LMFDB - Newform orbit 490.6.a.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [490,6,Mod(1,490)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(490, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("490.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$490 = 2 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	490.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-4,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$78.5880717084$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 70)
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 - 4 q^{2} - 5 q^{3} + 16 q^{4} - 25 q^{5} + 20 q^{6} - 64 q^{8} - 218 q^{9} + 100 q^{10} + 198 q^{11} - 80 q^{12} - 340 q^{13} + 125 q^{15} + 256 q^{16} + 1848 q^{17} + 872 q^{18} - 1210 q^{19} - 400 q^{20}+ \cdots - 43164 q^{99}+O(q^{100})

q - 4 * q^2 - 5 * q^3 + 16 * q^4 - 25 * q^5 + 20 * q^6 - 64 * q^8 - 218 * q^9 + 100 * q^10 + 198 * q^11 - 80 * q^12 - 340 * q^13 + 125 * q^15 + 256 * q^16 + 1848 * q^17 + 872 * q^18 - 1210 * q^19 - 400 * q^20 - 792 * q^22 + 2823 * q^23 + 320 * q^24 + 625 * q^25 + 1360 * q^26 + 2305 * q^27 - 4539 * q^29 - 500 * q^30 - 712 * q^31 - 1024 * q^32 - 990 * q^33 - 7392 * q^34 - 3488 * q^36 - 7324 * q^37 + 4840 * q^38 + 1700 * q^39 + 1600 * q^40 + 15633 * q^41 + 15827 * q^43 + 3168 * q^44 + 5450 * q^45 - 11292 * q^46 - 3192 * q^47 - 1280 * q^48 - 2500 * q^50 - 9240 * q^51 - 5440 * q^52 - 20046 * q^53 - 9220 * q^54 - 4950 * q^55 + 6050 * q^57 + 18156 * q^58 + 23046 * q^59 + 2000 * q^60 - 379 * q^61 + 2848 * q^62 + 4096 * q^64 + 8500 * q^65 + 3960 * q^66 - 35473 * q^67 + 29568 * q^68 - 14115 * q^69 + 71814 * q^71 + 13952 * q^72 + 31664 * q^73 + 29296 * q^74 - 3125 * q^75 - 19360 * q^76 - 6800 * q^78 + 8534 * q^79 - 6400 * q^80 + 41449 * q^81 - 62532 * q^82 - 106551 * q^83 - 46200 * q^85 - 63308 * q^86 + 22695 * q^87 - 12672 * q^88 - 12303 * q^89 - 21800 * q^90 + 45168 * q^92 + 3560 * q^93 + 12768 * q^94 + 30250 * q^95 + 5120 * q^96 - 102802 * q^97 - 43164 * 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

−4.00000

−5.00000

16.0000

−25.0000

20.0000

−64.0000

−218.000

100.000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$
$7$	$+1$

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	490.6.a.d		1
7.b	odd	2	1		490.6.a.f		1
7.c	even	3	2		70.6.e.a	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.6.e.a	✓	2	7.c	even	3	2
490.6.a.d		1	1.a	even	1	1	trivial
490.6.a.f		1	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} + 5$ T3 + 5 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(490))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 4$ T + 4
$3$	$T + 5$ T + 5
$5$	$T + 25$ T + 25
$7$	$T$ T
$11$	$T - 198$ T - 198
$13$	$T + 340$ T + 340
$17$	$T - 1848$ T - 1848
$19$	$T + 1210$ T + 1210
$23$	$T - 2823$ T - 2823
$29$	$T + 4539$ T + 4539
$31$	$T + 712$ T + 712
$37$	$T + 7324$ T + 7324
$41$	$T - 15633$ T - 15633
$43$	$T - 15827$ T - 15827
$47$	$T + 3192$ T + 3192
$53$	$T + 20046$ T + 20046
$59$	$T - 23046$ T - 23046
$61$	$T + 379$ T + 379
$67$	$T + 35473$ T + 35473
$71$	$T - 71814$ T - 71814
$73$	$T - 31664$ T - 31664
$79$	$T - 8534$ T - 8534
$83$	$T + 106551$ T + 106551
$89$	$T + 12303$ T + 12303
$97$	$T + 102802$ T + 102802
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