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Properties

Label 637.2.h.a
Level $637$
Weight $2$
Character orbit 637.h
Analytic conductor $5.086$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [637,2,Mod(165,637)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("637.165"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,-3,-2,3] 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - 3 \zeta_{6} q^{3} - q^{4} + 3 \zeta_{6} q^{5} - 3 \zeta_{6} q^{6} - 3 q^{8} + (6 \zeta_{6} - 6) q^{9} + 3 \zeta_{6} q^{10} + 3 \zeta_{6} q^{11} + 3 \zeta_{6} q^{12} + (4 \zeta_{6} - 1) q^{13} + \cdots - 18 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} - 2 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{8} - 6 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{12} + 2 q^{13} + 9 q^{15} - 2 q^{16} + 4 q^{17} - 6 q^{18} - q^{19} - 3 q^{20} + 3 q^{22} + 9 q^{24}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.500000 + 0.866025i
0.500000 − 0.866025i
1.00000 −1.50000 − 2.59808i −1.00000 1.50000 + 2.59808i −1.50000 − 2.59808i 0 −3.00000 −3.00000 + 5.19615i 1.50000 + 2.59808i
471.1 1.00000 −1.50000 + 2.59808i −1.00000 1.50000 − 2.59808i −1.50000 + 2.59808i 0 −3.00000 −3.00000 − 5.19615i 1.50000 − 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h 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.h.a 2
7.b odd 2 1 91.2.h.a yes 2
7.c even 3 1 637.2.f.a 2
7.c even 3 1 637.2.g.a 2
7.d odd 6 1 91.2.g.a ✓ 2
7.d odd 6 1 637.2.f.b 2
13.c even 3 1 637.2.g.a 2
21.c even 2 1 819.2.s.a 2
21.g even 6 1 819.2.n.c 2
91.g even 3 1 637.2.f.a 2
91.h even 3 1 inner 637.2.h.a 2
91.h even 3 1 8281.2.a.j 1
91.k even 6 1 8281.2.a.g 1
91.l odd 6 1 8281.2.a.c 1
91.m odd 6 1 637.2.f.b 2
91.m odd 6 1 1183.2.e.a 2
91.n odd 6 1 91.2.g.a ✓ 2
91.n odd 6 1 1183.2.e.a 2
91.p odd 6 1 1183.2.e.c 2
91.t odd 6 1 1183.2.e.c 2
91.v odd 6 1 91.2.h.a yes 2
91.v odd 6 1 8281.2.a.i 1
273.r even 6 1 819.2.s.a 2
273.bn even 6 1 819.2.n.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a ✓ 2 7.d odd 6 1
91.2.g.a ✓ 2 91.n odd 6 1
91.2.h.a yes 2 7.b odd 2 1
91.2.h.a yes 2 91.v odd 6 1
637.2.f.a 2 7.c even 3 1
637.2.f.a 2 91.g even 3 1
637.2.f.b 2 7.d odd 6 1
637.2.f.b 2 91.m odd 6 1
637.2.g.a 2 7.c even 3 1
637.2.g.a 2 13.c even 3 1
637.2.h.a 2 1.a even 1 1 trivial
637.2.h.a 2 91.h even 3 1 inner
819.2.n.c 2 21.g even 6 1
819.2.n.c 2 273.bn even 6 1
819.2.s.a 2 21.c even 2 1
819.2.s.a 2 273.r even 6 1
1183.2.e.a 2 91.m odd 6 1
1183.2.e.a 2 91.n odd 6 1
1183.2.e.c 2 91.p odd 6 1
1183.2.e.c 2 91.t odd 6 1
8281.2.a.c 1 91.l odd 6 1
8281.2.a.g 1 91.k even 6 1
8281.2.a.i 1 91.v odd 6 1
8281.2.a.j 1 91.h even 3 1

Hecke kernels

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

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 3T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 13 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$31$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$47$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$71$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$73$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
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