# Properties

 Label 3294.2.a.j Level $3294$ Weight $2$ Character orbit 3294.a Self dual yes Analytic conductor $26.303$ Analytic rank $1$ 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] = [3294,2,Mod(1,3294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3294, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3294.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3294 = 2 \cdot 3^{3} \cdot 61$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3294.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: $$26.3027224258$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^7 + q^8 $$q + q^{2} + q^{4} - q^{7} + q^{8} - q^{13} - q^{14} + q^{16} - 6 q^{17} - 4 q^{19} - 5 q^{25} - q^{26} - q^{28} + 3 q^{29} + 5 q^{31} + q^{32} - 6 q^{34} - 10 q^{37} - 4 q^{38} - 6 q^{41} + 5 q^{43} - 6 q^{49} - 5 q^{50} - q^{52} - 3 q^{53} - q^{56} + 3 q^{58} - 6 q^{59} + q^{61} + 5 q^{62} + q^{64} - 13 q^{67} - 6 q^{68} + 6 q^{71} + 11 q^{73} - 10 q^{74} - 4 q^{76} - 13 q^{79} - 6 q^{82} + 3 q^{83} + 5 q^{86} + 15 q^{89} + q^{91} - 19 q^{97} - 6 q^{98}+O(q^{100})$$ q + q^2 + q^4 - q^7 + q^8 - q^13 - q^14 + q^16 - 6 * q^17 - 4 * q^19 - 5 * q^25 - q^26 - q^28 + 3 * q^29 + 5 * q^31 + q^32 - 6 * q^34 - 10 * q^37 - 4 * q^38 - 6 * q^41 + 5 * q^43 - 6 * q^49 - 5 * q^50 - q^52 - 3 * q^53 - q^56 + 3 * q^58 - 6 * q^59 + q^61 + 5 * q^62 + q^64 - 13 * q^67 - 6 * q^68 + 6 * q^71 + 11 * q^73 - 10 * q^74 - 4 * q^76 - 13 * q^79 - 6 * q^82 + 3 * q^83 + 5 * q^86 + 15 * q^89 + q^91 - 19 * q^97 - 6 * q^98

## 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
1.00000 0 1.00000 0 0 −1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$+1$$
$$61$$ $$-1$$

## 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 3294.2.a.j yes 1
3.b odd 2 1 3294.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3294.2.a.c 1 3.b odd 2 1
3294.2.a.j yes 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3294))$$:

 $$T_{5}$$ T5 $$T_{7} + 1$$ T7 + 1 $$T_{11}$$ T11 $$T_{17} + 6$$ T17 + 6 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T + 1$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T - 3$$
$31$ $$T - 5$$
$37$ $$T + 10$$
$41$ $$T + 6$$
$43$ $$T - 5$$
$47$ $$T$$
$53$ $$T + 3$$
$59$ $$T + 6$$
$61$ $$T - 1$$
$67$ $$T + 13$$
$71$ $$T - 6$$
$73$ $$T - 11$$
$79$ $$T + 13$$
$83$ $$T - 3$$
$89$ $$T - 15$$
$97$ $$T + 19$$