# Properties

 Label 5625.2.a.j Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{15})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} + 4x + 1$$ x^4 - x^3 - 4*x^2 + 4*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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_{2}) q^{2} + \beta_1 q^{4} + (2 \beta_{3} + \beta_{2} + 2) q^{7} + ( - \beta_{2} + 1) q^{8}+O(q^{10})$$ q + (b3 + b2) * q^2 + b1 * q^4 + (2*b3 + b2 + 2) * q^7 + (-b2 + 1) * q^8 $$q + (\beta_{3} + \beta_{2}) q^{2} + \beta_1 q^{4} + (2 \beta_{3} + \beta_{2} + 2) q^{7} + ( - \beta_{2} + 1) q^{8} + (\beta_{2} + \beta_1 + 1) q^{11} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{13} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{14} + (\beta_{2} - 2 \beta_1 - 2) q^{16} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{19} + (4 \beta_{3} + 2 \beta_{2} + 3) q^{22} + (2 \beta_1 - 3) q^{23} + (7 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{26} + (3 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{28} + ( - \beta_{3} + \beta_{2} + \beta_1 + 6) q^{29} + (2 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{31} + ( - 5 \beta_{3} - 2 \beta_{2} - 2) q^{32} + (6 \beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{34} + ( - 4 \beta_{3} + \beta_{2} - 3 \beta_1 - 4) q^{37} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{38} + ( - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 1) q^{41} + ( - 6 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{43} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{44} + (\beta_{3} - \beta_{2} + 2) q^{46} + ( - \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 5) q^{47} + (5 \beta_{3} + 4 \beta_{2} + 3 \beta_1) q^{49} + ( - \beta_{3} + 3 \beta_1 + 2) q^{52} + ( - 5 \beta_{3} - 6 \beta_{2} - 1) q^{53} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{56} + (10 \beta_{3} + 7 \beta_{2} - \beta_1 + 3) q^{58} + ( - 5 \beta_{3} + 3 \beta_{2} - \beta_1 - 4) q^{59} + ( - 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 12) q^{61} + ( - 8 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{62} + (\beta_{3} - 4 \beta_{2} - \beta_1) q^{64} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{67} + ( - \beta_{2} + 4 \beta_1) q^{68} + (3 \beta_{3} - \beta_{2} - 6 \beta_1 + 10) q^{71} + (8 \beta_{3} - \beta_1 + 8) q^{73} + ( - 5 \beta_{3} - 7 \beta_{2} - 4 \beta_1 - 1) q^{74} + ( - \beta_{3} - \beta_1 + 2) q^{76} + (6 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 5) q^{77} + ( - 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 1) q^{79} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{82} + (5 \beta_{3} - \beta_{2} - 8 \beta_1 + 4) q^{83} + (7 \beta_{3} - 6 \beta_1 - 1) q^{86} + ( - 2 \beta_{3} + \beta_1 - 2) q^{88} + (4 \beta_{3} + \beta_{2} + 4) q^{89} + (7 \beta_{3} + 6 \beta_{2} + \beta_1 + 3) q^{91} + (2 \beta_{2} - 3 \beta_1 + 4) q^{92} + (11 \beta_{3} + 9 \beta_{2} - \beta_1 - 2) q^{94} + (4 \beta_{3} - 3 \beta_{2} - 8 \beta_1 + 8) q^{97} + (5 \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 11) q^{98}+O(q^{100})$$ q + (b3 + b2) * q^2 + b1 * q^4 + (2*b3 + b2 + 2) * q^7 + (-b2 + 1) * q^8 + (b2 + b1 + 1) * q^11 + (-2*b3 + b2 + 2*b1) * q^13 + (b3 + 2*b2 + 2*b1 + 2) * q^14 + (b2 - 2*b1 - 2) * q^16 + (-2*b3 + 2*b2 + b1) * q^17 + (-2*b3 + b2 + 2*b1 - 4) * q^19 + (4*b3 + 2*b2 + 3) * q^22 + (2*b1 - 3) * q^23 + (7*b3 + 2*b2 - 2*b1 + 4) * q^26 + (3*b3 + 2*b2 + b1 + 2) * q^28 + (-b3 + b2 + b1 + 6) * q^29 + (2*b3 + b2 - 3*b1 - 1) * q^31 + (-5*b3 - 2*b2 - 2) * q^32 + (6*b3 + b2 - 2*b1 + 5) * q^34 + (-4*b3 + b2 - 3*b1 - 4) * q^37 + (3*b3 - 2*b2 - 2*b1 + 4) * q^38 + (-2*b3 - 3*b2 + 3*b1 - 1) * q^41 + (-6*b3 - 2*b2 + 3*b1 - 3) * q^43 + (b3 + b2 + 2*b1 + 2) * q^44 + (b3 - b2 + 2) * q^46 + (-b3 - 3*b2 + 4*b1 + 5) * q^47 + (5*b3 + 4*b2 + 3*b1) * q^49 + (-b3 + 3*b1 + 2) * q^52 + (-5*b3 - 6*b2 - 1) * q^53 + (b3 - b2 - b1 + 1) * q^56 + (10*b3 + 7*b2 - b1 + 3) * q^58 + (-5*b3 + 3*b2 - b1 - 4) * q^59 + (-3*b3 - 2*b2 + b1 - 12) * q^61 + (-8*b3 - 4*b2 + 2*b1 - 1) * q^62 + (b3 - 4*b2 - b1) * q^64 + (-b3 - 4*b2 + 2*b1 + 2) * q^67 + (-b2 + 4*b1) * q^68 + (3*b3 - b2 - 6*b1 + 10) * q^71 + (8*b3 - b1 + 8) * q^73 + (-5*b3 - 7*b2 - 4*b1 - 1) * q^74 + (-b3 - b1 + 2) * q^76 + (6*b3 + 5*b2 + 2*b1 + 5) * q^77 + (-3*b3 + 4*b2 + 4*b1 - 1) * q^79 + (4*b3 + 2*b2 - 2*b1 - 3) * q^82 + (5*b3 - b2 - 8*b1 + 4) * q^83 + (7*b3 - 6*b1 - 1) * q^86 + (-2*b3 + b1 - 2) * q^88 + (4*b3 + b2 + 4) * q^89 + (7*b3 + 6*b2 + b1 + 3) * q^91 + (2*b2 - 3*b1 + 4) * q^92 + (11*b3 + 9*b2 - b1 - 2) * q^94 + (4*b3 - 3*b2 - 8*b1 + 8) * q^97 + (5*b3 + 3*b2 + 5*b1 + 11) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} + q^{4} + 5 q^{7} + 3 q^{8}+O(q^{10})$$ 4 * q - q^2 + q^4 + 5 * q^7 + 3 * q^8 $$4 q - q^{2} + q^{4} + 5 q^{7} + 3 q^{8} + 6 q^{11} + 7 q^{13} + 10 q^{14} - 9 q^{16} + 7 q^{17} - 9 q^{19} + 6 q^{22} - 10 q^{23} + 2 q^{26} + 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} - 10 q^{37} + 6 q^{38} + q^{43} + 9 q^{44} + 5 q^{46} + 23 q^{47} - 3 q^{49} + 13 q^{52} - 2 q^{58} - 4 q^{59} - 43 q^{61} + 10 q^{62} - 7 q^{64} + 8 q^{67} + 3 q^{68} + 27 q^{71} + 15 q^{73} - 5 q^{74} + 9 q^{76} + 15 q^{77} + 10 q^{79} - 20 q^{82} - 3 q^{83} - 24 q^{86} - 3 q^{88} + 9 q^{89} + 5 q^{91} + 15 q^{92} - 22 q^{94} + 13 q^{97} + 42 q^{98}+O(q^{100})$$ 4 * q - q^2 + q^4 + 5 * q^7 + 3 * q^8 + 6 * q^11 + 7 * q^13 + 10 * q^14 - 9 * q^16 + 7 * q^17 - 9 * q^19 + 6 * q^22 - 10 * q^23 + 2 * q^26 + 5 * q^28 + 28 * q^29 - 10 * q^31 + 7 * q^34 - 10 * q^37 + 6 * q^38 + q^43 + 9 * q^44 + 5 * q^46 + 23 * q^47 - 3 * q^49 + 13 * q^52 - 2 * q^58 - 4 * q^59 - 43 * q^61 + 10 * q^62 - 7 * q^64 + 8 * q^67 + 3 * q^68 + 27 * q^71 + 15 * q^73 - 5 * q^74 + 9 * q^76 + 15 * q^77 + 10 * q^79 - 20 * q^82 - 3 * q^83 - 24 * q^86 - 3 * q^88 + 9 * q^89 + 5 * q^91 + 15 * q^92 - 22 * q^94 + 13 * q^97 + 42 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{15} + \zeta_{15}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1

## 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
 1.33826 −0.209057 −1.95630 1.82709
−1.82709 0 1.33826 0 0 −1.44512 1.20906 0 0
1.2 −1.33826 0 −0.209057 0 0 1.27977 2.95630 0 0
1.3 0.209057 0 −1.95630 0 0 0.591023 −0.827091 0 0
1.4 1.95630 0 1.82709 0 0 4.57433 −0.338261 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.j 4
3.b odd 2 1 1875.2.a.g yes 4
5.b even 2 1 5625.2.a.m 4
15.d odd 2 1 1875.2.a.f 4
15.e even 4 2 1875.2.b.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.f 4 15.d odd 2 1
1875.2.a.g yes 4 3.b odd 2 1
1875.2.b.d 8 15.e even 4 2
5625.2.a.j 4 1.a even 1 1 trivial
5625.2.a.m 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{4} + T_{2}^{3} - 4T_{2}^{2} - 4T_{2} + 1$$ T2^4 + T2^3 - 4*T2^2 - 4*T2 + 1 $$T_{7}^{4} - 5T_{7}^{3} + 10T_{7} - 5$$ T7^4 - 5*T7^3 + 10*T7 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} - 4 T^{2} - 4 T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 5 T^{3} + 10 T - 5$$
$11$ $$T^{4} - 6 T^{3} + 6 T^{2} + 9 T - 9$$
$13$ $$T^{4} - 7 T^{3} - 6 T^{2} + 92 T - 89$$
$17$ $$T^{4} - 7 T^{3} - 16 T^{2} + 202 T - 359$$
$19$ $$T^{4} + 9 T^{3} + 6 T^{2} - 36 T - 9$$
$23$ $$T^{4} + 10 T^{3} + 20 T^{2} - 10 T - 5$$
$29$ $$T^{4} - 28 T^{3} + 284 T^{2} + \cdots + 1801$$
$31$ $$T^{4} + 10 T^{3} - 125 T - 125$$
$37$ $$T^{4} + 10 T^{3} - 90 T^{2} + \cdots - 1475$$
$41$ $$T^{4} - 70 T^{2} + 135 T + 145$$
$43$ $$T^{4} - T^{3} - 79 T^{2} - 341 T - 419$$
$47$ $$T^{4} - 23 T^{3} + 89 T^{2} + \cdots - 6089$$
$53$ $$T^{4} - 145 T^{2} + 270 T + 2995$$
$59$ $$T^{4} + 4 T^{3} - 154 T^{2} + \cdots + 1531$$
$61$ $$T^{4} + 43 T^{3} + 669 T^{2} + \cdots + 10261$$
$67$ $$T^{4} - 8 T^{3} - 61 T^{2} + 458 T + 151$$
$71$ $$T^{4} - 27 T^{3} + 134 T^{2} + \cdots + 271$$
$73$ $$T^{4} - 15 T^{3} - 60 T^{2} + \cdots + 2745$$
$79$ $$T^{4} - 10 T^{3} - 105 T^{2} + \cdots - 3155$$
$83$ $$T^{4} + 3 T^{3} - 246 T^{2} + \cdots + 13491$$
$89$ $$T^{4} - 9 T^{3} - 4 T^{2} + 96 T + 61$$
$97$ $$T^{4} - 13 T^{3} - 216 T^{2} + \cdots + 14701$$