# Properties

 Label 7800.2.a.l Level $7800$ Weight $2$ Character orbit 7800.a Self dual yes Analytic conductor $62.283$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + 5 q^{7} + q^{9}+O(q^{10})$$ q - q^3 + 5 * q^7 + q^9 $$q - q^{3} + 5 q^{7} + q^{9} + 5 q^{11} - q^{13} + 3 q^{17} - 4 q^{19} - 5 q^{21} - 5 q^{23} - q^{27} - 4 q^{29} - 5 q^{33} + 7 q^{37} + q^{39} + 11 q^{41} + 12 q^{43} - 6 q^{47} + 18 q^{49} - 3 q^{51} - q^{53} + 4 q^{57} + 12 q^{59} - 7 q^{61} + 5 q^{63} - 4 q^{67} + 5 q^{69} - 7 q^{71} + 14 q^{73} + 25 q^{77} - 5 q^{79} + q^{81} + 2 q^{83} + 4 q^{87} - 3 q^{89} - 5 q^{91} + q^{97} + 5 q^{99}+O(q^{100})$$ q - q^3 + 5 * q^7 + q^9 + 5 * q^11 - q^13 + 3 * q^17 - 4 * q^19 - 5 * q^21 - 5 * q^23 - q^27 - 4 * q^29 - 5 * q^33 + 7 * q^37 + q^39 + 11 * q^41 + 12 * q^43 - 6 * q^47 + 18 * q^49 - 3 * q^51 - q^53 + 4 * q^57 + 12 * q^59 - 7 * q^61 + 5 * q^63 - 4 * q^67 + 5 * q^69 - 7 * q^71 + 14 * q^73 + 25 * q^77 - 5 * q^79 + q^81 + 2 * q^83 + 4 * q^87 - 3 * q^89 - 5 * q^91 + q^97 + 5 * q^99

## 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
 0
0 −1.00000 0 0 0 5.00000 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.l 1
5.b even 2 1 7800.2.a.m 1
5.c odd 4 2 1560.2.l.a 2
15.e even 4 2 4680.2.l.c 2
20.e even 4 2 3120.2.l.b 2

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

 $$T_{7} - 5$$ T7 - 5 $$T_{11} - 5$$ T11 - 5 $$T_{17} - 3$$ T17 - 3 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 5$$
$11$ $$T - 5$$
$13$ $$T + 1$$
$17$ $$T - 3$$
$19$ $$T + 4$$
$23$ $$T + 5$$
$29$ $$T + 4$$
$31$ $$T$$
$37$ $$T - 7$$
$41$ $$T - 11$$
$43$ $$T - 12$$
$47$ $$T + 6$$
$53$ $$T + 1$$
$59$ $$T - 12$$
$61$ $$T + 7$$
$67$ $$T + 4$$
$71$ $$T + 7$$
$73$ $$T - 14$$
$79$ $$T + 5$$
$83$ $$T - 2$$
$89$ $$T + 3$$
$97$ $$T - 1$$