# Properties

 Label 1323.4.a.c Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ 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] = [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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + q^{4} + 6 q^{5} + 21 q^{8}+O(q^{10})$$ q - 3 * q^2 + q^4 + 6 * q^5 + 21 * q^8 $$q - 3 q^{2} + q^{4} + 6 q^{5} + 21 q^{8} - 18 q^{10} + 57 q^{11} + 62 q^{13} - 71 q^{16} + 12 q^{17} - 124 q^{19} + 6 q^{20} - 171 q^{22} - 156 q^{23} - 89 q^{25} - 186 q^{26} - 261 q^{29} - 109 q^{31} + 45 q^{32} - 36 q^{34} + 368 q^{37} + 372 q^{38} + 126 q^{40} - 54 q^{41} + 152 q^{43} + 57 q^{44} + 468 q^{46} + 78 q^{47} + 267 q^{50} + 62 q^{52} - 222 q^{53} + 342 q^{55} + 783 q^{58} - 285 q^{59} - 712 q^{61} + 327 q^{62} + 433 q^{64} + 372 q^{65} + 170 q^{67} + 12 q^{68} - 396 q^{71} - 475 q^{73} - 1104 q^{74} - 124 q^{76} - 163 q^{79} - 426 q^{80} + 162 q^{82} - 27 q^{83} + 72 q^{85} - 456 q^{86} + 1197 q^{88} - 642 q^{89} - 156 q^{92} - 234 q^{94} - 744 q^{95} + 1835 q^{97}+O(q^{100})$$ q - 3 * q^2 + q^4 + 6 * q^5 + 21 * q^8 - 18 * q^10 + 57 * q^11 + 62 * q^13 - 71 * q^16 + 12 * q^17 - 124 * q^19 + 6 * q^20 - 171 * q^22 - 156 * q^23 - 89 * q^25 - 186 * q^26 - 261 * q^29 - 109 * q^31 + 45 * q^32 - 36 * q^34 + 368 * q^37 + 372 * q^38 + 126 * q^40 - 54 * q^41 + 152 * q^43 + 57 * q^44 + 468 * q^46 + 78 * q^47 + 267 * q^50 + 62 * q^52 - 222 * q^53 + 342 * q^55 + 783 * q^58 - 285 * q^59 - 712 * q^61 + 327 * q^62 + 433 * q^64 + 372 * q^65 + 170 * q^67 + 12 * q^68 - 396 * q^71 - 475 * q^73 - 1104 * q^74 - 124 * q^76 - 163 * q^79 - 426 * q^80 + 162 * q^82 - 27 * q^83 + 72 * q^85 - 456 * q^86 + 1197 * q^88 - 642 * q^89 - 156 * q^92 - 234 * q^94 - 744 * q^95 + 1835 * 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 6.00000 0 0 21.0000 0 −18.0000
 $$n$$: 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.c 1
3.b odd 2 1 1323.4.a.l 1
7.b odd 2 1 1323.4.a.b 1
7.c even 3 2 189.4.e.d yes 2
21.c even 2 1 1323.4.a.m 1
21.h odd 6 2 189.4.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.a 2 21.h odd 6 2
189.4.e.d yes 2 7.c even 3 2
1323.4.a.b 1 7.b odd 2 1
1323.4.a.c 1 1.a even 1 1 trivial
1323.4.a.l 1 3.b odd 2 1
1323.4.a.m 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2} + 3$$ T2 + 3 $$T_{5} - 6$$ T5 - 6 $$T_{13} - 62$$ T13 - 62

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T$$
$5$ $$T - 6$$
$7$ $$T$$
$11$ $$T - 57$$
$13$ $$T - 62$$
$17$ $$T - 12$$
$19$ $$T + 124$$
$23$ $$T + 156$$
$29$ $$T + 261$$
$31$ $$T + 109$$
$37$ $$T - 368$$
$41$ $$T + 54$$
$43$ $$T - 152$$
$47$ $$T - 78$$
$53$ $$T + 222$$
$59$ $$T + 285$$
$61$ $$T + 712$$
$67$ $$T - 170$$
$71$ $$T + 396$$
$73$ $$T + 475$$
$79$ $$T + 163$$
$83$ $$T + 27$$
$89$ $$T + 642$$
$97$ $$T - 1835$$