Embedded newform 36.6.h.a.11.4

• Label: 36.6.h.a.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	36.11
Dual form			36.6.h.a.23.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.35087 - 1.83526i) q^{2} +(6.54818 - 14.1464i) q^{3} +(25.2636 + 19.6405i) q^{4} +(36.0599 - 20.8192i) q^{5} +(-61.0009 + 63.6781i) q^{6} +(117.288 + 67.7164i) q^{7} +(-99.1369 - 151.459i) q^{8} +(-157.243 - 185.267i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.35087	−	1.83526i	−0.945909	−	0.324432i
							\(3\)	6.54818	−	14.1464i	0.420066	−	0.907494i
\(5\)	36.0599	−	20.8192i	0.645059	−	0.372425i								−0.141502	−	0.989938i	\(-0.545193\pi\)
							0.786561	+	0.617513i	\(0.211860\pi\)
\(7\)	117.288	+	67.7164i	0.904710	+	0.522335i	0.878725	−	0.477328i	\(-0.158395\pi\)
							0.0259846	+	0.999662i	\(0.491728\pi\)
\(11\)	144.306	−	249.946i	0.359586	−	0.622822i	−0.628305	−	0.777967i	\(-0.716251\pi\)
							0.987892	+	0.155145i	\(0.0495844\pi\)
\(13\)	−399.564	−	692.066i	−0.655735	−	1.13577i	−0.981709	−	0.190388i	\(-0.939026\pi\)
							0.325974	−	0.945379i	\(-0.394308\pi\)
\(17\)		−	1426.41i		−	1.19708i	−0.801093	−	0.598540i	\(-0.795748\pi\)
							0.801093	−	0.598540i	\(-0.204252\pi\)
\(19\)			2191.46i			1.39267i	0.717716	+	0.696336i	\(0.245188\pi\)
							−0.717716	+	0.696336i	\(0.754812\pi\)
\(23\)	−276.033	−	478.103i	−0.108803	−	0.188453i	0.806483	−	0.591258i	\(-0.201368\pi\)
							−0.915286	+	0.402805i	\(0.868035\pi\)
\(29\)	2037.13	+	1176.14i	0.449804	+	0.259694i	0.707747	−	0.706466i	\(-0.249712\pi\)
							−0.257944	+	0.966160i	\(0.583045\pi\)
\(31\)	8140.74	−	4700.06i	1.52146	−	0.878414i	0.521778	−	0.853081i	\(-0.325269\pi\)
							0.999679	−	0.0253327i	\(-0.00806450\pi\)
\(37\)	2862.18			0.343711			0.171855	−	0.985122i	\(-0.445024\pi\)
							0.171855	+	0.985122i	\(0.445024\pi\)
\(41\)	−2498.01	+	1442.23i	−0.232078	+	0.133990i	−0.611530	−	0.791221i	\(-0.709446\pi\)
							0.379452	+	0.925211i	\(0.376112\pi\)
\(43\)	5637.32	+	3254.71i	0.464945	+	0.268436i	0.714121	−	0.700022i	\(-0.246827\pi\)
							−0.249176	+	0.968458i	\(0.580160\pi\)
\(47\)	−3575.66	+	6193.23i	−0.236109	+	0.408952i	−0.959594	−	0.281387i	\(-0.909205\pi\)
							0.723486	+	0.690339i	\(0.242539\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.11.4	✓	56
3.2	odd	2		108.6.h.a.35.25		56
4.3	odd	2	inner	36.6.h.a.11.7	yes	56
9.4	even	3		108.6.h.a.71.22		56
9.5	odd	6	inner	36.6.h.a.23.7	yes	56
12.11	even	2		108.6.h.a.35.22		56
36.23	even	6	inner	36.6.h.a.23.4	yes	56
36.31	odd	6		108.6.h.a.71.25		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.4	✓	56	1.1	even	1	trivial
36.6.h.a.11.7	yes	56	4.3	odd	2	inner
36.6.h.a.23.4	yes	56	36.23	even	6	inner
36.6.h.a.23.7	yes	56	9.5	odd	6	inner
108.6.h.a.35.22		56	12.11	even	2
108.6.h.a.35.25		56	3.2	odd	2
108.6.h.a.71.22		56	9.4	even	3
108.6.h.a.71.25		56	36.31	odd	6