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Properties

Label 80.11.b.a
Level $80$
Weight $11$
Character orbit 80.b
Analytic conductor $50.829$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [80,11,Mod(31,80)] 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("80.31"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 4793 x^{6} - 5744 x^{5} + 6951703 x^{4} - 6595252 x^{3} + 3794297756 x^{2} + \cdots + 655765351120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2}\cdot 5^{16} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{4} + 15 \beta_1) q^{7} + (\beta_{3} + 18 \beta_{2} - 36801) q^{9} + (\beta_{7} - \beta_{4} - 217 \beta_1) q^{11} + (\beta_{5} - 3 \beta_{3} + \cdots - 86660) q^{13}+ \cdots + ( - 53367 \beta_{7} + \cdots + 60606369 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 294408 q^{9} - 693280 q^{13} + 223440 q^{17} - 11463600 q^{21} + 15625000 q^{25} + 36140016 q^{29} + 166135200 q^{33} - 197596640 q^{37} - 558299616 q^{41} + 288750000 q^{45} - 609094408 q^{49} + 1711339680 q^{53}+ \cdots + 17150482160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} + 4793 x^{6} - 5744 x^{5} + 6951703 x^{4} - 6595252 x^{3} + 3794297756 x^{2} + \cdots + 655765351120 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 51193726562055 \nu^{7} + 168027712413121 \nu^{6} + \cdots + 31\!\cdots\!20 ) / 20\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 82493576621875 \nu^{7} + \cdots - 14\!\cdots\!00 ) / 20\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 499205411203575 \nu^{7} + \cdots + 11\!\cdots\!80 ) / 50\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 78\!\cdots\!95 \nu^{7} + \cdots - 19\!\cdots\!80 ) / 20\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!25 \nu^{7} + \cdots + 32\!\cdots\!80 ) / 20\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!45 \nu^{7} + \cdots + 33\!\cdots\!80 ) / 41\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!95 \nu^{7} + \cdots + 77\!\cdots\!80 ) / 41\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{4} - 5\beta_{2} + 377\beta _1 + 3125 ) / 12500 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - 2\beta_{4} + 625\beta_{3} + 11240\beta_{2} + 7004\beta _1 - 59887500 ) / 50000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 43236 \beta_{7} - 22086 \beta_{6} + 375 \beta_{5} + 155286 \beta_{4} + 6000 \beta_{3} + \cdots - 287950000 ) / 200000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14311 \beta_{7} - 4636 \beta_{6} - 18000 \beta_{5} + 38161 \beta_{4} - 946125 \beta_{3} + \cdots + 56588018750 ) / 25000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 55483268 \beta_{7} + 22096643 \beta_{6} - 838750 \beta_{5} - 200843393 \beta_{4} + \cdots + 693199400000 ) / 100000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 255970164 \beta_{7} + 81907914 \beta_{6} + 335548125 \beta_{5} - 736878414 \beta_{4} + \cdots - 536113312650000 ) / 100000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 293852637036 \beta_{7} - 105898683636 \beta_{6} + 5851195875 \beta_{5} + 995861922336 \beta_{4} + \cdots - 54\!\cdots\!00 ) / 200000 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.809017 − 52.0873i
−0.309017 + 32.5059i
−0.309017 + 26.3429i
0.809017 − 18.1335i
0.809017 + 18.1335i
−0.309017 − 26.3429i
−0.309017 − 32.5059i
0.809017 + 52.0873i
0 − 465.883i 0 −1397.54 0 − 12767.3i 0 −157998. 0
31.2 0 − 290.742i 0 1397.54 0 17192.0i 0 −25481.9 0
31.3 0 − 235.618i 0 1397.54 0 − 28733.8i 0 3533.03 0
31.4 0 − 162.191i 0 −1397.54 0 12257.5i 0 32743.1 0
31.5 0 162.191i 0 −1397.54 0 − 12257.5i 0 32743.1 0
31.6 0 235.618i 0 1397.54 0 28733.8i 0 3533.03 0
31.7 0 290.742i 0 1397.54 0 − 17192.0i 0 −25481.9 0
31.8 0 465.883i 0 −1397.54 0 12767.3i 0 −157998. 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.11.b.a ✓ 8
4.b odd 2 1 inner 80.11.b.a ✓ 8
8.b even 2 1 320.11.b.a 8
8.d odd 2 1 320.11.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.11.b.a ✓ 8 1.a even 1 1 trivial
80.11.b.a ✓ 8 4.b odd 2 1 inner
320.11.b.a 8 8.b even 2 1
320.11.b.a 8 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 383400T_{3}^{6} + 44483281200T_{3}^{4} + 1941626201040000T_{3}^{2} + 26794216966419360000 \) acting on \(S_{11}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{2} - 1953125)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 77\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 88\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 71\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 61\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 26\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
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