# Properties

 Label 7942.2.a.p Level $7942$ Weight $2$ Character orbit 7942.a Self dual yes Analytic conductor $63.417$ Analytic rank $1$ 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] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7942 = 2 \cdot 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7942.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: $$63.4171892853$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 3 q^{5} + 3 q^{7} + q^{8} - 3 q^{9}+O(q^{10})$$ q + q^2 + q^4 - 3 * q^5 + 3 * q^7 + q^8 - 3 * q^9 $$q + q^{2} + q^{4} - 3 q^{5} + 3 q^{7} + q^{8} - 3 q^{9} - 3 q^{10} - q^{11} - 2 q^{13} + 3 q^{14} + q^{16} + 4 q^{17} - 3 q^{18} - 3 q^{20} - q^{22} + 2 q^{23} + 4 q^{25} - 2 q^{26} + 3 q^{28} - 2 q^{31} + q^{32} + 4 q^{34} - 9 q^{35} - 3 q^{36} + 9 q^{37} - 3 q^{40} - 2 q^{41} - 8 q^{43} - q^{44} + 9 q^{45} + 2 q^{46} - 12 q^{47} + 2 q^{49} + 4 q^{50} - 2 q^{52} + 13 q^{53} + 3 q^{55} + 3 q^{56} - 10 q^{59} - 2 q^{62} - 9 q^{63} + q^{64} + 6 q^{65} - 12 q^{67} + 4 q^{68} - 9 q^{70} - 2 q^{71} - 3 q^{72} + 12 q^{73} + 9 q^{74} - 3 q^{77} + q^{79} - 3 q^{80} + 9 q^{81} - 2 q^{82} + 9 q^{83} - 12 q^{85} - 8 q^{86} - q^{88} - 6 q^{89} + 9 q^{90} - 6 q^{91} + 2 q^{92} - 12 q^{94} - 5 q^{97} + 2 q^{98} + 3 q^{99}+O(q^{100})$$ q + q^2 + q^4 - 3 * q^5 + 3 * q^7 + q^8 - 3 * q^9 - 3 * q^10 - q^11 - 2 * q^13 + 3 * q^14 + q^16 + 4 * q^17 - 3 * q^18 - 3 * q^20 - q^22 + 2 * q^23 + 4 * q^25 - 2 * q^26 + 3 * q^28 - 2 * q^31 + q^32 + 4 * q^34 - 9 * q^35 - 3 * q^36 + 9 * q^37 - 3 * q^40 - 2 * q^41 - 8 * q^43 - q^44 + 9 * q^45 + 2 * q^46 - 12 * q^47 + 2 * q^49 + 4 * q^50 - 2 * q^52 + 13 * q^53 + 3 * q^55 + 3 * q^56 - 10 * q^59 - 2 * q^62 - 9 * q^63 + q^64 + 6 * q^65 - 12 * q^67 + 4 * q^68 - 9 * q^70 - 2 * q^71 - 3 * q^72 + 12 * q^73 + 9 * q^74 - 3 * q^77 + q^79 - 3 * q^80 + 9 * q^81 - 2 * q^82 + 9 * q^83 - 12 * q^85 - 8 * q^86 - q^88 - 6 * q^89 + 9 * q^90 - 6 * q^91 + 2 * q^92 - 12 * q^94 - 5 * q^97 + 2 * q^98 + 3 * 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
1.00000 0 1.00000 −3.00000 0 3.00000 1.00000 −3.00000 −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$+1$$
$$19$$ $$-1$$

## 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 7942.2.a.p 1
19.b odd 2 1 7942.2.a.e 1
19.d odd 6 2 418.2.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.e 2 19.d odd 6 2
7942.2.a.e 1 19.b odd 2 1
7942.2.a.p 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7942))$$:

 $$T_{3}$$ T3 $$T_{5} + 3$$ T5 + 3 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T - 3$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T - 4$$
$19$ $$T$$
$23$ $$T - 2$$
$29$ $$T$$
$31$ $$T + 2$$
$37$ $$T - 9$$
$41$ $$T + 2$$
$43$ $$T + 8$$
$47$ $$T + 12$$
$53$ $$T - 13$$
$59$ $$T + 10$$
$61$ $$T$$
$67$ $$T + 12$$
$71$ $$T + 2$$
$73$ $$T - 12$$
$79$ $$T - 1$$
$83$ $$T - 9$$
$89$ $$T + 6$$
$97$ $$T + 5$$