(41\)	−6671.34	+	4847.01i	−0.619803	+	0.450313i	−0.852853	−	0.522152i	\(-0.825129\pi\)
							0.233050	+	0.972465i	\(0.425129\pi\)
\(43\)		−	1556.30i		−	0.128358i	−0.997938	−	0.0641789i	\(-0.979557\pi\)
							0.997938	−	0.0641789i	\(-0.0204428\pi\)
\(47\)	6056.57	+	8336.15i	0.399928	+	0.550454i	0.960726	−	0.277499i	\(-0.0895055\pi\)
							−0.560798	+	0.827953i	\(0.689505\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.f.a.2.13	yes	72
3.2	odd	2	inner	33.6.f.a.2.6	✓	72
11.6	odd	10	inner	33.6.f.a.17.6	yes	72
33.17	even	10	inner	33.6.f.a.17.13	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.2.6	✓	72	3.2	odd	2	inner
33.6.f.a.2.13	yes	72	1.1	even	1	trivial
33.6.f.a.17.6	yes	72	11.6	odd	10	inner
33.6.f.a.17.13	yes	72	33.17	even	10	inner