Properties

Label 1617.2.a.b
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $1$
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] = [1617,2,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(12.9118100068\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} + q^{9} - q^{11} + 2 q^{12} + 5 q^{13} - 4 q^{16} - 6 q^{17} - 2 q^{18} - 7 q^{19} + 2 q^{22} - 4 q^{23} - 5 q^{25} - 10 q^{26} + q^{27} - 2 q^{29} + 7 q^{31} + 8 q^{32} - q^{33} + 12 q^{34} + 2 q^{36} + 7 q^{37} + 14 q^{38} + 5 q^{39} - 4 q^{41} - 9 q^{43} - 2 q^{44} + 8 q^{46} - 6 q^{47} - 4 q^{48} + 10 q^{50} - 6 q^{51} + 10 q^{52} - 2 q^{53} - 2 q^{54} - 7 q^{57} + 4 q^{58} - 12 q^{59} + 2 q^{61} - 14 q^{62} - 8 q^{64} + 2 q^{66} + 7 q^{67} - 12 q^{68} - 4 q^{69} + 8 q^{71} + 5 q^{73} - 14 q^{74} - 5 q^{75} - 14 q^{76} - 10 q^{78} - 11 q^{79} + q^{81} + 8 q^{82} + 4 q^{83} + 18 q^{86} - 2 q^{87} - 6 q^{89} - 8 q^{92} + 7 q^{93} + 12 q^{94} + 8 q^{96} - 2 q^{97} - q^{99}+O(q^{100}) \) 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 1.00000 2.00000 0 −2.00000 0 0 1.00000 0
\(n\): 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 1617.2.a.b 1
3.b odd 2 1 4851.2.a.s 1
7.b odd 2 1 1617.2.a.a 1
7.d odd 6 2 231.2.i.c 2
21.c even 2 1 4851.2.a.r 1
21.g even 6 2 693.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.c 2 7.d odd 6 2
693.2.i.a 2 21.g even 6 2
1617.2.a.a 1 7.b odd 2 1
1617.2.a.b 1 1.a even 1 1 trivial
4851.2.a.r 1 21.c even 2 1
4851.2.a.s 1 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 5 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 7 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 7 \) Copy content Toggle raw display
$37$ \( T - 7 \) Copy content Toggle raw display
$41$ \( T + 4 \) Copy content Toggle raw display
$43$ \( T + 9 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 7 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T - 5 \) Copy content Toggle raw display
$79$ \( T + 11 \) Copy content Toggle raw display
$83$ \( T - 4 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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