Properties

Label 1305.4.a.b
Level $1305$
Weight $4$
Character orbit 1305.a
Self dual yes
Analytic conductor $76.997$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,4,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(76.9974925575\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - 7 q^{4} + 5 q^{5} - 14 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 7 q^{4} + 5 q^{5} - 14 q^{7} + 15 q^{8} - 5 q^{10} - 62 q^{11} + 42 q^{13} + 14 q^{14} + 41 q^{16} + 114 q^{17} - 70 q^{19} - 35 q^{20} + 62 q^{22} - 62 q^{23} + 25 q^{25} - 42 q^{26} + 98 q^{28} + 29 q^{29} + 142 q^{31} - 161 q^{32} - 114 q^{34} - 70 q^{35} + 146 q^{37} + 70 q^{38} + 75 q^{40} - 162 q^{41} + 352 q^{43} + 434 q^{44} + 62 q^{46} + 444 q^{47} - 147 q^{49} - 25 q^{50} - 294 q^{52} + 238 q^{53} - 310 q^{55} - 210 q^{56} - 29 q^{58} - 840 q^{59} + 2 q^{61} - 142 q^{62} - 167 q^{64} + 210 q^{65} - 154 q^{67} - 798 q^{68} + 70 q^{70} - 892 q^{71} - 38 q^{73} - 146 q^{74} + 490 q^{76} + 868 q^{77} + 1050 q^{79} + 205 q^{80} + 162 q^{82} + 778 q^{83} + 570 q^{85} - 352 q^{86} - 930 q^{88} - 1410 q^{89} - 588 q^{91} + 434 q^{92} - 444 q^{94} - 350 q^{95} + 466 q^{97} + 147 q^{98}+O(q^{100}) \) 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
−1.00000 0 −7.00000 5.00000 0 −14.0000 15.0000 0 −5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(29\) \( -1 \)

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 1305.4.a.b 1
3.b odd 2 1 145.4.a.a 1
12.b even 2 1 2320.4.a.f 1
15.d odd 2 1 725.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.4.a.a 1 3.b odd 2 1
725.4.a.a 1 15.d odd 2 1
1305.4.a.b 1 1.a even 1 1 trivial
2320.4.a.f 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1305))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 14 \) Copy content Toggle raw display
$11$ \( T + 62 \) Copy content Toggle raw display
$13$ \( T - 42 \) Copy content Toggle raw display
$17$ \( T - 114 \) Copy content Toggle raw display
$19$ \( T + 70 \) Copy content Toggle raw display
$23$ \( T + 62 \) Copy content Toggle raw display
$29$ \( T - 29 \) Copy content Toggle raw display
$31$ \( T - 142 \) Copy content Toggle raw display
$37$ \( T - 146 \) Copy content Toggle raw display
$41$ \( T + 162 \) Copy content Toggle raw display
$43$ \( T - 352 \) Copy content Toggle raw display
$47$ \( T - 444 \) Copy content Toggle raw display
$53$ \( T - 238 \) Copy content Toggle raw display
$59$ \( T + 840 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 154 \) Copy content Toggle raw display
$71$ \( T + 892 \) Copy content Toggle raw display
$73$ \( T + 38 \) Copy content Toggle raw display
$79$ \( T - 1050 \) Copy content Toggle raw display
$83$ \( T - 778 \) Copy content Toggle raw display
$89$ \( T + 1410 \) Copy content Toggle raw display
$97$ \( T - 466 \) Copy content Toggle raw display
show more
show less