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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(55,342)] 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("342.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-9,0,3,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(6\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \cdots - 1) q^{5} + (\zeta_{18}^{5} + \zeta_{18}^{4} + \cdots + 1) q^{7} - \zeta_{18}^{3} q^{8} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} + \cdots + 1) q^{10} + \cdots + (\zeta_{18}^{5} - 4 \zeta_{18}^{4} + \cdots + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{13} + 6 q^{14} + 3 q^{17} - 18 q^{19} - 3 q^{22} + 21 q^{23} + 9 q^{25} + 6 q^{26} - 3 q^{28} + 3 q^{29} + 9 q^{31} + 12 q^{34} - 18 q^{35} - 18 q^{37} + 6 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(\zeta_{18}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.173648 + 0.984808i
−0.766044 − 0.642788i
−0.173648 − 0.984808i
−0.766044 + 0.642788i
0.939693 − 0.342020i
0.939693 + 0.342020i
0.173648 − 0.984808i 0 −0.939693 − 0.342020i −3.20574 + 1.16679i 0 1.43969 + 2.49362i −0.500000 + 0.866025i 0 0.592396 + 3.35965i
73.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.386659 + 2.19285i 0 0.326352 + 0.565258i −0.500000 + 0.866025i 0 −1.70574 + 1.43128i
199.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i −3.20574 − 1.16679i 0 1.43969 − 2.49362i −0.500000 − 0.866025i 0 0.592396 − 3.35965i
253.1 0.766044 − 0.642788i 0 0.173648 − 0.984808i −0.386659 − 2.19285i 0 0.326352 − 0.565258i −0.500000 − 0.866025i 0 −1.70574 − 1.43128i
271.1 −0.939693 + 0.342020i 0 0.766044 − 0.642788i −0.907604 − 0.761570i 0 −0.266044 − 0.460802i −0.500000 + 0.866025i 0 1.11334 + 0.405223i
289.1 −0.939693 − 0.342020i 0 0.766044 + 0.642788i −0.907604 + 0.761570i 0 −0.266044 + 0.460802i −0.500000 − 0.866025i 0 1.11334 − 0.405223i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 55.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.u.a 6
3.b odd 2 1 114.2.i.d ✓ 6
12.b even 2 1 912.2.bo.f 6
19.e even 9 1 inner 342.2.u.a 6
19.e even 9 1 6498.2.a.bs 3
19.f odd 18 1 6498.2.a.bn 3
57.j even 18 1 2166.2.a.u 3
57.l odd 18 1 114.2.i.d ✓ 6
57.l odd 18 1 2166.2.a.o 3
228.v even 18 1 912.2.bo.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.d ✓ 6 3.b odd 2 1
114.2.i.d ✓ 6 57.l odd 18 1
342.2.u.a 6 1.a even 1 1 trivial
342.2.u.a 6 19.e even 9 1 inner
912.2.bo.f 6 12.b even 2 1
912.2.bo.f 6 228.v even 18 1
2166.2.a.o 3 57.l odd 18 1
2166.2.a.u 3 57.j even 18 1
6498.2.a.bn 3 19.f odd 18 1
6498.2.a.bs 3 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 9T_{5}^{5} + 36T_{5}^{4} + 90T_{5}^{3} + 162T_{5}^{2} + 162T_{5} + 81 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$19$ \( T^{6} + 18 T^{5} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 21 T^{5} + \cdots + 72361 \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} + \cdots + 45369 \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( (T^{3} + 9 T^{2} + \cdots - 361)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 15 T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + \cdots + 3249 \) Copy content Toggle raw display
$47$ \( T^{6} - 9 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
$53$ \( T^{6} + 12 T^{5} + \cdots + 94249 \) Copy content Toggle raw display
$59$ \( T^{6} + 27 T^{5} + \cdots + 289 \) Copy content Toggle raw display
$61$ \( T^{6} - 3 T^{5} + \cdots + 1682209 \) Copy content Toggle raw display
$67$ \( T^{6} - 21 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$71$ \( T^{6} + 39 T^{5} + \cdots + 201601 \) Copy content Toggle raw display
$73$ \( T^{6} - 36 T^{5} + \cdots + 1369 \) Copy content Toggle raw display
$79$ \( T^{6} + 45 T^{5} + \cdots + 4515625 \) Copy content Toggle raw display
$83$ \( T^{6} - 27 T^{5} + \cdots + 253009 \) Copy content Toggle raw display
$89$ \( T^{6} - 30 T^{5} + \cdots + 11449 \) Copy content Toggle raw display
$97$ \( T^{6} + 6 T^{5} + \cdots + 2809 \) Copy content Toggle raw display
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