Properties

 Label 7942.2.a.b Level $7942$ Weight $2$ Character orbit 7942.a Self dual yes Analytic conductor $63.417$ Analytic rank $2$ 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: $$2$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 2 q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - 2 * q^3 + q^4 - 3 * q^5 + 2 * q^6 - q^7 - q^8 + q^9 $$q - q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 2 q^{6} - q^{7} - q^{8} + q^{9} + 3 q^{10} + q^{11} - 2 q^{12} - 4 q^{13} + q^{14} + 6 q^{15} + q^{16} - q^{18} - 3 q^{20} + 2 q^{21} - q^{22} + 2 q^{24} + 4 q^{25} + 4 q^{26} + 4 q^{27} - q^{28} - 6 q^{29} - 6 q^{30} - 4 q^{31} - q^{32} - 2 q^{33} + 3 q^{35} + q^{36} - q^{37} + 8 q^{39} + 3 q^{40} - 6 q^{41} - 2 q^{42} - 4 q^{43} + q^{44} - 3 q^{45} + 6 q^{47} - 2 q^{48} - 6 q^{49} - 4 q^{50} - 4 q^{52} + 3 q^{53} - 4 q^{54} - 3 q^{55} + q^{56} + 6 q^{58} + 6 q^{60} - 4 q^{61} + 4 q^{62} - q^{63} + q^{64} + 12 q^{65} + 2 q^{66} - 4 q^{67} - 3 q^{70} - 6 q^{71} - q^{72} + 2 q^{73} + q^{74} - 8 q^{75} - q^{77} - 8 q^{78} + 11 q^{79} - 3 q^{80} - 11 q^{81} + 6 q^{82} - 3 q^{83} + 2 q^{84} + 4 q^{86} + 12 q^{87} - q^{88} - 18 q^{89} + 3 q^{90} + 4 q^{91} + 8 q^{93} - 6 q^{94} + 2 q^{96} - 19 q^{97} + 6 q^{98} + q^{99}+O(q^{100})$$ q - q^2 - 2 * q^3 + q^4 - 3 * q^5 + 2 * q^6 - q^7 - q^8 + q^9 + 3 * q^10 + q^11 - 2 * q^12 - 4 * q^13 + q^14 + 6 * q^15 + q^16 - q^18 - 3 * q^20 + 2 * q^21 - q^22 + 2 * q^24 + 4 * q^25 + 4 * q^26 + 4 * q^27 - q^28 - 6 * q^29 - 6 * q^30 - 4 * q^31 - q^32 - 2 * q^33 + 3 * q^35 + q^36 - q^37 + 8 * q^39 + 3 * q^40 - 6 * q^41 - 2 * q^42 - 4 * q^43 + q^44 - 3 * q^45 + 6 * q^47 - 2 * q^48 - 6 * q^49 - 4 * q^50 - 4 * q^52 + 3 * q^53 - 4 * q^54 - 3 * q^55 + q^56 + 6 * q^58 + 6 * q^60 - 4 * q^61 + 4 * q^62 - q^63 + q^64 + 12 * q^65 + 2 * q^66 - 4 * q^67 - 3 * q^70 - 6 * q^71 - q^72 + 2 * q^73 + q^74 - 8 * q^75 - q^77 - 8 * q^78 + 11 * q^79 - 3 * q^80 - 11 * q^81 + 6 * q^82 - 3 * q^83 + 2 * q^84 + 4 * q^86 + 12 * q^87 - q^88 - 18 * q^89 + 3 * q^90 + 4 * q^91 + 8 * q^93 - 6 * q^94 + 2 * q^96 - 19 * q^97 + 6 * q^98 + 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 −2.00000 1.00000 −3.00000 2.00000 −1.00000 −1.00000 1.00000 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.b 1
19.b odd 2 1 7942.2.a.r 1
19.c even 3 2 418.2.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.f 2 19.c even 3 2
7942.2.a.b 1 1.a even 1 1 trivial
7942.2.a.r 1 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7942))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{5} + 3$$ T5 + 3 $$T_{13} + 4$$ T13 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 2$$
$5$ $$T + 3$$
$7$ $$T + 1$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T + 1$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T - 6$$
$53$ $$T - 3$$
$59$ $$T$$
$61$ $$T + 4$$
$67$ $$T + 4$$
$71$ $$T + 6$$
$73$ $$T - 2$$
$79$ $$T - 11$$
$83$ $$T + 3$$
$89$ $$T + 18$$
$97$ $$T + 19$$