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Properties

Label 80.11.p.b
Level $80$
Weight $11$
Character orbit 80.p
Analytic conductor $50.829$
Analytic rank $0$
Dimension $2$
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 = [2,0,366] 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 183 i + 183) q^{3} + ( - 2500 i - 1875) q^{5} + (8407 i + 8407) q^{7} - 7929 i q^{9} + 173398 q^{11} + (232623 i - 232623) q^{13} + ( - 114375 i - 800625) q^{15} + (1880033 i + 1880033) q^{17} + \cdots - 1374872742 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 366 q^{3} - 3750 q^{5} + 16814 q^{7} + 346796 q^{11} - 465246 q^{13} - 1601250 q^{15} + 3760066 q^{17} + 6153924 q^{21} + 10456526 q^{23} - 5468750 q^{25} + 18709920 q^{27} + 20131996 q^{31} + 63463668 q^{33}+ \cdots - 1280722494 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(i\) \(1\) \(1\)

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
1.00000i
− 1.00000i
0 183.000 − 183.000i 0 −1875.00 − 2500.00i 0 8407.00 + 8407.00i 0 − 7929.00i 0
33.1 0 183.000 + 183.000i 0 −1875.00 + 2500.00i 0 8407.00 − 8407.00i 0 7929.00i 0
\(n\): e.g. 2-40 or 80-90
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.b 2
4.b odd 2 1 10.11.c.a ✓ 2
5.c odd 4 1 inner 80.11.p.b 2
12.b even 2 1 90.11.g.b 2
20.d odd 2 1 50.11.c.c 2
20.e even 4 1 10.11.c.a ✓ 2
20.e even 4 1 50.11.c.c 2
60.l odd 4 1 90.11.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.11.c.a ✓ 2 4.b odd 2 1
10.11.c.a ✓ 2 20.e even 4 1
50.11.c.c 2 20.d odd 2 1
50.11.c.c 2 20.e even 4 1
80.11.p.b 2 1.a even 1 1 trivial
80.11.p.b 2 5.c odd 4 1 inner
90.11.g.b 2 12.b even 2 1
90.11.g.b 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 366T_{3} + 66978 \) acting on \(S_{11}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 366T + 66978 \) Copy content Toggle raw display
$5$ \( T^{2} + 3750 T + 9765625 \) Copy content Toggle raw display
$7$ \( T^{2} - 16814 T + 141355298 \) Copy content Toggle raw display
$11$ \( (T - 173398)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 108226920258 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 7069048162178 \) Copy content Toggle raw display
$19$ \( T^{2} + 1213434433600 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 54669467994338 \) Copy content Toggle raw display
$29$ \( T^{2} + 614585747905600 \) Copy content Toggle raw display
$31$ \( (T - 10065998)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 63\!\cdots\!38 \) Copy content Toggle raw display
$41$ \( (T + 153003598)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 70\!\cdots\!98 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 59\!\cdots\!18 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 77\!\cdots\!78 \) Copy content Toggle raw display
$59$ \( T^{2} + 48\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T - 906185802)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 18\!\cdots\!78 \) Copy content Toggle raw display
$71$ \( (T - 3120877598)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 80\!\cdots\!38 \) Copy content Toggle raw display
$79$ \( T^{2} + 38\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 53\!\cdots\!18 \) Copy content Toggle raw display
$89$ \( T^{2} + 60\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 82\!\cdots\!18 \) Copy content Toggle raw display
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