# Properties

 Label 5625.2.a.t Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## 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: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: 8.8.5444000000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1$$ x^8 - 4*x^7 - 2*x^6 + 20*x^5 - 4*x^4 - 30*x^3 + 7*x^2 + 12*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{7} + (\beta_{3} - 2 \beta_{2} - 2) q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b2 - b1 + 1) * q^4 + (-b7 - b6 + b5 - b3 + b2 + 1) * q^7 + (b3 - 2*b2 - 2) * q^8 $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{7} + (\beta_{3} - 2 \beta_{2} - 2) q^{8} + (\beta_{7} + 2 \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{11} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 2) q^{13} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{14} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{16} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{17} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 3) q^{19} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 3) q^{22} + (2 \beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_1 - 1) q^{23} + ( - 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + 3 \beta_1 - 4) q^{26} + (2 \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{28} + ( - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{29} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_1 - 2) q^{31} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1 - 1) q^{32} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 2) q^{34} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 3) q^{37} + (\beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 5) q^{38} + ( - \beta_{7} + \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{41} + (2 \beta_{7} - 2 \beta_{3} + \beta_{2} + 4) q^{43} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 4) q^{44} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{5} - \beta_{2} - 3 \beta_1 + 1) q^{46} + (3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} + 3 \beta_1 - 2) q^{47} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{49} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \cdots + 6) q^{52}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{3} - 4 \beta_1 + 2) q^{98}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b2 - b1 + 1) * q^4 + (-b7 - b6 + b5 - b3 + b2 + 1) * q^7 + (b3 - 2*b2 - 2) * q^8 + (b7 + 2*b6 + b3 - b2 + b1 - 1) * q^11 + (-2*b5 - b4 + b3 - b1 + 2) * q^13 + (-b5 + b4 + b3 - b2 + b1 - 1) * q^14 + (b4 - 2*b3 + b2 - b1 + 2) * q^16 + (-b7 + b6 - b5 - b4 - b3 - 2*b1 - 1) * q^17 + (-2*b7 - b6 + b5 - b4 - 3) * q^19 + (b7 + b6 + b5 + 2*b4 - b3 + 2*b2 + 3) * q^22 + (2*b7 + b5 + 2*b4 + b1 - 1) * q^23 + (-2*b6 + b5 - 3*b4 + 3*b1 - 4) * q^26 + (2*b7 + b6 - b4 + b3 + b2 - b1 + 2) * q^28 + (-2*b6 - b5 + b4 - b3 - b2 + 1) * q^29 + (-b6 - b5 - b4 - 2*b1 - 2) * q^31 + (b5 - 2*b4 - b3 + b1 - 1) * q^32 + (2*b7 - 2*b6 + 2*b5 - b4 - 2*b2 - 2) * q^34 + (b7 + b6 + b5 + 2*b4 + 3*b3 - 2*b2 + 3) * q^37 + (b7 - b6 - b5 + 3*b4 + 2*b2 - 2*b1 + 5) * q^38 + (-b7 + b5 - b4 - 3*b3 + b2 + b1 - 1) * q^41 + (2*b7 - 2*b3 + b2 + 4) * q^43 + (-2*b7 - 2*b6 + b5 + b4 - 2*b2 + 2*b1 - 4) * q^44 + (-2*b7 + 3*b6 - b5 - b2 - 3*b1 + 1) * q^46 + (3*b7 + 3*b6 + 2*b5 + b4 + 2*b2 + 3*b1 - 2) * q^47 + (-2*b6 + 2*b5 + 2*b4 - 2*b3 - b2 + b1 + 1) * q^49 + (-2*b7 + b6 - 2*b5 + 2*b4 - 2*b3 + 3*b2 - 4*b1 + 6) * q^52 + (b7 - b6 - b5 - b4 + 3*b3 - 2*b2 - 7) * q^53 + (-b7 + 2*b6 - 2*b4 - b3 - b2 + b1 - 5) * q^56 + (-2*b7 - b6 - 5*b4 - b3 - 3*b1) * q^58 + (2*b7 + b5 + b4 + 2*b2 + 2*b1 - 2) * q^59 + (2*b7 - 2*b6 - 4*b5 + 4*b3 - 2*b2 - 3) * q^61 + (-b7 - b6 - b5 - 3*b4 - 2*b2 - 3*b1 - 2) * q^62 + (b6 - 3*b5 - b4 + 4*b3 - 2*b2 - 1) * q^64 + (4*b5 + 4*b4 - 2*b3 - b2 + 4) * q^67 + (-2*b7 + 2*b6 - 5*b5 + 2*b4 - 4*b1 + 4) * q^68 + (-b7 - 2*b6 - b5 - b4 + b3 + 3*b2 - 3) * q^71 + (b6 - 4*b5 - 4*b4 - 2*b2 - 4*b1 + 4) * q^73 + (2*b6 + b5 + 5*b4 - 2*b3 + 4*b2 + 4*b1 - 2) * q^74 + (2*b7 + 2*b6 - 2*b4 + 2*b3 - 5*b2 + 5*b1 - 6) * q^76 + (2*b6 - b5 + 4*b3 - b1 - 7) * q^77 + (2*b7 + 3*b6 + 3*b5 + 3*b4 - 2*b3 + 3*b2 + 3*b1 - 3) * q^79 + (b7 - b5 + b3 - 2*b2 - 2*b1 + 3) * q^82 + (-b7 + 3*b6 - b5 - b4 - 3*b3 + 2*b2 - b1) * q^83 + (-2*b7 + 2*b6 - 2*b5 - 4*b4 + b3 - 5*b2 + b1 - 7) * q^86 + (-2*b7 - 3*b6 - 2*b5 - 2*b4 + 2*b2 - 6*b1 + 6) * q^88 + (-b7 - 2*b6 + b5 - b4 - 2*b3 - 2*b2 - 2) * q^89 + (-2*b7 - 2*b5 - 4*b4 + 3*b2 - b1 - 3) * q^91 + (b7 - 3*b6 + 4*b5 - b4 - b3 + 3*b1 - 2) * q^92 + (5*b6 - b5 + 4*b4 + 2*b3 - 2*b2 + 3) * q^94 + (3*b6 + 2*b3 - 3*b2 + 2*b1) * q^97 + (-2*b7 + 2*b6 - 2*b5 - b3 - 4*b1 + 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8}+O(q^{10})$$ 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8 $$8 q - 4 q^{2} + 4 q^{4} + 8 q^{7} - 12 q^{8} - 2 q^{11} + 16 q^{13} - 6 q^{14} - 16 q^{17} - 14 q^{19} + 12 q^{22} - 14 q^{23} - 6 q^{26} + 16 q^{28} - 2 q^{29} - 22 q^{31} + 2 q^{32} - 12 q^{34} + 28 q^{37} + 16 q^{38} - 8 q^{41} + 20 q^{43} - 22 q^{44} - 2 q^{46} - 10 q^{47} + 16 q^{52} - 44 q^{53} - 30 q^{56} + 8 q^{58} - 14 q^{59} - 20 q^{61} - 16 q^{62} + 6 q^{64} + 16 q^{67} + 2 q^{68} - 16 q^{71} + 24 q^{73} - 26 q^{74} - 16 q^{76} - 46 q^{77} - 30 q^{79} + 16 q^{82} - 12 q^{83} - 32 q^{86} + 32 q^{88} - 16 q^{89} - 12 q^{91} + 2 q^{92} + 14 q^{94} + 16 q^{97} - 4 q^{98}+O(q^{100})$$ 8 * q - 4 * q^2 + 4 * q^4 + 8 * q^7 - 12 * q^8 - 2 * q^11 + 16 * q^13 - 6 * q^14 - 16 * q^17 - 14 * q^19 + 12 * q^22 - 14 * q^23 - 6 * q^26 + 16 * q^28 - 2 * q^29 - 22 * q^31 + 2 * q^32 - 12 * q^34 + 28 * q^37 + 16 * q^38 - 8 * q^41 + 20 * q^43 - 22 * q^44 - 2 * q^46 - 10 * q^47 + 16 * q^52 - 44 * q^53 - 30 * q^56 + 8 * q^58 - 14 * q^59 - 20 * q^61 - 16 * q^62 + 6 * q^64 + 16 * q^67 + 2 * q^68 - 16 * q^71 + 24 * q^73 - 26 * q^74 - 16 * q^76 - 46 * q^77 - 30 * q^79 + 16 * q^82 - 12 * q^83 - 32 * q^86 + 32 * q^88 - 16 * q^89 - 12 * q^91 + 2 * q^92 + 14 * q^94 + 16 * q^97 - 4 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3\nu + 1$$ v^3 - v^2 - 3*v + 1 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 1$$ v^4 - 2*v^3 - 3*v^2 + 4*v + 1 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 3\nu^{4} - \nu^{3} + 7\nu^{2} - 3\nu - 1$$ v^5 - 3*v^4 - v^3 + 7*v^2 - 3*v - 1 $$\beta_{6}$$ $$=$$ $$\nu^{6} - 3\nu^{5} - 3\nu^{4} + 11\nu^{3} + 3\nu^{2} - 9\nu - 2$$ v^6 - 3*v^5 - 3*v^4 + 11*v^3 + 3*v^2 - 9*v - 2 $$\beta_{7}$$ $$=$$ $$\nu^{7} - 4\nu^{6} - \nu^{5} + 16\nu^{4} - 5\nu^{3} - 16\nu^{2} + 5\nu + 2$$ v^7 - 4*v^6 - v^5 + 16*v^4 - 5*v^3 - 16*v^2 + 5*v + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4\beta _1 + 1$$ b3 + b2 + 4*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 7$$ b4 + 2*b3 + 5*b2 + 7*b1 + 7 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 3\beta_{4} + 7\beta_{3} + 9\beta_{2} + 21\beta _1 + 9$$ b5 + 3*b4 + 7*b3 + 9*b2 + 21*b1 + 9 $$\nu^{6}$$ $$=$$ $$\beta_{6} + 3\beta_{5} + 12\beta_{4} + 16\beta_{3} + 28\beta_{2} + 46\beta _1 + 33$$ b6 + 3*b5 + 12*b4 + 16*b3 + 28*b2 + 46*b1 + 33 $$\nu^{7}$$ $$=$$ $$\beta_{7} + 4\beta_{6} + 13\beta_{5} + 35\beta_{4} + 44\beta_{3} + 62\beta_{2} + 124\beta _1 + 64$$ b7 + 4*b6 + 13*b5 + 35*b4 + 44*b3 + 62*b2 + 124*b1 + 64

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.53767 −1.35083 −0.536547 −0.0898194 1.08982 1.53655 2.35083 2.53767
−2.53767 0 4.43979 0 0 1.04054 −6.19138 0 0
1.2 −2.35083 0 3.52640 0 0 3.48189 −3.58831 0 0
1.3 −1.53655 0 0.360976 0 0 1.49550 2.51844 0 0
1.4 −1.08982 0 −0.812294 0 0 −3.08724 3.06489 0 0
1.5 0.0898194 0 −1.99193 0 0 4.36070 −0.358553 0 0
1.6 0.536547 0 −1.71212 0 0 −2.57318 −1.99173 0 0
1.7 1.35083 0 −0.175259 0 0 1.59580 −2.93840 0 0
1.8 1.53767 0 0.364440 0 0 1.68601 −2.51496 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5625.2.a.t 8
3.b odd 2 1 1875.2.a.p 8
5.b even 2 1 5625.2.a.bd 8
15.d odd 2 1 1875.2.a.m 8
15.e even 4 2 1875.2.b.h 16
25.f odd 20 2 225.2.m.b 16
75.h odd 10 2 375.2.g.e 16
75.j odd 10 2 375.2.g.d 16
75.l even 20 2 75.2.i.a 16
75.l even 20 2 375.2.i.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.i.a 16 75.l even 20 2
225.2.m.b 16 25.f odd 20 2
375.2.g.d 16 75.j odd 10 2
375.2.g.e 16 75.h odd 10 2
375.2.i.c 16 75.l even 20 2
1875.2.a.m 8 15.d odd 2 1
1875.2.a.p 8 3.b odd 2 1
1875.2.b.h 16 15.e even 4 2
5625.2.a.t 8 1.a even 1 1 trivial
5625.2.a.bd 8 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{8} + 4T_{2}^{7} - 2T_{2}^{6} - 20T_{2}^{5} - 4T_{2}^{4} + 30T_{2}^{3} + 7T_{2}^{2} - 12T_{2} + 1$$ T2^8 + 4*T2^7 - 2*T2^6 - 20*T2^5 - 4*T2^4 + 30*T2^3 + 7*T2^2 - 12*T2 + 1 $$T_{7}^{8} - 8T_{7}^{7} + 4T_{7}^{6} + 108T_{7}^{5} - 254T_{7}^{4} - 160T_{7}^{3} + 1145T_{7}^{2} - 1340T_{7} + 505$$ T7^8 - 8*T7^7 + 4*T7^6 + 108*T7^5 - 254*T7^4 - 160*T7^3 + 1145*T7^2 - 1340*T7 + 505

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 4 T^{7} - 2 T^{6} - 20 T^{5} + \cdots + 1$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 8 T^{7} + 4 T^{6} + 108 T^{5} + \cdots + 505$$
$11$ $$T^{8} + 2 T^{7} - 43 T^{6} + \cdots + 5281$$
$13$ $$T^{8} - 16 T^{7} + 73 T^{6} + \cdots + 281$$
$17$ $$T^{8} + 16 T^{7} + 42 T^{6} + \cdots - 7339$$
$19$ $$T^{8} + 14 T^{7} + 11 T^{6} + \cdots + 2525$$
$23$ $$T^{8} + 14 T^{7} + 11 T^{6} + \cdots - 2095$$
$29$ $$T^{8} + 2 T^{7} - 106 T^{6} + \cdots - 395$$
$31$ $$T^{8} + 22 T^{7} + 169 T^{6} + \cdots + 125$$
$37$ $$T^{8} - 28 T^{7} + 204 T^{6} + \cdots + 93025$$
$41$ $$T^{8} + 8 T^{7} - 56 T^{6} + \cdots + 4705$$
$43$ $$T^{8} - 20 T^{7} + 6 T^{6} + \cdots + 22961$$
$47$ $$T^{8} + 10 T^{7} - 186 T^{6} + \cdots - 6057019$$
$53$ $$T^{8} + 44 T^{7} + 746 T^{6} + \cdots - 200995$$
$59$ $$T^{8} + 14 T^{7} - 29 T^{6} + \cdots - 3595$$
$61$ $$T^{8} + 20 T^{7} - 136 T^{6} + \cdots + 16604261$$
$67$ $$T^{8} - 16 T^{7} - 138 T^{6} + \cdots - 3739$$
$71$ $$T^{8} + 16 T^{7} - 78 T^{6} + \cdots - 159779$$
$73$ $$T^{8} - 24 T^{7} - 34 T^{6} + \cdots - 870295$$
$79$ $$T^{8} + 30 T^{7} + 145 T^{6} + \cdots - 1984975$$
$83$ $$T^{8} + 12 T^{7} - 188 T^{6} + \cdots + 48541$$
$89$ $$T^{8} + 16 T^{7} - 39 T^{6} - 1454 T^{5} + \cdots + 5$$
$97$ $$T^{8} - 16 T^{7} - 108 T^{6} + \cdots - 14719$$
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