Properties

Label 72.1.p.a
Level $72$
Weight $1$
Character orbit 72.p
Analytic conductor $0.036$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
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] = [72,1,Mod(43,72)]
 
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, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.43");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 72.p (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: \(0.0359326809096\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.648.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.41472.1

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{6} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{6} + q^{8} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{11} + q^{12} + \zeta_{6}^{2} q^{16} - q^{17} + q^{18} - q^{19} + \zeta_{6} q^{22} + \zeta_{6}^{2} q^{24} + \zeta_{6}^{2} q^{25} + q^{27} - \zeta_{6} q^{32} + \zeta_{6} q^{33} - \zeta_{6}^{2} q^{34} + \zeta_{6}^{2} q^{36} - \zeta_{6}^{2} q^{38} + \zeta_{6} q^{41} - \zeta_{6}^{2} q^{43} - q^{44} - \zeta_{6} q^{48} - \zeta_{6} q^{49} - \zeta_{6} q^{50} - \zeta_{6}^{2} q^{51} + \zeta_{6}^{2} q^{54} - \zeta_{6}^{2} q^{57} + \zeta_{6} q^{59} + q^{64} - q^{66} + \zeta_{6} q^{67} + \zeta_{6} q^{68} - \zeta_{6} q^{72} - q^{73} - \zeta_{6} q^{75} + \zeta_{6} q^{76} + \zeta_{6}^{2} q^{81} - q^{82} + 2 \zeta_{6}^{2} q^{83} + \zeta_{6} q^{86} - \zeta_{6}^{2} q^{88} + 2 q^{89} + q^{96} - \zeta_{6}^{2} q^{97} + q^{98} - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} - q^{4} - q^{6} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{3} - q^{4} - q^{6} + 2 q^{8} - q^{9} + q^{11} + 2 q^{12} - q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} + q^{22} - q^{24} - q^{25} + 2 q^{27} - q^{32} + q^{33} + q^{34} - q^{36} + q^{38} + q^{41} + q^{43} - 2 q^{44} - q^{48} - q^{49} - q^{50} + q^{51} - q^{54} + q^{57} + q^{59} + 2 q^{64} - 2 q^{66} + q^{67} + q^{68} - q^{72} - 2 q^{73} - q^{75} + q^{76} - q^{81} - 2 q^{82} - 2 q^{83} + q^{86} + q^{88} + 4 q^{89} + 2 q^{96} + q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i 0 1.00000 −0.500000 0.866025i 0
67.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.00000 −0.500000 + 0.866025i 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.c even 3 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.1.p.a 2
3.b odd 2 1 216.1.p.a 2
4.b odd 2 1 288.1.t.a 2
5.b even 2 1 1800.1.bk.d 2
5.c odd 4 2 1800.1.ba.b 4
7.b odd 2 1 3528.1.cg.a 2
7.c even 3 1 3528.1.ba.b 2
7.c even 3 1 3528.1.ce.a 2
7.d odd 6 1 3528.1.ba.a 2
7.d odd 6 1 3528.1.ce.b 2
8.b even 2 1 288.1.t.a 2
8.d odd 2 1 CM 72.1.p.a 2
9.c even 3 1 inner 72.1.p.a 2
9.c even 3 1 648.1.b.b 1
9.d odd 6 1 216.1.p.a 2
9.d odd 6 1 648.1.b.a 1
12.b even 2 1 864.1.t.a 2
16.e even 4 2 2304.1.o.c 4
16.f odd 4 2 2304.1.o.c 4
24.f even 2 1 216.1.p.a 2
24.h odd 2 1 864.1.t.a 2
36.f odd 6 1 288.1.t.a 2
36.f odd 6 1 2592.1.b.b 1
36.h even 6 1 864.1.t.a 2
36.h even 6 1 2592.1.b.a 1
40.e odd 2 1 1800.1.bk.d 2
40.k even 4 2 1800.1.ba.b 4
45.j even 6 1 1800.1.bk.d 2
45.k odd 12 2 1800.1.ba.b 4
56.e even 2 1 3528.1.cg.a 2
56.k odd 6 1 3528.1.ba.b 2
56.k odd 6 1 3528.1.ce.a 2
56.m even 6 1 3528.1.ba.a 2
56.m even 6 1 3528.1.ce.b 2
63.g even 3 1 3528.1.ce.a 2
63.h even 3 1 3528.1.ba.b 2
63.k odd 6 1 3528.1.ce.b 2
63.l odd 6 1 3528.1.cg.a 2
63.t odd 6 1 3528.1.ba.a 2
72.j odd 6 1 864.1.t.a 2
72.j odd 6 1 2592.1.b.a 1
72.l even 6 1 216.1.p.a 2
72.l even 6 1 648.1.b.a 1
72.n even 6 1 288.1.t.a 2
72.n even 6 1 2592.1.b.b 1
72.p odd 6 1 inner 72.1.p.a 2
72.p odd 6 1 648.1.b.b 1
144.v odd 12 2 2304.1.o.c 4
144.x even 12 2 2304.1.o.c 4
360.z odd 6 1 1800.1.bk.d 2
360.bo even 12 2 1800.1.ba.b 4
504.ba odd 6 1 3528.1.ce.a 2
504.be even 6 1 3528.1.cg.a 2
504.bf even 6 1 3528.1.ba.a 2
504.ce odd 6 1 3528.1.ba.b 2
504.cz even 6 1 3528.1.ce.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 1.a even 1 1 trivial
72.1.p.a 2 8.d odd 2 1 CM
72.1.p.a 2 9.c even 3 1 inner
72.1.p.a 2 72.p odd 6 1 inner
216.1.p.a 2 3.b odd 2 1
216.1.p.a 2 9.d odd 6 1
216.1.p.a 2 24.f even 2 1
216.1.p.a 2 72.l even 6 1
288.1.t.a 2 4.b odd 2 1
288.1.t.a 2 8.b even 2 1
288.1.t.a 2 36.f odd 6 1
288.1.t.a 2 72.n even 6 1
648.1.b.a 1 9.d odd 6 1
648.1.b.a 1 72.l even 6 1
648.1.b.b 1 9.c even 3 1
648.1.b.b 1 72.p odd 6 1
864.1.t.a 2 12.b even 2 1
864.1.t.a 2 24.h odd 2 1
864.1.t.a 2 36.h even 6 1
864.1.t.a 2 72.j odd 6 1
1800.1.ba.b 4 5.c odd 4 2
1800.1.ba.b 4 40.k even 4 2
1800.1.ba.b 4 45.k odd 12 2
1800.1.ba.b 4 360.bo even 12 2
1800.1.bk.d 2 5.b even 2 1
1800.1.bk.d 2 40.e odd 2 1
1800.1.bk.d 2 45.j even 6 1
1800.1.bk.d 2 360.z odd 6 1
2304.1.o.c 4 16.e even 4 2
2304.1.o.c 4 16.f odd 4 2
2304.1.o.c 4 144.v odd 12 2
2304.1.o.c 4 144.x even 12 2
2592.1.b.a 1 36.h even 6 1
2592.1.b.a 1 72.j odd 6 1
2592.1.b.b 1 36.f odd 6 1
2592.1.b.b 1 72.n even 6 1
3528.1.ba.a 2 7.d odd 6 1
3528.1.ba.a 2 56.m even 6 1
3528.1.ba.a 2 63.t odd 6 1
3528.1.ba.a 2 504.bf even 6 1
3528.1.ba.b 2 7.c even 3 1
3528.1.ba.b 2 56.k odd 6 1
3528.1.ba.b 2 63.h even 3 1
3528.1.ba.b 2 504.ce odd 6 1
3528.1.ce.a 2 7.c even 3 1
3528.1.ce.a 2 56.k odd 6 1
3528.1.ce.a 2 63.g even 3 1
3528.1.ce.a 2 504.ba odd 6 1
3528.1.ce.b 2 7.d odd 6 1
3528.1.ce.b 2 56.m even 6 1
3528.1.ce.b 2 63.k odd 6 1
3528.1.ce.b 2 504.cz even 6 1
3528.1.cg.a 2 7.b odd 2 1
3528.1.cg.a 2 56.e even 2 1
3528.1.cg.a 2 63.l odd 6 1
3528.1.cg.a 2 504.be even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - T + 1 \) Copy content Toggle raw display
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