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Properties

Label 1440.1.y.a
Level $1440$
Weight $1$
Character orbit 1440.y
Analytic conductor $0.719$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1440,1,Mod(433,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.433"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 0, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1440.y (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.718653618192\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.9000.2

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{5} + ( - \zeta_{8}^{2} + 1) q^{7} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} - \zeta_{8}^{2} q^{25} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{29} + (\zeta_{8}^{3} + \zeta_{8}) q^{35} - \zeta_{8}^{2} q^{49} + \cdots + (\zeta_{8}^{2} - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 4 q^{55} + 4 q^{73} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0.707107 + 0.707107i
−0.707107 − 0.707107i
0.707107 − 0.707107i
−0.707107 + 0.707107i
0 0 0 −0.707107 + 0.707107i 0 1.00000 − 1.00000i 0 0 0
433.2 0 0 0 0.707107 − 0.707107i 0 1.00000 − 1.00000i 0 0 0
1297.1 0 0 0 −0.707107 − 0.707107i 0 1.00000 + 1.00000i 0 0 0
1297.2 0 0 0 0.707107 + 0.707107i 0 1.00000 + 1.00000i 0 0 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
5.c odd 4 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
40.i odd 4 1 inner
120.w even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.1.y.a 4
3.b odd 2 1 inner 1440.1.y.a 4
4.b odd 2 1 360.1.u.a ✓ 4
5.c odd 4 1 inner 1440.1.y.a 4
8.b even 2 1 inner 1440.1.y.a 4
8.d odd 2 1 360.1.u.a ✓ 4
12.b even 2 1 360.1.u.a ✓ 4
15.e even 4 1 inner 1440.1.y.a 4
20.d odd 2 1 1800.1.u.b 4
20.e even 4 1 360.1.u.a ✓ 4
20.e even 4 1 1800.1.u.b 4
24.f even 2 1 360.1.u.a ✓ 4
24.h odd 2 1 CM 1440.1.y.a 4
36.f odd 6 2 3240.1.bv.c 8
36.h even 6 2 3240.1.bv.c 8
40.e odd 2 1 1800.1.u.b 4
40.i odd 4 1 inner 1440.1.y.a 4
40.k even 4 1 360.1.u.a ✓ 4
40.k even 4 1 1800.1.u.b 4
60.h even 2 1 1800.1.u.b 4
60.l odd 4 1 360.1.u.a ✓ 4
60.l odd 4 1 1800.1.u.b 4
72.l even 6 2 3240.1.bv.c 8
72.p odd 6 2 3240.1.bv.c 8
120.m even 2 1 1800.1.u.b 4
120.q odd 4 1 360.1.u.a ✓ 4
120.q odd 4 1 1800.1.u.b 4
120.w even 4 1 inner 1440.1.y.a 4
180.v odd 12 2 3240.1.bv.c 8
180.x even 12 2 3240.1.bv.c 8
360.bo even 12 2 3240.1.bv.c 8
360.bt odd 12 2 3240.1.bv.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.1.u.a ✓ 4 4.b odd 2 1
360.1.u.a ✓ 4 8.d odd 2 1
360.1.u.a ✓ 4 12.b even 2 1
360.1.u.a ✓ 4 20.e even 4 1
360.1.u.a ✓ 4 24.f even 2 1
360.1.u.a ✓ 4 40.k even 4 1
360.1.u.a ✓ 4 60.l odd 4 1
360.1.u.a ✓ 4 120.q odd 4 1
1440.1.y.a 4 1.a even 1 1 trivial
1440.1.y.a 4 3.b odd 2 1 inner
1440.1.y.a 4 5.c odd 4 1 inner
1440.1.y.a 4 8.b even 2 1 inner
1440.1.y.a 4 15.e even 4 1 inner
1440.1.y.a 4 24.h odd 2 1 CM
1440.1.y.a 4 40.i odd 4 1 inner
1440.1.y.a 4 120.w even 4 1 inner
1800.1.u.b 4 20.d odd 2 1
1800.1.u.b 4 20.e even 4 1
1800.1.u.b 4 40.e odd 2 1
1800.1.u.b 4 40.k even 4 1
1800.1.u.b 4 60.h even 2 1
1800.1.u.b 4 60.l odd 4 1
1800.1.u.b 4 120.m even 2 1
1800.1.u.b 4 120.q odd 4 1
3240.1.bv.c 8 36.f odd 6 2
3240.1.bv.c 8 36.h even 6 2
3240.1.bv.c 8 72.l even 6 2
3240.1.bv.c 8 72.p odd 6 2
3240.1.bv.c 8 180.v odd 12 2
3240.1.bv.c 8 180.x even 12 2
3240.1.bv.c 8 360.bo even 12 2
3240.1.bv.c 8 360.bt odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1440, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) 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
$59$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 16 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
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