Properties

Label 693.4.a.e
Level $693$
Weight $4$
Character orbit 693.a
Self dual yes
Analytic conductor $40.888$
Analytic rank $0$
Dimension $1$
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] = [693,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(40.8883236340\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 4 q^{4} - q^{5} - 7 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 4 q^{4} - q^{5} - 7 q^{7} - 24 q^{8} - 2 q^{10} + 11 q^{11} + 7 q^{13} - 14 q^{14} - 16 q^{16} + 14 q^{17} - 45 q^{19} + 4 q^{20} + 22 q^{22} + 88 q^{23} - 124 q^{25} + 14 q^{26} + 28 q^{28} + 69 q^{29} + 22 q^{31} + 160 q^{32} + 28 q^{34} + 7 q^{35} + 57 q^{37} - 90 q^{38} + 24 q^{40} + 380 q^{41} + 48 q^{43} - 44 q^{44} + 176 q^{46} + 385 q^{47} + 49 q^{49} - 248 q^{50} - 28 q^{52} + 672 q^{53} - 11 q^{55} + 168 q^{56} + 138 q^{58} + 469 q^{59} - 342 q^{61} + 44 q^{62} + 448 q^{64} - 7 q^{65} - 139 q^{67} - 56 q^{68} + 14 q^{70} - 132 q^{71} + 145 q^{73} + 114 q^{74} + 180 q^{76} - 77 q^{77} + 1244 q^{79} + 16 q^{80} + 760 q^{82} - 522 q^{83} - 14 q^{85} + 96 q^{86} - 264 q^{88} - 822 q^{89} - 49 q^{91} - 352 q^{92} + 770 q^{94} + 45 q^{95} + 272 q^{97} + 98 q^{98}+O(q^{100}) \) 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
2.00000 0 −4.00000 −1.00000 0 −7.00000 −24.0000 0 −2.00000
\(n\): 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 693.4.a.e 1
3.b odd 2 1 231.4.a.b 1
21.c even 2 1 1617.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.a.b 1 3.b odd 2 1
693.4.a.e 1 1.a even 1 1 trivial
1617.4.a.c 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(693))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T - 7 \) Copy content Toggle raw display
$17$ \( T - 14 \) Copy content Toggle raw display
$19$ \( T + 45 \) Copy content Toggle raw display
$23$ \( T - 88 \) Copy content Toggle raw display
$29$ \( T - 69 \) Copy content Toggle raw display
$31$ \( T - 22 \) Copy content Toggle raw display
$37$ \( T - 57 \) Copy content Toggle raw display
$41$ \( T - 380 \) Copy content Toggle raw display
$43$ \( T - 48 \) Copy content Toggle raw display
$47$ \( T - 385 \) Copy content Toggle raw display
$53$ \( T - 672 \) Copy content Toggle raw display
$59$ \( T - 469 \) Copy content Toggle raw display
$61$ \( T + 342 \) Copy content Toggle raw display
$67$ \( T + 139 \) Copy content Toggle raw display
$71$ \( T + 132 \) Copy content Toggle raw display
$73$ \( T - 145 \) Copy content Toggle raw display
$79$ \( T - 1244 \) Copy content Toggle raw display
$83$ \( T + 522 \) Copy content Toggle raw display
$89$ \( T + 822 \) Copy content Toggle raw display
$97$ \( T - 272 \) Copy content Toggle raw display
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