# Properties

 Label 7938.2.a.bx Level $7938$ Weight $2$ Character orbit 7938.a Self dual yes Analytic conductor $63.385$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 1$$ x^3 - x^2 - 4*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( - \beta_1 + 2) q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (-b1 + 2) * q^5 - q^8 $$q - q^{2} + q^{4} + ( - \beta_1 + 2) q^{5} - q^{8} + (\beta_1 - 2) q^{10} + (2 \beta_{2} + \beta_1) q^{11} + (\beta_1 - 1) q^{13} + q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + ( - \beta_{2} - \beta_1 - 1) q^{19} + ( - \beta_1 + 2) q^{20} + ( - 2 \beta_{2} - \beta_1) q^{22} + ( - \beta_{2} + \beta_1 - 3) q^{23} + (\beta_{2} - 3 \beta_1 + 2) q^{25} + ( - \beta_1 + 1) q^{26} + ( - \beta_{2} - 3 \beta_1 - 1) q^{29} + (\beta_{2} + 2 \beta_1 - 5) q^{31} - q^{32} + (2 \beta_{2} - 2 \beta_1) q^{34} + (2 \beta_{2} + 2 \beta_1 + 3) q^{37} + (\beta_{2} + \beta_1 + 1) q^{38} + (\beta_1 - 2) q^{40} + ( - \beta_{2} - \beta_1 + 4) q^{41} + (\beta_{2} - 2 \beta_1 - 5) q^{43} + (2 \beta_{2} + \beta_1) q^{44} + (\beta_{2} - \beta_1 + 3) q^{46} + ( - \beta_{2} + 2 \beta_1 - 2) q^{47} + ( - \beta_{2} + 3 \beta_1 - 2) q^{50} + (\beta_1 - 1) q^{52} + (\beta_{2} + 4 \beta_1 + 2) q^{53} + (3 \beta_{2} - \beta_1 - 1) q^{55} + (\beta_{2} + 3 \beta_1 + 1) q^{58} + ( - \beta_{2} - 5 \beta_1) q^{59} + ( - 3 \beta_{2} - 5 \beta_1 + 2) q^{61} + ( - \beta_{2} - 2 \beta_1 + 5) q^{62} + q^{64} + ( - \beta_{2} + 2 \beta_1 - 5) q^{65} + ( - 3 \beta_{2} + \beta_1 - 3) q^{67} + ( - 2 \beta_{2} + 2 \beta_1) q^{68} + (3 \beta_{2} - \beta_1 - 1) q^{71} + ( - 3 \beta_{2} - \beta_1 - 9) q^{73} + ( - 2 \beta_{2} - 2 \beta_1 - 3) q^{74} + ( - \beta_{2} - \beta_1 - 1) q^{76} + (4 \beta_{2} - 3 \beta_1) q^{79} + ( - \beta_1 + 2) q^{80} + (\beta_{2} + \beta_1 - 4) q^{82} + ( - 2 \beta_{2} - \beta_1 - 3) q^{83} + ( - 6 \beta_{2} + 4 \beta_1 - 8) q^{85} + ( - \beta_{2} + 2 \beta_1 + 5) q^{86} + ( - 2 \beta_{2} - \beta_1) q^{88} + (2 \beta_{2} - \beta_1 + 4) q^{89} + ( - \beta_{2} + \beta_1 - 3) q^{92} + (\beta_{2} - 2 \beta_1 + 2) q^{94} + ( - \beta_{2} + \beta_1) q^{95} + ( - 2 \beta_{2} - 10) q^{97}+O(q^{100})$$ q - q^2 + q^4 + (-b1 + 2) * q^5 - q^8 + (b1 - 2) * q^10 + (2*b2 + b1) * q^11 + (b1 - 1) * q^13 + q^16 + (-2*b2 + 2*b1) * q^17 + (-b2 - b1 - 1) * q^19 + (-b1 + 2) * q^20 + (-2*b2 - b1) * q^22 + (-b2 + b1 - 3) * q^23 + (b2 - 3*b1 + 2) * q^25 + (-b1 + 1) * q^26 + (-b2 - 3*b1 - 1) * q^29 + (b2 + 2*b1 - 5) * q^31 - q^32 + (2*b2 - 2*b1) * q^34 + (2*b2 + 2*b1 + 3) * q^37 + (b2 + b1 + 1) * q^38 + (b1 - 2) * q^40 + (-b2 - b1 + 4) * q^41 + (b2 - 2*b1 - 5) * q^43 + (2*b2 + b1) * q^44 + (b2 - b1 + 3) * q^46 + (-b2 + 2*b1 - 2) * q^47 + (-b2 + 3*b1 - 2) * q^50 + (b1 - 1) * q^52 + (b2 + 4*b1 + 2) * q^53 + (3*b2 - b1 - 1) * q^55 + (b2 + 3*b1 + 1) * q^58 + (-b2 - 5*b1) * q^59 + (-3*b2 - 5*b1 + 2) * q^61 + (-b2 - 2*b1 + 5) * q^62 + q^64 + (-b2 + 2*b1 - 5) * q^65 + (-3*b2 + b1 - 3) * q^67 + (-2*b2 + 2*b1) * q^68 + (3*b2 - b1 - 1) * q^71 + (-3*b2 - b1 - 9) * q^73 + (-2*b2 - 2*b1 - 3) * q^74 + (-b2 - b1 - 1) * q^76 + (4*b2 - 3*b1) * q^79 + (-b1 + 2) * q^80 + (b2 + b1 - 4) * q^82 + (-2*b2 - b1 - 3) * q^83 + (-6*b2 + 4*b1 - 8) * q^85 + (-b2 + 2*b1 + 5) * q^86 + (-2*b2 - b1) * q^88 + (2*b2 - b1 + 4) * q^89 + (-b2 + b1 - 3) * q^92 + (b2 - 2*b1 + 2) * q^94 + (-b2 + b1) * q^95 + (-2*b2 - 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} + 5 q^{5} - 3 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 + 5 * q^5 - 3 * q^8 $$3 q - 3 q^{2} + 3 q^{4} + 5 q^{5} - 3 q^{8} - 5 q^{10} - q^{11} - 2 q^{13} + 3 q^{16} + 4 q^{17} - 3 q^{19} + 5 q^{20} + q^{22} - 7 q^{23} + 2 q^{25} + 2 q^{26} - 5 q^{29} - 14 q^{31} - 3 q^{32} - 4 q^{34} + 9 q^{37} + 3 q^{38} - 5 q^{40} + 12 q^{41} - 18 q^{43} - q^{44} + 7 q^{46} - 3 q^{47} - 2 q^{50} - 2 q^{52} + 9 q^{53} - 7 q^{55} + 5 q^{58} - 4 q^{59} + 4 q^{61} + 14 q^{62} + 3 q^{64} - 12 q^{65} - 5 q^{67} + 4 q^{68} - 7 q^{71} - 25 q^{73} - 9 q^{74} - 3 q^{76} - 7 q^{79} + 5 q^{80} - 12 q^{82} - 8 q^{83} - 14 q^{85} + 18 q^{86} + q^{88} + 9 q^{89} - 7 q^{92} + 3 q^{94} + 2 q^{95} - 28 q^{97}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 + 5 * q^5 - 3 * q^8 - 5 * q^10 - q^11 - 2 * q^13 + 3 * q^16 + 4 * q^17 - 3 * q^19 + 5 * q^20 + q^22 - 7 * q^23 + 2 * q^25 + 2 * q^26 - 5 * q^29 - 14 * q^31 - 3 * q^32 - 4 * q^34 + 9 * q^37 + 3 * q^38 - 5 * q^40 + 12 * q^41 - 18 * q^43 - q^44 + 7 * q^46 - 3 * q^47 - 2 * q^50 - 2 * q^52 + 9 * q^53 - 7 * q^55 + 5 * q^58 - 4 * q^59 + 4 * q^61 + 14 * q^62 + 3 * q^64 - 12 * q^65 - 5 * q^67 + 4 * q^68 - 7 * q^71 - 25 * q^73 - 9 * q^74 - 3 * q^76 - 7 * q^79 + 5 * q^80 - 12 * q^82 - 8 * q^83 - 14 * q^85 + 18 * q^86 + q^88 + 9 * q^89 - 7 * q^92 + 3 * q^94 + 2 * q^95 - 28 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## 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
 2.46050 0.239123 −1.69963
−1.00000 0 1.00000 −0.460505 0 0 −1.00000 0 0.460505
1.2 −1.00000 0 1.00000 1.76088 0 0 −1.00000 0 −1.76088
1.3 −1.00000 0 1.00000 3.69963 0 0 −1.00000 0 −3.69963
 $$n$$: 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.bx 3
3.b odd 2 1 7938.2.a.by 3
7.b odd 2 1 7938.2.a.bu 3
7.d odd 6 2 1134.2.g.n 6
9.c even 3 2 2646.2.f.n 6
9.d odd 6 2 882.2.f.m 6
21.c even 2 1 7938.2.a.cb 3
21.g even 6 2 1134.2.g.k 6
63.g even 3 2 2646.2.h.p 6
63.h even 3 2 2646.2.e.o 6
63.i even 6 2 126.2.e.d 6
63.j odd 6 2 882.2.e.p 6
63.k odd 6 2 378.2.h.d 6
63.l odd 6 2 2646.2.f.o 6
63.n odd 6 2 882.2.h.o 6
63.o even 6 2 882.2.f.l 6
63.s even 6 2 126.2.h.c yes 6
63.t odd 6 2 378.2.e.c 6
252.n even 6 2 3024.2.t.g 6
252.r odd 6 2 1008.2.q.h 6
252.bj even 6 2 3024.2.q.h 6
252.bn odd 6 2 1008.2.t.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.d 6 63.i even 6 2
126.2.h.c yes 6 63.s even 6 2
378.2.e.c 6 63.t odd 6 2
378.2.h.d 6 63.k odd 6 2
882.2.e.p 6 63.j odd 6 2
882.2.f.l 6 63.o even 6 2
882.2.f.m 6 9.d odd 6 2
882.2.h.o 6 63.n odd 6 2
1008.2.q.h 6 252.r odd 6 2
1008.2.t.g 6 252.bn odd 6 2
1134.2.g.k 6 21.g even 6 2
1134.2.g.n 6 7.d odd 6 2
2646.2.e.o 6 63.h even 3 2
2646.2.f.n 6 9.c even 3 2
2646.2.f.o 6 63.l odd 6 2
2646.2.h.p 6 63.g even 3 2
3024.2.q.h 6 252.bj even 6 2
3024.2.t.g 6 252.n even 6 2
7938.2.a.bu 3 7.b odd 2 1
7938.2.a.bx 3 1.a even 1 1 trivial
7938.2.a.by 3 3.b odd 2 1
7938.2.a.cb 3 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7938))$$:

 $$T_{5}^{3} - 5T_{5}^{2} + 4T_{5} + 3$$ T5^3 - 5*T5^2 + 4*T5 + 3 $$T_{11}^{3} + T_{11}^{2} - 26T_{11} + 33$$ T11^3 + T11^2 - 26*T11 + 33 $$T_{13}^{3} + 2T_{13}^{2} - 3T_{13} - 3$$ T13^3 + 2*T13^2 - 3*T13 - 3 $$T_{17}^{3} - 4T_{17}^{2} - 44T_{17} + 168$$ T17^3 - 4*T17^2 - 44*T17 + 168 $$T_{23}^{3} + 7T_{23}^{2} + 4T_{23} - 3$$ T23^3 + 7*T23^2 + 4*T23 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 5 T^{2} + 4 T + 3$$
$7$ $$T^{3}$$
$11$ $$T^{3} + T^{2} - 26 T + 33$$
$13$ $$T^{3} + 2 T^{2} - 3 T - 3$$
$17$ $$T^{3} - 4 T^{2} - 44 T + 168$$
$19$ $$T^{3} + 3 T^{2} - 6 T - 7$$
$23$ $$T^{3} + 7 T^{2} + 4 T - 3$$
$29$ $$T^{3} + 5 T^{2} - 32 T + 33$$
$31$ $$T^{3} + 14 T^{2} + 45 T - 27$$
$37$ $$T^{3} - 9 T^{2} - 9 T + 73$$
$41$ $$T^{3} - 12 T^{2} + 39 T - 27$$
$43$ $$T^{3} + 18 T^{2} + 81 T + 1$$
$47$ $$T^{3} + 3 T^{2} - 24 T + 27$$
$53$ $$T^{3} - 9 T^{2} - 42 T - 9$$
$59$ $$T^{3} + 4 T^{2} - 101 T + 177$$
$61$ $$T^{3} - 4 T^{2} - 135 T + 717$$
$67$ $$T^{3} + 5 T^{2} - 58 T - 149$$
$71$ $$T^{3} + 7 T^{2} - 50 T - 99$$
$73$ $$T^{3} + 25 T^{2} + 152 T - 49$$
$79$ $$T^{3} + 7 T^{2} - 144 T - 771$$
$83$ $$T^{3} + 8 T^{2} - 5 T - 93$$
$89$ $$T^{3} - 9 T^{2} - 6 T + 63$$
$97$ $$T^{3} + 28 T^{2} + 236 T + 536$$