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Properties

Label 72.2.n.a
Level $72$
Weight $2$
Character orbit 72.n
Analytic conductor $0.575$
Analytic rank $0$
Dimension $4$
CM no
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(13,72)]
 
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([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.n (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: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} - 2 \zeta_{12} q^{5} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} + 1) q^{6} - 4 \zeta_{12}^{2} q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} - 2 \zeta_{12} q^{5} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} + 1) q^{6} - 4 \zeta_{12}^{2} q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + (2 \zeta_{12}^{3} + 2) q^{10} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{12} + 2 \zeta_{12} q^{13} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{14} + (4 \zeta_{12}^{2} - 2) q^{15} + 4 \zeta_{12}^{2} q^{16} + 5 q^{17} + ( - 3 \zeta_{12}^{3} + 3) q^{18} - \zeta_{12}^{3} q^{19} - 4 \zeta_{12}^{2} q^{20} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{21} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{22} + (2 \zeta_{12}^{2} - 2) q^{23} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{24} - \zeta_{12}^{2} q^{25} + ( - 2 \zeta_{12}^{3} - 2) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} - 8 \zeta_{12}^{3} q^{28} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{30} + ( - 4 \zeta_{12}^{2} + 4) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + ( - 3 \zeta_{12}^{2} - 3) q^{33} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 5 \zeta_{12}) q^{34} + 8 \zeta_{12}^{3} q^{35} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} - 2 \zeta_{12}^{3} q^{37} + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{38} + ( - 4 \zeta_{12}^{2} + 2) q^{39} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{40} + ( - 5 \zeta_{12}^{2} + 5) q^{41} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{42} + (11 \zeta_{12}^{3} - 11 \zeta_{12}) q^{43} + 6 q^{44} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{45} + ( - 2 \zeta_{12}^{3} + 2) q^{46} + 6 \zeta_{12}^{2} q^{47} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + (9 \zeta_{12}^{2} - 9) q^{49} + (\zeta_{12}^{2} + \zeta_{12} - 1) q^{50} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{51} + 4 \zeta_{12}^{2} q^{52} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 6) q^{54} - 6 q^{55} + (8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12}) q^{56} + (\zeta_{12}^{2} - 2) q^{57} - \zeta_{12} q^{59} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{60} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{61} + (4 \zeta_{12}^{3} - 4) q^{62} + 12 q^{63} + 8 \zeta_{12}^{3} q^{64} - 4 \zeta_{12}^{2} q^{65} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12} - 3) q^{66} + 3 \zeta_{12} q^{67} + 10 \zeta_{12} q^{68} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{69} + ( - 8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12}) q^{70} - 6 q^{71} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + 9 q^{73} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{74} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{75} + ( - 2 \zeta_{12}^{2} + 2) q^{76} - 12 \zeta_{12} q^{77} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{78} + 14 \zeta_{12}^{2} q^{79} - 8 \zeta_{12}^{3} q^{80} - 9 \zeta_{12}^{2} q^{81} + (5 \zeta_{12}^{3} - 5) q^{82} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{83} + (8 \zeta_{12}^{2} - 16) q^{84} - 10 \zeta_{12} q^{85} + ( - 11 \zeta_{12}^{2} + 11 \zeta_{12} + 11) q^{86} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12}) q^{88} - 14 q^{89} + (6 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{90} - 8 \zeta_{12}^{3} q^{91} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{92} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{93} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{94} + (2 \zeta_{12}^{2} - 2) q^{95} + (4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12} - 4) q^{96} - \zeta_{12}^{2} q^{97} + ( - 9 \zeta_{12}^{3} + 9) q^{98} + 9 \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 6 q^{6} - 8 q^{7} - 8 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 6 q^{6} - 8 q^{7} - 8 q^{8} - 6 q^{9} + 8 q^{10} - 8 q^{14} + 8 q^{16} + 20 q^{17} + 12 q^{18} - 8 q^{20} - 6 q^{22} - 4 q^{23} - 12 q^{24} - 2 q^{25} - 8 q^{26} + 12 q^{30} + 8 q^{31} + 8 q^{32} - 18 q^{33} - 10 q^{34} + 2 q^{38} - 8 q^{40} + 10 q^{41} + 24 q^{44} + 8 q^{46} + 12 q^{47} - 18 q^{49} - 2 q^{50} + 8 q^{52} - 18 q^{54} - 24 q^{55} + 16 q^{56} - 6 q^{57} - 16 q^{62} + 48 q^{63} - 8 q^{65} - 16 q^{70} - 24 q^{71} + 12 q^{72} + 36 q^{73} + 4 q^{74} + 4 q^{76} - 12 q^{78} + 28 q^{79} - 18 q^{81} - 20 q^{82} - 48 q^{84} + 22 q^{86} - 12 q^{88} - 56 q^{89} - 12 q^{90} + 12 q^{94} - 4 q^{95} - 2 q^{97} + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−1.36603 − 0.366025i −0.866025 − 1.50000i 1.73205 + 1.00000i −1.73205 − 1.00000i 0.633975 + 2.36603i −2.00000 − 3.46410i −2.00000 − 2.00000i −1.50000 + 2.59808i 2.00000 + 2.00000i
13.2 0.366025 − 1.36603i 0.866025 + 1.50000i −1.73205 − 1.00000i 1.73205 + 1.00000i 2.36603 − 0.633975i −2.00000 − 3.46410i −2.00000 + 2.00000i −1.50000 + 2.59808i 2.00000 − 2.00000i
61.1 −1.36603 + 0.366025i −0.866025 + 1.50000i 1.73205 − 1.00000i −1.73205 + 1.00000i 0.633975 − 2.36603i −2.00000 + 3.46410i −2.00000 + 2.00000i −1.50000 − 2.59808i 2.00000 − 2.00000i
61.2 0.366025 + 1.36603i 0.866025 − 1.50000i −1.73205 + 1.00000i 1.73205 − 1.00000i 2.36603 + 0.633975i −2.00000 + 3.46410i −2.00000 − 2.00000i −1.50000 − 2.59808i 2.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n 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.n.a ✓ 4
3.b odd 2 1 216.2.n.a 4
4.b odd 2 1 288.2.r.a 4
8.b even 2 1 inner 72.2.n.a ✓ 4
8.d odd 2 1 288.2.r.a 4
9.c even 3 1 inner 72.2.n.a ✓ 4
9.c even 3 1 648.2.d.d 2
9.d odd 6 1 216.2.n.a 4
9.d odd 6 1 648.2.d.a 2
12.b even 2 1 864.2.r.a 4
24.f even 2 1 864.2.r.a 4
24.h odd 2 1 216.2.n.a 4
36.f odd 6 1 288.2.r.a 4
36.f odd 6 1 2592.2.d.b 2
36.h even 6 1 864.2.r.a 4
36.h even 6 1 2592.2.d.a 2
72.j odd 6 1 216.2.n.a 4
72.j odd 6 1 648.2.d.a 2
72.l even 6 1 864.2.r.a 4
72.l even 6 1 2592.2.d.a 2
72.n even 6 1 inner 72.2.n.a ✓ 4
72.n even 6 1 648.2.d.d 2
72.p odd 6 1 288.2.r.a 4
72.p odd 6 1 2592.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.a ✓ 4 1.a even 1 1 trivial
72.2.n.a ✓ 4 8.b even 2 1 inner
72.2.n.a ✓ 4 9.c even 3 1 inner
72.2.n.a ✓ 4 72.n even 6 1 inner
216.2.n.a 4 3.b odd 2 1
216.2.n.a 4 9.d odd 6 1
216.2.n.a 4 24.h odd 2 1
216.2.n.a 4 72.j odd 6 1
288.2.r.a 4 4.b odd 2 1
288.2.r.a 4 8.d odd 2 1
288.2.r.a 4 36.f odd 6 1
288.2.r.a 4 72.p odd 6 1
648.2.d.a 2 9.d odd 6 1
648.2.d.a 2 72.j odd 6 1
648.2.d.d 2 9.c even 3 1
648.2.d.d 2 72.n even 6 1
864.2.r.a 4 12.b even 2 1
864.2.r.a 4 24.f even 2 1
864.2.r.a 4 36.h even 6 1
864.2.r.a 4 72.l even 6 1
2592.2.d.a 2 36.h even 6 1
2592.2.d.a 2 72.l even 6 1
2592.2.d.b 2 36.f odd 6 1
2592.2.d.b 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 4T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$13$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$17$ \( (T - 5)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$61$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( (T - 9)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T + 14)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
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