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Properties

Label 80.11.p.d
Level $80$
Weight $11$
Character orbit 80.p
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(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 = [8,0,-60] 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \( x^{8} + 1334x^{6} + 456089x^{4} + 43159076x^{2} + 31360000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 5)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 7 \beta_1 - 7) q^{3} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 665) q^{5} + (5 \beta_{7} - 5 \beta_{6} + \cdots + 1789) q^{7} + ( - 18 \beta_{7} + 144 \beta_{5} + \cdots + 11193 \beta_1) q^{9}+ \cdots + (4306086 \beta_{7} + \cdots - 5923226346 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 60 q^{3} - 5340 q^{5} + 14500 q^{7} + 233784 q^{11} + 433520 q^{13} + 3188580 q^{15} + 1045440 q^{17} - 20777784 q^{21} - 24737580 q^{23} - 35382400 q^{25} - 27386640 q^{27} - 258616 q^{31} + 11269320 q^{33}+ \cdots + 24688294760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 1334x^{6} + 456089x^{4} + 43159076x^{2} + 31360000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 1334\nu^{5} - 450489\nu^{3} - 39423876\nu ) / 33454400 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3 \nu^{7} - 112 \nu^{6} + 5458 \nu^{5} - 137200 \nu^{4} + 2991707 \nu^{3} - 33613888 \nu^{2} + \cdots - 450340800 ) / 6690880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{7} + 112 \nu^{6} + 5458 \nu^{5} + 137200 \nu^{4} + 2991707 \nu^{3} + 33613888 \nu^{2} + \cdots + 450340800 ) / 6690880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29 \nu^{7} - 1120 \nu^{6} + 53246 \nu^{5} - 1204000 \nu^{4} + 29466581 \nu^{3} + \cdots - 3529153600 ) / 50181600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29 \nu^{7} + 1120 \nu^{6} + 53246 \nu^{5} + 1204000 \nu^{4} + 29466581 \nu^{3} + \cdots + 3529153600 ) / 50181600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 1925\nu^{4} + 1065724\nu^{2} + 107527480 ) / 59740 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -841\nu^{7} - 1111534\nu^{5} - 373623849\nu^{3} - 35201745476\nu ) / 16727200 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{5} - 3\beta_{4} + 4\beta_{3} + 4\beta_{2} + 4\beta_1 ) / 120 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12\beta_{6} + 21\beta_{5} - 21\beta_{4} - 34\beta_{3} + 34\beta_{2} - 19976 ) / 60 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -78\beta_{7} + 555\beta_{5} + 555\beta_{4} - 629\beta_{3} - 629\beta_{2} + 133786\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -1334\beta_{6} - 3828\beta_{5} + 3828\beta_{4} + 5771\beta_{3} - 5771\beta_{2} + 2162674 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 87870\beta_{7} - 428712\beta_{5} - 428712\beta_{4} + 515501\beta_{3} + 515501\beta_{2} - 148909174\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3101706 \beta_{6} + 10916598 \beta_{5} - 10916598 \beta_{4} - 15210217 \beta_{3} + 15210217 \beta_{2} - 5070815438 ) / 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 82080438 \beta_{7} + 351448320 \beta_{5} + 351448320 \beta_{4} - 443744629 \beta_{3} + \cdots + 137332660886 \beta_1 ) / 30 \) 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
29.3902i
0.855727i
− 17.5804i
− 12.6655i
− 29.3902i
− 0.855727i
17.5804i
12.6655i
0 −246.218 + 246.218i 0 −3025.01 + 784.185i 0 8610.35 + 8610.35i 0 − 62197.5i 0
17.2 0 −70.6389 + 70.6389i 0 1004.90 − 2959.02i 0 −2611.74 − 2611.74i 0 49069.3i 0
17.3 0 13.8338 − 13.8338i 0 −721.162 + 3040.65i 0 13891.7 + 13891.7i 0 58666.3i 0
17.4 0 273.023 − 273.023i 0 71.2753 + 3124.19i 0 −12640.3 − 12640.3i 0 − 90034.1i 0
33.1 0 −246.218 − 246.218i 0 −3025.01 − 784.185i 0 8610.35 − 8610.35i 0 62197.5i 0
33.2 0 −70.6389 − 70.6389i 0 1004.90 + 2959.02i 0 −2611.74 + 2611.74i 0 − 49069.3i 0
33.3 0 13.8338 + 13.8338i 0 −721.162 − 3040.65i 0 13891.7 − 13891.7i 0 − 58666.3i 0
33.4 0 273.023 + 273.023i 0 71.2753 − 3124.19i 0 −12640.3 + 12640.3i 0 90034.1i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 17.4
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.d 8
4.b odd 2 1 5.11.c.a ✓ 8
5.c odd 4 1 inner 80.11.p.d 8
12.b even 2 1 45.11.g.a 8
20.d odd 2 1 25.11.c.a 8
20.e even 4 1 5.11.c.a ✓ 8
20.e even 4 1 25.11.c.a 8
60.l odd 4 1 45.11.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.11.c.a ✓ 8 4.b odd 2 1
5.11.c.a ✓ 8 20.e even 4 1
25.11.c.a 8 20.d odd 2 1
25.11.c.a 8 20.e even 4 1
45.11.g.a 8 12.b even 2 1
45.11.g.a 8 60.l odd 4 1
80.11.p.d 8 1.a even 1 1 trivial
80.11.p.d 8 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 60 T_{3}^{7} + 1800 T_{3}^{6} + 6802920 T_{3}^{5} + 18919661508 T_{3}^{4} + \cdots + 69\!\cdots\!16 \) 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
$3$ \( T^{8} + \cdots + 69\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 45\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 93\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 43\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
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