# Properties

 Label 2320.2.a.t 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: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1160) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} + q^{5} + (\beta_1 + 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 + q^5 + (b1 + 1) * q^7 + (b2 - b1 + 1) * q^9 $$q + ( - \beta_1 + 1) q^{3} + q^{5} + (\beta_1 + 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + (\beta_{2} + 3) q^{11} + ( - \beta_{2} - \beta_1) q^{13} + ( - \beta_1 + 1) q^{15} + (\beta_{2} + 1) q^{17} + (\beta_1 + 3) q^{19} + ( - \beta_{2} - \beta_1 - 2) q^{21} + ( - \beta_{2} + 2 \beta_1 + 1) q^{23} + q^{25} + (2 \beta_{2} + 2) q^{27} - q^{29} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{31} + (\beta_{2} - 5 \beta_1 + 4) q^{33} + (\beta_1 + 1) q^{35} + (\beta_{2} + 2 \beta_1 - 1) q^{37} + (2 \beta_1 + 2) q^{39} + ( - 3 \beta_{2} - \beta_1 - 2) q^{41} + (\beta_1 + 3) q^{43} + (\beta_{2} - \beta_1 + 1) q^{45} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{47} + (\beta_{2} + 3 \beta_1 - 3) q^{49} + (\beta_{2} - 3 \beta_1 + 2) q^{51} + ( - \beta_{2} + \beta_1 + 2) q^{53} + (\beta_{2} + 3) q^{55} + ( - \beta_{2} - 3 \beta_1) q^{57} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{59} + (\beta_{2} + 5 \beta_1 - 4) q^{61} + (\beta_1 - 3) q^{63} + ( - \beta_{2} - \beta_1) q^{65} + (\beta_{2} - 4 \beta_1 + 1) q^{67} + ( - 3 \beta_{2} + \beta_1 - 6) q^{69} + ( - \beta_{2} - 3 \beta_1 + 4) q^{71} + (2 \beta_{2} - \beta_1 + 5) q^{73} + ( - \beta_1 + 1) q^{75} + (\beta_{2} + 5 \beta_1 + 2) q^{77} + (\beta_{2} - 2 \beta_1 + 9) q^{79} + ( - \beta_{2} - 3 \beta_1 + 1) q^{81} + (2 \beta_{2} - \beta_1 + 1) q^{83} + (\beta_{2} + 1) q^{85} + (\beta_1 - 1) q^{87} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{89} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{91} + (\beta_{2} + 3 \beta_1 + 8) q^{93} + (\beta_1 + 3) q^{95} + ( - 2 \beta_{2} + \beta_1 - 5) q^{97} + (3 \beta_{2} - 6 \beta_1 + 11) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 + q^5 + (b1 + 1) * q^7 + (b2 - b1 + 1) * q^9 + (b2 + 3) * q^11 + (-b2 - b1) * q^13 + (-b1 + 1) * q^15 + (b2 + 1) * q^17 + (b1 + 3) * q^19 + (-b2 - b1 - 2) * q^21 + (-b2 + 2*b1 + 1) * q^23 + q^25 + (2*b2 + 2) * q^27 - q^29 + (-2*b2 - 3*b1 + 1) * q^31 + (b2 - 5*b1 + 4) * q^33 + (b1 + 1) * q^35 + (b2 + 2*b1 - 1) * q^37 + (2*b1 + 2) * q^39 + (-3*b2 - b1 - 2) * q^41 + (b1 + 3) * q^43 + (b2 - b1 + 1) * q^45 + (-2*b2 - 3*b1 + 1) * q^47 + (b2 + 3*b1 - 3) * q^49 + (b2 - 3*b1 + 2) * q^51 + (-b2 + b1 + 2) * q^53 + (b2 + 3) * q^55 + (-b2 - 3*b1) * q^57 + (-3*b2 + 3*b1 + 4) * q^59 + (b2 + 5*b1 - 4) * q^61 + (b1 - 3) * q^63 + (-b2 - b1) * q^65 + (b2 - 4*b1 + 1) * q^67 + (-3*b2 + b1 - 6) * q^69 + (-b2 - 3*b1 + 4) * q^71 + (2*b2 - b1 + 5) * q^73 + (-b1 + 1) * q^75 + (b2 + 5*b1 + 2) * q^77 + (b2 - 2*b1 + 9) * q^79 + (-b2 - 3*b1 + 1) * q^81 + (2*b2 - b1 + 1) * q^83 + (b2 + 1) * q^85 + (b1 - 1) * q^87 + (-2*b2 + 4*b1 - 4) * q^89 + (-2*b2 - 4*b1 - 2) * q^91 + (b2 + 3*b1 + 8) * q^93 + (b1 + 3) * q^95 + (-2*b2 + b1 - 5) * q^97 + (3*b2 - 6*b1 + 11) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 $$3 q + 2 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9} + 10 q^{11} - 2 q^{13} + 2 q^{15} + 4 q^{17} + 10 q^{19} - 8 q^{21} + 4 q^{23} + 3 q^{25} + 8 q^{27} - 3 q^{29} - 2 q^{31} + 8 q^{33} + 4 q^{35} + 8 q^{39} - 10 q^{41} + 10 q^{43} + 3 q^{45} - 2 q^{47} - 5 q^{49} + 4 q^{51} + 6 q^{53} + 10 q^{55} - 4 q^{57} + 12 q^{59} - 6 q^{61} - 8 q^{63} - 2 q^{65} - 20 q^{69} + 8 q^{71} + 16 q^{73} + 2 q^{75} + 12 q^{77} + 26 q^{79} - q^{81} + 4 q^{83} + 4 q^{85} - 2 q^{87} - 10 q^{89} - 12 q^{91} + 28 q^{93} + 10 q^{95} - 16 q^{97} + 30 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 + 3 * q^5 + 4 * q^7 + 3 * q^9 + 10 * q^11 - 2 * q^13 + 2 * q^15 + 4 * q^17 + 10 * q^19 - 8 * q^21 + 4 * q^23 + 3 * q^25 + 8 * q^27 - 3 * q^29 - 2 * q^31 + 8 * q^33 + 4 * q^35 + 8 * q^39 - 10 * q^41 + 10 * q^43 + 3 * q^45 - 2 * q^47 - 5 * q^49 + 4 * q^51 + 6 * q^53 + 10 * q^55 - 4 * q^57 + 12 * q^59 - 6 * q^61 - 8 * q^63 - 2 * q^65 - 20 * q^69 + 8 * q^71 + 16 * q^73 + 2 * q^75 + 12 * q^77 + 26 * q^79 - q^81 + 4 * q^83 + 4 * q^85 - 2 * q^87 - 10 * q^89 - 12 * q^91 + 28 * q^93 + 10 * q^95 - 16 * q^97 + 30 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 2\nu + 2$$ -v^2 + 2*v + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta _1 + 2$$ b1 + 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
 2.17009 −1.48119 0.311108
0 −1.70928 0 1.00000 0 3.70928 0 −0.0783777 0
1.2 0 0.806063 0 1.00000 0 1.19394 0 −2.35026 0
1.3 0 2.90321 0 1.00000 0 −0.903212 0 5.42864 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.t 3
4.b odd 2 1 1160.2.a.f 3
8.b even 2 1 9280.2.a.bi 3
8.d odd 2 1 9280.2.a.bs 3
20.d odd 2 1 5800.2.a.s 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.a.f 3 4.b odd 2 1
2320.2.a.t 3 1.a even 1 1 trivial
5800.2.a.s 3 20.d odd 2 1
9280.2.a.bi 3 8.b even 2 1
9280.2.a.bs 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2320))$$:

 $$T_{3}^{3} - 2T_{3}^{2} - 4T_{3} + 4$$ T3^3 - 2*T3^2 - 4*T3 + 4 $$T_{7}^{3} - 4T_{7}^{2} + 4$$ T7^3 - 4*T7^2 + 4 $$T_{11}^{3} - 10T_{11}^{2} + 24T_{11} + 4$$ T11^3 - 10*T11^2 + 24*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} + \cdots + 4$$
$5$ $$(T - 1)^{3}$$
$7$ $$T^{3} - 4T^{2} + 4$$
$11$ $$T^{3} - 10 T^{2} + \cdots + 4$$
$13$ $$T^{3} + 2 T^{2} + \cdots - 8$$
$17$ $$T^{3} - 4 T^{2} + \cdots + 20$$
$19$ $$T^{3} - 10 T^{2} + \cdots - 20$$
$23$ $$T^{3} - 4 T^{2} + \cdots + 116$$
$29$ $$(T + 1)^{3}$$
$31$ $$T^{3} + 2 T^{2} + \cdots + 116$$
$37$ $$T^{3} - 28T - 52$$
$41$ $$T^{3} + 10 T^{2} + \cdots - 536$$
$43$ $$T^{3} - 10 T^{2} + \cdots - 20$$
$47$ $$T^{3} + 2 T^{2} + \cdots + 116$$
$53$ $$T^{3} - 6 T^{2} + \cdots + 40$$
$59$ $$T^{3} - 12 T^{2} + \cdots + 944$$
$61$ $$T^{3} + 6 T^{2} + \cdots - 760$$
$67$ $$T^{3} - 100T - 268$$
$71$ $$T^{3} - 8 T^{2} + \cdots + 272$$
$73$ $$T^{3} - 16 T^{2} + \cdots + 100$$
$79$ $$T^{3} - 26 T^{2} + \cdots - 436$$
$83$ $$T^{3} - 4 T^{2} + \cdots + 68$$
$89$ $$T^{3} + 10 T^{2} + \cdots + 184$$
$97$ $$T^{3} + 16 T^{2} + \cdots - 100$$