Embedded newform 4056.2.a.b.1.1

• Label: 4056.2.a.b.1.1
• Level: $4056$
• Weight: $2$
• Character: 4056.1
• Self dual: yes
• Analytic conductor: $32.387$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(32.3873230598\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	4056.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -3.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +3.00000 q^{15} +3.00000 q^{17} +4.00000 q^{19} -4.00000 q^{21} -8.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -5.00000 q^{29} +8.00000 q^{31} +4.00000 q^{33} -12.0000 q^{35} -7.00000 q^{37} +9.00000 q^{41} +8.00000 q^{43} -3.00000 q^{45} +4.00000 q^{47} +9.00000 q^{49} -3.00000 q^{51} -5.00000 q^{53} +12.0000 q^{55} -4.00000 q^{57} -4.00000 q^{59} -5.00000 q^{61} +4.00000 q^{63} -8.00000 q^{67} +8.00000 q^{69} +4.00000 q^{71} -11.0000 q^{73} -4.00000 q^{75} -16.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -9.00000 q^{85} +5.00000 q^{87} +6.00000 q^{89} -8.00000 q^{93} -12.0000 q^{95} +14.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
			\(3\)	−1.00000	−0.577350
\(5\)	−3.00000	−1.34164				−0.670820	−	0.741620i	\(-0.734058\pi\)
			−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	4.00000	1.51186	0.755929	−	0.654654i	\(-0.227186\pi\)
			0.755929	+	0.654654i	\(0.227186\pi\)
\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
			−0.603023	+	0.797724i	\(0.706037\pi\)
\(13\)	0	0
			\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
0.363803	+	0.931476i				\(0.381478\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	−8.00000	−1.66812	−0.834058	−	0.551677i	\(-0.813988\pi\)
			−0.834058	+	0.551677i	\(0.813988\pi\)
\(29\)	−5.00000	−0.928477	−0.464238	−	0.885710i	\(-0.653672\pi\)
			−0.464238	+	0.885710i	\(0.653672\pi\)
\(31\)	8.00000	1.43684	0.718421	−	0.695608i	\(-0.244865\pi\)
			0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)	−7.00000	−1.15079	−0.575396	−	0.817875i	\(-0.695152\pi\)
			−0.575396	+	0.817875i	\(0.695152\pi\)
\(41\)	9.00000	1.40556	0.702782	−	0.711405i	\(-0.251941\pi\)
			0.702782	+	0.711405i	\(0.251941\pi\)
\(43\)	8.00000	1.21999	0.609994	−	0.792406i	\(-0.291172\pi\)
			0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	4.00000	0.583460	0.291730	−	0.956501i	\(-0.405769\pi\)
			0.291730	+	0.956501i	\(0.405769\pi\)
\(53\)	−5.00000	−0.686803	−0.343401	−	0.939189i	\(-0.611579\pi\)
			−0.343401	+	0.939189i	\(0.611579\pi\)
\(59\)	−4.00000	−0.520756	−0.260378	−	0.965507i	\(-0.583847\pi\)
			−0.260378	+	0.965507i	\(0.583847\pi\)
\(61\)	−5.00000	−0.640184	−0.320092	−	0.947386i	\(-0.603714\pi\)
			−0.320092	+	0.947386i	\(0.603714\pi\)
\(67\)	−8.00000	−0.977356	−0.488678	−	0.872464i	\(-0.662521\pi\)
			−0.488678	+	0.872464i	\(0.662521\pi\)
\(71\)	4.00000	0.474713	0.237356	−	0.971423i	\(-0.423719\pi\)
			0.237356	+	0.971423i	\(0.423719\pi\)
\(73\)	−11.0000	−1.28745	−0.643726	−	0.765256i	\(-0.722612\pi\)
			−0.643726	+	0.765256i	\(0.722612\pi\)
\(79\)	−4.00000	−0.450035	−0.225018	−	0.974355i	\(-0.572244\pi\)
			−0.225018	+	0.974355i	\(0.572244\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	6.00000	0.635999	0.317999	−	0.948091i	\(-0.396989\pi\)
			0.317999	+	0.948091i	\(0.396989\pi\)
\(97\)	14.0000	1.42148	0.710742	−	0.703452i	\(-0.248359\pi\)
			0.710742	+	0.703452i	\(0.248359\pi\)

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	4056.2.a.b.1.1		1
4.3	odd	2		8112.2.a.r.1.1		1
13.4	even	6		312.2.q.c.289.1	yes	2
13.5	odd	4		4056.2.c.b.337.2		2
13.8	odd	4		4056.2.c.b.337.1		2
13.10	even	6		312.2.q.c.217.1	✓	2
13.12	even	2		4056.2.a.j.1.1		1
39.17	odd	6		936.2.t.b.289.1		2
39.23	odd	6		936.2.t.b.217.1		2
52.23	odd	6		624.2.q.e.529.1		2
52.43	odd	6		624.2.q.e.289.1		2
52.51	odd	2		8112.2.a.bh.1.1		1
156.23	even	6		1872.2.t.a.1153.1		2
156.95	even	6		1872.2.t.a.289.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.c.217.1	✓	2	13.10	even	6
312.2.q.c.289.1	yes	2	13.4	even	6
624.2.q.e.289.1		2	52.43	odd	6
624.2.q.e.529.1		2	52.23	odd	6
936.2.t.b.217.1		2	39.23	odd	6
936.2.t.b.289.1		2	39.17	odd	6
1872.2.t.a.289.1		2	156.95	even	6
1872.2.t.a.1153.1		2	156.23	even	6
4056.2.a.b.1.1		1	1.1	even	1	trivial
4056.2.a.j.1.1		1	13.12	even	2
4056.2.c.b.337.1		2	13.8	odd	4
4056.2.c.b.337.2		2	13.5	odd	4
8112.2.a.r.1.1		1	4.3	odd	2
8112.2.a.bh.1.1		1	52.51	odd	2