# Properties

 Label 3330.2.d.e Level $3330$ Weight $2$ Character orbit 3330.d Analytic conductor $26.590$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + ( - 2 i + 1) q^{5} + 5 i q^{7} - i q^{8} +O(q^{10})$$ q + i * q^2 - q^4 + (-2*i + 1) * q^5 + 5*i * q^7 - i * q^8 $$q + i q^{2} - q^{4} + ( - 2 i + 1) q^{5} + 5 i q^{7} - i q^{8} + (i + 2) q^{10} - q^{11} - 5 q^{14} + q^{16} - 3 i q^{17} - 8 q^{19} + (2 i - 1) q^{20} - i q^{22} + 8 i q^{23} + ( - 4 i - 3) q^{25} - 5 i q^{28} - 5 q^{29} + 3 q^{31} + i q^{32} + 3 q^{34} + (5 i + 10) q^{35} - i q^{37} - 8 i q^{38} + ( - i - 2) q^{40} + 5 q^{41} - 9 i q^{43} + q^{44} - 8 q^{46} - 12 i q^{47} - 18 q^{49} + ( - 3 i + 4) q^{50} - 5 i q^{53} + (2 i - 1) q^{55} + 5 q^{56} - 5 i q^{58} - 4 q^{59} - 9 q^{61} + 3 i q^{62} - q^{64} + 8 i q^{67} + 3 i q^{68} + (10 i - 5) q^{70} - 6 q^{71} + 2 i q^{73} + q^{74} + 8 q^{76} - 5 i q^{77} + 8 q^{79} + ( - 2 i + 1) q^{80} + 5 i q^{82} + 4 i q^{83} + ( - 3 i - 6) q^{85} + 9 q^{86} + i q^{88} - 4 q^{89} - 8 i q^{92} + 12 q^{94} + (16 i - 8) q^{95} + 9 i q^{97} - 18 i q^{98} +O(q^{100})$$ q + i * q^2 - q^4 + (-2*i + 1) * q^5 + 5*i * q^7 - i * q^8 + (i + 2) * q^10 - q^11 - 5 * q^14 + q^16 - 3*i * q^17 - 8 * q^19 + (2*i - 1) * q^20 - i * q^22 + 8*i * q^23 + (-4*i - 3) * q^25 - 5*i * q^28 - 5 * q^29 + 3 * q^31 + i * q^32 + 3 * q^34 + (5*i + 10) * q^35 - i * q^37 - 8*i * q^38 + (-i - 2) * q^40 + 5 * q^41 - 9*i * q^43 + q^44 - 8 * q^46 - 12*i * q^47 - 18 * q^49 + (-3*i + 4) * q^50 - 5*i * q^53 + (2*i - 1) * q^55 + 5 * q^56 - 5*i * q^58 - 4 * q^59 - 9 * q^61 + 3*i * q^62 - q^64 + 8*i * q^67 + 3*i * q^68 + (10*i - 5) * q^70 - 6 * q^71 + 2*i * q^73 + q^74 + 8 * q^76 - 5*i * q^77 + 8 * q^79 + (-2*i + 1) * q^80 + 5*i * q^82 + 4*i * q^83 + (-3*i - 6) * q^85 + 9 * q^86 + i * q^88 - 4 * q^89 - 8*i * q^92 + 12 * q^94 + (16*i - 8) * q^95 + 9*i * q^97 - 18*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^5 $$2 q - 2 q^{4} + 2 q^{5} + 4 q^{10} - 2 q^{11} - 10 q^{14} + 2 q^{16} - 16 q^{19} - 2 q^{20} - 6 q^{25} - 10 q^{29} + 6 q^{31} + 6 q^{34} + 20 q^{35} - 4 q^{40} + 10 q^{41} + 2 q^{44} - 16 q^{46} - 36 q^{49} + 8 q^{50} - 2 q^{55} + 10 q^{56} - 8 q^{59} - 18 q^{61} - 2 q^{64} - 10 q^{70} - 12 q^{71} + 2 q^{74} + 16 q^{76} + 16 q^{79} + 2 q^{80} - 12 q^{85} + 18 q^{86} - 8 q^{89} + 24 q^{94} - 16 q^{95}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^10 - 2 * q^11 - 10 * q^14 + 2 * q^16 - 16 * q^19 - 2 * q^20 - 6 * q^25 - 10 * q^29 + 6 * q^31 + 6 * q^34 + 20 * q^35 - 4 * q^40 + 10 * q^41 + 2 * q^44 - 16 * q^46 - 36 * q^49 + 8 * q^50 - 2 * q^55 + 10 * q^56 - 8 * q^59 - 18 * q^61 - 2 * q^64 - 10 * q^70 - 12 * q^71 + 2 * q^74 + 16 * q^76 + 16 * q^79 + 2 * q^80 - 12 * q^85 + 18 * q^86 - 8 * q^89 + 24 * q^94 - 16 * q^95

## 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
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 1.00000 + 2.00000i 0 5.00000i 1.00000i 0 2.00000 1.00000i
1999.2 1.00000i 0 −1.00000 1.00000 2.00000i 0 5.00000i 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.e 2
3.b odd 2 1 1110.2.d.b 2
5.b even 2 1 inner 3330.2.d.e 2
15.d odd 2 1 1110.2.d.b 2
15.e even 4 1 5550.2.a.l 1
15.e even 4 1 5550.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.b 2 3.b odd 2 1
1110.2.d.b 2 15.d odd 2 1
3330.2.d.e 2 1.a even 1 1 trivial
3330.2.d.e 2 5.b even 2 1 inner
5550.2.a.l 1 15.e even 4 1
5550.2.a.be 1 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 $$S_{2}^{\mathrm{new}}(3330, [\chi])$$:

 $$T_{7}^{2} + 25$$ T7^2 + 25 $$T_{11} + 1$$ T11 + 1 $$T_{17}^{2} + 9$$ T17^2 + 9 $$T_{29} + 5$$ T29 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2} + 25$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T + 5)^{2}$$
$31$ $$(T - 3)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T - 5)^{2}$$
$43$ $$T^{2} + 81$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 25$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 9)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 4)^{2}$$
$97$ $$T^{2} + 81$$