# Properties

 Label 4002.2.a.bg Level $4002$ Weight $2$ Character orbit 4002.a Self dual yes Analytic conductor $31.956$ Analytic rank $1$ Dimension $7$ 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: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 13x^{5} + 16x^{4} + 19x^{3} - 8x^{2} - 10x - 2$$ x^7 - x^6 - 13*x^5 + 16*x^4 + 19*x^3 - 8*x^2 - 10*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} - \beta_{5} q^{5} + q^{6} + ( - \beta_{6} - \beta_{2} - 1) q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 - b5 * q^5 + q^6 + (-b6 - b2 - 1) * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} - \beta_{5} q^{5} + q^{6} + ( - \beta_{6} - \beta_{2} - 1) q^{7} - q^{8} + q^{9} + \beta_{5} q^{10} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{11} - q^{12} + ( - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{13}+ \cdots + ( - \beta_{6} - \beta_{4} - \beta_1) q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 - b5 * q^5 + q^6 + (-b6 - b2 - 1) * q^7 - q^8 + q^9 + b5 * q^10 + (-b6 - b4 - b1) * q^11 - q^12 + (-b6 - b5 + b3 - b1) * q^13 + (b6 + b2 + 1) * q^14 + b5 * q^15 + q^16 + (b6 + b5 + b4 + b1) * q^17 - q^18 + (b6 + b5 - b3 - 1) * q^19 - b5 * q^20 + (b6 + b2 + 1) * q^21 + (b6 + b4 + b1) * q^22 + q^23 + q^24 + (2*b6 + b2 + 2*b1) * q^25 + (b6 + b5 - b3 + b1) * q^26 - q^27 + (-b6 - b2 - 1) * q^28 + q^29 - b5 * q^30 + (b6 - b5 - b3 + 2*b2 - b1 - 2) * q^31 - q^32 + (b6 + b4 + b1) * q^33 + (-b6 - b5 - b4 - b1) * q^34 + (b6 + 3*b5 + b4 + 3*b1 + 2) * q^35 + q^36 + (2*b6 - b5 - b4 - b3 + b2 + 2*b1 - 3) * q^37 + (-b6 - b5 + b3 + 1) * q^38 + (b6 + b5 - b3 + b1) * q^39 + b5 * q^40 + (2*b5 + 2*b4 - b2 + 1) * q^41 + (-b6 - b2 - 1) * q^42 + (b6 + 3*b5 + 3*b4 + b2 + 2*b1) * q^43 + (-b6 - b4 - b1) * q^44 - b5 * q^45 - q^46 + (-b6 + b5 - b4 - 2*b2 - b1 - 2) * q^47 - q^48 + (3*b6 - b5 - 2*b4 + b3 + 3*b2 - b1 + 2) * q^49 + (-2*b6 - b2 - 2*b1) * q^50 + (-b6 - b5 - b4 - b1) * q^51 + (-b6 - b5 + b3 - b1) * q^52 + (2*b6 - b5 + b4 - 2*b1 + 1) * q^53 + q^54 + (b6 + b5 - b4 + b2 - b1 - 1) * q^55 + (b6 + b2 + 1) * q^56 + (-b6 - b5 + b3 + 1) * q^57 - q^58 + (-2*b6 + 2*b5 + b2 + 3) * q^59 + b5 * q^60 + (b6 - b4 + 2*b2 - b1) * q^61 + (-b6 + b5 + b3 - 2*b2 + b1 + 2) * q^62 + (-b6 - b2 - 1) * q^63 + q^64 + (b6 + b5 + b4 + b3 + b2 - b1 + 4) * q^65 + (-b6 - b4 - b1) * q^66 + (b6 - 2*b5 - 2*b4 + b3 - b2 - 3*b1 - 3) * q^67 + (b6 + b5 + b4 + b1) * q^68 - q^69 + (-b6 - 3*b5 - b4 - 3*b1 - 2) * q^70 + (-b6 + b5 - 2*b4 - b3 + b1) * q^71 - q^72 + (3*b4 - b3 + b2 + 2*b1 - 3) * q^73 + (-2*b6 + b5 + b4 + b3 - b2 - 2*b1 + 3) * q^74 + (-2*b6 - b2 - 2*b1) * q^75 + (b6 + b5 - b3 - 1) * q^76 + (-2*b6 + 2*b5 - b4 + b3 - b2 + 1) * q^77 + (-b6 - b5 + b3 - b1) * q^78 + (2*b5 + b4 - b3 - b2 - 5) * q^79 - b5 * q^80 + q^81 + (-2*b5 - 2*b4 + b2 - 1) * q^82 + (b4 + b3 + b1 - 1) * q^83 + (b6 + b2 + 1) * q^84 + (-3*b6 - b5 + b4 - 2*b2 - b1 - 4) * q^85 + (-b6 - 3*b5 - 3*b4 - b2 - 2*b1) * q^86 - q^87 + (b6 + b4 + b1) * q^88 + (-b4 + b3 + b2 + 3) * q^89 + b5 * q^90 + (-2*b6 + 4*b5 + b4 + b3 - b2 + 2*b1 + 1) * q^91 + q^92 + (-b6 + b5 + b3 - 2*b2 + b1 + 2) * q^93 + (b6 - b5 + b4 + 2*b2 + b1 + 2) * q^94 + (-b6 + b5 - b4 - b1 - 4) * q^95 + q^96 + (-2*b6 + 3*b5 - b3 + b1) * q^97 + (-3*b6 + b5 + 2*b4 - b3 - 3*b2 + b1 - 2) * q^98 + (-b6 - b4 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10})$$ 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9 $$7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} + 2 q^{5} + 7 q^{6} - 5 q^{7} - 7 q^{8} + 7 q^{9} - 2 q^{10} - q^{11} - 7 q^{12} + q^{13} + 5 q^{14} - 2 q^{15} + 7 q^{16} - q^{17} - 7 q^{18} - 9 q^{19} + 2 q^{20} + 5 q^{21} + q^{22} + 7 q^{23} + 7 q^{24} + q^{25} - q^{26} - 7 q^{27} - 5 q^{28} + 7 q^{29} + 2 q^{30} - 19 q^{31} - 7 q^{32} + q^{33} + q^{34} + 11 q^{35} + 7 q^{36} - 18 q^{37} + 9 q^{38} - q^{39} - 2 q^{40} + 4 q^{41} - 5 q^{42} - 9 q^{43} - q^{44} + 2 q^{45} - 7 q^{46} - 11 q^{47} - 7 q^{48} + 12 q^{49} - q^{50} + q^{51} + q^{52} + 8 q^{53} + 7 q^{54} - 11 q^{55} + 5 q^{56} + 9 q^{57} - 7 q^{58} + 12 q^{59} - 2 q^{60} - 5 q^{61} + 19 q^{62} - 5 q^{63} + 7 q^{64} + 23 q^{65} - q^{66} - 13 q^{67} - q^{68} - 7 q^{69} - 11 q^{70} - q^{71} - 7 q^{72} - 26 q^{73} + 18 q^{74} - q^{75} - 9 q^{76} + 6 q^{77} + q^{78} - 38 q^{79} + 2 q^{80} + 7 q^{81} - 4 q^{82} - 6 q^{83} + 5 q^{84} - 25 q^{85} + 9 q^{86} - 7 q^{87} + q^{88} + 20 q^{89} - 2 q^{90} + 2 q^{91} + 7 q^{92} + 19 q^{93} + 11 q^{94} - 31 q^{95} + 7 q^{96} - 8 q^{97} - 12 q^{98} - q^{99}+O(q^{100})$$ 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 + 2 * q^5 + 7 * q^6 - 5 * q^7 - 7 * q^8 + 7 * q^9 - 2 * q^10 - q^11 - 7 * q^12 + q^13 + 5 * q^14 - 2 * q^15 + 7 * q^16 - q^17 - 7 * q^18 - 9 * q^19 + 2 * q^20 + 5 * q^21 + q^22 + 7 * q^23 + 7 * q^24 + q^25 - q^26 - 7 * q^27 - 5 * q^28 + 7 * q^29 + 2 * q^30 - 19 * q^31 - 7 * q^32 + q^33 + q^34 + 11 * q^35 + 7 * q^36 - 18 * q^37 + 9 * q^38 - q^39 - 2 * q^40 + 4 * q^41 - 5 * q^42 - 9 * q^43 - q^44 + 2 * q^45 - 7 * q^46 - 11 * q^47 - 7 * q^48 + 12 * q^49 - q^50 + q^51 + q^52 + 8 * q^53 + 7 * q^54 - 11 * q^55 + 5 * q^56 + 9 * q^57 - 7 * q^58 + 12 * q^59 - 2 * q^60 - 5 * q^61 + 19 * q^62 - 5 * q^63 + 7 * q^64 + 23 * q^65 - q^66 - 13 * q^67 - q^68 - 7 * q^69 - 11 * q^70 - q^71 - 7 * q^72 - 26 * q^73 + 18 * q^74 - q^75 - 9 * q^76 + 6 * q^77 + q^78 - 38 * q^79 + 2 * q^80 + 7 * q^81 - 4 * q^82 - 6 * q^83 + 5 * q^84 - 25 * q^85 + 9 * q^86 - 7 * q^87 + q^88 + 20 * q^89 - 2 * q^90 + 2 * q^91 + 7 * q^92 + 19 * q^93 + 11 * q^94 - 31 * q^95 + 7 * q^96 - 8 * q^97 - 12 * q^98 - q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 13x^{5} + 16x^{4} + 19x^{3} - 8x^{2} - 10x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu^{6} - 7\nu^{5} - 47\nu^{4} + 100\nu^{3} + 4\nu^{2} - 44\nu - 6$$ 4*v^6 - 7*v^5 - 47*v^4 + 100*v^3 + 4*v^2 - 44*v - 6 $$\beta_{2}$$ $$=$$ $$-5\nu^{6} + 8\nu^{5} + 60\nu^{4} - 116\nu^{3} - 23\nu^{2} + 54\nu + 14$$ -5*v^6 + 8*v^5 + 60*v^4 - 116*v^3 - 23*v^2 + 54*v + 14 $$\beta_{3}$$ $$=$$ $$7\nu^{6} - 10\nu^{5} - 86\nu^{4} + 149\nu^{3} + 61\nu^{2} - 79\nu - 29$$ 7*v^6 - 10*v^5 - 86*v^4 + 149*v^3 + 61*v^2 - 79*v - 29 $$\beta_{4}$$ $$=$$ $$-7\nu^{6} + 11\nu^{5} + 85\nu^{4} - 161\nu^{3} - 45\nu^{2} + 87\nu + 26$$ -7*v^6 + 11*v^5 + 85*v^4 - 161*v^3 - 45*v^2 + 87*v + 26 $$\beta_{5}$$ $$=$$ $$8\nu^{6} - 12\nu^{5} - 98\nu^{4} + 177\nu^{3} + 64\nu^{2} - 95\nu - 35$$ 8*v^6 - 12*v^5 - 98*v^4 + 177*v^3 + 64*v^2 - 95*v - 35 $$\beta_{6}$$ $$=$$ $$-11\nu^{6} + 17\nu^{5} + 134\nu^{4} - 249\nu^{3} - 76\nu^{2} + 130\nu + 41$$ -11*v^6 + 17*v^5 + 134*v^4 - 249*v^3 - 76*v^2 + 130*v + 41
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2$$ (b5 + b4 + b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{6} + 3\beta_{5} - \beta_{3} - \beta_{2} + 8 ) / 2$$ (2*b6 + 3*b5 - b3 - b2 + 8) / 2 $$\nu^{3}$$ $$=$$ $$( 5\beta_{5} + 5\beta_{4} + 2\beta_{3} + 11\beta_{2} + 9\beta _1 + 3 ) / 2$$ (5*b5 + 5*b4 + 2*b3 + 11*b2 + 9*b1 + 3) / 2 $$\nu^{4}$$ $$=$$ $$( 24\beta_{6} + 26\beta_{5} - 7\beta_{4} - 9\beta_{3} - 18\beta_{2} - 5\beta _1 + 69 ) / 2$$ (24*b6 + 26*b5 - 7*b4 - 9*b3 - 18*b2 - 5*b1 + 69) / 2 $$\nu^{5}$$ $$=$$ $$( -8\beta_{6} + 30\beta_{5} + 47\beta_{4} + 33\beta_{3} + 122\beta_{2} + 95\beta _1 - 25 ) / 2$$ (-8*b6 + 30*b5 + 47*b4 + 33*b3 + 122*b2 + 95*b1 - 25) / 2 $$\nu^{6}$$ $$=$$ $$( 266\beta_{6} + 241\beta_{5} - 114\beta_{4} - 97\beta_{3} - 261\beta_{2} - 106\beta _1 + 698 ) / 2$$ (266*b6 + 241*b5 - 114*b4 - 97*b3 - 261*b2 - 106*b1 + 698) / 2

## 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
 2.99277 −3.51756 2.02256 −0.518095 0.932053 −0.571135 −0.340585
−1.00000 −1.00000 1.00000 −3.83000 1.00000 −4.91072 −1.00000 1.00000 3.83000
1.2 −1.00000 −1.00000 1.00000 −0.694170 1.00000 1.42319 −1.00000 1.00000 0.694170
1.3 −1.00000 −1.00000 1.00000 −0.662522 1.00000 −0.537195 −1.00000 1.00000 0.662522
1.4 −1.00000 −1.00000 1.00000 −0.324686 1.00000 2.40824 −1.00000 1.00000 0.324686
1.5 −1.00000 −1.00000 1.00000 1.78524 1.00000 0.541489 −1.00000 1.00000 −1.78524
1.6 −1.00000 −1.00000 1.00000 2.26164 1.00000 −5.18548 −1.00000 1.00000 −2.26164
1.7 −1.00000 −1.00000 1.00000 3.46450 1.00000 1.26047 −1.00000 1.00000 −3.46450
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 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.bg 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bg 7 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4002))$$:

 $$T_{5}^{7} - 2T_{5}^{6} - 16T_{5}^{5} + 34T_{5}^{4} + 29T_{5}^{3} - 42T_{5}^{2} - 40T_{5} - 8$$ T5^7 - 2*T5^6 - 16*T5^5 + 34*T5^4 + 29*T5^3 - 42*T5^2 - 40*T5 - 8 $$T_{7}^{7} + 5T_{7}^{6} - 18T_{7}^{5} - 52T_{7}^{4} + 172T_{7}^{3} - 96T_{7}^{2} - 48T_{7} + 32$$ T7^7 + 5*T7^6 - 18*T7^5 - 52*T7^4 + 172*T7^3 - 96*T7^2 - 48*T7 + 32 $$T_{11}^{7} + T_{11}^{6} - 37T_{11}^{5} - 47T_{11}^{4} + 248T_{11}^{3} + 204T_{11}^{2} - 456T_{11} - 176$$ T11^7 + T11^6 - 37*T11^5 - 47*T11^4 + 248*T11^3 + 204*T11^2 - 456*T11 - 176

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{7}$$
$3$ $$(T + 1)^{7}$$
$5$ $$T^{7} - 2 T^{6} + \cdots - 8$$
$7$ $$T^{7} + 5 T^{6} + \cdots + 32$$
$11$ $$T^{7} + T^{6} + \cdots - 176$$
$13$ $$T^{7} - T^{6} + \cdots - 6016$$
$17$ $$T^{7} + T^{6} + \cdots - 256$$
$19$ $$T^{7} + 9 T^{6} + \cdots - 1312$$
$23$ $$(T - 1)^{7}$$
$29$ $$(T - 1)^{7}$$
$31$ $$T^{7} + 19 T^{6} + \cdots + 589312$$
$37$ $$T^{7} + 18 T^{6} + \cdots + 3112$$
$41$ $$T^{7} - 4 T^{6} + \cdots + 134408$$
$43$ $$T^{7} + 9 T^{6} + \cdots - 1364288$$
$47$ $$T^{7} + 11 T^{6} + \cdots + 88832$$
$53$ $$T^{7} - 8 T^{6} + \cdots - 536128$$
$59$ $$T^{7} - 12 T^{6} + \cdots - 95296$$
$61$ $$T^{7} + 5 T^{6} + \cdots - 22768$$
$67$ $$T^{7} + 13 T^{6} + \cdots + 203776$$
$71$ $$T^{7} + T^{6} + \cdots + 183296$$
$73$ $$T^{7} + 26 T^{6} + \cdots - 836032$$
$79$ $$T^{7} + 38 T^{6} + \cdots + 86848$$
$83$ $$T^{7} + 6 T^{6} + \cdots - 2048$$
$89$ $$T^{7} - 20 T^{6} + \cdots + 512$$
$97$ $$T^{7} + 8 T^{6} + \cdots - 407552$$