Properties

Label 405.4.a.a
Level $405$
Weight $4$
Character orbit 405.a
Self dual yes
Analytic conductor $23.896$
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] = [405,4,Mod(1,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("405.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.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: \(23.8957735523\)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{2} + 17 q^{4} + 5 q^{5} + 9 q^{7} - 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} + 17 q^{4} + 5 q^{5} + 9 q^{7} - 45 q^{8} - 25 q^{10} - 8 q^{11} + 43 q^{13} - 45 q^{14} + 89 q^{16} - 122 q^{17} - 59 q^{19} + 85 q^{20} + 40 q^{22} - 213 q^{23} + 25 q^{25} - 215 q^{26} + 153 q^{28} + 224 q^{29} - 36 q^{31} - 85 q^{32} + 610 q^{34} + 45 q^{35} + 206 q^{37} + 295 q^{38} - 225 q^{40} + 413 q^{41} - 392 q^{43} - 136 q^{44} + 1065 q^{46} - 311 q^{47} - 262 q^{49} - 125 q^{50} + 731 q^{52} - 377 q^{53} - 40 q^{55} - 405 q^{56} - 1120 q^{58} + 337 q^{59} + 40 q^{61} + 180 q^{62} - 287 q^{64} + 215 q^{65} + 348 q^{67} - 2074 q^{68} - 225 q^{70} + 62 q^{71} - 1214 q^{73} - 1030 q^{74} - 1003 q^{76} - 72 q^{77} - 294 q^{79} + 445 q^{80} - 2065 q^{82} + 534 q^{83} - 610 q^{85} + 1960 q^{86} + 360 q^{88} - 810 q^{89} + 387 q^{91} - 3621 q^{92} + 1555 q^{94} - 295 q^{95} - 928 q^{97} + 1310 q^{98}+O(q^{100}) \) 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
−5.00000 0 17.0000 5.00000 0 9.00000 −45.0000 0 −25.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 405.4.a.a 1
3.b odd 2 1 405.4.a.b yes 1
5.b even 2 1 2025.4.a.f 1
9.c even 3 2 405.4.e.m 2
9.d odd 6 2 405.4.e.a 2
15.d odd 2 1 2025.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.a 1 1.a even 1 1 trivial
405.4.a.b yes 1 3.b odd 2 1
405.4.e.a 2 9.d odd 6 2
405.4.e.m 2 9.c even 3 2
2025.4.a.a 1 15.d odd 2 1
2025.4.a.f 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 5 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 9 \) Copy content Toggle raw display
$11$ \( T + 8 \) Copy content Toggle raw display
$13$ \( T - 43 \) Copy content Toggle raw display
$17$ \( T + 122 \) Copy content Toggle raw display
$19$ \( T + 59 \) Copy content Toggle raw display
$23$ \( T + 213 \) Copy content Toggle raw display
$29$ \( T - 224 \) Copy content Toggle raw display
$31$ \( T + 36 \) Copy content Toggle raw display
$37$ \( T - 206 \) Copy content Toggle raw display
$41$ \( T - 413 \) Copy content Toggle raw display
$43$ \( T + 392 \) Copy content Toggle raw display
$47$ \( T + 311 \) Copy content Toggle raw display
$53$ \( T + 377 \) Copy content Toggle raw display
$59$ \( T - 337 \) Copy content Toggle raw display
$61$ \( T - 40 \) Copy content Toggle raw display
$67$ \( T - 348 \) Copy content Toggle raw display
$71$ \( T - 62 \) Copy content Toggle raw display
$73$ \( T + 1214 \) Copy content Toggle raw display
$79$ \( T + 294 \) Copy content Toggle raw display
$83$ \( T - 534 \) Copy content Toggle raw display
$89$ \( T + 810 \) Copy content Toggle raw display
$97$ \( T + 928 \) Copy content Toggle raw display
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