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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 = [2,0,0,0,18340,0,680400] 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: \(\Q(\sqrt{22}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 24)
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 \(\beta = 1920\sqrt{22}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (19 \beta + 9170) q^{5} + ( - 203 \beta + 340200) q^{7} + (8646 \beta - 38860316) q^{11} + ( - 11546 \beta - 103940666) q^{13} + (200918 \beta + 836656846) q^{17} + ( - 653134 \beta + 1120443556) q^{19}+ \cdots + ( - 58194785300 \beta + 10\!\cdots\!74) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18340 q^{5} + 680400 q^{7} - 77720632 q^{11} - 207881332 q^{13} + 1673313692 q^{17} + 2240887112 q^{19} + 16655801552 q^{23} - 2312200850 q^{25} + 139493975892 q^{29} + 43133623424 q^{31} - 619372303200 q^{35}+ \cdots + 20\!\cdots\!48 q^{97}+O(q^{100}) \) 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
−4.69042
4.69042
0 0 0 −161936. 0 2.16834e6 0 0 0
1.2 0 0 0 180276. 0 −1.48794e6 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.u 2
3.b odd 2 1 48.16.a.h 2
4.b odd 2 1 72.16.a.f 2
12.b even 2 1 24.16.a.c ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.16.a.c ✓ 2 12.b even 2 1
48.16.a.h 2 3.b odd 2 1
72.16.a.f 2 4.b odd 2 1
144.16.a.u 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 18340T_{5} - 29193299900 \) 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
$5$ \( T^{2} + \cdots - 29193299900 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 3226346827200 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 45\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 7895007169244 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 25\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 33\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 38\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 62\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 58\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 60\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 11\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 94\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 30\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 59\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 60\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 75\!\cdots\!76 \) Copy content Toggle raw display
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