# Properties

 Label 1323.4.a.t Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 10 q^{4} + 4 \beta q^{5} + 2 \beta q^{8}+O(q^{10})$$ q + b * q^2 + 10 * q^4 + 4*b * q^5 + 2*b * q^8 $$q + \beta q^{2} + 10 q^{4} + 4 \beta q^{5} + 2 \beta q^{8} + 72 q^{10} + 4 \beta q^{11} - 29 q^{13} - 44 q^{16} + 12 \beta q^{17} - 29 q^{19} + 40 \beta q^{20} + 72 q^{22} + 20 \beta q^{23} + 163 q^{25} - 29 \beta q^{26} + 64 \beta q^{29} + 268 q^{31} - 60 \beta q^{32} + 216 q^{34} + 83 q^{37} - 29 \beta q^{38} + 144 q^{40} + 64 \beta q^{41} - 232 q^{43} + 40 \beta q^{44} + 360 q^{46} + 92 \beta q^{47} + 163 \beta q^{50} - 290 q^{52} + 72 \beta q^{53} + 288 q^{55} + 1152 q^{58} - 68 \beta q^{59} - 767 q^{61} + 268 \beta q^{62} - 728 q^{64} - 116 \beta q^{65} - 511 q^{67} + 120 \beta q^{68} + 168 \beta q^{71} - 137 q^{73} + 83 \beta q^{74} - 290 q^{76} - 475 q^{79} - 176 \beta q^{80} + 1152 q^{82} - 136 \beta q^{83} + 864 q^{85} - 232 \beta q^{86} + 144 q^{88} + 60 \beta q^{89} + 200 \beta q^{92} + 1656 q^{94} - 116 \beta q^{95} - 821 q^{97} +O(q^{100})$$ q + b * q^2 + 10 * q^4 + 4*b * q^5 + 2*b * q^8 + 72 * q^10 + 4*b * q^11 - 29 * q^13 - 44 * q^16 + 12*b * q^17 - 29 * q^19 + 40*b * q^20 + 72 * q^22 + 20*b * q^23 + 163 * q^25 - 29*b * q^26 + 64*b * q^29 + 268 * q^31 - 60*b * q^32 + 216 * q^34 + 83 * q^37 - 29*b * q^38 + 144 * q^40 + 64*b * q^41 - 232 * q^43 + 40*b * q^44 + 360 * q^46 + 92*b * q^47 + 163*b * q^50 - 290 * q^52 + 72*b * q^53 + 288 * q^55 + 1152 * q^58 - 68*b * q^59 - 767 * q^61 + 268*b * q^62 - 728 * q^64 - 116*b * q^65 - 511 * q^67 + 120*b * q^68 + 168*b * q^71 - 137 * q^73 + 83*b * q^74 - 290 * q^76 - 475 * q^79 - 176*b * q^80 + 1152 * q^82 - 136*b * q^83 + 864 * q^85 - 232*b * q^86 + 144 * q^88 + 60*b * q^89 + 200*b * q^92 + 1656 * q^94 - 116*b * q^95 - 821 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 20 q^{4}+O(q^{10})$$ 2 * q + 20 * q^4 $$2 q + 20 q^{4} + 144 q^{10} - 58 q^{13} - 88 q^{16} - 58 q^{19} + 144 q^{22} + 326 q^{25} + 536 q^{31} + 432 q^{34} + 166 q^{37} + 288 q^{40} - 464 q^{43} + 720 q^{46} - 580 q^{52} + 576 q^{55} + 2304 q^{58} - 1534 q^{61} - 1456 q^{64} - 1022 q^{67} - 274 q^{73} - 580 q^{76} - 950 q^{79} + 2304 q^{82} + 1728 q^{85} + 288 q^{88} + 3312 q^{94} - 1642 q^{97}+O(q^{100})$$ 2 * q + 20 * q^4 + 144 * q^10 - 58 * q^13 - 88 * q^16 - 58 * q^19 + 144 * q^22 + 326 * q^25 + 536 * q^31 + 432 * q^34 + 166 * q^37 + 288 * q^40 - 464 * q^43 + 720 * q^46 - 580 * q^52 + 576 * q^55 + 2304 * q^58 - 1534 * q^61 - 1456 * q^64 - 1022 * q^67 - 274 * q^73 - 580 * q^76 - 950 * q^79 + 2304 * q^82 + 1728 * q^85 + 288 * q^88 + 3312 * q^94 - 1642 * 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
 −1.41421 1.41421
−4.24264 0 10.0000 −16.9706 0 0 −8.48528 0 72.0000
1.2 4.24264 0 10.0000 16.9706 0 0 8.48528 0 72.0000
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.4.a.t 2
3.b odd 2 1 inner 1323.4.a.t 2
7.b odd 2 1 27.4.a.c 2
21.c even 2 1 27.4.a.c 2
28.d even 2 1 432.4.a.q 2
35.c odd 2 1 675.4.a.n 2
35.f even 4 2 675.4.b.i 4
56.e even 2 1 1728.4.a.bk 2
56.h odd 2 1 1728.4.a.bp 2
63.l odd 6 2 81.4.c.e 4
63.o even 6 2 81.4.c.e 4
84.h odd 2 1 432.4.a.q 2
105.g even 2 1 675.4.a.n 2
105.k odd 4 2 675.4.b.i 4
168.e odd 2 1 1728.4.a.bk 2
168.i even 2 1 1728.4.a.bp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.c 2 7.b odd 2 1
27.4.a.c 2 21.c even 2 1
81.4.c.e 4 63.l odd 6 2
81.4.c.e 4 63.o even 6 2
432.4.a.q 2 28.d even 2 1
432.4.a.q 2 84.h odd 2 1
675.4.a.n 2 35.c odd 2 1
675.4.a.n 2 105.g even 2 1
675.4.b.i 4 35.f even 4 2
675.4.b.i 4 105.k odd 4 2
1323.4.a.t 2 1.a even 1 1 trivial
1323.4.a.t 2 3.b odd 2 1 inner
1728.4.a.bk 2 56.e even 2 1
1728.4.a.bk 2 168.e odd 2 1
1728.4.a.bp 2 56.h odd 2 1
1728.4.a.bp 2 168.i even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 18$$ T2^2 - 18 $$T_{5}^{2} - 288$$ T5^2 - 288 $$T_{13} + 29$$ T13 + 29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 18$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 288$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 288$$
$13$ $$(T + 29)^{2}$$
$17$ $$T^{2} - 2592$$
$19$ $$(T + 29)^{2}$$
$23$ $$T^{2} - 7200$$
$29$ $$T^{2} - 73728$$
$31$ $$(T - 268)^{2}$$
$37$ $$(T - 83)^{2}$$
$41$ $$T^{2} - 73728$$
$43$ $$(T + 232)^{2}$$
$47$ $$T^{2} - 152352$$
$53$ $$T^{2} - 93312$$
$59$ $$T^{2} - 83232$$
$61$ $$(T + 767)^{2}$$
$67$ $$(T + 511)^{2}$$
$71$ $$T^{2} - 508032$$
$73$ $$(T + 137)^{2}$$
$79$ $$(T + 475)^{2}$$
$83$ $$T^{2} - 332928$$
$89$ $$T^{2} - 64800$$
$97$ $$(T + 821)^{2}$$