Properties

 Label 1323.4.a.n Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ 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] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + q^{4} + 12 q^{5} - 21 q^{8}+O(q^{10})$$ q + 3 * q^2 + q^4 + 12 * q^5 - 21 * q^8 $$q + 3 q^{2} + q^{4} + 12 q^{5} - 21 q^{8} + 36 q^{10} + 12 q^{11} + 61 q^{13} - 71 q^{16} + 117 q^{17} - 2 q^{19} + 12 q^{20} + 36 q^{22} - 75 q^{23} + 19 q^{25} + 183 q^{26} + 3 q^{29} - 263 q^{31} - 45 q^{32} + 351 q^{34} + 218 q^{37} - 6 q^{38} - 252 q^{40} + 246 q^{41} + 515 q^{43} + 12 q^{44} - 225 q^{46} - 318 q^{47} + 57 q^{50} + 61 q^{52} - 459 q^{53} + 144 q^{55} + 9 q^{58} + 255 q^{59} + 862 q^{61} - 789 q^{62} + 433 q^{64} + 732 q^{65} + 479 q^{67} + 117 q^{68} - 117 q^{71} + 430 q^{73} + 654 q^{74} - 2 q^{76} - 646 q^{79} - 852 q^{80} + 738 q^{82} + 348 q^{83} + 1404 q^{85} + 1545 q^{86} - 252 q^{88} + 585 q^{89} - 75 q^{92} - 954 q^{94} - 24 q^{95} + 376 q^{97}+O(q^{100})$$ q + 3 * q^2 + q^4 + 12 * q^5 - 21 * q^8 + 36 * q^10 + 12 * q^11 + 61 * q^13 - 71 * q^16 + 117 * q^17 - 2 * q^19 + 12 * q^20 + 36 * q^22 - 75 * q^23 + 19 * q^25 + 183 * q^26 + 3 * q^29 - 263 * q^31 - 45 * q^32 + 351 * q^34 + 218 * q^37 - 6 * q^38 - 252 * q^40 + 246 * q^41 + 515 * q^43 + 12 * q^44 - 225 * q^46 - 318 * q^47 + 57 * q^50 + 61 * q^52 - 459 * q^53 + 144 * q^55 + 9 * q^58 + 255 * q^59 + 862 * q^61 - 789 * q^62 + 433 * q^64 + 732 * q^65 + 479 * q^67 + 117 * q^68 - 117 * q^71 + 430 * q^73 + 654 * q^74 - 2 * q^76 - 646 * q^79 - 852 * q^80 + 738 * q^82 + 348 * q^83 + 1404 * q^85 + 1545 * q^86 - 252 * q^88 + 585 * q^89 - 75 * q^92 - 954 * q^94 - 24 * q^95 + 376 * 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
3.00000 0 1.00000 12.0000 0 0 −21.0000 0 36.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

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

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 1323.4.a.n 1
3.b odd 2 1 1323.4.a.a 1
7.b odd 2 1 189.4.a.d yes 1
21.c even 2 1 189.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.a.a 1 21.c even 2 1
189.4.a.d yes 1 7.b odd 2 1
1323.4.a.a 1 3.b odd 2 1
1323.4.a.n 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2} - 3$$ T2 - 3 $$T_{5} - 12$$ T5 - 12 $$T_{13} - 61$$ T13 - 61

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T$$
$5$ $$T - 12$$
$7$ $$T$$
$11$ $$T - 12$$
$13$ $$T - 61$$
$17$ $$T - 117$$
$19$ $$T + 2$$
$23$ $$T + 75$$
$29$ $$T - 3$$
$31$ $$T + 263$$
$37$ $$T - 218$$
$41$ $$T - 246$$
$43$ $$T - 515$$
$47$ $$T + 318$$
$53$ $$T + 459$$
$59$ $$T - 255$$
$61$ $$T - 862$$
$67$ $$T - 479$$
$71$ $$T + 117$$
$73$ $$T - 430$$
$79$ $$T + 646$$
$83$ $$T - 348$$
$89$ $$T - 585$$
$97$ $$T - 376$$