# Properties

 Label 4002.2.a.bb Level $4002$ Weight $2$ Character orbit 4002.a Self dual yes Analytic conductor $31.956$ 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] = [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: $$4$$ Coefficient field: 4.4.23252.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 2$$ x^4 - x^3 - 6*x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + (\beta_{3} - 1) q^{5} - q^{6} + ( - \beta_{3} - \beta_1) q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + (b3 - 1) * q^5 - q^6 + (-b3 - b1) * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} + (\beta_{3} - 1) q^{5} - q^{6} + ( - \beta_{3} - \beta_1) q^{7} - q^{8} + q^{9} + ( - \beta_{3} + 1) q^{10} + (\beta_1 - 3) q^{11} + q^{12} + ( - \beta_{2} - \beta_1) q^{13} + (\beta_{3} + \beta_1) q^{14} + (\beta_{3} - 1) q^{15} + q^{16} + (\beta_{3} + \beta_1) q^{17} - q^{18} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{19} + (\beta_{3} - 1) q^{20} + ( - \beta_{3} - \beta_1) q^{21} + ( - \beta_1 + 3) q^{22} + q^{23} - q^{24} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{25} + (\beta_{2} + \beta_1) q^{26} + q^{27} + ( - \beta_{3} - \beta_1) q^{28} - q^{29} + ( - \beta_{3} + 1) q^{30} + (\beta_{2} - \beta_1 - 4) q^{31} - q^{32} + (\beta_1 - 3) q^{33} + ( - \beta_{3} - \beta_1) q^{34} + (\beta_{3} - \beta_1 - 6) q^{35} + q^{36} + ( - 2 \beta_{2} + 3 \beta_1 + 3) q^{37} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{38} + ( - \beta_{2} - \beta_1) q^{39} + ( - \beta_{3} + 1) q^{40} + ( - 2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{41} + (\beta_{3} + \beta_1) q^{42} + (2 \beta_{3} - 6 \beta_1 + 2) q^{43} + (\beta_1 - 3) q^{44} + (\beta_{3} - 1) q^{45} - q^{46} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{47} + q^{48} + (2 \beta_{2} + 4 \beta_1 + 3) q^{49} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{50} + (\beta_{3} + \beta_1) q^{51} + ( - \beta_{2} - \beta_1) q^{52} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{53} - q^{54} + ( - 3 \beta_{3} + \beta_{2} + 4) q^{55} + (\beta_{3} + \beta_1) q^{56} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{57} + q^{58} + (\beta_{2} + 3 \beta_1 - 4) q^{59} + (\beta_{3} - 1) q^{60} + ( - \beta_1 - 3) q^{61} + ( - \beta_{2} + \beta_1 + 4) q^{62} + ( - \beta_{3} - \beta_1) q^{63} + q^{64} + (2 \beta_{3} - 3 \beta_1 - 3) q^{65} + ( - \beta_1 + 3) q^{66} + ( - 3 \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{67}+ \cdots + (\beta_1 - 3) q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 + (b3 - 1) * q^5 - q^6 + (-b3 - b1) * q^7 - q^8 + q^9 + (-b3 + 1) * q^10 + (b1 - 3) * q^11 + q^12 + (-b2 - b1) * q^13 + (b3 + b1) * q^14 + (b3 - 1) * q^15 + q^16 + (b3 + b1) * q^17 - q^18 + (-b3 + 2*b2 + b1 + 2) * q^19 + (b3 - 1) * q^20 + (-b3 - b1) * q^21 + (-b1 + 3) * q^22 + q^23 - q^24 + (-2*b3 - b2 + b1 + 1) * q^25 + (b2 + b1) * q^26 + q^27 + (-b3 - b1) * q^28 - q^29 + (-b3 + 1) * q^30 + (b2 - b1 - 4) * q^31 - q^32 + (b1 - 3) * q^33 + (-b3 - b1) * q^34 + (b3 - b1 - 6) * q^35 + q^36 + (-2*b2 + 3*b1 + 3) * q^37 + (b3 - 2*b2 - b1 - 2) * q^38 + (-b2 - b1) * q^39 + (-b3 + 1) * q^40 + (-2*b3 + b2 + 3*b1) * q^41 + (b3 + b1) * q^42 + (2*b3 - 6*b1 + 2) * q^43 + (b1 - 3) * q^44 + (b3 - 1) * q^45 - q^46 + (-b3 + 2*b2 - b1 + 2) * q^47 + q^48 + (2*b2 + 4*b1 + 3) * q^49 + (2*b3 + b2 - b1 - 1) * q^50 + (b3 + b1) * q^51 + (-b2 - b1) * q^52 + (-b3 - 2*b2 + 3*b1) * q^53 - q^54 + (-3*b3 + b2 + 4) * q^55 + (b3 + b1) * q^56 + (-b3 + 2*b2 + b1 + 2) * q^57 + q^58 + (b2 + 3*b1 - 4) * q^59 + (b3 - 1) * q^60 + (-b1 - 3) * q^61 + (-b2 + b1 + 4) * q^62 + (-b3 - b1) * q^63 + q^64 + (2*b3 - 3*b1 - 3) * q^65 + (-b1 + 3) * q^66 + (-3*b3 - 2*b2 + 2*b1 - 3) * q^67 + (b3 + b1) * q^68 + q^69 + (-b3 + b1 + 6) * q^70 + (4*b3 - b2 - 3*b1 - 4) * q^71 - q^72 + (3*b3 - b1 - 6) * q^73 + (2*b2 - 3*b1 - 3) * q^74 + (-2*b3 - b2 + b1 + 1) * q^75 + (-b3 + 2*b2 + b1 + 2) * q^76 + (3*b3 - 2*b2 + b1 - 4) * q^77 + (b2 + b1) * q^78 + (b3 - 3*b1 - 6) * q^79 + (b3 - 1) * q^80 + q^81 + (2*b3 - b2 - 3*b1) * q^82 + (-2*b3 + 2*b2 - 4*b1 - 4) * q^83 + (-b3 - b1) * q^84 + (-b3 + b1 + 6) * q^85 + (-2*b3 + 6*b1 - 2) * q^86 - q^87 + (-b1 + 3) * q^88 + (-b3 - 4*b2 + b1 - 6) * q^89 + (-b3 + 1) * q^90 + (-b3 + 2*b2 + 7*b1 + 6) * q^91 + q^92 + (b2 - b1 - 4) * q^93 + (b3 - 2*b2 + b1 - 2) * q^94 + (-b3 + 5*b1 - 2) * q^95 - q^96 + (-b3 - 4*b2 + b1 - 2) * q^97 + (-2*b2 - 4*b1 - 3) * q^98 + (b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 3 q^{10} - 11 q^{11} + 4 q^{12} - q^{13} + 2 q^{14} - 3 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 8 q^{19} - 3 q^{20} - 2 q^{21} + 11 q^{22} + 4 q^{23} - 4 q^{24} + 3 q^{25} + q^{26} + 4 q^{27} - 2 q^{28} - 4 q^{29} + 3 q^{30} - 17 q^{31} - 4 q^{32} - 11 q^{33} - 2 q^{34} - 24 q^{35} + 4 q^{36} + 15 q^{37} - 8 q^{38} - q^{39} + 3 q^{40} + q^{41} + 2 q^{42} + 4 q^{43} - 11 q^{44} - 3 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} + 16 q^{49} - 3 q^{50} + 2 q^{51} - q^{52} + 2 q^{53} - 4 q^{54} + 13 q^{55} + 2 q^{56} + 8 q^{57} + 4 q^{58} - 13 q^{59} - 3 q^{60} - 13 q^{61} + 17 q^{62} - 2 q^{63} + 4 q^{64} - 13 q^{65} + 11 q^{66} - 13 q^{67} + 2 q^{68} + 4 q^{69} + 24 q^{70} - 15 q^{71} - 4 q^{72} - 22 q^{73} - 15 q^{74} + 3 q^{75} + 8 q^{76} - 12 q^{77} + q^{78} - 26 q^{79} - 3 q^{80} + 4 q^{81} - q^{82} - 22 q^{83} - 2 q^{84} + 24 q^{85} - 4 q^{86} - 4 q^{87} + 11 q^{88} - 24 q^{89} + 3 q^{90} + 30 q^{91} + 4 q^{92} - 17 q^{93} - 6 q^{94} - 4 q^{95} - 4 q^{96} - 8 q^{97} - 16 q^{98} - 11 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 + 3 * q^10 - 11 * q^11 + 4 * q^12 - q^13 + 2 * q^14 - 3 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 8 * q^19 - 3 * q^20 - 2 * q^21 + 11 * q^22 + 4 * q^23 - 4 * q^24 + 3 * q^25 + q^26 + 4 * q^27 - 2 * q^28 - 4 * q^29 + 3 * q^30 - 17 * q^31 - 4 * q^32 - 11 * q^33 - 2 * q^34 - 24 * q^35 + 4 * q^36 + 15 * q^37 - 8 * q^38 - q^39 + 3 * q^40 + q^41 + 2 * q^42 + 4 * q^43 - 11 * q^44 - 3 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 + 16 * q^49 - 3 * q^50 + 2 * q^51 - q^52 + 2 * q^53 - 4 * q^54 + 13 * q^55 + 2 * q^56 + 8 * q^57 + 4 * q^58 - 13 * q^59 - 3 * q^60 - 13 * q^61 + 17 * q^62 - 2 * q^63 + 4 * q^64 - 13 * q^65 + 11 * q^66 - 13 * q^67 + 2 * q^68 + 4 * q^69 + 24 * q^70 - 15 * q^71 - 4 * q^72 - 22 * q^73 - 15 * q^74 + 3 * q^75 + 8 * q^76 - 12 * q^77 + q^78 - 26 * q^79 - 3 * q^80 + 4 * q^81 - q^82 - 22 * q^83 - 2 * q^84 + 24 * q^85 - 4 * q^86 - 4 * q^87 + 11 * q^88 - 24 * q^89 + 3 * q^90 + 30 * q^91 + 4 * q^92 - 17 * q^93 - 6 * q^94 - 4 * q^95 - 4 * q^96 - 8 * q^97 - 16 * q^98 - 11 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6x^{2} + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5\nu$$ v^3 - v^2 - 5*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 6\beta _1 + 3$$ b3 + b2 + 6*b1 + 3

## 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
 0.565882 −1.88474 2.95372 −0.634868
−1.00000 1.00000 1.00000 −3.96842 −1.00000 2.40254 −1.00000 1.00000 3.96842
1.2 −1.00000 1.00000 1.00000 −1.82358 −1.00000 2.70832 −1.00000 1.00000 1.82358
1.3 −1.00000 1.00000 1.00000 1.27661 −1.00000 −5.23034 −1.00000 1.00000 −1.27661
1.4 −1.00000 1.00000 1.00000 1.51539 −1.00000 −1.88053 −1.00000 1.00000 −1.51539
 $$n$$: 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.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bb 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4002))$$:

 $$T_{5}^{4} + 3T_{5}^{3} - 7T_{5}^{2} - 9T_{5} + 14$$ T5^4 + 3*T5^3 - 7*T5^2 - 9*T5 + 14 $$T_{7}^{4} + 2T_{7}^{3} - 20T_{7}^{2} - 4T_{7} + 64$$ T7^4 + 2*T7^3 - 20*T7^2 - 4*T7 + 64 $$T_{11}^{4} + 11T_{11}^{3} + 39T_{11}^{2} + 45T_{11} + 2$$ T11^4 + 11*T11^3 + 39*T11^2 + 45*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4} + 3 T^{3} + \cdots + 14$$
$7$ $$T^{4} + 2 T^{3} + \cdots + 64$$
$11$ $$T^{4} + 11 T^{3} + \cdots + 2$$
$13$ $$T^{4} + T^{3} + \cdots + 22$$
$17$ $$T^{4} - 2 T^{3} + \cdots + 64$$
$19$ $$T^{4} - 8 T^{3} + \cdots + 232$$
$23$ $$(T - 1)^{4}$$
$29$ $$(T + 1)^{4}$$
$31$ $$T^{4} + 17 T^{3} + \cdots - 56$$
$37$ $$T^{4} - 15 T^{3} + \cdots - 2672$$
$41$ $$T^{4} - T^{3} + \cdots + 434$$
$43$ $$T^{4} - 4 T^{3} + \cdots + 10352$$
$47$ $$T^{4} - 6 T^{3} + \cdots + 176$$
$53$ $$T^{4} - 2 T^{3} + \cdots + 56$$
$59$ $$T^{4} + 13 T^{3} + \cdots - 2404$$
$61$ $$T^{4} + 13 T^{3} + \cdots + 56$$
$67$ $$T^{4} + 13 T^{3} + \cdots - 9268$$
$71$ $$T^{4} + 15 T^{3} + \cdots - 3784$$
$73$ $$T^{4} + 22 T^{3} + \cdots - 472$$
$79$ $$T^{4} + 26 T^{3} + \cdots + 248$$
$83$ $$T^{4} + 22 T^{3} + \cdots - 10592$$
$89$ $$T^{4} + 24 T^{3} + \cdots - 3776$$
$97$ $$T^{4} + 8 T^{3} + \cdots + 6224$$