LMFDB - Newform orbit 90.14.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [90,14,Mod(1,90)] 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("90.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	90.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-64,0,4096,15625,0,-246868] 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:	$96.5078360567$
Analytic rank:	$1$
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 - 64 q^{2} + 4096 q^{4} + 15625 q^{5} - 246868 q^{7} - 262144 q^{8} - 1000000 q^{10} + 5813808 q^{11} - 13914094 q^{13} + 15799552 q^{14} + 16777216 q^{16} + 81264378 q^{17} - 44056660 q^{19} + 64000000 q^{20}+ \cdots + 2300492862912 q^{98}+O(q^{100}) $

q - 64 * q^2 + 4096 * q^4 + 15625 * q^5 - 246868 * q^7 - 262144 * q^8 - 1000000 * q^10 + 5813808 * q^11 - 13914094 * q^13 + 15799552 * q^14 + 16777216 * q^16 + 81264378 * q^17 - 44056660 * q^19 + 64000000 * q^20 - 372083712 * q^22 - 68823216 * q^23 + 244140625 * q^25 + 890502016 * q^26 - 1011171328 * q^28 - 1046213610 * q^29 - 9196464568 * q^31 - 1073741824 * q^32 - 5200920192 * q^34 - 3857312500 * q^35 + 17229030002 * q^37 + 2819626240 * q^38 - 4096000000 * q^40 + 17123113638 * q^41 - 12353902084 * q^43 + 23813357568 * q^44 + 4404685824 * q^46 + 118626784848 * q^47 - 35945200983 * q^49 - 15625000000 * q^50 - 56992129024 * q^52 + 78925851894 * q^53 + 90840750000 * q^55 + 64714964992 * q^56 + 66957671040 * q^58 - 75016072560 * q^59 - 207562346458 * q^61 + 588573732352 * q^62 + 68719476736 * q^64 - 217407718750 * q^65 + 617859006332 * q^67 + 332858892288 * q^68 + 246868000000 * q^70 - 1849245670272 * q^71 + 1449378197666 * q^73 - 1102657920128 * q^74 - 180456079360 * q^76 - 1435243153344 * q^77 + 3278875539560 * q^79 + 262144000000 * q^80 - 1095879272832 * q^82 - 576023687916 * q^83 + 1269755906250 * q^85 + 790649733376 * q^86 - 1524054884352 * q^88 - 1441392149490 * q^89 + 3434944557592 * q^91 - 281899892736 * q^92 - 7592114230272 * q^94 - 688385312500 * q^95 - 9161637335998 * q^97 + 2300492862912 * q^98

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

−64.0000

4096.00

15625.0

−246868.

−262144.

−1.00000e6

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	90.14.a.c		1
3.b	odd	2	1		30.14.a.f	✓	1
15.d	odd	2	1		150.14.a.a		1
15.e	even	4	2		150.14.c.e		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.14.a.f	✓	1	3.b	odd	2	1
90.14.a.c		1	1.a	even	1	1	trivial
150.14.a.a		1	15.d	odd	2	1
150.14.c.e		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{14}^{\mathrm{new}}(\Gamma_0(90))$:

$ T_{7} + 246868 $

T7 + 246868

$ T_{11} - 5813808 $

T11 - 5813808

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 64 $ T + 64
$3$	$ T $ T
$5$	$ T - 15625 $ T - 15625
$7$	$ T + 246868 $ T + 246868
$11$	$ T - 5813808 $ T - 5813808
$13$	$ T + 13914094 $ T + 13914094
$17$	$ T - 81264378 $ T - 81264378
$19$	$ T + 44056660 $ T + 44056660
$23$	$ T + 68823216 $ T + 68823216
$29$	$ T + 1046213610 $ T + 1046213610
$31$	$ T + 9196464568 $ T + 9196464568
$37$	$ T - 17229030002 $ T - 17229030002
$41$	$ T - 17123113638 $ T - 17123113638
$43$	$ T + 12353902084 $ T + 12353902084
$47$	$ T - 118626784848 $ T - 118626784848
$53$	$ T - 78925851894 $ T - 78925851894
$59$	$ T + 75016072560 $ T + 75016072560
$61$	$ T + 207562346458 $ T + 207562346458
$67$	$ T - 617859006332 $ T - 617859006332
$71$	$ T + 1849245670272 $ T + 1849245670272
$73$	$ T - 1449378197666 $ T - 1449378197666
$79$	$ T - 3278875539560 $ T - 3278875539560
$83$	$ T + 576023687916 $ T + 576023687916
$89$	$ T + 1441392149490 $ T + 1441392149490
$97$	$ T + 9161637335998 $ T + 9161637335998
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Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

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