# Properties

 Label 7938.2.a.u 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 $$q + q^{2} + q^{4} - 3 q^{5} + q^{8} - 3 q^{10} + 6 q^{11} - 2 q^{13} + q^{16} + 6 q^{17} + 7 q^{19} - 3 q^{20} + 6 q^{22} - 3 q^{23} + 4 q^{25} - 2 q^{26} - 6 q^{29} - 2 q^{31} + q^{32} + 6 q^{34} + 2 q^{37} + 7 q^{38} - 3 q^{40} + 2 q^{43} + 6 q^{44} - 3 q^{46} + 4 q^{50} - 2 q^{52} - 6 q^{53} - 18 q^{55} - 6 q^{58} - 5 q^{61} - 2 q^{62} + q^{64} + 6 q^{65} + 8 q^{67} + 6 q^{68} - 3 q^{71} - 2 q^{73} + 2 q^{74} + 7 q^{76} + 5 q^{79} - 3 q^{80} + 12 q^{83} - 18 q^{85} + 2 q^{86} + 6 q^{88} - 3 q^{92} - 21 q^{95} - 2 q^{97}+O(q^{100})$$ q + q^2 + q^4 - 3 * q^5 + q^8 - 3 * q^10 + 6 * q^11 - 2 * q^13 + q^16 + 6 * q^17 + 7 * q^19 - 3 * q^20 + 6 * q^22 - 3 * q^23 + 4 * q^25 - 2 * q^26 - 6 * q^29 - 2 * q^31 + q^32 + 6 * q^34 + 2 * q^37 + 7 * q^38 - 3 * q^40 + 2 * q^43 + 6 * q^44 - 3 * q^46 + 4 * q^50 - 2 * q^52 - 6 * q^53 - 18 * q^55 - 6 * q^58 - 5 * q^61 - 2 * q^62 + q^64 + 6 * q^65 + 8 * q^67 + 6 * q^68 - 3 * q^71 - 2 * q^73 + 2 * q^74 + 7 * q^76 + 5 * q^79 - 3 * q^80 + 12 * q^83 - 18 * q^85 + 2 * q^86 + 6 * q^88 - 3 * q^92 - 21 * q^95 - 2 * 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.u 1
3.b odd 2 1 7938.2.a.l 1
7.b odd 2 1 1134.2.a.h 1
9.c even 3 2 2646.2.f.c 2
9.d odd 6 2 882.2.f.h 2
21.c even 2 1 1134.2.a.a 1
28.d even 2 1 9072.2.a.w 1
63.g even 3 2 2646.2.h.a 2
63.h even 3 2 2646.2.e.j 2
63.i even 6 2 882.2.e.b 2
63.j odd 6 2 882.2.e.d 2
63.k odd 6 2 2646.2.h.e 2
63.l odd 6 2 378.2.f.a 2
63.n odd 6 2 882.2.h.f 2
63.o even 6 2 126.2.f.a 2
63.s even 6 2 882.2.h.j 2
63.t odd 6 2 2646.2.e.f 2
84.h odd 2 1 9072.2.a.c 1
252.s odd 6 2 1008.2.r.d 2
252.bi even 6 2 3024.2.r.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.a 2 63.o even 6 2
378.2.f.a 2 63.l odd 6 2
882.2.e.b 2 63.i even 6 2
882.2.e.d 2 63.j odd 6 2
882.2.f.h 2 9.d odd 6 2
882.2.h.f 2 63.n odd 6 2
882.2.h.j 2 63.s even 6 2
1008.2.r.d 2 252.s odd 6 2
1134.2.a.a 1 21.c even 2 1
1134.2.a.h 1 7.b odd 2 1
2646.2.e.f 2 63.t odd 6 2
2646.2.e.j 2 63.h even 3 2
2646.2.f.c 2 9.c even 3 2
2646.2.h.a 2 63.g even 3 2
2646.2.h.e 2 63.k odd 6 2
3024.2.r.a 2 252.bi even 6 2
7938.2.a.l 1 3.b odd 2 1
7938.2.a.u 1 1.a even 1 1 trivial
9072.2.a.c 1 84.h odd 2 1
9072.2.a.w 1 28.d even 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 $$T_{11} - 6$$ T11 - 6 $$T_{13} + 2$$ T13 + 2 $$T_{17} - 6$$ T17 - 6 $$T_{23} + 3$$ T23 + 3

## Hecke characteristic polynomials

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