Embedded newform 33.8.f.a.2.5

• Label: 33.8.f.a.2.5
• Level: $33$
• Weight: $8$
• Character: 33.2
• Analytic conductor: $10.309$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	33.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3087058410\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			2.5
Character	\(\chi\)	\(=\)	33.2
Dual form			33.8.f.a.17.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.00631 + 15.4078i) q^{2} +(43.8799 + 16.1726i) q^{3} +(-108.784 - 79.0360i) q^{4} +(190.716 - 61.9674i) q^{5} +(-468.861 + 595.129i) q^{6} +(-557.986 + 768.002i) q^{7} +(84.7229 - 61.5548i) q^{8} +(1663.89 + 1419.30i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 42 q^{3} - 1542 q^{4} - 1725 q^{6} - 10 q^{7} + 550 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.00631	+	15.4078i	−0.442499	+	1.36187i	0.442704	+	0.896668i	\(0.354019\pi\)
							−0.885203	+	0.465204i	\(0.845981\pi\)
\(3\)	43.8799	+	16.1726i	0.938299	+	0.345824i
							\(5\)	190.716	−	61.9674i	0.682327	−	0.221701i	0.0527131	−	0.998610i	\(-0.483213\pi\)
0.629614	+	0.776908i	\(0.283213\pi\)
\(7\)	−557.986	+	768.002i	−0.614865	+	0.846290i	−0.996967	−	0.0778309i	\(-0.975201\pi\)
							0.382101	+	0.924120i	\(0.375201\pi\)
\(11\)	−1161.25	+	4258.95i	−0.263058	+	0.964780i
							\(13\)	−2364.83	−	768.379i	−0.298537	−	0.0970004i	0.155919	−	0.987770i	\(-0.450166\pi\)
−0.454456	+	0.890769i	\(0.650166\pi\)
\(17\)	−173.365	−	533.564i	−0.00855837	−	0.0263400i	0.946686	−	0.322157i	\(-0.104408\pi\)
							−0.955245	+	0.295817i	\(0.904408\pi\)
\(19\)	−1996.13	−	2747.43i	−0.0667652	−	0.0918944i	0.774329	−	0.632784i	\(-0.218088\pi\)
							−0.841094	+	0.540889i	\(0.818088\pi\)
\(23\)		−	27004.8i		−	0.462799i	−0.972859	−	0.231400i	\(-0.925669\pi\)
							0.972859	−	0.231400i	\(-0.0743305\pi\)
\(29\)	207948.	+	151083.i	1.58329	+	1.15033i	0.912793	+	0.408422i	\(0.133921\pi\)
							0.670501	+	0.741909i	\(0.266079\pi\)
\(31\)	75509.0	−	232393.i	0.455232	−	1.40106i	−0.415631	−	0.909533i	\(-0.636439\pi\)
							0.870863	−	0.491526i	\(-0.163561\pi\)
\(37\)	92686.1	+	67340.4i	0.300821	+	0.218560i	0.727948	−	0.685632i	\(-0.240474\pi\)
							−0.427127	+	0.904192i	\(0.640474\pi\)
\(41\)	−479015.	+	348025.i	−1.08544	+	0.788619i	−0.978623	−	0.205660i	\(-0.934066\pi\)
							−0.106817	+	0.994279i	\(0.534066\pi\)
\(43\)		−	220955.i		−	0.423804i	−0.977291	−	0.211902i	\(-0.932034\pi\)
							0.977291	−	0.211902i	\(-0.0679657\pi\)
\(47\)	353185.	+	486118.i	0.496204	+	0.682966i	0.981517	−	0.191374i	\(-0.0612944\pi\)
							−0.485313	+	0.874340i	\(0.661294\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.f.a.2.5	✓	104
3.2	odd	2	inner	33.8.f.a.2.22	yes	104
11.6	odd	10	inner	33.8.f.a.17.22	yes	104
33.17	even	10	inner	33.8.f.a.17.5	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.f.a.2.5	✓	104	1.1	even	1	trivial
33.8.f.a.2.22	yes	104	3.2	odd	2	inner
33.8.f.a.17.5	yes	104	33.17	even	10	inner
33.8.f.a.17.22	yes	104	11.6	odd	10	inner