Newform orbit 2600.2.a.w

• Label: 2600.2.a.w
• Level: $2600$
• Weight: $2$
• Character orbit: 2600.a
• Self dual: yes
• Analytic conductor: $20.761$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$2600 = 2^{3} \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2600.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$20.7611045255$
Analytic rank:	$0$
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 520)
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 + 2) * q^3 + 2 * q^7 + (4*b + 3) * q^9 + 3*b * q^11 + q^13 + (2*b + 2) * q^17 + (-3*b - 4) * q^19 + (2*b + 4) * q^21 + (-5*b + 2) * q^23 + (8*b + 8) * q^27 + (-4*b - 4) * q^29 - b * q^31 + (6*b + 6) * q^33 + (-2*b + 4) * q^37 + (b + 2) * q^39 + (-2*b + 2) * q^41 + (-3*b + 6) * q^43 - 2 * q^47 - 3 * q^49 + (6*b + 8) * q^51 + (-2*b + 6) * q^53 + (-10*b - 14) * q^57 + (-b - 4) * q^59 + (-8*b + 4) * q^61 + (8*b + 6) * q^63 + (-2*b + 2) * q^67 + (-8*b - 6) * q^69 - 7*b * q^71 + (-2*b - 4) * q^73 + 6*b * q^77 + (2*b + 4) * q^79 + (12*b + 23) * q^81 + 2 * q^83 + (-12*b - 16) * q^87 - 10 * q^89 + 2 * q^91 + (-2*b - 2) * q^93 + (4*b - 6) * q^97 + (9*b + 24) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 4 * q^3 + 4 * q^7 + 6 * q^9 + 2 * q^13 + 4 * q^17 - 8 * q^19 + 8 * q^21 + 4 * q^23 + 16 * q^27 - 8 * q^29 + 12 * q^33 + 8 * q^37 + 4 * q^39 + 4 * q^41 + 12 * q^43 - 4 * q^47 - 6 * q^49 + 16 * q^51 + 12 * q^53 - 28 * q^57 - 8 * q^59 + 8 * q^61 + 12 * q^63 + 4 * q^67 - 12 * q^69 - 8 * q^73 + 8 * q^79 + 46 * q^81 + 4 * q^83 - 32 * q^87 - 20 * q^89 + 4 * q^91 - 4 * q^93 - 12 * q^97 + 48 * 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

0.585786

0

0

0

2.00000

0

−2.65685

0

1.2

0

3.41421

0

0

0

2.00000

0

8.65685

0

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$
$13$	$-1$

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	2600.2.a.w		2
4.b	odd	2	1		5200.2.a.bl		2
5.b	even	2	1		520.2.a.c	✓	2
5.c	odd	4	2		2600.2.d.i		4
15.d	odd	2	1		4680.2.a.w		2
20.d	odd	2	1		1040.2.a.n		2
40.e	odd	2	1		4160.2.a.u		2
40.f	even	2	1		4160.2.a.bn		2
60.h	even	2	1		9360.2.a.ck		2
65.d	even	2	1		6760.2.a.n		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
520.2.a.c	✓	2	5.b	even	2	1
1040.2.a.n		2	20.d	odd	2	1
2600.2.a.w		2	1.a	even	1	1	trivial
2600.2.d.i		4	5.c	odd	4	2
4160.2.a.u		2	40.e	odd	2	1
4160.2.a.bn		2	40.f	even	2	1
4680.2.a.w		2	15.d	odd	2	1
5200.2.a.bl		2	4.b	odd	2	1
6760.2.a.n		2	65.d	even	2	1
9360.2.a.ck		2	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2600))$:

$T_{3}^{2} - 4T_{3} + 2$

$T_{7} - 2$

$T_{11}^{2} - 18$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} - 4T + 2$
$5$	$T^{2}$
$7$	$(T - 2)^{2}$
$11$	$T^{2} - 18$