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Properties

Label 90.2.a.c
Level $90$
Weight $2$
Character orbit 90.a
Self dual yes
Analytic conductor $0.719$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [90,2,Mod(1,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 90.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: \(0.718653618192\)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} - 4 q^{7} + q^{8} + q^{10} + 2 q^{13} - 4 q^{14} + q^{16} - 6 q^{17} - 4 q^{19} + q^{20} + q^{25} + 2 q^{26} - 4 q^{28} + 6 q^{29} + 8 q^{31} + q^{32} - 6 q^{34} - 4 q^{35}+ \cdots + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
0
1.00000 0 1.00000 1.00000 0 −4.00000 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 90.2.a.c 1
3.b odd 2 1 30.2.a.a ✓ 1
4.b odd 2 1 720.2.a.j 1
5.b even 2 1 450.2.a.d 1
5.c odd 4 2 450.2.c.b 2
7.b odd 2 1 4410.2.a.z 1
8.b even 2 1 2880.2.a.a 1
8.d odd 2 1 2880.2.a.q 1
9.c even 3 2 810.2.e.b 2
9.d odd 6 2 810.2.e.l 2
12.b even 2 1 240.2.a.b 1
15.d odd 2 1 150.2.a.b 1
15.e even 4 2 150.2.c.a 2
20.d odd 2 1 3600.2.a.f 1
20.e even 4 2 3600.2.f.i 2
21.c even 2 1 1470.2.a.d 1
21.g even 6 2 1470.2.i.q 2
21.h odd 6 2 1470.2.i.o 2
24.f even 2 1 960.2.a.p 1
24.h odd 2 1 960.2.a.e 1
33.d even 2 1 3630.2.a.w 1
39.d odd 2 1 5070.2.a.w 1
39.f even 4 2 5070.2.b.k 2
48.i odd 4 2 3840.2.k.y 2
48.k even 4 2 3840.2.k.f 2
51.c odd 2 1 8670.2.a.g 1
60.h even 2 1 1200.2.a.k 1
60.l odd 4 2 1200.2.f.e 2
105.g even 2 1 7350.2.a.ct 1
120.i odd 2 1 4800.2.a.cq 1
120.m even 2 1 4800.2.a.d 1
120.q odd 4 2 4800.2.f.w 2
120.w even 4 2 4800.2.f.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a ✓ 1 3.b odd 2 1
90.2.a.c 1 1.a even 1 1 trivial
150.2.a.b 1 15.d odd 2 1
150.2.c.a 2 15.e even 4 2
240.2.a.b 1 12.b even 2 1
450.2.a.d 1 5.b even 2 1
450.2.c.b 2 5.c odd 4 2
720.2.a.j 1 4.b odd 2 1
810.2.e.b 2 9.c even 3 2
810.2.e.l 2 9.d odd 6 2
960.2.a.e 1 24.h odd 2 1
960.2.a.p 1 24.f even 2 1
1200.2.a.k 1 60.h even 2 1
1200.2.f.e 2 60.l odd 4 2
1470.2.a.d 1 21.c even 2 1
1470.2.i.o 2 21.h odd 6 2
1470.2.i.q 2 21.g even 6 2
2880.2.a.a 1 8.b even 2 1
2880.2.a.q 1 8.d odd 2 1
3600.2.a.f 1 20.d odd 2 1
3600.2.f.i 2 20.e even 4 2
3630.2.a.w 1 33.d even 2 1
3840.2.k.f 2 48.k even 4 2
3840.2.k.y 2 48.i odd 4 2
4410.2.a.z 1 7.b odd 2 1
4800.2.a.d 1 120.m even 2 1
4800.2.a.cq 1 120.i odd 2 1
4800.2.f.p 2 120.w even 4 2
4800.2.f.w 2 120.q odd 4 2
5070.2.a.w 1 39.d odd 2 1
5070.2.b.k 2 39.f even 4 2
7350.2.a.ct 1 105.g even 2 1
8670.2.a.g 1 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(90))\):

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 18 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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