LMFDB - Newform orbit 196.6.a.g

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 = [1,0,19] 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:	no (minimal twist has level 28)
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 + 19 q^{3} - 19 q^{5} + 118 q^{9} - 559 q^{11} - 282 q^{13} - 361 q^{15} - 1259 q^{17} + 1957 q^{19} - 2977 q^{23} - 2764 q^{25} - 2375 q^{27} - 62 q^{29} - 2037 q^{31} - 10621 q^{33} + 6023 q^{37} - 5358 q^{39}+ \cdots - 65962 q^{99}+O(q^{100}) $

q + 19 * q^3 - 19 * q^5 + 118 * q^9 - 559 * q^11 - 282 * q^13 - 361 * q^15 - 1259 * q^17 + 1957 * q^19 - 2977 * q^23 - 2764 * q^25 - 2375 * q^27 - 62 * q^29 - 2037 * q^31 - 10621 * q^33 + 6023 * q^37 - 5358 * q^39 + 2178 * q^41 + 23180 * q^43 - 2242 * q^45 - 26235 * q^47 - 23921 * q^51 + 30267 * q^53 + 10621 * q^55 + 37183 * q^57 - 44965 * q^59 - 27639 * q^61 + 5358 * q^65 - 58667 * q^67 - 56563 * q^69 - 9520 * q^71 + 6785 * q^73 - 52516 * q^75 - 16929 * q^79 - 73799 * q^81 + 59572 * q^83 + 23921 * q^85 - 1178 * q^87 + 51873 * q^89 - 38703 * q^93 - 37183 * q^95 - 134110 * q^97 - 65962 * 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

19.0000

−19.0000

118.000

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.g		1
4.b	odd	2	1		784.6.a.a		1
7.b	odd	2	1		196.6.a.b		1
7.c	even	3	2		196.6.e.b		2
7.d	odd	6	2		28.6.e.a	✓	2
21.g	even	6	2		252.6.k.c		2
28.d	even	2	1		784.6.a.k		1
28.f	even	6	2		112.6.i.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.6.e.a	✓	2	7.d	odd	6	2
112.6.i.a		2	28.f	even	6	2
196.6.a.b		1	7.b	odd	2	1
196.6.a.g		1	1.a	even	1	1	trivial
196.6.e.b		2	7.c	even	3	2
252.6.k.c		2	21.g	even	6	2
784.6.a.a		1	4.b	odd	2	1
784.6.a.k		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 19 $ T3 - 19 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(196))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 19 $ T - 19
$5$	$ T + 19 $ T + 19
$7$	$ T $ T
$11$	$ T + 559 $ T + 559
$13$	$ T + 282 $ T + 282
$17$	$ T + 1259 $ T + 1259
$19$	$ T - 1957 $ T - 1957
$23$	$ T + 2977 $ T + 2977
$29$	$ T + 62 $ T + 62
$31$	$ T + 2037 $ T + 2037
$37$	$ T - 6023 $ T - 6023
$41$	$ T - 2178 $ T - 2178
$43$	$ T - 23180 $ T - 23180
$47$	$ T + 26235 $ T + 26235
$53$	$ T - 30267 $ T - 30267
$59$	$ T + 44965 $ T + 44965
$61$	$ T + 27639 $ T + 27639
$67$	$ T + 58667 $ T + 58667
$71$	$ T + 9520 $ T + 9520
$73$	$ T - 6785 $ T - 6785
$79$	$ T + 16929 $ T + 16929
$83$	$ T - 59572 $ T - 59572
$89$	$ T - 51873 $ T - 51873
$97$	$ T + 134110 $ T + 134110
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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
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