LMFDB - Newform orbit 105.8.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [105,8,Mod(1,105)] 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("105.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 105 = 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	105.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.8004276758$
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 + 18 q^{2} - 27 q^{3} + 196 q^{4} - 125 q^{5} - 486 q^{6} + 343 q^{7} + 1224 q^{8} + 729 q^{9} - 2250 q^{10} - 8016 q^{11} - 5292 q^{12} - 1786 q^{13} + 6174 q^{14} + 3375 q^{15} - 3056 q^{16} + 8358 q^{17}+ \cdots - 5843664 q^{99}+O(q^{100}) $

q + 18 * q^2 - 27 * q^3 + 196 * q^4 - 125 * q^5 - 486 * q^6 + 343 * q^7 + 1224 * q^8 + 729 * q^9 - 2250 * q^10 - 8016 * q^11 - 5292 * q^12 - 1786 * q^13 + 6174 * q^14 + 3375 * q^15 - 3056 * q^16 + 8358 * q^17 + 13122 * q^18 - 5884 * q^19 - 24500 * q^20 - 9261 * q^21 - 144288 * q^22 - 77700 * q^23 - 33048 * q^24 + 15625 * q^25 - 32148 * q^26 - 19683 * q^27 + 67228 * q^28 + 155742 * q^29 + 60750 * q^30 - 310000 * q^31 - 211680 * q^32 + 216432 * q^33 + 150444 * q^34 - 42875 * q^35 + 142884 * q^36 - 433618 * q^37 - 105912 * q^38 + 48222 * q^39 - 153000 * q^40 + 357942 * q^41 - 166698 * q^42 - 724492 * q^43 - 1571136 * q^44 - 91125 * q^45 - 1398600 * q^46 + 175320 * q^47 + 82512 * q^48 + 117649 * q^49 + 281250 * q^50 - 225666 * q^51 - 350056 * q^52 + 132198 * q^53 - 354294 * q^54 + 1002000 * q^55 + 419832 * q^56 + 158868 * q^57 + 2803356 * q^58 + 2648628 * q^59 + 661500 * q^60 + 835478 * q^61 - 5580000 * q^62 + 250047 * q^63 - 3419072 * q^64 + 223250 * q^65 + 3895776 * q^66 + 3486308 * q^67 + 1638168 * q^68 + 2097900 * q^69 - 771750 * q^70 - 2872260 * q^71 + 892296 * q^72 + 5951882 * q^73 - 7805124 * q^74 - 421875 * q^75 - 1153264 * q^76 - 2749488 * q^77 + 867996 * q^78 - 1680904 * q^79 + 382000 * q^80 + 531441 * q^81 + 6442956 * q^82 + 3577524 * q^83 - 1815156 * q^84 - 1044750 * q^85 - 13040856 * q^86 - 4205034 * q^87 - 9811584 * q^88 - 6254826 * q^89 - 1640250 * q^90 - 612598 * q^91 - 15229200 * q^92 + 8370000 * q^93 + 3155760 * q^94 + 735500 * q^95 + 5715360 * q^96 - 5257054 * q^97 + 2117682 * q^98 - 5843664 * 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

18.0000

−27.0000

196.000

−125.000

−486.000

343.000

1224.00

729.000

−2250.00

Atkin-Lehner signs

$ p $	Sign
$3$	$ +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	105.8.a.b	✓	1
3.b	odd	2	1		315.8.a.a		1
5.b	even	2	1		525.8.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.8.a.b	✓	1	1.a	even	1	1	trivial
315.8.a.a		1	3.b	odd	2	1
525.8.a.a		1	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} - 18 $ T2 - 18 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(105))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 18 $ T - 18
$3$	$ T + 27 $ T + 27
$5$	$ T + 125 $ T + 125
$7$	$ T - 343 $ T - 343
$11$	$ T + 8016 $ T + 8016
$13$	$ T + 1786 $ T + 1786
$17$	$ T - 8358 $ T - 8358
$19$	$ T + 5884 $ T + 5884
$23$	$ T + 77700 $ T + 77700
$29$	$ T - 155742 $ T - 155742
$31$	$ T + 310000 $ T + 310000
$37$	$ T + 433618 $ T + 433618
$41$	$ T - 357942 $ T - 357942
$43$	$ T + 724492 $ T + 724492
$47$	$ T - 175320 $ T - 175320
$53$	$ T - 132198 $ T - 132198
$59$	$ T - 2648628 $ T - 2648628
$61$	$ T - 835478 $ T - 835478
$67$	$ T - 3486308 $ T - 3486308
$71$	$ T + 2872260 $ T + 2872260
$73$	$ T - 5951882 $ T - 5951882
$79$	$ T + 1680904 $ T + 1680904
$83$	$ T - 3577524 $ T - 3577524
$89$	$ T + 6254826 $ T + 6254826
$97$	$ T + 5257054 $ T + 5257054
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)

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