# Properties

 Label 4002.2.a.k Level $4002$ Weight $2$ Character orbit 4002.a Self dual yes Analytic conductor $31.956$ Analytic rank $1$ 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] = [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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - 4 q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} - 3 q^{17} + q^{18} + q^{19} + q^{20} - q^{21} - 4 q^{22} + q^{23} - q^{24} - 4 q^{25} - 2 q^{26} - q^{27} + q^{28} - q^{29} - q^{30} - 2 q^{31} + q^{32} + 4 q^{33} - 3 q^{34} + q^{35} + q^{36} - q^{37} + q^{38} + 2 q^{39} + q^{40} - 9 q^{41} - q^{42} - 5 q^{43} - 4 q^{44} + q^{45} + q^{46} + 7 q^{47} - q^{48} - 6 q^{49} - 4 q^{50} + 3 q^{51} - 2 q^{52} + 6 q^{53} - q^{54} - 4 q^{55} + q^{56} - q^{57} - q^{58} - 9 q^{59} - q^{60} - 10 q^{61} - 2 q^{62} + q^{63} + q^{64} - 2 q^{65} + 4 q^{66} - 8 q^{67} - 3 q^{68} - q^{69} + q^{70} - 2 q^{71} + q^{72} + 14 q^{73} - q^{74} + 4 q^{75} + q^{76} - 4 q^{77} + 2 q^{78} - 4 q^{79} + q^{80} + q^{81} - 9 q^{82} - 8 q^{83} - q^{84} - 3 q^{85} - 5 q^{86} + q^{87} - 4 q^{88} + 18 q^{89} + q^{90} - 2 q^{91} + q^{92} + 2 q^{93} + 7 q^{94} + q^{95} - q^{96} - 8 q^{97} - 6 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + q^7 + q^8 + q^9 + q^10 - 4 * q^11 - q^12 - 2 * q^13 + q^14 - q^15 + q^16 - 3 * q^17 + q^18 + q^19 + q^20 - q^21 - 4 * q^22 + q^23 - q^24 - 4 * q^25 - 2 * q^26 - q^27 + q^28 - q^29 - q^30 - 2 * q^31 + q^32 + 4 * q^33 - 3 * q^34 + q^35 + q^36 - q^37 + q^38 + 2 * q^39 + q^40 - 9 * q^41 - q^42 - 5 * q^43 - 4 * q^44 + q^45 + q^46 + 7 * q^47 - q^48 - 6 * q^49 - 4 * q^50 + 3 * q^51 - 2 * q^52 + 6 * q^53 - q^54 - 4 * q^55 + q^56 - q^57 - q^58 - 9 * q^59 - q^60 - 10 * q^61 - 2 * q^62 + q^63 + q^64 - 2 * q^65 + 4 * q^66 - 8 * q^67 - 3 * q^68 - q^69 + q^70 - 2 * q^71 + q^72 + 14 * q^73 - q^74 + 4 * q^75 + q^76 - 4 * q^77 + 2 * q^78 - 4 * q^79 + q^80 + q^81 - 9 * q^82 - 8 * q^83 - q^84 - 3 * q^85 - 5 * q^86 + q^87 - 4 * q^88 + 18 * q^89 + q^90 - 2 * q^91 + q^92 + 2 * q^93 + 7 * q^94 + q^95 - q^96 - 8 * q^97 - 6 * q^98 - 4 * 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
1.00000 −1.00000 1.00000 1.00000 −1.00000 1.00000 1.00000 1.00000 1.00000
 $$n$$: 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.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.k 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4002))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{7} - 1$$ T7 - 1 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T + 3$$
$19$ $$T - 1$$
$23$ $$T - 1$$
$29$ $$T + 1$$
$31$ $$T + 2$$
$37$ $$T + 1$$
$41$ $$T + 9$$
$43$ $$T + 5$$
$47$ $$T - 7$$
$53$ $$T - 6$$
$59$ $$T + 9$$
$61$ $$T + 10$$
$67$ $$T + 8$$
$71$ $$T + 2$$
$73$ $$T - 14$$
$79$ $$T + 4$$
$83$ $$T + 8$$
$89$ $$T - 18$$
$97$ $$T + 8$$