Properties

 Label 7938.2.a.bg Level $7938$ Weight $2$ Character orbit 7938.a Self dual yes Analytic conductor $63.385$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1134) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta - 2) q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b - 2) * q^5 - q^8 $$q - q^{2} + q^{4} + (\beta - 2) q^{5} - q^{8} + ( - \beta + 2) q^{10} + (3 \beta + 1) q^{11} + (2 \beta + 3) q^{13} + q^{16} - 7 q^{17} + ( - \beta + 1) q^{19} + (\beta - 2) q^{20} + ( - 3 \beta - 1) q^{22} + ( - 3 \beta + 1) q^{23} + ( - 4 \beta + 2) q^{25} + ( - 2 \beta - 3) q^{26} + ( - 2 \beta + 5) q^{29} + ( - 3 \beta - 3) q^{31} - q^{32} + 7 q^{34} + (5 \beta + 2) q^{37} + (\beta - 1) q^{38} + ( - \beta + 2) q^{40} + (2 \beta - 6) q^{41} + ( - 2 \beta + 2) q^{43} + (3 \beta + 1) q^{44} + (3 \beta - 1) q^{46} + ( - \beta - 3) q^{47} + (4 \beta - 2) q^{50} + (2 \beta + 3) q^{52} + ( - 2 \beta - 6) q^{53} + ( - 5 \beta + 7) q^{55} + (2 \beta - 5) q^{58} + ( - 3 \beta + 1) q^{59} + ( - 4 \beta + 3) q^{61} + (3 \beta + 3) q^{62} + q^{64} - \beta q^{65} + ( - \beta - 5) q^{67} - 7 q^{68} + ( - 2 \beta + 10) q^{71} + (\beta - 10) q^{73} + ( - 5 \beta - 2) q^{74} + ( - \beta + 1) q^{76} + ( - 7 \beta + 3) q^{79} + (\beta - 2) q^{80} + ( - 2 \beta + 6) q^{82} + ( - 9 \beta - 1) q^{83} + ( - 7 \beta + 14) q^{85} + (2 \beta - 2) q^{86} + ( - 3 \beta - 1) q^{88} + ( - 4 \beta - 3) q^{89} + ( - 3 \beta + 1) q^{92} + (\beta + 3) q^{94} + (3 \beta - 5) q^{95} + ( - 4 \beta - 4) q^{97} +O(q^{100})$$ q - q^2 + q^4 + (b - 2) * q^5 - q^8 + (-b + 2) * q^10 + (3*b + 1) * q^11 + (2*b + 3) * q^13 + q^16 - 7 * q^17 + (-b + 1) * q^19 + (b - 2) * q^20 + (-3*b - 1) * q^22 + (-3*b + 1) * q^23 + (-4*b + 2) * q^25 + (-2*b - 3) * q^26 + (-2*b + 5) * q^29 + (-3*b - 3) * q^31 - q^32 + 7 * q^34 + (5*b + 2) * q^37 + (b - 1) * q^38 + (-b + 2) * q^40 + (2*b - 6) * q^41 + (-2*b + 2) * q^43 + (3*b + 1) * q^44 + (3*b - 1) * q^46 + (-b - 3) * q^47 + (4*b - 2) * q^50 + (2*b + 3) * q^52 + (-2*b - 6) * q^53 + (-5*b + 7) * q^55 + (2*b - 5) * q^58 + (-3*b + 1) * q^59 + (-4*b + 3) * q^61 + (3*b + 3) * q^62 + q^64 - b * q^65 + (-b - 5) * q^67 - 7 * q^68 + (-2*b + 10) * q^71 + (b - 10) * q^73 + (-5*b - 2) * q^74 + (-b + 1) * q^76 + (-7*b + 3) * q^79 + (b - 2) * q^80 + (-2*b + 6) * q^82 + (-9*b - 1) * q^83 + (-7*b + 14) * q^85 + (2*b - 2) * q^86 + (-3*b - 1) * q^88 + (-4*b - 3) * q^89 + (-3*b + 1) * q^92 + (b + 3) * q^94 + (3*b - 5) * q^95 + (-4*b - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{16} - 14 q^{17} + 2 q^{19} - 4 q^{20} - 2 q^{22} + 2 q^{23} + 4 q^{25} - 6 q^{26} + 10 q^{29} - 6 q^{31} - 2 q^{32} + 14 q^{34} + 4 q^{37} - 2 q^{38} + 4 q^{40} - 12 q^{41} + 4 q^{43} + 2 q^{44} - 2 q^{46} - 6 q^{47} - 4 q^{50} + 6 q^{52} - 12 q^{53} + 14 q^{55} - 10 q^{58} + 2 q^{59} + 6 q^{61} + 6 q^{62} + 2 q^{64} - 10 q^{67} - 14 q^{68} + 20 q^{71} - 20 q^{73} - 4 q^{74} + 2 q^{76} + 6 q^{79} - 4 q^{80} + 12 q^{82} - 2 q^{83} + 28 q^{85} - 4 q^{86} - 2 q^{88} - 6 q^{89} + 2 q^{92} + 6 q^{94} - 10 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 4 * q^10 + 2 * q^11 + 6 * q^13 + 2 * q^16 - 14 * q^17 + 2 * q^19 - 4 * q^20 - 2 * q^22 + 2 * q^23 + 4 * q^25 - 6 * q^26 + 10 * q^29 - 6 * q^31 - 2 * q^32 + 14 * q^34 + 4 * q^37 - 2 * q^38 + 4 * q^40 - 12 * q^41 + 4 * q^43 + 2 * q^44 - 2 * q^46 - 6 * q^47 - 4 * q^50 + 6 * q^52 - 12 * q^53 + 14 * q^55 - 10 * q^58 + 2 * q^59 + 6 * q^61 + 6 * q^62 + 2 * q^64 - 10 * q^67 - 14 * q^68 + 20 * q^71 - 20 * q^73 - 4 * q^74 + 2 * q^76 + 6 * q^79 - 4 * q^80 + 12 * q^82 - 2 * q^83 + 28 * q^85 - 4 * q^86 - 2 * q^88 - 6 * q^89 + 2 * q^92 + 6 * q^94 - 10 * q^95 - 8 * 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
 −1.73205 1.73205
−1.00000 0 1.00000 −3.73205 0 0 −1.00000 0 3.73205
1.2 −1.00000 0 1.00000 −0.267949 0 0 −1.00000 0 0.267949
 $$n$$: 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.bg 2
3.b odd 2 1 7938.2.a.bt 2
7.b odd 2 1 1134.2.a.l 2
21.c even 2 1 1134.2.a.m yes 2
28.d even 2 1 9072.2.a.bp 2
63.l odd 6 2 1134.2.f.s 4
63.o even 6 2 1134.2.f.r 4
84.h odd 2 1 9072.2.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.l 2 7.b odd 2 1
1134.2.a.m yes 2 21.c even 2 1
1134.2.f.r 4 63.o even 6 2
1134.2.f.s 4 63.l odd 6 2
7938.2.a.bg 2 1.a even 1 1 trivial
7938.2.a.bt 2 3.b odd 2 1
9072.2.a.y 2 84.h odd 2 1
9072.2.a.bp 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7938))$$:

 $$T_{5}^{2} + 4T_{5} + 1$$ T5^2 + 4*T5 + 1 $$T_{11}^{2} - 2T_{11} - 26$$ T11^2 - 2*T11 - 26 $$T_{13}^{2} - 6T_{13} - 3$$ T13^2 - 6*T13 - 3 $$T_{17} + 7$$ T17 + 7 $$T_{23}^{2} - 2T_{23} - 26$$ T23^2 - 2*T23 - 26

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T - 26$$
$13$ $$T^{2} - 6T - 3$$
$17$ $$(T + 7)^{2}$$
$19$ $$T^{2} - 2T - 2$$
$23$ $$T^{2} - 2T - 26$$
$29$ $$T^{2} - 10T + 13$$
$31$ $$T^{2} + 6T - 18$$
$37$ $$T^{2} - 4T - 71$$
$41$ $$T^{2} + 12T + 24$$
$43$ $$T^{2} - 4T - 8$$
$47$ $$T^{2} + 6T + 6$$
$53$ $$T^{2} + 12T + 24$$
$59$ $$T^{2} - 2T - 26$$
$61$ $$T^{2} - 6T - 39$$
$67$ $$T^{2} + 10T + 22$$
$71$ $$T^{2} - 20T + 88$$
$73$ $$T^{2} + 20T + 97$$
$79$ $$T^{2} - 6T - 138$$
$83$ $$T^{2} + 2T - 242$$
$89$ $$T^{2} + 6T - 39$$
$97$ $$T^{2} + 8T - 32$$