# Properties

 Label 2320.2.d.f 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: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 145) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{7} + q^{9}+O(q^{10})$$ q - b1 * q^3 + (-b2 + b1) * q^5 - b3 * q^7 + q^9 $$q - \beta_1 q^{3} + ( - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{7} + q^{9} + ( - \beta_{2} + 3) q^{11} + (\beta_{3} - 3 \beta_1) q^{13} + (\beta_{3} + 2) q^{15} - \beta_1 q^{17} + (\beta_{2} - 5) q^{19} - 2 \beta_{2} q^{21} + (2 \beta_{3} + \beta_1) q^{23} + ( - 2 \beta_{3} + 1) q^{25} - 4 \beta_1 q^{27} - q^{29} + (\beta_{2} + 7) q^{31} + (\beta_{3} - 3 \beta_1) q^{33} + (2 \beta_{2} + 3 \beta_1) q^{35} + ( - 2 \beta_{3} - 3 \beta_1) q^{37} + (2 \beta_{2} - 6) q^{39} - 4 \beta_{2} q^{41} + ( - 2 \beta_{3} - 3 \beta_1) q^{43} + ( - \beta_{2} + \beta_1) q^{45} + \beta_1 q^{47} + q^{49} - 2 q^{51} + (3 \beta_{3} - \beta_1) q^{53} + ( - \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 3) q^{55} + ( - \beta_{3} + 5 \beta_1) q^{57} - 6 \beta_{2} q^{59} + (4 \beta_{2} - 4) q^{61} - \beta_{3} q^{63} + (3 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{65} + 3 \beta_1 q^{67} + (4 \beta_{2} + 2) q^{69} - 2 \beta_{2} q^{71} - 3 \beta_{3} q^{73} + ( - 4 \beta_{2} - \beta_1) q^{75} + ( - 3 \beta_{3} + 3 \beta_1) q^{77} + ( - 3 \beta_{2} + 1) q^{79} - 5 q^{81} + ( - 3 \beta_{3} - 2 \beta_1) q^{83} + (\beta_{3} + 2) q^{85} + \beta_1 q^{87} - 6 \beta_{2} q^{89} + ( - 6 \beta_{2} + 6) q^{91} + ( - \beta_{3} - 7 \beta_1) q^{93} + (\beta_{3} + 5 \beta_{2} - 5 \beta_1 - 3) q^{95} + (\beta_{3} + 6 \beta_1) q^{97} + ( - \beta_{2} + 3) q^{99}+O(q^{100})$$ q - b1 * q^3 + (-b2 + b1) * q^5 - b3 * q^7 + q^9 + (-b2 + 3) * q^11 + (b3 - 3*b1) * q^13 + (b3 + 2) * q^15 - b1 * q^17 + (b2 - 5) * q^19 - 2*b2 * q^21 + (2*b3 + b1) * q^23 + (-2*b3 + 1) * q^25 - 4*b1 * q^27 - q^29 + (b2 + 7) * q^31 + (b3 - 3*b1) * q^33 + (2*b2 + 3*b1) * q^35 + (-2*b3 - 3*b1) * q^37 + (2*b2 - 6) * q^39 - 4*b2 * q^41 + (-2*b3 - 3*b1) * q^43 + (-b2 + b1) * q^45 + b1 * q^47 + q^49 - 2 * q^51 + (3*b3 - b1) * q^53 + (-b3 - 3*b2 + 3*b1 + 3) * q^55 + (-b3 + 5*b1) * q^57 - 6*b2 * q^59 + (4*b2 - 4) * q^61 - b3 * q^63 + (3*b3 - 2*b2 - 3*b1 + 6) * q^65 + 3*b1 * q^67 + (4*b2 + 2) * q^69 - 2*b2 * q^71 - 3*b3 * q^73 + (-4*b2 - b1) * q^75 + (-3*b3 + 3*b1) * q^77 + (-3*b2 + 1) * q^79 - 5 * q^81 + (-3*b3 - 2*b1) * q^83 + (b3 + 2) * q^85 + b1 * q^87 - 6*b2 * q^89 + (-6*b2 + 6) * q^91 + (-b3 - 7*b1) * q^93 + (b3 + 5*b2 - 5*b1 - 3) * q^95 + (b3 + 6*b1) * q^97 + (-b2 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^9 $$4 q + 4 q^{9} + 12 q^{11} + 8 q^{15} - 20 q^{19} + 4 q^{25} - 4 q^{29} + 28 q^{31} - 24 q^{39} + 4 q^{49} - 8 q^{51} + 12 q^{55} - 16 q^{61} + 24 q^{65} + 8 q^{69} + 4 q^{79} - 20 q^{81} + 8 q^{85} + 24 q^{91} - 12 q^{95} + 12 q^{99}+O(q^{100})$$ 4 * q + 4 * q^9 + 12 * q^11 + 8 * q^15 - 20 * q^19 + 4 * q^25 - 4 * q^29 + 28 * q^31 - 24 * q^39 + 4 * q^49 - 8 * q^51 + 12 * q^55 - 16 * q^61 + 24 * q^65 + 8 * q^69 + 4 * q^79 - 20 * q^81 + 8 * q^85 + 24 * q^91 - 12 * q^95 + 12 * q^99

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

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

## 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
 0.517638i − 1.93185i − 0.517638i 1.93185i
0 1.41421i 0 −1.73205 + 1.41421i 0 2.44949i 0 1.00000 0
929.2 0 1.41421i 0 1.73205 + 1.41421i 0 2.44949i 0 1.00000 0
929.3 0 1.41421i 0 −1.73205 1.41421i 0 2.44949i 0 1.00000 0
929.4 0 1.41421i 0 1.73205 1.41421i 0 2.44949i 0 1.00000 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.f 4
4.b odd 2 1 145.2.b.b 4
5.b even 2 1 inner 2320.2.d.f 4
12.b even 2 1 1305.2.c.f 4
20.d odd 2 1 145.2.b.b 4
20.e even 4 2 725.2.a.f 4
60.h even 2 1 1305.2.c.f 4
60.l odd 4 2 6525.2.a.bj 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.b.b 4 4.b odd 2 1
145.2.b.b 4 20.d odd 2 1
725.2.a.f 4 20.e even 4 2
1305.2.c.f 4 12.b even 2 1
1305.2.c.f 4 60.h even 2 1
2320.2.d.f 4 1.a even 1 1 trivial
2320.2.d.f 4 5.b even 2 1 inner
6525.2.a.bj 4 60.l odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} + 2$$ T3^2 + 2 $$T_{7}^{2} + 6$$ T7^2 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 2)^{2}$$
$5$ $$T^{4} - 2T^{2} + 25$$
$7$ $$(T^{2} + 6)^{2}$$
$11$ $$(T^{2} - 6 T + 6)^{2}$$
$13$ $$T^{4} + 48T^{2} + 144$$
$17$ $$(T^{2} + 2)^{2}$$
$19$ $$(T^{2} + 10 T + 22)^{2}$$
$23$ $$T^{4} + 52T^{2} + 484$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} - 14 T + 46)^{2}$$
$37$ $$T^{4} + 84T^{2} + 36$$
$41$ $$(T^{2} - 48)^{2}$$
$43$ $$T^{4} + 84T^{2} + 36$$
$47$ $$(T^{2} + 2)^{2}$$
$53$ $$T^{4} + 112T^{2} + 2704$$
$59$ $$(T^{2} - 108)^{2}$$
$61$ $$(T^{2} + 8 T - 32)^{2}$$
$67$ $$(T^{2} + 18)^{2}$$
$71$ $$(T^{2} - 12)^{2}$$
$73$ $$(T^{2} + 54)^{2}$$
$79$ $$(T^{2} - 2 T - 26)^{2}$$
$83$ $$T^{4} + 124T^{2} + 2116$$
$89$ $$(T^{2} - 108)^{2}$$
$97$ $$T^{4} + 156T^{2} + 4356$$