# Properties

 Label 2646.2.a.i Level $2646$ Weight $2$ Character orbit 2646.a Self dual yes Analytic conductor $21.128$ 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] = [2646,2,Mod(1,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, 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("2646.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.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: $$21.1284163748$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 378) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + q^5 - q^8 $$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 5 q^{11} + q^{16} + 2 q^{17} + q^{19} + q^{20} + 5 q^{22} + q^{23} - 4 q^{25} - 4 q^{29} + 9 q^{31} - q^{32} - 2 q^{34} + 5 q^{37} - q^{38} - q^{40} - 9 q^{41} - 10 q^{43} - 5 q^{44} - q^{46} + 6 q^{47} + 4 q^{50} - 12 q^{53} - 5 q^{55} + 4 q^{58} - 14 q^{59} - 9 q^{62} + q^{64} - 8 q^{67} + 2 q^{68} + 13 q^{71} + 2 q^{73} - 5 q^{74} + q^{76} + 6 q^{79} + q^{80} + 9 q^{82} - 4 q^{83} + 2 q^{85} + 10 q^{86} + 5 q^{88} - 9 q^{89} + q^{92} - 6 q^{94} + q^{95} - 16 q^{97}+O(q^{100})$$ q - q^2 + q^4 + q^5 - q^8 - q^10 - 5 * q^11 + q^16 + 2 * q^17 + q^19 + q^20 + 5 * q^22 + q^23 - 4 * q^25 - 4 * q^29 + 9 * q^31 - q^32 - 2 * q^34 + 5 * q^37 - q^38 - q^40 - 9 * q^41 - 10 * q^43 - 5 * q^44 - q^46 + 6 * q^47 + 4 * q^50 - 12 * q^53 - 5 * q^55 + 4 * q^58 - 14 * q^59 - 9 * q^62 + q^64 - 8 * q^67 + 2 * q^68 + 13 * q^71 + 2 * q^73 - 5 * q^74 + q^76 + 6 * q^79 + q^80 + 9 * q^82 - 4 * q^83 + 2 * q^85 + 10 * q^86 + 5 * q^88 - 9 * q^89 + q^92 - 6 * q^94 + q^95 - 16 * 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
 0
−1.00000 0 1.00000 1.00000 0 0 −1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 2646.2.a.i 1
3.b odd 2 1 2646.2.a.v 1
7.b odd 2 1 378.2.a.c 1
21.c even 2 1 378.2.a.f yes 1
28.d even 2 1 3024.2.a.m 1
35.c odd 2 1 9450.2.a.dc 1
63.l odd 6 2 1134.2.f.n 2
63.o even 6 2 1134.2.f.c 2
84.h odd 2 1 3024.2.a.t 1
105.g even 2 1 9450.2.a.bx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.c 1 7.b odd 2 1
378.2.a.f yes 1 21.c even 2 1
1134.2.f.c 2 63.o even 6 2
1134.2.f.n 2 63.l odd 6 2
2646.2.a.i 1 1.a even 1 1 trivial
2646.2.a.v 1 3.b odd 2 1
3024.2.a.m 1 28.d even 2 1
3024.2.a.t 1 84.h odd 2 1
9450.2.a.bx 1 105.g even 2 1
9450.2.a.dc 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2646))$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{11} + 5$$ T11 + 5 $$T_{13}$$ T13 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 5$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T - 1$$
$23$ $$T - 1$$
$29$ $$T + 4$$
$31$ $$T - 9$$
$37$ $$T - 5$$
$41$ $$T + 9$$
$43$ $$T + 10$$
$47$ $$T - 6$$
$53$ $$T + 12$$
$59$ $$T + 14$$
$61$ $$T$$
$67$ $$T + 8$$
$71$ $$T - 13$$
$73$ $$T - 2$$
$79$ $$T - 6$$
$83$ $$T + 4$$
$89$ $$T + 9$$
$97$ $$T + 16$$