# Properties

 Label 2128.2.a.r Level $2128$ Weight $2$ Character orbit 2128.a Self dual yes Analytic conductor $16.992$ 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] = [2128,2,Mod(1,2128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2128, 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("2128.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2128 = 2^{4} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2128.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: $$16.9921655501$$ 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 2128.2.a.r 3
4.b odd 2 1 532.2.a.e 3
8.b even 2 1 8512.2.a.bl 3
8.d odd 2 1 8512.2.a.bn 3
12.b even 2 1 4788.2.a.o 3
28.d even 2 1 3724.2.a.i 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.e 3 4.b odd 2 1
2128.2.a.r 3 1.a even 1 1 trivial
3724.2.a.i 3 28.d even 2 1
4788.2.a.o 3 12.b even 2 1
8512.2.a.bl 3 8.b even 2 1
8512.2.a.bn 3 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2128))$$:

 $$T_{3}^{3} - T_{3}^{2} - 7T_{3} + 8$$ 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

## Hecke characteristic polynomials

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