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Properties

Label 720.6.a.ba
Level $720$
Weight $6$
Character orbit 720.a
Self dual yes
Analytic conductor $115.476$
Analytic rank $0$
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] = [720,6,Mod(1,720)] 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("720.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-50,0,8,0,0,0,488] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(115.476350265\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2161}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 540 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 120)
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 \(\beta = 4\sqrt{2161}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 25 q^{5} + (\beta + 4) q^{7} + (2 \beta + 244) q^{11} + (3 \beta + 306) q^{13} + ( - \beta - 430) q^{17} + (11 \beta + 48) q^{19} + (13 \beta + 1492) q^{23} + 625 q^{25} + (20 \beta - 1118) q^{29}+ \cdots + (468 \beta - 19630) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 50 q^{5} + 8 q^{7} + 488 q^{11} + 612 q^{13} - 860 q^{17} + 96 q^{19} + 2984 q^{23} + 1250 q^{25} - 2236 q^{29} - 9352 q^{31} - 200 q^{35} + 10612 q^{37} - 17156 q^{41} - 440 q^{43} - 16728 q^{47}+ \cdots - 39260 q^{97}+O(q^{100}) \) 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
−22.7433
23.7433
0 0 0 −25.0000 0 −181.946 0 0 0
1.2 0 0 0 −25.0000 0 189.946 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(5\) \( +1 \)

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 720.6.a.ba 2
3.b odd 2 1 240.6.a.p 2
4.b odd 2 1 360.6.a.k 2
12.b even 2 1 120.6.a.h ✓ 2
24.f even 2 1 960.6.a.be 2
24.h odd 2 1 960.6.a.bk 2
60.h even 2 1 600.6.a.l 2
60.l odd 4 2 600.6.f.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.6.a.h ✓ 2 12.b even 2 1
240.6.a.p 2 3.b odd 2 1
360.6.a.k 2 4.b odd 2 1
600.6.a.l 2 60.h even 2 1
600.6.f.m 4 60.l odd 4 2
720.6.a.ba 2 1.a even 1 1 trivial
960.6.a.be 2 24.f even 2 1
960.6.a.bk 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7}^{2} - 8T_{7} - 34560 \) Copy content Toggle raw display
\( T_{11}^{2} - 488T_{11} - 78768 \) 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
$5$ \( (T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8T - 34560 \) Copy content Toggle raw display
$11$ \( T^{2} - 488T - 78768 \) Copy content Toggle raw display
$13$ \( T^{2} - 612T - 217548 \) Copy content Toggle raw display
$17$ \( T^{2} + 860T + 150324 \) Copy content Toggle raw display
$19$ \( T^{2} - 96T - 4181392 \) Copy content Toggle raw display
$23$ \( T^{2} - 2984 T - 3617280 \) Copy content Toggle raw display
$29$ \( T^{2} + 2236 T - 12580476 \) Copy content Toggle raw display
$31$ \( T^{2} + 9352 T - 11362560 \) Copy content Toggle raw display
$37$ \( T^{2} - 10612 T - 5073900 \) Copy content Toggle raw display
$41$ \( T^{2} + 17156 T + 66805188 \) Copy content Toggle raw display
$43$ \( T^{2} + 440 T - 292602864 \) Copy content Toggle raw display
$47$ \( T^{2} + 16728 T + 40878080 \) Copy content Toggle raw display
$53$ \( T^{2} + 31484 T + 15321540 \) Copy content Toggle raw display
$59$ \( T^{2} + 61464 T + 843632208 \) Copy content Toggle raw display
$61$ \( T^{2} - 51596 T + 360023268 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 1210108912 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 2265987648 \) Copy content Toggle raw display
$73$ \( T^{2} - 58852 T + 281555076 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 2779010944 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 4662096912 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 2197457860 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 7187636924 \) Copy content Toggle raw display
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