# Properties

 Label 1520.2.a.q Level $1520$ Weight $2$ Character orbit 1520.a Self dual yes Analytic conductor $12.137$ 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] = [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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - q^{5} + (\beta_{2} - 2 \beta_1 + 1) q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 - q^5 + (b2 - 2*b1 + 1) * q^7 + b2 * q^9 $$q + \beta_1 q^{3} - q^{5} + (\beta_{2} - 2 \beta_1 + 1) q^{7} + \beta_{2} q^{9} - 2 \beta_{2} q^{11} + ( - 2 \beta_{2} + \beta_1 - 4) q^{13} - \beta_1 q^{15} + (\beta_{2} - 1) q^{17} - q^{19} + ( - \beta_{2} + 2 \beta_1 - 5) q^{21} + ( - \beta_{2} + 3) q^{23} + q^{25} + (\beta_{2} - 2 \beta_1 + 1) q^{27} + (\beta_{2} - 4 \beta_1 - 1) q^{29} + (2 \beta_{2} - 2) q^{31} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{33} + ( - \beta_{2} + 2 \beta_1 - 1) q^{35} + ( - \beta_{2} + \beta_1 - 7) q^{37} + ( - \beta_{2} - 6 \beta_1 + 1) q^{39} + (2 \beta_{2} + 2 \beta_1 - 8) q^{41} + ( - 2 \beta_{2} + 4 \beta_1 + 2) q^{43} - \beta_{2} q^{45} + 4 \beta_{2} q^{47} + (\beta_{2} - 6 \beta_1 + 6) q^{49} + (\beta_{2} + 1) q^{51} + (2 \beta_{2} - 7 \beta_1) q^{53} + 2 \beta_{2} q^{55} - \beta_1 q^{57} + (3 \beta_{2} + 4 \beta_1 - 5) q^{59} + (2 \beta_{2} + 2 \beta_1 - 6) q^{61} + ( - 2 \beta_{2} + 2) q^{63} + (2 \beta_{2} - \beta_1 + 4) q^{65} + ( - 4 \beta_{2} + \beta_1) q^{67} + ( - \beta_{2} + 2 \beta_1 - 1) q^{69} + (3 \beta_{2} + 4 \beta_1 - 3) q^{73} + \beta_1 q^{75} + (4 \beta_{2} - 4) q^{77} + ( - 2 \beta_1 - 8) q^{79} + ( - 4 \beta_{2} + 2 \beta_1 - 5) q^{81} + (2 \beta_1 + 4) q^{83} + ( - \beta_{2} + 1) q^{85} + ( - 3 \beta_{2} - 11) q^{87} + ( - 6 \beta_{2} + 6 \beta_1 - 4) q^{89} + ( - \beta_{2} + 10 \beta_1 - 13) q^{91} + (2 \beta_{2} + 2) q^{93} + q^{95} + ( - 3 \beta_{2} - \beta_1 + 3) q^{97} + (2 \beta_{2} - 4 \beta_1 - 8) q^{99}+O(q^{100})$$ q + b1 * q^3 - q^5 + (b2 - 2*b1 + 1) * q^7 + b2 * q^9 - 2*b2 * q^11 + (-2*b2 + b1 - 4) * q^13 - b1 * q^15 + (b2 - 1) * q^17 - q^19 + (-b2 + 2*b1 - 5) * q^21 + (-b2 + 3) * q^23 + q^25 + (b2 - 2*b1 + 1) * q^27 + (b2 - 4*b1 - 1) * q^29 + (2*b2 - 2) * q^31 + (-2*b2 - 2*b1 - 2) * q^33 + (-b2 + 2*b1 - 1) * q^35 + (-b2 + b1 - 7) * q^37 + (-b2 - 6*b1 + 1) * q^39 + (2*b2 + 2*b1 - 8) * q^41 + (-2*b2 + 4*b1 + 2) * q^43 - b2 * q^45 + 4*b2 * q^47 + (b2 - 6*b1 + 6) * q^49 + (b2 + 1) * q^51 + (2*b2 - 7*b1) * q^53 + 2*b2 * q^55 - b1 * q^57 + (3*b2 + 4*b1 - 5) * q^59 + (2*b2 + 2*b1 - 6) * q^61 + (-2*b2 + 2) * q^63 + (2*b2 - b1 + 4) * q^65 + (-4*b2 + b1) * q^67 + (-b2 + 2*b1 - 1) * q^69 + (3*b2 + 4*b1 - 3) * q^73 + b1 * q^75 + (4*b2 - 4) * q^77 + (-2*b1 - 8) * q^79 + (-4*b2 + 2*b1 - 5) * q^81 + (2*b1 + 4) * q^83 + (-b2 + 1) * q^85 + (-3*b2 - 11) * q^87 + (-6*b2 + 6*b1 - 4) * q^89 + (-b2 + 10*b1 - 13) * q^91 + (2*b2 + 2) * q^93 + q^95 + (-3*b2 - b1 + 3) * q^97 + (2*b2 - 4*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 3 q^{5} + q^{7}+O(q^{10})$$ 3 * q + q^3 - 3 * q^5 + q^7 $$3 q + q^{3} - 3 q^{5} + q^{7} - 11 q^{13} - q^{15} - 3 q^{17} - 3 q^{19} - 13 q^{21} + 9 q^{23} + 3 q^{25} + q^{27} - 7 q^{29} - 6 q^{31} - 8 q^{33} - q^{35} - 20 q^{37} - 3 q^{39} - 22 q^{41} + 10 q^{43} + 12 q^{49} + 3 q^{51} - 7 q^{53} - q^{57} - 11 q^{59} - 16 q^{61} + 6 q^{63} + 11 q^{65} + q^{67} - q^{69} - 5 q^{73} + q^{75} - 12 q^{77} - 26 q^{79} - 13 q^{81} + 14 q^{83} + 3 q^{85} - 33 q^{87} - 6 q^{89} - 29 q^{91} + 6 q^{93} + 3 q^{95} + 8 q^{97} - 28 q^{99}+O(q^{100})$$ 3 * q + q^3 - 3 * q^5 + q^7 - 11 * q^13 - q^15 - 3 * q^17 - 3 * q^19 - 13 * q^21 + 9 * q^23 + 3 * q^25 + q^27 - 7 * q^29 - 6 * q^31 - 8 * q^33 - q^35 - 20 * q^37 - 3 * q^39 - 22 * q^41 + 10 * q^43 + 12 * q^49 + 3 * q^51 - 7 * q^53 - q^57 - 11 * q^59 - 16 * q^61 + 6 * q^63 + 11 * q^65 + q^67 - q^69 - 5 * q^73 + q^75 - 12 * q^77 - 26 * q^79 - 13 * q^81 + 14 * q^83 + 3 * q^85 - 33 * q^87 - 6 * q^89 - 29 * q^91 + 6 * q^93 + 3 * q^95 + 8 * q^97 - 28 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 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
 −1.81361 0.470683 2.34292
0 −1.81361 0 −1.00000 0 4.91638 0 0.289169 0
1.2 0 0.470683 0 −1.00000 0 −2.71982 0 −2.77846 0
1.3 0 2.34292 0 −1.00000 0 −1.19656 0 2.48929 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.q 3
4.b odd 2 1 760.2.a.i 3
5.b even 2 1 7600.2.a.bp 3
8.b even 2 1 6080.2.a.br 3
8.d odd 2 1 6080.2.a.bx 3
12.b even 2 1 6840.2.a.bm 3
20.d odd 2 1 3800.2.a.w 3
20.e even 4 2 3800.2.d.n 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.i 3 4.b odd 2 1
1520.2.a.q 3 1.a even 1 1 trivial
3800.2.a.w 3 20.d odd 2 1
3800.2.d.n 6 20.e even 4 2
6080.2.a.br 3 8.b even 2 1
6080.2.a.bx 3 8.d odd 2 1
6840.2.a.bm 3 12.b even 2 1
7600.2.a.bp 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1520))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 4T + 2$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - T^{2} + \cdots - 16$$
$11$ $$T^{3} - 28T - 16$$
$13$ $$T^{3} + 11 T^{2} + \cdots - 86$$
$17$ $$T^{3} + 3 T^{2} + \cdots - 4$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} - 9 T^{2} + \cdots - 8$$
$29$ $$T^{3} + 7 T^{2} + \cdots - 292$$
$31$ $$T^{3} + 6 T^{2} + \cdots - 32$$
$37$ $$T^{3} + 20 T^{2} + \cdots + 244$$
$41$ $$T^{3} + 22 T^{2} + \cdots - 232$$
$43$ $$T^{3} - 10 T^{2} + \cdots + 352$$
$47$ $$T^{3} - 112T + 128$$
$53$ $$T^{3} + 7 T^{2} + \cdots - 1342$$
$59$ $$T^{3} + 11 T^{2} + \cdots - 1544$$
$61$ $$T^{3} + 16 T^{2} + \cdots - 352$$
$67$ $$T^{3} - T^{2} + \cdots - 262$$
$71$ $$T^{3}$$
$73$ $$T^{3} + 5 T^{2} + \cdots - 1228$$
$79$ $$T^{3} + 26 T^{2} + \cdots + 496$$
$83$ $$T^{3} - 14 T^{2} + \cdots - 16$$
$89$ $$T^{3} + 6 T^{2} + \cdots - 1256$$
$97$ $$T^{3} - 8 T^{2} + \cdots + 292$$