Embedded newform 7938.2.a.o.1.1

• Label: 7938.2.a.o.1.1
• Level: $7938$
• Weight: $2$
• Character: 7938.1
• Self dual: yes
• Analytic conductor: $63.385$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$7938 = 2 \cdot 3^{4} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7938.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$63.3852491245$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7938.1

$q$-expansion

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{8} -3.00000 q^{10} +3.00000 q^{11} +5.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} +5.00000 q^{19} +3.00000 q^{20} -3.00000 q^{22} +3.00000 q^{23} +4.00000 q^{25} -5.00000 q^{26} +3.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} -7.00000 q^{37} -5.00000 q^{38} -3.00000 q^{40} +9.00000 q^{41} +11.0000 q^{43} +3.00000 q^{44} -3.00000 q^{46} -4.00000 q^{50} +5.00000 q^{52} +3.00000 q^{53} +9.00000 q^{55} -3.00000 q^{58} -12.0000 q^{59} +2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +15.0000 q^{65} -4.00000 q^{67} -3.00000 q^{68} +11.0000 q^{73} +7.00000 q^{74} +5.00000 q^{76} +8.00000 q^{79} +3.00000 q^{80} -9.00000 q^{82} -3.00000 q^{83} -9.00000 q^{85} -11.0000 q^{86} -3.00000 q^{88} -15.0000 q^{89} +3.00000 q^{92} +15.0000 q^{95} -1.00000 q^{97} +O(q^{100})$

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
			$3$	0	0
$5$	3.00000	1.34164				0.670820	−	0.741620i	$-0.265942\pi$
			0.670820	+	0.741620i	$0.265942\pi$
$7$	0	0
			$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
0.452267	+	0.891883i				$0.350615\pi$
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
			0.693375	+	0.720577i	$0.256123\pi$
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
			−0.363803	+	0.931476i	$0.618522\pi$
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
			0.573539	+	0.819178i	$0.305570\pi$
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
			0.312772	+	0.949828i	$0.398743\pi$
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
			0.278543	+	0.960424i	$0.410149\pi$
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
			−0.359211	+	0.933257i	$0.616954\pi$
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
			−0.575396	+	0.817875i	$0.695152\pi$
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
			0.702782	+	0.711405i	$0.251941\pi$
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
			0.838742	+	0.544529i	$0.183292\pi$
$47$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.o.1.1		1
3.2	odd	2		7938.2.a.r.1.1		1
7.2	even	3		1134.2.g.f.487.1		2
7.4	even	3		1134.2.g.f.163.1		2
7.6	odd	2		7938.2.a.c.1.1		1
9.2	odd	6		882.2.f.e.589.1		2
9.4	even	3		2646.2.f.e.883.1		2
9.5	odd	6		882.2.f.e.295.1		2
9.7	even	3		2646.2.f.e.1765.1		2
21.2	odd	6		1134.2.g.d.487.1		2
21.11	odd	6		1134.2.g.d.163.1		2
21.20	even	2		7938.2.a.bd.1.1		1
63.2	odd	6		126.2.h.a.67.1	yes	2
63.4	even	3		378.2.h.b.289.1		2
63.5	even	6		882.2.e.h.655.1		2
63.11	odd	6		126.2.e.b.121.1	yes	2
63.13	odd	6		2646.2.f.i.883.1		2
63.16	even	3		378.2.h.b.361.1		2
63.20	even	6		882.2.f.a.589.1		2
63.23	odd	6		126.2.e.b.25.1	✓	2
63.25	even	3		378.2.e.a.37.1		2
63.31	odd	6		2646.2.h.f.667.1		2
63.32	odd	6		126.2.h.a.79.1	yes	2
63.34	odd	6		2646.2.f.i.1765.1		2
63.38	even	6		882.2.e.h.373.1		2
63.40	odd	6		2646.2.e.e.2125.1		2
63.41	even	6		882.2.f.a.295.1		2
63.47	even	6		882.2.h.e.67.1		2
63.52	odd	6		2646.2.e.e.1549.1		2
63.58	even	3		378.2.e.a.235.1		2
63.59	even	6		882.2.h.e.79.1		2
63.61	odd	6		2646.2.h.f.361.1		2
252.11	even	6		1008.2.q.e.625.1		2
252.23	even	6		1008.2.q.e.529.1		2
252.67	odd	6		3024.2.t.f.289.1		2
252.79	odd	6		3024.2.t.f.1873.1		2
252.95	even	6		1008.2.t.c.961.1		2
252.151	odd	6		3024.2.q.a.2305.1		2
252.191	even	6		1008.2.t.c.193.1		2
252.247	odd	6		3024.2.q.a.2881.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.e.b.25.1	✓	2	63.23	odd	6
126.2.e.b.121.1	yes	2	63.11	odd	6
126.2.h.a.67.1	yes	2	63.2	odd	6
126.2.h.a.79.1	yes	2	63.32	odd	6
378.2.e.a.37.1		2	63.25	even	3
378.2.e.a.235.1		2	63.58	even	3
378.2.h.b.289.1		2	63.4	even	3
378.2.h.b.361.1		2	63.16	even	3
882.2.e.h.373.1		2	63.38	even	6
882.2.e.h.655.1		2	63.5	even	6
882.2.f.a.295.1		2	63.41	even	6
882.2.f.a.589.1		2	63.20	even	6
882.2.f.e.295.1		2	9.5	odd	6
882.2.f.e.589.1		2	9.2	odd	6
882.2.h.e.67.1		2	63.47	even	6
882.2.h.e.79.1		2	63.59	even	6
1008.2.q.e.529.1		2	252.23	even	6
1008.2.q.e.625.1		2	252.11	even	6
1008.2.t.c.193.1		2	252.191	even	6
1008.2.t.c.961.1		2	252.95	even	6
1134.2.g.d.163.1		2	21.11	odd	6
1134.2.g.d.487.1		2	21.2	odd	6
1134.2.g.f.163.1		2	7.4	even	3
1134.2.g.f.487.1		2	7.2	even	3
2646.2.e.e.1549.1		2	63.52	odd	6
2646.2.e.e.2125.1		2	63.40	odd	6
2646.2.f.e.883.1		2	9.4	even	3
2646.2.f.e.1765.1		2	9.7	even	3
2646.2.f.i.883.1		2	63.13	odd	6
2646.2.f.i.1765.1		2	63.34	odd	6
2646.2.h.f.361.1		2	63.61	odd	6
2646.2.h.f.667.1		2	63.31	odd	6
3024.2.q.a.2305.1		2	252.151	odd	6
3024.2.q.a.2881.1		2	252.247	odd	6
3024.2.t.f.289.1		2	252.67	odd	6
3024.2.t.f.1873.1		2	252.79	odd	6
7938.2.a.c.1.1		1	7.6	odd	2
7938.2.a.o.1.1		1	1.1	even	1	trivial
7938.2.a.r.1.1		1	3.2	odd	2
7938.2.a.bd.1.1		1	21.20	even	2