# Properties

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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} - q^{5} + 3 q^{7}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 - q^5 + 3 * q^7 $$q - 2 q^{2} + 2 q^{4} - q^{5} + 3 q^{7} + 2 q^{10} + 3 q^{11} + 6 q^{13} - 6 q^{14} - 4 q^{16} - 3 q^{17} - 2 q^{20} - 6 q^{22} - 4 q^{23} - 4 q^{25} - 12 q^{26} + 6 q^{28} - 10 q^{29} - 2 q^{31} + 8 q^{32} + 6 q^{34} - 3 q^{35} - 8 q^{37} - 8 q^{41} - q^{43} + 6 q^{44} + 8 q^{46} - 3 q^{47} + 2 q^{49} + 8 q^{50} + 12 q^{52} - 6 q^{53} - 3 q^{55} + 20 q^{58} + 7 q^{61} + 4 q^{62} - 8 q^{64} - 6 q^{65} - 8 q^{67} - 6 q^{68} + 6 q^{70} + 12 q^{71} - 11 q^{73} + 16 q^{74} + 9 q^{77} + 4 q^{80} + 16 q^{82} - 4 q^{83} + 3 q^{85} + 2 q^{86} + 10 q^{89} + 18 q^{91} - 8 q^{92} + 6 q^{94} + 2 q^{97} - 4 q^{98}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 - q^5 + 3 * q^7 + 2 * q^10 + 3 * q^11 + 6 * q^13 - 6 * q^14 - 4 * q^16 - 3 * q^17 - 2 * q^20 - 6 * q^22 - 4 * q^23 - 4 * q^25 - 12 * q^26 + 6 * q^28 - 10 * q^29 - 2 * q^31 + 8 * q^32 + 6 * q^34 - 3 * q^35 - 8 * q^37 - 8 * q^41 - q^43 + 6 * q^44 + 8 * q^46 - 3 * q^47 + 2 * q^49 + 8 * q^50 + 12 * q^52 - 6 * q^53 - 3 * q^55 + 20 * q^58 + 7 * q^61 + 4 * q^62 - 8 * q^64 - 6 * q^65 - 8 * q^67 - 6 * q^68 + 6 * q^70 + 12 * q^71 - 11 * q^73 + 16 * q^74 + 9 * q^77 + 4 * q^80 + 16 * q^82 - 4 * q^83 + 3 * q^85 + 2 * q^86 + 10 * q^89 + 18 * q^91 - 8 * q^92 + 6 * q^94 + 2 * q^97 - 4 * 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
−2.00000 0 2.00000 −1.00000 0 3.00000 0 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$19$$ $$-1$$

## 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.a 1
3.b odd 2 1 1083.2.a.d 1
19.b odd 2 1 171.2.a.c 1
57.d even 2 1 57.2.a.b 1
76.d even 2 1 2736.2.a.h 1
95.d odd 2 1 4275.2.a.a 1
133.c even 2 1 8379.2.a.q 1
228.b odd 2 1 912.2.a.d 1
285.b even 2 1 1425.2.a.i 1
285.j odd 4 2 1425.2.c.a 2
399.h odd 2 1 2793.2.a.a 1
456.l odd 2 1 3648.2.a.y 1
456.p even 2 1 3648.2.a.h 1
627.b odd 2 1 6897.2.a.g 1
741.d even 2 1 9633.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.b 1 57.d even 2 1
171.2.a.c 1 19.b odd 2 1
912.2.a.d 1 228.b odd 2 1
1083.2.a.d 1 3.b odd 2 1
1425.2.a.i 1 285.b even 2 1
1425.2.c.a 2 285.j odd 4 2
2736.2.a.h 1 76.d even 2 1
2793.2.a.a 1 399.h odd 2 1
3249.2.a.a 1 1.a even 1 1 trivial
3648.2.a.h 1 456.p even 2 1
3648.2.a.y 1 456.l odd 2 1
4275.2.a.a 1 95.d odd 2 1
6897.2.a.g 1 627.b odd 2 1
8379.2.a.q 1 133.c even 2 1
9633.2.a.p 1 741.d even 2 1

## Hecke kernels

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

 $$T_{2} + 2$$ T2 + 2 $$T_{5} + 1$$ T5 + 1 $$T_{13} - 6$$ T13 - 6

## Hecke characteristic polynomials

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