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Properties

Label 72.2.l.a
Level $72$
Weight $2$
Character orbit 72.l
Analytic conductor $0.575$
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] = [72,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(0.574922894553\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{6} + 2 \beta_{3} q^{8} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{6} + 2 \beta_{3} q^{8} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{9} + ( - 3 \beta_{2} - \beta_1 - 3) q^{11} + ( - 2 \beta_{3} + 2) q^{12} + (4 \beta_{2} - 4) q^{16} + ( - 2 \beta_{3} + 6 \beta_{2} - 3) q^{17} + (\beta_{3} - 4) q^{18} + ( - 3 \beta_{3} + 6 \beta_1 - 1) q^{19} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{22} + ( - 4 \beta_{2} + 2 \beta_1 + 4) q^{24} + ( - 5 \beta_{2} + 5) q^{25} + (\beta_{3} + 5) q^{27} + (4 \beta_{3} - 4 \beta_1) q^{32} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 6) q^{33} + (6 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 4) q^{34} + (2 \beta_{2} - 4 \beta_1 - 2) q^{36} + (6 \beta_{2} - \beta_1 + 6) q^{38} + ( - 4 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 6) q^{41} + (6 \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 5) q^{43} + ( - 2 \beta_{3} - 12 \beta_{2} + 6) q^{44} + ( - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{48} - 7 \beta_{2} q^{49} + ( - 5 \beta_{3} + 5 \beta_1) q^{50} + ( - 6 \beta_{3} + 7 \beta_{2} + \beta_1 - 1) q^{51} + (2 \beta_{2} + 5 \beta_1 - 2) q^{54} + ( - 6 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 7) q^{57} + (5 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 6) q^{59} - 8 q^{64} + (5 \beta_{3} + 12 \beta_{2} - 6 \beta_1 - 8) q^{66} + ( - 3 \beta_{3} + 7 \beta_{2} - 3 \beta_1) q^{67} + ( - 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 12) q^{68} + (2 \beta_{3} - 8 \beta_{2} - 2 \beta_1) q^{72} + (6 \beta_{3} - 12 \beta_1 - 1) q^{73} + (5 \beta_{3} - 5 \beta_{2} - 5 \beta_1) q^{75} + (6 \beta_{3} - 2 \beta_{2} + 6 \beta_1) q^{76} + ( - 7 \beta_{2} - 4 \beta_1 + 7) q^{81} + ( - 3 \beta_{3} + 6 \beta_1 + 8) q^{82} + 2 \beta_1 q^{83} + (5 \beta_{3} + 6 \beta_{2} - 5 \beta_1 - 12) q^{86} + ( - 12 \beta_{3} - 4 \beta_{2} + 6 \beta_1 + 4) q^{88} + 4 \beta_{3} q^{89} + (4 \beta_{3} + 8) q^{96} + ( - 12 \beta_{3} + 5 \beta_{2} + 6 \beta_1 - 5) q^{97} - 7 \beta_{3} q^{98} + ( - 7 \beta_{3} - 6 \beta_{2} + 12 \beta_1 + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{4} - 4 q^{6} + 2 q^{9} - 18 q^{11} + 8 q^{12} - 8 q^{16} - 16 q^{18} - 4 q^{19} - 4 q^{22} + 8 q^{24} + 10 q^{25} + 20 q^{27} - 14 q^{33} + 8 q^{34} - 4 q^{36} + 36 q^{38} + 18 q^{41} - 10 q^{43} + 8 q^{48} - 14 q^{49} + 10 q^{51} - 4 q^{54} - 38 q^{57} + 18 q^{59} - 32 q^{64} - 8 q^{66} + 14 q^{67} - 36 q^{68} - 16 q^{72} - 4 q^{73} - 10 q^{75} - 4 q^{76} + 14 q^{81} + 32 q^{82} - 36 q^{86} + 8 q^{88} + 32 q^{96} - 10 q^{97} + 16 q^{99}+O(q^{100}) \) 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
−1.22474 − 0.707107i 1.72474 − 0.158919i 1.00000 + 1.73205i 0 −2.22474 − 1.02494i 0 − 2.82843i 2.94949 − 0.548188i 0
11.2 1.22474 + 0.707107i −0.724745 − 1.57313i 1.00000 + 1.73205i 0 0.224745 − 2.43916i 0 2.82843i −1.94949 + 2.28024i 0
59.1 −1.22474 + 0.707107i 1.72474 + 0.158919i 1.00000 − 1.73205i 0 −2.22474 + 1.02494i 0 2.82843i 2.94949 + 0.548188i 0
59.2 1.22474 − 0.707107i −0.724745 + 1.57313i 1.00000 − 1.73205i 0 0.224745 + 2.43916i 0 − 2.82843i −1.94949 − 2.28024i 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 72.2.l.a ✓ 4
3.b odd 2 1 216.2.l.a 4
4.b odd 2 1 288.2.p.a 4
8.b even 2 1 288.2.p.a 4
8.d odd 2 1 CM 72.2.l.a ✓ 4
9.c even 3 1 216.2.l.a 4
9.c even 3 1 648.2.f.a 4
9.d odd 6 1 inner 72.2.l.a ✓ 4
9.d odd 6 1 648.2.f.a 4
12.b even 2 1 864.2.p.a 4
24.f even 2 1 216.2.l.a 4
24.h odd 2 1 864.2.p.a 4
36.f odd 6 1 864.2.p.a 4
36.f odd 6 1 2592.2.f.a 4
36.h even 6 1 288.2.p.a 4
36.h even 6 1 2592.2.f.a 4
72.j odd 6 1 288.2.p.a 4
72.j odd 6 1 2592.2.f.a 4
72.l even 6 1 inner 72.2.l.a ✓ 4
72.l even 6 1 648.2.f.a 4
72.n even 6 1 864.2.p.a 4
72.n even 6 1 2592.2.f.a 4
72.p odd 6 1 216.2.l.a 4
72.p odd 6 1 648.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.a ✓ 4 1.a even 1 1 trivial
72.2.l.a ✓ 4 8.d odd 2 1 CM
72.2.l.a ✓ 4 9.d odd 6 1 inner
72.2.l.a ✓ 4 72.l even 6 1 inner
216.2.l.a 4 3.b odd 2 1
216.2.l.a 4 9.c even 3 1
216.2.l.a 4 24.f even 2 1
216.2.l.a 4 72.p odd 6 1
288.2.p.a 4 4.b odd 2 1
288.2.p.a 4 8.b even 2 1
288.2.p.a 4 36.h even 6 1
288.2.p.a 4 72.j odd 6 1
648.2.f.a 4 9.c even 3 1
648.2.f.a 4 9.d odd 6 1
648.2.f.a 4 72.l even 6 1
648.2.f.a 4 72.p odd 6 1
864.2.p.a 4 12.b even 2 1
864.2.p.a 4 24.h odd 2 1
864.2.p.a 4 36.f odd 6 1
864.2.p.a 4 72.n even 6 1
2592.2.f.a 4 36.f odd 6 1
2592.2.f.a 4 36.h even 6 1
2592.2.f.a 4 72.j odd 6 1
2592.2.f.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 \(S_{2}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + T^{2} - 6 T + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 18 T^{3} + 133 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 70T^{2} + 361 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 53)^{2} \) 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
$41$ \( T^{4} - 18 T^{3} + 103 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + 129 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 18 T^{3} + 85 T^{2} + \cdots + 529 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 14 T^{3} + 201 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 215)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 10 T^{3} + 291 T^{2} + \cdots + 36481 \) Copy content Toggle raw display
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