# Properties

 Label 1900.2.a.i Level $1900$ Weight $2$ Character orbit 1900.a Self dual yes Analytic conductor $15.172$ 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] = [1900,2,Mod(1,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, 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("1900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.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: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 1$$ x^3 - x^2 - 4*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 2 \beta_1 + 1) q^{7} + ( - \beta_1 + 2) q^{9}+O(q^{10})$$ q + (b2 + 1) * q^3 + (-b2 - 2*b1 + 1) * q^7 + (-b1 + 2) * q^9 $$q + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 2 \beta_1 + 1) q^{7} + ( - \beta_1 + 2) q^{9} + \beta_{2} q^{11} + (\beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{2} + 2 \beta_1 + 1) q^{17} + q^{19} + (2 \beta_{2} - 3 \beta_1 - 1) q^{21} + (\beta_1 + 5) q^{23} + ( - \beta_{2} - 2 \beta_1) q^{27} + (\beta_{2} - 2 \beta_1 + 2) q^{29} + (\beta_{2} + 3 \beta_1 - 1) q^{31} + ( - \beta_{2} - \beta_1 + 4) q^{33} + (\beta_{2} - 4) q^{37} + (\beta_1 + 4) q^{39} - 3 \beta_1 q^{41} + (2 \beta_{2} + 5) q^{43} + ( - \beta_{2} + 5 \beta_1 + 1) q^{47} + (3 \beta_1 + 6) q^{49} + (2 \beta_{2} + 5 \beta_1 - 5) q^{51} + ( - 4 \beta_1 + 1) q^{53} + (\beta_{2} + 1) q^{57} + (2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (\beta_{2} + 5 \beta_1 - 3) q^{61} + ( - 2 \beta_1 + 7) q^{63} + ( - 3 \beta_{2} + 4 \beta_1 + 4) q^{67} + (5 \beta_{2} + 2 \beta_1 + 4) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 - 1) q^{71} + ( - 2 \beta_{2} + \beta_1 + 4) q^{73} + (3 \beta_{2} - \beta_1 - 2) q^{77} + ( - 5 \beta_{2} - 1) q^{79} + (\beta_{2} - 8) q^{81} + ( - \beta_1 + 10) q^{83} + (\beta_{2} - 5 \beta_1 + 8) q^{87} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{89} + ( - 5 \beta_1 - 6) q^{91} + ( - 2 \beta_{2} + 5 \beta_1) q^{93} + (5 \beta_{2} + \beta_1 + 1) q^{97} + (2 \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^3 + (-b2 - 2*b1 + 1) * q^7 + (-b1 + 2) * q^9 + b2 * q^11 + (b2 + b1 + 1) * q^13 + (-b2 + 2*b1 + 1) * q^17 + q^19 + (2*b2 - 3*b1 - 1) * q^21 + (b1 + 5) * q^23 + (-b2 - 2*b1) * q^27 + (b2 - 2*b1 + 2) * q^29 + (b2 + 3*b1 - 1) * q^31 + (-b2 - b1 + 4) * q^33 + (b2 - 4) * q^37 + (b1 + 4) * q^39 - 3*b1 * q^41 + (2*b2 + 5) * q^43 + (-b2 + 5*b1 + 1) * q^47 + (3*b1 + 6) * q^49 + (2*b2 + 5*b1 - 5) * q^51 + (-4*b1 + 1) * q^53 + (b2 + 1) * q^57 + (2*b2 + 2*b1 - 2) * q^59 + (b2 + 5*b1 - 3) * q^61 + (-2*b1 + 7) * q^63 + (-3*b2 + 4*b1 + 4) * q^67 + (5*b2 + 2*b1 + 4) * q^69 + (-2*b2 - 2*b1 - 1) * q^71 + (-2*b2 + b1 + 4) * q^73 + (3*b2 - b1 - 2) * q^77 + (-5*b2 - 1) * q^79 + (b2 - 8) * q^81 + (-b1 + 10) * q^83 + (b2 - 5*b1 + 8) * q^87 + (-2*b2 + 3*b1 + 3) * q^89 + (-5*b1 - 6) * q^91 + (-2*b2 + 5*b1) * q^93 + (5*b2 + b1 + 1) * q^97 + (2*b2 - b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} + 2 q^{7} + 5 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9 $$3 q + 2 q^{3} + 2 q^{7} + 5 q^{9} - q^{11} + 3 q^{13} + 6 q^{17} + 3 q^{19} - 8 q^{21} + 16 q^{23} - q^{27} + 3 q^{29} - q^{31} + 12 q^{33} - 13 q^{37} + 13 q^{39} - 3 q^{41} + 13 q^{43} + 9 q^{47} + 21 q^{49} - 12 q^{51} - q^{53} + 2 q^{57} - 6 q^{59} - 5 q^{61} + 19 q^{63} + 19 q^{67} + 9 q^{69} - 3 q^{71} + 15 q^{73} - 10 q^{77} + 2 q^{79} - 25 q^{81} + 29 q^{83} + 18 q^{87} + 14 q^{89} - 23 q^{91} + 7 q^{93} - q^{97}+O(q^{100})$$ 3 * q + 2 * q^3 + 2 * q^7 + 5 * q^9 - q^11 + 3 * q^13 + 6 * q^17 + 3 * q^19 - 8 * q^21 + 16 * q^23 - q^27 + 3 * q^29 - q^31 + 12 * q^33 - 13 * q^37 + 13 * q^39 - 3 * q^41 + 13 * q^43 + 9 * q^47 + 21 * q^49 - 12 * q^51 - q^53 + 2 * q^57 - 6 * q^59 - 5 * q^61 + 19 * q^63 + 19 * q^67 + 9 * q^69 - 3 * q^71 + 15 * q^73 - 10 * q^77 + 2 * q^79 - 25 * q^81 + 29 * q^83 + 18 * q^87 + 14 * q^89 - 23 * q^91 + 7 * q^93 - q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3

## 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
 0.239123 2.46050 −1.69963
0 −2.18194 0 0 0 3.70370 0 1.76088 0
1.2 0 1.59358 0 0 0 −4.51459 0 −0.460505 0
1.3 0 2.58836 0 0 0 2.81089 0 3.69963 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 1900.2.a.i yes 3
4.b odd 2 1 7600.2.a.bl 3
5.b even 2 1 1900.2.a.g 3
5.c odd 4 2 1900.2.c.f 6
20.d odd 2 1 7600.2.a.ca 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.g 3 5.b even 2 1
1900.2.a.i yes 3 1.a even 1 1 trivial
1900.2.c.f 6 5.c odd 4 2
7600.2.a.bl 3 4.b odd 2 1
7600.2.a.ca 3 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1900))$$:

 $$T_{3}^{3} - 2T_{3}^{2} - 5T_{3} + 9$$ T3^3 - 2*T3^2 - 5*T3 + 9 $$T_{7}^{3} - 2T_{7}^{2} - 19T_{7} + 47$$ T7^3 - 2*T7^2 - 19*T7 + 47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 2 T^{2} - 5 T + 9$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 2 T^{2} - 19 T + 47$$
$11$ $$T^{3} + T^{2} - 6T + 3$$
$13$ $$T^{3} - 3 T^{2} - 6 T + 7$$
$17$ $$T^{3} - 6 T^{2} - 15 T + 99$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 16 T^{2} + 81 T - 129$$
$29$ $$T^{3} - 3 T^{2} - 24 T - 27$$
$31$ $$T^{3} + T^{2} - 40 T - 109$$
$37$ $$T^{3} + 13 T^{2} + 50 T + 59$$
$41$ $$T^{3} + 3 T^{2} - 36 T - 27$$
$43$ $$T^{3} - 13 T^{2} + 31 T + 69$$
$47$ $$T^{3} - 9 T^{2} - 96 T + 621$$
$53$ $$T^{3} + T^{2} - 69T + 3$$
$59$ $$T^{3} + 6 T^{2} - 24 T - 72$$
$61$ $$T^{3} + 5 T^{2} - 98 T - 489$$
$67$ $$T^{3} - 19 T^{2} - 26 T + 1323$$
$71$ $$T^{3} + 3 T^{2} - 33 T - 27$$
$73$ $$T^{3} - 15 T^{2} + 42 T + 49$$
$79$ $$T^{3} - 2 T^{2} - 157 T - 529$$
$83$ $$T^{3} - 29 T^{2} + 276 T - 861$$
$89$ $$T^{3} - 14 T^{2} - 9 T + 489$$
$97$ $$T^{3} + T^{2} - 154 T + 683$$