# Properties

 Label 7800.2.a.bq Level $7800$ Weight $2$ Character orbit 7800.a Self dual yes Analytic conductor $62.283$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.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: $$62.2833135766$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2700.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 15x - 20$$ x^3 - 15*x - 20 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta_1 q^{7} + q^{9}+O(q^{10})$$ q + q^3 - b1 * q^7 + q^9 $$q + q^{3} - \beta_1 q^{7} + q^{9} - \beta_1 q^{11} + q^{13} + ( - \beta_{2} + 2) q^{17} + ( - \beta_{2} + 1) q^{19} - \beta_1 q^{21} - 2 \beta_{2} q^{23} + q^{27} + q^{29} - \beta_1 q^{31} - \beta_1 q^{33} + (\beta_{2} + 3) q^{37} + q^{39} + (\beta_{2} + 1) q^{41} - 2 q^{43} + (\beta_{2} - \beta_1 + 1) q^{47} + (\beta_{2} + 2 \beta_1 + 3) q^{49} + ( - \beta_{2} + 2) q^{51} + (2 \beta_{2} + 1) q^{53} + ( - \beta_{2} + 1) q^{57} + (2 \beta_{2} - \beta_1 + 2) q^{59} + ( - 3 \beta_{2} - 2) q^{61} - \beta_1 q^{63} + ( - \beta_{2} - \beta_1 - 1) q^{67} - 2 \beta_{2} q^{69} + (\beta_{2} + 3) q^{71} + ( - 2 \beta_1 - 4) q^{73} + (\beta_{2} + 2 \beta_1 + 10) q^{77} + (\beta_{2} + 2 \beta_1 + 5) q^{79} + q^{81} + (2 \beta_{2} - \beta_1 - 2) q^{83} + q^{87} + (2 \beta_{2} + 2) q^{89} - \beta_1 q^{91} - \beta_1 q^{93} + (2 \beta_{2} + 4) q^{97} - \beta_1 q^{99}+O(q^{100})$$ q + q^3 - b1 * q^7 + q^9 - b1 * q^11 + q^13 + (-b2 + 2) * q^17 + (-b2 + 1) * q^19 - b1 * q^21 - 2*b2 * q^23 + q^27 + q^29 - b1 * q^31 - b1 * q^33 + (b2 + 3) * q^37 + q^39 + (b2 + 1) * q^41 - 2 * q^43 + (b2 - b1 + 1) * q^47 + (b2 + 2*b1 + 3) * q^49 + (-b2 + 2) * q^51 + (2*b2 + 1) * q^53 + (-b2 + 1) * q^57 + (2*b2 - b1 + 2) * q^59 + (-3*b2 - 2) * q^61 - b1 * q^63 + (-b2 - b1 - 1) * q^67 - 2*b2 * q^69 + (b2 + 3) * q^71 + (-2*b1 - 4) * q^73 + (b2 + 2*b1 + 10) * q^77 + (b2 + 2*b1 + 5) * q^79 + q^81 + (2*b2 - b1 - 2) * q^83 + q^87 + (2*b2 + 2) * q^89 - b1 * q^91 - b1 * q^93 + (2*b2 + 4) * q^97 - b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{9} + 3 q^{13} + 6 q^{17} + 3 q^{19} + 3 q^{27} + 3 q^{29} + 9 q^{37} + 3 q^{39} + 3 q^{41} - 6 q^{43} + 3 q^{47} + 9 q^{49} + 6 q^{51} + 3 q^{53} + 3 q^{57} + 6 q^{59} - 6 q^{61} - 3 q^{67} + 9 q^{71} - 12 q^{73} + 30 q^{77} + 15 q^{79} + 3 q^{81} - 6 q^{83} + 3 q^{87} + 6 q^{89} + 12 q^{97}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^9 + 3 * q^13 + 6 * q^17 + 3 * q^19 + 3 * q^27 + 3 * q^29 + 9 * q^37 + 3 * q^39 + 3 * q^41 - 6 * q^43 + 3 * q^47 + 9 * q^49 + 6 * q^51 + 3 * q^53 + 3 * q^57 + 6 * q^59 - 6 * q^61 - 3 * q^67 + 9 * q^71 - 12 * q^73 + 30 * q^77 + 15 * q^79 + 3 * q^81 - 6 * q^83 + 3 * q^87 + 6 * q^89 + 12 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 15x - 20$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 10$$ v^2 - 2*v - 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 10$$ b2 + 2*b1 + 10

## 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
 4.41883 −1.61323 −2.80560
0 1.00000 0 0 0 −4.41883 0 1.00000 0
1.2 0 1.00000 0 0 0 1.61323 0 1.00000 0
1.3 0 1.00000 0 0 0 2.80560 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$13$$ $$-1$$

## 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 7800.2.a.bq yes 3
5.b even 2 1 7800.2.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bk 3 5.b even 2 1
7800.2.a.bq yes 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7800))$$:

 $$T_{7}^{3} - 15T_{7} + 20$$ T7^3 - 15*T7 + 20 $$T_{11}^{3} - 15T_{11} + 20$$ T11^3 - 15*T11 + 20 $$T_{17}^{3} - 6T_{17}^{2} - 3T_{17} + 12$$ T17^3 - 6*T17^2 - 3*T17 + 12 $$T_{19}^{3} - 3T_{19}^{2} - 12T_{19} + 4$$ T19^3 - 3*T19^2 - 12*T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 15T + 20$$
$11$ $$T^{3} - 15T + 20$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} - 6 T^{2} + \cdots + 12$$
$19$ $$T^{3} - 3 T^{2} + \cdots + 4$$
$23$ $$T^{3} - 60T - 80$$
$29$ $$(T - 1)^{3}$$
$31$ $$T^{3} - 15T + 20$$
$37$ $$T^{3} - 9 T^{2} + \cdots + 28$$
$41$ $$T^{3} - 3 T^{2} + \cdots + 24$$
$43$ $$(T + 2)^{3}$$
$47$ $$T^{3} - 3 T^{2} + \cdots - 31$$
$53$ $$T^{3} - 3 T^{2} + \cdots + 139$$
$59$ $$T^{3} - 6 T^{2} + \cdots - 58$$
$61$ $$T^{3} + 6 T^{2} + \cdots - 532$$
$67$ $$T^{3} + 3 T^{2} + \cdots - 49$$
$71$ $$T^{3} - 9 T^{2} + \cdots + 28$$
$73$ $$T^{3} + 12 T^{2} + \cdots - 16$$
$79$ $$T^{3} - 15T^{2} + 100$$
$83$ $$T^{3} + 6 T^{2} + \cdots - 342$$
$89$ $$T^{3} - 6 T^{2} + \cdots + 192$$
$97$ $$T^{3} - 12 T^{2} + \cdots + 256$$