Embedded newform 1872.4.a.m.1.1

• Label: 1872.4.a.m.1.1
• Level: $1872$
• Weight: $4$
• Character: 1872.1
• Self dual: yes
• Analytic conductor: $110.452$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(110.451575531\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1872.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+12.0000 q^{5} -2.00000 q^{7} -36.0000 q^{11} +13.0000 q^{13} +78.0000 q^{17} -74.0000 q^{19} -96.0000 q^{23} +19.0000 q^{25} -18.0000 q^{29} +214.000 q^{31} -24.0000 q^{35} -286.000 q^{37} +384.000 q^{41} -524.000 q^{43} +300.000 q^{47} -339.000 q^{49} -558.000 q^{53} -432.000 q^{55} +576.000 q^{59} +74.0000 q^{61} +156.000 q^{65} -38.0000 q^{67} -456.000 q^{71} -682.000 q^{73} +72.0000 q^{77} -704.000 q^{79} -888.000 q^{83} +936.000 q^{85} +1020.00 q^{89} -26.0000 q^{91} -888.000 q^{95} +110.000 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	12.0000	1.07331				0.536656	−	0.843801i	\(-0.319687\pi\)
			0.536656	+	0.843801i	\(0.319687\pi\)
\(7\)	−2.00000	−0.107990	−0.0539949	−	0.998541i	\(-0.517195\pi\)
			−0.0539949	+	0.998541i	\(0.517195\pi\)
\(11\)	−36.0000	−0.986764	−0.493382	−	0.869813i	\(-0.664240\pi\)
			−0.493382	+	0.869813i	\(0.664240\pi\)
\(13\)	13.0000	0.277350
			\(17\)	78.0000	1.11281	0.556405	−	0.830911i	\(-0.312180\pi\)
0.556405	+	0.830911i				\(0.312180\pi\)
\(19\)	−74.0000	−0.893514	−0.446757	−	0.894655i	\(-0.647421\pi\)
			−0.446757	+	0.894655i	\(0.647421\pi\)
\(23\)	−96.0000	−0.870321	−0.435161	−	0.900353i	\(-0.643308\pi\)
			−0.435161	+	0.900353i	\(0.643308\pi\)
\(29\)	−18.0000	−0.115259	−0.0576296	−	0.998338i	\(-0.518354\pi\)
			−0.0576296	+	0.998338i	\(0.518354\pi\)
\(31\)	214.000	1.23986	0.619928	−	0.784659i	\(-0.287162\pi\)
			0.619928	+	0.784659i	\(0.287162\pi\)
\(37\)	−286.000	−1.27076	−0.635380	−	0.772200i	\(-0.719156\pi\)
			−0.635380	+	0.772200i	\(0.719156\pi\)
\(41\)	384.000	1.46270	0.731350	−	0.682002i	\(-0.238890\pi\)
			0.731350	+	0.682002i	\(0.238890\pi\)
\(43\)	−524.000	−1.85835	−0.929177	−	0.369634i	\(-0.879483\pi\)
			−0.929177	+	0.369634i	\(0.879483\pi\)
\(47\)	300.000	0.931053	0.465527	−	0.885034i	\(-0.345865\pi\)
			0.465527	+	0.885034i	\(0.345865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.m.1.1		1
3.2	odd	2		624.4.a.g.1.1		1
4.3	odd	2		117.4.a.a.1.1		1
12.11	even	2		39.4.a.a.1.1	✓	1
24.5	odd	2		2496.4.a.f.1.1		1
24.11	even	2		2496.4.a.o.1.1		1
52.51	odd	2		1521.4.a.f.1.1		1
60.59	even	2		975.4.a.e.1.1		1
84.83	odd	2		1911.4.a.f.1.1		1
156.47	odd	4		507.4.b.b.337.1		2
156.83	odd	4		507.4.b.b.337.2		2
156.155	even	2		507.4.a.c.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.a.1.1	✓	1	12.11	even	2
117.4.a.a.1.1		1	4.3	odd	2
507.4.a.c.1.1		1	156.155	even	2
507.4.b.b.337.1		2	156.47	odd	4
507.4.b.b.337.2		2	156.83	odd	4
624.4.a.g.1.1		1	3.2	odd	2
975.4.a.e.1.1		1	60.59	even	2
1521.4.a.f.1.1		1	52.51	odd	2
1872.4.a.m.1.1		1	1.1	even	1	trivial
1911.4.a.f.1.1		1	84.83	odd	2
2496.4.a.f.1.1		1	24.5	odd	2
2496.4.a.o.1.1		1	24.11	even	2