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Properties

Label 243.5.b.e
Level $243$
Weight $5$
Character orbit 243.b
Analytic conductor $25.119$
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] = [243,5,Mod(242,243)] 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("243.242"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 243.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,20,0,0,-110] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.1189010294\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 10 q^{4} - 16 \beta q^{5} - 55 q^{7} + 26 \beta q^{8} + 96 q^{10} - 38 \beta q^{11} - 166 q^{13} - 55 \beta q^{14} + 4 q^{16} + 108 \beta q^{17} - 442 q^{19} - 160 \beta q^{20} + 228 q^{22} + \cdots + 624 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{4} - 110 q^{7} + 192 q^{10} - 332 q^{13} + 8 q^{16} - 884 q^{19} + 456 q^{22} - 1822 q^{25} - 1100 q^{28} + 802 q^{31} - 1296 q^{34} + 1006 q^{37} + 4992 q^{40} - 4862 q^{43} - 4056 q^{46} + 1248 q^{49}+ \cdots - 25292 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
242.1
− 2.44949i
2.44949i
− 2.44949i 0 10.0000 39.1918i 0 −55.0000 − 63.6867i 0 96.0000
242.2 2.44949i 0 10.0000 − 39.1918i 0 −55.0000 63.6867i 0 96.0000
\(n\): e.g. 2-40 or 80-90
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 243.5.b.e ✓ 2
3.b odd 2 1 inner 243.5.b.e ✓ 2
9.c even 3 2 243.5.d.f 4
9.d odd 6 2 243.5.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.5.b.e ✓ 2 1.a even 1 1 trivial
243.5.b.e ✓ 2 3.b odd 2 1 inner
243.5.d.f 4 9.c even 3 2
243.5.d.f 4 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(243, [\chi])\):

\( T_{2}^{2} + 6 \) Copy content Toggle raw display
\( T_{7} + 55 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 6 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1536 \) Copy content Toggle raw display
$7$ \( (T + 55)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 8664 \) Copy content Toggle raw display
$13$ \( (T + 166)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 69984 \) Copy content Toggle raw display
$19$ \( (T + 442)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 685464 \) Copy content Toggle raw display
$29$ \( T^{2} + 2232600 \) Copy content Toggle raw display
$31$ \( (T - 401)^{2} \) Copy content Toggle raw display
$37$ \( (T - 503)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 490776 \) Copy content Toggle raw display
$43$ \( (T + 2431)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1038336 \) Copy content Toggle raw display
$53$ \( T^{2} + 3650400 \) Copy content Toggle raw display
$59$ \( T^{2} + 17136600 \) Copy content Toggle raw display
$61$ \( (T + 7225)^{2} \) Copy content Toggle raw display
$67$ \( (T - 3545)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 10549656 \) Copy content Toggle raw display
$73$ \( (T - 623)^{2} \) Copy content Toggle raw display
$79$ \( (T - 3158)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3801696 \) Copy content Toggle raw display
$89$ \( T^{2} + 119242584 \) Copy content Toggle raw display
$97$ \( (T + 12646)^{2} \) Copy content Toggle raw display
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