# Properties

 Label 5625.2.a.r Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.46840000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9$$ x^6 - x^5 - 11*x^4 + 8*x^3 + 31*x^2 - 15*x - 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{3} + \beta_{2}) q^{7} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (-b3 + b2) * q^7 + (b5 + b3 + b2 + b1 + 1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{3} + \beta_{2}) q^{7} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{5} - \beta_{3}) q^{11} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{13} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{14} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{16} + ( - \beta_{5} - \beta_{3} + 2 \beta_1) q^{17} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{19} + ( - 2 \beta_{4} - \beta_{2} - 1) q^{22} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{23} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 7) q^{26} + (2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{28} + (\beta_{2} - 5) q^{29} + (\beta_{5} - 2 \beta_{4} + \beta_{2}) q^{31} + (\beta_{5} + 2 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} + \beta_1 + 8) q^{32} + ( - 2 \beta_{4} + \beta_{2} + 7) q^{34} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2} - 2 \beta_1 + 5) q^{37} + ( - \beta_{5} + \beta_{4} - 7 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{38} + (\beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1 - 5) q^{41} + (2 \beta_{4} + \beta_{3} + 2 \beta_1) q^{43} + ( - \beta_{5} - 7 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{44} + ( - \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 2 \beta_1 - 6) q^{46} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{47} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{49} + (2 \beta_{4} + \beta_{3} - 4 \beta_1 + 6) q^{52} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 6) q^{53} + (\beta_{5} + \beta_{4} + 7 \beta_{3} + \beta_{2} + 5 \beta_1 + 7) q^{56} + (\beta_{5} + \beta_{3} + \beta_{2} - 4 \beta_1 + 1) q^{58} + (\beta_{5} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{59} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{61} + ( - \beta_{5} + \beta_{4} - 7 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{62} + (2 \beta_{5} + 4 \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 3) q^{64} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{67} + (\beta_{5} - 5 \beta_{3} + \beta_{2} + 4 \beta_1 + 1) q^{68} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{71} + (2 \beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1 + 6) q^{73} + ( - 3 \beta_{5} - \beta_{4} - 9 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 10) q^{74} + (2 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 6) q^{76} + ( - 6 \beta_{3} - 2 \beta_1) q^{77} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{79} + (\beta_{5} + 2 \beta_{4} + \beta_{3} - 4 \beta_1 - 6) q^{82} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{83} + (2 \beta_{5} + \beta_{4} + 8 \beta_{3} + 2 \beta_{2} + 8) q^{86} + ( - \beta_{5} - 4 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 8) q^{88} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{89} + ( - \beta_{5} + 4 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3) q^{91} + ( - 4 \beta_{4} + 6 \beta_{3} - \beta_{2} - 2 \beta_1 + 5) q^{92} + ( - 4 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 4) q^{94} + ( - 3 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 1) q^{97} + (\beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_1 + 6) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 + 2) * q^4 + (-b3 + b2) * q^7 + (b5 + b3 + b2 + b1 + 1) * q^8 + (-b5 - b3) * q^11 + (b3 + b2 - 2*b1 + 1) * q^13 + (b5 - b4 + b3 + b2 + b1 + 1) * q^14 + (b5 + 2*b4 + b3 + b2 + 2*b1 + 2) * q^16 + (-b5 - b3 + 2*b1) * q^17 + (-2*b4 + b3 + b2 + 1) * q^19 + (-2*b4 - b2 - 1) * q^22 + (b5 - 2*b4 + b3 + b2 - 2*b1 + 1) * q^23 + (b5 + b4 + b3 - b2 + 2*b1 - 7) * q^26 + (2*b4 - b3 + b2 + 2*b1 + 6) * q^28 + (b2 - 5) * q^29 + (b5 - 2*b4 + b2) * q^31 + (b5 + 2*b4 + 7*b3 + 2*b2 + b1 + 8) * q^32 + (-2*b4 + b2 + 7) * q^34 + (-b5 - 2*b4 - b2 - 2*b1 + 5) * q^37 + (-b5 + b4 - 7*b3 + b2 + 2*b1 + 1) * q^38 + (b5 + b3 + b2 - 2*b1 - 5) * q^41 + (2*b4 + b3 + 2*b1) * q^43 + (-b5 - 7*b3 - b2 - 2*b1 - 1) * q^44 + (-b5 + 2*b4 - 7*b3 + 2*b1 - 6) * q^46 + (-2*b5 + 2*b4 - 2*b3 - 2*b2 - 2) * q^47 + (-b5 + 2*b4 - b2 + 2*b1) * q^49 + (2*b4 + b3 - 4*b1 + 6) * q^52 + (-2*b5 - 2*b4 + 4*b3 + 6) * q^53 + (b5 + b4 + 7*b3 + b2 + 5*b1 + 7) * q^56 + (b5 + b3 + b2 - 4*b1 + 1) * q^58 + (b5 + b3 + 2*b2 - 2*b1 + 2) * q^59 + (-b5 + 2*b4 - 2*b3 - b2 + 2*b1 + 4) * q^61 + (-b5 + b4 - 7*b3 + 2*b2 + b1 + 2) * q^62 + (2*b5 + 4*b4 + 8*b3 + 2*b2 + 6*b1 + 3) * q^64 + (-b5 - 2*b4 - 2*b3 + b2) * q^67 + (b5 - 5*b3 + b2 + 4*b1 + 1) * q^68 + (-2*b5 - 2*b3 + b2 - 2*b1 + 1) * q^71 + (2*b5 + b3 + b2 - 2*b1 + 6) * q^73 + (-3*b5 - b4 - 9*b3 - 4*b2 + 4*b1 - 10) * q^74 + (2*b5 - 4*b4 + 3*b3 + 2*b1 + 6) * q^76 + (-6*b3 - 2*b1) * q^77 + (-b3 - 2*b2 + 2*b1 + 3) * q^79 + (b5 + 2*b4 + b3 - 4*b1 - 6) * q^82 + (b5 + b3 - 2*b2 - 2*b1 - 2) * q^83 + (2*b5 + b4 + 8*b3 + 2*b2 + 8) * q^86 + (-b5 - 4*b4 - b3 - 2*b2 - 2*b1 - 8) * q^88 + (-b5 + 2*b4 - b3 + 2*b2 + 4*b1 - 4) * q^89 + (-b5 + 4*b4 - b3 - 2*b2 + 3) * q^91 + (-4*b4 + 6*b3 - b2 - 2*b1 + 5) * q^92 + (-4*b4 + 6*b3 - 4*b2 - 4*b1 - 4) * q^94 + (-3*b5 - 2*b4 - 4*b3 - 1) * q^97 + (b5 - b4 + 7*b3 - b1 + 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10})$$ 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8 $$6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{14} + 17 q^{16} + 2 q^{17} - 2 q^{19} - 9 q^{22} - q^{23} - 37 q^{26} + 44 q^{28} - 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 22 q^{37} + 27 q^{38} - 33 q^{41} + 3 q^{43} + 11 q^{44} - 12 q^{46} - 6 q^{47} + 4 q^{49} + 33 q^{52} + 14 q^{53} + 30 q^{56} + q^{58} + 8 q^{59} + 34 q^{61} + 31 q^{62} + 12 q^{64} - 2 q^{67} + 27 q^{68} + 3 q^{71} + 36 q^{73} - 36 q^{74} + 27 q^{76} + 16 q^{77} + 25 q^{79} - 36 q^{82} - 12 q^{83} + 30 q^{86} - 56 q^{88} - 18 q^{89} + 28 q^{91} + 3 q^{92} - 50 q^{94} - 7 q^{97} + 15 q^{98}+O(q^{100})$$ 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8 + 4 * q^14 + 17 * q^16 + 2 * q^17 - 2 * q^19 - 9 * q^22 - q^23 - 37 * q^26 + 44 * q^28 - 31 * q^29 - 2 * q^31 + 33 * q^32 + 37 * q^34 + 22 * q^37 + 27 * q^38 - 33 * q^41 + 3 * q^43 + 11 * q^44 - 12 * q^46 - 6 * q^47 + 4 * q^49 + 33 * q^52 + 14 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 34 * q^61 + 31 * q^62 + 12 * q^64 - 2 * q^67 + 27 * q^68 + 3 * q^71 + 36 * q^73 - 36 * q^74 + 27 * q^76 + 16 * q^77 + 25 * q^79 - 36 * q^82 - 12 * q^83 + 30 * q^86 - 56 * q^88 - 18 * q^89 + 28 * q^91 + 3 * q^92 - 50 * q^94 - 7 * q^97 + 15 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 13\nu - 6 ) / 6$$ (v^5 - v^4 - 8*v^3 + 5*v^2 + 13*v - 6) / 6 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 3 ) / 2$$ (v^4 - v^3 - 6*v^2 + 3*v + 3) / 2 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + \nu^{4} + 14\nu^{3} - 11\nu^{2} - 43\nu + 24 ) / 6$$ (-v^5 + v^4 + 14*v^3 - 11*v^2 - 43*v + 24) / 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{3} + \beta_{2} + 5\beta _1 + 1$$ b5 + b3 + b2 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{5} + 2\beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 22$$ b5 + 2*b4 + b3 + 7*b2 + 2*b1 + 22 $$\nu^{5}$$ $$=$$ $$9\beta_{5} + 2\beta_{4} + 15\beta_{3} + 10\beta_{2} + 29\beta _1 + 16$$ 9*b5 + 2*b4 + 15*b3 + 10*b2 + 29*b1 + 16

## 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
 −2.38719 −2.02791 −0.364088 0.858825 2.13324 2.78712
−2.38719 0 3.69868 0 0 3.31671 −4.05506 0 0
1.2 −2.02791 0 2.11242 0 0 −0.505614 −0.227977 0 0
1.3 −0.364088 0 −1.86744 0 0 −2.24941 1.40809 0 0
1.4 0.858825 0 −1.26242 0 0 −3.88045 −2.80185 0 0
1.5 2.13324 0 2.55073 0 0 2.16876 1.17484 0 0
1.6 2.78712 0 5.76803 0 0 3.15000 10.5020 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.r 6
3.b odd 2 1 1875.2.a.i 6
5.b even 2 1 5625.2.a.o 6
15.d odd 2 1 1875.2.a.l yes 6
15.e even 4 2 1875.2.b.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.i 6 3.b odd 2 1
1875.2.a.l yes 6 15.d odd 2 1
1875.2.b.e 12 15.e even 4 2
5625.2.a.o 6 5.b even 2 1
5625.2.a.r 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

 $$T_{2}^{6} - T_{2}^{5} - 11T_{2}^{4} + 8T_{2}^{3} + 31T_{2}^{2} - 15T_{2} - 9$$ T2^6 - T2^5 - 11*T2^4 + 8*T2^3 + 31*T2^2 - 15*T2 - 9 $$T_{7}^{6} - 2T_{7}^{5} - 21T_{7}^{4} + 42T_{7}^{3} + 101T_{7}^{2} - 160T_{7} - 100$$ T7^6 - 2*T7^5 - 21*T7^4 + 42*T7^3 + 101*T7^2 - 160*T7 - 100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{5} - 11 T^{4} + 8 T^{3} + \cdots - 9$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6} - 2 T^{5} - 21 T^{4} + 42 T^{3} + \cdots - 100$$
$11$ $$T^{6} - 29 T^{4} + 8 T^{3} + 184 T^{2} + \cdots - 144$$
$13$ $$T^{6} - 56 T^{4} - 74 T^{3} + \cdots + 349$$
$17$ $$T^{6} - 2 T^{5} - 55 T^{4} + 80 T^{3} + \cdots - 576$$
$19$ $$T^{6} + 2 T^{5} - 74 T^{4} + \cdots - 5725$$
$23$ $$T^{6} + T^{5} - 69 T^{4} - 144 T^{3} + \cdots - 720$$
$29$ $$T^{6} + 31 T^{5} + 379 T^{4} + \cdots + 6480$$
$31$ $$T^{6} + 2 T^{5} - 86 T^{4} + \cdots + 3155$$
$37$ $$T^{6} - 22 T^{5} + 59 T^{4} + \cdots - 46100$$
$41$ $$T^{6} + 33 T^{5} + 399 T^{4} + \cdots + 720$$
$43$ $$T^{6} - 3 T^{5} - 76 T^{4} + \cdots - 1289$$
$47$ $$T^{6} + 6 T^{5} - 184 T^{4} + \cdots + 80064$$
$53$ $$T^{6} - 14 T^{5} - 164 T^{4} + \cdots + 14400$$
$59$ $$T^{6} - 8 T^{5} - 69 T^{4} + \cdots - 2880$$
$61$ $$T^{6} - 34 T^{5} + 406 T^{4} + \cdots - 72001$$
$67$ $$T^{6} + 2 T^{5} - 110 T^{4} - 540 T^{3} + \cdots + 59$$
$71$ $$T^{6} - 3 T^{5} - 225 T^{4} + \cdots - 12816$$
$73$ $$T^{6} - 36 T^{5} + 431 T^{4} + \cdots + 20380$$
$79$ $$T^{6} - 25 T^{5} + 150 T^{4} + \cdots + 2725$$
$83$ $$T^{6} + 12 T^{5} - 129 T^{4} + \cdots - 23616$$
$89$ $$T^{6} + 18 T^{5} - 219 T^{4} + \cdots - 42480$$
$97$ $$T^{6} + 7 T^{5} - 310 T^{4} + \cdots - 32291$$