# Properties

 Label 4002.2.a.l Level $4002$ Weight $2$ Character orbit 4002.a Self dual yes Analytic conductor $31.956$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4002 = 2 \cdot 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4002.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: $$31.9561308889$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} + 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 + 4 * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} + 4 q^{7} + q^{8} + q^{9} + 2 q^{10} - q^{12} - 2 q^{13} + 4 q^{14} - 2 q^{15} + q^{16} - 6 q^{17} + q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{21} + q^{23} - q^{24} - q^{25} - 2 q^{26} - q^{27} + 4 q^{28} + q^{29} - 2 q^{30} + 8 q^{31} + q^{32} - 6 q^{34} + 8 q^{35} + q^{36} + 10 q^{37} + 4 q^{38} + 2 q^{39} + 2 q^{40} + 10 q^{41} - 4 q^{42} + 4 q^{43} + 2 q^{45} + q^{46} - 8 q^{47} - q^{48} + 9 q^{49} - q^{50} + 6 q^{51} - 2 q^{52} - 6 q^{53} - q^{54} + 4 q^{56} - 4 q^{57} + q^{58} + 12 q^{59} - 2 q^{60} - 14 q^{61} + 8 q^{62} + 4 q^{63} + q^{64} - 4 q^{65} + 16 q^{67} - 6 q^{68} - q^{69} + 8 q^{70} - 8 q^{71} + q^{72} + 2 q^{73} + 10 q^{74} + q^{75} + 4 q^{76} + 2 q^{78} - 16 q^{79} + 2 q^{80} + q^{81} + 10 q^{82} - 12 q^{83} - 4 q^{84} - 12 q^{85} + 4 q^{86} - q^{87} + 10 q^{89} + 2 q^{90} - 8 q^{91} + q^{92} - 8 q^{93} - 8 q^{94} + 8 q^{95} - q^{96} + 6 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 + 4 * q^7 + q^8 + q^9 + 2 * q^10 - q^12 - 2 * q^13 + 4 * q^14 - 2 * q^15 + q^16 - 6 * q^17 + q^18 + 4 * q^19 + 2 * q^20 - 4 * q^21 + q^23 - q^24 - q^25 - 2 * q^26 - q^27 + 4 * q^28 + q^29 - 2 * q^30 + 8 * q^31 + q^32 - 6 * q^34 + 8 * q^35 + q^36 + 10 * q^37 + 4 * q^38 + 2 * q^39 + 2 * q^40 + 10 * q^41 - 4 * q^42 + 4 * q^43 + 2 * q^45 + q^46 - 8 * q^47 - q^48 + 9 * q^49 - q^50 + 6 * q^51 - 2 * q^52 - 6 * q^53 - q^54 + 4 * q^56 - 4 * q^57 + q^58 + 12 * q^59 - 2 * q^60 - 14 * q^61 + 8 * q^62 + 4 * q^63 + q^64 - 4 * q^65 + 16 * q^67 - 6 * q^68 - q^69 + 8 * q^70 - 8 * q^71 + q^72 + 2 * q^73 + 10 * q^74 + q^75 + 4 * q^76 + 2 * q^78 - 16 * q^79 + 2 * q^80 + q^81 + 10 * q^82 - 12 * q^83 - 4 * q^84 - 12 * q^85 + 4 * q^86 - q^87 + 10 * q^89 + 2 * q^90 - 8 * q^91 + q^92 - 8 * q^93 - 8 * q^94 + 8 * q^95 - q^96 + 6 * q^97 + 9 * q^98

## 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
1.00000 −1.00000 1.00000 2.00000 −1.00000 4.00000 1.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$+1$$
$$23$$ $$-1$$
$$29$$ $$-1$$

## 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 4002.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.l 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4002))$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7} - 4$$ T7 - 4 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 2$$
$7$ $$T - 4$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T + 6$$
$19$ $$T - 4$$
$23$ $$T - 1$$
$29$ $$T - 1$$
$31$ $$T - 8$$
$37$ $$T - 10$$
$41$ $$T - 10$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T + 6$$
$59$ $$T - 12$$
$61$ $$T + 14$$
$67$ $$T - 16$$
$71$ $$T + 8$$
$73$ $$T - 2$$
$79$ $$T + 16$$
$83$ $$T + 12$$
$89$ $$T - 10$$
$97$ $$T - 6$$
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