LMFDB - Newform orbit 294.4.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,4,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, 4, 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, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,-3,4,6,6,0,-8,9,-12,-30] 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:	$17.3465615417$
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 - 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{5} + 6 q^{6} - 8 q^{8} + 9 q^{9} - 12 q^{10} - 30 q^{11} - 12 q^{12} - 53 q^{13} - 18 q^{15} + 16 q^{16} + 84 q^{17} - 18 q^{18} + 97 q^{19} + 24 q^{20} + 60 q^{22}+ \cdots - 270 q^{99}+O(q^{100}) $

q - 2 * q^2 - 3 * q^3 + 4 * q^4 + 6 * q^5 + 6 * q^6 - 8 * q^8 + 9 * q^9 - 12 * q^10 - 30 * q^11 - 12 * q^12 - 53 * q^13 - 18 * q^15 + 16 * q^16 + 84 * q^17 - 18 * q^18 + 97 * q^19 + 24 * q^20 + 60 * q^22 + 84 * q^23 + 24 * q^24 - 89 * q^25 + 106 * q^26 - 27 * q^27 - 180 * q^29 + 36 * q^30 - 179 * q^31 - 32 * q^32 + 90 * q^33 - 168 * q^34 + 36 * q^36 - 145 * q^37 - 194 * q^38 + 159 * q^39 - 48 * q^40 - 126 * q^41 - 325 * q^43 - 120 * q^44 + 54 * q^45 - 168 * q^46 + 366 * q^47 - 48 * q^48 + 178 * q^50 - 252 * q^51 - 212 * q^52 - 768 * q^53 + 54 * q^54 - 180 * q^55 - 291 * q^57 + 360 * q^58 + 264 * q^59 - 72 * q^60 - 818 * q^61 + 358 * q^62 + 64 * q^64 - 318 * q^65 - 180 * q^66 - 523 * q^67 + 336 * q^68 - 252 * q^69 - 342 * q^71 - 72 * q^72 + 43 * q^73 + 290 * q^74 + 267 * q^75 + 388 * q^76 - 318 * q^78 - 1171 * q^79 + 96 * q^80 + 81 * q^81 + 252 * q^82 + 810 * q^83 + 504 * q^85 + 650 * q^86 + 540 * q^87 + 240 * q^88 + 600 * q^89 - 108 * q^90 + 336 * q^92 + 537 * q^93 - 732 * q^94 + 582 * q^95 + 96 * q^96 - 386 * q^97 - 270 * 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

−2.00000

−3.00000

4.00000

6.00000

−8.00000

9.00000

−12.0000

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	294.4.a.c		1
3.b	odd	2	1		882.4.a.l		1
4.b	odd	2	1		2352.4.a.bf		1
7.b	odd	2	1		294.4.a.d		1
7.c	even	3	2		294.4.e.i		2
7.d	odd	6	2		42.4.e.a	✓	2
21.c	even	2	1		882.4.a.o		1
21.g	even	6	2		126.4.g.b		2
21.h	odd	6	2		882.4.g.g		2
28.d	even	2	1		2352.4.a.f		1
28.f	even	6	2		336.4.q.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.e.a	✓	2	7.d	odd	6	2
126.4.g.b		2	21.g	even	6	2
294.4.a.c		1	1.a	even	1	1	trivial
294.4.a.d		1	7.b	odd	2	1
294.4.e.i		2	7.c	even	3	2
336.4.q.f		2	28.f	even	6	2
882.4.a.l		1	3.b	odd	2	1
882.4.a.o		1	21.c	even	2	1
882.4.g.g		2	21.h	odd	6	2
2352.4.a.f		1	28.d	even	2	1
2352.4.a.bf		1	4.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 $S_{4}^{\mathrm{new}}(\Gamma_0(294))$:

$ T_{5} - 6 $

T5 - 6

$ T_{11} + 30 $

T11 + 30

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T + 3 $ T + 3
$5$	$ T - 6 $ T - 6
$7$	$ T $ T
$11$	$ T + 30 $ T + 30
$13$	$ T + 53 $ T + 53
$17$	$ T - 84 $ T - 84
$19$	$ T - 97 $ T - 97
$23$	$ T - 84 $ T - 84
$29$	$ T + 180 $ T + 180
$31$	$ T + 179 $ T + 179
$37$	$ T + 145 $ T + 145
$41$	$ T + 126 $ T + 126
$43$	$ T + 325 $ T + 325
$47$	$ T - 366 $ T - 366
$53$	$ T + 768 $ T + 768
$59$	$ T - 264 $ T - 264
$61$	$ T + 818 $ T + 818
$67$	$ T + 523 $ T + 523
$71$	$ T + 342 $ T + 342
$73$	$ T - 43 $ T - 43
$79$	$ T + 1171 $ T + 1171
$83$	$ T - 810 $ T - 810
$89$	$ T - 600 $ T - 600
$97$	$ T + 386 $ T + 386
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Classical	Maass
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

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