# Properties

 Label 500.1.b.a Level $500$ Weight $1$ Character orbit 500.b Analytic conductor $0.250$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -20 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$500 = 2^{2} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 500.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.249532506317$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - \beta_1 q^{3} - q^{4} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1) q^{7} - \beta_{3} q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + b3 * q^2 - b1 * q^3 - q^4 + b2 * q^6 + (-b3 - b1) * q^7 - b3 * q^8 + b2 * q^9 $$q + \beta_{3} q^{2} - \beta_1 q^{3} - q^{4} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1) q^{7} - \beta_{3} q^{8} + \beta_{2} q^{9} + \beta_1 q^{12} + (\beta_{2} + 1) q^{14} + q^{16} + \beta_1 q^{18} - q^{21} + (\beta_{3} + \beta_1) q^{23} - \beta_{2} q^{24} - \beta_{3} q^{27} + (\beta_{3} + \beta_1) q^{28} - \beta_{2} q^{29} + \beta_{3} q^{32} - \beta_{2} q^{36} + \beta_{2} q^{41} - \beta_{3} q^{42} - \beta_1 q^{43} + ( - \beta_{2} - 1) q^{46} + \beta_1 q^{47} - \beta_1 q^{48} + ( - \beta_{2} - 1) q^{49} + q^{54} + ( - \beta_{2} - 1) q^{56} - \beta_1 q^{58} + ( - \beta_{2} - 1) q^{61} - \beta_{3} q^{63} - q^{64} + 2 \beta_{3} q^{67} + q^{69} - \beta_1 q^{72} + \beta_1 q^{82} + (\beta_{3} + \beta_1) q^{83} + q^{84} + \beta_{2} q^{86} + (\beta_{3} - \beta_1) q^{87} + (\beta_{2} + 1) q^{89} + ( - \beta_{3} - \beta_1) q^{92} - \beta_{2} q^{94} + \beta_{2} q^{96} + ( - \beta_{3} - \beta_1) q^{98}+O(q^{100})$$ q + b3 * q^2 - b1 * q^3 - q^4 + b2 * q^6 + (-b3 - b1) * q^7 - b3 * q^8 + b2 * q^9 + b1 * q^12 + (b2 + 1) * q^14 + q^16 + b1 * q^18 - q^21 + (b3 + b1) * q^23 - b2 * q^24 - b3 * q^27 + (b3 + b1) * q^28 - b2 * q^29 + b3 * q^32 - b2 * q^36 + b2 * q^41 - b3 * q^42 - b1 * q^43 + (-b2 - 1) * q^46 + b1 * q^47 - b1 * q^48 + (-b2 - 1) * q^49 + q^54 + (-b2 - 1) * q^56 - b1 * q^58 + (-b2 - 1) * q^61 - b3 * q^63 - q^64 + 2*b3 * q^67 + q^69 - b1 * q^72 + b1 * q^82 + (b3 + b1) * q^83 + q^84 + b2 * q^86 + (b3 - b1) * q^87 + (b2 + 1) * q^89 + (-b3 - b1) * q^92 - b2 * q^94 + b2 * q^96 + (-b3 - b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 2 * q^6 - 2 * q^9 $$4 q - 4 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{14} + 4 q^{16} - 4 q^{21} + 2 q^{24} + 2 q^{29} + 2 q^{36} - 2 q^{41} - 2 q^{46} - 2 q^{49} + 4 q^{54} - 2 q^{56} - 2 q^{61} - 4 q^{64} + 4 q^{69} + 4 q^{84} - 2 q^{86} + 2 q^{89} + 2 q^{94} - 2 q^{96}+O(q^{100})$$ 4 * q - 4 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^14 + 4 * q^16 - 4 * q^21 + 2 * q^24 + 2 * q^29 + 2 * q^36 - 2 * q^41 - 2 * q^46 - 2 * q^49 + 4 * q^54 - 2 * q^56 - 2 * q^61 - 4 * q^64 + 4 * q^69 + 4 * q^84 - 2 * q^86 + 2 * q^89 + 2 * q^94 - 2 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*b1

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 1.61803i − 0.618034i 0.618034i − 1.61803i
1.00000i 1.61803i −1.00000 0 −1.61803 0.618034i 1.00000i −1.61803 0
251.2 1.00000i 0.618034i −1.00000 0 0.618034 1.61803i 1.00000i 0.618034 0
251.3 1.00000i 0.618034i −1.00000 0 0.618034 1.61803i 1.00000i 0.618034 0
251.4 1.00000i 1.61803i −1.00000 0 −1.61803 0.618034i 1.00000i −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 $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
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 500.1.b.a 4
4.b odd 2 1 inner 500.1.b.a 4
5.b even 2 1 inner 500.1.b.a 4
5.c odd 4 1 500.1.d.a 2
5.c odd 4 1 500.1.d.b 2
20.d odd 2 1 CM 500.1.b.a 4
20.e even 4 1 500.1.d.a 2
20.e even 4 1 500.1.d.b 2
25.d even 5 2 2500.1.j.c 8
25.d even 5 2 2500.1.j.d 8
25.e even 10 2 2500.1.j.c 8
25.e even 10 2 2500.1.j.d 8
25.f odd 20 2 2500.1.h.a 4
25.f odd 20 2 2500.1.h.b 4
25.f odd 20 2 2500.1.h.c 4
25.f odd 20 2 2500.1.h.d 4
100.h odd 10 2 2500.1.j.c 8
100.h odd 10 2 2500.1.j.d 8
100.j odd 10 2 2500.1.j.c 8
100.j odd 10 2 2500.1.j.d 8
100.l even 20 2 2500.1.h.a 4
100.l even 20 2 2500.1.h.b 4
100.l even 20 2 2500.1.h.c 4
100.l even 20 2 2500.1.h.d 4

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(500, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 3T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 3T^{2} + 1$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 3T^{2} + 1$$
$29$ $$(T^{2} - T - 1)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + T - 1)^{2}$$
$43$ $$T^{4} + 3T^{2} + 1$$
$47$ $$T^{4} + 3T^{2} + 1$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + T - 1)^{2}$$
$67$ $$(T^{2} + 4)^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 3T^{2} + 1$$
$89$ $$(T^{2} - T - 1)^{2}$$
$97$ $$T^{4}$$