LMFDB - Newform orbit 324.4.a.c

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.

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

$\beta_{1}$	$=$	$6\nu - 2$ 6*v - 2
$\beta_{2}$	$=$	$3\nu^{2} - 3\nu - 14$ 3v^2 - 3v - 14

Display $\nu^j$ in terms of $\beta_i$

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

Display $\beta_i$ in terms of $\nu^j$

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

−13.8439

30.7080

1.2

−4.89803

−10.6545

1.3

12.7419

−14.0535

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 $T_{5}^{3} + 6T_{5}^{2} - 171T_{5} - 864$ T5^3 + 6*T5^2 - 171*T5 - 864 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(324))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3}$ T^3
$5$	$T^{3} + 6 T^{2} + \cdots - 864$ T^3 + 6T^2 - 171T - 864
$7$	$T^{3} - 6 T^{2} + \cdots - 4598$ T^3 - 6T^2 - 609T - 4598
$11$	$T^{3} + 51 T^{2} + \cdots - 66231$ T^3 + 51T^2 - 1197T - 66231
$13$	$T^{3} + 12 T^{2} + \cdots - 64466$ T^3 + 12T^2 - 5271T - 64466
$17$	$T^{3} + 111 T^{2} + \cdots - 577476$ T^3 + 111T^2 - 6624T - 577476
$19$	$T^{3} - 15 T^{2} + \cdots - 216368$ T^3 - 15T^2 - 7512T - 216368
$23$	$T^{3} + 210 T^{2} + \cdots + 2322$ T^3 + 210T^2 + 10359T + 2322
$29$	$T^{3} + 456 T^{2} + \cdots + 2879658$ T^3 + 456T^2 + 64737T + 2879658
$31$	$T^{3} + 48 T^{2} + \cdots - 3054788$ T^3 + 48T^2 - 34953T - 3054788
$37$	$T^{3} + 48 T^{2} + \cdots - 682352$ T^3 + 48T^2 - 20508T - 682352
$41$	$T^{3} + 897 T^{2} + \cdots + 11796543$ T^3 + 897T^2 + 223551T + 11796543
$43$	$T^{3} + 129 T^{2} + \cdots - 1425149$ T^3 + 129T^2 - 186909T - 1425149
$47$	$T^{3} + 522 T^{2} + \cdots - 64558782$ T^3 + 522T^2 - 161433T - 64558782
$53$	$T^{3} + 1104 T^{2} + \cdots + 11853648$ T^3 + 1104T^2 + 317700T + 11853648
$59$	$T^{3} + 453 T^{2} + \cdots - 96892713$ T^3 + 453T^2 - 291285T - 96892713
$61$	$T^{3} - 402 T^{2} + \cdots + 1209736$ T^3 - 402T^2 + 7941T + 1209736
$67$	$T^{3} - 213 T^{2} + \cdots - 3095063$ T^3 - 213T^2 - 80133T - 3095063
$71$	$T^{3} - 60 T^{2} + \cdots + 113211648$ T^3 - 60T^2 - 423072T + 113211648
$73$	$T^{3} - 375 T^{2} + \cdots + 158369284$ T^3 - 375T^2 - 381048T + 158369284
$79$	$T^{3} + 552 T^{2} + \cdots - 17848772$ T^3 + 552T^2 - 21849T - 17848772
$83$	$T^{3} - 612 T^{2} + \cdots + 3478788$ T^3 - 612T^2 - 117693T + 3478788
$89$	$T^{3} + 462 T^{2} + \cdots + 170122248$ T^3 + 462T^2 - 774180T + 170122248
$97$	$T^{3} + 93 T^{2} + \cdots + 86400523$ T^3 + 93T^2 - 1039641T + 86400523
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