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Properties

Label 1080.4.q.e
Level $1080$
Weight $4$
Character orbit 1080.q
Analytic conductor $63.722$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1080,4,Mod(361,1080)] 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("1080.361"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,0,-50,0,22] 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: \(63.7220628062\)
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: \( x^{20} + 1542 x^{18} + 976429 x^{16} + 327887620 x^{14} + 62946909772 x^{12} + 6953278937404 x^{10} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{25} \)
Twist minimal: no (minimal twist has level 360)
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.

\(f(q)\) \(=\) \( q + (5 \beta_{2} - 5) q^{5} + (\beta_{3} + 2 \beta_{2}) q^{7} + (\beta_{6} - 5 \beta_{2}) q^{11} + ( - \beta_{10} + 2 \beta_{2} - 2) q^{13} + ( - \beta_{13} - \beta_{12} + \beta_{11} + \cdots + 4) q^{17}+ \cdots + ( - 3 \beta_{18} + \cdots - 132 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 50 q^{5} + 22 q^{7} - 50 q^{11} - 16 q^{13} + 68 q^{17} + 212 q^{19} - 50 q^{23} - 250 q^{25} + 64 q^{29} - 22 q^{31} - 220 q^{35} + 600 q^{37} + 198 q^{41} - 382 q^{43} - 128 q^{47} - 1230 q^{49}+ \cdots - 1354 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 1542 x^{18} + 976429 x^{16} + 327887620 x^{14} + 62946909772 x^{12} + 6953278937404 x^{10} + \cdots + 45\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 96\!\cdots\!57 \nu^{19} + \cdots - 41\!\cdots\!04 ) / 45\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 96\!\cdots\!57 \nu^{19} + \cdots + 22\!\cdots\!28 ) / 45\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 96\!\cdots\!57 \nu^{19} + \cdots + 41\!\cdots\!04 ) / 45\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 89\!\cdots\!13 \nu^{19} + \cdots - 10\!\cdots\!72 ) / 28\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 89\!\cdots\!13 \nu^{19} + \cdots - 10\!\cdots\!72 ) / 28\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!29 \nu^{19} + \cdots + 24\!\cdots\!48 ) / 28\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19\!\cdots\!29 \nu^{19} + \cdots + 24\!\cdots\!48 ) / 28\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!14 \nu^{19} + \cdots - 61\!\cdots\!40 ) / 35\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!14 \nu^{19} + \cdots - 61\!\cdots\!40 ) / 35\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!49 \nu^{19} + \cdots + 88\!\cdots\!36 ) / 35\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!49 \nu^{19} + \cdots + 88\!\cdots\!36 ) / 35\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27\!\cdots\!79 \nu^{19} + \cdots + 21\!\cdots\!52 ) / 70\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 27\!\cdots\!79 \nu^{19} + \cdots + 21\!\cdots\!52 ) / 70\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!95 \nu^{19} + \cdots - 25\!\cdots\!08 ) / 94\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!95 \nu^{19} + \cdots - 25\!\cdots\!08 ) / 94\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 20\!\cdots\!86 \nu^{19} + \cdots - 12\!\cdots\!92 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 20\!\cdots\!86 \nu^{19} + \cdots - 12\!\cdots\!92 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 57\!\cdots\!51 \nu^{19} + \cdots - 88\!\cdots\!84 ) / 28\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 57\!\cdots\!51 \nu^{19} + \cdots + 88\!\cdots\!84 ) / 28\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{19} - 3 \beta_{18} - 4 \beta_{17} - 4 \beta_{16} - \beta_{15} - \beta_{14} - 4 \beta_{13} + \cdots - 458 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{19} - 9 \beta_{18} + 73 \beta_{17} - 73 \beta_{16} + 89 \beta_{15} - 89 \beta_{14} + \cdots + 1688 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3165 \beta_{19} + 3165 \beta_{18} + 3395 \beta_{17} + 3395 \beta_{16} + 1114 \beta_{15} + \cdots + 377242 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3683 \beta_{19} + 3683 \beta_{18} - 29649 \beta_{17} + 29649 \beta_{16} - 42048 \beta_{15} + \cdots - 580844 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 381966 \beta_{19} - 381966 \beta_{18} - 300398 \beta_{17} - 300398 \beta_{16} - 77870 \beta_{15} + \cdots - 39351724 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1638424 \beta_{19} - 1638424 \beta_{18} + 10870400 \beta_{17} - 10870400 \beta_{16} + \cdots + 206920912 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 140665612 \beta_{19} + 140665612 \beta_{18} + 80075590 \beta_{17} + 80075590 \beta_{16} + \cdots + 13080743644 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 759164986 \beta_{19} + 759164986 \beta_{18} - 3958908372 \beta_{17} + 3958908372 \beta_{16} + \cdots - 79236886492 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 156473455074 \beta_{19} - 156473455074 \beta_{18} - 64963016980 \beta_{17} - 64963016980 \beta_{16} + \cdots - 13545691061732 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 346678225262 \beta_{19} - 346678225262 \beta_{18} + 1445710453468 \beta_{17} - 1445710453468 \beta_{16} + \cdots + 31753315424764 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 19390607723090 \beta_{19} + 19390607723090 \beta_{18} + 5910991027440 \beta_{17} + \cdots + 15\!\cdots\!32 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 51289898110010 \beta_{19} + 51289898110010 \beta_{18} - 176381937138320 \beta_{17} + \cdots - 43\!\cdots\!84 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 72\!\cdots\!34 \beta_{19} + \cdots - 57\!\cdots\!72 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 66\!\cdots\!14 \beta_{19} + \cdots + 53\!\cdots\!08 ) / 9 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 80\!\cdots\!14 \beta_{19} + \cdots + 62\!\cdots\!16 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 28\!\cdots\!54 \beta_{19} + \cdots - 21\!\cdots\!64 ) / 9 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 10\!\cdots\!90 \beta_{19} + \cdots - 77\!\cdots\!56 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 11\!\cdots\!70 \beta_{19} + \cdots + 89\!\cdots\!72 ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
− 18.7823i
− 14.9187i
− 7.80140i
− 6.59757i
− 1.44843i
1.23574i
6.11710i
13.8440i
15.7380i
19.5418i
18.7823i
14.9187i
7.80140i
6.59757i
1.44843i
− 1.23574i
− 6.11710i
− 13.8440i
− 15.7380i
− 19.5418i
0 0 0 −2.50000 + 4.33013i 0 −15.7660 − 27.3074i 0 0 0
361.2 0 0 0 −2.50000 + 4.33013i 0 −12.4200 − 21.5121i 0 0 0
361.3 0 0 0 −2.50000 + 4.33013i 0 −6.25621 − 10.8361i 0 0 0
361.4 0 0 0 −2.50000 + 4.33013i 0 −5.21366 − 9.03032i 0 0 0
361.5 0 0 0 −2.50000 + 4.33013i 0 −0.754378 − 1.30662i 0 0 0
361.6 0 0 0 −2.50000 + 4.33013i 0 1.57018 + 2.71963i 0 0 0
361.7 0 0 0 −2.50000 + 4.33013i 0 5.79756 + 10.0417i 0 0 0
361.8 0 0 0 −2.50000 + 4.33013i 0 12.4893 + 21.6321i 0 0 0
361.9 0 0 0 −2.50000 + 4.33013i 0 14.1295 + 24.4730i 0 0 0
361.10 0 0 0 −2.50000 + 4.33013i 0 17.4237 + 30.1787i 0 0 0
721.1 0 0 0 −2.50000 − 4.33013i 0 −15.7660 + 27.3074i 0 0 0
721.2 0 0 0 −2.50000 − 4.33013i 0 −12.4200 + 21.5121i 0 0 0
721.3 0 0 0 −2.50000 − 4.33013i 0 −6.25621 + 10.8361i 0 0 0
721.4 0 0 0 −2.50000 − 4.33013i 0 −5.21366 + 9.03032i 0 0 0
721.5 0 0 0 −2.50000 − 4.33013i 0 −0.754378 + 1.30662i 0 0 0
721.6 0 0 0 −2.50000 − 4.33013i 0 1.57018 − 2.71963i 0 0 0
721.7 0 0 0 −2.50000 − 4.33013i 0 5.79756 − 10.0417i 0 0 0
721.8 0 0 0 −2.50000 − 4.33013i 0 12.4893 − 21.6321i 0 0 0
721.9 0 0 0 −2.50000 − 4.33013i 0 14.1295 − 24.4730i 0 0 0
721.10 0 0 0 −2.50000 − 4.33013i 0 17.4237 − 30.1787i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.10
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 1080.4.q.e 20
3.b odd 2 1 360.4.q.e ✓ 20
9.c even 3 1 inner 1080.4.q.e 20
9.d odd 6 1 360.4.q.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.q.e ✓ 20 3.b odd 2 1
360.4.q.e ✓ 20 9.d odd 6 1
1080.4.q.e 20 1.a even 1 1 trivial
1080.4.q.e 20 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} - 22 T_{7}^{19} + 2572 T_{7}^{18} - 31512 T_{7}^{17} + 3795924 T_{7}^{16} + \cdots + 19\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(1080, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{10} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 78\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 69\!\cdots\!28)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 34\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 39\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 68\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 53\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 48\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 63\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
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