Embedded newform 2142.2.a.l.1.1

• Label: 2142.2.a.l.1.1
• Level: $2142$
• Weight: $2$
• Character: 2142.1
• Self dual: yes
• Analytic conductor: $17.104$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2142.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(17.1039561130\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 238)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2142.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -4.00000 q^{10} +4.00000 q^{11} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -6.00000 q^{19} -4.00000 q^{20} +4.00000 q^{22} +11.0000 q^{25} -4.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} -4.00000 q^{35} -10.0000 q^{37} -6.00000 q^{38} -4.00000 q^{40} -6.00000 q^{41} +4.00000 q^{44} -4.00000 q^{47} +1.00000 q^{49} +11.0000 q^{50} -4.00000 q^{52} -14.0000 q^{53} -16.0000 q^{55} +1.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} -12.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +16.0000 q^{65} +4.00000 q^{67} +1.00000 q^{68} -4.00000 q^{70} +8.00000 q^{71} +2.00000 q^{73} -10.0000 q^{74} -6.00000 q^{76} +4.00000 q^{77} -4.00000 q^{80} -6.00000 q^{82} -10.0000 q^{83} -4.00000 q^{85} +4.00000 q^{88} -10.0000 q^{89} -4.00000 q^{91} -4.00000 q^{94} +24.0000 q^{95} +6.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	0.707107
			\(3\)	0	0
\(5\)	−4.00000	−1.78885				−0.894427	−	0.447214i	\(-0.852416\pi\)
			−0.894427	+	0.447214i	\(0.852416\pi\)
\(7\)	1.00000	0.377964
			\(11\)	4.00000	1.20605	0.603023	−	0.797724i	\(-0.293963\pi\)
0.603023	+	0.797724i				\(0.293963\pi\)
\(13\)	−4.00000	−1.10940	−0.554700	−	0.832050i	\(-0.687167\pi\)
			−0.554700	+	0.832050i	\(0.687167\pi\)
\(17\)	1.00000	0.242536
			\(19\)	−6.00000	−1.37649	−0.688247	−	0.725476i	\(-0.741620\pi\)
−0.688247	+	0.725476i				\(0.741620\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−10.0000	−1.64399	−0.821995	−	0.569495i	\(-0.807139\pi\)
			−0.821995	+	0.569495i	\(0.807139\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	−4.00000	−0.583460	−0.291730	−	0.956501i	\(-0.594231\pi\)
			−0.291730	+	0.956501i	\(0.594231\pi\)
\(53\)	−14.0000	−1.92305	−0.961524	−	0.274721i	\(-0.911414\pi\)
			−0.961524	+	0.274721i	\(0.911414\pi\)
\(59\)	6.00000	0.781133	0.390567	−	0.920575i	\(-0.372279\pi\)
			0.390567	+	0.920575i	\(0.372279\pi\)
\(61\)	−12.0000	−1.53644	−0.768221	−	0.640184i	\(-0.778858\pi\)
			−0.768221	+	0.640184i	\(0.778858\pi\)
\(67\)	4.00000	0.488678	0.244339	−	0.969690i	\(-0.421429\pi\)
			0.244339	+	0.969690i	\(0.421429\pi\)
\(71\)	8.00000	0.949425	0.474713	−	0.880141i	\(-0.342552\pi\)
			0.474713	+	0.880141i	\(0.342552\pi\)
\(73\)	2.00000	0.234082	0.117041	−	0.993127i	\(-0.462659\pi\)
			0.117041	+	0.993127i	\(0.462659\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	−10.0000	−1.09764	−0.548821	−	0.835940i	\(-0.684923\pi\)
			−0.548821	+	0.835940i	\(0.684923\pi\)
\(89\)	−10.0000	−1.06000	−0.529999	−	0.847998i	\(-0.677808\pi\)
			−0.529999	+	0.847998i	\(0.677808\pi\)
\(97\)	6.00000	0.609208	0.304604	−	0.952479i	\(-0.401476\pi\)
			0.304604	+	0.952479i	\(0.401476\pi\)

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	2142.2.a.l.1.1		1
3.2	odd	2		238.2.a.b.1.1	✓	1
12.11	even	2		1904.2.a.b.1.1		1
15.14	odd	2		5950.2.a.k.1.1		1
21.20	even	2		1666.2.a.b.1.1		1
24.5	odd	2		7616.2.a.a.1.1		1
24.11	even	2		7616.2.a.i.1.1		1
51.50	odd	2		4046.2.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.a.b.1.1	✓	1	3.2	odd	2
1666.2.a.b.1.1		1	21.20	even	2
1904.2.a.b.1.1		1	12.11	even	2
2142.2.a.l.1.1		1	1.1	even	1	trivial
4046.2.a.b.1.1		1	51.50	odd	2
5950.2.a.k.1.1		1	15.14	odd	2
7616.2.a.a.1.1		1	24.5	odd	2
7616.2.a.i.1.1		1	24.11	even	2