# Properties

 Label 1520.2.a.s Level $1520$ Weight $2$ Character orbit 1520.a Self dual yes Analytic conductor $12.137$ Analytic rank $0$ 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] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.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: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 760) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + q^{5} - \beta_1 q^{7} + ( - \beta_1 + 1) q^{9}+O(q^{10})$$ q - b2 * q^3 + q^5 - b1 * q^7 + (-b1 + 1) * q^9 $$q - \beta_{2} q^{3} + q^{5} - \beta_1 q^{7} + ( - \beta_1 + 1) q^{9} + (\beta_{2} + 2) q^{13} - \beta_{2} q^{15} + (\beta_1 + 2) q^{17} - q^{19} + ( - 2 \beta_{2} - \beta_1) q^{21} + ( - 2 \beta_{2} + \beta_1) q^{23} + q^{25} - \beta_1 q^{27} + (\beta_1 + 6) q^{29} + 2 \beta_1 q^{31} - \beta_1 q^{35} + ( - \beta_{2} - \beta_1 + 2) q^{37} + ( - 2 \beta_{2} + \beta_1 - 4) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{41} + (2 \beta_{2} + 4) q^{43} + ( - \beta_1 + 1) q^{45} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 2 \beta_{2} + \beta_1 + 1) q^{49} + \beta_1 q^{51} + ( - \beta_{2} + 2 \beta_1 + 2) q^{53} + \beta_{2} q^{57} + (4 \beta_{2} - \beta_1 - 4) q^{59} + ( - 2 \beta_{2} + 2) q^{61} + ( - 2 \beta_{2} + 8) q^{63} + (\beta_{2} + 2) q^{65} + (5 \beta_{2} - 2 \beta_1) q^{67} + (2 \beta_{2} - \beta_1 + 8) q^{69} - 4 \beta_1 q^{71} + (4 \beta_{2} - \beta_1 + 2) q^{73} - \beta_{2} q^{75} + 2 \beta_{2} q^{79} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{81} + (2 \beta_1 + 4) q^{83} + (\beta_1 + 2) q^{85} + ( - 4 \beta_{2} + \beta_1) q^{87} + (2 \beta_{2} - 2 \beta_1 + 2) q^{89} + (2 \beta_{2} - \beta_1) q^{91} + (4 \beta_{2} + 2 \beta_1) q^{93} - q^{95} + (\beta_{2} - 3 \beta_1 - 2) q^{97}+O(q^{100})$$ q - b2 * q^3 + q^5 - b1 * q^7 + (-b1 + 1) * q^9 + (b2 + 2) * q^13 - b2 * q^15 + (b1 + 2) * q^17 - q^19 + (-2*b2 - b1) * q^21 + (-2*b2 + b1) * q^23 + q^25 - b1 * q^27 + (b1 + 6) * q^29 + 2*b1 * q^31 - b1 * q^35 + (-b2 - b1 + 2) * q^37 + (-2*b2 + b1 - 4) * q^39 + (-2*b2 + 2*b1 + 2) * q^41 + (2*b2 + 4) * q^43 + (-b1 + 1) * q^45 + (2*b2 + 2*b1) * q^47 + (-2*b2 + b1 + 1) * q^49 + b1 * q^51 + (-b2 + 2*b1 + 2) * q^53 + b2 * q^57 + (4*b2 - b1 - 4) * q^59 + (-2*b2 + 2) * q^61 + (-2*b2 + 8) * q^63 + (b2 + 2) * q^65 + (5*b2 - 2*b1) * q^67 + (2*b2 - b1 + 8) * q^69 - 4*b1 * q^71 + (4*b2 - b1 + 2) * q^73 - b2 * q^75 + 2*b2 * q^79 + (-2*b2 + 2*b1 - 3) * q^81 + (2*b1 + 4) * q^83 + (b1 + 2) * q^85 + (-4*b2 + b1) * q^87 + (2*b2 - 2*b1 + 2) * q^89 + (2*b2 - b1) * q^91 + (4*b2 + 2*b1) * q^93 - q^95 + (b2 - 3*b1 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9}+O(q^{10})$$ 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9 $$3 q + q^{3} + 3 q^{5} + q^{7} + 4 q^{9} + 5 q^{13} + q^{15} + 5 q^{17} - 3 q^{19} + 3 q^{21} + q^{23} + 3 q^{25} + q^{27} + 17 q^{29} - 2 q^{31} + q^{35} + 8 q^{37} - 11 q^{39} + 6 q^{41} + 10 q^{43} + 4 q^{45} - 4 q^{47} + 4 q^{49} - q^{51} + 5 q^{53} - q^{57} - 15 q^{59} + 8 q^{61} + 26 q^{63} + 5 q^{65} - 3 q^{67} + 23 q^{69} + 4 q^{71} + 3 q^{73} + q^{75} - 2 q^{79} - 9 q^{81} + 10 q^{83} + 5 q^{85} + 3 q^{87} + 6 q^{89} - q^{91} - 6 q^{93} - 3 q^{95} - 4 q^{97}+O(q^{100})$$ 3 * q + q^3 + 3 * q^5 + q^7 + 4 * q^9 + 5 * q^13 + q^15 + 5 * q^17 - 3 * q^19 + 3 * q^21 + q^23 + 3 * q^25 + q^27 + 17 * q^29 - 2 * q^31 + q^35 + 8 * q^37 - 11 * q^39 + 6 * q^41 + 10 * q^43 + 4 * q^45 - 4 * q^47 + 4 * q^49 - q^51 + 5 * q^53 - q^57 - 15 * q^59 + 8 * q^61 + 26 * q^63 + 5 * q^65 - 3 * q^67 + 23 * q^69 + 4 * q^71 + 3 * q^73 + q^75 - 2 * q^79 - 9 * q^81 + 10 * q^83 + 5 * q^85 + 3 * q^87 + 6 * q^89 - q^91 - 6 * q^93 - 3 * q^95 - 4 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ v^2 + v - 3 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 2$$ (-b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 6 ) / 2$$ (b2 + b1 + 6) / 2

## 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
 −1.86081 2.11491 −0.254102
0 −2.32340 0 1.00000 0 1.39821 0 2.39821 0
1.2 0 0.642074 0 1.00000 0 −3.58774 0 −2.58774 0
1.3 0 2.68133 0 1.00000 0 3.18953 0 4.18953 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
$$2$$ $$+1$$
$$5$$ $$-1$$
$$19$$ $$+1$$

## 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 1520.2.a.s 3
4.b odd 2 1 760.2.a.j 3
5.b even 2 1 7600.2.a.bq 3
8.b even 2 1 6080.2.a.bq 3
8.d odd 2 1 6080.2.a.bv 3
12.b even 2 1 6840.2.a.bg 3
20.d odd 2 1 3800.2.a.x 3
20.e even 4 2 3800.2.d.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.j 3 4.b odd 2 1
1520.2.a.s 3 1.a even 1 1 trivial
3800.2.a.x 3 20.d odd 2 1
3800.2.d.l 6 20.e even 4 2
6080.2.a.bq 3 8.b even 2 1
6080.2.a.bv 3 8.d odd 2 1
6840.2.a.bg 3 12.b even 2 1
7600.2.a.bq 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1520))$$:

 $$T_{3}^{3} - T_{3}^{2} - 6T_{3} + 4$$ T3^3 - T3^2 - 6*T3 + 4 $$T_{7}^{3} - T_{7}^{2} - 12T_{7} + 16$$ T7^3 - T7^2 - 12*T7 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 6T + 4$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - T^{2} + \cdots + 16$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 5 T^{2} + \cdots + 4$$
$17$ $$T^{3} - 5 T^{2} + \cdots + 4$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} - T^{2} + \cdots + 64$$
$29$ $$T^{3} - 17 T^{2} + \cdots - 124$$
$31$ $$T^{3} + 2 T^{2} + \cdots - 128$$
$37$ $$T^{3} - 8T^{2} + 8$$
$41$ $$T^{3} - 6 T^{2} + \cdots + 56$$
$43$ $$T^{3} - 10 T^{2} + \cdots + 32$$
$47$ $$T^{3} + 4 T^{2} + \cdots + 128$$
$53$ $$T^{3} - 5 T^{2} + \cdots - 52$$
$59$ $$T^{3} + 15 T^{2} + \cdots - 784$$
$61$ $$T^{3} - 8 T^{2} + \cdots + 64$$
$67$ $$T^{3} + 3 T^{2} + \cdots - 1052$$
$71$ $$T^{3} - 4 T^{2} + \cdots + 1024$$
$73$ $$T^{3} - 3 T^{2} + \cdots - 292$$
$79$ $$T^{3} + 2 T^{2} + \cdots - 32$$
$83$ $$T^{3} - 10 T^{2} + \cdots + 32$$
$89$ $$T^{3} - 6 T^{2} + \cdots + 184$$
$97$ $$T^{3} + 4 T^{2} + \cdots + 296$$