# Properties

 Label 3249.2.a.c Level $3249$ Weight $2$ Character orbit 3249.a Self dual yes Analytic conductor $25.943$ 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] = [3249,2,Mod(1,3249)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3249, 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("3249.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3249 = 3^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3249.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: $$25.9433956167$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + q^{7} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + q^7 + 3 * q^8 $$q - q^{2} - q^{4} + q^{7} + 3 q^{8} + 2 q^{11} + 5 q^{13} - q^{14} - q^{16} + 4 q^{17} - 2 q^{22} + 4 q^{23} - 5 q^{25} - 5 q^{26} - q^{28} + 8 q^{29} - 3 q^{31} - 5 q^{32} - 4 q^{34} + 3 q^{37} + 12 q^{41} - q^{43} - 2 q^{44} - 4 q^{46} + 6 q^{47} - 6 q^{49} + 5 q^{50} - 5 q^{52} - 4 q^{53} + 3 q^{56} - 8 q^{58} - 10 q^{59} - 13 q^{61} + 3 q^{62} + 7 q^{64} + 11 q^{67} - 4 q^{68} - 6 q^{71} - 11 q^{73} - 3 q^{74} + 2 q^{77} + q^{79} - 12 q^{82} + q^{86} + 6 q^{88} + 6 q^{89} + 5 q^{91} - 4 q^{92} - 6 q^{94} + 2 q^{97} + 6 q^{98}+O(q^{100})$$ q - q^2 - q^4 + q^7 + 3 * q^8 + 2 * q^11 + 5 * q^13 - q^14 - q^16 + 4 * q^17 - 2 * q^22 + 4 * q^23 - 5 * q^25 - 5 * q^26 - q^28 + 8 * q^29 - 3 * q^31 - 5 * q^32 - 4 * q^34 + 3 * q^37 + 12 * q^41 - q^43 - 2 * q^44 - 4 * q^46 + 6 * q^47 - 6 * q^49 + 5 * q^50 - 5 * q^52 - 4 * q^53 + 3 * q^56 - 8 * q^58 - 10 * q^59 - 13 * q^61 + 3 * q^62 + 7 * q^64 + 11 * q^67 - 4 * q^68 - 6 * q^71 - 11 * q^73 - 3 * q^74 + 2 * q^77 + q^79 - 12 * q^82 + q^86 + 6 * q^88 + 6 * q^89 + 5 * q^91 - 4 * q^92 - 6 * q^94 + 2 * 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 3.00000 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
$$3$$ $$-1$$
$$19$$ $$1$$

## 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 3249.2.a.c 1
3.b odd 2 1 1083.2.a.c 1
19.b odd 2 1 3249.2.a.f 1
19.c even 3 2 171.2.f.a 2
57.d even 2 1 1083.2.a.b 1
57.h odd 6 2 57.2.e.a 2
76.g odd 6 2 2736.2.s.j 2
228.m even 6 2 912.2.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.a 2 57.h odd 6 2
171.2.f.a 2 19.c even 3 2
912.2.q.a 2 228.m even 6 2
1083.2.a.b 1 57.d even 2 1
1083.2.a.c 1 3.b odd 2 1
2736.2.s.j 2 76.g odd 6 2
3249.2.a.c 1 1.a even 1 1 trivial
3249.2.a.f 1 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3249))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5}$$ T5 $$T_{13} - 5$$ T13 - 5

## Hecke characteristic polynomials

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