Properties

Label 714.2.a.g
Level $714$
Weight $2$
Character orbit 714.a
Self dual yes
Analytic conductor $5.701$
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] = [714,2,Mod(1,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, 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("714.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.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: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + 3 q^{5} - q^{6} - q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + 3 q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + 3 q^{10} + q^{11} - q^{12} + q^{13} - q^{14} - 3 q^{15} + q^{16} - q^{17} + q^{18} + 6 q^{19} + 3 q^{20} + q^{21} + q^{22} - 2 q^{23} - q^{24} + 4 q^{25} + q^{26} - q^{27} - q^{28} - 2 q^{29} - 3 q^{30} + q^{32} - q^{33} - q^{34} - 3 q^{35} + q^{36} + 5 q^{37} + 6 q^{38} - q^{39} + 3 q^{40} + 4 q^{41} + q^{42} - 9 q^{43} + q^{44} + 3 q^{45} - 2 q^{46} - q^{48} + q^{49} + 4 q^{50} + q^{51} + q^{52} + 11 q^{53} - q^{54} + 3 q^{55} - q^{56} - 6 q^{57} - 2 q^{58} + 4 q^{59} - 3 q^{60} + 6 q^{61} - q^{63} + q^{64} + 3 q^{65} - q^{66} - 11 q^{67} - q^{68} + 2 q^{69} - 3 q^{70} - 12 q^{71} + q^{72} - 5 q^{73} + 5 q^{74} - 4 q^{75} + 6 q^{76} - q^{77} - q^{78} - 15 q^{79} + 3 q^{80} + q^{81} + 4 q^{82} + q^{83} + q^{84} - 3 q^{85} - 9 q^{86} + 2 q^{87} + q^{88} + 9 q^{89} + 3 q^{90} - q^{91} - 2 q^{92} + 18 q^{95} - q^{96} - 9 q^{97} + q^{98} + q^{99}+O(q^{100}) \) 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
1.00000 −1.00000 1.00000 3.00000 −1.00000 −1.00000 1.00000 1.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(17\) \(1\)

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 714.2.a.g 1
3.b odd 2 1 2142.2.a.a 1
4.b odd 2 1 5712.2.a.bb 1
7.b odd 2 1 4998.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.a.g 1 1.a even 1 1 trivial
2142.2.a.a 1 3.b odd 2 1
4998.2.a.bj 1 7.b odd 2 1
5712.2.a.bb 1 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(714))\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{11} - 1 \) 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
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T + 1 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T + 2 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 5 \) Copy content Toggle raw display
$41$ \( T - 4 \) Copy content Toggle raw display
$43$ \( T + 9 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 11 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T + 11 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T + 5 \) Copy content Toggle raw display
$79$ \( T + 15 \) Copy content Toggle raw display
$83$ \( T - 1 \) Copy content Toggle raw display
$89$ \( T - 9 \) Copy content Toggle raw display
$97$ \( T + 9 \) Copy content Toggle raw display
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