# Properties

 Label 3330.2.d.n Level $3330$ Weight $2$ Character orbit 3330.d Analytic conductor $26.590$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.5161984.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ x^6 - 4*x^3 + 25*x^2 - 20*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} - q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} - \beta_{4} q^{8}+O(q^{10})$$ q + b4 * q^2 - q^4 + (b3 + b1) * q^5 + (b4 + b2 - b1) * q^7 - b4 * q^8 $$q + \beta_{4} q^{2} - q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} - \beta_{4} q^{8} + (\beta_{5} + \beta_{2}) q^{10} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + (\beta_{5} + \beta_{4}) q^{13} + ( - \beta_{2} - \beta_1 - 1) q^{14} + q^{16} + (2 \beta_{5} + 3 \beta_{4} + \beta_{2} - \beta_1) q^{17} + (\beta_{3} + \beta_{2} + \beta_1 - 3) q^{19} + ( - \beta_{3} - \beta_1) q^{20} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{22} + (\beta_{5} - 3 \beta_{4} - \beta_{2} + \beta_1) q^{23} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{25} + ( - \beta_{3} - 1) q^{26} + ( - \beta_{4} - \beta_{2} + \beta_1) q^{28} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{29} + (\beta_{3} - \beta_{2} - \beta_1) q^{31} + \beta_{4} q^{32} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{34} + (2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{35} - \beta_{4} q^{37} + (\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_1) q^{38} + ( - \beta_{5} - \beta_{2}) q^{40} + ( - 3 \beta_{3} + 2) q^{41} + ( - 3 \beta_{5} + \beta_{2} - \beta_1) q^{43} + (2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{44} + ( - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{46} + (2 \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1) q^{47} + 2 \beta_{3} q^{49} + (\beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{50} + ( - \beta_{5} - \beta_{4}) q^{52} + (2 \beta_{5} - 3 \beta_{4}) q^{53} + (\beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1 - 7) q^{55} + (\beta_{2} + \beta_1 + 1) q^{56} + ( - \beta_{5} - 6 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{58} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 2) q^{59} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{61} + (\beta_{5} - \beta_{2} + \beta_1) q^{62} - q^{64} + (\beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{65} + ( - 2 \beta_{5} - 10 \beta_{4}) q^{67} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{2} + \beta_1) q^{68} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_1 - 1) q^{70} + (2 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{71} + (\beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{73} + q^{74} + ( - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{76} + ( - 4 \beta_{5} - \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{77} + (\beta_{2} + \beta_1 + 4) q^{79} + (\beta_{3} + \beta_1) q^{80} + ( - 3 \beta_{5} + 2 \beta_{4}) q^{82} + ( - 3 \beta_{5} - 7 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{83} + (4 \beta_{5} + 7 \beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 1) q^{85} + (3 \beta_{3} - \beta_{2} - \beta_1) q^{86} + (2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{88} + ( - 3 \beta_{3} + 1) q^{89} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{91} + ( - \beta_{5} + 3 \beta_{4} + \beta_{2} - \beta_1) q^{92} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{94} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{95} + (7 \beta_{5} + 2 \beta_{2} - 2 \beta_1) q^{97} + 2 \beta_{5} q^{98}+O(q^{100})$$ q + b4 * q^2 - q^4 + (b3 + b1) * q^5 + (b4 + b2 - b1) * q^7 - b4 * q^8 + (b5 + b2) * q^10 + (-2*b3 - b2 - b1 + 1) * q^11 + (b5 + b4) * q^13 + (-b2 - b1 - 1) * q^14 + q^16 + (2*b5 + 3*b4 + b2 - b1) * q^17 + (b3 + b2 + b1 - 3) * q^19 + (-b3 - b1) * q^20 + (-2*b5 + b4 - b2 + b1) * q^22 + (b5 - 3*b4 - b2 + b1) * q^23 + (-b5 - b4 + b3 + 2*b2 + 2) * q^25 + (-b3 - 1) * q^26 + (-b4 - b2 + b1) * q^28 + (-b3 - 2*b2 - 2*b1 - 6) * q^29 + (b3 - b2 - b1) * q^31 + b4 * q^32 + (-2*b3 - b2 - b1 - 3) * q^34 + (2*b5 + b4 - b3 - b2 + 3) * q^35 - b4 * q^37 + (b5 - 3*b4 + b2 - b1) * q^38 + (-b5 - b2) * q^40 + (-3*b3 + 2) * q^41 + (-3*b5 + b2 - b1) * q^43 + (2*b3 + b2 + b1 - 1) * q^44 + (-b3 + b2 + b1 + 3) * q^46 + (2*b5 + 2*b4 + b2 - b1) * q^47 + 2*b3 * q^49 + (b5 + 2*b4 + b3 - 2*b1 + 1) * q^50 + (-b5 - b4) * q^52 + (2*b5 - 3*b4) * q^53 + (b5 + b4 - 2*b2 + b1 - 7) * q^55 + (b2 + b1 + 1) * q^56 + (-b5 - 6*b4 - 2*b2 + 2*b1) * q^58 + (-2*b3 - 3*b2 - 3*b1 - 2) * q^59 + (-3*b3 - 3*b2 - 3*b1) * q^61 + (b5 - b2 + b1) * q^62 - q^64 + (b5 + 3*b4 + b3 + 2*b2 - b1 - 1) * q^65 + (-2*b5 - 10*b4) * q^67 + (-2*b5 - 3*b4 - b2 + b1) * q^68 + (-b5 + 3*b4 - 2*b3 + b1 - 1) * q^70 + (2*b3 - 3*b2 - 3*b1) * q^71 + (b5 - b4 - b2 + b1) * q^73 + q^74 + (-b3 - b2 - b1 + 3) * q^76 + (-4*b5 - b4 + 2*b2 - 2*b1) * q^77 + (b2 + b1 + 4) * q^79 + (b3 + b1) * q^80 + (-3*b5 + 2*b4) * q^82 + (-3*b5 - 7*b4 - 2*b2 + 2*b1) * q^83 + (4*b5 + 7*b4 + b3 + 3*b2 - 2*b1 + 1) * q^85 + (3*b3 - b2 - b1) * q^86 + (2*b5 - b4 + b2 - b1) * q^88 + (-3*b3 + 1) * q^89 + (-3*b3 - b2 - b1 + 1) * q^91 + (-b5 + 3*b4 + b2 - b1) * q^92 + (-2*b3 - b2 - b1 - 2) * q^94 + (-2*b4 - 2*b3 + b2 - 4*b1 + 4) * q^95 + (7*b5 + 2*b2 - 2*b1) * q^97 + 2*b5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 2 q^{5}+O(q^{10})$$ 6 * q - 6 * q^4 - 2 * q^5 $$6 q - 6 q^{4} - 2 q^{5} + 10 q^{11} - 6 q^{14} + 6 q^{16} - 20 q^{19} + 2 q^{20} + 10 q^{25} - 4 q^{26} - 34 q^{29} - 2 q^{31} - 14 q^{34} + 20 q^{35} + 18 q^{41} - 10 q^{44} + 20 q^{46} - 4 q^{49} + 4 q^{50} - 42 q^{55} + 6 q^{56} - 8 q^{59} + 6 q^{61} - 6 q^{64} - 8 q^{65} - 2 q^{70} - 4 q^{71} + 6 q^{74} + 20 q^{76} + 24 q^{79} - 2 q^{80} + 4 q^{85} - 6 q^{86} + 12 q^{89} + 12 q^{91} - 8 q^{94} + 28 q^{95}+O(q^{100})$$ 6 * q - 6 * q^4 - 2 * q^5 + 10 * q^11 - 6 * q^14 + 6 * q^16 - 20 * q^19 + 2 * q^20 + 10 * q^25 - 4 * q^26 - 34 * q^29 - 2 * q^31 - 14 * q^34 + 20 * q^35 + 18 * q^41 - 10 * q^44 + 20 * q^46 - 4 * q^49 + 4 * q^50 - 42 * q^55 + 6 * q^56 - 8 * q^59 + 6 * q^61 - 6 * q^64 - 8 * q^65 - 2 * q^70 - 4 * q^71 + 6 * q^74 + 20 * q^76 + 24 * q^79 - 2 * q^80 + 4 * q^85 - 6 * q^86 + 12 * q^89 + 12 * q^91 - 8 * q^94 + 28 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121$$ (-5*v^5 - 2*v^4 - 25*v^3 + 10*v^2 - 121*v + 100) / 121 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121$$ (7*v^5 + 27*v^4 + 35*v^3 - 14*v^2 + 223) / 121 $$\beta_{4}$$ $$=$$ $$( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242$$ (-25*v^5 - 10*v^4 - 4*v^3 + 50*v^2 - 605*v + 258) / 242 $$\beta_{5}$$ $$=$$ $$( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242$$ (-65*v^5 - 26*v^4 + 38*v^3 + 372*v^2 - 1331*v + 574) / 242
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1$$ b5 - 3*b4 + b2 - b1 $$\nu^{3}$$ $$=$$ $$2\beta_{4} - 5\beta_{2} + 2$$ 2*b4 - 5*b2 + 2 $$\nu^{4}$$ $$=$$ $$5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15$$ 5*b3 + 7*b2 + 7*b1 - 15 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16$$ 2*b5 - 16*b4 - 2*b3 - 29*b1 + 16

## 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
 1.32001 + 1.32001i −1.75233 − 1.75233i 0.432320 + 0.432320i 1.32001 − 1.32001i −1.75233 + 1.75233i 0.432320 − 0.432320i
1.00000i 0 −1.00000 −1.80487 + 1.32001i 0 3.64002i 1.00000i 0 1.32001 + 1.80487i
1999.2 1.00000i 0 −1.00000 −1.38900 1.75233i 0 2.50466i 1.00000i 0 −1.75233 + 1.38900i
1999.3 1.00000i 0 −1.00000 2.19388 + 0.432320i 0 1.86464i 1.00000i 0 0.432320 2.19388i
1999.4 1.00000i 0 −1.00000 −1.80487 1.32001i 0 3.64002i 1.00000i 0 1.32001 1.80487i
1999.5 1.00000i 0 −1.00000 −1.38900 + 1.75233i 0 2.50466i 1.00000i 0 −1.75233 1.38900i
1999.6 1.00000i 0 −1.00000 2.19388 0.432320i 0 1.86464i 1.00000i 0 0.432320 + 2.19388i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1999.6 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.n 6
3.b odd 2 1 1110.2.d.i 6
5.b even 2 1 inner 3330.2.d.n 6
15.d odd 2 1 1110.2.d.i 6
15.e even 4 1 5550.2.a.cf 3
15.e even 4 1 5550.2.a.cg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.i 6 3.b odd 2 1
1110.2.d.i 6 15.d odd 2 1
3330.2.d.n 6 1.a even 1 1 trivial
3330.2.d.n 6 5.b even 2 1 inner
5550.2.a.cf 3 15.e even 4 1
5550.2.a.cg 3 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 $$S_{2}^{\mathrm{new}}(3330, [\chi])$$:

 $$T_{7}^{6} + 23T_{7}^{4} + 151T_{7}^{2} + 289$$ T7^6 + 23*T7^4 + 151*T7^2 + 289 $$T_{11}^{3} - 5T_{11}^{2} - 11T_{11} + 59$$ T11^3 - 5*T11^2 - 11*T11 + 59 $$T_{17}^{6} + 55T_{17}^{4} + 23T_{17}^{2} + 1$$ T17^6 + 55*T17^4 + 23*T17^2 + 1 $$T_{29}^{3} + 17T_{29}^{2} + 66T_{29} - 50$$ T29^3 + 17*T29^2 + 66*T29 - 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 2 T^{5} - 3 T^{4} - 24 T^{3} + \cdots + 125$$
$7$ $$T^{6} + 23 T^{4} + 151 T^{2} + \cdots + 289$$
$11$ $$(T^{3} - 5 T^{2} - 11 T + 59)^{2}$$
$13$ $$T^{6} + 14 T^{4} + 57 T^{2} + 64$$
$17$ $$T^{6} + 55 T^{4} + 23 T^{2} + 1$$
$19$ $$(T^{3} + 10 T^{2} + 25 T + 8)^{2}$$
$23$ $$T^{6} + 82 T^{4} + 401 T^{2} + \cdots + 256$$
$29$ $$(T^{3} + 17 T^{2} + 66 T - 50)^{2}$$
$31$ $$(T^{3} + T^{2} - 24 T + 20)^{2}$$
$37$ $$(T^{2} + 1)^{3}$$
$41$ $$(T^{3} - 9 T^{2} - 30 T + 34)^{2}$$
$43$ $$T^{6} + 185 T^{4} + 6280 T^{2} + \cdots + 59536$$
$47$ $$T^{6} + 44 T^{4} + 132 T^{2} + \cdots + 64$$
$53$ $$T^{6} + 91 T^{4} + 467 T^{2} + \cdots + 121$$
$59$ $$(T^{3} + 4 T^{2} - 62 T - 232)^{2}$$
$61$ $$(T^{3} - 3 T^{2} - 72 T + 108)^{2}$$
$67$ $$T^{6} + 312 T^{4} + 25232 T^{2} + \cdots + 295936$$
$71$ $$(T^{3} + 2 T^{2} - 162 T + 148)^{2}$$
$73$ $$T^{6} + 54 T^{4} + 377 T^{2} + 4$$
$79$ $$(T^{3} - 12 T^{2} + 38 T - 16)^{2}$$
$83$ $$T^{6} + 206 T^{4} + 1897 T^{2} + \cdots + 1936$$
$89$ $$(T^{3} - 6 T^{2} - 45 T - 4)^{2}$$
$97$ $$T^{6} + 493 T^{4} + 63872 T^{2} + \cdots + 1085764$$