# Properties

 Label 625.2.e.e Level $625$ Weight $2$ Character orbit 625.e Analytic conductor $4.991$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(124,625)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("625.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.e (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{20}^{5} + \zeta_{20}) q^{2} + ( - \zeta_{20}^{7} + \cdots + 2 \zeta_{20}) q^{3}+ \cdots + (3 \zeta_{20}^{4} - \zeta_{20}^{2} + 3) q^{9}+O(q^{10})$$ q + (z^5 + z) * q^2 + (-z^7 + z^5 + 2*z) * q^3 + (z^2 - 1) * q^4 + (2*z^6 + z^4 + 2*z^2) * q^6 + (-3*z^7 + z^5 - 3*z^3) * q^7 + (-z^7 - z^5 + z^3 + z) * q^8 + (3*z^4 - z^2 + 3) * q^9 $$q + (\zeta_{20}^{5} + \zeta_{20}) q^{2} + ( - \zeta_{20}^{7} + \cdots + 2 \zeta_{20}) q^{3}+ \cdots + ( - \zeta_{20}^{6} + \zeta_{20}^{4} - 3) q^{99}+O(q^{100})$$ q + (z^5 + z) * q^2 + (-z^7 + z^5 + 2*z) * q^3 + (z^2 - 1) * q^4 + (2*z^6 + z^4 + 2*z^2) * q^6 + (-3*z^7 + z^5 - 3*z^3) * q^7 + (-z^7 - z^5 + z^3 + z) * q^8 + (3*z^4 - z^2 + 3) * q^9 + (z^4 - z^2) * q^11 + (z^7 + z^3 - z) * q^12 + (3*z^7 + 4*z^3 - 4*z) * q^13 + (-5*z^6 + 3*z^4 - 3*z^2 + 5) * q^14 + (3*z^4 + 3) * q^16 + (-z^7 + 5*z^5 - 5*z^3 + z) * q^17 + (2*z^7 + 3*z^5 + 2*z^3) * q^18 + (3*z^6 - 4*z^4 + 3*z^2) * q^19 + (-7*z^6 - 5*z^2 + 5) * q^21 - z * q^22 + (3*z^5 - 3*z^3 + 3*z) * q^23 + (-3*z^6 + 3*z^4 + 4) * q^24 + (3*z^6 - 3*z^4 - 7) * q^26 + (2*z^5 - z^3 + 2*z) * q^27 + (z^7 - z^5 + 3*z) * q^28 + (-4*z^6 + 2*z^2 - 2) * q^29 + 2*z^4 * q^31 + (-z^7 + 5*z^5 - z^3) * q^32 + (z^7 - z) * q^33 + (z^4 - 4*z^2 + 1) * q^34 + (3*z^6 - 4*z^4 + 4*z^2 - 3) * q^36 + 3*z^7 * q^37 + (2*z^7 - z^3 + z) * q^38 + (10*z^6 - 3*z^4 + 3*z^2 - 10) * q^39 + (-4*z^4 - 4) * q^41 + (-12*z^7 + 5*z^5 - 5*z^3 + 12*z) * q^42 + (-4*z^7 + 6*z^5 - 4*z^3) * q^43 + (z^6 - 2*z^4 + z^2) * q^44 + 3*z^6 * q^46 + (z^7 - z^5 - 11*z) * q^47 + (6*z^5 + 3*z^3 + 6*z) * q^48 + (3*z^6 - 3*z^4 - 6) * q^49 + (3*z^6 - 3*z^4 - 2) * q^51 + (z^5 - 5*z^3 + z) * q^52 + (-4*z^7 + 4*z^5 + 6*z) * q^53 + (3*z^6 + z^2 - 1) * q^54 + (-z^6 - 7*z^4 - z^2) * q^56 + (2*z^7 - z^5 + 2*z^3) * q^57 + (-2*z^7 - 2*z^5 + 2*z^3 + 2*z) * q^58 + (-8*z^4 - z^2 - 8) * q^59 + (-7*z^6 + 4*z^4 - 4*z^2 + 7) * q^61 + (2*z^7 + 2*z^3 - 2*z) * q^62 + (-13*z^7 - 3*z^3 + 3*z) * q^63 + (-3*z^6 + z^4 - z^2 + 3) * q^64 + (-z^4 - z^2 - 1) * q^66 + (9*z^7 - 5*z^5 + 5*z^3 - 9*z) * q^67 + (5*z^7 - 9*z^5 + 5*z^3) * q^68 + (3*z^6 + 3*z^2) * q^69 + (-z^6 + 7*z^2 - 7) * q^71 + (-z^7 + z^5 + 8*z) * q^72 + (-3*z^5 + 5*z^3 - 3*z) * q^73 + (3*z^6 - 3*z^4 - 3) * q^74 + (-4*z^6 + 4*z^4 - 3) * q^76 + (-z^5 + 4*z^3 - z) * q^77 + (10*z^7 - 10*z^5 - 17*z) * q^78 + (-3*z^6 - 6*z^2 + 6) * q^79 + (-6*z^6 + 4*z^4 - 6*z^2) * q^81 + (-4*z^7 - 4*z^5 - 4*z^3) * q^82 + (8*z^7 - 8*z) * q^83 + (2*z^4 + 3*z^2 + 2) * q^84 + (-2*z^6 + 4*z^4 - 4*z^2 + 2) * q^86 + (-6*z^7 - 2*z^3 + 2*z) * q^87 + (2*z^7 - z^3 + z) * q^88 + (-3*z^6 + 2*z^4 - 2*z^2 + 3) * q^89 + (5*z^4 + 13*z^2 + 5) * q^91 + (3*z^7 - 6*z^5 + 6*z^3 - 3*z) * q^92 + (2*z^7 + 2*z^5 + 2*z^3) * q^93 + (-11*z^6 - z^4 - 11*z^2) * q^94 + (7*z^6 + 3*z^2 - 3) * q^96 + (-z^7 + z^5 + 10*z) * q^97 + (-6*z^5 - 3*z^3 - 6*z) * q^98 + (-z^6 + z^4 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{4} + 6 q^{6} + 16 q^{9}+O(q^{10})$$ 8 * q - 6 * q^4 + 6 * q^6 + 16 * q^9 $$8 q - 6 q^{4} + 6 q^{6} + 16 q^{9} - 4 q^{11} + 18 q^{14} + 18 q^{16} + 20 q^{19} + 16 q^{21} + 20 q^{24} - 44 q^{26} - 20 q^{29} - 4 q^{31} - 2 q^{34} - 2 q^{36} - 48 q^{39} - 24 q^{41} + 8 q^{44} + 6 q^{46} - 36 q^{49} - 4 q^{51} + 10 q^{56} - 50 q^{59} + 26 q^{61} + 14 q^{64} - 8 q^{66} + 12 q^{69} - 44 q^{71} - 12 q^{74} - 40 q^{76} + 30 q^{79} - 32 q^{81} + 18 q^{84} - 4 q^{86} + 10 q^{89} + 56 q^{91} - 42 q^{94} - 4 q^{96} - 28 q^{99}+O(q^{100})$$ 8 * q - 6 * q^4 + 6 * q^6 + 16 * q^9 - 4 * q^11 + 18 * q^14 + 18 * q^16 + 20 * q^19 + 16 * q^21 + 20 * q^24 - 44 * q^26 - 20 * q^29 - 4 * q^31 - 2 * q^34 - 2 * q^36 - 48 * q^39 - 24 * q^41 + 8 * q^44 + 6 * q^46 - 36 * q^49 - 4 * q^51 + 10 * q^56 - 50 * q^59 + 26 * q^61 + 14 * q^64 - 8 * q^66 + 12 * q^69 - 44 * q^71 - 12 * q^74 - 40 * q^76 + 30 * q^79 - 32 * q^81 + 18 * q^84 - 4 * q^86 + 10 * q^89 + 56 * q^91 - 42 * q^94 - 4 * q^96 - 28 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{20}^{6}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 −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 0.190983i −0.224514 + 0.309017i −1.30902 0.951057i 0 0.190983 0.138757i 2.85410i 1.31433 + 1.80902i 0.881966 + 2.71441i 0
124.2 0.587785 + 0.190983i 0.224514 0.309017i −1.30902 0.951057i 0 0.190983 0.138757i 2.85410i −1.31433 1.80902i 0.881966 + 2.71441i 0
249.1 −0.951057 + 1.30902i −2.48990 + 0.809017i −0.190983 0.587785i 0 1.30902 4.02874i 3.85410i −2.12663 0.690983i 3.11803 2.26538i 0
249.2 0.951057 1.30902i 2.48990 0.809017i −0.190983 0.587785i 0 1.30902 4.02874i 3.85410i 2.12663 + 0.690983i 3.11803 2.26538i 0
374.1 −0.951057 1.30902i −2.48990 0.809017i −0.190983 + 0.587785i 0 1.30902 + 4.02874i 3.85410i −2.12663 + 0.690983i 3.11803 + 2.26538i 0
374.2 0.951057 + 1.30902i 2.48990 + 0.809017i −0.190983 + 0.587785i 0 1.30902 + 4.02874i 3.85410i 2.12663 0.690983i 3.11803 + 2.26538i 0
499.1 −0.587785 + 0.190983i −0.224514 0.309017i −1.30902 + 0.951057i 0 0.190983 + 0.138757i 2.85410i 1.31433 1.80902i 0.881966 2.71441i 0
499.2 0.587785 0.190983i 0.224514 + 0.309017i −1.30902 + 0.951057i 0 0.190983 + 0.138757i 2.85410i −1.31433 + 1.80902i 0.881966 2.71441i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 124.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.e.e 8
5.b even 2 1 inner 625.2.e.e 8
5.c odd 4 1 625.2.d.c 4
5.c odd 4 1 625.2.d.i 4
25.d even 5 1 625.2.b.b 4
25.d even 5 1 inner 625.2.e.e 8
25.d even 5 2 625.2.e.f 8
25.e even 10 1 625.2.b.b 4
25.e even 10 1 inner 625.2.e.e 8
25.e even 10 2 625.2.e.f 8
25.f odd 20 1 625.2.a.a 2
25.f odd 20 1 625.2.a.d yes 2
25.f odd 20 1 625.2.d.c 4
25.f odd 20 2 625.2.d.e 4
25.f odd 20 2 625.2.d.f 4
25.f odd 20 1 625.2.d.i 4
75.l even 20 1 5625.2.a.c 2
75.l even 20 1 5625.2.a.e 2
100.l even 20 1 10000.2.a.b 2
100.l even 20 1 10000.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.a 2 25.f odd 20 1
625.2.a.d yes 2 25.f odd 20 1
625.2.b.b 4 25.d even 5 1
625.2.b.b 4 25.e even 10 1
625.2.d.c 4 5.c odd 4 1
625.2.d.c 4 25.f odd 20 1
625.2.d.e 4 25.f odd 20 2
625.2.d.f 4 25.f odd 20 2
625.2.d.i 4 5.c odd 4 1
625.2.d.i 4 25.f odd 20 1
625.2.e.e 8 1.a even 1 1 trivial
625.2.e.e 8 5.b even 2 1 inner
625.2.e.e 8 25.d even 5 1 inner
625.2.e.e 8 25.e even 10 1 inner
625.2.e.f 8 25.d even 5 2
625.2.e.f 8 25.e even 10 2
5625.2.a.c 2 75.l even 20 1
5625.2.a.e 2 75.l even 20 1
10000.2.a.b 2 100.l even 20 1
10000.2.a.m 2 100.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(625, [\chi])$$:

 $$T_{2}^{8} + T_{2}^{6} + 6T_{2}^{4} - 4T_{2}^{2} + 1$$ T2^8 + T2^6 + 6*T2^4 - 4*T2^2 + 1 $$T_{3}^{8} - 11T_{3}^{6} + 46T_{3}^{4} + 4T_{3}^{2} + 1$$ T3^8 - 11*T3^6 + 46*T3^4 + 4*T3^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{6} + 6 T^{4} + \cdots + 1$$
$3$ $$T^{8} - 11 T^{6} + \cdots + 1$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 23 T^{2} + 121)^{2}$$
$11$ $$(T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2}$$
$13$ $$T^{8} - T^{6} + \cdots + 130321$$
$17$ $$T^{8} + 31 T^{6} + \cdots + 14641$$
$19$ $$(T^{4} - 10 T^{3} + \cdots + 25)^{2}$$
$23$ $$T^{8} - 36 T^{6} + \cdots + 6561$$
$29$ $$(T^{4} + 10 T^{3} + \cdots + 400)^{2}$$
$31$ $$(T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2}$$
$37$ $$T^{8} - 9 T^{6} + \cdots + 6561$$
$41$ $$(T^{4} + 12 T^{3} + \cdots + 256)^{2}$$
$43$ $$(T^{4} + 72 T^{2} + 16)^{2}$$
$47$ $$T^{8} - 164 T^{6} + \cdots + 141158161$$
$53$ $$T^{8} - 116 T^{6} + \cdots + 256$$
$59$ $$(T^{4} + 25 T^{3} + \cdots + 3025)^{2}$$
$61$ $$(T^{4} - 13 T^{3} + \cdots + 361)^{2}$$
$67$ $$T^{8} - 209 T^{6} + \cdots + 707281$$
$71$ $$(T^{4} + 22 T^{3} + \cdots + 1681)^{2}$$
$73$ $$T^{8} - 76 T^{6} + \cdots + 1$$
$79$ $$(T^{4} - 15 T^{3} + \cdots + 2025)^{2}$$
$83$ $$T^{8} - 256 T^{6} + \cdots + 16777216$$
$89$ $$(T^{4} - 5 T^{3} + 10 T^{2} + 25)^{2}$$
$97$ $$T^{8} - 139 T^{6} + \cdots + 62742241$$