Properties

 Label 7942.2.a.q 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} - 3 q^{9}+O(q^{10})$$ q + q^2 + q^4 + q^5 + q^7 + q^8 - 3 * q^9 $$q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} - 3 q^{9} + q^{10} + q^{11} + 2 q^{13} + q^{14} + q^{16} - 3 q^{18} + q^{20} + q^{22} - 6 q^{23} - 4 q^{25} + 2 q^{26} + q^{28} - 8 q^{29} - 10 q^{31} + q^{32} + q^{35} - 3 q^{36} + 3 q^{37} + q^{40} - 2 q^{41} - 8 q^{43} + q^{44} - 3 q^{45} - 6 q^{46} - 6 q^{49} - 4 q^{50} + 2 q^{52} - 9 q^{53} + q^{55} + q^{56} - 8 q^{58} - 14 q^{59} - 10 q^{62} - 3 q^{63} + q^{64} + 2 q^{65} + 12 q^{67} + q^{70} + 6 q^{71} - 3 q^{72} + 3 q^{74} + q^{77} - 11 q^{79} + q^{80} + 9 q^{81} - 2 q^{82} + 7 q^{83} - 8 q^{86} + q^{88} + 14 q^{89} - 3 q^{90} + 2 q^{91} - 6 q^{92} + 5 q^{97} - 6 q^{98} - 3 q^{99}+O(q^{100})$$ q + q^2 + q^4 + q^5 + q^7 + q^8 - 3 * q^9 + q^10 + q^11 + 2 * q^13 + q^14 + q^16 - 3 * q^18 + q^20 + q^22 - 6 * q^23 - 4 * q^25 + 2 * q^26 + q^28 - 8 * q^29 - 10 * q^31 + q^32 + q^35 - 3 * q^36 + 3 * q^37 + q^40 - 2 * q^41 - 8 * q^43 + q^44 - 3 * q^45 - 6 * q^46 - 6 * q^49 - 4 * q^50 + 2 * q^52 - 9 * q^53 + q^55 + q^56 - 8 * q^58 - 14 * q^59 - 10 * q^62 - 3 * q^63 + q^64 + 2 * q^65 + 12 * q^67 + q^70 + 6 * q^71 - 3 * q^72 + 3 * q^74 + q^77 - 11 * q^79 + q^80 + 9 * q^81 - 2 * q^82 + 7 * q^83 - 8 * q^86 + q^88 + 14 * q^89 - 3 * q^90 + 2 * q^91 - 6 * q^92 + 5 * q^97 - 6 * 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 1.00000 0 1.00000 1.00000 −3.00000 1.00000
 $$n$$: 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.q 1
19.b odd 2 1 7942.2.a.f 1
19.c even 3 2 418.2.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.b 2 19.c even 3 2
7942.2.a.f 1 19.b odd 2 1
7942.2.a.q 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 $$T_{5} - 1$$ T5 - 1 $$T_{13} - 2$$ T13 - 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T - 1$$
$13$ $$T - 2$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 6$$
$29$ $$T + 8$$
$31$ $$T + 10$$
$37$ $$T - 3$$
$41$ $$T + 2$$
$43$ $$T + 8$$
$47$ $$T$$
$53$ $$T + 9$$
$59$ $$T + 14$$
$61$ $$T$$
$67$ $$T - 12$$
$71$ $$T - 6$$
$73$ $$T$$
$79$ $$T + 11$$
$83$ $$T - 7$$
$89$ $$T - 14$$
$97$ $$T - 5$$