# Properties

 Label 1323.4.a.d 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 27) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + q^{4} + 15 q^{5} + 21 q^{8}+O(q^{10})$$ q - 3 * q^2 + q^4 + 15 * q^5 + 21 * q^8 $$q - 3 q^{2} + q^{4} + 15 q^{5} + 21 q^{8} - 45 q^{10} + 15 q^{11} - 20 q^{13} - 71 q^{16} + 72 q^{17} - 2 q^{19} + 15 q^{20} - 45 q^{22} - 114 q^{23} + 100 q^{25} + 60 q^{26} - 30 q^{29} - 101 q^{31} + 45 q^{32} - 216 q^{34} - 430 q^{37} + 6 q^{38} + 315 q^{40} - 30 q^{41} + 110 q^{43} + 15 q^{44} + 342 q^{46} - 330 q^{47} - 300 q^{50} - 20 q^{52} - 621 q^{53} + 225 q^{55} + 90 q^{58} - 660 q^{59} + 376 q^{61} + 303 q^{62} + 433 q^{64} - 300 q^{65} - 250 q^{67} + 72 q^{68} + 360 q^{71} - 785 q^{73} + 1290 q^{74} - 2 q^{76} + 488 q^{79} - 1065 q^{80} + 90 q^{82} + 489 q^{83} + 1080 q^{85} - 330 q^{86} + 315 q^{88} - 450 q^{89} - 114 q^{92} + 990 q^{94} - 30 q^{95} + 1105 q^{97}+O(q^{100})$$ q - 3 * q^2 + q^4 + 15 * q^5 + 21 * q^8 - 45 * q^10 + 15 * q^11 - 20 * q^13 - 71 * q^16 + 72 * q^17 - 2 * q^19 + 15 * q^20 - 45 * q^22 - 114 * q^23 + 100 * q^25 + 60 * q^26 - 30 * q^29 - 101 * q^31 + 45 * q^32 - 216 * q^34 - 430 * q^37 + 6 * q^38 + 315 * q^40 - 30 * q^41 + 110 * q^43 + 15 * q^44 + 342 * q^46 - 330 * q^47 - 300 * q^50 - 20 * q^52 - 621 * q^53 + 225 * q^55 + 90 * q^58 - 660 * q^59 + 376 * q^61 + 303 * q^62 + 433 * q^64 - 300 * q^65 - 250 * q^67 + 72 * q^68 + 360 * q^71 - 785 * q^73 + 1290 * q^74 - 2 * q^76 + 488 * q^79 - 1065 * q^80 + 90 * q^82 + 489 * q^83 + 1080 * q^85 - 330 * q^86 + 315 * q^88 - 450 * q^89 - 114 * q^92 + 990 * q^94 - 30 * q^95 + 1105 * 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 15.0000 0 0 21.0000 0 −45.0000
 $$n$$: 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.d 1
3.b odd 2 1 1323.4.a.k 1
7.b odd 2 1 27.4.a.a 1
21.c even 2 1 27.4.a.b yes 1
28.d even 2 1 432.4.a.a 1
35.c odd 2 1 675.4.a.j 1
35.f even 4 2 675.4.b.b 2
56.e even 2 1 1728.4.a.bd 1
56.h odd 2 1 1728.4.a.bc 1
63.l odd 6 2 81.4.c.c 2
63.o even 6 2 81.4.c.a 2
84.h odd 2 1 432.4.a.n 1
105.g even 2 1 675.4.a.a 1
105.k odd 4 2 675.4.b.a 2
168.e odd 2 1 1728.4.a.d 1
168.i even 2 1 1728.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.a 1 7.b odd 2 1
27.4.a.b yes 1 21.c even 2 1
81.4.c.a 2 63.o even 6 2
81.4.c.c 2 63.l odd 6 2
432.4.a.a 1 28.d even 2 1
432.4.a.n 1 84.h odd 2 1
675.4.a.a 1 105.g even 2 1
675.4.a.j 1 35.c odd 2 1
675.4.b.a 2 105.k odd 4 2
675.4.b.b 2 35.f even 4 2
1323.4.a.d 1 1.a even 1 1 trivial
1323.4.a.k 1 3.b odd 2 1
1728.4.a.c 1 168.i even 2 1
1728.4.a.d 1 168.e odd 2 1
1728.4.a.bc 1 56.h odd 2 1
1728.4.a.bd 1 56.e 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 $$T_{5} - 15$$ T5 - 15 $$T_{13} + 20$$ T13 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T$$
$5$ $$T - 15$$
$7$ $$T$$
$11$ $$T - 15$$
$13$ $$T + 20$$
$17$ $$T - 72$$
$19$ $$T + 2$$
$23$ $$T + 114$$
$29$ $$T + 30$$
$31$ $$T + 101$$
$37$ $$T + 430$$
$41$ $$T + 30$$
$43$ $$T - 110$$
$47$ $$T + 330$$
$53$ $$T + 621$$
$59$ $$T + 660$$
$61$ $$T - 376$$
$67$ $$T + 250$$
$71$ $$T - 360$$
$73$ $$T + 785$$
$79$ $$T - 488$$
$83$ $$T - 489$$
$89$ $$T + 450$$
$97$ $$T - 1105$$