Properties

Label 740.1.cb.a
Level $740$
Weight $1$
Character orbit 740.cb
Analytic conductor $0.369$
Analytic rank $0$
Dimension $12$
Projective image $D_{36}$
CM discriminant -4
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,1,Mod(87,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 9, 11]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.87");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 740.cb (of order \(36\), degree \(12\), 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: \(0.369308109348\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{36}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{36}^{8} q^{2} + \zeta_{36}^{16} q^{4} + \zeta_{36}^{14} q^{5} + \zeta_{36}^{6} q^{8} + \zeta_{36}^{11} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{36}^{8} q^{2} + \zeta_{36}^{16} q^{4} + \zeta_{36}^{14} q^{5} + \zeta_{36}^{6} q^{8} + \zeta_{36}^{11} q^{9} + \zeta_{36}^{4} q^{10} + (\zeta_{36}^{13} - \zeta_{36}) q^{13} - \zeta_{36}^{14} q^{16} + (\zeta_{36}^{15} + \zeta_{36}^{7}) q^{17} + \zeta_{36} q^{18} - \zeta_{36}^{12} q^{20} - \zeta_{36}^{10} q^{25} + (\zeta_{36}^{9} + \zeta_{36}^{3}) q^{26} + ( - \zeta_{36}^{13} + \zeta_{36}^{8}) q^{29} - \zeta_{36}^{4} q^{32} + ( - \zeta_{36}^{15} + \zeta_{36}^{5}) q^{34} - \zeta_{36}^{9} q^{36} - \zeta_{36}^{16} q^{37} - \zeta_{36}^{2} q^{40} + (\zeta_{36}^{12} + \zeta_{36}^{2}) q^{41} - \zeta_{36}^{7} q^{45} - \zeta_{36}^{17} q^{49} - q^{50} + ( - \zeta_{36}^{17} - \zeta_{36}^{11}) q^{52} + (\zeta_{36}^{17} - \zeta_{36}^{2}) q^{53} + ( - \zeta_{36}^{16} - \zeta_{36}^{3}) q^{58} + ( - \zeta_{36}^{5} - 1) q^{61} + \zeta_{36}^{12} q^{64} + ( - \zeta_{36}^{15} - \zeta_{36}^{9}) q^{65} + ( - \zeta_{36}^{13} - \zeta_{36}^{5}) q^{68} + \zeta_{36}^{17} q^{72} + ( - \zeta_{36}^{6} + \zeta_{36}^{3}) q^{73} - \zeta_{36}^{6} q^{74} + \zeta_{36}^{10} q^{80} - \zeta_{36}^{4} q^{81} + ( - \zeta_{36}^{10} + \zeta_{36}^{2}) q^{82} + ( - \zeta_{36}^{11} - \zeta_{36}^{3}) q^{85} + (\zeta_{36}^{10} + \zeta_{36}^{9}) q^{89} + \zeta_{36}^{15} q^{90} + ( - \zeta_{36}^{14} + \zeta_{36}^{10}) q^{97} - \zeta_{36}^{7} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{8} + 6 q^{20} - 6 q^{41} - 12 q^{50} - 12 q^{61} - 6 q^{64} - 6 q^{73} - 6 q^{74}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(-\zeta_{36}^{7}\) \(-\zeta_{36}^{9}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
87.1
0.984808 0.173648i
0.342020 0.939693i
−0.342020 0.939693i
−0.642788 0.766044i
0.642788 + 0.766044i
0.342020 + 0.939693i
−0.342020 + 0.939693i
−0.984808 + 0.173648i
−0.984808 0.173648i
−0.642788 + 0.766044i
0.642788 0.766044i
0.984808 + 0.173648i
−0.173648 + 0.984808i 0 −0.939693 0.342020i −0.766044 0.642788i 0 0 0.500000 0.866025i −0.342020 0.939693i 0.766044 0.642788i
143.1 0.939693 0.342020i 0 0.766044 0.642788i −0.173648 + 0.984808i 0 0 0.500000 0.866025i 0.642788 0.766044i 0.173648 + 0.984808i
163.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.173648 0.984808i 0 0 0.500000 + 0.866025i −0.642788 0.766044i 0.173648 0.984808i
167.1 −0.766044 0.642788i 0 0.173648 + 0.984808i 0.939693 0.342020i 0 0 0.500000 0.866025i 0.984808 + 0.173648i −0.939693 0.342020i
203.1 −0.766044 0.642788i 0 0.173648 + 0.984808i 0.939693 0.342020i 0 0 0.500000 0.866025i −0.984808 0.173648i −0.939693 0.342020i
207.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i −0.173648 0.984808i 0 0 0.500000 + 0.866025i 0.642788 + 0.766044i 0.173648 0.984808i
227.1 0.939693 0.342020i 0 0.766044 0.642788i −0.173648 + 0.984808i 0 0 0.500000 0.866025i −0.642788 + 0.766044i 0.173648 + 0.984808i
283.1 −0.173648 + 0.984808i 0 −0.939693 0.342020i −0.766044 0.642788i 0 0 0.500000 0.866025i 0.342020 + 0.939693i 0.766044 0.642788i
387.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i −0.766044 + 0.642788i 0 0 0.500000 + 0.866025i 0.342020 0.939693i 0.766044 + 0.642788i
483.1 −0.766044 + 0.642788i 0 0.173648 0.984808i 0.939693 + 0.342020i 0 0 0.500000 + 0.866025i 0.984808 0.173648i −0.939693 + 0.342020i
627.1 −0.766044 + 0.642788i 0 0.173648 0.984808i 0.939693 + 0.342020i 0 0 0.500000 + 0.866025i −0.984808 + 0.173648i −0.939693 + 0.342020i
723.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i −0.766044 + 0.642788i 0 0 0.500000 + 0.866025i −0.342020 + 0.939693i 0.766044 + 0.642788i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 87.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
185.z even 36 1 inner
740.cb odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.1.cb.a yes 12
4.b odd 2 1 CM 740.1.cb.a yes 12
5.b even 2 1 3700.1.dd.a 12
5.c odd 4 1 740.1.bw.a 12
5.c odd 4 1 3700.1.cy.a 12
20.d odd 2 1 3700.1.dd.a 12
20.e even 4 1 740.1.bw.a 12
20.e even 4 1 3700.1.cy.a 12
37.i odd 36 1 740.1.bw.a 12
148.q even 36 1 740.1.bw.a 12
185.z even 36 1 inner 740.1.cb.a yes 12
185.ba odd 36 1 3700.1.cy.a 12
185.bc even 36 1 3700.1.dd.a 12
740.bw odd 36 1 3700.1.dd.a 12
740.ca even 36 1 3700.1.cy.a 12
740.cb odd 36 1 inner 740.1.cb.a yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.1.bw.a 12 5.c odd 4 1
740.1.bw.a 12 20.e even 4 1
740.1.bw.a 12 37.i odd 36 1
740.1.bw.a 12 148.q even 36 1
740.1.cb.a yes 12 1.a even 1 1 trivial
740.1.cb.a yes 12 4.b odd 2 1 CM
740.1.cb.a yes 12 185.z even 36 1 inner
740.1.cb.a yes 12 740.cb odd 36 1 inner
3700.1.cy.a 12 5.c odd 4 1
3700.1.cy.a 12 20.e even 4 1
3700.1.cy.a 12 185.ba odd 36 1
3700.1.cy.a 12 740.ca even 36 1
3700.1.dd.a 12 5.b even 2 1
3700.1.dd.a 12 20.d odd 2 1
3700.1.dd.a 12 185.bc even 36 1
3700.1.dd.a 12 740.bw odd 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(740, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 27T^{6} + 729 \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} + 14 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} + 12 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} + 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{6} - 3 T^{4} + 9 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
show more
show less