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Properties

Label 800.6.a.y
Level $800$
Weight $6$
Character orbit 800.a
Self dual yes
Analytic conductor $128.307$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [800,6,Mod(1,800)] 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("800.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,228,0,0,0,128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 207x^{4} + 419x^{3} + 5927x^{2} - 6137x - 8328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 5^{2} \)
Twist minimal: yes
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} + \beta_1) q^{7} + (\beta_{4} + 38) q^{9} + ( - \beta_{3} - 3 \beta_1) q^{11} + (\beta_{5} - \beta_{4} + 21) q^{13} + (2 \beta_{5} + \beta_{4} + 135) q^{17} + ( - 2 \beta_{3} + 8 \beta_{2} - 19 \beta_1) q^{19}+ \cdots + (13 \beta_{3} - 202 \beta_{2} - 7664 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 228 q^{9} + 128 q^{13} + 814 q^{17} + 884 q^{21} + 4152 q^{29} - 5058 q^{33} - 348 q^{37} + 10334 q^{41} + 1454 q^{49} + 33924 q^{53} - 38450 q^{57} + 25796 q^{61} + 5796 q^{69} + 59218 q^{73} - 49052 q^{77}+ \cdots - 250052 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 207x^{4} + 419x^{3} + 5927x^{2} - 6137x - 8328 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 56\nu^{5} - 140\nu^{4} - 11344\nu^{3} + 17156\nu^{2} + 283250\nu - 144489 ) / 1749 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -16\nu^{5} + 40\nu^{4} + 5240\nu^{3} - 7900\nu^{2} - 382756\nu + 192696 ) / 1749 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 4\nu^{2} - 4\nu - 280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{4} - 8\nu^{3} - 720\nu^{2} + 724\nu + 8249 ) / 11 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 2\beta _1 + 282 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{4} + 7\beta_{3} + 2\beta_{2} + 610\beta _1 + 844 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{5} + 183\beta_{4} + 7\beta_{3} + 2\beta_{2} + 608\beta _1 + 42993 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 55\beta_{5} + 910\beta_{4} + 1453\beta_{3} + 665\beta_{2} + 105151\beta _1 + 213559 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−12.7478
−5.29401
−0.794658
1.79466
6.29401
13.7478
0 −26.4955 0 0 0 −35.9189 0 459.013 0
1.2 0 −11.5880 0 0 0 89.6769 0 −108.718 0
1.3 0 −2.58932 0 0 0 −204.489 0 −236.295 0
1.4 0 2.58932 0 0 0 204.489 0 −236.295 0
1.5 0 11.5880 0 0 0 −89.6769 0 −108.718 0
1.6 0 26.4955 0 0 0 35.9189 0 459.013 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.a.y yes 6
4.b odd 2 1 inner 800.6.a.y yes 6
5.b even 2 1 800.6.a.x ✓ 6
5.c odd 4 2 800.6.c.p 12
20.d odd 2 1 800.6.a.x ✓ 6
20.e even 4 2 800.6.c.p 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.6.a.x ✓ 6 5.b even 2 1
800.6.a.x ✓ 6 20.d odd 2 1
800.6.a.y yes 6 1.a even 1 1 trivial
800.6.a.y yes 6 4.b odd 2 1 inner
800.6.c.p 12 5.c odd 4 2
800.6.c.p 12 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3}^{6} - 843T_{3}^{4} + 99875T_{3}^{2} - 632025 \) Copy content Toggle raw display
\( T_{11}^{6} - 781187T_{11}^{4} + 150735355475T_{11}^{2} - 7280148286726225 \) Copy content Toggle raw display
\( T_{13}^{3} - 64T_{13}^{2} - 831280T_{13} - 13557632 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 843 T^{4} + \cdots - 632025 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots - 433859342400 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 72\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{3} - 64 T^{2} + \cdots - 13557632)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 407 T^{2} + \cdots + 3248837483)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 13\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2076 T^{2} + \cdots + 43086643200)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{3} + 174 T^{2} + \cdots + 388929363400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 5167 T^{2} + \cdots + 41293330275)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots - 4332967674264)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 14177606744200)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 59\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 11131332226389)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 25\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 74566113251375)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 32041374433688)^{2} \) Copy content Toggle raw display
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