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Properties

Label 504.3.g.b
Level $504$
Weight $3$
Character orbit 504.g
Analytic conductor $13.733$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,3,Mod(379,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.379");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 504.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7330053238\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.292213762624.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{5} - \beta_{3} - \beta_{2} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{5} - \beta_{3} - \beta_{2} + \cdots - 1) q^{5}+ \cdots + 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 5 q^{4} - 13 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + 5 q^{4} - 13 q^{8} + 16 q^{10} + 32 q^{11} - 7 q^{14} - 71 q^{16} + 80 q^{17} + 56 q^{19} + 108 q^{20} + 66 q^{22} - 16 q^{25} - 24 q^{26} + 7 q^{28} + 19 q^{32} + 74 q^{34} - 56 q^{35} + 14 q^{38} + 84 q^{40} - 128 q^{41} - 50 q^{44} - 152 q^{46} - 56 q^{49} - 33 q^{50} + 132 q^{52} + 49 q^{56} + 24 q^{58} - 104 q^{59} - 120 q^{62} - 55 q^{64} + 72 q^{65} + 304 q^{67} + 190 q^{68} + 56 q^{70} - 112 q^{73} - 8 q^{74} + 70 q^{76} - 124 q^{80} + 450 q^{82} - 72 q^{83} - 210 q^{86} - 486 q^{88} + 512 q^{89} - 56 q^{91} + 472 q^{92} + 472 q^{94} + 64 q^{97} + 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 4\nu^{5} - 22\nu^{4} - 12\nu^{3} + 32\nu^{2} + 32\nu - 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 5\nu^{6} - 4\nu^{5} + 6\nu^{4} - 4\nu^{3} + 32\nu^{2} + 32\nu + 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} + 12\nu^{5} - 6\nu^{4} - 12\nu^{3} + 96\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} - \nu^{6} - 4\nu^{5} + 50\nu^{4} - 28\nu^{3} - 96\nu^{2} - 96\nu + 512 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} + 3\nu^{6} - 12\nu^{5} - 38\nu^{4} + 164\nu^{3} + 128\nu^{2} - 96\nu - 640 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 2\beta_{4} - 3\beta_{3} + \beta_{2} + \beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} + 8\beta_{5} - 2\beta_{4} - 5\beta_{3} + \beta_{2} - 5\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + 3\beta_{6} + 10\beta_{5} + 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 9\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -6\beta_{7} - 21\beta_{6} - 14\beta_{5} + 24\beta_{4} - 27\beta_{3} - 9\beta_{2} - 9\beta _1 - 33 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
1.85837 + 0.739226i
1.85837 − 0.739226i
1.37098 + 1.45617i
1.37098 − 1.45617i
−1.05468 + 1.69931i
−1.05468 − 1.69931i
−1.67467 + 1.09337i
−1.67467 − 1.09337i
−1.85837 − 0.739226i 0 2.90709 + 2.74751i 3.46547i 0 2.64575i −3.37142 − 7.25490i 0 2.56177 − 6.44013i
379.2 −1.85837 + 0.739226i 0 2.90709 − 2.74751i − 3.46547i 0 − 2.64575i −3.37142 + 7.25490i 0 2.56177 + 6.44013i
379.3 −1.37098 − 1.45617i 0 −0.240837 + 3.99274i − 6.26788i 0 − 2.64575i 6.14428 − 5.12327i 0 −9.12707 + 8.59313i
379.4 −1.37098 + 1.45617i 0 −0.240837 − 3.99274i 6.26788i 0 2.64575i 6.14428 + 5.12327i 0 −9.12707 − 8.59313i
379.5 1.05468 − 1.69931i 0 −1.77532 − 3.58445i 4.88287i 0 − 2.64575i −7.96347 − 0.763618i 0 8.29751 + 5.14984i
379.6 1.05468 + 1.69931i 0 −1.77532 + 3.58445i − 4.88287i 0 2.64575i −7.96347 + 0.763618i 0 8.29751 − 5.14984i
379.7 1.67467 − 1.09337i 0 1.60906 − 3.66209i 5.73252i 0 2.64575i −1.30939 − 7.89212i 0 6.26779 + 9.60010i
379.8 1.67467 + 1.09337i 0 1.60906 + 3.66209i − 5.73252i 0 − 2.64575i −1.30939 + 7.89212i 0 6.26779 − 9.60010i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.3.g.b 8
3.b odd 2 1 56.3.g.b ✓ 8
4.b odd 2 1 2016.3.g.b 8
8.b even 2 1 2016.3.g.b 8
8.d odd 2 1 inner 504.3.g.b 8
12.b even 2 1 224.3.g.b 8
21.c even 2 1 392.3.g.m 8
21.g even 6 2 392.3.k.n 16
21.h odd 6 2 392.3.k.o 16
24.f even 2 1 56.3.g.b ✓ 8
24.h odd 2 1 224.3.g.b 8
48.i odd 4 2 1792.3.d.j 16
48.k even 4 2 1792.3.d.j 16
84.h odd 2 1 1568.3.g.m 8
168.e odd 2 1 392.3.g.m 8
168.i even 2 1 1568.3.g.m 8
168.v even 6 2 392.3.k.o 16
168.be odd 6 2 392.3.k.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.b ✓ 8 3.b odd 2 1
56.3.g.b ✓ 8 24.f even 2 1
224.3.g.b 8 12.b even 2 1
224.3.g.b 8 24.h odd 2 1
392.3.g.m 8 21.c even 2 1
392.3.g.m 8 168.e odd 2 1
392.3.k.n 16 21.g even 6 2
392.3.k.n 16 168.be odd 6 2
392.3.k.o 16 21.h odd 6 2
392.3.k.o 16 168.v even 6 2
504.3.g.b 8 1.a even 1 1 trivial
504.3.g.b 8 8.d odd 2 1 inner
1568.3.g.m 8 84.h odd 2 1
1568.3.g.m 8 168.i even 2 1
1792.3.d.j 16 48.i odd 4 2
1792.3.d.j 16 48.k even 4 2
2016.3.g.b 8 4.b odd 2 1
2016.3.g.b 8 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 108T_{5}^{6} + 4164T_{5}^{4} + 66944T_{5}^{2} + 369664 \) acting on \(S_{3}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 108 T^{6} + \cdots + 369664 \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{3} + \cdots - 864)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 908 T^{6} + \cdots + 133448704 \) Copy content Toggle raw display
$17$ \( (T^{4} - 40 T^{3} + \cdots - 752)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 28 T^{3} + \cdots + 1288)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 15607005184 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 13389266944 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 61917364224 \) Copy content Toggle raw display
$37$ \( T^{8} + 7440 T^{6} + \cdots + 554696704 \) Copy content Toggle raw display
$41$ \( (T^{4} + 64 T^{3} + \cdots + 837776)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2716 T^{2} + \cdots - 217952)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 7542537191424 \) Copy content Toggle raw display
$53$ \( T^{8} + 3552 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
$59$ \( (T^{4} + 52 T^{3} + \cdots - 11252856)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 3223777158144 \) Copy content Toggle raw display
$67$ \( (T^{4} - 152 T^{3} + \cdots - 791808)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 221437256269824 \) Copy content Toggle raw display
$73$ \( (T^{4} + 56 T^{3} + \cdots - 1726704)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 89369947930624 \) Copy content Toggle raw display
$83$ \( (T^{4} + 36 T^{3} + \cdots + 3180872)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 256 T^{3} + \cdots - 618736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 32 T^{3} + \cdots + 9539216)^{2} \) Copy content Toggle raw display
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