LMFDB - Newform orbit 196.6.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [196,6,Mod(1,196)] 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("196.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 196 = 2^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.4352286833$
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 + 16 q^{3} - 16 q^{5} + 13 q^{9} - 76 q^{11} - 880 q^{13} - 256 q^{15} + 1056 q^{17} - 1936 q^{19} + 936 q^{23} - 2869 q^{25} - 3680 q^{27} - 3982 q^{29} - 1568 q^{31} - 1216 q^{33} + 4938 q^{37} - 14080 q^{39}+ \cdots - 988 q^{99}+O(q^{100}) $

q + 16 * q^3 - 16 * q^5 + 13 * q^9 - 76 * q^11 - 880 * q^13 - 256 * q^15 + 1056 * q^17 - 1936 * q^19 + 936 * q^23 - 2869 * q^25 - 3680 * q^27 - 3982 * q^29 - 1568 * q^31 - 1216 * q^33 + 4938 * q^37 - 14080 * q^39 + 15840 * q^41 - 16412 * q^43 - 208 * q^45 + 20768 * q^47 + 16896 * q^51 - 37402 * q^53 + 1216 * q^55 - 30976 * q^57 - 21136 * q^59 + 2992 * q^61 + 14080 * q^65 - 45836 * q^67 + 14976 * q^69 - 49840 * q^71 + 56320 * q^73 - 45904 * q^75 + 40744 * q^79 - 62039 * q^81 - 112464 * q^83 - 16896 * q^85 - 63712 * q^87 - 64256 * q^89 - 25088 * q^93 + 30976 * q^95 + 2272 * q^97 - 988 * 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

−16.0000

13.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -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	196.6.a.f	yes	1
4.b	odd	2	1		784.6.a.b		1
7.b	odd	2	1		196.6.a.c	✓	1
7.c	even	3	2		196.6.e.c		2
7.d	odd	6	2		196.6.e.h		2
28.d	even	2	1		784.6.a.j		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
196.6.a.c	✓	1	7.b	odd	2	1
196.6.a.f	yes	1	1.a	even	1	1	trivial
196.6.e.c		2	7.c	even	3	2
196.6.e.h		2	7.d	odd	6	2
784.6.a.b		1	4.b	odd	2	1
784.6.a.j		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 16 $ T3 - 16 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(196))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 16 $ T - 16
$5$	$ T + 16 $ T + 16
$7$	$ T $ T
$11$	$ T + 76 $ T + 76
$13$	$ T + 880 $ T + 880
$17$	$ T - 1056 $ T - 1056
$19$	$ T + 1936 $ T + 1936
$23$	$ T - 936 $ T - 936
$29$	$ T + 3982 $ T + 3982
$31$	$ T + 1568 $ T + 1568
$37$	$ T - 4938 $ T - 4938
$41$	$ T - 15840 $ T - 15840
$43$	$ T + 16412 $ T + 16412
$47$	$ T - 20768 $ T - 20768
$53$	$ T + 37402 $ T + 37402
$59$	$ T + 21136 $ T + 21136
$61$	$ T - 2992 $ T - 2992
$67$	$ T + 45836 $ T + 45836
$71$	$ T + 49840 $ T + 49840
$73$	$ T - 56320 $ T - 56320
$79$	$ T - 40744 $ T - 40744
$83$	$ T + 112464 $ T + 112464
$89$	$ T + 64256 $ T + 64256
$97$	$ T - 2272 $ T - 2272
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Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	196.a (trivial)

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