Properties

Label 387.2.y.a
Level $387$
Weight $2$
Character orbit 387.y
Analytic conductor $3.090$
Analytic rank $0$
Dimension $12$
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] = [387,2,Mod(10,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.y (of order \(21\), 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: \(3.09021055822\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{21}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{21}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{21}^{3} q^{4} + ( - \zeta_{21}^{11} + \zeta_{21}^{10} + \zeta_{21}^{7} + 3 \zeta_{21}^{6} - \zeta_{21}^{5} + \zeta_{21}^{3} - \zeta_{21}^{2} + 3 \zeta_{21} + 1) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{21}^{3} q^{4} + ( - \zeta_{21}^{11} + \zeta_{21}^{10} + \zeta_{21}^{7} + 3 \zeta_{21}^{6} - \zeta_{21}^{5} + \zeta_{21}^{3} - \zeta_{21}^{2} + 3 \zeta_{21} + 1) q^{7} + ( - 5 \zeta_{21}^{11} + \zeta_{21}^{9} - \zeta_{21}^{8} + \zeta_{21}^{6} - 4 \zeta_{21}^{5} - 2 \zeta_{21}^{4} + \zeta_{21}^{3} + \cdots + 1) q^{13} + \cdots + (3 \zeta_{21}^{11} - 8 \zeta_{21}^{10} - 3 \zeta_{21}^{9} + 3 \zeta_{21}^{8} - 3 \zeta_{21}^{6} - 8 \zeta_{21}^{5} + \cdots - 3) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{4} - q^{7} - 7 q^{13} - 8 q^{16} + 8 q^{19} - 5 q^{25} + 2 q^{28} + 70 q^{31} + 11 q^{37} - 8 q^{43} - 55 q^{49} - 56 q^{52} + 14 q^{61} + 16 q^{64} + 5 q^{67} - 7 q^{73} - 16 q^{76} - 4 q^{79} + 7 q^{91} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(173\)
\(\chi(n)\) \(\zeta_{21}^{11}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−0.988831 + 0.149042i
0.955573 0.294755i
0.0747301 0.997204i
−0.733052 + 0.680173i
0.826239 + 0.563320i
0.365341 + 0.930874i
−0.988831 0.149042i
−0.733052 0.680173i
0.955573 + 0.294755i
0.0747301 + 0.997204i
0.826239 0.563320i
0.365341 0.930874i
0 0 1.80194 0.867767i 0 0 −1.71951 2.97828i 0 0 0
100.1 0 0 −1.24698 + 1.56366i 0 0 2.42168 4.19447i 0 0 0
109.1 0 0 0.445042 1.94986i 0 0 −1.57775 2.73274i 0 0 0
154.1 0 0 −1.24698 1.56366i 0 0 −2.64420 + 4.57988i 0 0 0
181.1 0 0 0.445042 1.94986i 0 0 0.676779 1.17221i 0 0 0
253.1 0 0 1.80194 + 0.867767i 0 0 2.34300 + 4.05820i 0 0 0
271.1 0 0 1.80194 + 0.867767i 0 0 −1.71951 + 2.97828i 0 0 0
289.1 0 0 −1.24698 + 1.56366i 0 0 −2.64420 4.57988i 0 0 0
298.1 0 0 −1.24698 1.56366i 0 0 2.42168 + 4.19447i 0 0 0
316.1 0 0 0.445042 + 1.94986i 0 0 −1.57775 + 2.73274i 0 0 0
325.1 0 0 0.445042 + 1.94986i 0 0 0.676779 + 1.17221i 0 0 0
361.1 0 0 1.80194 0.867767i 0 0 2.34300 4.05820i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
43.g even 21 1 inner
129.o odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.y.a 12
3.b odd 2 1 CM 387.2.y.a 12
43.g even 21 1 inner 387.2.y.a 12
129.o odd 42 1 inner 387.2.y.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.y.a 12 1.a even 1 1 trivial
387.2.y.a 12 3.b odd 2 1 CM
387.2.y.a 12 43.g even 21 1 inner
387.2.y.a 12 129.o odd 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + 49 T^{10} + \cdots + 3108169 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 7 T^{11} + 294 T^{9} + \cdots + 82369 \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} - 552 T^{9} + \cdots + 51566761 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 70 T^{11} + \cdots + 2241170281 \) Copy content Toggle raw display
$37$ \( T^{12} - 11 T^{11} + \cdots + 3129619249 \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} + 8 T^{11} + \cdots + 6321363049 \) 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
$61$ \( T^{12} - 14 T^{11} + \cdots + 63775946521 \) Copy content Toggle raw display
$67$ \( T^{12} - 5 T^{11} + \cdots + 658842279481 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + 7 T^{11} + \cdots + 31218295969 \) Copy content Toggle raw display
$79$ \( T^{12} + 4 T^{11} + \cdots + 4799433759121 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + 28 T^{11} + \cdots + 765024867649 \) Copy content Toggle raw display
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