Embedded newform 2366.2.a.j.1.1

• Label: 2366.2.a.j.1.1
• Level: $2366$
• Weight: $2$
• Character: 2366.1
• Self dual: yes
• Analytic conductor: $18.893$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$2366 = 2 \cdot 7 \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2366.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$18.8926051182$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2366.1

$q$-expansion

$f(q)$

$=$

$q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{12} -1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{21} -2.00000 q^{24} -5.00000 q^{25} +4.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} -6.00000 q^{41} +2.00000 q^{42} +8.00000 q^{43} +12.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -5.00000 q^{50} -12.0000 q^{51} +6.00000 q^{53} +4.00000 q^{54} -1.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} +6.00000 q^{59} +8.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +10.0000 q^{75} -2.00000 q^{76} +8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{82} +6.00000 q^{83} +2.00000 q^{84} +8.00000 q^{86} +12.0000 q^{87} +6.00000 q^{89} -8.00000 q^{93} +12.0000 q^{94} -2.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} +O(q^{100})$

Coefficient data

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

(See $a_n$ instead)

$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$
$2$	1.00000	0.707107
			$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
−0.577350	+	0.816497i				$0.695913\pi$
$5$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$7$	−1.00000	−0.377964
			$11$	0	0		−	1.00000i	$-0.5\pi$
		1.00000i				$0.5\pi$
$13$	0	0
			$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
0.727607	+	0.685994i				$0.240633\pi$
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
			−0.229416	+	0.973329i	$0.573682\pi$
$23$	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	$0.383046\pi$
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
			−0.164399	+	0.986394i	$0.552568\pi$
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
			−0.468521	+	0.883452i	$0.655213\pi$
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
			0.609994	+	0.792406i	$0.291172\pi$
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
			0.875190	+	0.483779i	$0.160736\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.a.j.1.1		1
13.5	odd	4		2366.2.d.b.337.1		2
13.8	odd	4		2366.2.d.b.337.2		2
13.12	even	2		14.2.a.a.1.1	✓	1
39.38	odd	2		126.2.a.b.1.1		1
52.51	odd	2		112.2.a.c.1.1		1
65.12	odd	4		350.2.c.d.99.1		2
65.38	odd	4		350.2.c.d.99.2		2
65.64	even	2		350.2.a.f.1.1		1
91.12	odd	6		98.2.c.a.67.1		2
91.25	even	6		98.2.c.b.79.1		2
91.38	odd	6		98.2.c.a.79.1		2
91.51	even	6		98.2.c.b.67.1		2
91.90	odd	2		98.2.a.a.1.1		1
104.51	odd	2		448.2.a.a.1.1		1
104.77	even	2		448.2.a.g.1.1		1
117.25	even	6		1134.2.f.l.757.1		2
117.38	odd	6		1134.2.f.f.757.1		2
117.77	odd	6		1134.2.f.f.379.1		2
117.103	even	6		1134.2.f.l.379.1		2
143.142	odd	2		1694.2.a.e.1.1		1
156.155	even	2		1008.2.a.h.1.1		1
195.38	even	4		3150.2.g.j.2899.1		2
195.77	even	4		3150.2.g.j.2899.2		2
195.194	odd	2		3150.2.a.i.1.1		1
208.51	odd	4		1792.2.b.g.897.2		2
208.77	even	4		1792.2.b.c.897.1		2
208.155	odd	4		1792.2.b.g.897.1		2
208.181	even	4		1792.2.b.c.897.2		2
221.220	even	2		4046.2.a.f.1.1		1
247.246	odd	2		5054.2.a.c.1.1		1
260.103	even	4		2800.2.g.h.449.2		2
260.207	even	4		2800.2.g.h.449.1		2
260.259	odd	2		2800.2.a.g.1.1		1
273.38	even	6		882.2.g.d.667.1		2
273.116	odd	6		882.2.g.c.667.1		2
273.194	even	6		882.2.g.d.361.1		2
273.233	odd	6		882.2.g.c.361.1		2
273.272	even	2		882.2.a.i.1.1		1
299.298	odd	2		7406.2.a.a.1.1		1
312.77	odd	2		4032.2.a.w.1.1		1
312.155	even	2		4032.2.a.r.1.1		1
364.51	odd	6		784.2.i.c.753.1		2
364.103	even	6		784.2.i.i.753.1		2
364.207	odd	6		784.2.i.c.177.1		2
364.311	even	6		784.2.i.i.177.1		2
364.363	even	2		784.2.a.b.1.1		1
455.272	even	4		2450.2.c.c.99.1		2
455.363	even	4		2450.2.c.c.99.2		2
455.454	odd	2		2450.2.a.t.1.1		1
728.181	odd	2		3136.2.a.e.1.1		1
728.363	even	2		3136.2.a.z.1.1		1
1092.1091	odd	2		7056.2.a.bd.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	13.12	even	2
98.2.a.a.1.1		1	91.90	odd	2
98.2.c.a.67.1		2	91.12	odd	6
98.2.c.a.79.1		2	91.38	odd	6
98.2.c.b.67.1		2	91.51	even	6
98.2.c.b.79.1		2	91.25	even	6
112.2.a.c.1.1		1	52.51	odd	2
126.2.a.b.1.1		1	39.38	odd	2
350.2.a.f.1.1		1	65.64	even	2
350.2.c.d.99.1		2	65.12	odd	4
350.2.c.d.99.2		2	65.38	odd	4
448.2.a.a.1.1		1	104.51	odd	2
448.2.a.g.1.1		1	104.77	even	2
784.2.a.b.1.1		1	364.363	even	2
784.2.i.c.177.1		2	364.207	odd	6
784.2.i.c.753.1		2	364.51	odd	6
784.2.i.i.177.1		2	364.311	even	6
784.2.i.i.753.1		2	364.103	even	6
882.2.a.i.1.1		1	273.272	even	2
882.2.g.c.361.1		2	273.233	odd	6
882.2.g.c.667.1		2	273.116	odd	6
882.2.g.d.361.1		2	273.194	even	6
882.2.g.d.667.1		2	273.38	even	6
1008.2.a.h.1.1		1	156.155	even	2
1134.2.f.f.379.1		2	117.77	odd	6
1134.2.f.f.757.1		2	117.38	odd	6
1134.2.f.l.379.1		2	117.103	even	6
1134.2.f.l.757.1		2	117.25	even	6
1694.2.a.e.1.1		1	143.142	odd	2
1792.2.b.c.897.1		2	208.77	even	4
1792.2.b.c.897.2		2	208.181	even	4
1792.2.b.g.897.1		2	208.155	odd	4
1792.2.b.g.897.2		2	208.51	odd	4
2366.2.a.j.1.1		1	1.1	even	1	trivial
2366.2.d.b.337.1		2	13.5	odd	4
2366.2.d.b.337.2		2	13.8	odd	4
2450.2.a.t.1.1		1	455.454	odd	2
2450.2.c.c.99.1		2	455.272	even	4
2450.2.c.c.99.2		2	455.363	even	4
2800.2.a.g.1.1		1	260.259	odd	2
2800.2.g.h.449.1		2	260.207	even	4
2800.2.g.h.449.2		2	260.103	even	4
3136.2.a.e.1.1		1	728.181	odd	2
3136.2.a.z.1.1		1	728.363	even	2
3150.2.a.i.1.1		1	195.194	odd	2
3150.2.g.j.2899.1		2	195.38	even	4
3150.2.g.j.2899.2		2	195.77	even	4
4032.2.a.r.1.1		1	312.155	even	2
4032.2.a.w.1.1		1	312.77	odd	2
4046.2.a.f.1.1		1	221.220	even	2
5054.2.a.c.1.1		1	247.246	odd	2
7056.2.a.bd.1.1		1	1092.1091	odd	2
7406.2.a.a.1.1		1	299.298	odd	2