# Properties

 Label 5625.2.a.h Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + ( - 4 \beta + 2) q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 + (-4*b + 2) * q^7 - 3 * q^8 $$q + q^{2} - q^{4} + ( - 4 \beta + 2) q^{7} - 3 q^{8} - 2 \beta q^{11} + (\beta - 5) q^{13} + ( - 4 \beta + 2) q^{14} - q^{16} + ( - 3 \beta + 2) q^{17} - 2 \beta q^{19} - 2 \beta q^{22} + ( - 4 \beta + 2) q^{23} + (\beta - 5) q^{26} + (4 \beta - 2) q^{28} + (\beta - 6) q^{29} + (2 \beta + 4) q^{31} + 5 q^{32} + ( - 3 \beta + 2) q^{34} - 5 \beta q^{37} - 2 \beta q^{38} + ( - \beta + 3) q^{41} + (6 \beta - 4) q^{43} + 2 \beta q^{44} + ( - 4 \beta + 2) q^{46} + ( - 2 \beta - 2) q^{47} + 13 q^{49} + ( - \beta + 5) q^{52} + ( - \beta + 3) q^{53} + (12 \beta - 6) q^{56} + (\beta - 6) q^{58} + 4 q^{59} + ( - \beta + 1) q^{61} + (2 \beta + 4) q^{62} + 7 q^{64} + ( - 2 \beta - 2) q^{67} + (3 \beta - 2) q^{68} + ( - 2 \beta + 4) q^{71} + ( - 5 \beta + 5) q^{73} - 5 \beta q^{74} + 2 \beta q^{76} + (4 \beta + 8) q^{77} + ( - \beta + 3) q^{82} + ( - 4 \beta + 10) q^{83} + (6 \beta - 4) q^{86} + 6 \beta q^{88} + ( - \beta - 6) q^{89} + (18 \beta - 14) q^{91} + (4 \beta - 2) q^{92} + ( - 2 \beta - 2) q^{94} + ( - 3 \beta - 4) q^{97} + 13 q^{98} +O(q^{100})$$ q + q^2 - q^4 + (-4*b + 2) * q^7 - 3 * q^8 - 2*b * q^11 + (b - 5) * q^13 + (-4*b + 2) * q^14 - q^16 + (-3*b + 2) * q^17 - 2*b * q^19 - 2*b * q^22 + (-4*b + 2) * q^23 + (b - 5) * q^26 + (4*b - 2) * q^28 + (b - 6) * q^29 + (2*b + 4) * q^31 + 5 * q^32 + (-3*b + 2) * q^34 - 5*b * q^37 - 2*b * q^38 + (-b + 3) * q^41 + (6*b - 4) * q^43 + 2*b * q^44 + (-4*b + 2) * q^46 + (-2*b - 2) * q^47 + 13 * q^49 + (-b + 5) * q^52 + (-b + 3) * q^53 + (12*b - 6) * q^56 + (b - 6) * q^58 + 4 * q^59 + (-b + 1) * q^61 + (2*b + 4) * q^62 + 7 * q^64 + (-2*b - 2) * q^67 + (3*b - 2) * q^68 + (-2*b + 4) * q^71 + (-5*b + 5) * q^73 - 5*b * q^74 + 2*b * q^76 + (4*b + 8) * q^77 + (-b + 3) * q^82 + (-4*b + 10) * q^83 + (6*b - 4) * q^86 + 6*b * q^88 + (-b - 6) * q^89 + (18*b - 14) * q^91 + (4*b - 2) * q^92 + (-2*b - 2) * q^94 + (-3*b - 4) * q^97 + 13 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{4} - 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 $$2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} - 2 q^{11} - 9 q^{13} - 2 q^{16} + q^{17} - 2 q^{19} - 2 q^{22} - 9 q^{26} - 11 q^{29} + 10 q^{31} + 10 q^{32} + q^{34} - 5 q^{37} - 2 q^{38} + 5 q^{41} - 2 q^{43} + 2 q^{44} - 6 q^{47} + 26 q^{49} + 9 q^{52} + 5 q^{53} - 11 q^{58} + 8 q^{59} + q^{61} + 10 q^{62} + 14 q^{64} - 6 q^{67} - q^{68} + 6 q^{71} + 5 q^{73} - 5 q^{74} + 2 q^{76} + 20 q^{77} + 5 q^{82} + 16 q^{83} - 2 q^{86} + 6 q^{88} - 13 q^{89} - 10 q^{91} - 6 q^{94} - 11 q^{97} + 26 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 - 2 * q^11 - 9 * q^13 - 2 * q^16 + q^17 - 2 * q^19 - 2 * q^22 - 9 * q^26 - 11 * q^29 + 10 * q^31 + 10 * q^32 + q^34 - 5 * q^37 - 2 * q^38 + 5 * q^41 - 2 * q^43 + 2 * q^44 - 6 * q^47 + 26 * q^49 + 9 * q^52 + 5 * q^53 - 11 * q^58 + 8 * q^59 + q^61 + 10 * q^62 + 14 * q^64 - 6 * q^67 - q^68 + 6 * q^71 + 5 * q^73 - 5 * q^74 + 2 * q^76 + 20 * q^77 + 5 * q^82 + 16 * q^83 - 2 * q^86 + 6 * q^88 - 13 * q^89 - 10 * q^91 - 6 * q^94 - 11 * q^97 + 26 * q^98

## 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
 1.61803 −0.618034
1.00000 0 −1.00000 0 0 −4.47214 −3.00000 0 0
1.2 1.00000 0 −1.00000 0 0 4.47214 −3.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 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.h 2
3.b odd 2 1 1875.2.a.a 2
5.b even 2 1 5625.2.a.a 2
15.d odd 2 1 1875.2.a.d 2
15.e even 4 2 1875.2.b.b 4
25.e even 10 2 225.2.h.a 4
75.h odd 10 2 75.2.g.a 4
75.j odd 10 2 375.2.g.a 4
75.l even 20 4 375.2.i.a 8

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

## Hecke kernels

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

 $$T_{2} - 1$$ T2 - 1 $$T_{7}^{2} - 20$$ T7^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 20$$
$11$ $$T^{2} + 2T - 4$$
$13$ $$T^{2} + 9T + 19$$
$17$ $$T^{2} - T - 11$$
$19$ $$T^{2} + 2T - 4$$
$23$ $$T^{2} - 20$$
$29$ $$T^{2} + 11T + 29$$
$31$ $$T^{2} - 10T + 20$$
$37$ $$T^{2} + 5T - 25$$
$41$ $$T^{2} - 5T + 5$$
$43$ $$T^{2} + 2T - 44$$
$47$ $$T^{2} + 6T + 4$$
$53$ $$T^{2} - 5T + 5$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} - T - 1$$
$67$ $$T^{2} + 6T + 4$$
$71$ $$T^{2} - 6T + 4$$
$73$ $$T^{2} - 5T - 25$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 16T + 44$$
$89$ $$T^{2} + 13T + 41$$
$97$ $$T^{2} + 11T + 19$$