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Properties

Label 160.4.c.c
Level $160$
Weight $4$
Character orbit 160.c
Analytic conductor $9.440$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $4$

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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] = [160,4,Mod(129,160)] 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("160.129"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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)
 

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.

\(f(q)\) \(=\) \( q + (2 \beta_{3} - \beta_1) q^{3} - 5 \beta_{2} q^{5} + ( - 5 \beta_{3} - 6 \beta_1) q^{7} + ( - 14 \beta_{2} - 27) q^{9} + ( - 15 \beta_{3} - 20 \beta_1) q^{15} + (86 \beta_{2} + 118) q^{21} + (\beta_{3} - 64 \beta_1) q^{23}+ \cdots + 1386 q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 108 q^{9} + 472 q^{21} + 500 q^{25} - 1224 q^{29} + 1400 q^{45} - 1372 q^{49} - 616 q^{69} + 1004 q^{81} + 5544 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{3} + 10\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 5\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
− 0.618034i
1.61803i
− 1.61803i
0.618034i
0 − 9.23607i 0 −11.1803 0 33.5967i 0 −58.3050 0
129.2 0 − 4.76393i 0 11.1803 0 − 15.5967i 0 4.30495 0
129.3 0 4.76393i 0 11.1803 0 15.5967i 0 4.30495 0
129.4 0 9.23607i 0 −11.1803 0 − 33.5967i 0 −58.3050 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.4.c.c ✓ 4
3.b odd 2 1 1440.4.f.g 4
4.b odd 2 1 inner 160.4.c.c ✓ 4
5.b even 2 1 inner 160.4.c.c ✓ 4
5.c odd 4 1 800.4.a.l 2
5.c odd 4 1 800.4.a.t 2
8.b even 2 1 320.4.c.f 4
8.d odd 2 1 320.4.c.f 4
12.b even 2 1 1440.4.f.g 4
15.d odd 2 1 1440.4.f.g 4
20.d odd 2 1 CM 160.4.c.c ✓ 4
20.e even 4 1 800.4.a.l 2
20.e even 4 1 800.4.a.t 2
40.e odd 2 1 320.4.c.f 4
40.f even 2 1 320.4.c.f 4
40.i odd 4 1 1600.4.a.cb 2
40.i odd 4 1 1600.4.a.cp 2
40.k even 4 1 1600.4.a.cb 2
40.k even 4 1 1600.4.a.cp 2
60.h even 2 1 1440.4.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.c ✓ 4 1.a even 1 1 trivial
160.4.c.c ✓ 4 4.b odd 2 1 inner
160.4.c.c ✓ 4 5.b even 2 1 inner
160.4.c.c ✓ 4 20.d odd 2 1 CM
320.4.c.f 4 8.b even 2 1
320.4.c.f 4 8.d odd 2 1
320.4.c.f 4 40.e odd 2 1
320.4.c.f 4 40.f even 2 1
800.4.a.l 2 5.c odd 4 1
800.4.a.l 2 20.e even 4 1
800.4.a.t 2 5.c odd 4 1
800.4.a.t 2 20.e even 4 1
1440.4.f.g 4 3.b odd 2 1
1440.4.f.g 4 12.b even 2 1
1440.4.f.g 4 15.d odd 2 1
1440.4.f.g 4 60.h even 2 1
1600.4.a.cb 2 40.i odd 4 1
1600.4.a.cb 2 40.k even 4 1
1600.4.a.cp 2 40.i odd 4 1
1600.4.a.cp 2 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 108T_{3}^{2} + 1936 \) acting on \(S_{4}^{\mathrm{new}}(160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 1372 T^{2} + 274576 \) Copy content Toggle raw display
$11$ \( T^{4} \) 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
$23$ \( T^{4} + 48668 T^{2} + 235806736 \) Copy content Toggle raw display
$29$ \( (T + 306)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 212180)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 318028 T^{2} + 302899216 \) Copy content Toggle raw display
$47$ \( T^{4} + 415292 T^{2} + 699285136 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 1620)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 1203052 T^{2} + 1628176 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 1280781104656 \) Copy content Toggle raw display
$89$ \( (T - 1386)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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