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Properties

Label 1260.1.u.b
Level $1260$
Weight $1$
Character orbit 1260.u
Analytic conductor $0.629$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -84
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,1,Mod(307,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.307");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1260.u (of order \(4\), 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: \(0.628821915918\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.220500.3

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + q^{5} - \zeta_{8} q^{7} - \zeta_{8} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + q^{5} - \zeta_{8} q^{7} - \zeta_{8} q^{8} - \zeta_{8}^{3} q^{10} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} - q^{14} - q^{16} + (\zeta_{8}^{2} - 1) q^{17} + (\zeta_{8}^{3} + \zeta_{8}) q^{19} - \zeta_{8}^{2} q^{20} + ( - \zeta_{8}^{2} - 1) q^{22} + q^{25} + \zeta_{8}^{3} q^{28} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{31} + \zeta_{8}^{3} q^{32} + (\zeta_{8}^{3} + \zeta_{8}) q^{34} - \zeta_{8} q^{35} + (\zeta_{8}^{2} + 1) q^{37} + (\zeta_{8}^{2} + 1) q^{38} - \zeta_{8} q^{40} - \zeta_{8}^{2} q^{41} + (\zeta_{8}^{3} - \zeta_{8}) q^{44} + \zeta_{8}^{2} q^{49} - \zeta_{8}^{3} q^{50} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{55} + \zeta_{8}^{2} q^{56} + ( - \zeta_{8}^{2} + 1) q^{62} + \zeta_{8}^{2} q^{64} + (\zeta_{8}^{2} + 1) q^{68} - q^{70} + (\zeta_{8}^{3} + \zeta_{8}) q^{71} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{74} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{76} + (\zeta_{8}^{2} - 1) q^{77} - q^{80} - 2 \zeta_{8} q^{82} + (\zeta_{8}^{2} - 1) q^{85} + (\zeta_{8}^{2} - 1) q^{88} + (\zeta_{8}^{3} + \zeta_{8}) q^{95} + \zeta_{8} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 4 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{22} + 4 q^{25} + 4 q^{37} + 4 q^{38} + 4 q^{62} + 4 q^{68} - 4 q^{70} - 4 q^{77} - 4 q^{80} - 4 q^{85} - 4 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{8}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.707107 − 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 − 0.707107i
−0.707107 + 0.707107i 0 − 1.00000i 1.00000 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 −0.707107 + 0.707107i
307.2 0.707107 − 0.707107i 0 − 1.00000i 1.00000 0 −0.707107 − 0.707107i −0.707107 − 0.707107i 0 0.707107 − 0.707107i
1063.1 −0.707107 − 0.707107i 0 1.00000i 1.00000 0 0.707107 − 0.707107i 0.707107 − 0.707107i 0 −0.707107 − 0.707107i
1063.2 0.707107 + 0.707107i 0 1.00000i 1.00000 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0.707107 + 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
4.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
60.l odd 4 1 inner
140.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.1.u.b yes 4
3.b odd 2 1 1260.1.u.a ✓ 4
4.b odd 2 1 inner 1260.1.u.b yes 4
5.c odd 4 1 1260.1.u.a ✓ 4
7.b odd 2 1 1260.1.u.a ✓ 4
12.b even 2 1 1260.1.u.a ✓ 4
15.e even 4 1 inner 1260.1.u.b yes 4
20.e even 4 1 1260.1.u.a ✓ 4
21.c even 2 1 inner 1260.1.u.b yes 4
28.d even 2 1 1260.1.u.a ✓ 4
35.f even 4 1 inner 1260.1.u.b yes 4
60.l odd 4 1 inner 1260.1.u.b yes 4
84.h odd 2 1 CM 1260.1.u.b yes 4
105.k odd 4 1 1260.1.u.a ✓ 4
140.j odd 4 1 inner 1260.1.u.b yes 4
420.w even 4 1 1260.1.u.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.u.a ✓ 4 3.b odd 2 1
1260.1.u.a ✓ 4 5.c odd 4 1
1260.1.u.a ✓ 4 7.b odd 2 1
1260.1.u.a ✓ 4 12.b even 2 1
1260.1.u.a ✓ 4 20.e even 4 1
1260.1.u.a ✓ 4 28.d even 2 1
1260.1.u.a ✓ 4 105.k odd 4 1
1260.1.u.a ✓ 4 420.w even 4 1
1260.1.u.b yes 4 1.a even 1 1 trivial
1260.1.u.b yes 4 4.b odd 2 1 inner
1260.1.u.b yes 4 15.e even 4 1 inner
1260.1.u.b yes 4 21.c even 2 1 inner
1260.1.u.b yes 4 35.f even 4 1 inner
1260.1.u.b yes 4 60.l odd 4 1 inner
1260.1.u.b yes 4 84.h odd 2 1 CM
1260.1.u.b yes 4 140.j odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + 2T_{17} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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