Properties

Label 693.4.a.b
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 77)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{2} + q^{4} - 12 q^{5} + 7 q^{7} + 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} + q^{4} - 12 q^{5} + 7 q^{7} + 21 q^{8} + 36 q^{10} - 11 q^{11} + 38 q^{13} - 21 q^{14} - 71 q^{16} + 48 q^{17} - 70 q^{19} - 12 q^{20} + 33 q^{22} - 12 q^{23} + 19 q^{25} - 114 q^{26} + 7 q^{28} - 126 q^{29} - 70 q^{31} + 45 q^{32} - 144 q^{34} - 84 q^{35} - 358 q^{37} + 210 q^{38} - 252 q^{40} + 216 q^{41} + 344 q^{43} - 11 q^{44} + 36 q^{46} - 390 q^{47} + 49 q^{49} - 57 q^{50} + 38 q^{52} - 438 q^{53} + 132 q^{55} + 147 q^{56} + 378 q^{58} + 552 q^{59} + 830 q^{61} + 210 q^{62} + 433 q^{64} - 456 q^{65} - 196 q^{67} + 48 q^{68} + 252 q^{70} - 648 q^{71} - 16 q^{73} + 1074 q^{74} - 70 q^{76} - 77 q^{77} + 1352 q^{79} + 852 q^{80} - 648 q^{82} - 90 q^{83} - 576 q^{85} - 1032 q^{86} - 231 q^{88} - 1146 q^{89} + 266 q^{91} - 12 q^{92} + 1170 q^{94} + 840 q^{95} - 70 q^{97} - 147 q^{98}+O(q^{100}) \) 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 −12.0000 0 7.00000 21.0000 0 36.0000
\(n\): 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.b 1
3.b odd 2 1 77.4.a.a 1
12.b even 2 1 1232.4.a.d 1
15.d odd 2 1 1925.4.a.a 1
21.c even 2 1 539.4.a.a 1
33.d even 2 1 847.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.a.a 1 3.b odd 2 1
539.4.a.a 1 21.c even 2 1
693.4.a.b 1 1.a even 1 1 trivial
847.4.a.a 1 33.d even 2 1
1232.4.a.d 1 12.b even 2 1
1925.4.a.a 1 15.d odd 2 1

Hecke kernels

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

\( T_{2} + 3 \) Copy content Toggle raw display
\( T_{5} + 12 \) 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
$5$ \( T + 12 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T - 38 \) Copy content Toggle raw display
$17$ \( T - 48 \) Copy content Toggle raw display
$19$ \( T + 70 \) Copy content Toggle raw display
$23$ \( T + 12 \) Copy content Toggle raw display
$29$ \( T + 126 \) Copy content Toggle raw display
$31$ \( T + 70 \) Copy content Toggle raw display
$37$ \( T + 358 \) Copy content Toggle raw display
$41$ \( T - 216 \) Copy content Toggle raw display
$43$ \( T - 344 \) Copy content Toggle raw display
$47$ \( T + 390 \) Copy content Toggle raw display
$53$ \( T + 438 \) Copy content Toggle raw display
$59$ \( T - 552 \) Copy content Toggle raw display
$61$ \( T - 830 \) Copy content Toggle raw display
$67$ \( T + 196 \) Copy content Toggle raw display
$71$ \( T + 648 \) Copy content Toggle raw display
$73$ \( T + 16 \) Copy content Toggle raw display
$79$ \( T - 1352 \) Copy content Toggle raw display
$83$ \( T + 90 \) Copy content Toggle raw display
$89$ \( T + 1146 \) Copy content Toggle raw display
$97$ \( T + 70 \) Copy content Toggle raw display
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