LMFDB - Newform orbit 390.4.a.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [390,4,Mod(1,390)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(390, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("390.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	390.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,-3,4,-5,-6,8,8,9,-10,12] 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:	yes
Analytic conductor:	$23.0107449022$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 + 2 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} - 6 q^{6} + 8 q^{7} + 8 q^{8} + 9 q^{9} - 10 q^{10} + 12 q^{11} - 12 q^{12} + 13 q^{13} + 16 q^{14} + 15 q^{15} + 16 q^{16} - 42 q^{17} + 18 q^{18} - 52 q^{19}+ \cdots + 108 q^{99}+O(q^{100}) $

q + 2 * q^2 - 3 * q^3 + 4 * q^4 - 5 * q^5 - 6 * q^6 + 8 * q^7 + 8 * q^8 + 9 * q^9 - 10 * q^10 + 12 * q^11 - 12 * q^12 + 13 * q^13 + 16 * q^14 + 15 * q^15 + 16 * q^16 - 42 * q^17 + 18 * q^18 - 52 * q^19 - 20 * q^20 - 24 * q^21 + 24 * q^22 + 132 * q^23 - 24 * q^24 + 25 * q^25 + 26 * q^26 - 27 * q^27 + 32 * q^28 + 282 * q^29 + 30 * q^30 + 116 * q^31 + 32 * q^32 - 36 * q^33 - 84 * q^34 - 40 * q^35 + 36 * q^36 + 398 * q^37 - 104 * q^38 - 39 * q^39 - 40 * q^40 + 174 * q^41 - 48 * q^42 - 76 * q^43 + 48 * q^44 - 45 * q^45 + 264 * q^46 + 456 * q^47 - 48 * q^48 - 279 * q^49 + 50 * q^50 + 126 * q^51 + 52 * q^52 + 150 * q^53 - 54 * q^54 - 60 * q^55 + 64 * q^56 + 156 * q^57 + 564 * q^58 - 156 * q^59 + 60 * q^60 + 230 * q^61 + 232 * q^62 + 72 * q^63 + 64 * q^64 - 65 * q^65 - 72 * q^66 - 592 * q^67 - 168 * q^68 - 396 * q^69 - 80 * q^70 + 408 * q^71 + 72 * q^72 - 730 * q^73 + 796 * q^74 - 75 * q^75 - 208 * q^76 + 96 * q^77 - 78 * q^78 + 728 * q^79 - 80 * q^80 + 81 * q^81 + 348 * q^82 + 36 * q^83 - 96 * q^84 + 210 * q^85 - 152 * q^86 - 846 * q^87 + 96 * q^88 - 1482 * q^89 - 90 * q^90 + 104 * q^91 + 528 * q^92 - 348 * q^93 + 912 * q^94 + 260 * q^95 - 96 * q^96 + 1742 * q^97 - 558 * q^98 + 108 * 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

2.00000

−3.00000

4.00000

−5.00000

−6.00000

8.00000

9.00000

−10.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$5$	$ +1 $
$13$	$ -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	390.4.a.i	✓	1
3.b	odd	2	1		1170.4.a.g		1
5.b	even	2	1		1950.4.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.4.a.i	✓	1	1.a	even	1	1	trivial
1170.4.a.g		1	3.b	odd	2	1
1950.4.a.e		1	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(390))$:

$ T_{7} - 8 $

T7 - 8

$ T_{11} - 12 $

T11 - 12

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T + 3 $ T + 3
$5$	$ T + 5 $ T + 5
$7$	$ T - 8 $ T - 8
$11$	$ T - 12 $ T - 12
$13$	$ T - 13 $ T - 13
$17$	$ T + 42 $ T + 42
$19$	$ T + 52 $ T + 52
$23$	$ T - 132 $ T - 132
$29$	$ T - 282 $ T - 282
$31$	$ T - 116 $ T - 116
$37$	$ T - 398 $ T - 398
$41$	$ T - 174 $ T - 174
$43$	$ T + 76 $ T + 76
$47$	$ T - 456 $ T - 456
$53$	$ T - 150 $ T - 150
$59$	$ T + 156 $ T + 156
$61$	$ T - 230 $ T - 230
$67$	$ T + 592 $ T + 592
$71$	$ T - 408 $ T - 408
$73$	$ T + 730 $ T + 730
$79$	$ T - 728 $ T - 728
$83$	$ T - 36 $ T - 36
$89$	$ T + 1482 $ T + 1482
$97$	$ T - 1742 $ T - 1742
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Classical	Maass
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
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