LMFDB - Newform orbit 468.4.a.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [468,4,Mod(1,468)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(468, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("468.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$468 = 2^{2} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	468.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:	$27.6128938827$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5126992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 25x^{2} + 117$ x^4 - 25*x^2 + 117
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{5}\cdot 3$
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{5} - \beta_{3} q^{7} + ( - \beta_{2} + \beta_1) q^{11} + 13 q^{13} + (2 \beta_{2} + 4 \beta_1) q^{17} + ( - \beta_{3} + 24) q^{19} + 2 \beta_{2} q^{23} + (4 \beta_{3} + 75) q^{25} - 2 \beta_{2} q^{29}+ \cdots + ( - 52 \beta_{3} + 318) q^{97}+O(q^{100})$ q + b1 * q^5 - b3 * q^7 + (-b2 + b1) * q^11 + 13 * q^13 + (2b2 + 4b1) * q^17 + (-b3 + 24) * q^19 + 2b2 q^23 + (4b3 + 75) q^25 - 2b2 q^29 + (3b3 + 104) q^31 + (-8b2 - 14b1) * q^35 + (8b3 + 186) q^37 + (6b2 - 5b1) * q^41 + (-10b3 + 156) q^43 + (b2 - 23b1) q^47 + 285 * q^49 + (-8b2 + 10b1) * q^53 + (-14b3 + 236) q^55 + (17b2 - b1) q^59 + (-16b3 + 366) q^61 + 13b1 q^65 + (9b3 + 392) q^67 + (13b2 + 17b1) * q^71 + 518 * q^73 + (-22b2 + 40b1) * q^77 + 308 * q^79 + (-21b2 - 39b1) * q^83 + (52b3 + 728) q^85 + (6b2 + 73b1) * q^89 - 13b3 q^91 + (-8b2 + 10b1) * q^95 + (-52b3 + 318) q^97
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 52 q^{13} + 96 q^{19} + 300 q^{25} + 416 q^{31} + 744 q^{37} + 624 q^{43} + 1140 q^{49} + 944 q^{55} + 1464 q^{61} + 1568 q^{67} + 2072 q^{73} + 1232 q^{79} + 2912 q^{85} + 1272 q^{97}+O(q^{100})$ 4 * q + 52 * q^13 + 96 * q^19 + 300 * q^25 + 416 * q^31 + 744 * q^37 + 624 * q^43 + 1140 * q^49 + 944 * q^55 + 1464 * q^61 + 1568 * q^67 + 2072 * q^73 + 1232 * q^79 + 2912 * q^85 + 1272 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} - 25x^{2} + 117$ x^4 - 25*x^2 + 117 :

$\beta_{1}$	$=$	$4\nu$ 4*v
$\beta_{2}$	$=$	$2\nu^{3} - 32\nu$ 2v^3 - 32v
$\beta_{3}$	$=$	$4\nu^{2} - 50$ 4*v^2 - 50

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

$\nu$	$=$	$( \beta_1 ) / 4$ (b1) / 4
$\nu^{2}$	$=$	$( \beta_{3} + 50 ) / 4$ (b3 + 50) / 4
$\nu^{3}$	$=$	$( \beta_{2} + 8\beta_1 ) / 2$ (b2 + 8*b1) / 2

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

−4.33186
−2.49700
2.49700
4.33186

−17.3274

−25.0599

1.2

−9.98801

25.0599

1.3

9.98801

25.0599

1.4

17.3274

−25.0599

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$
$13$	$-1$

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	468.4.a.h	✓	4
3.b	odd	2	1	inner	468.4.a.h	✓	4
4.b	odd	2	1		1872.4.a.bp		4
12.b	even	2	1		1872.4.a.bp		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
468.4.a.h	✓	4	1.a	even	1	1	trivial
468.4.a.h	✓	4	3.b	odd	2	1	inner
1872.4.a.bp		4	4.b	odd	2	1
1872.4.a.bp		4	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{4} - 400T_{5}^{2} + 29952$ T5^4 - 400*T5^2 + 29952 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(468))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4}$ T^4
$5$	$T^{4} - 400 T^{2} + 29952$ T^4 - 400*T^2 + 29952
$7$	$(T^{2} - 628)^{2}$ (T^2 - 628)^2
$11$	$T^{4} - 3496 T^{2} + 151632$ T^4 - 3496*T^2 + 151632
$13$	$(T - 13)^{4}$ (T - 13)^4
$17$	$T^{4} - 17056 T^{2} + 45556992$ T^4 - 17056*T^2 + 45556992
$19$	$(T^{2} - 48 T - 52)^{2}$ (T^2 - 48*T - 52)^2
$23$	$T^{4} - 11808 T^{2} + 21835008$ T^4 - 11808*T^2 + 21835008
$29$	$T^{4} - 11808 T^{2} + 21835008$ T^4 - 11808*T^2 + 21835008
$31$	$(T^{2} - 208 T + 5164)^{2}$ (T^2 - 208*T + 5164)^2
$37$	$(T^{2} - 372 T - 5596)^{2}$ (T^2 - 372*T - 5596)^2
$41$	$T^{4} - 120592 T^{2} + 382457088$ T^4 - 120592*T^2 + 382457088
$43$	$(T^{2} - 312 T - 38464)^{2}$ (T^2 - 312*T - 38464)^2
$47$	$T^{4} + \cdots + 10881788112$ T^4 - 217864*T^2 + 10881788112
$53$	$T^{4} - 240448 T^{2} + 80990208$ T^4 - 240448*T^2 + 80990208
$59$	$T^{4} + \cdots + 107019845712$ T^4 - 855976*T^2 + 107019845712
$61$	$(T^{2} - 732 T - 26812)^{2}$ (T^2 - 732*T - 26812)^2
$67$	$(T^{2} - 784 T + 102796)^{2}$ (T^2 - 784*T + 102796)^2
$71$	$T^{4} + \cdots + 79116337872$ T^4 - 582664*T^2 + 79116337872
$73$	$(T - 518)^{4}$ (T - 518)^4
$79$	$(T - 308)^{4}$ (T - 308)^4
$83$	$T^{4} + \cdots + 559558623312$ T^4 - 1792296*T^2 + 559558623312
$89$	$T^{4} + \cdots + 378108327168$ T^4 - 2174800*T^2 + 378108327168
$97$	$(T^{2} - 636 T - 1596988)^{2}$ (T^2 - 636*T - 1596988)^2
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