# Properties

 Label 3344.2.a.k Level $3344$ Weight $2$ Character orbit 3344.a Self dual yes Analytic conductor $26.702$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 2 q^{5} + (\beta - 2) q^{7} + (\beta + 1) q^{9} +O(q^{10})$$ q - b * q^3 + 2 * q^5 + (b - 2) * q^7 + (b + 1) * q^9 $$q - \beta q^{3} + 2 q^{5} + (\beta - 2) q^{7} + (\beta + 1) q^{9} - q^{11} + (\beta - 2) q^{13} - 2 \beta q^{15} + ( - \beta + 2) q^{17} + q^{19} + (\beta - 4) q^{21} + (\beta - 4) q^{23} - q^{25} + (\beta - 4) q^{27} + ( - 3 \beta + 2) q^{29} - 2 q^{31} + \beta q^{33} + (2 \beta - 4) q^{35} - 2 \beta q^{37} + (\beta - 4) q^{39} + 2 q^{41} + (2 \beta + 2) q^{45} - 8 q^{47} + ( - 3 \beta + 1) q^{49} + ( - \beta + 4) q^{51} + 5 \beta q^{53} - 2 q^{55} - \beta q^{57} + 3 \beta q^{59} + (4 \beta - 4) q^{61} + 2 q^{63} + (2 \beta - 4) q^{65} + 3 \beta q^{67} + (3 \beta - 4) q^{69} - 6 q^{71} + ( - 3 \beta - 2) q^{73} + \beta q^{75} + ( - \beta + 2) q^{77} + 4 q^{79} - 7 q^{81} + ( - 4 \beta - 4) q^{83} + ( - 2 \beta + 4) q^{85} + (\beta + 12) q^{87} + (2 \beta - 6) q^{89} + ( - 3 \beta + 8) q^{91} + 2 \beta q^{93} + 2 q^{95} + ( - 2 \beta - 2) q^{97} + ( - \beta - 1) q^{99} +O(q^{100})$$ q - b * q^3 + 2 * q^5 + (b - 2) * q^7 + (b + 1) * q^9 - q^11 + (b - 2) * q^13 - 2*b * q^15 + (-b + 2) * q^17 + q^19 + (b - 4) * q^21 + (b - 4) * q^23 - q^25 + (b - 4) * q^27 + (-3*b + 2) * q^29 - 2 * q^31 + b * q^33 + (2*b - 4) * q^35 - 2*b * q^37 + (b - 4) * q^39 + 2 * q^41 + (2*b + 2) * q^45 - 8 * q^47 + (-3*b + 1) * q^49 + (-b + 4) * q^51 + 5*b * q^53 - 2 * q^55 - b * q^57 + 3*b * q^59 + (4*b - 4) * q^61 + 2 * q^63 + (2*b - 4) * q^65 + 3*b * q^67 + (3*b - 4) * q^69 - 6 * q^71 + (-3*b - 2) * q^73 + b * q^75 + (-b + 2) * q^77 + 4 * q^79 - 7 * q^81 + (-4*b - 4) * q^83 + (-2*b + 4) * q^85 + (b + 12) * q^87 + (2*b - 6) * q^89 + (-3*b + 8) * q^91 + 2*b * q^93 + 2 * q^95 + (-2*b - 2) * q^97 + (-b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - q^3 + 4 * q^5 - 3 * q^7 + 3 * q^9 $$2 q - q^{3} + 4 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} - 3 q^{13} - 2 q^{15} + 3 q^{17} + 2 q^{19} - 7 q^{21} - 7 q^{23} - 2 q^{25} - 7 q^{27} + q^{29} - 4 q^{31} + q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 4 q^{41} + 6 q^{45} - 16 q^{47} - q^{49} + 7 q^{51} + 5 q^{53} - 4 q^{55} - q^{57} + 3 q^{59} - 4 q^{61} + 4 q^{63} - 6 q^{65} + 3 q^{67} - 5 q^{69} - 12 q^{71} - 7 q^{73} + q^{75} + 3 q^{77} + 8 q^{79} - 14 q^{81} - 12 q^{83} + 6 q^{85} + 25 q^{87} - 10 q^{89} + 13 q^{91} + 2 q^{93} + 4 q^{95} - 6 q^{97} - 3 q^{99}+O(q^{100})$$ 2 * q - q^3 + 4 * q^5 - 3 * q^7 + 3 * q^9 - 2 * q^11 - 3 * q^13 - 2 * q^15 + 3 * q^17 + 2 * q^19 - 7 * q^21 - 7 * q^23 - 2 * q^25 - 7 * q^27 + q^29 - 4 * q^31 + q^33 - 6 * q^35 - 2 * q^37 - 7 * q^39 + 4 * q^41 + 6 * q^45 - 16 * q^47 - q^49 + 7 * q^51 + 5 * q^53 - 4 * q^55 - q^57 + 3 * q^59 - 4 * q^61 + 4 * q^63 - 6 * q^65 + 3 * q^67 - 5 * q^69 - 12 * q^71 - 7 * q^73 + q^75 + 3 * q^77 + 8 * q^79 - 14 * q^81 - 12 * q^83 + 6 * q^85 + 25 * q^87 - 10 * q^89 + 13 * q^91 + 2 * q^93 + 4 * q^95 - 6 * q^97 - 3 * 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
 2.56155 −1.56155
0 −2.56155 0 2.00000 0 0.561553 0 3.56155 0
1.2 0 1.56155 0 2.00000 0 −3.56155 0 −0.561553 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.k 2
4.b odd 2 1 418.2.a.e 2
12.b even 2 1 3762.2.a.y 2
44.c even 2 1 4598.2.a.bj 2
76.d even 2 1 7942.2.a.x 2

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

 $$T_{3}^{2} + T_{3} - 4$$ T3^2 + T3 - 4 $$T_{5} - 2$$ T5 - 2 $$T_{7}^{2} + 3T_{7} - 2$$ T7^2 + 3*T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 4$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2} + 3T - 2$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 3T - 2$$
$17$ $$T^{2} - 3T - 2$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 7T + 8$$
$29$ $$T^{2} - T - 38$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 2T - 16$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2}$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} - 5T - 100$$
$59$ $$T^{2} - 3T - 36$$
$61$ $$T^{2} + 4T - 64$$
$67$ $$T^{2} - 3T - 36$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 7T - 26$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 12T - 32$$
$89$ $$T^{2} + 10T + 8$$
$97$ $$T^{2} + 6T - 8$$