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Properties

Label 144.9.e.d
Level $144$
Weight $9$
Character orbit 144.e
Analytic conductor $58.663$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,9,Mod(17,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.17"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 144.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,7064,0,0,0,0,0,-83648] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(58.6625198488\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content 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: \( 3 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 55 \beta q^{5} + 3532 q^{7} + 4756 \beta q^{11} - 41824 q^{13} + 22341 \beta q^{17} + 36304 q^{19} - 97492 \beta q^{23} + 336175 q^{25} + 63449 \beta q^{29} + 471196 q^{31} + 194260 \beta q^{35} - 3007402 q^{37} + \cdots + 20431328 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7064 q^{7} - 83648 q^{13} + 72608 q^{19} + 672350 q^{25} + 942392 q^{31} - 6014804 q^{37} - 7247440 q^{43} + 13420446 q^{49} - 9416880 q^{55} - 10881260 q^{61} + 12243152 q^{67} - 98062304 q^{73}+ \cdots + 40862656 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 − 233.345i 0 3532.00 0 0 0
17.2 0 0 0 233.345i 0 3532.00 0 0 0
\(n\): e.g. 2-40 or 80-90
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 144.9.e.d 2
3.b odd 2 1 inner 144.9.e.d 2
4.b odd 2 1 18.9.b.a ✓ 2
12.b even 2 1 18.9.b.a ✓ 2
20.d odd 2 1 450.9.d.b 2
20.e even 4 2 450.9.b.a 4
36.f odd 6 2 162.9.d.d 4
36.h even 6 2 162.9.d.d 4
60.h even 2 1 450.9.d.b 2
60.l odd 4 2 450.9.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.b.a ✓ 2 4.b odd 2 1
18.9.b.a ✓ 2 12.b even 2 1
144.9.e.d 2 1.a even 1 1 trivial
144.9.e.d 2 3.b odd 2 1 inner
162.9.d.d 4 36.f odd 6 2
162.9.d.d 4 36.h even 6 2
450.9.b.a 4 20.e even 4 2
450.9.b.a 4 60.l odd 4 2
450.9.d.b 2 20.d odd 2 1
450.9.d.b 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 54450 \) acting on \(S_{9}^{\mathrm{new}}(144, [\chi])\). 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
$5$ \( T^{2} + 54450 \) Copy content Toggle raw display
$7$ \( (T - 3532)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 407151648 \) Copy content Toggle raw display
$13$ \( (T + 41824)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 8984165058 \) Copy content Toggle raw display
$19$ \( (T - 36304)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 171084421152 \) Copy content Toggle raw display
$29$ \( T^{2} + 72463960818 \) Copy content Toggle raw display
$31$ \( (T - 471196)^{2} \) Copy content Toggle raw display
$37$ \( (T + 3007402)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2942383814658 \) Copy content Toggle raw display
$43$ \( (T + 3623720)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 36175677760800 \) Copy content Toggle raw display
$53$ \( T^{2} + 105609828713202 \) Copy content Toggle raw display
$59$ \( T^{2} + 7225898804352 \) Copy content Toggle raw display
$61$ \( (T + 5440630)^{2} \) Copy content Toggle raw display
$67$ \( (T - 6121576)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 449188925797152 \) Copy content Toggle raw display
$73$ \( (T + 49031152)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8357756)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 26\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{2} + 11\!\cdots\!22 \) Copy content Toggle raw display
$97$ \( (T - 20431328)^{2} \) Copy content Toggle raw display
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