Properties

Label 72.3.b.c
Level $72$
Weight $3$
Character orbit 72.b
Analytic conductor $1.962$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,3,Mod(19,72)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 72.b (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: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{2} + 1) q^{4} + (2 \beta_{3} + \beta_1) q^{5} + 2 \beta_{2} q^{7} + (2 \beta_{3} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{2} + 1) q^{4} + (2 \beta_{3} + \beta_1) q^{5} + 2 \beta_{2} q^{7} + (2 \beta_{3} + 2 \beta_1) q^{8} + ( - 2 \beta_{2} - 6) q^{10} + (4 \beta_{3} - 2 \beta_1) q^{11} + 4 \beta_{2} q^{13} + ( - 2 \beta_{3} - 4 \beta_1) q^{14} + ( - 2 \beta_{2} - 14) q^{16} + ( - 8 \beta_{3} + 4 \beta_1) q^{17} + 8 q^{19} + ( - 4 \beta_{3} + 4 \beta_1) q^{20} + ( - 4 \beta_{2} + 20) q^{22} + ( - 16 \beta_{3} - 8 \beta_1) q^{23} + q^{25} + ( - 4 \beta_{3} - 8 \beta_1) q^{26} + (2 \beta_{2} + 30) q^{28} + (10 \beta_{3} + 5 \beta_1) q^{29} - 2 \beta_{2} q^{31} + ( - 12 \beta_{3} + 4 \beta_1) q^{32} + (8 \beta_{2} - 40) q^{34} + (12 \beta_{3} - 6 \beta_1) q^{35} - 12 \beta_{2} q^{37} + 8 \beta_{3} q^{38} + (4 \beta_{2} - 36) q^{40} + ( - 8 \beta_{3} + 4 \beta_1) q^{41} - 40 q^{43} + (24 \beta_{3} + 8 \beta_1) q^{44} + (16 \beta_{2} + 48) q^{46} + (16 \beta_{3} + 8 \beta_1) q^{47} - 11 q^{49} + \beta_{3} q^{50} + (4 \beta_{2} + 60) q^{52} + (6 \beta_{3} + 3 \beta_1) q^{53} - 16 \beta_{2} q^{55} + (28 \beta_{3} - 4 \beta_1) q^{56} + ( - 10 \beta_{2} - 30) q^{58} + ( - 8 \beta_{3} + 4 \beta_1) q^{59} - 4 \beta_{2} q^{61} + (2 \beta_{3} + 4 \beta_1) q^{62} + (12 \beta_{2} - 44) q^{64} + (24 \beta_{3} - 12 \beta_1) q^{65} + 80 q^{67} + ( - 48 \beta_{3} - 16 \beta_1) q^{68} + ( - 12 \beta_{2} + 60) q^{70} - 10 q^{73} + (12 \beta_{3} + 24 \beta_1) q^{74} + ( - 8 \beta_{2} + 8) q^{76} + ( - 40 \beta_{3} - 20 \beta_1) q^{77} + 14 \beta_{2} q^{79} + ( - 40 \beta_{3} - 8 \beta_1) q^{80} + (8 \beta_{2} - 40) q^{82} + ( - 44 \beta_{3} + 22 \beta_1) q^{83} + 32 \beta_{2} q^{85} - 40 \beta_{3} q^{86} + ( - 24 \beta_{2} - 40) q^{88} + (16 \beta_{3} - 8 \beta_1) q^{89} - 120 q^{91} + (32 \beta_{3} - 32 \beta_1) q^{92} + ( - 16 \beta_{2} - 48) q^{94} + (16 \beta_{3} + 8 \beta_1) q^{95} + 50 q^{97} - 11 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 24 q^{10} - 56 q^{16} + 32 q^{19} + 80 q^{22} + 4 q^{25} + 120 q^{28} - 160 q^{34} - 144 q^{40} - 160 q^{43} + 192 q^{46} - 44 q^{49} + 240 q^{52} - 120 q^{58} - 176 q^{64} + 320 q^{67} + 240 q^{70} - 40 q^{73} + 32 q^{76} - 160 q^{82} - 160 q^{88} - 480 q^{91} - 192 q^{94} + 200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 2\nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} + \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.58114 1.22474i
1.58114 + 1.22474i
−1.58114 1.22474i
−1.58114 + 1.22474i
−1.58114 1.22474i 0 1.00000 + 3.87298i 4.89898i 0 7.74597i 3.16228 7.34847i 0 −6.00000 + 7.74597i
19.2 −1.58114 + 1.22474i 0 1.00000 3.87298i 4.89898i 0 7.74597i 3.16228 + 7.34847i 0 −6.00000 7.74597i
19.3 1.58114 1.22474i 0 1.00000 3.87298i 4.89898i 0 7.74597i −3.16228 7.34847i 0 −6.00000 7.74597i
19.4 1.58114 + 1.22474i 0 1.00000 + 3.87298i 4.89898i 0 7.74597i −3.16228 + 7.34847i 0 −6.00000 + 7.74597i
\(n\): 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
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.3.b.c 4
3.b odd 2 1 inner 72.3.b.c 4
4.b odd 2 1 288.3.b.c 4
8.b even 2 1 288.3.b.c 4
8.d odd 2 1 inner 72.3.b.c 4
12.b even 2 1 288.3.b.c 4
16.e even 4 2 2304.3.g.y 8
16.f odd 4 2 2304.3.g.y 8
24.f even 2 1 inner 72.3.b.c 4
24.h odd 2 1 288.3.b.c 4
48.i odd 4 2 2304.3.g.y 8
48.k even 4 2 2304.3.g.y 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.b.c 4 1.a even 1 1 trivial
72.3.b.c 4 3.b odd 2 1 inner
72.3.b.c 4 8.d odd 2 1 inner
72.3.b.c 4 24.f even 2 1 inner
288.3.b.c 4 4.b odd 2 1
288.3.b.c 4 8.b even 2 1
288.3.b.c 4 12.b even 2 1
288.3.b.c 4 24.h odd 2 1
2304.3.g.y 8 16.e even 4 2
2304.3.g.y 8 16.f odd 4 2
2304.3.g.y 8 48.i odd 4 2
2304.3.g.y 8 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 24 \) acting on \(S_{3}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 160)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 240)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 640)^{2} \) Copy content Toggle raw display
$19$ \( (T - 8)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1536)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2160)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 640)^{2} \) Copy content Toggle raw display
$43$ \( (T + 40)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 1536)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 640)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 240)^{2} \) Copy content Toggle raw display
$67$ \( (T - 80)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T + 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2940)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 19360)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 2560)^{2} \) Copy content Toggle raw display
$97$ \( (T - 50)^{4} \) Copy content Toggle raw display
show more
show less