LMFDB - Newform orbit 2252.1.d.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2252,1,Mod(1125,2252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2252.1125"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$2252 = 2^{2} \cdot 563$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2252.d (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.12389440845$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{54})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 1$ x^9 - 9x^7 + 27x^5 - 30x^3 + 9x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{27}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{27} - \cdots)$

$q$ -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 $1,\beta_1,\ldots,\beta_{8}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{3} + \beta_{4} q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{6} + \beta_{3}) q^{11} + \beta_{6} q^{13} - \beta_{7} q^{17} + (\beta_{7} - \beta_{2}) q^{19} + ( - \beta_{5} - \beta_{3}) q^{21}+ \cdots + ( - \beta_{8} - \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100})$ q - b1 * q^3 + b4 * q^7 + (b2 + 1) * q^9 + (-b6 + b3) * q^11 + b6 * q^13 - b7 * q^17 + (b7 - b2) * q^19 + (-b5 - b3) * q^21 + (b5 - b4) * q^23 + q^25 + (-b3 - b1) * q^27 + (b7 + b5 - b4 - b2) * q^33 + (-b7 - b5) * q^39 + (-b8 + b1) * q^47 + (b8 + 1) * q^49 + (b8 + b6) * q^51 + (-b8 - b6 + b3 + b1) * q^57 - b5 * q^59 + (-b8 + b1) * q^61 + (b6 + b4 + b2) * q^63 - b7 * q^67 + (-b6 + b5 - b4 + b3) * q^69 + b8 * q^71 - b1 * q^75 + (b8 + b7 - b2) * q^77 + (b4 + b2 + 1) * q^81 + (-b8 + b2 + b1) * q^91 + (-b8 - b6 + b5 - b4 + b3 + b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$9 q + 9 q^{9} + 9 q^{25} + 9 q^{49} + 9 q^{81}+O(q^{100})$ 9 * q + 9 * q^9 + 9 * q^25 + 9 * q^49 + 9 * q^81

Basis of coefficient ring in terms of $\nu = \zeta_{54} + \zeta_{54}^{-1}$ :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 2$ v^2 - 2
$\beta_{3}$	$=$	$\nu^{3} - 3\nu$ v^3 - 3*v
$\beta_{4}$	$=$	$\nu^{4} - 4\nu^{2} + 2$ v^4 - 4*v^2 + 2
$\beta_{5}$	$=$	$\nu^{5} - 5\nu^{3} + 5\nu$ v^5 - 5v^3 + 5v
$\beta_{6}$	$=$	$\nu^{6} - 6\nu^{4} + 9\nu^{2} - 2$ v^6 - 6v^4 + 9v^2 - 2
$\beta_{7}$	$=$	$\nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu$ v^7 - 7v^5 + 14v^3 - 7*v
$\beta_{8}$	$=$	$\nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2$ v^8 - 8v^6 + 20v^4 - 16*v^2 + 2

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 2$ b2 + 2
$\nu^{3}$	$=$	$\beta_{3} + 3\beta_1$ b3 + 3*b1
$\nu^{4}$	$=$	$\beta_{4} + 4\beta_{2} + 6$ b4 + 4*b2 + 6
$\nu^{5}$	$=$	$\beta_{5} + 5\beta_{3} + 10\beta_1$ b5 + 5b3 + 10b1
$\nu^{6}$	$=$	$\beta_{6} + 6\beta_{4} + 15\beta_{2} + 20$ b6 + 6b4 + 15b2 + 20
$\nu^{7}$	$=$	$\beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1$ b7 + 7b5 + 21b3 + 35*b1
$\nu^{8}$	$=$	$\beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70$ b8 + 8b6 + 28b4 + 56*b2 + 70

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2252\mathbb{Z}\right)^\times$ .

$n$	$565$	$1127$
$\chi(n)$	$-1$	$1$

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}

1125.1

1.98648
1.67098
1.37248
0.573606
0.116290
−0.792160
−1.19432
−1.78727
−1.94609

−1.98648

1.78727

2.94609

1125.2

−1.67098

−1.37248

1.79216

1125.3

−1.37248

−1.98648

0.883710

1125.4

−0.573606

0.792160

−0.670976

1125.5

−0.116290

1.94609

−0.986477

1125.6

0.792160

−0.116290

−0.372483

1125.7

1.19432

−1.67098

0.426394

1125.8

1.78727

−0.573606

2.19432

1125.9

1.94609

1.19432

2.78727

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
563.b	odd	2	1	CM by $\Q(\sqrt{-563})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2252.1.d.a	✓	9
563.b	odd	2	1	CM	2252.1.d.a	✓	9

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2252.1.d.a	✓	9	1.a	even	1	1	trivial
2252.1.d.a	✓	9	563.b	odd	2	1	CM

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2252, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{9}$ T^9
$3$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$5$	$T^{9}$ T^9
$7$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$11$	$(T^{3} - 3 T + 1)^{3}$ (T^3 - 3*T + 1)^3
$13$	$(T^{3} - 3 T + 1)^{3}$ (T^3 - 3*T + 1)^3
$17$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$19$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$23$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$29$	$T^{9}$ T^9
$31$	$T^{9}$ T^9
$37$	$T^{9}$ T^9
$41$	$T^{9}$ T^9
$43$	$T^{9}$ T^9
$47$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$53$	$T^{9}$ T^9
$59$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$61$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$67$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$71$	$T^{9} - 9 T^{7} + \cdots + 1$ T^9 - 9T^7 + 27T^5 - 30T^3 + 9T + 1
$73$	$T^{9}$ T^9
$79$	$T^{9}$ T^9
$83$	$T^{9}$ T^9
$89$	$T^{9}$ T^9
$97$	$T^{9}$ T^9
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