# Properties

 Label 2500.1.b.a Level $2500$ Weight $1$ Character orbit 2500.b Self dual yes Analytic conductor $1.248$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -4 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2500 = 2^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2500.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.24766253158$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 100) Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.6250000.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{4} - q^{8} + q^{9} + ( 1 - \beta ) q^{13} + q^{16} + \beta q^{17} - q^{18} + ( -1 + \beta ) q^{26} -\beta q^{29} - q^{32} -\beta q^{34} + q^{36} + \beta q^{37} + ( -1 + \beta ) q^{41} + q^{49} + ( 1 - \beta ) q^{52} + ( 1 - \beta ) q^{53} + \beta q^{58} + ( -1 + \beta ) q^{61} + q^{64} + \beta q^{68} - q^{72} + ( 1 - \beta ) q^{73} -\beta q^{74} + q^{81} + ( 1 - \beta ) q^{82} -\beta q^{89} + \beta q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{9} + q^{13} + 2 q^{16} + q^{17} - 2 q^{18} - q^{26} - q^{29} - 2 q^{32} - q^{34} + 2 q^{36} + q^{37} - q^{41} + 2 q^{49} + q^{52} + q^{53} + q^{58} - q^{61} + 2 q^{64} + q^{68} - 2 q^{72} + q^{73} - q^{74} + 2 q^{81} + q^{82} - q^{89} + q^{97} - 2 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times$$.

 $$n$$ $$1251$$ $$1877$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1251.1
 1.61803 −0.618034
−1.00000 0 1.00000 0 0 0 −1.00000 1.00000 0
1251.2 −1.00000 0 1.00000 0 0 0 −1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2500.1.b.a 2
4.b odd 2 1 CM 2500.1.b.a 2
5.b even 2 1 2500.1.b.b 2
5.c odd 4 2 2500.1.d.a 4
20.d odd 2 1 2500.1.b.b 2
20.e even 4 2 2500.1.d.a 4
25.d even 5 2 500.1.j.a 4
25.d even 5 2 2500.1.j.b 4
25.e even 10 2 100.1.j.a 4
25.e even 10 2 2500.1.j.a 4
25.f odd 20 4 500.1.h.a 8
25.f odd 20 4 2500.1.h.e 8
75.h odd 10 2 900.1.x.a 4
100.h odd 10 2 100.1.j.a 4
100.h odd 10 2 2500.1.j.a 4
100.j odd 10 2 500.1.j.a 4
100.j odd 10 2 2500.1.j.b 4
100.l even 20 4 500.1.h.a 8
100.l even 20 4 2500.1.h.e 8
200.o even 10 2 1600.1.bh.a 4
200.s odd 10 2 1600.1.bh.a 4
300.r even 10 2 900.1.x.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.1.j.a 4 25.e even 10 2
100.1.j.a 4 100.h odd 10 2
500.1.h.a 8 25.f odd 20 4
500.1.h.a 8 100.l even 20 4
500.1.j.a 4 25.d even 5 2
500.1.j.a 4 100.j odd 10 2
900.1.x.a 4 75.h odd 10 2
900.1.x.a 4 300.r even 10 2
1600.1.bh.a 4 200.o even 10 2
1600.1.bh.a 4 200.s odd 10 2
2500.1.b.a 2 1.a even 1 1 trivial
2500.1.b.a 2 4.b odd 2 1 CM
2500.1.b.b 2 5.b even 2 1
2500.1.b.b 2 20.d odd 2 1
2500.1.d.a 4 5.c odd 4 2
2500.1.d.a 4 20.e even 4 2
2500.1.h.e 8 25.f odd 20 4
2500.1.h.e 8 100.l even 20 4
2500.1.j.a 4 25.e even 10 2
2500.1.j.a 4 100.h odd 10 2
2500.1.j.b 4 25.d even 5 2
2500.1.j.b 4 100.j odd 10 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} - T_{13} - 1$$ acting on $$S_{1}^{\mathrm{new}}(2500, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$-1 - T + T^{2}$$
$17$ $$-1 - T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$-1 + T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$-1 - T + T^{2}$$
$41$ $$-1 + T + T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$-1 - T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$-1 + T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-1 - T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$-1 + T + T^{2}$$
$97$ $$-1 - T + T^{2}$$