# Properties

 Label 7800.2.a.bp Level $7800$ Weight $2$ Character orbit 7800.a Self dual yes Analytic conductor $62.283$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1560) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{9}+O(q^{10})$$ q + q^3 + q^9 $$q + q^{3} + q^{9} + ( - \beta_1 - 2) q^{11} + q^{13} + (\beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{2} - \beta_1 - 1) q^{19} + (\beta_{2} + \beta_1 - 3) q^{23} + q^{27} + ( - 2 \beta_{2} - 4) q^{29} + (2 \beta_1 + 4) q^{31} + ( - \beta_1 - 2) q^{33} + (2 \beta_1 + 2) q^{37} + q^{39} + (\beta_{2} - 2 \beta_1 - 3) q^{41} + ( - \beta_{2} - \beta_1 + 1) q^{43} + (\beta_{2} + 2 \beta_1 - 5) q^{47} - 7 q^{49} + (\beta_{2} + \beta_1 - 1) q^{51} + ( - 2 \beta_{2} - 4) q^{53} + ( - \beta_{2} - \beta_1 - 1) q^{57} + (3 \beta_1 - 2) q^{59} + (\beta_{2} - \beta_1 + 1) q^{61} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{67} + (\beta_{2} + \beta_1 - 3) q^{69} + (\beta_{2} + 3) q^{71} + ( - 4 \beta_1 - 2) q^{73} + (\beta_{2} + \beta_1 + 7) q^{79} + q^{81} + (\beta_{2} - 2 \beta_1 - 1) q^{83} + ( - 2 \beta_{2} - 4) q^{87} + (\beta_{2} - 2 \beta_1 - 7) q^{89} + (2 \beta_1 + 4) q^{93} - 6 q^{97} + ( - \beta_1 - 2) q^{99}+O(q^{100})$$ q + q^3 + q^9 + (-b1 - 2) * q^11 + q^13 + (b2 + b1 - 1) * q^17 + (-b2 - b1 - 1) * q^19 + (b2 + b1 - 3) * q^23 + q^27 + (-2*b2 - 4) * q^29 + (2*b1 + 4) * q^31 + (-b1 - 2) * q^33 + (2*b1 + 2) * q^37 + q^39 + (b2 - 2*b1 - 3) * q^41 + (-b2 - b1 + 1) * q^43 + (b2 + 2*b1 - 5) * q^47 - 7 * q^49 + (b2 + b1 - 1) * q^51 + (-2*b2 - 4) * q^53 + (-b2 - b1 - 1) * q^57 + (3*b1 - 2) * q^59 + (b2 - b1 + 1) * q^61 + (-2*b2 + 2*b1 - 2) * q^67 + (b2 + b1 - 3) * q^69 + (b2 + 3) * q^71 + (-4*b1 - 2) * q^73 + (b2 + b1 + 7) * q^79 + q^81 + (b2 - 2*b1 - 1) * q^83 + (-2*b2 - 4) * q^87 + (b2 - 2*b1 - 7) * q^89 + (2*b1 + 4) * q^93 - 6 * q^97 + (-b1 - 2) * 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 $$3 q + 3 q^{3} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 4 q^{17} - 2 q^{19} - 10 q^{23} + 3 q^{27} - 10 q^{29} + 12 q^{31} - 6 q^{33} + 6 q^{37} + 3 q^{39} - 10 q^{41} + 4 q^{43} - 16 q^{47} - 21 q^{49} - 4 q^{51} - 10 q^{53} - 2 q^{57} - 6 q^{59} + 2 q^{61} - 4 q^{67} - 10 q^{69} + 8 q^{71} - 6 q^{73} + 20 q^{79} + 3 q^{81} - 4 q^{83} - 10 q^{87} - 22 q^{89} + 12 q^{93} - 18 q^{97} - 6 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^9 - 6 * q^11 + 3 * q^13 - 4 * q^17 - 2 * q^19 - 10 * q^23 + 3 * q^27 - 10 * q^29 + 12 * q^31 - 6 * q^33 + 6 * q^37 + 3 * q^39 - 10 * q^41 + 4 * q^43 - 16 * q^47 - 21 * q^49 - 4 * q^51 - 10 * q^53 - 2 * q^57 - 6 * q^59 + 2 * q^61 - 4 * q^67 - 10 * q^69 + 8 * q^71 - 6 * q^73 + 20 * q^79 + 3 * q^81 - 4 * q^83 - 10 * q^87 - 22 * q^89 + 12 * q^93 - 18 * q^97 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 3\nu + 3$$ -v^2 + 3*v + 3
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 3\beta _1 + 9 ) / 2$$ (b2 + 3*b1 + 9) / 2

## 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
 3.12489 −1.76156 −0.363328
0 1.00000 0 0 0 0 0 1.00000 0
1.2 0 1.00000 0 0 0 0 0 1.00000 0
1.3 0 1.00000 0 0 0 0 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.bp 3
5.b even 2 1 7800.2.a.bj 3
5.c odd 4 2 1560.2.l.c 6
15.e even 4 2 4680.2.l.e 6
20.e even 4 2 3120.2.l.m 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.c 6 5.c odd 4 2
3120.2.l.m 6 20.e even 4 2
4680.2.l.e 6 15.e even 4 2
7800.2.a.bj 3 5.b even 2 1
7800.2.a.bp 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7800))$$:

 $$T_{7}$$ T7 $$T_{11}^{3} + 6T_{11}^{2} + 2T_{11} - 20$$ T11^3 + 6*T11^2 + 2*T11 - 20 $$T_{17}^{3} + 4T_{17}^{2} - 20T_{17} - 64$$ T17^3 + 4*T17^2 - 20*T17 - 64 $$T_{19}^{3} + 2T_{19}^{2} - 24T_{19} + 16$$ T19^3 + 2*T19^2 - 24*T19 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3}$$
$11$ $$T^{3} + 6 T^{2} + \cdots - 20$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} + 4 T^{2} + \cdots - 64$$
$19$ $$T^{3} + 2 T^{2} + \cdots + 16$$
$23$ $$T^{3} + 10 T^{2} + \cdots - 80$$
$29$ $$T^{3} + 10 T^{2} + \cdots - 472$$
$31$ $$T^{3} - 12 T^{2} + \cdots + 160$$
$37$ $$T^{3} - 6 T^{2} + \cdots + 136$$
$41$ $$T^{3} + 10 T^{2} + \cdots - 332$$
$43$ $$T^{3} - 4 T^{2} + \cdots + 64$$
$47$ $$T^{3} + 16 T^{2} + \cdots - 256$$
$53$ $$T^{3} + 10 T^{2} + \cdots - 472$$
$59$ $$T^{3} + 6 T^{2} + \cdots + 44$$
$61$ $$T^{3} - 2 T^{2} + \cdots + 32$$
$67$ $$T^{3} + 4 T^{2} + \cdots - 256$$
$71$ $$T^{3} - 8 T^{2} + \cdots + 64$$
$73$ $$T^{3} + 6 T^{2} + \cdots - 824$$
$79$ $$T^{3} - 20 T^{2} + \cdots - 160$$
$83$ $$T^{3} + 4 T^{2} + \cdots - 232$$
$89$ $$T^{3} + 22 T^{2} + \cdots - 244$$
$97$ $$(T + 6)^{3}$$