LMFDB - Newform orbit 525.6.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,6,Mod(1,525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("525.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$84.2015054018$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
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 - 10 q^{2} - 9 q^{3} + 68 q^{4} + 90 q^{6} + 49 q^{7} - 360 q^{8} + 81 q^{9} + 92 q^{11} - 612 q^{12} - 670 q^{13} - 490 q^{14} + 1424 q^{16} + 222 q^{17} - 810 q^{18} - 908 q^{19} - 441 q^{21} - 920 q^{22}+ \cdots + 7452 q^{99}+O(q^{100}) $

q - 10 * q^2 - 9 * q^3 + 68 * q^4 + 90 * q^6 + 49 * q^7 - 360 * q^8 + 81 * q^9 + 92 * q^11 - 612 * q^12 - 670 * q^13 - 490 * q^14 + 1424 * q^16 + 222 * q^17 - 810 * q^18 - 908 * q^19 - 441 * q^21 - 920 * q^22 + 1176 * q^23 + 3240 * q^24 + 6700 * q^26 - 729 * q^27 + 3332 * q^28 + 1118 * q^29 + 3696 * q^31 - 2720 * q^32 - 828 * q^33 - 2220 * q^34 + 5508 * q^36 - 4182 * q^37 + 9080 * q^38 + 6030 * q^39 - 6662 * q^41 + 4410 * q^42 + 3700 * q^43 + 6256 * q^44 - 11760 * q^46 + 7056 * q^47 - 12816 * q^48 + 2401 * q^49 - 1998 * q^51 - 45560 * q^52 + 37578 * q^53 + 7290 * q^54 - 17640 * q^56 + 8172 * q^57 - 11180 * q^58 + 32700 * q^59 - 10802 * q^61 - 36960 * q^62 + 3969 * q^63 - 18368 * q^64 + 8280 * q^66 - 64996 * q^67 + 15096 * q^68 - 10584 * q^69 - 61320 * q^71 - 29160 * q^72 - 38922 * q^73 + 41820 * q^74 - 61744 * q^76 + 4508 * q^77 - 60300 * q^78 - 88096 * q^79 + 6561 * q^81 + 66620 * q^82 - 71892 * q^83 - 29988 * q^84 - 37000 * q^86 - 10062 * q^87 - 33120 * q^88 + 111818 * q^89 - 32830 * q^91 + 79968 * q^92 - 33264 * q^93 - 70560 * q^94 + 24480 * q^96 + 150846 * q^97 - 24010 * q^98 + 7452 * 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

−10.0000

−9.00000

68.0000

90.0000

49.0000

−360.000

81.0000

Atkin-Lehner signs

$ p $	Sign
$3$	$ +1 $
$5$	$ +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	525.6.a.a		1
5.b	even	2	1		21.6.a.d	✓	1
5.c	odd	4	2		525.6.d.a		2
15.d	odd	2	1		63.6.a.a		1
20.d	odd	2	1		336.6.a.a		1
35.c	odd	2	1		147.6.a.g		1
35.i	odd	6	2		147.6.e.b		2
35.j	even	6	2		147.6.e.a		2
60.h	even	2	1		1008.6.a.bc		1
105.g	even	2	1		441.6.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.6.a.d	✓	1	5.b	even	2	1
63.6.a.a		1	15.d	odd	2	1
147.6.a.g		1	35.c	odd	2	1
147.6.e.a		2	35.j	even	6	2
147.6.e.b		2	35.i	odd	6	2
336.6.a.a		1	20.d	odd	2	1
441.6.a.b		1	105.g	even	2	1
525.6.a.a		1	1.a	even	1	1	trivial
525.6.d.a		2	5.c	odd	4	2
1008.6.a.bc		1	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} + 10 $ T2 + 10 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(525))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 10 $ T + 10
$3$	$ T + 9 $ T + 9
$5$	$ T $ T
$7$	$ T - 49 $ T - 49
$11$	$ T - 92 $ T - 92
$13$	$ T + 670 $ T + 670
$17$	$ T - 222 $ T - 222
$19$	$ T + 908 $ T + 908
$23$	$ T - 1176 $ T - 1176
$29$	$ T - 1118 $ T - 1118
$31$	$ T - 3696 $ T - 3696
$37$	$ T + 4182 $ T + 4182
$41$	$ T + 6662 $ T + 6662
$43$	$ T - 3700 $ T - 3700
$47$	$ T - 7056 $ T - 7056
$53$	$ T - 37578 $ T - 37578
$59$	$ T - 32700 $ T - 32700
$61$	$ T + 10802 $ T + 10802
$67$	$ T + 64996 $ T + 64996
$71$	$ T + 61320 $ T + 61320
$73$	$ T + 38922 $ T + 38922
$79$	$ T + 88096 $ T + 88096
$83$	$ T + 71892 $ T + 71892
$89$	$ T - 111818 $ T - 111818
$97$	$ T - 150846 $ T - 150846
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

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