# Properties

 Label 1323.4.a.p Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

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: $$2$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \beta - 1) q^{2} + ( - 4 \beta + 1) q^{4} + (\beta - 11) q^{5} + ( - 10 \beta - 9) q^{8}+O(q^{10})$$ q + (2*b - 1) * q^2 + (-4*b + 1) * q^4 + (b - 11) * q^5 + (-10*b - 9) * q^8 $$q + (2 \beta - 1) q^{2} + ( - 4 \beta + 1) q^{4} + (\beta - 11) q^{5} + ( - 10 \beta - 9) q^{8} + ( - 23 \beta + 15) q^{10} + ( - 6 \beta + 31) q^{11} + (17 \beta + 32) q^{13} + (24 \beta - 39) q^{16} + ( - 18 \beta + 16) q^{17} + (22 \beta - 81) q^{19} + (45 \beta - 19) q^{20} + (68 \beta - 55) q^{22} + (61 \beta + 85) q^{23} + ( - 22 \beta - 2) q^{25} + (47 \beta + 36) q^{26} + ( - 19 \beta - 64) q^{29} + (119 \beta - 89) q^{31} + ( - 22 \beta + 207) q^{32} + (50 \beta - 88) q^{34} + ( - 33 \beta - 175) q^{37} + ( - 184 \beta + 169) q^{38} + (101 \beta + 79) q^{40} + (72 \beta - 29) q^{41} + ( - 82 \beta + 378) q^{43} + ( - 130 \beta + 79) q^{44} + (109 \beta + 159) q^{46} + ( - 236 \beta + 90) q^{47} + (18 \beta - 86) q^{50} + ( - 111 \beta - 104) q^{52} + ( - 24 \beta + 544) q^{53} + (97 \beta - 353) q^{55} + ( - 109 \beta - 12) q^{58} + ( - 388 \beta - 254) q^{59} + (6 \beta + 76) q^{61} + ( - 297 \beta + 565) q^{62} + (244 \beta + 17) q^{64} + ( - 155 \beta - 318) q^{65} + ( - 533 \beta + 234) q^{67} + ( - 82 \beta + 160) q^{68} + (511 \beta + 7) q^{71} + ( - 38 \beta - 394) q^{73} + ( - 317 \beta + 43) q^{74} + (346 \beta - 257) q^{76} + ( - 226 \beta - 238) q^{79} + ( - 303 \beta + 477) q^{80} + ( - 130 \beta + 317) q^{82} + (71 \beta - 746) q^{83} + (214 \beta - 212) q^{85} + (838 \beta - 706) q^{86} + ( - 256 \beta - 159) q^{88} + ( - 846 \beta + 115) q^{89} + ( - 279 \beta - 403) q^{92} + (416 \beta - 1034) q^{94} + ( - 323 \beta + 935) q^{95} + ( - 280 \beta + 320) q^{97}+O(q^{100})$$ q + (2*b - 1) * q^2 + (-4*b + 1) * q^4 + (b - 11) * q^5 + (-10*b - 9) * q^8 + (-23*b + 15) * q^10 + (-6*b + 31) * q^11 + (17*b + 32) * q^13 + (24*b - 39) * q^16 + (-18*b + 16) * q^17 + (22*b - 81) * q^19 + (45*b - 19) * q^20 + (68*b - 55) * q^22 + (61*b + 85) * q^23 + (-22*b - 2) * q^25 + (47*b + 36) * q^26 + (-19*b - 64) * q^29 + (119*b - 89) * q^31 + (-22*b + 207) * q^32 + (50*b - 88) * q^34 + (-33*b - 175) * q^37 + (-184*b + 169) * q^38 + (101*b + 79) * q^40 + (72*b - 29) * q^41 + (-82*b + 378) * q^43 + (-130*b + 79) * q^44 + (109*b + 159) * q^46 + (-236*b + 90) * q^47 + (18*b - 86) * q^50 + (-111*b - 104) * q^52 + (-24*b + 544) * q^53 + (97*b - 353) * q^55 + (-109*b - 12) * q^58 + (-388*b - 254) * q^59 + (6*b + 76) * q^61 + (-297*b + 565) * q^62 + (244*b + 17) * q^64 + (-155*b - 318) * q^65 + (-533*b + 234) * q^67 + (-82*b + 160) * q^68 + (511*b + 7) * q^71 + (-38*b - 394) * q^73 + (-317*b + 43) * q^74 + (346*b - 257) * q^76 + (-226*b - 238) * q^79 + (-303*b + 477) * q^80 + (-130*b + 317) * q^82 + (71*b - 746) * q^83 + (214*b - 212) * q^85 + (838*b - 706) * q^86 + (-256*b - 159) * q^88 + (-846*b + 115) * q^89 + (-279*b - 403) * q^92 + (416*b - 1034) * q^94 + (-323*b + 935) * q^95 + (-280*b + 320) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 22 q^{5} - 18 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 22 * q^5 - 18 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 22 q^{5} - 18 q^{8} + 30 q^{10} + 62 q^{11} + 64 q^{13} - 78 q^{16} + 32 q^{17} - 162 q^{19} - 38 q^{20} - 110 q^{22} + 170 q^{23} - 4 q^{25} + 72 q^{26} - 128 q^{29} - 178 q^{31} + 414 q^{32} - 176 q^{34} - 350 q^{37} + 338 q^{38} + 158 q^{40} - 58 q^{41} + 756 q^{43} + 158 q^{44} + 318 q^{46} + 180 q^{47} - 172 q^{50} - 208 q^{52} + 1088 q^{53} - 706 q^{55} - 24 q^{58} - 508 q^{59} + 152 q^{61} + 1130 q^{62} + 34 q^{64} - 636 q^{65} + 468 q^{67} + 320 q^{68} + 14 q^{71} - 788 q^{73} + 86 q^{74} - 514 q^{76} - 476 q^{79} + 954 q^{80} + 634 q^{82} - 1492 q^{83} - 424 q^{85} - 1412 q^{86} - 318 q^{88} + 230 q^{89} - 806 q^{92} - 2068 q^{94} + 1870 q^{95} + 640 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 22 * q^5 - 18 * q^8 + 30 * q^10 + 62 * q^11 + 64 * q^13 - 78 * q^16 + 32 * q^17 - 162 * q^19 - 38 * q^20 - 110 * q^22 + 170 * q^23 - 4 * q^25 + 72 * q^26 - 128 * q^29 - 178 * q^31 + 414 * q^32 - 176 * q^34 - 350 * q^37 + 338 * q^38 + 158 * q^40 - 58 * q^41 + 756 * q^43 + 158 * q^44 + 318 * q^46 + 180 * q^47 - 172 * q^50 - 208 * q^52 + 1088 * q^53 - 706 * q^55 - 24 * q^58 - 508 * q^59 + 152 * q^61 + 1130 * q^62 + 34 * q^64 - 636 * q^65 + 468 * q^67 + 320 * q^68 + 14 * q^71 - 788 * q^73 + 86 * q^74 - 514 * q^76 - 476 * q^79 + 954 * q^80 + 634 * q^82 - 1492 * q^83 - 424 * q^85 - 1412 * q^86 - 318 * q^88 + 230 * q^89 - 806 * q^92 - 2068 * q^94 + 1870 * q^95 + 640 * 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
−3.82843 0 6.65685 −12.4142 0 0 5.14214 0 47.5269
1.2 1.82843 0 −4.65685 −9.58579 0 0 −23.1421 0 −17.5269
 $$n$$: 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.p 2
3.b odd 2 1 1323.4.a.w yes 2
7.b odd 2 1 1323.4.a.q yes 2
21.c even 2 1 1323.4.a.v yes 2

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

 $$T_{2}^{2} + 2T_{2} - 7$$ T2^2 + 2*T2 - 7 $$T_{5}^{2} + 22T_{5} + 119$$ T5^2 + 22*T5 + 119 $$T_{13}^{2} - 64T_{13} + 446$$ T13^2 - 64*T13 + 446

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 7$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 22T + 119$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 62T + 889$$
$13$ $$T^{2} - 64T + 446$$
$17$ $$T^{2} - 32T - 392$$
$19$ $$T^{2} + 162T + 5593$$
$23$ $$T^{2} - 170T - 217$$
$29$ $$T^{2} + 128T + 3374$$
$31$ $$T^{2} + 178T - 20401$$
$37$ $$T^{2} + 350T + 28447$$
$41$ $$T^{2} + 58T - 9527$$
$43$ $$T^{2} - 756T + 129436$$
$47$ $$T^{2} - 180T - 103292$$
$53$ $$T^{2} - 1088 T + 294784$$
$59$ $$T^{2} + 508T - 236572$$
$61$ $$T^{2} - 152T + 5704$$
$67$ $$T^{2} - 468T - 513422$$
$71$ $$T^{2} - 14T - 522193$$
$73$ $$T^{2} + 788T + 152348$$
$79$ $$T^{2} + 476T - 45508$$
$83$ $$T^{2} + 1492 T + 546434$$
$89$ $$T^{2} - 230 T - 1418207$$
$97$ $$T^{2} - 640T - 54400$$
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