Properties

Label 693.4.a.c
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)
 
\(f(q)\) \(=\) \( q - 3 q^{2} + q^{4} + 14 q^{5} - 7 q^{7} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} + q^{4} + 14 q^{5} - 7 q^{7} + 21 q^{8} - 42 q^{10} + 11 q^{11} + 2 q^{13} + 21 q^{14} - 71 q^{16} + 74 q^{17} + 14 q^{20} - 33 q^{22} + 148 q^{23} + 71 q^{25} - 6 q^{26} - 7 q^{28} - 26 q^{29} + 112 q^{31} + 45 q^{32} - 222 q^{34} - 98 q^{35} - 98 q^{37} + 294 q^{40} + 10 q^{41} + 208 q^{43} + 11 q^{44} - 444 q^{46} - 460 q^{47} + 49 q^{49} - 213 q^{50} + 2 q^{52} - 258 q^{53} + 154 q^{55} - 147 q^{56} + 78 q^{58} + 204 q^{59} + 178 q^{61} - 336 q^{62} + 433 q^{64} + 28 q^{65} - 924 q^{67} + 74 q^{68} + 294 q^{70} + 748 q^{71} - 230 q^{73} + 294 q^{74} - 77 q^{77} - 456 q^{79} - 994 q^{80} - 30 q^{82} + 228 q^{83} + 1036 q^{85} - 624 q^{86} + 231 q^{88} + 198 q^{89} - 14 q^{91} + 148 q^{92} + 1380 q^{94} + 562 q^{97} - 147 q^{98}+O(q^{100}) \) 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
−3.00000 0 1.00000 14.0000 0 −7.00000 21.0000 0 −42.0000
\(n\): 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.c 1
3.b odd 2 1 231.4.a.d 1
21.c even 2 1 1617.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.a.d 1 3.b odd 2 1
693.4.a.c 1 1.a even 1 1 trivial
1617.4.a.f 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(693))\):

\( T_{2} + 3 \) Copy content Toggle raw display
\( T_{5} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 14 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T - 11 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 74 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T - 148 \) Copy content Toggle raw display
$29$ \( T + 26 \) Copy content Toggle raw display
$31$ \( T - 112 \) Copy content Toggle raw display
$37$ \( T + 98 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T - 208 \) Copy content Toggle raw display
$47$ \( T + 460 \) Copy content Toggle raw display
$53$ \( T + 258 \) Copy content Toggle raw display
$59$ \( T - 204 \) Copy content Toggle raw display
$61$ \( T - 178 \) Copy content Toggle raw display
$67$ \( T + 924 \) Copy content Toggle raw display
$71$ \( T - 748 \) Copy content Toggle raw display
$73$ \( T + 230 \) Copy content Toggle raw display
$79$ \( T + 456 \) Copy content Toggle raw display
$83$ \( T - 228 \) Copy content Toggle raw display
$89$ \( T - 198 \) Copy content Toggle raw display
$97$ \( T - 562 \) Copy content Toggle raw display
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