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Properties

Label 387.2.h.c
Level $387$
Weight $2$
Character orbit 387.h
Analytic conductor $3.090$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [387,2,Mod(208,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.h (of order \(3\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 129)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 2 q^{4} + ( - 2 \zeta_{6} + 2) q^{5} - 3 \zeta_{6} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 2 q^{4} + ( - 2 \zeta_{6} + 2) q^{5} - 3 \zeta_{6} q^{7} + ( - 4 \zeta_{6} + 4) q^{10} + 6 q^{11} - \zeta_{6} q^{13} - 6 \zeta_{6} q^{14} - 4 q^{16} + 6 \zeta_{6} q^{17} + (4 \zeta_{6} - 4) q^{19} + ( - 4 \zeta_{6} + 4) q^{20} + 12 q^{22} + (4 \zeta_{6} - 4) q^{23} + \zeta_{6} q^{25} - 2 \zeta_{6} q^{26} - 6 \zeta_{6} q^{28} - 6 \zeta_{6} q^{29} + (5 \zeta_{6} - 5) q^{31} - 8 q^{32} + 12 \zeta_{6} q^{34} - 6 q^{35} + (3 \zeta_{6} - 3) q^{37} + (8 \zeta_{6} - 8) q^{38} - 2 q^{41} + ( - 6 \zeta_{6} - 1) q^{43} + 12 q^{44} + (8 \zeta_{6} - 8) q^{46} + 8 q^{47} + (2 \zeta_{6} - 2) q^{49} + 2 \zeta_{6} q^{50} - 2 \zeta_{6} q^{52} + ( - 10 \zeta_{6} + 10) q^{53} + ( - 12 \zeta_{6} + 12) q^{55} - 12 \zeta_{6} q^{58} - 2 \zeta_{6} q^{61} + (10 \zeta_{6} - 10) q^{62} - 8 q^{64} - 2 q^{65} + (9 \zeta_{6} - 9) q^{67} + 12 \zeta_{6} q^{68} - 12 q^{70} + 2 \zeta_{6} q^{71} + 7 \zeta_{6} q^{73} + (6 \zeta_{6} - 6) q^{74} + (8 \zeta_{6} - 8) q^{76} - 18 \zeta_{6} q^{77} - 4 \zeta_{6} q^{79} + (8 \zeta_{6} - 8) q^{80} - 4 q^{82} + (12 \zeta_{6} - 12) q^{83} + 12 q^{85} + ( - 12 \zeta_{6} - 2) q^{86} + ( - 8 \zeta_{6} + 8) q^{89} + (3 \zeta_{6} - 3) q^{91} + (8 \zeta_{6} - 8) q^{92} + 16 q^{94} + 8 \zeta_{6} q^{95} + 10 q^{97} + (4 \zeta_{6} - 4) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 4 q^{4} + 2 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 4 q^{4} + 2 q^{5} - 3 q^{7} + 4 q^{10} + 12 q^{11} - q^{13} - 6 q^{14} - 8 q^{16} + 6 q^{17} - 4 q^{19} + 4 q^{20} + 24 q^{22} - 4 q^{23} + q^{25} - 2 q^{26} - 6 q^{28} - 6 q^{29} - 5 q^{31} - 16 q^{32} + 12 q^{34} - 12 q^{35} - 3 q^{37} - 8 q^{38} - 4 q^{41} - 8 q^{43} + 24 q^{44} - 8 q^{46} + 16 q^{47} - 2 q^{49} + 2 q^{50} - 2 q^{52} + 10 q^{53} + 12 q^{55} - 12 q^{58} - 2 q^{61} - 10 q^{62} - 16 q^{64} - 4 q^{65} - 9 q^{67} + 12 q^{68} - 24 q^{70} + 2 q^{71} + 7 q^{73} - 6 q^{74} - 8 q^{76} - 18 q^{77} - 4 q^{79} - 8 q^{80} - 8 q^{82} - 12 q^{83} + 24 q^{85} - 16 q^{86} + 8 q^{89} - 3 q^{91} - 8 q^{92} + 32 q^{94} + 8 q^{95} + 20 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(46\) \(173\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
0.500000 − 0.866025i
0.500000 + 0.866025i
2.00000 0 2.00000 1.00000 + 1.73205i 0 −1.50000 + 2.59808i 0 0 2.00000 + 3.46410i
307.1 2.00000 0 2.00000 1.00000 − 1.73205i 0 −1.50000 − 2.59808i 0 0 2.00000 − 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.h.c 2
3.b odd 2 1 129.2.e.a ✓ 2
12.b even 2 1 2064.2.q.f 2
43.c even 3 1 inner 387.2.h.c 2
129.f odd 6 1 129.2.e.a ✓ 2
129.f odd 6 1 5547.2.a.a 1
129.h even 6 1 5547.2.a.d 1
516.p even 6 1 2064.2.q.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.2.e.a ✓ 2 3.b odd 2 1
129.2.e.a ✓ 2 129.f odd 6 1
387.2.h.c 2 1.a even 1 1 trivial
387.2.h.c 2 43.c even 3 1 inner
2064.2.q.f 2 12.b even 2 1
2064.2.q.f 2 516.p even 6 1
5547.2.a.a 1 129.f odd 6 1
5547.2.a.d 1 129.h even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 43 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$71$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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