Embedded newform 36.6.h.a.11.20

• Label: 36.6.h.a.11.20
• Level: $36$
• Weight: $6$
• Character: 36.11
• Analytic conductor: $5.774$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	36.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.77381751327\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.20
Character	\(\chi\)	\(=\)	36.11
Dual form			36.6.h.a.23.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.65969 - 4.31354i) q^{2} +(-9.73802 - 12.1726i) q^{3} +(-5.21330 - 31.5725i) q^{4} +(-51.8636 + 29.9435i) q^{5} +(-88.1450 - 2.54244i) q^{6} +(33.8827 + 19.5622i) q^{7} +(-155.268 - 93.0578i) q^{8} +(-53.3420 + 237.073i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\)	\(19\)	\(29\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.65969	−	4.31354i	0.646948	−	0.762534i
							\(3\)	−9.73802	−	12.1726i	−0.624694	−	0.780870i
\(5\)	−51.8636	+	29.9435i	−0.927765	+	0.535645i								−0.886104	−	0.463486i	\(-0.846598\pi\)
							−0.0416611	+	0.999132i	\(0.513265\pi\)
\(7\)	33.8827	+	19.5622i	0.261356	+	0.150894i	0.624953	−	0.780662i	\(-0.285118\pi\)
							−0.363597	+	0.931556i	\(0.618451\pi\)
\(11\)	108.537	−	187.992i	0.270456	−	0.468443i	−0.698523	−	0.715588i	\(-0.746159\pi\)
							0.968979	+	0.247144i	\(0.0794922\pi\)
\(13\)	−532.156	−	921.721i	−0.873334	−	1.51266i	−0.858527	−	0.512769i	\(-0.828620\pi\)
							−0.0148071	−	0.999890i	\(-0.504713\pi\)
\(17\)			315.283i			0.264593i	0.991210	+	0.132297i	\(0.0422351\pi\)
							−0.991210	+	0.132297i	\(0.957765\pi\)
\(19\)		−	1889.72i		−	1.20092i	−0.799655	−	0.600460i	\(-0.794984\pi\)
							0.799655	−	0.600460i	\(-0.205016\pi\)
\(23\)	−1200.19	−	2078.79i	−0.473074	−	0.819389i	0.526451	−	0.850206i	\(-0.323522\pi\)
							−0.999525	+	0.0308168i	\(0.990189\pi\)
\(29\)	5185.33	+	2993.75i	1.14494	+	0.661030i	0.947648	−	0.319315i	\(-0.103453\pi\)
							0.197289	+	0.980345i	\(0.436786\pi\)
\(31\)	6949.56	−	4012.33i	1.29883	−	0.749881i	0.318629	−	0.947879i	\(-0.396778\pi\)
							0.980202	+	0.197999i	\(0.0634442\pi\)
\(37\)	5243.91			0.629725			0.314862	−	0.949137i	\(-0.398042\pi\)
							0.314862	+	0.949137i	\(0.398042\pi\)
\(41\)	−7589.92	+	4382.04i	−0.705144	+	0.407115i	−0.809260	−	0.587450i	\(-0.800132\pi\)
							0.104117	+	0.994565i	\(0.466798\pi\)
\(43\)	−8780.43	−	5069.38i	−0.724176	−	0.418103i	0.0921115	−	0.995749i	\(-0.470638\pi\)
							−0.816288	+	0.577645i	\(0.803972\pi\)
\(47\)	−7272.14	+	12595.7i	−0.480195	+	0.831721i	−0.999742	−	0.0227202i	\(-0.992767\pi\)
							0.519547	+	0.854442i	\(0.326101\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.11.20	yes	56
3.2	odd	2		108.6.h.a.35.9		56
4.3	odd	2	inner	36.6.h.a.11.11	✓	56
9.4	even	3		108.6.h.a.71.18		56
9.5	odd	6	inner	36.6.h.a.23.11	yes	56
12.11	even	2		108.6.h.a.35.18		56
36.23	even	6	inner	36.6.h.a.23.20	yes	56
36.31	odd	6		108.6.h.a.71.9		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.11	✓	56	4.3	odd	2	inner
36.6.h.a.11.20	yes	56	1.1	even	1	trivial
36.6.h.a.23.11	yes	56	9.5	odd	6	inner
36.6.h.a.23.20	yes	56	36.23	even	6	inner
108.6.h.a.35.9		56	3.2	odd	2
108.6.h.a.35.18		56	12.11	even	2
108.6.h.a.71.9		56	36.31	odd	6
108.6.h.a.71.18		56	9.4	even	3