# Properties

 Label 3332.2.a.n Level $3332$ Weight $2$ Character orbit 3332.a Self dual yes Analytic conductor $26.606$ 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] = [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: $$0$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + ( - \beta + 2) q^{5} + (3 \beta + 1) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (-b + 2) * q^5 + (3*b + 1) * q^9 $$q + (\beta + 1) q^{3} + ( - \beta + 2) q^{5} + (3 \beta + 1) q^{9} + ( - 2 \beta + 2) q^{11} + ( - 2 \beta + 4) q^{13} - q^{15} + q^{17} + 6 q^{19} + 2 \beta q^{23} + ( - 3 \beta + 2) q^{25} + (4 \beta + 7) q^{27} + (2 \beta - 6) q^{29} + (\beta + 8) q^{31} + ( - 2 \beta - 4) q^{33} + ( - 4 \beta + 2) q^{37} - 2 q^{39} + (5 \beta - 3) q^{41} + (\beta - 3) q^{43} + (2 \beta - 7) q^{45} - 10 q^{47} + (\beta + 1) q^{51} + (\beta + 2) q^{53} + ( - 4 \beta + 10) q^{55} + (6 \beta + 6) q^{57} + 4 \beta q^{59} + ( - \beta + 3) q^{61} + ( - 6 \beta + 14) q^{65} + (\beta + 4) q^{67} + (4 \beta + 6) q^{69} - 2 q^{71} + ( - 5 \beta + 7) q^{73} + ( - 4 \beta - 7) q^{75} - 6 q^{79} + (6 \beta + 16) q^{81} + (4 \beta - 4) q^{83} + ( - \beta + 2) q^{85} - 2 \beta q^{87} + 2 q^{89} + (10 \beta + 11) q^{93} + ( - 6 \beta + 12) q^{95} + (3 \beta + 6) q^{97} + ( - 2 \beta - 16) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + (-b + 2) * q^5 + (3*b + 1) * q^9 + (-2*b + 2) * q^11 + (-2*b + 4) * q^13 - q^15 + q^17 + 6 * q^19 + 2*b * q^23 + (-3*b + 2) * q^25 + (4*b + 7) * q^27 + (2*b - 6) * q^29 + (b + 8) * q^31 + (-2*b - 4) * q^33 + (-4*b + 2) * q^37 - 2 * q^39 + (5*b - 3) * q^41 + (b - 3) * q^43 + (2*b - 7) * q^45 - 10 * q^47 + (b + 1) * q^51 + (b + 2) * q^53 + (-4*b + 10) * q^55 + (6*b + 6) * q^57 + 4*b * q^59 + (-b + 3) * q^61 + (-6*b + 14) * q^65 + (b + 4) * q^67 + (4*b + 6) * q^69 - 2 * q^71 + (-5*b + 7) * q^73 + (-4*b - 7) * q^75 - 6 * q^79 + (6*b + 16) * q^81 + (4*b - 4) * q^83 + (-b + 2) * q^85 - 2*b * q^87 + 2 * q^89 + (10*b + 11) * q^93 + (-6*b + 12) * q^95 + (3*b + 6) * q^97 + (-2*b - 16) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 3 q^{5} + 5 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9 $$2 q + 3 q^{3} + 3 q^{5} + 5 q^{9} + 2 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 12 q^{19} + 2 q^{23} + q^{25} + 18 q^{27} - 10 q^{29} + 17 q^{31} - 10 q^{33} - 4 q^{39} - q^{41} - 5 q^{43} - 12 q^{45} - 20 q^{47} + 3 q^{51} + 5 q^{53} + 16 q^{55} + 18 q^{57} + 4 q^{59} + 5 q^{61} + 22 q^{65} + 9 q^{67} + 16 q^{69} - 4 q^{71} + 9 q^{73} - 18 q^{75} - 12 q^{79} + 38 q^{81} - 4 q^{83} + 3 q^{85} - 2 q^{87} + 4 q^{89} + 32 q^{93} + 18 q^{95} + 15 q^{97} - 34 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9 + 2 * q^11 + 6 * q^13 - 2 * q^15 + 2 * q^17 + 12 * q^19 + 2 * q^23 + q^25 + 18 * q^27 - 10 * q^29 + 17 * q^31 - 10 * q^33 - 4 * q^39 - q^41 - 5 * q^43 - 12 * q^45 - 20 * q^47 + 3 * q^51 + 5 * q^53 + 16 * q^55 + 18 * q^57 + 4 * q^59 + 5 * q^61 + 22 * q^65 + 9 * q^67 + 16 * q^69 - 4 * q^71 + 9 * q^73 - 18 * q^75 - 12 * q^79 + 38 * q^81 - 4 * q^83 + 3 * q^85 - 2 * q^87 + 4 * q^89 + 32 * q^93 + 18 * q^95 + 15 * q^97 - 34 * 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
 −1.30278 2.30278
0 −0.302776 0 3.30278 0 0 0 −2.90833 0
1.2 0 3.30278 0 −0.302776 0 0 0 7.90833 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.n 2
7.b odd 2 1 476.2.a.a 2
21.c even 2 1 4284.2.a.p 2
28.d even 2 1 1904.2.a.l 2
56.e even 2 1 7616.2.a.m 2
56.h odd 2 1 7616.2.a.z 2
119.d odd 2 1 8092.2.a.n 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T - 1$$
$5$ $$T^{2} - 3T - 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T - 12$$
$13$ $$T^{2} - 6T - 4$$
$17$ $$(T - 1)^{2}$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} - 2T - 12$$
$29$ $$T^{2} + 10T + 12$$
$31$ $$T^{2} - 17T + 69$$
$37$ $$T^{2} - 52$$
$41$ $$T^{2} + T - 81$$
$43$ $$T^{2} + 5T + 3$$
$47$ $$(T + 10)^{2}$$
$53$ $$T^{2} - 5T + 3$$
$59$ $$T^{2} - 4T - 48$$
$61$ $$T^{2} - 5T + 3$$
$67$ $$T^{2} - 9T + 17$$
$71$ $$(T + 2)^{2}$$
$73$ $$T^{2} - 9T - 61$$
$79$ $$(T + 6)^{2}$$
$83$ $$T^{2} + 4T - 48$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} - 15T + 27$$