Embedded newform 1728.2.a.t.1.1

• Label: 1728.2.a.t.1.1
• Level: $1728$
• Weight: $2$
• Character: 1728.1
• Self dual: yes
• Analytic conductor: $13.798$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(13.7981494693\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 864)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1728.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{5} +3.00000 q^{7} +3.00000 q^{11} +4.00000 q^{17} -6.00000 q^{19} +6.00000 q^{23} -4.00000 q^{25} +2.00000 q^{29} +9.00000 q^{31} +3.00000 q^{35} +2.00000 q^{37} -10.0000 q^{41} -6.00000 q^{43} +6.00000 q^{47} +2.00000 q^{49} -13.0000 q^{53} +3.00000 q^{55} +12.0000 q^{59} -8.00000 q^{61} -6.00000 q^{67} +12.0000 q^{71} +9.00000 q^{73} +9.00000 q^{77} +3.00000 q^{83} +4.00000 q^{85} +14.0000 q^{89} -6.00000 q^{95} -9.00000 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
			\(3\)	0	0
\(5\)	1.00000	0.447214				0.223607	−	0.974679i	\(-0.428217\pi\)
			0.223607	+	0.974679i	\(0.428217\pi\)
\(7\)	3.00000	1.13389	0.566947	−	0.823754i	\(-0.308125\pi\)
			0.566947	+	0.823754i	\(0.308125\pi\)
\(11\)	3.00000	0.904534	0.452267	−	0.891883i	\(-0.350615\pi\)
			0.452267	+	0.891883i	\(0.350615\pi\)
\(13\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(17\)	4.00000	0.970143	0.485071	−	0.874475i	\(-0.338794\pi\)
			0.485071	+	0.874475i	\(0.338794\pi\)
\(19\)	−6.00000	−1.37649	−0.688247	−	0.725476i	\(-0.741620\pi\)
			−0.688247	+	0.725476i	\(0.741620\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	9.00000	1.61645	0.808224	−	0.588875i	\(-0.200429\pi\)
			0.808224	+	0.588875i	\(0.200429\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	−10.0000	−1.56174	−0.780869	−	0.624695i	\(-0.785223\pi\)
			−0.780869	+	0.624695i	\(0.785223\pi\)
\(43\)	−6.00000	−0.914991	−0.457496	−	0.889212i	\(-0.651253\pi\)
			−0.457496	+	0.889212i	\(0.651253\pi\)
\(47\)	6.00000	0.875190	0.437595	−	0.899172i	\(-0.355830\pi\)
			0.437595	+	0.899172i	\(0.355830\pi\)
\(53\)	−13.0000	−1.78569	−0.892844	−	0.450367i	\(-0.851293\pi\)
			−0.892844	+	0.450367i	\(0.851293\pi\)
\(59\)	12.0000	1.56227	0.781133	−	0.624364i	\(-0.214642\pi\)
			0.781133	+	0.624364i	\(0.214642\pi\)
\(61\)	−8.00000	−1.02430	−0.512148	−	0.858898i	\(-0.671150\pi\)
			−0.512148	+	0.858898i	\(0.671150\pi\)
\(67\)	−6.00000	−0.733017	−0.366508	−	0.930415i	\(-0.619447\pi\)
			−0.366508	+	0.930415i	\(0.619447\pi\)
\(71\)	12.0000	1.42414	0.712069	−	0.702109i	\(-0.247758\pi\)
			0.712069	+	0.702109i	\(0.247758\pi\)
\(73\)	9.00000	1.05337	0.526685	−	0.850060i	\(-0.323435\pi\)
			0.526685	+	0.850060i	\(0.323435\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	3.00000	0.329293	0.164646	−	0.986353i	\(-0.447352\pi\)
			0.164646	+	0.986353i	\(0.447352\pi\)
\(89\)	14.0000	1.48400	0.741999	−	0.670402i	\(-0.233878\pi\)
			0.741999	+	0.670402i	\(0.233878\pi\)
\(97\)	−9.00000	−0.913812	−0.456906	−	0.889515i	\(-0.651042\pi\)
			−0.456906	+	0.889515i	\(0.651042\pi\)

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	1728.2.a.t.1.1		1
3.2	odd	2		1728.2.a.k.1.1		1
4.3	odd	2		1728.2.a.q.1.1		1
8.3	odd	2		864.2.a.e.1.1	✓	1
8.5	even	2		864.2.a.f.1.1	yes	1
12.11	even	2		1728.2.a.j.1.1		1
24.5	odd	2		864.2.a.h.1.1	yes	1
24.11	even	2		864.2.a.g.1.1	yes	1
72.5	odd	6		2592.2.i.i.865.1		2
72.11	even	6		2592.2.i.l.1729.1		2
72.13	even	6		2592.2.i.m.865.1		2
72.29	odd	6		2592.2.i.i.1729.1		2
72.43	odd	6		2592.2.i.p.1729.1		2
72.59	even	6		2592.2.i.l.865.1		2
72.61	even	6		2592.2.i.m.1729.1		2
72.67	odd	6		2592.2.i.p.865.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.a.e.1.1	✓	1	8.3	odd	2
864.2.a.f.1.1	yes	1	8.5	even	2
864.2.a.g.1.1	yes	1	24.11	even	2
864.2.a.h.1.1	yes	1	24.5	odd	2
1728.2.a.j.1.1		1	12.11	even	2
1728.2.a.k.1.1		1	3.2	odd	2
1728.2.a.q.1.1		1	4.3	odd	2
1728.2.a.t.1.1		1	1.1	even	1	trivial
2592.2.i.i.865.1		2	72.5	odd	6
2592.2.i.i.1729.1		2	72.29	odd	6
2592.2.i.l.865.1		2	72.59	even	6
2592.2.i.l.1729.1		2	72.11	even	6
2592.2.i.m.865.1		2	72.13	even	6
2592.2.i.m.1729.1		2	72.61	even	6
2592.2.i.p.865.1		2	72.67	odd	6
2592.2.i.p.1729.1		2	72.43	odd	6