Embedded newform 36.6.h.a.11.22

• Label: 36.6.h.a.11.22
• 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			11.22
Character	\(\chi\)	\(=\)	36.11
Dual form			36.6.h.a.23.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.95514 + 4.04436i) q^{2} +(15.4848 - 1.79474i) q^{3} +(-0.713689 + 31.9920i) q^{4} +(-77.8058 + 44.9212i) q^{5} +(68.5032 + 55.5276i) q^{6} +(124.034 + 71.6112i) q^{7} +(-132.210 + 123.647i) q^{8} +(236.558 - 55.5825i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.95514	+	4.04436i	0.699177	+	0.714949i
							\(3\)	15.4848	−	1.79474i	0.993350	−	0.115133i
\(5\)	−77.8058	+	44.9212i	−1.39183	+	0.803575i								−0.993518	−	0.113675i	\(-0.963738\pi\)
							−0.398314	+	0.917249i	\(0.630405\pi\)
\(7\)	124.034	+	71.6112i	0.956745	+	0.552377i	0.895170	−	0.445725i	\(-0.147054\pi\)
							0.0615753	+	0.998102i	\(0.480388\pi\)
\(11\)	278.231	−	481.910i	0.693303	−	1.20084i	−0.277446	−	0.960741i	\(-0.589488\pi\)
							0.970749	−	0.240095i	\(-0.0771787\pi\)
\(13\)	179.796	+	311.416i	0.295068	+	0.511072i	0.975001	−	0.222202i	\(-0.0713245\pi\)
							−0.679933	+	0.733274i	\(0.737991\pi\)
\(17\)		−	1076.42i		−	0.903356i	−0.892181	−	0.451678i	\(-0.850826\pi\)
							0.892181	−	0.451678i	\(-0.149174\pi\)
\(19\)			348.135i			0.221240i	0.993863	+	0.110620i	\(0.0352837\pi\)
							−0.993863	+	0.110620i	\(0.964716\pi\)
\(23\)	−1469.06	−	2544.48i	−0.579054	−	1.00295i	−0.995588	−	0.0938304i	\(-0.970089\pi\)
							0.416535	−	0.909120i	\(-0.363244\pi\)
\(29\)	3331.19	+	1923.26i	0.735536	+	0.424662i	0.820444	−	0.571727i	\(-0.193726\pi\)
							−0.0849079	+	0.996389i	\(0.527060\pi\)
\(31\)	2346.41	−	1354.70i	0.438531	−	0.253186i	−0.264443	−	0.964401i	\(-0.585188\pi\)
							0.702974	+	0.711215i	\(0.251855\pi\)
\(37\)	−1992.90			−0.239321			−0.119661	−	0.992815i	\(-0.538181\pi\)
							−0.119661	+	0.992815i	\(0.538181\pi\)
\(41\)	−13432.0	+	7754.99i	−1.24791	+	0.720479i	−0.970692	−	0.240329i	\(-0.922745\pi\)
							−0.277215	+	0.960808i	\(0.589411\pi\)
\(43\)	12474.1	+	7201.91i	1.02882	+	0.593987i	0.916645	−	0.399702i	\(-0.130887\pi\)
							0.112170	+	0.993689i	\(0.464220\pi\)
\(47\)	1232.97	−	2135.56i	0.0814153	−	0.141015i	−0.822443	−	0.568848i	\(-0.807389\pi\)
							0.903858	+	0.427832i	\(0.140723\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.11.22	✓	56
3.2	odd	2		108.6.h.a.35.7		56
4.3	odd	2	inner	36.6.h.a.11.26	yes	56
9.4	even	3		108.6.h.a.71.3		56
9.5	odd	6	inner	36.6.h.a.23.26	yes	56
12.11	even	2		108.6.h.a.35.3		56
36.23	even	6	inner	36.6.h.a.23.22	yes	56
36.31	odd	6		108.6.h.a.71.7		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.22	✓	56	1.1	even	1	trivial
36.6.h.a.11.26	yes	56	4.3	odd	2	inner
36.6.h.a.23.22	yes	56	36.23	even	6	inner
36.6.h.a.23.26	yes	56	9.5	odd	6	inner
108.6.h.a.35.3		56	12.11	even	2
108.6.h.a.35.7		56	3.2	odd	2
108.6.h.a.71.3		56	9.4	even	3
108.6.h.a.71.7		56	36.31	odd	6