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Properties

Label 504.6.s.b
Level $504$
Weight $6$
Character orbit 504.s
Analytic conductor $80.833$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,6,Mod(289,504)] 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("504.289"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 504.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,-81] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(80.8334451857\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 200 x^{8} - 198 x^{7} + 34197 x^{6} - 16185 x^{5} + 1170401 x^{4} + 2020497 x^{3} + \cdots + 13068225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4} - 16 \beta_{3} - 16) q^{5} + (\beta_{8} - \beta_{6} + \beta_{5} + \cdots + 15) q^{7} + (\beta_{9} + \beta_{8} - \beta_{7} + \cdots + 1) q^{11} + ( - 3 \beta_{9} - 6 \beta_{8} + \cdots - 72) q^{13}+ \cdots + (37 \beta_{9} + 74 \beta_{8} + \cdots - 65728) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 81 q^{5} + 116 q^{7} + 361 q^{11} - 684 q^{13} - 1809 q^{17} + 1277 q^{19} + 911 q^{23} - 3940 q^{25} - 10884 q^{29} + 2187 q^{31} + 16845 q^{35} + 8181 q^{37} + 33156 q^{41} + 12664 q^{43} - 16101 q^{47}+ \cdots - 656548 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 200 x^{8} - 198 x^{7} + 34197 x^{6} - 16185 x^{5} + 1170401 x^{4} + 2020497 x^{3} + \cdots + 13068225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2025121474 \nu^{9} - 10800435830 \nu^{8} + 43579563150 \nu^{7} - 1409134397898 \nu^{6} + \cdots + 28\!\cdots\!70 ) / 75\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12994831946063 \nu^{9} - 1220135688085 \nu^{8} + \cdots - 27\!\cdots\!35 ) / 27\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30337577204812 \nu^{9} - 585628022039268 \nu^{8} + \cdots - 19\!\cdots\!05 ) / 28\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19\!\cdots\!89 \nu^{9} + \cdots + 46\!\cdots\!05 ) / 68\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\!\cdots\!16 \nu^{9} + \cdots + 50\!\cdots\!85 ) / 34\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!22 \nu^{9} + \cdots + 29\!\cdots\!15 ) / 34\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26\!\cdots\!22 \nu^{9} + \cdots + 18\!\cdots\!20 ) / 34\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 42\!\cdots\!46 \nu^{9} + \cdots + 51\!\cdots\!85 ) / 34\!\cdots\!15 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{4} + 319\beta_{3} - \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{9} + 24 \beta_{8} + 9 \beta_{7} + 18 \beta_{6} + 12 \beta_{5} + 159 \beta_{4} + 12 \beta_{3} + \cdots + 1416 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 353 \beta_{9} - 50 \beta_{8} - 50 \beta_{7} + 403 \beta_{6} - 635 \beta_{5} - 635 \beta_{4} + \cdots - 90895 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2100 \beta_{9} - 894 \beta_{8} + 1497 \beta_{7} - 2703 \beta_{6} + 16797 \beta_{5} + 1206 \beta_{4} + \cdots - 2100 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 69390 \beta_{9} + 138780 \beta_{8} - 58697 \beta_{7} - 117394 \beta_{6} + 69390 \beta_{5} + \cdots + 14521808 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 62442 \beta_{9} - 732987 \beta_{8} - 732987 \beta_{7} + 670545 \beta_{6} - 6522444 \beta_{5} + \cdots - 137009430 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2012355 \beta_{9} - 21292951 \beta_{8} + 11652653 \beta_{7} + 7627943 \beta_{6} + 8008931 \beta_{5} + \cdots - 2012355 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 141714183 \beta_{9} + 283428366 \beta_{8} - 15771312 \beta_{7} - 31542624 \beta_{6} + \cdots + 29597846781 ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
3.36475 − 5.82792i
−6.63412 + 11.4906i
6.16047 − 10.6702i
−2.57166 + 4.45425i
−0.319443 + 0.553292i
3.36475 + 5.82792i
−6.63412 − 11.4906i
6.16047 + 10.6702i
−2.57166 − 4.45425i
−0.319443 − 0.553292i
0 0 0 −48.8402 + 84.5938i 0 −122.707 − 41.8327i 0 0 0
289.2 0 0 0 −39.9714 + 69.2326i 0 113.410 − 62.8111i 0 0 0
289.3 0 0 0 10.0228 − 17.3600i 0 102.547 + 79.3170i 0 0 0
289.4 0 0 0 13.0334 − 22.5744i 0 −32.5264 + 125.495i 0 0 0
289.5 0 0 0 25.2555 − 43.7438i 0 −2.72289 − 129.613i 0 0 0
361.1 0 0 0 −48.8402 − 84.5938i 0 −122.707 + 41.8327i 0 0 0
361.2 0 0 0 −39.9714 − 69.2326i 0 113.410 + 62.8111i 0 0 0
361.3 0 0 0 10.0228 + 17.3600i 0 102.547 − 79.3170i 0 0 0
361.4 0 0 0 13.0334 + 22.5744i 0 −32.5264 − 125.495i 0 0 0
361.5 0 0 0 25.2555 + 43.7438i 0 −2.72289 + 129.613i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.6.s.b 10
3.b odd 2 1 56.6.i.b ✓ 10
7.c even 3 1 inner 504.6.s.b 10
12.b even 2 1 112.6.i.f 10
21.c even 2 1 392.6.i.o 10
21.g even 6 1 392.6.a.j 5
21.g even 6 1 392.6.i.o 10
21.h odd 6 1 56.6.i.b ✓ 10
21.h odd 6 1 392.6.a.k 5
84.j odd 6 1 784.6.a.bl 5
84.n even 6 1 112.6.i.f 10
84.n even 6 1 784.6.a.bk 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.i.b ✓ 10 3.b odd 2 1
56.6.i.b ✓ 10 21.h odd 6 1
112.6.i.f 10 12.b even 2 1
112.6.i.f 10 84.n even 6 1
392.6.a.j 5 21.g even 6 1
392.6.a.k 5 21.h odd 6 1
392.6.i.o 10 21.c even 2 1
392.6.i.o 10 21.g even 6 1
504.6.s.b 10 1.a even 1 1 trivial
504.6.s.b 10 7.c even 3 1 inner
784.6.a.bk 5 84.n even 6 1
784.6.a.bl 5 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 81 T_{5}^{9} + 13063 T_{5}^{8} + 22102 T_{5}^{7} + 46920445 T_{5}^{6} + \cdots + 42\!\cdots\!01 \) acting on \(S_{6}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 13\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 40\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 165262458987360)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 15\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 32\!\cdots\!89 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 32\!\cdots\!72)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 74\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 13\!\cdots\!40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 34\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 64\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 50\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 16\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 15\!\cdots\!48)^{2} \) Copy content Toggle raw display
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