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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [180,8,Mod(109,180)] 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("180.109"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 180.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,330] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(56.2293045871\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{1129})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 565x^{2} + 79524 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 60)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} - \beta_{2} + \cdots + 82) q^{5} + (\beta_{3} + \beta_{2} - 114 \beta_1) q^{7} + (17 \beta_{3} - 17 \beta_{2} - 3154) q^{11} + (15 \beta_{3} + 15 \beta_{2} + 825 \beta_1) q^{13} + (87 \beta_{3} + 87 \beta_{2} + 3848 \beta_1) q^{17}+ \cdots + (17720 \beta_{3} + \cdots - 3630534 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 330 q^{5} - 12684 q^{11} - 11280 q^{19} - 201600 q^{25} - 159516 q^{29} + 305416 q^{31} + 219180 q^{35} + 1522800 q^{41} + 2360436 q^{49} - 2006080 q^{55} - 4222356 q^{59} + 1574888 q^{61} + 2172300 q^{65}+ \cdots + 6520800 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 565x^{2} + 79524 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 566\nu ) / 141 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 705\nu^{2} + 1271\nu + 199233 ) / 141 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} - 705\nu^{2} + 1271\nu - 199233 ) / 141 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} - 2826 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -283\beta_{3} - 283\beta_{2} + 1271\beta_1 ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
17.3003i
− 17.3003i
16.3003i
− 16.3003i
0 0 0 −1.50149 − 279.504i 0 621.003i 0 0 0
109.2 0 0 0 −1.50149 + 279.504i 0 − 621.003i 0 0 0
109.3 0 0 0 166.501 − 224.504i 0 − 284.997i 0 0 0
109.4 0 0 0 166.501 + 224.504i 0 284.997i 0 0 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.8.d.c 4
3.b odd 2 1 60.8.d.b ✓ 4
5.b even 2 1 inner 180.8.d.c 4
12.b even 2 1 240.8.f.c 4
15.d odd 2 1 60.8.d.b ✓ 4
15.e even 4 1 300.8.a.j 2
15.e even 4 1 300.8.a.k 2
60.h even 2 1 240.8.f.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.8.d.b ✓ 4 3.b odd 2 1
60.8.d.b ✓ 4 15.d odd 2 1
180.8.d.c 4 1.a even 1 1 trivial
180.8.d.c 4 5.b even 2 1 inner
240.8.f.c 4 12.b even 2 1
240.8.f.c 4 60.h even 2 1
300.8.a.j 2 15.e even 4 1
300.8.a.k 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 466868T_{7}^{2} + 31323336256 \) acting on \(S_{8}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 6103515625 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 31323336256 \) Copy content Toggle raw display
$11$ \( (T^{2} + 6342 T + 1898216)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 23410114560000 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 984791261755456 \) Copy content Toggle raw display
$19$ \( (T^{2} + 5640 T - 483840000)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 79758 T - 23936383584)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 152708 T - 52210714784)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{2} - 761400 T + 132110549900)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{2} + \cdots - 1410030476104)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 787444 T + 65990395684)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 10803266611200)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 2846846202624)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots + 27181827906500)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
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