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Properties

Label 637.2.e.l
Level $637$
Weight $2$
Character orbit 637.e
Analytic conductor $5.086$
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] = [637,2,Mod(79,637)] 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("637.79"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,2,4,-6,5] 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.4406832.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{2} + \beta_1) q^{2} + (\beta_{5} - \beta_{4} - \beta_{3} + 1) q^{3} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots - 2) q^{4} + (2 \beta_{4} - \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 3 \beta_{2}) q^{6}+ \cdots + (2 \beta_{3} - 7 \beta_{2} - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 4 q^{3} - 6 q^{4} + 5 q^{5} - 4 q^{6} - 12 q^{8} - 11 q^{9} - 14 q^{10} + 4 q^{11} + 18 q^{12} + 6 q^{13} + 4 q^{15} - 4 q^{16} + 4 q^{17} - 8 q^{18} + 7 q^{19} - 32 q^{20} - 16 q^{22} - q^{23}+ \cdots - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 6\nu^{4} - 36\nu^{3} + 24\nu^{2} + 5\nu - 30 ) / 149 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{5} + 36\nu^{4} - 67\nu^{3} + 144\nu^{2} + 30\nu + 416 ) / 149 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30\nu^{5} - 31\nu^{4} + 186\nu^{3} + 174\nu^{2} + 744\nu + 155 ) / 149 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 89\nu^{5} - 87\nu^{4} + 522\nu^{3} + 695\nu^{2} + 2088\nu + 435 ) / 149 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 6\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{5} + 19\beta_{4} + 6\beta_{3} - 12\beta _1 - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -12\beta_{5} + 42\beta_{4} + 43\beta_{2} - 43\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
1.43310 + 2.48220i
−0.105378 − 0.182520i
−0.827721 − 1.43366i
1.43310 − 2.48220i
−0.105378 + 0.182520i
−0.827721 + 1.43366i
−0.933099 + 1.61618i 1.67445 + 2.90023i −0.741348 − 1.28405i −0.433099 + 0.750150i −6.24970 0 −0.965392 −4.10755 + 7.11448i −0.808249 − 1.39993i
79.2 0.605378 − 1.04855i −0.872413 − 1.51106i 0.267035 + 0.462518i 1.10538 − 1.91457i −2.11256 0 3.06814 −0.0222090 + 0.0384672i −1.33834 − 2.31808i
79.3 1.32772 − 2.29968i 1.19797 + 2.07494i −2.52569 − 4.37462i 1.82772 − 3.16571i 6.36226 0 −8.10275 −1.37024 + 2.37333i −4.85341 − 8.40635i
508.1 −0.933099 − 1.61618i 1.67445 − 2.90023i −0.741348 + 1.28405i −0.433099 − 0.750150i −6.24970 0 −0.965392 −4.10755 − 7.11448i −0.808249 + 1.39993i
508.2 0.605378 + 1.04855i −0.872413 + 1.51106i 0.267035 − 0.462518i 1.10538 + 1.91457i −2.11256 0 3.06814 −0.0222090 − 0.0384672i −1.33834 + 2.31808i
508.3 1.32772 + 2.29968i 1.19797 − 2.07494i −2.52569 + 4.37462i 1.82772 + 3.16571i 6.36226 0 −8.10275 −1.37024 − 2.37333i −4.85341 + 8.40635i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.3
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 637.2.e.l 6
7.b odd 2 1 637.2.e.k 6
7.c even 3 1 637.2.a.h ✓ 3
7.c even 3 1 inner 637.2.e.l 6
7.d odd 6 1 637.2.a.i yes 3
7.d odd 6 1 637.2.e.k 6
21.g even 6 1 5733.2.a.bd 3
21.h odd 6 1 5733.2.a.be 3
91.r even 6 1 8281.2.a.bh 3
91.s odd 6 1 8281.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.h ✓ 3 7.c even 3 1
637.2.a.i yes 3 7.d odd 6 1
637.2.e.k 6 7.b odd 2 1
637.2.e.k 6 7.d odd 6 1
637.2.e.l 6 1.a even 1 1 trivial
637.2.e.l 6 7.c even 3 1 inner
5733.2.a.bd 3 21.g even 6 1
5733.2.a.be 3 21.h odd 6 1
8281.2.a.bh 3 91.r even 6 1
8281.2.a.bk 3 91.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{6} - 2T_{2}^{5} + 8T_{2}^{4} - 4T_{2}^{3} + 28T_{2}^{2} - 24T_{2} + 36 \) Copy content Toggle raw display
\( T_{3}^{6} - 4T_{3}^{5} + 18T_{3}^{4} - 20T_{3}^{3} + 60T_{3}^{2} - 28T_{3} + 196 \) Copy content Toggle raw display
\( T_{5}^{6} - 5T_{5}^{5} + 22T_{5}^{4} - 29T_{5}^{3} + 44T_{5}^{2} + 21T_{5} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} + \cdots + 196 \) Copy content Toggle raw display
$5$ \( T^{6} - 5 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} - 4 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T - 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + \cdots + 196 \) Copy content Toggle raw display
$19$ \( T^{6} - 7 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$23$ \( T^{6} + T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$29$ \( (T^{3} + 7 T^{2} - 13 T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$37$ \( T^{6} - 10 T^{5} + \cdots + 6724 \) Copy content Toggle raw display
$41$ \( (T^{3} + 6 T^{2} + \cdots - 504)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 9 T^{2} + \cdots + 101)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 17 T^{5} + \cdots + 21609 \) Copy content Toggle raw display
$53$ \( T^{6} + 13 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{6} - 22 T^{5} + \cdots + 63504 \) Copy content Toggle raw display
$61$ \( T^{6} + 24 T^{5} + \cdots + 50176 \) Copy content Toggle raw display
$67$ \( T^{6} - 14 T^{5} + \cdots + 419904 \) Copy content Toggle raw display
$71$ \( (T^{3} - 4 T^{2} + \cdots + 194)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 5 T^{5} + \cdots + 2436721 \) Copy content Toggle raw display
$79$ \( T^{6} + T^{5} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( (T^{3} + 23 T^{2} + \cdots + 203)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 11 T^{5} + \cdots + 441 \) Copy content Toggle raw display
$97$ \( (T^{3} - 3 T^{2} - 71 T + 7)^{2} \) Copy content Toggle raw display
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