# Properties

 Label 324.4.a.c Level $324$ Weight $4$ Character orbit 324.a Self dual yes Analytic conductor $19.117$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(1,324)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, 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("324.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 324.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: $$19.1166188419$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1509.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 7x + 4$$ x^3 - x^2 - 7*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 36) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 2) q^{5} + (\beta_{2} - \beta_1 + 2) q^{7}+O(q^{10})$$ q + (-b2 - 2) * q^5 + (b2 - b1 + 2) * q^7 $$q + ( - \beta_{2} - 2) q^{5} + (\beta_{2} - \beta_1 + 2) q^{7} + (2 \beta_{2} + 3 \beta_1 - 17) q^{11} + (5 \beta_{2} + 4 \beta_1 - 4) q^{13} + (\beta_{2} - 6 \beta_1 - 37) q^{17} + ( - 7 \beta_{2} - 2 \beta_1 + 5) q^{19} + ( - 5 \beta_{2} - 3 \beta_1 - 70) q^{23} + ( - 3 \beta_{2} - 6 \beta_1 + 1) q^{25} + (5 \beta_{2} - 152) q^{29} + ( - 15 \beta_{2} - 3 \beta_1 - 16) q^{31} + (\beta_{2} + 15 \beta_1 - 184) q^{35} + (10 \beta_{2} + 8 \beta_1 - 16) q^{37} + (14 \beta_{2} + 12 \beta_1 - 299) q^{41} + (27 \beta_1 - 43) q^{43} + ( - 39 \beta_{2} - 21 \beta_1 - 174) q^{47} + (13 \beta_{2} - 22 \beta_1 + 75) q^{49} + ( - 22 \beta_{2} - 368) q^{53} + (33 \beta_{2} - 15 \beta_1 - 36) q^{55} + (34 \beta_{2} - 15 \beta_1 - 151) q^{59} + (3 \beta_{2} - 12 \beta_1 + 134) q^{61} + (37 \beta_{2} - 6 \beta_1 - 370) q^{65} + ( - 24 \beta_{2} - 3 \beta_1 + 71) q^{67} + (52 \beta_{2} + 12 \beta_1 + 20) q^{71} + ( - 21 \beta_{2} + 30 \beta_1 + 125) q^{73} + ( - 65 \beta_{2} + 12 \beta_1 - 376) q^{77} + (23 \beta_{2} - 5 \beta_1 - 184) q^{79} + (15 \beta_{2} + 33 \beta_1 + 204) q^{83} + (30 \beta_{2} + 60 \beta_1 - 396) q^{85} + ( - 44 \beta_{2} - 60 \beta_1 - 154) q^{89} + ( - 75 \beta_{2} - 33 \beta_1 - 44) q^{91} + ( - 44 \beta_{2} - 24 \beta_1 + 728) q^{95} + ( - 70 \beta_{2} - 56 \beta_1 - 31) q^{97}+O(q^{100})$$ q + (-b2 - 2) * q^5 + (b2 - b1 + 2) * q^7 + (2*b2 + 3*b1 - 17) * q^11 + (5*b2 + 4*b1 - 4) * q^13 + (b2 - 6*b1 - 37) * q^17 + (-7*b2 - 2*b1 + 5) * q^19 + (-5*b2 - 3*b1 - 70) * q^23 + (-3*b2 - 6*b1 + 1) * q^25 + (5*b2 - 152) * q^29 + (-15*b2 - 3*b1 - 16) * q^31 + (b2 + 15*b1 - 184) * q^35 + (10*b2 + 8*b1 - 16) * q^37 + (14*b2 + 12*b1 - 299) * q^41 + (27*b1 - 43) * q^43 + (-39*b2 - 21*b1 - 174) * q^47 + (13*b2 - 22*b1 + 75) * q^49 + (-22*b2 - 368) * q^53 + (33*b2 - 15*b1 - 36) * q^55 + (34*b2 - 15*b1 - 151) * q^59 + (3*b2 - 12*b1 + 134) * q^61 + (37*b2 - 6*b1 - 370) * q^65 + (-24*b2 - 3*b1 + 71) * q^67 + (52*b2 + 12*b1 + 20) * q^71 + (-21*b2 + 30*b1 + 125) * q^73 + (-65*b2 + 12*b1 - 376) * q^77 + (23*b2 - 5*b1 - 184) * q^79 + (15*b2 + 33*b1 + 204) * q^83 + (30*b2 + 60*b1 - 396) * q^85 + (-44*b2 - 60*b1 - 154) * q^89 + (-75*b2 - 33*b1 - 44) * q^91 + (-44*b2 - 24*b1 + 728) * q^95 + (-70*b2 - 56*b1 - 31) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{5} + 6 q^{7}+O(q^{10})$$ 3 * q - 6 * q^5 + 6 * q^7 $$3 q - 6 q^{5} + 6 q^{7} - 51 q^{11} - 12 q^{13} - 111 q^{17} + 15 q^{19} - 210 q^{23} + 3 q^{25} - 456 q^{29} - 48 q^{31} - 552 q^{35} - 48 q^{37} - 897 q^{41} - 129 q^{43} - 522 q^{47} + 225 q^{49} - 1104 q^{53} - 108 q^{55} - 453 q^{59} + 402 q^{61} - 1110 q^{65} + 213 q^{67} + 60 q^{71} + 375 q^{73} - 1128 q^{77} - 552 q^{79} + 612 q^{83} - 1188 q^{85} - 462 q^{89} - 132 q^{91} + 2184 q^{95} - 93 q^{97}+O(q^{100})$$ 3 * q - 6 * q^5 + 6 * q^7 - 51 * q^11 - 12 * q^13 - 111 * q^17 + 15 * q^19 - 210 * q^23 + 3 * q^25 - 456 * q^29 - 48 * q^31 - 552 * q^35 - 48 * q^37 - 897 * q^41 - 129 * q^43 - 522 * q^47 + 225 * q^49 - 1104 * q^53 - 108 * q^55 - 453 * q^59 + 402 * q^61 - 1110 * q^65 + 213 * q^67 + 60 * q^71 + 375 * q^73 - 1128 * q^77 - 552 * q^79 + 612 * q^83 - 1188 * q^85 - 462 * q^89 - 132 * q^91 + 2184 * q^95 - 93 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$6\nu - 2$$ 6*v - 2 $$\beta_{2}$$ $$=$$ $$3\nu^{2} - 3\nu - 14$$ 3*v^2 - 3*v - 14
 $$\nu$$ $$=$$ $$( \beta _1 + 2 ) / 6$$ (b1 + 2) / 6 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + \beta _1 + 30 ) / 6$$ (2*b2 + b1 + 30) / 6

## 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
 −2.47735 2.92542 0.551929
0 0 0 −13.8439 0 30.7080 0 0 0
1.2 0 0 0 −4.89803 0 −10.6545 0 0 0
1.3 0 0 0 12.7419 0 −14.0535 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$+1$$

## 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 324.4.a.c 3
3.b odd 2 1 324.4.a.d 3
4.b odd 2 1 1296.4.a.v 3
9.c even 3 2 36.4.e.a 6
9.d odd 6 2 108.4.e.a 6
12.b even 2 1 1296.4.a.w 3
36.f odd 6 2 144.4.i.d 6
36.h even 6 2 432.4.i.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.e.a 6 9.c even 3 2
108.4.e.a 6 9.d odd 6 2
144.4.i.d 6 36.f odd 6 2
324.4.a.c 3 1.a even 1 1 trivial
324.4.a.d 3 3.b odd 2 1
432.4.i.d 6 36.h even 6 2
1296.4.a.v 3 4.b odd 2 1
1296.4.a.w 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} + 6T_{5}^{2} - 171T_{5} - 864$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(324))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 6 T^{2} + \cdots - 864$$
$7$ $$T^{3} - 6 T^{2} + \cdots - 4598$$
$11$ $$T^{3} + 51 T^{2} + \cdots - 66231$$
$13$ $$T^{3} + 12 T^{2} + \cdots - 64466$$
$17$ $$T^{3} + 111 T^{2} + \cdots - 577476$$
$19$ $$T^{3} - 15 T^{2} + \cdots - 216368$$
$23$ $$T^{3} + 210 T^{2} + \cdots + 2322$$
$29$ $$T^{3} + 456 T^{2} + \cdots + 2879658$$
$31$ $$T^{3} + 48 T^{2} + \cdots - 3054788$$
$37$ $$T^{3} + 48 T^{2} + \cdots - 682352$$
$41$ $$T^{3} + 897 T^{2} + \cdots + 11796543$$
$43$ $$T^{3} + 129 T^{2} + \cdots - 1425149$$
$47$ $$T^{3} + 522 T^{2} + \cdots - 64558782$$
$53$ $$T^{3} + 1104 T^{2} + \cdots + 11853648$$
$59$ $$T^{3} + 453 T^{2} + \cdots - 96892713$$
$61$ $$T^{3} - 402 T^{2} + \cdots + 1209736$$
$67$ $$T^{3} - 213 T^{2} + \cdots - 3095063$$
$71$ $$T^{3} - 60 T^{2} + \cdots + 113211648$$
$73$ $$T^{3} - 375 T^{2} + \cdots + 158369284$$
$79$ $$T^{3} + 552 T^{2} + \cdots - 17848772$$
$83$ $$T^{3} - 612 T^{2} + \cdots + 3478788$$
$89$ $$T^{3} + 462 T^{2} + \cdots + 170122248$$
$97$ $$T^{3} + 93 T^{2} + \cdots + 86400523$$
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