# Properties

 Label 3420.2.a.h Level $3420$ Weight $2$ Character orbit 3420.a Self dual yes Analytic conductor $27.309$ 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] = [3420,2,Mod(1,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, 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("3420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3420.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: $$27.3088374913$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 380) 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.

 $$f(q)$$ $$=$$ $$q - q^{5} + 2 q^{7}+O(q^{10})$$ q - q^5 + 2 * q^7 $$q - q^{5} + 2 q^{7} - 2 \beta q^{11} + (\beta - 1) q^{13} - 2 \beta q^{17} + q^{19} - 2 \beta q^{23} + q^{25} + 2 \beta q^{29} + (2 \beta + 2) q^{31} - 2 q^{35} + ( - \beta + 5) q^{37} + 6 q^{41} + (4 \beta + 2) q^{43} + (4 \beta - 6) q^{47} - 3 q^{49} + ( - \beta + 9) q^{53} + 2 \beta q^{55} + 4 \beta q^{59} + ( - 6 \beta + 2) q^{61} + ( - \beta + 1) q^{65} + ( - \beta + 5) q^{67} + (2 \beta + 6) q^{71} + ( - 2 \beta - 4) q^{73} - 4 \beta q^{77} + ( - 4 \beta - 4) q^{79} + 2 \beta q^{83} + 2 \beta q^{85} + ( - 2 \beta + 12) q^{89} + (2 \beta - 2) q^{91} - q^{95} + (9 \beta - 1) q^{97} +O(q^{100})$$ q - q^5 + 2 * q^7 - 2*b * q^11 + (b - 1) * q^13 - 2*b * q^17 + q^19 - 2*b * q^23 + q^25 + 2*b * q^29 + (2*b + 2) * q^31 - 2 * q^35 + (-b + 5) * q^37 + 6 * q^41 + (4*b + 2) * q^43 + (4*b - 6) * q^47 - 3 * q^49 + (-b + 9) * q^53 + 2*b * q^55 + 4*b * q^59 + (-6*b + 2) * q^61 + (-b + 1) * q^65 + (-b + 5) * q^67 + (2*b + 6) * q^71 + (-2*b - 4) * q^73 - 4*b * q^77 + (-4*b - 4) * q^79 + 2*b * q^83 + 2*b * q^85 + (-2*b + 12) * q^89 + (2*b - 2) * q^91 - q^95 + (9*b - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + 4 * q^7 $$2 q - 2 q^{5} + 4 q^{7} - 2 q^{13} + 2 q^{19} + 2 q^{25} + 4 q^{31} - 4 q^{35} + 10 q^{37} + 12 q^{41} + 4 q^{43} - 12 q^{47} - 6 q^{49} + 18 q^{53} + 4 q^{61} + 2 q^{65} + 10 q^{67} + 12 q^{71} - 8 q^{73} - 8 q^{79} + 24 q^{89} - 4 q^{91} - 2 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 4 * q^7 - 2 * q^13 + 2 * q^19 + 2 * q^25 + 4 * q^31 - 4 * q^35 + 10 * q^37 + 12 * q^41 + 4 * q^43 - 12 * q^47 - 6 * q^49 + 18 * q^53 + 4 * q^61 + 2 * q^65 + 10 * q^67 + 12 * q^71 - 8 * q^73 - 8 * q^79 + 24 * q^89 - 4 * q^91 - 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
 1.73205 −1.73205
0 0 0 −1.00000 0 2.00000 0 0 0
1.2 0 0 0 −1.00000 0 2.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 3420.2.a.h 2
3.b odd 2 1 380.2.a.d 2
12.b even 2 1 1520.2.a.l 2
15.d odd 2 1 1900.2.a.d 2
15.e even 4 2 1900.2.c.e 4
24.f even 2 1 6080.2.a.bj 2
24.h odd 2 1 6080.2.a.z 2
57.d even 2 1 7220.2.a.h 2
60.h even 2 1 7600.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.d 2 3.b odd 2 1
1520.2.a.l 2 12.b even 2 1
1900.2.a.d 2 15.d odd 2 1
1900.2.c.e 4 15.e even 4 2
3420.2.a.h 2 1.a even 1 1 trivial
6080.2.a.z 2 24.h odd 2 1
6080.2.a.bj 2 24.f even 2 1
7220.2.a.h 2 57.d even 2 1
7600.2.a.bf 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3420))$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11}^{2} - 12$$ T11^2 - 12 $$T_{13}^{2} + 2T_{13} - 2$$ T13^2 + 2*T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} - 12$$
$13$ $$T^{2} + 2T - 2$$
$17$ $$T^{2} - 12$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 12$$
$29$ $$T^{2} - 12$$
$31$ $$T^{2} - 4T - 8$$
$37$ $$T^{2} - 10T + 22$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} - 4T - 44$$
$47$ $$T^{2} + 12T - 12$$
$53$ $$T^{2} - 18T + 78$$
$59$ $$T^{2} - 48$$
$61$ $$T^{2} - 4T - 104$$
$67$ $$T^{2} - 10T + 22$$
$71$ $$T^{2} - 12T + 24$$
$73$ $$T^{2} + 8T + 4$$
$79$ $$T^{2} + 8T - 32$$
$83$ $$T^{2} - 12$$
$89$ $$T^{2} - 24T + 132$$
$97$ $$T^{2} + 2T - 242$$