# Properties

 Label 1323.4.a.be Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $6$ 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: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 40x^{4} + 453x^{2} - 1278$$ x^6 - 40*x^4 + 453*x^2 - 1278 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} + 5) q^{4} - \beta_{4} q^{5} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 + 5) * q^4 - b4 * q^5 + (-b4 + b2 + b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} + 5) q^{4} - \beta_{4} q^{5} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{8} + ( - 3 \beta_{5} + \beta_{3} + 4) q^{10} + (\beta_{4} - 2 \beta_{2} - 4 \beta_1) q^{11} + (5 \beta_{5} + 2 \beta_{3} - 11) q^{13} + (4 \beta_{5} + 3 \beta_{3} - 27) q^{16} + (5 \beta_{4} - 4 \beta_1) q^{17} + ( - 11 \beta_{3} + 14) q^{19} + (\beta_{4} - 2 \beta_{2} + 8 \beta_1) q^{20} + ( - 11 \beta_{5} - 23 \beta_{3} - 48) q^{22} + (6 \beta_{4} + 4 \beta_{2} - 10 \beta_1) q^{23} + ( - 2 \beta_{5} - 12 \beta_{3} - 3) q^{25} + (8 \beta_{4} + 7 \beta_{2} - 3 \beta_1) q^{26} + ( - 9 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{29} + ( - 11 \beta_{5} + 13 \beta_{3} + 13) q^{31} + (13 \beta_{4} - \beta_{2} - 23 \beta_1) q^{32} + (15 \beta_{5} - 9 \beta_{3} - 72) q^{34} + (\beta_{5} - 12 \beta_{3} - 185) q^{37} + (11 \beta_{4} - 11 \beta_{2} - 30 \beta_1) q^{38} + (13 \beta_{5} - 19 \beta_{3} + 76) q^{40} + (15 \beta_{4} + 4 \beta_{2} + 52 \beta_1) q^{41} + (8 \beta_{5} - 2 \beta_{3} - 263) q^{43} + ( - 7 \beta_{4} - 18 \beta_{2} - 108 \beta_1) q^{44} + (46 \beta_{5} + 20 \beta_{3} - 170) q^{46} + (29 \beta_{4} - 14 \beta_{2} - 26 \beta_1) q^{47} + (8 \beta_{4} - 14 \beta_{2} - 51 \beta_1) q^{50} + (33 \beta_{5} + 36 \beta_{3} - 11) q^{52} + ( - 8 \beta_{4} + 12 \beta_{2} - 26 \beta_1) q^{53} + ( - 22 \beta_{5} + 24 \beta_{3} - 38) q^{55} + ( - 41 \beta_{5} - 11 \beta_{3} + 18) q^{58} + ( - 23 \beta_{4} + 26 \beta_{2} - 40 \beta_1) q^{59} + ( - 63 \beta_{5} + \beta_{3} - 127) q^{61} + ( - 35 \beta_{4} + 2 \beta_{2} + 65 \beta_1) q^{62} + ( - 69 \beta_{3} - 131) q^{64} + (13 \beta_{4} + 16 \beta_{2} - 94 \beta_1) q^{65} + ( - 54 \beta_{5} - 19 \beta_{3} - 323) q^{67} + ( - \beta_{4} + 6 \beta_{2} - 76 \beta_1) q^{68} + ( - 51 \beta_{4} + 18 \beta_{2} - 12 \beta_1) q^{71} + ( - 20 \beta_{5} + 55 \beta_{3} + 388) q^{73} + (14 \beta_{4} - 11 \beta_{2} - 233 \beta_1) q^{74} + ( - 44 \beta_{5} - 52 \beta_{3} - 502) q^{76} + (61 \beta_{5} - 64 \beta_{3} + 165) q^{79} + (37 \beta_{4} + 10 \beta_{2} - 64 \beta_1) q^{80} + (73 \beta_{5} + 73 \beta_{3} + 600) q^{82} + (28 \beta_{4} - 14 \beta_{2} + 318 \beta_1) q^{83} + (22 \beta_{5} + 56 \beta_{3} - 626) q^{85} + (18 \beta_{4} + 6 \beta_{2} - 271 \beta_1) q^{86} + ( - 59 \beta_{5} - 79 \beta_{3} - 920) q^{88} + ( - 51 \beta_{4} - 26 \beta_{2} - 174 \beta_1) q^{89} + (24 \beta_{4} + 34 \beta_{2} - 10 \beta_1) q^{92} + ( - 11 \beta_{5} - 181 \beta_{3} - 398) q^{94} + ( - 80 \beta_{4} + 22 \beta_{2} - 88 \beta_1) q^{95} + ( - 72 \beta_{5} - 154 \beta_{3} + 189) q^{97}+O(q^{100})$$ q + b1 * q^2 + (b3 + 5) * q^4 - b4 * q^5 + (-b4 + b2 + b1) * q^8 + (-3*b5 + b3 + 4) * q^10 + (b4 - 2*b2 - 4*b1) * q^11 + (5*b5 + 2*b3 - 11) * q^13 + (4*b5 + 3*b3 - 27) * q^16 + (5*b4 - 4*b1) * q^17 + (-11*b3 + 14) * q^19 + (b4 - 2*b2 + 8*b1) * q^20 + (-11*b5 - 23*b3 - 48) * q^22 + (6*b4 + 4*b2 - 10*b1) * q^23 + (-2*b5 - 12*b3 - 3) * q^25 + (8*b4 + 7*b2 - 3*b1) * q^26 + (-9*b4 - 2*b2 - 2*b1) * q^29 + (-11*b5 + 13*b3 + 13) * q^31 + (13*b4 - b2 - 23*b1) * q^32 + (15*b5 - 9*b3 - 72) * q^34 + (b5 - 12*b3 - 185) * q^37 + (11*b4 - 11*b2 - 30*b1) * q^38 + (13*b5 - 19*b3 + 76) * q^40 + (15*b4 + 4*b2 + 52*b1) * q^41 + (8*b5 - 2*b3 - 263) * q^43 + (-7*b4 - 18*b2 - 108*b1) * q^44 + (46*b5 + 20*b3 - 170) * q^46 + (29*b4 - 14*b2 - 26*b1) * q^47 + (8*b4 - 14*b2 - 51*b1) * q^50 + (33*b5 + 36*b3 - 11) * q^52 + (-8*b4 + 12*b2 - 26*b1) * q^53 + (-22*b5 + 24*b3 - 38) * q^55 + (-41*b5 - 11*b3 + 18) * q^58 + (-23*b4 + 26*b2 - 40*b1) * q^59 + (-63*b5 + b3 - 127) * q^61 + (-35*b4 + 2*b2 + 65*b1) * q^62 + (-69*b3 - 131) * q^64 + (13*b4 + 16*b2 - 94*b1) * q^65 + (-54*b5 - 19*b3 - 323) * q^67 + (-b4 + 6*b2 - 76*b1) * q^68 + (-51*b4 + 18*b2 - 12*b1) * q^71 + (-20*b5 + 55*b3 + 388) * q^73 + (14*b4 - 11*b2 - 233*b1) * q^74 + (-44*b5 - 52*b3 - 502) * q^76 + (61*b5 - 64*b3 + 165) * q^79 + (37*b4 + 10*b2 - 64*b1) * q^80 + (73*b5 + 73*b3 + 600) * q^82 + (28*b4 - 14*b2 + 318*b1) * q^83 + (22*b5 + 56*b3 - 626) * q^85 + (18*b4 + 6*b2 - 271*b1) * q^86 + (-59*b5 - 79*b3 - 920) * q^88 + (-51*b4 - 26*b2 - 174*b1) * q^89 + (24*b4 + 34*b2 - 10*b1) * q^92 + (-11*b5 - 181*b3 - 398) * q^94 + (-80*b4 + 22*b2 - 88*b1) * q^95 + (-72*b5 - 154*b3 + 189) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 32 q^{4}+O(q^{10})$$ 6 * q + 32 * q^4 $$6 q + 32 q^{4} + 20 q^{10} - 52 q^{13} - 148 q^{16} + 62 q^{19} - 356 q^{22} - 46 q^{25} + 82 q^{31} - 420 q^{34} - 1132 q^{37} + 444 q^{40} - 1566 q^{43} - 888 q^{46} + 72 q^{52} - 224 q^{55} + 4 q^{58} - 886 q^{61} - 924 q^{64} - 2084 q^{67} + 2398 q^{73} - 3204 q^{76} + 984 q^{79} + 3892 q^{82} - 3600 q^{85} - 5796 q^{88} - 2772 q^{94} + 682 q^{97}+O(q^{100})$$ 6 * q + 32 * q^4 + 20 * q^10 - 52 * q^13 - 148 * q^16 + 62 * q^19 - 356 * q^22 - 46 * q^25 + 82 * q^31 - 420 * q^34 - 1132 * q^37 + 444 * q^40 - 1566 * q^43 - 888 * q^46 + 72 * q^52 - 224 * q^55 + 4 * q^58 - 886 * q^61 - 924 * q^64 - 2084 * q^67 + 2398 * q^73 - 3204 * q^76 + 984 * q^79 + 3892 * q^82 - 3600 * q^85 - 5796 * q^88 - 2772 * q^94 + 682 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 40x^{4} + 453x^{2} - 1278$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 19\nu^{3} - 6\nu ) / 12$$ (v^5 - 19*v^3 - 6*v) / 12 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 13$$ v^2 - 13 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 31\nu^{3} + 198\nu ) / 12$$ (v^5 - 31*v^3 + 198*v) / 12 $$\beta_{5}$$ $$=$$ $$( \nu^{4} - 27\nu^{2} + 130 ) / 4$$ (v^4 - 27*v^2 + 130) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 13$$ b3 + 13 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{2} + 17\beta_1$$ -b4 + b2 + 17*b1 $$\nu^{4}$$ $$=$$ $$4\beta_{5} + 27\beta_{3} + 221$$ 4*b5 + 27*b3 + 221 $$\nu^{5}$$ $$=$$ $$-19\beta_{4} + 31\beta_{2} + 329\beta_1$$ -19*b4 + 31*b2 + 329*b1

## 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
 −4.70753 −3.68757 −2.05936 2.05936 3.68757 4.70753
−4.70753 0 14.1608 0.830453 0 0 −29.0024 0 −3.90938
1.2 −3.68757 0 5.59820 −11.8719 0 0 8.85681 0 43.7785
1.3 −2.05936 0 −3.75905 14.5041 0 0 24.2161 0 −29.8691
1.4 2.05936 0 −3.75905 −14.5041 0 0 −24.2161 0 −29.8691
1.5 3.68757 0 5.59820 11.8719 0 0 −8.85681 0 43.7785
1.6 4.70753 0 14.1608 −0.830453 0 0 29.0024 0 −3.90938
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.be 6
3.b odd 2 1 inner 1323.4.a.be 6
7.b odd 2 1 1323.4.a.bd 6
7.d odd 6 2 189.4.e.e 12
21.c even 2 1 1323.4.a.bd 6
21.g even 6 2 189.4.e.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.e 12 7.d odd 6 2
189.4.e.e 12 21.g even 6 2
1323.4.a.bd 6 7.b odd 2 1
1323.4.a.bd 6 21.c even 2 1
1323.4.a.be 6 1.a even 1 1 trivial
1323.4.a.be 6 3.b odd 2 1 inner

## Hecke kernels

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

 $$T_{2}^{6} - 40T_{2}^{4} + 453T_{2}^{2} - 1278$$ T2^6 - 40*T2^4 + 453*T2^2 - 1278 $$T_{5}^{6} - 352T_{5}^{4} + 29892T_{5}^{2} - 20448$$ T5^6 - 352*T5^4 + 29892*T5^2 - 20448 $$T_{13}^{3} + 26T_{13}^{2} - 3211T_{13} + 35818$$ T13^3 + 26*T13^2 - 3211*T13 + 35818

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 40 T^{4} + \cdots - 1278$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 352 T^{4} + \cdots - 20448$$
$7$ $$T^{6}$$
$11$ $$T^{6} + \cdots - 2041855488$$
$13$ $$(T^{3} + 26 T^{2} + \cdots + 35818)^{2}$$
$17$ $$T^{6} + \cdots - 4889546208$$
$19$ $$(T^{3} - 31 T^{2} + \cdots + 71044)^{2}$$
$23$ $$T^{6} + \cdots - 54061976448$$
$29$ $$T^{6} + \cdots - 1224041183712$$
$31$ $$(T^{3} - 41 T^{2} + \cdots + 2204979)^{2}$$
$37$ $$(T^{3} + 566 T^{2} + \cdots + 4245738)^{2}$$
$41$ $$T^{6} + \cdots - 69555350983392$$
$43$ $$(T^{3} + 783 T^{2} + \cdots + 15519697)^{2}$$
$47$ $$T^{6} + \cdots - 554088448396512$$
$53$ $$T^{6} + \cdots - 248222191554432$$
$59$ $$T^{6} + \cdots - 26\!\cdots\!72$$
$61$ $$(T^{3} + 443 T^{2} + \cdots - 219938513)^{2}$$
$67$ $$(T^{3} + 1042 T^{2} + \cdots - 181536376)^{2}$$
$71$ $$T^{6} + \cdots - 17\!\cdots\!28$$
$73$ $$(T^{3} - 1199 T^{2} + \cdots + 138593088)^{2}$$
$79$ $$(T^{3} - 492 T^{2} + \cdots - 61449092)^{2}$$
$83$ $$T^{6} + \cdots - 28\!\cdots\!92$$
$89$ $$T^{6} + \cdots - 51\!\cdots\!68$$
$97$ $$(T^{3} - 341 T^{2} + \cdots + 1582590381)^{2}$$