Properties

 Label 8512.2.a.bn Level $8512$ Weight $2$ Character orbit 8512.a Self dual yes Analytic conductor $67.969$ 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] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8512 = 2^{6} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8512.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: $$67.9686622005$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.733.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x + 8$$ x^3 - x^2 - 7*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 532) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{2} + \beta_1 - 1) q^{5} - q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b2 + b1 - 1) * q^5 - q^7 + (b2 + 2) * q^9 $$q + \beta_1 q^{3} + (\beta_{2} + \beta_1 - 1) q^{5} - q^{7} + (\beta_{2} + 2) q^{9} + (\beta_{2} + 1) q^{11} + (\beta_{2} + \beta_1 - 3) q^{13} + (2 \beta_{2} + \beta_1 + 2) q^{15} + (\beta_{2} + 3) q^{17} + q^{19} - \beta_1 q^{21} + (\beta_{2} + \beta_1 + 1) q^{23} + ( - \beta_{2} + \beta_1 + 2) q^{25} + (\beta_{2} + \beta_1 - 3) q^{27} + ( - \beta_{2} - 2 \beta_1 - 3) q^{29} + ( - \beta_{2} - 2 \beta_1 - 1) q^{31} + (\beta_{2} + 3 \beta_1 - 3) q^{33} + ( - \beta_{2} - \beta_1 + 1) q^{35} + (\beta_{2} - \beta_1 - 1) q^{37} + (2 \beta_{2} - \beta_1 + 2) q^{39} + ( - \beta_{2} + 2 \beta_1 + 3) q^{41} + (2 \beta_{2} - 2 \beta_1 + 2) q^{43} + (3 \beta_1 + 2) q^{45} + (3 \beta_{2} + \beta_1 - 1) q^{47} + q^{49} + (\beta_{2} + 5 \beta_1 - 3) q^{51} + ( - 3 \beta_1 - 2) q^{53} + ( - \beta_{2} + 2 \beta_1 + 3) q^{55} + \beta_1 q^{57} + (3 \beta_{2} + 3 \beta_1 + 3) q^{59} + (\beta_{2} + 5 \beta_1 - 1) q^{61} + ( - \beta_{2} - 2) q^{63} + ( - 3 \beta_{2} - \beta_1 + 9) q^{65} + (\beta_{2} + 3) q^{67} + (2 \beta_{2} + 3 \beta_1 + 2) q^{69} + (\beta_{2} + 3 \beta_1 - 1) q^{71} + (2 \beta_{2} + 3 \beta_1 - 8) q^{73} + 8 q^{75} + ( - \beta_{2} - 1) q^{77} + (4 \beta_{2} + 2) q^{79} + ( - \beta_{2} - \beta_1 - 4) q^{81} + ( - 2 \beta_{2} + \beta_1 + 2) q^{83} + (\beta_{2} + 4 \beta_1 + 1) q^{85} + ( - 3 \beta_{2} - 5 \beta_1 - 7) q^{87} + (2 \beta_{2} - 6) q^{89} + ( - \beta_{2} - \beta_1 + 3) q^{91} + ( - 3 \beta_{2} - 3 \beta_1 - 7) q^{93} + (\beta_{2} + \beta_1 - 1) q^{95} + (5 \beta_{2} + \beta_1 + 9) q^{97} + (\beta_{2} - \beta_1 + 9) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b2 + b1 - 1) * q^5 - q^7 + (b2 + 2) * q^9 + (b2 + 1) * q^11 + (b2 + b1 - 3) * q^13 + (2*b2 + b1 + 2) * q^15 + (b2 + 3) * q^17 + q^19 - b1 * q^21 + (b2 + b1 + 1) * q^23 + (-b2 + b1 + 2) * q^25 + (b2 + b1 - 3) * q^27 + (-b2 - 2*b1 - 3) * q^29 + (-b2 - 2*b1 - 1) * q^31 + (b2 + 3*b1 - 3) * q^33 + (-b2 - b1 + 1) * q^35 + (b2 - b1 - 1) * q^37 + (2*b2 - b1 + 2) * q^39 + (-b2 + 2*b1 + 3) * q^41 + (2*b2 - 2*b1 + 2) * q^43 + (3*b1 + 2) * q^45 + (3*b2 + b1 - 1) * q^47 + q^49 + (b2 + 5*b1 - 3) * q^51 + (-3*b1 - 2) * q^53 + (-b2 + 2*b1 + 3) * q^55 + b1 * q^57 + (3*b2 + 3*b1 + 3) * q^59 + (b2 + 5*b1 - 1) * q^61 + (-b2 - 2) * q^63 + (-3*b2 - b1 + 9) * q^65 + (b2 + 3) * q^67 + (2*b2 + 3*b1 + 2) * q^69 + (b2 + 3*b1 - 1) * q^71 + (2*b2 + 3*b1 - 8) * q^73 + 8 * q^75 + (-b2 - 1) * q^77 + (4*b2 + 2) * q^79 + (-b2 - b1 - 4) * q^81 + (-2*b2 + b1 + 2) * q^83 + (b2 + 4*b1 + 1) * q^85 + (-3*b2 - 5*b1 - 7) * q^87 + (2*b2 - 6) * q^89 + (-b2 - b1 + 3) * q^91 + (-3*b2 - 3*b1 - 7) * q^93 + (b2 + b1 - 1) * q^95 + (5*b2 + b1 + 9) * q^97 + (b2 - b1 + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10})$$ 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 $$3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 8 q^{13} + 7 q^{15} + 9 q^{17} + 3 q^{19} - q^{21} + 4 q^{23} + 7 q^{25} - 8 q^{27} - 11 q^{29} - 5 q^{31} - 6 q^{33} + 2 q^{35} - 4 q^{37} + 5 q^{39} + 11 q^{41} + 4 q^{43} + 9 q^{45} - 2 q^{47} + 3 q^{49} - 4 q^{51} - 9 q^{53} + 11 q^{55} + q^{57} + 12 q^{59} + 2 q^{61} - 6 q^{63} + 26 q^{65} + 9 q^{67} + 9 q^{69} - 21 q^{73} + 24 q^{75} - 3 q^{77} + 6 q^{79} - 13 q^{81} + 7 q^{83} + 7 q^{85} - 26 q^{87} - 18 q^{89} + 8 q^{91} - 24 q^{93} - 2 q^{95} + 28 q^{97} + 26 q^{99}+O(q^{100})$$ 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 8 * q^13 + 7 * q^15 + 9 * q^17 + 3 * q^19 - q^21 + 4 * q^23 + 7 * q^25 - 8 * q^27 - 11 * q^29 - 5 * q^31 - 6 * q^33 + 2 * q^35 - 4 * q^37 + 5 * q^39 + 11 * q^41 + 4 * q^43 + 9 * q^45 - 2 * q^47 + 3 * q^49 - 4 * q^51 - 9 * q^53 + 11 * q^55 + q^57 + 12 * q^59 + 2 * q^61 - 6 * q^63 + 26 * q^65 + 9 * q^67 + 9 * q^69 - 21 * q^73 + 24 * q^75 - 3 * q^77 + 6 * q^79 - 13 * q^81 + 7 * q^83 + 7 * q^85 - 26 * q^87 - 18 * q^89 + 8 * q^91 - 24 * q^93 - 2 * q^95 + 28 * q^97 + 26 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ b2 + 5

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
 −2.69639 1.17819 2.51820
0 −2.69639 0 −1.42586 0 −1.00000 0 4.27053 0
1.2 0 1.17819 0 −3.43366 0 −1.00000 0 −1.61186 0
1.3 0 2.51820 0 2.85952 0 −1.00000 0 3.34132 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$7$$ $$+1$$
$$19$$ $$-1$$

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 8512.2.a.bn 3
4.b odd 2 1 8512.2.a.bl 3
8.b even 2 1 532.2.a.e 3
8.d odd 2 1 2128.2.a.r 3
24.h odd 2 1 4788.2.a.o 3
56.h odd 2 1 3724.2.a.i 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.e 3 8.b even 2 1
2128.2.a.r 3 8.d odd 2 1
3724.2.a.i 3 56.h odd 2 1
4788.2.a.o 3 24.h odd 2 1
8512.2.a.bl 3 4.b odd 2 1
8512.2.a.bn 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8512))$$:

 $$T_{3}^{3} - T_{3}^{2} - 7T_{3} + 8$$ T3^3 - T3^2 - 7*T3 + 8 $$T_{5}^{3} + 2T_{5}^{2} - 9T_{5} - 14$$ T5^3 + 2*T5^2 - 9*T5 - 14 $$T_{11}^{3} - 3T_{11}^{2} - 7T_{11} + 20$$ T11^3 - 3*T11^2 - 7*T11 + 20 $$T_{23}^{3} - 4T_{23}^{2} - 5T_{23} + 4$$ T23^3 - 4*T23^2 - 5*T23 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 7T + 8$$
$5$ $$T^{3} + 2 T^{2} + \cdots - 14$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} - 3 T^{2} + \cdots + 20$$
$13$ $$T^{3} + 8 T^{2} + \cdots - 16$$
$17$ $$T^{3} - 9 T^{2} + \cdots + 14$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 4 T^{2} + \cdots + 4$$
$29$ $$T^{3} + 11 T^{2} + \cdots - 2$$
$31$ $$T^{3} + 5 T^{2} + \cdots + 4$$
$37$ $$T^{3} + 4 T^{2} + \cdots - 50$$
$41$ $$T^{3} - 11 T^{2} + \cdots + 280$$
$43$ $$T^{3} - 4 T^{2} + \cdots - 32$$
$47$ $$T^{3} + 2 T^{2} + \cdots + 184$$
$53$ $$T^{3} + 9 T^{2} + \cdots - 322$$
$59$ $$T^{3} - 12 T^{2} + \cdots + 108$$
$61$ $$T^{3} - 2 T^{2} + \cdots + 202$$
$67$ $$T^{3} - 9 T^{2} + \cdots + 14$$
$71$ $$T^{3} - 55T - 58$$
$73$ $$T^{3} + 21 T^{2} + \cdots - 302$$
$79$ $$T^{3} - 6 T^{2} + \cdots + 1016$$
$83$ $$T^{3} - 7 T^{2} + \cdots + 100$$
$89$ $$T^{3} + 18 T^{2} + \cdots + 64$$
$97$ $$T^{3} - 28 T^{2} + \cdots + 2536$$