Newform orbit 2600.2.d.a

• Label: 2600.2.d.a
• Level: $2600$
• Weight: $2$
• Character orbit: 2600.d
• Analytic conductor: $20.761$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$2600 = 2^{3} \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2600.d (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 3*i * q^3 - 2*i * q^7 - 6 * q^9 - 2 * q^11 - i * q^13 - 2*i * q^17 - 4 * q^19 + 6 * q^21 + i * q^23 - 9*i * q^27 - q^29 - 2 * q^31 - 6*i * q^33 + 3 * q^39 + 12 * q^41 - 5*i * q^43 - 4*i * q^47 + 3 * q^49 + 6 * q^51 - 9*i * q^53 - 12*i * q^57 - 8 * q^59 + 7 * q^61 + 12*i * q^63 - 14*i * q^67 - 3 * q^69 - 6*i * q^73 + 4*i * q^77 + 15 * q^79 + 9 * q^81 - 4*i * q^83 - 3*i * q^87 + 18 * q^89 - 2 * q^91 - 6*i * q^93 - 16*i * q^97 + 12 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 12 q^{9} - 4 q^{11} - 8 q^{19} + 12 q^{21} - 2 q^{29} - 4 q^{31} + 6 q^{39} + 24 q^{41} + 6 q^{49} + 12 q^{51} - 16 q^{59} + 14 q^{61} - 6 q^{69} + 30 q^{79} + 18 q^{81} + 36 q^{89} - 4 q^{91} + 24 q^{99}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1249.1

	−	1.00000i
		1.00000i

0

−

3.00000i

0

0

0

2.00000i

0

−6.00000

0

1249.2

0

3.00000i

0

0

0

−

2.00000i

0

−6.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2600.2.d.a		2
5.b	even	2	1	inner	2600.2.d.a		2
5.c	odd	4	1		2600.2.a.a	✓	1
5.c	odd	4	1		2600.2.a.m	yes	1
20.e	even	4	1		5200.2.a.a		1
20.e	even	4	1		5200.2.a.bk		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2600.2.a.a	✓	1	5.c	odd	4	1
2600.2.a.m	yes	1	5.c	odd	4	1
2600.2.d.a		2	1.a	even	1	1	trivial
2600.2.d.a		2	5.b	even	2	1	inner
5200.2.a.a		1	20.e	even	4	1
5200.2.a.bk		1	20.e	even	4	1

Hecke kernels

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

$T_{3}^{2} + 9$

$T_{7}^{2} + 4$

$T_{11} + 2$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 9$
$5$	$T^{2}$
$7$	$T^{2} + 4$
$11$	$(T + 2)^{2}$