Embedded newform 36.6.h.a.23.7

• Label: 36.6.h.a.23.7
• 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			23.7
Character	\(\chi\)	\(=\)	36.23
Dual form			36.6.h.a.11.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.26482 + 3.71636i) q^{2} +(-6.54818 - 14.1464i) q^{3} +(4.37736 - 31.6992i) q^{4} +(36.0599 + 20.8192i) q^{5} +(80.5000 + 35.9966i) q^{6} +(-117.288 + 67.7164i) q^{7} +(99.1369 + 151.459i) q^{8} +(-157.243 + 185.267i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.26482	+	3.71636i	−0.753921	+	0.656966i
							\(3\)	−6.54818	−	14.1464i	−0.420066	−	0.907494i
\(5\)	36.0599	+	20.8192i	0.645059	+	0.372425i								0.786561	−	0.617513i	\(-0.211860\pi\)
							−0.141502	+	0.989938i	\(0.545193\pi\)
\(7\)	−117.288	+	67.7164i	−0.904710	+	0.522335i	−0.878725	−	0.477328i	\(-0.841605\pi\)
							−0.0259846	+	0.999662i	\(0.508272\pi\)
\(11\)	−144.306	−	249.946i	−0.359586	−	0.622822i	0.628305	−	0.777967i	\(-0.283749\pi\)
							−0.987892	+	0.155145i	\(0.950416\pi\)
\(13\)	−399.564	+	692.066i	−0.655735	+	1.13577i	0.325974	+	0.945379i	\(0.394308\pi\)
							−0.981709	+	0.190388i	\(0.939026\pi\)
\(17\)			1426.41i			1.19708i	0.801093	+	0.598540i	\(0.204252\pi\)
							−0.801093	+	0.598540i	\(0.795748\pi\)
\(19\)			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	\(-0.798632\pi\)
							0.915286	+	0.402805i	\(0.131965\pi\)
\(29\)	2037.13	−	1176.14i	0.449804	−	0.259694i	−0.257944	−	0.966160i	\(-0.583045\pi\)
							0.707747	+	0.706466i	\(0.249712\pi\)
\(31\)	−8140.74	−	4700.06i	−1.52146	−	0.878414i	−0.999679	−	0.0253327i	\(-0.991935\pi\)
							−0.521778	−	0.853081i	\(-0.674731\pi\)
\(37\)	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	0.379452	−	0.925211i	\(-0.376112\pi\)
							−0.611530	+	0.791221i	\(0.709446\pi\)
\(43\)	−5637.32	+	3254.71i	−0.464945	+	0.268436i	−0.714121	−	0.700022i	\(-0.753173\pi\)
							0.249176	+	0.968458i	\(0.419840\pi\)
\(47\)	3575.66	+	6193.23i	0.236109	+	0.408952i	0.959594	−	0.281387i	\(-0.0907945\pi\)
							−0.723486	+	0.690339i	\(0.757461\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.4	✓	56	9.2	odd	6	inner
36.6.h.a.11.7	yes	56	36.11	even	6	inner
36.6.h.a.23.4	yes	56	4.3	odd	2	inner
36.6.h.a.23.7	yes	56	1.1	even	1	trivial
108.6.h.a.35.22		56	36.7	odd	6
108.6.h.a.35.25		56	9.7	even	3
108.6.h.a.71.22		56	3.2	odd	2
108.6.h.a.71.25		56	12.11	even	2