# Properties

 Label 1904.2.a.r Level $1904$ Weight $2$ Character orbit 1904.a Self dual yes Analytic conductor $15.204$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.13448.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 7x^{2} + 2$$ x^4 - 7*x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 1) q^{3} + ( - \beta_{2} + 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10})$$ q + (-b3 - 1) * q^3 + (-b2 + 1) * q^5 - q^7 + (b2 - b1 + 2) * q^9 $$q + ( - \beta_{3} - 1) q^{3} + ( - \beta_{2} + 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} - \beta_1 q^{11} + 2 q^{13} + ( - \beta_{3} + \beta_{2}) q^{15} + q^{17} + (\beta_{3} + 1) q^{21} + ( - 2 \beta_{3} - 2) q^{23} + (\beta_{3} - 4 \beta_{2} + \beta_1) q^{25} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{29}+ \cdots + (4 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 14) q^{99}+O(q^{100})$$ q + (-b3 - 1) * q^3 + (-b2 + 1) * q^5 - q^7 + (b2 - b1 + 2) * q^9 - b1 * q^11 + 2 * q^13 + (-b3 + b2) * q^15 + q^17 + (b3 + 1) * q^21 + (-2*b3 - 2) * q^23 + (b3 - 4*b2 + b1) * q^25 + (-b3 - b2 + 2*b1 - 2) * q^27 + (-2*b3 + 2*b2 - b1 + 2) * q^29 + (-3*b2 - b1 + 1) * q^31 + (-2*b3 + 2*b1 - 2) * q^33 + (b2 - 1) * q^35 + (b1 + 2) * q^37 + (-2*b3 - 2) * q^39 + (-b3 - 2*b2 + b1 + 3) * q^41 + (3*b3 + b1 + 3) * q^43 + (-b3 + 3*b2 - b1) * q^45 + (2*b3 - 4*b2 - 2) * q^47 + q^49 + (-b3 - 1) * q^51 + (3*b2 - b1 + 5) * q^53 + (2*b2 + 2) * q^55 + 2*b1 * q^59 + (-b3 + 1) * q^61 + (-b2 + b1 - 2) * q^63 + (-2*b2 + 2) * q^65 + (2*b3 + b2 - b1 + 3) * q^67 + (2*b2 - 2*b1 + 10) * q^69 + (4*b2 - 2*b1 + 4) * q^71 + (-3*b3 + 2*b2 + b1 - 3) * q^73 + (3*b3 + 3*b2 - b1 + 2) * q^75 + b1 * q^77 + (-2*b3 + 4*b2 + 2) * q^79 + (5*b3 - b2 - 2*b1 + 5) * q^81 + (6*b3 - 4*b2 + 2) * q^83 + (-b2 + 1) * q^85 + (-6*b3 + 2) * q^87 + (-4*b3 + 2*b2 - 2*b1) * q^89 - 2 * q^91 + (-3*b3 + 3*b2 + 2*b1) * q^93 + (-2*b3 - 3*b2 + b1 + 1) * q^97 + (4*b3 + 2*b2 - 3*b1 + 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{3} + 5 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q - 3 * q^3 + 5 * q^5 - 4 * q^7 + 7 * q^9 $$4 q - 3 q^{3} + 5 q^{5} - 4 q^{7} + 7 q^{9} + 8 q^{13} + 4 q^{17} + 3 q^{21} - 6 q^{23} + 3 q^{25} - 6 q^{27} + 8 q^{29} + 7 q^{31} - 6 q^{33} - 5 q^{35} + 8 q^{37} - 6 q^{39} + 15 q^{41} + 9 q^{43} - 2 q^{45} - 6 q^{47} + 4 q^{49} - 3 q^{51} + 17 q^{53} + 6 q^{55} + 5 q^{61} - 7 q^{63} + 10 q^{65} + 9 q^{67} + 38 q^{69} + 12 q^{71} - 11 q^{73} + 2 q^{75} + 6 q^{79} + 16 q^{81} + 6 q^{83} + 5 q^{85} + 14 q^{87} + 2 q^{89} - 8 q^{91} + 9 q^{97} + 50 q^{99}+O(q^{100})$$ 4 * q - 3 * q^3 + 5 * q^5 - 4 * q^7 + 7 * q^9 + 8 * q^13 + 4 * q^17 + 3 * q^21 - 6 * q^23 + 3 * q^25 - 6 * q^27 + 8 * q^29 + 7 * q^31 - 6 * q^33 - 5 * q^35 + 8 * q^37 - 6 * q^39 + 15 * q^41 + 9 * q^43 - 2 * q^45 - 6 * q^47 + 4 * q^49 - 3 * q^51 + 17 * q^53 + 6 * q^55 + 5 * q^61 - 7 * q^63 + 10 * q^65 + 9 * q^67 + 38 * q^69 + 12 * q^71 - 11 * q^73 + 2 * q^75 + 6 * q^79 + 16 * q^81 + 6 * q^83 + 5 * q^85 + 14 * q^87 + 2 * q^89 - 8 * q^91 + 9 * q^97 + 50 * q^99

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

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + \nu^{2} - 6\nu - 4 ) / 2$$ (v^3 + v^2 - 6*v - 4) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 6\nu - 4 ) / 2$$ (-v^3 + v^2 + 6*v - 4) / 2
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4$$ b3 + b2 + 4 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 3\beta_1$$ -b3 + b2 + 3*b1

## 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
 −2.58874 2.58874 0.546295 −0.546295
0 −3.25886 0 0.557299 0 −1.00000 0 7.62018 0
1.2 0 −1.44270 0 −1.25886 0 −1.00000 0 −0.918614 0
1.3 0 −0.706585 0 4.40815 0 −1.00000 0 −2.50074 0
1.4 0 2.40815 0 1.29341 0 −1.00000 0 2.79917 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.r 4
4.b odd 2 1 952.2.a.h 4
8.b even 2 1 7616.2.a.bo 4
8.d odd 2 1 7616.2.a.bi 4
12.b even 2 1 8568.2.a.bg 4
28.d even 2 1 6664.2.a.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.a.h 4 4.b odd 2 1
1904.2.a.r 4 1.a even 1 1 trivial
6664.2.a.n 4 28.d even 2 1
7616.2.a.bi 4 8.d odd 2 1
7616.2.a.bo 4 8.b even 2 1
8568.2.a.bg 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1904))$$:

 $$T_{3}^{4} + 3T_{3}^{3} - 5T_{3}^{2} - 16T_{3} - 8$$ T3^4 + 3*T3^3 - 5*T3^2 - 16*T3 - 8 $$T_{5}^{4} - 5T_{5}^{3} + T_{5}^{2} + 8T_{5} - 4$$ T5^4 - 5*T5^3 + T5^2 + 8*T5 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3 T^{3} + \cdots - 8$$
$5$ $$T^{4} - 5 T^{3} + \cdots - 4$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4} - 28T^{2} + 32$$
$13$ $$(T - 2)^{4}$$
$17$ $$(T - 1)^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 6 T^{3} + \cdots - 128$$
$29$ $$T^{4} - 8 T^{3} + \cdots - 80$$
$31$ $$T^{4} - 7 T^{3} + \cdots - 1600$$
$37$ $$T^{4} - 8 T^{3} + \cdots - 64$$
$41$ $$T^{4} - 15 T^{3} + \cdots - 1132$$
$43$ $$T^{4} - 9 T^{3} + \cdots - 1168$$
$47$ $$T^{4} + 6 T^{3} + \cdots + 640$$
$53$ $$T^{4} - 17 T^{3} + \cdots - 2500$$
$59$ $$T^{4} - 112T^{2} + 512$$
$61$ $$T^{4} - 5 T^{3} + \cdots - 4$$
$67$ $$T^{4} - 9 T^{3} + \cdots + 80$$
$71$ $$T^{4} - 12 T^{3} + \cdots - 2560$$
$73$ $$T^{4} + 11 T^{3} + \cdots + 3284$$
$79$ $$T^{4} - 6 T^{3} + \cdots + 640$$
$83$ $$T^{4} - 6 T^{3} + \cdots + 14464$$
$89$ $$T^{4} - 2 T^{3} + \cdots + 2000$$
$97$ $$T^{4} - 9 T^{3} + \cdots + 1460$$