# Properties

 Label 3360.2.a.x Level $3360$ Weight $2$ Character orbit 3360.a Self dual yes Analytic conductor $26.830$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(1,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3360.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: $$26.8297350792$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 + q^5 - q^7 + q^9 $$q + q^{3} + q^{5} - q^{7} + q^{9} + 4 q^{11} - 6 q^{13} + q^{15} + 6 q^{17} + 4 q^{19} - q^{21} + 4 q^{23} + q^{25} + q^{27} - 2 q^{29} - 8 q^{31} + 4 q^{33} - q^{35} + 6 q^{37} - 6 q^{39} + 6 q^{41} - 8 q^{43} + q^{45} + q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} + 10 q^{61} - q^{63} - 6 q^{65} - 8 q^{67} + 4 q^{69} - 12 q^{71} - 14 q^{73} + q^{75} - 4 q^{77} + 16 q^{79} + q^{81} + 12 q^{83} + 6 q^{85} - 2 q^{87} + 14 q^{89} + 6 q^{91} - 8 q^{93} + 4 q^{95} + 18 q^{97} + 4 q^{99}+O(q^{100})$$ q + q^3 + q^5 - q^7 + q^9 + 4 * q^11 - 6 * q^13 + q^15 + 6 * q^17 + 4 * q^19 - q^21 + 4 * q^23 + q^25 + q^27 - 2 * q^29 - 8 * q^31 + 4 * q^33 - q^35 + 6 * q^37 - 6 * q^39 + 6 * q^41 - 8 * q^43 + q^45 + q^49 + 6 * q^51 + 6 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 + 10 * q^61 - q^63 - 6 * q^65 - 8 * q^67 + 4 * q^69 - 12 * q^71 - 14 * q^73 + q^75 - 4 * q^77 + 16 * q^79 + q^81 + 12 * q^83 + 6 * q^85 - 2 * q^87 + 14 * q^89 + 6 * q^91 - 8 * q^93 + 4 * q^95 + 18 * q^97 + 4 * 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
 0
0 1.00000 0 1.00000 0 −1.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 3360.2.a.x yes 1
4.b odd 2 1 3360.2.a.h 1
8.b even 2 1 6720.2.a.c 1
8.d odd 2 1 6720.2.a.bx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.h 1 4.b odd 2 1
3360.2.a.x yes 1 1.a even 1 1 trivial
6720.2.a.c 1 8.b even 2 1
6720.2.a.bx 1 8.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3360))$$:

 $$T_{11} - 4$$ T11 - 4 $$T_{13} + 6$$ T13 + 6 $$T_{17} - 6$$ T17 - 6 $$T_{19} - 4$$ T19 - 4 $$T_{23} - 4$$ T23 - 4

## Hecke characteristic polynomials

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