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Properties

Label 504.6.a.t
Level $504$
Weight $6$
Character orbit 504.a
Self dual yes
Analytic conductor $80.833$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,6,Mod(1,504)] 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("504.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,64,0,98] 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: \(80.8334451857\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{249}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 168)
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 \(\beta = 2\sqrt{249}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 32) q^{5} + 49 q^{7} + (15 \beta - 270) q^{11} + ( - 16 \beta + 102) q^{13} + ( - 51 \beta - 152) q^{17} + (74 \beta - 560) q^{19} + (71 \beta + 470) q^{23} + ( - 64 \beta - 1105) q^{25}+ \cdots + (2446 \beta - 54194) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{5} + 98 q^{7} - 540 q^{11} + 204 q^{13} - 304 q^{17} - 1120 q^{19} + 940 q^{23} - 2210 q^{25} - 932 q^{29} - 16408 q^{31} + 3136 q^{35} - 1764 q^{37} + 2552 q^{41} - 24632 q^{43} + 36760 q^{47}+ \cdots - 108388 q^{97}+O(q^{100}) \) 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
8.38987
−7.38987
0 0 0 0.440532 0 49.0000 0 0 0
1.2 0 0 0 63.5595 0 49.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

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 504.6.a.t 2
3.b odd 2 1 168.6.a.h ✓ 2
4.b odd 2 1 1008.6.a.bu 2
12.b even 2 1 336.6.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.a.h ✓ 2 3.b odd 2 1
336.6.a.w 2 12.b even 2 1
504.6.a.t 2 1.a even 1 1 trivial
1008.6.a.bu 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(504))\):

\( T_{5}^{2} - 64T_{5} + 28 \) Copy content Toggle raw display
\( T_{11}^{2} + 540T_{11} - 151200 \) 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
$5$ \( T^{2} - 64T + 28 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 540T - 151200 \) Copy content Toggle raw display
$13$ \( T^{2} - 204T - 244572 \) Copy content Toggle raw display
$17$ \( T^{2} + 304 T - 2567492 \) Copy content Toggle raw display
$19$ \( T^{2} + 1120 T - 5140496 \) Copy content Toggle raw display
$23$ \( T^{2} - 940 T - 4799936 \) Copy content Toggle raw display
$29$ \( T^{2} + 932T + 117556 \) Copy content Toggle raw display
$31$ \( T^{2} + 16408 T + 65198080 \) Copy content Toggle raw display
$37$ \( T^{2} + 1764 T - 21038460 \) Copy content Toggle raw display
$41$ \( T^{2} - 2552 T - 279204980 \) Copy content Toggle raw display
$43$ \( T^{2} + 24632 T + 150902992 \) Copy content Toggle raw display
$47$ \( T^{2} - 36760 T + 269454976 \) Copy content Toggle raw display
$53$ \( T^{2} + 9164 T - 309454172 \) Copy content Toggle raw display
$59$ \( T^{2} - 39888 T - 65500368 \) Copy content Toggle raw display
$61$ \( T^{2} + 25084 T - 361897100 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 2233348368 \) Copy content Toggle raw display
$71$ \( T^{2} - 35508 T - 616144128 \) Copy content Toggle raw display
$73$ \( T^{2} + 17188 T - 277567820 \) Copy content Toggle raw display
$79$ \( T^{2} + 95800 T + 758016256 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 2407745328 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 1901622836 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 3021994700 \) Copy content Toggle raw display
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