# Properties

 Label 3344.2.a.n 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} - q^{5} + (\beta + 2) q^{7} + 2 \beta q^{9}+O(q^{10})$$ q + (b + 1) * q^3 - q^5 + (b + 2) * q^7 + 2*b * q^9 $$q + (\beta + 1) q^{3} - q^{5} + (\beta + 2) q^{7} + 2 \beta q^{9} + q^{11} + (3 \beta - 2) q^{13} + ( - \beta - 1) q^{15} + (\beta + 2) q^{17} + q^{19} + (3 \beta + 4) q^{21} + 3 q^{23} - 4 q^{25} + ( - \beta + 1) q^{27} + ( - 3 \beta - 2) q^{29} + (\beta + 5) q^{31} + (\beta + 1) q^{33} + ( - \beta - 2) q^{35} + (5 \beta + 3) q^{37} + (\beta + 4) q^{39} + ( - 4 \beta + 4) q^{41} + (4 \beta - 6) q^{43} - 2 \beta q^{45} + ( - 2 \beta - 6) q^{47} + (4 \beta - 1) q^{49} + (3 \beta + 4) q^{51} + ( - 6 \beta + 4) q^{53} - q^{55} + (\beta + 1) q^{57} + ( - \beta + 3) q^{59} + ( - 5 \beta - 4) q^{61} + (4 \beta + 4) q^{63} + ( - 3 \beta + 2) q^{65} + (\beta + 9) q^{67} + (3 \beta + 3) q^{69} + (\beta + 11) q^{71} + ( - 6 \beta + 4) q^{73} + ( - 4 \beta - 4) q^{75} + (\beta + 2) q^{77} + ( - \beta + 16) q^{79} + ( - 6 \beta - 1) q^{81} + ( - \beta - 2) q^{83} + ( - \beta - 2) q^{85} + ( - 5 \beta - 8) q^{87} + (7 \beta - 5) q^{89} + (4 \beta + 2) q^{91} + (6 \beta + 7) q^{93} - q^{95} + (\beta + 1) q^{97} + 2 \beta q^{99} +O(q^{100})$$ q + (b + 1) * q^3 - q^5 + (b + 2) * q^7 + 2*b * q^9 + q^11 + (3*b - 2) * q^13 + (-b - 1) * q^15 + (b + 2) * q^17 + q^19 + (3*b + 4) * q^21 + 3 * q^23 - 4 * q^25 + (-b + 1) * q^27 + (-3*b - 2) * q^29 + (b + 5) * q^31 + (b + 1) * q^33 + (-b - 2) * q^35 + (5*b + 3) * q^37 + (b + 4) * q^39 + (-4*b + 4) * q^41 + (4*b - 6) * q^43 - 2*b * q^45 + (-2*b - 6) * q^47 + (4*b - 1) * q^49 + (3*b + 4) * q^51 + (-6*b + 4) * q^53 - q^55 + (b + 1) * q^57 + (-b + 3) * q^59 + (-5*b - 4) * q^61 + (4*b + 4) * q^63 + (-3*b + 2) * q^65 + (b + 9) * q^67 + (3*b + 3) * q^69 + (b + 11) * q^71 + (-6*b + 4) * q^73 + (-4*b - 4) * q^75 + (b + 2) * q^77 + (-b + 16) * q^79 + (-6*b - 1) * q^81 + (-b - 2) * q^83 + (-b - 2) * q^85 + (-5*b - 8) * q^87 + (7*b - 5) * q^89 + (4*b + 2) * q^91 + (6*b + 7) * q^93 - q^95 + (b + 1) * q^97 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 $$2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} + 8 q^{21} + 6 q^{23} - 8 q^{25} + 2 q^{27} - 4 q^{29} + 10 q^{31} + 2 q^{33} - 4 q^{35} + 6 q^{37} + 8 q^{39} + 8 q^{41} - 12 q^{43} - 12 q^{47} - 2 q^{49} + 8 q^{51} + 8 q^{53} - 2 q^{55} + 2 q^{57} + 6 q^{59} - 8 q^{61} + 8 q^{63} + 4 q^{65} + 18 q^{67} + 6 q^{69} + 22 q^{71} + 8 q^{73} - 8 q^{75} + 4 q^{77} + 32 q^{79} - 2 q^{81} - 4 q^{83} - 4 q^{85} - 16 q^{87} - 10 q^{89} + 4 q^{91} + 14 q^{93} - 2 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^19 + 8 * q^21 + 6 * q^23 - 8 * q^25 + 2 * q^27 - 4 * q^29 + 10 * q^31 + 2 * q^33 - 4 * q^35 + 6 * q^37 + 8 * q^39 + 8 * q^41 - 12 * q^43 - 12 * q^47 - 2 * q^49 + 8 * q^51 + 8 * q^53 - 2 * q^55 + 2 * q^57 + 6 * q^59 - 8 * q^61 + 8 * q^63 + 4 * q^65 + 18 * q^67 + 6 * q^69 + 22 * q^71 + 8 * q^73 - 8 * q^75 + 4 * q^77 + 32 * q^79 - 2 * q^81 - 4 * q^83 - 4 * q^85 - 16 * q^87 - 10 * q^89 + 4 * q^91 + 14 * q^93 - 2 * q^95 + 2 * q^97

## 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
 −1.41421 1.41421
0 −0.414214 0 −1.00000 0 0.585786 0 −2.82843 0
1.2 0 2.41421 0 −1.00000 0 3.41421 0 2.82843 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.n 2
4.b odd 2 1 209.2.a.b 2
12.b even 2 1 1881.2.a.d 2
20.d odd 2 1 5225.2.a.f 2
44.c even 2 1 2299.2.a.f 2
76.d even 2 1 3971.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.b 2 4.b odd 2 1
1881.2.a.d 2 12.b even 2 1
2299.2.a.f 2 44.c even 2 1
3344.2.a.n 2 1.a even 1 1 trivial
3971.2.a.d 2 76.d even 2 1
5225.2.a.f 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3344))$$:

 $$T_{3}^{2} - 2T_{3} - 1$$ T3^2 - 2*T3 - 1 $$T_{5} + 1$$ T5 + 1 $$T_{7}^{2} - 4T_{7} + 2$$ T7^2 - 4*T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 4T + 2$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 4T - 14$$
$17$ $$T^{2} - 4T + 2$$
$19$ $$(T - 1)^{2}$$
$23$ $$(T - 3)^{2}$$
$29$ $$T^{2} + 4T - 14$$
$31$ $$T^{2} - 10T + 23$$
$37$ $$T^{2} - 6T - 41$$
$41$ $$T^{2} - 8T - 16$$
$43$ $$T^{2} + 12T + 4$$
$47$ $$T^{2} + 12T + 28$$
$53$ $$T^{2} - 8T - 56$$
$59$ $$T^{2} - 6T + 7$$
$61$ $$T^{2} + 8T - 34$$
$67$ $$T^{2} - 18T + 79$$
$71$ $$T^{2} - 22T + 119$$
$73$ $$T^{2} - 8T - 56$$
$79$ $$T^{2} - 32T + 254$$
$83$ $$T^{2} + 4T + 2$$
$89$ $$T^{2} + 10T - 73$$
$97$ $$T^{2} - 2T - 1$$