# Properties

 Label 3330.2.d.l 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: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ 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
 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
1.00000i 0 −1.00000 1.00000 + 2.00000i 0 2.46410i 1.00000i 0 2.00000 1.00000i
1999.2 1.00000i 0 −1.00000 1.00000 + 2.00000i 0 4.46410i 1.00000i 0 2.00000 1.00000i
1999.3 1.00000i 0 −1.00000 1.00000 2.00000i 0 4.46410i 1.00000i 0 2.00000 + 1.00000i
1999.4 1.00000i 0 −1.00000 1.00000 2.00000i 0 2.46410i 1.00000i 0 2.00000 + 1.00000i
 $$n$$: 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.l yes 4
3.b odd 2 1 3330.2.d.i 4
5.b even 2 1 inner 3330.2.d.l yes 4
15.d odd 2 1 3330.2.d.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3330.2.d.i 4 3.b odd 2 1
3330.2.d.i 4 15.d odd 2 1
3330.2.d.l yes 4 1.a even 1 1 trivial
3330.2.d.l yes 4 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{4} + 26T_{7}^{2} + 121$$ T7^4 + 26*T7^2 + 121 $$T_{11}^{2} - 3$$ T11^2 - 3 $$T_{17}^{4} + 42T_{17}^{2} + 9$$ T17^4 + 42*T17^2 + 9 $$T_{29} + 3$$ T29 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 2 T + 5)^{2}$$
$7$ $$T^{4} + 26T^{2} + 121$$
$11$ $$(T^{2} - 3)^{2}$$
$13$ $$(T^{2} + 16)^{2}$$
$17$ $$T^{4} + 42T^{2} + 9$$
$19$ $$(T^{2} - 48)^{2}$$
$23$ $$T^{4} + 32T^{2} + 64$$
$29$ $$(T + 3)^{4}$$
$31$ $$(T^{2} + 8 T + 13)^{2}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T^{2} + 8 T - 11)^{2}$$
$43$ $$T^{4} + 62T^{2} + 529$$
$47$ $$T^{4} + 152T^{2} + 2704$$
$53$ $$T^{4} + 14T^{2} + 1$$
$59$ $$(T^{2} + 12 T + 24)^{2}$$
$61$ $$(T^{2} - 27)^{2}$$
$67$ $$T^{4} + 32T^{2} + 64$$
$71$ $$(T^{2} - 12 T - 12)^{2}$$
$73$ $$T^{4} + 248T^{2} + 8464$$
$79$ $$(T^{2} - 16 T + 52)^{2}$$
$83$ $$T^{4} + 96T^{2} + 576$$
$89$ $$(T^{2} + 12 T + 24)^{2}$$
$97$ $$T^{4} + 494 T^{2} + 57121$$