Properties

Label 309.2.a.a
Level $309$
Weight $2$
Character orbit 309.a
Self dual yes
Analytic conductor $2.467$
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] = [309,2,Mod(1,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("309.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 309.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: \(2.46737742246\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + q^{9} + q^{10} - 2 q^{11} - q^{12} - 5 q^{13} + 2 q^{14} - q^{15} - q^{16} - q^{18} - 8 q^{19} + q^{20} - 2 q^{21} + 2 q^{22} + q^{23} + 3 q^{24} - 4 q^{25} + 5 q^{26} + q^{27} + 2 q^{28} - 2 q^{29} + q^{30} + 5 q^{31} - 5 q^{32} - 2 q^{33} + 2 q^{35} - q^{36} + 2 q^{37} + 8 q^{38} - 5 q^{39} - 3 q^{40} + 8 q^{41} + 2 q^{42} - 11 q^{43} + 2 q^{44} - q^{45} - q^{46} - 2 q^{47} - q^{48} - 3 q^{49} + 4 q^{50} + 5 q^{52} + 10 q^{53} - q^{54} + 2 q^{55} - 6 q^{56} - 8 q^{57} + 2 q^{58} - 11 q^{59} + q^{60} - 5 q^{61} - 5 q^{62} - 2 q^{63} + 7 q^{64} + 5 q^{65} + 2 q^{66} + 11 q^{67} + q^{69} - 2 q^{70} + 16 q^{71} + 3 q^{72} + 12 q^{73} - 2 q^{74} - 4 q^{75} + 8 q^{76} + 4 q^{77} + 5 q^{78} + 6 q^{79} + q^{80} + q^{81} - 8 q^{82} + q^{83} + 2 q^{84} + 11 q^{86} - 2 q^{87} - 6 q^{88} - 6 q^{89} + q^{90} + 10 q^{91} - q^{92} + 5 q^{93} + 2 q^{94} + 8 q^{95} - 5 q^{96} - 7 q^{97} + 3 q^{98} - 2 q^{99}+O(q^{100}) \) 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 −1.00000 −1.00000 −2.00000 3.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(103\) \(-1\)

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 309.2.a.a 1
3.b odd 2 1 927.2.a.a 1
4.b odd 2 1 4944.2.a.c 1
5.b even 2 1 7725.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.2.a.a 1 1.a even 1 1 trivial
927.2.a.a 1 3.b odd 2 1
4944.2.a.c 1 4.b odd 2 1
7725.2.a.k 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(309))\). 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
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 5 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T + 11 \) Copy content Toggle raw display
$47$ \( T + 2 \) Copy content Toggle raw display
$53$ \( T - 10 \) Copy content Toggle raw display
$59$ \( T + 11 \) Copy content Toggle raw display
$61$ \( T + 5 \) Copy content Toggle raw display
$67$ \( T - 11 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T - 12 \) Copy content Toggle raw display
$79$ \( T - 6 \) Copy content Toggle raw display
$83$ \( T - 1 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
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