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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,2,Mod(28,171)] 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("171.28"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
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: \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 57)
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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{5} + \beta_{2} + \cdots - 1) q^{2} + (2 \beta_{9} - \beta_{7} + \beta_{6} + \cdots - 1) q^{4} + (\beta_{11} + \beta_{8} + \cdots - \beta_{3}) q^{5} + ( - \beta_{10} + \beta_{9} + \beta_{6} + \cdots - 2) q^{7}+ \cdots + ( - \beta_{10} + \beta_{9} - \beta_{8} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} - 3 q^{4} - 6 q^{5} - 9 q^{7} - 3 q^{8} - 18 q^{10} + 9 q^{11} - 3 q^{13} + 3 q^{14} + 33 q^{16} - 12 q^{17} - 9 q^{19} + 18 q^{20} - 24 q^{22} - 9 q^{23} - 12 q^{25} + 30 q^{28} + 3 q^{32}+ \cdots + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} + \cdots + 57 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + \cdots + 1060 ) / 218 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} + \cdots - 1706 ) / 218 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} + \cdots - 2263 ) / 218 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + \cdots + 524 ) / 218 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + \cdots + 555 ) / 218 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + \cdots - 83 ) / 218 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + \cdots + 26 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + \cdots + 4350 ) / 218 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} + \cdots - 2201 ) / 218 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} + \cdots - 5194 ) / 218 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + \cdots - 5469 ) / 218 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} + \cdots - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} + \cdots + 32 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + \cdots + 77 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} + \cdots - 121 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} + \cdots - 535 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + \cdots + 257 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + \cdots + 3359 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} + \cdots + 1451 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} + \cdots - 18736 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-\beta_{3} + \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
0.500000 + 1.96356i
0.500000 − 0.677980i
0.500000 − 1.96356i
0.500000 + 0.677980i
0.500000 − 2.42499i
0.500000 + 1.74095i
0.500000 + 2.42499i
0.500000 − 1.74095i
0.500000 + 0.168222i
0.500000 + 1.80139i
0.500000 − 0.168222i
0.500000 − 1.80139i
−0.131669 − 0.746734i 0 1.33911 − 0.487396i 2.97104 + 1.08137i 0 −1.89771 + 3.28694i −1.29853 − 2.24912i 0 0.416301 − 2.36096i
28.2 0.458021 + 2.59757i 0 −4.65819 + 1.69544i −1.91800 − 0.698096i 0 −1.30802 + 2.26556i −3.89993 − 6.75488i 0 0.934866 − 5.30189i
55.1 −0.131669 + 0.746734i 0 1.33911 + 0.487396i 2.97104 − 1.08137i 0 −1.89771 − 3.28694i −1.29853 + 2.24912i 0 0.416301 + 2.36096i
55.2 0.458021 − 2.59757i 0 −4.65819 − 1.69544i −1.91800 + 0.698096i 0 −1.30802 − 2.26556i −3.89993 + 6.75488i 0 0.934866 + 5.30189i
73.1 −1.22451 − 1.02748i 0 0.0964003 + 0.546713i 0.174371 − 0.988909i 0 −1.28482 − 2.22537i −1.15479 + 2.00015i 0 −1.22961 + 1.03176i
73.2 0.958464 + 0.804247i 0 −0.0754558 − 0.427931i −0.755623 + 4.28535i 0 0.898157 + 1.55565i 1.52303 − 2.63796i 0 −4.17072 + 3.49965i
82.1 −1.22451 + 1.02748i 0 0.0964003 − 0.546713i 0.174371 + 0.988909i 0 −1.28482 + 2.22537i −1.15479 − 2.00015i 0 −1.22961 − 1.03176i
82.2 0.958464 − 0.804247i 0 −0.0754558 + 0.427931i −0.755623 − 4.28535i 0 0.898157 − 1.55565i 1.52303 + 2.63796i 0 −4.17072 − 3.49965i
100.1 −0.791518 + 0.288089i 0 −0.988583 + 0.829520i −1.30800 − 1.09754i 0 −1.96517 − 3.40377i 1.38582 − 2.40031i 0 1.35149 + 0.491903i
100.2 2.23121 − 0.812094i 0 2.78672 − 2.33833i −2.16379 − 1.81563i 0 1.05756 + 1.83175i 1.94440 − 3.36780i 0 −6.30233 − 2.29386i
118.1 −0.791518 − 0.288089i 0 −0.988583 − 0.829520i −1.30800 + 1.09754i 0 −1.96517 + 3.40377i 1.38582 + 2.40031i 0 1.35149 − 0.491903i
118.2 2.23121 + 0.812094i 0 2.78672 + 2.33833i −2.16379 + 1.81563i 0 1.05756 − 1.83175i 1.94440 + 3.36780i 0 −6.30233 + 2.29386i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 28.2
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 171.2.u.e 12
3.b odd 2 1 57.2.i.b ✓ 12
12.b even 2 1 912.2.bo.j 12
19.e even 9 1 inner 171.2.u.e 12
19.e even 9 1 3249.2.a.bh 6
19.f odd 18 1 3249.2.a.bg 6
57.j even 18 1 1083.2.a.p 6
57.l odd 18 1 57.2.i.b ✓ 12
57.l odd 18 1 1083.2.a.q 6
228.v even 18 1 912.2.bo.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.i.b ✓ 12 3.b odd 2 1
57.2.i.b ✓ 12 57.l odd 18 1
171.2.u.e 12 1.a even 1 1 trivial
171.2.u.e 12 19.e even 9 1 inner
912.2.bo.j 12 12.b even 2 1
912.2.bo.j 12 228.v even 18 1
1083.2.a.p 6 57.j even 18 1
1083.2.a.q 6 57.l odd 18 1
3249.2.a.bg 6 19.f odd 18 1
3249.2.a.bh 6 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 3 T_{2}^{11} + 6 T_{2}^{10} - 8 T_{2}^{9} - 9 T_{2}^{8} + 9 T_{2}^{7} + 99 T_{2}^{6} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots + 18496 \) Copy content Toggle raw display
$7$ \( T^{12} + 9 T^{11} + \cdots + 145161 \) Copy content Toggle raw display
$11$ \( T^{12} - 9 T^{11} + \cdots + 207936 \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 3249 \) Copy content Toggle raw display
$17$ \( T^{12} + 12 T^{11} + \cdots + 41062464 \) Copy content Toggle raw display
$19$ \( T^{12} + 9 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + 9 T^{11} + \cdots + 2483776 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{10} + \cdots + 87616 \) Copy content Toggle raw display
$31$ \( T^{12} + 36 T^{10} + \cdots + 185761 \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} + \cdots + 2467)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 18 T^{11} + \cdots + 87616 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 587917009 \) Copy content Toggle raw display
$47$ \( T^{12} - 9 T^{11} + \cdots + 46656 \) Copy content Toggle raw display
$53$ \( T^{12} - 30 T^{11} + \cdots + 1871424 \) Copy content Toggle raw display
$59$ \( T^{12} - 30 T^{11} + \cdots + 1617984 \) Copy content Toggle raw display
$61$ \( T^{12} - 21 T^{11} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1254293056 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 241118784 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1056705049 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 4606201161 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 75809912896 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21486869056 \) Copy content Toggle raw display
$97$ \( T^{12} + 27 T^{11} + \cdots + 128881 \) Copy content Toggle raw display
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