Embedded newform 33.8.e.a.25.5

• Label: 33.8.e.a.25.5
• Level: $33$
• Weight: $8$
• Character: 33.25
• 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			25.5
Character	\(\chi\)	\(=\)	33.25
Dual form			33.8.e.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.67280 - 4.84807i) q^{2} +(-8.34346 - 25.6785i) q^{3} +(-18.5317 + 57.0347i) q^{4} +(269.190 + 195.578i) q^{5} +(-180.166 - 130.898i) q^{6} +(-36.4929 + 112.314i) q^{7} +(479.095 + 1474.50i) q^{8} +(-589.773 + 428.495i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 22 q^{2} + 189 q^{3} - 692 q^{4} - 777 q^{5} + 594 q^{6} - 83 q^{7} + 3568 q^{8} - 5103 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.67280	−	4.84807i	0.589798	−	0.428513i	−0.252445	−	0.967611i	\(-0.581235\pi\)
							0.842243	+	0.539098i	\(0.181235\pi\)
\(3\)	−8.34346	−	25.6785i	−0.178411	−	0.549093i
							\(5\)	269.190	+	195.578i	0.963083	+	0.699720i	0.953865	−	0.300237i	\(-0.0970658\pi\)
0.00921791	+	0.999958i	\(0.497066\pi\)
\(7\)	−36.4929	+	112.314i	−0.0402129	+	0.123762i	−0.969148	−	0.246481i	\(-0.920726\pi\)
							0.928935	+	0.370243i	\(0.120726\pi\)
\(11\)	4083.84	+	1676.14i	0.925111	+	0.379696i
							\(13\)	6585.24	−	4784.46i	0.831324	−	0.603992i	−0.0886098	−	0.996066i	\(-0.528242\pi\)
0.919934	+	0.392074i	\(0.128242\pi\)
\(17\)	1949.84	+	1416.64i	0.0962560	+	0.0699341i	0.634872	−	0.772617i	\(-0.281053\pi\)
							−0.538616	+	0.842551i	\(0.681053\pi\)
\(19\)	4248.60	+	13075.8i	0.142104	+	0.437353i	0.996627	−	0.0820613i	\(-0.0261503\pi\)
							−0.854523	+	0.519414i	\(0.826150\pi\)
\(23\)	20856.3			0.357430			0.178715	−	0.983901i	\(-0.442806\pi\)
							0.178715	+	0.983901i	\(0.442806\pi\)
\(29\)	−40052.4	+	123269.i	−0.304955	+	0.938554i	0.674739	+	0.738056i	\(0.264256\pi\)
							−0.979694	+	0.200498i	\(0.935744\pi\)
\(31\)	−106523.	+	77393.2i	−0.642208	+	0.466591i	−0.860608	−	0.509268i	\(-0.829916\pi\)
							0.218400	+	0.975859i	\(0.429916\pi\)
\(37\)	11937.4	−	36739.6i	0.0387440	−	0.119242i	−0.929814	−	0.368030i	\(-0.880032\pi\)
							0.968558	+	0.248788i	\(0.0800324\pi\)
\(41\)	−166928.	−	513752.i	−0.378256	−	1.16415i	−0.941255	−	0.337696i	\(-0.890352\pi\)
							0.562999	−	0.826458i	\(-0.309648\pi\)
\(43\)	−540279.			−1.03628			−0.518141	−	0.855295i	\(-0.673376\pi\)
							−0.518141	+	0.855295i	\(0.673376\pi\)
\(47\)	−325407.	−	1.00150e6i	−0.457177	−	1.40705i	−0.868560	−	0.495583i	\(-0.834954\pi\)
							0.411383	−	0.911462i	\(-0.365046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.e.a.25.5	yes	28
3.2	odd	2		99.8.f.c.91.3		28
11.2	odd	10		363.8.a.p.1.10		14
11.4	even	5	inner	33.8.e.a.4.5	✓	28
11.9	even	5		363.8.a.q.1.5		14
33.26	odd	10		99.8.f.c.37.3		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.a.4.5	✓	28	11.4	even	5	inner
33.8.e.a.25.5	yes	28	1.1	even	1	trivial
99.8.f.c.37.3		28	33.26	odd	10
99.8.f.c.91.3		28	3.2	odd	2
363.8.a.p.1.10		14	11.2	odd	10
363.8.a.q.1.5		14	11.9	even	5