# Properties

 Label 7938.2.a.b Level $7938$ Weight $2$ Character orbit 7938.a Self dual yes Analytic conductor $63.385$ Analytic rank $0$ 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] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, 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("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7938 = 2 \cdot 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7938.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: $$63.3852491245$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 3 q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - 3 * q^5 - q^8 $$q - q^{2} + q^{4} - 3 q^{5} - q^{8} + 3 q^{10} - 3 q^{11} + q^{13} + q^{16} - 3 q^{17} + 7 q^{19} - 3 q^{20} + 3 q^{22} - 9 q^{23} + 4 q^{25} - q^{26} + 3 q^{29} - 8 q^{31} - q^{32} + 3 q^{34} - q^{37} - 7 q^{38} + 3 q^{40} - 3 q^{41} - q^{43} - 3 q^{44} + 9 q^{46} - 4 q^{50} + q^{52} + 3 q^{53} + 9 q^{55} - 3 q^{58} - 2 q^{61} + 8 q^{62} + q^{64} - 3 q^{65} - 4 q^{67} - 3 q^{68} + 12 q^{71} - 11 q^{73} + q^{74} + 7 q^{76} - 16 q^{79} - 3 q^{80} + 3 q^{82} + 9 q^{83} + 9 q^{85} + q^{86} + 3 q^{88} - 3 q^{89} - 9 q^{92} - 21 q^{95} + q^{97}+O(q^{100})$$ q - q^2 + q^4 - 3 * q^5 - q^8 + 3 * q^10 - 3 * q^11 + q^13 + q^16 - 3 * q^17 + 7 * q^19 - 3 * q^20 + 3 * q^22 - 9 * q^23 + 4 * q^25 - q^26 + 3 * q^29 - 8 * q^31 - q^32 + 3 * q^34 - q^37 - 7 * q^38 + 3 * q^40 - 3 * q^41 - q^43 - 3 * q^44 + 9 * q^46 - 4 * q^50 + q^52 + 3 * q^53 + 9 * q^55 - 3 * q^58 - 2 * q^61 + 8 * q^62 + q^64 - 3 * q^65 - 4 * q^67 - 3 * q^68 + 12 * q^71 - 11 * q^73 + q^74 + 7 * q^76 - 16 * q^79 - 3 * q^80 + 3 * q^82 + 9 * q^83 + 9 * q^85 + q^86 + 3 * q^88 - 3 * q^89 - 9 * q^92 - 21 * q^95 + q^97

## 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 0 −1.00000 0 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
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## 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 7938.2.a.b 1
3.b odd 2 1 7938.2.a.be 1
7.b odd 2 1 7938.2.a.m 1
7.d odd 6 2 1134.2.g.e 2
9.c even 3 2 882.2.f.g 2
9.d odd 6 2 2646.2.f.a 2
21.c even 2 1 7938.2.a.t 1
21.g even 6 2 1134.2.g.c 2
63.g even 3 2 882.2.h.i 2
63.h even 3 2 882.2.e.c 2
63.i even 6 2 378.2.e.b 2
63.j odd 6 2 2646.2.e.g 2
63.k odd 6 2 126.2.h.b yes 2
63.l odd 6 2 882.2.f.i 2
63.n odd 6 2 2646.2.h.d 2
63.o even 6 2 2646.2.f.d 2
63.s even 6 2 378.2.h.a 2
63.t odd 6 2 126.2.e.a 2
252.n even 6 2 1008.2.t.f 2
252.r odd 6 2 3024.2.q.f 2
252.bj even 6 2 1008.2.q.a 2
252.bn odd 6 2 3024.2.t.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 63.t odd 6 2
126.2.h.b yes 2 63.k odd 6 2
378.2.e.b 2 63.i even 6 2
378.2.h.a 2 63.s even 6 2
882.2.e.c 2 63.h even 3 2
882.2.f.g 2 9.c even 3 2
882.2.f.i 2 63.l odd 6 2
882.2.h.i 2 63.g even 3 2
1008.2.q.a 2 252.bj even 6 2
1008.2.t.f 2 252.n even 6 2
1134.2.g.c 2 21.g even 6 2
1134.2.g.e 2 7.d odd 6 2
2646.2.e.g 2 63.j odd 6 2
2646.2.f.a 2 9.d odd 6 2
2646.2.f.d 2 63.o even 6 2
2646.2.h.d 2 63.n odd 6 2
3024.2.q.f 2 252.r odd 6 2
3024.2.t.a 2 252.bn odd 6 2
7938.2.a.b 1 1.a even 1 1 trivial
7938.2.a.m 1 7.b odd 2 1
7938.2.a.t 1 21.c even 2 1
7938.2.a.be 1 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7938))$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{11} + 3$$ T11 + 3 $$T_{13} - 1$$ T13 - 1 $$T_{17} + 3$$ T17 + 3 $$T_{23} + 9$$ T23 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T$$
$11$ $$T + 3$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T - 7$$
$23$ $$T + 9$$
$29$ $$T - 3$$
$31$ $$T + 8$$
$37$ $$T + 1$$
$41$ $$T + 3$$
$43$ $$T + 1$$
$47$ $$T$$
$53$ $$T - 3$$
$59$ $$T$$
$61$ $$T + 2$$
$67$ $$T + 4$$
$71$ $$T - 12$$
$73$ $$T + 11$$
$79$ $$T + 16$$
$83$ $$T - 9$$
$89$ $$T + 3$$
$97$ $$T - 1$$