Embedded newform 36.5.f.a.7.20

• Label: 36.5.f.a.7.20
• 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			7.20
Character	\(\chi\)	\(=\)	36.7
Dual form			36.5.f.a.31.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.68138 - 1.56444i) q^{2} +(4.25251 + 7.93197i) q^{3} +(11.1051 - 11.5186i) q^{4} +(1.01545 + 1.75881i) q^{5} +(28.0642 + 22.5478i) q^{6} +(20.0352 + 11.5673i) q^{7} +(22.8618 - 59.7774i) q^{8} +(-44.8324 + 67.4615i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - q^{2} - q^{4} - 2 q^{5} + 15 q^{6} + 122 q^{8} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.68138	−	1.56444i	0.920344	−	0.391110i
							\(3\)	4.25251	+	7.93197i	0.472501	+	0.881330i
\(5\)	1.01545	+	1.75881i	0.0406181	+	0.0703525i								0.885620	−	0.464411i	\(-0.153734\pi\)
							−0.845002	+	0.534764i	\(0.820401\pi\)
\(7\)	20.0352	+	11.5673i	0.408882	+	0.236068i	0.690309	−	0.723514i	\(-0.257474\pi\)
							−0.281427	+	0.959583i	\(0.590808\pi\)
\(11\)	4.32958	+	2.49968i	0.0357816	+	0.0206585i	0.517784	−	0.855511i	\(-0.326757\pi\)
							−0.482002	+	0.876170i	\(0.660090\pi\)
\(13\)	−137.824	−	238.718i	−0.815527	−	1.41253i	−0.908949	−	0.416908i	\(-0.863114\pi\)
							0.0934218	−	0.995627i	\(-0.470219\pi\)
\(17\)	−266.009			−0.920447			−0.460224	−	0.887803i	\(-0.652231\pi\)
							−0.460224	+	0.887803i	\(0.652231\pi\)
\(19\)			367.194i			1.01716i	0.861015	+	0.508579i	\(0.169829\pi\)
							−0.861015	+	0.508579i	\(0.830171\pi\)
\(23\)	−544.473	+	314.352i	−1.02925	+	0.594238i	−0.916770	−	0.399416i	\(-0.869213\pi\)
							−0.112481	+	0.993654i	\(0.535880\pi\)
\(29\)	−319.481	+	553.357i	−0.379882	+	0.657975i	−0.991045	−	0.133530i	\(-0.957369\pi\)
							0.611163	+	0.791505i	\(0.290702\pi\)
\(31\)	1191.19	−	687.735i	1.23953	−	0.715645i	0.270535	−	0.962710i	\(-0.412799\pi\)
							0.968999	+	0.247065i	\(0.0794660\pi\)
\(37\)	1466.19			1.07099			0.535496	−	0.844538i	\(-0.320125\pi\)
							0.535496	+	0.844538i	\(0.320125\pi\)
\(41\)	593.019	+	1027.14i	0.352777	+	0.611028i	0.986735	−	0.162339i	\(-0.0519040\pi\)
							−0.633958	+	0.773368i	\(0.718571\pi\)
\(43\)	1430.33	+	825.804i	0.773572	+	0.446622i	0.834147	−	0.551542i	\(-0.185960\pi\)
							−0.0605755	+	0.998164i	\(0.519294\pi\)
\(47\)	307.864	+	177.745i	0.139368	+	0.0804642i	0.568063	−	0.822985i	\(-0.307693\pi\)
							−0.428695	+	0.903449i	\(0.641026\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.5.f.a.7.20	yes	44
3.2	odd	2		108.5.f.a.19.3		44
4.3	odd	2	inner	36.5.f.a.7.11	✓	44
9.2	odd	6		324.5.d.e.163.18		22
9.4	even	3	inner	36.5.f.a.31.11	yes	44
9.5	odd	6		108.5.f.a.91.12		44
9.7	even	3		324.5.d.f.163.5		22
12.11	even	2		108.5.f.a.19.12		44
36.7	odd	6		324.5.d.f.163.6		22
36.11	even	6		324.5.d.e.163.17		22
36.23	even	6		108.5.f.a.91.3		44
36.31	odd	6	inner	36.5.f.a.31.20	yes	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.5.f.a.7.11	✓	44	4.3	odd	2	inner
36.5.f.a.7.20	yes	44	1.1	even	1	trivial
36.5.f.a.31.11	yes	44	9.4	even	3	inner
36.5.f.a.31.20	yes	44	36.31	odd	6	inner
108.5.f.a.19.3		44	3.2	odd	2
108.5.f.a.19.12		44	12.11	even	2
108.5.f.a.91.3		44	36.23	even	6
108.5.f.a.91.12		44	9.5	odd	6
324.5.d.e.163.17		22	36.11	even	6
324.5.d.e.163.18		22	9.2	odd	6
324.5.d.f.163.5		22	9.7	even	3
324.5.d.f.163.6		22	36.7	odd	6