LMFDB - Newform orbit 66.10.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [66,10,Mod(1,66)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(66, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("66.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 66 = 2 \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	66.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,16,81,256,-1054,1296,-3640] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.9923651868$
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} - 1054 q^{5} + 1296 q^{6} - 3640 q^{7} + 4096 q^{8} + 6561 q^{9} - 16864 q^{10} - 14641 q^{11} + 20736 q^{12} - 29142 q^{13} - 58240 q^{14} - 85374 q^{15} + 65536 q^{16}+ \cdots - 96059601 q^{99}+O(q^{100}) $

q + 16 * q^2 + 81 * q^3 + 256 * q^4 - 1054 * q^5 + 1296 * q^6 - 3640 * q^7 + 4096 * q^8 + 6561 * q^9 - 16864 * q^10 - 14641 * q^11 + 20736 * q^12 - 29142 * q^13 - 58240 * q^14 - 85374 * q^15 + 65536 * q^16 - 413218 * q^17 + 104976 * q^18 - 464024 * q^19 - 269824 * q^20 - 294840 * q^21 - 234256 * q^22 - 568700 * q^23 + 331776 * q^24 - 842209 * q^25 - 466272 * q^26 + 531441 * q^27 - 931840 * q^28 + 1740738 * q^29 - 1365984 * q^30 - 2235240 * q^31 + 1048576 * q^32 - 1185921 * q^33 - 6611488 * q^34 + 3836560 * q^35 + 1679616 * q^36 - 5730762 * q^37 - 7424384 * q^38 - 2360502 * q^39 - 4317184 * q^40 - 1462098 * q^41 - 4717440 * q^42 + 2021968 * q^43 - 3748096 * q^44 - 6915294 * q^45 - 9099200 * q^46 - 2708052 * q^47 + 5308416 * q^48 - 27104007 * q^49 - 13475344 * q^50 - 33470658 * q^51 - 7460352 * q^52 - 29309734 * q^53 + 8503056 * q^54 + 15431614 * q^55 - 14909440 * q^56 - 37585944 * q^57 + 27851808 * q^58 - 3160212 * q^59 - 21855744 * q^60 + 91013146 * q^61 - 35763840 * q^62 - 23882040 * q^63 + 16777216 * q^64 + 30715668 * q^65 - 18974736 * q^66 + 80652364 * q^67 - 105783808 * q^68 - 46064700 * q^69 + 61384960 * q^70 - 350770740 * q^71 + 26873856 * q^72 - 172526526 * q^73 - 91692192 * q^74 - 68218929 * q^75 - 118790144 * q^76 + 53293240 * q^77 - 37768032 * q^78 - 55281976 * q^79 - 69074944 * q^80 + 43046721 * q^81 - 23393568 * q^82 + 207382444 * q^83 - 75479040 * q^84 + 435531772 * q^85 + 32351488 * q^86 + 140999778 * q^87 - 59969536 * q^88 + 1114520170 * q^89 - 110644704 * q^90 + 106076880 * q^91 - 145587200 * q^92 - 181054440 * q^93 - 43328832 * q^94 + 489081296 * q^95 + 84934656 * q^96 + 1164373042 * q^97 - 433664112 * q^98 - 96059601 * 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

−1054.00

1296.00

−3640.00

4096.00

6561.00

−16864.0

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$11$	$ +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	66.10.a.b	✓	1
3.b	odd	2	1		198.10.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
66.10.a.b	✓	1	1.a	even	1	1	trivial
198.10.a.e		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 1054 $ T5 + 1054 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(66))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 16 $ T - 16
$3$	$ T - 81 $ T - 81
$5$	$ T + 1054 $ T + 1054
$7$	$ T + 3640 $ T + 3640
$11$	$ T + 14641 $ T + 14641
$13$	$ T + 29142 $ T + 29142
$17$	$ T + 413218 $ T + 413218
$19$	$ T + 464024 $ T + 464024
$23$	$ T + 568700 $ T + 568700
$29$	$ T - 1740738 $ T - 1740738
$31$	$ T + 2235240 $ T + 2235240
$37$	$ T + 5730762 $ T + 5730762
$41$	$ T + 1462098 $ T + 1462098
$43$	$ T - 2021968 $ T - 2021968
$47$	$ T + 2708052 $ T + 2708052
$53$	$ T + 29309734 $ T + 29309734
$59$	$ T + 3160212 $ T + 3160212
$61$	$ T - 91013146 $ T - 91013146
$67$	$ T - 80652364 $ T - 80652364
$71$	$ T + 350770740 $ T + 350770740
$73$	$ T + 172526526 $ T + 172526526
$79$	$ T + 55281976 $ T + 55281976
$83$	$ T - 207382444 $ T - 207382444
$89$	$ T - 1114520170 $ T - 1114520170
$97$	$ T - 1164373042 $ T - 1164373042
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Classical	Maass
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 66 = 2 \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	66.a (trivial)

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