LMFDB - Newform orbit 294.8.a.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,8,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, 8, 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, 8); N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-8,27,64,18] 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:	$91.8411974923$
Analytic rank:	$0$
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 - 8 q^{2} + 27 q^{3} + 64 q^{4} + 18 q^{5} - 216 q^{6} - 512 q^{8} + 729 q^{9} - 144 q^{10} + 8172 q^{11} + 1728 q^{12} + 14242 q^{13} + 486 q^{15} + 4096 q^{16} + 21462 q^{17} - 5832 q^{18} + 5884 q^{19}+ \cdots + 5957388 q^{99}+O(q^{100})

q - 8 * q^2 + 27 * q^3 + 64 * q^4 + 18 * q^5 - 216 * q^6 - 512 * q^8 + 729 * q^9 - 144 * q^10 + 8172 * q^11 + 1728 * q^12 + 14242 * q^13 + 486 * q^15 + 4096 * q^16 + 21462 * q^17 - 5832 * q^18 + 5884 * q^19 + 1152 * q^20 - 65376 * q^22 - 98784 * q^23 - 13824 * q^24 - 77801 * q^25 - 113936 * q^26 + 19683 * q^27 + 165174 * q^29 - 3888 * q^30 + 241312 * q^31 - 32768 * q^32 + 220644 * q^33 - 171696 * q^34 + 46656 * q^36 + 185438 * q^37 - 47072 * q^38 + 384534 * q^39 - 9216 * q^40 - 59682 * q^41 - 809308 * q^43 + 523008 * q^44 + 13122 * q^45 + 790272 * q^46 - 942096 * q^47 + 110592 * q^48 + 622408 * q^50 + 579474 * q^51 + 911488 * q^52 + 226398 * q^53 - 157464 * q^54 + 147096 * q^55 + 158868 * q^57 - 1321392 * q^58 + 2205732 * q^59 + 31104 * q^60 + 1156690 * q^61 - 1930496 * q^62 + 262144 * q^64 + 256356 * q^65 - 1765152 * q^66 - 3740404 * q^67 + 1373568 * q^68 - 2667168 * q^69 - 2593296 * q^71 - 373248 * q^72 + 1038742 * q^73 - 1483504 * q^74 - 2100627 * q^75 + 376576 * q^76 - 3076272 * q^78 + 2280032 * q^79 + 73728 * q^80 + 531441 * q^81 + 477456 * q^82 + 283404 * q^83 + 386316 * q^85 + 6474464 * q^86 + 4459698 * q^87 - 4184064 * q^88 + 5227230 * q^89 - 104976 * q^90 - 6322176 * q^92 + 6515424 * q^93 + 7536768 * q^94 + 105912 * q^95 - 884736 * q^96 - 6168770 * q^97 + 5957388 * 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

−8.00000

27.0000

64.0000

18.0000

−216.000

−512.000

729.000

−144.000

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	294.8.a.h		1
7.b	odd	2	1		42.8.a.b	✓	1
7.c	even	3	2		294.8.e.i		2
7.d	odd	6	2		294.8.e.o		2
21.c	even	2	1		126.8.a.f		1
28.d	even	2	1		336.8.a.h		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.8.a.b	✓	1	7.b	odd	2	1
126.8.a.f		1	21.c	even	2	1
294.8.a.h		1	1.a	even	1	1	trivial
294.8.e.i		2	7.c	even	3	2
294.8.e.o		2	7.d	odd	6	2
336.8.a.h		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} - 18$ T5 - 18 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(294))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 8$ T + 8
$3$	$T - 27$ T - 27
$5$	$T - 18$ T - 18
$7$	$T$ T
$11$	$T - 8172$ T - 8172
$13$	$T - 14242$ T - 14242
$17$	$T - 21462$ T - 21462
$19$	$T - 5884$ T - 5884
$23$	$T + 98784$ T + 98784
$29$	$T - 165174$ T - 165174
$31$	$T - 241312$ T - 241312
$37$	$T - 185438$ T - 185438
$41$	$T + 59682$ T + 59682
$43$	$T + 809308$ T + 809308
$47$	$T + 942096$ T + 942096
$53$	$T - 226398$ T - 226398
$59$	$T - 2205732$ T - 2205732
$61$	$T - 1156690$ T - 1156690
$67$	$T + 3740404$ T + 3740404
$71$	$T + 2593296$ T + 2593296
$73$	$T - 1038742$ T - 1038742
$79$	$T - 2280032$ T - 2280032
$83$	$T - 283404$ T - 283404
$89$	$T - 5227230$ T - 5227230
$97$	$T + 6168770$ T + 6168770
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