Properties

Label 315.4.a.e
Level $315$
Weight $4$
Character orbit 315.a
Self dual yes
Analytic conductor $18.586$
Analytic rank $0$
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] = [315,4,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, 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("315.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(18.5856016518\)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{2} + q^{4} + 5 q^{5} + 7 q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + q^{4} + 5 q^{5} + 7 q^{7} - 21 q^{8} + 15 q^{10} + 60 q^{11} + 38 q^{13} + 21 q^{14} - 71 q^{16} - 84 q^{17} + 110 q^{19} + 5 q^{20} + 180 q^{22} + 120 q^{23} + 25 q^{25} + 114 q^{26} + 7 q^{28} + 162 q^{29} + 236 q^{31} - 45 q^{32} - 252 q^{34} + 35 q^{35} - 376 q^{37} + 330 q^{38} - 105 q^{40} - 126 q^{41} - 34 q^{43} + 60 q^{44} + 360 q^{46} - 6 q^{47} + 49 q^{49} + 75 q^{50} + 38 q^{52} + 582 q^{53} + 300 q^{55} - 147 q^{56} + 486 q^{58} + 492 q^{59} - 880 q^{61} + 708 q^{62} + 433 q^{64} + 190 q^{65} - 826 q^{67} - 84 q^{68} + 105 q^{70} - 666 q^{71} - 826 q^{73} - 1128 q^{74} + 110 q^{76} + 420 q^{77} - 592 q^{79} - 355 q^{80} - 378 q^{82} + 792 q^{83} - 420 q^{85} - 102 q^{86} - 1260 q^{88} + 1002 q^{89} + 266 q^{91} + 120 q^{92} - 18 q^{94} + 550 q^{95} + 1442 q^{97} + 147 q^{98}+O(q^{100}) \) 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
3.00000 0 1.00000 5.00000 0 7.00000 −21.0000 0 15.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(7\) \( -1 \)

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 315.4.a.e yes 1
3.b odd 2 1 315.4.a.b 1
5.b even 2 1 1575.4.a.c 1
7.b odd 2 1 2205.4.a.p 1
15.d odd 2 1 1575.4.a.i 1
21.c even 2 1 2205.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.a.b 1 3.b odd 2 1
315.4.a.e yes 1 1.a even 1 1 trivial
1575.4.a.c 1 5.b even 2 1
1575.4.a.i 1 15.d odd 2 1
2205.4.a.f 1 21.c even 2 1
2205.4.a.p 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T - 60 \) Copy content Toggle raw display
$13$ \( T - 38 \) Copy content Toggle raw display
$17$ \( T + 84 \) Copy content Toggle raw display
$19$ \( T - 110 \) Copy content Toggle raw display
$23$ \( T - 120 \) Copy content Toggle raw display
$29$ \( T - 162 \) Copy content Toggle raw display
$31$ \( T - 236 \) Copy content Toggle raw display
$37$ \( T + 376 \) Copy content Toggle raw display
$41$ \( T + 126 \) Copy content Toggle raw display
$43$ \( T + 34 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T - 582 \) Copy content Toggle raw display
$59$ \( T - 492 \) Copy content Toggle raw display
$61$ \( T + 880 \) Copy content Toggle raw display
$67$ \( T + 826 \) Copy content Toggle raw display
$71$ \( T + 666 \) Copy content Toggle raw display
$73$ \( T + 826 \) Copy content Toggle raw display
$79$ \( T + 592 \) Copy content Toggle raw display
$83$ \( T - 792 \) Copy content Toggle raw display
$89$ \( T - 1002 \) Copy content Toggle raw display
$97$ \( T - 1442 \) Copy content Toggle raw display
show more
show less