LMFDB - Newform orbit 245.10.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,10,Mod(1,245)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$N$	$=$	$245 = 5 \cdot 7^{2}$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	245.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$126.183779860$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 35)
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 + 28 q^{2} + 116 q^{3} + 272 q^{4} - 625 q^{5} + 3248 q^{6} - 6720 q^{8} - 6227 q^{9} - 17500 q^{10} - 25548 q^{11} + 31552 q^{12} + 42306 q^{13} - 72500 q^{15} - 327424 q^{16} + 526342 q^{17} - 174356 q^{18}+ \cdots + 159087396 q^{99}+O(q^{100})

q + 28 * q^2 + 116 * q^3 + 272 * q^4 - 625 * q^5 + 3248 * q^6 - 6720 * q^8 - 6227 * q^9 - 17500 * q^10 - 25548 * q^11 + 31552 * q^12 + 42306 * q^13 - 72500 * q^15 - 327424 * q^16 + 526342 * q^17 - 174356 * q^18 + 350060 * q^19 - 170000 * q^20 - 715344 * q^22 - 621976 * q^23 - 779520 * q^24 + 390625 * q^25 + 1184568 * q^26 - 3005560 * q^27 + 6720430 * q^29 - 2030000 * q^30 + 6412208 * q^31 - 5727232 * q^32 - 2963568 * q^33 + 14737576 * q^34 - 1693744 * q^36 - 2317682 * q^37 + 9801680 * q^38 + 4907496 * q^39 + 4200000 * q^40 + 10224678 * q^41 + 30114004 * q^43 - 6949056 * q^44 + 3891875 * q^45 - 17415328 * q^46 + 23644912 * q^47 - 37981184 * q^48 + 10937500 * q^50 + 61055672 * q^51 + 11507232 * q^52 + 57292654 * q^53 - 84155680 * q^54 + 15967500 * q^55 + 40606960 * q^57 + 188172040 * q^58 - 84934780 * q^59 - 19720000 * q^60 - 14677822 * q^61 + 179541824 * q^62 + 7278592 * q^64 - 26441250 * q^65 - 82979904 * q^66 - 244557812 * q^67 + 143165024 * q^68 - 72149216 * q^69 + 61901952 * q^71 + 41845440 * q^72 + 283763726 * q^73 - 64895096 * q^74 + 45312500 * q^75 + 95216320 * q^76 + 137409888 * q^78 + 276107480 * q^79 + 204640000 * q^80 - 226078919 * q^81 + 286290984 * q^82 + 72995956 * q^83 - 328963750 * q^85 + 843192112 * q^86 + 779569880 * q^87 + 171682560 * q^88 + 896368470 * q^89 + 108972500 * q^90 - 169177472 * q^92 + 743816128 * q^93 + 662057536 * q^94 - 218787500 * q^95 - 664358912 * q^96 - 1205809578 * q^97 + 159087396 * 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

28.0000

116.000

272.000

−625.000

3248.00

−6720.00

−6227.00

−17500.0

Atkin-Lehner signs

$p$	Sign
$5$	$+1$
$7$	$-1$

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	245.10.a.b		1
7.b	odd	2	1		35.10.a.a	✓	1
21.c	even	2	1		315.10.a.a		1
35.c	odd	2	1		175.10.a.a		1
35.f	even	4	2		175.10.b.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.10.a.a	✓	1	7.b	odd	2	1
175.10.a.a		1	35.c	odd	2	1
175.10.b.a		2	35.f	even	4	2
245.10.a.b		1	1.a	even	1	1	trivial
315.10.a.a		1	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{10}^{\mathrm{new}}(\Gamma_0(245))$ :

T_{2} - 28

T_{3} - 116

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 28$ T - 28
$3$	$T - 116$ T - 116
$5$	$T + 625$ T + 625
$7$	$T$ T
$11$	$T + 25548$ T + 25548
$13$	$T - 42306$ T - 42306
$17$	$T - 526342$ T - 526342
$19$	$T - 350060$ T - 350060
$23$	$T + 621976$ T + 621976
$29$	$T - 6720430$ T - 6720430
$31$	$T - 6412208$ T - 6412208
$37$	$T + 2317682$ T + 2317682
$41$	$T - 10224678$ T - 10224678
$43$	$T - 30114004$ T - 30114004
$47$	$T - 23644912$ T - 23644912
$53$	$T - 57292654$ T - 57292654
$59$	$T + 84934780$ T + 84934780
$61$	$T + 14677822$ T + 14677822
$67$	$T + 244557812$ T + 244557812
$71$	$T - 61901952$ T - 61901952
$73$	$T - 283763726$ T - 283763726
$79$	$T - 276107480$ T - 276107480
$83$	$T - 72995956$ T - 72995956
$89$	$T - 896368470$ T - 896368470
$97$	$T + 1205809578$ T + 1205809578
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