LMFDB - Newform orbit 26.10.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [26,10,Mod(1,26)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(26, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("26.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 26 = 2 \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	26.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,16,75] 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:	$13.3909317403$
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^{2} + 75 q^{3} + 256 q^{4} - 1979 q^{5} + 1200 q^{6} - 10115 q^{7} + 4096 q^{8} - 14058 q^{9} - 31664 q^{10} + 18850 q^{11} + 19200 q^{12} + 28561 q^{13} - 161840 q^{14} - 148425 q^{15} + 65536 q^{16}+ \cdots - 264993300 q^{99}+O(q^{100}) $

q + 16 * q^2 + 75 * q^3 + 256 * q^4 - 1979 * q^5 + 1200 * q^6 - 10115 * q^7 + 4096 * q^8 - 14058 * q^9 - 31664 * q^10 + 18850 * q^11 + 19200 * q^12 + 28561 * q^13 - 161840 * q^14 - 148425 * q^15 + 65536 * q^16 - 142403 * q^17 - 224928 * q^18 + 83302 * q^19 - 506624 * q^20 - 758625 * q^21 + 301600 * q^22 - 536544 * q^23 + 307200 * q^24 + 1963316 * q^25 + 456976 * q^26 - 2530575 * q^27 - 2589440 * q^28 - 2600442 * q^29 - 2374800 * q^30 - 2214004 * q^31 + 1048576 * q^32 + 1413750 * q^33 - 2278448 * q^34 + 20017585 * q^35 - 3598848 * q^36 + 18099241 * q^37 + 1332832 * q^38 + 2142075 * q^39 - 8105984 * q^40 + 26812240 * q^41 - 12138000 * q^42 - 42253475 * q^43 + 4825600 * q^44 + 27820782 * q^45 - 8584704 * q^46 + 35914993 * q^47 + 4915200 * q^48 + 61959618 * q^49 + 31413056 * q^50 - 10680225 * q^51 + 7311616 * q^52 - 66514064 * q^53 - 40489200 * q^54 - 37304150 * q^55 - 41431040 * q^56 + 6247650 * q^57 - 41607072 * q^58 - 108164002 * q^59 - 37996800 * q^60 - 207449912 * q^61 - 35424064 * q^62 + 142196670 * q^63 + 16777216 * q^64 - 56522219 * q^65 + 22620000 * q^66 + 193015514 * q^67 - 36455168 * q^68 - 40240800 * q^69 + 320281360 * q^70 - 201833497 * q^71 - 57581568 * q^72 - 121628110 * q^73 + 289587856 * q^74 + 147248700 * q^75 + 21325312 * q^76 - 190667750 * q^77 + 34273200 * q^78 + 112871912 * q^79 - 129695744 * q^80 + 86910489 * q^81 + 428995840 * q^82 + 308254212 * q^83 - 194208000 * q^84 + 281815537 * q^85 - 676055600 * q^86 - 195033150 * q^87 + 77209600 * q^88 - 6374870 * q^89 + 445132512 * q^90 - 288894515 * q^91 - 137355264 * q^92 - 166050300 * q^93 + 574639888 * q^94 - 164854658 * q^95 + 78643200 * q^96 + 871266886 * q^97 + 991353888 * q^98 - 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

16.0000

75.0000

256.000

−1979.00

1200.00

−10115.0

4096.00

−14058.0

−31664.0

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	26.10.a.c	✓	1
3.b	odd	2	1		234.10.a.a		1
4.b	odd	2	1		208.10.a.b		1
13.b	even	2	1		338.10.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.10.a.c	✓	1	1.a	even	1	1	trivial
208.10.a.b		1	4.b	odd	2	1
234.10.a.a		1	3.b	odd	2	1
338.10.a.b		1	13.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 75 $ T3 - 75 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(26))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 16 $ T - 16
$3$	$ 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:	\( N \)	\(=\)	\( 26 = 2 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	26.a (trivial)

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