# Properties

 Label 7800.2.a.bg 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: 3.3.788.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x - 3$$ x^3 - x^2 - 7*x - 3 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 $$q - q^{3} - \beta_1 q^{7} + q^{9} + (\beta_{2} - \beta_1 - 1) q^{11} + q^{13} + q^{17} + (\beta_{2} + 2 \beta_1 - 3) q^{19} + \beta_1 q^{21} + (\beta_{2} + 2 \beta_1 - 1) q^{23} - q^{27} + ( - 2 \beta_{2} - 2 \beta_1 + 1) q^{29} + ( - 3 \beta_{2} - \beta_1 - 1) q^{31} + ( - \beta_{2} + \beta_1 + 1) q^{33} + ( - \beta_{2} + 5) q^{37} - q^{39} + ( - 3 \beta_{2} + 1) q^{41} + ( - \beta_{2} - 1) q^{43} + (2 \beta_{2} + 3 \beta_1 - 2) q^{47} + (\beta_{2} + 2 \beta_1 - 3) q^{49} - q^{51} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{53} + ( - \beta_{2} - 2 \beta_1 + 3) q^{57} + ( - 2 \beta_{2} + \beta_1 - 2) q^{59} + (2 \beta_{2} + 2 \beta_1 - 3) q^{61} - \beta_1 q^{63} + (\beta_{2} - \beta_1 + 1) q^{67} + ( - \beta_{2} - 2 \beta_1 + 1) q^{69} + (\beta_{2} - 2 \beta_1 - 3) q^{71} + (3 \beta_{2} + 2 \beta_1 - 5) q^{73} + (2 \beta_{2} + 2 \beta_1 + 5) q^{77} + ( - \beta_{2} - 5) q^{79} + q^{81} + ( - 2 \beta_{2} + \beta_1 + 4) q^{83} + (2 \beta_{2} + 2 \beta_1 - 1) q^{87} + (2 \beta_{2} + 2 \beta_1 - 4) q^{89} - \beta_1 q^{91} + (3 \beta_{2} + \beta_1 + 1) q^{93} + (2 \beta_1 + 8) q^{97} + (\beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q - q^3 - b1 * q^7 + q^9 + (b2 - b1 - 1) * q^11 + q^13 + q^17 + (b2 + 2*b1 - 3) * q^19 + b1 * q^21 + (b2 + 2*b1 - 1) * q^23 - q^27 + (-2*b2 - 2*b1 + 1) * q^29 + (-3*b2 - b1 - 1) * q^31 + (-b2 + b1 + 1) * q^33 + (-b2 + 5) * q^37 - q^39 + (-3*b2 + 1) * q^41 + (-b2 - 1) * q^43 + (2*b2 + 3*b1 - 2) * q^47 + (b2 + 2*b1 - 3) * q^49 - q^51 + (-3*b2 + 2*b1 + 2) * q^53 + (-b2 - 2*b1 + 3) * q^57 + (-2*b2 + b1 - 2) * q^59 + (2*b2 + 2*b1 - 3) * q^61 - b1 * q^63 + (b2 - b1 + 1) * q^67 + (-b2 - 2*b1 + 1) * q^69 + (b2 - 2*b1 - 3) * q^71 + (3*b2 + 2*b1 - 5) * q^73 + (2*b2 + 2*b1 + 5) * q^77 + (-b2 - 5) * q^79 + q^81 + (-2*b2 + b1 + 4) * q^83 + (2*b2 + 2*b1 - 1) * q^87 + (2*b2 + 2*b1 - 4) * q^89 - b1 * q^91 + (3*b2 + b1 + 1) * q^93 + (2*b1 + 8) * q^97 + (b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - q^7 + 3 * q^9 $$3 q - 3 q^{3} - q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} - 6 q^{19} + q^{21} - 3 q^{27} - q^{29} - 7 q^{31} + 3 q^{33} + 14 q^{37} - 3 q^{39} - 4 q^{43} - q^{47} - 6 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{57} - 7 q^{59} - 5 q^{61} - q^{63} + 3 q^{67} - 10 q^{71} - 10 q^{73} + 19 q^{77} - 16 q^{79} + 3 q^{81} + 11 q^{83} + q^{87} - 8 q^{89} - q^{91} + 7 q^{93} + 26 q^{97} - 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 3 * q^17 - 6 * q^19 + q^21 - 3 * q^27 - q^29 - 7 * q^31 + 3 * q^33 + 14 * q^37 - 3 * q^39 - 4 * q^43 - q^47 - 6 * q^49 - 3 * q^51 + 5 * q^53 + 6 * q^57 - 7 * q^59 - 5 * q^61 - q^63 + 3 * q^67 - 10 * q^71 - 10 * q^73 + 19 * q^77 - 16 * q^79 + 3 * q^81 + 11 * q^83 + q^87 - 8 * q^89 - q^91 + 7 * q^93 + 26 * q^97 - 3 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 4$$ v^2 - 2*v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 4$$ b2 + 2*b1 + 4

## 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
 3.35386 −0.476452 −1.87740
0 −1.00000 0 0 0 −3.35386 0 1.00000 0
1.2 0 −1.00000 0 0 0 0.476452 0 1.00000 0
1.3 0 −1.00000 0 0 0 1.87740 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.bg 3
5.b even 2 1 7800.2.a.br yes 3

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

 $$T_{7}^{3} + T_{7}^{2} - 7T_{7} + 3$$ T7^3 + T7^2 - 7*T7 + 3 $$T_{11}^{3} + 3T_{11}^{2} - 17T_{11} - 53$$ T11^3 + 3*T11^2 - 17*T11 - 53 $$T_{17} - 1$$ T17 - 1 $$T_{19}^{3} + 6T_{19}^{2} - 20T_{19} - 100$$ T19^3 + 6*T19^2 - 20*T19 - 100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3} + T^{2} - 7T + 3$$
$11$ $$T^{3} + 3 T^{2} + \cdots - 53$$
$13$ $$(T - 1)^{3}$$
$17$ $$(T - 1)^{3}$$
$19$ $$T^{3} + 6 T^{2} + \cdots - 100$$
$23$ $$T^{3} - 32T - 44$$
$29$ $$T^{3} + T^{2} + \cdots - 93$$
$31$ $$T^{3} + 7 T^{2} + \cdots - 425$$
$37$ $$T^{3} - 14 T^{2} + \cdots - 60$$
$41$ $$T^{3} - 84T - 52$$
$43$ $$T^{3} + 4 T^{2} + \cdots - 12$$
$47$ $$T^{3} + T^{2} + \cdots - 89$$
$53$ $$T^{3} - 5 T^{2} + \cdots + 781$$
$59$ $$T^{3} + 7 T^{2} + \cdots + 9$$
$61$ $$T^{3} + 5 T^{2} + \cdots - 9$$
$67$ $$T^{3} - 3 T^{2} + \cdots - 15$$
$71$ $$T^{3} + 10 T^{2} + \cdots - 180$$
$73$ $$T^{3} + 10 T^{2} + \cdots + 52$$
$79$ $$T^{3} + 16 T^{2} + \cdots + 100$$
$83$ $$T^{3} - 11 T^{2} + \cdots + 255$$
$89$ $$T^{3} + 8 T^{2} + \cdots - 48$$
$97$ $$T^{3} - 26 T^{2} + \cdots - 440$$