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Properties

Label 990.2.i.f
Level $990$
Weight $2$
Character orbit 990.i
Analytic conductor $7.905$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [990,2,Mod(331,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("990.331"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,3,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - \beta_{3}) q^{3} + (\beta_1 - 1) q^{4} + ( - \beta_1 + 1) q^{5} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{6} + ( - \beta_{2} + 2 \beta_1) q^{7} - q^{8} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{9}+ \cdots + ( - \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 3 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 3 q^{11} + 3 q^{13} - 6 q^{14} - 3 q^{16} + 6 q^{17} - 6 q^{19} + 3 q^{20} - 3 q^{22} - 6 q^{23} - 3 q^{25} + 6 q^{26} + 27 q^{27} - 12 q^{28}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{18}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{18}^{5} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{18}^{5} + \zeta_{18}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \) Copy content Toggle raw display

Display \(\zeta_{18}^j\) in terms of \(\beta_i\)

\(\zeta_{18}\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{2}\)\(=\) \( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{18}^{4}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{18}^{5}\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{18}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/990\mathbb{Z}\right)^\times\).

\(n\) \(397\) \(541\) \(551\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
331.1
−0.766044 − 0.642788i
0.939693 − 0.342020i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
−0.173648 − 0.984808i
0.500000 − 0.866025i −1.11334 − 1.32683i −0.500000 − 0.866025i 0.500000 + 0.866025i −1.70574 + 0.300767i 0.826352 − 1.43128i −1.00000 −0.520945 + 2.95442i 1.00000
331.2 0.500000 − 0.866025i −0.592396 + 1.62760i −0.500000 − 0.866025i 0.500000 + 0.866025i 1.11334 + 1.32683i 0.233956 − 0.405223i −1.00000 −2.29813 − 1.92836i 1.00000
331.3 0.500000 − 0.866025i 1.70574 − 0.300767i −0.500000 − 0.866025i 0.500000 + 0.866025i 0.592396 − 1.62760i 1.93969 − 3.35965i −1.00000 2.81908 − 1.02606i 1.00000
661.1 0.500000 + 0.866025i −1.11334 + 1.32683i −0.500000 + 0.866025i 0.500000 − 0.866025i −1.70574 − 0.300767i 0.826352 + 1.43128i −1.00000 −0.520945 − 2.95442i 1.00000
661.2 0.500000 + 0.866025i −0.592396 − 1.62760i −0.500000 + 0.866025i 0.500000 − 0.866025i 1.11334 − 1.32683i 0.233956 + 0.405223i −1.00000 −2.29813 + 1.92836i 1.00000
661.3 0.500000 + 0.866025i 1.70574 + 0.300767i −0.500000 + 0.866025i 0.500000 − 0.866025i 0.592396 + 1.62760i 1.93969 + 3.35965i −1.00000 2.81908 + 1.02606i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.3
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 990.2.i.f ✓ 6
3.b odd 2 1 2970.2.i.e 6
9.c even 3 1 inner 990.2.i.f ✓ 6
9.c even 3 1 8910.2.a.y 3
9.d odd 6 1 2970.2.i.e 6
9.d odd 6 1 8910.2.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.i.f ✓ 6 1.a even 1 1 trivial
990.2.i.f ✓ 6 9.c even 3 1 inner
2970.2.i.e 6 3.b odd 2 1
2970.2.i.e 6 9.d odd 6 1
8910.2.a.y 3 9.c even 3 1
8910.2.a.bm 3 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 6T_{7}^{5} + 27T_{7}^{4} - 48T_{7}^{3} + 63T_{7}^{2} - 27T_{7} + 9 \) acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} - 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} - 3 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$17$ \( (T^{3} - 3 T^{2} - 24 T + 53)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 3 T^{2} - 24 T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 2601 \) Copy content Toggle raw display
$29$ \( T^{6} + 15 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$37$ \( (T^{3} + 12 T^{2} + 27 T - 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} + 21 T^{4} + \cdots + 289 \) Copy content Toggle raw display
$53$ \( (T^{3} - 3 T^{2} - 24 T + 53)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 18 T^{5} + \cdots + 1369 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$67$ \( T^{6} - 18 T^{5} + \cdots + 128881 \) Copy content Toggle raw display
$71$ \( (T^{3} - 18 T^{2} + \cdots + 136)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 9 T^{2} + \cdots + 739)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 15 T^{5} + \cdots + 7921 \) Copy content Toggle raw display
$83$ \( T^{6} + 3 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$89$ \( (T^{3} - 252 T - 72)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 21 T^{5} + \cdots + 25281 \) Copy content Toggle raw display
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