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Properties

Label 3871.1.cc.a
Level $3871$
Weight $1$
Character orbit 3871.cc
Analytic conductor $1.932$
Analytic rank $0$
Dimension $24$
Projective image $D_{78}$
CM discriminant -7
Inner twists $4$

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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] = [3871,1,Mod(197,3871)] 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("3871.197"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3871, base_ring=CyclotomicField(78)) chi = DirichletCharacter(H, H._module([0, 35])) 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.cc (of order \(78\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.93188066390\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{39})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{78}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{78} - \cdots)\)

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + (\zeta_{78}^{36} - \zeta_{78}^{19}) q^{2} + (\zeta_{78}^{38} + \cdots + \zeta_{78}^{16}) q^{4} + ( - \zeta_{78}^{35} + \cdots - \zeta_{78}^{13}) q^{8} + \zeta_{78}^{4} q^{9} + ( - \zeta_{78}^{26} - \zeta_{78}^{12}) q^{11}+ \cdots + ( - \zeta_{78}^{30} - \zeta_{78}^{16}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - q^{2} - 15 q^{8} + q^{9} + 14 q^{11} - q^{16} + 2 q^{18} + 2 q^{22} + q^{23} - q^{25} + 2 q^{46} + q^{50} + 13 q^{53} - 15 q^{64} - 2 q^{67} + 27 q^{72} - 2 q^{79} + q^{81} + 13 q^{86} - 25 q^{88}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1030\) \(2845\)
\(\chi(n)\) \(-\zeta_{78}^{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−0.996757 + 0.0804666i
0.428693 − 0.903450i
−0.996757 − 0.0804666i
−0.919979 − 0.391967i
−0.632445 + 0.774605i
−0.632445 − 0.774605i
0.948536 − 0.316668i
0.948536 + 0.316668i
0.692724 + 0.721202i
0.799443 + 0.600742i
−0.845190 − 0.534466i
−0.200026 − 0.979791i
−0.0402659 − 0.999189i
0.799443 − 0.600742i
−0.200026 + 0.979791i
−0.0402659 + 0.999189i
0.987050 − 0.160411i
−0.845190 + 0.534466i
0.428693 + 0.903450i
−0.919979 + 0.391967i
−1.01121 + 0.759873i 0 0.166916 − 0.576262i 0 0 0 −0.179438 − 0.473138i 0.948536 − 0.316668i 0
344.1 −1.81613 − 0.773781i 0 2.00687 + 2.08938i 0 0 0 −1.32800 − 3.50165i −0.200026 + 0.979791i 0
393.1 −1.01121 − 0.759873i 0 0.166916 + 0.576262i 0 0 0 −0.179438 + 0.473138i 0.948536 + 0.316668i 0
442.1 −0.554631 − 0.0447744i 0 −0.681440 − 0.110745i 0 0 0 0.913255 + 0.225097i −0.0402659 + 0.999189i 0
540.1 1.31415 − 1.36817i 0 −0.104647 − 2.59678i 0 0 0 −2.27038 − 2.01139i −0.919979 + 0.391967i 0
638.1 1.31415 + 1.36817i 0 −0.104647 + 2.59678i 0 0 0 −2.27038 + 2.01139i −0.919979 − 0.391967i 0
785.1 1.55512 + 0.983395i 0 1.02262 + 2.15513i 0 0 0 −0.307268 + 2.53058i 0.278217 − 0.960518i 0
932.1 1.55512 − 0.983395i 0 1.02262 − 2.15513i 0 0 0 −0.307268 − 2.53058i 0.278217 + 0.960518i 0
1030.1 −1.66849 − 0.271156i 0 1.76180 + 0.588174i 0 0 0 −1.28330 − 0.673526i −0.996757 − 0.0804666i 0
1324.1 0.593932 − 1.25168i 0 −0.581513 − 0.712225i 0 0 0 0.108331 − 0.0267011i −0.845190 + 0.534466i 0
1373.1 0.398754 + 1.95323i 0 −2.73611 + 1.16575i 0 0 0 −2.23556 − 3.23877i −0.632445 + 0.774605i 0
1569.1 −0.0643806 − 1.59759i 0 −1.55139 + 0.125241i 0 0 0 0.107239 + 0.883189i 0.692724 − 0.721202i 0
2010.1 0.813261 − 0.271506i 0 −0.211765 + 0.159131i 0 0 0 −0.616065 + 0.892525i 0.987050 − 0.160411i 0
2108.1 0.593932 + 1.25168i 0 −0.581513 + 0.712225i 0 0 0 0.108331 + 0.0267011i −0.845190 − 0.534466i 0
2255.1 −0.0643806 + 1.59759i 0 −1.55139 − 0.125241i 0 0 0 0.107239 − 0.883189i 0.692724 + 0.721202i 0
2598.1 0.813261 + 0.271506i 0 −0.211765 − 0.159131i 0 0 0 −0.616065 − 0.892525i 0.987050 + 0.160411i 0
2745.1 −0.111301 + 0.384257i 0 0.709925 + 0.448929i 0 0 0 −0.550962 + 0.488110i 0.799443 − 0.600742i 0
2794.1 0.398754 − 1.95323i 0 −2.73611 − 1.16575i 0 0 0 −2.23556 + 3.23877i −0.632445 − 0.774605i 0
2892.1 −1.81613 + 0.773781i 0 2.00687 − 2.08938i 0 0 0 −1.32800 + 3.50165i −0.200026 − 0.979791i 0
3039.1 −0.554631 + 0.0447744i 0 −0.681440 + 0.110745i 0 0 0 0.913255 − 0.225097i −0.0402659 − 0.999189i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
79.h odd 78 1 inner
553.bl even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3871.1.cc.a ✓ 24
7.b odd 2 1 CM 3871.1.cc.a ✓ 24
7.c even 3 1 3871.1.bx.a 24
7.c even 3 1 3871.1.cg.a 24
7.d odd 6 1 3871.1.bx.a 24
7.d odd 6 1 3871.1.cg.a 24
79.h odd 78 1 inner 3871.1.cc.a ✓ 24
553.bd odd 78 1 3871.1.cg.a 24
553.be even 78 1 3871.1.cg.a 24
553.bf even 78 1 3871.1.bx.a 24
553.bl even 78 1 inner 3871.1.cc.a ✓ 24
553.bm odd 78 1 3871.1.bx.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3871.1.bx.a 24 7.c even 3 1
3871.1.bx.a 24 7.d odd 6 1
3871.1.bx.a 24 553.bf even 78 1
3871.1.bx.a 24 553.bm odd 78 1
3871.1.cc.a ✓ 24 1.a even 1 1 trivial
3871.1.cc.a ✓ 24 7.b odd 2 1 CM
3871.1.cc.a ✓ 24 79.h odd 78 1 inner
3871.1.cc.a ✓ 24 553.bl even 78 1 inner
3871.1.cg.a 24 7.c even 3 1
3871.1.cg.a 24 7.d odd 6 1
3871.1.cg.a 24 553.bd odd 78 1
3871.1.cg.a 24 553.be even 78 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3871, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} + T^{23} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{24} \) Copy content Toggle raw display
$5$ \( T^{24} \) Copy content Toggle raw display
$7$ \( T^{24} \) Copy content Toggle raw display
$11$ \( T^{24} - 14 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{24} \) Copy content Toggle raw display
$17$ \( T^{24} \) Copy content Toggle raw display
$19$ \( T^{24} \) Copy content Toggle raw display
$23$ \( T^{24} - T^{23} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{24} - 26 T^{21} + \cdots + 169 \) Copy content Toggle raw display
$31$ \( T^{24} \) Copy content Toggle raw display
$37$ \( T^{24} - 26 T^{21} + \cdots + 169 \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( T^{24} + 26 T^{19} + \cdots + 169 \) Copy content Toggle raw display
$47$ \( T^{24} \) Copy content Toggle raw display
$53$ \( T^{24} - 13 T^{23} + \cdots + 169 \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( T^{24} \) Copy content Toggle raw display
$67$ \( T^{24} + 2 T^{23} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( (T^{12} + 13 T^{5} + \cdots + 13)^{2} \) Copy content Toggle raw display
$73$ \( T^{24} \) Copy content Toggle raw display
$79$ \( (T^{12} + T^{11} + T^{10} + \cdots + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( T^{24} \) Copy content Toggle raw display
$97$ \( T^{24} \) Copy content Toggle raw display
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