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Properties

Label 3871.1.m.f
Level $3871$
Weight $1$
Character orbit 3871.m
Analytic conductor $1.932$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -79
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3871,1,Mod(1500,3871)] 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("3871.1500"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3871, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3871 = 7^{2} \cdot 79 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3871.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-4,0,-4,0] 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: \(1.93188066390\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.6879707136000000000000.5
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8x^{14} + 45x^{12} + 128x^{10} + 264x^{8} + 212x^{6} + 125x^{4} + 12x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{7} + \beta_{4}) q^{2} + ( - \beta_{8} - \beta_{7}) q^{4} + \beta_{3} q^{5} + q^{8} + (\beta_{8} - 1) q^{9} + ( - \beta_{15} + \beta_{11} + \cdots - \beta_{3}) q^{10} + ( - \beta_{10} + \beta_{6}) q^{11}+ \cdots - \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} - 4 q^{4} + 16 q^{8} - 8 q^{9} - 4 q^{18} - 8 q^{25} + 8 q^{32} + 8 q^{36} + 8 q^{50} - 8 q^{64} + 4 q^{65} - 8 q^{72} + 8 q^{79} - 8 q^{81} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 8x^{14} + 45x^{12} + 128x^{10} + 264x^{8} + 212x^{6} + 125x^{4} + 12x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\nu^{14} + 73\nu^{12} + 400\nu^{10} + 1000\nu^{8} + 1940\nu^{6} + 520\nu^{4} + 50\nu^{2} - 1587 ) / 836 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\nu^{15} + 73\nu^{13} + 400\nu^{11} + 1000\nu^{9} + 1940\nu^{7} + 520\nu^{5} + 50\nu^{3} - 2423\nu ) / 836 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{14} + 22\nu^{12} + 120\nu^{10} + 300\nu^{8} + 560\nu^{6} + 156\nu^{4} + 15\nu^{2} - 122 ) / 76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43\nu^{15} + 315\nu^{13} + 1720\nu^{11} + 4300\nu^{9} + 8100\nu^{7} + 2236\nu^{5} + 215\nu^{3} - 3765\nu ) / 836 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 48\nu^{14} + 357\nu^{12} + 1920\nu^{10} + 4800\nu^{8} + 8696\nu^{6} + 2496\nu^{4} + 240\nu^{2} - 967 ) / 836 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -39\nu^{14} - 310\nu^{12} - 1736\nu^{10} - 4890\nu^{8} - 9920\nu^{6} - 7440\nu^{4} - 3363\nu^{2} - 62 ) / 418 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -85\nu^{14} - 670\nu^{12} - 3752\nu^{10} - 10480\nu^{8} - 21440\nu^{6} - 16080\nu^{4} - 10105\nu^{2} - 134 ) / 836 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -85\nu^{15} - 670\nu^{13} - 3752\nu^{11} - 10480\nu^{9} - 21440\nu^{7} - 16080\nu^{5} - 10105\nu^{3} - 134\nu ) / 836 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -43\nu^{14} - 359\nu^{12} - 2050\nu^{10} - 6104\nu^{8} - 12852\nu^{6} - 11652\nu^{4} - 6155\nu^{2} - 591 ) / 418 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -62\nu^{15} - 457\nu^{13} - 2480\nu^{11} - 6200\nu^{9} - 11478\nu^{7} - 3224\nu^{5} - 310\nu^{3} + 2619\nu ) / 418 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -160\nu^{14} - 1267\nu^{12} - 7104\nu^{10} - 19960\nu^{8} - 40940\nu^{6} - 31640\nu^{4} - 19324\nu^{2} - 1855 ) / 836 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 245 \nu^{15} - 1937 \nu^{13} - 10856 \nu^{11} - 30440 \nu^{9} - 62380 \nu^{7} - 47720 \nu^{5} + \cdots - 2825 \nu ) / 836 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19\nu^{15} + 150\nu^{13} + 840\nu^{11} + 2350\nu^{9} + 4800\nu^{7} + 3600\nu^{5} + 2105\nu^{3} + 30\nu ) / 38 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 552 \nu^{15} - 4375 \nu^{13} - 24500 \nu^{11} - 68708 \nu^{9} - 140000 \nu^{7} - 105000 \nu^{5} + \cdots - 875 \nu ) / 836 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - 2\beta_{8} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - 3\beta_{9} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{12} + 7\beta_{8} + \beta_{7} + \beta_{4} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{14} - 5\beta_{13} + 10\beta_{9} + \beta_{5} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 6\beta_{4} + 15\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{11} - 8\beta_{5} + 22\beta_{3} + 34\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 56\beta_{12} + 8\beta_{10} - 98\beta_{8} - 29\beta_{7} - 8\beta_{6} - 56\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8\beta_{15} + 37\beta_{14} + 85\beta_{13} - 125\beta_{9} - 85\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -210\beta_{12} - 45\beta_{10} + 372\beta_{8} + 130\beta_{7} + 130\beta_{4} - 242 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 45 \beta_{15} - 175 \beta_{14} - 340 \beta_{13} + 45 \beta_{11} + 452 \beta_{9} + 220 \beta_{5} + \cdots - 407 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 220\beta_{6} - 560\beta_{4} + 792\beta_{2} + 859 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -220\beta_{11} - 1000\beta_{5} + 1572\beta_{3} + 1431\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3003\beta_{12} + 1000\beta_{10} - 5434\beta_{8} - 2352\beta_{7} - 1000\beta_{6} - 3003\beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1000\beta_{15} + 3352\beta_{14} + 5355\beta_{13} - 6085\beta_{9} - 5355\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3871\mathbb{Z}\right)^\times\).

\(n\) \(1030\) \(2845\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1500.1
−0.987688 + 1.71073i
0.156434 − 0.270952i
−0.156434 + 0.270952i
0.987688 − 1.71073i
0.453990 − 0.786335i
−0.891007 + 1.54327i
0.891007 − 1.54327i
−0.453990 + 0.786335i
−0.987688 − 1.71073i
0.156434 + 0.270952i
−0.156434 − 0.270952i
0.987688 + 1.71073i
0.453990 + 0.786335i
−0.891007 − 1.54327i
0.891007 + 1.54327i
−0.453990 − 0.786335i
−0.809017 + 1.40126i 0 −0.809017 − 1.40126i −0.891007 + 1.54327i 0 0 1.00000 −0.500000 + 0.866025i −1.44168 − 2.49706i
1500.2 −0.809017 + 1.40126i 0 −0.809017 − 1.40126i −0.453990 + 0.786335i 0 0 1.00000 −0.500000 + 0.866025i −0.734572 − 1.27232i
1500.3 −0.809017 + 1.40126i 0 −0.809017 − 1.40126i 0.453990 − 0.786335i 0 0 1.00000 −0.500000 + 0.866025i 0.734572 + 1.27232i
1500.4 −0.809017 + 1.40126i 0 −0.809017 − 1.40126i 0.891007 − 1.54327i 0 0 1.00000 −0.500000 + 0.866025i 1.44168 + 2.49706i
1500.5 0.309017 − 0.535233i 0 0.309017 + 0.535233i −0.987688 + 1.71073i 0 0 1.00000 −0.500000 + 0.866025i 0.610425 + 1.05729i
1500.6 0.309017 − 0.535233i 0 0.309017 + 0.535233i −0.156434 + 0.270952i 0 0 1.00000 −0.500000 + 0.866025i 0.0966818 + 0.167458i
1500.7 0.309017 − 0.535233i 0 0.309017 + 0.535233i 0.156434 − 0.270952i 0 0 1.00000 −0.500000 + 0.866025i −0.0966818 − 0.167458i
1500.8 0.309017 − 0.535233i 0 0.309017 + 0.535233i 0.987688 − 1.71073i 0 0 1.00000 −0.500000 + 0.866025i −0.610425 − 1.05729i
3791.1 −0.809017 − 1.40126i 0 −0.809017 + 1.40126i −0.891007 − 1.54327i 0 0 1.00000 −0.500000 − 0.866025i −1.44168 + 2.49706i
3791.2 −0.809017 − 1.40126i 0 −0.809017 + 1.40126i −0.453990 − 0.786335i 0 0 1.00000 −0.500000 − 0.866025i −0.734572 + 1.27232i
3791.3 −0.809017 − 1.40126i 0 −0.809017 + 1.40126i 0.453990 + 0.786335i 0 0 1.00000 −0.500000 − 0.866025i 0.734572 − 1.27232i
3791.4 −0.809017 − 1.40126i 0 −0.809017 + 1.40126i 0.891007 + 1.54327i 0 0 1.00000 −0.500000 − 0.866025i 1.44168 − 2.49706i
3791.5 0.309017 + 0.535233i 0 0.309017 − 0.535233i −0.987688 − 1.71073i 0 0 1.00000 −0.500000 − 0.866025i 0.610425 − 1.05729i
3791.6 0.309017 + 0.535233i 0 0.309017 − 0.535233i −0.156434 − 0.270952i 0 0 1.00000 −0.500000 − 0.866025i 0.0966818 − 0.167458i
3791.7 0.309017 + 0.535233i 0 0.309017 − 0.535233i 0.156434 + 0.270952i 0 0 1.00000 −0.500000 − 0.866025i −0.0966818 + 0.167458i
3791.8 0.309017 + 0.535233i 0 0.309017 − 0.535233i 0.987688 + 1.71073i 0 0 1.00000 −0.500000 − 0.866025i −0.610425 + 1.05729i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1500.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.b odd 2 1 CM by \(\Q(\sqrt{-79}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
553.d even 2 1 inner
553.l even 6 1 inner
553.m odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3871.1.m.f 16
7.b odd 2 1 inner 3871.1.m.f 16
7.c even 3 1 3871.1.c.e ✓ 8
7.c even 3 1 inner 3871.1.m.f 16
7.d odd 6 1 3871.1.c.e ✓ 8
7.d odd 6 1 inner 3871.1.m.f 16
79.b odd 2 1 CM 3871.1.m.f 16
553.d even 2 1 inner 3871.1.m.f 16
553.l even 6 1 3871.1.c.e ✓ 8
553.l even 6 1 inner 3871.1.m.f 16
553.m odd 6 1 3871.1.c.e ✓ 8
553.m odd 6 1 inner 3871.1.m.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3871.1.c.e ✓ 8 7.c even 3 1
3871.1.c.e ✓ 8 7.d odd 6 1
3871.1.c.e ✓ 8 553.l even 6 1
3871.1.c.e ✓ 8 553.m odd 6 1
3871.1.m.f 16 1.a even 1 1 trivial
3871.1.m.f 16 7.b odd 2 1 inner
3871.1.m.f 16 7.c even 3 1 inner
3871.1.m.f 16 7.d odd 6 1 inner
3871.1.m.f 16 79.b odd 2 1 CM
3871.1.m.f 16 553.d even 2 1 inner
3871.1.m.f 16 553.l even 6 1 inner
3871.1.m.f 16 553.m odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3871, [\chi])\):

\( T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{16} + 8T_{5}^{14} + 45T_{5}^{12} + 128T_{5}^{10} + 264T_{5}^{8} + 212T_{5}^{6} + 125T_{5}^{4} + 12T_{5}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + 2 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 5 T^{6} + 20 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{8} + 5 T^{6} + 20 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{8} + 5 T^{6} + 20 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$89$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{8} - 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
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