LMFDB - Newform orbit 78.4.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [78,4,Mod(1,78)] 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("78.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(78, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	78.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,3,4,-16] 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:	$4.60214898045$
Analytic rank:	$1$
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 - 2 q^{2} + 3 q^{3} + 4 q^{4} - 16 q^{5} - 6 q^{6} - 8 q^{7} - 8 q^{8} + 9 q^{9} + 32 q^{10} - 38 q^{11} + 12 q^{12} - 13 q^{13} + 16 q^{14} - 48 q^{15} + 16 q^{16} - 78 q^{17} - 18 q^{18} - 72 q^{19}+ \cdots - 342 q^{99}+O(q^{100}) $

q - 2 * q^2 + 3 * q^3 + 4 * q^4 - 16 * q^5 - 6 * q^6 - 8 * q^7 - 8 * q^8 + 9 * q^9 + 32 * q^10 - 38 * q^11 + 12 * q^12 - 13 * q^13 + 16 * q^14 - 48 * q^15 + 16 * q^16 - 78 * q^17 - 18 * q^18 - 72 * q^19 - 64 * q^20 - 24 * q^21 + 76 * q^22 - 52 * q^23 - 24 * q^24 + 131 * q^25 + 26 * q^26 + 27 * q^27 - 32 * q^28 + 242 * q^29 + 96 * q^30 + 76 * q^31 - 32 * q^32 - 114 * q^33 + 156 * q^34 + 128 * q^35 + 36 * q^36 + 342 * q^37 + 144 * q^38 - 39 * q^39 + 128 * q^40 - 336 * q^41 + 48 * q^42 + 76 * q^43 - 152 * q^44 - 144 * q^45 + 104 * q^46 + 94 * q^47 + 48 * q^48 - 279 * q^49 - 262 * q^50 - 234 * q^51 - 52 * q^52 - 450 * q^53 - 54 * q^54 + 608 * q^55 + 64 * q^56 - 216 * q^57 - 484 * q^58 + 854 * q^59 - 192 * q^60 - 110 * q^61 - 152 * q^62 - 72 * q^63 + 64 * q^64 + 208 * q^65 + 228 * q^66 - 908 * q^67 - 312 * q^68 - 156 * q^69 - 256 * q^70 + 838 * q^71 - 72 * q^72 - 970 * q^73 - 684 * q^74 + 393 * q^75 - 288 * q^76 + 304 * q^77 + 78 * q^78 - 352 * q^79 - 256 * q^80 + 81 * q^81 + 672 * q^82 + 474 * q^83 - 96 * q^84 + 1248 * q^85 - 152 * q^86 + 726 * q^87 + 304 * q^88 - 1452 * q^89 + 288 * q^90 + 104 * q^91 - 208 * q^92 + 228 * q^93 - 188 * q^94 + 1152 * q^95 - 96 * q^96 - 562 * q^97 + 558 * q^98 - 342 * 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

−16.0000

−6.00000

−8.00000

9.00000

32.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ -1 $
$13$	$ +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	78.4.a.b	✓	1
3.b	odd	2	1		234.4.a.j		1
4.b	odd	2	1		624.4.a.a		1
5.b	even	2	1		1950.4.a.k		1
8.b	even	2	1		2496.4.a.h		1
8.d	odd	2	1		2496.4.a.p		1
12.b	even	2	1		1872.4.a.p		1
13.b	even	2	1		1014.4.a.k		1
13.d	odd	4	2		1014.4.b.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.4.a.b	✓	1	1.a	even	1	1	trivial
234.4.a.j		1	3.b	odd	2	1
624.4.a.a		1	4.b	odd	2	1
1014.4.a.k		1	13.b	even	2	1
1014.4.b.f		2	13.d	odd	4	2
1872.4.a.p		1	12.b	even	2	1
1950.4.a.k		1	5.b	even	2	1
2496.4.a.h		1	8.b	even	2	1
2496.4.a.p		1	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(78))$:

$ T_{5} + 16 $

T5 + 16

$ T_{7} + 8 $

T7 + 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T - 3 $ T - 3
$5$	$ T + 16 $ T + 16
$7$	$ T + 8 $ T + 8
$11$	$ T + 38 $ T + 38
$13$	$ T + 13 $ T + 13
$17$	$ T + 78 $ T + 78
$19$	$ T + 72 $ T + 72
$23$	$ T + 52 $ T + 52
$29$	$ T - 242 $ T - 242
$31$	$ T - 76 $ T - 76
$37$	$ T - 342 $ T - 342
$41$	$ T + 336 $ T + 336
$43$	$ T - 76 $ T - 76
$47$	$ T - 94 $ T - 94
$53$	$ T + 450 $ T + 450
$59$	$ T - 854 $ T - 854
$61$	$ T + 110 $ T + 110
$67$	$ T + 908 $ T + 908
$71$	$ T - 838 $ T - 838
$73$	$ T + 970 $ T + 970
$79$	$ T + 352 $ T + 352
$83$	$ T - 474 $ T - 474
$89$	$ T + 1452 $ T + 1452
$97$	$ T + 562 $ T + 562
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	78.a (trivial)

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