Properties

Label 180.3.o.a
Level $180$
Weight $3$
Character orbit 180.o
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
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] = [180,3,Mod(41,180)]
 
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, 5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.41");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.o (of order \(6\), 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: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta_{2} - 3) q^{3} + (\beta_{3} - \beta_1) q^{5} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{7} - 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta_{2} - 3) q^{3} + (\beta_{3} - \beta_1) q^{5} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{7} - 9 \beta_{2} q^{9} + (\beta_{2} - 6 \beta_1 + 1) q^{11} + ( - 2 \beta_{3} + 10 \beta_{2} - 2 \beta_1) q^{13} - 3 \beta_{3} q^{15} + ( - 6 \beta_{3} + 2 \beta_{2} - 1) q^{17} + (2 \beta_{3} - 4 \beta_1 - 19) q^{19} + ( - 6 \beta_{3} + 12 \beta_{2} - 6 \beta_1) q^{21} + (12 \beta_{3} - 4 \beta_{2} - 12 \beta_1 + 8) q^{23} + ( - 5 \beta_{2} + 5) q^{25} + 27 q^{27} + ( - 10 \beta_{2} - 12 \beta_1 - 10) q^{29} + (14 \beta_{3} - 2 \beta_{2} + 14 \beta_1) q^{31} + ( - 18 \beta_{3} + 3 \beta_{2} + 18 \beta_1 - 6) q^{33} + (4 \beta_{3} - 20 \beta_{2} + 10) q^{35} + 14 q^{37} + ( - 6 \beta_{3} + 12 \beta_1 - 30) q^{39} + (15 \beta_{2} - 30) q^{41} + (16 \beta_{3} + 11 \beta_{2} - 8 \beta_1 - 11) q^{43} + 9 \beta_1 q^{45} + (34 \beta_{2} - 6 \beta_1 + 34) q^{47} + (16 \beta_{3} - 27 \beta_{2} + 16 \beta_1) q^{49} + ( - 3 \beta_{2} + 18 \beta_1 - 3) q^{51} + ( - 6 \beta_{3} + 28 \beta_{2} - 14) q^{53} + (\beta_{3} - 2 \beta_1 + 30) q^{55} + ( - 12 \beta_{3} - 57 \beta_{2} + 6 \beta_1 + 57) q^{57} + ( - 6 \beta_{3} + 29 \beta_{2} + 6 \beta_1 - 58) q^{59} + ( - 12 \beta_{3} - 22 \beta_{2} + 6 \beta_1 + 22) q^{61} + ( - 18 \beta_{3} + 36 \beta_1 - 36) q^{63} + (10 \beta_{2} - 10 \beta_1 + 10) q^{65} + (24 \beta_{3} + 7 \beta_{2} + 24 \beta_1) q^{67} + ( - 36 \beta_{3} + 24 \beta_{2} - 12) q^{69} + ( - 18 \beta_{3} - 72 \beta_{2} + 36) q^{71} + ( - 6 \beta_{3} + 12 \beta_1 - 37) q^{73} + 15 \beta_{2} q^{75} + (30 \beta_{3} - 64 \beta_{2} - 30 \beta_1 + 128) q^{77} + ( - 12 \beta_{3} + 80 \beta_{2} + 6 \beta_1 - 80) q^{79} + (81 \beta_{2} - 81) q^{81} + ( - 52 \beta_{2} - 52) q^{83} + ( - \beta_{3} + 30 \beta_{2} - \beta_1) q^{85} + ( - 36 \beta_{3} - 30 \beta_{2} + 36 \beta_1 + 60) q^{87} + (24 \beta_{3} + 48 \beta_{2} - 24) q^{89} + (28 \beta_{3} - 56 \beta_1 + 100) q^{91} + (42 \beta_{3} - 84 \beta_1 + 6) q^{93} + ( - 19 \beta_{3} - 10 \beta_{2} + 19 \beta_1 + 20) q^{95} + ( - 52 \beta_{3} - \beta_{2} + 26 \beta_1 + 1) q^{97} + (54 \beta_{3} - 18 \beta_{2} + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} + 8 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} + 8 q^{7} - 18 q^{9} + 6 q^{11} + 20 q^{13} - 76 q^{19} + 24 q^{21} + 24 q^{23} + 10 q^{25} + 108 q^{27} - 60 q^{29} - 4 q^{31} - 18 q^{33} + 56 q^{37} - 120 q^{39} - 90 q^{41} - 22 q^{43} + 204 q^{47} - 54 q^{49} - 18 q^{51} + 120 q^{55} + 114 q^{57} - 174 q^{59} + 44 q^{61} - 144 q^{63} + 60 q^{65} + 14 q^{67} - 148 q^{73} + 30 q^{75} + 384 q^{77} - 160 q^{79} - 162 q^{81} - 312 q^{83} + 60 q^{85} + 180 q^{87} + 400 q^{91} + 24 q^{93} + 60 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
1.93649 1.11803i
−1.93649 + 1.11803i
1.93649 + 1.11803i
−1.93649 1.11803i
0 −1.50000 2.59808i 0 −1.93649 1.11803i 0 −1.87298 3.24410i 0 −4.50000 + 7.79423i 0
41.2 0 −1.50000 2.59808i 0 1.93649 + 1.11803i 0 5.87298 + 10.1723i 0 −4.50000 + 7.79423i 0
101.1 0 −1.50000 + 2.59808i 0 −1.93649 + 1.11803i 0 −1.87298 + 3.24410i 0 −4.50000 7.79423i 0
101.2 0 −1.50000 + 2.59808i 0 1.93649 1.11803i 0 5.87298 10.1723i 0 −4.50000 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.o.a 4
3.b odd 2 1 540.3.o.a 4
4.b odd 2 1 720.3.bs.a 4
5.b even 2 1 900.3.p.b 4
5.c odd 4 2 900.3.u.b 8
9.c even 3 1 540.3.o.a 4
9.c even 3 1 1620.3.g.a 4
9.d odd 6 1 inner 180.3.o.a 4
9.d odd 6 1 1620.3.g.a 4
12.b even 2 1 2160.3.bs.a 4
15.d odd 2 1 2700.3.p.a 4
15.e even 4 2 2700.3.u.a 8
36.f odd 6 1 2160.3.bs.a 4
36.h even 6 1 720.3.bs.a 4
45.h odd 6 1 900.3.p.b 4
45.j even 6 1 2700.3.p.a 4
45.k odd 12 2 2700.3.u.a 8
45.l even 12 2 900.3.u.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.a 4 1.a even 1 1 trivial
180.3.o.a 4 9.d odd 6 1 inner
540.3.o.a 4 3.b odd 2 1
540.3.o.a 4 9.c even 3 1
720.3.bs.a 4 4.b odd 2 1
720.3.bs.a 4 36.h even 6 1
900.3.p.b 4 5.b even 2 1
900.3.p.b 4 45.h odd 6 1
900.3.u.b 8 5.c odd 4 2
900.3.u.b 8 45.l even 12 2
1620.3.g.a 4 9.c even 3 1
1620.3.g.a 4 9.d odd 6 1
2160.3.bs.a 4 12.b even 2 1
2160.3.bs.a 4 36.f odd 6 1
2700.3.p.a 4 15.d odd 2 1
2700.3.p.a 4 45.j even 6 1
2700.3.u.a 8 15.e even 4 2
2700.3.u.a 8 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 8T_{7}^{3} + 108T_{7}^{2} + 352T_{7} + 1936 \) acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 5T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + 108 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} - 165 T^{2} + \cdots + 31329 \) Copy content Toggle raw display
$13$ \( T^{4} - 20 T^{3} + 360 T^{2} + \cdots + 1600 \) Copy content Toggle raw display
$17$ \( T^{4} + 366 T^{2} + 31329 \) Copy content Toggle raw display
$19$ \( (T^{2} + 38 T + 301)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 24 T^{3} - 480 T^{2} + \cdots + 451584 \) Copy content Toggle raw display
$29$ \( T^{4} + 60 T^{3} + 780 T^{2} + \cdots + 176400 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} + 2952 T^{2} + \cdots + 8620096 \) Copy content Toggle raw display
$37$ \( (T - 14)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 45 T + 675)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 22 T^{3} + 1323 T^{2} + \cdots + 703921 \) Copy content Toggle raw display
$47$ \( T^{4} - 204 T^{3} + \cdots + 10810944 \) Copy content Toggle raw display
$53$ \( T^{4} + 1536 T^{2} + 166464 \) Copy content Toggle raw display
$59$ \( T^{4} + 174 T^{3} + 12435 T^{2} + \cdots + 5489649 \) Copy content Toggle raw display
$61$ \( T^{4} - 44 T^{3} + 1992 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + 8787 T^{2} + \cdots + 73805281 \) Copy content Toggle raw display
$71$ \( T^{4} + 11016 T^{2} + \cdots + 5143824 \) Copy content Toggle raw display
$73$ \( (T^{2} + 74 T + 829)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 160 T^{3} + \cdots + 34339600 \) Copy content Toggle raw display
$83$ \( (T^{2} + 156 T + 8112)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 9216 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + 10143 T^{2} + \cdots + 102799321 \) Copy content Toggle raw display
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