Properties

Label 78.2.e.a
Level $78$
Weight $2$
Character orbit 78.e
Analytic conductor $0.623$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [78,2,Mod(55,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, 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("78.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.e (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: \(0.622833135766\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + 3 q^{5} + \zeta_{6} q^{6} - 2 \zeta_{6} q^{7} - q^{8} - \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{10} + (6 \zeta_{6} - 6) q^{11} + \cdots + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} + 6 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - q^{9} + 3 q^{10} - 6 q^{11} + 2 q^{12} - 7 q^{13} - 4 q^{14} - 3 q^{15} - q^{16} + 3 q^{17} - 2 q^{18} - 2 q^{19} - 3 q^{20} + 4 q^{21}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 3.00000 0.500000 0.866025i −1.00000 + 1.73205i −1.00000 −0.500000 + 0.866025i 1.50000 + 2.59808i
61.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 3.00000 0.500000 + 0.866025i −1.00000 1.73205i −1.00000 −0.500000 0.866025i 1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.e.a 2
3.b odd 2 1 234.2.h.a 2
4.b odd 2 1 624.2.q.g 2
5.b even 2 1 1950.2.i.m 2
5.c odd 4 2 1950.2.z.g 4
12.b even 2 1 1872.2.t.c 2
13.b even 2 1 1014.2.e.a 2
13.c even 3 1 inner 78.2.e.a 2
13.c even 3 1 1014.2.a.c 1
13.d odd 4 2 1014.2.i.b 4
13.e even 6 1 1014.2.a.f 1
13.e even 6 1 1014.2.e.a 2
13.f odd 12 2 1014.2.b.c 2
13.f odd 12 2 1014.2.i.b 4
39.h odd 6 1 3042.2.a.h 1
39.i odd 6 1 234.2.h.a 2
39.i odd 6 1 3042.2.a.i 1
39.k even 12 2 3042.2.b.h 2
52.i odd 6 1 8112.2.a.c 1
52.j odd 6 1 624.2.q.g 2
52.j odd 6 1 8112.2.a.m 1
65.n even 6 1 1950.2.i.m 2
65.q odd 12 2 1950.2.z.g 4
156.p even 6 1 1872.2.t.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 1.a even 1 1 trivial
78.2.e.a 2 13.c even 3 1 inner
234.2.h.a 2 3.b odd 2 1
234.2.h.a 2 39.i odd 6 1
624.2.q.g 2 4.b odd 2 1
624.2.q.g 2 52.j odd 6 1
1014.2.a.c 1 13.c even 3 1
1014.2.a.f 1 13.e even 6 1
1014.2.b.c 2 13.f odd 12 2
1014.2.e.a 2 13.b even 2 1
1014.2.e.a 2 13.e even 6 1
1014.2.i.b 4 13.d odd 4 2
1014.2.i.b 4 13.f odd 12 2
1872.2.t.c 2 12.b even 2 1
1872.2.t.c 2 156.p even 6 1
1950.2.i.m 2 5.b even 2 1
1950.2.i.m 2 65.n even 6 1
1950.2.z.g 4 5.c odd 4 2
1950.2.z.g 4 65.q odd 12 2
3042.2.a.h 1 39.h odd 6 1
3042.2.a.i 1 39.i odd 6 1
3042.2.b.h 2 39.k even 12 2
8112.2.a.c 1 52.i odd 6 1
8112.2.a.m 1 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3 \) acting on \(S_{2}^{\mathrm{new}}(78, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 13)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
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