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Properties

Label 3751.1.cd.a
Level $3751$
Weight $1$
Character orbit 3751.cd
Analytic conductor $1.872$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3751,1,Mod(112,3751)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3751.112"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3751 = 11^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3751.cd (of order \(30\), degree \(8\), not 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: \(1.87199286239\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 341)
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{30}^{11} - \zeta_{30}^{5}) q^{3} - \zeta_{30}^{3} q^{4} + ( - \zeta_{30}^{11} - \zeta_{30}^{3}) q^{5} + (\zeta_{30}^{10} - \zeta_{30}^{7} - \zeta_{30}) q^{9} + (\zeta_{30}^{14} + \zeta_{30}^{8}) q^{12} + \cdots + ( - \zeta_{30}^{13} + \zeta_{30}^{2}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{3} - 2 q^{4} - q^{5} - 2 q^{9} + 2 q^{12} + 4 q^{15} - 2 q^{16} - q^{20} - 3 q^{23} + 2 q^{27} + q^{31} - 2 q^{36} - q^{37} - 8 q^{45} + 2 q^{47} + 2 q^{48} + q^{49} - 6 q^{53} - q^{59}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3751\mathbb{Z}\right)^\times\).

\(n\) \(2421\) \(2543\)
\(\chi(n)\) \(-\zeta_{30}^{13}\) \(-\zeta_{30}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
112.1
0.913545 − 0.406737i
−0.104528 + 0.994522i
−0.978148 − 0.207912i
0.669131 + 0.743145i
0.913545 + 0.406737i
0.669131 − 0.743145i
−0.104528 − 0.994522i
−0.978148 + 0.207912i
0 −0.604528 + 0.128496i 0.309017 − 0.951057i 0.204489 + 0.0434654i 0 0 0 −0.564602 + 0.251377i 0
475.1 0 0.413545 + 0.459289i 0.309017 − 0.951057i 1.22256 − 1.35779i 0 0 0 0.0646021 − 0.614648i 0
1371.1 0 0.169131 − 1.60917i −0.809017 − 0.587785i −0.139886 − 1.33093i 0 0 0 −1.58268 − 0.336408i 0
1812.1 0 −1.47815 − 0.658114i −0.809017 + 0.587785i −1.78716 + 0.795697i 0 0 0 1.08268 + 1.20243i 0
1909.1 0 −0.604528 − 0.128496i 0.309017 + 0.951057i 0.204489 − 0.0434654i 0 0 0 −0.564602 − 0.251377i 0
2097.1 0 −1.47815 + 0.658114i −0.809017 − 0.587785i −1.78716 − 0.795697i 0 0 0 1.08268 − 1.20243i 0
2756.1 0 0.413545 − 0.459289i 0.309017 + 0.951057i 1.22256 + 1.35779i 0 0 0 0.0646021 + 0.614648i 0
3264.1 0 0.169131 + 1.60917i −0.809017 + 0.587785i −0.139886 + 1.33093i 0 0 0 −1.58268 + 0.336408i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 112.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
341.bk even 15 1 inner
341.bz odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3751.1.cd.a 8
11.b odd 2 1 CM 3751.1.cd.a 8
11.c even 5 1 341.1.by.a ✓ 8
11.c even 5 1 3751.1.br.a 8
11.c even 5 1 3751.1.bx.a 8
11.c even 5 1 3751.1.ce.a 8
11.d odd 10 1 341.1.by.a ✓ 8
11.d odd 10 1 3751.1.br.a 8
11.d odd 10 1 3751.1.bx.a 8
11.d odd 10 1 3751.1.ce.a 8
31.g even 15 1 3751.1.bx.a 8
33.f even 10 1 3069.1.jl.a 8
33.h odd 10 1 3069.1.jl.a 8
341.bg even 15 1 3751.1.ce.a 8
341.bj even 15 1 3751.1.br.a 8
341.bk even 15 1 inner 3751.1.cd.a 8
341.bl even 15 1 341.1.by.a ✓ 8
341.bn odd 30 1 341.1.by.a ✓ 8
341.bt odd 30 1 3751.1.ce.a 8
341.by odd 30 1 3751.1.bx.a 8
341.bz odd 30 1 inner 3751.1.cd.a 8
341.ca odd 30 1 3751.1.br.a 8
1023.ct odd 30 1 3069.1.jl.a 8
1023.ef even 30 1 3069.1.jl.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
341.1.by.a ✓ 8 11.c even 5 1
341.1.by.a ✓ 8 11.d odd 10 1
341.1.by.a ✓ 8 341.bl even 15 1
341.1.by.a ✓ 8 341.bn odd 30 1
3069.1.jl.a 8 33.f even 10 1
3069.1.jl.a 8 33.h odd 10 1
3069.1.jl.a 8 1023.ct odd 30 1
3069.1.jl.a 8 1023.ef even 30 1
3751.1.br.a 8 11.c even 5 1
3751.1.br.a 8 11.d odd 10 1
3751.1.br.a 8 341.bj even 15 1
3751.1.br.a 8 341.ca odd 30 1
3751.1.bx.a 8 11.c even 5 1
3751.1.bx.a 8 11.d odd 10 1
3751.1.bx.a 8 31.g even 15 1
3751.1.bx.a 8 341.by odd 30 1
3751.1.cd.a 8 1.a even 1 1 trivial
3751.1.cd.a 8 11.b odd 2 1 CM
3751.1.cd.a 8 341.bk even 15 1 inner
3751.1.cd.a 8 341.bz odd 30 1 inner
3751.1.ce.a 8 11.c even 5 1
3751.1.ce.a 8 11.d odd 10 1
3751.1.ce.a 8 341.bg even 15 1
3751.1.ce.a 8 341.bt odd 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3751, [\chi])\).

Hecke characteristic polynomials

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