Embedded newform 36.6.h.a.23.11

• Label: 36.6.h.a.23.11
• Level: $36$
• Weight: $6$
• Character: 36.23
• 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			23.11
Character	\(\chi\)	\(=\)	36.23
Dual form			36.6.h.a.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90579 - 5.32616i) q^{2} +(9.73802 - 12.1726i) q^{3} +(-24.7359 + 20.3011i) q^{4} +(-51.8636 - 29.9435i) q^{5} +(-83.3916 - 28.6679i) q^{6} +(-33.8827 + 19.5622i) q^{7} +(155.268 + 93.0578i) q^{8} +(-53.3420 - 237.073i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(19\)	\(29\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90579	−	5.32616i	−0.336899	−	0.941541i
							\(3\)	9.73802	−	12.1726i	0.624694	−	0.780870i
\(5\)	−51.8636	−	29.9435i	−0.927765	−	0.535645i								−0.0416611	−	0.999132i	\(-0.513265\pi\)
							−0.886104	+	0.463486i	\(0.846598\pi\)
\(7\)	−33.8827	+	19.5622i	−0.261356	+	0.150894i	−0.624953	−	0.780662i	\(-0.714882\pi\)
							0.363597	+	0.931556i	\(0.381549\pi\)
\(11\)	−108.537	−	187.992i	−0.270456	−	0.468443i	0.698523	−	0.715588i	\(-0.253841\pi\)
							−0.968979	+	0.247144i	\(0.920508\pi\)
\(13\)	−532.156	+	921.721i	−0.873334	+	1.51266i	−0.0148071	+	0.999890i	\(0.504713\pi\)
							−0.858527	+	0.512769i	\(0.828620\pi\)
\(17\)		−	315.283i		−	0.264593i	−0.991210	−	0.132297i	\(-0.957765\pi\)
							0.991210	−	0.132297i	\(-0.0422351\pi\)
\(19\)		−	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	\(-0.676478\pi\)
							0.999525	+	0.0308168i	\(0.00981084\pi\)
\(29\)	5185.33	−	2993.75i	1.14494	−	0.661030i	0.197289	−	0.980345i	\(-0.436786\pi\)
							0.947648	+	0.319315i	\(0.103453\pi\)
\(31\)	−6949.56	−	4012.33i	−1.29883	−	0.749881i	−0.318629	−	0.947879i	\(-0.603222\pi\)
							−0.980202	+	0.197999i	\(0.936556\pi\)
\(37\)	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	0.104117	−	0.994565i	\(-0.466798\pi\)
							−0.809260	+	0.587450i	\(0.800132\pi\)
\(43\)	8780.43	−	5069.38i	0.724176	−	0.418103i	−0.0921115	−	0.995749i	\(-0.529362\pi\)
							0.816288	+	0.577645i	\(0.196028\pi\)
\(47\)	7272.14	+	12595.7i	0.480195	+	0.831721i	0.999742	−	0.0227202i	\(-0.00723270\pi\)
							−0.519547	+	0.854442i	\(0.673899\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.11	✓	56	36.11	even	6	inner
36.6.h.a.11.20	yes	56	9.2	odd	6	inner
36.6.h.a.23.11	yes	56	1.1	even	1	trivial
36.6.h.a.23.20	yes	56	4.3	odd	2	inner
108.6.h.a.35.9		56	9.7	even	3
108.6.h.a.35.18		56	36.7	odd	6
108.6.h.a.71.9		56	12.11	even	2
108.6.h.a.71.18		56	3.2	odd	2