# Properties

 Label 2500.1.j.c Level $2500$ Weight $1$ Character orbit 2500.j Analytic conductor $1.248$ Analytic rank $0$ Dimension $8$ Projective image $D_{5}$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2500 = 2^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2500.j (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.24766253158$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 500) Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.250000.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{20} q^{2} + ( - \zeta_{20}^{3} + \zeta_{20}) q^{3} + \zeta_{20}^{2} q^{4} + (\zeta_{20}^{4} - \zeta_{20}^{2}) q^{6} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{7} - \zeta_{20}^{3} q^{8} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{9} +O(q^{10})$$ q - z * q^2 + (-z^3 + z) * q^3 + z^2 * q^4 + (z^4 - z^2) * q^6 + (z^7 + z^3) * q^7 - z^3 * q^8 + (z^6 - z^4 + z^2) * q^9 $$q - \zeta_{20} q^{2} + ( - \zeta_{20}^{3} + \zeta_{20}) q^{3} + \zeta_{20}^{2} q^{4} + (\zeta_{20}^{4} - \zeta_{20}^{2}) q^{6} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{7} - \zeta_{20}^{3} q^{8} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{9} + ( - \zeta_{20}^{5} + \zeta_{20}^{3}) q^{12} + ( - \zeta_{20}^{8} - \zeta_{20}^{4}) q^{14} + \zeta_{20}^{4} q^{16} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3}) q^{18} + (\zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} + 1) q^{21} + (\zeta_{20}^{9} - \zeta_{20}^{3}) q^{23} + (\zeta_{20}^{6} - \zeta_{20}^{4}) q^{24} + ( - \zeta_{20}^{9} + \zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3}) q^{27} + (\zeta_{20}^{9} + \zeta_{20}^{5}) q^{28} + ( - \zeta_{20}^{8} + \zeta_{20}^{6}) q^{29} - \zeta_{20}^{5} q^{32} + (\zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4}) q^{36} + (\zeta_{20}^{8} + 1) q^{41} + ( - \zeta_{20}^{9} + \zeta_{20}^{7} - \zeta_{20}^{5} - \zeta_{20}) q^{42} + (\zeta_{20}^{9} + \zeta_{20}) q^{43} + (\zeta_{20}^{4} + 1) q^{46} + (\zeta_{20}^{3} - \zeta_{20}) q^{47} + ( - \zeta_{20}^{7} + \zeta_{20}^{5}) q^{48} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{49} + ( - \zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{54} + ( - \zeta_{20}^{6} + 1) q^{56} + (\zeta_{20}^{9} - \zeta_{20}^{7}) q^{58} + (\zeta_{20}^{8} + \zeta_{20}^{4}) q^{61} + (2 \zeta_{20}^{9} - \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}) q^{63} + \zeta_{20}^{6} q^{64} + \zeta_{20}^{3} q^{67} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{69} + ( - \zeta_{20}^{9} + \zeta_{20}^{7} - \zeta_{20}^{5}) q^{72} + (\zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{81} + ( - \zeta_{20}^{9} - \zeta_{20}) q^{82} + (\zeta_{20}^{5} + \zeta_{20}) q^{83} + ( - \zeta_{20}^{8} + \zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{84} + ( - \zeta_{20}^{2} + 1) q^{86} + ( - 2 \zeta_{20}^{9} + \zeta_{20}^{7} - \zeta_{20}) q^{87} + ( - \zeta_{20}^{8} - \zeta_{20}^{4}) q^{89} + ( - \zeta_{20}^{5} - \zeta_{20}) q^{92} + ( - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{94} + (\zeta_{20}^{8} - \zeta_{20}^{6}) q^{96} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}) q^{98} +O(q^{100})$$ q - z * q^2 + (-z^3 + z) * q^3 + z^2 * q^4 + (z^4 - z^2) * q^6 + (z^7 + z^3) * q^7 - z^3 * q^8 + (z^6 - z^4 + z^2) * q^9 + (-z^5 + z^3) * q^12 + (-z^8 - z^4) * q^14 + z^4 * q^16 + (-z^7 + z^5 - z^3) * q^18 + (z^8 - z^6 + z^4 + 1) * q^21 + (z^9 - z^3) * q^23 + (z^6 - z^4) * q^24 + (-z^9 + z^7 - z^5 + z^3) * q^27 + (z^9 + z^5) * q^28 + (-z^8 + z^6) * q^29 - z^5 * q^32 + (z^8 - z^6 + z^4) * q^36 + (z^8 + 1) * q^41 + (-z^9 + z^7 - z^5 - z) * q^42 + (z^9 + z) * q^43 + (z^4 + 1) * q^46 + (z^3 - z) * q^47 + (-z^7 + z^5) * q^48 + (z^6 - z^4 - 1) * q^49 + (-z^8 + z^6 - z^4 - 1) * q^54 + (-z^6 + 1) * q^56 + (z^9 - z^7) * q^58 + (z^8 + z^4) * q^61 + (2*z^9 - z^7 + z^5 - z^3 + z) * q^63 + z^6 * q^64 + z^3 * q^67 + (z^6 - z^4 + z^2 - 1) * q^69 + (-z^9 + z^7 - z^5) * q^72 + (z^8 - z^6 + z^4 - z^2 - 1) * q^81 + (-z^9 - z) * q^82 + (z^5 + z) * q^83 + (-z^8 + z^6 + z^2 - 1) * q^84 + (-z^2 + 1) * q^86 + (-2*z^9 + z^7 - z) * q^87 + (-z^8 - z^4) * q^89 + (-z^5 - z) * q^92 + (-z^4 + z^2) * q^94 + (z^8 - z^6) * q^96 + (-z^7 + z^5 + z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{4} - 4 q^{6} + 6 q^{9}+O(q^{10})$$ 8 * q + 2 * q^4 - 4 * q^6 + 6 * q^9 $$8 q + 2 q^{4} - 4 q^{6} + 6 q^{9} + 4 q^{14} - 2 q^{16} + 2 q^{21} + 4 q^{24} + 4 q^{29} - 6 q^{36} + 6 q^{41} + 6 q^{46} - 4 q^{49} - 2 q^{54} + 6 q^{56} - 4 q^{61} + 2 q^{64} - 2 q^{69} - 2 q^{84} + 6 q^{86} + 4 q^{89} + 4 q^{94} - 4 q^{96}+O(q^{100})$$ 8 * q + 2 * q^4 - 4 * q^6 + 6 * q^9 + 4 * q^14 - 2 * q^16 + 2 * q^21 + 4 * q^24 + 4 * q^29 - 6 * q^36 + 6 * q^41 + 6 * q^46 - 4 * q^49 - 2 * q^54 + 6 * q^56 - 4 * q^61 + 2 * q^64 - 2 * q^69 - 2 * q^84 + 6 * q^86 + 4 * q^89 + 4 * q^94 - 4 * q^96

## Character values

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

 $$n$$ $$1251$$ $$1877$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{20}^{2}$$

## 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
 0.587785 + 0.809017i −0.587785 − 0.809017i 0.951057 − 0.309017i −0.951057 + 0.309017i 0.951057 + 0.309017i −0.951057 − 0.309017i 0.587785 − 0.809017i −0.587785 + 0.809017i
−0.587785 0.809017i 1.53884 + 0.500000i −0.309017 + 0.951057i 0 −0.500000 1.53884i 0.618034i 0.951057 0.309017i 1.30902 + 0.951057i 0
251.2 0.587785 + 0.809017i −1.53884 0.500000i −0.309017 + 0.951057i 0 −0.500000 1.53884i 0.618034i −0.951057 + 0.309017i 1.30902 + 0.951057i 0
751.1 −0.951057 + 0.309017i 0.363271 + 0.500000i 0.809017 0.587785i 0 −0.500000 0.363271i 1.61803i −0.587785 + 0.809017i 0.190983 0.587785i 0
751.2 0.951057 0.309017i −0.363271 0.500000i 0.809017 0.587785i 0 −0.500000 0.363271i 1.61803i 0.587785 0.809017i 0.190983 0.587785i 0
1751.1 −0.951057 0.309017i 0.363271 0.500000i 0.809017 + 0.587785i 0 −0.500000 + 0.363271i 1.61803i −0.587785 0.809017i 0.190983 + 0.587785i 0
1751.2 0.951057 + 0.309017i −0.363271 + 0.500000i 0.809017 + 0.587785i 0 −0.500000 + 0.363271i 1.61803i 0.587785 + 0.809017i 0.190983 + 0.587785i 0
2251.1 −0.587785 + 0.809017i 1.53884 0.500000i −0.309017 0.951057i 0 −0.500000 + 1.53884i 0.618034i 0.951057 + 0.309017i 1.30902 0.951057i 0
2251.2 0.587785 0.809017i −1.53884 + 0.500000i −0.309017 0.951057i 0 −0.500000 + 1.53884i 0.618034i −0.951057 0.309017i 1.30902 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2251.2 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
25.d even 5 1 inner
25.e even 10 1 inner
100.h odd 10 1 inner
100.j odd 10 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.1.b.a 4 25.d even 5 1
500.1.b.a 4 25.e even 10 1
500.1.b.a 4 100.h odd 10 1
500.1.b.a 4 100.j odd 10 1
500.1.d.a 2 25.f odd 20 1
500.1.d.a 2 100.l even 20 1
500.1.d.b 2 25.f odd 20 1
500.1.d.b 2 100.l even 20 1
2500.1.h.a 4 5.c odd 4 1
2500.1.h.a 4 20.e even 4 1
2500.1.h.a 4 25.f odd 20 1
2500.1.h.a 4 100.l even 20 1
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 5.c odd 4 1
2500.1.h.d 4 20.e even 4 1
2500.1.h.d 4 25.f odd 20 1
2500.1.h.d 4 100.l even 20 1
2500.1.j.c 8 1.a even 1 1 trivial
2500.1.j.c 8 4.b odd 2 1 inner
2500.1.j.c 8 5.b even 2 1 inner
2500.1.j.c 8 20.d odd 2 1 CM
2500.1.j.c 8 25.d even 5 1 inner
2500.1.j.c 8 25.e even 10 1 inner
2500.1.j.c 8 100.h odd 10 1 inner
2500.1.j.c 8 100.j odd 10 1 inner
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 can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2500, [\chi])$$:

 $$T_{3}^{8} - 4T_{3}^{6} + 6T_{3}^{4} + T_{3}^{2} + 1$$ T3^8 - 4*T3^6 + 6*T3^4 + T3^2 + 1 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$3$ $$T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 3 T^{2} + 1)^{2}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1$$
$29$ $$(T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$(T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2}$$
$43$ $$(T^{4} + 3 T^{2} + 1)^{2}$$
$47$ $$T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$(T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1)^{2}$$
$67$ $$T^{8} - 4 T^{6} + 16 T^{4} - 64 T^{2} + \cdots + 256$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1$$
$89$ $$(T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2}$$
$97$ $$T^{8}$$