# Properties

 Label 7942.2.a.i 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - q^{8} - 2 q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - 2 * q^5 - q^6 - 3 * q^7 - q^8 - 2 * q^9 $$q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - q^{8} - 2 q^{9} + 2 q^{10} - q^{11} + q^{12} - q^{13} + 3 q^{14} - 2 q^{15} + q^{16} - 7 q^{17} + 2 q^{18} - 2 q^{20} - 3 q^{21} + q^{22} - 5 q^{23} - q^{24} - q^{25} + q^{26} - 5 q^{27} - 3 q^{28} - q^{29} + 2 q^{30} - 10 q^{31} - q^{32} - q^{33} + 7 q^{34} + 6 q^{35} - 2 q^{36} + 6 q^{37} - q^{39} + 2 q^{40} - 6 q^{41} + 3 q^{42} - 4 q^{43} - q^{44} + 4 q^{45} + 5 q^{46} + q^{48} + 2 q^{49} + q^{50} - 7 q^{51} - q^{52} + q^{53} + 5 q^{54} + 2 q^{55} + 3 q^{56} + q^{58} - 3 q^{59} - 2 q^{60} - 12 q^{61} + 10 q^{62} + 6 q^{63} + q^{64} + 2 q^{65} + q^{66} - 3 q^{67} - 7 q^{68} - 5 q^{69} - 6 q^{70} + 10 q^{71} + 2 q^{72} + 3 q^{73} - 6 q^{74} - q^{75} + 3 q^{77} + q^{78} - 8 q^{79} - 2 q^{80} + q^{81} + 6 q^{82} + 8 q^{83} - 3 q^{84} + 14 q^{85} + 4 q^{86} - q^{87} + q^{88} + 8 q^{89} - 4 q^{90} + 3 q^{91} - 5 q^{92} - 10 q^{93} - q^{96} - 8 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - 2 * q^5 - q^6 - 3 * q^7 - q^8 - 2 * q^9 + 2 * q^10 - q^11 + q^12 - q^13 + 3 * q^14 - 2 * q^15 + q^16 - 7 * q^17 + 2 * q^18 - 2 * q^20 - 3 * q^21 + q^22 - 5 * q^23 - q^24 - q^25 + q^26 - 5 * q^27 - 3 * q^28 - q^29 + 2 * q^30 - 10 * q^31 - q^32 - q^33 + 7 * q^34 + 6 * q^35 - 2 * q^36 + 6 * q^37 - q^39 + 2 * q^40 - 6 * q^41 + 3 * q^42 - 4 * q^43 - q^44 + 4 * q^45 + 5 * q^46 + q^48 + 2 * q^49 + q^50 - 7 * q^51 - q^52 + q^53 + 5 * q^54 + 2 * q^55 + 3 * q^56 + q^58 - 3 * q^59 - 2 * q^60 - 12 * q^61 + 10 * q^62 + 6 * q^63 + q^64 + 2 * q^65 + q^66 - 3 * q^67 - 7 * q^68 - 5 * q^69 - 6 * q^70 + 10 * q^71 + 2 * q^72 + 3 * q^73 - 6 * q^74 - q^75 + 3 * q^77 + q^78 - 8 * q^79 - 2 * q^80 + q^81 + 6 * q^82 + 8 * q^83 - 3 * q^84 + 14 * q^85 + 4 * q^86 - q^87 + q^88 + 8 * q^89 - 4 * q^90 + 3 * q^91 - 5 * q^92 - 10 * q^93 - q^96 - 8 * q^97 - 2 * q^98 + 2 * 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 1.00000 1.00000 −2.00000 −1.00000 −3.00000 −1.00000 −2.00000 2.00000
 $$n$$: 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.i 1
19.b odd 2 1 418.2.a.a 1
57.d even 2 1 3762.2.a.g 1
76.d even 2 1 3344.2.a.h 1
209.d even 2 1 4598.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.a 1 19.b odd 2 1
3344.2.a.h 1 76.d even 2 1
3762.2.a.g 1 57.d even 2 1
4598.2.a.b 1 209.d even 2 1
7942.2.a.i 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7942))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5} + 2$$ T5 + 2 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

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