Embedded newform 36.6.h.a.23.17

• Label: 36.6.h.a.23.17
• 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			23.17
Character	\(\chi\)	\(=\)	36.23
Dual form			36.6.h.a.11.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.72594 - 5.38713i) q^{2} +(-3.10718 + 15.2756i) q^{3} +(-26.0423 - 18.5957i) q^{4} +(-20.1454 - 11.6310i) q^{5} +(76.9290 + 43.1036i) q^{6} +(-156.832 + 90.5473i) q^{7} +(-145.125 + 108.198i) q^{8} +(-223.691 - 94.9283i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.72594	−	5.38713i	0.305106	−	0.952318i
							\(3\)	−3.10718	+	15.2756i	−0.199326	+	0.979933i
\(5\)	−20.1454	−	11.6310i	−0.360372	−	0.208061i								0.308872	−	0.951104i	\(-0.400049\pi\)
							−0.669244	+	0.743043i	\(0.733382\pi\)
\(7\)	−156.832	+	90.5473i	−1.20974	+	0.698442i	−0.962702	−	0.270565i	\(-0.912790\pi\)
							−0.247035	+	0.969007i	\(0.579456\pi\)
\(11\)	41.2642	+	71.4718i	0.102823	+	0.178095i	0.912847	−	0.408302i	\(-0.133879\pi\)
							−0.810023	+	0.586398i	\(0.800546\pi\)
\(13\)	−70.1922	+	121.577i	−0.115194	+	0.199522i	−0.917857	−	0.396910i	\(-0.870082\pi\)
							0.802663	+	0.596433i	\(0.203416\pi\)
\(17\)			901.814i			0.756824i	0.925637	+	0.378412i	\(0.123530\pi\)
							−0.925637	+	0.378412i	\(0.876470\pi\)
\(19\)		−	2363.52i		−	1.50202i	−0.660290	−	0.751011i	\(-0.729567\pi\)
							0.660290	−	0.751011i	\(-0.270433\pi\)
\(23\)	160.190	−	277.458i	0.0631417	−	0.109365i	−0.832726	−	0.553685i	\(-0.813221\pi\)
							0.895868	+	0.444320i	\(0.146555\pi\)
\(29\)	−7037.98	+	4063.38i	−1.55401	+	0.897207i	−0.556198	+	0.831050i	\(0.687740\pi\)
							−0.997809	+	0.0661569i	\(0.978926\pi\)
\(31\)	1533.72	+	885.495i	0.286644	+	0.165494i	0.636427	−	0.771337i	\(-0.280412\pi\)
							−0.349784	+	0.936831i	\(0.613745\pi\)
\(37\)	13713.9			1.64686			0.823429	−	0.567419i	\(-0.192058\pi\)
							0.823429	+	0.567419i	\(0.192058\pi\)
\(41\)	−4374.37	−	2525.55i	−0.406402	−	0.234637i	0.282840	−	0.959167i	\(-0.408723\pi\)
							−0.689243	+	0.724530i	\(0.742057\pi\)
\(43\)	−17261.6	+	9966.01i	−1.42367	+	0.821959i	−0.996611	−	0.0822625i	\(-0.973785\pi\)
							−0.427064	+	0.904221i	\(0.640452\pi\)
\(47\)	6620.98	+	11467.9i	0.437197	+	0.757248i	0.997472	−	0.0710587i	\(-0.0226378\pi\)
							−0.560275	+	0.828307i	\(0.689304\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.23.17	yes	56
3.2	odd	2		108.6.h.a.71.12		56
4.3	odd	2	inner	36.6.h.a.23.27	yes	56
9.2	odd	6	inner	36.6.h.a.11.27	yes	56
9.7	even	3		108.6.h.a.35.2		56
12.11	even	2		108.6.h.a.71.2		56
36.7	odd	6		108.6.h.a.35.12		56
36.11	even	6	inner	36.6.h.a.11.17	✓	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.17	✓	56	36.11	even	6	inner
36.6.h.a.11.27	yes	56	9.2	odd	6	inner
36.6.h.a.23.17	yes	56	1.1	even	1	trivial
36.6.h.a.23.27	yes	56	4.3	odd	2	inner
108.6.h.a.35.2		56	9.7	even	3
108.6.h.a.35.12		56	36.7	odd	6
108.6.h.a.71.2		56	12.11	even	2
108.6.h.a.71.12		56	3.2	odd	2