# Properties

 Label 3344.2.a.t Level $3344$ Weight $2$ Character orbit 3344.a Self dual yes Analytic conductor $26.702$ Analytic rank $1$ Dimension $5$ 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: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.246832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ x^5 - 2*x^4 - 5*x^3 + 6*x^2 + 7*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + (\beta_{2} - \beta_1 - 1) q^{5} + (\beta_{2} + \beta_1 - 2) q^{7} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{9}+O(q^{10})$$ q - b3 * q^3 + (b2 - b1 - 1) * q^5 + (b2 + b1 - 2) * q^7 + (-b4 - b3 - 2*b2 + 2) * q^9 $$q - \beta_{3} q^{3} + (\beta_{2} - \beta_1 - 1) q^{5} + (\beta_{2} + \beta_1 - 2) q^{7} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{9} - q^{11} + (\beta_{4} - \beta_1 + 1) q^{13} + (2 \beta_{4} + 3 \beta_{3} + \beta_{2} - 2) q^{15} + (2 \beta_{3} + 2 \beta_1 - 2) q^{17} + q^{19} + (2 \beta_{3} - \beta_{2} + 2) q^{21} + ( - 3 \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 1) q^{23}+ \cdots + (\beta_{4} + \beta_{3} + 2 \beta_{2} - 2) q^{99}+O(q^{100})$$ q - b3 * q^3 + (b2 - b1 - 1) * q^5 + (b2 + b1 - 2) * q^7 + (-b4 - b3 - 2*b2 + 2) * q^9 - q^11 + (b4 - b1 + 1) * q^13 + (2*b4 + 3*b3 + b2 - 2) * q^15 + (2*b3 + 2*b1 - 2) * q^17 + q^19 + (2*b3 - b2 + 2) * q^21 + (-3*b4 - b3 - b2 - b1 + 1) * q^23 + (b4 - b3 - 2*b2 + 2) * q^25 + (-b4 - b3 - 2*b2 + b1 + 3) * q^27 + (-b4 + 2*b3 - b2 + 3*b1 + 1) * q^29 + (b3 + b2 - 2*b1 - 2) * q^31 + b3 * q^33 + (b4 + b3 - 3*b2 + b1 + 2) * q^35 + (-3*b4 - b3 - b2 + b1 + 1) * q^37 + (-b4 - b3 + b2 - b1) * q^39 + (b4 - 4*b3 - 2*b2 - 3*b1 + 3) * q^41 + (b4 + 3*b3 - 2*b2 - 4) * q^43 + (4*b3 + 3*b2 + b1 - 8) * q^45 + (4*b4 + 2*b3 + 2*b2 + 2) * q^47 + (b4 + 3*b3 + 2*b1 - 1) * q^49 + (2*b3 + 2*b2 - 6) * q^51 + (4*b4 + 2*b3 + 2*b2 - 2*b1 - 4) * q^53 + (-b2 + b1 + 1) * q^55 - b3 * q^57 + (-b4 + b3 + b2 + 5*b1 - 3) * q^59 + (2*b3 + 2*b2 + 2*b1 - 4) * q^61 + (b4 - b3 + b2 - 3*b1 - 4) * q^63 + (-2*b3 + 3*b2 - 2*b1) * q^65 + (5*b3 - b2 + 4*b1 - 4) * q^67 + (5*b4 + b3 - b2 + 3*b1 - 3) * q^69 + (2*b4 + 3*b3 - 2*b2 - 2*b1 - 4) * q^71 + (-2*b4 + 2*b2 - 6*b1 + 2) * q^73 + (-5*b4 - 6*b3 - 2*b2 - b1 + 7) * q^75 + (-b2 - b1 + 2) * q^77 + (-2*b1 - 8) * q^79 + (b4 - 3*b3 + 3*b2 + b1 - 1) * q^81 + (-b4 - 3*b3 - 4*b2 - 2*b1 + 6) * q^83 + (-4*b4 - 4*b3 - 4*b2 + 2*b1) * q^85 + (-2*b3 + b2 + b1 - 6) * q^87 + (-b4 - b3 - b2 + b1 - 5) * q^89 + (-3*b4 - 2*b3 - b2 - b1 - 3) * q^91 + (4*b4 + 6*b3 + 4*b2 - 9) * q^93 + (b2 - b1 - 1) * q^95 + (-3*b4 + b3 - 5*b2 + b1 + 5) * q^97 + (b4 + b3 + 2*b2 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10})$$ 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 $$5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39} + 2 q^{41} - 20 q^{43} - 28 q^{45} + 20 q^{47} + 3 q^{49} - 24 q^{51} - 14 q^{53} + 5 q^{55} - q^{57} - 3 q^{59} - 10 q^{61} - 24 q^{63} - 9 q^{67} - 5 q^{69} - 23 q^{71} + 18 q^{75} + 6 q^{77} - 44 q^{79} + q^{81} + 14 q^{83} - 12 q^{85} - 28 q^{87} - 27 q^{89} - 24 q^{91} - 27 q^{93} - 5 q^{95} + 15 q^{97} - 4 q^{99}+O(q^{100})$$ 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 3$$ v^3 - 2*v^2 - 3*v + 3 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 3\nu^{3} - 2\nu^{2} + 7\nu + 1$$ v^4 - 3*v^3 - 2*v^2 + 7*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 5\beta _1 + 1$$ b3 + 2*b2 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 3\beta_{3} + 8\beta_{2} + 10\beta _1 + 6$$ b4 + 3*b3 + 8*b2 + 10*b1 + 6

## 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
 −1.15351 0.245526 2.71457 −1.51908 1.71250
0 −2.26452 0 0.637602 0 −2.66942 0 2.12805 0
1.2 0 −2.15766 0 −3.43077 0 −3.93972 0 1.65548 0
1.3 0 −0.121872 0 −1.06025 0 3.36889 0 −2.98515 0
1.4 0 0.563416 0 2.34577 0 −1.69239 0 −2.68256 0
1.5 0 2.98063 0 −3.49235 0 −1.06736 0 5.88418 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.t 5
4.b odd 2 1 209.2.a.c 5
12.b even 2 1 1881.2.a.k 5
20.d odd 2 1 5225.2.a.h 5
44.c even 2 1 2299.2.a.n 5
76.d even 2 1 3971.2.a.h 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.c 5 4.b odd 2 1
1881.2.a.k 5 12.b even 2 1
2299.2.a.n 5 44.c even 2 1
3344.2.a.t 5 1.a even 1 1 trivial
3971.2.a.h 5 76.d even 2 1
5225.2.a.h 5 20.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{5} + T_{3}^{4} - 9T_{3}^{3} - 11T_{3}^{2} + 7T_{3} + 1$$ T3^5 + T3^4 - 9*T3^3 - 11*T3^2 + 7*T3 + 1 $$T_{5}^{5} + 5T_{5}^{4} - 3T_{5}^{3} - 33T_{5}^{2} - 9T_{5} + 19$$ T5^5 + 5*T5^4 - 3*T5^3 - 33*T5^2 - 9*T5 + 19 $$T_{7}^{5} + 6T_{7}^{4} - T_{7}^{3} - 62T_{7}^{2} - 119T_{7} - 64$$ T7^5 + 6*T7^4 - T7^3 - 62*T7^2 - 119*T7 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} + T^{4} - 9 T^{3} + \cdots + 1$$
$5$ $$T^{5} + 5 T^{4} + \cdots + 19$$
$7$ $$T^{5} + 6 T^{4} + \cdots - 64$$
$11$ $$(T + 1)^{5}$$
$13$ $$T^{5} - 4 T^{4} + \cdots + 2$$
$17$ $$T^{5} + 4 T^{4} + \cdots - 64$$
$19$ $$(T - 1)^{5}$$
$23$ $$T^{5} + 3 T^{4} + \cdots + 784$$
$29$ $$T^{5} - 10 T^{4} + \cdots + 490$$
$31$ $$T^{5} + 11 T^{4} + \cdots + 757$$
$37$ $$T^{5} - T^{4} + \cdots - 3088$$
$41$ $$T^{5} - 2 T^{4} + \cdots - 4112$$
$43$ $$T^{5} + 20 T^{4} + \cdots - 11266$$
$47$ $$T^{5} - 20 T^{4} + \cdots - 13184$$
$53$ $$T^{5} + 14 T^{4} + \cdots + 30304$$
$59$ $$T^{5} + 3 T^{4} + \cdots + 2000$$
$61$ $$T^{5} + 10 T^{4} + \cdots - 736$$
$67$ $$T^{5} + 9 T^{4} + \cdots - 17689$$
$71$ $$T^{5} + 23 T^{4} + \cdots - 19081$$
$73$ $$T^{5} - 340 T^{3} + \cdots + 155392$$
$79$ $$T^{5} + 44 T^{4} + \cdots + 36800$$
$83$ $$T^{5} - 14 T^{4} + \cdots + 3908$$
$89$ $$T^{5} + 27 T^{4} + \cdots + 320$$
$97$ $$T^{5} - 15 T^{4} + \cdots - 37456$$