Embedded newform 234.2.a.a.1.1

• Label: 234.2.a.a.1.1
• Level: $234$
• Weight: $2$
• Character: 234.1
• Self dual: yes
• Analytic conductor: $1.868$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	234.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.86849940730\)
Analytic rank:	\(1\)
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	\(\chi\)	\(=\)	234.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{25} +1.00000 q^{26} -2.00000 q^{28} -8.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} +4.00000 q^{35} +6.00000 q^{37} +6.00000 q^{38} +2.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} +8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -1.00000 q^{52} +12.0000 q^{53} +8.00000 q^{55} +2.00000 q^{56} +8.00000 q^{58} +4.00000 q^{59} +10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -2.00000 q^{67} -4.00000 q^{70} -16.0000 q^{71} +14.0000 q^{73} -6.00000 q^{74} -6.00000 q^{76} +8.00000 q^{77} -4.00000 q^{79} -2.00000 q^{80} -6.00000 q^{82} -12.0000 q^{83} +8.00000 q^{86} +4.00000 q^{88} -6.00000 q^{89} +2.00000 q^{91} +4.00000 q^{92} -8.00000 q^{94} +12.0000 q^{95} -10.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(5\)	−2.00000	−0.894427				−0.447214	−	0.894427i	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
			−0.377964	+	0.925820i	\(0.623376\pi\)
\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
			−0.603023	+	0.797724i	\(0.706037\pi\)
\(13\)	−1.00000	−0.277350
			\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(19\)	−6.00000	−1.37649	−0.688247	−	0.725476i	\(-0.741620\pi\)
			−0.688247	+	0.725476i	\(0.741620\pi\)
\(23\)	4.00000	0.834058	0.417029	−	0.908893i	\(-0.363071\pi\)
			0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	−8.00000	−1.48556	−0.742781	−	0.669534i	\(-0.766494\pi\)
			−0.742781	+	0.669534i	\(0.766494\pi\)
\(31\)	−2.00000	−0.359211	−0.179605	−	0.983739i	\(-0.557482\pi\)
			−0.179605	+	0.983739i	\(0.557482\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	8.00000	1.16692	0.583460	−	0.812142i	\(-0.301699\pi\)
			0.583460	+	0.812142i	\(0.301699\pi\)
\(53\)	12.0000	1.64833	0.824163	−	0.566352i	\(-0.191646\pi\)
			0.824163	+	0.566352i	\(0.191646\pi\)
\(59\)	4.00000	0.520756	0.260378	−	0.965507i	\(-0.416153\pi\)
			0.260378	+	0.965507i	\(0.416153\pi\)
\(61\)	10.0000	1.28037	0.640184	−	0.768221i	\(-0.278858\pi\)
			0.640184	+	0.768221i	\(0.278858\pi\)
\(67\)	−2.00000	−0.244339	−0.122169	−	0.992509i	\(-0.538985\pi\)
			−0.122169	+	0.992509i	\(0.538985\pi\)
\(71\)	−16.0000	−1.89885	−0.949425	−	0.313993i	\(-0.898333\pi\)
			−0.949425	+	0.313993i	\(0.898333\pi\)
\(73\)	14.0000	1.63858	0.819288	−	0.573382i	\(-0.194369\pi\)
			0.819288	+	0.573382i	\(0.194369\pi\)
\(79\)	−4.00000	−0.450035	−0.225018	−	0.974355i	\(-0.572244\pi\)
			−0.225018	+	0.974355i	\(0.572244\pi\)
\(83\)	−12.0000	−1.31717	−0.658586	−	0.752506i	\(-0.728845\pi\)
			−0.658586	+	0.752506i	\(0.728845\pi\)
\(89\)	−6.00000	−0.635999	−0.317999	−	0.948091i	\(-0.603011\pi\)
			−0.317999	+	0.948091i	\(0.603011\pi\)
\(97\)	−10.0000	−1.01535	−0.507673	−	0.861550i	\(-0.669494\pi\)
			−0.507673	+	0.861550i	\(0.669494\pi\)

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	234.2.a.a.1.1	✓	1
3.2	odd	2		234.2.a.d.1.1	yes	1
4.3	odd	2		1872.2.a.g.1.1		1
5.2	odd	4		5850.2.e.d.5149.1		2
5.3	odd	4		5850.2.e.d.5149.2		2
5.4	even	2		5850.2.a.bv.1.1		1
8.3	odd	2		7488.2.a.bu.1.1		1
8.5	even	2		7488.2.a.bp.1.1		1
9.2	odd	6		2106.2.e.e.1405.1		2
9.4	even	3		2106.2.e.z.703.1		2
9.5	odd	6		2106.2.e.e.703.1		2
9.7	even	3		2106.2.e.z.1405.1		2
12.11	even	2		1872.2.a.p.1.1		1
13.5	odd	4		3042.2.b.c.1351.2		2
13.8	odd	4		3042.2.b.c.1351.1		2
13.12	even	2		3042.2.a.o.1.1		1
15.2	even	4		5850.2.e.bd.5149.2		2
15.8	even	4		5850.2.e.bd.5149.1		2
15.14	odd	2		5850.2.a.v.1.1		1
24.5	odd	2		7488.2.a.j.1.1		1
24.11	even	2		7488.2.a.s.1.1		1
39.5	even	4		3042.2.b.b.1351.1		2
39.8	even	4		3042.2.b.b.1351.2		2
39.38	odd	2		3042.2.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.a.a.1.1	✓	1	1.1	even	1	trivial
234.2.a.d.1.1	yes	1	3.2	odd	2
1872.2.a.g.1.1		1	4.3	odd	2
1872.2.a.p.1.1		1	12.11	even	2
2106.2.e.e.703.1		2	9.5	odd	6
2106.2.e.e.1405.1		2	9.2	odd	6
2106.2.e.z.703.1		2	9.4	even	3
2106.2.e.z.1405.1		2	9.7	even	3
3042.2.a.b.1.1		1	39.38	odd	2
3042.2.a.o.1.1		1	13.12	even	2
3042.2.b.b.1351.1		2	39.5	even	4
3042.2.b.b.1351.2		2	39.8	even	4
3042.2.b.c.1351.1		2	13.8	odd	4
3042.2.b.c.1351.2		2	13.5	odd	4
5850.2.a.v.1.1		1	15.14	odd	2
5850.2.a.bv.1.1		1	5.4	even	2
5850.2.e.d.5149.1		2	5.2	odd	4
5850.2.e.d.5149.2		2	5.3	odd	4
5850.2.e.bd.5149.1		2	15.8	even	4
5850.2.e.bd.5149.2		2	15.2	even	4
7488.2.a.j.1.1		1	24.5	odd	2
7488.2.a.s.1.1		1	24.11	even	2
7488.2.a.bp.1.1		1	8.5	even	2
7488.2.a.bu.1.1		1	8.3	odd	2