# Properties

 Label 4004.2.a.c Level $4004$ Weight $2$ Character orbit 4004.a Self dual yes Analytic conductor $31.972$ Analytic rank $0$ Dimension $1$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

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

## 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
0 2.00000 0 4.00000 0 1.00000 0 1.00000 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.c 1

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T - 4$$
$7$ $$T - 1$$
$11$ $$T - 1$$
$13$ $$T + 1$$
$17$ $$T - 2$$
$19$ $$T - 4$$
$23$ $$T - 8$$
$29$ $$T + 10$$
$31$ $$T - 8$$
$37$ $$T - 2$$
$41$ $$T + 10$$
$43$ $$T + 10$$
$47$ $$T + 12$$
$53$ $$T + 10$$
$59$ $$T - 12$$
$61$ $$T - 2$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T - 10$$
$83$ $$T + 4$$
$89$ $$T - 12$$
$97$ $$T + 8$$