Embedded newform 33.6.e.a.31.1

• Label: 33.6.e.a.31.1
• Level: $33$
• Weight: $6$
• Character: 33.31
• Analytic conductor: $5.293$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.29266605383\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 3*x^19 + 142*x^18 - 389*x^17 + 14927*x^16 - 6599*x^15 + 1399353*x^14 + 1969563*x^13 + 129076450*x^12 + 14028509*x^11 + 4158720649*x^10 - 7877307064*x^9 + 107078602608*x^8 - 53496436514*x^7 + 2527225278875*x^6 - 6173773919508*x^5 + 63231264058241*x^4 - 118816044595930*x^3 + 398041877081860*x^2 - 161017247606600*x + 25735126080400
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{4}\cdot 11^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			31.1
Root			\(-3.00337 - 9.24343i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.31
Dual form			33.6.e.a.16.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.69436 + 8.29237i) q^{2} +(-7.28115 + 5.29007i) q^{3} +(-35.6154 - 25.8761i) q^{4} +(-23.0548 - 70.9555i) q^{5} +(-24.2492 - 74.6314i) q^{6} +(46.3463 + 33.6726i) q^{7} +(84.8093 - 61.6176i) q^{8} +(25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.69436	+	8.29237i	−0.476299	+	1.46590i	0.367898	+	0.929866i	\(0.380077\pi\)
							−0.844197	+	0.536032i	\(0.819923\pi\)
\(3\)	−7.28115	+	5.29007i	−0.467086	+	0.339358i
							\(5\)	−23.0548	−	70.9555i	−0.412417	−	1.26929i	−0.914541	−	0.404494i	\(-0.867448\pi\)
0.502123	−	0.864796i	\(-0.332552\pi\)
\(7\)	46.3463	+	33.6726i	0.357495	+	0.259735i	0.752007	−	0.659156i	\(-0.229086\pi\)
							−0.394511	+	0.918891i	\(0.629086\pi\)
\(11\)	−338.498	+	215.570i	−0.843479	+	0.537162i
							\(13\)	287.812	−	885.794i	0.472335	−	1.45370i	−0.377183	−	0.926139i	\(-0.623107\pi\)
0.849518	−	0.527560i	\(-0.176893\pi\)
\(17\)	−303.985	−	935.571i	−0.255112	−	0.785153i	−0.993808	−	0.111114i	\(-0.964558\pi\)
							0.738696	−	0.674039i	\(-0.235442\pi\)
\(19\)	−365.049	+	265.224i	−0.231989	+	0.168550i	−0.697707	−	0.716383i	\(-0.745796\pi\)
							0.465718	+	0.884933i	\(0.345796\pi\)
\(23\)	−4608.25			−1.81642			−0.908210	−	0.418515i	\(-0.862551\pi\)
							−0.908210	+	0.418515i	\(0.862551\pi\)
\(29\)	−1847.35	−	1342.18i	−0.407900	−	0.296357i	0.364851	−	0.931066i	\(-0.381120\pi\)
							−0.772751	+	0.634709i	\(0.781120\pi\)
\(31\)	−400.413	+	1232.34i	−0.0748348	+	0.230318i	−0.981476	−	0.191584i	\(-0.938637\pi\)
							0.906641	+	0.421902i	\(0.138637\pi\)
\(37\)	2592.40	+	1883.49i	0.311313	+	0.226182i	0.732460	−	0.680810i	\(-0.238372\pi\)
							−0.421146	+	0.906993i	\(0.638372\pi\)
\(41\)	2443.29	−	1775.15i	0.226994	−	0.164921i	−0.468475	−	0.883477i	\(-0.655196\pi\)
							0.695469	+	0.718556i	\(0.255196\pi\)
\(43\)	−13692.5			−1.12930			−0.564652	−	0.825329i	\(-0.690990\pi\)
							−0.564652	+	0.825329i	\(0.690990\pi\)
\(47\)	−14644.0	+	10639.5i	−0.966973	+	0.702547i	−0.954760	−	0.297379i	\(-0.903888\pi\)
							−0.0122130	+	0.999925i	\(0.503888\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.31.1	yes	20
3.2	odd	2		99.6.f.c.64.5		20
11.4	even	5		363.6.a.u.1.2		10
11.5	even	5	inner	33.6.e.a.16.1	✓	20
11.7	odd	10		363.6.a.q.1.9		10
33.5	odd	10		99.6.f.c.82.5		20
33.26	odd	10		1089.6.a.bh.1.9		10
33.29	even	10		1089.6.a.bl.1.2		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.16.1	✓	20	11.5	even	5	inner
33.6.e.a.31.1	yes	20	1.1	even	1	trivial
99.6.f.c.64.5		20	3.2	odd	2
99.6.f.c.82.5		20	33.5	odd	10
363.6.a.q.1.9		10	11.7	odd	10
363.6.a.u.1.2		10	11.4	even	5
1089.6.a.bh.1.9		10	33.26	odd	10
1089.6.a.bl.1.2		10	33.29	even	10