Embedded newform 448.2.a.d.1.1

• Label: 448.2.a.d.1.1
• Level: $448$
• Weight: $2$
• Character: 448.1
• Self dual: yes
• Analytic conductor: $3.577$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(3.57729801055\)
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	\(\chi\)	\(=\)	448.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(5\)	−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	\(-0.293963\pi\)
0.603023	+	0.797724i				\(0.293963\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	−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	\(-0.244865\pi\)
			0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	2.00000	0.312348	0.156174	−	0.987730i	\(-0.450084\pi\)
			0.156174	+	0.987730i	\(0.450084\pi\)
\(43\)	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	448.2.a.d.1.1		1
3.2	odd	2		4032.2.a.bb.1.1		1
4.3	odd	2		448.2.a.e.1.1		1
7.6	odd	2		3136.2.a.q.1.1		1
8.3	odd	2		112.2.a.b.1.1		1
8.5	even	2		56.2.a.a.1.1	✓	1
12.11	even	2		4032.2.a.bk.1.1		1
16.3	odd	4		1792.2.b.d.897.2		2
16.5	even	4		1792.2.b.i.897.1		2
16.11	odd	4		1792.2.b.d.897.1		2
16.13	even	4		1792.2.b.i.897.2		2
24.5	odd	2		504.2.a.c.1.1		1
24.11	even	2		1008.2.a.d.1.1		1
28.27	even	2		3136.2.a.p.1.1		1
40.3	even	4		2800.2.g.p.449.1		2
40.13	odd	4		1400.2.g.g.449.2		2
40.19	odd	2		2800.2.a.p.1.1		1
40.27	even	4		2800.2.g.p.449.2		2
40.29	even	2		1400.2.a.g.1.1		1
40.37	odd	4		1400.2.g.g.449.1		2
56.3	even	6		784.2.i.g.177.1		2
56.5	odd	6		392.2.i.d.361.1		2
56.11	odd	6		784.2.i.e.177.1		2
56.13	odd	2		392.2.a.d.1.1		1
56.19	even	6		784.2.i.g.753.1		2
56.27	even	2		784.2.a.e.1.1		1
56.37	even	6		392.2.i.c.361.1		2
56.45	odd	6		392.2.i.d.177.1		2
56.51	odd	6		784.2.i.e.753.1		2
56.53	even	6		392.2.i.c.177.1		2
88.21	odd	2		6776.2.a.g.1.1		1
104.77	even	2		9464.2.a.c.1.1		1
168.5	even	6		3528.2.s.e.361.1		2
168.53	odd	6		3528.2.s.t.3313.1		2
168.83	odd	2		7056.2.a.bo.1.1		1
168.101	even	6		3528.2.s.e.3313.1		2
168.125	even	2		3528.2.a.x.1.1		1
168.149	odd	6		3528.2.s.t.361.1		2
280.69	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	8.5	even	2
112.2.a.b.1.1		1	8.3	odd	2
392.2.a.d.1.1		1	56.13	odd	2
392.2.i.c.177.1		2	56.53	even	6
392.2.i.c.361.1		2	56.37	even	6
392.2.i.d.177.1		2	56.45	odd	6
392.2.i.d.361.1		2	56.5	odd	6
448.2.a.d.1.1		1	1.1	even	1	trivial
448.2.a.e.1.1		1	4.3	odd	2
504.2.a.c.1.1		1	24.5	odd	2
784.2.a.e.1.1		1	56.27	even	2
784.2.i.e.177.1		2	56.11	odd	6
784.2.i.e.753.1		2	56.51	odd	6
784.2.i.g.177.1		2	56.3	even	6
784.2.i.g.753.1		2	56.19	even	6
1008.2.a.d.1.1		1	24.11	even	2
1400.2.a.g.1.1		1	40.29	even	2
1400.2.g.g.449.1		2	40.37	odd	4
1400.2.g.g.449.2		2	40.13	odd	4
1792.2.b.d.897.1		2	16.11	odd	4
1792.2.b.d.897.2		2	16.3	odd	4
1792.2.b.i.897.1		2	16.5	even	4
1792.2.b.i.897.2		2	16.13	even	4
2800.2.a.p.1.1		1	40.19	odd	2
2800.2.g.p.449.1		2	40.3	even	4
2800.2.g.p.449.2		2	40.27	even	4
3136.2.a.p.1.1		1	28.27	even	2
3136.2.a.q.1.1		1	7.6	odd	2
3528.2.a.x.1.1		1	168.125	even	2
3528.2.s.e.361.1		2	168.5	even	6
3528.2.s.e.3313.1		2	168.101	even	6
3528.2.s.t.361.1		2	168.149	odd	6
3528.2.s.t.3313.1		2	168.53	odd	6
4032.2.a.bb.1.1		1	3.2	odd	2
4032.2.a.bk.1.1		1	12.11	even	2
6776.2.a.g.1.1		1	88.21	odd	2
7056.2.a.bo.1.1		1	168.83	odd	2
9464.2.a.c.1.1		1	104.77	even	2
9800.2.a.u.1.1		1	280.69	odd	2