Embedded newform 1008.2.a.d.1.1

• Label: 1008.2.a.d.1.1
• Level: $1008$
• Weight: $2$
• Character: 1008.1
• Self dual: yes
• Analytic conductor: $8.049$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.04892052375\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1008.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{5} +1.00000 q^{7} -4.00000 q^{11} +2.00000 q^{13} +6.00000 q^{17} -8.00000 q^{19} -1.00000 q^{25} -6.00000 q^{29} -8.00000 q^{31} -2.00000 q^{35} -2.00000 q^{37} -2.00000 q^{41} +4.00000 q^{43} -8.00000 q^{47} +1.00000 q^{49} -6.00000 q^{53} +8.00000 q^{55} -6.00000 q^{61} -4.00000 q^{65} +4.00000 q^{67} -8.00000 q^{71} +10.0000 q^{73} -4.00000 q^{77} -16.0000 q^{79} +8.00000 q^{83} -12.0000 q^{85} +6.00000 q^{89} +2.00000 q^{91} +16.0000 q^{95} -6.00000 q^{97} +O(q^{100})\)

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
\(5\)	−2.00000	−0.894427				−0.447214	−	0.894427i	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	1.00000	0.377964
			\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
−0.603023	+	0.797724i				\(0.706037\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	−8.00000	−1.83533	−0.917663	−	0.397360i	\(-0.869927\pi\)
			−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	−2.00000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	−2.00000	−0.312348	−0.156174	−	0.987730i	\(-0.549916\pi\)
			−0.156174	+	0.987730i	\(0.549916\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	−8.00000	−1.16692	−0.583460	−	0.812142i	\(-0.698301\pi\)
			−0.583460	+	0.812142i	\(0.698301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.2.a.d.1.1		1
3.2	odd	2		112.2.a.b.1.1		1
4.3	odd	2		504.2.a.c.1.1		1
7.6	odd	2		7056.2.a.bo.1.1		1
8.3	odd	2		4032.2.a.bb.1.1		1
8.5	even	2		4032.2.a.bk.1.1		1
12.11	even	2		56.2.a.a.1.1	✓	1
15.2	even	4		2800.2.g.p.449.2		2
15.8	even	4		2800.2.g.p.449.1		2
15.14	odd	2		2800.2.a.p.1.1		1
21.2	odd	6		784.2.i.e.753.1		2
21.5	even	6		784.2.i.g.753.1		2
21.11	odd	6		784.2.i.e.177.1		2
21.17	even	6		784.2.i.g.177.1		2
21.20	even	2		784.2.a.e.1.1		1
24.5	odd	2		448.2.a.e.1.1		1
24.11	even	2		448.2.a.d.1.1		1
28.3	even	6		3528.2.s.e.3313.1		2
28.11	odd	6		3528.2.s.t.3313.1		2
28.19	even	6		3528.2.s.e.361.1		2
28.23	odd	6		3528.2.s.t.361.1		2
28.27	even	2		3528.2.a.x.1.1		1
48.5	odd	4		1792.2.b.d.897.2		2
48.11	even	4		1792.2.b.i.897.2		2
48.29	odd	4		1792.2.b.d.897.1		2
48.35	even	4		1792.2.b.i.897.1		2
60.23	odd	4		1400.2.g.g.449.2		2
60.47	odd	4		1400.2.g.g.449.1		2
60.59	even	2		1400.2.a.g.1.1		1
84.11	even	6		392.2.i.c.177.1		2
84.23	even	6		392.2.i.c.361.1		2
84.47	odd	6		392.2.i.d.361.1		2
84.59	odd	6		392.2.i.d.177.1		2
84.83	odd	2		392.2.a.d.1.1		1
132.131	odd	2		6776.2.a.g.1.1		1
156.155	even	2		9464.2.a.c.1.1		1
168.83	odd	2		3136.2.a.q.1.1		1
168.125	even	2		3136.2.a.p.1.1		1
420.419	odd	2		9800.2.a.u.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.a.a.1.1	✓	1	12.11	even	2
112.2.a.b.1.1		1	3.2	odd	2
392.2.a.d.1.1		1	84.83	odd	2
392.2.i.c.177.1		2	84.11	even	6
392.2.i.c.361.1		2	84.23	even	6
392.2.i.d.177.1		2	84.59	odd	6
392.2.i.d.361.1		2	84.47	odd	6
448.2.a.d.1.1		1	24.11	even	2
448.2.a.e.1.1		1	24.5	odd	2
504.2.a.c.1.1		1	4.3	odd	2
784.2.a.e.1.1		1	21.20	even	2
784.2.i.e.177.1		2	21.11	odd	6
784.2.i.e.753.1		2	21.2	odd	6
784.2.i.g.177.1		2	21.17	even	6
784.2.i.g.753.1		2	21.5	even	6
1008.2.a.d.1.1		1	1.1	even	1	trivial
1400.2.a.g.1.1		1	60.59	even	2
1400.2.g.g.449.1		2	60.47	odd	4
1400.2.g.g.449.2		2	60.23	odd	4
1792.2.b.d.897.1		2	48.29	odd	4
1792.2.b.d.897.2		2	48.5	odd	4
1792.2.b.i.897.1		2	48.35	even	4
1792.2.b.i.897.2		2	48.11	even	4
2800.2.a.p.1.1		1	15.14	odd	2
2800.2.g.p.449.1		2	15.8	even	4
2800.2.g.p.449.2		2	15.2	even	4
3136.2.a.p.1.1		1	168.125	even	2
3136.2.a.q.1.1		1	168.83	odd	2
3528.2.a.x.1.1		1	28.27	even	2
3528.2.s.e.361.1		2	28.19	even	6
3528.2.s.e.3313.1		2	28.3	even	6
3528.2.s.t.361.1		2	28.23	odd	6
3528.2.s.t.3313.1		2	28.11	odd	6
4032.2.a.bb.1.1		1	8.3	odd	2
4032.2.a.bk.1.1		1	8.5	even	2
6776.2.a.g.1.1		1	132.131	odd	2
7056.2.a.bo.1.1		1	7.6	odd	2
9464.2.a.c.1.1		1	156.155	even	2
9800.2.a.u.1.1		1	420.419	odd	2