# Properties

 Label 2320.2.a.q 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: 3.3.469.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 4$$ x^3 - x^2 - 5*x + 4 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + q^{5} + \beta_1 q^{7} + ( - 2 \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q - b2 * q^3 + q^5 + b1 * q^7 + (-2*b2 + b1 + 1) * q^9 $$q - \beta_{2} q^{3} + q^{5} + \beta_1 q^{7} + ( - 2 \beta_{2} + \beta_1 + 1) q^{9} - 2 \beta_1 q^{11} + ( - \beta_{2} - 2 \beta_1 + 2) q^{13} - \beta_{2} q^{15} + (\beta_1 - 2) q^{17} + 2 \beta_{2} q^{19} + ( - \beta_{2} - \beta_1) q^{21} + \beta_{2} q^{23} + q^{25} + ( - 3 \beta_{2} + \beta_1 + 8) q^{27} + q^{29} + ( - 2 \beta_{2} + 3 \beta_1) q^{31} + (2 \beta_{2} + 2 \beta_1) q^{33} + \beta_1 q^{35} + (2 \beta_1 - 2) q^{37} + ( - 2 \beta_{2} + 3 \beta_1 + 4) q^{39} + (2 \beta_1 + 2) q^{41} - 5 \beta_1 q^{43} + ( - 2 \beta_{2} + \beta_1 + 1) q^{45} + 8 q^{47} + (\beta_{2} - 3) q^{49} + (\beta_{2} - \beta_1) q^{51} + ( - 3 \beta_1 + 2) q^{53} - 2 \beta_1 q^{55} + (4 \beta_{2} - 2 \beta_1 - 8) q^{57} + (3 \beta_{2} + 2 \beta_1) q^{59} + (\beta_{2} + 2) q^{61} + ( - \beta_{2} - \beta_1 + 4) q^{63} + ( - \beta_{2} - 2 \beta_1 + 2) q^{65} + 4 \beta_{2} q^{67} + (2 \beta_{2} - \beta_1 - 4) q^{69} - 4 \beta_{2} q^{71} + (4 \beta_{2} - 3 \beta_1 + 6) q^{73} - \beta_{2} q^{75} + ( - 2 \beta_{2} - 8) q^{77} + (3 \beta_{2} - 6 \beta_1) q^{79} + ( - 9 \beta_{2} - \beta_1 + 9) q^{81} + ( - 2 \beta_{2} + 8) q^{83} + (\beta_1 - 2) q^{85} - \beta_{2} q^{87} + (2 \beta_{2} + 4 \beta_1 + 2) q^{89} + ( - 3 \beta_{2} + \beta_1 - 8) q^{91} + ( - 7 \beta_{2} - \beta_1 + 8) q^{93} + 2 \beta_{2} q^{95} + ( - \beta_{2} + 2 \beta_1 + 6) q^{97} + (2 \beta_{2} + 2 \beta_1 - 8) q^{99}+O(q^{100})$$ q - b2 * q^3 + q^5 + b1 * q^7 + (-2*b2 + b1 + 1) * q^9 - 2*b1 * q^11 + (-b2 - 2*b1 + 2) * q^13 - b2 * q^15 + (b1 - 2) * q^17 + 2*b2 * q^19 + (-b2 - b1) * q^21 + b2 * q^23 + q^25 + (-3*b2 + b1 + 8) * q^27 + q^29 + (-2*b2 + 3*b1) * q^31 + (2*b2 + 2*b1) * q^33 + b1 * q^35 + (2*b1 - 2) * q^37 + (-2*b2 + 3*b1 + 4) * q^39 + (2*b1 + 2) * q^41 - 5*b1 * q^43 + (-2*b2 + b1 + 1) * q^45 + 8 * q^47 + (b2 - 3) * q^49 + (b2 - b1) * q^51 + (-3*b1 + 2) * q^53 - 2*b1 * q^55 + (4*b2 - 2*b1 - 8) * q^57 + (3*b2 + 2*b1) * q^59 + (b2 + 2) * q^61 + (-b2 - b1 + 4) * q^63 + (-b2 - 2*b1 + 2) * q^65 + 4*b2 * q^67 + (2*b2 - b1 - 4) * q^69 - 4*b2 * q^71 + (4*b2 - 3*b1 + 6) * q^73 - b2 * q^75 + (-2*b2 - 8) * q^77 + (3*b2 - 6*b1) * q^79 + (-9*b2 - b1 + 9) * q^81 + (-2*b2 + 8) * q^83 + (b1 - 2) * q^85 - b2 * q^87 + (2*b2 + 4*b1 + 2) * q^89 + (-3*b2 + b1 - 8) * q^91 + (-7*b2 - b1 + 8) * q^93 + 2*b2 * q^95 + (-b2 + 2*b1 + 6) * q^97 + (2*b2 + 2*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 3 q^{5} + q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q + q^3 + 3 * q^5 + q^7 + 6 * q^9 $$3 q + q^{3} + 3 q^{5} + q^{7} + 6 q^{9} - 2 q^{11} + 5 q^{13} + q^{15} - 5 q^{17} - 2 q^{19} - q^{23} + 3 q^{25} + 28 q^{27} + 3 q^{29} + 5 q^{31} + q^{35} - 4 q^{37} + 17 q^{39} + 8 q^{41} - 5 q^{43} + 6 q^{45} + 24 q^{47} - 10 q^{49} - 2 q^{51} + 3 q^{53} - 2 q^{55} - 30 q^{57} - q^{59} + 5 q^{61} + 12 q^{63} + 5 q^{65} - 4 q^{67} - 15 q^{69} + 4 q^{71} + 11 q^{73} + q^{75} - 22 q^{77} - 9 q^{79} + 35 q^{81} + 26 q^{83} - 5 q^{85} + q^{87} + 8 q^{89} - 20 q^{91} + 30 q^{93} - 2 q^{95} + 21 q^{97} - 24 q^{99}+O(q^{100})$$ 3 * q + q^3 + 3 * q^5 + q^7 + 6 * q^9 - 2 * q^11 + 5 * q^13 + q^15 - 5 * q^17 - 2 * q^19 - q^23 + 3 * q^25 + 28 * q^27 + 3 * q^29 + 5 * q^31 + q^35 - 4 * q^37 + 17 * q^39 + 8 * q^41 - 5 * q^43 + 6 * q^45 + 24 * q^47 - 10 * q^49 - 2 * q^51 + 3 * q^53 - 2 * q^55 - 30 * q^57 - q^59 + 5 * q^61 + 12 * q^63 + 5 * q^65 - 4 * q^67 - 15 * q^69 + 4 * q^71 + 11 * q^73 + q^75 - 22 * q^77 - 9 * q^79 + 35 * q^81 + 26 * q^83 - 5 * q^85 + q^87 + 8 * q^89 - 20 * q^91 + 30 * q^93 - 2 * q^95 + 21 * q^97 - 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 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
 2.39138 −2.16425 0.772866
0 −1.71871 0 1.00000 0 2.39138 0 −0.0460370 0
1.2 0 −0.683969 0 1.00000 0 −2.16425 0 −2.53219 0
1.3 0 3.40268 0 1.00000 0 0.772866 0 8.57822 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.q 3
4.b odd 2 1 290.2.a.d 3
8.b even 2 1 9280.2.a.bn 3
8.d odd 2 1 9280.2.a.bp 3
12.b even 2 1 2610.2.a.w 3
20.d odd 2 1 1450.2.a.r 3
20.e even 4 2 1450.2.b.j 6
116.d odd 2 1 8410.2.a.w 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.a.d 3 4.b odd 2 1
1450.2.a.r 3 20.d odd 2 1
1450.2.b.j 6 20.e even 4 2
2320.2.a.q 3 1.a even 1 1 trivial
2610.2.a.w 3 12.b even 2 1
8410.2.a.w 3 116.d odd 2 1
9280.2.a.bn 3 8.b even 2 1
9280.2.a.bp 3 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2320))$$:

 $$T_{3}^{3} - T_{3}^{2} - 7T_{3} - 4$$ T3^3 - T3^2 - 7*T3 - 4 $$T_{7}^{3} - T_{7}^{2} - 5T_{7} + 4$$ T7^3 - T7^2 - 5*T7 + 4 $$T_{11}^{3} + 2T_{11}^{2} - 20T_{11} - 32$$ T11^3 + 2*T11^2 - 20*T11 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 7T - 4$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - T^{2} - 5T + 4$$
$11$ $$T^{3} + 2 T^{2} + \cdots - 32$$
$13$ $$T^{3} - 5 T^{2} + \cdots + 98$$
$17$ $$T^{3} + 5 T^{2} + \cdots - 2$$
$19$ $$T^{3} + 2 T^{2} + \cdots + 32$$
$23$ $$T^{3} + T^{2} - 7T + 4$$
$29$ $$(T - 1)^{3}$$
$31$ $$T^{3} - 5 T^{2} + \cdots + 268$$
$37$ $$T^{3} + 4 T^{2} + \cdots - 8$$
$41$ $$T^{3} - 8T^{2} + 56$$
$43$ $$T^{3} + 5 T^{2} + \cdots - 500$$
$47$ $$(T - 8)^{3}$$
$53$ $$T^{3} - 3 T^{2} + \cdots - 14$$
$59$ $$T^{3} + T^{2} + \cdots - 196$$
$61$ $$T^{3} - 5T^{2} + T + 14$$
$67$ $$T^{3} + 4 T^{2} + \cdots + 256$$
$71$ $$T^{3} - 4 T^{2} + \cdots - 256$$
$73$ $$T^{3} - 11 T^{2} + \cdots + 862$$
$79$ $$T^{3} + 9 T^{2} + \cdots - 2052$$
$83$ $$T^{3} - 26 T^{2} + \cdots - 448$$
$89$ $$T^{3} - 8 T^{2} + \cdots - 136$$
$97$ $$T^{3} - 21 T^{2} + \cdots - 98$$