Embedded newform 36.6.h.a.23.16

• Label: 36.6.h.a.23.16
• 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			23.16
Character	\(\chi\)	\(=\)	36.23
Dual form			36.6.h.a.11.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.253656 + 5.65116i) q^{2} +(-2.08778 + 15.4480i) q^{3} +(-31.8713 + 2.86690i) q^{4} +(84.3255 + 48.6853i) q^{5} +(-87.8289 - 7.87989i) q^{6} +(-87.9661 + 50.7872i) q^{7} +(-24.2857 - 179.383i) q^{8} +(-234.282 - 64.5040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.253656	+	5.65116i	0.0448405	+	0.998994i
							\(3\)	−2.08778	+	15.4480i	−0.133931	+	0.990991i
\(5\)	84.3255	+	48.6853i	1.50846	+	0.870910i								0.999951	+	0.00985262i	\(0.00313624\pi\)
							0.508508	+	0.861057i	\(0.330197\pi\)
\(7\)	−87.9661	+	50.7872i	−0.678532	+	0.391750i	−0.799302	−	0.600930i	\(-0.794797\pi\)
							0.120770	+	0.992681i	\(0.461464\pi\)
\(11\)	−73.2291	−	126.837i	−0.182474	−	0.316055i	0.760248	−	0.649633i	\(-0.225077\pi\)
							−0.942723	+	0.333578i	\(0.891744\pi\)
\(13\)	402.832	−	697.726i	0.661098	−	1.14506i	−0.319229	−	0.947678i	\(-0.603424\pi\)
							0.980327	−	0.197378i	\(-0.0632427\pi\)
\(17\)			1238.37i			1.03927i	0.854389	+	0.519633i	\(0.173931\pi\)
							−0.854389	+	0.519633i	\(0.826069\pi\)
\(19\)			2616.12i			1.66254i	0.555866	+	0.831272i	\(0.312387\pi\)
							−0.555866	+	0.831272i	\(0.687613\pi\)
\(23\)	84.5912	−	146.516i	0.0333431	−	0.0577519i	−0.848872	−	0.528598i	\(-0.822718\pi\)
							0.882215	+	0.470846i	\(0.156051\pi\)
\(29\)	1952.82	−	1127.46i	0.431189	−	0.248947i	−0.268664	−	0.963234i	\(-0.586582\pi\)
							0.699853	+	0.714287i	\(0.253249\pi\)
\(31\)	1787.03	+	1031.74i	0.333985	+	0.192826i	0.657609	−	0.753359i	\(-0.271568\pi\)
							−0.323624	+	0.946186i	\(0.604901\pi\)
\(37\)	445.632			0.0535146			0.0267573	−	0.999642i	\(-0.491482\pi\)
							0.0267573	+	0.999642i	\(0.491482\pi\)
\(41\)	−1720.61	−	993.393i	−0.159853	−	0.0922914i	0.417939	−	0.908475i	\(-0.362752\pi\)
							−0.577793	+	0.816184i	\(0.696086\pi\)
\(43\)	3464.26	−	2000.09i	0.285719	−	0.164960i	−0.350291	−	0.936641i	\(-0.613917\pi\)
							0.636010	+	0.771681i	\(0.280584\pi\)
\(47\)	−3539.24	−	6130.14i	−0.233703	−	0.404786i	0.725192	−	0.688547i	\(-0.241751\pi\)
							−0.958895	+	0.283761i	\(0.908418\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.23.16	yes	56
3.2	odd	2		108.6.h.a.71.13		56
4.3	odd	2	inner	36.6.h.a.23.6	yes	56
9.2	odd	6	inner	36.6.h.a.11.6	✓	56
9.7	even	3		108.6.h.a.35.23		56
12.11	even	2		108.6.h.a.71.23		56
36.7	odd	6		108.6.h.a.35.13		56
36.11	even	6	inner	36.6.h.a.11.16	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.6	✓	56	9.2	odd	6	inner
36.6.h.a.11.16	yes	56	36.11	even	6	inner
36.6.h.a.23.6	yes	56	4.3	odd	2	inner
36.6.h.a.23.16	yes	56	1.1	even	1	trivial
108.6.h.a.35.13		56	36.7	odd	6
108.6.h.a.35.23		56	9.7	even	3
108.6.h.a.71.13		56	3.2	odd	2
108.6.h.a.71.23		56	12.11	even	2