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Properties

Label 27.7.b.a
Level $27$
Weight $7$
Character orbit 27.b
Analytic conductor $6.211$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,7,Mod(26,27)] 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("27.26"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.21146025774\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-10}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 3\sqrt{-10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 26 q^{4} + 14 \beta q^{5} - 403 q^{7} + 38 \beta q^{8} - 1260 q^{10} - 158 \beta q^{11} - 961 q^{13} - 403 \beta q^{14} - 5084 q^{16} + 1014 \beta q^{17} + 8021 q^{19} - 364 \beta q^{20} + \cdots + 44760 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 52 q^{4} - 806 q^{7} - 2520 q^{10} - 1922 q^{13} - 10168 q^{16} + 16042 q^{19} + 28440 q^{22} - 4030 q^{25} + 20956 q^{28} + 97708 q^{31} - 182520 q^{34} + 48334 q^{37} - 95760 q^{40} - 121604 q^{43}+ \cdots - 2394626 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 3.16228i
3.16228i
− 9.48683i 0 −26.0000 − 132.816i 0 −403.000 − 360.500i 0 −1260.00
26.2 9.48683i 0 −26.0000 132.816i 0 −403.000 360.500i 0 −1260.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.7.b.a ✓ 2
3.b odd 2 1 inner 27.7.b.a ✓ 2
4.b odd 2 1 432.7.e.g 2
9.c even 3 2 81.7.d.c 4
9.d odd 6 2 81.7.d.c 4
12.b even 2 1 432.7.e.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.7.b.a ✓ 2 1.a even 1 1 trivial
27.7.b.a ✓ 2 3.b odd 2 1 inner
81.7.d.c 4 9.c even 3 2
81.7.d.c 4 9.d odd 6 2
432.7.e.g 2 4.b odd 2 1
432.7.e.g 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 90 \) acting on \(S_{7}^{\mathrm{new}}(27, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 90 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 17640 \) Copy content Toggle raw display
$7$ \( (T + 403)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2246760 \) Copy content Toggle raw display
$13$ \( (T + 961)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 92537640 \) Copy content Toggle raw display
$19$ \( (T - 8021)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 112493160 \) Copy content Toggle raw display
$29$ \( T^{2} + 6084000 \) Copy content Toggle raw display
$31$ \( (T - 48854)^{2} \) Copy content Toggle raw display
$37$ \( (T - 24167)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 5008644000 \) Copy content Toggle raw display
$43$ \( (T + 60802)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 728974440 \) Copy content Toggle raw display
$53$ \( T^{2} + 18943385760 \) Copy content Toggle raw display
$59$ \( T^{2} + 9514757160 \) Copy content Toggle raw display
$61$ \( (T - 272999)^{2} \) Copy content Toggle raw display
$67$ \( (T + 85579)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 116795571840 \) Copy content Toggle raw display
$73$ \( (T + 152737)^{2} \) Copy content Toggle raw display
$79$ \( (T + 74059)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 9326916000 \) Copy content Toggle raw display
$89$ \( T^{2} + 1424896404840 \) Copy content Toggle raw display
$97$ \( (T + 1197313)^{2} \) Copy content Toggle raw display
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