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Properties

Label 80.11.p.c
Level $80$
Weight $11$
Character orbit 80.p
Analytic conductor $50.829$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(17,80)] 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("80.17"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 21 \beta_1 - 21) q^{3} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots + 910) q^{5} + (5 \beta_{5} - 5 \beta_{4} + \cdots - 2244) q^{7} + ( - 30 \beta_{5} + 61 \beta_{3} + \cdots + 59175 \beta_1) q^{9}+ \cdots + (2576190 \beta_{5} + \cdots + 77635512 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 128 q^{3} + 5460 q^{5} - 13512 q^{7} - 647832 q^{11} - 742902 q^{13} - 1577720 q^{15} - 755118 q^{17} + 12277112 q^{21} + 15052992 q^{23} + 42644850 q^{25} + 47998120 q^{27} - 153847152 q^{31} - 173025784 q^{33}+ \cdots + 33281088582 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 68121\nu^{5} + 149814\nu^{4} + 329476\nu^{3} - 39101454\nu^{2} + 4554477405\nu - 10394598718 ) / 10016360270 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60546049 \nu^{5} - 58729384 \nu^{4} + 638377544 \nu^{3} - 84835233476 \nu^{2} + \cdots - 18141291569772 ) / 30049080810 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 60844529 \nu^{5} - 325695636 \nu^{4} - 716280824 \nu^{3} + 85006560996 \nu^{2} + \cdots + 822290385952 ) / 30049080810 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -31840\nu^{5} - 738610\nu^{4} - 8310240\nu^{3} + 18276160\nu^{2} - 20819537159 ) / 10470063 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 517677579 \nu^{5} - 1138493986 \nu^{4} + 16684615276 \nu^{3} + 797964943846 \nu^{2} + \cdots + 67978379651882 ) / 30049080810 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - 10\beta_{3} + 3\beta _1 + 3 ) / 400 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 25\beta_{3} - 25\beta_{2} + 17383\beta_1 ) / 100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 261\beta_{5} - 261\beta_{4} + 2610\beta_{2} - 228817\beta _1 + 228817 ) / 400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{4} - 3980\beta_{3} - 3980\beta_{2} - 2268051 ) / 50 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13165\beta_{5} - 13165\beta_{4} + 159202\beta_{3} + 19926521\beta _1 + 19926521 ) / 80 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−12.3957 + 12.3957i
10.1043 − 10.1043i
2.29143 − 2.29143i
−12.3957 − 12.3957i
10.1043 + 10.1043i
2.29143 + 2.29143i
0 −326.149 + 326.149i 0 3124.94 + 19.4459i 0 1507.13 + 1507.13i 0 − 153697.i 0
17.2 0 −4.29207 + 4.29207i 0 −2978.33 − 946.124i 0 −21284.7 − 21284.7i 0 59012.2i 0
17.3 0 266.441 − 266.441i 0 2583.39 − 1758.32i 0 13021.6 + 13021.6i 0 − 82932.3i 0
33.1 0 −326.149 − 326.149i 0 3124.94 − 19.4459i 0 1507.13 − 1507.13i 0 153697.i 0
33.2 0 −4.29207 − 4.29207i 0 −2978.33 + 946.124i 0 −21284.7 + 21284.7i 0 − 59012.2i 0
33.3 0 266.441 + 266.441i 0 2583.39 + 1758.32i 0 13021.6 − 13021.6i 0 82932.3i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 17.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.11.p.c 6
4.b odd 2 1 10.11.c.c ✓ 6
5.c odd 4 1 inner 80.11.p.c 6
12.b even 2 1 90.11.g.c 6
20.d odd 2 1 50.11.c.e 6
20.e even 4 1 10.11.c.c ✓ 6
20.e even 4 1 50.11.c.e 6
60.l odd 4 1 90.11.g.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.11.c.c ✓ 6 4.b odd 2 1
10.11.c.c ✓ 6 20.e even 4 1
50.11.c.e 6 20.d odd 2 1
50.11.c.e 6 20.e even 4 1
80.11.p.c 6 1.a even 1 1 trivial
80.11.p.c 6 5.c odd 4 1 inner
90.11.g.c 6 12.b even 2 1
90.11.g.c 6 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 128T_{3}^{5} + 8192T_{3}^{4} - 20688696T_{3}^{3} + 30028037796T_{3}^{2} + 258527462832T_{3} + 1112900707872 \) acting on \(S_{11}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots + 1112900707872 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( (T^{3} + \cdots - 26\!\cdots\!52)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 77\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 71\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 10\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 28\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots + 19\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 31\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 89\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 85\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 15\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 29\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 58\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 16\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
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