Embedded newform 2352.4.a.bw.1.2

• Label: 2352.4.a.bw.1.2
• Level: $2352$
• Weight: $4$
• Character: 2352.1
• Self dual: yes
• Analytic conductor: $138.772$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(138.772492334\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{8})^+\)
Defining polynomial:	\(x^{2} - 2\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			\(1.41421\) of defining polynomial
Character	\(\chi\)	\(=\)	2352.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -4.58579 q^{5} +9.00000 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q + 6q^{3} - 12q^{5} + 18q^{9} + O(q^{10}) \)

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
			\(3\)	3.00000	0.577350
\(5\)	−4.58579	−0.410165				−0.205083	−	0.978745i	\(-0.565746\pi\)
			−0.205083	+	0.978745i	\(0.565746\pi\)
\(7\)	0	0
			\(11\)	6.48528	0.177762	0.0888812	−	0.996042i	\(-0.471671\pi\)
0.0888812	+	0.996042i				\(0.471671\pi\)
\(13\)	−45.2132	−0.964607	−0.482303	−	0.876004i	\(-0.660200\pi\)
			−0.482303	+	0.876004i	\(0.660200\pi\)
\(17\)	−81.5563	−1.16355	−0.581774	−	0.813350i	\(-0.697641\pi\)
			−0.581774	+	0.813350i	\(0.697641\pi\)
\(19\)	−5.05382	−0.0610225	−0.0305112	−	0.999534i	\(-0.509714\pi\)
			−0.0305112	+	0.999534i	\(0.509714\pi\)
\(23\)	−106.250	−0.963244	−0.481622	−	0.876379i	\(-0.659952\pi\)
			−0.481622	+	0.876379i	\(0.659952\pi\)
\(29\)	−268.132	−1.71693	−0.858463	−	0.512875i	\(-0.828580\pi\)
			−0.858463	+	0.512875i	\(0.828580\pi\)
\(31\)	292.368	1.69390	0.846948	−	0.531676i	\(-0.178438\pi\)
			0.846948	+	0.531676i	\(0.178438\pi\)
\(37\)	114.558	0.509008	0.254504	−	0.967072i	\(-0.418088\pi\)
			0.254504	+	0.967072i	\(0.418088\pi\)
\(41\)	161.605	0.615573	0.307786	−	0.951456i	\(-0.400412\pi\)
			0.307786	+	0.951456i	\(0.400412\pi\)
\(43\)	471.294	1.67143	0.835716	−	0.549162i	\(-0.185053\pi\)
			0.835716	+	0.549162i	\(0.185053\pi\)
\(47\)	346.004	1.07383	0.536914	−	0.843637i	\(-0.319590\pi\)
			0.536914	+	0.843637i	\(0.319590\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bw.1.2		2
4.3	odd	2		294.4.a.l.1.2	✓	2
7.6	odd	2		2352.4.a.bu.1.1		2
12.11	even	2		882.4.a.bb.1.1		2
28.3	even	6		294.4.e.k.79.2		4
28.11	odd	6		294.4.e.m.79.1		4
28.19	even	6		294.4.e.k.67.2		4
28.23	odd	6		294.4.e.m.67.1		4
28.27	even	2		294.4.a.o.1.1	yes	2
84.11	even	6		882.4.g.be.667.2		4
84.23	even	6		882.4.g.be.361.2		4
84.47	odd	6		882.4.g.bk.361.1		4
84.59	odd	6		882.4.g.bk.667.1		4
84.83	odd	2		882.4.a.t.1.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.4.a.l.1.2	✓	2	4.3	odd	2
294.4.a.o.1.1	yes	2	28.27	even	2
294.4.e.k.67.2		4	28.19	even	6
294.4.e.k.79.2		4	28.3	even	6
294.4.e.m.67.1		4	28.23	odd	6
294.4.e.m.79.1		4	28.11	odd	6
882.4.a.t.1.2		2	84.83	odd	2
882.4.a.bb.1.1		2	12.11	even	2
882.4.g.be.361.2		4	84.23	even	6
882.4.g.be.667.2		4	84.11	even	6
882.4.g.bk.361.1		4	84.47	odd	6
882.4.g.bk.667.1		4	84.59	odd	6
2352.4.a.bu.1.1		2	7.6	odd	2
2352.4.a.bw.1.2		2	1.1	even	1	trivial