Embedded newform 2816.2.g.a.1407.2

• Label: 2816.2.g.a.1407.2
• Level: $2816$
• Weight: $2$
• Character: 2816.1407
• Analytic conductor: $22.486$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -11
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.4858732092\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{11})\)
Defining polynomial:	\( x^{4} - 5x^{2} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 176)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1407.2
Root			\(-1.65831 - 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	2816.1407
Dual form			2816.2.g.a.1407.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-3.31662 q^{3} +3.00000i q^{5} +8.00000 q^{9} -3.31662 q^{11} -9.94987i q^{15} +3.31662i q^{23} -4.00000 q^{25} -16.5831 q^{27} +9.94987i q^{31} +11.0000 q^{33} +7.00000i q^{37} +24.0000i q^{45} -6.63325i q^{47} -7.00000 q^{49} +6.00000i q^{53} -9.94987i q^{55} +3.31662 q^{59} +9.94987 q^{67} -11.0000i q^{69} +16.5831i q^{71} +13.2665 q^{75} +31.0000 q^{81} +9.00000 q^{89} -33.0000i q^{93} -17.0000 q^{97} -26.5330 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{9} - 16 q^{25} + 44 q^{33} - 28 q^{49} + 124 q^{81} + 36 q^{89} - 68 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times\).

\(n\)	\(1025\)	\(1541\)	\(2047\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−3.31662	−1.91485	−0.957427	−	0.288675i	\(-0.906785\pi\)
−0.957427	+	0.288675i				\(0.906785\pi\)
\(5\)	3.00000i	1.34164i	0.741620	+	0.670820i	\(0.234058\pi\)
			−0.741620	+	0.670820i	\(0.765942\pi\)
\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(11\)	−3.31662	−1.00000
			\(13\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	3.31662i	0.691564i	0.938315	+	0.345782i	\(0.112386\pi\)
			−0.938315	+	0.345782i	\(0.887614\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	9.94987i	1.78705i	0.449013	+	0.893525i	\(0.351776\pi\)
			−0.449013	+	0.893525i	\(0.648224\pi\)
\(37\)	7.00000i	1.15079i	0.817875	+	0.575396i	\(0.195152\pi\)
			−0.817875	+	0.575396i	\(0.804848\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(47\)	− 6.63325i	− 0.967559i	−0.875190	−	0.483779i	\(-0.839264\pi\)
			0.875190	−	0.483779i	\(-0.160736\pi\)
\(53\)	6.00000i	0.824163i	0.911147	+	0.412082i	\(0.135198\pi\)
			−0.911147	+	0.412082i	\(0.864802\pi\)
\(59\)	3.31662	0.431788	0.215894	−	0.976417i	\(-0.430733\pi\)
			0.215894	+	0.976417i	\(0.430733\pi\)
\(61\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(67\)	9.94987	1.21557	0.607785	−	0.794101i	\(-0.292058\pi\)
			0.607785	+	0.794101i	\(0.292058\pi\)
\(71\)	16.5831i	1.96805i	0.178017	+	0.984027i	\(0.443032\pi\)
			−0.178017	+	0.984027i	\(0.556968\pi\)
\(73\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(89\)	9.00000	0.953998	0.476999	−	0.878904i	\(-0.341725\pi\)
			0.476999	+	0.878904i	\(0.341725\pi\)
\(97\)	−17.0000	−1.72609	−0.863044	−	0.505128i	\(-0.831445\pi\)
			−0.863044	+	0.505128i	\(0.831445\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.g.a.1407.2		4
4.3	odd	2	inner	2816.2.g.a.1407.4		4
8.3	odd	2	inner	2816.2.g.a.1407.1		4
8.5	even	2	inner	2816.2.g.a.1407.3		4
11.10	odd	2	CM	2816.2.g.a.1407.2		4
16.3	odd	4		176.2.e.a.175.2	yes	2
16.5	even	4		704.2.e.a.703.2		2
16.11	odd	4		704.2.e.a.703.1		2
16.13	even	4		176.2.e.a.175.1	✓	2
44.43	even	2	inner	2816.2.g.a.1407.4		4
48.29	odd	4		1584.2.o.a.703.1		2
48.35	even	4		1584.2.o.a.703.2		2
88.21	odd	2	inner	2816.2.g.a.1407.3		4
88.43	even	2	inner	2816.2.g.a.1407.1		4
176.21	odd	4		704.2.e.a.703.2		2
176.43	even	4		704.2.e.a.703.1		2
176.109	odd	4		176.2.e.a.175.1	✓	2
176.131	even	4		176.2.e.a.175.2	yes	2
528.131	odd	4		1584.2.o.a.703.2		2
528.461	even	4		1584.2.o.a.703.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
176.2.e.a.175.1	✓	2	16.13	even	4
176.2.e.a.175.1	✓	2	176.109	odd	4
176.2.e.a.175.2	yes	2	16.3	odd	4
176.2.e.a.175.2	yes	2	176.131	even	4
704.2.e.a.703.1		2	16.11	odd	4
704.2.e.a.703.1		2	176.43	even	4
704.2.e.a.703.2		2	16.5	even	4
704.2.e.a.703.2		2	176.21	odd	4
1584.2.o.a.703.1		2	48.29	odd	4
1584.2.o.a.703.1		2	528.461	even	4
1584.2.o.a.703.2		2	48.35	even	4
1584.2.o.a.703.2		2	528.131	odd	4
2816.2.g.a.1407.1		4	8.3	odd	2	inner
2816.2.g.a.1407.1		4	88.43	even	2	inner
2816.2.g.a.1407.2		4	1.1	even	1	trivial
2816.2.g.a.1407.2		4	11.10	odd	2	CM
2816.2.g.a.1407.3		4	8.5	even	2	inner
2816.2.g.a.1407.3		4	88.21	odd	2	inner
2816.2.g.a.1407.4		4	4.3	odd	2	inner
2816.2.g.a.1407.4		4	44.43	even	2	inner