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Properties

Label 864.5.e.g
Level $864$
Weight $5$
Character orbit 864.e
Analytic conductor $89.312$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,5,Mod(161,864)] 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("864.161"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 864.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(89.3116481044\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 22 x^{14} - 60 x^{13} + 313 x^{12} + 1368 x^{11} + 1844 x^{10} - 4788 x^{9} - 11779 x^{8} + \cdots + 16848900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{56}\cdot 3^{24} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{4} q^{7} + (\beta_{5} - \beta_{3} - \beta_{2}) q^{11} + (\beta_{10} - 2) q^{13} + (\beta_{9} - \beta_1) q^{17} + ( - \beta_{14} + \beta_{4}) q^{19} + ( - \beta_{6} + \beta_{5} + \cdots - \beta_{2}) q^{23}+ \cdots + ( - 13 \beta_{15} - 24 \beta_{10} + \cdots - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{13} + 704 q^{25} - 2624 q^{37} + 1728 q^{49} + 3264 q^{61} - 5424 q^{73} + 4704 q^{85} - 80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 22 x^{14} - 60 x^{13} + 313 x^{12} + 1368 x^{11} + 1844 x^{10} - 4788 x^{9} - 11779 x^{8} + \cdots + 16848900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\!\cdots\!32 \nu^{15} + \cdots + 27\!\cdots\!80 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5413982430824 \nu^{15} - 44464970174057 \nu^{14} + 172482854620380 \nu^{13} + \cdots - 26\!\cdots\!84 ) / 70\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 844355027236632 \nu^{15} + \cdots + 48\!\cdots\!60 ) / 14\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!75 \nu^{15} + \cdots - 10\!\cdots\!20 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!76 \nu^{15} + \cdots - 41\!\cdots\!70 ) / 10\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!84 \nu^{15} + \cdots - 25\!\cdots\!40 ) / 43\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 66\!\cdots\!33 \nu^{15} + \cdots + 46\!\cdots\!40 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 46\!\cdots\!00 \nu^{15} + \cdots + 69\!\cdots\!40 ) / 97\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27\!\cdots\!16 \nu^{15} + \cdots + 73\!\cdots\!20 ) / 38\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!69 \nu^{15} + \cdots - 17\!\cdots\!04 ) / 39\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 57\!\cdots\!96 \nu^{15} + \cdots + 59\!\cdots\!40 ) / 59\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 42\!\cdots\!12 \nu^{15} + \cdots + 40\!\cdots\!00 ) / 36\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 40\!\cdots\!81 \nu^{15} + \cdots + 16\!\cdots\!10 ) / 29\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24\!\cdots\!19 \nu^{15} + \cdots + 24\!\cdots\!20 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 32\!\cdots\!40 \nu^{15} + \cdots + 25\!\cdots\!20 ) / 14\!\cdots\!95 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4\beta_{15} - 9\beta_{12} + 8\beta_{7} - 12\beta_{6} + 24\beta_{5} + 72\beta_{4} - 3\beta_{3} + 20\beta_{2} ) / 3456 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 29 \beta_{15} - 12 \beta_{14} + 20 \beta_{13} - 12 \beta_{12} - 7 \beta_{11} + 24 \beta_{10} + \cdots + 9504 ) / 3456 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 26 \beta_{15} - 42 \beta_{14} + 30 \beta_{13} - 42 \beta_{12} + 3 \beta_{11} + 60 \beta_{9} + \cdots + 38880 ) / 3456 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{15} + 56 \beta_{14} + 120 \beta_{13} - 58 \beta_{12} - 7 \beta_{11} + 24 \beta_{10} + \cdots - 20448 ) / 1152 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 439 \beta_{15} - 306 \beta_{14} + 1170 \beta_{13} + 1332 \beta_{12} - 330 \beta_{11} + \cdots - 51840 ) / 3456 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7007 \beta_{15} - 1656 \beta_{14} + 4768 \beta_{13} + 6777 \beta_{12} - 497 \beta_{11} + \cdots - 4381344 ) / 3456 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 28640 \beta_{15} - 16800 \beta_{14} + 16296 \beta_{13} + 63255 \beta_{12} - 2436 \beta_{11} + \cdots - 22752576 ) / 3456 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 66979 \beta_{15} - 34864 \beta_{14} + 3792 \beta_{13} + 102236 \beta_{12} - 1489 \beta_{11} + \cdots - 43552800 ) / 1152 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 874223 \beta_{15} - 527166 \beta_{14} - 98670 \beta_{13} + 1519101 \beta_{12} + 30120 \beta_{11} + \cdots - 600276096 ) / 3456 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3418576 \beta_{15} - 2279316 \beta_{14} - 1826980 \beta_{13} + 6320103 \beta_{12} + \cdots - 2404083456 ) / 3456 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 12392560 \beta_{15} - 7690650 \beta_{14} - 11810370 \beta_{13} + 19812765 \beta_{12} + \cdots - 8342525664 ) / 3456 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3407373 \beta_{15} - 2934632 \beta_{14} - 11429696 \beta_{13} + 7570174 \beta_{12} + \cdots - 2426655744 ) / 576 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 54054230 \beta_{15} + 45334620 \beta_{14} - 338340912 \beta_{13} - 121046571 \beta_{12} + \cdots + 36656193600 ) / 3456 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1088240107 \beta_{15} + 722832084 \beta_{14} - 1405766756 \beta_{13} - 1882570896 \beta_{12} + \cdots + 758068159392 ) / 3456 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 8231976664 \beta_{15} + 5761642386 \beta_{14} - 5074497798 \beta_{13} - 14721293634 \beta_{12} + \cdots + 5678571860640 ) / 3456 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−3.06658 − 1.75061i
4.48080 + 1.75061i
0.389689 − 1.84121i
−1.80390 + 1.84121i
0.542596 − 2.34121i
−1.95681 + 2.34121i
3.05115 − 1.25061i
−1.63694 + 1.25061i
3.05115 + 1.25061i
−1.63694 − 1.25061i
0.542596 + 2.34121i
−1.95681 − 2.34121i
0.389689 + 1.84121i
−1.80390 − 1.84121i
−3.06658 + 1.75061i
4.48080 − 1.75061i
0 0 0 − 33.5878i 0 −14.6267 0 0 0
161.2 0 0 0 − 33.5878i 0 14.6267 0 0 0
161.3 0 0 0 − 26.0821i 0 −48.1712 0 0 0
161.4 0 0 0 − 26.0821i 0 48.1712 0 0 0
161.5 0 0 0 − 20.7560i 0 −63.5469 0 0 0
161.6 0 0 0 − 20.7560i 0 63.5469 0 0 0
161.7 0 0 0 − 9.20720i 0 −58.8505 0 0 0
161.8 0 0 0 − 9.20720i 0 58.8505 0 0 0
161.9 0 0 0 9.20720i 0 −58.8505 0 0 0
161.10 0 0 0 9.20720i 0 58.8505 0 0 0
161.11 0 0 0 20.7560i 0 −63.5469 0 0 0
161.12 0 0 0 20.7560i 0 63.5469 0 0 0
161.13 0 0 0 26.0821i 0 −48.1712 0 0 0
161.14 0 0 0 26.0821i 0 48.1712 0 0 0
161.15 0 0 0 33.5878i 0 −14.6267 0 0 0
161.16 0 0 0 33.5878i 0 14.6267 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.5.e.g ✓ 16
3.b odd 2 1 inner 864.5.e.g ✓ 16
4.b odd 2 1 inner 864.5.e.g ✓ 16
12.b even 2 1 inner 864.5.e.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.5.e.g ✓ 16 1.a even 1 1 trivial
864.5.e.g ✓ 16 3.b odd 2 1 inner
864.5.e.g ✓ 16 4.b odd 2 1 inner
864.5.e.g ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(864, [\chi])\):

\( T_{5}^{8} + 2324T_{5}^{6} + 1736358T_{5}^{4} + 461728148T_{5}^{2} + 28027891777 \) Copy content Toggle raw display
\( T_{7}^{8} - 10036T_{7}^{6} + 33494406T_{7}^{4} - 39170019316T_{7}^{2} + 6943209056737 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 2324 T^{6} + \cdots + 28027891777)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 6943209056737)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 25\!\cdots\!61)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8 T^{3} + \cdots + 45786256)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 11\!\cdots\!12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 73\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 16\!\cdots\!73)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 4258963571776)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 10\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 34\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 90\!\cdots\!16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 10\!\cdots\!53)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 33\!\cdots\!64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 23410220749056)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 63\!\cdots\!28)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 110225586445281)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 29\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 60\!\cdots\!61)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 15\!\cdots\!12)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 10\!\cdots\!97)^{4} \) Copy content Toggle raw display
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