# Properties

 Label 9405.2.a.v Level $9405$ Weight $2$ Character orbit 9405.a Self dual yes Analytic conductor $75.099$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

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: $$5$$ Coefficient field: $$\Q(\zeta_{22})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1$$ x^5 - x^4 - 4*x^3 + 3*x^2 + 3*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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} + \beta_{3} + \beta_1) q^{2} + (\beta_{4} + \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - 3) q^{7} + (\beta_{4} + 2 \beta_{2}) q^{8}+O(q^{10})$$ q + (-b4 + b3 + b1) * q^2 + (b4 + b1 + 1) * q^4 - q^5 + (-b4 + 2*b3 - b2 - 3) * q^7 + (b4 + 2*b2) * q^8 $$q + ( - \beta_{4} + \beta_{3} + \beta_1) q^{2} + (\beta_{4} + \beta_1 + 1) q^{4} - q^{5} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - 3) q^{7} + (\beta_{4} + 2 \beta_{2}) q^{8} + (\beta_{4} - \beta_{3} - \beta_1) q^{10} + q^{11} + ( - \beta_{3} + 3 \beta_{2} + 1) q^{13} + (3 \beta_{4} - \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{14}+ \cdots + ( - 11 \beta_{4} + 4 \beta_{3} + \cdots - 6) q^{98}+O(q^{100})$$ q + (-b4 + b3 + b1) * q^2 + (b4 + b1 + 1) * q^4 - q^5 + (-b4 + 2*b3 - b2 - 3) * q^7 + (b4 + 2*b2) * q^8 + (b4 - b3 - b1) * q^10 + q^11 + (-b3 + 3*b2 + 1) * q^13 + (3*b4 - b3 - 2*b2 - 3*b1 + 1) * q^14 + (2*b4 - b3 + 3*b2 - 2*b1 + 1) * q^16 + (2*b4 + b3 + b2 + 1) * q^17 + q^19 + (-b4 - b1 - 1) * q^20 + (-b4 + b3 + b1) * q^22 + (-b4 + 2*b1 + 1) * q^23 + q^25 + (4*b4 - 3*b3 + 3*b2 + b1 + 5) * q^26 + (-4*b4 + b3 - 2*b1 - 5) * q^28 + (b4 - 3*b3 + 2*b2 - 5*b1) * q^29 + (b3 + b2 - 5*b1) * q^31 + (2*b4 - b3 - b2 - b1 + 1) * q^32 + (2*b4 + 3*b3 + 3*b2 + b1 + 1) * q^34 + (b4 - 2*b3 + b2 + 3) * q^35 + (4*b3 + 2*b1 - 3) * q^37 + (-b4 + b3 + b1) * q^38 + (-b4 - 2*b2) * q^40 + (-3*b4 + b3 - 3*b2 + 3*b1 - 5) * q^41 + (6*b4 + 3*b2 + b1 - 1) * q^43 + (b4 + b1 + 1) * q^44 + (-b4 + b2 + 3*b1 + 3) * q^46 + (2*b4 + b3 - 4*b1 + 5) * q^47 + (b4 - 3*b3 - 2*b2 + 2*b1 + 4) * q^49 + (-b4 + b3 + b1) * q^50 + (-2*b4 + 5*b3 + 2*b2 + 6*b1 - 2) * q^52 + (-6*b4 + 7*b3 - 3*b2 + 4*b1 - 3) * q^53 - q^55 + (-6*b3 - 2*b2 - b1 + 1) * q^56 + (b4 - 4*b3 - 2*b2 - 5*b1 - 5) * q^58 + (-2*b4 - b3 + b2 + 2*b1 + 3) * q^59 + (-3*b3 + b1 + 1) * q^61 + (3*b4 - 4*b2 - 5*b1 - 2) * q^62 + (-8*b4 + 5*b3 - 6*b2 + 4*b1 - 8) * q^64 + (b3 - 3*b2 - 1) * q^65 + (5*b3 + b2 + 3*b1 - 7) * q^67 + (4*b4 + b3 + 4*b2 + 2*b1 + 6) * q^68 + (-3*b4 + b3 + 2*b2 + 3*b1 - 1) * q^70 + (-2*b4 + b2 + 1) * q^71 + (2*b4 - b3 + 2*b2 - b1 - 2) * q^73 + (7*b4 + b3 + 2*b2 - b1 + 6) * q^74 + (b4 + b1 + 1) * q^76 + (-b4 + 2*b3 - b2 - 3) * q^77 + (-2*b4 - 2*b3 + 4*b2 + 2*b1 + 1) * q^79 + (-2*b4 + b3 - 3*b2 + 2*b1 - 1) * q^80 + (-4*b3 - 3*b2 - 2*b1 + 1) * q^82 + (5*b3 - b2 + 2*b1 + 5) * q^83 + (-2*b4 - b3 - b2 - 1) * q^85 + (7*b4 + 2*b3 + 10*b2 + 1) * q^86 + (b4 + 2*b2) * q^88 + (6*b4 + b3 + b2 + 2*b1 + 4) * q^89 + (7*b4 - 7*b3 - b2 - 4*b1 - 1) * q^91 + (b4 + b3 + 3*b2 + 2*b1 + 4) * q^92 + (-4*b4 + 8*b3 - 2*b2 + b1 - 5) * q^94 - q^95 + (-3*b4 + 5*b3 + 3*b1 - 5) * q^97 + (-11*b4 + 4*b3 + b2 + 6*b1 - 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8}+O(q^{10})$$ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 $$5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31} + 2 q^{32} + 4 q^{34} + 11 q^{35} - 9 q^{37} + 3 q^{38} + 3 q^{40} - 15 q^{41} - 13 q^{43} + 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{49} + 3 q^{50} + q^{52} + 5 q^{53} - 5 q^{55} - 33 q^{58} + 17 q^{59} + 3 q^{61} - 14 q^{62} - 17 q^{64} - q^{65} - 28 q^{67} + 25 q^{68} + 6 q^{71} - 16 q^{73} + 21 q^{74} + 5 q^{76} - 11 q^{77} + 3 q^{79} + 3 q^{80} + 2 q^{82} + 33 q^{83} - 3 q^{85} - 10 q^{86} - 3 q^{88} + 16 q^{89} - 22 q^{91} + 19 q^{92} - 10 q^{94} - 5 q^{95} - 14 q^{97} - 10 q^{98}+O(q^{100})$$ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{22} + \zeta_{22}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 4\nu^{2} + 2$$ v^4 - 4*v^2 + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 4\beta_{2} + 6$$ b4 + 4*b2 + 6

## 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
 0.284630 −1.68251 −0.830830 1.30972 1.91899
−2.22871 0 2.96714 −1.00000 0 −4.42518 −2.15546 0 2.22871
1.2 −0.0881559 0 −1.99223 −1.00000 0 −1.95185 0.351939 0 0.0881559
1.3 1.37279 0 −0.115460 −1.00000 0 2.43232 −2.90407 0 −1.37279
1.4 1.54620 0 0.390736 −1.00000 0 −4.16140 −2.48825 0 −1.54620
1.5 2.39788 0 3.74982 −1.00000 0 −2.89389 4.19584 0 −2.39788
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.v 5
3.b odd 2 1 1045.2.a.d 5
15.d odd 2 1 5225.2.a.j 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.d 5 3.b odd 2 1
5225.2.a.j 5 15.d odd 2 1
9405.2.a.v 5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9405))$$:

 $$T_{2}^{5} - 3T_{2}^{4} - 3T_{2}^{3} + 15T_{2}^{2} - 10T_{2} - 1$$ T2^5 - 3*T2^4 - 3*T2^3 + 15*T2^2 - 10*T2 - 1 $$T_{7}^{5} + 11T_{7}^{4} + 33T_{7}^{3} - 22T_{7}^{2} - 231T_{7} - 253$$ T7^5 + 11*T7^4 + 33*T7^3 - 22*T7^2 - 231*T7 - 253 $$T_{13}^{5} - T_{13}^{4} - 37T_{13}^{3} + 47T_{13}^{2} + 322T_{13} - 529$$ T13^5 - T13^4 - 37*T13^3 + 47*T13^2 + 322*T13 - 529 $$T_{17}^{5} - 3T_{17}^{4} - 25T_{17}^{3} + 59T_{17}^{2} - 10T_{17} - 23$$ T17^5 - 3*T17^4 - 25*T17^3 + 59*T17^2 - 10*T17 - 23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - 3 T^{4} + \cdots - 1$$
$3$ $$T^{5}$$
$5$ $$(T + 1)^{5}$$
$7$ $$T^{5} + 11 T^{4} + \cdots - 253$$
$11$ $$(T - 1)^{5}$$
$13$ $$T^{5} - T^{4} + \cdots - 529$$
$17$ $$T^{5} - 3 T^{4} + \cdots - 23$$
$19$ $$(T - 1)^{5}$$
$23$ $$T^{5} - 8 T^{4} + \cdots + 1$$
$29$ $$T^{5} + 11 T^{4} + \cdots - 1441$$
$31$ $$T^{5} + 5 T^{4} + \cdots + 10649$$
$37$ $$T^{5} + 9 T^{4} + \cdots + 3917$$
$41$ $$T^{5} + 15 T^{4} + \cdots - 593$$
$43$ $$T^{5} + 13 T^{4} + \cdots + 28753$$
$47$ $$T^{5} - 20 T^{4} + \cdots + 989$$
$53$ $$T^{5} - 5 T^{4} + \cdots - 3917$$
$59$ $$T^{5} - 17 T^{4} + \cdots - 197$$
$61$ $$T^{5} - 3 T^{4} + \cdots + 241$$
$67$ $$T^{5} + 28 T^{4} + \cdots - 11881$$
$71$ $$T^{5} - 6 T^{4} + \cdots + 23$$
$73$ $$T^{5} + 16 T^{4} + \cdots + 23$$
$79$ $$T^{5} - 3 T^{4} + \cdots - 15203$$
$83$ $$T^{5} - 33 T^{4} + \cdots + 737$$
$89$ $$T^{5} - 16 T^{4} + \cdots - 9791$$
$97$ $$T^{5} + 14 T^{4} + \cdots + 3323$$