↑

Properties

Label 180.2.n.c
Level $180$
Weight $2$
Character orbit 180.n
Analytic conductor $1.437$
Analytic rank $0$
Dimension $8$
CM discriminant -20
Inner twists $8$

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] = [180,2,Mod(59,180)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("180.59"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] 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: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.8
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_1 q^{3} + 2 \beta_{3} q^{4} - \beta_{2} q^{5} + ( - \beta_{6} - \beta_{2} - 1) q^{6} + (\beta_{7} + 2 \beta_{5} + 3 \beta_{4}) q^{7} - 2 \beta_{4} q^{8} + (\beta_{6} + 2 \beta_{3} - 2) q^{9}+ \cdots + (6 \beta_{7} + 6 \beta_{4} - 6 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 8 q^{6} - 8 q^{9} + 36 q^{14} - 16 q^{16} + 20 q^{21} - 8 q^{24} + 20 q^{25} - 36 q^{29} - 20 q^{30} - 32 q^{36} - 72 q^{41} + 40 q^{45} - 8 q^{46} - 36 q^{49} + 28 q^{54} + 72 q^{56} + 16 q^{61}+ \cdots + 16 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 14\nu^{4} - 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{6} + 7\nu^{4} + 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{7} - 7\nu^{5} + 35\nu^{3} - 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 7\nu^{5} - 35\nu^{3} - 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{6} - 14\nu^{4} + 7\nu^{2} - 162 ) / 63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 4\nu^{5} - 7\nu^{3} - 36\nu ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{5} + 3\beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{7} + 7\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{6} - 7\beta_{2} - 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -21\beta_{5} - 21\beta_{4} - 29\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−0.178197 + 1.72286i
1.40294 − 1.01575i
−1.40294 + 1.01575i
0.178197 − 1.72286i
−0.178197 − 1.72286i
1.40294 + 1.01575i
−1.40294 − 1.01575i
0.178197 + 1.72286i
−1.22474 + 0.707107i −0.178197 + 1.72286i 1.00000 − 1.73205i −1.93649 − 1.11803i −1.00000 − 2.23607i −2.62769 − 4.55129i 2.82843i −2.93649 − 0.614017i 3.16228
59.2 −1.22474 + 0.707107i 1.40294 − 1.01575i 1.00000 − 1.73205i 1.93649 + 1.11803i −1.00000 + 2.23607i −1.04655 − 1.81267i 2.82843i 0.936492 − 2.85008i −3.16228
59.3 1.22474 − 0.707107i −1.40294 + 1.01575i 1.00000 − 1.73205i 1.93649 + 1.11803i −1.00000 + 2.23607i 1.04655 + 1.81267i − 2.82843i 0.936492 − 2.85008i 3.16228
59.4 1.22474 − 0.707107i 0.178197 − 1.72286i 1.00000 − 1.73205i −1.93649 − 1.11803i −1.00000 − 2.23607i 2.62769 + 4.55129i − 2.82843i −2.93649 − 0.614017i −3.16228
119.1 −1.22474 − 0.707107i −0.178197 − 1.72286i 1.00000 + 1.73205i −1.93649 + 1.11803i −1.00000 + 2.23607i −2.62769 + 4.55129i − 2.82843i −2.93649 + 0.614017i 3.16228
119.2 −1.22474 − 0.707107i 1.40294 + 1.01575i 1.00000 + 1.73205i 1.93649 − 1.11803i −1.00000 − 2.23607i −1.04655 + 1.81267i − 2.82843i 0.936492 + 2.85008i −3.16228
119.3 1.22474 + 0.707107i −1.40294 − 1.01575i 1.00000 + 1.73205i 1.93649 − 1.11803i −1.00000 − 2.23607i 1.04655 − 1.81267i 2.82843i 0.936492 + 2.85008i 3.16228
119.4 1.22474 + 0.707107i 0.178197 + 1.72286i 1.00000 + 1.73205i −1.93649 + 1.11803i −1.00000 + 2.23607i 2.62769 − 4.55129i 2.82843i −2.93649 + 0.614017i −3.16228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner
45.h odd 6 1 inner
180.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.2.n.c ✓ 8
3.b odd 2 1 540.2.n.c 8
4.b odd 2 1 inner 180.2.n.c ✓ 8
5.b even 2 1 inner 180.2.n.c ✓ 8
5.c odd 4 2 900.2.r.a 8
9.c even 3 1 540.2.n.c 8
9.d odd 6 1 inner 180.2.n.c ✓ 8
12.b even 2 1 540.2.n.c 8
15.d odd 2 1 540.2.n.c 8
20.d odd 2 1 CM 180.2.n.c ✓ 8
20.e even 4 2 900.2.r.a 8
36.f odd 6 1 540.2.n.c 8
36.h even 6 1 inner 180.2.n.c ✓ 8
45.h odd 6 1 inner 180.2.n.c ✓ 8
45.j even 6 1 540.2.n.c 8
45.l even 12 2 900.2.r.a 8
60.h even 2 1 540.2.n.c 8
180.n even 6 1 inner 180.2.n.c ✓ 8
180.p odd 6 1 540.2.n.c 8
180.v odd 12 2 900.2.r.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.n.c ✓ 8 1.a even 1 1 trivial
180.2.n.c ✓ 8 4.b odd 2 1 inner
180.2.n.c ✓ 8 5.b even 2 1 inner
180.2.n.c ✓ 8 9.d odd 6 1 inner
180.2.n.c ✓ 8 20.d odd 2 1 CM
180.2.n.c ✓ 8 36.h even 6 1 inner
180.2.n.c ✓ 8 45.h odd 6 1 inner
180.2.n.c ✓ 8 180.n even 6 1 inner
540.2.n.c 8 3.b odd 2 1
540.2.n.c 8 9.c even 3 1
540.2.n.c 8 12.b even 2 1
540.2.n.c 8 15.d odd 2 1
540.2.n.c 8 36.f odd 6 1
540.2.n.c 8 45.j even 6 1
540.2.n.c 8 60.h even 2 1
540.2.n.c 8 180.p odd 6 1
900.2.r.a 8 5.c odd 4 2
900.2.r.a 8 20.e even 4 2
900.2.r.a 8 45.l even 12 2
900.2.r.a 8 180.v odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 32T_{7}^{6} + 903T_{7}^{4} + 3872T_{7}^{2} + 14641 \) acting on \(S_{2}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 32 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 136 T^{6} + \cdots + 20151121 \) Copy content Toggle raw display
$29$ \( (T^{4} + 18 T^{3} + \cdots + 49)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 36 T^{3} + \cdots + 10609)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 184 T^{6} + \cdots + 3418801 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + \cdots + 14161)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 152 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - 256 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$89$ \( (T^{4} + 214 T^{2} + 2809)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
show more
show less