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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 = [12,0,0,0,0,0,-6,6] 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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 237x^{8} + 1312x^{6} + 5283x^{4} + 11049x^{2} + 16129 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{11} - \beta_{4}) q^{2} - \beta_{6} q^{4} + (\beta_{11} - \beta_{9} + \cdots - \beta_{2}) q^{5} + ( - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{3} + 1) q^{8} + ( - \beta_{8} - \beta_{5} + \cdots - \beta_1) q^{10}+ \cdots + ( - 2 \beta_{10} + \beta_{8} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{7} + 6 q^{8} + 6 q^{11} - 6 q^{17} - 12 q^{19} - 12 q^{20} + 12 q^{23} - 18 q^{25} + 12 q^{26} + 6 q^{31} + 6 q^{34} + 12 q^{35} + 24 q^{37} - 18 q^{38} - 30 q^{41} + 36 q^{43} - 18 q^{46} + 24 q^{47}+ \cdots - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 237x^{8} + 1312x^{6} + 5283x^{4} + 11049x^{2} + 16129 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2291\nu^{11} - 36666\nu^{9} - 482769\nu^{7} - 2213185\nu^{5} - 10761471\nu^{3} - 22506813\nu ) / 14861667 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2291\nu^{10} - 36666\nu^{8} - 482769\nu^{6} - 2213185\nu^{4} - 10761471\nu^{2} - 7645146 ) / 14861667 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -41156\nu^{10} - 1543067\nu^{8} - 21142697\nu^{6} - 189912199\nu^{4} - 589122112\nu^{2} - 1347625321 ) / 44585001 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 41156 \nu^{11} - 1543067 \nu^{9} - 21142697 \nu^{7} - 189912199 \nu^{5} + \cdots - 1347625321 \nu ) / 44585001 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 138862 \nu^{10} - 3050569 \nu^{8} - 39340177 \nu^{6} - 256113644 \nu^{4} - 759110054 \nu^{2} - 1472101577 ) / 44585001 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 138862 \nu^{11} + 3050569 \nu^{9} + 39340177 \nu^{7} + 256113644 \nu^{5} + 759110054 \nu^{3} + 1472101577 \nu ) / 44585001 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -167470\nu^{10} - 1983982\nu^{8} - 22819837\nu^{6} - 11627711\nu^{4} + 7214074\nu^{2} + 651433165 ) / 44585001 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -167470\nu^{11} - 1983982\nu^{9} - 22819837\nu^{7} - 11627711\nu^{5} + 7214074\nu^{3} + 651433165\nu ) / 44585001 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 237152 \nu^{11} + 3343541 \nu^{9} + 39895049 \nu^{7} + 107128846 \nu^{5} + 255584545 \nu^{3} - 217329893 \nu ) / 44585001 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 278308\nu^{10} + 4886608\nu^{8} + 61037746\nu^{6} + 297041045\nu^{4} + 844706657\nu^{2} + 1130295428 ) / 44585001 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 2\beta_{8} - \beta_{6} + 3\beta_{4} - 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + 2\beta_{9} + \beta_{7} + 2\beta_{5} - 6\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -44\beta_{11} - 44\beta_{8} - 29\beta_{6} - 29\beta_{4} + 50\beta_{3} - 50 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -44\beta_{10} - 44\beta_{9} + 29\beta_{7} + 15\beta_{5} + 50\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 348\beta_{11} + 183\beta_{8} + 531\beta_{6} - 183\beta_{4} + 505 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 348\beta_{10} + 183\beta_{9} - 531\beta_{7} - 531\beta_{5} + 505\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2116\beta_{11} + 3995\beta_{8} - 2116\beta_{6} + 6111\beta_{4} - 5502\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2116\beta_{10} + 3995\beta_{9} + 2116\beta_{7} + 3995\beta_{5} - 5502\beta_{2} - 5502\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -69389\beta_{11} - 69389\beta_{8} - 45317\beta_{6} - 45317\beta_{4} + 61451\beta_{3} - 61451 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -69389\beta_{10} - 69389\beta_{9} + 45317\beta_{7} + 24072\beta_{5} + 61451\beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-\beta_{6}\)

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
1.68009 − 2.91001i
−1.68009 + 2.91001i
−0.925476 + 1.60297i
0.925476 − 1.60297i
1.68009 + 2.91001i
−1.68009 − 2.91001i
−0.925476 − 1.60297i
0.925476 + 1.60297i
0.905969 − 1.56918i
−0.905969 + 1.56918i
0.905969 + 1.56918i
−0.905969 − 1.56918i
−0.173648 + 0.984808i 0 −0.939693 − 0.342020i −0.156911 + 0.0571108i 0 1.18009 + 2.04398i 0.500000 − 0.866025i 0 −0.0289959 − 0.164444i
55.2 −0.173648 + 0.984808i 0 −0.939693 − 0.342020i 2.03630 − 0.741151i 0 −2.18009 − 3.77603i 0.500000 − 0.866025i 0 0.376292 + 2.13406i
73.1 −0.766044 − 0.642788i 0 0.173648 + 0.984808i −0.666084 + 3.77755i 0 −1.42548 − 2.46900i 0.500000 − 0.866025i 0 2.93841 − 2.46562i
73.2 −0.766044 − 0.642788i 0 0.173648 + 0.984808i 0.318787 − 1.80793i 0 0.425476 + 0.736946i 0.500000 − 0.866025i 0 −1.40632 + 1.18004i
199.1 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i −0.156911 − 0.0571108i 0 1.18009 − 2.04398i 0.500000 + 0.866025i 0 −0.0289959 + 0.164444i
199.2 −0.173648 − 0.984808i 0 −0.939693 + 0.342020i 2.03630 + 0.741151i 0 −2.18009 + 3.77603i 0.500000 + 0.866025i 0 0.376292 − 2.13406i
253.1 −0.766044 + 0.642788i 0 0.173648 − 0.984808i −0.666084 − 3.77755i 0 −1.42548 + 2.46900i 0.500000 + 0.866025i 0 2.93841 + 2.46562i
253.2 −0.766044 + 0.642788i 0 0.173648 − 0.984808i 0.318787 + 1.80793i 0 0.425476 − 0.736946i 0.500000 + 0.866025i 0 −1.40632 − 1.18004i
271.1 0.939693 − 0.342020i 0 0.766044 − 0.642788i −3.37468 − 2.83169i 0 0.405969 + 0.703159i 0.500000 − 0.866025i 0 −4.13966 − 1.50671i
271.2 0.939693 − 0.342020i 0 0.766044 − 0.642788i 1.84259 + 1.54612i 0 −1.40597 − 2.43521i 0.500000 − 0.866025i 0 2.26027 + 0.822671i
289.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i −3.37468 + 2.83169i 0 0.405969 − 0.703159i 0.500000 + 0.866025i 0 −4.13966 + 1.50671i
289.2 0.939693 + 0.342020i 0 0.766044 + 0.642788i 1.84259 − 1.54612i 0 −1.40597 + 2.43521i 0.500000 + 0.866025i 0 2.26027 − 0.822671i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 55.2
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.g yes 12
3.b odd 2 1 342.2.u.f ✓ 12
19.e even 9 1 inner 342.2.u.g yes 12
19.e even 9 1 6498.2.a.cb 6
19.f odd 18 1 6498.2.a.cd 6
57.j even 18 1 6498.2.a.cc 6
57.l odd 18 1 342.2.u.f ✓ 12
57.l odd 18 1 6498.2.a.ce 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.u.f ✓ 12 3.b odd 2 1
342.2.u.f ✓ 12 57.l odd 18 1
342.2.u.g yes 12 1.a even 1 1 trivial
342.2.u.g yes 12 19.e even 9 1 inner
6498.2.a.cb 6 19.e even 9 1
6498.2.a.cc 6 57.j even 18 1
6498.2.a.cd 6 19.f odd 18 1
6498.2.a.ce 6 57.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 9 T_{5}^{10} - 44 T_{5}^{9} + 144 T_{5}^{8} - 738 T_{5}^{7} + 3826 T_{5}^{6} - 11052 T_{5}^{5} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 9 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + \cdots + 3249 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 33074001 \) Copy content Toggle raw display
$13$ \( T^{12} + 27 T^{10} + \cdots + 32041 \) Copy content Toggle raw display
$17$ \( T^{12} + 6 T^{11} + \cdots + 1896129 \) Copy content Toggle raw display
$19$ \( T^{12} + 12 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + \cdots + 8346321 \) Copy content Toggle raw display
$29$ \( T^{12} + 81 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} + \cdots + 1369 \) Copy content Toggle raw display
$37$ \( (T^{6} - 12 T^{5} + \cdots + 15101)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 30 T^{11} + \cdots + 2047761 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 1850634361 \) Copy content Toggle raw display
$47$ \( T^{12} - 24 T^{11} + \cdots + 76055841 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 183349245249 \) Copy content Toggle raw display
$59$ \( T^{12} + 42 T^{11} + \cdots + 33074001 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 2994387841 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 21549946401 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1326634929 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1937144169 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 229189321 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 191850201 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2338399449 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 64313467201 \) Copy content Toggle raw display
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