Properties

 Label 1323.4.a.y Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.3576.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 15x + 3$$ x^3 - x^2 - 15*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - 2 \beta_1 + 6) q^{4} + (\beta_{2} + 2 \beta_1 + 1) q^{5} + ( - \beta_{2} + 8 \beta_1 - 28) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 - 2*b1 + 6) * q^4 + (b2 + 2*b1 + 1) * q^5 + (-b2 + 8*b1 - 28) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} - 2 \beta_1 + 6) q^{4} + (\beta_{2} + 2 \beta_1 + 1) q^{5} + ( - \beta_{2} + 8 \beta_1 - 28) q^{8} + (3 \beta_{2} + 3 \beta_1 + 28) q^{10} + ( - 2 \beta_{2} - 38) q^{11} + ( - 3 \beta_{2} + 6 \beta_1 - 4) q^{13} + ( - \beta_{2} - 34 \beta_1 + 64) q^{16} + ( - 10 \beta_{2} - 4 \beta_1 - 19) q^{17} + (4 \beta_{2} - 8 \beta_1 + 18) q^{19} + ( - 2 \beta_{2} + 24 \beta_1 + 34) q^{20} + ( - 2 \beta_{2} - 50 \beta_1) q^{22} + (5 \beta_{2} - 10 \beta_1 - 58) q^{23} + (5 \beta_{2} + 26 \beta_1 + 16) q^{25} + (3 \beta_{2} - 34 \beta_1 + 84) q^{26} + (3 \beta_{2} - 58 \beta_1 - 78) q^{29} + ( - 11 \beta_{2} - 50 \beta_1 + 124) q^{31} + ( - 27 \beta_{2} + 62 \beta_1 - 252) q^{32} + ( - 14 \beta_{2} - 71 \beta_1 - 56) q^{34} + (3 \beta_{2} + 30 \beta_1 + 189) q^{37} + ( - 4 \beta_{2} + 58 \beta_1 - 112) q^{38} + ( - 2 \beta_{2} - 50 \beta_1 + 112) q^{40} + ( - 19 \beta_{2} - 38 \beta_1 + 143) q^{41} + ( - 28 \beta_{2} - 16 \beta_1 - 29) q^{43} + ( - 36 \beta_{2} + 88 \beta_1 - 396) q^{44} + ( - 5 \beta_{2} - 8 \beta_1 - 140) q^{46} + (21 \beta_{2} - 6 \beta_1 + 255) q^{47} + (31 \beta_{2} - 6 \beta_1 + 364) q^{50} + ( - 7 \beta_{2} + 122 \beta_1 - 444) q^{52} + ( - 43 \beta_{2} + 26 \beta_1 + 38) q^{53} + ( - 34 \beta_{2} - 112 \beta_1 - 206) q^{55} + ( - 55 \beta_{2} + 56 \beta_1 - 812) q^{58} + (2 \beta_{2} + 44 \beta_1 + 83) q^{59} + (40 \beta_{2} - 116 \beta_1 - 90) q^{61} + ( - 61 \beta_{2} + 158 \beta_1 - 700) q^{62} + (43 \beta_{2} - 266 \beta_1 + 356) q^{64} + (20 \beta_{2} - 44 \beta_1 - 88) q^{65} + (5 \beta_{2} - 46 \beta_1 - 240) q^{67} + ( - 5 \beta_{2} + 34 \beta_1 - 842) q^{68} + (25 \beta_{2} + 130 \beta_1 - 308) q^{71} + (20 \beta_{2} + 140 \beta_1 + 134) q^{73} + (33 \beta_{2} + 147 \beta_1 + 420) q^{74} + (22 \beta_{2} - 188 \beta_1 + 668) q^{76} + (41 \beta_{2} + 170 \beta_1 + 271) q^{79} + ( - 36 \beta_{2} + 8 \beta_1 - 972) q^{80} + ( - 57 \beta_{2} + 105 \beta_1 - 532) q^{82} + (41 \beta_{2} + 122 \beta_1 + 221) q^{83} + ( - 11 \beta_{2} - 230 \beta_1 - 971) q^{85} + ( - 44 \beta_{2} - 165 \beta_1 - 224) q^{86} + (68 \beta_{2} - 388 \beta_1 + 1232) q^{88} + (91 \beta_{2} - 58 \beta_1 - 26) q^{89} + ( - 53 \beta_{2} - 74 \beta_1 + 352) q^{92} + (15 \beta_{2} + 393 \beta_1 - 84) q^{94} + ( - 14 \beta_{2} + 84 \beta_1 + 130) q^{95} + (110 \beta_{2} + 68 \beta_1 - 178) q^{97}+O(q^{100})$$ q + b1 * q^2 + (b2 - 2*b1 + 6) * q^4 + (b2 + 2*b1 + 1) * q^5 + (-b2 + 8*b1 - 28) * q^8 + (3*b2 + 3*b1 + 28) * q^10 + (-2*b2 - 38) * q^11 + (-3*b2 + 6*b1 - 4) * q^13 + (-b2 - 34*b1 + 64) * q^16 + (-10*b2 - 4*b1 - 19) * q^17 + (4*b2 - 8*b1 + 18) * q^19 + (-2*b2 + 24*b1 + 34) * q^20 + (-2*b2 - 50*b1) * q^22 + (5*b2 - 10*b1 - 58) * q^23 + (5*b2 + 26*b1 + 16) * q^25 + (3*b2 - 34*b1 + 84) * q^26 + (3*b2 - 58*b1 - 78) * q^29 + (-11*b2 - 50*b1 + 124) * q^31 + (-27*b2 + 62*b1 - 252) * q^32 + (-14*b2 - 71*b1 - 56) * q^34 + (3*b2 + 30*b1 + 189) * q^37 + (-4*b2 + 58*b1 - 112) * q^38 + (-2*b2 - 50*b1 + 112) * q^40 + (-19*b2 - 38*b1 + 143) * q^41 + (-28*b2 - 16*b1 - 29) * q^43 + (-36*b2 + 88*b1 - 396) * q^44 + (-5*b2 - 8*b1 - 140) * q^46 + (21*b2 - 6*b1 + 255) * q^47 + (31*b2 - 6*b1 + 364) * q^50 + (-7*b2 + 122*b1 - 444) * q^52 + (-43*b2 + 26*b1 + 38) * q^53 + (-34*b2 - 112*b1 - 206) * q^55 + (-55*b2 + 56*b1 - 812) * q^58 + (2*b2 + 44*b1 + 83) * q^59 + (40*b2 - 116*b1 - 90) * q^61 + (-61*b2 + 158*b1 - 700) * q^62 + (43*b2 - 266*b1 + 356) * q^64 + (20*b2 - 44*b1 - 88) * q^65 + (5*b2 - 46*b1 - 240) * q^67 + (-5*b2 + 34*b1 - 842) * q^68 + (25*b2 + 130*b1 - 308) * q^71 + (20*b2 + 140*b1 + 134) * q^73 + (33*b2 + 147*b1 + 420) * q^74 + (22*b2 - 188*b1 + 668) * q^76 + (41*b2 + 170*b1 + 271) * q^79 + (-36*b2 + 8*b1 - 972) * q^80 + (-57*b2 + 105*b1 - 532) * q^82 + (41*b2 + 122*b1 + 221) * q^83 + (-11*b2 - 230*b1 - 971) * q^85 + (-44*b2 - 165*b1 - 224) * q^86 + (68*b2 - 388*b1 + 1232) * q^88 + (91*b2 - 58*b1 - 26) * q^89 + (-53*b2 - 74*b1 + 352) * q^92 + (15*b2 + 393*b1 - 84) * q^94 + (-14*b2 + 84*b1 + 130) * q^95 + (110*b2 + 68*b1 - 178) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 21 q^{4} + 2 q^{5} - 93 q^{8}+O(q^{10})$$ 3 * q - q^2 + 21 * q^4 + 2 * q^5 - 93 * q^8 $$3 q - q^{2} + 21 q^{4} + 2 q^{5} - 93 q^{8} + 84 q^{10} - 116 q^{11} - 21 q^{13} + 225 q^{16} - 63 q^{17} + 66 q^{19} + 76 q^{20} + 48 q^{22} - 159 q^{23} + 27 q^{25} + 289 q^{26} - 173 q^{29} + 411 q^{31} - 845 q^{32} - 111 q^{34} + 540 q^{37} - 398 q^{38} + 384 q^{40} + 448 q^{41} - 99 q^{43} - 1312 q^{44} - 417 q^{46} + 792 q^{47} + 1129 q^{50} - 1461 q^{52} + 45 q^{53} - 540 q^{55} - 2547 q^{58} + 207 q^{59} - 114 q^{61} - 2319 q^{62} + 1377 q^{64} - 200 q^{65} - 669 q^{67} - 2565 q^{68} - 1029 q^{71} + 282 q^{73} + 1146 q^{74} + 2214 q^{76} + 684 q^{79} - 2960 q^{80} - 1758 q^{82} + 582 q^{83} - 2694 q^{85} - 551 q^{86} + 4152 q^{88} + 71 q^{89} + 1077 q^{92} - 630 q^{94} + 292 q^{95} - 492 q^{97}+O(q^{100})$$ 3 * q - q^2 + 21 * q^4 + 2 * q^5 - 93 * q^8 + 84 * q^10 - 116 * q^11 - 21 * q^13 + 225 * q^16 - 63 * q^17 + 66 * q^19 + 76 * q^20 + 48 * q^22 - 159 * q^23 + 27 * q^25 + 289 * q^26 - 173 * q^29 + 411 * q^31 - 845 * q^32 - 111 * q^34 + 540 * q^37 - 398 * q^38 + 384 * q^40 + 448 * q^41 - 99 * q^43 - 1312 * q^44 - 417 * q^46 + 792 * q^47 + 1129 * q^50 - 1461 * q^52 + 45 * q^53 - 540 * q^55 - 2547 * q^58 + 207 * q^59 - 114 * q^61 - 2319 * q^62 + 1377 * q^64 - 200 * q^65 - 669 * q^67 - 2565 * q^68 - 1029 * q^71 + 282 * q^73 + 1146 * q^74 + 2214 * q^76 + 684 * q^79 - 2960 * q^80 - 1758 * q^82 + 582 * q^83 - 2694 * q^85 - 551 * q^86 + 4152 * q^88 + 71 * q^89 + 1077 * q^92 - 630 * q^94 + 292 * q^95 - 492 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 15x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} - 11 ) / 2$$ (v^2 - 11) / 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{2} + 6\nu + 9 ) / 2$$ (-v^2 + 6*v + 9) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 3$$ (b2 + b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$2\beta _1 + 11$$ 2*b1 + 11

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
 0.197906 −3.51298 4.31507
−5.48042 0 22.0350 −4.88670 0 0 −76.9175 0 26.7812
1.2 0.670500 0 −7.55043 −9.86843 0 0 −10.4266 0 −6.61678
1.3 3.80992 0 6.51546 16.7551 0 0 −5.65596 0 63.8356
 $$n$$: 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.y 3
3.b odd 2 1 1323.4.a.z 3
7.b odd 2 1 189.4.a.j 3
21.c even 2 1 189.4.a.k yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.a.j 3 7.b odd 2 1
189.4.a.k yes 3 21.c even 2 1
1323.4.a.y 3 1.a even 1 1 trivial
1323.4.a.z 3 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2}^{3} + T_{2}^{2} - 22T_{2} + 14$$ T2^3 + T2^2 - 22*T2 + 14 $$T_{5}^{3} - 2T_{5}^{2} - 199T_{5} - 808$$ T5^3 - 2*T5^2 - 199*T5 - 808 $$T_{13}^{3} + 21T_{13}^{2} - 1824T_{13} - 10592$$ T13^3 + 21*T13^2 - 1824*T13 - 10592

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} + \cdots + 14$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 2 T^{2} + \cdots - 808$$
$7$ $$T^{3}$$
$11$ $$T^{3} + 116 T^{2} + \cdots + 35488$$
$13$ $$T^{3} + 21 T^{2} + \cdots - 10592$$
$17$ $$T^{3} + 63 T^{2} + \cdots - 555039$$
$19$ $$T^{3} - 66 T^{2} + \cdots + 59656$$
$23$ $$T^{3} + 159 T^{2} + \cdots - 154548$$
$29$ $$T^{3} + 173 T^{2} + \cdots - 10752188$$
$31$ $$T^{3} - 411 T^{2} + \cdots + 11998368$$
$37$ $$T^{3} - 540 T^{2} + \cdots - 2250234$$
$41$ $$T^{3} - 448 T^{2} + \cdots + 13925734$$
$43$ $$T^{3} + 99 T^{2} + \cdots - 8005583$$
$47$ $$T^{3} - 792 T^{2} + \cdots + 861678$$
$53$ $$T^{3} - 45 T^{2} + \cdots - 39848868$$
$59$ $$T^{3} - 207 T^{2} + \cdots + 3479247$$
$61$ $$T^{3} + 114 T^{2} + \cdots - 101463944$$
$67$ $$T^{3} + 669 T^{2} + \cdots - 4658224$$
$71$ $$T^{3} + 1029 T^{2} + \cdots - 183653568$$
$73$ $$T^{3} - 282 T^{2} + \cdots - 7203384$$
$79$ $$T^{3} - 684 T^{2} + \cdots - 65522918$$
$83$ $$T^{3} - 582 T^{2} + \cdots - 48300948$$
$89$ $$T^{3} - 71 T^{2} + \cdots + 437197844$$
$97$ $$T^{3} + 492 T^{2} + \cdots + 10780128$$