# Properties

 Label 2646.2.a.n Level $2646$ Weight $2$ Character orbit 2646.a Self dual yes Analytic conductor $21.128$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 3 q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + 3 * q^5 - q^8 $$q - q^{2} + q^{4} + 3 q^{5} - q^{8} - 3 q^{10} + 3 q^{11} + 4 q^{13} + q^{16} + 6 q^{17} + 7 q^{19} + 3 q^{20} - 3 q^{22} - 3 q^{23} + 4 q^{25} - 4 q^{26} - 5 q^{31} - q^{32} - 6 q^{34} - 7 q^{37} - 7 q^{38} - 3 q^{40} + 9 q^{41} - 10 q^{43} + 3 q^{44} + 3 q^{46} - 6 q^{47} - 4 q^{50} + 4 q^{52} + 12 q^{53} + 9 q^{55} + 6 q^{59} - 8 q^{61} + 5 q^{62} + q^{64} + 12 q^{65} - 4 q^{67} + 6 q^{68} + 9 q^{71} - 2 q^{73} + 7 q^{74} + 7 q^{76} - 10 q^{79} + 3 q^{80} - 9 q^{82} + 18 q^{85} + 10 q^{86} - 3 q^{88} - 15 q^{89} - 3 q^{92} + 6 q^{94} + 21 q^{95} - 8 q^{97}+O(q^{100})$$ q - q^2 + q^4 + 3 * q^5 - q^8 - 3 * q^10 + 3 * q^11 + 4 * q^13 + q^16 + 6 * q^17 + 7 * q^19 + 3 * q^20 - 3 * q^22 - 3 * q^23 + 4 * q^25 - 4 * q^26 - 5 * q^31 - q^32 - 6 * q^34 - 7 * q^37 - 7 * q^38 - 3 * q^40 + 9 * q^41 - 10 * q^43 + 3 * q^44 + 3 * q^46 - 6 * q^47 - 4 * q^50 + 4 * q^52 + 12 * q^53 + 9 * q^55 + 6 * q^59 - 8 * q^61 + 5 * q^62 + q^64 + 12 * q^65 - 4 * q^67 + 6 * q^68 + 9 * q^71 - 2 * q^73 + 7 * q^74 + 7 * q^76 - 10 * q^79 + 3 * q^80 - 9 * q^82 + 18 * q^85 + 10 * q^86 - 3 * q^88 - 15 * q^89 - 3 * q^92 + 6 * q^94 + 21 * q^95 - 8 * 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 3.00000 0 0 −1.00000 0 −3.00000
 $$n$$: 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.n 1
3.b odd 2 1 2646.2.a.q 1
7.b odd 2 1 378.2.a.b 1
21.c even 2 1 378.2.a.g yes 1
28.d even 2 1 3024.2.a.c 1
35.c odd 2 1 9450.2.a.cu 1
63.l odd 6 2 1134.2.f.o 2
63.o even 6 2 1134.2.f.b 2
84.h odd 2 1 3024.2.a.bb 1
105.g even 2 1 9450.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.b 1 7.b odd 2 1
378.2.a.g yes 1 21.c even 2 1
1134.2.f.b 2 63.o even 6 2
1134.2.f.o 2 63.l odd 6 2
2646.2.a.n 1 1.a even 1 1 trivial
2646.2.a.q 1 3.b odd 2 1
3024.2.a.c 1 28.d even 2 1
3024.2.a.bb 1 84.h odd 2 1
9450.2.a.h 1 105.g even 2 1
9450.2.a.cu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2646))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{11} - 3$$ T11 - 3 $$T_{13} - 4$$ T13 - 4 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 3$$
$7$ $$T$$
$11$ $$T - 3$$
$13$ $$T - 4$$
$17$ $$T - 6$$
$19$ $$T - 7$$
$23$ $$T + 3$$
$29$ $$T$$
$31$ $$T + 5$$
$37$ $$T + 7$$
$41$ $$T - 9$$
$43$ $$T + 10$$
$47$ $$T + 6$$
$53$ $$T - 12$$
$59$ $$T - 6$$
$61$ $$T + 8$$
$67$ $$T + 4$$
$71$ $$T - 9$$
$73$ $$T + 2$$
$79$ $$T + 10$$
$83$ $$T$$
$89$ $$T + 15$$
$97$ $$T + 8$$