# Properties

 Label 3344.2.a.l Level $3344$ Weight $2$ Character orbit 3344.a Self dual yes Analytic conductor $26.702$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - \beta + 2) q^{5} + ( - \beta + 3) q^{7} + (\beta + 2) q^{9}+O(q^{10})$$ q + b * q^3 + (-b + 2) * q^5 + (-b + 3) * q^7 + (b + 2) * q^9 $$q + \beta q^{3} + ( - \beta + 2) q^{5} + ( - \beta + 3) q^{7} + (\beta + 2) q^{9} - q^{11} + (\beta + 3) q^{13} + (\beta - 5) q^{15} + (2 \beta - 4) q^{17} - q^{19} + (2 \beta - 5) q^{21} + ( - 2 \beta - 2) q^{23} + ( - 3 \beta + 4) q^{25} + 5 q^{27} + (\beta + 4) q^{29} + ( - \beta + 9) q^{31} - \beta q^{33} + ( - 4 \beta + 11) q^{35} + 8 q^{37} + (4 \beta + 5) q^{39} + (\beta + 1) q^{41} + (3 \beta - 2) q^{43} + ( - \beta - 1) q^{45} + (4 \beta - 2) q^{47} + ( - 5 \beta + 7) q^{49} + ( - 2 \beta + 10) q^{51} + (2 \beta + 2) q^{53} + (\beta - 2) q^{55} - \beta q^{57} + 2 q^{61} + q^{63} + ( - 2 \beta + 1) q^{65} + ( - \beta + 3) q^{67} + ( - 4 \beta - 10) q^{69} + ( - 3 \beta + 6) q^{71} + ( - 2 \beta + 6) q^{73} + (\beta - 15) q^{75} + (\beta - 3) q^{77} + ( - 2 \beta + 2) q^{79} + (2 \beta - 6) q^{81} + ( - \beta + 2) q^{83} + (6 \beta - 18) q^{85} + (5 \beta + 5) q^{87} + ( - 2 \beta - 8) q^{89} + ( - \beta + 4) q^{91} + (8 \beta - 5) q^{93} + (\beta - 2) q^{95} + 8 q^{97} + ( - \beta - 2) q^{99} +O(q^{100})$$ q + b * q^3 + (-b + 2) * q^5 + (-b + 3) * q^7 + (b + 2) * q^9 - q^11 + (b + 3) * q^13 + (b - 5) * q^15 + (2*b - 4) * q^17 - q^19 + (2*b - 5) * q^21 + (-2*b - 2) * q^23 + (-3*b + 4) * q^25 + 5 * q^27 + (b + 4) * q^29 + (-b + 9) * q^31 - b * q^33 + (-4*b + 11) * q^35 + 8 * q^37 + (4*b + 5) * q^39 + (b + 1) * q^41 + (3*b - 2) * q^43 + (-b - 1) * q^45 + (4*b - 2) * q^47 + (-5*b + 7) * q^49 + (-2*b + 10) * q^51 + (2*b + 2) * q^53 + (b - 2) * q^55 - b * q^57 + 2 * q^61 + q^63 + (-2*b + 1) * q^65 + (-b + 3) * q^67 + (-4*b - 10) * q^69 + (-3*b + 6) * q^71 + (-2*b + 6) * q^73 + (b - 15) * q^75 + (b - 3) * q^77 + (-2*b + 2) * q^79 + (2*b - 6) * q^81 + (-b + 2) * q^83 + (6*b - 18) * q^85 + (5*b + 5) * q^87 + (-2*b - 8) * q^89 + (-b + 4) * q^91 + (8*b - 5) * q^93 + (b - 2) * q^95 + 8 * q^97 + (-b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q + q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9 $$2 q + q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} - 2 q^{11} + 7 q^{13} - 9 q^{15} - 6 q^{17} - 2 q^{19} - 8 q^{21} - 6 q^{23} + 5 q^{25} + 10 q^{27} + 9 q^{29} + 17 q^{31} - q^{33} + 18 q^{35} + 16 q^{37} + 14 q^{39} + 3 q^{41} - q^{43} - 3 q^{45} + 9 q^{49} + 18 q^{51} + 6 q^{53} - 3 q^{55} - q^{57} + 4 q^{61} + 2 q^{63} + 5 q^{67} - 24 q^{69} + 9 q^{71} + 10 q^{73} - 29 q^{75} - 5 q^{77} + 2 q^{79} - 10 q^{81} + 3 q^{83} - 30 q^{85} + 15 q^{87} - 18 q^{89} + 7 q^{91} - 2 q^{93} - 3 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100})$$ 2 * q + q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9 - 2 * q^11 + 7 * q^13 - 9 * q^15 - 6 * q^17 - 2 * q^19 - 8 * q^21 - 6 * q^23 + 5 * q^25 + 10 * q^27 + 9 * q^29 + 17 * q^31 - q^33 + 18 * q^35 + 16 * q^37 + 14 * q^39 + 3 * q^41 - q^43 - 3 * q^45 + 9 * q^49 + 18 * q^51 + 6 * q^53 - 3 * q^55 - q^57 + 4 * q^61 + 2 * q^63 + 5 * q^67 - 24 * q^69 + 9 * q^71 + 10 * q^73 - 29 * q^75 - 5 * q^77 + 2 * q^79 - 10 * q^81 + 3 * q^83 - 30 * q^85 + 15 * q^87 - 18 * q^89 + 7 * q^91 - 2 * q^93 - 3 * q^95 + 16 * q^97 - 5 * q^99

## 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
 −1.79129 2.79129
0 −1.79129 0 3.79129 0 4.79129 0 0.208712 0
1.2 0 2.79129 0 −0.791288 0 0.208712 0 4.79129 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.l 2
4.b odd 2 1 418.2.a.f 2
12.b even 2 1 3762.2.a.s 2
44.c even 2 1 4598.2.a.y 2
76.d even 2 1 7942.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.f 2 4.b odd 2 1
3344.2.a.l 2 1.a even 1 1 trivial
3762.2.a.s 2 12.b even 2 1
4598.2.a.y 2 44.c even 2 1
7942.2.a.w 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3344))$$:

 $$T_{3}^{2} - T_{3} - 5$$ T3^2 - T3 - 5 $$T_{5}^{2} - 3T_{5} - 3$$ T5^2 - 3*T5 - 3 $$T_{7}^{2} - 5T_{7} + 1$$ T7^2 - 5*T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 5$$
$5$ $$T^{2} - 3T - 3$$
$7$ $$T^{2} - 5T + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 7T + 7$$
$17$ $$T^{2} + 6T - 12$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 6T - 12$$
$29$ $$T^{2} - 9T + 15$$
$31$ $$T^{2} - 17T + 67$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 3T - 3$$
$43$ $$T^{2} + T - 47$$
$47$ $$T^{2} - 84$$
$53$ $$T^{2} - 6T - 12$$
$59$ $$T^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} - 5T + 1$$
$71$ $$T^{2} - 9T - 27$$
$73$ $$T^{2} - 10T + 4$$
$79$ $$T^{2} - 2T - 20$$
$83$ $$T^{2} - 3T - 3$$
$89$ $$T^{2} + 18T + 60$$
$97$ $$(T - 8)^{2}$$