# Properties

 Label 7942.2.a.j 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} + 2 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + 2 * q^3 + q^4 - 2 * q^6 + 2 * q^7 - q^8 + q^9 $$q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{11} + 2 q^{12} + q^{13} - 2 q^{14} + q^{16} - 6 q^{17} - q^{18} + 4 q^{21} - q^{22} - 2 q^{24} - 5 q^{25} - q^{26} - 4 q^{27} + 2 q^{28} - 9 q^{29} - 8 q^{31} - q^{32} + 2 q^{33} + 6 q^{34} + q^{36} + 10 q^{37} + 2 q^{39} + 6 q^{41} - 4 q^{42} + 5 q^{43} + q^{44} - 3 q^{47} + 2 q^{48} - 3 q^{49} + 5 q^{50} - 12 q^{51} + q^{52} + 6 q^{53} + 4 q^{54} - 2 q^{56} + 9 q^{58} - 6 q^{59} - 13 q^{61} + 8 q^{62} + 2 q^{63} + q^{64} - 2 q^{66} + 10 q^{67} - 6 q^{68} - 9 q^{71} - q^{72} - 4 q^{73} - 10 q^{74} - 10 q^{75} + 2 q^{77} - 2 q^{78} - 8 q^{79} - 11 q^{81} - 6 q^{82} - 9 q^{83} + 4 q^{84} - 5 q^{86} - 18 q^{87} - q^{88} - 15 q^{89} + 2 q^{91} - 16 q^{93} + 3 q^{94} - 2 q^{96} + q^{97} + 3 q^{98} + q^{99}+O(q^{100})$$ q - q^2 + 2 * q^3 + q^4 - 2 * q^6 + 2 * q^7 - q^8 + q^9 + q^11 + 2 * q^12 + q^13 - 2 * q^14 + q^16 - 6 * q^17 - q^18 + 4 * q^21 - q^22 - 2 * q^24 - 5 * q^25 - q^26 - 4 * q^27 + 2 * q^28 - 9 * q^29 - 8 * q^31 - q^32 + 2 * q^33 + 6 * q^34 + q^36 + 10 * q^37 + 2 * q^39 + 6 * q^41 - 4 * q^42 + 5 * q^43 + q^44 - 3 * q^47 + 2 * q^48 - 3 * q^49 + 5 * q^50 - 12 * q^51 + q^52 + 6 * q^53 + 4 * q^54 - 2 * q^56 + 9 * q^58 - 6 * q^59 - 13 * q^61 + 8 * q^62 + 2 * q^63 + q^64 - 2 * q^66 + 10 * q^67 - 6 * q^68 - 9 * q^71 - q^72 - 4 * q^73 - 10 * q^74 - 10 * q^75 + 2 * q^77 - 2 * q^78 - 8 * q^79 - 11 * q^81 - 6 * q^82 - 9 * q^83 + 4 * q^84 - 5 * q^86 - 18 * q^87 - q^88 - 15 * q^89 + 2 * q^91 - 16 * q^93 + 3 * q^94 - 2 * q^96 + q^97 + 3 * 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 0 −2.00000 2.00000 −1.00000 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
$$2$$ $$+1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.d 2 19.d odd 6 2
7942.2.a.j 1 1.a even 1 1 trivial
7942.2.a.l 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 $$T_{5}$$ T5 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T - 1$$
$13$ $$T - 1$$
$17$ $$T + 6$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 9$$
$31$ $$T + 8$$
$37$ $$T - 10$$
$41$ $$T - 6$$
$43$ $$T - 5$$
$47$ $$T + 3$$
$53$ $$T - 6$$
$59$ $$T + 6$$
$61$ $$T + 13$$
$67$ $$T - 10$$
$71$ $$T + 9$$
$73$ $$T + 4$$
$79$ $$T + 8$$
$83$ $$T + 9$$
$89$ $$T + 15$$
$97$ $$T - 1$$