# Properties

 Label 57.2.d.a Level $57$ Weight $2$ Character orbit 57.d Analytic conductor $0.455$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,2,Mod(56,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("57.56");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 57.d (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 9$$ x^4 + 4*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - \beta_1 q^{3} + \beta_{3} q^{5} + ( - \beta_{3} - 1) q^{6} + q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} - 2) q^{9}+O(q^{10})$$ q - b2 * q^2 - b1 * q^3 + b3 * q^5 + (-b3 - 1) * q^6 + q^7 + 2*b2 * q^8 + (b3 - 2) * q^9 $$q - \beta_{2} q^{2} - \beta_1 q^{3} + \beta_{3} q^{5} + ( - \beta_{3} - 1) q^{6} + q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} - 2) q^{9} + (\beta_{2} + 2 \beta_1) q^{10} - \beta_{3} q^{11} + (\beta_{2} + 2 \beta_1) q^{13} - \beta_{2} q^{14} + ( - 3 \beta_{2} - \beta_1) q^{15} - 4 q^{16} - 3 \beta_{3} q^{17} + (3 \beta_{2} + 2 \beta_1) q^{18} + ( - \beta_{2} - 2 \beta_1 + 3) q^{19} - \beta_1 q^{21} + ( - \beta_{2} - 2 \beta_1) q^{22} + 2 \beta_{3} q^{23} + (2 \beta_{3} + 2) q^{24} + 2 \beta_{3} q^{26} + ( - 3 \beta_{2} + \beta_1) q^{27} + 4 \beta_{2} q^{29} + ( - \beta_{3} + 5) q^{30} + ( - \beta_{2} - 2 \beta_1) q^{31} + (3 \beta_{2} + \beta_1) q^{33} + ( - 3 \beta_{2} - 6 \beta_1) q^{34} + \beta_{3} q^{35} + (3 \beta_{2} + 6 \beta_1) q^{37} + ( - 2 \beta_{3} - 3 \beta_{2}) q^{38} + ( - \beta_{3} + 5) q^{39} + ( - 2 \beta_{2} - 4 \beta_1) q^{40} - 7 \beta_{2} q^{41} + ( - \beta_{3} - 1) q^{42} - 5 q^{43} + ( - 2 \beta_{3} - 5) q^{45} + (2 \beta_{2} + 4 \beta_1) q^{46} + \beta_{3} q^{47} + 4 \beta_1 q^{48} - 6 q^{49} + (9 \beta_{2} + 3 \beta_1) q^{51} + 3 \beta_{2} q^{53} + (\beta_{3} + 7) q^{54} + 5 q^{55} + 2 \beta_{2} q^{56} + (\beta_{3} - 3 \beta_1 - 5) q^{57} - 8 q^{58} + \beta_{2} q^{59} - q^{61} - 2 \beta_{3} q^{62} + (\beta_{3} - 2) q^{63} + 8 q^{64} + 5 \beta_{2} q^{65} + (\beta_{3} - 5) q^{66} + (2 \beta_{2} + 4 \beta_1) q^{67} + ( - 6 \beta_{2} - 2 \beta_1) q^{69} + (\beta_{2} + 2 \beta_1) q^{70} - 3 \beta_{2} q^{71} + ( - 6 \beta_{2} - 4 \beta_1) q^{72} - 3 q^{73} + 6 \beta_{3} q^{74} - \beta_{3} q^{77} + ( - 6 \beta_{2} - 2 \beta_1) q^{78} + ( - 4 \beta_{2} - 8 \beta_1) q^{79} - 4 \beta_{3} q^{80} + ( - 4 \beta_{3} - 1) q^{81} + 14 q^{82} - 4 \beta_{3} q^{83} + 15 q^{85} + 5 \beta_{2} q^{86} + (4 \beta_{3} + 4) q^{87} + (2 \beta_{2} + 4 \beta_1) q^{88} - 9 \beta_{2} q^{89} + (3 \beta_{2} - 4 \beta_1) q^{90} + (\beta_{2} + 2 \beta_1) q^{91} + (\beta_{3} - 5) q^{93} + (\beta_{2} + 2 \beta_1) q^{94} + (3 \beta_{3} - 5 \beta_{2}) q^{95} + (\beta_{2} + 2 \beta_1) q^{97} + 6 \beta_{2} q^{98} + (2 \beta_{3} + 5) q^{99}+O(q^{100})$$ q - b2 * q^2 - b1 * q^3 + b3 * q^5 + (-b3 - 1) * q^6 + q^7 + 2*b2 * q^8 + (b3 - 2) * q^9 + (b2 + 2*b1) * q^10 - b3 * q^11 + (b2 + 2*b1) * q^13 - b2 * q^14 + (-3*b2 - b1) * q^15 - 4 * q^16 - 3*b3 * q^17 + (3*b2 + 2*b1) * q^18 + (-b2 - 2*b1 + 3) * q^19 - b1 * q^21 + (-b2 - 2*b1) * q^22 + 2*b3 * q^23 + (2*b3 + 2) * q^24 + 2*b3 * q^26 + (-3*b2 + b1) * q^27 + 4*b2 * q^29 + (-b3 + 5) * q^30 + (-b2 - 2*b1) * q^31 + (3*b2 + b1) * q^33 + (-3*b2 - 6*b1) * q^34 + b3 * q^35 + (3*b2 + 6*b1) * q^37 + (-2*b3 - 3*b2) * q^38 + (-b3 + 5) * q^39 + (-2*b2 - 4*b1) * q^40 - 7*b2 * q^41 + (-b3 - 1) * q^42 - 5 * q^43 + (-2*b3 - 5) * q^45 + (2*b2 + 4*b1) * q^46 + b3 * q^47 + 4*b1 * q^48 - 6 * q^49 + (9*b2 + 3*b1) * q^51 + 3*b2 * q^53 + (b3 + 7) * q^54 + 5 * q^55 + 2*b2 * q^56 + (b3 - 3*b1 - 5) * q^57 - 8 * q^58 + b2 * q^59 - q^61 - 2*b3 * q^62 + (b3 - 2) * q^63 + 8 * q^64 + 5*b2 * q^65 + (b3 - 5) * q^66 + (2*b2 + 4*b1) * q^67 + (-6*b2 - 2*b1) * q^69 + (b2 + 2*b1) * q^70 - 3*b2 * q^71 + (-6*b2 - 4*b1) * q^72 - 3 * q^73 + 6*b3 * q^74 - b3 * q^77 + (-6*b2 - 2*b1) * q^78 + (-4*b2 - 8*b1) * q^79 - 4*b3 * q^80 + (-4*b3 - 1) * q^81 + 14 * q^82 - 4*b3 * q^83 + 15 * q^85 + 5*b2 * q^86 + (4*b3 + 4) * q^87 + (2*b2 + 4*b1) * q^88 - 9*b2 * q^89 + (3*b2 - 4*b1) * q^90 + (b2 + 2*b1) * q^91 + (b3 - 5) * q^93 + (b2 + 2*b1) * q^94 + (3*b3 - 5*b2) * q^95 + (b2 + 2*b1) * q^97 + 6*b2 * q^98 + (2*b3 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{6} + 4 q^{7} - 8 q^{9}+O(q^{10})$$ 4 * q - 4 * q^6 + 4 * q^7 - 8 * q^9 $$4 q - 4 q^{6} + 4 q^{7} - 8 q^{9} - 16 q^{16} + 12 q^{19} + 8 q^{24} + 20 q^{30} + 20 q^{39} - 4 q^{42} - 20 q^{43} - 20 q^{45} - 24 q^{49} + 28 q^{54} + 20 q^{55} - 20 q^{57} - 32 q^{58} - 4 q^{61} - 8 q^{63} + 32 q^{64} - 20 q^{66} - 12 q^{73} - 4 q^{81} + 56 q^{82} + 60 q^{85} + 16 q^{87} - 20 q^{93} + 20 q^{99}+O(q^{100})$$ 4 * q - 4 * q^6 + 4 * q^7 - 8 * q^9 - 16 * q^16 + 12 * q^19 + 8 * q^24 + 20 * q^30 + 20 * q^39 - 4 * q^42 - 20 * q^43 - 20 * q^45 - 24 * q^49 + 28 * q^54 + 20 * q^55 - 20 * q^57 - 32 * q^58 - 4 * q^61 - 8 * q^63 + 32 * q^64 - 20 * q^66 - 12 * q^73 - 4 * q^81 + 56 * q^82 + 60 * q^85 + 16 * q^87 - 20 * q^93 + 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + \nu ) / 3$$ (v^3 + v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 2$$ b3 - 2 $$\nu^{3}$$ $$=$$ $$3\beta_{2} - \beta_1$$ 3*b2 - b1

## Character values

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

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
56.1
 −0.707107 + 1.58114i −0.707107 − 1.58114i 0.707107 + 1.58114i 0.707107 − 1.58114i
−1.41421 0.707107 1.58114i 0 2.23607i −1.00000 + 2.23607i 1.00000 2.82843 −2.00000 2.23607i 3.16228i
56.2 −1.41421 0.707107 + 1.58114i 0 2.23607i −1.00000 2.23607i 1.00000 2.82843 −2.00000 + 2.23607i 3.16228i
56.3 1.41421 −0.707107 1.58114i 0 2.23607i −1.00000 2.23607i 1.00000 −2.82843 −2.00000 + 2.23607i 3.16228i
56.4 1.41421 −0.707107 + 1.58114i 0 2.23607i −1.00000 + 2.23607i 1.00000 −2.82843 −2.00000 2.23607i 3.16228i
 $$n$$: 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
3.b odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.d.a 4
3.b odd 2 1 inner 57.2.d.a 4
4.b odd 2 1 912.2.f.f 4
12.b even 2 1 912.2.f.f 4
19.b odd 2 1 inner 57.2.d.a 4
57.d even 2 1 inner 57.2.d.a 4
76.d even 2 1 912.2.f.f 4
228.b odd 2 1 912.2.f.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.d.a 4 1.a even 1 1 trivial
57.2.d.a 4 3.b odd 2 1 inner
57.2.d.a 4 19.b odd 2 1 inner
57.2.d.a 4 57.d even 2 1 inner
912.2.f.f 4 4.b odd 2 1
912.2.f.f 4 12.b even 2 1
912.2.f.f 4 76.d even 2 1
912.2.f.f 4 228.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(57, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4} + 4T^{2} + 9$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T - 1)^{4}$$
$11$ $$(T^{2} + 5)^{2}$$
$13$ $$(T^{2} + 10)^{2}$$
$17$ $$(T^{2} + 45)^{2}$$
$19$ $$(T^{2} - 6 T + 19)^{2}$$
$23$ $$(T^{2} + 20)^{2}$$
$29$ $$(T^{2} - 32)^{2}$$
$31$ $$(T^{2} + 10)^{2}$$
$37$ $$(T^{2} + 90)^{2}$$
$41$ $$(T^{2} - 98)^{2}$$
$43$ $$(T + 5)^{4}$$
$47$ $$(T^{2} + 5)^{2}$$
$53$ $$(T^{2} - 18)^{2}$$
$59$ $$(T^{2} - 2)^{2}$$
$61$ $$(T + 1)^{4}$$
$67$ $$(T^{2} + 40)^{2}$$
$71$ $$(T^{2} - 18)^{2}$$
$73$ $$(T + 3)^{4}$$
$79$ $$(T^{2} + 160)^{2}$$
$83$ $$(T^{2} + 80)^{2}$$
$89$ $$(T^{2} - 162)^{2}$$
$97$ $$(T^{2} + 10)^{2}$$