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Properties

Label 180.3.f.h
Level $180$
Weight $3$
Character orbit 180.f
Analytic conductor $4.905$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,3,Mod(19,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.f (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.389136420864.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 24x^{4} + 80x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + (\beta_{4} + \beta_1 - 1) q^{5} + ( - \beta_{5} - \beta_1) q^{7} + ( - \beta_{7} + \beta_{6} + \cdots - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + (\beta_{4} + \beta_1 - 1) q^{5} + ( - \beta_{5} - \beta_1) q^{7} + ( - \beta_{7} + \beta_{6} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - 12 \beta_{7} - 12 \beta_{6} + \cdots - 9 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{4} - 4 q^{5} - 42 q^{10} - 20 q^{14} - 46 q^{16} - 52 q^{20} + 32 q^{25} - 84 q^{26} + 184 q^{29} + 12 q^{34} - 6 q^{40} + 256 q^{41} - 348 q^{44} + 112 q^{46} - 24 q^{49} - 72 q^{50} + 244 q^{56} + 304 q^{61} - 10 q^{64} - 168 q^{65} - 104 q^{70} + 252 q^{74} - 24 q^{76} + 308 q^{80} + 24 q^{85} + 280 q^{86} - 560 q^{89} + 376 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 24x^{4} + 80x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 3\nu^{4} + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 4\nu^{6} + 5\nu^{5} + 20\nu^{4} + 24\nu^{3} + 32\nu^{2} + 80\nu + 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 3\nu^{5} - 16\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 5\nu^{5} + 8\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 5\nu^{5} + 20\nu^{4} - 24\nu^{3} + 32\nu^{2} - 80\nu + 192 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{4} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{7} - \beta_{6} + 4\beta_{5} + 3\beta_{4} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{7} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7\beta_{7} - 19\beta_{6} - 20\beta_{5} - 7\beta_{4} - 2\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.52274 − 1.29664i
−1.52274 + 1.29664i
−0.656712 − 1.88911i
−0.656712 + 1.88911i
0.656712 − 1.88911i
0.656712 + 1.88911i
1.52274 − 1.29664i
1.52274 + 1.29664i
−1.52274 − 1.29664i 0 0.637459 + 3.94888i −4.27492 − 2.59328i 0 −0.837253 4.14959 − 6.83966i 0 3.14704 + 9.49190i
19.2 −1.52274 + 1.29664i 0 0.637459 − 3.94888i −4.27492 + 2.59328i 0 −0.837253 4.14959 + 6.83966i 0 3.14704 − 9.49190i
19.3 −0.656712 − 1.88911i 0 −3.13746 + 2.48120i 3.27492 − 3.77822i 0 9.55505 6.74766 + 4.29756i 0 −9.28814 − 3.70547i
19.4 −0.656712 + 1.88911i 0 −3.13746 − 2.48120i 3.27492 + 3.77822i 0 9.55505 6.74766 − 4.29756i 0 −9.28814 + 3.70547i
19.5 0.656712 − 1.88911i 0 −3.13746 − 2.48120i 3.27492 − 3.77822i 0 −9.55505 −6.74766 + 4.29756i 0 −4.98678 − 8.66787i
19.6 0.656712 + 1.88911i 0 −3.13746 + 2.48120i 3.27492 + 3.77822i 0 −9.55505 −6.74766 − 4.29756i 0 −4.98678 + 8.66787i
19.7 1.52274 − 1.29664i 0 0.637459 − 3.94888i −4.27492 − 2.59328i 0 0.837253 −4.14959 − 6.83966i 0 −9.87212 + 1.59414i
19.8 1.52274 + 1.29664i 0 0.637459 + 3.94888i −4.27492 + 2.59328i 0 0.837253 −4.14959 + 6.83966i 0 −9.87212 − 1.59414i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.f.h 8
3.b odd 2 1 60.3.f.b ✓ 8
4.b odd 2 1 inner 180.3.f.h 8
5.b even 2 1 inner 180.3.f.h 8
5.c odd 4 2 900.3.c.r 8
12.b even 2 1 60.3.f.b ✓ 8
15.d odd 2 1 60.3.f.b ✓ 8
15.e even 4 2 300.3.c.f 8
20.d odd 2 1 inner 180.3.f.h 8
20.e even 4 2 900.3.c.r 8
24.f even 2 1 960.3.j.e 8
24.h odd 2 1 960.3.j.e 8
60.h even 2 1 60.3.f.b ✓ 8
60.l odd 4 2 300.3.c.f 8
120.i odd 2 1 960.3.j.e 8
120.m even 2 1 960.3.j.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.b ✓ 8 3.b odd 2 1
60.3.f.b ✓ 8 12.b even 2 1
60.3.f.b ✓ 8 15.d odd 2 1
60.3.f.b ✓ 8 60.h even 2 1
180.3.f.h 8 1.a even 1 1 trivial
180.3.f.h 8 4.b odd 2 1 inner
180.3.f.h 8 5.b even 2 1 inner
180.3.f.h 8 20.d odd 2 1 inner
300.3.c.f 8 15.e even 4 2
300.3.c.f 8 60.l odd 4 2
900.3.c.r 8 5.c odd 4 2
900.3.c.r 8 20.e even 4 2
960.3.j.e 8 24.f even 2 1
960.3.j.e 8 24.h odd 2 1
960.3.j.e 8 120.i odd 2 1
960.3.j.e 8 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\):

\( T_{7}^{4} - 92T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{4} + 84T_{13}^{2} + 1536 \) Copy content Toggle raw display
\( T_{23}^{4} - 368T_{23}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 92 T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 348 T^{2} + 24576)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 84 T^{2} + 1536)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1044 T^{2} + 221184)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1008 T^{2} + 221184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 368 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 46 T + 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2304 T^{2} + 393216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 756 T^{2} + 124416)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 64 T - 1028)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2528 T^{2} + 1364224)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 3296 T^{2} + 614656)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 756 T^{2} + 124416)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 16668 T^{2} + 69033984)^{2} \) Copy content Toggle raw display
$61$ \( (T - 38)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 9392 T^{2} + 7573504)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 17136 T^{2} + 884736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4176 T^{2} + 3538944)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2304 T^{2} + 393216)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 4064 T^{2} + 2027776)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 140 T + 3988)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 45888 T^{2} + 495550464)^{2} \) Copy content Toggle raw display
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