# Properties

 Label 2500.1.h.a Level $2500$ Weight $1$ Character orbit 2500.h Analytic conductor $1.248$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -20 Inner twists $4$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.24766253158$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 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.

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

## Character values

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

 $$n$$ $$1251$$ $$1877$$ $$\chi(n)$$ $$-1$$ $$\zeta_{10}^{3}$$

## 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
 −0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i
0.309017 + 0.951057i −0.500000 + 0.363271i −0.809017 + 0.587785i 0 −0.500000 0.363271i −1.61803 −0.809017 0.587785i −0.190983 + 0.587785i 0
999.1 −0.809017 + 0.587785i −0.500000 + 1.53884i 0.309017 0.951057i 0 −0.500000 1.53884i 0.618034 0.309017 + 0.951057i −1.30902 0.951057i 0
1499.1 −0.809017 0.587785i −0.500000 1.53884i 0.309017 + 0.951057i 0 −0.500000 + 1.53884i 0.618034 0.309017 0.951057i −1.30902 + 0.951057i 0
1999.1 0.309017 0.951057i −0.500000 0.363271i −0.809017 0.587785i 0 −0.500000 + 0.363271i −1.61803 −0.809017 + 0.587785i −0.190983 0.587785i 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 $$\Q(\sqrt{-5})$$
25.d even 5 1 inner
100.h 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.h.a 4
4.b odd 2 1 2500.1.h.d 4
5.b even 2 1 2500.1.h.d 4
5.c odd 4 2 2500.1.j.c 8
20.d odd 2 1 CM 2500.1.h.a 4
20.e even 4 2 2500.1.j.c 8
25.d even 5 1 500.1.d.b 2
25.d even 5 1 inner 2500.1.h.a 4
25.d even 5 2 2500.1.h.b 4
25.e even 10 1 500.1.d.a 2
25.e even 10 2 2500.1.h.c 4
25.e even 10 1 2500.1.h.d 4
25.f odd 20 2 500.1.b.a 4
25.f odd 20 2 2500.1.j.c 8
25.f odd 20 4 2500.1.j.d 8
100.h odd 10 1 500.1.d.b 2
100.h odd 10 1 inner 2500.1.h.a 4
100.h odd 10 2 2500.1.h.b 4
100.j odd 10 1 500.1.d.a 2
100.j odd 10 2 2500.1.h.c 4
100.j odd 10 1 2500.1.h.d 4
100.l even 20 2 500.1.b.a 4
100.l even 20 2 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 25.f odd 20 2
500.1.b.a 4 100.l even 20 2
500.1.d.a 2 25.e even 10 1
500.1.d.a 2 100.j odd 10 1
500.1.d.b 2 25.d even 5 1
500.1.d.b 2 100.h odd 10 1
2500.1.h.a 4 1.a even 1 1 trivial
2500.1.h.a 4 20.d odd 2 1 CM
2500.1.h.a 4 25.d even 5 1 inner
2500.1.h.a 4 100.h odd 10 1 inner
2500.1.h.b 4 25.d even 5 2
2500.1.h.b 4 100.h odd 10 2
2500.1.h.c 4 25.e even 10 2
2500.1.h.c 4 100.j odd 10 2
2500.1.h.d 4 4.b odd 2 1
2500.1.h.d 4 5.b even 2 1
2500.1.h.d 4 25.e even 10 1
2500.1.h.d 4 100.j odd 10 1
2500.1.j.c 8 5.c odd 4 2
2500.1.j.c 8 20.e even 4 2
2500.1.j.c 8 25.f odd 20 2
2500.1.j.c 8 100.l even 20 2
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 $$T_{3}^{4} + 2T_{3}^{3} + 4T_{3}^{2} + 3T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2500, [\chi])$$.

## Hecke characteristic polynomials

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