Embedded newform 1764.2.k.b.361.1

• Label: 1764.2.k.b.361.1
• Level: $1764$
• Weight: $2$
• Character: 1764.361
• Analytic conductor: $14.086$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.0856109166\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1764.361
Dual form			1764.2.k.b.1549.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{5} +(-1.50000 + 2.59808i) q^{11} -2.00000 q^{13} +(-1.50000 + 2.59808i) q^{17} +(-0.500000 - 0.866025i) q^{19} +(1.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +6.00000 q^{29} +(-3.50000 + 6.06218i) q^{31} +(0.500000 + 0.866025i) q^{37} +6.00000 q^{41} -4.00000 q^{43} +(4.50000 + 7.79423i) q^{47} +(1.50000 - 2.59808i) q^{53} +9.00000 q^{55} +(-4.50000 + 7.79423i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(3.00000 + 5.19615i) q^{65} +(3.50000 - 6.06218i) q^{67} +(-0.500000 + 0.866025i) q^{73} +(6.50000 + 11.2583i) q^{79} +12.0000 q^{83} +9.00000 q^{85} +(-7.50000 - 12.9904i) q^{89} +(-1.50000 + 2.59808i) q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 3 * q^5 - 3 * q^11 - 4 * q^13 - 3 * q^17 - q^19 + 3 * q^23 - 4 * q^25 + 12 * q^29 - 7 * q^31 + q^37 + 12 * q^41 - 8 * q^43 + 9 * q^47 + 3 * q^53 + 18 * q^55 - 9 * q^59 - q^61 + 6 * q^65 + 7 * q^67 - q^73 + 13 * q^79 + 24 * q^83 + 18 * q^85 - 15 * q^89 - 3 * q^95 + 20 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\)	\(785\)	\(883\)	\(1081\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		0			0
							\(3\)		0			0
\(5\)	−1.50000	−	2.59808i	−0.670820	−	1.16190i								−0.977672	−	0.210138i	\(-0.932609\pi\)
							0.306851	−	0.951757i	\(-0.400725\pi\)
\(7\)		0			0
							\(11\)	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	\(-0.982718\pi\)
0.546259	+	0.837616i	\(0.316051\pi\)
\(13\)	−2.00000			−0.554700			−0.277350	−	0.960769i	\(-0.589456\pi\)
							−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	\(-0.951855\pi\)
							0.624780	+	0.780801i	\(0.285189\pi\)
\(19\)	−0.500000	−	0.866025i	−0.114708	−	0.198680i	0.802955	−	0.596040i	\(-0.203260\pi\)
							−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	1.50000	+	2.59808i	0.312772	+	0.541736i	0.978961	−	0.204046i	\(-0.0654092\pi\)
							−0.666190	+	0.745782i	\(0.732076\pi\)
\(29\)	6.00000			1.11417			0.557086	−	0.830455i	\(-0.311919\pi\)
							0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−3.50000	+	6.06218i	−0.628619	+	1.08880i	0.359211	+	0.933257i	\(0.383046\pi\)
							−0.987829	+	0.155543i	\(0.950287\pi\)
\(37\)	0.500000	+	0.866025i	0.0821995	+	0.142374i	0.904194	−	0.427121i	\(-0.140472\pi\)
							−0.821995	+	0.569495i	\(0.807139\pi\)
\(41\)	6.00000			0.937043			0.468521	−	0.883452i	\(-0.344787\pi\)
							0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−4.00000			−0.609994			−0.304997	−	0.952353i	\(-0.598656\pi\)
							−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	4.50000	+	7.79423i	0.656392	+	1.13691i	0.981543	+	0.191243i	\(0.0612518\pi\)
							−0.325150	+	0.945662i	\(0.605415\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.k.b.361.1		2
3.2	odd	2		196.2.e.a.165.1		2
7.2	even	3	inner	1764.2.k.b.1549.1		2
7.3	odd	6		1764.2.a.a.1.1		1
7.4	even	3		1764.2.a.j.1.1		1
7.5	odd	6		252.2.k.c.37.1		2
7.6	odd	2		252.2.k.c.109.1		2
12.11	even	2		784.2.i.d.753.1		2
21.2	odd	6		196.2.e.a.177.1		2
21.5	even	6		28.2.e.a.9.1	✓	2
21.11	odd	6		196.2.a.a.1.1		1
21.17	even	6		196.2.a.b.1.1		1
21.20	even	2		28.2.e.a.25.1	yes	2
28.3	even	6		7056.2.a.f.1.1		1
28.11	odd	6		7056.2.a.bw.1.1		1
28.19	even	6		1008.2.s.p.289.1		2
28.27	even	2		1008.2.s.p.865.1		2
63.5	even	6		2268.2.l.h.541.1		2
63.13	odd	6		2268.2.i.h.865.1		2
63.20	even	6		2268.2.l.h.109.1		2
63.34	odd	6		2268.2.l.a.109.1		2
63.40	odd	6		2268.2.l.a.541.1		2
63.41	even	6		2268.2.i.a.865.1		2
63.47	even	6		2268.2.i.a.2053.1		2
63.61	odd	6		2268.2.i.h.2053.1		2
84.11	even	6		784.2.a.g.1.1		1
84.23	even	6		784.2.i.d.177.1		2
84.47	odd	6		112.2.i.b.65.1		2
84.59	odd	6		784.2.a.d.1.1		1
84.83	odd	2		112.2.i.b.81.1		2
105.17	odd	12		4900.2.e.i.2549.1		2
105.32	even	12		4900.2.e.h.2549.2		2
105.38	odd	12		4900.2.e.i.2549.2		2
105.47	odd	12		700.2.r.b.149.1		4
105.53	even	12		4900.2.e.h.2549.1		2
105.59	even	6		4900.2.a.g.1.1		1
105.62	odd	4		700.2.r.b.249.2		4
105.68	odd	12		700.2.r.b.149.2		4
105.74	odd	6		4900.2.a.n.1.1		1
105.83	odd	4		700.2.r.b.249.1		4
105.89	even	6		700.2.i.c.401.1		2
105.104	even	2		700.2.i.c.501.1		2
168.5	even	6		448.2.i.e.65.1		2
168.11	even	6		3136.2.a.k.1.1		1
168.53	odd	6		3136.2.a.v.1.1		1
168.59	odd	6		3136.2.a.s.1.1		1
168.83	odd	2		448.2.i.c.193.1		2
168.101	even	6		3136.2.a.h.1.1		1
168.125	even	2		448.2.i.e.193.1		2
168.131	odd	6		448.2.i.c.65.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.e.a.9.1	✓	2	21.5	even	6
28.2.e.a.25.1	yes	2	21.20	even	2
112.2.i.b.65.1		2	84.47	odd	6
112.2.i.b.81.1		2	84.83	odd	2
196.2.a.a.1.1		1	21.11	odd	6
196.2.a.b.1.1		1	21.17	even	6
196.2.e.a.165.1		2	3.2	odd	2
196.2.e.a.177.1		2	21.2	odd	6
252.2.k.c.37.1		2	7.5	odd	6
252.2.k.c.109.1		2	7.6	odd	2
448.2.i.c.65.1		2	168.131	odd	6
448.2.i.c.193.1		2	168.83	odd	2
448.2.i.e.65.1		2	168.5	even	6
448.2.i.e.193.1		2	168.125	even	2
700.2.i.c.401.1		2	105.89	even	6
700.2.i.c.501.1		2	105.104	even	2
700.2.r.b.149.1		4	105.47	odd	12
700.2.r.b.149.2		4	105.68	odd	12
700.2.r.b.249.1		4	105.83	odd	4
700.2.r.b.249.2		4	105.62	odd	4
784.2.a.d.1.1		1	84.59	odd	6
784.2.a.g.1.1		1	84.11	even	6
784.2.i.d.177.1		2	84.23	even	6
784.2.i.d.753.1		2	12.11	even	2
1008.2.s.p.289.1		2	28.19	even	6
1008.2.s.p.865.1		2	28.27	even	2
1764.2.a.a.1.1		1	7.3	odd	6
1764.2.a.j.1.1		1	7.4	even	3
1764.2.k.b.361.1		2	1.1	even	1	trivial
1764.2.k.b.1549.1		2	7.2	even	3	inner
2268.2.i.a.865.1		2	63.41	even	6
2268.2.i.a.2053.1		2	63.47	even	6
2268.2.i.h.865.1		2	63.13	odd	6
2268.2.i.h.2053.1		2	63.61	odd	6
2268.2.l.a.109.1		2	63.34	odd	6
2268.2.l.a.541.1		2	63.40	odd	6
2268.2.l.h.109.1		2	63.20	even	6
2268.2.l.h.541.1		2	63.5	even	6
3136.2.a.h.1.1		1	168.101	even	6
3136.2.a.k.1.1		1	168.11	even	6
3136.2.a.s.1.1		1	168.59	odd	6
3136.2.a.v.1.1		1	168.53	odd	6
4900.2.a.g.1.1		1	105.59	even	6
4900.2.a.n.1.1		1	105.74	odd	6
4900.2.e.h.2549.1		2	105.53	even	12
4900.2.e.h.2549.2		2	105.32	even	12
4900.2.e.i.2549.1		2	105.17	odd	12
4900.2.e.i.2549.2		2	105.38	odd	12
7056.2.a.f.1.1		1	28.3	even	6
7056.2.a.bw.1.1		1	28.11	odd	6