# Properties

 Label 9405.2.a.z Level $9405$ Weight $2$ Character orbit 9405.a Self dual yes Analytic conductor $75.099$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9405.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: $$75.0993031010$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.7281497.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1$$ x^6 - 2*x^5 - 5*x^4 + 7*x^3 + 6*x^2 - 2*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1045) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + (\beta_{5} + \beta_{4}) q^{4} + q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{7} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{8}+O(q^{10})$$ q + b4 * q^2 + (b5 + b4) * q^4 + q^5 + (-b3 - b1 + 1) * q^7 + (b5 + b4 + b1 + 1) * q^8 $$q + \beta_{4} q^{2} + (\beta_{5} + \beta_{4}) q^{4} + q^{5} + ( - \beta_{3} - \beta_1 + 1) q^{7} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{8} + \beta_{4} q^{10} + q^{11} + (\beta_{4} + \beta_{3} - 1) q^{13} + (\beta_{4} + 2 \beta_{2} + 1) q^{14} + ( - \beta_{5} + 2 \beta_{4} + \cdots + \beta_1) q^{16}+ \cdots + ( - 2 \beta_{4} - 3 \beta_{3} + \cdots + 2) q^{98}+O(q^{100})$$ q + b4 * q^2 + (b5 + b4) * q^4 + q^5 + (-b3 - b1 + 1) * q^7 + (b5 + b4 + b1 + 1) * q^8 + b4 * q^10 + q^11 + (b4 + b3 - 1) * q^13 + (b4 + 2*b2 + 1) * q^14 + (-b5 + 2*b4 - b2 + b1) * q^16 + (-b4 + b3 + 2*b2 - 2*b1 + 1) * q^17 - q^19 + (b5 + b4) * q^20 + b4 * q^22 + (b5 - b4 + 2*b3 - 2*b2 - b1) * q^23 + q^25 + (b5 - b2 + 2) * q^26 + (b5 + 2*b2 - 2*b1 + 2) * q^28 + (-b5 + b3 - b2 - 3*b1 + 3) * q^29 + (-b4 + b3 - 3*b1 - 2) * q^31 + (-b4 + b3 - 2*b2 - b1 + 1) * q^32 + (-b5 - 2*b4 - 2*b3 + 3*b2 - 4*b1 + 2) * q^34 + (-b3 - b1 + 1) * q^35 + (b5 + 3*b4 + b3 - b2 + b1 - 2) * q^37 - b4 * q^38 + (b5 + b4 + b1 + 1) * q^40 + (b5 + 2*b4 - b3 + 2*b1 + 2) * q^41 + (b5 + b4 - b3 + 2*b2 + b1) * q^43 + (b5 + b4) * q^44 + (-b5 + 3*b4 + 2*b3 - 3*b2 + 5*b1 - 4) * q^46 + (-b5 + 2*b4 - 2*b2 + 2*b1 + 2) * q^47 + (b5 - b3 + 3*b2 - 1) * q^49 + b4 * q^50 + (3*b4 - b3 - b2 + 3*b1) * q^52 + (b4 - b3 - 2*b1 + 3) * q^53 + q^55 + (-2*b3 - 3*b1 + 1) * q^56 + (2*b4 + b3 + b2 + b1 + 3) * q^58 + (2*b5 + b4 - b3 + 2*b2 + 2*b1 + 3) * q^59 + (3*b4 - b3 - 3*b2 + 2*b1 + 1) * q^61 + (-b5 - 3*b4 + 2*b2 + 1) * q^62 + (b5 - 2*b4 + 2*b3 + 2*b1 - 3) * q^64 + (b4 + b3 - 1) * q^65 + (b5 + 2*b4 - 2*b3 - b2 + 6*b1 - 2) * q^67 + (-2*b5 - 3*b4 - 5*b3 + 5*b2 - 3*b1 + 2) * q^68 + (b4 + 2*b2 + 1) * q^70 + (-2*b5 + 2*b4 + b2 + 3*b1 + 1) * q^71 + (b5 + 2*b4 + 2*b2 + b1 - 1) * q^73 + (3*b5 + 4*b4 + b3 - 3*b2 + 3*b1 + 3) * q^74 + (-b5 - b4) * q^76 + (-b3 - b1 + 1) * q^77 + (b5 - 3*b4 + 3*b3 + 3*b2 - b1 + 2) * q^79 + (-b5 + 2*b4 - b2 + b1) * q^80 + (2*b5 + 6*b4 - b2 + b1 + 1) * q^82 + (-2*b5 + 5*b4 - 3*b3 + 4*b1 - 1) * q^83 + (-b4 + b3 + 2*b2 - 2*b1 + 1) * q^85 + (b5 + b4 - 2*b3 + 2*b2 - 3*b1 + 2) * q^86 + (b5 + b4 + b1 + 1) * q^88 + (3*b5 + 2*b4 - b2 + 3*b1 + 1) * q^89 + (b4 + 2*b3 + b2 - 3) * q^91 + (b5 + 2*b4 - b3 - 6*b2 + 7*b1 - 1) * q^92 + (2*b5 + 4*b4 + 2*b3 - 4*b2 + 3*b1 + 1) * q^94 - q^95 + (-4*b5 - 5*b4 - 2*b3 - 2*b2 - 2*b1 + 3) * q^97 + (-2*b4 - 3*b3 + 4*b2 - 5*b1 + 2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10})$$ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 $$6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100})$$ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 2\nu - 1$$ v^5 - 2*v^4 - 4*v^3 + 6*v^2 + 2*v - 1 $$\beta_{4}$$ $$=$$ $$\nu^{5} - 2\nu^{4} - 5\nu^{3} + 7\nu^{2} + 5\nu - 1$$ v^5 - 2*v^4 - 5*v^3 + 7*v^2 + 5*v - 1 $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 3\nu^{4} + 3\nu^{3} - 11\nu^{2} + 4$$ -v^5 + 3*v^4 + 3*v^3 - 11*v^2 + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 2$$ -b4 + b3 + b2 + 4*b1 + 2 $$\nu^{4}$$ $$=$$ $$\beta_{5} - \beta_{4} + 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 9$$ b5 - b4 + 2*b3 + 6*b2 + 7*b1 + 9 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - 6\beta_{4} + 9\beta_{3} + 10\beta_{2} + 22\beta _1 + 15$$ 2*b5 - 6*b4 + 9*b3 + 10*b2 + 22*b1 + 15

## 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
 −0.326248 2.59744 1.77015 −0.748369 −1.79049 0.497517
−1.73890 0 1.02379 1.00000 0 2.22757 1.69754 0 −1.73890
1.2 −1.21244 0 −0.529980 1.00000 0 −3.37010 3.06746 0 −1.21244
1.3 −0.205229 0 −1.95788 1.00000 0 2.33231 0.812271 0 −0.205229
1.4 0.412130 0 −1.83015 1.00000 0 0.0703171 −1.57852 0 0.412130
1.5 2.23198 0 2.98176 1.00000 0 4.13295 2.19127 0 2.23198
1.6 2.51246 0 4.31247 1.00000 0 −0.393051 5.80998 0 2.51246
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$+1$$

## 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 9405.2.a.z 6
3.b odd 2 1 1045.2.a.f 6
15.d odd 2 1 5225.2.a.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.f 6 3.b odd 2 1
5225.2.a.l 6 15.d odd 2 1
9405.2.a.z 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9405))$$:

 $$T_{2}^{6} - 2T_{2}^{5} - 6T_{2}^{4} + 8T_{2}^{3} + 11T_{2}^{2} - 3T_{2} - 1$$ T2^6 - 2*T2^5 - 6*T2^4 + 8*T2^3 + 11*T2^2 - 3*T2 - 1 $$T_{7}^{6} - 5T_{7}^{5} - 7T_{7}^{4} + 58T_{7}^{3} - 53T_{7}^{2} - 25T_{7} + 2$$ T7^6 - 5*T7^5 - 7*T7^4 + 58*T7^3 - 53*T7^2 - 25*T7 + 2 $$T_{13}^{6} + 5T_{13}^{5} - 7T_{13}^{4} - 39T_{13}^{3} + 22T_{13}^{2} + 21T_{13} + 2$$ T13^6 + 5*T13^5 - 7*T13^4 - 39*T13^3 + 22*T13^2 + 21*T13 + 2 $$T_{17}^{6} + T_{17}^{5} - 73T_{17}^{4} - 25T_{17}^{3} + 1226T_{17}^{2} - 1329T_{17} - 1360$$ T17^6 + T17^5 - 73*T17^4 - 25*T17^3 + 1226*T17^2 - 1329*T17 - 1360

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 2 T^{5} + \cdots - 1$$
$3$ $$T^{6}$$
$5$ $$(T - 1)^{6}$$
$7$ $$T^{6} - 5 T^{5} + \cdots + 2$$
$11$ $$(T - 1)^{6}$$
$13$ $$T^{6} + 5 T^{5} + \cdots + 2$$
$17$ $$T^{6} + T^{5} + \cdots - 1360$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} + 4 T^{5} + \cdots - 8912$$
$29$ $$T^{6} - 9 T^{5} + \cdots + 3074$$
$31$ $$T^{6} + 21 T^{5} + \cdots - 20$$
$37$ $$T^{6} + 3 T^{5} + \cdots - 592$$
$41$ $$T^{6} - 23 T^{5} + \cdots - 4210$$
$43$ $$T^{6} - 7 T^{5} + \cdots - 326$$
$47$ $$T^{6} - 18 T^{5} + \cdots - 620$$
$53$ $$T^{6} - 17 T^{5} + \cdots - 4084$$
$59$ $$T^{6} - 29 T^{5} + \cdots - 54968$$
$61$ $$T^{6} - 17 T^{5} + \cdots + 2294$$
$67$ $$T^{6} - 8 T^{5} + \cdots - 73006$$
$71$ $$T^{6} - 12 T^{5} + \cdots - 5912$$
$73$ $$T^{6} - 2 T^{5} + \cdots - 4000$$
$79$ $$T^{6} - 3 T^{5} + \cdots + 28264$$
$83$ $$T^{6} - 11 T^{5} + \cdots - 82582$$
$89$ $$T^{6} - 22 T^{5} + \cdots + 3286$$
$97$ $$T^{6} + 2 T^{5} + \cdots - 967268$$
show more
show less