Properties

 Label 3330.2.d.d Level $3330$ Weight $2$ Character orbit 3330.d Analytic conductor $26.590$ Analytic rank $0$ 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: $$0$$ 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.

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

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

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

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} + 5$$ T11 + 5 $$T_{17}^{2} + 1$$ T17^2 + 1 $$T_{29} + 3$$ T29 + 3

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 5)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 1$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 3)^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T - 1)^{2}$$
$43$ $$T^{2} + 49$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2} + 9$$
$59$ $$(T + 8)^{2}$$
$61$ $$(T - 5)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} + 1$$