Properties

Label 693.2.a.j
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: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 3) q^{4} - 3 q^{5} + q^{7} + (2 \beta + 5) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 3) q^{4} - 3 q^{5} + q^{7} + (2 \beta + 5) q^{8} - 3 \beta q^{10} + q^{11} + q^{13} + \beta q^{14} + (5 \beta + 4) q^{16} + (2 \beta - 4) q^{17} + (2 \beta - 3) q^{19} + ( - 3 \beta - 9) q^{20} + \beta q^{22} + ( - 2 \beta + 2) q^{23} + 4 q^{25} + \beta q^{26} + (\beta + 3) q^{28} + ( - 4 \beta + 1) q^{29} - 2 \beta q^{31} + (5 \beta + 15) q^{32} + ( - 2 \beta + 10) q^{34} - 3 q^{35} + q^{37} + ( - \beta + 10) q^{38} + ( - 6 \beta - 15) q^{40} + ( - 4 \beta + 4) q^{41} + ( - 2 \beta - 2) q^{43} + (\beta + 3) q^{44} - 10 q^{46} + ( - 2 \beta - 5) q^{47} + q^{49} + 4 \beta q^{50} + (\beta + 3) q^{52} + ( - 2 \beta + 6) q^{53} - 3 q^{55} + (2 \beta + 5) q^{56} + ( - 3 \beta - 20) q^{58} + (2 \beta - 1) q^{59} + 10 q^{61} + ( - 2 \beta - 10) q^{62} + (10 \beta + 17) q^{64} - 3 q^{65} + ( - 2 \beta + 5) q^{67} + (4 \beta - 2) q^{68} - 3 \beta q^{70} + (4 \beta - 4) q^{71} + 7 q^{73} + \beta q^{74} + (5 \beta + 1) q^{76} + q^{77} - 4 \beta q^{79} + ( - 15 \beta - 12) q^{80} - 20 q^{82} + ( - 2 \beta + 8) q^{83} + ( - 6 \beta + 12) q^{85} + ( - 4 \beta - 10) q^{86} + (2 \beta + 5) q^{88} + (4 \beta - 2) q^{89} + q^{91} + ( - 6 \beta - 4) q^{92} + ( - 7 \beta - 10) q^{94} + ( - 6 \beta + 9) q^{95} + ( - 2 \beta - 6) q^{97} + \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 7 q^{4} - 6 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 7 q^{4} - 6 q^{5} + 2 q^{7} + 12 q^{8} - 3 q^{10} + 2 q^{11} + 2 q^{13} + q^{14} + 13 q^{16} - 6 q^{17} - 4 q^{19} - 21 q^{20} + q^{22} + 2 q^{23} + 8 q^{25} + q^{26} + 7 q^{28} - 2 q^{29} - 2 q^{31} + 35 q^{32} + 18 q^{34} - 6 q^{35} + 2 q^{37} + 19 q^{38} - 36 q^{40} + 4 q^{41} - 6 q^{43} + 7 q^{44} - 20 q^{46} - 12 q^{47} + 2 q^{49} + 4 q^{50} + 7 q^{52} + 10 q^{53} - 6 q^{55} + 12 q^{56} - 43 q^{58} + 20 q^{61} - 22 q^{62} + 44 q^{64} - 6 q^{65} + 8 q^{67} - 3 q^{70} - 4 q^{71} + 14 q^{73} + q^{74} + 7 q^{76} + 2 q^{77} - 4 q^{79} - 39 q^{80} - 40 q^{82} + 14 q^{83} + 18 q^{85} - 24 q^{86} + 12 q^{88} + 2 q^{91} - 14 q^{92} - 27 q^{94} + 12 q^{95} - 14 q^{97} + q^{98}+O(q^{100}) \) 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
−1.79129
2.79129
−1.79129 0 1.20871 −3.00000 0 1.00000 1.41742 0 5.37386
1.2 2.79129 0 5.79129 −3.00000 0 1.00000 10.5826 0 −8.37386
\(n\): 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.j 2
3.b odd 2 1 231.2.a.b 2
7.b odd 2 1 4851.2.a.ba 2
11.b odd 2 1 7623.2.a.bf 2
12.b even 2 1 3696.2.a.bl 2
15.d odd 2 1 5775.2.a.bn 2
21.c even 2 1 1617.2.a.o 2
33.d even 2 1 2541.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 3.b odd 2 1
693.2.a.j 2 1.a even 1 1 trivial
1617.2.a.o 2 21.c even 2 1
2541.2.a.z 2 33.d even 2 1
3696.2.a.bl 2 12.b even 2 1
4851.2.a.ba 2 7.b odd 2 1
5775.2.a.bn 2 15.d odd 2 1
7623.2.a.bf 2 11.b odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 5 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T - 17 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 20 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 83 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 20 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 80 \) Copy content Toggle raw display
$43$ \( T^{2} + 6T - 12 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T + 15 \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 21 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 5 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 80 \) Copy content Toggle raw display
$73$ \( (T - 7)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 80 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 28 \) Copy content Toggle raw display
$89$ \( T^{2} - 84 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 28 \) Copy content Toggle raw display
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