Embedded newform 1323.4.a.t.1.2

• Label: 1323.4.a.t.1.2
• Level: $1323$
• Weight: $4$
• Character: 1323.1
• Self dual: yes
• Analytic conductor: $78.060$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(78.0595269376\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{8})^+\)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			\(1.41421\) of defining polynomial
Character	\(\chi\)	\(=\)	1323.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+4.24264 q^{2} +10.0000 q^{4} +16.9706 q^{5} +8.48528 q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	4.24264	1.50000	0.750000	−	0.661438i	\(-0.230053\pi\)
			0.750000	+	0.661438i	\(0.230053\pi\)
\(3\)	0	0
			\(5\)	16.9706	1.51789	0.758947	−	0.651153i	\(-0.225714\pi\)
0.758947	+	0.651153i				\(0.225714\pi\)
\(7\)	0	0
			\(11\)	16.9706	0.465165	0.232583	−	0.972577i	\(-0.425282\pi\)
0.232583	+	0.972577i				\(0.425282\pi\)
\(13\)	−29.0000	−0.618704	−0.309352	−	0.950948i	\(-0.600112\pi\)
			−0.309352	+	0.950948i	\(0.600112\pi\)
\(17\)	50.9117	0.726347	0.363173	−	0.931722i	\(-0.381693\pi\)
			0.363173	+	0.931722i	\(0.381693\pi\)
\(19\)	−29.0000	−0.350161	−0.175080	−	0.984554i	\(-0.556019\pi\)
			−0.175080	+	0.984554i	\(0.556019\pi\)
\(23\)	84.8528	0.769262	0.384631	−	0.923070i	\(-0.374329\pi\)
			0.384631	+	0.923070i	\(0.374329\pi\)
\(29\)	271.529	1.73868	0.869339	−	0.494216i	\(-0.164545\pi\)
			0.869339	+	0.494216i	\(0.164545\pi\)
\(31\)	268.000	1.55272	0.776358	−	0.630292i	\(-0.217065\pi\)
			0.776358	+	0.630292i	\(0.217065\pi\)
\(37\)	83.0000	0.368787	0.184393	−	0.982853i	\(-0.440968\pi\)
			0.184393	+	0.982853i	\(0.440968\pi\)
\(41\)	271.529	1.03429	0.517143	−	0.855899i	\(-0.326996\pi\)
			0.517143	+	0.855899i	\(0.326996\pi\)
\(43\)	−232.000	−0.822783	−0.411391	−	0.911459i	\(-0.634957\pi\)
			−0.411391	+	0.911459i	\(0.634957\pi\)
\(47\)	390.323	1.21137	0.605686	−	0.795704i	\(-0.292899\pi\)
			0.605686	+	0.795704i	\(0.292899\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.t.1.2		2
3.2	odd	2	inner	1323.4.a.t.1.1		2
7.6	odd	2		27.4.a.c.1.2	yes	2
21.20	even	2		27.4.a.c.1.1	✓	2
28.27	even	2		432.4.a.q.1.1		2
35.13	even	4		675.4.b.i.649.1		4
35.27	even	4		675.4.b.i.649.4		4
35.34	odd	2		675.4.a.n.1.1		2
56.13	odd	2		1728.4.a.bp.1.2		2
56.27	even	2		1728.4.a.bk.1.2		2
63.13	odd	6		81.4.c.e.55.1		4
63.20	even	6		81.4.c.e.28.2		4
63.34	odd	6		81.4.c.e.28.1		4
63.41	even	6		81.4.c.e.55.2		4
84.83	odd	2		432.4.a.q.1.2		2
105.62	odd	4		675.4.b.i.649.2		4
105.83	odd	4		675.4.b.i.649.3		4
105.104	even	2		675.4.a.n.1.2		2
168.83	odd	2		1728.4.a.bk.1.1		2
168.125	even	2		1728.4.a.bp.1.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.a.c.1.1	✓	2	21.20	even	2
27.4.a.c.1.2	yes	2	7.6	odd	2
81.4.c.e.28.1		4	63.34	odd	6
81.4.c.e.28.2		4	63.20	even	6
81.4.c.e.55.1		4	63.13	odd	6
81.4.c.e.55.2		4	63.41	even	6
432.4.a.q.1.1		2	28.27	even	2
432.4.a.q.1.2		2	84.83	odd	2
675.4.a.n.1.1		2	35.34	odd	2
675.4.a.n.1.2		2	105.104	even	2
675.4.b.i.649.1		4	35.13	even	4
675.4.b.i.649.2		4	105.62	odd	4
675.4.b.i.649.3		4	105.83	odd	4
675.4.b.i.649.4		4	35.27	even	4
1323.4.a.t.1.1		2	3.2	odd	2	inner
1323.4.a.t.1.2		2	1.1	even	1	trivial
1728.4.a.bk.1.1		2	168.83	odd	2
1728.4.a.bk.1.2		2	56.27	even	2
1728.4.a.bp.1.1		2	168.125	even	2
1728.4.a.bp.1.2		2	56.13	odd	2