Embedded newform 33.6.e.a.25.1

• Label: 33.6.e.a.25.1
• Level: $33$
• Weight: $6$
• Character: 33.25
• 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			25.1
Root			\(-7.12695 - 5.17803i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.25
Dual form			33.6.e.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.93597 + 5.76582i) q^{2} +(2.78115 + 8.55951i) q^{3} +(19.8464 - 61.0809i) q^{4} +(-37.2750 - 27.0819i) q^{5} +(-71.4237 - 51.8924i) q^{6} +(37.2644 - 114.688i) q^{7} +(97.6803 + 300.629i) q^{8} +(-65.5304 + 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 45 q^{3} - 124 q^{4} - 33 q^{5} - 18 q^{6} - 335 q^{7} - 472 q^{8} - 405 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.93597	+	5.76582i	−1.40289	+	1.01926i	−0.408586	+	0.912720i	\(0.633978\pi\)
							−0.994308	+	0.106542i	\(0.966022\pi\)
\(3\)	2.78115	+	8.55951i	0.178411	+	0.549093i
							\(5\)	−37.2750	−	27.0819i	−0.666796	−	0.484456i	0.202155	−	0.979354i	\(-0.435206\pi\)
−0.868951	+	0.494898i	\(0.835206\pi\)
\(7\)	37.2644	−	114.688i	0.287441	−	0.884653i	−0.698215	−	0.715888i	\(-0.746022\pi\)
							0.985656	−	0.168765i	\(-0.0539779\pi\)
\(11\)	319.427	+	242.935i	0.795957	+	0.605354i
							\(13\)	488.591	−	354.982i	0.801839	−	0.582570i	−0.109614	−	0.993974i	\(-0.534962\pi\)
0.911453	+	0.411404i	\(0.134962\pi\)
\(17\)	−1125.77	−	817.922i	−0.944776	−	0.686420i	0.00478977	−	0.999989i	\(-0.498475\pi\)
							−0.949565	+	0.313569i	\(0.898475\pi\)
\(19\)	−258.169	−	794.562i	−0.164066	−	0.504944i	0.834900	−	0.550402i	\(-0.185525\pi\)
							−0.998966	+	0.0454574i	\(0.985525\pi\)
\(23\)	3264.14			1.28662			0.643309	−	0.765607i	\(-0.277561\pi\)
							0.643309	+	0.765607i	\(0.277561\pi\)
\(29\)	2614.37	−	8046.20i	0.577261	−	1.77663i	−0.0510888	−	0.998694i	\(-0.516269\pi\)
							0.628349	−	0.777931i	\(-0.283731\pi\)
\(31\)	5178.44	−	3762.36i	0.967821	−	0.703163i	0.0128669	−	0.999917i	\(-0.495904\pi\)
							0.954954	+	0.296754i	\(0.0959042\pi\)
\(37\)	−3631.02	+	11175.1i	−0.436038	+	1.34199i	0.455982	+	0.889989i	\(0.349288\pi\)
							−0.892019	+	0.451997i	\(0.850712\pi\)
\(41\)	−3290.02	−	10125.6i	−0.305660	−	0.940725i	−0.979430	−	0.201784i	\(-0.935326\pi\)
							0.673770	−	0.738941i	\(-0.264674\pi\)
\(43\)	−2640.34			−0.217765			−0.108883	−	0.994055i	\(-0.534727\pi\)
							−0.108883	+	0.994055i	\(0.534727\pi\)
\(47\)	−4454.65	−	13710.0i	−0.294150	−	0.905300i	−0.983506	−	0.180877i	\(-0.942106\pi\)
							0.689356	−	0.724423i	\(-0.257894\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.6.e.a.25.1	yes	20
3.2	odd	2		99.6.f.c.91.5		20
11.2	odd	10		363.6.a.q.1.2		10
11.4	even	5	inner	33.6.e.a.4.1	✓	20
11.9	even	5		363.6.a.u.1.9		10
33.2	even	10		1089.6.a.bl.1.9		10
33.20	odd	10		1089.6.a.bh.1.2		10
33.26	odd	10		99.6.f.c.37.5		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.4.1	✓	20	11.4	even	5	inner
33.6.e.a.25.1	yes	20	1.1	even	1	trivial
99.6.f.c.37.5		20	33.26	odd	10
99.6.f.c.91.5		20	3.2	odd	2
363.6.a.q.1.2		10	11.2	odd	10
363.6.a.u.1.9		10	11.9	even	5
1089.6.a.bh.1.2		10	33.20	odd	10
1089.6.a.bl.1.9		10	33.2	even	10