# Properties

 Label 1904.2.a.o Level $1904$ Weight $2$ Character orbit 1904.a Self dual yes Analytic conductor $15.204$ Analytic rank $1$ 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] = [1904,2,Mod(1,1904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1904, 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("1904.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1904 = 2^{4} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1904.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: $$15.2035165449$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 952) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{3} + ( - \beta_{2} - 2) q^{5} - q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^3 + (-b2 - 2) * q^5 - q^7 + (b2 + 2*b1 + 1) * q^9 $$q + (\beta_1 + 1) q^{3} + ( - \beta_{2} - 2) q^{5} - q^{7} + (\beta_{2} + 2 \beta_1 + 1) q^{9} - 2 \beta_1 q^{11} + (2 \beta_{2} - 2 \beta_1) q^{13} + ( - \beta_{2} - 3 \beta_1 - 3) q^{15} - q^{17} - 2 \beta_{2} q^{19} + ( - \beta_1 - 1) q^{21} + (2 \beta_{2} - 2 \beta_1 + 2) q^{23} + (2 \beta_{2} + \beta_1 + 2) q^{25} + (3 \beta_{2} + \beta_1 + 5) q^{27} + ( - 2 \beta_{2} - 4) q^{29} + (\beta_{2} - 4 \beta_1 - 2) q^{31} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{33} + (\beta_{2} + 2) q^{35} + ( - 2 \beta_{2} - 8) q^{37} - 4 q^{39} + (2 \beta_{2} + \beta_1 - 3) q^{41} + ( - 2 \beta_{2} + 3 \beta_1 - 3) q^{43} + ( - \beta_{2} - 7 \beta_1 - 7) q^{45} + (4 \beta_1 + 4) q^{47} + q^{49} + ( - \beta_1 - 1) q^{51} + (3 \beta_{2} + 2 \beta_1 - 2) q^{53} + (6 \beta_1 + 2) q^{55} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{57} + (4 \beta_{2} - 4 \beta_1 + 4) q^{59} + (\beta_1 - 9) q^{61} + ( - \beta_{2} - 2 \beta_1 - 1) q^{63} + (4 \beta_1 - 4) q^{65} + ( - 3 \beta_{2} - 6) q^{67} + (2 \beta_1 - 2) q^{69} + ( - 2 \beta_{2} + 8 \beta_1) q^{71} + ( - 2 \beta_{2} - 5 \beta_1 - 1) q^{73} + (3 \beta_{2} + 5 \beta_1 + 7) q^{75} + 2 \beta_1 q^{77} - 4 q^{79} + (\beta_{2} + 3 \beta_1 + 8) q^{81} + ( - 2 \beta_{2} + 4) q^{83} + (\beta_{2} + 2) q^{85} + ( - 2 \beta_{2} - 6 \beta_1 - 6) q^{87} + (6 \beta_{2} - 2 \beta_1) q^{89} + ( - 2 \beta_{2} + 2 \beta_1) q^{91} + ( - 3 \beta_{2} - 5 \beta_1 - 13) q^{93} + (2 \beta_1 + 6) q^{95} + ( - 3 \beta_{2} + 6 \beta_1 + 2) q^{97} + ( - 4 \beta_{2} - 4 \beta_1 - 14) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^3 + (-b2 - 2) * q^5 - q^7 + (b2 + 2*b1 + 1) * q^9 - 2*b1 * q^11 + (2*b2 - 2*b1) * q^13 + (-b2 - 3*b1 - 3) * q^15 - q^17 - 2*b2 * q^19 + (-b1 - 1) * q^21 + (2*b2 - 2*b1 + 2) * q^23 + (2*b2 + b1 + 2) * q^25 + (3*b2 + b1 + 5) * q^27 + (-2*b2 - 4) * q^29 + (b2 - 4*b1 - 2) * q^31 + (-2*b2 - 2*b1 - 6) * q^33 + (b2 + 2) * q^35 + (-2*b2 - 8) * q^37 - 4 * q^39 + (2*b2 + b1 - 3) * q^41 + (-2*b2 + 3*b1 - 3) * q^43 + (-b2 - 7*b1 - 7) * q^45 + (4*b1 + 4) * q^47 + q^49 + (-b1 - 1) * q^51 + (3*b2 + 2*b1 - 2) * q^53 + (6*b1 + 2) * q^55 + (-2*b2 - 2*b1 - 2) * q^57 + (4*b2 - 4*b1 + 4) * q^59 + (b1 - 9) * q^61 + (-b2 - 2*b1 - 1) * q^63 + (4*b1 - 4) * q^65 + (-3*b2 - 6) * q^67 + (2*b1 - 2) * q^69 + (-2*b2 + 8*b1) * q^71 + (-2*b2 - 5*b1 - 1) * q^73 + (3*b2 + 5*b1 + 7) * q^75 + 2*b1 * q^77 - 4 * q^79 + (b2 + 3*b1 + 8) * q^81 + (-2*b2 + 4) * q^83 + (b2 + 2) * q^85 + (-2*b2 - 6*b1 - 6) * q^87 + (6*b2 - 2*b1) * q^89 + (-2*b2 + 2*b1) * q^91 + (-3*b2 - 5*b1 - 13) * q^93 + (2*b1 + 6) * q^95 + (-3*b2 + 6*b1 + 2) * q^97 + (-4*b2 - 4*b1 - 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 5 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - 5 * q^5 - 3 * q^7 + 2 * q^9 $$3 q + 3 q^{3} - 5 q^{5} - 3 q^{7} + 2 q^{9} - 2 q^{13} - 8 q^{15} - 3 q^{17} + 2 q^{19} - 3 q^{21} + 4 q^{23} + 4 q^{25} + 12 q^{27} - 10 q^{29} - 7 q^{31} - 16 q^{33} + 5 q^{35} - 22 q^{37} - 12 q^{39} - 11 q^{41} - 7 q^{43} - 20 q^{45} + 12 q^{47} + 3 q^{49} - 3 q^{51} - 9 q^{53} + 6 q^{55} - 4 q^{57} + 8 q^{59} - 27 q^{61} - 2 q^{63} - 12 q^{65} - 15 q^{67} - 6 q^{69} + 2 q^{71} - q^{73} + 18 q^{75} - 12 q^{79} + 23 q^{81} + 14 q^{83} + 5 q^{85} - 16 q^{87} - 6 q^{89} + 2 q^{91} - 36 q^{93} + 18 q^{95} + 9 q^{97} - 38 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - 5 * q^5 - 3 * q^7 + 2 * q^9 - 2 * q^13 - 8 * q^15 - 3 * q^17 + 2 * q^19 - 3 * q^21 + 4 * q^23 + 4 * q^25 + 12 * q^27 - 10 * q^29 - 7 * q^31 - 16 * q^33 + 5 * q^35 - 22 * q^37 - 12 * q^39 - 11 * q^41 - 7 * q^43 - 20 * q^45 + 12 * q^47 + 3 * q^49 - 3 * q^51 - 9 * q^53 + 6 * q^55 - 4 * q^57 + 8 * q^59 - 27 * q^61 - 2 * q^63 - 12 * q^65 - 15 * q^67 - 6 * q^69 + 2 * q^71 - q^73 + 18 * q^75 - 12 * q^79 + 23 * q^81 + 14 * q^83 + 5 * q^85 - 16 * q^87 - 6 * q^89 + 2 * q^91 - 36 * q^93 + 18 * q^95 + 9 * q^97 - 38 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 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
 −1.86081 −0.254102 2.11491
0 −0.860806 0 −2.46260 0 −1.00000 0 −2.25901 0
1.2 0 0.745898 0 0.935432 0 −1.00000 0 −2.44364 0
1.3 0 3.11491 0 −3.47283 0 −1.00000 0 6.70265 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
$$2$$ $$1$$
$$7$$ $$1$$
$$17$$ $$1$$

## 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 1904.2.a.o 3
4.b odd 2 1 952.2.a.c 3
8.b even 2 1 7616.2.a.bb 3
8.d odd 2 1 7616.2.a.bh 3
12.b even 2 1 8568.2.a.be 3
28.d even 2 1 6664.2.a.m 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.a.c 3 4.b odd 2 1
1904.2.a.o 3 1.a even 1 1 trivial
6664.2.a.m 3 28.d even 2 1
7616.2.a.bb 3 8.b even 2 1
7616.2.a.bh 3 8.d odd 2 1
8568.2.a.be 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1904))$$:

 $$T_{3}^{3} - 3T_{3}^{2} - T_{3} + 2$$ T3^3 - 3*T3^2 - T3 + 2 $$T_{5}^{3} + 5T_{5}^{2} + 3T_{5} - 8$$ T5^3 + 5*T5^2 + 3*T5 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 3T^{2} - T + 2$$
$5$ $$T^{3} + 5 T^{2} + \cdots - 8$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} - 16T + 8$$
$13$ $$T^{3} + 2 T^{2} + \cdots - 32$$
$17$ $$(T + 1)^{3}$$
$19$ $$T^{3} - 2 T^{2} + \cdots - 16$$
$23$ $$T^{3} - 4 T^{2} + \cdots + 16$$
$29$ $$T^{3} + 10 T^{2} + \cdots - 64$$
$31$ $$T^{3} + 7 T^{2} + \cdots - 208$$
$37$ $$T^{3} + 22 T^{2} + \cdots + 208$$
$41$ $$T^{3} + 11 T^{2} + \cdots - 74$$
$43$ $$T^{3} + 7 T^{2} + \cdots + 8$$
$47$ $$T^{3} - 12 T^{2} + \cdots + 128$$
$53$ $$T^{3} + 9 T^{2} + \cdots - 326$$
$59$ $$T^{3} - 8 T^{2} + \cdots + 128$$
$61$ $$T^{3} + 27 T^{2} + \cdots + 692$$
$67$ $$T^{3} + 15 T^{2} + \cdots - 216$$
$71$ $$T^{3} - 2 T^{2} + \cdots + 848$$
$73$ $$T^{3} + T^{2} + \cdots + 658$$
$79$ $$(T + 4)^{3}$$
$83$ $$T^{3} - 14 T^{2} + \cdots - 32$$
$89$ $$T^{3} + 6 T^{2} + \cdots + 512$$
$97$ $$T^{3} - 9 T^{2} + \cdots + 1006$$