# Properties

 Label 3330.2.d.h 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: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

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

## 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
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
1.00000i 0 −1.00000 −2.22474 + 0.224745i 0 1.44949i 1.00000i 0 0.224745 + 2.22474i
1999.2 1.00000i 0 −1.00000 0.224745 2.22474i 0 3.44949i 1.00000i 0 −2.22474 0.224745i
1999.3 1.00000i 0 −1.00000 −2.22474 0.224745i 0 1.44949i 1.00000i 0 0.224745 2.22474i
1999.4 1.00000i 0 −1.00000 0.224745 + 2.22474i 0 3.44949i 1.00000i 0 −2.22474 + 0.224745i
 $$n$$: 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.h 4
3.b odd 2 1 1110.2.d.h 4
5.b even 2 1 inner 3330.2.d.h 4
15.d odd 2 1 1110.2.d.h 4
15.e even 4 1 5550.2.a.bt 2
15.e even 4 1 5550.2.a.cc 2

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

 $$T_{7}^{4} + 14T_{7}^{2} + 25$$ T7^4 + 14*T7^2 + 25 $$T_{11}^{2} - 2T_{11} - 5$$ T11^2 - 2*T11 - 5 $$T_{17}^{4} + 14T_{17}^{2} + 25$$ T17^4 + 14*T17^2 + 25 $$T_{29}^{2} - 12T_{29} + 30$$ T29^2 - 12*T29 + 30

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 4 T^{3} + 8 T^{2} + 20 T + 25$$
$7$ $$T^{4} + 14T^{2} + 25$$
$11$ $$(T^{2} - 2 T - 5)^{2}$$
$13$ $$T^{4} + 30T^{2} + 9$$
$17$ $$T^{4} + 14T^{2} + 25$$
$19$ $$(T + 3)^{4}$$
$23$ $$(T^{2} + 1)^{2}$$
$29$ $$(T^{2} - 12 T + 30)^{2}$$
$31$ $$(T + 2)^{4}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T^{2} + 4 T - 2)^{2}$$
$43$ $$T^{4} + 56T^{2} + 400$$
$47$ $$T^{4} + 44T^{2} + 100$$
$53$ $$(T^{2} + 9)^{2}$$
$59$ $$(T^{2} - 8 T - 38)^{2}$$
$61$ $$(T^{2} + 20 T + 76)^{2}$$
$67$ $$T^{4} + 56T^{2} + 400$$
$71$ $$(T^{2} + 12 T - 18)^{2}$$
$73$ $$T^{4} + 194T^{2} + 9025$$
$79$ $$(T^{2} + 12 T - 18)^{2}$$
$83$ $$T^{4} + 14T^{2} + 25$$
$89$ $$(T^{2} + 14 T - 5)^{2}$$
$97$ $$T^{4} + 140T^{2} + 1444$$