# Properties

 Label 1323.4.a.o Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

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

 $$f(q)$$ $$=$$ $$q + (\beta - 3) q^{2} + ( - 6 \beta + 4) q^{4} + ( - 4 \beta + 3) q^{5} + (14 \beta - 6) q^{8}+O(q^{10})$$ q + (b - 3) * q^2 + (-6*b + 4) * q^4 + (-4*b + 3) * q^5 + (14*b - 6) * q^8 $$q + (\beta - 3) q^{2} + ( - 6 \beta + 4) q^{4} + ( - 4 \beta + 3) q^{5} + (14 \beta - 6) q^{8} + (15 \beta - 21) q^{10} + ( - 16 \beta - 24) q^{11} + ( - 24 \beta - 26) q^{13} + 28 q^{16} + (28 \beta + 15) q^{17} + (60 \beta - 32) q^{19} + ( - 34 \beta + 84) q^{20} + (24 \beta + 24) q^{22} + ( - 68 \beta - 30) q^{23} + ( - 24 \beta - 68) q^{25} + (46 \beta + 6) q^{26} + (36 \beta - 180) q^{29} + ( - 72 \beta + 70) q^{31} + ( - 84 \beta - 36) q^{32} + ( - 69 \beta + 39) q^{34} + (72 \beta - 115) q^{37} + ( - 212 \beta + 276) q^{38} + (66 \beta - 186) q^{40} + ( - 204 \beta + 117) q^{41} + ( - 12 \beta - 469) q^{43} + (80 \beta + 192) q^{44} + (174 \beta - 114) q^{46} + (176 \beta + 309) q^{47} + (4 \beta + 132) q^{50} + (60 \beta + 328) q^{52} + ( - 304 \beta + 210) q^{53} + (48 \beta + 120) q^{55} + ( - 288 \beta + 648) q^{58} + ( - 80 \beta + 141) q^{59} + ( - 288 \beta + 16) q^{61} + (286 \beta - 426) q^{62} + (216 \beta - 368) q^{64} + (32 \beta + 210) q^{65} + (216 \beta + 272) q^{67} + (22 \beta - 444) q^{68} + (120 \beta + 252) q^{71} + ( - 300 \beta + 382) q^{73} + ( - 331 \beta + 561) q^{74} + (432 \beta - 1208) q^{76} + ( - 540 \beta + 119) q^{79} + ( - 112 \beta + 84) q^{80} + (729 \beta - 963) q^{82} + ( - 432 \beta - 261) q^{83} + (24 \beta - 291) q^{85} + ( - 433 \beta + 1371) q^{86} + ( - 240 \beta - 528) q^{88} + ( - 352 \beta + 354) q^{89} + ( - 92 \beta + 1104) q^{92} + ( - 219 \beta - 399) q^{94} + (308 \beta - 816) q^{95} + (228 \beta - 332) q^{97}+O(q^{100})$$ q + (b - 3) * q^2 + (-6*b + 4) * q^4 + (-4*b + 3) * q^5 + (14*b - 6) * q^8 + (15*b - 21) * q^10 + (-16*b - 24) * q^11 + (-24*b - 26) * q^13 + 28 * q^16 + (28*b + 15) * q^17 + (60*b - 32) * q^19 + (-34*b + 84) * q^20 + (24*b + 24) * q^22 + (-68*b - 30) * q^23 + (-24*b - 68) * q^25 + (46*b + 6) * q^26 + (36*b - 180) * q^29 + (-72*b + 70) * q^31 + (-84*b - 36) * q^32 + (-69*b + 39) * q^34 + (72*b - 115) * q^37 + (-212*b + 276) * q^38 + (66*b - 186) * q^40 + (-204*b + 117) * q^41 + (-12*b - 469) * q^43 + (80*b + 192) * q^44 + (174*b - 114) * q^46 + (176*b + 309) * q^47 + (4*b + 132) * q^50 + (60*b + 328) * q^52 + (-304*b + 210) * q^53 + (48*b + 120) * q^55 + (-288*b + 648) * q^58 + (-80*b + 141) * q^59 + (-288*b + 16) * q^61 + (286*b - 426) * q^62 + (216*b - 368) * q^64 + (32*b + 210) * q^65 + (216*b + 272) * q^67 + (22*b - 444) * q^68 + (120*b + 252) * q^71 + (-300*b + 382) * q^73 + (-331*b + 561) * q^74 + (432*b - 1208) * q^76 + (-540*b + 119) * q^79 + (-112*b + 84) * q^80 + (729*b - 963) * q^82 + (-432*b - 261) * q^83 + (24*b - 291) * q^85 + (-433*b + 1371) * q^86 + (-240*b - 528) * q^88 + (-352*b + 354) * q^89 + (-92*b + 1104) * q^92 + (-219*b - 399) * q^94 + (308*b - 816) * q^95 + (228*b - 332) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{2} + 8 q^{4} + 6 q^{5} - 12 q^{8}+O(q^{10})$$ 2 * q - 6 * q^2 + 8 * q^4 + 6 * q^5 - 12 * q^8 $$2 q - 6 q^{2} + 8 q^{4} + 6 q^{5} - 12 q^{8} - 42 q^{10} - 48 q^{11} - 52 q^{13} + 56 q^{16} + 30 q^{17} - 64 q^{19} + 168 q^{20} + 48 q^{22} - 60 q^{23} - 136 q^{25} + 12 q^{26} - 360 q^{29} + 140 q^{31} - 72 q^{32} + 78 q^{34} - 230 q^{37} + 552 q^{38} - 372 q^{40} + 234 q^{41} - 938 q^{43} + 384 q^{44} - 228 q^{46} + 618 q^{47} + 264 q^{50} + 656 q^{52} + 420 q^{53} + 240 q^{55} + 1296 q^{58} + 282 q^{59} + 32 q^{61} - 852 q^{62} - 736 q^{64} + 420 q^{65} + 544 q^{67} - 888 q^{68} + 504 q^{71} + 764 q^{73} + 1122 q^{74} - 2416 q^{76} + 238 q^{79} + 168 q^{80} - 1926 q^{82} - 522 q^{83} - 582 q^{85} + 2742 q^{86} - 1056 q^{88} + 708 q^{89} + 2208 q^{92} - 798 q^{94} - 1632 q^{95} - 664 q^{97}+O(q^{100})$$ 2 * q - 6 * q^2 + 8 * q^4 + 6 * q^5 - 12 * q^8 - 42 * q^10 - 48 * q^11 - 52 * q^13 + 56 * q^16 + 30 * q^17 - 64 * q^19 + 168 * q^20 + 48 * q^22 - 60 * q^23 - 136 * q^25 + 12 * q^26 - 360 * q^29 + 140 * q^31 - 72 * q^32 + 78 * q^34 - 230 * q^37 + 552 * q^38 - 372 * q^40 + 234 * q^41 - 938 * q^43 + 384 * q^44 - 228 * q^46 + 618 * q^47 + 264 * q^50 + 656 * q^52 + 420 * q^53 + 240 * q^55 + 1296 * q^58 + 282 * q^59 + 32 * q^61 - 852 * q^62 - 736 * q^64 + 420 * q^65 + 544 * q^67 - 888 * q^68 + 504 * q^71 + 764 * q^73 + 1122 * q^74 - 2416 * q^76 + 238 * q^79 + 168 * q^80 - 1926 * q^82 - 522 * q^83 - 582 * q^85 + 2742 * q^86 - 1056 * q^88 + 708 * q^89 + 2208 * q^92 - 798 * q^94 - 1632 * q^95 - 664 * 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
 −1.73205 1.73205
−4.73205 0 14.3923 9.92820 0 0 −30.2487 0 −46.9808
1.2 −1.26795 0 −6.39230 −3.92820 0 0 18.2487 0 4.98076
 $$n$$: 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.o 2
3.b odd 2 1 1323.4.a.x 2
7.b odd 2 1 189.4.a.e 2
21.c even 2 1 189.4.a.i yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.a.e 2 7.b odd 2 1
189.4.a.i yes 2 21.c even 2 1
1323.4.a.o 2 1.a even 1 1 trivial
1323.4.a.x 2 3.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 6T_{2} + 6$$ T2^2 + 6*T2 + 6 $$T_{5}^{2} - 6T_{5} - 39$$ T5^2 - 6*T5 - 39 $$T_{13}^{2} + 52T_{13} - 1052$$ T13^2 + 52*T13 - 1052

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 6T + 6$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6T - 39$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 48T - 192$$
$13$ $$T^{2} + 52T - 1052$$
$17$ $$T^{2} - 30T - 2127$$
$19$ $$T^{2} + 64T - 9776$$
$23$ $$T^{2} + 60T - 12972$$
$29$ $$T^{2} + 360T + 28512$$
$31$ $$T^{2} - 140T - 10652$$
$37$ $$T^{2} + 230T - 2327$$
$41$ $$T^{2} - 234T - 111159$$
$43$ $$T^{2} + 938T + 219529$$
$47$ $$T^{2} - 618T + 2553$$
$53$ $$T^{2} - 420T - 233148$$
$59$ $$T^{2} - 282T + 681$$
$61$ $$T^{2} - 32T - 248576$$
$67$ $$T^{2} - 544T - 65984$$
$71$ $$T^{2} - 504T + 20304$$
$73$ $$T^{2} - 764T - 124076$$
$79$ $$T^{2} - 238T - 860639$$
$83$ $$T^{2} + 522T - 491751$$
$89$ $$T^{2} - 708T - 246396$$
$97$ $$T^{2} + 664T - 45728$$