Embedded newform 42.2.a.a.1.1

• Label: 42.2.a.a.1.1
• Level: $42$
• Weight: $2$
• Character: 42.1
• Self dual: yes
• Analytic conductor: $0.335$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.335371688489\)
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	\(\chi\)	\(=\)	42.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} +1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} -6.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +8.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -12.0000 q^{65} +4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -10.0000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +4.00000 q^{77} -6.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} +1.00000 q^{84} -4.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -6.00000 q^{91} +8.00000 q^{92} +8.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
			\(3\)	−1.00000	−0.577350
\(5\)	−2.00000	−0.894427				−0.447214	−	0.894427i	\(-0.647584\pi\)
			−0.447214	+	0.894427i	\(0.647584\pi\)
\(7\)	−1.00000	−0.377964
			\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
−0.603023	+	0.797724i				\(0.706037\pi\)
\(13\)	6.00000	1.66410	0.832050	−	0.554700i	\(-0.187167\pi\)
			0.832050	+	0.554700i	\(0.187167\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	−4.00000	−0.917663	−0.458831	−	0.888523i	\(-0.651732\pi\)
			−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	8.00000	1.66812	0.834058	−	0.551677i	\(-0.186012\pi\)
			0.834058	+	0.551677i	\(0.186012\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	−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	\(0.655213\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	6.00000	0.824163	0.412082	−	0.911147i	\(-0.364802\pi\)
			0.412082	+	0.911147i	\(0.364802\pi\)
\(59\)	4.00000	0.520756	0.260378	−	0.965507i	\(-0.416153\pi\)
			0.260378	+	0.965507i	\(0.416153\pi\)
\(61\)	6.00000	0.768221	0.384111	−	0.923287i	\(-0.374508\pi\)
			0.384111	+	0.923287i	\(0.374508\pi\)
\(67\)	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	\(0.342552\pi\)
\(73\)	10.0000	1.17041	0.585206	−	0.810885i	\(-0.301014\pi\)
			0.585206	+	0.810885i	\(0.301014\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	−4.00000	−0.439057	−0.219529	−	0.975606i	\(-0.570452\pi\)
			−0.219529	+	0.975606i	\(0.570452\pi\)
\(89\)	−6.00000	−0.635999	−0.317999	−	0.948091i	\(-0.603011\pi\)
			−0.317999	+	0.948091i	\(0.603011\pi\)
\(97\)	−14.0000	−1.42148	−0.710742	−	0.703452i	\(-0.751641\pi\)
			−0.710742	+	0.703452i	\(0.751641\pi\)

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	42.2.a.a.1.1	✓	1
3.2	odd	2		126.2.a.a.1.1		1
4.3	odd	2		336.2.a.d.1.1		1
5.2	odd	4		1050.2.g.a.799.2		2
5.3	odd	4		1050.2.g.a.799.1		2
5.4	even	2		1050.2.a.i.1.1		1
7.2	even	3		294.2.e.c.67.1		2
7.3	odd	6		294.2.e.a.79.1		2
7.4	even	3		294.2.e.c.79.1		2
7.5	odd	6		294.2.e.a.67.1		2
7.6	odd	2		294.2.a.g.1.1		1
8.3	odd	2		1344.2.a.i.1.1		1
8.5	even	2		1344.2.a.q.1.1		1
9.2	odd	6		1134.2.f.j.757.1		2
9.4	even	3		1134.2.f.g.379.1		2
9.5	odd	6		1134.2.f.j.379.1		2
9.7	even	3		1134.2.f.g.757.1		2
11.10	odd	2		5082.2.a.d.1.1		1
12.11	even	2		1008.2.a.j.1.1		1
13.12	even	2		7098.2.a.f.1.1		1
15.2	even	4		3150.2.g.r.2899.1		2
15.8	even	4		3150.2.g.r.2899.2		2
15.14	odd	2		3150.2.a.bo.1.1		1
16.3	odd	4		5376.2.c.e.2689.2		2
16.5	even	4		5376.2.c.bc.2689.2		2
16.11	odd	4		5376.2.c.e.2689.1		2
16.13	even	4		5376.2.c.bc.2689.1		2
20.19	odd	2		8400.2.a.k.1.1		1
21.2	odd	6		882.2.g.h.361.1		2
21.5	even	6		882.2.g.j.361.1		2
21.11	odd	6		882.2.g.h.667.1		2
21.17	even	6		882.2.g.j.667.1		2
21.20	even	2		882.2.a.b.1.1		1
24.5	odd	2		4032.2.a.e.1.1		1
24.11	even	2		4032.2.a.m.1.1		1
28.3	even	6		2352.2.q.n.961.1		2
28.11	odd	6		2352.2.q.i.961.1		2
28.19	even	6		2352.2.q.n.1537.1		2
28.23	odd	6		2352.2.q.i.1537.1		2
28.27	even	2		2352.2.a.l.1.1		1
35.34	odd	2		7350.2.a.f.1.1		1
56.13	odd	2		9408.2.a.n.1.1		1
56.27	even	2		9408.2.a.bw.1.1		1
84.83	odd	2		7056.2.a.k.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.a.a.1.1	✓	1	1.1	even	1	trivial
126.2.a.a.1.1		1	3.2	odd	2
294.2.a.g.1.1		1	7.6	odd	2
294.2.e.a.67.1		2	7.5	odd	6
294.2.e.a.79.1		2	7.3	odd	6
294.2.e.c.67.1		2	7.2	even	3
294.2.e.c.79.1		2	7.4	even	3
336.2.a.d.1.1		1	4.3	odd	2
882.2.a.b.1.1		1	21.20	even	2
882.2.g.h.361.1		2	21.2	odd	6
882.2.g.h.667.1		2	21.11	odd	6
882.2.g.j.361.1		2	21.5	even	6
882.2.g.j.667.1		2	21.17	even	6
1008.2.a.j.1.1		1	12.11	even	2
1050.2.a.i.1.1		1	5.4	even	2
1050.2.g.a.799.1		2	5.3	odd	4
1050.2.g.a.799.2		2	5.2	odd	4
1134.2.f.g.379.1		2	9.4	even	3
1134.2.f.g.757.1		2	9.7	even	3
1134.2.f.j.379.1		2	9.5	odd	6
1134.2.f.j.757.1		2	9.2	odd	6
1344.2.a.i.1.1		1	8.3	odd	2
1344.2.a.q.1.1		1	8.5	even	2
2352.2.a.l.1.1		1	28.27	even	2
2352.2.q.i.961.1		2	28.11	odd	6
2352.2.q.i.1537.1		2	28.23	odd	6
2352.2.q.n.961.1		2	28.3	even	6
2352.2.q.n.1537.1		2	28.19	even	6
3150.2.a.bo.1.1		1	15.14	odd	2
3150.2.g.r.2899.1		2	15.2	even	4
3150.2.g.r.2899.2		2	15.8	even	4
4032.2.a.e.1.1		1	24.5	odd	2
4032.2.a.m.1.1		1	24.11	even	2
5082.2.a.d.1.1		1	11.10	odd	2
5376.2.c.e.2689.1		2	16.11	odd	4
5376.2.c.e.2689.2		2	16.3	odd	4
5376.2.c.bc.2689.1		2	16.13	even	4
5376.2.c.bc.2689.2		2	16.5	even	4
7056.2.a.k.1.1		1	84.83	odd	2
7098.2.a.f.1.1		1	13.12	even	2
7350.2.a.f.1.1		1	35.34	odd	2
8400.2.a.k.1.1		1	20.19	odd	2
9408.2.a.n.1.1		1	56.13	odd	2
9408.2.a.bw.1.1		1	56.27	even	2