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Properties

Label 2523.1.j.c
Level $2523$
Weight $1$
Character orbit 2523.j
Analytic conductor $1.259$
Analytic rank $0$
Dimension $12$
Projective image $D_{5}$
CM discriminant -3
Inner twists $12$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2523,1,Mod(605,2523)] 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("2523.605"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2523, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 8])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2523 = 3 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2523.j (of order \(14\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.25914102687\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: 12.0.4413675765625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.6365529.1

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

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{11} + \beta_{9} + \beta_{7} + \cdots - 1) q^{4} + ( - \beta_{5} + \beta_{4}) q^{7} - \beta_{9} q^{9} - q^{12} + \beta_{6} q^{13} - \beta_{7} q^{16} + \beta_{10} q^{19} + ( - \beta_{9} - \beta_{8}) q^{21}+ \cdots + (\beta_{11} - \beta_{10} + \beta_{9} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} - 2 q^{4} + q^{7} - 2 q^{9} - 12 q^{12} + q^{13} - 2 q^{16} - q^{19} - q^{21} - 2 q^{25} + 2 q^{27} - 6 q^{28} - q^{31} - 2 q^{36} - q^{37} - q^{39} - q^{43} + 2 q^{48} - q^{49}+ \cdots - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 2x^{10} - 3x^{9} + 5x^{8} - 8x^{7} + 13x^{6} + 8x^{5} + 5x^{4} + 3x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 8 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 21\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - 21\nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 34\nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} + 34\nu^{3} ) / 13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{10} - 55\nu^{3} ) / 13 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{11} - 55\nu^{4} ) / 13 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{11} - 89\nu^{4} ) / 13 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{11} + 6 \nu^{10} - 9 \nu^{9} + 15 \nu^{8} - 24 \nu^{7} + 39 \nu^{6} - 65 \nu^{5} + 15 \nu^{4} + \cdots + 3 ) / 13 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8 \nu^{11} - 8 \nu^{10} + 16 \nu^{9} - 24 \nu^{8} + 40 \nu^{7} - 65 \nu^{6} + 104 \nu^{5} + 64 \nu^{4} + \cdots + 8 ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} + 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} - 5\beta_{10} - 3\beta_{9} - 3\beta_{7} + 3\beta_{5} + 3\beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{11} - 8\beta_{10} - 8\beta_{8} + 8\beta_{6} - 8\beta_{4} + 8\beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13\beta_{3} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -21\beta_{5} - 34\beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -34\beta_{7} - 55\beta_{6} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 55\beta_{9} - 89\beta_{8} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2523\mathbb{Z}\right)^\times\).

\(n\) \(842\) \(1684\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{9} + \beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
605.1
0.360046 + 1.57747i
−0.137526 − 0.602539i
−0.556829 + 0.268155i
1.45780 − 0.702039i
−0.556829 − 0.268155i
1.45780 + 0.702039i
−1.00883 − 1.26503i
0.385338 + 0.483198i
−1.00883 + 1.26503i
0.385338 − 0.483198i
0.360046 − 1.57747i
−0.137526 + 0.602539i
0 0.900969 − 0.433884i −0.900969 − 0.433884i 0 0 −0.556829 + 0.268155i 0 0.623490 − 0.781831i 0
605.2 0 0.900969 − 0.433884i −0.900969 − 0.433884i 0 0 1.45780 − 0.702039i 0 0.623490 − 0.781831i 0
1031.1 0 −0.623490 + 0.781831i 0.623490 + 0.781831i 0 0 −1.00883 + 1.26503i 0 −0.222521 − 0.974928i 0
1031.2 0 −0.623490 + 0.781831i 0.623490 + 0.781831i 0 0 0.385338 − 0.483198i 0 −0.222521 − 0.974928i 0
1412.1 0 −0.623490 − 0.781831i 0.623490 − 0.781831i 0 0 −1.00883 − 1.26503i 0 −0.222521 + 0.974928i 0
1412.2 0 −0.623490 − 0.781831i 0.623490 − 0.781831i 0 0 0.385338 + 0.483198i 0 −0.222521 + 0.974928i 0
1415.1 0 0.222521 − 0.974928i −0.222521 − 0.974928i 0 0 −0.137526 + 0.602539i 0 −0.900969 − 0.433884i 0
1415.2 0 0.222521 − 0.974928i −0.222521 − 0.974928i 0 0 0.360046 − 1.57747i 0 −0.900969 − 0.433884i 0
1619.1 0 0.222521 + 0.974928i −0.222521 + 0.974928i 0 0 −0.137526 − 0.602539i 0 −0.900969 + 0.433884i 0
1619.2 0 0.222521 + 0.974928i −0.222521 + 0.974928i 0 0 0.360046 + 1.57747i 0 −0.900969 + 0.433884i 0
2327.1 0 0.900969 + 0.433884i −0.900969 + 0.433884i 0 0 −0.556829 − 0.268155i 0 0.623490 + 0.781831i 0
2327.2 0 0.900969 + 0.433884i −0.900969 + 0.433884i 0 0 1.45780 + 0.702039i 0 0.623490 + 0.781831i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 605.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
29.d even 7 5 inner
87.j odd 14 5 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2523.1.j.c 12
3.b odd 2 1 CM 2523.1.j.c 12
29.b even 2 1 2523.1.j.a 12
29.c odd 4 2 2523.1.h.c 24
29.d even 7 1 2523.1.b.a ✓ 2
29.d even 7 5 inner 2523.1.j.c 12
29.e even 14 1 2523.1.b.c yes 2
29.e even 14 5 2523.1.j.a 12
29.f odd 28 2 2523.1.d.a 4
29.f odd 28 10 2523.1.h.c 24
87.d odd 2 1 2523.1.j.a 12
87.f even 4 2 2523.1.h.c 24
87.h odd 14 1 2523.1.b.c yes 2
87.h odd 14 5 2523.1.j.a 12
87.j odd 14 1 2523.1.b.a ✓ 2
87.j odd 14 5 inner 2523.1.j.c 12
87.k even 28 2 2523.1.d.a 4
87.k even 28 10 2523.1.h.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2523.1.b.a ✓ 2 29.d even 7 1
2523.1.b.a ✓ 2 87.j odd 14 1
2523.1.b.c yes 2 29.e even 14 1
2523.1.b.c yes 2 87.h odd 14 1
2523.1.d.a 4 29.f odd 28 2
2523.1.d.a 4 87.k even 28 2
2523.1.h.c 24 29.c odd 4 2
2523.1.h.c 24 29.f odd 28 10
2523.1.h.c 24 87.f even 4 2
2523.1.h.c 24 87.k even 28 10
2523.1.j.a 12 29.b even 2 1
2523.1.j.a 12 29.e even 14 5
2523.1.j.a 12 87.d odd 2 1
2523.1.j.a 12 87.h odd 14 5
2523.1.j.c 12 1.a even 1 1 trivial
2523.1.j.c 12 3.b odd 2 1 CM
2523.1.j.c 12 29.d even 7 5 inner
2523.1.j.c 12 87.j odd 14 5 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2523, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{19}^{12} + T_{19}^{11} + 2 T_{19}^{10} + 3 T_{19}^{9} + 5 T_{19}^{8} + 8 T_{19}^{7} + 13 T_{19}^{6} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
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