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Properties

Label 144.10.a.q
Level $144$
Weight $10$
Character orbit 144.a
Self dual yes
Analytic conductor $74.165$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,10,Mod(1,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,736] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(74.1651604076\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{886}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 886 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 48\sqrt{886}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 368) q^{5} + (4 \beta + 1476) q^{7} + ( - 12 \beta - 3392) q^{11} + ( - 80 \beta - 29726) q^{13} + ( - 214 \beta - 115616) q^{17} + ( - 328 \beta - 125960) q^{19} + ( - 968 \beta + 26752) q^{23}+ \cdots + ( - 487616 \beta - 1018022386) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 736 q^{5} + 2952 q^{7} - 6784 q^{11} - 59452 q^{13} - 231232 q^{17} - 251920 q^{19} + 53504 q^{23} + 447286 q^{25} - 5094624 q^{29} + 1715336 q^{31} + 17417088 q^{35} - 4403796 q^{37} - 34452672 q^{41}+ \cdots - 2036044772 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−29.7658
29.7658
0 0 0 −1060.76 0 −4239.02 0 0 0
1.2 0 0 0 1796.76 0 7191.02 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 144.10.a.q 2
3.b odd 2 1 144.10.a.o 2
4.b odd 2 1 72.10.a.g yes 2
12.b even 2 1 72.10.a.f ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.10.a.f ✓ 2 12.b even 2 1
72.10.a.g yes 2 4.b odd 2 1
144.10.a.o 2 3.b odd 2 1
144.10.a.q 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 736T_{5} - 1905920 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(144))\). 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
$5$ \( T^{2} - 736 T - 1905920 \) Copy content Toggle raw display
$7$ \( T^{2} - 2952 T - 30482928 \) Copy content Toggle raw display
$11$ \( T^{2} + 6784 T - 282447872 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 12180966524 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 80118330368 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 203750031296 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 1912072650752 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 3544361798400 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 18579145268720 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 11895761192220 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 126138951060480 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 117062329135040 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 61\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 32\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 15\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 77\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 59\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 37\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 55\!\cdots\!32 \) Copy content Toggle raw display
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