# Properties

 Label 500.1.p.a Level $500$ Weight $1$ Character orbit 500.p Analytic conductor $0.250$ Analytic rank $0$ Dimension $20$ Projective image $D_{25}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$500 = 2^{2} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 500.p (of order $$50$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.249532506317$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\Q(\zeta_{50})$$ Defining polynomial: $$x^{20} - x^{15} + x^{10} - x^{5} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{25}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{25} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{50}^{13} q^{2} -\zeta_{50} q^{4} -\zeta_{50}^{9} q^{5} + \zeta_{50}^{14} q^{8} -\zeta_{50}^{7} q^{9} +O(q^{10})$$ $$q -\zeta_{50}^{13} q^{2} -\zeta_{50} q^{4} -\zeta_{50}^{9} q^{5} + \zeta_{50}^{14} q^{8} -\zeta_{50}^{7} q^{9} + \zeta_{50}^{22} q^{10} + ( -\zeta_{50}^{3} - \zeta_{50}^{11} ) q^{13} + \zeta_{50}^{2} q^{16} + ( \zeta_{50}^{6} - \zeta_{50}^{17} ) q^{17} + \zeta_{50}^{20} q^{18} + \zeta_{50}^{10} q^{20} + \zeta_{50}^{18} q^{25} + ( \zeta_{50}^{16} + \zeta_{50}^{24} ) q^{26} + ( \zeta_{50}^{4} + \zeta_{50}^{8} ) q^{29} -\zeta_{50}^{15} q^{32} + ( -\zeta_{50}^{5} - \zeta_{50}^{19} ) q^{34} + \zeta_{50}^{8} q^{36} + ( -\zeta_{50}^{5} + \zeta_{50}^{24} ) q^{37} -\zeta_{50}^{23} q^{40} + ( -\zeta_{50}^{21} - \zeta_{50}^{23} ) q^{41} + \zeta_{50}^{16} q^{45} + \zeta_{50}^{10} q^{49} + \zeta_{50}^{6} q^{50} + ( \zeta_{50}^{4} + \zeta_{50}^{12} ) q^{52} + ( \zeta_{50}^{2} - \zeta_{50}^{15} ) q^{53} + ( -\zeta_{50}^{17} - \zeta_{50}^{21} ) q^{58} + ( \zeta_{50}^{12} - \zeta_{50}^{19} ) q^{61} -\zeta_{50}^{3} q^{64} + ( \zeta_{50}^{12} + \zeta_{50}^{20} ) q^{65} + ( -\zeta_{50}^{7} + \zeta_{50}^{18} ) q^{68} -\zeta_{50}^{21} q^{72} + ( -\zeta_{50}^{19} + \zeta_{50}^{22} ) q^{73} + ( \zeta_{50}^{12} + \zeta_{50}^{18} ) q^{74} -\zeta_{50}^{11} q^{80} + \zeta_{50}^{14} q^{81} + ( -\zeta_{50}^{9} - \zeta_{50}^{11} ) q^{82} + ( -\zeta_{50} - \zeta_{50}^{15} ) q^{85} + ( \zeta_{50}^{20} - \zeta_{50}^{21} ) q^{89} + \zeta_{50}^{4} q^{90} + ( \zeta_{50}^{14} - \zeta_{50}^{23} ) q^{97} -\zeta_{50}^{23} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + O(q^{10})$$ $$20q - 5q^{18} - 5q^{20} - 5q^{32} - 5q^{34} - 5q^{37} - 5q^{49} - 5q^{53} - 5q^{65} - 5q^{85} - 5q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$251$$ $$377$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{50}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −0.0627905 + 0.998027i 0.992115 − 0.125333i 0.187381 + 0.982287i −0.0627905 − 0.998027i 0.637424 − 0.770513i −0.876307 − 0.481754i 0.425779 − 0.904827i −0.535827 + 0.844328i −0.968583 + 0.248690i 0.425779 + 0.904827i −0.876307 + 0.481754i 0.929776 − 0.368125i 0.929776 + 0.368125i 0.187381 − 0.982287i 0.992115 + 0.125333i −0.968583 − 0.248690i −0.535827 − 0.844328i −0.728969 + 0.684547i −0.728969 − 0.684547i 0.637424 + 0.770513i
0.728969 0.684547i 0 0.0627905 0.998027i 0.535827 0.844328i 0 0 −0.637424 0.770513i −0.425779 + 0.904827i −0.187381 0.982287i
31.1 0.0627905 + 0.998027i 0 −0.992115 + 0.125333i −0.425779 + 0.904827i 0 0 −0.187381 0.982287i −0.637424 + 0.770513i −0.929776 0.368125i
71.1 −0.637424 + 0.770513i 0 −0.187381 0.982287i −0.992115 + 0.125333i 0 0 0.876307 + 0.481754i 0.968583 + 0.248690i 0.535827 0.844328i
91.1 0.728969 + 0.684547i 0 0.0627905 + 0.998027i 0.535827 + 0.844328i 0 0 −0.637424 + 0.770513i −0.425779 0.904827i −0.187381 + 0.982287i
111.1 −0.425779 0.904827i 0 −0.637424 + 0.770513i 0.0627905 + 0.998027i 0 0 0.968583 + 0.248690i −0.992115 0.125333i 0.876307 0.481754i
131.1 0.968583 + 0.248690i 0 0.876307 + 0.481754i −0.187381 0.982287i 0 0 0.728969 + 0.684547i −0.929776 0.368125i 0.0627905 0.998027i
171.1 0.535827 + 0.844328i 0 −0.425779 + 0.904827i 0.728969 0.684547i 0 0 −0.992115 + 0.125333i 0.0627905 + 0.998027i 0.968583 + 0.248690i
191.1 0.876307 0.481754i 0 0.535827 0.844328i −0.929776 0.368125i 0 0 0.0627905 0.998027i 0.728969 0.684547i −0.992115 + 0.125333i
211.1 −0.992115 + 0.125333i 0 0.968583 0.248690i −0.637424 0.770513i 0 0 −0.929776 + 0.368125i −0.187381 0.982287i 0.728969 + 0.684547i
231.1 0.535827 0.844328i 0 −0.425779 0.904827i 0.728969 + 0.684547i 0 0 −0.992115 0.125333i 0.0627905 0.998027i 0.968583 0.248690i
271.1 0.968583 0.248690i 0 0.876307 0.481754i −0.187381 + 0.982287i 0 0 0.728969 0.684547i −0.929776 + 0.368125i 0.0627905 + 0.998027i
291.1 −0.187381 0.982287i 0 −0.929776 + 0.368125i 0.968583 0.248690i 0 0 0.535827 + 0.844328i 0.876307 + 0.481754i −0.425779 0.904827i
311.1 −0.187381 + 0.982287i 0 −0.929776 0.368125i 0.968583 + 0.248690i 0 0 0.535827 0.844328i 0.876307 0.481754i −0.425779 + 0.904827i
331.1 −0.637424 0.770513i 0 −0.187381 + 0.982287i −0.992115 0.125333i 0 0 0.876307 0.481754i 0.968583 0.248690i 0.535827 + 0.844328i
371.1 0.0627905 0.998027i 0 −0.992115 0.125333i −0.425779 0.904827i 0 0 −0.187381 + 0.982287i −0.637424 0.770513i −0.929776 + 0.368125i
391.1 −0.992115 0.125333i 0 0.968583 + 0.248690i −0.637424 + 0.770513i 0 0 −0.929776 0.368125i −0.187381 + 0.982287i 0.728969 0.684547i
411.1 0.876307 + 0.481754i 0 0.535827 + 0.844328i −0.929776 + 0.368125i 0 0 0.0627905 + 0.998027i 0.728969 + 0.684547i −0.992115 0.125333i
431.1 −0.929776 + 0.368125i 0 0.728969 0.684547i 0.876307 0.481754i 0 0 −0.425779 + 0.904827i 0.535827 + 0.844328i −0.637424 + 0.770513i
471.1 −0.929776 0.368125i 0 0.728969 + 0.684547i 0.876307 + 0.481754i 0 0 −0.425779 0.904827i 0.535827 0.844328i −0.637424 0.770513i
491.1 −0.425779 + 0.904827i 0 −0.637424 0.770513i 0.0627905 0.998027i 0 0 0.968583 0.248690i −0.992115 + 0.125333i 0.876307 + 0.481754i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 491.1 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 $$\Q(\sqrt{-1})$$
125.g even 25 1 inner
500.p odd 50 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.1.p.a 20
4.b odd 2 1 CM 500.1.p.a 20
5.b even 2 1 2500.1.p.a 20
5.c odd 4 2 2500.1.n.a 40
20.d odd 2 1 2500.1.p.a 20
20.e even 4 2 2500.1.n.a 40
125.g even 25 1 inner 500.1.p.a 20
125.h even 50 1 2500.1.p.a 20
125.i odd 100 2 2500.1.n.a 40
500.n odd 50 1 2500.1.p.a 20
500.p odd 50 1 inner 500.1.p.a 20
500.r even 100 2 2500.1.n.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.1.p.a 20 1.a even 1 1 trivial
500.1.p.a 20 4.b odd 2 1 CM
500.1.p.a 20 125.g even 25 1 inner
500.1.p.a 20 500.p odd 50 1 inner
2500.1.n.a 40 5.c odd 4 2
2500.1.n.a 40 20.e even 4 2
2500.1.n.a 40 125.i odd 100 2
2500.1.n.a 40 500.r even 100 2
2500.1.p.a 20 5.b even 2 1
2500.1.p.a 20 20.d odd 2 1
2500.1.p.a 20 125.h even 50 1
2500.1.p.a 20 500.n odd 50 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{5} + T^{10} + T^{15} + T^{20}$$
$3$ $$T^{20}$$
$5$ $$1 + T^{5} + T^{10} + T^{15} + T^{20}$$
$7$ $$T^{20}$$
$11$ $$T^{20}$$
$13$ $$1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20}$$
$17$ $$1 + 10 T + 85 T^{2} + 200 T^{3} - 25 T^{4} - 122 T^{5} + 385 T^{6} - 615 T^{7} + 675 T^{8} - 225 T^{9} - 246 T^{10} + 145 T^{11} + 20 T^{12} - 75 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$19$ $$T^{20}$$
$23$ $$T^{20}$$
$29$ $$1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$31$ $$T^{20}$$
$37$ $$1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20}$$
$41$ $$1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$43$ $$T^{20}$$
$47$ $$T^{20}$$
$53$ $$1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20}$$
$59$ $$T^{20}$$
$61$ $$1 - 15 T + 85 T^{2} - 175 T^{3} + 600 T^{4} - 747 T^{5} + 360 T^{6} - 640 T^{7} + 800 T^{8} - 100 T^{9} + 379 T^{10} - 505 T^{11} - 5 T^{12} - 75 T^{13} + 150 T^{14} + 2 T^{15} + 5 T^{16} - 20 T^{17} + T^{20}$$
$67$ $$T^{20}$$
$71$ $$T^{20}$$
$73$ $$1 + 10 T + 110 T^{2} + 575 T^{3} + 1850 T^{4} + 3628 T^{5} + 4160 T^{6} + 2685 T^{7} + 1050 T^{8} + 525 T^{9} + 379 T^{10} + 145 T^{11} + 45 T^{12} + 50 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$
$79$ $$T^{20}$$
$83$ $$T^{20}$$
$89$ $$1 - 10 T + 70 T^{2} - 45 T^{3} - 340 T^{4} - 252 T^{5} + 665 T^{6} + 1520 T^{7} + 1655 T^{8} + 1235 T^{9} + 629 T^{10} + 205 T^{11} + 165 T^{12} + 185 T^{13} + 170 T^{14} + 122 T^{15} + 70 T^{16} + 35 T^{17} + 15 T^{18} + 5 T^{19} + T^{20}$$
$97$ $$1 + 10 T + 85 T^{2} + 200 T^{3} - 25 T^{4} - 122 T^{5} + 385 T^{6} - 615 T^{7} + 675 T^{8} - 225 T^{9} - 246 T^{10} + 145 T^{11} + 20 T^{12} - 75 T^{13} + 25 T^{14} + 2 T^{15} + 5 T^{16} + 5 T^{17} + T^{20}$$