# Properties

 Label 2600.2.a.bc Level $2600$ Weight $2$ Character orbit 2600.a Self dual yes Analytic conductor $20.761$ 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] = [2600,2,Mod(1,2600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2600, 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("2600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2600 = 2^{3} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2600.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: $$20.7611045255$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.10592240.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 13x^{3} + 24x^{2} + 34x - 50$$ x^5 - 2*x^4 - 13*x^3 + 24*x^2 + 34*x - 50 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 520) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{3} + \beta_{2}) q^{7} + (\beta_{3} + \beta_{2} + 3) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b3 + b2) * q^7 + (b3 + b2 + 3) * q^9 $$q + \beta_1 q^{3} + (\beta_{3} + \beta_{2}) q^{7} + (\beta_{3} + \beta_{2} + 3) q^{9} + (\beta_{4} - \beta_{2} - \beta_1 + 2) q^{11} + q^{13} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{17} + ( - \beta_{4} + \beta_{2} - \beta_1 + 4) q^{19} + (\beta_{4} - \beta_{3} + \beta_{2} + \cdots - 1) q^{21}+ \cdots + (\beta_{4} + 2 \beta_{3} - 5 \beta_{2} + \cdots + 10) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b3 + b2) * q^7 + (b3 + b2 + 3) * q^9 + (b4 - b2 - b1 + 2) * q^11 + q^13 + (-b4 + b3 + b2 - 1) * q^17 + (-b4 + b2 - b1 + 4) * q^19 + (b4 - b3 + b2 + 2*b1 - 1) * q^21 + (-b4 + b3 + b2 + b1 + 3) * q^23 + (b4 - b3 + b2 + 2*b1 - 1) * q^27 + (b3 - 2*b2 + 1) * q^29 + (-b4 - b2 - b1) * q^31 + (2*b4 - b3 - 4*b2 + 2*b1 - 5) * q^33 + (2*b4 - b3 - b2) * q^37 + b1 * q^39 + (2*b3 - 2) * q^41 + (-2*b2 + b1) * q^43 + (-b3 - b2 + 4) * q^47 + (-2*b4 + b2 + 4) * q^49 + (-2*b3 + b2 - 1) * q^51 + (2*b2 + 2*b1 - 2) * q^53 + (-2*b4 - b3 + 2*b2 + 4*b1 - 7) * q^57 + (b4 - 2*b3 - b2 + b1 + 2) * q^59 + (-3*b3 - 2*b2 + 2*b1 + 1) * q^61 + (-2*b4 + 3*b3 + 4*b2 + 11) * q^63 + (-2*b4 + b3 - 2*b1 + 1) * q^67 + (-b3 + 2*b2 + 4*b1 + 5) * q^69 + (-b3 - 2*b2 - 3*b1 + 5) * q^71 + (-b3 + 2*b2 + 2*b1 - 3) * q^73 + (-2*b4 + 2*b3 - 2*b2 - 2*b1 + 4) * q^77 + (2*b2 - 2*b1 + 2) * q^79 + (-2*b4 + 3*b3 + 4*b2 + 2) * q^81 + (-3*b3 - 5) * q^83 + (4*b4 - 4*b3 - 8*b2 + 2) * q^87 + (2*b4 - 2*b3 - 2) * q^89 + (b3 + b2) * q^91 + (-3*b3 - 4*b2 - 2*b1 - 5) * q^93 + (2*b3 + 2*b2 - 2*b1 + 2) * q^97 + (b4 + 2*b3 - 5*b2 - 5*b1 + 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 2 q^{3} + 15 q^{9}+O(q^{10})$$ 5 * q + 2 * q^3 + 15 * q^9 $$5 q + 2 q^{3} + 15 q^{9} + 8 q^{11} + 5 q^{13} - 6 q^{17} + 18 q^{19} + 2 q^{21} + 16 q^{23} + 2 q^{27} + 2 q^{29} - 4 q^{31} - 22 q^{33} + 2 q^{37} + 2 q^{39} - 12 q^{41} + 20 q^{47} + 19 q^{49} - 2 q^{51} - 4 q^{53} - 26 q^{57} + 14 q^{59} + 10 q^{61} + 54 q^{63} - 2 q^{67} + 36 q^{69} + 18 q^{71} - 8 q^{73} + 10 q^{77} + 8 q^{79} + 9 q^{81} - 22 q^{83} + 10 q^{87} - 6 q^{89} - 30 q^{93} + 6 q^{97} + 34 q^{99}+O(q^{100})$$ 5 * q + 2 * q^3 + 15 * q^9 + 8 * q^11 + 5 * q^13 - 6 * q^17 + 18 * q^19 + 2 * q^21 + 16 * q^23 + 2 * q^27 + 2 * q^29 - 4 * q^31 - 22 * q^33 + 2 * q^37 + 2 * q^39 - 12 * q^41 + 20 * q^47 + 19 * q^49 - 2 * q^51 - 4 * q^53 - 26 * q^57 + 14 * q^59 + 10 * q^61 + 54 * q^63 - 2 * q^67 + 36 * q^69 + 18 * q^71 - 8 * q^73 + 10 * q^77 + 8 * q^79 + 9 * q^81 - 22 * q^83 + 10 * q^87 - 6 * q^89 - 30 * q^93 + 6 * q^97 + 34 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 13x^{3} + 24x^{2} + 34x - 50$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 2\nu^{3} - 10\nu^{2} - 16\nu + 15 ) / 5$$ (v^4 + 2*v^3 - 10*v^2 - 16*v + 15) / 5 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} - 2\nu^{3} + 15\nu^{2} + 16\nu - 45 ) / 5$$ (-v^4 - 2*v^3 + 15*v^2 + 16*v - 45) / 5 $$\beta_{4}$$ $$=$$ $$( -2\nu^{4} + \nu^{3} + 25\nu^{2} - 8\nu - 55 ) / 5$$ (-2*v^4 + v^3 + 25*v^2 - 8*v - 55) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 6$$ b3 + b2 + 6 $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} + 8\beta _1 - 1$$ b4 - b3 + b2 + 8*b1 - 1 $$\nu^{4}$$ $$=$$ $$-2\beta_{4} + 12\beta_{3} + 13\beta_{2} + 47$$ -2*b4 + 12*b3 + 13*b2 + 47

## 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
 −3.05748 −1.76881 1.16232 2.57527 3.08870
0 −3.05748 0 0 0 3.34821 0 6.34821 0
1.2 0 −1.76881 0 0 0 −2.87131 0 0.128689 0
1.3 0 1.16232 0 0 0 −4.64901 0 −1.64901 0
1.4 0 2.57527 0 0 0 0.632021 0 3.63202 0
1.5 0 3.08870 0 0 0 3.54010 0 6.54010 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## 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 2600.2.a.bc 5
4.b odd 2 1 5200.2.a.cm 5
5.b even 2 1 2600.2.a.bb 5
5.c odd 4 2 520.2.d.c 10
15.e even 4 2 4680.2.l.i 10
20.d odd 2 1 5200.2.a.cn 5
20.e even 4 2 1040.2.d.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.d.c 10 5.c odd 4 2
1040.2.d.f 10 20.e even 4 2
2600.2.a.bb 5 5.b even 2 1
2600.2.a.bc 5 1.a even 1 1 trivial
4680.2.l.i 10 15.e even 4 2
5200.2.a.cm 5 4.b odd 2 1
5200.2.a.cn 5 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2600))$$:

 $$T_{3}^{5} - 2T_{3}^{4} - 13T_{3}^{3} + 24T_{3}^{2} + 34T_{3} - 50$$ T3^5 - 2*T3^4 - 13*T3^3 + 24*T3^2 + 34*T3 - 50 $$T_{7}^{5} - 27T_{7}^{3} + 14T_{7}^{2} + 160T_{7} - 100$$ T7^5 - 27*T7^3 + 14*T7^2 + 160*T7 - 100 $$T_{11}^{5} - 8T_{11}^{4} - 20T_{11}^{3} + 222T_{11}^{2} + 72T_{11} - 1544$$ T11^5 - 8*T11^4 - 20*T11^3 + 222*T11^2 + 72*T11 - 1544

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 2 T^{4} + \cdots - 50$$
$5$ $$T^{5}$$
$7$ $$T^{5} - 27 T^{3} + \cdots - 100$$
$11$ $$T^{5} - 8 T^{4} + \cdots - 1544$$
$13$ $$(T - 1)^{5}$$
$17$ $$T^{5} + 6 T^{4} + \cdots - 72$$
$19$ $$T^{5} - 18 T^{4} + \cdots + 5368$$
$23$ $$T^{5} - 16 T^{4} + \cdots + 1616$$
$29$ $$T^{5} - 2 T^{4} + \cdots - 9232$$
$31$ $$T^{5} + 4 T^{4} + \cdots + 1944$$
$37$ $$T^{5} - 2 T^{4} + \cdots + 1660$$
$41$ $$T^{5} + 12 T^{4} + \cdots + 5120$$
$43$ $$T^{5} - 89 T^{3} + \cdots + 5010$$
$47$ $$T^{5} - 20 T^{4} + \cdots - 60$$
$53$ $$T^{5} + 4 T^{4} + \cdots - 1408$$
$59$ $$T^{5} - 14 T^{4} + \cdots + 4008$$
$61$ $$T^{5} - 10 T^{4} + \cdots + 1472$$
$67$ $$T^{5} + 2 T^{4} + \cdots + 36304$$
$71$ $$T^{5} - 18 T^{4} + \cdots - 5770$$
$73$ $$T^{5} + 8 T^{4} + \cdots + 9232$$
$79$ $$T^{5} - 8 T^{4} + \cdots - 14720$$
$83$ $$T^{5} + 22 T^{4} + \cdots + 13472$$
$89$ $$T^{5} + 6 T^{4} + \cdots + 2208$$
$97$ $$T^{5} - 6 T^{4} + \cdots + 160$$