# Properties

 Label 2320.2.a.l Level $2320$ Weight $2$ Character orbit 2320.a Self dual yes Analytic conductor $18.525$ 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] = [2320,2,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, 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("2320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2320 = 2^{4} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2320.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: $$18.5252932689$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 290) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{3} - q^{5} + (\beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^3 - q^5 + (b2 - b1 - 1) * q^7 + (b2 - b1 + 2) * q^9 $$q + (\beta_1 - 1) q^{3} - q^{5} + (\beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} - 2 \beta_1 q^{11} + (2 \beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_1 + 1) q^{15} + (\beta_{2} + \beta_1 + 1) q^{17} + (2 \beta_{2} + 2 \beta_1) q^{19} + ( - 3 \beta_{2} - 4) q^{21} + (2 \beta_{2} + \beta_1 + 5) q^{23} + q^{25} + ( - 3 \beta_{2} - 4) q^{27} - q^{29} + ( - \beta_{2} - \beta_1 + 3) q^{31} + ( - 2 \beta_{2} - 8) q^{33} + ( - \beta_{2} + \beta_1 + 1) q^{35} + ( - 2 \beta_{2} + 2 \beta_1) q^{37} + ( - 3 \beta_{2} + 3 \beta_1 + 1) q^{39} + (2 \beta_{2} - 2 \beta_1 - 4) q^{41} + ( - \beta_{2} - \beta_1 - 3) q^{43} + ( - \beta_{2} + \beta_1 - 2) q^{45} + 4 \beta_1 q^{47} + ( - \beta_1 + 6) q^{49} + ( - \beta_{2} + 2 \beta_1 + 2) q^{51} + (3 \beta_{2} + 3 \beta_1 + 3) q^{53} + 2 \beta_1 q^{55} + ( - 2 \beta_{2} + 2 \beta_1 + 6) q^{57} + (2 \beta_{2} + 3 \beta_1 + 5) q^{59} + ( - 3 \beta_1 + 5) q^{61} + (3 \beta_{2} - 4 \beta_1 + 10) q^{63} + ( - 2 \beta_{2} - \beta_1 - 1) q^{65} + (2 \beta_{2} + 4 \beta_1 - 6) q^{67} + ( - 3 \beta_{2} + 7 \beta_1 - 3) q^{69} + 12 q^{71} + (\beta_{2} + \beta_1 - 3) q^{73} + (\beta_1 - 1) q^{75} + (4 \beta_{2} + 2 \beta_1 + 10) q^{77} + ( - 4 \beta_{2} - 3 \beta_1 - 3) q^{79} + (3 \beta_{2} - 4 \beta_1 + 1) q^{81} + (2 \beta_{2} + 2 \beta_1 - 4) q^{83} + ( - \beta_{2} - \beta_1 - 1) q^{85} + ( - \beta_1 + 1) q^{87} + 2 \beta_1 q^{89} + ( - 3 \beta_{2} - 8 \beta_1 + 8) q^{91} + (\beta_{2} + 2 \beta_1 - 6) q^{93} + ( - 2 \beta_{2} - 2 \beta_1) q^{95} + ( - 2 \beta_{2} - \beta_1 + 3) q^{97} + (4 \beta_{2} - 4 \beta_1 + 10) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^3 - q^5 + (b2 - b1 - 1) * q^7 + (b2 - b1 + 2) * q^9 - 2*b1 * q^11 + (2*b2 + b1 + 1) * q^13 + (-b1 + 1) * q^15 + (b2 + b1 + 1) * q^17 + (2*b2 + 2*b1) * q^19 + (-3*b2 - 4) * q^21 + (2*b2 + b1 + 5) * q^23 + q^25 + (-3*b2 - 4) * q^27 - q^29 + (-b2 - b1 + 3) * q^31 + (-2*b2 - 8) * q^33 + (-b2 + b1 + 1) * q^35 + (-2*b2 + 2*b1) * q^37 + (-3*b2 + 3*b1 + 1) * q^39 + (2*b2 - 2*b1 - 4) * q^41 + (-b2 - b1 - 3) * q^43 + (-b2 + b1 - 2) * q^45 + 4*b1 * q^47 + (-b1 + 6) * q^49 + (-b2 + 2*b1 + 2) * q^51 + (3*b2 + 3*b1 + 3) * q^53 + 2*b1 * q^55 + (-2*b2 + 2*b1 + 6) * q^57 + (2*b2 + 3*b1 + 5) * q^59 + (-3*b1 + 5) * q^61 + (3*b2 - 4*b1 + 10) * q^63 + (-2*b2 - b1 - 1) * q^65 + (2*b2 + 4*b1 - 6) * q^67 + (-3*b2 + 7*b1 - 3) * q^69 + 12 * q^71 + (b2 + b1 - 3) * q^73 + (b1 - 1) * q^75 + (4*b2 + 2*b1 + 10) * q^77 + (-4*b2 - 3*b1 - 3) * q^79 + (3*b2 - 4*b1 + 1) * q^81 + (2*b2 + 2*b1 - 4) * q^83 + (-b2 - b1 - 1) * q^85 + (-b1 + 1) * q^87 + 2*b1 * q^89 + (-3*b2 - 8*b1 + 8) * q^91 + (b2 + 2*b1 - 6) * q^93 + (-2*b2 - 2*b1) * q^95 + (-2*b2 - b1 + 3) * q^97 + (4*b2 - 4*b1 + 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 $$3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 6 q^{9} + 3 q^{13} + 3 q^{15} + 3 q^{17} - 12 q^{21} + 15 q^{23} + 3 q^{25} - 12 q^{27} - 3 q^{29} + 9 q^{31} - 24 q^{33} + 3 q^{35} + 3 q^{39} - 12 q^{41} - 9 q^{43} - 6 q^{45} + 18 q^{49} + 6 q^{51} + 9 q^{53} + 18 q^{57} + 15 q^{59} + 15 q^{61} + 30 q^{63} - 3 q^{65} - 18 q^{67} - 9 q^{69} + 36 q^{71} - 9 q^{73} - 3 q^{75} + 30 q^{77} - 9 q^{79} + 3 q^{81} - 12 q^{83} - 3 q^{85} + 3 q^{87} + 24 q^{91} - 18 q^{93} + 9 q^{97} + 30 q^{99}+O(q^{100})$$ 3 * q - 3 * q^3 - 3 * q^5 - 3 * q^7 + 6 * q^9 + 3 * q^13 + 3 * q^15 + 3 * q^17 - 12 * q^21 + 15 * q^23 + 3 * q^25 - 12 * q^27 - 3 * q^29 + 9 * q^31 - 24 * q^33 + 3 * q^35 + 3 * q^39 - 12 * q^41 - 9 * q^43 - 6 * q^45 + 18 * q^49 + 6 * q^51 + 9 * q^53 + 18 * q^57 + 15 * q^59 + 15 * q^61 + 30 * q^63 - 3 * q^65 - 18 * q^67 - 9 * q^69 + 36 * q^71 - 9 * q^73 - 3 * q^75 + 30 * q^77 - 9 * q^79 + 3 * q^81 - 12 * q^83 - 3 * q^85 + 3 * q^87 + 24 * q^91 - 18 * q^93 + 9 * q^97 + 30 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4

## 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
 −2.14510 −0.523976 2.66908
0 −3.14510 0 −1.00000 0 3.89167 0 6.89167 0
1.2 0 −1.52398 0 −1.00000 0 −3.67750 0 −0.677496 0
1.3 0 1.66908 0 −1.00000 0 −3.21417 0 −0.214175 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$29$$ $$1$$

## 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 2320.2.a.l 3
4.b odd 2 1 290.2.a.e 3
8.b even 2 1 9280.2.a.by 3
8.d odd 2 1 9280.2.a.bf 3
12.b even 2 1 2610.2.a.x 3
20.d odd 2 1 1450.2.a.p 3
20.e even 4 2 1450.2.b.l 6
116.d odd 2 1 8410.2.a.v 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.a.e 3 4.b odd 2 1
1450.2.a.p 3 20.d odd 2 1
1450.2.b.l 6 20.e even 4 2
2320.2.a.l 3 1.a even 1 1 trivial
2610.2.a.x 3 12.b even 2 1
8410.2.a.v 3 116.d odd 2 1
9280.2.a.bf 3 8.d odd 2 1
9280.2.a.by 3 8.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2320))$$:

 $$T_{3}^{3} + 3T_{3}^{2} - 3T_{3} - 8$$ T3^3 + 3*T3^2 - 3*T3 - 8 $$T_{7}^{3} + 3T_{7}^{2} - 15T_{7} - 46$$ T7^3 + 3*T7^2 - 15*T7 - 46 $$T_{11}^{3} - 24T_{11} + 24$$ T11^3 - 24*T11 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 3 T^{2} + \cdots - 8$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} + 3 T^{2} + \cdots - 46$$
$11$ $$T^{3} - 24T + 24$$
$13$ $$T^{3} - 3 T^{2} + \cdots + 118$$
$17$ $$T^{3} - 3 T^{2} + \cdots + 18$$
$19$ $$T^{3} - 48T + 56$$
$23$ $$T^{3} - 15 T^{2} + \cdots + 138$$
$29$ $$(T + 1)^{3}$$
$31$ $$T^{3} - 9 T^{2} + \cdots + 2$$
$37$ $$T^{3} - 72T + 232$$
$41$ $$T^{3} + 12 T^{2} + \cdots - 456$$
$43$ $$T^{3} + 9 T^{2} + \cdots - 16$$
$47$ $$T^{3} - 96T - 192$$
$53$ $$T^{3} - 9 T^{2} + \cdots + 486$$
$59$ $$T^{3} - 15 T^{2} + \cdots + 168$$
$61$ $$T^{3} - 15 T^{2} + \cdots + 226$$
$67$ $$T^{3} + 18T^{2} - 736$$
$71$ $$(T - 12)^{3}$$
$73$ $$T^{3} + 9 T^{2} + \cdots - 2$$
$79$ $$T^{3} + 9 T^{2} + \cdots - 1102$$
$83$ $$T^{3} + 12T^{2} - 72$$
$89$ $$T^{3} - 24T - 24$$
$97$ $$T^{3} - 9 T^{2} + \cdots - 2$$