Properties

Label 72.5.e.a
Level $72$
Weight $5$
Character orbit 72.e
Analytic conductor $7.443$
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] = [72,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.17");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(7.44263734204\)
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: \(\Z[a_1, \ldots, a_{23}]\)
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.

\(f(q)\) \(=\) \( q + 11 \beta q^{5} - 60 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 11 \beta q^{5} - 60 q^{7} + 44 \beta q^{11} - 176 q^{13} + 385 \beta q^{17} - 592 q^{19} - 364 \beta q^{23} + 383 q^{25} + 357 \beta q^{29} - 268 q^{31} - 660 \beta q^{35} + 2118 q^{37} - 321 \beta q^{41} + 1672 q^{43} - 2748 \beta q^{47} + 1199 q^{49} + 917 \beta q^{53} - 968 q^{55} + 2744 \beta q^{59} - 1862 q^{61} - 1936 \beta q^{65} + 2168 q^{67} + 4444 \beta q^{71} - 6704 q^{73} - 2640 \beta q^{77} - 9236 q^{79} + 6196 \beta q^{83} - 8470 q^{85} + 5775 \beta q^{89} + 10560 q^{91} - 6512 \beta q^{95} + 2240 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 120 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 120 q^{7} - 352 q^{13} - 1184 q^{19} + 766 q^{25} - 536 q^{31} + 4236 q^{37} + 3344 q^{43} + 2398 q^{49} - 1936 q^{55} - 3724 q^{61} + 4336 q^{67} - 13408 q^{73} - 18472 q^{79} - 16940 q^{85} + 21120 q^{91} + 4480 q^{97}+O(q^{100}) \) 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 15.5563i 0 −60.0000 0 0 0
17.2 0 0 0 15.5563i 0 −60.0000 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 72.5.e.a 2
3.b odd 2 1 inner 72.5.e.a 2
4.b odd 2 1 144.5.e.d 2
8.b even 2 1 576.5.e.b 2
8.d odd 2 1 576.5.e.i 2
9.c even 3 2 648.5.m.b 4
9.d odd 6 2 648.5.m.b 4
12.b even 2 1 144.5.e.d 2
24.f even 2 1 576.5.e.i 2
24.h odd 2 1 576.5.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.5.e.a 2 1.a even 1 1 trivial
72.5.e.a 2 3.b odd 2 1 inner
144.5.e.d 2 4.b odd 2 1
144.5.e.d 2 12.b even 2 1
576.5.e.b 2 8.b even 2 1
576.5.e.b 2 24.h odd 2 1
576.5.e.i 2 8.d odd 2 1
576.5.e.i 2 24.f even 2 1
648.5.m.b 4 9.c even 3 2
648.5.m.b 4 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 242 \) acting on \(S_{5}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 242 \) Copy content Toggle raw display
$7$ \( (T + 60)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 3872 \) Copy content Toggle raw display
$13$ \( (T + 176)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 296450 \) Copy content Toggle raw display
$19$ \( (T + 592)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 264992 \) Copy content Toggle raw display
$29$ \( T^{2} + 254898 \) Copy content Toggle raw display
$31$ \( (T + 268)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2118)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 206082 \) Copy content Toggle raw display
$43$ \( (T - 1672)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 15103008 \) Copy content Toggle raw display
$53$ \( T^{2} + 1681778 \) Copy content Toggle raw display
$59$ \( T^{2} + 15059072 \) Copy content Toggle raw display
$61$ \( (T + 1862)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2168)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 39498272 \) Copy content Toggle raw display
$73$ \( (T + 6704)^{2} \) Copy content Toggle raw display
$79$ \( (T + 9236)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 76780832 \) Copy content Toggle raw display
$89$ \( T^{2} + 66701250 \) Copy content Toggle raw display
$97$ \( (T - 2240)^{2} \) Copy content Toggle raw display
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