↑

Properties

Label 1376.1.by.a
Level $1376$
Weight $1$
Character orbit 1376.by
Analytic conductor $0.687$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -8
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1376,1,Mod(15,1376)] 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("1376.15"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1376, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 21, 26])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1376 = 2^{5} \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1376.by (of order \(42\), degree \(12\), 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: \(0.686713457383\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 344)
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{42}^{17} + \zeta_{42}^{5}) q^{3} + ( - \zeta_{42}^{13} + \cdots - \zeta_{42}) q^{9} + (\zeta_{42}^{19} + \zeta_{42}^{11}) q^{11} + (\zeta_{42}^{20} + \zeta_{42}^{18}) q^{17} + ( - \zeta_{42}^{12} + \zeta_{42}^{7}) q^{19}+ \cdots + ( - \zeta_{42}^{20} - \zeta_{42}^{12} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 3 q^{9} - 2 q^{11} - q^{17} + 8 q^{19} + q^{25} + 8 q^{27} - 9 q^{33} + 2 q^{41} - q^{43} - 6 q^{49} + 3 q^{51} - 2 q^{57} - 2 q^{59} + q^{67} - q^{73} - 10 q^{75} - 2 q^{81} + 5 q^{83}+ \cdots - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1376\mathbb{Z}\right)^\times\).

\(n\) \(517\) \(1119\) \(1121\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{42}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−0.733052 − 0.680173i
0.365341 + 0.930874i
−0.988831 + 0.149042i
0.955573 − 0.294755i
0.0747301 − 0.997204i
−0.733052 + 0.680173i
0.0747301 + 0.997204i
0.365341 − 0.930874i
0.955573 + 0.294755i
−0.988831 − 0.149042i
0.826239 − 0.563320i
0.826239 + 0.563320i
0 0.162592 − 0.414278i 0 0 0 0 0 0.587862 + 0.545456i 0
111.1 0 −1.03030 − 0.702449i 0 0 0 0 0 0.202749 + 0.516596i 0
143.1 0 −0.0931869 − 1.24349i 0 0 0 0 0 −0.548760 + 0.0827122i 0
239.1 0 −0.440071 + 0.0663300i 0 0 0 0 0 −0.766310 + 0.236375i 0
271.1 0 −1.32091 + 1.22563i 0 0 0 0 0 0.167917 − 2.24070i 0
367.1 0 0.162592 + 0.414278i 0 0 0 0 0 0.587862 − 0.545456i 0
655.1 0 −1.32091 − 1.22563i 0 0 0 0 0 0.167917 + 2.24070i 0
719.1 0 −1.03030 + 0.702449i 0 0 0 0 0 0.202749 − 0.516596i 0
783.1 0 −0.440071 − 0.0663300i 0 0 0 0 0 −0.766310 − 0.236375i 0
943.1 0 −0.0931869 + 1.24349i 0 0 0 0 0 −0.548760 − 0.0827122i 0
1135.1 0 1.72188 − 0.531130i 0 0 0 0 0 1.85654 − 1.26577i 0
1199.1 0 1.72188 + 0.531130i 0 0 0 0 0 1.85654 + 1.26577i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
43.g even 21 1 inner
344.be odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1376.1.by.a 12
4.b odd 2 1 344.1.be.a ✓ 12
8.b even 2 1 344.1.be.a ✓ 12
8.d odd 2 1 CM 1376.1.by.a 12
12.b even 2 1 3096.1.eq.a 12
24.h odd 2 1 3096.1.eq.a 12
43.g even 21 1 inner 1376.1.by.a 12
172.o odd 42 1 344.1.be.a ✓ 12
344.z even 42 1 344.1.be.a ✓ 12
344.be odd 42 1 inner 1376.1.by.a 12
516.bb even 42 1 3096.1.eq.a 12
1032.cf odd 42 1 3096.1.eq.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
344.1.be.a ✓ 12 4.b odd 2 1
344.1.be.a ✓ 12 8.b even 2 1
344.1.be.a ✓ 12 172.o odd 42 1
344.1.be.a ✓ 12 344.z even 42 1
1376.1.by.a 12 1.a even 1 1 trivial
1376.1.by.a 12 8.d odd 2 1 CM
1376.1.by.a 12 43.g even 21 1 inner
1376.1.by.a 12 344.be odd 42 1 inner
3096.1.eq.a 12 12.b even 2 1
3096.1.eq.a 12 24.h odd 2 1
3096.1.eq.a 12 516.bb even 42 1
3096.1.eq.a 12 1032.cf odd 42 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1376, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} - 2 T^{11} + \cdots + 1 \) 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
$59$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{12} \) 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
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} - 5 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
show more
show less