# Properties

 Label 1323.4.a.bf Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $6$ Inner twists $2$

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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: $$x^{6} - 42x^{4} + 369x^{2} - 112$$ x^6 - 42*x^4 + 369*x^2 - 112 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 3^{4}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 6) q^{4} + ( - \beta_{3} - 2 \beta_1) q^{5} + (\beta_{5} + 2 \beta_{3} + 8 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 6) * q^4 + (-b3 - 2*b1) * q^5 + (b5 + 2*b3 + 8*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 6) q^{4} + ( - \beta_{3} - 2 \beta_1) q^{5} + (\beta_{5} + 2 \beta_{3} + 8 \beta_1) q^{8} + ( - \beta_{4} - 4 \beta_{2} - 30) q^{10} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{4} - \beta_{2} - 18) q^{13} + (3 \beta_{4} + 8 \beta_{2} + 70) q^{16} + (3 \beta_{5} + 2 \beta_{3} + 7 \beta_1) q^{17} + (\beta_{4} - 5 \beta_{2} - 33) q^{19} + ( - 6 \beta_{5} - 8 \beta_{3} - 49 \beta_1) q^{20} + ( - 3 \beta_{2} - 14) q^{22} + ( - \beta_{5} - 5 \beta_{3} - 15 \beta_1) q^{23} + (3 \beta_{4} + 9 \beta_{2} + 70) q^{25} + ( - 3 \beta_{5} - 10 \beta_{3} - 23 \beta_1) q^{26} + ( - 7 \beta_{5} + 4 \beta_{3} - 21 \beta_1) q^{29} + (5 \beta_{4} - 7 \beta_{2} + 15) q^{31} + (6 \beta_{5} + 24 \beta_{3} + 71 \beta_1) q^{32} + (5 \beta_{4} + 23 \beta_{2} + 108) q^{34} + ( - 6 \beta_{4} - 4 \beta_{2} - 67) q^{37} + ( - 3 \beta_{5} - 2 \beta_{3} - 88 \beta_1) q^{38} + ( - 6 \beta_{4} - 57 \beta_{2} - 474) q^{40} + (6 \beta_{5} - 3 \beta_{3} - 54 \beta_1) q^{41} + ( - 3 \beta_{4} - 11 \beta_{2} - 110) q^{43} + (5 \beta_{5} - 14 \beta_{3} - 36 \beta_1) q^{44} + ( - 6 \beta_{4} - 29 \beta_{2} - 222) q^{46} + ( - 3 \beta_{5} + 4 \beta_{3} + 59 \beta_1) q^{47} + (15 \beta_{5} + 42 \beta_{3} + 145 \beta_1) q^{50} + ( - 5 \beta_{4} - 47 \beta_{2} - 204) q^{52} + ( - 8 \beta_{5} + 20 \beta_{3} - 20 \beta_1) q^{53} + ( - 4 \beta_{4} + 26 \beta_{2} - 141) q^{55} + ( - 3 \beta_{4} - 41 \beta_{2} - 300) q^{58} + ( - 3 \beta_{5} + 28 \beta_{3} - 109 \beta_1) q^{59} + (10 \beta_{4} - 2 \beta_{2} - 192) q^{61} + (3 \beta_{5} + 26 \beta_{3} - 80 \beta_1) q^{62} + (6 \beta_{4} + 79 \beta_{2} + 494) q^{64} + ( - 3 \beta_{5} + 24 \beta_{3} + 135 \beta_1) q^{65} + ( - 3 \beta_{4} + 9 \beta_{2} + 154) q^{67} + (9 \beta_{5} + 70 \beta_{3} + 257 \beta_1) q^{68} + ( - 21 \beta_{3} - 14 \beta_1) q^{71} + ( - 7 \beta_{4} - 55 \beta_{2} - 210) q^{73} + ( - 16 \beta_{5} - 56 \beta_{3} - 77 \beta_1) q^{74} + ( - 13 \beta_{4} - 64 \beta_{2} - 978) q^{76} + ( - 9 \beta_{4} - 45 \beta_{2} - 250) q^{79} + ( - 21 \beta_{5} - 98 \beta_{3} - 622 \beta_1) q^{80} + (3 \beta_{4} - 36 \beta_{2} - 750) q^{82} + (39 \beta_{5} + 52 \beta_{3} - 19 \beta_1) q^{83} + (3 \beta_{4} - 99 \beta_{2} - 372) q^{85} + ( - 17 \beta_{5} - 46 \beta_{3} - 205 \beta_1) q^{86} + ( - 9 \beta_{4} - 20 \beta_{2} - 410) q^{88} + (24 \beta_{5} - 67 \beta_{3} + 160 \beta_1) q^{89} + ( - 33 \beta_{5} - 66 \beta_{3} - 362 \beta_1) q^{92} + (\beta_{4} + 55 \beta_{2} + 828) q^{94} + (39 \beta_{5} + 39 \beta_{3} + 189 \beta_1) q^{95} + (10 \beta_{4} + 70 \beta_{2} - 552) q^{97}+O(q^{100})$$ q + b1 * q^2 + (b2 + 6) * q^4 + (-b3 - 2*b1) * q^5 + (b5 + 2*b3 + 8*b1) * q^8 + (-b4 - 4*b2 - 30) * q^10 + (-b5 + b3 - b1) * q^11 + (-b4 - b2 - 18) * q^13 + (3*b4 + 8*b2 + 70) * q^16 + (3*b5 + 2*b3 + 7*b1) * q^17 + (b4 - 5*b2 - 33) * q^19 + (-6*b5 - 8*b3 - 49*b1) * q^20 + (-3*b2 - 14) * q^22 + (-b5 - 5*b3 - 15*b1) * q^23 + (3*b4 + 9*b2 + 70) * q^25 + (-3*b5 - 10*b3 - 23*b1) * q^26 + (-7*b5 + 4*b3 - 21*b1) * q^29 + (5*b4 - 7*b2 + 15) * q^31 + (6*b5 + 24*b3 + 71*b1) * q^32 + (5*b4 + 23*b2 + 108) * q^34 + (-6*b4 - 4*b2 - 67) * q^37 + (-3*b5 - 2*b3 - 88*b1) * q^38 + (-6*b4 - 57*b2 - 474) * q^40 + (6*b5 - 3*b3 - 54*b1) * q^41 + (-3*b4 - 11*b2 - 110) * q^43 + (5*b5 - 14*b3 - 36*b1) * q^44 + (-6*b4 - 29*b2 - 222) * q^46 + (-3*b5 + 4*b3 + 59*b1) * q^47 + (15*b5 + 42*b3 + 145*b1) * q^50 + (-5*b4 - 47*b2 - 204) * q^52 + (-8*b5 + 20*b3 - 20*b1) * q^53 + (-4*b4 + 26*b2 - 141) * q^55 + (-3*b4 - 41*b2 - 300) * q^58 + (-3*b5 + 28*b3 - 109*b1) * q^59 + (10*b4 - 2*b2 - 192) * q^61 + (3*b5 + 26*b3 - 80*b1) * q^62 + (6*b4 + 79*b2 + 494) * q^64 + (-3*b5 + 24*b3 + 135*b1) * q^65 + (-3*b4 + 9*b2 + 154) * q^67 + (9*b5 + 70*b3 + 257*b1) * q^68 + (-21*b3 - 14*b1) * q^71 + (-7*b4 - 55*b2 - 210) * q^73 + (-16*b5 - 56*b3 - 77*b1) * q^74 + (-13*b4 - 64*b2 - 978) * q^76 + (-9*b4 - 45*b2 - 250) * q^79 + (-21*b5 - 98*b3 - 622*b1) * q^80 + (3*b4 - 36*b2 - 750) * q^82 + (39*b5 + 52*b3 - 19*b1) * q^83 + (3*b4 - 99*b2 - 372) * q^85 + (-17*b5 - 46*b3 - 205*b1) * q^86 + (-9*b4 - 20*b2 - 410) * q^88 + (24*b5 - 67*b3 + 160*b1) * q^89 + (-33*b5 - 66*b3 - 362*b1) * q^92 + (b4 + 55*b2 + 828) * q^94 + (39*b5 + 39*b3 + 189*b1) * q^95 + (10*b4 + 70*b2 - 552) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 36 q^{4}+O(q^{10})$$ 6 * q + 36 * q^4 $$6 q + 36 q^{4} - 180 q^{10} - 108 q^{13} + 420 q^{16} - 198 q^{19} - 84 q^{22} + 420 q^{25} + 90 q^{31} + 648 q^{34} - 402 q^{37} - 2844 q^{40} - 660 q^{43} - 1332 q^{46} - 1224 q^{52} - 846 q^{55} - 1800 q^{58} - 1152 q^{61} + 2964 q^{64} + 924 q^{67} - 1260 q^{73} - 5868 q^{76} - 1500 q^{79} - 4500 q^{82} - 2232 q^{85} - 2460 q^{88} + 4968 q^{94} - 3312 q^{97}+O(q^{100})$$ 6 * q + 36 * q^4 - 180 * q^10 - 108 * q^13 + 420 * q^16 - 198 * q^19 - 84 * q^22 + 420 * q^25 + 90 * q^31 + 648 * q^34 - 402 * q^37 - 2844 * q^40 - 660 * q^43 - 1332 * q^46 - 1224 * q^52 - 846 * q^55 - 1800 * q^58 - 1152 * q^61 + 2964 * q^64 + 924 * q^67 - 1260 * q^73 - 5868 * q^76 - 1500 * q^79 - 4500 * q^82 - 2232 * q^85 - 2460 * q^88 + 4968 * q^94 - 3312 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 42x^{4} + 369x^{2} - 112$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 14$$ v^2 - 14 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 38\nu^{3} + 265\nu ) / 12$$ (v^5 - 38*v^3 + 265*v) / 12 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - 32\nu^{2} + 106 ) / 3$$ (v^4 - 32*v^2 + 106) / 3 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} + 44\nu^{3} - 409\nu ) / 6$$ (-v^5 + 44*v^3 - 409*v) / 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 14$$ b2 + 14 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2\beta_{3} + 24\beta_1$$ b5 + 2*b3 + 24*b1 $$\nu^{4}$$ $$=$$ $$3\beta_{4} + 32\beta_{2} + 342$$ 3*b4 + 32*b2 + 342 $$\nu^{5}$$ $$=$$ $$38\beta_{5} + 88\beta_{3} + 647\beta_1$$ 38*b5 + 88*b3 + 647*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
 −5.45019 −3.46131 −0.560992 0.560992 3.46131 5.45019
−5.45019 0 21.7046 19.3432 0 0 −74.6929 0 −105.424
1.2 −3.46131 0 3.98067 −6.55599 0 0 13.9122 0 22.6923
1.3 −0.560992 0 −7.68529 12.9561 0 0 8.79932 0 −7.26827
1.4 0.560992 0 −7.68529 −12.9561 0 0 −8.79932 0 −7.26827
1.5 3.46131 0 3.98067 6.55599 0 0 −13.9122 0 22.6923
1.6 5.45019 0 21.7046 −19.3432 0 0 74.6929 0 −105.424
 $$n$$: 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.bf 6
3.b odd 2 1 inner 1323.4.a.bf 6
7.b odd 2 1 1323.4.a.bg yes 6
21.c even 2 1 1323.4.a.bg yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.4.a.bf 6 1.a even 1 1 trivial
1323.4.a.bf 6 3.b odd 2 1 inner
1323.4.a.bg yes 6 7.b odd 2 1
1323.4.a.bg yes 6 21.c even 2 1

## Hecke kernels

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

 $$T_{2}^{6} - 42T_{2}^{4} + 369T_{2}^{2} - 112$$ T2^6 - 42*T2^4 + 369*T2^2 - 112 $$T_{5}^{6} - 585T_{5}^{4} + 86103T_{5}^{2} - 2699487$$ T5^6 - 585*T5^4 + 86103*T5^2 - 2699487 $$T_{13}^{3} + 54T_{13}^{2} - 684T_{13} - 48168$$ T13^3 + 54*T13^2 - 684*T13 - 48168

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 42 T^{4} + \cdots - 112$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 585 T^{4} + \cdots - 2699487$$
$7$ $$T^{6}$$
$11$ $$T^{6} - 2457 T^{4} + \cdots - 1539727$$
$13$ $$(T^{3} + 54 T^{2} + \cdots - 48168)^{2}$$
$17$ $$T^{6} + \cdots - 143993106432$$
$19$ $$(T^{3} + 99 T^{2} + \cdots - 450387)^{2}$$
$23$ $$T^{6} + \cdots - 16248512287$$
$29$ $$T^{6} + \cdots - 1629346543552$$
$31$ $$(T^{3} - 45 T^{2} + \cdots - 1675863)^{2}$$
$37$ $$(T^{3} + 201 T^{2} + \cdots - 8763417)^{2}$$
$41$ $$T^{6} + \cdots - 249467996315727$$
$43$ $$(T^{3} + 330 T^{2} + \cdots - 819288)^{2}$$
$47$ $$T^{6} + \cdots - 135023220900288$$
$53$ $$T^{6} + \cdots - 38601195716608$$
$59$ $$T^{6} + \cdots - 11\!\cdots\!48$$
$61$ $$(T^{3} + 576 T^{2} + \cdots - 9597312)^{2}$$
$67$ $$(T^{3} - 462 T^{2} + \cdots + 4527656)^{2}$$
$71$ $$T^{6} + \cdots - 232630513987207$$
$73$ $$(T^{3} + 630 T^{2} + \cdots + 78997464)^{2}$$
$79$ $$(T^{3} + 750 T^{2} + \cdots + 20170072)^{2}$$
$83$ $$T^{6} + \cdots - 99\!\cdots\!12$$
$89$ $$T^{6} + \cdots - 12\!\cdots\!07$$
$97$ $$(T^{3} + 1656 T^{2} + \cdots - 913012992)^{2}$$
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