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Properties

Label 72.4.l.a
Level $72$
Weight $4$
Character orbit 72.l
Analytic conductor $4.248$
Analytic rank $0$
Dimension $4$
CM discriminant -8
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(11,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.l (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: \(4.24813752041\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} + (5 \beta_{2} + \beta_1 - 5) q^{3} + 8 \beta_{2} q^{4} + (10 \beta_{3} + 4 \beta_{2} - 10 \beta_1) q^{6} + 16 \beta_{3} q^{8} + (10 \beta_{3} - 23 \beta_{2} - 10 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{2} + (5 \beta_{2} + \beta_1 - 5) q^{3} + 8 \beta_{2} q^{4} + (10 \beta_{3} + 4 \beta_{2} - 10 \beta_1) q^{6} + 16 \beta_{3} q^{8} + (10 \beta_{3} - 23 \beta_{2} - 10 \beta_1) q^{9} + ( - 9 \beta_{2} + 25 \beta_1 - 9) q^{11} + (8 \beta_{3} - 40) q^{12} + (64 \beta_{2} - 64) q^{16} + (38 \beta_{3} - 90 \beta_{2} + 45) q^{17} + ( - 46 \beta_{3} - 40) q^{18} + ( - 45 \beta_{3} + 90 \beta_1 + 53) q^{19} + ( - 18 \beta_{3} + 100 \beta_{2} - 18 \beta_1) q^{22} + (32 \beta_{2} - 80 \beta_1 - 32) q^{24} + ( - 125 \beta_{2} + 125) q^{25} + ( - 73 \beta_{3} + 95) q^{27} + (128 \beta_{3} - 128 \beta_1) q^{32} + (116 \beta_{3} + 5 \beta_{2} + \cdots + 90) q^{33}+ \cdots + ( - 665 \beta_{3} + 414 \beta_{2} + \cdots - 707) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{3} + 16 q^{4} + 8 q^{6} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{3} + 16 q^{4} + 8 q^{6} - 46 q^{9} - 54 q^{11} - 160 q^{12} - 128 q^{16} - 160 q^{18} + 212 q^{19} + 200 q^{22} - 64 q^{24} + 250 q^{25} + 380 q^{27} + 370 q^{33} - 304 q^{34} + 368 q^{36} + 1080 q^{38} - 1566 q^{41} + 290 q^{43} - 640 q^{48} - 686 q^{49} + 1198 q^{51} + 584 q^{54} + 10 q^{57} - 2538 q^{59} - 2048 q^{64} - 2000 q^{66} - 70 q^{67} + 2160 q^{68} - 640 q^{72} + 860 q^{73} + 1250 q^{75} + 848 q^{76} - 658 q^{81} + 320 q^{82} + 4104 q^{86} - 1600 q^{88} - 1024 q^{96} + 1910 q^{97} - 2000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

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

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 − 0.707107i
−2.44949 − 1.41421i −3.72474 + 3.62302i 4.00000 + 6.92820i 0 14.2474 − 3.60697i 0 − 22.6274i 0.747449 − 26.9897i 0
11.2 2.44949 + 1.41421i −1.27526 + 5.03723i 4.00000 + 6.92820i 0 −10.2474 + 10.5352i 0 22.6274i −23.7474 − 12.8475i 0
59.1 −2.44949 + 1.41421i −3.72474 − 3.62302i 4.00000 − 6.92820i 0 14.2474 + 3.60697i 0 22.6274i 0.747449 + 26.9897i 0
59.2 2.44949 − 1.41421i −1.27526 − 5.03723i 4.00000 − 6.92820i 0 −10.2474 − 10.5352i 0 − 22.6274i −23.7474 + 12.8475i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
9.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.4.l.a ✓ 4
3.b odd 2 1 216.4.l.a 4
4.b odd 2 1 288.4.p.a 4
8.b even 2 1 288.4.p.a 4
8.d odd 2 1 CM 72.4.l.a ✓ 4
9.c even 3 1 216.4.l.a 4
9.d odd 6 1 inner 72.4.l.a ✓ 4
12.b even 2 1 864.4.p.a 4
24.f even 2 1 216.4.l.a 4
24.h odd 2 1 864.4.p.a 4
36.f odd 6 1 864.4.p.a 4
36.h even 6 1 288.4.p.a 4
72.j odd 6 1 288.4.p.a 4
72.l even 6 1 inner 72.4.l.a ✓ 4
72.n even 6 1 864.4.p.a 4
72.p odd 6 1 216.4.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.l.a ✓ 4 1.a even 1 1 trivial
72.4.l.a ✓ 4 8.d odd 2 1 CM
72.4.l.a ✓ 4 9.d odd 6 1 inner
72.4.l.a ✓ 4 72.l even 6 1 inner
216.4.l.a 4 3.b odd 2 1
216.4.l.a 4 9.c even 3 1
216.4.l.a 4 24.f even 2 1
216.4.l.a 4 72.p odd 6 1
288.4.p.a 4 4.b odd 2 1
288.4.p.a 4 8.b even 2 1
288.4.p.a 4 36.h even 6 1
288.4.p.a 4 72.j odd 6 1
864.4.p.a 4 12.b even 2 1
864.4.p.a 4 24.h odd 2 1
864.4.p.a 4 36.f odd 6 1
864.4.p.a 4 72.n even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{4} + 10 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 54 T^{3} + \cdots + 1014049 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 17926 T^{2} + 10156969 \) Copy content Toggle raw display
$19$ \( (T^{2} - 106 T - 9341)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 41437894969 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 23845845241 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 260443853569 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 805307017321 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 430 T - 982151)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 215897763904 \) Copy content Toggle raw display
$89$ \( (T^{2} + 1767200)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 828247426561 \) Copy content Toggle raw display
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