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Properties

Label 243.7.b.c
Level $243$
Weight $7$
Character orbit 243.b
Analytic conductor $55.903$
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] = [243,7,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, 7); 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, 7, names="a")
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 7 \)
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,-166,0,0,190] 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: \(55.9031423196\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 7 \)
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 \(\beta = 7\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - 83 q^{4} - 20 \beta q^{5} + 95 q^{7} + 19 \beta q^{8} - 2940 q^{10} + 152 \beta q^{11} - 589 q^{13} - 95 \beta q^{14} - 2519 q^{16} + 228 \beta q^{17} + 8315 q^{19} + 1660 \beta q^{20} + \cdots + 108624 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 166 q^{4} + 190 q^{7} - 5880 q^{10} - 1178 q^{13} - 5038 q^{16} + 16630 q^{19} + 44688 q^{22} - 86350 q^{25} - 15770 q^{28} - 70226 q^{31} + 67032 q^{34} - 124490 q^{37} + 111720 q^{40} - 23354 q^{43}+ \cdots - 2708162 q^{97}+O(q^{100}) \) 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
0.500000 + 0.866025i
0.500000 − 0.866025i
− 12.1244i 0 −83.0000 − 242.487i 0 95.0000 230.363i 0 −2940.00
242.2 12.1244i 0 −83.0000 242.487i 0 95.0000 − 230.363i 0 −2940.00
\(n\): 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 243.7.b.c ✓ 2
3.b odd 2 1 inner 243.7.b.c ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.7.b.c ✓ 2 1.a even 1 1 trivial
243.7.b.c ✓ 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(243, [\chi])\):

\( T_{2}^{2} + 147 \) Copy content Toggle raw display
\( T_{7} - 95 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 147 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 58800 \) Copy content Toggle raw display
$7$ \( (T - 95)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3396288 \) Copy content Toggle raw display
$13$ \( (T + 589)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 7641648 \) Copy content Toggle raw display
$19$ \( (T - 8315)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 104713392 \) Copy content Toggle raw display
$29$ \( T^{2} + 975459072 \) Copy content Toggle raw display
$31$ \( (T + 35113)^{2} \) Copy content Toggle raw display
$37$ \( (T + 62245)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3029905200 \) Copy content Toggle raw display
$43$ \( (T + 11677)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 19616959488 \) Copy content Toggle raw display
$53$ \( T^{2} + 67521601728 \) Copy content Toggle raw display
$59$ \( T^{2} + 449159088 \) Copy content Toggle raw display
$61$ \( (T - 218906)^{2} \) Copy content Toggle raw display
$67$ \( (T - 128186)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 20663016192 \) Copy content Toggle raw display
$73$ \( (T + 469726)^{2} \) Copy content Toggle raw display
$79$ \( (T + 225625)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 44977599408 \) Copy content Toggle raw display
$89$ \( T^{2} + 499362624432 \) Copy content Toggle raw display
$97$ \( (T + 1354081)^{2} \) Copy content Toggle raw display
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