Embedded newform 33.8.e.a.4.1

• Label: 33.8.e.a.4.1
• Level: $33$
• Weight: $8$
• Character: 33.4
• Analytic conductor: $10.309$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3087058410\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	33.4
Dual form			33.8.e.a.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.2616 - 13.2678i) q^{2} +(-8.34346 + 25.6785i) q^{3} +(117.897 + 362.849i) q^{4} +(-299.083 + 217.296i) q^{5} +(493.063 - 358.231i) q^{6} +(233.965 + 720.070i) q^{7} +(1768.39 - 5442.55i) q^{8} +(-589.773 - 428.495i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.2616	−	13.2678i	−1.61411	−	1.17272i	−0.847952	−	0.530073i	\(-0.822165\pi\)
							−0.766161	−	0.642649i	\(-0.777835\pi\)
\(3\)	−8.34346	+	25.6785i	−0.178411	+	0.549093i
							\(5\)	−299.083	+	217.296i	−1.07003	+	0.777423i	−0.975918	−	0.218138i	\(-0.930002\pi\)
−0.0941134	+	0.995561i	\(0.530002\pi\)
\(7\)	233.965	+	720.070i	0.257815	+	0.793472i	0.993262	+	0.115890i	\(0.0369720\pi\)
							−0.735447	+	0.677582i	\(0.763028\pi\)
\(11\)	−2654.48	+	3527.17i	−0.601318	+	0.799010i
							\(13\)	1647.29	+	1196.82i	0.207954	+	0.151088i	0.686888	−	0.726764i	\(-0.258976\pi\)
−0.478933	+	0.877851i	\(0.658976\pi\)
\(17\)	−3803.71	+	2763.56i	−0.187774	+	0.136426i	−0.677701	−	0.735338i	\(-0.737024\pi\)
							0.489927	+	0.871764i	\(0.337024\pi\)
\(19\)	−150.016	+	461.703i	−0.00501766	+	0.0154428i	−0.953534	−	0.301285i	\(-0.902584\pi\)
							0.948516	+	0.316728i	\(0.102584\pi\)
\(23\)	10368.5			0.177693			0.0888465	−	0.996045i	\(-0.471682\pi\)
							0.0888465	+	0.996045i	\(0.471682\pi\)
\(29\)	−37363.7	−	114994.i	−0.284484	−	0.875551i	−0.986553	−	0.163442i	\(-0.947740\pi\)
							0.702069	−	0.712109i	\(-0.252260\pi\)
\(31\)	−83629.9	−	60760.7i	−0.504192	−	0.366317i	0.306424	−	0.951895i	\(-0.400867\pi\)
							−0.810616	+	0.585578i	\(0.800867\pi\)
\(37\)	−137856.	−	424276.i	−0.447423	−	1.37703i	−0.879805	−	0.475335i	\(-0.842327\pi\)
							0.432382	−	0.901691i	\(-0.357673\pi\)
\(41\)	41466.7	−	127621.i	0.0939627	−	0.289188i	−0.893019	−	0.450018i	\(-0.851417\pi\)
							0.986982	+	0.160831i	\(0.0514174\pi\)
\(43\)	−412551.			−0.791293			−0.395647	−	0.918403i	\(-0.629480\pi\)
							−0.395647	+	0.918403i	\(0.629480\pi\)
\(47\)	−212827.	+	655015.i	−0.299009	+	0.920256i	0.682836	+	0.730572i	\(0.260746\pi\)
							−0.981845	+	0.189684i	\(0.939254\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.a.4.1	✓	28
3.2	odd	2		99.8.f.c.37.7		28
11.3	even	5	inner	33.8.e.a.25.1	yes	28
11.5	even	5		363.8.a.q.1.14		14
11.6	odd	10		363.8.a.p.1.1		14
33.14	odd	10		99.8.f.c.91.7		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.a.4.1	✓	28	1.1	even	1	trivial
33.8.e.a.25.1	yes	28	11.3	even	5	inner
99.8.f.c.37.7		28	3.2	odd	2
99.8.f.c.91.7		28	33.14	odd	10
363.8.a.p.1.1		14	11.6	odd	10
363.8.a.q.1.14		14	11.5	even	5