LMFDB - Newform orbit 42.10.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [42,10,Mod(1,42)] 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("42.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	42.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,16,81,256,-1634] 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:	$21.6315051189$
Analytic rank:	$1$
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 + 16 q^{2} + 81 q^{3} + 256 q^{4} - 1634 q^{5} + 1296 q^{6} - 2401 q^{7} + 4096 q^{8} + 6561 q^{9} - 26144 q^{10} - 71164 q^{11} + 20736 q^{12} - 102402 q^{13} - 38416 q^{14} - 132354 q^{15} + 65536 q^{16}+ \cdots - 466907004 q^{99}+O(q^{100}) $

q + 16 * q^2 + 81 * q^3 + 256 * q^4 - 1634 * q^5 + 1296 * q^6 - 2401 * q^7 + 4096 * q^8 + 6561 * q^9 - 26144 * q^10 - 71164 * q^11 + 20736 * q^12 - 102402 * q^13 - 38416 * q^14 - 132354 * q^15 + 65536 * q^16 - 181798 * q^17 + 104976 * q^18 + 592964 * q^19 - 418304 * q^20 - 194481 * q^21 - 1138624 * q^22 - 754528 * q^23 + 331776 * q^24 + 716831 * q^25 - 1638432 * q^26 + 531441 * q^27 - 614656 * q^28 - 3968162 * q^29 - 2117664 * q^30 - 1068480 * q^31 + 1048576 * q^32 - 5764284 * q^33 - 2908768 * q^34 + 3923234 * q^35 + 1679616 * q^36 - 6329434 * q^37 + 9487424 * q^38 - 8294562 * q^39 - 6692864 * q^40 + 32715234 * q^41 - 3111696 * q^42 - 19074724 * q^43 - 18217984 * q^44 - 10720674 * q^45 - 12072448 * q^46 - 58195200 * q^47 + 5308416 * q^48 + 5764801 * q^49 + 11469296 * q^50 - 14725638 * q^51 - 26214912 * q^52 + 61610790 * q^53 + 8503056 * q^54 + 116281976 * q^55 - 9834496 * q^56 + 48030084 * q^57 - 63490592 * q^58 + 26642572 * q^59 - 33882624 * q^60 + 156889854 * q^61 - 17095680 * q^62 - 15752961 * q^63 + 16777216 * q^64 + 167324868 * q^65 - 92228544 * q^66 + 120969508 * q^67 - 46540288 * q^68 - 61116768 * q^69 + 62771744 * q^70 - 51310048 * q^71 + 26873856 * q^72 + 199480570 * q^73 - 101270944 * q^74 + 58063311 * q^75 + 151798784 * q^76 + 170864764 * q^77 - 132712992 * q^78 + 16131696 * q^79 - 107085824 * q^80 + 43046721 * q^81 + 523443744 * q^82 + 323632628 * q^83 - 49787136 * q^84 + 297057932 * q^85 - 305195584 * q^86 - 321421122 * q^87 - 291487744 * q^88 - 797470830 * q^89 - 171530784 * q^90 + 245867202 * q^91 - 193159168 * q^92 - 86546880 * q^93 - 931123200 * q^94 - 968903176 * q^95 + 84934656 * q^96 - 1043298158 * q^97 + 92236816 * q^98 - 466907004 * 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

16.0000

81.0000

256.000

−1634.00

1296.00

−2401.00

4096.00

6561.00

−26144.0

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -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	42.10.a.f	✓	1
3.b	odd	2	1		126.10.a.c		1
4.b	odd	2	1		336.10.a.b		1
7.b	odd	2	1		294.10.a.h		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.10.a.f	✓	1	1.a	even	1	1	trivial
126.10.a.c		1	3.b	odd	2	1
294.10.a.h		1	7.b	odd	2	1
336.10.a.b		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 1634 $ T5 + 1634 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(42))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 16 $ T - 16
$3$	$ T - 81 $ T - 81
$5$	$ T + 1634 $ T + 1634
$7$	$ T + 2401 $ T + 2401
$11$	$ T + 71164 $ T + 71164
$13$	$ T + 102402 $ T + 102402
$17$	$ T + 181798 $ T + 181798
$19$	$ T - 592964 $ T - 592964
$23$	$ T + 754528 $ T + 754528
$29$	$ T + 3968162 $ T + 3968162
$31$	$ T + 1068480 $ T + 1068480
$37$	$ T + 6329434 $ T + 6329434
$41$	$ T - 32715234 $ T - 32715234
$43$	$ T + 19074724 $ T + 19074724
$47$	$ T + 58195200 $ T + 58195200
$53$	$ T - 61610790 $ T - 61610790
$59$	$ T - 26642572 $ T - 26642572
$61$	$ T - 156889854 $ T - 156889854
$67$	$ T - 120969508 $ T - 120969508
$71$	$ T + 51310048 $ T + 51310048
$73$	$ T - 199480570 $ T - 199480570
$79$	$ T - 16131696 $ T - 16131696
$83$	$ T - 323632628 $ T - 323632628
$89$	$ T + 797470830 $ T + 797470830
$97$	$ T + 1043298158 $ T + 1043298158
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Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)

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