# Properties

 Label 57.2.j.a Level $57$ Weight $2$ Character orbit 57.j Analytic conductor $0.455$ Analytic rank $0$ Dimension $6$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,2,Mod(2,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("57.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 57.j (of order $$18$$, degree $$6$$, 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: $$0.455147291521$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

## $q$-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 primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{18}^{5} - \zeta_{18}^{2}) q^{3} + 2 \zeta_{18} q^{4} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 3 \zeta_{18}) q^{7} - 3 \zeta_{18}^{4} q^{9} +O(q^{10})$$ q + (2*z^5 - z^2) * q^3 + 2*z * q^4 + (-3*z^5 + 2*z^4 + 2*z^2 - 3*z) * q^7 - 3*z^4 * q^9 $$q + (2 \zeta_{18}^{5} - \zeta_{18}^{2}) q^{3} + 2 \zeta_{18} q^{4} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 3 \zeta_{18}) q^{7} - 3 \zeta_{18}^{4} q^{9} + (2 \zeta_{18}^{3} - 4) q^{12} + ( - \zeta_{18}^{5} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 1) q^{13} + 4 \zeta_{18}^{2} q^{16} + (5 \zeta_{18}^{3} - 2) q^{19} + (5 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - \zeta_{18} + 4) q^{21} + ( - 5 \zeta_{18}^{4} + 5 \zeta_{18}) q^{25} + (3 \zeta_{18}^{3} + 3) q^{27} + (4 \zeta_{18}^{5} - 2 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 6) q^{28} + (5 \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} - 5 \zeta_{18}) q^{31} - 6 \zeta_{18}^{5} q^{36} + (7 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 4 \zeta_{18}^{2} + 4 \zeta_{18}) q^{37} + ( - 5 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 7 \zeta_{18}^{2} + 7 \zeta_{18}) q^{39} + (7 \zeta_{18}^{4} - \zeta_{18}^{3} - 6 \zeta_{18} - 6) q^{43} + (4 \zeta_{18}^{4} - 8 \zeta_{18}) q^{48} + ( - 8 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + 7 \zeta_{18}^{3} + 5 \zeta_{18}^{2} + 3 \zeta_{18} - 7) q^{49} + ( - 6 \zeta_{18}^{4} - 8 \zeta_{18}^{3} - 2 \zeta_{18} + 2) q^{52} + (\zeta_{18}^{5} - 8 \zeta_{18}^{2}) q^{57} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{3} - 5 \zeta_{18}^{2} - 9) q^{61} + (3 \zeta_{18}^{5} - 6 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 3) q^{63} + 8 \zeta_{18}^{3} q^{64} + ( - 2 \zeta_{18}^{5} - 7 \zeta_{18}^{3} + 9 \zeta_{18}^{2} + 9) q^{67} + (8 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} + 8) q^{73} + (10 \zeta_{18}^{3} - 5) q^{75} + (10 \zeta_{18}^{4} - 4 \zeta_{18}) q^{76} + ( - 7 \zeta_{18}^{4} - 10 \zeta_{18}^{3} + 10 \zeta_{18} + 7) q^{79} + (9 \zeta_{18}^{5} - 9 \zeta_{18}^{2}) q^{81} + (10 \zeta_{18}^{5} - 10 \zeta_{18}^{4} - 2 \zeta_{18}^{2} + 8 \zeta_{18}) q^{84} + (6 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + 6 \zeta_{18}^{3} - 11 \zeta_{18}^{2} - \zeta_{18} + 5) q^{91} + ( - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{3} - 7 \zeta_{18} + 11) q^{93} + ( - 3 \zeta_{18}^{5} + 6 \zeta_{18}^{2}) q^{97} +O(q^{100})$$ q + (2*z^5 - z^2) * q^3 + 2*z * q^4 + (-3*z^5 + 2*z^4 + 2*z^2 - 3*z) * q^7 - 3*z^4 * q^9 + (2*z^3 - 4) * q^12 + (-z^5 - 3*z^3 - 3*z^2 - 1) * q^13 + 4*z^2 * q^16 + (5*z^3 - 2) * q^19 + (5*z^4 - 5*z^3 - z + 4) * q^21 + (-5*z^4 + 5*z) * q^25 + (3*z^3 + 3) * q^27 + (4*z^5 - 2*z^3 - 6*z^2 + 6) * q^28 + (5*z^5 - z^4 + z^2 - 5*z) * q^31 - 6*z^5 * q^36 + (7*z^5 + 3*z^4 - 4*z^2 + 4*z) * q^37 + (-5*z^5 - 2*z^4 + 7*z^2 + 7*z) * q^39 + (7*z^4 - z^3 - 6*z - 6) * q^43 + (4*z^4 - 8*z) * q^48 + (-8*z^5 - 8*z^4 + 7*z^3 + 5*z^2 + 3*z - 7) * q^49 + (-6*z^4 - 8*z^3 - 2*z + 2) * q^52 + (z^5 - 8*z^2) * q^57 + (-4*z^5 + 4*z^3 - 5*z^2 - 9) * q^61 + (3*z^5 - 6*z^3 + 6*z^2 - 3) * q^63 + 8*z^3 * q^64 + (-2*z^5 - 7*z^3 + 9*z^2 + 9) * q^67 + (8*z^4 + z^3 + z + 8) * q^73 + (10*z^3 - 5) * q^75 + (10*z^4 - 4*z) * q^76 + (-7*z^4 - 10*z^3 + 10*z + 7) * q^79 + (9*z^5 - 9*z^2) * q^81 + (10*z^5 - 10*z^4 - 2*z^2 + 8*z) * q^84 + (6*z^5 + 2*z^4 + 6*z^3 - 11*z^2 - z + 5) * q^91 + (-4*z^4 - 4*z^3 - 7*z + 11) * q^93 + (-3*z^5 + 6*z^2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q - 18 q^{12} - 15 q^{13} + 3 q^{19} + 9 q^{21} + 27 q^{27} + 30 q^{28} - 39 q^{43} - 21 q^{49} - 12 q^{52} - 42 q^{61} - 36 q^{63} + 24 q^{64} + 33 q^{67} + 51 q^{73} + 12 q^{79} + 48 q^{91} + 54 q^{93}+O(q^{100})$$ 6 * q - 18 * q^12 - 15 * q^13 + 3 * q^19 + 9 * q^21 + 27 * q^27 + 30 * q^28 - 39 * q^43 - 21 * q^49 - 12 * q^52 - 42 * q^61 - 36 * q^63 + 24 * q^64 + 33 * q^67 + 51 * q^73 + 12 * q^79 + 48 * q^91 + 54 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$-1$$ $$\zeta_{18}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.939693 + 0.342020i −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i −0.173648 − 0.984808i −0.766044 − 0.642788i
0 −1.11334 + 1.32683i 1.87939 + 0.684040i 0 0 −0.418748 0.725293i 0 −0.520945 2.95442i 0
14.1 0 1.70574 + 0.300767i −1.53209 + 1.28558i 0 0 −2.05303 3.55596i 0 2.81908 + 1.02606i 0
29.1 0 −1.11334 1.32683i 1.87939 0.684040i 0 0 −0.418748 + 0.725293i 0 −0.520945 + 2.95442i 0
32.1 0 −0.592396 + 1.62760i −0.347296 + 1.96962i 0 0 2.47178 4.28125i 0 −2.29813 1.92836i 0
41.1 0 −0.592396 1.62760i −0.347296 1.96962i 0 0 2.47178 + 4.28125i 0 −2.29813 + 1.92836i 0
53.1 0 1.70574 0.300767i −1.53209 1.28558i 0 0 −2.05303 + 3.55596i 0 2.81908 1.02606i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
19.f odd 18 1 inner
57.j even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.j.a 6
3.b odd 2 1 CM 57.2.j.a 6
4.b odd 2 1 912.2.cc.a 6
12.b even 2 1 912.2.cc.a 6
19.e even 9 1 1083.2.d.a 6
19.f odd 18 1 inner 57.2.j.a 6
19.f odd 18 1 1083.2.d.a 6
57.j even 18 1 inner 57.2.j.a 6
57.j even 18 1 1083.2.d.a 6
57.l odd 18 1 1083.2.d.a 6
76.k even 18 1 912.2.cc.a 6
228.u odd 18 1 912.2.cc.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.j.a 6 1.a even 1 1 trivial
57.2.j.a 6 3.b odd 2 1 CM
57.2.j.a 6 19.f odd 18 1 inner
57.2.j.a 6 57.j even 18 1 inner
912.2.cc.a 6 4.b odd 2 1
912.2.cc.a 6 12.b even 2 1
912.2.cc.a 6 76.k even 18 1
912.2.cc.a 6 228.u odd 18 1
1083.2.d.a 6 19.e even 9 1
1083.2.d.a 6 19.f odd 18 1
1083.2.d.a 6 57.j even 18 1
1083.2.d.a 6 57.l odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(57, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 9T^{3} + 27$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 21 T^{4} + 34 T^{3} + \cdots + 289$$
$11$ $$T^{6}$$
$13$ $$T^{6} + 15 T^{5} + 114 T^{4} + \cdots + 8427$$
$17$ $$T^{6}$$
$19$ $$(T^{2} - T + 19)^{3}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6} - 93 T^{4} + 8649 T^{2} + \cdots + 118803$$
$37$ $$T^{6} + 222 T^{4} + 12321 T^{2} + \cdots + 98283$$
$41$ $$T^{6}$$
$43$ $$T^{6} + 39 T^{5} + 636 T^{4} + \cdots + 5041$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$T^{6} + 42 T^{5} + 771 T^{4} + \cdots + 516961$$
$67$ $$T^{6} - 33 T^{5} + 564 T^{4} + \cdots + 189003$$
$71$ $$T^{6}$$
$73$ $$T^{6} - 51 T^{5} + 1086 T^{4} + \cdots + 73441$$
$79$ $$T^{6} - 12 T^{5} + 285 T^{4} + \cdots + 48387$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6} - 243 T^{3} + 19683$$