Properties

Label 72.4.d.a
Level $72$
Weight $4$
Character orbit 72.d
Analytic conductor $4.248$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,4,Mod(37,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), 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: \(4.24813752041\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 8 q^{4} - 7 \beta q^{5} - 34 q^{7} - 8 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 8 q^{4} - 7 \beta q^{5} - 34 q^{7} - 8 \beta q^{8} + 56 q^{10} + 2 \beta q^{11} - 34 \beta q^{14} + 64 q^{16} + 56 \beta q^{20} - 16 q^{22} - 267 q^{25} + 272 q^{28} - 79 \beta q^{29} - 70 q^{31} + 64 \beta q^{32} + 238 \beta q^{35} - 448 q^{40} - 16 \beta q^{44} + 813 q^{49} - 267 \beta q^{50} - 205 \beta q^{53} + 112 q^{55} + 272 \beta q^{56} + 632 q^{58} - 196 \beta q^{59} - 70 \beta q^{62} - 512 q^{64} - 1904 q^{70} - 322 q^{73} - 68 \beta q^{77} + 1370 q^{79} - 448 \beta q^{80} + 434 \beta q^{83} + 128 q^{88} - 574 q^{97} + 813 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} - 68 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{4} - 68 q^{7} + 112 q^{10} + 128 q^{16} - 32 q^{22} - 534 q^{25} + 544 q^{28} - 140 q^{31} - 896 q^{40} + 1626 q^{49} + 224 q^{55} + 1264 q^{58} - 1024 q^{64} - 3808 q^{70} - 644 q^{73} + 2740 q^{79} + 256 q^{88} - 1148 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.41421i
1.41421i
2.82843i 0 −8.00000 19.7990i 0 −34.0000 22.6274i 0 56.0000
37.2 2.82843i 0 −8.00000 19.7990i 0 −34.0000 22.6274i 0 56.0000
\(n\): 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.4.d.a 2
3.b odd 2 1 inner 72.4.d.a 2
4.b odd 2 1 288.4.d.b 2
8.b even 2 1 inner 72.4.d.a 2
8.d odd 2 1 288.4.d.b 2
12.b even 2 1 288.4.d.b 2
16.e even 4 2 2304.4.a.bo 2
16.f odd 4 2 2304.4.a.u 2
24.f even 2 1 288.4.d.b 2
24.h odd 2 1 CM 72.4.d.a 2
48.i odd 4 2 2304.4.a.bo 2
48.k even 4 2 2304.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.d.a 2 1.a even 1 1 trivial
72.4.d.a 2 3.b odd 2 1 inner
72.4.d.a 2 8.b even 2 1 inner
72.4.d.a 2 24.h odd 2 1 CM
288.4.d.b 2 4.b odd 2 1
288.4.d.b 2 8.d odd 2 1
288.4.d.b 2 12.b even 2 1
288.4.d.b 2 24.f even 2 1
2304.4.a.u 2 16.f odd 4 2
2304.4.a.u 2 48.k even 4 2
2304.4.a.bo 2 16.e even 4 2
2304.4.a.bo 2 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 392 \) acting on \(S_{4}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 392 \) Copy content Toggle raw display
$7$ \( (T + 34)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 32 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 49928 \) Copy content Toggle raw display
$31$ \( (T + 70)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 336200 \) Copy content Toggle raw display
$59$ \( T^{2} + 307328 \) 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
$73$ \( (T + 322)^{2} \) Copy content Toggle raw display
$79$ \( (T - 1370)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1506848 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 574)^{2} \) Copy content Toggle raw display
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