LMFDB - Newform orbit 880.4.b.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,4,Mod(529,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.529"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$880 = 2^{4} \cdot 5 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	880.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$51.9216808051$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 5$ x^2 - x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 $\beta = \frac{1}{2}(1 + \sqrt{-19})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - 2 \beta + 1) q^{3} - 5 \beta q^{5} + ( - 4 \beta + 2) q^{7} + 8 q^{9} + 11 q^{11} + (32 \beta - 16) q^{13} + (5 \beta - 50) q^{15} + (12 \beta - 6) q^{17} - 68 q^{19} - 38 q^{21} + ( - 54 \beta + 27) q^{23} + \cdots + 88 q^{99} +O(q^{100})$ q + (-2b + 1) q^3 - 5b q^5 + (-4b + 2) q^7 + 8 * q^9 + 11 * q^11 + (32b - 16) q^13 + (5b - 50) q^15 + (12b - 6) q^17 - 68 * q^19 - 38 * q^21 + (-54b + 27) q^23 + (25b - 125) q^25 + (-70b + 35) q^27 - 260 * q^29 - 175 * q^31 + (-22b + 11) q^33 + (10b - 100) q^35 + (78b - 39) q^37 + 304 * q^39 - 380 * q^41 + (-140b + 70) q^43 - 40b q^45 + (140b - 70) q^47 + 267 * q^49 + 114 * q^51 + (208b - 104) q^53 - 55b q^55 + (136b - 68) q^57 - 143 * q^59 + 676 * q^61 + (-32b + 16) q^63 + (-80b + 800) q^65 + (242b - 121) q^67 - 513 * q^69 - 1035 * q^71 + (152b - 76) q^73 + (225b + 125) q^75 + (-44b + 22) q^77 + 218 * q^79 - 449 * q^81 + (348b - 174) q^83 + (-30b + 300) q^85 + (520b - 260) q^87 - 1279 * q^89 + 608 * q^91 + (350b - 175) q^93 + 340b q^95 + (-354b + 177) q^97 + 88 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 5 q^{5} + 16 q^{9} + 22 q^{11} - 95 q^{15} - 136 q^{19} - 76 q^{21} - 225 q^{25} - 520 q^{29} - 350 q^{31} - 190 q^{35} + 608 q^{39} - 760 q^{41} - 40 q^{45} + 534 q^{49} + 228 q^{51} - 55 q^{55}+ \cdots + 176 q^{99}+O(q^{100})$ 2 * q - 5 * q^5 + 16 * q^9 + 22 * q^11 - 95 * q^15 - 136 * q^19 - 76 * q^21 - 225 * q^25 - 520 * q^29 - 350 * q^31 - 190 * q^35 + 608 * q^39 - 760 * q^41 - 40 * q^45 + 534 * q^49 + 228 * q^51 - 55 * q^55 - 286 * q^59 + 1352 * q^61 + 1520 * q^65 - 1026 * q^69 - 2070 * q^71 + 475 * q^75 + 436 * q^79 - 898 * q^81 + 570 * q^85 - 2558 * q^89 + 1216 * q^91 + 340 * q^95 + 176 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$ .

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$1$	$1$

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}

529.1

0.500000	+	2.17945i
0.500000	−	2.17945i

−

4.35890i

−2.50000

−

10.8972i

−

8.71780i

8.00000

529.2

4.35890i

−2.50000

10.8972i

8.71780i

8.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	880.4.b.b		2
4.b	odd	2	1		220.4.b.a	✓	2
5.b	even	2	1	inner	880.4.b.b		2
12.b	even	2	1		1980.4.c.a		2
20.d	odd	2	1		220.4.b.a	✓	2
20.e	even	4	2		1100.4.a.f		2
60.h	even	2	1		1980.4.c.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
220.4.b.a	✓	2	4.b	odd	2	1
220.4.b.a	✓	2	20.d	odd	2	1
880.4.b.b		2	1.a	even	1	1	trivial
880.4.b.b		2	5.b	even	2	1	inner
1100.4.a.f		2	20.e	even	4	2
1980.4.c.a		2	12.b	even	2	1
1980.4.c.a		2	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 19$ T3^2 + 19 acting on $S_{4}^{\mathrm{new}}(880, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 19$ T^2 + 19
$5$	$T^{2} + 5T + 125$ T^2 + 5*T + 125
$7$	$T^{2} + 76$ T^2 + 76
$11$	$(T - 11)^{2}$ (T - 11)^2
$13$	$T^{2} + 4864$ T^2 + 4864
$17$	$T^{2} + 684$ T^2 + 684
$19$	$(T + 68)^{2}$ (T + 68)^2
$23$	$T^{2} + 13851$ T^2 + 13851
$29$	$(T + 260)^{2}$ (T + 260)^2
$31$	$(T + 175)^{2}$ (T + 175)^2
$37$	$T^{2} + 28899$ T^2 + 28899
$41$	$(T + 380)^{2}$ (T + 380)^2
$43$	$T^{2} + 93100$ T^2 + 93100
$47$	$T^{2} + 93100$ T^2 + 93100
$53$	$T^{2} + 205504$ T^2 + 205504
$59$	$(T + 143)^{2}$ (T + 143)^2
$61$	$(T - 676)^{2}$ (T - 676)^2
$67$	$T^{2} + 278179$ T^2 + 278179
$71$	$(T + 1035)^{2}$ (T + 1035)^2
$73$	$T^{2} + 109744$ T^2 + 109744
$79$	$(T - 218)^{2}$ (T - 218)^2
$83$	$T^{2} + 575244$ T^2 + 575244
$89$	$(T + 1279)^{2}$ (T + 1279)^2
$97$	$T^{2} + 595251$ T^2 + 595251
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