# Properties

 Label 625.2.e.b 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: 8.0.484000000.6 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121$$ x^8 - x^6 + 16*x^4 - 66*x^2 + 121 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 125) 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + \beta_1) q^{2} - \beta_1 q^{3} + (3 \beta_{5} - 3 \beta_{3} + \beta_{2} + 1) q^{4} + ( - 3 \beta_{5} + 2 \beta_{3} - 3) q^{6} + (\beta_{7} + \beta_{4} + \beta_1) q^{7} + (2 \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{3} - \beta_{2} - 1) q^{9}+O(q^{10})$$ q + (-b4 + b1) * q^2 - b1 * q^3 + (3*b5 - 3*b3 + b2 + 1) * q^4 + (-3*b5 + 2*b3 - 3) * q^6 + (b7 + b4 + b1) * q^7 + (2*b7 + b6 + 2*b4 - b1) * q^8 + (b3 - b2 - 1) * q^9 $$q + ( - \beta_{4} + \beta_1) q^{2} - \beta_1 q^{3} + (3 \beta_{5} - 3 \beta_{3} + \beta_{2} + 1) q^{4} + ( - 3 \beta_{5} + 2 \beta_{3} - 3) q^{6} + (\beta_{7} + \beta_{4} + \beta_1) q^{7} + (2 \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{8}+ \cdots + ( - 2 \beta_{5} - 2 \beta_{2} - 2) q^{99}+O(q^{100})$$ q + (-b4 + b1) * q^2 - b1 * q^3 + (3*b5 - 3*b3 + b2 + 1) * q^4 + (-3*b5 + 2*b3 - 3) * q^6 + (b7 + b4 + b1) * q^7 + (2*b7 + b6 + 2*b4 - b1) * q^8 + (b3 - b2 - 1) * q^9 + 2*b5 * q^11 + (-2*b7 - b6) * q^12 + 2*b7 * q^13 + (3*b5 + b3 - b2) * q^14 + (4*b3 - b2 - 4) * q^16 + (2*b7 + 2*b4) * q^17 + b4 * q^18 + (2*b5 + 4*b3 + 2) * q^19 + (b5 - b3 + 5*b2 + 5) * q^21 + (2*b6 + 2*b4 - 2*b1) * q^22 + (-b7 + b6 + b4 - 2*b1) * q^23 + (-b5 - b2 + 7) * q^24 + (4*b5 + 4*b2 - 6) * q^26 + (2*b7 - 2*b6 - b4 + 3*b1) * q^27 + (b6 + b4 - 2*b1) * q^28 + (-4*b5 + 4*b3 - b2 - 1) * q^29 - 2*b3 * q^31 + (-b7 - b1) * q^32 + (-2*b7 - 2*b4) * q^33 + (6*b3 - 2*b2 - 6) * q^34 + (-2*b5 + b3 - b2) * q^36 + (2*b7 + 2*b6) * q^37 + (-4*b7 + 2*b6) * q^38 + (-4*b5 + 6*b3 - 6*b2) * q^39 + (5*b3 - 3*b2 - 5) * q^41 + (-4*b7 + b6 - 4*b4 - b1) * q^42 + (2*b7 - 3*b4 + 2*b1) * q^43 + (-4*b5 + 6*b3 - 4) * q^44 + (-7*b5 + 7*b3 - 2*b2 - 2) * q^46 + (-2*b6 - 2*b4 + 3*b1) * q^47 + (-b7 + b6 - 4*b4 + 3*b1) * q^48 + (-5*b5 - 5*b2 - 2) * q^49 + (2*b5 + 2*b2 + 8) * q^51 + (-4*b7 + 4*b6 + 6*b4 - 10*b1) * q^52 + 4*b1 * q^53 + (9*b5 - 9*b3 + b2 + 1) * q^54 + (-7*b5 + b3 - 7) * q^56 + (-2*b7 - 6*b4 - 2*b1) * q^57 + (-3*b7 - 4*b6 - 3*b4 + 4*b1) * q^58 + (-4*b3 + 2*b2 + 4) * q^59 + (-3*b5 + 5*b3 - 5*b2) * q^61 + 2*b7 * q^62 + (b7 - 2*b6) * q^63 + (b5 - 6*b3 + 6*b2) * q^64 + (-6*b3 + 2*b2 + 6) * q^66 + (2*b7 + 2*b4) * q^67 + (-4*b7 + 2*b4 - 4*b1) * q^68 + (2*b5 - 5*b3 + 2) * q^69 + (8*b5 - 8*b3 - 2*b2 - 2) * q^71 + (-2*b6 - 2*b4 + b1) * q^72 + (4*b7 - 4*b6 + 6*b4 - 2*b1) * q^73 + (10*b5 + 10*b2 - 4) * q^74 + (-6*b5 - 6*b2 + 2) * q^76 + (2*b7 - 2*b6 + 2*b4) * q^77 + (-4*b6 - 4*b4 + 10*b1) * q^78 + (6*b5 - 6*b3 + 4*b2 + 4) * q^79 + (2*b5 + 8*b3 + 2) * q^81 + (-2*b7 + 5*b4 - 2*b1) * q^82 + (-3*b7 - b6 - 3*b4 + b1) * q^83 + (-2*b3 - 3*b2 + 2) * q^84 + (16*b5 - 13*b3 + 13*b2) * q^86 + (3*b7 + b6) * q^87 + (-2*b7 - 4*b6) * q^88 + (7*b5 + b3 - b2) * q^89 + (-2*b3 + 8*b2 + 2) * q^91 + (-3*b7 - 5*b6 - 3*b4 + 5*b1) * q^92 + 2*b4 * q^93 + (7*b5 - 12*b3 + 7) * q^94 + (2*b5 - 2*b3 - b2 - 1) * q^96 + (-2*b6 - 2*b4 - 2*b1) * q^97 + (5*b7 - 5*b6 - 3*b4 + 8*b1) * q^98 + (-2*b5 - 2*b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{4} - 14 q^{6} - 4 q^{9}+O(q^{10})$$ 8 * q - 6 * q^4 - 14 * q^6 - 4 * q^9 $$8 q - 6 q^{4} - 14 q^{6} - 4 q^{9} - 4 q^{11} - 2 q^{14} - 22 q^{16} + 20 q^{19} + 26 q^{21} + 60 q^{24} - 64 q^{26} + 10 q^{29} - 4 q^{31} - 32 q^{34} + 8 q^{36} + 32 q^{39} - 24 q^{41} - 12 q^{44} + 16 q^{46} + 4 q^{49} + 56 q^{51} - 30 q^{54} - 40 q^{56} + 20 q^{59} + 26 q^{61} - 26 q^{64} + 32 q^{66} + 2 q^{69} - 44 q^{71} - 72 q^{74} + 40 q^{76} + 28 q^{81} + 18 q^{84} - 84 q^{86} - 10 q^{89} - 4 q^{91} + 18 q^{94} - 14 q^{96} - 8 q^{99}+O(q^{100})$$ 8 * q - 6 * q^4 - 14 * q^6 - 4 * q^9 - 4 * q^11 - 2 * q^14 - 22 * q^16 + 20 * q^19 + 26 * q^21 + 60 * q^24 - 64 * q^26 + 10 * q^29 - 4 * q^31 - 32 * q^34 + 8 * q^36 + 32 * q^39 - 24 * q^41 - 12 * q^44 + 16 * q^46 + 4 * q^49 + 56 * q^51 - 30 * q^54 - 40 * q^56 + 20 * q^59 + 26 * q^61 - 26 * q^64 + 32 * q^66 + 2 * q^69 - 44 * q^71 - 72 * q^74 + 40 * q^76 + 28 * q^81 + 18 * q^84 - 84 * q^86 - 10 * q^89 - 4 * q^91 + 18 * q^94 - 14 * q^96 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -7\nu^{6} - 37\nu^{4} - 629\nu^{2} - 363 ) / 1991$$ (-7*v^6 - 37*v^4 - 629*v^2 - 363) / 1991 $$\beta_{3}$$ $$=$$ $$( -28\nu^{6} - 148\nu^{4} - 525\nu^{2} + 539 ) / 1991$$ (-28*v^6 - 148*v^4 - 525*v^2 + 539) / 1991 $$\beta_{4}$$ $$=$$ $$( -28\nu^{7} - 148\nu^{5} - 525\nu^{3} + 539\nu ) / 1991$$ (-28*v^7 - 148*v^5 - 525*v^3 + 539*v) / 1991 $$\beta_{5}$$ $$=$$ $$( 40\nu^{6} - 73\nu^{4} + 750\nu^{2} - 2761 ) / 1991$$ (40*v^6 - 73*v^4 + 750*v^2 - 2761) / 1991 $$\beta_{6}$$ $$=$$ $$( 61\nu^{7} + 38\nu^{5} + 646\nu^{3} - 1672\nu ) / 1991$$ (61*v^7 + 38*v^5 + 646*v^3 - 1672*v) / 1991 $$\beta_{7}$$ $$=$$ $$( 68\nu^{7} + 75\nu^{5} + 1275\nu^{3} - 3300\nu ) / 1991$$ (68*v^7 + 75*v^5 + 1275*v^3 - 3300*v) / 1991
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4\beta_{2} - 1$$ b3 - 4*b2 - 1 $$\nu^{3}$$ $$=$$ $$4\beta_{7} - 4\beta_{6} + \beta_{4} + 3\beta_1$$ 4*b7 - 4*b6 + b4 + 3*b1 $$\nu^{4}$$ $$=$$ $$-7\beta_{5} - 10\beta_{3} - 7$$ -7*b5 - 10*b3 - 7 $$\nu^{5}$$ $$=$$ $$-7\beta_{7} - 17\beta_{4} - 7\beta_1$$ -7*b7 - 17*b4 - 7*b1 $$\nu^{6}$$ $$=$$ $$37\beta_{5} - 37\beta_{3} + 75\beta_{2} + 75$$ 37*b5 - 37*b3 + 75*b2 + 75 $$\nu^{7}$$ $$=$$ $$-38\beta_{7} + 75\beta_{6}$$ -38*b7 + 75*b6

## Character values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$1 + \beta_{2} - \beta_{3} + \beta_{5}$$

## 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
 −1.26313 + 1.73855i 1.26313 − 1.73855i −1.46782 + 0.476925i 1.46782 − 0.476925i −1.46782 − 0.476925i 1.46782 + 0.476925i −1.26313 − 1.73855i 1.26313 + 1.73855i
−1.26313 0.410415i 1.26313 1.73855i −0.190983 0.138757i 0 −2.30902 + 1.67760i 3.47709i 1.74560 + 2.40261i −0.500000 1.53884i 0
124.2 1.26313 + 0.410415i −1.26313 + 1.73855i −0.190983 0.138757i 0 −2.30902 + 1.67760i 3.47709i −1.74560 2.40261i −0.500000 1.53884i 0
249.1 −1.46782 + 2.02029i 1.46782 0.476925i −1.30902 4.02874i 0 −1.19098 + 3.66547i 0.953850i 5.31064 + 1.72553i −0.500000 + 0.363271i 0
249.2 1.46782 2.02029i −1.46782 + 0.476925i −1.30902 4.02874i 0 −1.19098 + 3.66547i 0.953850i −5.31064 1.72553i −0.500000 + 0.363271i 0
374.1 −1.46782 2.02029i 1.46782 + 0.476925i −1.30902 + 4.02874i 0 −1.19098 3.66547i 0.953850i 5.31064 1.72553i −0.500000 0.363271i 0
374.2 1.46782 + 2.02029i −1.46782 0.476925i −1.30902 + 4.02874i 0 −1.19098 3.66547i 0.953850i −5.31064 + 1.72553i −0.500000 0.363271i 0
499.1 −1.26313 + 0.410415i 1.26313 + 1.73855i −0.190983 + 0.138757i 0 −2.30902 1.67760i 3.47709i 1.74560 2.40261i −0.500000 + 1.53884i 0
499.2 1.26313 0.410415i −1.26313 1.73855i −0.190983 + 0.138757i 0 −2.30902 1.67760i 3.47709i −1.74560 + 2.40261i −0.500000 + 1.53884i 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.b 8
5.b even 2 1 inner 625.2.e.b 8
5.c odd 4 2 625.2.d.l 8
25.d even 5 1 125.2.b.a 4
25.d even 5 1 inner 625.2.e.b 8
25.d even 5 2 625.2.e.h 8
25.e even 10 1 125.2.b.a 4
25.e even 10 1 inner 625.2.e.b 8
25.e even 10 2 625.2.e.h 8
25.f odd 20 2 125.2.a.c 4
25.f odd 20 4 625.2.d.k 8
25.f odd 20 2 625.2.d.l 8
75.h odd 10 1 1125.2.b.a 4
75.j odd 10 1 1125.2.b.a 4
75.l even 20 2 1125.2.a.k 4
100.h odd 10 1 2000.2.c.c 4
100.j odd 10 1 2000.2.c.c 4
100.l even 20 2 2000.2.a.o 4
175.s even 20 2 6125.2.a.o 4
200.v even 20 2 8000.2.a.bk 4
200.x odd 20 2 8000.2.a.bj 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.a.c 4 25.f odd 20 2
125.2.b.a 4 25.d even 5 1
125.2.b.a 4 25.e even 10 1
625.2.d.k 8 25.f odd 20 4
625.2.d.l 8 5.c odd 4 2
625.2.d.l 8 25.f odd 20 2
625.2.e.b 8 1.a even 1 1 trivial
625.2.e.b 8 5.b even 2 1 inner
625.2.e.b 8 25.d even 5 1 inner
625.2.e.b 8 25.e even 10 1 inner
625.2.e.h 8 25.d even 5 2
625.2.e.h 8 25.e even 10 2
1125.2.a.k 4 75.l even 20 2
1125.2.b.a 4 75.h odd 10 1
1125.2.b.a 4 75.j odd 10 1
2000.2.a.o 4 100.l even 20 2
2000.2.c.c 4 100.h odd 10 1
2000.2.c.c 4 100.j odd 10 1
6125.2.a.o 4 175.s even 20 2
8000.2.a.bj 4 200.x odd 20 2
8000.2.a.bk 4 200.v even 20 2

## Hecke kernels

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

 $$T_{2}^{8} + T_{2}^{6} + 31T_{2}^{4} - 99T_{2}^{2} + 121$$ T2^8 + T2^6 + 31*T2^4 - 99*T2^2 + 121 $$T_{3}^{8} - T_{3}^{6} + 16T_{3}^{4} - 66T_{3}^{2} + 121$$ T3^8 - T3^6 + 16*T3^4 - 66*T3^2 + 121

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{6} + \cdots + 121$$
$3$ $$T^{8} - T^{6} + \cdots + 121$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 13 T^{2} + 11)^{2}$$
$11$ $$(T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2}$$
$13$ $$T^{8} + 4 T^{6} + \cdots + 30976$$
$17$ $$T^{8} - 4 T^{6} + \cdots + 30976$$
$19$ $$(T^{4} - 10 T^{3} + \cdots + 400)^{2}$$
$23$ $$T^{8} + 9 T^{6} + \cdots + 121$$
$29$ $$(T^{4} - 5 T^{3} + 40 T^{2} + \cdots + 25)^{2}$$
$31$ $$(T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2}$$
$37$ $$T^{8} + 36 T^{6} + \cdots + 30976$$
$41$ $$(T^{4} + 12 T^{3} + \cdots + 961)^{2}$$
$43$ $$(T^{4} + 107 T^{2} + 1331)^{2}$$
$47$ $$T^{8} - 69 T^{6} + \cdots + 121$$
$53$ $$T^{8} - 16 T^{6} + \cdots + 7929856$$
$59$ $$(T^{4} - 10 T^{3} + \cdots + 400)^{2}$$
$61$ $$(T^{4} - 13 T^{3} + \cdots + 961)^{2}$$
$67$ $$T^{8} - 4 T^{6} + \cdots + 30976$$
$71$ $$(T^{4} + 22 T^{3} + \cdots + 13456)^{2}$$
$73$ $$T^{8} - 396 T^{6} + \cdots + 453519616$$
$79$ $$(T^{4} + 40 T^{2} + \cdots + 400)^{2}$$
$83$ $$T^{8} - 66 T^{6} + \cdots + 1771561$$
$89$ $$(T^{4} + 5 T^{3} + \cdots + 3025)^{2}$$
$97$ $$T^{8} + 76 T^{6} + \cdots + 30976$$