# Properties

 Label 4002.2.a.bi Level $4002$ Weight $2$ Character orbit 4002.a Self dual yes Analytic conductor $31.956$ Analytic rank $1$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600$$ x^8 - 24*x^6 - 3*x^5 + 194*x^4 + 39*x^3 - 607*x^2 - 104*x + 600 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - \beta_{4} q^{5} - q^{6} + (\beta_{6} - 1) q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - b4 * q^5 - q^6 + (b6 - 1) * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - \beta_{4} q^{5} - q^{6} + (\beta_{6} - 1) q^{7} - q^{8} + q^{9} + \beta_{4} q^{10} + ( - \beta_{2} - \beta_1 - 1) q^{11} + q^{12} + ( - \beta_{7} + \beta_{4} + \beta_{2} - 1) q^{13} + ( - \beta_{6} + 1) q^{14} - \beta_{4} q^{15} + q^{16} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{17}+ \cdots + ( - \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - b4 * q^5 - q^6 + (b6 - 1) * q^7 - q^8 + q^9 + b4 * q^10 + (-b2 - b1 - 1) * q^11 + q^12 + (-b7 + b4 + b2 - 1) * q^13 + (-b6 + 1) * q^14 - b4 * q^15 + q^16 + (b7 - b6 - b5 + b3 + b1) * q^17 - q^18 + (-b6 + b5 + b4 - 1) * q^19 - b4 * q^20 + (b6 - 1) * q^21 + (b2 + b1 + 1) * q^22 - q^23 - q^24 + (-b6 + b4 - 2*b3 + b1 + 1) * q^25 + (b7 - b4 - b2 + 1) * q^26 + q^27 + (b6 - 1) * q^28 + q^29 + b4 * q^30 + (b7 + b4 + b2 - 1) * q^31 - q^32 + (-b2 - b1 - 1) * q^33 + (-b7 + b6 + b5 - b3 - b1) * q^34 + (b7 - b6 + b4 + b3 + b1) * q^35 + q^36 + (b5 + b4 - b3 + b2 - 3) * q^37 + (b6 - b5 - b4 + 1) * q^38 + (-b7 + b4 + b2 - 1) * q^39 + b4 * q^40 + (-b7 + b5 - b3 - b2 - b1 - 3) * q^41 + (-b6 + 1) * q^42 + (b7 + b4 - 2*b2 - 1) * q^43 + (-b2 - b1 - 1) * q^44 - b4 * q^45 + q^46 + (-b7 - b6 + b4 - b3 + 2*b2 + b1 - 4) * q^47 + q^48 + (-b5 - b4 + b3 + b1 + 3) * q^49 + (b6 - b4 + 2*b3 - b1 - 1) * q^50 + (b7 - b6 - b5 + b3 + b1) * q^51 + (-b7 + b4 + b2 - 1) * q^52 + (b7 - b6 - b5 - 2*b4 + 2*b1 + 2) * q^53 - q^54 + (-b6 - b5 + b4 + 2*b3 + b2 + 1) * q^55 + (-b6 + 1) * q^56 + (-b6 + b5 + b4 - 1) * q^57 - q^58 + (2*b7 - b6 + b4 - b1 - 4) * q^59 - b4 * q^60 + (-b7 + b6 - b5 - b4 + b3 + b1 + 2) * q^61 + (-b7 - b4 - b2 + 1) * q^62 + (b6 - 1) * q^63 + q^64 + (b5 + b4 - 2*b3 - b2 - b1 - 5) * q^65 + (b2 + b1 + 1) * q^66 + (-2*b6 + b5 - 2*b4 + b3 + b1 + 2) * q^67 + (b7 - b6 - b5 + b3 + b1) * q^68 - q^69 + (-b7 + b6 - b4 - b3 - b1) * q^70 + (-b7 + 2*b5 - b4 + b2 - 2*b1 - 5) * q^71 - q^72 + (-b7 - b6 - b5 + 2*b4 + 2) * q^73 + (-b5 - b4 + b3 - b2 + 3) * q^74 + (-b6 + b4 - 2*b3 + b1 + 1) * q^75 + (-b6 + b5 + b4 - 1) * q^76 + (-2*b7 - 2*b6 + 3*b4 - b3 + 3*b2 + 2*b1 - 3) * q^77 + (b7 - b4 - b2 + 1) * q^78 + (2*b6 - b5 + b3 - b2 - 2*b1 - 1) * q^79 - b4 * q^80 + q^81 + (b7 - b5 + b3 + b2 + b1 + 3) * q^82 + (-b7 + b4 + b3 - 2*b2 - b1 - 5) * q^83 + (b6 - 1) * q^84 + (b7 + 3*b6 + b4 + b3 - 2*b2 - 3*b1 - 4) * q^85 + (-b7 - b4 + 2*b2 + 1) * q^86 + q^87 + (b2 + b1 + 1) * q^88 + (-b7 + b6 + b4 - 2*b1 - 8) * q^89 + b4 * q^90 + (-2*b7 + 2*b6 + b4 - b3 - b2 - 4*b1 - 5) * q^91 - q^92 + (b7 + b4 + b2 - 1) * q^93 + (b7 + b6 - b4 + b3 - 2*b2 - b1 + 4) * q^94 + (-b7 + 3*b6 + b4 + b3 - b1 - 8) * q^95 - q^96 + (2*b7 - b6 - 2*b4 + 3*b3 - 2*b2 + b1 + 3) * q^97 + (b5 + b4 - b3 - b1 - 3) * q^98 + (-b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10})$$ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 $$8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 7 q^{11} + 8 q^{12} - 3 q^{13} + 6 q^{14} - 3 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{21} + 7 q^{22} - 8 q^{23} - 8 q^{24} + 13 q^{25} + 3 q^{26} + 8 q^{27} - 6 q^{28} + 8 q^{29} + 3 q^{30} - q^{31} - 8 q^{32} - 7 q^{33} + 12 q^{34} - 6 q^{35} + 8 q^{36} - 11 q^{37} + 4 q^{38} - 3 q^{39} + 3 q^{40} - 17 q^{41} + 6 q^{42} - 10 q^{43} - 7 q^{44} - 3 q^{45} + 8 q^{46} - 26 q^{47} + 8 q^{48} + 10 q^{49} - 13 q^{50} - 12 q^{51} - 3 q^{52} - 2 q^{53} - 8 q^{54} + q^{55} + 6 q^{56} - 4 q^{57} - 8 q^{58} - 25 q^{59} - 3 q^{60} + 3 q^{61} + q^{62} - 6 q^{63} + 8 q^{64} - 25 q^{65} + 7 q^{66} + q^{67} - 12 q^{68} - 8 q^{69} + 6 q^{70} - 27 q^{71} - 8 q^{72} + 16 q^{73} + 11 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{77} + 3 q^{78} - 6 q^{79} - 3 q^{80} + 8 q^{81} + 17 q^{82} - 44 q^{83} - 6 q^{84} - 20 q^{85} + 10 q^{86} + 8 q^{87} + 7 q^{88} - 52 q^{89} + 3 q^{90} - 18 q^{91} - 8 q^{92} - q^{93} + 26 q^{94} - 56 q^{95} - 8 q^{96} - 4 q^{97} - 10 q^{98} - 7 q^{99}+O(q^{100})$$ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 + 3 * q^10 - 7 * q^11 + 8 * q^12 - 3 * q^13 + 6 * q^14 - 3 * q^15 + 8 * q^16 - 12 * q^17 - 8 * q^18 - 4 * q^19 - 3 * q^20 - 6 * q^21 + 7 * q^22 - 8 * q^23 - 8 * q^24 + 13 * q^25 + 3 * q^26 + 8 * q^27 - 6 * q^28 + 8 * q^29 + 3 * q^30 - q^31 - 8 * q^32 - 7 * q^33 + 12 * q^34 - 6 * q^35 + 8 * q^36 - 11 * q^37 + 4 * q^38 - 3 * q^39 + 3 * q^40 - 17 * q^41 + 6 * q^42 - 10 * q^43 - 7 * q^44 - 3 * q^45 + 8 * q^46 - 26 * q^47 + 8 * q^48 + 10 * q^49 - 13 * q^50 - 12 * q^51 - 3 * q^52 - 2 * q^53 - 8 * q^54 + q^55 + 6 * q^56 - 4 * q^57 - 8 * q^58 - 25 * q^59 - 3 * q^60 + 3 * q^61 + q^62 - 6 * q^63 + 8 * q^64 - 25 * q^65 + 7 * q^66 + q^67 - 12 * q^68 - 8 * q^69 + 6 * q^70 - 27 * q^71 - 8 * q^72 + 16 * q^73 + 11 * q^74 + 13 * q^75 - 4 * q^76 - 16 * q^77 + 3 * q^78 - 6 * q^79 - 3 * q^80 + 8 * q^81 + 17 * q^82 - 44 * q^83 - 6 * q^84 - 20 * q^85 + 10 * q^86 + 8 * q^87 + 7 * q^88 - 52 * q^89 + 3 * q^90 - 18 * q^91 - 8 * q^92 - q^93 + 26 * q^94 - 56 * q^95 - 8 * q^96 - 4 * q^97 - 10 * q^98 - 7 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600$$ :

 $$\beta_{1}$$ $$=$$ $$( -11\nu^{7} - 79\nu^{6} + 173\nu^{5} + 1466\nu^{4} - 180\nu^{3} - 8105\nu^{2} - 2752\nu + 12248 ) / 1048$$ (-11*v^7 - 79*v^6 + 173*v^5 + 1466*v^4 - 180*v^3 - 8105*v^2 - 2752*v + 12248) / 1048 $$\beta_{2}$$ $$=$$ $$( 9\nu^{7} + 17\nu^{6} - 213\nu^{5} - 342\nu^{4} + 1362\nu^{3} + 1701\nu^{2} - 1726\nu - 732 ) / 524$$ (9*v^7 + 17*v^6 - 213*v^5 - 342*v^4 + 1362*v^3 + 1701*v^2 - 1726*v - 732) / 524 $$\beta_{3}$$ $$=$$ $$( -21\nu^{7} + 135\nu^{6} + 235\nu^{5} - 2346\nu^{4} + 228\nu^{3} + 11489\nu^{2} - 3920\nu - 15584 ) / 1048$$ (-21*v^7 + 135*v^6 + 235*v^5 - 2346*v^4 + 228*v^3 + 11489*v^2 - 3920*v - 15584) / 1048 $$\beta_{4}$$ $$=$$ $$( 57\nu^{7} - 67\nu^{6} - 1087\nu^{5} + 978\nu^{4} + 6268\nu^{3} - 3637\nu^{2} - 11368\nu + 4272 ) / 1048$$ (57*v^7 - 67*v^6 - 1087*v^5 + 978*v^4 + 6268*v^3 - 3637*v^2 - 11368*v + 4272) / 1048 $$\beta_{5}$$ $$=$$ $$( 57\nu^{7} - 67\nu^{6} - 1087\nu^{5} + 978\nu^{4} + 6268\nu^{3} - 3637\nu^{2} - 9272\nu + 4272 ) / 1048$$ (57*v^7 - 67*v^6 - 1087*v^5 + 978*v^4 + 6268*v^3 - 3637*v^2 - 9272*v + 4272) / 1048 $$\beta_{6}$$ $$=$$ $$( -20\nu^{7} + 35\nu^{6} + 386\nu^{5} - 681\nu^{4} - 2197\nu^{3} + 3949\nu^{2} + 3690\nu - 6408 ) / 262$$ (-20*v^7 + 35*v^6 + 386*v^5 - 681*v^4 - 2197*v^3 + 3949*v^2 + 3690*v - 6408) / 262 $$\beta_{7}$$ $$=$$ $$( 103\nu^{7} - 213\nu^{6} - 2001\nu^{5} + 3422\nu^{4} + 11308\nu^{3} - 14331\nu^{2} - 17104\nu + 15552 ) / 1048$$ (103*v^7 - 213*v^6 - 2001*v^5 + 3422*v^4 + 11308*v^3 - 14331*v^2 - 17104*v + 15552) / 1048
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} ) / 2$$ (b5 - b4) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - \beta_{4} + \beta_{3} - 2\beta_{2} - \beta _1 + 13 ) / 2$$ (b7 - b4 + b3 - 2*b2 - b1 + 13) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} + 8\beta_{5} - 5\beta_{4} + \beta_{3} - 2\beta_{2} + \beta _1 + 3 ) / 2$$ (-b7 + 8*b5 - 5*b4 + b3 - 2*b2 + b1 + 3) / 2 $$\nu^{4}$$ $$=$$ $$( 11\beta_{7} - 4\beta_{6} + \beta_{5} - 16\beta_{4} + 13\beta_{3} - 26\beta_{2} - 13\beta _1 + 109 ) / 2$$ (11*b7 - 4*b6 + b5 - 16*b4 + 13*b3 - 26*b2 - 13*b1 + 109) / 2 $$\nu^{5}$$ $$=$$ $$( -20\beta_{7} + 77\beta_{5} - 27\beta_{4} + 6\beta_{3} - 32\beta_{2} + 8\beta _1 + 44 ) / 2$$ (-20*b7 + 77*b5 - 27*b4 + 6*b3 - 32*b2 + 8*b1 + 44) / 2 $$\nu^{6}$$ $$=$$ $$( 99\beta_{7} - 72\beta_{6} + 23\beta_{5} - 188\beta_{4} + 147\beta_{3} - 286\beta_{2} - 153\beta _1 + 1017 ) / 2$$ (99*b7 - 72*b6 + 23*b5 - 188*b4 + 147*b3 - 286*b2 - 153*b1 + 1017) / 2 $$\nu^{7}$$ $$=$$ $$-140\beta_{7} - 8\beta_{6} + 399\beta_{5} - 69\beta_{4} + 9\beta_{3} - 204\beta_{2} + 11\beta _1 + 257$$ -140*b7 - 8*b6 + 399*b5 - 69*b4 + 9*b3 - 204*b2 + 11*b1 + 257

## 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
 −2.45069 −1.57534 −2.15995 1.98746 3.34107 2.85533 −3.18819 1.19031
−1.00000 1.00000 1.00000 −4.40973 −1.00000 −0.551242 −1.00000 1.00000 4.40973
1.2 −1.00000 1.00000 1.00000 −2.69359 −1.00000 −3.87906 −1.00000 1.00000 2.69359
1.3 −1.00000 1.00000 1.00000 −1.69873 −1.00000 3.41134 −1.00000 1.00000 1.69873
1.4 −1.00000 1.00000 1.00000 −0.881455 −1.00000 0.253486 −1.00000 1.00000 0.881455
1.5 −1.00000 1.00000 1.00000 −0.464023 −1.00000 −2.35800 −1.00000 1.00000 0.464023
1.6 −1.00000 1.00000 1.00000 1.27098 −1.00000 3.55617 −1.00000 1.00000 −1.27098
1.7 −1.00000 1.00000 1.00000 1.60812 −1.00000 −3.37648 −1.00000 1.00000 −1.60812
1.8 −1.00000 1.00000 1.00000 4.26843 −1.00000 −3.05622 −1.00000 1.00000 −4.26843
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.bi 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bi 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4002))$$:

 $$T_{5}^{8} + 3T_{5}^{7} - 22T_{5}^{6} - 66T_{5}^{5} + 67T_{5}^{4} + 231T_{5}^{3} - 18T_{5}^{2} - 204T_{5} - 72$$ T5^8 + 3*T5^7 - 22*T5^6 - 66*T5^5 + 67*T5^4 + 231*T5^3 - 18*T5^2 - 204*T5 - 72 $$T_{7}^{8} + 6T_{7}^{7} - 15T_{7}^{6} - 144T_{7}^{5} - 80T_{7}^{4} + 844T_{7}^{3} + 1400T_{7}^{2} + 224T_{7} - 160$$ T7^8 + 6*T7^7 - 15*T7^6 - 144*T7^5 - 80*T7^4 + 844*T7^3 + 1400*T7^2 + 224*T7 - 160 $$T_{11}^{8} + 7T_{11}^{7} - 39T_{11}^{6} - 315T_{11}^{5} + 266T_{11}^{4} + 3846T_{11}^{3} + 2376T_{11}^{2} - 8424T_{11} - 3888$$ T11^8 + 7*T11^7 - 39*T11^6 - 315*T11^5 + 266*T11^4 + 3846*T11^3 + 2376*T11^2 - 8424*T11 - 3888

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{8}$$
$3$ $$(T - 1)^{8}$$
$5$ $$T^{8} + 3 T^{7} + \cdots - 72$$
$7$ $$T^{8} + 6 T^{7} + \cdots - 160$$
$11$ $$T^{8} + 7 T^{7} + \cdots - 3888$$
$13$ $$T^{8} + 3 T^{7} + \cdots + 1264$$
$17$ $$T^{8} + 12 T^{7} + \cdots - 89568$$
$19$ $$T^{8} + 4 T^{7} + \cdots + 178784$$
$23$ $$(T + 1)^{8}$$
$29$ $$(T - 1)^{8}$$
$31$ $$T^{8} + T^{7} + \cdots - 10368$$
$37$ $$T^{8} + 11 T^{7} + \cdots + 18680$$
$41$ $$T^{8} + 17 T^{7} + \cdots + 30024$$
$43$ $$T^{8} + 10 T^{7} + \cdots - 222912$$
$47$ $$T^{8} + 26 T^{7} + \cdots + 182016$$
$53$ $$T^{8} + 2 T^{7} + \cdots + 3939840$$
$59$ $$T^{8} + 25 T^{7} + \cdots - 42129216$$
$61$ $$T^{8} - 3 T^{7} + \cdots - 14816$$
$67$ $$T^{8} - T^{7} + \cdots + 7438336$$
$71$ $$T^{8} + 27 T^{7} + \cdots - 6184320$$
$73$ $$T^{8} - 16 T^{7} + \cdots + 640384$$
$79$ $$T^{8} + 6 T^{7} + \cdots + 152000$$
$83$ $$T^{8} + 44 T^{7} + \cdots + 8418048$$
$89$ $$T^{8} + 52 T^{7} + \cdots - 27648$$
$97$ $$T^{8} + 4 T^{7} + \cdots - 2388480$$