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Properties

Label 90.2.e.c
Level $90$
Weight $2$
Character orbit 90.e
Analytic conductor $0.719$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,2,Mod(31,90)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 90.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.718653618192\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} - \beta_1 + 1) q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{2} + 1) q^{5} + (\beta_{3} + \beta_{2}) q^{6} + ( - \beta_{3} + 2 \beta_1 - 1) q^{7} - q^{8} + (\beta_{3} - \beta_1 - 2) q^{9}+ \cdots + ( - \beta_{3} + \beta_{2} + 5 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} + q^{6} - q^{7} - 4 q^{8} - 10 q^{9} + 4 q^{10} + 3 q^{11} - q^{12} + 2 q^{13} + q^{14} + q^{15} - 2 q^{16} - 18 q^{17} - 5 q^{18} + 2 q^{19} + 2 q^{20}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.68614 + 0.396143i
−1.18614 − 1.26217i
−1.18614 + 1.26217i
1.68614 − 0.396143i
0.500000 + 0.866025i 0.500000 − 1.65831i −0.500000 + 0.866025i 0.500000 − 0.866025i 1.68614 − 0.396143i 1.18614 + 2.05446i −1.00000 −2.50000 − 1.65831i 1.00000
31.2 0.500000 + 0.866025i 0.500000 + 1.65831i −0.500000 + 0.866025i 0.500000 − 0.866025i −1.18614 + 1.26217i −1.68614 − 2.92048i −1.00000 −2.50000 + 1.65831i 1.00000
61.1 0.500000 − 0.866025i 0.500000 − 1.65831i −0.500000 − 0.866025i 0.500000 + 0.866025i −1.18614 − 1.26217i −1.68614 + 2.92048i −1.00000 −2.50000 − 1.65831i 1.00000
61.2 0.500000 − 0.866025i 0.500000 + 1.65831i −0.500000 − 0.866025i 0.500000 + 0.866025i 1.68614 + 0.396143i 1.18614 − 2.05446i −1.00000 −2.50000 + 1.65831i 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.2.e.c ✓ 4
3.b odd 2 1 270.2.e.c 4
4.b odd 2 1 720.2.q.f 4
5.b even 2 1 450.2.e.j 4
5.c odd 4 2 450.2.j.g 8
9.c even 3 1 inner 90.2.e.c ✓ 4
9.c even 3 1 810.2.a.i 2
9.d odd 6 1 270.2.e.c 4
9.d odd 6 1 810.2.a.k 2
12.b even 2 1 2160.2.q.f 4
15.d odd 2 1 1350.2.e.l 4
15.e even 4 2 1350.2.j.f 8
36.f odd 6 1 720.2.q.f 4
36.f odd 6 1 6480.2.a.be 2
36.h even 6 1 2160.2.q.f 4
36.h even 6 1 6480.2.a.bn 2
45.h odd 6 1 1350.2.e.l 4
45.h odd 6 1 4050.2.a.bo 2
45.j even 6 1 450.2.e.j 4
45.j even 6 1 4050.2.a.bw 2
45.k odd 12 2 450.2.j.g 8
45.k odd 12 2 4050.2.c.v 4
45.l even 12 2 1350.2.j.f 8
45.l even 12 2 4050.2.c.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c ✓ 4 1.a even 1 1 trivial
90.2.e.c ✓ 4 9.c even 3 1 inner
270.2.e.c 4 3.b odd 2 1
270.2.e.c 4 9.d odd 6 1
450.2.e.j 4 5.b even 2 1
450.2.e.j 4 45.j even 6 1
450.2.j.g 8 5.c odd 4 2
450.2.j.g 8 45.k odd 12 2
720.2.q.f 4 4.b odd 2 1
720.2.q.f 4 36.f odd 6 1
810.2.a.i 2 9.c even 3 1
810.2.a.k 2 9.d odd 6 1
1350.2.e.l 4 15.d odd 2 1
1350.2.e.l 4 45.h odd 6 1
1350.2.j.f 8 15.e even 4 2
1350.2.j.f 8 45.l even 12 2
2160.2.q.f 4 12.b even 2 1
2160.2.q.f 4 36.h even 6 1
4050.2.a.bo 2 45.h odd 6 1
4050.2.a.bw 2 45.j even 6 1
4050.2.c.v 4 45.k odd 12 2
4050.2.c.ba 4 45.l even 12 2
6480.2.a.be 2 36.f odd 6 1
6480.2.a.bn 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + T_{7}^{3} + 9T_{7}^{2} - 8T_{7} + 64 \) acting on \(S_{2}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{4} - 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9 T + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( (T + 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 17 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( T^{4} - 9 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( (T^{2} - 132)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$61$ \( T^{4} + T^{3} + \cdots + 5476 \) Copy content Toggle raw display
$67$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 11 T - 44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 9 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$89$ \( (T^{2} - 15 T - 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 11 T^{3} + \cdots + 484 \) Copy content Toggle raw display
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