LMFDB - Newform orbit 528.4.b.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [528,4,Mod(65,528)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(528, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("528.65"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$528 = 2^{4} \cdot 3 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	528.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-8,0,0,0,0,0,-10,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$31.1530084830$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 2x^{2} - 3x + 9$ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}\cdot 3^{2}$
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -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.

$f(q)$	$=$	$q + ( - \beta_1 - 2) q^{3} + ( - 2 \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - 5 \beta_{3} + 3 \beta_{2} + \cdots - 1) q^{9} + 11 \beta_{3} q^{11} + (12 \beta_{3} - 6 \beta_{2} + \cdots + 62) q^{15} + ( - 17 \beta_{3} + 12 \beta_{2} + \cdots + 6) q^{23}+ \cdots + (77 \beta_{3} + 132 \beta_{2} + \cdots + 550) q^{99}+O(q^{100})$ q + (-b1 - 2) * q^3 + (-2b2 + 2b1 - 1) * q^5 + (-5b3 + 3b2 + 5b1 - 1) q^9 + 11b3 q^11 + (12b3 - 6b2 - 11b1 + 62) q^15 + (-17b3 + 12b2 - 12b1 + 6) q^23 + (-4b3 + 32b2 + 40b1 - 146) q^25 + (37b3 - 68) q^27 + (-b3 + 8b2 + 10b1 - 166) * q^31 + (-33b3 - 33b2 - 11b1 + 44) q^33 + (-2b3 + 16b2 + 20b1 - 209) q^37 + (-85b3 - 3b2 - 59b1 + 19) q^45 - 194b3 q^47 + 343 * q^49 + 68b3 q^53 + (-11b3 + 88b2 + 110b1 - 198) q^55 + (-163b3 - 80b2 + 80b1 - 40) q^59 + (-17b3 + 136b2 + 170b1 + 276) q^67 + (-21b3 + 87b2 + 83b1 - 440) q^69 + (-223b3 - 68b2 + 68b1 - 34) q^71 + (180b3 - 108b2 + 126b1 - 324) q^75 + (-111b3 - 111b2 + 31b1 + 284) q^81 + (-166b3 - 186b2 + 186b1 - 93) q^89 + (45b3 - 27b2 + 161b1 + 178) q^93 + (-32b3 + 256b2 + 320b1 + 145) q^97 + (77b3 + 132b2 - 77b1 + 550) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 8 q^{3} - 10 q^{9} + 260 q^{15} - 648 q^{25} - 272 q^{27} - 680 q^{31} + 242 q^{33} - 868 q^{37} + 82 q^{45} + 1372 q^{49} - 968 q^{55} + 832 q^{67} - 1934 q^{69} - 1080 q^{75} + 1358 q^{81} + 766 q^{93}+ \cdots + 1936 q^{99}+O(q^{100})$ 4 * q - 8 * q^3 - 10 * q^9 + 260 * q^15 - 648 * q^25 - 272 * q^27 - 680 * q^31 + 242 * q^33 - 868 * q^37 + 82 * q^45 + 1372 * q^49 - 968 * q^55 + 832 * q^67 - 1934 * q^69 - 1080 * q^75 + 1358 * q^81 + 766 * q^93 + 68 * q^97 + 1936 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} - x^{3} - 2x^{2} - 3x + 9$ x^4 - x^3 - 2*x^2 - 3*x + 9 :

$\beta_{1}$	$=$	$( -\nu^{3} + 10\nu^{2} + 2\nu - 9 ) / 6$ (-v^3 + 10v^2 + 2v - 9) / 6
$\beta_{2}$	$=$	$( -2\nu^{3} - 4\nu^{2} + 10\nu + 9 ) / 3$ (-2v^3 - 4v^2 + 10*v + 9) / 3
$\beta_{3}$	$=$	$-\nu^{3} + 4$ -v^3 + 4

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

$\nu$	$=$	$( -4\beta_{3} + 5\beta_{2} + 4\beta _1 + 7 ) / 18$ (-4b3 + 5b2 + 4*b1 + 7) / 18
$\nu^{2}$	$=$	$( -\beta_{3} - \beta_{2} + 10\beta _1 + 22 ) / 18$ (-b3 - b2 + 10*b1 + 22) / 18
$\nu^{3}$	$=$	$-\beta_{3} + 4$ -b3 + 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$ .

$n$	$133$	$145$	$353$	$463$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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}

65.1

1.68614	+	0.396143i
1.68614	−	0.396143i
−1.18614	−	1.26217i
−1.18614	+	1.26217i

−4.87228

−

1.80579i

8.95521i

20.4783

17.5966i

65.2

−4.87228

1.80579i

−

8.95521i

20.4783

−

17.5966i

65.3

0.872281

−

5.12241i

22.2217i

−25.4783

−

8.93637i

65.4

0.872281

5.12241i

−

22.2217i

−25.4783

8.93637i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11})$
3.b	odd	2	1	inner
33.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	528.4.b.b		4
3.b	odd	2	1	inner	528.4.b.b		4
4.b	odd	2	1		132.4.b.a	✓	4
11.b	odd	2	1	CM	528.4.b.b		4
12.b	even	2	1		132.4.b.a	✓	4
33.d	even	2	1	inner	528.4.b.b		4
44.c	even	2	1		132.4.b.a	✓	4
132.d	odd	2	1		132.4.b.a	✓	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
132.4.b.a	✓	4	4.b	odd	2	1
132.4.b.a	✓	4	12.b	even	2	1
132.4.b.a	✓	4	44.c	even	2	1
132.4.b.a	✓	4	132.d	odd	2	1
528.4.b.b		4	1.a	even	1	1	trivial
528.4.b.b		4	3.b	odd	2	1	inner
528.4.b.b		4	11.b	odd	2	1	CM
528.4.b.b		4	33.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(528, [\chi])$ :

T_{5}^{4} + 574T_{5}^{2} + 39601

T5^4 + 574*T5^2 + 39601

T_{17}

T17

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} + 8 T^{3} + \cdots + 729$ T^4 + 8T^3 + 37T^2 + 216*T + 729
$5$	$T^{4} + 574 T^{2} + 39601$ T^4 + 574*T^2 + 39601
$7$	$T^{4}$ T^4
$11$	$(T^{2} + 1331)^{2}$ (T^2 + 1331)^2
$13$	$T^{4}$ T^4
$17$	$T^{4}$ T^4
$19$	$T^{4}$ T^4
$23$	$T^{4} + 35998 T^{2} + 253009$ T^4 + 35998*T^2 + 253009
$29$	$T^{4}$ T^4
$31$	$(T^{2} + 340 T + 26227)^{2}$ (T^2 + 340*T + 26227)^2
$37$	$(T^{2} + 434 T + 36397)^{2}$ (T^2 + 434*T + 36397)^2
$41$	$T^{4}$ T^4
$43$	$T^{4}$ T^4
$47$	$(T^{2} + 413996)^{2}$ (T^2 + 413996)^2
$53$	$(T^{2} + 50864)^{2}$ (T^2 + 50864)^2
$59$	$T^{4} + \cdots + 97982146441$ T^4 + 929158*T^2 + 97982146441
$61$	$T^{4}$ T^4
$67$	$(T^{2} - 416 T - 729233)^{2}$ (T^2 - 416*T - 729233)^2
$71$	$T^{4} + 1090366 T^{2} + 276656689$ T^4 + 1090366*T^2 + 276656689
$73$	$T^{4}$ T^4
$79$	$T^{4}$ T^4
$83$	$T^{4}$ T^4
$89$	$T^{4} + \cdots + 4398696652249$ T^4 + 4212214*T^2 + 4398696652249
$97$	$(T^{2} - 34 T - 2736863)^{2}$ (T^2 - 34*T - 2736863)^2
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