Embedded newform 36.5.f.a.7.12

• Label: 36.5.f.a.7.12
• Level: $36$
• Weight: $5$
• Character: 36.7
• Analytic conductor: $3.721$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	36.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.72131867102\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.12
Character	\(\chi\)	\(=\)	36.7
Dual form			36.5.f.a.31.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0678484 - 3.99942i) q^{2} +(-2.72970 + 8.57606i) q^{3} +(-15.9908 - 0.542709i) q^{4} +(16.6139 + 28.7760i) q^{5} +(34.1141 + 11.4991i) q^{6} +(39.9759 + 23.0801i) q^{7} +(-3.25547 + 63.9171i) q^{8} +(-66.0975 - 46.8201i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - q^{2} - q^{4} - 2 q^{5} + 15 q^{6} + 122 q^{8} - 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(19\)	\(29\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0678484	−	3.99942i	0.0169621	−	0.999856i
							\(3\)	−2.72970	+	8.57606i	−0.303300	+	0.952895i
\(5\)	16.6139	+	28.7760i	0.664554	+	1.15104i								0.979406	+	0.201901i	\(0.0647118\pi\)
							−0.314852	+	0.949141i	\(0.601955\pi\)
\(7\)	39.9759	+	23.0801i	0.815835	+	0.471023i	0.848978	−	0.528428i	\(-0.177218\pi\)
							−0.0331429	+	0.999451i	\(0.510552\pi\)
\(11\)	−63.6109	−	36.7258i	−0.525710	−	0.303519i	0.213558	−	0.976930i	\(-0.431495\pi\)
							−0.739268	+	0.673412i	\(0.764828\pi\)
\(13\)	151.520	+	262.440i	0.896566	+	1.55290i	0.831855	+	0.554993i	\(0.187279\pi\)
							0.0647110	+	0.997904i	\(0.479387\pi\)
\(17\)	−182.019			−0.629823			−0.314912	−	0.949121i	\(-0.601975\pi\)
							−0.314912	+	0.949121i	\(0.601975\pi\)
\(19\)		−	314.215i		−	0.870402i	−0.900333	−	0.435201i	\(-0.856677\pi\)
							0.900333	−	0.435201i	\(-0.143323\pi\)
\(23\)	−290.919	+	167.962i	−0.549941	+	0.317509i	−0.749098	−	0.662459i	\(-0.769513\pi\)
							0.199157	+	0.979968i	\(0.436180\pi\)
\(29\)	357.370	−	618.983i	0.424935	−	0.736008i	−0.571480	−	0.820616i	\(-0.693630\pi\)
							0.996414	+	0.0846079i	\(0.0269638\pi\)
\(31\)	985.186	−	568.798i	1.02517	−	0.591881i	0.109571	−	0.993979i	\(-0.465052\pi\)
							0.915597	+	0.402098i	\(0.131719\pi\)
\(37\)	1008.45			0.736632			0.368316	−	0.929701i	\(-0.379934\pi\)
							0.368316	+	0.929701i	\(0.379934\pi\)
\(41\)	−557.553	−	965.709i	−0.331679	−	0.574485i	0.651162	−	0.758939i	\(-0.274282\pi\)
							−0.982841	+	0.184454i	\(0.940948\pi\)
\(43\)	2182.06	+	1259.82i	1.18013	+	0.681349i	0.956045	−	0.293220i	\(-0.0947268\pi\)
							0.224087	+	0.974569i	\(0.428060\pi\)
\(47\)	980.476	+	566.078i	0.443855	+	0.256260i	0.705232	−	0.708977i	\(-0.250843\pi\)
							−0.261376	+	0.965237i	\(0.584176\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.5.f.a.7.12	✓	44
3.2	odd	2		108.5.f.a.19.11		44
4.3	odd	2	inner	36.5.f.a.7.18	yes	44
9.2	odd	6		324.5.d.e.163.19		22
9.4	even	3	inner	36.5.f.a.31.18	yes	44
9.5	odd	6		108.5.f.a.91.5		44
9.7	even	3		324.5.d.f.163.4		22
12.11	even	2		108.5.f.a.19.5		44
36.7	odd	6		324.5.d.f.163.3		22
36.11	even	6		324.5.d.e.163.20		22
36.23	even	6		108.5.f.a.91.11		44
36.31	odd	6	inner	36.5.f.a.31.12	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.5.f.a.7.12	✓	44	1.1	even	1	trivial
36.5.f.a.7.18	yes	44	4.3	odd	2	inner
36.5.f.a.31.12	yes	44	36.31	odd	6	inner
36.5.f.a.31.18	yes	44	9.4	even	3	inner
108.5.f.a.19.5		44	12.11	even	2
108.5.f.a.19.11		44	3.2	odd	2
108.5.f.a.91.5		44	9.5	odd	6
108.5.f.a.91.11		44	36.23	even	6
324.5.d.e.163.19		22	9.2	odd	6
324.5.d.e.163.20		22	36.11	even	6
324.5.d.f.163.3		22	36.7	odd	6
324.5.d.f.163.4		22	9.7	even	3