LMFDB - Newform orbit 208.10.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,10,Mod(1,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,273] 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:	$107.127453922$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
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 + 273 q^{3} + 1015 q^{5} - 3955 q^{7} + 54846 q^{9} + 50998 q^{11} - 28561 q^{13} + 277095 q^{15} + 509757 q^{17} + 626574 q^{19} - 1079715 q^{21} - 653524 q^{23} - 922900 q^{25} + 9599499 q^{27} - 4943006 q^{29}+ \cdots + 2797036308 q^{99}+O(q^{100}) $

q + 273 * q^3 + 1015 * q^5 - 3955 * q^7 + 54846 * q^9 + 50998 * q^11 - 28561 * q^13 + 277095 * q^15 + 509757 * q^17 + 626574 * q^19 - 1079715 * q^21 - 653524 * q^23 - 922900 * q^25 + 9599499 * q^27 - 4943006 * q^29 - 4071700 * q^31 + 13922454 * q^33 - 4014325 * q^35 + 2348883 * q^37 - 7797153 * q^39 - 13350960 * q^41 + 7834847 * q^43 + 55668690 * q^45 + 39637681 * q^47 - 24711582 * q^49 + 139163661 * q^51 + 73200924 * q^53 + 51762970 * q^55 + 171054702 * q^57 + 141141614 * q^59 - 132061256 * q^61 - 216915930 * q^63 - 28989415 * q^65 + 185673110 * q^67 - 178412052 * q^69 - 224452625 * q^71 - 172523674 * q^73 - 251951700 * q^75 - 201697090 * q^77 + 643288156 * q^79 + 1541129409 * q^81 - 720077280 * q^83 + 517403355 * q^85 - 1349440638 * q^87 - 73028106 * q^89 + 112958755 * q^91 - 1111574100 * q^93 + 635972610 * q^95 - 15879778 * q^97 + 2797036308 * 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

273.000

1015.00

−3955.00

54846.0

Atkin-Lehner signs

$ p $	Sign
$2$	$ -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	208.10.a.c		1
4.b	odd	2	1		26.10.a.a	✓	1
12.b	even	2	1		234.10.a.b		1
52.b	odd	2	1		338.10.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.10.a.a	✓	1	4.b	odd	2	1
208.10.a.c		1	1.a	even	1	1	trivial
234.10.a.b		1	12.b	even	2	1
338.10.a.c		1	52.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 273 $ T3 - 273 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(208))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 273 $ T - 273
$5$	$ T - 1015 $ T - 1015
$7$	$ T + 3955 $ T + 3955
$11$	$ T - 50998 $ T - 50998
$13$	$ T + 28561 $ T + 28561
$17$	$ T - 509757 $ T - 509757
$19$	$ T - 626574 $ T - 626574
$23$	$ T + 653524 $ T + 653524
$29$	$ T + 4943006 $ T + 4943006
$31$	$ T + 4071700 $ T + 4071700
$37$	$ T - 2348883 $ T - 2348883
$41$	$ T + 13350960 $ T + 13350960
$43$	$ T - 7834847 $ T - 7834847
$47$	$ T - 39637681 $ T - 39637681
$53$	$ T - 73200924 $ T - 73200924
$59$	$ T - 141141614 $ T - 141141614
$61$	$ T + 132061256 $ T + 132061256
$67$	$ T - 185673110 $ T - 185673110
$71$	$ T + 224452625 $ T + 224452625
$73$	$ T + 172523674 $ T + 172523674
$79$	$ T - 643288156 $ T - 643288156
$83$	$ T + 720077280 $ T + 720077280
$89$	$ T + 73028106 $ T + 73028106
$97$	$ T + 15879778 $ T + 15879778
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Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

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