LMFDB - Newform orbit 294.10.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,10,Mod(1,294)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, 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("294.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-16,81,256,76] 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:	$151.420535832$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
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} + 76 q^{5} - 1296 q^{6} - 4096 q^{8} + 6561 q^{9} - 1216 q^{10} + 38386 q^{11} + 20736 q^{12} - 98298 q^{13} + 6156 q^{15} + 65536 q^{16} - 104524 q^{17} - 104976 q^{18}+ \cdots + 251850546 q^{99}+O(q^{100})

q - 16 * q^2 + 81 * q^3 + 256 * q^4 + 76 * q^5 - 1296 * q^6 - 4096 * q^8 + 6561 * q^9 - 1216 * q^10 + 38386 * q^11 + 20736 * q^12 - 98298 * q^13 + 6156 * q^15 + 65536 * q^16 - 104524 * q^17 - 104976 * q^18 + 420580 * q^19 + 19456 * q^20 - 614176 * q^22 - 139118 * q^23 - 331776 * q^24 - 1947349 * q^25 + 1572768 * q^26 + 531441 * q^27 - 1916290 * q^29 - 98496 * q^30 + 6379488 * q^31 - 1048576 * q^32 + 3109266 * q^33 + 1672384 * q^34 + 1679616 * q^36 - 6629278 * q^37 - 6729280 * q^38 - 7962138 * q^39 - 311296 * q^40 + 6692112 * q^41 - 23269732 * q^43 + 9826816 * q^44 + 498636 * q^45 + 2225888 * q^46 + 22000596 * q^47 + 5308416 * q^48 + 31157584 * q^50 - 8466444 * q^51 - 25164288 * q^52 + 18919770 * q^53 - 8503056 * q^54 + 2917336 * q^55 + 34066980 * q^57 + 30660640 * q^58 - 179035544 * q^59 + 1575936 * q^60 + 19797786 * q^61 - 102071808 * q^62 + 16777216 * q^64 - 7470648 * q^65 - 49748256 * q^66 - 263015240 * q^67 - 26758144 * q^68 - 11268558 * q^69 + 22447678 * q^71 - 26873856 * q^72 - 11023774 * q^73 + 106068448 * q^74 - 157735269 * q^75 + 107668480 * q^76 + 127394208 * q^78 - 284917908 * q^79 + 4980736 * q^80 + 43046721 * q^81 - 107073792 * q^82 + 226865924 * q^83 - 7943824 * q^85 + 372315712 * q^86 - 155219490 * q^87 - 157229056 * q^88 + 191377296 * q^89 - 7978176 * q^90 - 35614208 * q^92 + 516738528 * q^93 - 352009536 * q^94 + 31964080 * q^95 - 84934656 * q^96 + 1162236578 * q^97 + 251850546 * 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

76.0000

−1296.00

−4096.00

6561.00

−1216.00

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	294.10.a.f		1
7.b	odd	2	1		42.10.a.a	✓	1
21.c	even	2	1		126.10.a.f		1
28.d	even	2	1		336.10.a.f		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.10.a.a	✓	1	7.b	odd	2	1
126.10.a.f		1	21.c	even	2	1
294.10.a.f		1	1.a	even	1	1	trivial
336.10.a.f		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} - 76$ T5 - 76 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(294))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 16$ T + 16
$3$	$T - 81$ T - 81
$5$	$T - 76$ T - 76
$7$	$T$ T
$11$	$T - 38386$ T - 38386
$13$	$T + 98298$ T + 98298
$17$	$T + 104524$ T + 104524
$19$	$T - 420580$ T - 420580
$23$	$T + 139118$ T + 139118
$29$	$T + 1916290$ T + 1916290
$31$	$T - 6379488$ T - 6379488
$37$	$T + 6629278$ T + 6629278
$41$	$T - 6692112$ T - 6692112
$43$	$T + 23269732$ T + 23269732
$47$	$T - 22000596$ T - 22000596
$53$	$T - 18919770$ T - 18919770
$59$	$T + 179035544$ T + 179035544
$61$	$T - 19797786$ T - 19797786
$67$	$T + 263015240$ T + 263015240
$71$	$T - 22447678$ T - 22447678
$73$	$T + 11023774$ T + 11023774
$79$	$T + 284917908$ T + 284917908
$83$	$T - 226865924$ T - 226865924
$89$	$T - 191377296$ T - 191377296
$97$	$T - 1162236578$ T - 1162236578
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