LMFDB - Newform orbit 240.10.a.i

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [240,10,Mod(1,240)] 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("240.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	240.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,81,0,625,0,7168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$123.608600679$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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 + 81 q^{3} + 625 q^{5} + 7168 q^{7} + 6561 q^{9} + 83748 q^{11} + 128126 q^{13} + 50625 q^{15} + 560802 q^{17} + 577660 q^{19} + 580608 q^{21} - 2431296 q^{23} + 390625 q^{25} + 531441 q^{27} + 5791710 q^{29}+ \cdots + 549470628 q^{99}+O(q^{100}) $

q + 81 * q^3 + 625 * q^5 + 7168 * q^7 + 6561 * q^9 + 83748 * q^11 + 128126 * q^13 + 50625 * q^15 + 560802 * q^17 + 577660 * q^19 + 580608 * q^21 - 2431296 * q^23 + 390625 * q^25 + 531441 * q^27 + 5791710 * q^29 - 4145312 * q^31 + 6783588 * q^33 + 4480000 * q^35 - 7011658 * q^37 + 10378206 * q^39 - 8881398 * q^41 + 15730684 * q^43 + 4100625 * q^45 - 60552072 * q^47 + 11026617 * q^49 + 45424962 * q^51 + 30273366 * q^53 + 52342500 * q^55 + 46790460 * q^57 - 45957660 * q^59 + 37595102 * q^61 + 47029248 * q^63 + 80078750 * q^65 - 196784012 * q^67 - 196934976 * q^69 - 56047992 * q^71 - 159688054 * q^73 + 31640625 * q^75 + 600305664 * q^77 - 201923360 * q^79 + 43046721 * q^81 + 362955444 * q^83 + 350501250 * q^85 + 469128510 * q^87 - 272479110 * q^89 + 918407168 * q^91 - 335770272 * q^93 + 361037500 * q^95 - 600852478 * q^97 + 549470628 * 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

81.0000

625.000

7168.00

6561.00

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$5$	$ -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	240.10.a.i		1
4.b	odd	2	1		30.10.a.b	✓	1
12.b	even	2	1		90.10.a.f		1
20.d	odd	2	1		150.10.a.k		1
20.e	even	4	2		150.10.c.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.10.a.b	✓	1	4.b	odd	2	1
90.10.a.f		1	12.b	even	2	1
150.10.a.k		1	20.d	odd	2	1
150.10.c.f		2	20.e	even	4	2
240.10.a.i		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} - 7168 $ T7 - 7168 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(240))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 81 $ T - 81
$5$	$ T - 625 $ T - 625
$7$	$ T - 7168 $ T - 7168
$11$	$ T - 83748 $ T - 83748
$13$	$ T - 128126 $ T - 128126
$17$	$ T - 560802 $ T - 560802
$19$	$ T - 577660 $ T - 577660
$23$	$ T + 2431296 $ T + 2431296
$29$	$ T - 5791710 $ T - 5791710
$31$	$ T + 4145312 $ T + 4145312
$37$	$ T + 7011658 $ T + 7011658
$41$	$ T + 8881398 $ T + 8881398
$43$	$ T - 15730684 $ T - 15730684
$47$	$ T + 60552072 $ T + 60552072
$53$	$ T - 30273366 $ T - 30273366
$59$	$ T + 45957660 $ T + 45957660
$61$	$ T - 37595102 $ T - 37595102
$67$	$ T + 196784012 $ T + 196784012
$71$	$ T + 56047992 $ T + 56047992
$73$	$ T + 159688054 $ T + 159688054
$79$	$ T + 201923360 $ T + 201923360
$83$	$ T - 362955444 $ T - 362955444
$89$	$ T + 272479110 $ T + 272479110
$97$	$ T + 600852478 $ T + 600852478
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Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

\(n\):		e.g. 2-40 or 80-90
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