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Properties

Label 1480.1.by.b
Level $1480$
Weight $1$
Character orbit 1480.by
Analytic conductor $0.739$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -40
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1480,1,Mod(1099,1480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1480, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1480.1099");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1480 = 2^{3} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1480.by (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: \(0.738616218697\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.54760.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - \zeta_{6}^{2} q^{5} - \zeta_{6}^{2} q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + \zeta_{6}^{2} q^{4} - \zeta_{6}^{2} q^{5} - \zeta_{6}^{2} q^{7} - q^{8} - \zeta_{6} q^{9} + q^{10} - q^{11} + \zeta_{6}^{2} q^{13} + 2 q^{14} - \zeta_{6} q^{16} - \zeta_{6}^{2} q^{18} - \zeta_{6}^{2} q^{19} + \zeta_{6} q^{20} - \zeta_{6} q^{22} + q^{23} - \zeta_{6} q^{25} - q^{26} + 2 \zeta_{6} q^{28} - \zeta_{6}^{2} q^{32} - 2 \zeta_{6} q^{35} + q^{36} + \zeta_{6} q^{37} + q^{38} + \zeta_{6}^{2} q^{40} + \zeta_{6}^{2} q^{41} - \zeta_{6}^{2} q^{44} - q^{45} + \zeta_{6} q^{46} + q^{47} - 3 \zeta_{6} q^{49} - \zeta_{6}^{2} q^{50} - \zeta_{6} q^{52} + \zeta_{6} q^{53} + \zeta_{6}^{2} q^{55} + 2 \zeta_{6}^{2} q^{56} + \zeta_{6} q^{59} - 2 q^{63} + q^{64} + \zeta_{6} q^{65} - 2 \zeta_{6}^{2} q^{70} + \zeta_{6} q^{72} + \zeta_{6}^{2} q^{74} + \zeta_{6} q^{76} + 2 \zeta_{6}^{2} q^{77} - q^{80} + \zeta_{6}^{2} q^{81} - 2 q^{82} + q^{88} + \zeta_{6} q^{89} - \zeta_{6} q^{90} + 2 \zeta_{6} q^{91} + \zeta_{6}^{2} q^{92} + \zeta_{6} q^{94} - \zeta_{6} q^{95} - 3 \zeta_{6}^{2} q^{98} + \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{9} + 2 q^{10} - 2 q^{11} - q^{13} + 4 q^{14} - q^{16} + q^{18} + q^{19} + q^{20} - q^{22} + 2 q^{23} - q^{25} - 2 q^{26} + 2 q^{28} + q^{32} - 2 q^{35} + 2 q^{36} + q^{37} + 2 q^{38} - q^{40} - 2 q^{41} + q^{44} - 2 q^{45} + q^{46} + 2 q^{47} - 3 q^{49} + q^{50} - q^{52} + 2 q^{53} - q^{55} - 2 q^{56} + q^{59} - 4 q^{63} + 2 q^{64} + q^{65} + 2 q^{70} + q^{72} - q^{74} + q^{76} - 2 q^{77} - 2 q^{80} - q^{81} - 4 q^{82} + 2 q^{88} + q^{89} - q^{90} + 2 q^{91} - q^{92} + q^{94} - q^{95} + 3 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1480\mathbb{Z}\right)^\times\).

\(n\) \(297\) \(741\) \(1001\) \(1111\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1099.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 − 0.866025i 0 1.00000 − 1.73205i −1.00000 −0.500000 − 0.866025i 1.00000
1379.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.500000 + 0.866025i 0 1.00000 + 1.73205i −1.00000 −0.500000 + 0.866025i 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
37.c even 3 1 inner
1480.by odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1480.1.by.b yes 2
5.b even 2 1 1480.1.by.a ✓ 2
8.d odd 2 1 1480.1.by.a ✓ 2
37.c even 3 1 inner 1480.1.by.b yes 2
40.e odd 2 1 CM 1480.1.by.b yes 2
185.n even 6 1 1480.1.by.a ✓ 2
296.p odd 6 1 1480.1.by.a ✓ 2
1480.by odd 6 1 inner 1480.1.by.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.1.by.a ✓ 2 5.b even 2 1
1480.1.by.a ✓ 2 8.d odd 2 1
1480.1.by.a ✓ 2 185.n even 6 1
1480.1.by.a ✓ 2 296.p odd 6 1
1480.1.by.b yes 2 1.a even 1 1 trivial
1480.1.by.b yes 2 37.c even 3 1 inner
1480.1.by.b yes 2 40.e odd 2 1 CM
1480.1.by.b yes 2 1480.by odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 2T_{7} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1480, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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