Embedded newform 36.6.h.a.11.15

• Label: 36.6.h.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	36.11
Dual form			36.6.h.a.23.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.216257 - 5.65272i) q^{2} +(-13.9499 + 6.95697i) q^{3} +(-31.9065 - 2.44488i) q^{4} +(-0.143132 + 0.0826372i) q^{5} +(36.3090 + 80.3595i) q^{6} +(192.634 + 111.217i) q^{7} +(-20.7202 + 179.830i) q^{8} +(146.201 - 194.098i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.216257	−	5.65272i	0.0382292	−	0.999269i
							\(3\)	−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	0.999871	−	0.0160647i	\(-0.00511378\pi\)
							0.486023	+	0.873946i	\(0.338447\pi\)
\(11\)	−329.552	+	570.801i	−0.821188	+	1.42234i	0.0836100	+	0.996499i	\(0.473355\pi\)
							−0.904798	+	0.425841i	\(0.859978\pi\)
\(13\)	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	\(0.262069\pi\)
							−0.679795	+	0.733402i	\(0.737931\pi\)
\(23\)	−1327.18	−	2298.74i	−0.523129	−	0.906086i	−0.999638	−	0.0269164i	\(-0.991431\pi\)
							0.476509	−	0.879170i	\(-0.341902\pi\)
\(29\)	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	−0.383412	−	0.923577i	\(-0.625251\pi\)
							0.608136	+	0.793833i	\(0.291918\pi\)
\(37\)	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	0.596631	−	0.802516i	\(-0.296506\pi\)
							−0.396684	+	0.917955i	\(0.629839\pi\)
\(47\)	4973.66	−	8614.64i	0.328422	−	0.568843i	−0.653777	−	0.756687i	\(-0.726817\pi\)
							0.982199	+	0.187844i	\(0.0601500\pi\)

(See \(a_n\) instead)

Twists

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

    

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