Embedded newform 33.4.e.b.25.2

• Label: 33.4.e.b.25.2
• Level: $33$
• Weight: $4$
• Character: 33.25
• Analytic conductor: $1.947$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94706303019\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.682515625.5
Defining polynomial:	\( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			25.2
Root			\(-0.390899 + 1.20306i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.25
Dual form			33.4.e.b.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.523388 - 0.380264i) q^{2} +(0.927051 + 2.85317i) q^{3} +(-2.34280 + 7.21040i) q^{4} +(9.01441 + 6.54935i) q^{5} +(1.57016 + 1.14079i) q^{6} +(8.07696 - 24.8583i) q^{7} +(3.11499 + 9.58696i) q^{8} +(-7.28115 + 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} - 6 q^{3} - 16 q^{4} + 9 q^{5} - 18 q^{6} + 3 q^{7} + 36 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.523388	−	0.380264i	0.185046	−	0.134443i	−0.491406	−	0.870931i	\(-0.663517\pi\)
							0.676452	+	0.736487i	\(0.263517\pi\)
\(3\)	0.927051	+	2.85317i	0.178411	+	0.549093i
							\(5\)	9.01441	+	6.54935i	0.806273	+	0.585792i	0.912748	−	0.408524i	\(-0.133956\pi\)
−0.106475	+	0.994315i	\(0.533956\pi\)
\(7\)	8.07696	−	24.8583i	0.436115	−	1.34222i	−0.455825	−	0.890069i	\(-0.650656\pi\)
							0.891940	−	0.452154i	\(-0.149344\pi\)
\(11\)	−36.0937	+	5.31471i	−0.989332	+	0.145677i
							\(13\)	43.2976	−	31.4576i	0.923738	−	0.671135i	−0.0207134	−	0.999785i	\(-0.506594\pi\)
0.944452	+	0.328650i	\(0.106594\pi\)
\(17\)	−18.9494	−	13.7676i	−0.270348	−	0.196419i	0.444349	−	0.895854i	\(-0.353435\pi\)
							−0.714697	+	0.699435i	\(0.753435\pi\)
\(19\)	−21.8916	−	67.3756i	−0.264331	−	0.813527i	−0.991847	−	0.127436i	\(-0.959325\pi\)
							0.727516	−	0.686091i	\(-0.240675\pi\)
\(23\)	164.114			1.48783			0.743914	−	0.668275i	\(-0.232967\pi\)
							0.743914	+	0.668275i	\(0.232967\pi\)
\(29\)	−67.5495	+	207.896i	−0.432539	+	1.33122i	0.463049	+	0.886333i	\(0.346755\pi\)
							−0.895588	+	0.444885i	\(0.853245\pi\)
\(31\)	62.0698	−	45.0964i	0.359615	−	0.261276i	−0.393276	−	0.919420i	\(-0.628658\pi\)
							0.752892	+	0.658145i	\(0.228658\pi\)
\(37\)	−87.5569	+	269.472i	−0.389034	+	1.19732i	0.544477	+	0.838776i	\(0.316728\pi\)
							−0.933511	+	0.358548i	\(0.883272\pi\)
\(41\)	1.50043	+	4.61784i	0.00571530	+	0.0175899i	0.953874	−	0.300209i	\(-0.0970564\pi\)
							−0.948158	+	0.317799i	\(0.897056\pi\)
\(43\)	−333.848			−1.18398			−0.591992	−	0.805944i	\(-0.701658\pi\)
							−0.591992	+	0.805944i	\(0.701658\pi\)
\(47\)	−121.601	−	374.250i	−0.377390	−	1.16149i	−0.941852	−	0.336029i	\(-0.890916\pi\)
							0.564461	−	0.825459i	\(-0.309084\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.4.e.b.25.2	yes	8
3.2	odd	2		99.4.f.b.91.1		8
11.2	odd	10		363.4.a.t.1.3		4
11.4	even	5	inner	33.4.e.b.4.2	✓	8
11.9	even	5		363.4.a.p.1.2		4
33.2	even	10		1089.4.a.z.1.2		4
33.20	odd	10		1089.4.a.bg.1.3		4
33.26	odd	10		99.4.f.b.37.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.b.4.2	✓	8	11.4	even	5	inner
33.4.e.b.25.2	yes	8	1.1	even	1	trivial
99.4.f.b.37.1		8	33.26	odd	10
99.4.f.b.91.1		8	3.2	odd	2
363.4.a.p.1.2		4	11.9	even	5
363.4.a.t.1.3		4	11.2	odd	10
1089.4.a.z.1.2		4	33.2	even	10
1089.4.a.bg.1.3		4	33.20	odd	10