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Properties

Label 720.3.bs.a
Level $720$
Weight $3$
Character orbit 720.bs
Analytic conductor $19.619$
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] = [720,3,Mod(401,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("720.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 720.bs (of order \(6\), degree \(2\), not 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: \(19.6185790339\)
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: no (minimal twist has level 180)
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.

\(f(q)\) \(=\) \( q + 3 \beta_{2} q^{3} + \beta_1 q^{5} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{7} + (9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_{2} q^{3} + \beta_1 q^{5} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{7} + (9 \beta_{2} - 9) q^{9} + (6 \beta_{3} + \beta_{2} - 6 \beta_1 - 2) q^{11} + ( - 4 \beta_{3} - 10 \beta_{2} + \cdots + 10) q^{13}+ \cdots + ( - 54 \beta_{3} - 18 \beta_{2} + 9) q^{99}+O(q^{100}) \) 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

Display 100 coefficients 10 coefficients

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

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

\(\nu\)\(=\) \( \beta_1 \) 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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−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
401.2 0 1.50000 + 2.59808i 0 1.93649 + 1.11803i 0 −5.87298 − 10.1723i 0 −4.50000 + 7.79423i 0
641.1 0 1.50000 − 2.59808i 0 −1.93649 + 1.11803i 0 1.87298 − 3.24410i 0 −4.50000 − 7.79423i 0
641.2 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 720.3.bs.a 4
3.b odd 2 1 2160.3.bs.a 4
4.b odd 2 1 180.3.o.a ✓ 4
9.c even 3 1 2160.3.bs.a 4
9.d odd 6 1 inner 720.3.bs.a 4
12.b even 2 1 540.3.o.a 4
20.d odd 2 1 900.3.p.b 4
20.e even 4 2 900.3.u.b 8
36.f odd 6 1 540.3.o.a 4
36.f odd 6 1 1620.3.g.a 4
36.h even 6 1 180.3.o.a ✓ 4
36.h even 6 1 1620.3.g.a 4
60.h even 2 1 2700.3.p.a 4
60.l odd 4 2 2700.3.u.a 8
180.n even 6 1 900.3.p.b 4
180.p odd 6 1 2700.3.p.a 4
180.v odd 12 2 900.3.u.b 8
180.x even 12 2 2700.3.u.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.a ✓ 4 4.b odd 2 1
180.3.o.a ✓ 4 36.h even 6 1
540.3.o.a 4 12.b even 2 1
540.3.o.a 4 36.f odd 6 1
720.3.bs.a 4 1.a even 1 1 trivial
720.3.bs.a 4 9.d odd 6 1 inner
900.3.p.b 4 20.d odd 2 1
900.3.p.b 4 180.n even 6 1
900.3.u.b 8 20.e even 4 2
900.3.u.b 8 180.v odd 12 2
1620.3.g.a 4 36.f odd 6 1
1620.3.g.a 4 36.h even 6 1
2160.3.bs.a 4 3.b odd 2 1
2160.3.bs.a 4 9.c even 3 1
2700.3.p.a 4 60.h even 2 1
2700.3.p.a 4 180.p odd 6 1
2700.3.u.a 8 60.l odd 4 2
2700.3.u.a 8 180.x even 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 \(S_{3}^{\mathrm{new}}(720, [\chi])\). 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
$7$ \( T^{4} + 8 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 31329 \) Copy content Toggle raw display
$13$ \( T^{4} - 20 T^{3} + \cdots + 1600 \) 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
$23$ \( T^{4} + 24 T^{3} + \cdots + 451584 \) Copy content Toggle raw display
$29$ \( T^{4} + 60 T^{3} + \cdots + 176400 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 8620096 \) 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
$43$ \( T^{4} - 22 T^{3} + \cdots + 703921 \) 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
$59$ \( T^{4} - 174 T^{3} + \cdots + 5489649 \) Copy content Toggle raw display
$61$ \( T^{4} - 44 T^{3} + \cdots + 3136 \) Copy content Toggle raw display
$67$ \( T^{4} + 14 T^{3} + \cdots + 73805281 \) Copy content Toggle raw display
$71$ \( T^{4} + 11016 T^{2} + 5143824 \) 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
$89$ \( T^{4} + 9216 T^{2} + 1327104 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + \cdots + 102799321 \) Copy content Toggle raw display
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