# Properties

 Label 3192.2.a.q Level $3192$ Weight $2$ Character orbit 3192.a Self dual yes Analytic conductor $25.488$ 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] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3192.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: $$25.4882483252$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + (\beta + 1) q^{5} + q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (b + 1) * q^5 + q^7 + q^9 $$q - q^{3} + (\beta + 1) q^{5} + q^{7} + q^{9} + ( - 2 \beta - 2) q^{13} + ( - \beta - 1) q^{15} + (\beta - 3) q^{17} - q^{19} - q^{21} + (2 \beta - 1) q^{25} - q^{27} + ( - 5 \beta - 1) q^{29} + 2 q^{31} + (\beta + 1) q^{35} - 10 q^{37} + (2 \beta + 2) q^{39} + ( - 4 \beta - 2) q^{41} + (4 \beta - 2) q^{43} + (\beta + 1) q^{45} + ( - 5 \beta - 3) q^{47} + q^{49} + ( - \beta + 3) q^{51} + ( - \beta - 5) q^{53} + q^{57} + 8 q^{59} - 2 q^{61} + q^{63} + ( - 4 \beta - 8) q^{65} + ( - 2 \beta - 10) q^{67} + (\beta + 3) q^{71} + 6 \beta q^{73} + ( - 2 \beta + 1) q^{75} + 4 q^{79} + q^{81} + (3 \beta + 5) q^{83} - 2 \beta q^{85} + (5 \beta + 1) q^{87} + (4 \beta - 2) q^{89} + ( - 2 \beta - 2) q^{91} - 2 q^{93} + ( - \beta - 1) q^{95} + 2 q^{97} +O(q^{100})$$ q - q^3 + (b + 1) * q^5 + q^7 + q^9 + (-2*b - 2) * q^13 + (-b - 1) * q^15 + (b - 3) * q^17 - q^19 - q^21 + (2*b - 1) * q^25 - q^27 + (-5*b - 1) * q^29 + 2 * q^31 + (b + 1) * q^35 - 10 * q^37 + (2*b + 2) * q^39 + (-4*b - 2) * q^41 + (4*b - 2) * q^43 + (b + 1) * q^45 + (-5*b - 3) * q^47 + q^49 + (-b + 3) * q^51 + (-b - 5) * q^53 + q^57 + 8 * q^59 - 2 * q^61 + q^63 + (-4*b - 8) * q^65 + (-2*b - 10) * q^67 + (b + 3) * q^71 + 6*b * q^73 + (-2*b + 1) * q^75 + 4 * q^79 + q^81 + (3*b + 5) * q^83 - 2*b * q^85 + (5*b + 1) * q^87 + (4*b - 2) * q^89 + (-2*b - 2) * q^91 - 2 * q^93 + (-b - 1) * q^95 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 2 q^{35} - 20 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45} - 6 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 2 q^{57} + 16 q^{59} - 4 q^{61} + 2 q^{63} - 16 q^{65} - 20 q^{67} + 6 q^{71} + 2 q^{75} + 8 q^{79} + 2 q^{81} + 10 q^{83} + 2 q^{87} - 4 q^{89} - 4 q^{91} - 4 q^{93} - 2 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 + 2 * q^35 - 20 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 + 2 * q^45 - 6 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 2 * q^63 - 16 * q^65 - 20 * q^67 + 6 * q^71 + 2 * q^75 + 8 * q^79 + 2 * q^81 + 10 * q^83 + 2 * q^87 - 4 * q^89 - 4 * q^91 - 4 * q^93 - 2 * 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
 −1.73205 1.73205
0 −1.00000 0 −0.732051 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 2.73205 0 1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$19$$ $$1$$

## 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 3192.2.a.q 2
3.b odd 2 1 9576.2.a.bj 2
4.b odd 2 1 6384.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.q 2 1.a even 1 1 trivial
6384.2.a.bs 2 4.b odd 2 1
9576.2.a.bj 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3192))$$:

 $$T_{5}^{2} - 2T_{5} - 2$$ T5^2 - 2*T5 - 2 $$T_{11}$$ T11 $$T_{17}^{2} + 6T_{17} + 6$$ T17^2 + 6*T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 2T - 2$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4T - 8$$
$17$ $$T^{2} + 6T + 6$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 2T - 74$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} + 4T - 44$$
$43$ $$T^{2} + 4T - 44$$
$47$ $$T^{2} + 6T - 66$$
$53$ $$T^{2} + 10T + 22$$
$59$ $$(T - 8)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 20T + 88$$
$71$ $$T^{2} - 6T + 6$$
$73$ $$T^{2} - 108$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} - 10T - 2$$
$89$ $$T^{2} + 4T - 44$$
$97$ $$(T - 2)^{2}$$