Embedded newform 1456.4.a.b.1.1

• Label: 1456.4.a.b.1.1
• Level: $1456$
• Weight: $4$
• Character: 1456.1
• Self dual: yes
• Analytic conductor: $85.907$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(85.9067809684\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1456.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-7.00000 q^{3} -7.00000 q^{7} +22.0000 q^{9} -39.0000 q^{11} +13.0000 q^{13} +24.0000 q^{17} -38.0000 q^{19} +49.0000 q^{21} -39.0000 q^{23} -125.000 q^{25} +35.0000 q^{27} -96.0000 q^{29} -227.000 q^{31} +273.000 q^{33} +425.000 q^{37} -91.0000 q^{39} -105.000 q^{41} -344.000 q^{43} -99.0000 q^{47} +49.0000 q^{49} -168.000 q^{51} -540.000 q^{53} +266.000 q^{57} -114.000 q^{59} -565.000 q^{61} -154.000 q^{63} +385.000 q^{67} +273.000 q^{69} +156.000 q^{71} -673.000 q^{73} +875.000 q^{75} +273.000 q^{77} -749.000 q^{79} -839.000 q^{81} +1044.00 q^{83} +672.000 q^{87} -690.000 q^{89} -91.0000 q^{91} +1589.00 q^{93} +317.000 q^{97} -858.000 q^{99} +O(q^{100})\)

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
			\(3\)	−7.00000	−1.34715	−0.673575	−	0.739119i	\(-0.735242\pi\)
−0.673575	+	0.739119i				\(0.735242\pi\)
\(5\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	−7.00000	−0.377964
			\(11\)	−39.0000	−1.06899	−0.534497	−	0.845170i	\(-0.679499\pi\)
−0.534497	+	0.845170i				\(0.679499\pi\)
\(13\)	13.0000	0.277350
			\(17\)	24.0000	0.342403	0.171202	−	0.985236i	\(-0.445235\pi\)
0.171202	+	0.985236i				\(0.445235\pi\)
\(19\)	−38.0000	−0.458831	−0.229416	−	0.973329i	\(-0.573682\pi\)
			−0.229416	+	0.973329i	\(0.573682\pi\)
\(23\)	−39.0000	−0.353568	−0.176784	−	0.984250i	\(-0.556569\pi\)
			−0.176784	+	0.984250i	\(0.556569\pi\)
\(29\)	−96.0000	−0.614716	−0.307358	−	0.951594i	\(-0.599445\pi\)
			−0.307358	+	0.951594i	\(0.599445\pi\)
\(31\)	−227.000	−1.31517	−0.657587	−	0.753378i	\(-0.728423\pi\)
			−0.657587	+	0.753378i	\(0.728423\pi\)
\(37\)	425.000	1.88837	0.944183	−	0.329420i	\(-0.106853\pi\)
			0.944183	+	0.329420i	\(0.106853\pi\)
\(41\)	−105.000	−0.399957	−0.199979	−	0.979800i	\(-0.564087\pi\)
			−0.199979	+	0.979800i	\(0.564087\pi\)
\(43\)	−344.000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	−99.0000	−0.307248	−0.153624	−	0.988129i	\(-0.549094\pi\)
			−0.153624	+	0.988129i	\(0.549094\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.4.a.b.1.1		1
4.3	odd	2		182.4.a.a.1.1	✓	1
12.11	even	2		1638.4.a.j.1.1		1
28.27	even	2		1274.4.a.a.1.1		1
52.51	odd	2		2366.4.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.a.a.1.1	✓	1	4.3	odd	2
1274.4.a.a.1.1		1	28.27	even	2
1456.4.a.b.1.1		1	1.1	even	1	trivial
1638.4.a.j.1.1		1	12.11	even	2
2366.4.a.g.1.1		1	52.51	odd	2