# Properties

 Label 2320.2.d.d Level $2320$ Weight $2$ Character orbit 2320.d Analytic conductor $18.525$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(929,2320)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2320.929");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2320 = 2^{4} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2320.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$18.5252932689$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 290) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{5} + \beta_{2} q^{7} + (\beta_{3} - 1) q^{9}+O(q^{10})$$ q - b1 * q^3 + (-b2 + b1) * q^5 + b2 * q^7 + (b3 - 1) * q^9 $$q - \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{5} + \beta_{2} q^{7} + (\beta_{3} - 1) q^{9} - 2 q^{11} + \beta_1 q^{13} + ( - \beta_{3} + 3) q^{15} + ( - \beta_{2} - 2 \beta_1) q^{17} - 2 q^{19} + q^{21} + ( - 2 \beta_{2} + \beta_1) q^{23} - 5 q^{25} + ( - \beta_{2} + \beta_1) q^{27} + q^{29} + \beta_{3} q^{31} + 2 \beta_1 q^{33} + (\beta_{3} + 2) q^{35} + ( - 4 \beta_{2} + 4 \beta_1) q^{37} + ( - \beta_{3} + 4) q^{39} + 2 \beta_{3} q^{41} + ( - \beta_{2} - 2 \beta_1) q^{43} + ( - 2 \beta_{2} - 3 \beta_1) q^{45} + 4 \beta_1 q^{47} + ( - \beta_{3} + 4) q^{49} + (2 \beta_{3} - 9) q^{51} + (3 \beta_{2} + 4 \beta_1) q^{53} + (2 \beta_{2} - 2 \beta_1) q^{55} + 2 \beta_1 q^{57} + (\beta_{3} - 1) q^{59} + ( - \beta_{3} - 7) q^{61} + (3 \beta_{2} - \beta_1) q^{63} + (\beta_{3} - 3) q^{65} + ( - 4 \beta_{2} + 4 \beta_1) q^{67} + ( - \beta_{3} + 2) q^{69} + (2 \beta_{3} - 2) q^{71} + ( - 3 \beta_{2} + 6 \beta_1) q^{73} + 5 \beta_1 q^{75} - 2 \beta_{2} q^{77} + (3 \beta_{3} - 5) q^{79} + 2 \beta_{3} q^{81} + ( - 6 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{3} + 4) q^{85} - \beta_1 q^{87} - 2 q^{89} - q^{91} + ( - \beta_{2} + 3 \beta_1) q^{93} + (2 \beta_{2} - 2 \beta_1) q^{95} + ( - 2 \beta_{2} - 5 \beta_1) q^{97} + ( - 2 \beta_{3} + 2) q^{99}+O(q^{100})$$ q - b1 * q^3 + (-b2 + b1) * q^5 + b2 * q^7 + (b3 - 1) * q^9 - 2 * q^11 + b1 * q^13 + (-b3 + 3) * q^15 + (-b2 - 2*b1) * q^17 - 2 * q^19 + q^21 + (-2*b2 + b1) * q^23 - 5 * q^25 + (-b2 + b1) * q^27 + q^29 + b3 * q^31 + 2*b1 * q^33 + (b3 + 2) * q^35 + (-4*b2 + 4*b1) * q^37 + (-b3 + 4) * q^39 + 2*b3 * q^41 + (-b2 - 2*b1) * q^43 + (-2*b2 - 3*b1) * q^45 + 4*b1 * q^47 + (-b3 + 4) * q^49 + (2*b3 - 9) * q^51 + (3*b2 + 4*b1) * q^53 + (2*b2 - 2*b1) * q^55 + 2*b1 * q^57 + (b3 - 1) * q^59 + (-b3 - 7) * q^61 + (3*b2 - b1) * q^63 + (b3 - 3) * q^65 + (-4*b2 + 4*b1) * q^67 + (-b3 + 2) * q^69 + (2*b3 - 2) * q^71 + (-3*b2 + 6*b1) * q^73 + 5*b1 * q^75 - 2*b2 * q^77 + (3*b3 - 5) * q^79 + 2*b3 * q^81 + (-6*b2 + 2*b1) * q^83 + (-3*b3 + 4) * q^85 - b1 * q^87 - 2 * q^89 - q^91 + (-b2 + 3*b1) * q^93 + (2*b2 - 2*b1) * q^95 + (-2*b2 - 5*b1) * q^97 + (-2*b3 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^9 $$4 q - 2 q^{9} - 8 q^{11} + 10 q^{15} - 8 q^{19} + 4 q^{21} - 20 q^{25} + 4 q^{29} + 2 q^{31} + 10 q^{35} + 14 q^{39} + 4 q^{41} + 14 q^{49} - 32 q^{51} - 2 q^{59} - 30 q^{61} - 10 q^{65} + 6 q^{69} - 4 q^{71} - 14 q^{79} + 4 q^{81} + 10 q^{85} - 8 q^{89} - 4 q^{91} + 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^9 - 8 * q^11 + 10 * q^15 - 8 * q^19 + 4 * q^21 - 20 * q^25 + 4 * q^29 + 2 * q^31 + 10 * q^35 + 14 * q^39 + 4 * q^41 + 14 * q^49 - 32 * q^51 - 2 * q^59 - 30 * q^61 - 10 * q^65 + 6 * q^69 - 4 * q^71 - 14 * q^79 + 4 * q^81 + 10 * q^85 - 8 * q^89 - 4 * q^91 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + \nu$$ v^3 + v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 5\nu$$ 2*v^3 + 5*v $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 5$$ 3*v^2 + 5
 $$\nu$$ $$=$$ $$( \beta_{2} - 2\beta_1 ) / 3$$ (b2 - 2*b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 5 ) / 3$$ (b3 - 5) / 3 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 5\beta_1 ) / 3$$ (-b2 + 5*b1) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$$.

 $$n$$ $$321$$ $$581$$ $$1857$$ $$2031$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
929.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
0 2.61803i 0 2.23607i 0 0.381966i 0 −3.85410 0
929.2 0 0.381966i 0 2.23607i 0 2.61803i 0 2.85410 0
929.3 0 0.381966i 0 2.23607i 0 2.61803i 0 2.85410 0
929.4 0 2.61803i 0 2.23607i 0 0.381966i 0 −3.85410 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2320.2.d.d 4
4.b odd 2 1 290.2.b.a 4
5.b even 2 1 inner 2320.2.d.d 4
12.b even 2 1 2610.2.e.e 4
20.d odd 2 1 290.2.b.a 4
20.e even 4 1 1450.2.a.k 2
20.e even 4 1 1450.2.a.l 2
60.h even 2 1 2610.2.e.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.b.a 4 4.b odd 2 1
290.2.b.a 4 20.d odd 2 1
1450.2.a.k 2 20.e even 4 1
1450.2.a.l 2 20.e even 4 1
2320.2.d.d 4 1.a even 1 1 trivial
2320.2.d.d 4 5.b even 2 1 inner
2610.2.e.e 4 12.b even 2 1
2610.2.e.e 4 60.h even 2 1

## Hecke kernels

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

 $$T_{3}^{4} + 7T_{3}^{2} + 1$$ T3^4 + 7*T3^2 + 1 $$T_{7}^{4} + 7T_{7}^{2} + 1$$ T7^4 + 7*T7^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 7T^{2} + 1$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$T^{4} + 7T^{2} + 1$$
$11$ $$(T + 2)^{4}$$
$13$ $$T^{4} + 7T^{2} + 1$$
$17$ $$T^{4} + 43T^{2} + 361$$
$19$ $$(T + 2)^{4}$$
$23$ $$T^{4} + 27T^{2} + 81$$
$29$ $$(T - 1)^{4}$$
$31$ $$(T^{2} - T - 11)^{2}$$
$37$ $$(T^{2} + 80)^{2}$$
$41$ $$(T^{2} - 2 T - 44)^{2}$$
$43$ $$T^{4} + 43T^{2} + 361$$
$47$ $$T^{4} + 112T^{2} + 256$$
$53$ $$T^{4} + 223 T^{2} + 11881$$
$59$ $$(T^{2} + T - 11)^{2}$$
$61$ $$(T^{2} + 15 T + 45)^{2}$$
$67$ $$(T^{2} + 80)^{2}$$
$71$ $$(T^{2} + 2 T - 44)^{2}$$
$73$ $$T^{4} + 243T^{2} + 6561$$
$79$ $$(T^{2} + 7 T - 89)^{2}$$
$83$ $$T^{4} + 232T^{2} + 1936$$
$89$ $$(T + 2)^{4}$$
$97$ $$T^{4} + 243T^{2} + 9801$$