# Properties

 Label 2320.2.d.g Level $2320$ Weight $2$ Character orbit 2320.d Analytic conductor $18.525$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.84345856.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 13x^{4} + 41x^{2} + 1$$ x^6 + 13*x^4 + 41*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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} - \beta_1 q^{5} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10})$$ q + b3 * q^3 - b1 * q^5 + (b3 + b2 + b1 + 1) * q^7 + (-b2 + b1 - 2) * q^9 $$q + \beta_{3} q^{3} - \beta_1 q^{5} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} + \beta_1 - 2) q^{9} + (\beta_{5} + \beta_1 + 2) q^{11} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{5} + 2 \beta_{3} - 1) q^{15} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{17} + (\beta_{5} + \beta_{2} + 3) q^{19} + (2 \beta_{5} + 2 \beta_1 - 2) q^{21} + ( - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1 + 1) q^{25} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{27} + q^{29} + ( - \beta_{5} - \beta_1 - 4) q^{31} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{33} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{35} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{37} + (4 \beta_{5} + \beta_{2} + 3 \beta_1 - 1) q^{39} + ( - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{43} + (\beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 - 6) q^{45} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{47} + (2 \beta_{5} + 2 \beta_1 + 1) q^{49} + ( - 2 \beta_{5} - 2 \beta_{2} - 8) q^{51} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{53} + (2 \beta_{5} + \beta_{3} - \beta_1 - 3) q^{55} + ( - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{57} + 10 q^{59} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{61} + ( - 2 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 3) q^{63} + (\beta_{5} + 3 \beta_{3} + 5 \beta_{2} + 3 \beta_1 + 6) q^{65} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{67} + 2 q^{69} - 2 q^{71} + (\beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 3) q^{73} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1) q^{75} + ( - \beta_{4} - 3 \beta_{3}) q^{77} + ( - \beta_{5} - 2 \beta_{2} + \beta_1) q^{79} + ( - 2 \beta_{5} - 2 \beta_{2} + 1) q^{81} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{83} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 5) q^{85} + \beta_{3} q^{87} + (2 \beta_{5} + 2 \beta_1 + 6) q^{89} + (2 \beta_{5} - 2 \beta_{2} + 4 \beta_1 - 6) q^{91} + (\beta_{4} - \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 3) q^{93} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{95} + (3 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{97} + ( - 5 \beta_{5} - 3 \beta_{2} - 2 \beta_1 + 1) q^{99}+O(q^{100})$$ q + b3 * q^3 - b1 * q^5 + (b3 + b2 + b1 + 1) * q^7 + (-b2 + b1 - 2) * q^9 + (b5 + b1 + 2) * q^11 + (b4 + b3 + b2 + b1 + 1) * q^13 + (-b5 + 2*b3 - 1) * q^15 + (b3 - b2 - b1 - 1) * q^17 + (b5 + b2 + 3) * q^19 + (2*b5 + 2*b1 - 2) * q^21 + (-b4 + b2 + b1 + 1) * q^23 + (-b5 - b4 - b2 - b1 + 1) * q^25 + (-2*b3 - b2 - b1 - 1) * q^27 + q^29 + (-b5 - b1 - 4) * q^31 + (-b4 - b3 - 3*b2 - 3*b1 - 3) * q^33 + (b4 + 2*b3 + b2 + b1 + 3) * q^35 + (b3 + b2 + b1 + 1) * q^37 + (4*b5 + b2 + 3*b1 - 1) * q^39 + (-3*b3 - 2*b2 - 2*b1 - 2) * q^43 + (b5 + b4 + b2 + 2*b1 - 6) * q^45 + (-b3 - 2*b2 - 2*b1 - 2) * q^47 + (2*b5 + 2*b1 + 1) * q^49 + (-2*b5 - 2*b2 - 8) * q^51 + (-b4 - b3 - 3*b2 - 3*b1 - 3) * q^53 + (2*b5 + b3 - b1 - 3) * q^55 + (-b4 + 3*b3 - 2*b2 - 2*b1 - 2) * q^57 + 10 * q^59 + (-2*b2 + 2*b1 + 2) * q^61 + (-2*b4 - 5*b3 - 3*b2 - 3*b1 - 3) * q^63 + (b5 + 3*b3 + 5*b2 + 3*b1 + 6) * q^65 + (b4 + 2*b3 - b2 - b1 - 1) * q^67 + 2 * q^69 - 2 * q^71 + (b4 + 2*b3 + 3*b2 + 3*b1 + 3) * q^73 + (-3*b5 + b4 + 3*b3 + b2 + b1) * q^75 + (-b4 - 3*b3) * q^77 + (-b5 - 2*b2 + b1) * q^79 + (-2*b5 - 2*b2 + 1) * q^81 + (b3 - b2 - b1 - 1) * q^83 + (-2*b5 - b4 + 2*b3 - b2 - b1 - 5) * q^85 + b3 * q^87 + (2*b5 + 2*b1 + 6) * q^89 + (2*b5 - 2*b2 + 4*b1 - 6) * q^91 + (b4 - b3 + 3*b2 + 3*b1 + 3) * q^93 + (b5 - b4 + b3 - b2 - 2*b1 + 3) * q^95 + (3*b4 - 2*b3 + b2 + b1 + 1) * q^97 + (-5*b5 - 3*b2 - 2*b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{5} - 12 q^{9}+O(q^{10})$$ 6 * q + 3 * q^5 - 12 * q^9 $$6 q + 3 q^{5} - 12 q^{9} + 10 q^{11} - 7 q^{15} + 16 q^{19} - 16 q^{21} + 11 q^{25} + 6 q^{29} - 22 q^{31} + 12 q^{35} - 14 q^{39} - 44 q^{45} + 2 q^{49} - 44 q^{51} - 13 q^{55} + 60 q^{59} + 12 q^{61} + 13 q^{65} + 12 q^{69} - 12 q^{71} - 9 q^{75} + 2 q^{79} + 10 q^{81} - 26 q^{85} + 32 q^{89} - 40 q^{91} + 28 q^{95} + 16 q^{99}+O(q^{100})$$ 6 * q + 3 * q^5 - 12 * q^9 + 10 * q^11 - 7 * q^15 + 16 * q^19 - 16 * q^21 + 11 * q^25 + 6 * q^29 - 22 * q^31 + 12 * q^35 - 14 * q^39 - 44 * q^45 + 2 * q^49 - 44 * q^51 - 13 * q^55 + 60 * q^59 + 12 * q^61 + 13 * q^65 + 12 * q^69 - 12 * q^71 - 9 * q^75 + 2 * q^79 + 10 * q^81 - 26 * q^85 + 32 * q^89 - 40 * q^91 + 28 * q^95 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 13x^{4} + 41x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + \nu^{4} + 14\nu^{3} + 6\nu^{2} + 47\nu - 5 ) / 4$$ (v^5 + v^4 + 14*v^3 + 6*v^2 + 47*v - 5) / 4 $$\beta_{2}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 14\nu^{3} - 6\nu^{2} + 47\nu + 1 ) / 4$$ (v^5 - v^4 + 14*v^3 - 6*v^2 + 47*v + 1) / 4 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} - 12\nu^{3} - 33\nu ) / 2$$ (-v^5 - 12*v^3 - 33*v) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 12\nu^{3} + 37\nu ) / 2$$ (v^5 + 12*v^3 + 37*v) / 2 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 14\nu^{3} + 10\nu^{2} - 47\nu + 15 ) / 4$$ (-v^5 + v^4 - 14*v^3 + 10*v^2 - 47*v + 15) / 4
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} ) / 2$$ (b4 + b3) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{2} - 4$$ b5 + b2 - 4 $$\nu^{3}$$ $$=$$ $$( -7\beta_{4} - 5\beta_{3} + 2\beta_{2} + 2\beta _1 + 2 ) / 2$$ (-7*b4 - 5*b3 + 2*b2 + 2*b1 + 2) / 2 $$\nu^{4}$$ $$=$$ $$-6\beta_{5} - 8\beta_{2} + 2\beta _1 + 27$$ -6*b5 - 8*b2 + 2*b1 + 27 $$\nu^{5}$$ $$=$$ $$( 51\beta_{4} + 23\beta_{3} - 24\beta_{2} - 24\beta _1 - 24 ) / 2$$ (51*b4 + 23*b3 - 24*b2 - 24*b1 - 24) / 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
 − 2.30229i 0.156785i − 2.77035i 2.77035i − 0.156785i 2.30229i
0 2.89028i 0 2.17686 + 0.511167i 0 3.91261i 0 −5.35371 0
929.2 0 2.56387i 0 1.28672 1.82876i 0 1.09364i 0 −3.57344 0
929.3 0 0.269894i 0 −1.96358 1.06975i 0 1.86960i 0 2.92716 0
929.4 0 0.269894i 0 −1.96358 + 1.06975i 0 1.86960i 0 2.92716 0
929.5 0 2.56387i 0 1.28672 + 1.82876i 0 1.09364i 0 −3.57344 0
929.6 0 2.89028i 0 2.17686 0.511167i 0 3.91261i 0 −5.35371 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 929.6 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.g 6
4.b odd 2 1 145.2.b.c 6
5.b even 2 1 inner 2320.2.d.g 6
12.b even 2 1 1305.2.c.h 6
20.d odd 2 1 145.2.b.c 6
20.e even 4 2 725.2.a.l 6
60.h even 2 1 1305.2.c.h 6
60.l odd 4 2 6525.2.a.bt 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.b.c 6 4.b odd 2 1
145.2.b.c 6 20.d odd 2 1
725.2.a.l 6 20.e even 4 2
1305.2.c.h 6 12.b even 2 1
1305.2.c.h 6 60.h even 2 1
2320.2.d.g 6 1.a even 1 1 trivial
2320.2.d.g 6 5.b even 2 1 inner
6525.2.a.bt 6 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 $$S_{2}^{\mathrm{new}}(2320, [\chi])$$:

 $$T_{3}^{6} + 15T_{3}^{4} + 56T_{3}^{2} + 4$$ T3^6 + 15*T3^4 + 56*T3^2 + 4 $$T_{7}^{6} + 20T_{7}^{4} + 76T_{7}^{2} + 64$$ T7^6 + 20*T7^4 + 76*T7^2 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 15 T^{4} + 56 T^{2} + 4$$
$5$ $$T^{6} - 3 T^{5} - T^{4} + 14 T^{3} + \cdots + 125$$
$7$ $$T^{6} + 20 T^{4} + 76 T^{2} + 64$$
$11$ $$(T^{3} - 5 T^{2} - 6 T + 38)^{2}$$
$13$ $$T^{6} + 59 T^{4} + 1048 T^{2} + \cdots + 5776$$
$17$ $$T^{6} + 48 T^{4} + 380 T^{2} + \cdots + 784$$
$19$ $$(T^{3} - 8 T^{2} + 6 T + 8)^{2}$$
$23$ $$T^{6} + 56 T^{4} + 60 T^{2} + 16$$
$29$ $$(T - 1)^{6}$$
$31$ $$(T^{3} + 11 T^{2} + 26 T - 22)^{2}$$
$37$ $$T^{6} + 20 T^{4} + 76 T^{2} + 64$$
$41$ $$T^{6}$$
$43$ $$T^{6} + 127 T^{4} + 1400 T^{2} + \cdots + 196$$
$47$ $$T^{6} + 63 T^{4} + 1304 T^{2} + \cdots + 8836$$
$53$ $$T^{6} + 187 T^{4} + 7640 T^{2} + \cdots + 5776$$
$59$ $$(T - 10)^{6}$$
$61$ $$(T^{3} - 6 T^{2} - 64 T - 64)^{2}$$
$67$ $$T^{6} + 140 T^{4} + 6316 T^{2} + \cdots + 92416$$
$71$ $$(T + 2)^{6}$$
$73$ $$T^{6} + 188 T^{4} + 8556 T^{2} + \cdots + 3136$$
$79$ $$(T^{3} - T^{2} - 54 T - 98)^{2}$$
$83$ $$T^{6} + 48 T^{4} + 380 T^{2} + \cdots + 784$$
$89$ $$(T^{3} - 16 T^{2} + 28 T + 304)^{2}$$
$97$ $$T^{6} + 476 T^{4} + 52396 T^{2} + \cdots + 7744$$