Embedded newform 36.6.h.a.23.6

• Label: 36.6.h.a.23.6
• 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			23.6
Character	\(\chi\)	\(=\)	36.23
Dual form			36.6.h.a.11.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.76722 - 3.04525i) q^{2} +(2.08778 - 15.4480i) q^{3} +(13.4528 + 29.0348i) q^{4} +(84.3255 + 48.6853i) q^{5} +(-56.9960 + 67.2863i) q^{6} +(87.9661 - 50.7872i) q^{7} +(24.2857 - 179.383i) q^{8} +(-234.282 - 64.5040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.76722	−	3.04525i	−0.842734	−	0.538330i
							\(3\)	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	−0.120770	−	0.992681i	\(-0.538536\pi\)
							0.799302	+	0.600930i	\(0.205203\pi\)
\(11\)	73.2291	+	126.837i	0.182474	+	0.316055i	0.942723	−	0.333578i	\(-0.108256\pi\)
							−0.760248	+	0.649633i	\(0.774923\pi\)
\(13\)	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	\(-0.687613\pi\)
							0.555866	−	0.831272i	\(-0.312387\pi\)
\(23\)	−84.5912	+	146.516i	−0.0333431	+	0.0577519i	−0.882215	−	0.470846i	\(-0.843949\pi\)
							0.848872	+	0.528598i	\(0.177282\pi\)
\(29\)	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	0.323624	−	0.946186i	\(-0.395099\pi\)
							−0.657609	+	0.753359i	\(0.728432\pi\)
\(37\)	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	−0.636010	−	0.771681i	\(-0.719416\pi\)
							0.350291	+	0.936641i	\(0.386083\pi\)
\(47\)	3539.24	+	6130.14i	0.233703	+	0.404786i	0.958895	−	0.283761i	\(-0.0915822\pi\)
							−0.725192	+	0.688547i	\(0.758249\pi\)

(See \(a_n\) instead)

Twists

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

    

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