LMFDB - Newform orbit 288.4.d.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [288,4,Mod(145,288)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("288.145"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	288.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,68] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$16.9925500817$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 72)
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 $\beta = 2\sqrt{-2}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 7 \beta q^{5} + 34 q^{7} + 2 \beta q^{11} - 267 q^{25} + 79 \beta q^{29} + 70 q^{31} + 238 \beta q^{35} + 813 q^{49} + 205 \beta q^{53} - 112 q^{55} - 196 \beta q^{59} - 322 q^{73} + 68 \beta q^{77} + \cdots - 574 q^{97} +O(q^{100}) $ q + 7b q^5 + 34 * q^7 + 2b q^11 - 267 * q^25 + 79b q^29 + 70 * q^31 + 238b q^35 + 813 * q^49 + 205b q^53 - 112 * q^55 - 196b q^59 - 322 * q^73 + 68b q^77 - 1370 * q^79 + 434b q^83 - 574 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 68 q^{7} - 534 q^{25} + 140 q^{31} + 1626 q^{49} - 224 q^{55} - 644 q^{73} - 2740 q^{79} - 1148 q^{97}+O(q^{100}) $ 2 * q + 68 * q^7 - 534 * q^25 + 140 * q^31 + 1626 * q^49 - 224 * q^55 - 644 * q^73 - 2740 * q^79 - 1148 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$1$

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} $

145.1

	−	1.41421i
		1.41421i

−

19.7990i

34.0000

145.2

19.7990i

34.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by $\Q(\sqrt{-6}) $
3.b	odd	2	1	inner
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.4.d.b		2
3.b	odd	2	1	inner	288.4.d.b		2
4.b	odd	2	1		72.4.d.a	✓	2
8.b	even	2	1	inner	288.4.d.b		2
8.d	odd	2	1		72.4.d.a	✓	2
12.b	even	2	1		72.4.d.a	✓	2
16.e	even	4	2		2304.4.a.u		2
16.f	odd	4	2		2304.4.a.bo		2
24.f	even	2	1		72.4.d.a	✓	2
24.h	odd	2	1	CM	288.4.d.b		2
48.i	odd	4	2		2304.4.a.u		2
48.k	even	4	2		2304.4.a.bo		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.4.d.a	✓	2	4.b	odd	2	1
72.4.d.a	✓	2	8.d	odd	2	1
72.4.d.a	✓	2	12.b	even	2	1
72.4.d.a	✓	2	24.f	even	2	1
288.4.d.b		2	1.a	even	1	1	trivial
288.4.d.b		2	3.b	odd	2	1	inner
288.4.d.b		2	8.b	even	2	1	inner
288.4.d.b		2	24.h	odd	2	1	CM
2304.4.a.u		2	16.e	even	4	2
2304.4.a.u		2	48.i	odd	4	2
2304.4.a.bo		2	16.f	odd	4	2
2304.4.a.bo		2	48.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{2} + 392 $ T5^2 + 392 acting on $S_{4}^{\mathrm{new}}(288, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 392 $ T^2 + 392
$7$	$ (T - 34)^{2} $ (T - 34)^2
$11$	$ T^{2} + 32 $ T^2 + 32
$13$	$ T^{2} $ T^2
$17$	$ T^{2} $ T^2
$19$	$ T^{2} $ T^2
$23$	$ T^{2} $ T^2
$29$	$ T^{2} + 49928 $ T^2 + 49928
$31$	$ (T - 70)^{2} $ (T - 70)^2
$37$	$ T^{2} $ T^2
$41$	$ T^{2} $ T^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} $ T^2
$53$	$ T^{2} + 336200 $ T^2 + 336200
$59$	$ T^{2} + 307328 $ T^2 + 307328
$61$	$ T^{2} $ T^2
$67$	$ T^{2} $ T^2
$71$	$ T^{2} $ T^2
$73$	$ (T + 322)^{2} $ (T + 322)^2
$79$	$ (T + 1370)^{2} $ (T + 1370)^2
$83$	$ T^{2} + 1506848 $ T^2 + 1506848
$89$	$ T^{2} $ T^2
$97$	$ (T + 574)^{2} $ (T + 574)^2
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Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	288.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 7 \beta q^{5} + 34 q^{7} + 2 \beta q^{11} - 267 q^{25} + 79 \beta q^{29} + 70 q^{31} + 238 \beta q^{35} + 813 q^{49} + 205 \beta q^{53} - 112 q^{55} - 196 \beta q^{59} - 322 q^{73} + 68 \beta q^{77} + \cdots - 574 q^{97} +O(q^{100}) \) q + 7b q^5 + 34 * q^7 + 2b q^11 - 267 * q^25 + 79b q^29 + 70 * q^31 + 238b q^35 + 813 * q^49 + 205b q^53 - 112 * q^55 - 196b q^59 - 322 * q^73 + 68b q^77 - 1370 * q^79 + 434b q^83 - 574 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 68 q^{7} - 534 q^{25} + 140 q^{31} + 1626 q^{49} - 224 q^{55} - 644 q^{73} - 2740 q^{79} - 1148 q^{97}+O(q^{100}) \) 2 * q + 68 * q^7 - 534 * q^25 + 140 * q^31 + 1626 * q^49 - 224 * q^55 - 644 * q^73 - 2740 * q^79 - 1148 * q^97

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