LMFDB - Newform orbit 1445.4.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1445 = 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1445.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,8,-7,5] 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:	yes
Analytic conductor:	$85.2577599583$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
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)

$f(q)$

$=$

$ q - q^{2} + 8 q^{3} - 7 q^{4} + 5 q^{5} - 8 q^{6} + 14 q^{7} + 15 q^{8} + 37 q^{9} - 5 q^{10} - 20 q^{11} - 56 q^{12} - 58 q^{13} - 14 q^{14} + 40 q^{15} + 41 q^{16} - 37 q^{18} - 80 q^{19} - 35 q^{20}+ \cdots - 740 q^{99}+O(q^{100}) $

q - q^2 + 8 * q^3 - 7 * q^4 + 5 * q^5 - 8 * q^6 + 14 * q^7 + 15 * q^8 + 37 * q^9 - 5 * q^10 - 20 * q^11 - 56 * q^12 - 58 * q^13 - 14 * q^14 + 40 * q^15 + 41 * q^16 - 37 * q^18 - 80 * q^19 - 35 * q^20 + 112 * q^21 + 20 * q^22 - 118 * q^23 + 120 * q^24 + 25 * q^25 + 58 * q^26 + 80 * q^27 - 98 * q^28 + 126 * q^29 - 40 * q^30 + 70 * q^31 - 161 * q^32 - 160 * q^33 + 70 * q^35 - 259 * q^36 - 134 * q^37 + 80 * q^38 - 464 * q^39 + 75 * q^40 + 100 * q^41 - 112 * q^42 - 272 * q^43 + 140 * q^44 + 185 * q^45 + 118 * q^46 - 464 * q^47 + 328 * q^48 - 147 * q^49 - 25 * q^50 + 406 * q^52 - 642 * q^53 - 80 * q^54 - 100 * q^55 + 210 * q^56 - 640 * q^57 - 126 * q^58 + 180 * q^59 - 280 * q^60 - 110 * q^61 - 70 * q^62 + 518 * q^63 - 167 * q^64 - 290 * q^65 + 160 * q^66 - 924 * q^67 - 944 * q^69 - 70 * q^70 + 90 * q^71 + 555 * q^72 + 828 * q^73 + 134 * q^74 + 200 * q^75 + 560 * q^76 - 280 * q^77 + 464 * q^78 + 1334 * q^79 + 205 * q^80 - 359 * q^81 - 100 * q^82 - 552 * q^83 - 784 * q^84 + 272 * q^86 + 1008 * q^87 - 300 * q^88 + 1490 * q^89 - 185 * q^90 - 812 * q^91 + 826 * q^92 + 560 * q^93 + 464 * q^94 - 400 * q^95 - 1288 * q^96 + 1376 * q^97 + 147 * q^98 - 740 * q^99

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

−1.00000

8.00000

−7.00000

5.00000

−8.00000

14.0000

15.0000

37.0000

−5.00000

Atkin-Lehner signs

$ p $	Sign
$5$	$ -1 $
$17$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1445.4.a.e		1
17.b	even	2	1		1445.4.a.d		1
17.c	even	4	2		85.4.d.a	✓	2
51.f	odd	4	2		765.4.g.a		2
85.f	odd	4	2		425.4.c.a		2
85.i	odd	4	2		425.4.c.b		2
85.j	even	4	2		425.4.d.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.4.d.a	✓	2	17.c	even	4	2
425.4.c.a		2	85.f	odd	4	2
425.4.c.b		2	85.i	odd	4	2
425.4.d.a		2	85.j	even	4	2
765.4.g.a		2	51.f	odd	4	2
1445.4.a.d		1	17.b	even	2	1
1445.4.a.e		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1445))$:

$ T_{2} + 1 $

T2 + 1

$ T_{3} - 8 $

T3 - 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T - 8 $ T - 8
$5$	$ T - 5 $ T - 5
$7$	$ T - 14 $ T - 14
$11$	$ T + 20 $ T + 20
$13$	$ T + 58 $ T + 58
$17$	$ T $ T
$19$	$ T + 80 $ T + 80
$23$	$ T + 118 $ T + 118
$29$	$ T - 126 $ T - 126
$31$	$ T - 70 $ T - 70
$37$	$ T + 134 $ T + 134
$41$	$ T - 100 $ T - 100
$43$	$ T + 272 $ T + 272
$47$	$ T + 464 $ T + 464
$53$	$ T + 642 $ T + 642
$59$	$ T - 180 $ T - 180
$61$	$ T + 110 $ T + 110
$67$	$ T + 924 $ T + 924
$71$	$ T - 90 $ T - 90
$73$	$ T - 828 $ T - 828
$79$	$ T - 1334 $ T - 1334
$83$	$ T + 552 $ T + 552
$89$	$ T - 1490 $ T - 1490
$97$	$ T - 1376 $ T - 1376
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Level:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1445.a (trivial)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
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