# Properties

 Label 1881.2.a.k Level $1881$ Weight $2$ Character orbit 1881.a Self dual yes Analytic conductor $15.020$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1881 = 3^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1881.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: $$15.0198606202$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.246832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ x^5 - 2*x^4 - 5*x^3 + 6*x^2 + 7*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + ( - \beta_{2} - \beta_1 + 2) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8}+O(q^{10})$$ q - b2 * q^2 + (b4 + b3 + b2 + b1) * q^4 + (-b2 + b1 + 1) * q^5 + (-b2 - b1 + 2) * q^7 + (-b2 - 2*b1) * q^8 $$q - \beta_{2} q^{2} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + ( - \beta_{2} - \beta_1 + 2) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} + (\beta_{4} - \beta_{2} - \beta_1 + 3) q^{10} - q^{11} + (\beta_{4} - \beta_1 + 1) q^{13} + (\beta_{4} + 2 \beta_{3} + 3 \beta_1 + 1) q^{14} + ( - \beta_{4} + \beta_{3} + \cdots + 3 \beta_1) q^{16}+ \cdots + (3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \cdots + 3) q^{98}+O(q^{100})$$ q - b2 * q^2 + (b4 + b3 + b2 + b1) * q^4 + (-b2 + b1 + 1) * q^5 + (-b2 - b1 + 2) * q^7 + (-b2 - 2*b1) * q^8 + (b4 - b2 - b1 + 3) * q^10 - q^11 + (b4 - b1 + 1) * q^13 + (b4 + 2*b3 + 3*b1 + 1) * q^14 + (-b4 + b3 + b2 + 3*b1) * q^16 + (-2*b3 - 2*b1 + 2) * q^17 - q^19 + (b4 + 3*b3 + 2*b1) * q^20 + b2 * q^22 + (-3*b4 - b3 - b2 - b1 + 1) * q^23 + (b4 - b3 - 2*b2 + 2) * q^25 + (2*b3 - b2 + 3*b1) * q^26 + (2*b4 - 3*b2 - 3*b1) * q^28 + (b4 - 2*b3 + b2 - 3*b1 - 1) * q^29 + (-b3 - b2 + 2*b1 + 2) * q^31 + (-4*b3 - b2 - 4*b1) * q^32 + (-2*b4 + 4*b1 - 2) * q^34 + (b4 + b3 - 3*b2 + b1 + 2) * q^35 + (-3*b4 - b3 - b2 + b1 + 1) * q^37 + b2 * q^38 + (b4 + 2*b3 - b2 - b1 - 3) * q^40 + (-b4 + 4*b3 + 2*b2 + 3*b1 - 3) * q^41 + (-b4 - 3*b3 + 2*b2 + 4) * q^43 + (-b4 - b3 - b2 - b1) * q^44 + (-2*b3 + 4*b2 - 2) * q^46 + (4*b4 + 2*b3 + 2*b2 + 2) * q^47 + (b4 + 3*b3 + 2*b1 - 1) * q^49 + (b4 + 2*b3 - b2 + 3*b1 + 5) * q^50 + (b4 - 2*b2 - 3*b1 + 3) * q^52 + (-4*b4 - 2*b3 - 2*b2 + 2*b1 + 4) * q^53 + (b2 - b1 - 1) * q^55 + (b4 + 4*b3 + 4*b2 + 5*b1 + 3) * q^56 + (-3*b4 + b3 + 2*b2 + 6*b1 - 4) * q^58 + (-b4 + b3 + b2 + 5*b1 - 3) * q^59 + (2*b3 + 2*b2 + 2*b1 - 4) * q^61 + (-2*b3 - 3*b2 - 3*b1 + 4) * q^62 + (-b4 - b3 + 3*b2 + 3*b1 - 2) * q^64 + (2*b3 - 3*b2 + 2*b1) * q^65 + (-5*b3 + b2 - 4*b1 + 4) * q^67 + (-2*b3 - 6*b1 - 2) * q^68 + (4*b4 + 4*b3 - b2 + 2*b1 + 8) * q^70 + (2*b4 + 3*b3 - 2*b2 - 2*b1 - 4) * q^71 + (-2*b4 + 2*b2 - 6*b1 + 2) * q^73 + (-4*b3 + 2*b2 - 4*b1) * q^74 + (-b4 - b3 - b2 - b1) * q^76 + (b2 + b1 - 2) * q^77 + (2*b1 + 8) * q^79 + (b4 - b3 + 4*b2 + 2) * q^80 + (2*b4 - 2*b3 - b2 - 9*b1 - 2) * q^82 + (-b4 - 3*b3 - 4*b2 - 2*b1 + 6) * q^83 + (-4*b4 - 4*b3 - 4*b2 + 2*b1) * q^85 + (-5*b4 - 6*b3 - 5*b2 - 3*b1 - 5) * q^86 + (b2 + 2*b1) * q^88 + (b4 + b3 + b2 - b1 + 5) * q^89 + (3*b4 + 2*b3 + b2 + b1 + 3) * q^91 + (-4*b3 - 2*b1 - 10) * q^92 + (4*b3 - 8*b2 + 2*b1) * q^94 + (b2 - b1 - 1) * q^95 + (-3*b4 + b3 - 5*b2 + b1 + 5) * q^97 + (3*b4 + 2*b3 - 2*b2 - 3*b1 + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10})$$ 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 $$5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29} + 11 q^{31} - 14 q^{32} - 4 q^{34} + 8 q^{35} + q^{37} + 2 q^{38} - 16 q^{40} - 2 q^{41} + 20 q^{43} - 6 q^{44} - 4 q^{46} + 20 q^{47} + 3 q^{49} + 32 q^{50} + 6 q^{52} + 14 q^{53} - 5 q^{55} + 38 q^{56} - 6 q^{58} - 3 q^{59} - 10 q^{61} + 6 q^{62} + 9 q^{67} - 24 q^{68} + 50 q^{70} - 23 q^{71} - 8 q^{74} - 6 q^{76} - 6 q^{77} + 44 q^{79} + 18 q^{80} - 30 q^{82} + 14 q^{83} - 12 q^{85} - 52 q^{86} + 6 q^{88} + 27 q^{89} + 24 q^{91} - 58 q^{92} - 8 q^{94} - 5 q^{95} + 15 q^{97} + 10 q^{98}+O(q^{100})$$ 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 3$$ v^3 - 2*v^2 - 3*v + 3 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 3\nu^{3} - 2\nu^{2} + 7\nu + 1$$ v^4 - 3*v^3 - 2*v^2 + 7*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 5\beta _1 + 1$$ b3 + 2*b2 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 3\beta_{3} + 8\beta_{2} + 10\beta _1 + 6$$ b4 + 3*b3 + 8*b2 + 10*b1 + 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.71457 −1.51908 −1.15351 1.71250 0.245526
−2.65432 0 5.04540 1.06025 0 −3.36889 −8.08346 0 −2.81425
1.2 −1.82669 0 1.33679 −2.34577 0 1.69239 1.21147 0 4.28499
1.3 −0.484093 0 −1.76565 −0.637602 0 2.66942 1.82293 0 0.308658
1.4 0.779856 0 −1.39182 3.49235 0 1.06736 −2.64513 0 2.72353
1.5 2.18524 0 2.77529 3.43077 0 3.93972 1.69419 0 7.49706
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$1$$
$$19$$ $$1$$

## 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 1881.2.a.k 5
3.b odd 2 1 209.2.a.c 5
12.b even 2 1 3344.2.a.t 5
15.d odd 2 1 5225.2.a.h 5
33.d even 2 1 2299.2.a.n 5
57.d even 2 1 3971.2.a.h 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.c 5 3.b odd 2 1
1881.2.a.k 5 1.a even 1 1 trivial
2299.2.a.n 5 33.d even 2 1
3344.2.a.t 5 12.b even 2 1
3971.2.a.h 5 57.d even 2 1
5225.2.a.h 5 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1881))$$:

 $$T_{2}^{5} + 2T_{2}^{4} - 6T_{2}^{3} - 10T_{2}^{2} + 5T_{2} + 4$$ T2^5 + 2*T2^4 - 6*T2^3 - 10*T2^2 + 5*T2 + 4 $$T_{5}^{5} - 5T_{5}^{4} - 3T_{5}^{3} + 33T_{5}^{2} - 9T_{5} - 19$$ T5^5 - 5*T5^4 - 3*T5^3 + 33*T5^2 - 9*T5 - 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} + 2 T^{4} + \cdots + 4$$
$3$ $$T^{5}$$
$5$ $$T^{5} - 5 T^{4} + \cdots - 19$$
$7$ $$T^{5} - 6 T^{4} + \cdots + 64$$
$11$ $$(T + 1)^{5}$$
$13$ $$T^{5} - 4 T^{4} + \cdots + 2$$
$17$ $$T^{5} - 4 T^{4} + \cdots + 64$$
$19$ $$(T + 1)^{5}$$
$23$ $$T^{5} + 3 T^{4} + \cdots + 784$$
$29$ $$T^{5} + 10 T^{4} + \cdots - 490$$
$31$ $$T^{5} - 11 T^{4} + \cdots - 757$$
$37$ $$T^{5} - T^{4} + \cdots - 3088$$
$41$ $$T^{5} + 2 T^{4} + \cdots + 4112$$
$43$ $$T^{5} - 20 T^{4} + \cdots + 11266$$
$47$ $$T^{5} - 20 T^{4} + \cdots - 13184$$
$53$ $$T^{5} - 14 T^{4} + \cdots - 30304$$
$59$ $$T^{5} + 3 T^{4} + \cdots + 2000$$
$61$ $$T^{5} + 10 T^{4} + \cdots - 736$$
$67$ $$T^{5} - 9 T^{4} + \cdots + 17689$$
$71$ $$T^{5} + 23 T^{4} + \cdots - 19081$$
$73$ $$T^{5} - 340 T^{3} + \cdots + 155392$$
$79$ $$T^{5} - 44 T^{4} + \cdots - 36800$$
$83$ $$T^{5} - 14 T^{4} + \cdots + 3908$$
$89$ $$T^{5} - 27 T^{4} + \cdots - 320$$
$97$ $$T^{5} - 15 T^{4} + \cdots - 37456$$
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