# Properties

 Label 3332.2.a.h 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: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} - 2 \beta q^{5} + ( - 2 \beta + 1) q^{9} +O(q^{10})$$ q + (b - 1) * q^3 - 2*b * q^5 + (-2*b + 1) * q^9 $$q + (\beta - 1) q^{3} - 2 \beta q^{5} + ( - 2 \beta + 1) q^{9} + ( - \beta - 3) q^{11} + (2 \beta - 2) q^{13} + (2 \beta - 6) q^{15} + q^{17} + ( - 2 \beta - 2) q^{19} + ( - \beta - 3) q^{23} + 7 q^{25} - 4 q^{27} - 2 \beta q^{29} + (3 \beta + 1) q^{31} - 2 \beta q^{33} + (2 \beta + 8) q^{37} + ( - 4 \beta + 8) q^{39} + 6 q^{41} + (6 \beta + 2) q^{43} + ( - 2 \beta + 12) q^{45} - 4 \beta q^{47} + (\beta - 1) q^{51} + ( - 4 \beta + 6) q^{53} + (6 \beta + 6) q^{55} - 4 q^{57} + ( - 2 \beta - 6) q^{59} + (2 \beta + 4) q^{61} + (4 \beta - 12) q^{65} + ( - 4 \beta + 8) q^{67} - 2 \beta q^{69} + (3 \beta - 3) q^{71} - 2 q^{73} + (7 \beta - 7) q^{75} + (3 \beta - 7) q^{79} + (2 \beta + 1) q^{81} + (2 \beta + 6) q^{83} - 2 \beta q^{85} + (2 \beta - 6) q^{87} + ( - 2 \beta - 6) q^{89} + ( - 2 \beta + 8) q^{93} + (4 \beta + 12) q^{95} + ( - 4 \beta - 2) q^{97} + (5 \beta + 3) q^{99} +O(q^{100})$$ q + (b - 1) * q^3 - 2*b * q^5 + (-2*b + 1) * q^9 + (-b - 3) * q^11 + (2*b - 2) * q^13 + (2*b - 6) * q^15 + q^17 + (-2*b - 2) * q^19 + (-b - 3) * q^23 + 7 * q^25 - 4 * q^27 - 2*b * q^29 + (3*b + 1) * q^31 - 2*b * q^33 + (2*b + 8) * q^37 + (-4*b + 8) * q^39 + 6 * q^41 + (6*b + 2) * q^43 + (-2*b + 12) * q^45 - 4*b * q^47 + (b - 1) * q^51 + (-4*b + 6) * q^53 + (6*b + 6) * q^55 - 4 * q^57 + (-2*b - 6) * q^59 + (2*b + 4) * q^61 + (4*b - 12) * q^65 + (-4*b + 8) * q^67 - 2*b * q^69 + (3*b - 3) * q^71 - 2 * q^73 + (7*b - 7) * q^75 + (3*b - 7) * q^79 + (2*b + 1) * q^81 + (2*b + 6) * q^83 - 2*b * q^85 + (2*b - 6) * q^87 + (-2*b - 6) * q^89 + (-2*b + 8) * q^93 + (4*b + 12) * q^95 + (-4*b - 2) * q^97 + (5*b + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} - 6 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 4 q^{19} - 6 q^{23} + 14 q^{25} - 8 q^{27} + 2 q^{31} + 16 q^{37} + 16 q^{39} + 12 q^{41} + 4 q^{43} + 24 q^{45} - 2 q^{51} + 12 q^{53} + 12 q^{55} - 8 q^{57} - 12 q^{59} + 8 q^{61} - 24 q^{65} + 16 q^{67} - 6 q^{71} - 4 q^{73} - 14 q^{75} - 14 q^{79} + 2 q^{81} + 12 q^{83} - 12 q^{87} - 12 q^{89} + 16 q^{93} + 24 q^{95} - 4 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^11 - 4 * q^13 - 12 * q^15 + 2 * q^17 - 4 * q^19 - 6 * q^23 + 14 * q^25 - 8 * q^27 + 2 * q^31 + 16 * q^37 + 16 * q^39 + 12 * q^41 + 4 * q^43 + 24 * q^45 - 2 * q^51 + 12 * q^53 + 12 * q^55 - 8 * q^57 - 12 * q^59 + 8 * q^61 - 24 * q^65 + 16 * q^67 - 6 * q^71 - 4 * q^73 - 14 * q^75 - 14 * q^79 + 2 * q^81 + 12 * q^83 - 12 * q^87 - 12 * q^89 + 16 * q^93 + 24 * q^95 - 4 * q^97 + 6 * 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
 −1.73205 1.73205
0 −2.73205 0 3.46410 0 0 0 4.46410 0
1.2 0 0.732051 0 −3.46410 0 0 0 −2.46410 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.h 2
7.b odd 2 1 68.2.a.a 2
21.c even 2 1 612.2.a.e 2
28.d even 2 1 272.2.a.e 2
35.c odd 2 1 1700.2.a.d 2
35.f even 4 2 1700.2.e.c 4
56.e even 2 1 1088.2.a.t 2
56.h odd 2 1 1088.2.a.p 2
77.b even 2 1 8228.2.a.k 2
84.h odd 2 1 2448.2.a.y 2
119.d odd 2 1 1156.2.a.a 2
119.f odd 4 2 1156.2.b.c 4
119.l odd 8 4 1156.2.e.d 8
119.p even 16 8 1156.2.h.f 16
140.c even 2 1 6800.2.a.bh 2
168.e odd 2 1 9792.2.a.cs 2
168.i even 2 1 9792.2.a.cr 2
476.e even 2 1 4624.2.a.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.a.a 2 7.b odd 2 1
272.2.a.e 2 28.d even 2 1
612.2.a.e 2 21.c even 2 1
1088.2.a.p 2 56.h odd 2 1
1088.2.a.t 2 56.e even 2 1
1156.2.a.a 2 119.d odd 2 1
1156.2.b.c 4 119.f odd 4 2
1156.2.e.d 8 119.l odd 8 4
1156.2.h.f 16 119.p even 16 8
1700.2.a.d 2 35.c odd 2 1
1700.2.e.c 4 35.f even 4 2
2448.2.a.y 2 84.h odd 2 1
3332.2.a.h 2 1.a even 1 1 trivial
4624.2.a.x 2 476.e even 2 1
6800.2.a.bh 2 140.c even 2 1
8228.2.a.k 2 77.b even 2 1
9792.2.a.cr 2 168.i even 2 1
9792.2.a.cs 2 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3332))$$:

 $$T_{3}^{2} + 2T_{3} - 2$$ T3^2 + 2*T3 - 2 $$T_{5}^{2} - 12$$ T5^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 2$$
$5$ $$T^{2} - 12$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 6T + 6$$
$13$ $$T^{2} + 4T - 8$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2} + 4T - 8$$
$23$ $$T^{2} + 6T + 6$$
$29$ $$T^{2} - 12$$
$31$ $$T^{2} - 2T - 26$$
$37$ $$T^{2} - 16T + 52$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} - 4T - 104$$
$47$ $$T^{2} - 48$$
$53$ $$T^{2} - 12T - 12$$
$59$ $$T^{2} + 12T + 24$$
$61$ $$T^{2} - 8T + 4$$
$67$ $$T^{2} - 16T + 16$$
$71$ $$T^{2} + 6T - 18$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} + 14T + 22$$
$83$ $$T^{2} - 12T + 24$$
$89$ $$T^{2} + 12T + 24$$
$97$ $$T^{2} + 4T - 44$$