# Properties

 Label 1323.4.a.bl Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $8$ Inner twists $2$

# 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136$$ x^8 - 54*x^6 + 887*x^4 - 4176*x^2 + 3136 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$7$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{4} - \beta_{2} + 6) q^{4} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{5} + ( - 2 \beta_{7} + 2 \beta_{5} + \cdots + 5 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b4 - b2 + 6) * q^4 + (-b5 - b3 - b1) * q^5 + (-2*b7 + 2*b5 - b3 + 5*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{4} - \beta_{2} + 6) q^{4} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{5} + ( - 2 \beta_{7} + 2 \beta_{5} + \cdots + 5 \beta_1) q^{8}+ \cdots + (20 \beta_{6} + 60 \beta_{4} + \cdots - 684) q^{97}+O(q^{100})$$ q + b1 * q^2 + (b4 - b2 + 6) * q^4 + (-b5 - b3 - b1) * q^5 + (-2*b7 + 2*b5 - b3 + 5*b1) * q^8 + (-3*b4 - 19*b2 - 18) * q^10 + (-2*b7 - b5 + 5*b3 + 7*b1) * q^11 + (2*b6 + 12*b2 - 42) * q^13 + (3*b6 + 3*b4 + 3*b2 + 27) * q^16 + (9*b7 - 5*b5 + 2*b3 - 11*b1) * q^17 + (-6*b6 - 6*b4 + 17*b2 + 33) * q^19 + (6*b7 + 2*b5 - 11*b3 - 31*b1) * q^20 + (-6*b6 + 7*b4 + 59*b2 + 64) * q^22 + (3*b7 + 6*b5 + 15*b3 - 16*b1) * q^23 + (-6*b6 - 2*b4 + 23*b2 + 18) * q^25 + (8*b5 - 6*b3 - 52*b1) * q^26 + (6*b7 + 18*b5 - 18*b3 - 26*b1) * q^29 + (4*b6 + 12*b4 + 68*b2 - 9) * q^31 + (10*b7 + 2*b5 - 16*b3 - 7*b1) * q^32 + (-7*b6 - 30*b4 - 141) * q^34 + (18*b6 - 6*b4 - 123*b2 + 71) * q^37 + (12*b7 - 36*b5 + 71*b3 + 21*b1) * q^38 + (13*b6 - 9*b4 + 35*b2 - 207) * q^40 + (-15*b7 - 14*b5 + 5*b3 - 8*b1) * q^41 + (-6*b6 - 8*b4 - 91*b2 - 238) * q^43 + (2*b7 - 2*b5 + 73*b3 + 87*b1) * q^44 + (-9*b6 - 7*b4 + 259*b2 - 209) * q^46 + (-15*b7 + 13*b5 - 36*b3 - 47*b1) * q^47 + (4*b7 - 28*b5 + 77*b3 + 34*b1) * q^50 + (-2*b6 - 36*b4 - 80*b2 - 318) * q^52 + (-20*b7 + 14*b5 - 64*b3 + 34*b1) * q^53 + (22*b6 - 30*b4 - 153*b2 - 183) * q^55 + (36*b6 + 4*b4 - 124*b2 - 148) * q^58 + (-3*b7 + 51*b5 - 88*b3 - 57*b1) * q^59 + (10*b6 - 24*b4 - 104*b2 - 312) * q^61 + (-24*b7 + 40*b5 + 32*b3 + 55*b1) * q^62 + (-6*b6 - 37*b4 - 239*b2 - 192) * q^64 + (-40*b7 + 64*b5 + 58*b3 + 28*b1) * q^65 + (6*b6 + 20*b4 - 146*b2 + 10) * q^67 + (-12*b7 - 48*b5 + 47*b3 - 228*b1) * q^68 + (-24*b7 + 9*b5 + 81*b3 + 29*b1) * q^71 + (-40*b6 + 12*b4 + 211*b2 - 222) * q^73 + (12*b7 + 60*b5 - 285*b3 - 61*b1) * q^74 + (-59*b6 - 15*b4 + 609*b2 - 363) * q^76 + (6*b6 - 20*b4 + 5*b2 + 286) * q^79 + (-30*b7 + 18*b5 + 6*b3 - 87*b1) * q^80 + (-19*b6 - 21*b4 + 9*b2 - 315) * q^82 + (42*b5 - 70*b3 - 66*b1) * q^83 + (-54*b6 + 26*b4 - 275*b2 + 452) * q^85 + (16*b7 - 40*b5 - 37*b3 - 264*b1) * q^86 + (-27*b6 + 25*b4 + 449*b2 + 485) * q^88 + (15*b7 + 46*b5 + 143*b3 - 56*b1) * q^89 + (-10*b7 - 98*b5 + 220*b3 - 85*b1) * q^92 + (49*b6 - 6*b4 - 364*b2 - 549) * q^94 + (37*b7 - 100*b5 - 145*b3 + 18*b1) * q^95 + (20*b6 + 60*b4 - 144*b2 - 684) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 44 q^{4}+O(q^{10})$$ 8 * q + 44 * q^4 $$8 q + 44 q^{4} - 132 q^{10} - 336 q^{13} + 204 q^{16} + 288 q^{19} + 484 q^{22} + 152 q^{25} - 120 q^{31} - 1008 q^{34} + 592 q^{37} - 1620 q^{40} - 1872 q^{43} - 1644 q^{46} - 2400 q^{52} - 1344 q^{55} - 1200 q^{58} - 2400 q^{61} - 1388 q^{64} - 1824 q^{73} - 2844 q^{76} + 2368 q^{79} - 2436 q^{82} + 3512 q^{85} + 3780 q^{88} - 4368 q^{94} - 5712 q^{97}+O(q^{100})$$ 8 * q + 44 * q^4 - 132 * q^10 - 336 * q^13 + 204 * q^16 + 288 * q^19 + 484 * q^22 + 152 * q^25 - 120 * q^31 - 1008 * q^34 + 592 * q^37 - 1620 * q^40 - 1872 * q^43 - 1644 * q^46 - 2400 * q^52 - 1344 * q^55 - 1200 * q^58 - 2400 * q^61 - 1388 * q^64 - 1824 * q^73 - 2844 * q^76 + 2368 * q^79 - 2436 * q^82 + 3512 * q^85 + 3780 * q^88 - 4368 * q^94 - 5712 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{6} + 38\nu^{4} - 367\nu^{2} + 680 ) / 264$$ (-v^6 + 38*v^4 - 367*v^2 + 680) / 264 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 38\nu^{5} - 367\nu^{3} + 680\nu ) / 264$$ (-v^7 + 38*v^5 - 367*v^3 + 680*v) / 264 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} + 38\nu^{4} - 103\nu^{2} - 3016 ) / 264$$ (-v^6 + 38*v^4 - 103*v^2 - 3016) / 264 $$\beta_{5}$$ $$=$$ $$( -7\nu^{7} + 354\nu^{5} - 4945\nu^{3} + 13032\nu ) / 1056$$ (-7*v^7 + 354*v^5 - 4945*v^3 + 13032*v) / 1056 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 6\nu^{4} - 821\nu^{2} + 2796 ) / 132$$ (v^6 + 6*v^4 - 821*v^2 + 2796) / 132 $$\beta_{7}$$ $$=$$ $$( -5\nu^{7} + 278\nu^{5} - 4739\nu^{3} + 22760\nu ) / 1056$$ (-5*v^7 + 278*v^5 - 4739*v^3 + 22760*v) / 1056
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{2} + 14$$ b4 - b2 + 14 $$\nu^{3}$$ $$=$$ $$-2\beta_{7} + 2\beta_{5} - \beta_{3} + 21\beta_1$$ -2*b7 + 2*b5 - b3 + 21*b1 $$\nu^{4}$$ $$=$$ $$3\beta_{6} + 27\beta_{4} - 21\beta_{2} + 299$$ 3*b6 + 27*b4 - 21*b2 + 299 $$\nu^{5}$$ $$=$$ $$-54\beta_{7} + 66\beta_{5} - 48\beta_{3} + 473\beta_1$$ -54*b7 + 66*b5 - 48*b3 + 473*b1 $$\nu^{6}$$ $$=$$ $$114\beta_{6} + 659\beta_{4} - 695\beta_{2} + 6904$$ 114*b6 + 659*b4 - 695*b2 + 6904 $$\nu^{7}$$ $$=$$ $$-1318\beta_{7} + 1774\beta_{5} - 1721\beta_{3} + 10947\beta_1$$ -1318*b7 + 1774*b5 - 1721*b3 + 10947*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
 −5.00098 −4.67291 −2.49657 −0.959848 0.959848 2.49657 4.67291 5.00098
−5.00098 0 17.0098 3.98256 0 0 −45.0578 0 −19.9167
1.2 −4.67291 0 13.8361 15.5408 0 0 −27.2713 0 −72.6208
1.3 −2.49657 0 −1.76715 −14.5857 0 0 24.3844 0 36.4142
1.4 −0.959848 0 −7.07869 10.2898 0 0 14.4733 0 −9.87663
1.5 0.959848 0 −7.07869 −10.2898 0 0 −14.4733 0 −9.87663
1.6 2.49657 0 −1.76715 14.5857 0 0 −24.3844 0 36.4142
1.7 4.67291 0 13.8361 −15.5408 0 0 27.2713 0 −72.6208
1.8 5.00098 0 17.0098 −3.98256 0 0 45.0578 0 −19.9167
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.bl 8
3.b odd 2 1 inner 1323.4.a.bl 8
7.b odd 2 1 1323.4.a.bm yes 8
21.c even 2 1 1323.4.a.bm yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.4.a.bl 8 1.a even 1 1 trivial
1323.4.a.bl 8 3.b odd 2 1 inner
1323.4.a.bm yes 8 7.b odd 2 1
1323.4.a.bm yes 8 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2}^{8} - 54T_{2}^{6} + 887T_{2}^{4} - 4176T_{2}^{2} + 3136$$ T2^8 - 54*T2^6 + 887*T2^4 - 4176*T2^2 + 3136 $$T_{5}^{8} - 576T_{5}^{6} + 108362T_{5}^{4} - 7017984T_{5}^{2} + 86285521$$ T5^8 - 576*T5^6 + 108362*T5^4 - 7017984*T5^2 + 86285521 $$T_{13}^{4} + 168T_{13}^{3} + 8328T_{13}^{2} + 100608T_{13} - 932864$$ T13^4 + 168*T13^3 + 8328*T13^2 + 100608*T13 - 932864

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 54 T^{6} + \cdots + 3136$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 576 T^{6} + \cdots + 86285521$$
$7$ $$T^{8}$$
$11$ $$T^{8} - 4864 T^{6} + \cdots + 625681$$
$13$ $$(T^{4} + 168 T^{3} + \cdots - 932864)^{2}$$
$17$ $$T^{8} + \cdots + 47\!\cdots\!96$$
$19$ $$(T^{4} - 144 T^{3} + \cdots - 53761631)^{2}$$
$23$ $$T^{8} + \cdots + 12\!\cdots\!41$$
$29$ $$T^{8} + \cdots + 50\!\cdots\!36$$
$31$ $$(T^{4} + 60 T^{3} + \cdots - 21279671)^{2}$$
$37$ $$(T^{4} - 296 T^{3} + \cdots - 2024956199)^{2}$$
$41$ $$T^{8} + \cdots + 20\!\cdots\!61$$
$43$ $$(T^{4} + 936 T^{3} + \cdots - 1639683644)^{2}$$
$47$ $$T^{8} + \cdots + 69\!\cdots\!36$$
$53$ $$T^{8} + \cdots + 19\!\cdots\!76$$
$59$ $$T^{8} + \cdots + 16\!\cdots\!56$$
$61$ $$(T^{4} + 1200 T^{3} + \cdots + 451950064)^{2}$$
$67$ $$(T^{4} - 179656 T^{2} + \cdots - 798350336)^{2}$$
$71$ $$T^{8} + \cdots + 67\!\cdots\!61$$
$73$ $$(T^{4} + 912 T^{3} + \cdots - 33168592604)^{2}$$
$79$ $$(T^{4} - 1184 T^{3} + \cdots + 898048804)^{2}$$
$83$ $$T^{8} + \cdots + 45\!\cdots\!56$$
$89$ $$T^{8} + \cdots + 61\!\cdots\!21$$
$97$ $$(T^{4} + 2856 T^{3} + \cdots - 356770409984)^{2}$$
show more
show less