Embedded newform 36.7.f.a.7.7

• Label: 36.7.f.a.7.7
• Level: $36$
• Weight: $7$
• Character: 36.7
• Analytic conductor: $8.282$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	36.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.28194701031\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.7
Character	\(\chi\)	\(=\)	36.7
Dual form			36.7.f.a.31.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.68379 + 4.39624i) q^{2} +(-3.84106 - 26.7254i) q^{3} +(25.3462 - 58.7671i) q^{4} +(-60.0263 - 103.969i) q^{5} +(143.164 + 161.741i) q^{6} +(148.400 + 85.6789i) q^{7} +(88.9451 + 504.215i) q^{8} +(-699.493 + 205.307i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - q^{2} - q^{4} - 2 q^{5} - 321 q^{6} + 266 q^{8} + 456 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.68379	+	4.39624i	−0.835474	+	0.549530i
							\(3\)	−3.84106	−	26.7254i	−0.142261	−	0.989829i
\(5\)	−60.0263	−	103.969i	−0.480210	−	0.831748i								0.519532	−	0.854451i	\(-0.326106\pi\)
							−0.999742	+	0.0227027i	\(0.992773\pi\)
\(7\)	148.400	+	85.6789i	0.432654	+	0.249793i	0.700477	−	0.713675i	\(-0.252971\pi\)
							−0.267823	+	0.963468i	\(0.586304\pi\)
\(11\)	−556.498	−	321.294i	−0.418105	−	0.241393i	0.276161	−	0.961111i	\(-0.410938\pi\)
							−0.694266	+	0.719718i	\(0.744271\pi\)
\(13\)	210.147	+	363.985i	0.0956518	+	0.165674i	0.909880	−	0.414870i	\(-0.136173\pi\)
							−0.814229	+	0.580544i	\(0.802840\pi\)
\(17\)	−8924.71			−1.81655			−0.908275	−	0.418373i	\(-0.862600\pi\)
							−0.908275	+	0.418373i	\(0.862600\pi\)
\(19\)			9169.05i			1.33679i	0.743806	+	0.668395i	\(0.233019\pi\)
							−0.743806	+	0.668395i	\(0.766981\pi\)
\(23\)	4394.30	−	2537.05i	0.361166	−	0.208519i	−0.308426	−	0.951248i	\(-0.599802\pi\)
							0.669592	+	0.742729i	\(0.266469\pi\)
\(29\)	−22981.5	+	39805.1i	−0.942289	+	1.63209i	−0.181198	+	0.983447i	\(0.557998\pi\)
							−0.761091	+	0.648646i	\(0.775336\pi\)
\(31\)	23861.4	−	13776.4i	0.800959	−	0.462434i	−0.0428475	−	0.999082i	\(-0.513643\pi\)
							0.843806	+	0.536648i	\(0.180310\pi\)
\(37\)	−28037.2			−0.553516			−0.276758	−	0.960940i	\(-0.589260\pi\)
							−0.276758	+	0.960940i	\(0.589260\pi\)
\(41\)	−28075.9	−	48628.9i	−0.407363	−	0.705574i	0.587230	−	0.809420i	\(-0.300218\pi\)
							−0.994593	+	0.103846i	\(0.966885\pi\)
\(43\)	−47309.3	−	27314.0i	−0.595033	−	0.343542i	0.172052	−	0.985088i	\(-0.444960\pi\)
							−0.767085	+	0.641545i	\(0.778294\pi\)
\(47\)	−65546.4	−	37843.2i	−0.631328	−	0.364498i	0.149938	−	0.988695i	\(-0.452093\pi\)
							−0.781266	+	0.624198i	\(0.785426\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.7.f.a.7.7	✓	68
3.2	odd	2		108.7.f.a.19.28		68
4.3	odd	2	inner	36.7.f.a.7.18	yes	68
9.4	even	3	inner	36.7.f.a.31.18	yes	68
9.5	odd	6		108.7.f.a.91.17		68
12.11	even	2		108.7.f.a.19.17		68
36.23	even	6		108.7.f.a.91.28		68
36.31	odd	6	inner	36.7.f.a.31.7	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.7.f.a.7.7	✓	68	1.1	even	1	trivial
36.7.f.a.7.18	yes	68	4.3	odd	2	inner
36.7.f.a.31.7	yes	68	36.31	odd	6	inner
36.7.f.a.31.18	yes	68	9.4	even	3	inner
108.7.f.a.19.17		68	12.11	even	2
108.7.f.a.19.28		68	3.2	odd	2
108.7.f.a.91.17		68	9.5	odd	6
108.7.f.a.91.28		68	36.23	even	6