LMFDB - Newform orbit 208.10.a.b

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,-75] 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:	$1$
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 - 75 q^{3} - 1979 q^{5} + 10115 q^{7} - 14058 q^{9} - 18850 q^{11} + 28561 q^{13} + 148425 q^{15} - 142403 q^{17} - 83302 q^{19} - 758625 q^{21} + 536544 q^{23} + 1963316 q^{25} + 2530575 q^{27} - 2600442 q^{29}+ \cdots + 264993300 q^{99}+O(q^{100}) $

q - 75 * q^3 - 1979 * q^5 + 10115 * q^7 - 14058 * q^9 - 18850 * q^11 + 28561 * q^13 + 148425 * q^15 - 142403 * q^17 - 83302 * q^19 - 758625 * q^21 + 536544 * q^23 + 1963316 * q^25 + 2530575 * q^27 - 2600442 * q^29 + 2214004 * q^31 + 1413750 * q^33 - 20017585 * q^35 + 18099241 * q^37 - 2142075 * q^39 + 26812240 * q^41 + 42253475 * q^43 + 27820782 * q^45 - 35914993 * q^47 + 61959618 * q^49 + 10680225 * q^51 - 66514064 * q^53 + 37304150 * q^55 + 6247650 * q^57 + 108164002 * q^59 - 207449912 * q^61 - 142196670 * q^63 - 56522219 * q^65 - 193015514 * q^67 - 40240800 * q^69 + 201833497 * q^71 - 121628110 * q^73 - 147248700 * q^75 - 190667750 * q^77 - 112871912 * q^79 + 86910489 * q^81 - 308254212 * q^83 + 281815537 * q^85 + 195033150 * q^87 - 6374870 * q^89 + 288894515 * q^91 - 166050300 * q^93 + 164854658 * q^95 + 871266886 * q^97 + 264993300 * 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

−75.0000

−1979.00

10115.0

−14058.0

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.b		1
4.b	odd	2	1		26.10.a.c	✓	1
12.b	even	2	1		234.10.a.a		1
52.b	odd	2	1		338.10.a.b		1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 75 $ T3 + 75 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(208))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 75 $ T + 75
$5$	$ T + 1979 $ T + 1979
$7$	$ T - 10115 $ T - 10115
$11$	$ T + 18850 $ T + 18850
$13$	$ T - 28561 $ T - 28561
$17$	$ T + 142403 $ T + 142403
$19$	$ T + 83302 $ T + 83302
$23$	$ T - 536544 $ T - 536544
$29$	$ T + 2600442 $ T + 2600442
$31$	$ T - 2214004 $ T - 2214004
$37$	$ T - 18099241 $ T - 18099241
$41$	$ T - 26812240 $ T - 26812240
$43$	$ T - 42253475 $ T - 42253475
$47$	$ T + 35914993 $ T + 35914993
$53$	$ T + 66514064 $ T + 66514064
$59$	$ T - 108164002 $ T - 108164002
$61$	$ T + 207449912 $ T + 207449912
$67$	$ T + 193015514 $ T + 193015514
$71$	$ T - 201833497 $ T - 201833497
$73$	$ T + 121628110 $ T + 121628110
$79$	$ T + 112871912 $ T + 112871912
$83$	$ T + 308254212 $ T + 308254212
$89$	$ T + 6374870 $ T + 6374870
$97$	$ T - 871266886 $ T - 871266886
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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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