Properties

Label 693.2.a.k
Level $693$
Weight $2$
Character orbit 693.a
Self dual yes
Analytic conductor $5.534$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(1,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.a (trivial)

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: yes
Analytic conductor: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + 3 \beta q^{4} + q^{5} + q^{7} + (4 \beta + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + 3 \beta q^{4} + q^{5} + q^{7} + (4 \beta + 1) q^{8} + (\beta + 1) q^{10} - q^{11} + ( - 2 \beta - 1) q^{13} + (\beta + 1) q^{14} + (3 \beta + 5) q^{16} + ( - 4 \beta + 4) q^{17} + ( - 4 \beta + 1) q^{19} + 3 \beta q^{20} + ( - \beta - 1) q^{22} + ( - 4 \beta + 6) q^{23} - 4 q^{25} + ( - 5 \beta - 3) q^{26} + 3 \beta q^{28} + (2 \beta + 3) q^{29} + (4 \beta + 2) q^{31} + (3 \beta + 6) q^{32} - 4 \beta q^{34} + q^{35} + (4 \beta - 3) q^{37} + ( - 7 \beta - 3) q^{38} + (4 \beta + 1) q^{40} + ( - 4 \beta + 2) q^{41} - 4 q^{43} - 3 \beta q^{44} + ( - 2 \beta + 2) q^{46} + (2 \beta + 1) q^{47} + q^{49} + ( - 4 \beta - 4) q^{50} + ( - 9 \beta - 6) q^{52} + (4 \beta + 8) q^{53} - q^{55} + (4 \beta + 1) q^{56} + (7 \beta + 5) q^{58} + ( - 10 \beta + 9) q^{59} + ( - 4 \beta + 6) q^{61} + (10 \beta + 6) q^{62} + (6 \beta - 1) q^{64} + ( - 2 \beta - 1) q^{65} + ( - 10 \beta + 3) q^{67} - 12 q^{68} + (\beta + 1) q^{70} + (4 \beta - 2) q^{71} + ( - 2 \beta - 11) q^{73} + (5 \beta + 1) q^{74} + ( - 9 \beta - 12) q^{76} - q^{77} + ( - 4 \beta + 2) q^{79} + (3 \beta + 5) q^{80} + ( - 6 \beta - 2) q^{82} + (12 \beta - 4) q^{83} + ( - 4 \beta + 4) q^{85} + ( - 4 \beta - 4) q^{86} + ( - 4 \beta - 1) q^{88} - 10 q^{89} + ( - 2 \beta - 1) q^{91} + (6 \beta - 12) q^{92} + (5 \beta + 3) q^{94} + ( - 4 \beta + 1) q^{95} - 8 q^{97} + (\beta + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{8} + 3 q^{10} - 2 q^{11} - 4 q^{13} + 3 q^{14} + 13 q^{16} + 4 q^{17} - 2 q^{19} + 3 q^{20} - 3 q^{22} + 8 q^{23} - 8 q^{25} - 11 q^{26} + 3 q^{28} + 8 q^{29} + 8 q^{31} + 15 q^{32} - 4 q^{34} + 2 q^{35} - 2 q^{37} - 13 q^{38} + 6 q^{40} - 8 q^{43} - 3 q^{44} + 2 q^{46} + 4 q^{47} + 2 q^{49} - 12 q^{50} - 21 q^{52} + 20 q^{53} - 2 q^{55} + 6 q^{56} + 17 q^{58} + 8 q^{59} + 8 q^{61} + 22 q^{62} + 4 q^{64} - 4 q^{65} - 4 q^{67} - 24 q^{68} + 3 q^{70} - 24 q^{73} + 7 q^{74} - 33 q^{76} - 2 q^{77} + 13 q^{80} - 10 q^{82} + 4 q^{83} + 4 q^{85} - 12 q^{86} - 6 q^{88} - 20 q^{89} - 4 q^{91} - 18 q^{92} + 11 q^{94} - 2 q^{95} - 16 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
0.381966 0 −1.85410 1.00000 0 1.00000 −1.47214 0 0.381966
1.2 2.61803 0 4.85410 1.00000 0 1.00000 7.47214 0 2.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.a.k yes 2
3.b odd 2 1 693.2.a.e 2
7.b odd 2 1 4851.2.a.bf 2
11.b odd 2 1 7623.2.a.v 2
21.c even 2 1 4851.2.a.u 2
33.d even 2 1 7623.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.a.e 2 3.b odd 2 1
693.2.a.k yes 2 1.a even 1 1 trivial
4851.2.a.u 2 21.c even 2 1
4851.2.a.bf 2 7.b odd 2 1
7623.2.a.v 2 11.b odd 2 1
7623.2.a.bw 2 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(693))\):

\( T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 11 \) Copy content Toggle raw display
$31$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$41$ \( T^{2} - 20 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 1 \) Copy content Toggle raw display
$53$ \( T^{2} - 20T + 80 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T - 109 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 121 \) Copy content Toggle raw display
$71$ \( T^{2} - 20 \) Copy content Toggle raw display
$73$ \( T^{2} + 24T + 139 \) Copy content Toggle raw display
$79$ \( T^{2} - 20 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 176 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( (T + 8)^{2} \) Copy content Toggle raw display
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