# Properties

 Label 7800.2.a.y Level $7800$ Weight $2$ Character orbit 7800.a Self dual yes Analytic conductor $62.283$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{7} + q^{9}+O(q^{10})$$ q - q^3 + b * q^7 + q^9 $$q - q^{3} + \beta q^{7} + q^{9} + \beta q^{11} + q^{13} + ( - 2 \beta + 1) q^{17} - 2 \beta q^{19} - \beta q^{21} + 2 q^{23} - q^{27} + (4 \beta - 3) q^{29} + ( - 3 \beta - 4) q^{31} - \beta q^{33} + (2 \beta - 4) q^{37} - q^{39} + (2 \beta - 6) q^{41} + (4 \beta + 2) q^{43} + ( - 5 \beta + 2) q^{47} - 4 q^{49} + (2 \beta - 1) q^{51} + ( - 4 \beta - 5) q^{53} + 2 \beta q^{57} + \beta q^{59} + ( - 2 \beta - 9) q^{61} + \beta q^{63} + ( - \beta + 6) q^{67} - 2 q^{69} + ( - 6 \beta - 4) q^{71} - 2 \beta q^{73} + 3 q^{77} + 14 q^{79} + q^{81} + (\beta + 12) q^{83} + ( - 4 \beta + 3) q^{87} + ( - 4 \beta + 4) q^{89} + \beta q^{91} + (3 \beta + 4) q^{93} + (4 \beta - 2) q^{97} + \beta q^{99} +O(q^{100})$$ q - q^3 + b * q^7 + q^9 + b * q^11 + q^13 + (-2*b + 1) * q^17 - 2*b * q^19 - b * q^21 + 2 * q^23 - q^27 + (4*b - 3) * q^29 + (-3*b - 4) * q^31 - b * q^33 + (2*b - 4) * q^37 - q^39 + (2*b - 6) * q^41 + (4*b + 2) * q^43 + (-5*b + 2) * q^47 - 4 * q^49 + (2*b - 1) * q^51 + (-4*b - 5) * q^53 + 2*b * q^57 + b * q^59 + (-2*b - 9) * q^61 + b * q^63 + (-b + 6) * q^67 - 2 * q^69 + (-6*b - 4) * q^71 - 2*b * q^73 + 3 * q^77 + 14 * q^79 + q^81 + (b + 12) * q^83 + (-4*b + 3) * q^87 + (-4*b + 4) * q^89 + b * q^91 + (3*b + 4) * q^93 + (4*b - 2) * q^97 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} + 2 q^{13} + 2 q^{17} + 4 q^{23} - 2 q^{27} - 6 q^{29} - 8 q^{31} - 8 q^{37} - 2 q^{39} - 12 q^{41} + 4 q^{43} + 4 q^{47} - 8 q^{49} - 2 q^{51} - 10 q^{53} - 18 q^{61} + 12 q^{67} - 4 q^{69} - 8 q^{71} + 6 q^{77} + 28 q^{79} + 2 q^{81} + 24 q^{83} + 6 q^{87} + 8 q^{89} + 8 q^{93} - 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 + 2 * q^13 + 2 * q^17 + 4 * q^23 - 2 * q^27 - 6 * q^29 - 8 * q^31 - 8 * q^37 - 2 * q^39 - 12 * q^41 + 4 * q^43 + 4 * q^47 - 8 * q^49 - 2 * q^51 - 10 * q^53 - 18 * q^61 + 12 * q^67 - 4 * q^69 - 8 * q^71 + 6 * q^77 + 28 * q^79 + 2 * q^81 + 24 * q^83 + 6 * q^87 + 8 * q^89 + 8 * q^93 - 4 * q^97

## 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
 −1.73205 1.73205
0 −1.00000 0 0 0 −1.73205 0 1.00000 0
1.2 0 −1.00000 0 0 0 1.73205 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.y 2
5.b even 2 1 7800.2.a.bc yes 2

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

 $$T_{7}^{2} - 3$$ T7^2 - 3 $$T_{11}^{2} - 3$$ T11^2 - 3 $$T_{17}^{2} - 2T_{17} - 11$$ T17^2 - 2*T17 - 11 $$T_{19}^{2} - 12$$ T19^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 3$$
$11$ $$T^{2} - 3$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 2T - 11$$
$19$ $$T^{2} - 12$$
$23$ $$(T - 2)^{2}$$
$29$ $$T^{2} + 6T - 39$$
$31$ $$T^{2} + 8T - 11$$
$37$ $$T^{2} + 8T + 4$$
$41$ $$T^{2} + 12T + 24$$
$43$ $$T^{2} - 4T - 44$$
$47$ $$T^{2} - 4T - 71$$
$53$ $$T^{2} + 10T - 23$$
$59$ $$T^{2} - 3$$
$61$ $$T^{2} + 18T + 69$$
$67$ $$T^{2} - 12T + 33$$
$71$ $$T^{2} + 8T - 92$$
$73$ $$T^{2} - 12$$
$79$ $$(T - 14)^{2}$$
$83$ $$T^{2} - 24T + 141$$
$89$ $$T^{2} - 8T - 32$$
$97$ $$T^{2} + 4T - 44$$