↑

Properties

Label 80.11.b.b
Level $80$
Weight $11$
Character orbit 80.b
Analytic conductor $50.829$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

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 = [12] 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 531 x^{10} - 5928 x^{9} + 212132 x^{8} - 2347716 x^{7} + 50127183 x^{6} - 614836428 x^{5} + \cdots + 4542627246336 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{66}\cdot 3^{4}\cdot 5^{14}\cdot 89^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - 5 \beta_{3} q^{5} + (\beta_{5} - \beta_{4} + \cdots + 16 \beta_1) q^{7} + (\beta_{9} - \beta_{6} - 28 \beta_{3} - 2969) q^{9} + (2 \beta_{10} - \beta_{7} + \cdots - 154 \beta_1) q^{11}+ \cdots + ( - 84546 \beta_{10} + \cdots + 18098376 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 35628 q^{9} + 972144 q^{13} + 5541528 q^{17} + 13213368 q^{21} + 23437500 q^{25} - 16009176 q^{29} - 97301616 q^{33} + 161981040 q^{37} + 95999184 q^{41} + 131625000 q^{45} - 1035679596 q^{49} - 345298608 q^{53}+ \cdots + 23581939464 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 531 x^{10} - 5928 x^{9} + 212132 x^{8} - 2347716 x^{7} + 50127183 x^{6} - 614836428 x^{5} + \cdots + 4542627246336 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 99\!\cdots\!67 \nu^{11} + \cdots - 32\!\cdots\!12 ) / 80\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!39 \nu^{11} + \cdots + 12\!\cdots\!84 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 67\!\cdots\!25 \nu^{11} + \cdots - 40\!\cdots\!00 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19\!\cdots\!79 \nu^{11} + \cdots + 24\!\cdots\!84 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 88\!\cdots\!13 \nu^{11} + \cdots - 40\!\cdots\!08 ) / 52\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!39 \nu^{11} + \cdots - 43\!\cdots\!24 ) / 73\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 79\!\cdots\!57 \nu^{11} + \cdots - 58\!\cdots\!52 ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 45\!\cdots\!63 \nu^{11} + \cdots + 16\!\cdots\!48 ) / 98\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 45\!\cdots\!89 \nu^{11} + \cdots - 28\!\cdots\!04 ) / 98\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32\!\cdots\!87 \nu^{11} + \cdots - 18\!\cdots\!72 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 93\!\cdots\!93 \nu^{11} + \cdots - 73\!\cdots\!48 ) / 73\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 400 \beta_{11} - 5226 \beta_{10} + 1600 \beta_{9} + 1400 \beta_{8} - 5058 \beta_{7} + \cdots + 573087 \beta_1 ) / 34176000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 1180 \beta_{11} + 18594 \beta_{10} + 2940 \beta_{9} + 5020 \beta_{8} + 4962 \beta_{7} + \cdots - 1512288000 ) / 17088000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 27930\beta_{11} - 132190\beta_{9} - 87520\beta_{8} - 113585\beta_{6} + 3818908\beta_{3} + 6331104000 ) / 4272000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 587830 \beta_{11} - 5917038 \beta_{10} + 3021490 \beta_{9} + 2234070 \beta_{8} + \cdots - 405278944000 ) / 17088000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3391340 \beta_{11} + 23382453 \beta_{10} + 18846620 \beta_{9} + 10243660 \beta_{8} + \cdots - 1139274048000 ) / 3417600 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 120271075 \beta_{11} - 781835325 \beta_{9} - 406442875 \beta_{8} - 933646750 \beta_{6} + \cdots + 64461070944000 ) / 4272000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 11012594640 \beta_{11} - 45390044442 \beta_{10} + 73103880320 \beta_{9} + \cdots - 43\!\cdots\!00 ) / 34176000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 91198717940 \beta_{11} + 222472727886 \beta_{10} + 687776542820 \beta_{9} + \cdots - 44\!\cdots\!00 ) / 17088000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 939401432410 \beta_{11} - 7280993257630 \beta_{9} - 2671714941440 \beta_{8} + \cdots + 39\!\cdots\!00 ) / 4272000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 6715172334470 \beta_{11} + 2956803872706 \beta_{10} + 56999862846210 \beta_{9} + \cdots - 31\!\cdots\!00 ) / 3417600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 661178311811700 \beta_{11} + \cdots - 29\!\cdots\!00 ) / 17088000 \) 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
3.01668 + 5.22505i
3.65023 + 6.32239i
−9.75043 + 16.8882i
5.78008 − 10.0114i
6.10020 − 10.5658i
−8.79676 − 15.2364i
−8.79676 + 15.2364i
6.10020 + 10.5658i
5.78008 + 10.0114i
−9.75043 − 16.8882i
3.65023 − 6.32239i
3.01668 − 5.22505i
0 − 386.213i 0 1397.54 0 17612.2i 0 −90111.9 0
31.2 0 − 327.560i 0 −1397.54 0 − 15439.6i 0 −48246.2 0
31.3 0 − 305.480i 0 −1397.54 0 27826.9i 0 −34269.0 0
31.4 0 − 96.4705i 0 1397.54 0 − 3736.09i 0 49742.4 0
31.5 0 − 94.7976i 0 −1397.54 0 − 19879.9i 0 50062.4 0
31.6 0 − 63.5672i 0 1397.54 0 − 21923.0i 0 55008.2 0
31.7 0 63.5672i 0 1397.54 0 21923.0i 0 55008.2 0
31.8 0 94.7976i 0 −1397.54 0 19879.9i 0 50062.4 0
31.9 0 96.4705i 0 1397.54 0 3736.09i 0 49742.4 0
31.10 0 305.480i 0 −1397.54 0 − 27826.9i 0 −34269.0 0
31.11 0 327.560i 0 −1397.54 0 15439.6i 0 −48246.2 0
31.12 0 386.213i 0 1397.54 0 − 17612.2i 0 −90111.9 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 31.12
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.b ✓ 12
4.b odd 2 1 inner 80.11.b.b ✓ 12
8.b even 2 1 320.11.b.b 12
8.d odd 2 1 320.11.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.11.b.b ✓ 12 1.a even 1 1 trivial
80.11.b.b ✓ 12 4.b odd 2 1 inner
320.11.b.b 12 8.b even 2 1
320.11.b.b 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 372108 T_{3}^{10} + 47905596000 T_{3}^{8} + \cdots + 50\!\cdots\!64 \) acting on \(S_{11}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( (T^{2} - 1953125)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 77\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 16\!\cdots\!36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 18\!\cdots\!64)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 68\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 28\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 35\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 48\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 91\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 26\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 88\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 30\!\cdots\!64)^{2} \) Copy content Toggle raw display
show more
show less