↑

Properties

Label 27.12.c.a
Level $27$
Weight $12$
Character orbit 27.c
Analytic conductor $20.745$
Analytic rank $0$
Dimension $20$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,12,Mod(10,27)] 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("27.10"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 27.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.7452658751\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 9863 x^{18} + 40416552 x^{16} + 89424581388 x^{14} + 116167273852206 x^{12} + \cdots + 59\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{65} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta_{11} - \beta_{2}) q^{2} + (\beta_{12} + 921 \beta_{11} + \cdots - 921) q^{4} + ( - \beta_{13} - 723 \beta_{11} + \cdots + 723) q^{5} + (\beta_{14} + 891 \beta_{11} + 133 \beta_{2}) q^{7}+ \cdots + ( - 127423 \beta_{10} + \cdots - 65300339953) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 33 q^{2} - 9217 q^{4} + 7230 q^{5} + 8512 q^{7} + 29118 q^{8} + 4092 q^{10} + 112776 q^{11} + 279706 q^{13} + 3901584 q^{14} - 7342081 q^{16} - 27765792 q^{17} + 7029400 q^{19} + 34163508 q^{20}+ \cdots - 1310123604078 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 9863 x^{18} + 40416552 x^{16} + 89424581388 x^{14} + 116167273852206 x^{12} + \cdots + 59\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 31\!\cdots\!77 \nu^{18} + \cdots - 84\!\cdots\!28 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31\!\cdots\!77 \nu^{18} + \cdots + 84\!\cdots\!28 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31\!\cdots\!77 \nu^{18} + \cdots + 92\!\cdots\!28 ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 69\!\cdots\!61 \nu^{18} + \cdots + 11\!\cdots\!80 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17\!\cdots\!67 \nu^{18} + \cdots + 36\!\cdots\!88 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48\!\cdots\!63 \nu^{18} + \cdots - 67\!\cdots\!12 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!51 \nu^{18} + \cdots + 55\!\cdots\!56 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!19 \nu^{18} + \cdots + 37\!\cdots\!36 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!27 \nu^{18} + \cdots - 39\!\cdots\!04 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!69 \nu^{18} + \cdots + 16\!\cdots\!32 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 55\!\cdots\!73 \nu^{19} + \cdots + 59\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 15\!\cdots\!51 \nu^{19} + \cdots - 17\!\cdots\!60 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 75\!\cdots\!89 \nu^{19} + \cdots + 12\!\cdots\!60 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 45\!\cdots\!55 \nu^{19} + \cdots - 18\!\cdots\!56 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\!\cdots\!15 \nu^{19} + \cdots + 14\!\cdots\!24 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 59\!\cdots\!11 \nu^{19} + \cdots + 14\!\cdots\!92 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 60\!\cdots\!63 \nu^{19} + \cdots - 62\!\cdots\!88 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 12\!\cdots\!81 \nu^{19} + \cdots - 45\!\cdots\!08 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 11\!\cdots\!89 \nu^{19} + \cdots - 21\!\cdots\!04 ) / 20\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 5\beta_{2} + 5\beta _1 - 2960 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{16} - 2 \beta_{14} - 16 \beta_{12} - 32204 \beta_{11} + \beta_{6} - \beta_{5} - 8 \beta_{3} + \cdots + 16102 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 10 \beta_{10} + 4 \beta_{9} - 15 \beta_{8} + 8 \beta_{7} + 14 \beta_{6} + 21 \beta_{5} + 343 \beta_{4} + \cdots + 14815766 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1040 \beta_{19} + 1920 \beta_{18} - 368 \beta_{17} - 16784 \beta_{16} + 952 \beta_{15} + \cdots - 180935110 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 35224 \beta_{10} - 8896 \beta_{9} + 56556 \beta_{8} - 27560 \beta_{7} - 64594 \beta_{6} + \cdots - 28714305124 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4444560 \beta_{19} - 6988048 \beta_{18} + 1325416 \beta_{17} + 40642610 \beta_{16} + \cdots + 577255805686 ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 286899702 \beta_{10} + 39256364 \beta_{9} - 485021673 \beta_{8} + 247334128 \beta_{7} + \cdots + 180759302557266 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 41771245472 \beta_{19} + 57946264048 \beta_{18} - 11567084712 \beta_{17} - 289306488684 \beta_{16} + \cdots - 51\!\cdots\!02 ) / 81 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 716787286200 \beta_{10} - 22263917968 \beta_{9} + 1265532513732 \beta_{8} - 687840762768 \beta_{7} + \cdots - 39\!\cdots\!32 ) / 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 116802460776608 \beta_{19} - 145925699124064 \beta_{18} + 32859015609264 \beta_{17} + \cdots + 14\!\cdots\!02 ) / 81 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 577769913650794 \beta_{10} - 41533208603996 \beta_{9} + \cdots + 30\!\cdots\!30 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 10\!\cdots\!76 \beta_{19} + \cdots - 13\!\cdots\!10 ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 41\!\cdots\!44 \beta_{10} + \cdots - 21\!\cdots\!52 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 80\!\cdots\!56 \beta_{19} + \cdots + 11\!\cdots\!22 ) / 81 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 98\!\cdots\!38 \beta_{10} + \cdots + 50\!\cdots\!82 ) / 27 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 20\!\cdots\!68 \beta_{19} + \cdots - 29\!\cdots\!06 ) / 81 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 26\!\cdots\!24 \beta_{10} + \cdots - 13\!\cdots\!96 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 57\!\cdots\!44 \beta_{19} + \cdots + 84\!\cdots\!74 ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
49.9602i
37.0247i
31.4671i
15.2468i
1.73070i
− 5.09438i
− 20.7390i
− 26.4646i
− 39.8650i
− 44.9985i
− 49.9602i
− 37.0247i
− 31.4671i
− 15.2468i
− 1.73070i
5.09438i
20.7390i
26.4646i
39.8650i
44.9985i
−41.7668 − 72.3422i 0 −2464.93 + 4269.38i 2698.81 − 4674.47i 0 23240.9 + 40254.5i 240732. 0 −450882.
10.2 −30.5643 − 52.9389i 0 −844.353 + 1462.46i 1684.05 − 2916.87i 0 −5398.44 − 9350.38i −21963.1 0 −205888.
10.3 −25.7513 − 44.6025i 0 −302.258 + 523.527i −4464.02 + 7731.91i 0 −5876.40 − 10178.2i −74343.1 0 459817.
10.4 −11.7041 − 20.2721i 0 750.028 − 1299.09i −1606.12 + 2781.88i 0 −185.973 − 322.114i −83053.6 0 75192.8
10.5 0.00116679 + 0.00202094i 0 1024.00 − 1773.62i 5478.49 − 9489.02i 0 −31655.6 − 54829.1i 9.55835 0 25.5690
10.6 5.91186 + 10.2396i 0 954.100 − 1652.55i 2014.61 − 3489.40i 0 34855.3 + 60371.1i 46777.0 0 47640.3
10.7 19.4605 + 33.7066i 0 266.578 − 461.726i −1691.47 + 2929.71i 0 −16014.5 − 27738.0i 100461. 0 −131667.
10.8 24.4190 + 42.2949i 0 −168.574 + 291.979i −5966.72 + 10334.7i 0 19533.0 + 33832.1i 83554.5 0 −582805.
10.9 36.0241 + 62.3956i 0 −1571.47 + 2721.87i 5310.88 − 9198.71i 0 26579.3 + 46036.7i −78888.8 0 765278.
10.10 40.4699 + 70.0959i 0 −2251.62 + 3899.92i 156.497 − 271.061i 0 −40821.5 − 70704.9i −198726. 0 25333.7
19.1 −41.7668 + 72.3422i 0 −2464.93 − 4269.38i 2698.81 + 4674.47i 0 23240.9 − 40254.5i 240732. 0 −450882.
19.2 −30.5643 + 52.9389i 0 −844.353 − 1462.46i 1684.05 + 2916.87i 0 −5398.44 + 9350.38i −21963.1 0 −205888.
19.3 −25.7513 + 44.6025i 0 −302.258 − 523.527i −4464.02 − 7731.91i 0 −5876.40 + 10178.2i −74343.1 0 459817.
19.4 −11.7041 + 20.2721i 0 750.028 + 1299.09i −1606.12 − 2781.88i 0 −185.973 + 322.114i −83053.6 0 75192.8
19.5 0.00116679 − 0.00202094i 0 1024.00 + 1773.62i 5478.49 + 9489.02i 0 −31655.6 + 54829.1i 9.55835 0 25.5690
19.6 5.91186 − 10.2396i 0 954.100 + 1652.55i 2014.61 + 3489.40i 0 34855.3 − 60371.1i 46777.0 0 47640.3
19.7 19.4605 − 33.7066i 0 266.578 + 461.726i −1691.47 − 2929.71i 0 −16014.5 + 27738.0i 100461. 0 −131667.
19.8 24.4190 − 42.2949i 0 −168.574 − 291.979i −5966.72 − 10334.7i 0 19533.0 − 33832.1i 83554.5 0 −582805.
19.9 36.0241 − 62.3956i 0 −1571.47 − 2721.87i 5310.88 + 9198.71i 0 26579.3 − 46036.7i −78888.8 0 765278.
19.10 40.4699 − 70.0959i 0 −2251.62 − 3899.92i 156.497 + 271.061i 0 −40821.5 + 70704.9i −198726. 0 25333.7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.12.c.a 20
3.b odd 2 1 9.12.c.a ✓ 20
9.c even 3 1 inner 27.12.c.a 20
9.c even 3 1 81.12.a.c 10
9.d odd 6 1 9.12.c.a ✓ 20
9.d odd 6 1 81.12.a.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.12.c.a ✓ 20 3.b odd 2 1
9.12.c.a ✓ 20 9.d odd 6 1
27.12.c.a 20 1.a even 1 1 trivial
27.12.c.a 20 9.c even 3 1 inner
81.12.a.c 10 9.c even 3 1
81.12.a.e 10 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 46\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 13\!\cdots\!52)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 36\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 94\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 29\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 64\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 53\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 68\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 45\!\cdots\!01 \) Copy content Toggle raw display
show more
show less