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Properties

Label 381.1.t.a
Level $381$
Weight $1$
Character orbit 381.t
Analytic conductor $0.190$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [381,1,Mod(38,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("381.38");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 381.t (of order \(42\), degree \(12\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.190143769813\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
comment: defining polynomial
 
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q - \zeta_{42}^{11} q^{3} - \zeta_{42}^{9} q^{4} + ( - \zeta_{42}^{17} + \zeta_{42}^{14}) q^{7} - \zeta_{42} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{42}^{11} q^{3} - \zeta_{42}^{9} q^{4} + ( - \zeta_{42}^{17} + \zeta_{42}^{14}) q^{7} - \zeta_{42} q^{9} + \zeta_{42}^{20} q^{12} + (\zeta_{42}^{16} - \zeta_{42}^{15}) q^{13} + \zeta_{42}^{18} q^{16} + ( - \zeta_{42}^{13} + \zeta_{42}^{8}) q^{19} + ( - \zeta_{42}^{7} + \zeta_{42}^{4}) q^{21} - \zeta_{42}^{3} q^{25} + \zeta_{42}^{12} q^{27} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{28} + ( - \zeta_{42}^{19} + \zeta_{42}^{6}) q^{31} + \zeta_{42}^{10} q^{36} + (\zeta_{42}^{12} + \zeta_{42}^{2}) q^{37} + (\zeta_{42}^{6} - \zeta_{42}^{5}) q^{39} + ( - \zeta_{42}^{19} + \zeta_{42}^{10}) q^{43} + \zeta_{42}^{8} q^{48} + ( - \zeta_{42}^{13} + \cdots - \zeta_{42}^{7}) q^{49} + \cdots + (\zeta_{42}^{20} + \zeta_{42}^{14}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{3} - 2 q^{4} - 5 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{3} - 2 q^{4} - 5 q^{7} + q^{9} + q^{12} - q^{13} - 2 q^{16} + 2 q^{19} - 5 q^{21} - 2 q^{25} - 2 q^{27} + 2 q^{28} - q^{31} + q^{36} - q^{37} - q^{39} + 2 q^{43} + q^{48} - 4 q^{49} - q^{52} - q^{57} + 2 q^{61} - 4 q^{63} - 2 q^{64} - q^{67} - 5 q^{73} - 6 q^{75} + 2 q^{76} - q^{79} + q^{81} + 2 q^{84} - 2 q^{91} - q^{93} - 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/381\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(130\)
\(\chi(n)\) \(-1\) \(-\zeta_{42}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1
0.0747301 + 0.997204i
−0.988831 + 0.149042i
0.955573 − 0.294755i
0.365341 + 0.930874i
−0.733052 + 0.680173i
−0.733052 − 0.680173i
0.826239 + 0.563320i
0.365341 − 0.930874i
0.955573 + 0.294755i
−0.988831 − 0.149042i
0.826239 − 0.563320i
0.0747301 − 0.997204i
0 −0.733052 − 0.680173i 0.623490 + 0.781831i 0 0 0.455573 + 1.16078i 0 0.0747301 + 0.997204i 0
47.1 0 0.0747301 + 0.997204i −0.222521 + 0.974928i 0 0 0.326239 − 0.302705i 0 −0.988831 + 0.149042i 0
50.1 0 −0.988831 + 0.149042i −0.900969 − 0.433884i 0 0 −0.134659 + 1.79690i 0 0.955573 − 0.294755i 0
122.1 0 0.826239 + 0.563320i −0.222521 − 0.974928i 0 0 −0.425270 + 0.131178i 0 0.365341 + 0.930874i 0
152.1 0 0.365341 + 0.930874i −0.900969 + 0.433884i 0 0 −1.48883 + 1.01507i 0 −0.733052 + 0.680173i 0
188.1 0 0.365341 − 0.930874i −0.900969 − 0.433884i 0 0 −1.48883 − 1.01507i 0 −0.733052 − 0.680173i 0
200.1 0 0.955573 + 0.294755i 0.623490 − 0.781831i 0 0 −1.23305 + 0.185853i 0 0.826239 + 0.563320i 0
203.1 0 0.826239 − 0.563320i −0.222521 + 0.974928i 0 0 −0.425270 − 0.131178i 0 0.365341 − 0.930874i 0
221.1 0 −0.988831 − 0.149042i −0.900969 + 0.433884i 0 0 −0.134659 − 1.79690i 0 0.955573 + 0.294755i 0
227.1 0 0.0747301 − 0.997204i −0.222521 − 0.974928i 0 0 0.326239 + 0.302705i 0 −0.988831 − 0.149042i 0
341.1 0 0.955573 − 0.294755i 0.623490 + 0.781831i 0 0 −1.23305 − 0.185853i 0 0.826239 − 0.563320i 0
371.1 0 −0.733052 + 0.680173i 0.623490 − 0.781831i 0 0 0.455573 − 1.16078i 0 0.0747301 − 0.997204i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
127.i even 21 1 inner
381.t odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.1.t.a ✓ 12
3.b odd 2 1 CM 381.1.t.a ✓ 12
127.i even 21 1 inner 381.1.t.a ✓ 12
381.t odd 42 1 inner 381.1.t.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.1.t.a ✓ 12 1.a even 1 1 trivial
381.1.t.a ✓ 12 3.b odd 2 1 CM
381.1.t.a ✓ 12 127.i even 21 1 inner
381.1.t.a ✓ 12 381.t odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(381, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 5 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} - T^{5} - 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} - 2 T^{11} + \cdots + 1 \) 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
$73$ \( T^{12} + 5 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + 5 T^{11} + \cdots + 1 \) Copy content Toggle raw display
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