Properties

Label 684.2.bb.a
Level $684$
Weight $2$
Character orbit 684.bb
Analytic conductor $5.462$
Analytic rank $1$
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] = [684,2,Mod(293,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.bb (of order \(6\), 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: \(5.46176749826\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 2) q^{3} - \zeta_{6} q^{7} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 2) q^{3} - \zeta_{6} q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + (\zeta_{6} - 2) q^{11} + (6 \zeta_{6} - 3) q^{13} + ( - \zeta_{6} - 1) q^{17} + ( - 2 \zeta_{6} - 3) q^{19} + (\zeta_{6} + 1) q^{21} + ( - 10 \zeta_{6} + 5) q^{23} + (5 \zeta_{6} - 5) q^{25} + (6 \zeta_{6} - 3) q^{27} + (6 \zeta_{6} - 6) q^{29} + ( - 5 \zeta_{6} - 5) q^{31} + ( - 3 \zeta_{6} + 3) q^{33} - 9 \zeta_{6} q^{39} - 6 \zeta_{6} q^{41} - q^{43} + ( - 2 \zeta_{6} - 2) q^{47} + ( - 6 \zeta_{6} + 6) q^{49} + 3 q^{51} - 9 \zeta_{6} q^{53} + ( - \zeta_{6} + 8) q^{57} - 12 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} - 3 q^{63} + (10 \zeta_{6} - 5) q^{67} + 15 \zeta_{6} q^{69} + (15 \zeta_{6} - 15) q^{71} + (11 \zeta_{6} - 11) q^{73} + ( - 10 \zeta_{6} + 5) q^{75} + (\zeta_{6} + 1) q^{77} + (18 \zeta_{6} - 9) q^{79} - 9 \zeta_{6} q^{81} + ( - 3 \zeta_{6} + 6) q^{83} + ( - 12 \zeta_{6} + 6) q^{87} + 3 \zeta_{6} q^{89} + ( - 3 \zeta_{6} + 6) q^{91} + 15 q^{93} + (10 \zeta_{6} - 5) q^{97} + (6 \zeta_{6} - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{17} - 8 q^{19} + 3 q^{21} - 5 q^{25} - 6 q^{29} - 15 q^{31} + 3 q^{33} - 9 q^{39} - 6 q^{41} - 2 q^{43} - 6 q^{47} + 6 q^{49} + 6 q^{51} - 9 q^{53} + 15 q^{57} - 12 q^{59} + 2 q^{61} - 6 q^{63} + 15 q^{69} - 15 q^{71} - 11 q^{73} + 3 q^{77} - 9 q^{81} + 9 q^{83} + 3 q^{89} + 9 q^{91} + 30 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(\zeta_{6}\) \(1\) \(1 - \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.50000 + 0.866025i 0 0 0 −0.500000 0.866025i 0 1.50000 2.59808i 0
677.1 0 −1.50000 0.866025i 0 0 0 −0.500000 + 0.866025i 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.bb.a yes 2
3.b odd 2 1 2052.2.bb.a 2
9.c even 3 1 2052.2.n.a 2
9.d odd 6 1 684.2.n.a 2
19.d odd 6 1 684.2.n.a 2
57.f even 6 1 2052.2.n.a 2
171.i odd 6 1 2052.2.bb.a 2
171.t even 6 1 inner 684.2.bb.a yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.n.a 2 9.d odd 6 1
684.2.n.a 2 19.d odd 6 1
684.2.bb.a yes 2 1.a even 1 1 trivial
684.2.bb.a yes 2 171.t even 6 1 inner
2052.2.n.a 2 9.c even 3 1
2052.2.n.a 2 57.f even 6 1
2052.2.bb.a 2 3.b odd 2 1
2052.2.bb.a 2 171.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$13$ \( T^{2} + 27 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 75 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 15T + 75 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$53$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 75 \) Copy content Toggle raw display
$71$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} + 243 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 27 \) Copy content Toggle raw display
$89$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + 75 \) Copy content Toggle raw display
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