# Properties

 Label 4004.2.a.e Level $4004$ Weight $2$ Character orbit 4004.a Self dual yes Analytic conductor $31.972$ Analytic rank $1$ 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] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4004.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: $$31.9721009693$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{15})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} + 4x + 1$$ x^4 - x^3 - 4*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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_{2}) q^{3} + (\beta_{3} - \beta_1 + 1) q^{5} - q^{7} + (\beta_1 - 1) q^{9}+O(q^{10})$$ q + (-b3 - b2) * q^3 + (b3 - b1 + 1) * q^5 - q^7 + (b1 - 1) * q^9 $$q + ( - \beta_{3} - \beta_{2}) q^{3} + (\beta_{3} - \beta_1 + 1) q^{5} - q^{7} + (\beta_1 - 1) q^{9} + q^{11} - q^{13} + (2 \beta_{3} - \beta_1 + 1) q^{15} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{17} + ( - 2 \beta_{3} + \beta_{2} - 2) q^{19} + (\beta_{3} + \beta_{2}) q^{21} - \beta_{2} q^{23} + ( - \beta_{3} - \beta_{2} - 3) q^{25} + (2 \beta_{3} + 3 \beta_{2} - 1) q^{27} + 3 \beta_1 q^{29} + (2 \beta_1 + 1) q^{31} + ( - \beta_{3} - \beta_{2}) q^{33} + ( - \beta_{3} + \beta_1 - 1) q^{35} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{37} + (\beta_{3} + \beta_{2}) q^{39} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 1) q^{41} + (4 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{43} + (\beta_1 - 2) q^{45} + ( - 2 \beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{47} + q^{49} + ( - 4 \beta_{3} + \beta_1 - 5) q^{51} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 5) q^{53} + (\beta_{3} - \beta_1 + 1) q^{55} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{57} + (6 \beta_{3} + 6 \beta_{2} - \beta_1 - 1) q^{59} + (3 \beta_{3} - \beta_{2} + \beta_1 - 3) q^{61} + ( - \beta_1 + 1) q^{63} + ( - \beta_{3} + \beta_1 - 1) q^{65} + (4 \beta_1 - 5) q^{67} + (\beta_{3} + 2) q^{69} + ( - 4 \beta_{3} - 6) q^{71} + (\beta_{3} - 5 \beta_{2} - \beta_1 + 3) q^{73} + (3 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{75} - q^{77} + (7 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 6) q^{79} + (\beta_{2} - 5 \beta_1 - 3) q^{81} + (5 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{83} + ( - \beta_{3} + 3 \beta_{2} - 4) q^{85} + ( - 6 \beta_{3} - 3 \beta_{2} - 3) q^{87} + ( - 8 \beta_{3} - 3 \beta_{2} + \cdots - 10) q^{89}+ \cdots + (\beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b3 - b2) * q^3 + (b3 - b1 + 1) * q^5 - q^7 + (b1 - 1) * q^9 + q^11 - q^13 + (2*b3 - b1 + 1) * q^15 + (-b3 + 2*b2 + b1 - 1) * q^17 + (-2*b3 + b2 - 2) * q^19 + (b3 + b2) * q^21 - b2 * q^23 + (-b3 - b2 - 3) * q^25 + (2*b3 + 3*b2 - 1) * q^27 + 3*b1 * q^29 + (2*b1 + 1) * q^31 + (-b3 - b2) * q^33 + (-b3 + b1 - 1) * q^35 + (-b3 + 3*b2 - b1) * q^37 + (b3 + b2) * q^39 + (2*b3 - 3*b2 - 3*b1 + 1) * q^41 + (4*b3 + 2*b2 - b1 + 1) * q^43 + (b1 - 2) * q^45 + (-2*b3 + b2 + 4*b1 - 4) * q^47 + q^49 + (-4*b3 + b1 - 5) * q^51 + (-b3 + 2*b2 - 4*b1 - 5) * q^53 + (b3 - b1 + 1) * q^55 + (-b3 + 2*b2 + 2*b1 - 2) * q^57 + (6*b3 + 6*b2 - b1 - 1) * q^59 + (3*b3 - b2 + b1 - 3) * q^61 + (-b1 + 1) * q^63 + (-b3 + b1 - 1) * q^65 + (4*b1 - 5) * q^67 + (b3 + 2) * q^69 + (-4*b3 - 6) * q^71 + (b3 - 5*b2 - b1 + 3) * q^73 + (3*b3 + 3*b2 + b1 + 2) * q^75 - q^77 + (7*b3 + 2*b2 - 3*b1 + 6) * q^79 + (b2 - 5*b1 - 3) * q^81 + (5*b3 + b2 - b1 - 1) * q^83 + (-b3 + 3*b2 - 4) * q^85 + (-6*b3 - 3*b2 - 3) * q^87 + (-8*b3 - 3*b2 + 4*b1 - 10) * q^89 + q^91 + (-5*b3 - 3*b2 - 2) * q^93 + (-b3 + 3*b2 - 3) * q^95 + (10*b3 + b2 + b1 + 8) * q^97 + (b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10})$$ 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9 $$4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} + q^{17} - 3 q^{19} - q^{21} - q^{23} - 11 q^{25} - 5 q^{27} + 3 q^{29} + 6 q^{31} + q^{33} - q^{35} + 4 q^{37} - q^{39} - 6 q^{41} - 3 q^{43} - 7 q^{45} - 7 q^{47} + 4 q^{49} - 11 q^{51} - 20 q^{53} + q^{55} - 2 q^{57} - 11 q^{59} - 18 q^{61} + 3 q^{63} - q^{65} - 16 q^{67} + 6 q^{69} - 16 q^{71} + 4 q^{73} + 6 q^{75} - 4 q^{77} + 9 q^{79} - 16 q^{81} - 14 q^{83} - 11 q^{85} - 3 q^{87} - 23 q^{89} + 4 q^{91} - q^{93} - 7 q^{95} + 14 q^{97} - 3 q^{99}+O(q^{100})$$ 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9 + 4 * q^11 - 4 * q^13 - q^15 + q^17 - 3 * q^19 - q^21 - q^23 - 11 * q^25 - 5 * q^27 + 3 * q^29 + 6 * q^31 + q^33 - q^35 + 4 * q^37 - q^39 - 6 * q^41 - 3 * q^43 - 7 * q^45 - 7 * q^47 + 4 * q^49 - 11 * q^51 - 20 * q^53 + q^55 - 2 * q^57 - 11 * q^59 - 18 * q^61 + 3 * q^63 - q^65 - 16 * q^67 + 6 * q^69 - 16 * q^71 + 4 * q^73 + 6 * q^75 - 4 * q^77 + 9 * q^79 - 16 * q^81 - 14 * q^83 - 11 * q^85 - 3 * q^87 - 23 * q^89 + 4 * q^91 - q^93 - 7 * q^95 + 14 * q^97 - 3 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{15} + \zeta_{15}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 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
 1.82709 −1.95630 −0.209057 1.33826
0 −1.95630 0 −0.209057 0 −1.00000 0 0.827091 0
1.2 0 −0.209057 0 1.33826 0 −1.00000 0 −2.95630 0
1.3 0 1.33826 0 1.82709 0 −1.00000 0 −1.20906 0
1.4 0 1.82709 0 −1.95630 0 −1.00000 0 0.338261 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
$$2$$ $$-1$$
$$7$$ $$+1$$
$$11$$ $$-1$$
$$13$$ $$+1$$

## 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 4004.2.a.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.e 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - T_{3}^{3} - 4T_{3}^{2} + 4T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4004))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{3} - 4 T^{2} + \cdots + 1$$
$5$ $$T^{4} - T^{3} - 4 T^{2} + \cdots + 1$$
$7$ $$(T + 1)^{4}$$
$11$ $$(T - 1)^{4}$$
$13$ $$(T + 1)^{4}$$
$17$ $$T^{4} - T^{3} + \cdots - 59$$
$19$ $$T^{4} + 3 T^{3} + \cdots + 31$$
$23$ $$T^{4} + T^{3} - 4 T^{2} + \cdots + 1$$
$29$ $$T^{4} - 3 T^{3} + \cdots + 81$$
$31$ $$T^{4} - 6 T^{3} + \cdots - 29$$
$37$ $$T^{4} - 4 T^{3} + \cdots + 31$$
$41$ $$T^{4} + 6 T^{3} + \cdots - 659$$
$43$ $$T^{4} + 3 T^{3} + \cdots + 1$$
$47$ $$T^{4} + 7 T^{3} + \cdots + 811$$
$53$ $$T^{4} + 20 T^{3} + \cdots - 6605$$
$59$ $$T^{4} + 11 T^{3} + \cdots + 2311$$
$61$ $$T^{4} + 18 T^{3} + \cdots - 29$$
$67$ $$T^{4} + 16 T^{3} + \cdots + 61$$
$71$ $$(T^{2} + 8 T - 4)^{2}$$
$73$ $$T^{4} - 4 T^{3} + \cdots + 421$$
$79$ $$T^{4} - 9 T^{3} + \cdots - 2169$$
$83$ $$T^{4} + 14 T^{3} + \cdots + 31$$
$89$ $$T^{4} + 23 T^{3} + \cdots - 10679$$
$97$ $$T^{4} - 14 T^{3} + \cdots + 12211$$