Embedded newform 1296.2.i.i.865.1

• Label: 1296.2.i.i.865.1
• Level: $1296$
• Weight: $2$
• Character: 1296.865
• Analytic conductor: $10.349$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3486121020\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			865.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	1296.865
Dual form			1296.2.i.i.433.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{7} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(1135\)	\(1217\)
\(\chi(n)\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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
\(5\)		0			0									0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(7\)	−0.500000	+	0.866025i	−0.188982	+	0.327327i	−0.944911	−	0.327327i	\(-0.893852\pi\)
							0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(13\)	−2.50000	−	4.33013i	−0.693375	−	1.20096i	−0.970725	−	0.240192i	\(-0.922790\pi\)
							0.277350	−	0.960769i	\(-0.410544\pi\)
\(17\)		0			0			−	1.00000i	\(-0.5\pi\)
									1.00000i	\(0.5\pi\)
\(19\)	7.00000			1.60591			0.802955	−	0.596040i	\(-0.203260\pi\)
							0.802955	+	0.596040i	\(0.203260\pi\)
\(23\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(29\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)
\(31\)	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	\(-0.283621\pi\)
							−0.987829	+	0.155543i	\(0.950287\pi\)
\(37\)	11.0000			1.80839			0.904194	−	0.427121i	\(-0.140472\pi\)
							0.904194	+	0.427121i	\(0.140472\pi\)
\(41\)		0			0		0.866025	−	0.500000i	\(-0.166667\pi\)
							−0.866025	+	0.500000i	\(0.833333\pi\)
\(43\)	4.00000	−	6.92820i	0.609994	−	1.05654i	−0.381246	−	0.924473i	\(-0.624505\pi\)
							0.991241	−	0.132068i	\(-0.0421616\pi\)
\(47\)		0			0		−0.866025	−	0.500000i	\(-0.833333\pi\)
							0.866025	+	0.500000i	\(0.166667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.2.i.i.865.1		2
3.2	odd	2	CM	1296.2.i.i.865.1		2
4.3	odd	2		81.2.c.a.55.1		2
9.2	odd	6		432.2.a.e.1.1		1
9.4	even	3	inner	1296.2.i.i.433.1		2
9.5	odd	6	inner	1296.2.i.i.433.1		2
9.7	even	3		432.2.a.e.1.1		1
12.11	even	2		81.2.c.a.55.1		2
36.7	odd	6		27.2.a.a.1.1	✓	1
36.11	even	6		27.2.a.a.1.1	✓	1
36.23	even	6		81.2.c.a.28.1		2
36.31	odd	6		81.2.c.a.28.1		2
72.11	even	6		1728.2.a.n.1.1		1
72.29	odd	6		1728.2.a.o.1.1		1
72.43	odd	6		1728.2.a.n.1.1		1
72.61	even	6		1728.2.a.o.1.1		1
108.7	odd	18		729.2.e.f.82.1		6
108.11	even	18		729.2.e.f.568.1		6
108.23	even	18		729.2.e.f.163.1		6
108.31	odd	18		729.2.e.f.163.1		6
108.43	odd	18		729.2.e.f.568.1		6
108.47	even	18		729.2.e.f.82.1		6
108.59	even	18		729.2.e.f.406.1		6
108.67	odd	18		729.2.e.f.649.1		6
108.79	odd	18		729.2.e.f.325.1		6
108.83	even	18		729.2.e.f.325.1		6
108.95	even	18		729.2.e.f.649.1		6
108.103	odd	18		729.2.e.f.406.1		6
180.7	even	12		675.2.b.f.649.1		2
180.43	even	12		675.2.b.f.649.2		2
180.47	odd	12		675.2.b.f.649.1		2
180.79	odd	6		675.2.a.e.1.1		1
180.83	odd	12		675.2.b.f.649.2		2
180.119	even	6		675.2.a.e.1.1		1
252.83	odd	6		1323.2.a.i.1.1		1
252.223	even	6		1323.2.a.i.1.1		1
396.43	even	6		3267.2.a.f.1.1		1
396.263	odd	6		3267.2.a.f.1.1		1
468.155	even	6		4563.2.a.e.1.1		1
468.259	odd	6		4563.2.a.e.1.1		1
612.407	even	6		7803.2.a.k.1.1		1
612.475	odd	6		7803.2.a.k.1.1		1
684.151	even	6		9747.2.a.f.1.1		1
684.227	odd	6		9747.2.a.f.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.a.a.1.1	✓	1	36.7	odd	6
27.2.a.a.1.1	✓	1	36.11	even	6
81.2.c.a.28.1		2	36.23	even	6
81.2.c.a.28.1		2	36.31	odd	6
81.2.c.a.55.1		2	4.3	odd	2
81.2.c.a.55.1		2	12.11	even	2
432.2.a.e.1.1		1	9.2	odd	6
432.2.a.e.1.1		1	9.7	even	3
675.2.a.e.1.1		1	180.79	odd	6
675.2.a.e.1.1		1	180.119	even	6
675.2.b.f.649.1		2	180.7	even	12
675.2.b.f.649.1		2	180.47	odd	12
675.2.b.f.649.2		2	180.43	even	12
675.2.b.f.649.2		2	180.83	odd	12
729.2.e.f.82.1		6	108.7	odd	18
729.2.e.f.82.1		6	108.47	even	18
729.2.e.f.163.1		6	108.23	even	18
729.2.e.f.163.1		6	108.31	odd	18
729.2.e.f.325.1		6	108.79	odd	18
729.2.e.f.325.1		6	108.83	even	18
729.2.e.f.406.1		6	108.59	even	18
729.2.e.f.406.1		6	108.103	odd	18
729.2.e.f.568.1		6	108.11	even	18
729.2.e.f.568.1		6	108.43	odd	18
729.2.e.f.649.1		6	108.67	odd	18
729.2.e.f.649.1		6	108.95	even	18
1296.2.i.i.433.1		2	9.4	even	3	inner
1296.2.i.i.433.1		2	9.5	odd	6	inner
1296.2.i.i.865.1		2	1.1	even	1	trivial
1296.2.i.i.865.1		2	3.2	odd	2	CM
1323.2.a.i.1.1		1	252.83	odd	6
1323.2.a.i.1.1		1	252.223	even	6
1728.2.a.n.1.1		1	72.11	even	6
1728.2.a.n.1.1		1	72.43	odd	6
1728.2.a.o.1.1		1	72.29	odd	6
1728.2.a.o.1.1		1	72.61	even	6
3267.2.a.f.1.1		1	396.43	even	6
3267.2.a.f.1.1		1	396.263	odd	6
4563.2.a.e.1.1		1	468.155	even	6
4563.2.a.e.1.1		1	468.259	odd	6
7803.2.a.k.1.1		1	612.407	even	6
7803.2.a.k.1.1		1	612.475	odd	6
9747.2.a.f.1.1		1	684.151	even	6
9747.2.a.f.1.1		1	684.227	odd	6