Properties

 Label 3344.2.a.q Level $3344$ Weight $2$ Character orbit 3344.a Self dual yes Analytic conductor $26.702$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3344 = 2^{4} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3344.a (trivial)

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: yes Analytic conductor: $$26.7019744359$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{2} + \beta_1 + 2) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^3 + (-b2 - b1 - 1) * q^5 + (-b2 + b1 + 2) * q^7 + (b2 + b1 + 1) * q^9 $$q - \beta_1 q^{3} + ( - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{2} + \beta_1 + 2) q^{7} + (\beta_{2} + \beta_1 + 1) q^{9} + q^{11} + (\beta_{2} - \beta_1) q^{13} + (3 \beta_1 + 3) q^{15} + ( - \beta_{2} - 3) q^{17} + q^{19} + ( - 2 \beta_{2} - 2 \beta_1 - 5) q^{21} + (\beta_{2} + 5) q^{23} + (3 \beta_1 + 4) q^{25} - 3 q^{27} + (2 \beta_{2} + \beta_1 + 2) q^{29} + ( - 3 \beta_1 + 1) q^{31} - \beta_1 q^{33} + ( - 3 \beta_{2} - 6 \beta_1) q^{35} + ( - 2 \beta_{2} - 2) q^{37} + (2 \beta_{2} + 5) q^{39} + ( - 2 \beta_{2} - 3 \beta_1 - 3) q^{41} + ( - \beta_{2} - 3 \beta_1 + 7) q^{43} + ( - 3 \beta_1 - 9) q^{45} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{47} + ( - 2 \beta_{2} + \beta_1 + 9) q^{49} + ( - \beta_{2} + 4 \beta_1 - 1) q^{51} + (\beta_{2} - 4 \beta_1 - 3) q^{53} + ( - \beta_{2} - \beta_1 - 1) q^{55} - \beta_1 q^{57} + ( - \beta_{2} - 2 \beta_1 + 1) q^{59} + (2 \beta_{2} + 8) q^{61} + (3 \beta_{2} + 6 \beta_1) q^{63} + (\beta_{2} + 4 \beta_1 - 2) q^{65} + (3 \beta_{2} - \beta_1) q^{67} + (\beta_{2} - 6 \beta_1 + 1) q^{69} + (\beta_{2} + 5 \beta_1 - 7) q^{71} + ( - \beta_{2} - 1) q^{73} + ( - 3 \beta_{2} - 7 \beta_1 - 12) q^{75} + ( - \beta_{2} + \beta_1 + 2) q^{77} + (4 \beta_{2} + 2 \beta_1 + 2) q^{79} + ( - 3 \beta_{2} - 3) q^{81} + ( - \beta_{2} + \beta_1 + 11) q^{83} + (2 \beta_{2} + 2 \beta_1 + 8) q^{85} + (\beta_{2} - 5 \beta_1 - 2) q^{87} + (2 \beta_{2} + 2 \beta_1 + 2) q^{89} + (\beta_1 - 12) q^{91} + (3 \beta_{2} + 2 \beta_1 + 12) q^{93} + ( - \beta_{2} - \beta_1 - 1) q^{95} + ( - 4 \beta_{2} + 4) q^{97} + (\beta_{2} + \beta_1 + 1) q^{99}+O(q^{100})$$ q - b1 * q^3 + (-b2 - b1 - 1) * q^5 + (-b2 + b1 + 2) * q^7 + (b2 + b1 + 1) * q^9 + q^11 + (b2 - b1) * q^13 + (3*b1 + 3) * q^15 + (-b2 - 3) * q^17 + q^19 + (-2*b2 - 2*b1 - 5) * q^21 + (b2 + 5) * q^23 + (3*b1 + 4) * q^25 - 3 * q^27 + (2*b2 + b1 + 2) * q^29 + (-3*b1 + 1) * q^31 - b1 * q^33 + (-3*b2 - 6*b1) * q^35 + (-2*b2 - 2) * q^37 + (2*b2 + 5) * q^39 + (-2*b2 - 3*b1 - 3) * q^41 + (-b2 - 3*b1 + 7) * q^43 + (-3*b1 - 9) * q^45 + (-2*b2 - 4*b1 + 4) * q^47 + (-2*b2 + b1 + 9) * q^49 + (-b2 + 4*b1 - 1) * q^51 + (b2 - 4*b1 - 3) * q^53 + (-b2 - b1 - 1) * q^55 - b1 * q^57 + (-b2 - 2*b1 + 1) * q^59 + (2*b2 + 8) * q^61 + (3*b2 + 6*b1) * q^63 + (b2 + 4*b1 - 2) * q^65 + (3*b2 - b1) * q^67 + (b2 - 6*b1 + 1) * q^69 + (b2 + 5*b1 - 7) * q^71 + (-b2 - 1) * q^73 + (-3*b2 - 7*b1 - 12) * q^75 + (-b2 + b1 + 2) * q^77 + (4*b2 + 2*b1 + 2) * q^79 + (-3*b2 - 3) * q^81 + (-b2 + b1 + 11) * q^83 + (2*b2 + 2*b1 + 8) * q^85 + (b2 - 5*b1 - 2) * q^87 + (2*b2 + 2*b1 + 2) * q^89 + (b1 - 12) * q^91 + (3*b2 + 2*b1 + 12) * q^93 + (-b2 - b1 - 1) * q^95 + (-4*b2 + 4) * q^97 + (b2 + b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 $$3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45} + 12 q^{47} + 27 q^{49} - 3 q^{51} - 9 q^{53} - 3 q^{55} + 3 q^{59} + 24 q^{61} - 6 q^{65} + 3 q^{69} - 21 q^{71} - 3 q^{73} - 36 q^{75} + 6 q^{77} + 6 q^{79} - 9 q^{81} + 33 q^{83} + 24 q^{85} - 6 q^{87} + 6 q^{89} - 36 q^{91} + 36 q^{93} - 3 q^{95} + 12 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.66908 −0.523976 −2.14510
0 −2.66908 0 −4.12398 0 4.21417 0 4.12398 0
1.2 0 0.523976 0 2.72545 0 4.67750 0 −2.72545 0
1.3 0 2.14510 0 −1.60147 0 −2.89167 0 1.60147 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3344.2.a.q 3
4.b odd 2 1 418.2.a.g 3
12.b even 2 1 3762.2.a.bg 3
44.c even 2 1 4598.2.a.bo 3
76.d even 2 1 7942.2.a.bi 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.g 3 4.b odd 2 1
3344.2.a.q 3 1.a even 1 1 trivial
3762.2.a.bg 3 12.b even 2 1
4598.2.a.bo 3 44.c even 2 1
7942.2.a.bi 3 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3344))$$:

 $$T_{3}^{3} - 6T_{3} + 3$$ T3^3 - 6*T3 + 3 $$T_{5}^{3} + 3T_{5}^{2} - 9T_{5} - 18$$ T5^3 + 3*T5^2 - 9*T5 - 18 $$T_{7}^{3} - 6T_{7}^{2} - 6T_{7} + 57$$ T7^3 - 6*T7^2 - 6*T7 + 57

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 6T + 3$$
$5$ $$T^{3} + 3 T^{2} - 9 T - 18$$
$7$ $$T^{3} - 6 T^{2} - 6 T + 57$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} - 18T - 29$$
$17$ $$T^{3} + 9 T^{2} + 18 T - 4$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 15 T^{2} + 66 T - 76$$
$29$ $$T^{3} - 6 T^{2} - 24 T + 147$$
$31$ $$T^{3} - 3 T^{2} - 51 T + 134$$
$37$ $$T^{3} + 6 T^{2} - 24 T - 96$$
$41$ $$T^{3} + 9 T^{2} - 45 T - 122$$
$43$ $$T^{3} - 21 T^{2} + 93 T + 184$$
$47$ $$T^{3} - 12 T^{2} - 60 T + 672$$
$53$ $$T^{3} + 9 T^{2} - 90 T - 452$$
$59$ $$T^{3} - 3 T^{2} - 24 T + 64$$
$61$ $$T^{3} - 24 T^{2} + 156 T - 192$$
$67$ $$T^{3} - 96T - 123$$
$71$ $$T^{3} + 21 T^{2} + 3 T - 1306$$
$73$ $$T^{3} + 3 T^{2} - 6 T - 12$$
$79$ $$T^{3} - 6 T^{2} - 132 T + 944$$
$83$ $$T^{3} - 33 T^{2} + 345 T - 1104$$
$89$ $$T^{3} - 6 T^{2} - 36 T + 144$$
$97$ $$T^{3} - 12 T^{2} - 96 T + 256$$