Embedded newform 36.6.h.a.11.5

• Label: 36.6.h.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	36.11
Dual form			36.6.h.a.23.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.78727 + 3.01364i) q^{2} +(13.9499 - 6.95697i) q^{3} +(13.8359 - 28.8542i) q^{4} +(-0.143132 + 0.0826372i) q^{5} +(-45.8163 + 75.3450i) q^{6} +(-192.634 - 111.217i) q^{7} +(20.7202 + 179.830i) q^{8} +(146.201 - 194.098i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.78727	+	3.01364i	−0.846278	+	0.532742i
							\(3\)	13.9499	−	6.95697i	0.894889	−	0.446290i
\(5\)	−0.143132	+	0.0826372i	−0.00256042	+	0.00147826i								−0.501280	−	0.865285i	\(-0.667137\pi\)
							0.498719	+	0.866764i	\(0.333804\pi\)
\(7\)	−192.634	−	111.217i	−1.48589	−	0.857881i	−0.486023	−	0.873946i	\(-0.661553\pi\)
							−0.999871	+	0.0160647i	\(0.994886\pi\)
\(11\)	329.552	−	570.801i	0.821188	−	1.42234i	−0.0836100	−	0.996499i	\(-0.526645\pi\)
							0.904798	−	0.425841i	\(-0.140022\pi\)
\(13\)	43.8997	+	76.0365i	0.0720449	+	0.124785i	0.899797	−	0.436308i	\(-0.143714\pi\)
							−0.827752	+	0.561093i	\(0.810381\pi\)
\(17\)		−	85.1645i		−	0.0714721i	−0.999361	−	0.0357360i	\(-0.988622\pi\)
							0.999361	−	0.0357360i	\(-0.0113776\pi\)
\(19\)		−	2308.11i		−	1.46680i	−0.679795	−	0.733402i	\(-0.737931\pi\)
							0.679795	−	0.733402i	\(-0.262069\pi\)
\(23\)	1327.18	+	2298.74i	0.523129	+	0.906086i	0.999638	+	0.0269164i	\(0.00856881\pi\)
							−0.476509	+	0.879170i	\(0.658098\pi\)
\(29\)	1337.58	+	772.253i	0.295342	+	0.170516i	0.640348	−	0.768085i	\(-0.278790\pi\)
							−0.345007	+	0.938600i	\(0.612123\pi\)
\(31\)	−1202.41	+	694.212i	−0.224724	+	0.129744i	−0.608136	−	0.793833i	\(-0.708082\pi\)
							0.383412	+	0.923577i	\(0.374749\pi\)
\(37\)	4404.71			0.528948			0.264474	−	0.964393i	\(-0.414802\pi\)
							0.264474	+	0.964393i	\(0.414802\pi\)
\(41\)	−3145.98	+	1816.33i	−0.292278	+	0.168747i	−0.638969	−	0.769233i	\(-0.720639\pi\)
							0.346691	+	0.937979i	\(0.387305\pi\)
\(43\)	−2424.30	−	1399.67i	−0.199947	−	0.115440i	0.396684	−	0.917955i	\(-0.370161\pi\)
							−0.596631	+	0.802516i	\(0.703494\pi\)
\(47\)	−4973.66	+	8614.64i	−0.328422	+	0.568843i	−0.982199	−	0.187844i	\(-0.939850\pi\)
							0.653777	+	0.756687i	\(0.273183\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	36.6.h.a.11.5	✓	56
3.2	odd	2		108.6.h.a.35.24		56
4.3	odd	2	inner	36.6.h.a.11.15	yes	56
9.4	even	3		108.6.h.a.71.14		56
9.5	odd	6	inner	36.6.h.a.23.15	yes	56
12.11	even	2		108.6.h.a.35.14		56
36.23	even	6	inner	36.6.h.a.23.5	yes	56
36.31	odd	6		108.6.h.a.71.24		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.h.a.11.5	✓	56	1.1	even	1	trivial
36.6.h.a.11.15	yes	56	4.3	odd	2	inner
36.6.h.a.23.5	yes	56	36.23	even	6	inner
36.6.h.a.23.15	yes	56	9.5	odd	6	inner
108.6.h.a.35.14		56	12.11	even	2
108.6.h.a.35.24		56	3.2	odd	2
108.6.h.a.71.14		56	9.4	even	3
108.6.h.a.71.24		56	36.31	odd	6