# Properties

 Label 7938.2.a.bm 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: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta + 1) q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b + 1) * q^5 - q^8 $$q - q^{2} + q^{4} + (\beta + 1) q^{5} - q^{8} + ( - \beta - 1) q^{10} - 2 q^{11} + 2 \beta q^{13} + q^{16} + 2 q^{17} + ( - \beta - 5) q^{19} + (\beta + 1) q^{20} + 2 q^{22} + q^{23} + (2 \beta + 2) q^{25} - 2 \beta q^{26} + ( - 2 \beta + 2) q^{29} - 6 q^{31} - q^{32} - 2 q^{34} + ( - 4 \beta + 2) q^{37} + (\beta + 5) q^{38} + ( - \beta - 1) q^{40} - 4 \beta q^{41} + ( - 2 \beta + 2) q^{43} - 2 q^{44} - q^{46} - 4 \beta q^{47} + ( - 2 \beta - 2) q^{50} + 2 \beta q^{52} + ( - 2 \beta + 6) q^{53} + ( - 2 \beta - 2) q^{55} + (2 \beta - 2) q^{58} - 2 q^{59} + ( - \beta - 9) q^{61} + 6 q^{62} + q^{64} + (2 \beta + 12) q^{65} + (2 \beta - 8) q^{67} + 2 q^{68} + ( - 2 \beta - 5) q^{71} + ( - 2 \beta + 2) q^{73} + (4 \beta - 2) q^{74} + ( - \beta - 5) q^{76} + (2 \beta + 3) q^{79} + (\beta + 1) q^{80} + 4 \beta q^{82} + 2 q^{83} + (2 \beta + 2) q^{85} + (2 \beta - 2) q^{86} + 2 q^{88} + (2 \beta - 12) q^{89} + q^{92} + 4 \beta q^{94} + ( - 6 \beta - 11) q^{95} + (2 \beta + 2) q^{97} +O(q^{100})$$ q - q^2 + q^4 + (b + 1) * q^5 - q^8 + (-b - 1) * q^10 - 2 * q^11 + 2*b * q^13 + q^16 + 2 * q^17 + (-b - 5) * q^19 + (b + 1) * q^20 + 2 * q^22 + q^23 + (2*b + 2) * q^25 - 2*b * q^26 + (-2*b + 2) * q^29 - 6 * q^31 - q^32 - 2 * q^34 + (-4*b + 2) * q^37 + (b + 5) * q^38 + (-b - 1) * q^40 - 4*b * q^41 + (-2*b + 2) * q^43 - 2 * q^44 - q^46 - 4*b * q^47 + (-2*b - 2) * q^50 + 2*b * q^52 + (-2*b + 6) * q^53 + (-2*b - 2) * q^55 + (2*b - 2) * q^58 - 2 * q^59 + (-b - 9) * q^61 + 6 * q^62 + q^64 + (2*b + 12) * q^65 + (2*b - 8) * q^67 + 2 * q^68 + (-2*b - 5) * q^71 + (-2*b + 2) * q^73 + (4*b - 2) * q^74 + (-b - 5) * q^76 + (2*b + 3) * q^79 + (b + 1) * q^80 + 4*b * q^82 + 2 * q^83 + (2*b + 2) * q^85 + (2*b - 2) * q^86 + 2 * q^88 + (2*b - 12) * q^89 + q^92 + 4*b * q^94 + (-6*b - 11) * q^95 + (2*b + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} - 4 q^{11} + 2 q^{16} + 4 q^{17} - 10 q^{19} + 2 q^{20} + 4 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{29} - 12 q^{31} - 2 q^{32} - 4 q^{34} + 4 q^{37} + 10 q^{38} - 2 q^{40} + 4 q^{43} - 4 q^{44} - 2 q^{46} - 4 q^{50} + 12 q^{53} - 4 q^{55} - 4 q^{58} - 4 q^{59} - 18 q^{61} + 12 q^{62} + 2 q^{64} + 24 q^{65} - 16 q^{67} + 4 q^{68} - 10 q^{71} + 4 q^{73} - 4 q^{74} - 10 q^{76} + 6 q^{79} + 2 q^{80} + 4 q^{83} + 4 q^{85} - 4 q^{86} + 4 q^{88} - 24 q^{89} + 2 q^{92} - 22 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 - 2 * q^10 - 4 * q^11 + 2 * q^16 + 4 * q^17 - 10 * q^19 + 2 * q^20 + 4 * q^22 + 2 * q^23 + 4 * q^25 + 4 * q^29 - 12 * q^31 - 2 * q^32 - 4 * q^34 + 4 * q^37 + 10 * q^38 - 2 * q^40 + 4 * q^43 - 4 * q^44 - 2 * q^46 - 4 * q^50 + 12 * q^53 - 4 * q^55 - 4 * q^58 - 4 * q^59 - 18 * q^61 + 12 * q^62 + 2 * q^64 + 24 * q^65 - 16 * q^67 + 4 * q^68 - 10 * q^71 + 4 * q^73 - 4 * q^74 - 10 * q^76 + 6 * q^79 + 2 * q^80 + 4 * q^83 + 4 * q^85 - 4 * q^86 + 4 * q^88 - 24 * q^89 + 2 * q^92 - 22 * q^95 + 4 * 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
 −2.44949 2.44949
−1.00000 0 1.00000 −1.44949 0 0 −1.00000 0 1.44949
1.2 −1.00000 0 1.00000 3.44949 0 0 −1.00000 0 −3.44949
 $$n$$: 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.bm 2
3.b odd 2 1 7938.2.a.bn 2
7.b odd 2 1 1134.2.a.i 2
9.c even 3 2 2646.2.f.k 4
9.d odd 6 2 882.2.f.j 4
21.c even 2 1 1134.2.a.p 2
28.d even 2 1 9072.2.a.bd 2
63.g even 3 2 2646.2.h.n 4
63.h even 3 2 2646.2.e.k 4
63.i even 6 2 882.2.e.m 4
63.j odd 6 2 882.2.e.n 4
63.k odd 6 2 2646.2.h.m 4
63.l odd 6 2 378.2.f.d 4
63.n odd 6 2 882.2.h.l 4
63.o even 6 2 126.2.f.c 4
63.s even 6 2 882.2.h.k 4
63.t odd 6 2 2646.2.e.l 4
84.h odd 2 1 9072.2.a.bk 2
252.s odd 6 2 1008.2.r.e 4
252.bi even 6 2 3024.2.r.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 63.o even 6 2
378.2.f.d 4 63.l odd 6 2
882.2.e.m 4 63.i even 6 2
882.2.e.n 4 63.j odd 6 2
882.2.f.j 4 9.d odd 6 2
882.2.h.k 4 63.s even 6 2
882.2.h.l 4 63.n odd 6 2
1008.2.r.e 4 252.s odd 6 2
1134.2.a.i 2 7.b odd 2 1
1134.2.a.p 2 21.c even 2 1
2646.2.e.k 4 63.h even 3 2
2646.2.e.l 4 63.t odd 6 2
2646.2.f.k 4 9.c even 3 2
2646.2.h.m 4 63.k odd 6 2
2646.2.h.n 4 63.g even 3 2
3024.2.r.e 4 252.bi even 6 2
7938.2.a.bm 2 1.a even 1 1 trivial
7938.2.a.bn 2 3.b odd 2 1
9072.2.a.bd 2 28.d even 2 1
9072.2.a.bk 2 84.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 2T_{5} - 5$$ T5^2 - 2*T5 - 5 $$T_{11} + 2$$ T11 + 2 $$T_{13}^{2} - 24$$ T13^2 - 24 $$T_{17} - 2$$ T17 - 2 $$T_{23} - 1$$ T23 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T - 5$$
$7$ $$T^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} - 24$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 10T + 19$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} - 4T - 20$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} - 4T - 92$$
$41$ $$T^{2} - 96$$
$43$ $$T^{2} - 4T - 20$$
$47$ $$T^{2} - 96$$
$53$ $$T^{2} - 12T + 12$$
$59$ $$(T + 2)^{2}$$
$61$ $$T^{2} + 18T + 75$$
$67$ $$T^{2} + 16T + 40$$
$71$ $$T^{2} + 10T + 1$$
$73$ $$T^{2} - 4T - 20$$
$79$ $$T^{2} - 6T - 15$$
$83$ $$(T - 2)^{2}$$
$89$ $$T^{2} + 24T + 120$$
$97$ $$T^{2} - 4T - 20$$