Properties

 Label 6498.2.a.bu Level $6498$ Weight $2$ Character orbit 6498.a Self dual yes Analytic conductor $51.887$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6498, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6498 = 2 \cdot 3^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6498.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: $$51.8867912334$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + ( - \beta_1 + 2) q^{5} + ( - \beta_{2} - 2 \beta_1 + 1) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + (-b1 + 2) * q^5 + (-b2 - 2*b1 + 1) * q^7 + q^8 $$q + q^{2} + q^{4} + ( - \beta_1 + 2) q^{5} + ( - \beta_{2} - 2 \beta_1 + 1) q^{7} + q^{8} + ( - \beta_1 + 2) q^{10} + (2 \beta_{2} - 3 \beta_1) q^{11} + (2 \beta_{2} - \beta_1 - 2) q^{13} + ( - \beta_{2} - 2 \beta_1 + 1) q^{14} + q^{16} + ( - \beta_{2} + 2) q^{17} + ( - \beta_1 + 2) q^{20} + (2 \beta_{2} - 3 \beta_1) q^{22} + ( - 6 \beta_{2} + 3 \beta_1 - 2) q^{23} + (\beta_{2} - 4 \beta_1 + 1) q^{25} + (2 \beta_{2} - \beta_1 - 2) q^{26} + ( - \beta_{2} - 2 \beta_1 + 1) q^{28} + ( - \beta_{2} + \beta_1 + 2) q^{29} + (\beta_{2} - \beta_1 - 5) q^{31} + q^{32} + ( - \beta_{2} + 2) q^{34} + ( - 4 \beta_1 + 7) q^{35} + (4 \beta_1 + 1) q^{37} + ( - \beta_1 + 2) q^{40} + (3 \beta_{2} - 2 \beta_1 + 3) q^{41} + ( - 2 \beta_{2} + 5 \beta_1 + 3) q^{43} + (2 \beta_{2} - 3 \beta_1) q^{44} + ( - 6 \beta_{2} + 3 \beta_1 - 2) q^{46} + (3 \beta_{2} + \beta_1 + 3) q^{47} + (\beta_{2} + \beta_1 + 8) q^{49} + (\beta_{2} - 4 \beta_1 + 1) q^{50} + (2 \beta_{2} - \beta_1 - 2) q^{52} + (2 \beta_1 + 9) q^{53} + (7 \beta_{2} - 8 \beta_1 + 4) q^{55} + ( - \beta_{2} - 2 \beta_1 + 1) q^{56} + ( - \beta_{2} + \beta_1 + 2) q^{58} + (\beta_{2} + 4 \beta_1 + 6) q^{59} + ( - 3 \beta_{2} + \beta_1 + 1) q^{61} + (\beta_{2} - \beta_1 - 5) q^{62} + q^{64} + (5 \beta_{2} - 2 \beta_1 - 4) q^{65} + ( - 11 \beta_{2} + 5 \beta_1 - 4) q^{67} + ( - \beta_{2} + 2) q^{68} + ( - 4 \beta_1 + 7) q^{70} + ( - 2 \beta_{2} + 5 \beta_1 + 7) q^{71} + ( - 3 \beta_{2} + \beta_1 + 5) q^{73} + (4 \beta_1 + 1) q^{74} + (10 \beta_{2} - 6 \beta_1 + 7) q^{77} + (5 \beta_{2} - 4 \beta_1 - 5) q^{79} + ( - \beta_1 + 2) q^{80} + (3 \beta_{2} - 2 \beta_1 + 3) q^{82} + ( - 4 \beta_{2} + \beta_1 + 1) q^{83} + ( - 2 \beta_{2} - \beta_1 + 5) q^{85} + ( - 2 \beta_{2} + 5 \beta_1 + 3) q^{86} + (2 \beta_{2} - 3 \beta_1) q^{88} + ( - 7 \beta_{2} + 7 \beta_1 + 1) q^{89} + (8 \beta_{2} - 2 \beta_1 - 5) q^{91} + ( - 6 \beta_{2} + 3 \beta_1 - 2) q^{92} + (3 \beta_{2} + \beta_1 + 3) q^{94} + ( - 8 \beta_{2} + 7 \beta_1 - 4) q^{97} + (\beta_{2} + \beta_1 + 8) q^{98}+O(q^{100})$$ q + q^2 + q^4 + (-b1 + 2) * q^5 + (-b2 - 2*b1 + 1) * q^7 + q^8 + (-b1 + 2) * q^10 + (2*b2 - 3*b1) * q^11 + (2*b2 - b1 - 2) * q^13 + (-b2 - 2*b1 + 1) * q^14 + q^16 + (-b2 + 2) * q^17 + (-b1 + 2) * q^20 + (2*b2 - 3*b1) * q^22 + (-6*b2 + 3*b1 - 2) * q^23 + (b2 - 4*b1 + 1) * q^25 + (2*b2 - b1 - 2) * q^26 + (-b2 - 2*b1 + 1) * q^28 + (-b2 + b1 + 2) * q^29 + (b2 - b1 - 5) * q^31 + q^32 + (-b2 + 2) * q^34 + (-4*b1 + 7) * q^35 + (4*b1 + 1) * q^37 + (-b1 + 2) * q^40 + (3*b2 - 2*b1 + 3) * q^41 + (-2*b2 + 5*b1 + 3) * q^43 + (2*b2 - 3*b1) * q^44 + (-6*b2 + 3*b1 - 2) * q^46 + (3*b2 + b1 + 3) * q^47 + (b2 + b1 + 8) * q^49 + (b2 - 4*b1 + 1) * q^50 + (2*b2 - b1 - 2) * q^52 + (2*b1 + 9) * q^53 + (7*b2 - 8*b1 + 4) * q^55 + (-b2 - 2*b1 + 1) * q^56 + (-b2 + b1 + 2) * q^58 + (b2 + 4*b1 + 6) * q^59 + (-3*b2 + b1 + 1) * q^61 + (b2 - b1 - 5) * q^62 + q^64 + (5*b2 - 2*b1 - 4) * q^65 + (-11*b2 + 5*b1 - 4) * q^67 + (-b2 + 2) * q^68 + (-4*b1 + 7) * q^70 + (-2*b2 + 5*b1 + 7) * q^71 + (-3*b2 + b1 + 5) * q^73 + (4*b1 + 1) * q^74 + (10*b2 - 6*b1 + 7) * q^77 + (5*b2 - 4*b1 - 5) * q^79 + (-b1 + 2) * q^80 + (3*b2 - 2*b1 + 3) * q^82 + (-4*b2 + b1 + 1) * q^83 + (-2*b2 - b1 + 5) * q^85 + (-2*b2 + 5*b1 + 3) * q^86 + (2*b2 - 3*b1) * q^88 + (-7*b2 + 7*b1 + 1) * q^89 + (8*b2 - 2*b1 - 5) * q^91 + (-6*b2 + 3*b1 - 2) * q^92 + (3*b2 + b1 + 3) * q^94 + (-8*b2 + 7*b1 - 4) * q^97 + (b2 + b1 + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 $$3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} + 6 q^{20} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} + 6 q^{29} - 15 q^{31} + 3 q^{32} + 6 q^{34} + 21 q^{35} + 3 q^{37} + 6 q^{40} + 9 q^{41} + 9 q^{43} - 6 q^{46} + 9 q^{47} + 24 q^{49} + 3 q^{50} - 6 q^{52} + 27 q^{53} + 12 q^{55} + 3 q^{56} + 6 q^{58} + 18 q^{59} + 3 q^{61} - 15 q^{62} + 3 q^{64} - 12 q^{65} - 12 q^{67} + 6 q^{68} + 21 q^{70} + 21 q^{71} + 15 q^{73} + 3 q^{74} + 21 q^{77} - 15 q^{79} + 6 q^{80} + 9 q^{82} + 3 q^{83} + 15 q^{85} + 9 q^{86} + 3 q^{89} - 15 q^{91} - 6 q^{92} + 9 q^{94} - 12 q^{97} + 24 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^10 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 + 6 * q^20 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 + 6 * q^29 - 15 * q^31 + 3 * q^32 + 6 * q^34 + 21 * q^35 + 3 * q^37 + 6 * q^40 + 9 * q^41 + 9 * q^43 - 6 * q^46 + 9 * q^47 + 24 * q^49 + 3 * q^50 - 6 * q^52 + 27 * q^53 + 12 * q^55 + 3 * q^56 + 6 * q^58 + 18 * q^59 + 3 * q^61 - 15 * q^62 + 3 * q^64 - 12 * q^65 - 12 * q^67 + 6 * q^68 + 21 * q^70 + 21 * q^71 + 15 * q^73 + 3 * q^74 + 21 * q^77 - 15 * q^79 + 6 * q^80 + 9 * q^82 + 3 * q^83 + 15 * q^85 + 9 * q^86 + 3 * q^89 - 15 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^97 + 24 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

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
 1.87939 −0.347296 −1.53209
1.00000 0 1.00000 0.120615 0 −4.29086 1.00000 0 0.120615
1.2 1.00000 0 1.00000 2.34730 0 3.57398 1.00000 0 2.34730
1.3 1.00000 0 1.00000 3.53209 0 3.71688 1.00000 0 3.53209
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

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 6498.2.a.bu 3
3.b odd 2 1 2166.2.a.p 3
19.b odd 2 1 6498.2.a.bp 3
19.f odd 18 2 342.2.u.b 6
57.d even 2 1 2166.2.a.r 3
57.j even 18 2 114.2.i.c 6
228.u odd 18 2 912.2.bo.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.c 6 57.j even 18 2
342.2.u.b 6 19.f odd 18 2
912.2.bo.d 6 228.u odd 18 2
2166.2.a.p 3 3.b odd 2 1
2166.2.a.r 3 57.d even 2 1
6498.2.a.bp 3 19.b odd 2 1
6498.2.a.bu 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6498))$$:

 $$T_{5}^{3} - 6T_{5}^{2} + 9T_{5} - 1$$ T5^3 - 6*T5^2 + 9*T5 - 1 $$T_{7}^{3} - 3T_{7}^{2} - 18T_{7} + 57$$ T7^3 - 3*T7^2 - 18*T7 + 57 $$T_{11}^{3} - 21T_{11} - 37$$ T11^3 - 21*T11 - 37 $$T_{13}^{3} + 6T_{13}^{2} + 3T_{13} - 1$$ T13^3 + 6*T13^2 + 3*T13 - 1 $$T_{29}^{3} - 6T_{29}^{2} + 9T_{29} - 1$$ T29^3 - 6*T29^2 + 9*T29 - 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 6 T^{2} + 9 T - 1$$
$7$ $$T^{3} - 3 T^{2} - 18 T + 57$$
$11$ $$T^{3} - 21T - 37$$
$13$ $$T^{3} + 6 T^{2} + 3 T - 1$$
$17$ $$T^{3} - 6 T^{2} + 9 T - 3$$
$19$ $$T^{3}$$
$23$ $$T^{3} + 6 T^{2} - 69 T - 397$$
$29$ $$T^{3} - 6 T^{2} + 9 T - 1$$
$31$ $$T^{3} + 15 T^{2} + 72 T + 109$$
$37$ $$T^{3} - 3 T^{2} - 45 T - 17$$
$41$ $$T^{3} - 9 T^{2} + 6 T + 53$$
$43$ $$T^{3} - 9 T^{2} - 30 T + 251$$
$47$ $$T^{3} - 9 T^{2} - 12 T + 71$$
$53$ $$T^{3} - 27 T^{2} + 231 T - 629$$
$59$ $$T^{3} - 18 T^{2} + 45 T - 9$$
$61$ $$T^{3} - 3 T^{2} - 18 T - 17$$
$67$ $$T^{3} + 12 T^{2} - 225 T - 2649$$
$71$ $$T^{3} - 21 T^{2} + 90 T + 163$$
$73$ $$T^{3} - 15 T^{2} + 54 T - 57$$
$79$ $$T^{3} + 15 T^{2} + 12 T - 181$$
$83$ $$T^{3} - 3 T^{2} - 36 T - 51$$
$89$ $$T^{3} - 3 T^{2} - 144 T + 489$$
$97$ $$T^{3} + 12 T^{2} - 123 T - 467$$