Properties

Label 72.22.a.a
Level $72$
Weight $22$
Character orbit 72.a
Self dual yes
Analytic conductor $201.224$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,22,Mod(1,72)]
 
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, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 72.a (trivial)

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: yes
Analytic conductor: \(201.223687887\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{537541}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 134385 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 4032\sqrt{537541}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 5 \beta - 10974310) q^{5} + (233 \beta - 329725704) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \beta - 10974310) q^{5} + (233 \beta - 329725704) q^{7} + (41046 \beta + 14955448436) q^{11} + ( - 133514 \beta + 2234433406) q^{13} + (1743998 \beta + 8832702360910) q^{17} + ( - 1299182 \beta - 11233989648748) q^{19} + (50101454 \beta - 60497636524792) q^{23} + (109743100 \beta - 137931254777425) q^{25} + (187750985 \beta + 16\!\cdots\!42) q^{29}+ \cdots + ( - 107227390371524 \beta - 60\!\cdots\!10) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 21948620 q^{5} - 659451408 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 21948620 q^{5} - 659451408 q^{7} + 29910896872 q^{11} + 4468866812 q^{13} + 17665404721820 q^{17} - 22467979297496 q^{19} - 120995273049584 q^{23} - 275862509554850 q^{25} + 33\!\cdots\!84 q^{29}+ \cdots - 12\!\cdots\!20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
367.086
−366.086
0 0 0 −2.57551e7 0 3.59057e8 0 0 0
1.2 0 0 0 3.80644e6 0 −1.01851e9 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.22.a.a 2
3.b odd 2 1 24.22.a.a 2
12.b even 2 1 48.22.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.a.a 2 3.b odd 2 1
48.22.a.h 2 12.b even 2 1
72.22.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 21948620T_{5} - 98034943473500 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(72))\). 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
$5$ \( T^{2} + 21948620 T - 98034943473500 \) Copy content Toggle raw display
$7$ \( T^{2} + 659451408 T - 36\!\cdots\!60 \) Copy content Toggle raw display
$11$ \( T^{2} - 29910896872 T - 14\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{2} - 4468866812 T - 15\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{2} - 17665404721820 T + 51\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{2} + 22467979297496 T + 11\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{2} + 120995273049584 T - 18\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 70\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 40\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 29\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 91\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 26\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 12\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 43\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 32\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 86\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 28\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 21\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 66\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
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