LMFDB - Newform orbit 6.10.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6,10,Mod(1,6)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("6.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$N$	$=$	$6 = 2 \cdot 3$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	6.a (trivial)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.09021501698$
Analytic rank:	$0$
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} + 2694 q^{5} - 1296 q^{6} - 3544 q^{7} - 4096 q^{8} + 6561 q^{9} - 43104 q^{10} + 29580 q^{11} + 20736 q^{12} - 44818 q^{13} + 56704 q^{14} + 218214 q^{15} + 65536 q^{16}+ \cdots + 194074380 q^{99}+O(q^{100})

q - 16 * q^2 + 81 * q^3 + 256 * q^4 + 2694 * q^5 - 1296 * q^6 - 3544 * q^7 - 4096 * q^8 + 6561 * q^9 - 43104 * q^10 + 29580 * q^11 + 20736 * q^12 - 44818 * q^13 + 56704 * q^14 + 218214 * q^15 + 65536 * q^16 - 101934 * q^17 - 104976 * q^18 - 895084 * q^19 + 689664 * q^20 - 287064 * q^21 - 473280 * q^22 - 1113000 * q^23 - 331776 * q^24 + 5304511 * q^25 + 717088 * q^26 + 531441 * q^27 - 907264 * q^28 - 2357346 * q^29 - 3491424 * q^30 + 175808 * q^31 - 1048576 * q^32 + 2395980 * q^33 + 1630944 * q^34 - 9547536 * q^35 + 1679616 * q^36 - 2919418 * q^37 + 14321344 * q^38 - 3630258 * q^39 - 11034624 * q^40 + 26218794 * q^41 + 4593024 * q^42 - 18762964 * q^43 + 7572480 * q^44 + 17675334 * q^45 + 17808000 * q^46 - 20966160 * q^47 + 5308416 * q^48 - 27793671 * q^49 - 84872176 * q^50 - 8256654 * q^51 - 11473408 * q^52 + 57251574 * q^53 - 8503056 * q^54 + 79688520 * q^55 + 14516224 * q^56 - 72501804 * q^57 + 37717536 * q^58 + 33587580 * q^59 + 55862784 * q^60 + 82260830 * q^61 - 2812928 * q^62 - 23252184 * q^63 + 16777216 * q^64 - 120739692 * q^65 - 38335680 * q^66 - 188455804 * q^67 - 26095104 * q^68 - 90153000 * q^69 + 152760576 * q^70 + 80924040 * q^71 - 26873856 * q^72 - 236140918 * q^73 + 46710688 * q^74 + 429665391 * q^75 - 229141504 * q^76 - 104831520 * q^77 + 58084128 * q^78 + 526909808 * q^79 + 176553984 * q^80 + 43046721 * q^81 - 419500704 * q^82 + 18346452 * q^83 - 73488384 * q^84 - 274610196 * q^85 + 300207424 * q^86 - 190945026 * q^87 - 121159680 * q^88 + 690643098 * q^89 - 282805344 * q^90 + 158834992 * q^91 - 284928000 * q^92 + 14240448 * q^93 + 335458560 * q^94 - 2411356296 * q^95 - 84934656 * q^96 - 438251038 * q^97 + 444698736 * q^98 + 194074380 * 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

2694.00

−1296.00

−3544.00

−4096.00

6561.00

−43104.0

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$

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	6.10.a.a	✓	1
3.b	odd	2	1		18.10.a.c		1
4.b	odd	2	1		48.10.a.d		1
5.b	even	2	1		150.10.a.h		1
5.c	odd	4	2		150.10.c.d		2
7.b	odd	2	1		294.10.a.a		1
8.b	even	2	1		192.10.a.a		1
8.d	odd	2	1		192.10.a.h		1
9.c	even	3	2		162.10.c.f		2
9.d	odd	6	2		162.10.c.e		2
12.b	even	2	1		144.10.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.10.a.a	✓	1	1.a	even	1	1	trivial
18.10.a.c		1	3.b	odd	2	1
48.10.a.d		1	4.b	odd	2	1
144.10.a.a		1	12.b	even	2	1
150.10.a.h		1	5.b	even	2	1
150.10.c.d		2	5.c	odd	4	2
162.10.c.e		2	9.d	odd	6	2
162.10.c.f		2	9.c	even	3	2
192.10.a.a		1	8.b	even	2	1
192.10.a.h		1	8.d	odd	2	1
294.10.a.a		1	7.b	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{10}^{\mathrm{new}}(\Gamma_0(6))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 16$ T + 16
$3$	$T - 81$ T - 81
$5$	$T - 2694$ T - 2694
$7$	$T + 3544$ T + 3544
$11$	$T - 29580$ T - 29580
$13$	$T + 44818$ T + 44818
$17$	$T + 101934$ T + 101934
$19$	$T + 895084$ T + 895084
$23$	$T + 1113000$ T + 1113000
$29$	$T + 2357346$ T + 2357346
$31$	$T - 175808$ T - 175808
$37$	$T + 2919418$ T + 2919418
$41$	$T - 26218794$ T - 26218794
$43$	$T + 18762964$ T + 18762964
$47$	$T + 20966160$ T + 20966160
$53$	$T - 57251574$ T - 57251574
$59$	$T - 33587580$ T - 33587580
$61$	$T - 82260830$ T - 82260830
$67$	$T + 188455804$ T + 188455804
$71$	$T - 80924040$ T - 80924040
$73$	$T + 236140918$ T + 236140918
$79$	$T - 526909808$ T - 526909808
$83$	$T - 18346452$ T - 18346452
$89$	$T - 690643098$ T - 690643098
$97$	$T + 438251038$ T + 438251038
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