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Properties

Label 72.2.i.b
Level $72$
Weight $2$
Character orbit 72.i
Analytic conductor $0.575$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,2,Mod(25,72)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.i (of order \(3\), 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{5} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{7} + (\beta_{3} + 3 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{5} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{7} + (\beta_{3} + 3 \beta_{2}) q^{9} - \beta_{2} q^{11} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{13} + (3 \beta_{2} + \beta_1 - 6) q^{15} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{19} + (\beta_{3} - 6 \beta_{2} - 2 \beta_1 + 3) q^{21} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{23} + ( - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{25} + ( - 2 \beta_{3} + 2 \beta_1 + 3) q^{27} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{29} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{31} + (\beta_{3} - \beta_1) q^{33} + (\beta_{3} + \beta_{2} + \beta_1 + 7) q^{35} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{37} + ( - 2 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{39} + (4 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 7) q^{41} + ( - 2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 2) q^{43} + ( - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{45} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{47} + ( - 6 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 5) q^{49} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 3) q^{51} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 6) q^{53} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{55} + ( - \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 3) q^{57} + (7 \beta_{2} - 7) q^{59} + (\beta_{3} - 2 \beta_1 + 1) q^{61} + (5 \beta_{3} - 6 \beta_{2} - 4 \beta_1 + 3) q^{63} + (3 \beta_{3} + 8 \beta_{2} - 6 \beta_1 + 3) q^{65} + (4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1) q^{67} + ( - 2 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{69} - 4 q^{71} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 2) q^{73} + (4 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 3) q^{75} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{77} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 1) q^{79} + (6 \beta_{2} + 5 \beta_1 - 6) q^{81} + ( - \beta_{3} - 12 \beta_{2} + 2 \beta_1 - 1) q^{83} + (4 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 8) q^{85} + ( - \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 3) q^{87} + 6 q^{89} + (\beta_{3} + \beta_{2} + \beta_1 - 5) q^{91} + ( - 4 \beta_{3} + 3 \beta_{2} + \beta_1 - 6) q^{93} + ( - 8 \beta_{3} - 12 \beta_{2} + 4 \beta_1 + 8) q^{95} + (2 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 2) q^{97} + ( - 3 \beta_{2} - \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{3} + q^{5} - 3 q^{7} + 5 q^{9} - 2 q^{11} - 5 q^{13} - 17 q^{15} - 10 q^{17} + 14 q^{19} - 3 q^{21} - 5 q^{23} - 7 q^{25} + 16 q^{27} + 3 q^{29} - 7 q^{31} - 2 q^{33} + 30 q^{35} + 12 q^{37} + 19 q^{39} + 12 q^{41} - 8 q^{43} + 5 q^{45} - 3 q^{47} - 7 q^{49} - 19 q^{51} - 20 q^{53} - 2 q^{55} - 13 q^{57} - 14 q^{59} + q^{61} - 9 q^{63} + 19 q^{65} - 4 q^{67} + 19 q^{69} - 16 q^{71} - 14 q^{73} - 7 q^{75} - 3 q^{77} + 7 q^{79} - 7 q^{81} - 25 q^{83} + 14 q^{85} + 3 q^{87} + 24 q^{89} - 18 q^{91} - 13 q^{93} + 20 q^{95} + 8 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−1.18614 + 1.26217i
1.68614 − 0.396143i
−1.18614 − 1.26217i
1.68614 + 0.396143i
0 −1.18614 + 1.26217i 0 1.68614 + 2.92048i 0 0.686141 − 1.18843i 0 −0.186141 − 2.99422i 0
25.2 0 1.68614 − 0.396143i 0 −1.18614 − 2.05446i 0 −2.18614 + 3.78651i 0 2.68614 − 1.33591i 0
49.1 0 −1.18614 − 1.26217i 0 1.68614 − 2.92048i 0 0.686141 + 1.18843i 0 −0.186141 + 2.99422i 0
49.2 0 1.68614 + 0.396143i 0 −1.18614 + 2.05446i 0 −2.18614 − 3.78651i 0 2.68614 + 1.33591i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.i.b ✓ 4
3.b odd 2 1 216.2.i.b 4
4.b odd 2 1 144.2.i.d 4
8.b even 2 1 576.2.i.j 4
8.d odd 2 1 576.2.i.l 4
9.c even 3 1 inner 72.2.i.b ✓ 4
9.c even 3 1 648.2.a.f 2
9.d odd 6 1 216.2.i.b 4
9.d odd 6 1 648.2.a.g 2
12.b even 2 1 432.2.i.d 4
24.f even 2 1 1728.2.i.j 4
24.h odd 2 1 1728.2.i.i 4
36.f odd 6 1 144.2.i.d 4
36.f odd 6 1 1296.2.a.n 2
36.h even 6 1 432.2.i.d 4
36.h even 6 1 1296.2.a.p 2
72.j odd 6 1 1728.2.i.i 4
72.j odd 6 1 5184.2.a.bp 2
72.l even 6 1 1728.2.i.j 4
72.l even 6 1 5184.2.a.bo 2
72.n even 6 1 576.2.i.j 4
72.n even 6 1 5184.2.a.bt 2
72.p odd 6 1 576.2.i.l 4
72.p odd 6 1 5184.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.b ✓ 4 1.a even 1 1 trivial
72.2.i.b ✓ 4 9.c even 3 1 inner
144.2.i.d 4 4.b odd 2 1
144.2.i.d 4 36.f odd 6 1
216.2.i.b 4 3.b odd 2 1
216.2.i.b 4 9.d odd 6 1
432.2.i.d 4 12.b even 2 1
432.2.i.d 4 36.h even 6 1
576.2.i.j 4 8.b even 2 1
576.2.i.j 4 72.n even 6 1
576.2.i.l 4 8.d odd 2 1
576.2.i.l 4 72.p odd 6 1
648.2.a.f 2 9.c even 3 1
648.2.a.g 2 9.d odd 6 1
1296.2.a.n 2 36.f odd 6 1
1296.2.a.p 2 36.h even 6 1
1728.2.i.i 4 24.h odd 2 1
1728.2.i.i 4 72.j odd 6 1
1728.2.i.j 4 24.f even 2 1
1728.2.i.j 4 72.l even 6 1
5184.2.a.bo 2 72.l even 6 1
5184.2.a.bp 2 72.j odd 6 1
5184.2.a.bs 2 72.p odd 6 1
5184.2.a.bt 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - T_{5}^{3} + 9T_{5}^{2} + 8T_{5} + 64 \) acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 2 T^{2} - 3 T + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + 9 T^{2} + 8 T + 64 \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 5 T^{3} + 27 T^{2} - 10 T + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 5 T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 5 T^{3} + 27 T^{2} - 10 T + 4 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36 \) Copy content Toggle raw display
$31$ \( T^{4} + 7 T^{3} + 45 T^{2} + 28 T + 16 \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 141 T^{2} - 36 T + 9 \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} + 81 T^{2} - 136 T + 289 \) Copy content Toggle raw display
$47$ \( T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36 \) Copy content Toggle raw display
$53$ \( (T^{2} + 10 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - T^{3} + 9 T^{2} + 8 T + 64 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + 45 T^{2} - 116 T + 841 \) Copy content Toggle raw display
$71$ \( (T + 4)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 7 T - 62)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 7 T^{3} + 45 T^{2} - 28 T + 16 \) Copy content Toggle raw display
$83$ \( T^{4} + 25 T^{3} + 477 T^{2} + \cdots + 21904 \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} + 81 T^{2} + 136 T + 289 \) Copy content Toggle raw display
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