# Properties

 Label 57.2.e.a Level $57$ Weight $2$ Character orbit 57.e Analytic conductor $0.455$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,2,Mod(7,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("57.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 57.e (of order $$3$$, degree $$2$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{6} + q^{7} - 3 q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 + z * q^4 + z * q^6 + q^7 - 3 * q^8 - z * q^9 $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{6} + q^{7} - 3 q^{8} - \zeta_{6} q^{9} - 2 q^{11} + q^{12} - 5 \zeta_{6} q^{13} + (\zeta_{6} - 1) q^{14} + ( - \zeta_{6} + 1) q^{16} + ( - 4 \zeta_{6} + 4) q^{17} + q^{18} + (2 \zeta_{6} - 5) q^{19} + ( - \zeta_{6} + 1) q^{21} + ( - 2 \zeta_{6} + 2) q^{22} + 4 \zeta_{6} q^{23} + (3 \zeta_{6} - 3) q^{24} + 5 \zeta_{6} q^{25} + 5 q^{26} - q^{27} + \zeta_{6} q^{28} + 8 \zeta_{6} q^{29} - 3 q^{31} - 5 \zeta_{6} q^{32} + (2 \zeta_{6} - 2) q^{33} + 4 \zeta_{6} q^{34} + ( - \zeta_{6} + 1) q^{36} + 3 q^{37} + ( - 5 \zeta_{6} + 3) q^{38} - 5 q^{39} + ( - 12 \zeta_{6} + 12) q^{41} + \zeta_{6} q^{42} + ( - \zeta_{6} + 1) q^{43} - 2 \zeta_{6} q^{44} - 4 q^{46} + 6 \zeta_{6} q^{47} - \zeta_{6} q^{48} - 6 q^{49} - 5 q^{50} - 4 \zeta_{6} q^{51} + ( - 5 \zeta_{6} + 5) q^{52} - 4 \zeta_{6} q^{53} + ( - \zeta_{6} + 1) q^{54} - 3 q^{56} + (5 \zeta_{6} - 3) q^{57} - 8 q^{58} + (10 \zeta_{6} - 10) q^{59} + 13 \zeta_{6} q^{61} + ( - 3 \zeta_{6} + 3) q^{62} - \zeta_{6} q^{63} + 7 q^{64} - 2 \zeta_{6} q^{66} - 11 \zeta_{6} q^{67} + 4 q^{68} + 4 q^{69} + (6 \zeta_{6} - 6) q^{71} + 3 \zeta_{6} q^{72} + ( - 11 \zeta_{6} + 11) q^{73} + (3 \zeta_{6} - 3) q^{74} + 5 q^{75} + ( - 3 \zeta_{6} - 2) q^{76} - 2 q^{77} + ( - 5 \zeta_{6} + 5) q^{78} + (\zeta_{6} - 1) q^{79} + (\zeta_{6} - 1) q^{81} + 12 \zeta_{6} q^{82} + q^{84} + \zeta_{6} q^{86} + 8 q^{87} + 6 q^{88} + 6 \zeta_{6} q^{89} - 5 \zeta_{6} q^{91} + (4 \zeta_{6} - 4) q^{92} + (3 \zeta_{6} - 3) q^{93} - 6 q^{94} - 5 q^{96} + (2 \zeta_{6} - 2) q^{97} + ( - 6 \zeta_{6} + 6) q^{98} + 2 \zeta_{6} q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (-z + 1) * q^3 + z * q^4 + z * q^6 + q^7 - 3 * q^8 - z * q^9 - 2 * q^11 + q^12 - 5*z * q^13 + (z - 1) * q^14 + (-z + 1) * q^16 + (-4*z + 4) * q^17 + q^18 + (2*z - 5) * q^19 + (-z + 1) * q^21 + (-2*z + 2) * q^22 + 4*z * q^23 + (3*z - 3) * q^24 + 5*z * q^25 + 5 * q^26 - q^27 + z * q^28 + 8*z * q^29 - 3 * q^31 - 5*z * q^32 + (2*z - 2) * q^33 + 4*z * q^34 + (-z + 1) * q^36 + 3 * q^37 + (-5*z + 3) * q^38 - 5 * q^39 + (-12*z + 12) * q^41 + z * q^42 + (-z + 1) * q^43 - 2*z * q^44 - 4 * q^46 + 6*z * q^47 - z * q^48 - 6 * q^49 - 5 * q^50 - 4*z * q^51 + (-5*z + 5) * q^52 - 4*z * q^53 + (-z + 1) * q^54 - 3 * q^56 + (5*z - 3) * q^57 - 8 * q^58 + (10*z - 10) * q^59 + 13*z * q^61 + (-3*z + 3) * q^62 - z * q^63 + 7 * q^64 - 2*z * q^66 - 11*z * q^67 + 4 * q^68 + 4 * q^69 + (6*z - 6) * q^71 + 3*z * q^72 + (-11*z + 11) * q^73 + (3*z - 3) * q^74 + 5 * q^75 + (-3*z - 2) * q^76 - 2 * q^77 + (-5*z + 5) * q^78 + (z - 1) * q^79 + (z - 1) * q^81 + 12*z * q^82 + q^84 + z * q^86 + 8 * q^87 + 6 * q^88 + 6*z * q^89 - 5*z * q^91 + (4*z - 4) * q^92 + (3*z - 3) * q^93 - 6 * q^94 - 5 * q^96 + (2*z - 2) * q^97 + (-6*z + 6) * q^98 + 2*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} - 6 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 + q^3 + q^4 + q^6 + 2 * q^7 - 6 * q^8 - q^9 $$2 q - q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} - 6 q^{8} - q^{9} - 4 q^{11} + 2 q^{12} - 5 q^{13} - q^{14} + q^{16} + 4 q^{17} + 2 q^{18} - 8 q^{19} + q^{21} + 2 q^{22} + 4 q^{23} - 3 q^{24} + 5 q^{25} + 10 q^{26} - 2 q^{27} + q^{28} + 8 q^{29} - 6 q^{31} - 5 q^{32} - 2 q^{33} + 4 q^{34} + q^{36} + 6 q^{37} + q^{38} - 10 q^{39} + 12 q^{41} + q^{42} + q^{43} - 2 q^{44} - 8 q^{46} + 6 q^{47} - q^{48} - 12 q^{49} - 10 q^{50} - 4 q^{51} + 5 q^{52} - 4 q^{53} + q^{54} - 6 q^{56} - q^{57} - 16 q^{58} - 10 q^{59} + 13 q^{61} + 3 q^{62} - q^{63} + 14 q^{64} - 2 q^{66} - 11 q^{67} + 8 q^{68} + 8 q^{69} - 6 q^{71} + 3 q^{72} + 11 q^{73} - 3 q^{74} + 10 q^{75} - 7 q^{76} - 4 q^{77} + 5 q^{78} - q^{79} - q^{81} + 12 q^{82} + 2 q^{84} + q^{86} + 16 q^{87} + 12 q^{88} + 6 q^{89} - 5 q^{91} - 4 q^{92} - 3 q^{93} - 12 q^{94} - 10 q^{96} - 2 q^{97} + 6 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q - q^2 + q^3 + q^4 + q^6 + 2 * q^7 - 6 * q^8 - q^9 - 4 * q^11 + 2 * q^12 - 5 * q^13 - q^14 + q^16 + 4 * q^17 + 2 * q^18 - 8 * q^19 + q^21 + 2 * q^22 + 4 * q^23 - 3 * q^24 + 5 * q^25 + 10 * q^26 - 2 * q^27 + q^28 + 8 * q^29 - 6 * q^31 - 5 * q^32 - 2 * q^33 + 4 * q^34 + q^36 + 6 * q^37 + q^38 - 10 * q^39 + 12 * q^41 + q^42 + q^43 - 2 * q^44 - 8 * q^46 + 6 * q^47 - q^48 - 12 * q^49 - 10 * q^50 - 4 * q^51 + 5 * q^52 - 4 * q^53 + q^54 - 6 * q^56 - q^57 - 16 * q^58 - 10 * q^59 + 13 * q^61 + 3 * q^62 - q^63 + 14 * q^64 - 2 * q^66 - 11 * q^67 + 8 * q^68 + 8 * q^69 - 6 * q^71 + 3 * q^72 + 11 * q^73 - 3 * q^74 + 10 * q^75 - 7 * q^76 - 4 * q^77 + 5 * q^78 - q^79 - q^81 + 12 * q^82 + 2 * q^84 + q^86 + 16 * q^87 + 12 * q^88 + 6 * q^89 - 5 * q^91 - 4 * q^92 - 3 * q^93 - 12 * q^94 - 10 * q^96 - 2 * q^97 + 6 * q^98 + 2 * q^99

## Character values

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

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 0 0.500000 0.866025i 1.00000 −3.00000 −0.500000 + 0.866025i 0
49.1 −0.500000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 −3.00000 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.e.a 2
3.b odd 2 1 171.2.f.a 2
4.b odd 2 1 912.2.q.a 2
12.b even 2 1 2736.2.s.j 2
19.c even 3 1 inner 57.2.e.a 2
19.c even 3 1 1083.2.a.c 1
19.d odd 6 1 1083.2.a.b 1
57.f even 6 1 3249.2.a.f 1
57.h odd 6 1 171.2.f.a 2
57.h odd 6 1 3249.2.a.c 1
76.g odd 6 1 912.2.q.a 2
228.m even 6 1 2736.2.s.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.a 2 1.a even 1 1 trivial
57.2.e.a 2 19.c even 3 1 inner
171.2.f.a 2 3.b odd 2 1
171.2.f.a 2 57.h odd 6 1
912.2.q.a 2 4.b odd 2 1
912.2.q.a 2 76.g odd 6 1
1083.2.a.b 1 19.d odd 6 1
1083.2.a.c 1 19.c even 3 1
2736.2.s.j 2 12.b even 2 1
2736.2.s.j 2 228.m even 6 1
3249.2.a.c 1 57.h odd 6 1
3249.2.a.f 1 57.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(57, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2} - 4T + 16$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} - 8T + 64$$
$31$ $$(T + 3)^{2}$$
$37$ $$(T - 3)^{2}$$
$41$ $$T^{2} - 12T + 144$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} + 4T + 16$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} + T + 1$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} + 2T + 4$$