Embedded newform 36.6.h.a.11.2

• Label: 36.6.h.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	36.11
Dual form			36.6.h.a.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.50860 + 1.28660i) q^{2} +(-13.2152 - 8.26786i) q^{3} +(28.6893 - 14.1747i) q^{4} +(38.3603 - 22.1474i) q^{5} +(83.4349 + 28.5416i) q^{6} +(-102.940 - 59.4323i) q^{7} +(-139.801 + 114.994i) q^{8} +(106.285 + 218.523i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.50860	+	1.28660i	−0.973792	+	0.227440i
							\(3\)	−13.2152	−	8.26786i	−0.847758	−	0.530383i
\(5\)	38.3603	−	22.1474i	0.686211	−	0.396184i								−0.115980	−	0.993252i	\(-0.537001\pi\)
							0.802191	+	0.597068i	\(0.203668\pi\)
\(7\)	−102.940	−	59.4323i	−0.794032	−	0.458435i	0.0473481	−	0.998878i	\(-0.484923\pi\)
							−0.841380	+	0.540444i	\(0.818256\pi\)
\(11\)	−349.253	+	604.924i	−0.870279	+	1.50737i	−0.00857110	+	0.999963i	\(0.502728\pi\)
							−0.861708	+	0.507404i	\(0.830605\pi\)
\(13\)	−271.774	−	470.726i	−0.446014	−	0.772520i	0.552108	−	0.833773i	\(-0.313824\pi\)
							−0.998122	+	0.0612530i	\(0.980490\pi\)
\(17\)			1884.06i			1.58114i	0.612369	+	0.790572i	\(0.290217\pi\)
							−0.612369	+	0.790572i	\(0.709783\pi\)
\(19\)			961.023i			0.610731i	0.952235	+	0.305366i	\(0.0987786\pi\)
							−0.952235	+	0.305366i	\(0.901221\pi\)
\(23\)	−566.279	−	980.824i	−0.223208	−	0.386608i	0.732572	−	0.680690i	\(-0.238320\pi\)
							−0.955780	+	0.294081i	\(0.904986\pi\)
\(29\)	173.063	+	99.9180i	0.0382128	+	0.0220622i	0.518985	−	0.854784i	\(-0.326310\pi\)
							−0.480772	+	0.876846i	\(0.659643\pi\)
\(31\)	225.699	−	130.308i	0.0421819	−	0.0243537i	−0.478761	−	0.877945i	\(-0.658914\pi\)
							0.520943	+	0.853592i	\(0.325581\pi\)
\(37\)	−11136.8			−1.33738			−0.668690	−	0.743541i	\(-0.733145\pi\)
							−0.668690	+	0.743541i	\(0.733145\pi\)
\(41\)	−8446.51	+	4876.59i	−0.784725	+	0.453061i	−0.838102	−	0.545513i	\(-0.816335\pi\)
							0.0533772	+	0.998574i	\(0.483001\pi\)
\(43\)	5303.93	+	3062.23i	0.437448	+	0.252561i	0.702515	−	0.711669i	\(-0.252061\pi\)
							−0.265066	+	0.964230i	\(0.585394\pi\)
\(47\)	−589.363	+	1020.81i	−0.0389169	+	0.0674061i	−0.884828	−	0.465918i	\(-0.845724\pi\)
							0.845911	+	0.533324i	\(0.179057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.11.2	✓	56
3.2	odd	2		108.6.h.a.35.27		56
4.3	odd	2	inner	36.6.h.a.11.13	yes	56
9.4	even	3		108.6.h.a.71.16		56
9.5	odd	6	inner	36.6.h.a.23.13	yes	56
12.11	even	2		108.6.h.a.35.16		56
36.23	even	6	inner	36.6.h.a.23.2	yes	56
36.31	odd	6		108.6.h.a.71.27		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.2	✓	56	1.1	even	1	trivial
36.6.h.a.11.13	yes	56	4.3	odd	2	inner
36.6.h.a.23.2	yes	56	36.23	even	6	inner
36.6.h.a.23.13	yes	56	9.5	odd	6	inner
108.6.h.a.35.16		56	12.11	even	2
108.6.h.a.35.27		56	3.2	odd	2
108.6.h.a.71.16		56	9.4	even	3
108.6.h.a.71.27		56	36.31	odd	6