Newform orbit 3920.2.a.bq

• Label: 3920.2.a.bq
• Level: $3920$
• Weight: $2$
• Character orbit: 3920.a
• Self dual: yes
• Analytic conductor: $31.301$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$3920 = 2^{4} \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3920.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$31.3013575923$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2})$
Defining polynomial:	$x^{2} - 2$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{2}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (b - 1) * q^3 + q^5 - 2*b * q^9 + (2*b - 2) * q^11 + (-2*b - 2) * q^13 + (b - 1) * q^15 + (2*b + 2) * q^17 - 2*b * q^19 + (-b + 1) * q^23 + q^25 + (-b - 1) * q^27 - q^29 + 6 * q^31 + (-4*b + 6) * q^33 - 2 * q^39 + (-2*b - 5) * q^41 + (b - 5) * q^43 - 2*b * q^45 - 2 * q^47 + 2 * q^51 + (2*b - 4) * q^53 + (2*b - 2) * q^55 + (2*b - 4) * q^57 + (-6*b + 4) * q^59 + (6*b - 3) * q^61 + (-2*b - 2) * q^65 + (b - 11) * q^67 + (2*b - 3) * q^69 + (-6*b + 4) * q^71 + (-2*b + 2) * q^73 + (b - 1) * q^75 + (-2*b - 12) * q^79 + (6*b - 1) * q^81 + (-9*b - 1) * q^83 + (2*b + 2) * q^85 + (-b + 1) * q^87 + (-4*b - 3) * q^89 + (6*b - 6) * q^93 - 2*b * q^95 + (4*b + 6) * q^97 + (4*b - 8) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 2 * q^3 + 2 * q^5 - 4 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^29 + 12 * q^31 + 12 * q^33 - 4 * q^39 - 10 * q^41 - 10 * q^43 - 4 * q^47 + 4 * q^51 - 8 * q^53 - 4 * q^55 - 8 * q^57 + 8 * q^59 - 6 * q^61 - 4 * q^65 - 22 * q^67 - 6 * q^69 + 8 * q^71 + 4 * q^73 - 2 * q^75 - 24 * q^79 - 2 * q^81 - 2 * q^83 + 4 * q^85 + 2 * q^87 - 6 * q^89 - 12 * q^93 + 12 * q^97 - 16 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

Label  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

−1.41421
1.41421

0

−2.41421

0

1.00000

0

0

0

2.82843

0

1.2

0

0.414214

0

1.00000

0

0

0

−2.82843

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$
$7$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3920.2.a.bq		2
4.b	odd	2	1		245.2.a.h		2
7.b	odd	2	1		3920.2.a.bv		2
7.c	even	3	2		560.2.q.k		4
12.b	even	2	1		2205.2.a.n		2
20.d	odd	2	1		1225.2.a.k		2
20.e	even	4	2		1225.2.b.g		4
28.d	even	2	1		245.2.a.g		2
28.f	even	6	2		245.2.e.e		4
28.g	odd	6	2		35.2.e.a	✓	4
84.h	odd	2	1		2205.2.a.q		2
84.n	even	6	2		315.2.j.e		4
140.c	even	2	1		1225.2.a.m		2
140.j	odd	4	2		1225.2.b.h		4
140.p	odd	6	2		175.2.e.c		4
140.w	even	12	4		175.2.k.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.2.e.a	✓	4	28.g	odd	6	2
175.2.e.c		4	140.p	odd	6	2
175.2.k.a		8	140.w	even	12	4
245.2.a.g		2	28.d	even	2	1
245.2.a.h		2	4.b	odd	2	1
245.2.e.e		4	28.f	even	6	2
315.2.j.e		4	84.n	even	6	2
560.2.q.k		4	7.c	even	3	2
1225.2.a.k		2	20.d	odd	2	1
1225.2.a.m		2	140.c	even	2	1
1225.2.b.g		4	20.e	even	4	2
1225.2.b.h		4	140.j	odd	4	2
2205.2.a.n		2	12.b	even	2	1
2205.2.a.q		2	84.h	odd	2	1
3920.2.a.bq		2	1.a	even	1	1	trivial
3920.2.a.bv		2	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3920))$:

$T_{3}^{2} + 2T_{3} - 1$

$T_{11}^{2} + 4T_{11} - 4$

$T_{13}^{2} + 4T_{13} - 4$

$T_{17}^{2} - 4T_{17} - 4$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 2T - 1$
$5$	$(T - 1)^{2}$
$7$	$T^{2}$
$11$	$T^{2} + 4T - 4$