# Properties

 Label 2320.2.g.e Level $2320$ Weight $2$ Character orbit 2320.g Analytic conductor $18.525$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(1681,2320)]

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, 1]))

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

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

chi := DirichletCharacter("2320.1681");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2320 = 2^{4} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2320.g (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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - q^{5} + (\beta_{2} + 1) q^{7} + q^{9}+O(q^{10})$$ q - b1 * q^3 - q^5 + (b2 + 1) * q^7 + q^9 $$q - \beta_1 q^{3} - q^{5} + (\beta_{2} + 1) q^{7} + q^{9} + (\beta_{3} - 2 \beta_1) q^{11} + (2 \beta_{2} + 2) q^{13} + \beta_1 q^{15} + (2 \beta_{3} - \beta_1) q^{17} + 3 \beta_1 q^{19} + ( - \beta_{3} - \beta_1) q^{21} + ( - 3 \beta_{2} + 3) q^{23} + q^{25} - 4 \beta_1 q^{27} + (3 \beta_{2} + \beta_1) q^{29} + 3 \beta_1 q^{31} + (2 \beta_{2} - 4) q^{33} + ( - \beta_{2} - 1) q^{35} + 3 \beta_1 q^{37} + ( - 2 \beta_{3} - 2 \beta_1) q^{39} + (3 \beta_{3} - \beta_1) q^{41} + 3 \beta_1 q^{43} - q^{45} + (4 \beta_{3} + \beta_1) q^{47} + (2 \beta_{2} - 3) q^{49} + (4 \beta_{2} - 2) q^{51} + ( - \beta_{3} + 2 \beta_1) q^{55} + 6 q^{57} - 6 q^{59} + ( - 3 \beta_{3} - 3 \beta_1) q^{61} + (\beta_{2} + 1) q^{63} + ( - 2 \beta_{2} - 2) q^{65} + (7 \beta_{2} + 1) q^{67} + (3 \beta_{3} - 3 \beta_1) q^{69} - 6 q^{71} + ( - 3 \beta_{3} - 6 \beta_1) q^{73} - \beta_1 q^{75} + ( - \beta_{3} + \beta_1) q^{77} + (3 \beta_{3} - 6 \beta_1) q^{79} - 5 q^{81} + (3 \beta_{2} + 3) q^{83} + ( - 2 \beta_{3} + \beta_1) q^{85} + ( - 3 \beta_{3} + 2) q^{87} + (2 \beta_{3} + 2 \beta_1) q^{89} + (4 \beta_{2} + 8) q^{91} + 6 q^{93} - 3 \beta_1 q^{95} + 3 \beta_{3} q^{97} + (\beta_{3} - 2 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^3 - q^5 + (b2 + 1) * q^7 + q^9 + (b3 - 2*b1) * q^11 + (2*b2 + 2) * q^13 + b1 * q^15 + (2*b3 - b1) * q^17 + 3*b1 * q^19 + (-b3 - b1) * q^21 + (-3*b2 + 3) * q^23 + q^25 - 4*b1 * q^27 + (3*b2 + b1) * q^29 + 3*b1 * q^31 + (2*b2 - 4) * q^33 + (-b2 - 1) * q^35 + 3*b1 * q^37 + (-2*b3 - 2*b1) * q^39 + (3*b3 - b1) * q^41 + 3*b1 * q^43 - q^45 + (4*b3 + b1) * q^47 + (2*b2 - 3) * q^49 + (4*b2 - 2) * q^51 + (-b3 + 2*b1) * q^55 + 6 * q^57 - 6 * q^59 + (-3*b3 - 3*b1) * q^61 + (b2 + 1) * q^63 + (-2*b2 - 2) * q^65 + (7*b2 + 1) * q^67 + (3*b3 - 3*b1) * q^69 - 6 * q^71 + (-3*b3 - 6*b1) * q^73 - b1 * q^75 + (-b3 + b1) * q^77 + (3*b3 - 6*b1) * q^79 - 5 * q^81 + (3*b2 + 3) * q^83 + (-2*b3 + b1) * q^85 + (-3*b3 + 2) * q^87 + (2*b3 + 2*b1) * q^89 + (4*b2 + 8) * q^91 + 6 * q^93 - 3*b1 * q^95 + 3*b3 * q^97 + (b3 - 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^5 + 4 * q^7 + 4 * q^9 $$4 q - 4 q^{5} + 4 q^{7} + 4 q^{9} + 8 q^{13} + 12 q^{23} + 4 q^{25} - 16 q^{33} - 4 q^{35} - 4 q^{45} - 12 q^{49} - 8 q^{51} + 24 q^{57} - 24 q^{59} + 4 q^{63} - 8 q^{65} + 4 q^{67} - 24 q^{71} - 20 q^{81} + 12 q^{83} + 8 q^{87} + 32 q^{91} + 24 q^{93}+O(q^{100})$$ 4 * q - 4 * q^5 + 4 * q^7 + 4 * q^9 + 8 * q^13 + 12 * q^23 + 4 * q^25 - 16 * q^33 - 4 * q^35 - 4 * q^45 - 12 * q^49 - 8 * q^51 + 24 * q^57 - 24 * q^59 + 4 * q^63 - 8 * q^65 + 4 * q^67 - 24 * q^71 - 20 * q^81 + 12 * q^83 + 8 * q^87 + 32 * q^91 + 24 * q^93

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1681.1
 − 1.93185i 0.517638i 1.93185i − 0.517638i
0 1.41421i 0 −1.00000 0 −0.732051 0 1.00000 0
1681.2 0 1.41421i 0 −1.00000 0 2.73205 0 1.00000 0
1681.3 0 1.41421i 0 −1.00000 0 −0.732051 0 1.00000 0
1681.4 0 1.41421i 0 −1.00000 0 2.73205 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
29.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.g.e 4
4.b odd 2 1 145.2.c.a 4
12.b even 2 1 1305.2.d.a 4
20.d odd 2 1 725.2.c.d 4
20.e even 4 2 725.2.d.b 8
29.b even 2 1 inner 2320.2.g.e 4
116.d odd 2 1 145.2.c.a 4
116.e even 4 2 4205.2.a.g 4
348.b even 2 1 1305.2.d.a 4
580.e odd 2 1 725.2.c.d 4
580.o even 4 2 725.2.d.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.c.a 4 4.b odd 2 1
145.2.c.a 4 116.d odd 2 1
725.2.c.d 4 20.d odd 2 1
725.2.c.d 4 580.e odd 2 1
725.2.d.b 8 20.e even 4 2
725.2.d.b 8 580.o even 4 2
1305.2.d.a 4 12.b even 2 1
1305.2.d.a 4 348.b even 2 1
2320.2.g.e 4 1.a even 1 1 trivial
2320.2.g.e 4 29.b even 2 1 inner
4205.2.a.g 4 116.e even 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 $$T_{7}^{2} - 2T_{7} - 2$$ T7^2 - 2*T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 2)^{2}$$
$5$ $$(T + 1)^{4}$$
$7$ $$(T^{2} - 2 T - 2)^{2}$$
$11$ $$T^{4} + 28T^{2} + 4$$
$13$ $$(T^{2} - 4 T - 8)^{2}$$
$17$ $$T^{4} + 52T^{2} + 484$$
$19$ $$(T^{2} + 18)^{2}$$
$23$ $$(T^{2} - 6 T - 18)^{2}$$
$29$ $$T^{4} - 50T^{2} + 841$$
$31$ $$(T^{2} + 18)^{2}$$
$37$ $$(T^{2} + 18)^{2}$$
$41$ $$T^{4} + 112T^{2} + 2704$$
$43$ $$(T^{2} + 18)^{2}$$
$47$ $$T^{4} + 196T^{2} + 8836$$
$53$ $$T^{4}$$
$59$ $$(T + 6)^{4}$$
$61$ $$T^{4} + 144T^{2} + 1296$$
$67$ $$(T^{2} - 2 T - 146)^{2}$$
$71$ $$(T + 6)^{4}$$
$73$ $$T^{4} + 252T^{2} + 324$$
$79$ $$T^{4} + 252T^{2} + 324$$
$83$ $$(T^{2} - 6 T - 18)^{2}$$
$89$ $$T^{4} + 64T^{2} + 256$$
$97$ $$(T^{2} + 54)^{2}$$