# Properties

 Label 500.1.d.a Level $500$ Weight $1$ Character orbit 500.d Self dual yes Analytic conductor $0.250$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -20 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$500 = 2^{2} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 500.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.249532506317$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.250000.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( - \beta + 1) q^{3} + q^{4} + (\beta - 1) q^{6} + \beta q^{7} - q^{8} + ( - \beta + 1) q^{9}+O(q^{10})$$ q - q^2 + (-b + 1) * q^3 + q^4 + (b - 1) * q^6 + b * q^7 - q^8 + (-b + 1) * q^9 $$q - q^{2} + ( - \beta + 1) q^{3} + q^{4} + (\beta - 1) q^{6} + \beta q^{7} - q^{8} + ( - \beta + 1) q^{9} + ( - \beta + 1) q^{12} - \beta q^{14} + q^{16} + (\beta - 1) q^{18} - q^{21} + \beta q^{23} + (\beta - 1) q^{24} + q^{27} + \beta q^{28} + (\beta - 1) q^{29} - q^{32} + ( - \beta + 1) q^{36} + (\beta - 1) q^{41} + q^{42} + ( - \beta + 1) q^{43} - \beta q^{46} + ( - \beta + 1) q^{47} + ( - \beta + 1) q^{48} + \beta q^{49} - q^{54} - \beta q^{56} + ( - \beta + 1) q^{58} - \beta q^{61} - q^{63} + q^{64} - 2 q^{67} - q^{69} + (\beta - 1) q^{72} + ( - \beta + 1) q^{82} + \beta q^{83} - q^{84} + (\beta - 1) q^{86} + (\beta - 2) q^{87} - \beta q^{89} + \beta q^{92} + (\beta - 1) q^{94} + (\beta - 1) q^{96} - \beta q^{98} +O(q^{100})$$ q - q^2 + (-b + 1) * q^3 + q^4 + (b - 1) * q^6 + b * q^7 - q^8 + (-b + 1) * q^9 + (-b + 1) * q^12 - b * q^14 + q^16 + (b - 1) * q^18 - q^21 + b * q^23 + (b - 1) * q^24 + q^27 + b * q^28 + (b - 1) * q^29 - q^32 + (-b + 1) * q^36 + (b - 1) * q^41 + q^42 + (-b + 1) * q^43 - b * q^46 + (-b + 1) * q^47 + (-b + 1) * q^48 + b * q^49 - q^54 - b * q^56 + (-b + 1) * q^58 - b * q^61 - q^63 + q^64 - 2 * q^67 - q^69 + (b - 1) * q^72 + (-b + 1) * q^82 + b * q^83 - q^84 + (b - 1) * q^86 + (b - 2) * q^87 - b * q^89 + b * q^92 + (b - 1) * q^94 + (b - 1) * q^96 - b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + q^9 $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + q^{7} - 2 q^{8} + q^{9} + q^{12} - q^{14} + 2 q^{16} - q^{18} - 2 q^{21} + q^{23} - q^{24} + 2 q^{27} + q^{28} - q^{29} - 2 q^{32} + q^{36} - q^{41} + 2 q^{42} + q^{43} - q^{46} + q^{47} + q^{48} + q^{49} - 2 q^{54} - q^{56} + q^{58} - q^{61} - 2 q^{63} + 2 q^{64} - 4 q^{67} - 2 q^{69} - q^{72} + q^{82} + q^{83} - 2 q^{84} - q^{86} - 3 q^{87} - q^{89} + q^{92} - q^{94} - q^{96} - q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + q^3 + 2 * q^4 - q^6 + q^7 - 2 * q^8 + q^9 + q^12 - q^14 + 2 * q^16 - q^18 - 2 * q^21 + q^23 - q^24 + 2 * q^27 + q^28 - q^29 - 2 * q^32 + q^36 - q^41 + 2 * q^42 + q^43 - q^46 + q^47 + q^48 + q^49 - 2 * q^54 - q^56 + q^58 - q^61 - 2 * q^63 + 2 * q^64 - 4 * q^67 - 2 * q^69 - q^72 + q^82 + q^83 - 2 * q^84 - q^86 - 3 * q^87 - q^89 + q^92 - q^94 - q^96 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/500\mathbb{Z}\right)^\times$$.

 $$n$$ $$251$$ $$377$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
499.1
 1.61803 −0.618034
−1.00000 −0.618034 1.00000 0 0.618034 1.61803 −1.00000 −0.618034 0
499.2 −1.00000 1.61803 1.00000 0 −1.61803 −0.618034 −1.00000 1.61803 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.1.d.a 2
4.b odd 2 1 500.1.d.b 2
5.b even 2 1 500.1.d.b 2
5.c odd 4 2 500.1.b.a 4
20.d odd 2 1 CM 500.1.d.a 2
20.e even 4 2 500.1.b.a 4
25.d even 5 2 2500.1.h.c 4
25.d even 5 2 2500.1.h.d 4
25.e even 10 2 2500.1.h.a 4
25.e even 10 2 2500.1.h.b 4
25.f odd 20 4 2500.1.j.c 8
25.f odd 20 4 2500.1.j.d 8
100.h odd 10 2 2500.1.h.c 4
100.h odd 10 2 2500.1.h.d 4
100.j odd 10 2 2500.1.h.a 4
100.j odd 10 2 2500.1.h.b 4
100.l even 20 4 2500.1.j.c 8
100.l even 20 4 2500.1.j.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.1.b.a 4 5.c odd 4 2
500.1.b.a 4 20.e even 4 2
500.1.d.a 2 1.a even 1 1 trivial
500.1.d.a 2 20.d odd 2 1 CM
500.1.d.b 2 4.b odd 2 1
500.1.d.b 2 5.b even 2 1
2500.1.h.a 4 25.e even 10 2
2500.1.h.a 4 100.j odd 10 2
2500.1.h.b 4 25.e even 10 2
2500.1.h.b 4 100.j odd 10 2
2500.1.h.c 4 25.d even 5 2
2500.1.h.c 4 100.h odd 10 2
2500.1.h.d 4 25.d even 5 2
2500.1.h.d 4 100.h odd 10 2
2500.1.j.c 8 25.f odd 20 4
2500.1.j.c 8 100.l even 20 4
2500.1.j.d 8 25.f odd 20 4
2500.1.j.d 8 100.l even 20 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} - 1$$ acting on $$S_{1}^{\mathrm{new}}(500, [\chi])$$.

## Hecke characteristic polynomials

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