Properties

Label 6897.2.a.a
Level $6897$
Weight $2$
Character orbit 6897.a
Self dual yes
Analytic conductor $55.073$
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] = [6897,2,Mod(1,6897)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6897, 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("6897.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6897.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: \(55.0728222741\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} + 3 q^{8} + q^{9} + 2 q^{10} - q^{12} - 6 q^{13} - 2 q^{15} - q^{16} + 6 q^{17} - q^{18} + q^{19} + 2 q^{20} + 4 q^{23} + 3 q^{24} - q^{25} + 6 q^{26} + q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{31} - 5 q^{32} - 6 q^{34} - q^{36} - 10 q^{37} - q^{38} - 6 q^{39} - 6 q^{40} + 2 q^{41} + 4 q^{43} - 2 q^{45} - 4 q^{46} + 12 q^{47} - q^{48} - 7 q^{49} + q^{50} + 6 q^{51} + 6 q^{52} - 6 q^{53} - q^{54} + q^{57} + 2 q^{58} - 12 q^{59} + 2 q^{60} + 2 q^{61} - 8 q^{62} + 7 q^{64} + 12 q^{65} - 4 q^{67} - 6 q^{68} + 4 q^{69} + 3 q^{72} - 10 q^{73} + 10 q^{74} - q^{75} - q^{76} + 6 q^{78} + 2 q^{80} + q^{81} - 2 q^{82} - 16 q^{83} - 12 q^{85} - 4 q^{86} - 2 q^{87} - 2 q^{89} + 2 q^{90} - 4 q^{92} + 8 q^{93} - 12 q^{94} - 2 q^{95} - 5 q^{96} + 10 q^{97} + 7 q^{98}+O(q^{100}) \) 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
−1.00000 1.00000 −1.00000 −2.00000 −1.00000 0 3.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(11\) \( -1 \)
\(19\) \( -1 \)

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 6897.2.a.a 1
11.b odd 2 1 57.2.a.c 1
33.d even 2 1 171.2.a.a 1
44.c even 2 1 912.2.a.b 1
55.d odd 2 1 1425.2.a.a 1
55.e even 4 2 1425.2.c.g 2
77.b even 2 1 2793.2.a.i 1
88.b odd 2 1 3648.2.a.o 1
88.g even 2 1 3648.2.a.bf 1
132.d odd 2 1 2736.2.a.s 1
143.d odd 2 1 9633.2.a.h 1
165.d even 2 1 4275.2.a.m 1
209.d even 2 1 1083.2.a.a 1
231.h odd 2 1 8379.2.a.e 1
627.b odd 2 1 3249.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.c 1 11.b odd 2 1
171.2.a.a 1 33.d even 2 1
912.2.a.b 1 44.c even 2 1
1083.2.a.a 1 209.d even 2 1
1425.2.a.a 1 55.d odd 2 1
1425.2.c.g 2 55.e even 4 2
2736.2.a.s 1 132.d odd 2 1
2793.2.a.i 1 77.b even 2 1
3249.2.a.g 1 627.b odd 2 1
3648.2.a.o 1 88.b odd 2 1
3648.2.a.bf 1 88.g even 2 1
4275.2.a.m 1 165.d even 2 1
6897.2.a.a 1 1.a even 1 1 trivial
8379.2.a.e 1 231.h odd 2 1
9633.2.a.h 1 143.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6897))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 16 \) Copy content Toggle raw display
$89$ \( T + 2 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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