LMFDB - Newform orbit 490.8.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [490,8,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, 8, 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, 8); N := Newforms(S);

Level:	$N$	$=$	$490 = 2 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	490.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,8,93] 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:	$153.068662487$
Analytic rank:	$0$
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 + 8 q^{2} + 93 q^{3} + 64 q^{4} - 125 q^{5} + 744 q^{6} + 512 q^{8} + 6462 q^{9} - 1000 q^{10} - 2167 q^{11} + 5952 q^{12} + 1661 q^{13} - 11625 q^{15} + 4096 q^{16} + 35771 q^{17} + 51696 q^{18} - 20222 q^{19}+ \cdots - 14003154 q^{99}+O(q^{100})

q + 8 * q^2 + 93 * q^3 + 64 * q^4 - 125 * q^5 + 744 * q^6 + 512 * q^8 + 6462 * q^9 - 1000 * q^10 - 2167 * q^11 + 5952 * q^12 + 1661 * q^13 - 11625 * q^15 + 4096 * q^16 + 35771 * q^17 + 51696 * q^18 - 20222 * q^19 - 8000 * q^20 - 17336 * q^22 - 42130 * q^23 + 47616 * q^24 + 15625 * q^25 + 13288 * q^26 + 397575 * q^27 - 111789 * q^29 - 93000 * q^30 + 269504 * q^31 + 32768 * q^32 - 201531 * q^33 + 286168 * q^34 + 413568 * q^36 + 532774 * q^37 - 161776 * q^38 + 154473 * q^39 - 64000 * q^40 - 158056 * q^41 - 521874 * q^43 - 138688 * q^44 - 807750 * q^45 - 337040 * q^46 + 939733 * q^47 + 380928 * q^48 + 125000 * q^50 + 3326703 * q^51 + 106304 * q^52 - 408384 * q^53 + 3180600 * q^54 + 270875 * q^55 - 1880646 * q^57 - 894312 * q^58 + 522172 * q^59 - 744000 * q^60 - 350080 * q^61 + 2156032 * q^62 + 262144 * q^64 - 207625 * q^65 - 1612248 * q^66 - 3931176 * q^67 + 2289344 * q^68 - 3918090 * q^69 + 1194016 * q^71 + 3308544 * q^72 - 998350 * q^73 + 4262192 * q^74 + 1453125 * q^75 - 1294208 * q^76 + 1235784 * q^78 - 2120709 * q^79 - 512000 * q^80 + 22842081 * q^81 - 1264448 * q^82 + 1746708 * q^83 - 4471375 * q^85 - 4174992 * q^86 - 10396377 * q^87 - 1109504 * q^88 + 10077740 * q^89 - 6462000 * q^90 - 2696320 * q^92 + 25063872 * q^93 + 7517864 * q^94 + 2527750 * q^95 + 3047424 * q^96 + 6238295 * q^97 - 14003154 * 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

8.00000

93.0000

64.0000

−125.000

744.000

512.000

6462.00

−1000.00

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	490.8.a.e		1
7.b	odd	2	1		70.8.a.a	✓	1
28.d	even	2	1		560.8.a.b		1
35.c	odd	2	1		350.8.a.e		1
35.f	even	4	2		350.8.c.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.8.a.a	✓	1	7.b	odd	2	1
350.8.a.e		1	35.c	odd	2	1
350.8.c.a		2	35.f	even	4	2
490.8.a.e		1	1.a	even	1	1	trivial
560.8.a.b		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 93$ T3 - 93 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(490))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 8$ T - 8
$3$	$T - 93$ T - 93
$5$	$T + 125$ T + 125
$7$	$T$ T
$11$	$T + 2167$ T + 2167
$13$	$T - 1661$ T - 1661
$17$	$T - 35771$ T - 35771
$19$	$T + 20222$ T + 20222
$23$	$T + 42130$ T + 42130
$29$	$T + 111789$ T + 111789
$31$	$T - 269504$ T - 269504
$37$	$T - 532774$ T - 532774
$41$	$T + 158056$ T + 158056
$43$	$T + 521874$ T + 521874
$47$	$T - 939733$ T - 939733
$53$	$T + 408384$ T + 408384
$59$	$T - 522172$ T - 522172
$61$	$T + 350080$ T + 350080
$67$	$T + 3931176$ T + 3931176
$71$	$T - 1194016$ T - 1194016
$73$	$T + 998350$ T + 998350
$79$	$T + 2120709$ T + 2120709
$83$	$T - 1746708$ T - 1746708
$89$	$T - 10077740$ T - 10077740
$97$	$T - 6238295$ T - 6238295
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