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Properties

Label 3872.1.r.b
Level $3872$
Weight $1$
Character orbit 3872.r
Analytic conductor $1.932$
Analytic rank $0$
Dimension $16$
Projective image $D_{12}$
CM discriminant -4
Inner twists $16$

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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] = [3872,1,Mod(161,3872)] 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("3872.161"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3872, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 7])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3872 = 2^{5} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3872.r (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.93237972891\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: 16.0.6879707136000000000000.7
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of 12.0.18698185645162496.1

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{14} - \beta_{10} + \cdots + \beta_{2}) q^{5} + \beta_{4} q^{9} - \beta_{12} q^{13} - \beta_{7} q^{17} + ( - 2 \beta_{13} + 2 \beta_{8} + \cdots - 2) q^{25} + (\beta_{15} + \beta_{12} + \cdots - \beta_{3}) q^{29}+ \cdots + \beta_{6} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{9} - 8 q^{25} - 4 q^{37} - 4 q^{49} - 4 q^{53} - 4 q^{81} - 16 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 362 ) / 209 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 780\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} + 780\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{13} + 780\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{12} - 1351\nu^{2} ) / 209 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -4\nu^{13} - 2911\nu^{3} ) / 209 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{14} - 2911\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -4\nu^{15} - 2911\nu^{5} ) / 209 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{14} - 5042\nu^{4} ) / 209 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{15} - 723\nu^{5} ) / 19 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 60\nu^{15} - 225\nu^{13} + 840\nu^{11} - 3135\nu^{9} + 11704\nu^{7} - 225\nu^{5} + 60\nu^{3} - 15\nu ) / 209 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 56\nu^{14} - 224\nu^{12} + 840\nu^{10} - 3135\nu^{8} + 11704\nu^{6} - 3136\nu^{4} + 840\nu^{2} - 224 ) / 209 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 104\nu^{14} - 390\nu^{12} + 1456\nu^{10} - 5434\nu^{8} + 20273\nu^{6} - 390\nu^{4} + 104\nu^{2} - 26 ) / 209 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 164\nu^{15} - 615\nu^{13} + 2296\nu^{11} - 8569\nu^{9} + 31977\nu^{7} - 615\nu^{5} + 164\nu^{3} - 41\nu ) / 209 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{10} - 7\beta_{8} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - 11\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{14} + 26\beta_{13} - 26\beta_{8} - 26\beta_{4} + 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -15\beta_{15} + 41\beta_{12} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -56\beta_{14} + 97\beta_{13} - 56\beta_{10} + 56\beta_{6} + 56\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -56\beta_{15} + 153\beta_{12} - 56\beta_{11} + 153\beta_{9} + 56\beta_{7} + 209\beta_{5} + 56\beta_{3} - 209\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 209\beta_{2} - 362 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 209\beta_{3} - 780\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -780\beta_{6} - 1351\beta_{4} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -780\beta_{7} - 2911\beta_{5} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -2911\beta_{10} + 5042\beta_{8} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -2911\beta_{11} + 7953\beta_{9} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3872\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(1695\) \(2785\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−1.83730 − 0.596975i
1.83730 + 0.596975i
−0.492303 − 0.159959i
0.492303 + 0.159959i
−1.83730 + 0.596975i
1.83730 − 0.596975i
−0.492303 + 0.159959i
0.492303 − 0.159959i
0.304260 + 0.418778i
−0.304260 − 0.418778i
1.13551 + 1.56290i
−1.13551 − 1.56290i
0.304260 − 0.418778i
−0.304260 + 0.418778i
1.13551 − 1.56290i
−1.13551 + 1.56290i
0 0 0 −1.40126 + 1.01807i 0 0 0 0.809017 + 0.587785i 0
161.2 0 0 0 −1.40126 + 1.01807i 0 0 0 0.809017 + 0.587785i 0
161.3 0 0 0 1.40126 − 1.01807i 0 0 0 0.809017 + 0.587785i 0
161.4 0 0 0 1.40126 − 1.01807i 0 0 0 0.809017 + 0.587785i 0
481.1 0 0 0 −1.40126 − 1.01807i 0 0 0 0.809017 − 0.587785i 0
481.2 0 0 0 −1.40126 − 1.01807i 0 0 0 0.809017 − 0.587785i 0
481.3 0 0 0 1.40126 + 1.01807i 0 0 0 0.809017 − 0.587785i 0
481.4 0 0 0 1.40126 + 1.01807i 0 0 0 0.809017 − 0.587785i 0
3137.1 0 0 0 −0.535233 − 1.64728i 0 0 0 −0.309017 + 0.951057i 0
3137.2 0 0 0 −0.535233 − 1.64728i 0 0 0 −0.309017 + 0.951057i 0
3137.3 0 0 0 0.535233 + 1.64728i 0 0 0 −0.309017 + 0.951057i 0
3137.4 0 0 0 0.535233 + 1.64728i 0 0 0 −0.309017 + 0.951057i 0
3361.1 0 0 0 −0.535233 + 1.64728i 0 0 0 −0.309017 − 0.951057i 0
3361.2 0 0 0 −0.535233 + 1.64728i 0 0 0 −0.309017 − 0.951057i 0
3361.3 0 0 0 0.535233 − 1.64728i 0 0 0 −0.309017 − 0.951057i 0
3361.4 0 0 0 0.535233 − 1.64728i 0 0 0 −0.309017 − 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
44.c even 2 1 inner
44.g even 10 3 inner
44.h odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3872.1.r.b 16
4.b odd 2 1 CM 3872.1.r.b 16
11.b odd 2 1 inner 3872.1.r.b 16
11.c even 5 1 3872.1.h.b ✓ 4
11.c even 5 3 inner 3872.1.r.b 16
11.d odd 10 1 3872.1.h.b ✓ 4
11.d odd 10 3 inner 3872.1.r.b 16
44.c even 2 1 inner 3872.1.r.b 16
44.g even 10 1 3872.1.h.b ✓ 4
44.g even 10 3 inner 3872.1.r.b 16
44.h odd 10 1 3872.1.h.b ✓ 4
44.h odd 10 3 inner 3872.1.r.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3872.1.h.b ✓ 4 11.c even 5 1
3872.1.h.b ✓ 4 11.d odd 10 1
3872.1.h.b ✓ 4 44.g even 10 1
3872.1.h.b ✓ 4 44.h odd 10 1
3872.1.r.b 16 1.a even 1 1 trivial
3872.1.r.b 16 4.b odd 2 1 CM
3872.1.r.b 16 11.b odd 2 1 inner
3872.1.r.b 16 11.c even 5 3 inner
3872.1.r.b 16 11.d odd 10 3 inner
3872.1.r.b 16 44.c even 2 1 inner
3872.1.r.b 16 44.g even 10 3 inner
3872.1.r.b 16 44.h odd 10 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 3T_{5}^{6} + 9T_{5}^{4} + 27T_{5}^{2} + 81 \) acting on \(S_{1}^{\mathrm{new}}(3872, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} - 2 T^{6} + 4 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( (T + 1)^{16} \) Copy content Toggle raw display
$97$ \( (T^{8} + 3 T^{6} + 9 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
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