↑

Properties

Label 1296.4.a.q
Level $1296$
Weight $4$
Character orbit 1296.a
Self dual yes
Analytic conductor $76.466$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1296,4,Mod(1,1296)] 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("1296.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,8,0,30] 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: \(76.4664753674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{201}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 648)
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 \(\beta = \sqrt{201}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 4) q^{5} + ( - \beta + 15) q^{7} + (3 \beta + 23) q^{11} + ( - \beta + 46) q^{13} + (8 \beta + 11) q^{17} + ( - 5 \beta + 43) q^{19} + (\beta + 29) q^{23} + (8 \beta + 92) q^{25} + ( - \beta + 54) q^{29}+ \cdots + (50 \beta + 236) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{5} + 30 q^{7} + 46 q^{11} + 92 q^{13} + 22 q^{17} + 86 q^{19} + 58 q^{23} + 184 q^{25} + 108 q^{29} - 136 q^{31} - 282 q^{35} - 180 q^{37} + 672 q^{41} + 70 q^{43} - 852 q^{47} + 166 q^{49} + 668 q^{53}+ \cdots + 472 q^{97}+O(q^{100}) \) 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
−6.58872
7.58872
0 0 0 −10.1774 0 29.1774 0 0 0
1.2 0 0 0 18.1774 0 0.822553 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 1296.4.a.q 2
3.b odd 2 1 1296.4.a.m 2
4.b odd 2 1 648.4.a.f yes 2
12.b even 2 1 648.4.a.c ✓ 2
36.f odd 6 2 648.4.i.n 4
36.h even 6 2 648.4.i.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.c ✓ 2 12.b even 2 1
648.4.a.f yes 2 4.b odd 2 1
648.4.i.n 4 36.f odd 6 2
648.4.i.t 4 36.h even 6 2
1296.4.a.m 2 3.b odd 2 1
1296.4.a.q 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 8T_{5} - 185 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\). 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
$5$ \( T^{2} - 8T - 185 \) Copy content Toggle raw display
$7$ \( T^{2} - 30T + 24 \) Copy content Toggle raw display
$11$ \( T^{2} - 46T - 1280 \) Copy content Toggle raw display
$13$ \( T^{2} - 92T + 1915 \) Copy content Toggle raw display
$17$ \( T^{2} - 22T - 12743 \) Copy content Toggle raw display
$19$ \( T^{2} - 86T - 3176 \) Copy content Toggle raw display
$23$ \( T^{2} - 58T + 640 \) Copy content Toggle raw display
$29$ \( T^{2} - 108T + 2715 \) Copy content Toggle raw display
$31$ \( T^{2} + 136T - 46832 \) Copy content Toggle raw display
$37$ \( T^{2} + 180T + 7899 \) Copy content Toggle raw display
$41$ \( T^{2} - 672T + 112092 \) Copy content Toggle raw display
$43$ \( T^{2} - 70T - 105104 \) Copy content Toggle raw display
$47$ \( T^{2} + 852T + 180672 \) Copy content Toggle raw display
$53$ \( T^{2} - 668T + 60100 \) Copy content Toggle raw display
$59$ \( T^{2} + 548T + 67840 \) Copy content Toggle raw display
$61$ \( T^{2} - 284T - 37925 \) Copy content Toggle raw display
$67$ \( T^{2} + 1162 T + 313240 \) Copy content Toggle raw display
$71$ \( T^{2} - 806T + 160600 \) Copy content Toggle raw display
$73$ \( T^{2} + 1510 T + 504901 \) Copy content Toggle raw display
$79$ \( T^{2} - 22T - 337760 \) Copy content Toggle raw display
$83$ \( T^{2} + 1540 T + 302656 \) Copy content Toggle raw display
$89$ \( (T - 1323)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 472T - 446804 \) Copy content Toggle raw display
show more
show less