Embedded newform 238.2.a.b.1.1

• Label: 238.2.a.b.1.1
• Level: $238$
• Weight: $2$
• Character: 238.1
• Self dual: yes
• Analytic conductor: $1.900$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 238 = 2 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	238.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.90043956811\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	238.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} -4.00000 q^{11} +2.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} +8.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +4.00000 q^{20} +2.00000 q^{21} +4.00000 q^{22} -2.00000 q^{24} +11.0000 q^{25} +4.00000 q^{26} -4.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -8.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -8.00000 q^{33} +1.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} +6.00000 q^{38} -8.00000 q^{39} -4.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} -4.00000 q^{44} +4.00000 q^{45} +4.00000 q^{47} +2.00000 q^{48} +1.00000 q^{49} -11.0000 q^{50} -2.00000 q^{51} -4.00000 q^{52} +14.0000 q^{53} +4.00000 q^{54} -16.0000 q^{55} -1.00000 q^{56} -12.0000 q^{57} -6.00000 q^{58} -6.00000 q^{59} +8.00000 q^{60} -12.0000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -16.0000 q^{65} +8.00000 q^{66} +4.00000 q^{67} -1.00000 q^{68} -4.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} +10.0000 q^{74} +22.0000 q^{75} -6.00000 q^{76} -4.00000 q^{77} +8.00000 q^{78} +4.00000 q^{80} -11.0000 q^{81} -6.00000 q^{82} +10.0000 q^{83} +2.00000 q^{84} -4.00000 q^{85} +12.0000 q^{87} +4.00000 q^{88} +10.0000 q^{89} -4.00000 q^{90} -4.00000 q^{91} +8.00000 q^{93} -4.00000 q^{94} -24.0000 q^{95} -2.00000 q^{96} +6.00000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	2.00000	1.15470	0.577350	−	0.816497i	\(-0.304087\pi\)
0.577350	+	0.816497i				\(0.304087\pi\)
\(5\)	4.00000	1.78885	0.894427	−	0.447214i	\(-0.147584\pi\)
			0.894427	+	0.447214i	\(0.147584\pi\)
\(7\)	1.00000	0.377964
			\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
−0.603023	+	0.797724i				\(0.706037\pi\)
\(13\)	−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	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	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	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	4.00000	0.583460	0.291730	−	0.956501i	\(-0.405769\pi\)
			0.291730	+	0.956501i	\(0.405769\pi\)
\(53\)	14.0000	1.92305	0.961524	−	0.274721i	\(-0.0885855\pi\)
			0.961524	+	0.274721i	\(0.0885855\pi\)
\(59\)	−6.00000	−0.781133	−0.390567	−	0.920575i	\(-0.627721\pi\)
			−0.390567	+	0.920575i	\(0.627721\pi\)
\(61\)	−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	\(-0.657448\pi\)
			−0.474713	+	0.880141i	\(0.657448\pi\)
\(73\)	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	\(-0.315077\pi\)
			0.548821	+	0.835940i	\(0.315077\pi\)
\(89\)	10.0000	1.06000	0.529999	−	0.847998i	\(-0.322192\pi\)
			0.529999	+	0.847998i	\(0.322192\pi\)
\(97\)	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	238.2.a.b.1.1	✓	1
3.2	odd	2		2142.2.a.l.1.1		1
4.3	odd	2		1904.2.a.b.1.1		1
5.4	even	2		5950.2.a.k.1.1		1
7.6	odd	2		1666.2.a.b.1.1		1
8.3	odd	2		7616.2.a.i.1.1		1
8.5	even	2		7616.2.a.a.1.1		1
17.16	even	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	1.1	even	1	trivial
1666.2.a.b.1.1		1	7.6	odd	2
1904.2.a.b.1.1		1	4.3	odd	2
2142.2.a.l.1.1		1	3.2	odd	2
4046.2.a.b.1.1		1	17.16	even	2
5950.2.a.k.1.1		1	5.4	even	2
7616.2.a.a.1.1		1	8.5	even	2
7616.2.a.i.1.1		1	8.3	odd	2