Properties

Label 72.3.e.a
Level $72$
Weight $3$
Character orbit 72.e
Analytic conductor $1.962$
Analytic rank $0$
Dimension $2$
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] = [72,3,Mod(17,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 72.e (of order \(2\), degree \(1\), 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: \(1.96185790339\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \beta q^{5} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \beta q^{5} + 12 q^{7} - 4 \beta q^{11} - 8 q^{13} + 7 \beta q^{17} - 16 q^{19} - 28 \beta q^{23} - 25 q^{25} - 21 \beta q^{29} - 4 q^{31} + 60 \beta q^{35} + 30 q^{37} - 15 \beta q^{41} - 8 q^{43} - 12 \beta q^{47} + 95 q^{49} + 35 \beta q^{53} + 40 q^{55} + 56 \beta q^{59} - 14 q^{61} - 40 \beta q^{65} - 88 q^{67} - 20 \beta q^{71} - 80 q^{73} - 48 \beta q^{77} + 100 q^{79} - 92 \beta q^{83} - 70 q^{85} + 105 \beta q^{89} - 96 q^{91} - 80 \beta q^{95} - 112 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{7} - 16 q^{13} - 32 q^{19} - 50 q^{25} - 8 q^{31} + 60 q^{37} - 16 q^{43} + 190 q^{49} + 80 q^{55} - 28 q^{61} - 176 q^{67} - 160 q^{73} + 200 q^{79} - 140 q^{85} - 192 q^{91} - 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.41421i
1.41421i
0 0 0 7.07107i 0 12.0000 0 0 0
17.2 0 0 0 7.07107i 0 12.0000 0 0 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.3.e.a 2
3.b odd 2 1 inner 72.3.e.a 2
4.b odd 2 1 144.3.e.a 2
5.b even 2 1 1800.3.l.a 2
5.c odd 4 2 1800.3.c.a 4
7.b odd 2 1 3528.3.d.a 2
8.b even 2 1 576.3.e.h 2
8.d odd 2 1 576.3.e.a 2
9.c even 3 2 648.3.m.a 4
9.d odd 6 2 648.3.m.a 4
12.b even 2 1 144.3.e.a 2
15.d odd 2 1 1800.3.l.a 2
15.e even 4 2 1800.3.c.a 4
16.e even 4 2 2304.3.h.a 4
16.f odd 4 2 2304.3.h.h 4
20.d odd 2 1 3600.3.l.l 2
20.e even 4 2 3600.3.c.c 4
21.c even 2 1 3528.3.d.a 2
24.f even 2 1 576.3.e.a 2
24.h odd 2 1 576.3.e.h 2
36.f odd 6 2 1296.3.q.k 4
36.h even 6 2 1296.3.q.k 4
48.i odd 4 2 2304.3.h.a 4
48.k even 4 2 2304.3.h.h 4
60.h even 2 1 3600.3.l.l 2
60.l odd 4 2 3600.3.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.e.a 2 1.a even 1 1 trivial
72.3.e.a 2 3.b odd 2 1 inner
144.3.e.a 2 4.b odd 2 1
144.3.e.a 2 12.b even 2 1
576.3.e.a 2 8.d odd 2 1
576.3.e.a 2 24.f even 2 1
576.3.e.h 2 8.b even 2 1
576.3.e.h 2 24.h odd 2 1
648.3.m.a 4 9.c even 3 2
648.3.m.a 4 9.d odd 6 2
1296.3.q.k 4 36.f odd 6 2
1296.3.q.k 4 36.h even 6 2
1800.3.c.a 4 5.c odd 4 2
1800.3.c.a 4 15.e even 4 2
1800.3.l.a 2 5.b even 2 1
1800.3.l.a 2 15.d odd 2 1
2304.3.h.a 4 16.e even 4 2
2304.3.h.a 4 48.i odd 4 2
2304.3.h.h 4 16.f odd 4 2
2304.3.h.h 4 48.k even 4 2
3528.3.d.a 2 7.b odd 2 1
3528.3.d.a 2 21.c even 2 1
3600.3.c.c 4 20.e even 4 2
3600.3.c.c 4 60.l odd 4 2
3600.3.l.l 2 20.d odd 2 1
3600.3.l.l 2 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 50 \) Copy content Toggle raw display
$7$ \( (T - 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 32 \) Copy content Toggle raw display
$13$ \( (T + 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 98 \) Copy content Toggle raw display
$19$ \( (T + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1568 \) Copy content Toggle raw display
$29$ \( T^{2} + 882 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T - 30)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 450 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 288 \) Copy content Toggle raw display
$53$ \( T^{2} + 2450 \) Copy content Toggle raw display
$59$ \( T^{2} + 6272 \) Copy content Toggle raw display
$61$ \( (T + 14)^{2} \) Copy content Toggle raw display
$67$ \( (T + 88)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 800 \) Copy content Toggle raw display
$73$ \( (T + 80)^{2} \) Copy content Toggle raw display
$79$ \( (T - 100)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16928 \) Copy content Toggle raw display
$89$ \( T^{2} + 22050 \) Copy content Toggle raw display
$97$ \( (T + 112)^{2} \) Copy content Toggle raw display
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