Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,16,Mod(1,144)] 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("144.1"); S:= CuspForms(chi, 16); 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, 0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,140260,0,-126192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(205.478647344\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{58}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 8)
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 \(\beta = 5760\sqrt{58}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \beta + 70130) q^{5} + (10 \beta - 63096) q^{7} + ( - 1923 \beta + 10341316) q^{11} + ( - 260 \beta + 249903238) q^{13} + (19544 \beta - 1569758450) q^{17} + (31757 \beta + 237334276) q^{19} + (45250 \beta - 20388001304) q^{23}+ \cdots + (28846490824 \beta + 203817795209378) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 140260 q^{5} - 126192 q^{7} + 20682632 q^{11} + 499806476 q^{13} - 3139516900 q^{17} + 474668552 q^{19} - 40776002608 q^{23} + 10378903150 q^{25} - 35253157356 q^{29} + 34389193280 q^{31} + 145094219040 q^{35}+ \cdots + 407635590418756 q^{97}+O(q^{100}) \) 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
−7.61577
7.61577
0 0 0 −105337. 0 −501765. 0 0 0
1.2 0 0 0 245597. 0 375573. 0 0 0
\(n\): e.g. 2-40 or 80-90
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 144.16.a.v 2
3.b odd 2 1 16.16.a.f 2
4.b odd 2 1 72.16.a.g 2
12.b even 2 1 8.16.a.c ✓ 2
24.f even 2 1 64.16.a.n 2
24.h odd 2 1 64.16.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.a.c ✓ 2 12.b even 2 1
16.16.a.f 2 3.b odd 2 1
64.16.a.l 2 24.h odd 2 1
64.16.a.n 2 24.f even 2 1
72.16.a.g 2 4.b odd 2 1
144.16.a.v 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 140260T_{5} - 25870595900 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(144))\). 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
$5$ \( T^{2} + \cdots - 25870595900 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 188448974784 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 70\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 18\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 87\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 88\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 33\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 33\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 26\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 31\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 73\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 15\!\cdots\!16 \) Copy content Toggle raw display
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