Embedded newform 36.6.h.a.23.13

• Label: 36.6.h.a.23.13
• 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			23.13
Character	\(\chi\)	\(=\)	36.23
Dual form			36.6.h.a.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.64008 + 5.41389i) q^{2} +(13.2152 - 8.26786i) q^{3} +(-26.6203 - 17.7584i) q^{4} +(38.3603 + 22.1474i) q^{5} +(23.0872 + 85.1057i) q^{6} +(102.940 - 59.4323i) q^{7} +(139.801 - 114.994i) q^{8} +(106.285 - 218.523i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\)	\(19\)	\(29\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.64008	+	5.41389i	−0.289927	+	0.957049i
							\(3\)	13.2152	−	8.26786i	0.847758	−	0.530383i
\(5\)	38.3603	+	22.1474i	0.686211	+	0.396184i								0.802191	−	0.597068i	\(-0.203668\pi\)
							−0.115980	+	0.993252i	\(0.537001\pi\)
\(7\)	102.940	−	59.4323i	0.794032	−	0.458435i	−0.0473481	−	0.998878i	\(-0.515077\pi\)
							0.841380	+	0.540444i	\(0.181744\pi\)
\(11\)	349.253	+	604.924i	0.870279	+	1.50737i	0.861708	+	0.507404i	\(0.169395\pi\)
							0.00857110	+	0.999963i	\(0.497272\pi\)
\(13\)	−271.774	+	470.726i	−0.446014	+	0.772520i	−0.998122	−	0.0612530i	\(-0.980490\pi\)
							0.552108	+	0.833773i	\(0.313824\pi\)
\(17\)		−	1884.06i		−	1.58114i	−0.612369	−	0.790572i	\(-0.709783\pi\)
							0.612369	−	0.790572i	\(-0.290217\pi\)
\(19\)			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	\(-0.761680\pi\)
							0.955780	+	0.294081i	\(0.0950136\pi\)
\(29\)	173.063	−	99.9180i	0.0382128	−	0.0220622i	−0.480772	−	0.876846i	\(-0.659643\pi\)
							0.518985	+	0.854784i	\(0.326310\pi\)
\(31\)	−225.699	−	130.308i	−0.0421819	−	0.0243537i	0.478761	−	0.877945i	\(-0.341086\pi\)
							−0.520943	+	0.853592i	\(0.674419\pi\)
\(37\)	−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	0.0533772	−	0.998574i	\(-0.483001\pi\)
							−0.838102	+	0.545513i	\(0.816335\pi\)
\(43\)	−5303.93	+	3062.23i	−0.437448	+	0.252561i	−0.702515	−	0.711669i	\(-0.747939\pi\)
							0.265066	+	0.964230i	\(0.414606\pi\)
\(47\)	589.363	+	1020.81i	0.0389169	+	0.0674061i	0.884828	−	0.465918i	\(-0.154276\pi\)
							−0.845911	+	0.533324i	\(0.820943\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.2	✓	56	9.2	odd	6	inner
36.6.h.a.11.13	yes	56	36.11	even	6	inner
36.6.h.a.23.2	yes	56	4.3	odd	2	inner
36.6.h.a.23.13	yes	56	1.1	even	1	trivial
108.6.h.a.35.16		56	36.7	odd	6
108.6.h.a.35.27		56	9.7	even	3
108.6.h.a.71.16		56	3.2	odd	2
108.6.h.a.71.27		56	12.11	even	2