LMFDB - Newform orbit 294.6.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,6,Mod(1,294)] 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("294.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-4,-9,16,-26,36,0,-64,81,104,-358] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.1528430250$
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 - 4 q^{2} - 9 q^{3} + 16 q^{4} - 26 q^{5} + 36 q^{6} - 64 q^{8} + 81 q^{9} + 104 q^{10} - 358 q^{11} - 144 q^{12} - 332 q^{13} + 234 q^{15} + 256 q^{16} - 126 q^{17} - 324 q^{18} + 2200 q^{19} - 416 q^{20}+ \cdots - 28998 q^{99}+O(q^{100}) $

q - 4 * q^2 - 9 * q^3 + 16 * q^4 - 26 * q^5 + 36 * q^6 - 64 * q^8 + 81 * q^9 + 104 * q^10 - 358 * q^11 - 144 * q^12 - 332 * q^13 + 234 * q^15 + 256 * q^16 - 126 * q^17 - 324 * q^18 + 2200 * q^19 - 416 * q^20 + 1432 * q^22 - 2142 * q^23 + 576 * q^24 - 2449 * q^25 + 1328 * q^26 - 729 * q^27 - 3610 * q^29 - 936 * q^30 - 5668 * q^31 - 1024 * q^32 + 3222 * q^33 + 504 * q^34 + 1296 * q^36 - 2922 * q^37 - 8800 * q^38 + 2988 * q^39 + 1664 * q^40 + 2142 * q^41 + 6388 * q^43 - 5728 * q^44 - 2106 * q^45 + 8568 * q^46 + 6520 * q^47 - 2304 * q^48 + 9796 * q^50 + 1134 * q^51 - 5312 * q^52 - 10702 * q^53 + 2916 * q^54 + 9308 * q^55 - 19800 * q^57 + 14440 * q^58 - 42524 * q^59 + 3744 * q^60 + 44840 * q^61 + 22672 * q^62 + 4096 * q^64 + 8632 * q^65 - 12888 * q^66 - 1448 * q^67 - 2016 * q^68 + 19278 * q^69 - 4402 * q^71 - 5184 * q^72 - 20500 * q^73 + 11688 * q^74 + 22041 * q^75 + 35200 * q^76 - 11952 * q^78 + 65236 * q^79 - 6656 * q^80 + 6561 * q^81 - 8568 * q^82 + 102804 * q^83 + 3276 * q^85 - 25552 * q^86 + 32490 * q^87 + 22912 * q^88 + 128006 * q^89 + 8424 * q^90 - 34272 * q^92 + 51012 * q^93 - 26080 * q^94 - 57200 * q^95 + 9216 * q^96 + 113324 * q^97 - 28998 * 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

−4.00000

−9.00000

16.0000

−26.0000

36.0000

−64.0000

81.0000

104.000

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	294.6.a.a	✓	1
3.b	odd	2	1		882.6.a.t		1
7.b	odd	2	1		294.6.a.g	yes	1
7.c	even	3	2		294.6.e.q		2
7.d	odd	6	2		294.6.e.j		2
21.c	even	2	1		882.6.a.p		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
294.6.a.a	✓	1	1.a	even	1	1	trivial
294.6.a.g	yes	1	7.b	odd	2	1
294.6.e.j		2	7.d	odd	6	2
294.6.e.q		2	7.c	even	3	2
882.6.a.p		1	21.c	even	2	1
882.6.a.t		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(294))$:

$ T_{5} + 26 $

T5 + 26

$ T_{11} + 358 $

T11 + 358

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 4 $ T + 4
$3$	$ T + 9 $ T + 9
$5$	$ T + 26 $ T + 26
$7$	$ T $ T
$11$	$ T + 358 $ T + 358
$13$	$ T + 332 $ T + 332
$17$	$ T + 126 $ T + 126
$19$	$ T - 2200 $ T - 2200
$23$	$ T + 2142 $ T + 2142
$29$	$ T + 3610 $ T + 3610
$31$	$ T + 5668 $ T + 5668
$37$	$ T + 2922 $ T + 2922
$41$	$ T - 2142 $ T - 2142
$43$	$ T - 6388 $ T - 6388
$47$	$ T - 6520 $ T - 6520
$53$	$ T + 10702 $ T + 10702
$59$	$ T + 42524 $ T + 42524
$61$	$ T - 44840 $ T - 44840
$67$	$ T + 1448 $ T + 1448
$71$	$ T + 4402 $ T + 4402
$73$	$ T + 20500 $ T + 20500
$79$	$ T - 65236 $ T - 65236
$83$	$ T - 102804 $ T - 102804
$89$	$ T - 128006 $ T - 128006
$97$	$ T - 113324 $ T - 113324
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Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

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