# Properties

 Label 5625.2.a.b Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $1$ 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: $$1$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1875) 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta - 1) q^{4} - 2 q^{7} + (2 \beta - 1) q^{8} +O(q^{10})$$ q - b * q^2 + (b - 1) * q^4 - 2 * q^7 + (2*b - 1) * q^8 $$q - \beta q^{2} + (\beta - 1) q^{4} - 2 q^{7} + (2 \beta - 1) q^{8} + 3 q^{11} - q^{13} + 2 \beta q^{14} - 3 \beta q^{16} + (2 \beta + 1) q^{17} + ( - 6 \beta + 3) q^{19} - 3 \beta q^{22} + (\beta - 7) q^{23} + \beta q^{26} + ( - 2 \beta + 2) q^{28} + (\beta + 2) q^{29} + (6 \beta - 1) q^{31} + ( - \beta + 5) q^{32} + ( - 3 \beta - 2) q^{34} - 2 q^{37} + (3 \beta + 6) q^{38} + ( - \beta + 11) q^{41} + (\beta - 9) q^{43} + (3 \beta - 3) q^{44} + (6 \beta - 1) q^{46} + ( - 2 \beta + 8) q^{47} - 3 q^{49} + ( - \beta + 1) q^{52} + ( - 2 \beta - 8) q^{53} + ( - 4 \beta + 2) q^{56} + ( - 3 \beta - 1) q^{58} + ( - 8 \beta + 9) q^{59} + (6 \beta - 1) q^{61} + ( - 5 \beta - 6) q^{62} + (2 \beta + 1) q^{64} + ( - 10 \beta + 3) q^{67} + (\beta + 1) q^{68} + ( - 5 \beta - 2) q^{71} + (6 \beta + 6) q^{73} + 2 \beta q^{74} + (3 \beta - 9) q^{76} - 6 q^{77} + (3 \beta - 14) q^{79} + ( - 10 \beta + 1) q^{82} - 9 q^{83} + (8 \beta - 1) q^{86} + (6 \beta - 3) q^{88} + (10 \beta - 5) q^{89} + 2 q^{91} + ( - 7 \beta + 8) q^{92} + ( - 6 \beta + 2) q^{94} + (3 \beta - 1) q^{97} + 3 \beta q^{98} +O(q^{100})$$ q - b * q^2 + (b - 1) * q^4 - 2 * q^7 + (2*b - 1) * q^8 + 3 * q^11 - q^13 + 2*b * q^14 - 3*b * q^16 + (2*b + 1) * q^17 + (-6*b + 3) * q^19 - 3*b * q^22 + (b - 7) * q^23 + b * q^26 + (-2*b + 2) * q^28 + (b + 2) * q^29 + (6*b - 1) * q^31 + (-b + 5) * q^32 + (-3*b - 2) * q^34 - 2 * q^37 + (3*b + 6) * q^38 + (-b + 11) * q^41 + (b - 9) * q^43 + (3*b - 3) * q^44 + (6*b - 1) * q^46 + (-2*b + 8) * q^47 - 3 * q^49 + (-b + 1) * q^52 + (-2*b - 8) * q^53 + (-4*b + 2) * q^56 + (-3*b - 1) * q^58 + (-8*b + 9) * q^59 + (6*b - 1) * q^61 + (-5*b - 6) * q^62 + (2*b + 1) * q^64 + (-10*b + 3) * q^67 + (b + 1) * q^68 + (-5*b - 2) * q^71 + (6*b + 6) * q^73 + 2*b * q^74 + (3*b - 9) * q^76 - 6 * q^77 + (3*b - 14) * q^79 + (-10*b + 1) * q^82 - 9 * q^83 + (8*b - 1) * q^86 + (6*b - 3) * q^88 + (10*b - 5) * q^89 + 2 * q^91 + (-7*b + 8) * q^92 + (-6*b + 2) * q^94 + (3*b - 1) * q^97 + 3*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 4 q^{7}+O(q^{10})$$ 2 * q - q^2 - q^4 - 4 * q^7 $$2 q - q^{2} - q^{4} - 4 q^{7} + 6 q^{11} - 2 q^{13} + 2 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{22} - 13 q^{23} + q^{26} + 2 q^{28} + 5 q^{29} + 4 q^{31} + 9 q^{32} - 7 q^{34} - 4 q^{37} + 15 q^{38} + 21 q^{41} - 17 q^{43} - 3 q^{44} + 4 q^{46} + 14 q^{47} - 6 q^{49} + q^{52} - 18 q^{53} - 5 q^{58} + 10 q^{59} + 4 q^{61} - 17 q^{62} + 4 q^{64} - 4 q^{67} + 3 q^{68} - 9 q^{71} + 18 q^{73} + 2 q^{74} - 15 q^{76} - 12 q^{77} - 25 q^{79} - 8 q^{82} - 18 q^{83} + 6 q^{86} + 4 q^{91} + 9 q^{92} - 2 q^{94} + q^{97} + 3 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - 4 * q^7 + 6 * q^11 - 2 * q^13 + 2 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^22 - 13 * q^23 + q^26 + 2 * q^28 + 5 * q^29 + 4 * q^31 + 9 * q^32 - 7 * q^34 - 4 * q^37 + 15 * q^38 + 21 * q^41 - 17 * q^43 - 3 * q^44 + 4 * q^46 + 14 * q^47 - 6 * q^49 + q^52 - 18 * q^53 - 5 * q^58 + 10 * q^59 + 4 * q^61 - 17 * q^62 + 4 * q^64 - 4 * q^67 + 3 * q^68 - 9 * q^71 + 18 * q^73 + 2 * q^74 - 15 * q^76 - 12 * q^77 - 25 * q^79 - 8 * q^82 - 18 * q^83 + 6 * q^86 + 4 * q^91 + 9 * q^92 - 2 * q^94 + q^97 + 3 * 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
−1.61803 0 0.618034 0 0 −2.00000 2.23607 0 0
1.2 0.618034 0 −1.61803 0 0 −2.00000 −2.23607 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.b 2
3.b odd 2 1 1875.2.a.c yes 2
5.b even 2 1 5625.2.a.g 2
15.d odd 2 1 1875.2.a.b 2
15.e even 4 2 1875.2.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.b 2 15.d odd 2 1
1875.2.a.c yes 2 3.b odd 2 1
1875.2.b.a 4 15.e even 4 2
5625.2.a.b 2 1.a even 1 1 trivial
5625.2.a.g 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 4T - 1$$
$19$ $$T^{2} - 45$$
$23$ $$T^{2} + 13T + 41$$
$29$ $$T^{2} - 5T + 5$$
$31$ $$T^{2} - 4T - 41$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} - 21T + 109$$
$43$ $$T^{2} + 17T + 71$$
$47$ $$T^{2} - 14T + 44$$
$53$ $$T^{2} + 18T + 76$$
$59$ $$T^{2} - 10T - 55$$
$61$ $$T^{2} - 4T - 41$$
$67$ $$T^{2} + 4T - 121$$
$71$ $$T^{2} + 9T - 11$$
$73$ $$T^{2} - 18T + 36$$
$79$ $$T^{2} + 25T + 145$$
$83$ $$(T + 9)^{2}$$
$89$ $$T^{2} - 125$$
$97$ $$T^{2} - T - 11$$