# Properties

 Label 3332.2.a.j Level $3332$ Weight $2$ Character orbit 3332.a Self dual yes Analytic conductor $26.606$ 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] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3332.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: $$26.6061539535$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 476) 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + ( - \beta + 1) q^{5} + \beta q^{9} +O(q^{10})$$ q - b * q^3 + (-b + 1) * q^5 + b * q^9 $$q - \beta q^{3} + ( - \beta + 1) q^{5} + \beta q^{9} + (2 \beta - 4) q^{13} + 3 q^{15} - q^{17} + (2 \beta - 4) q^{19} + ( - \beta - 1) q^{25} + (2 \beta - 3) q^{27} + (\beta - 3) q^{31} + 2 \beta q^{37} + (2 \beta - 6) q^{39} + 3 \beta q^{41} + ( - \beta + 6) q^{43} - 3 q^{45} + (2 \beta - 2) q^{47} + \beta q^{51} + ( - \beta - 5) q^{53} + (2 \beta - 6) q^{57} + (4 \beta - 4) q^{59} + ( - 3 \beta + 4) q^{61} + (4 \beta - 10) q^{65} + ( - \beta - 3) q^{67} + 6 \beta q^{71} + ( - 3 \beta + 4) q^{73} + (2 \beta + 3) q^{75} + ( - 2 \beta + 10) q^{79} + ( - 2 \beta - 6) q^{81} - 6 q^{83} + (\beta - 1) q^{85} + ( - 2 \beta + 14) q^{89} + (2 \beta - 3) q^{93} + (4 \beta - 10) q^{95} + ( - \beta + 5) q^{97} +O(q^{100})$$ q - b * q^3 + (-b + 1) * q^5 + b * q^9 + (2*b - 4) * q^13 + 3 * q^15 - q^17 + (2*b - 4) * q^19 + (-b - 1) * q^25 + (2*b - 3) * q^27 + (b - 3) * q^31 + 2*b * q^37 + (2*b - 6) * q^39 + 3*b * q^41 + (-b + 6) * q^43 - 3 * q^45 + (2*b - 2) * q^47 + b * q^51 + (-b - 5) * q^53 + (2*b - 6) * q^57 + (4*b - 4) * q^59 + (-3*b + 4) * q^61 + (4*b - 10) * q^65 + (-b - 3) * q^67 + 6*b * q^71 + (-3*b + 4) * q^73 + (2*b + 3) * q^75 + (-2*b + 10) * q^79 + (-2*b - 6) * q^81 - 6 * q^83 + (b - 1) * q^85 + (-2*b + 14) * q^89 + (2*b - 3) * q^93 + (4*b - 10) * q^95 + (-b + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{5} + q^{9}+O(q^{10})$$ 2 * q - q^3 + q^5 + q^9 $$2 q - q^{3} + q^{5} + q^{9} - 6 q^{13} + 6 q^{15} - 2 q^{17} - 6 q^{19} - 3 q^{25} - 4 q^{27} - 5 q^{31} + 2 q^{37} - 10 q^{39} + 3 q^{41} + 11 q^{43} - 6 q^{45} - 2 q^{47} + q^{51} - 11 q^{53} - 10 q^{57} - 4 q^{59} + 5 q^{61} - 16 q^{65} - 7 q^{67} + 6 q^{71} + 5 q^{73} + 8 q^{75} + 18 q^{79} - 14 q^{81} - 12 q^{83} - q^{85} + 26 q^{89} - 4 q^{93} - 16 q^{95} + 9 q^{97}+O(q^{100})$$ 2 * q - q^3 + q^5 + q^9 - 6 * q^13 + 6 * q^15 - 2 * q^17 - 6 * q^19 - 3 * q^25 - 4 * q^27 - 5 * q^31 + 2 * q^37 - 10 * q^39 + 3 * q^41 + 11 * q^43 - 6 * q^45 - 2 * q^47 + q^51 - 11 * q^53 - 10 * q^57 - 4 * q^59 + 5 * q^61 - 16 * q^65 - 7 * q^67 + 6 * q^71 + 5 * q^73 + 8 * q^75 + 18 * q^79 - 14 * q^81 - 12 * q^83 - q^85 + 26 * q^89 - 4 * q^93 - 16 * q^95 + 9 * 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
 2.30278 −1.30278
0 −2.30278 0 −1.30278 0 0 0 2.30278 0
1.2 0 1.30278 0 2.30278 0 0 0 −1.30278 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$17$$ $$1$$

## 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 3332.2.a.j 2
7.b odd 2 1 476.2.a.d 2
21.c even 2 1 4284.2.a.n 2
28.d even 2 1 1904.2.a.h 2
56.e even 2 1 7616.2.a.v 2
56.h odd 2 1 7616.2.a.q 2
119.d odd 2 1 8092.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.a.d 2 7.b odd 2 1
1904.2.a.h 2 28.d even 2 1
3332.2.a.j 2 1.a even 1 1 trivial
4284.2.a.n 2 21.c even 2 1
7616.2.a.q 2 56.h odd 2 1
7616.2.a.v 2 56.e even 2 1
8092.2.a.k 2 119.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3332))$$:

 $$T_{3}^{2} + T_{3} - 3$$ T3^2 + T3 - 3 $$T_{5}^{2} - T_{5} - 3$$ T5^2 - T5 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 3$$
$5$ $$T^{2} - T - 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T - 4$$
$17$ $$(T + 1)^{2}$$
$19$ $$T^{2} + 6T - 4$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 5T + 3$$
$37$ $$T^{2} - 2T - 12$$
$41$ $$T^{2} - 3T - 27$$
$43$ $$T^{2} - 11T + 27$$
$47$ $$T^{2} + 2T - 12$$
$53$ $$T^{2} + 11T + 27$$
$59$ $$T^{2} + 4T - 48$$
$61$ $$T^{2} - 5T - 23$$
$67$ $$T^{2} + 7T + 9$$
$71$ $$T^{2} - 6T - 108$$
$73$ $$T^{2} - 5T - 23$$
$79$ $$T^{2} - 18T + 68$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 26T + 156$$
$97$ $$T^{2} - 9T + 17$$