Embedded newform 33.6.f.a.17.13

• Label: 33.6.f.a.17.13
• Level: $33$
• Weight: $6$
• Character: 33.17
• 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			17.13
Character	\(\chi\)	\(=\)	33.17
Dual form			33.6.f.a.2.13

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.f.a.2.6	✓	72	11.2	odd	10	inner
33.6.f.a.2.13	yes	72	33.2	even	10	inner
33.6.f.a.17.6	yes	72	3.2	odd	2	inner
33.6.f.a.17.13	yes	72	1.1	even	1	trivial