Properties

 Label 2320.2.a.v Level $2320$ Weight $2$ Character orbit 2320.a Self dual yes Analytic conductor $18.525$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.3145252.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 11x^{3} + 9x^{2} + 22x - 11$$ x^5 - 2*x^4 - 11*x^3 + 9*x^2 + 22*x - 11 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{4} - \beta_{2} + 2) q^{9}+O(q^{10})$$ q - b3 * q^3 - q^5 + (b2 - 1) * q^7 + (b4 - b2 + 2) * q^9 $$q - \beta_{3} q^{3} - q^{5} + (\beta_{2} - 1) q^{7} + (\beta_{4} - \beta_{2} + 2) q^{9} + (\beta_{4} + 2) q^{11} + ( - \beta_1 + 1) q^{13} + \beta_{3} q^{15} + (\beta_{2} + 1) q^{17} + ( - \beta_{3} + \beta_1 - 1) q^{19} + (3 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{21} + ( - \beta_{3} - 2 \beta_1 - 2) q^{23} + q^{25} + ( - \beta_{4} - 4 \beta_{3} + \cdots - \beta_1) q^{27}+ \cdots + (3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 18) q^{99}+O(q^{100})$$ q - b3 * q^3 - q^5 + (b2 - 1) * q^7 + (b4 - b2 + 2) * q^9 + (b4 + 2) * q^11 + (-b1 + 1) * q^13 + b3 * q^15 + (b2 + 1) * q^17 + (-b3 + b1 - 1) * q^19 + (3*b3 + b2 + 2*b1 + 1) * q^21 + (-b3 - 2*b1 - 2) * q^23 + q^25 + (-b4 - 4*b3 - b2 - b1) * q^27 + q^29 + (-2*b3 - b2 + 2*b1 - 3) * q^31 + (-b4 - 5*b3 + b1 + 1) * q^33 + (-b2 + 1) * q^35 + (-b3 - 3*b1 + 3) * q^37 + (-b3 - b2 - b1) * q^39 + (-b4 - b3 - b1 + 1) * q^41 + (b4 + 3*b3 + b2 - b1 - 2) * q^43 + (-b4 + b2 - 2) * q^45 + (-b4 + 2*b1 - 2) * q^47 + (-2*b3 - 2*b2 + b1 + 4) * q^49 + (b3 + b2 + 2*b1 + 1) * q^51 + (-b4 - 2*b3 + b2 + 1) * q^53 + (-b4 - 2) * q^55 + (b4 + b3 + b1 + 5) * q^57 + (b1 + 5) * q^59 + (2*b3 + 2*b2 + b1 + 5) * q^61 + (-3*b4 + b3 + 3*b2 + 4*b1 - 11) * q^63 + (b1 - 1) * q^65 + (b4 + 2*b3 - 4*b1 + 4) * q^67 + (b4 + 2*b3 - 3*b2 - 2*b1 + 5) * q^69 + (b4 + b3 + 2*b2 - 3*b1 + 3) * q^71 + (b4 + b3 - b2 + b1 + 4) * q^73 - b3 * q^75 + (-3*b4 - b3 + 2*b2 + 5*b1 - 1) * q^77 + (-b3 - 4*b1 + 2) * q^79 + (2*b4 + b3 - 3*b2 - 4*b1 + 12) * q^81 + (3*b3 - 2*b2 + 3*b1 + 1) * q^83 + (-b2 - 1) * q^85 - b3 * q^87 + (-2*b3 - 2) * q^89 + (-b4 + 2*b3 + b2 - b1 - 2) * q^91 + (2*b4 + b3 - b2 + 9) * q^93 + (b3 - b1 + 1) * q^95 + (-3*b3 + 2*b2 - 2*b1 + 6) * q^97 + (3*b4 + 2*b3 - 4*b2 + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + q^{3} - 5 q^{5} - 7 q^{7} + 12 q^{9}+O(q^{10})$$ 5 * q + q^3 - 5 * q^5 - 7 * q^7 + 12 * q^9 $$5 q + q^{3} - 5 q^{5} - 7 q^{7} + 12 q^{9} + 10 q^{11} + 3 q^{13} - q^{15} + 3 q^{17} - 2 q^{19} + 4 q^{21} - 13 q^{23} + 5 q^{25} + 4 q^{27} + 5 q^{29} - 7 q^{31} + 12 q^{33} + 7 q^{35} + 10 q^{37} + q^{39} + 4 q^{41} - 17 q^{43} - 12 q^{45} - 6 q^{47} + 28 q^{49} + 6 q^{51} + 5 q^{53} - 10 q^{55} + 26 q^{57} + 27 q^{59} + 21 q^{61} - 54 q^{63} - 3 q^{65} + 10 q^{67} + 25 q^{69} + 4 q^{71} + 23 q^{73} + q^{75} + 2 q^{77} + 3 q^{79} + 57 q^{81} + 12 q^{83} - 3 q^{85} + q^{87} - 8 q^{89} - 16 q^{91} + 46 q^{93} + 2 q^{95} + 25 q^{97} + 96 q^{99}+O(q^{100})$$ 5 * q + q^3 - 5 * q^5 - 7 * q^7 + 12 * q^9 + 10 * q^11 + 3 * q^13 - q^15 + 3 * q^17 - 2 * q^19 + 4 * q^21 - 13 * q^23 + 5 * q^25 + 4 * q^27 + 5 * q^29 - 7 * q^31 + 12 * q^33 + 7 * q^35 + 10 * q^37 + q^39 + 4 * q^41 - 17 * q^43 - 12 * q^45 - 6 * q^47 + 28 * q^49 + 6 * q^51 + 5 * q^53 - 10 * q^55 + 26 * q^57 + 27 * q^59 + 21 * q^61 - 54 * q^63 - 3 * q^65 + 10 * q^67 + 25 * q^69 + 4 * q^71 + 23 * q^73 + q^75 + 2 * q^77 + 3 * q^79 + 57 * q^81 + 12 * q^83 - 3 * q^85 + q^87 - 8 * q^89 - 16 * q^91 + 46 * q^93 + 2 * q^95 + 25 * q^97 + 96 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 11x^{3} + 9x^{2} + 22x - 11$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 9\nu - 11 ) / 4$$ (v^4 - 3*v^3 - 4*v^2 + 9*v - 11) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 3\nu^{3} - 8\nu^{2} + 13\nu + 9 ) / 4$$ (v^4 - 3*v^3 - 8*v^2 + 13*v + 9) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + \nu^{3} - 16\nu^{2} - 15\nu + 25 ) / 4$$ (v^4 + v^3 - 16*v^2 - 15*v + 25) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta _1 + 5$$ -b3 + b2 + b1 + 5 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 3\beta_{3} + 2\beta_{2} + 9\beta _1 + 6$$ b4 - 3*b3 + 2*b2 + 9*b1 + 6 $$\nu^{4}$$ $$=$$ $$3\beta_{4} - 13\beta_{3} + 14\beta_{2} + 22\beta _1 + 49$$ 3*b4 - 13*b3 + 14*b2 + 22*b1 + 49

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
 0.465076 1.58684 −2.28850 3.90462 −1.66804
0 −3.26516 0 −1.00000 0 −2.98362 0 7.66125 0
1.2 0 −0.959449 0 −1.00000 0 −4.10933 0 −2.07946 0
1.3 0 −0.184115 0 −1.00000 0 1.70985 0 −2.96610 0
1.4 0 2.08906 0 −1.00000 0 3.25238 0 1.36419 0
1.5 0 3.31966 0 −1.00000 0 −4.86927 0 8.02012 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.v 5
4.b odd 2 1 1160.2.a.h 5
8.b even 2 1 9280.2.a.ci 5
8.d odd 2 1 9280.2.a.ck 5
20.d odd 2 1 5800.2.a.u 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1160.2.a.h 5 4.b odd 2 1
2320.2.a.v 5 1.a even 1 1 trivial
5800.2.a.u 5 20.d odd 2 1
9280.2.a.ci 5 8.b even 2 1
9280.2.a.ck 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2320))$$:

 $$T_{3}^{5} - T_{3}^{4} - 13T_{3}^{3} + 10T_{3}^{2} + 24T_{3} + 4$$ T3^5 - T3^4 - 13*T3^3 + 10*T3^2 + 24*T3 + 4 $$T_{7}^{5} + 7T_{7}^{4} - 7T_{7}^{3} - 106T_{7}^{2} - 36T_{7} + 332$$ T7^5 + 7*T7^4 - 7*T7^3 - 106*T7^2 - 36*T7 + 332 $$T_{11}^{5} - 10T_{11}^{4} - 4T_{11}^{3} + 276T_{11}^{2} - 616T_{11} - 176$$ T11^5 - 10*T11^4 - 4*T11^3 + 276*T11^2 - 616*T11 - 176

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - T^{4} - 13 T^{3} + \cdots + 4$$
$5$ $$(T + 1)^{5}$$
$7$ $$T^{5} + 7 T^{4} + \cdots + 332$$
$11$ $$T^{5} - 10 T^{4} + \cdots - 176$$
$13$ $$T^{5} - 3 T^{4} + \cdots - 8$$
$17$ $$T^{5} - 3 T^{4} + \cdots + 116$$
$19$ $$T^{5} + 2 T^{4} + \cdots + 16$$
$23$ $$T^{5} + 13 T^{4} + \cdots + 3268$$
$29$ $$(T - 1)^{5}$$
$31$ $$T^{5} + 7 T^{4} + \cdots - 1900$$
$37$ $$T^{5} - 10 T^{4} + \cdots + 3280$$
$41$ $$T^{5} - 4 T^{4} + \cdots - 3296$$
$43$ $$T^{5} + 17 T^{4} + \cdots + 6644$$
$47$ $$T^{5} + 6 T^{4} + \cdots + 2816$$
$53$ $$T^{5} - 5 T^{4} + \cdots + 1864$$
$59$ $$T^{5} - 27 T^{4} + \cdots - 2896$$
$61$ $$T^{5} - 21 T^{4} + \cdots + 22808$$
$67$ $$T^{5} - 10 T^{4} + \cdots - 79808$$
$71$ $$T^{5} - 4 T^{4} + \cdots - 1024$$
$73$ $$T^{5} - 23 T^{4} + \cdots + 2644$$
$79$ $$T^{5} - 3 T^{4} + \cdots + 25156$$
$83$ $$T^{5} - 12 T^{4} + \cdots + 33008$$
$89$ $$T^{5} + 8 T^{4} + \cdots + 800$$
$97$ $$T^{5} - 25 T^{4} + \cdots - 126116$$