# Properties

 Label 3330.2.d.j Level $3330$ Weight $2$ Character orbit 3330.d Analytic conductor $26.590$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1999,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3330.1999");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.d (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$26.5901838731$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1110) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} - 2 \beta_1 q^{7} - \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 - q^4 - b2 * q^5 - 2*b1 * q^7 - b1 * q^8 $$q + \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} - 2 \beta_1 q^{7} - \beta_1 q^{8} + \beta_{3} q^{10} + (\beta_{3} - 3) q^{11} + 2 \beta_{2} q^{13} + 2 q^{14} + q^{16} - 2 \beta_{2} q^{17} + ( - \beta_{3} + 5) q^{19} + \beta_{2} q^{20} + (\beta_{2} - 3 \beta_1) q^{22} - 4 \beta_1 q^{23} - 5 q^{25} - 2 \beta_{3} q^{26} + 2 \beta_1 q^{28} + 4 q^{29} + (2 \beta_{3} + 2) q^{31} + \beta_1 q^{32} + 2 \beta_{3} q^{34} - 2 \beta_{3} q^{35} - \beta_1 q^{37} + ( - \beta_{2} + 5 \beta_1) q^{38} - \beta_{3} q^{40} - 2 \beta_{3} q^{41} + ( - 2 \beta_{2} + 6 \beta_1) q^{43} + ( - \beta_{3} + 3) q^{44} + 4 q^{46} + ( - 3 \beta_{2} + 3 \beta_1) q^{47} + 3 q^{49} - 5 \beta_1 q^{50} - 2 \beta_{2} q^{52} + (2 \beta_{2} - 8 \beta_1) q^{53} + (3 \beta_{2} - 5 \beta_1) q^{55} - 2 q^{56} + 4 \beta_1 q^{58} + ( - 4 \beta_{3} - 2) q^{59} + ( - \beta_{3} + 3) q^{61} + (2 \beta_{2} + 2 \beta_1) q^{62} - q^{64} + 10 q^{65} + ( - 2 \beta_{2} + 6 \beta_1) q^{67} + 2 \beta_{2} q^{68} - 2 \beta_{2} q^{70} + ( - 2 \beta_{3} - 2) q^{71} + ( - 2 \beta_{2} + 6 \beta_1) q^{73} + q^{74} + (\beta_{3} - 5) q^{76} + ( - 2 \beta_{2} + 6 \beta_1) q^{77} + ( - 2 \beta_{3} - 2) q^{79} - \beta_{2} q^{80} - 2 \beta_{2} q^{82} + ( - 6 \beta_{2} + 2 \beta_1) q^{83} - 10 q^{85} + (2 \beta_{3} - 6) q^{86} + ( - \beta_{2} + 3 \beta_1) q^{88} + ( - 2 \beta_{3} - 12) q^{89} + 4 \beta_{3} q^{91} + 4 \beta_1 q^{92} + (3 \beta_{3} - 3) q^{94} + ( - 5 \beta_{2} + 5 \beta_1) q^{95} + ( - 5 \beta_{2} - 3 \beta_1) q^{97} + 3 \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 - q^4 - b2 * q^5 - 2*b1 * q^7 - b1 * q^8 + b3 * q^10 + (b3 - 3) * q^11 + 2*b2 * q^13 + 2 * q^14 + q^16 - 2*b2 * q^17 + (-b3 + 5) * q^19 + b2 * q^20 + (b2 - 3*b1) * q^22 - 4*b1 * q^23 - 5 * q^25 - 2*b3 * q^26 + 2*b1 * q^28 + 4 * q^29 + (2*b3 + 2) * q^31 + b1 * q^32 + 2*b3 * q^34 - 2*b3 * q^35 - b1 * q^37 + (-b2 + 5*b1) * q^38 - b3 * q^40 - 2*b3 * q^41 + (-2*b2 + 6*b1) * q^43 + (-b3 + 3) * q^44 + 4 * q^46 + (-3*b2 + 3*b1) * q^47 + 3 * q^49 - 5*b1 * q^50 - 2*b2 * q^52 + (2*b2 - 8*b1) * q^53 + (3*b2 - 5*b1) * q^55 - 2 * q^56 + 4*b1 * q^58 + (-4*b3 - 2) * q^59 + (-b3 + 3) * q^61 + (2*b2 + 2*b1) * q^62 - q^64 + 10 * q^65 + (-2*b2 + 6*b1) * q^67 + 2*b2 * q^68 - 2*b2 * q^70 + (-2*b3 - 2) * q^71 + (-2*b2 + 6*b1) * q^73 + q^74 + (b3 - 5) * q^76 + (-2*b2 + 6*b1) * q^77 + (-2*b3 - 2) * q^79 - b2 * q^80 - 2*b2 * q^82 + (-6*b2 + 2*b1) * q^83 - 10 * q^85 + (2*b3 - 6) * q^86 + (-b2 + 3*b1) * q^88 + (-2*b3 - 12) * q^89 + 4*b3 * q^91 + 4*b1 * q^92 + (3*b3 - 3) * q^94 + (-5*b2 + 5*b1) * q^95 + (-5*b2 - 3*b1) * q^97 + 3*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} - 12 q^{11} + 8 q^{14} + 4 q^{16} + 20 q^{19} - 20 q^{25} + 16 q^{29} + 8 q^{31} + 12 q^{44} + 16 q^{46} + 12 q^{49} - 8 q^{56} - 8 q^{59} + 12 q^{61} - 4 q^{64} + 40 q^{65} - 8 q^{71} + 4 q^{74} - 20 q^{76} - 8 q^{79} - 40 q^{85} - 24 q^{86} - 48 q^{89} - 12 q^{94}+O(q^{100})$$ 4 * q - 4 * q^4 - 12 * q^11 + 8 * q^14 + 4 * q^16 + 20 * q^19 - 20 * q^25 + 16 * q^29 + 8 * q^31 + 12 * q^44 + 16 * q^46 + 12 * q^49 - 8 * q^56 - 8 * q^59 + 12 * q^61 - 4 * q^64 + 40 * q^65 - 8 * q^71 + 4 * q^74 - 20 * q^76 - 8 * q^79 - 40 * q^85 - 24 * q^86 - 48 * q^89 - 12 * q^94

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1999.1
 1.61803i − 0.618034i 0.618034i − 1.61803i
1.00000i 0 −1.00000 2.23607i 0 2.00000i 1.00000i 0 −2.23607
1999.2 1.00000i 0 −1.00000 2.23607i 0 2.00000i 1.00000i 0 2.23607
1999.3 1.00000i 0 −1.00000 2.23607i 0 2.00000i 1.00000i 0 2.23607
1999.4 1.00000i 0 −1.00000 2.23607i 0 2.00000i 1.00000i 0 −2.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.d.j 4
3.b odd 2 1 1110.2.d.f 4
5.b even 2 1 inner 3330.2.d.j 4
15.d odd 2 1 1110.2.d.f 4
15.e even 4 1 5550.2.a.bu 2
15.e even 4 1 5550.2.a.bz 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.f 4 3.b odd 2 1
1110.2.d.f 4 15.d odd 2 1
3330.2.d.j 4 1.a even 1 1 trivial
3330.2.d.j 4 5.b even 2 1 inner
5550.2.a.bu 2 15.e even 4 1
5550.2.a.bz 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3330, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}^{2} + 6T_{11} + 4$$ T11^2 + 6*T11 + 4 $$T_{17}^{2} + 20$$ T17^2 + 20 $$T_{29} - 4$$ T29 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 6 T + 4)^{2}$$
$13$ $$(T^{2} + 20)^{2}$$
$17$ $$(T^{2} + 20)^{2}$$
$19$ $$(T^{2} - 10 T + 20)^{2}$$
$23$ $$(T^{2} + 16)^{2}$$
$29$ $$(T - 4)^{4}$$
$31$ $$(T^{2} - 4 T - 16)^{2}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T^{2} - 20)^{2}$$
$43$ $$T^{4} + 112T^{2} + 256$$
$47$ $$T^{4} + 108T^{2} + 1296$$
$53$ $$T^{4} + 168T^{2} + 1936$$
$59$ $$(T^{2} + 4 T - 76)^{2}$$
$61$ $$(T^{2} - 6 T + 4)^{2}$$
$67$ $$T^{4} + 112T^{2} + 256$$
$71$ $$(T^{2} + 4 T - 16)^{2}$$
$73$ $$T^{4} + 112T^{2} + 256$$
$79$ $$(T^{2} + 4 T - 16)^{2}$$
$83$ $$T^{4} + 368 T^{2} + 30976$$
$89$ $$(T^{2} + 24 T + 124)^{2}$$
$97$ $$T^{4} + 268 T^{2} + 13456$$