Properties

Label 180.2.i.b
Level $180$
Weight $2$
Character orbit 180.i
Analytic conductor $1.437$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,2,Mod(61,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.i (of order \(3\), degree \(2\), 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: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{2} + 1) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{2} + 1) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{7} + (\beta_{3} + \beta_{2} + 1) q^{9} + (2 \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{11} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 2) q^{13} - \beta_{4} q^{15} + ( - \beta_{5} + \beta_{2} - 2 \beta_1 + 1) q^{17} + (\beta_{5} - \beta_{2} + 2 \beta_1 + 1) q^{19} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 5) q^{21} + ( - \beta_{5} + \beta_{2} + \beta_1 + 1) q^{23} + \beta_{2} q^{25} + ( - \beta_{5} - \beta_{4} + 5 \beta_{2} - \beta_1 + 3) q^{27} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{29} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{31} + ( - \beta_{5} + 2 \beta_{3} - 5 \beta_{2} - 1) q^{33} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{35} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_1 + 3) q^{37} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 4 \beta_{2} - 5) q^{39} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{41} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{43} + (\beta_{5} + \beta_{2}) q^{45} + (\beta_{5} - 3 \beta_{4} + 5 \beta_{2} + 2 \beta_1 - 1) q^{47} + (\beta_{5} + 2 \beta_{4} - \beta_{3} - 7 \beta_{2} - 6) q^{49} + (\beta_{5} + \beta_{3} - \beta_{2} + 4) q^{51} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 - 2) q^{53} + ( - \beta_{5} + \beta_{2} - 2 \beta_1 + 1) q^{55} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{57} + (2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 - 2) q^{59} + ( - 2 \beta_{5} + 4 \beta_{4} + \beta_{3} + 7 \beta_{2} - 2 \beta_1 + 1) q^{61} + ( - \beta_{3} + 8 \beta_{2} + 3 \beta_1 - 1) q^{63} + (2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{65} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1 - 3) q^{67} + (\beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 1) q^{69} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_1 + 7) q^{71} + 8 q^{73} + ( - \beta_{4} + \beta_1) q^{75} + ( - \beta_{5} - 6 \beta_{4} + 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{77} + 2 \beta_{2} q^{79} + (2 \beta_{5} - 4 \beta_{4} - \beta_{2} + 2 \beta_1 + 3) q^{81} + (\beta_{5} - \beta_{4} - \beta_{3} - 7 \beta_{2}) q^{83} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2}) q^{85} + ( - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 10) q^{87} - 3 q^{89} + ( - 3 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 10 \beta_1 + 3) q^{91} + (2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + 5) q^{93} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{95} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 6 \beta_{2}) q^{97} + ( - \beta_{5} + 6 \beta_{4} - \beta_{3} + 7 \beta_{2} - 6 \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + 3 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{3} + 3 q^{5} - 3 q^{7} + 5 q^{9} - 6 q^{13} + q^{15} + 12 q^{19} - 20 q^{21} + 3 q^{23} - 3 q^{25} + 2 q^{27} + 3 q^{29} - 6 q^{31} + 12 q^{33} - 6 q^{35} + 24 q^{37} - 20 q^{39} - 3 q^{41} - 6 q^{43} - 2 q^{45} - 15 q^{47} - 18 q^{49} + 30 q^{51} - 12 q^{53} - 32 q^{57} - 6 q^{59} - 21 q^{61} - 29 q^{63} + 6 q^{65} - 9 q^{67} + 15 q^{69} + 48 q^{71} + 48 q^{73} + 2 q^{75} - 6 q^{77} - 6 q^{79} + 29 q^{81} + 21 q^{83} + 42 q^{87} - 18 q^{89} + 16 q^{93} + 6 q^{95} - 18 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} + \nu^{4} + 2\nu^{3} + 27\nu^{2} - 12\nu - 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 2\nu^{4} + 2\nu^{3} + 6\nu + 18 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{5} + \nu^{4} + 11\nu^{3} + 15\nu + 72 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 5\beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 4\beta_{4} - \beta_{2} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{5} + \beta_{4} - 8\beta_{2} + 4\beta _1 + 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
1.71903 0.211943i
0.403374 + 1.68443i
−1.62241 0.606458i
1.71903 + 0.211943i
0.403374 1.68443i
−1.62241 + 0.606458i
0 −1.71903 + 0.211943i 0 0.500000 + 0.866025i 0 −1.36710 + 2.36788i 0 2.91016 0.728674i 0
61.2 0 −0.403374 1.68443i 0 0.500000 + 0.866025i 0 1.91751 3.32123i 0 −2.67458 + 1.35891i 0
61.3 0 1.62241 + 0.606458i 0 0.500000 + 0.866025i 0 −2.05042 + 3.55142i 0 2.26442 + 1.96784i 0
121.1 0 −1.71903 0.211943i 0 0.500000 0.866025i 0 −1.36710 2.36788i 0 2.91016 + 0.728674i 0
121.2 0 −0.403374 + 1.68443i 0 0.500000 0.866025i 0 1.91751 + 3.32123i 0 −2.67458 1.35891i 0
121.3 0 1.62241 0.606458i 0 0.500000 0.866025i 0 −2.05042 3.55142i 0 2.26442 1.96784i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.2.i.b 6
3.b odd 2 1 540.2.i.b 6
4.b odd 2 1 720.2.q.k 6
5.b even 2 1 900.2.i.c 6
5.c odd 4 2 900.2.s.c 12
9.c even 3 1 inner 180.2.i.b 6
9.c even 3 1 1620.2.a.i 3
9.d odd 6 1 540.2.i.b 6
9.d odd 6 1 1620.2.a.j 3
12.b even 2 1 2160.2.q.i 6
15.d odd 2 1 2700.2.i.c 6
15.e even 4 2 2700.2.s.c 12
36.f odd 6 1 720.2.q.k 6
36.f odd 6 1 6480.2.a.bt 3
36.h even 6 1 2160.2.q.i 6
36.h even 6 1 6480.2.a.bw 3
45.h odd 6 1 2700.2.i.c 6
45.h odd 6 1 8100.2.a.u 3
45.j even 6 1 900.2.i.c 6
45.j even 6 1 8100.2.a.v 3
45.k odd 12 2 900.2.s.c 12
45.k odd 12 2 8100.2.d.p 6
45.l even 12 2 2700.2.s.c 12
45.l even 12 2 8100.2.d.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.b 6 1.a even 1 1 trivial
180.2.i.b 6 9.c even 3 1 inner
540.2.i.b 6 3.b odd 2 1
540.2.i.b 6 9.d odd 6 1
720.2.q.k 6 4.b odd 2 1
720.2.q.k 6 36.f odd 6 1
900.2.i.c 6 5.b even 2 1
900.2.i.c 6 45.j even 6 1
900.2.s.c 12 5.c odd 4 2
900.2.s.c 12 45.k odd 12 2
1620.2.a.i 3 9.c even 3 1
1620.2.a.j 3 9.d odd 6 1
2160.2.q.i 6 12.b even 2 1
2160.2.q.i 6 36.h even 6 1
2700.2.i.c 6 15.d odd 2 1
2700.2.i.c 6 45.h odd 6 1
2700.2.s.c 12 15.e even 4 2
2700.2.s.c 12 45.l even 12 2
6480.2.a.bt 3 36.f odd 6 1
6480.2.a.bw 3 36.h even 6 1
8100.2.a.u 3 45.h odd 6 1
8100.2.a.v 3 45.j even 6 1
8100.2.d.o 6 45.l even 12 2
8100.2.d.p 6 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 3T_{7}^{5} + 24T_{7}^{4} + 41T_{7}^{3} + 354T_{7}^{2} + 645T_{7} + 1849 \) acting on \(S_{2}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} - 2 T^{4} - 3 T^{3} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} + 24 T^{4} + \cdots + 1849 \) Copy content Toggle raw display
$11$ \( T^{6} + 24 T^{4} - 72 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + 48 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$17$ \( (T^{3} - 24 T + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 6 T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + 24 T^{4} + 63 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} + 78 T^{4} + \cdots + 77841 \) Copy content Toggle raw display
$31$ \( T^{6} + 6 T^{5} + 48 T^{4} - 64 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( (T^{3} - 12 T^{2} - 36 T + 436)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + 90 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} + 48 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$47$ \( T^{6} + 15 T^{5} + 186 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} - 60 T + 72)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + 96 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$61$ \( T^{6} + 21 T^{5} + 378 T^{4} + \cdots + 167281 \) Copy content Toggle raw display
$67$ \( T^{6} + 9 T^{5} + 102 T^{4} + \cdots + 22801 \) Copy content Toggle raw display
$71$ \( (T^{3} - 24 T^{2} + 108 T + 324)^{2} \) Copy content Toggle raw display
$73$ \( (T - 8)^{6} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$83$ \( T^{6} - 21 T^{5} + 312 T^{4} + \cdots + 59049 \) Copy content Toggle raw display
$89$ \( (T + 3)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 18 T^{5} + 288 T^{4} + \cdots + 179776 \) Copy content Toggle raw display
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