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Properties

Label 624.6.a.j
Level $624$
Weight $6$
Character orbit 624.a
Self dual yes
Analytic conductor $100.080$
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] = [624,6,Mod(1,624)] 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("624.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 624.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-18,0,10,0,-138] 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: \(100.079503563\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3241}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 810 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 78)
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 \(\beta = \sqrt{3241}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 q^{3} + ( - \beta + 5) q^{5} + ( - \beta - 69) q^{7} + 81 q^{9} + ( - 8 \beta - 272) q^{11} + 169 q^{13} + (9 \beta - 45) q^{15} + (20 \beta + 614) q^{17} + (29 \beta + 261) q^{19} + (9 \beta + 621) q^{21}+ \cdots + ( - 648 \beta - 22032) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{3} + 10 q^{5} - 138 q^{7} + 162 q^{9} - 544 q^{11} + 338 q^{13} - 90 q^{15} + 1228 q^{17} + 522 q^{19} + 1242 q^{21} + 1632 q^{23} + 282 q^{25} - 1458 q^{27} + 2208 q^{29} - 874 q^{31} + 4896 q^{33}+ \cdots - 44064 q^{99}+O(q^{100}) \) 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
28.9649
−27.9649
0 −9.00000 0 −51.9298 0 −125.930 0 81.0000 0
1.2 0 −9.00000 0 61.9298 0 −12.0702 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(13\) \( -1 \)

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 624.6.a.j 2
4.b odd 2 1 78.6.a.h ✓ 2
12.b even 2 1 234.6.a.i 2
52.b odd 2 1 1014.6.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.6.a.h ✓ 2 4.b odd 2 1
234.6.a.i 2 12.b even 2 1
624.6.a.j 2 1.a even 1 1 trivial
1014.6.a.i 2 52.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(624))\):

\( T_{5}^{2} - 10T_{5} - 3216 \) Copy content Toggle raw display
\( T_{7}^{2} + 138T_{7} + 1520 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 10T - 3216 \) Copy content Toggle raw display
$7$ \( T^{2} + 138T + 1520 \) Copy content Toggle raw display
$11$ \( T^{2} + 544T - 133440 \) Copy content Toggle raw display
$13$ \( (T - 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1228 T - 919404 \) Copy content Toggle raw display
$19$ \( T^{2} - 522 T - 2657560 \) Copy content Toggle raw display
$23$ \( T^{2} - 1632 T - 6801408 \) Copy content Toggle raw display
$29$ \( T^{2} - 2208 T - 12898980 \) Copy content Toggle raw display
$31$ \( T^{2} + 874 T - 3338480 \) Copy content Toggle raw display
$37$ \( T^{2} + 2160 T - 2580196 \) Copy content Toggle raw display
$41$ \( T^{2} + 12638 T - 52636440 \) Copy content Toggle raw display
$43$ \( T^{2} - 17760 T - 8315536 \) Copy content Toggle raw display
$47$ \( T^{2} + 4300 T - 540189600 \) Copy content Toggle raw display
$53$ \( T^{2} + 12020 T + 35290404 \) Copy content Toggle raw display
$59$ \( T^{2} - 27532 T - 19593600 \) Copy content Toggle raw display
$61$ \( T^{2} + 38016 T - 941435332 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1816784488 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 1055598480 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 2256976540 \) Copy content Toggle raw display
$79$ \( T^{2} - 62168 T - 64734080 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 2367643392 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 2883762000 \) Copy content Toggle raw display
$97$ \( T^{2} - 4896 T - 890714212 \) Copy content Toggle raw display
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