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Properties

Label 531.2.f.a
Level $531$
Weight $2$
Character orbit 531.f
Analytic conductor $4.240$
Analytic rank $0$
Dimension $12$
CM discriminant -59
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,2,Mod(176,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.176");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 531.f (of order \(6\), degree \(2\), 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: \(4.24005634733\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7x^{9} + 22x^{6} - 189x^{3} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + 2 \beta_{5} q^{4} + (\beta_{10} + \beta_{2}) q^{5} + (\beta_{9} + \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{11} - \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + 2 \beta_{5} q^{4} + (\beta_{10} + \beta_{2}) q^{5} + (\beta_{9} + \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots + (3 \beta_{11} - 4 \beta_{9} + \cdots - 12) q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{4} - 3 q^{15} - 24 q^{16} - 30 q^{21} + 30 q^{25} - 21 q^{27} + 33 q^{45} - 42 q^{49} + 78 q^{57} - 12 q^{60} - 12 q^{63} - 96 q^{64} + 54 q^{68} + 24 q^{75} - 30 q^{84} + 51 q^{87} - 108 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 7x^{9} + 22x^{6} - 189x^{3} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} + 88\nu^{6} - 22\nu^{3} + 189 ) / 594 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{11} + 154\nu^{8} - 484\nu^{5} - 2133\nu^{2} ) / 5346 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{9} - 22\nu^{6} + 154\nu^{3} - 729 ) / 594 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{11} - 55\nu^{8} - 209\nu^{5} + 1323\nu^{2} ) / 2673 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{10} - 22\nu^{7} + 154\nu^{4} - 729\nu ) / 594 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -14\nu^{9} + 44\nu^{6} - 11\nu^{3} + 1458 ) / 297 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9\nu^{11} - \nu^{10} + 88\nu^{7} - 22\nu^{4} - 315\nu^{2} + 189\nu ) / 1782 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -9\nu^{11} - 28\nu^{10} + 88\nu^{7} - 22\nu^{4} + 315\nu^{2} + 2916\nu ) / 1782 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34\nu^{11} - 22\nu^{8} + 451\nu^{5} - 4347\nu^{2} ) / 2673 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 3\beta_{10} + 4\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - 6\beta_{6} - 4\beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{5} + 7\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -7\beta_{11} + 21\beta_{9} + \beta_{7} - 7\beta_{6} - 7\beta_{4} - 7\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{11} - 21\beta_{6} + 20\beta_{4} + 20\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -22\beta_{8} + 22\beta_{3} + 101 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -44\beta_{11} - 66\beta_{10} + 66\beta_{9} - 44\beta_{6} - 44\beta_{4} - 44\beta_{2} + 101\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 66\beta_{11} + 66\beta_{6} + 66\beta_{4} + 101\beta_{2} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\) \(119\) \(415\)
\(\chi(n)\) \(\beta_{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
176.1
−0.971419 − 1.43400i
−1.58090 − 0.707650i
1.40329 − 1.01527i
1.72759 − 0.124276i
−0.756168 + 1.55827i
0.177605 + 1.72292i
−0.971419 + 1.43400i
−1.58090 + 0.707650i
1.40329 + 1.01527i
1.72759 + 0.124276i
−0.756168 − 1.55827i
0.177605 − 1.72292i
0 −1.72759 + 0.124276i 1.00000 − 1.73205i −1.74002 − 1.00460i 0 −1.02627 − 1.77756i 0 2.96911 − 0.429394i 0
176.2 0 −1.40329 + 1.01527i 1.00000 − 1.73205i 3.70969 + 2.14179i 0 2.22334 + 3.85094i 0 0.938451 − 2.84944i 0
176.3 0 −0.177605 − 1.72292i 1.00000 − 1.73205i −2.81851 − 1.62727i 0 0.130331 + 0.225740i 0 −2.93691 + 0.611998i 0
176.4 0 0.756168 − 1.55827i 1.00000 − 1.73205i 3.86655 + 2.23235i 0 −1.59875 − 2.76912i 0 −1.85642 − 2.35663i 0
176.5 0 0.971419 + 1.43400i 1.00000 − 1.73205i −2.12653 − 1.22775i 0 2.62502 + 4.54668i 0 −1.11269 + 2.78602i 0
176.6 0 1.58090 + 0.707650i 1.00000 − 1.73205i −0.891177 − 0.514521i 0 −2.35367 − 4.07668i 0 1.99846 + 2.23744i 0
353.1 0 −1.72759 − 0.124276i 1.00000 + 1.73205i −1.74002 + 1.00460i 0 −1.02627 + 1.77756i 0 2.96911 + 0.429394i 0
353.2 0 −1.40329 − 1.01527i 1.00000 + 1.73205i 3.70969 − 2.14179i 0 2.22334 − 3.85094i 0 0.938451 + 2.84944i 0
353.3 0 −0.177605 + 1.72292i 1.00000 + 1.73205i −2.81851 + 1.62727i 0 0.130331 − 0.225740i 0 −2.93691 − 0.611998i 0
353.4 0 0.756168 + 1.55827i 1.00000 + 1.73205i 3.86655 − 2.23235i 0 −1.59875 + 2.76912i 0 −1.85642 + 2.35663i 0
353.5 0 0.971419 − 1.43400i 1.00000 + 1.73205i −2.12653 + 1.22775i 0 2.62502 − 4.54668i 0 −1.11269 − 2.78602i 0
353.6 0 1.58090 − 0.707650i 1.00000 + 1.73205i −0.891177 + 0.514521i 0 −2.35367 + 4.07668i 0 1.99846 − 2.23744i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 176.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)
9.d odd 6 1 inner
531.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.2.f.a ✓ 12
9.d odd 6 1 inner 531.2.f.a ✓ 12
59.b odd 2 1 CM 531.2.f.a ✓ 12
531.f even 6 1 inner 531.2.f.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.2.f.a ✓ 12 1.a even 1 1 trivial
531.2.f.a ✓ 12 9.d odd 6 1 inner
531.2.f.a ✓ 12 59.b odd 2 1 CM
531.2.f.a ✓ 12 531.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(531, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 7 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} - 30 T^{10} + \cdots + 99856 \) Copy content Toggle raw display
$7$ \( T^{12} + 42 T^{10} + \cdots + 35344 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{4} + 43 T^{2} + 64)^{3} \) Copy content Toggle raw display
$19$ \( (T^{6} - 114 T^{4} + \cdots - 6416)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} - 174 T^{10} + \cdots + 795664 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 42726543616 \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1759634704 \) Copy content Toggle raw display
$59$ \( (T^{4} - 59 T^{2} + 3481)^{3} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( (T^{4} + 367 T^{2} + 23716)^{3} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 611355099664 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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