Properties

Label 315.4.a.a
Level $315$
Weight $4$
Character orbit 315.a
Self dual yes
Analytic conductor $18.586$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: no (minimal twist has level 105)
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)
 
\(f(q)\) \(=\) \( q - 5 q^{2} + 17 q^{4} - 5 q^{5} + 7 q^{7} - 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{2} + 17 q^{4} - 5 q^{5} + 7 q^{7} - 45 q^{8} + 25 q^{10} - 12 q^{11} + 30 q^{13} - 35 q^{14} + 89 q^{16} + 134 q^{17} - 92 q^{19} - 85 q^{20} + 60 q^{22} - 112 q^{23} + 25 q^{25} - 150 q^{26} + 119 q^{28} + 58 q^{29} - 224 q^{31} - 85 q^{32} - 670 q^{34} - 35 q^{35} - 146 q^{37} + 460 q^{38} + 225 q^{40} - 18 q^{41} + 340 q^{43} - 204 q^{44} + 560 q^{46} - 208 q^{47} + 49 q^{49} - 125 q^{50} + 510 q^{52} + 754 q^{53} + 60 q^{55} - 315 q^{56} - 290 q^{58} - 380 q^{59} + 718 q^{61} + 1120 q^{62} - 287 q^{64} - 150 q^{65} + 412 q^{67} + 2278 q^{68} + 175 q^{70} + 960 q^{71} + 1066 q^{73} + 730 q^{74} - 1564 q^{76} - 84 q^{77} + 896 q^{79} - 445 q^{80} + 90 q^{82} - 436 q^{83} - 670 q^{85} - 1700 q^{86} + 540 q^{88} + 1038 q^{89} + 210 q^{91} - 1904 q^{92} + 1040 q^{94} + 460 q^{95} - 702 q^{97} - 245 q^{98}+O(q^{100}) \) 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 7.00000 −45.0000 0 25.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.a 1
3.b odd 2 1 105.4.a.b 1
5.b even 2 1 1575.4.a.l 1
7.b odd 2 1 2205.4.a.b 1
12.b even 2 1 1680.4.a.u 1
15.d odd 2 1 525.4.a.a 1
15.e even 4 2 525.4.d.a 2
21.c even 2 1 735.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.b 1 3.b odd 2 1
315.4.a.a 1 1.a even 1 1 trivial
525.4.a.a 1 15.d odd 2 1
525.4.d.a 2 15.e even 4 2
735.4.a.j 1 21.c even 2 1
1575.4.a.l 1 5.b even 2 1
1680.4.a.u 1 12.b even 2 1
2205.4.a.b 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\). 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
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 12 \) Copy content Toggle raw display
$13$ \( T - 30 \) Copy content Toggle raw display
$17$ \( T - 134 \) Copy content Toggle raw display
$19$ \( T + 92 \) Copy content Toggle raw display
$23$ \( T + 112 \) Copy content Toggle raw display
$29$ \( T - 58 \) Copy content Toggle raw display
$31$ \( T + 224 \) Copy content Toggle raw display
$37$ \( T + 146 \) Copy content Toggle raw display
$41$ \( T + 18 \) Copy content Toggle raw display
$43$ \( T - 340 \) Copy content Toggle raw display
$47$ \( T + 208 \) Copy content Toggle raw display
$53$ \( T - 754 \) Copy content Toggle raw display
$59$ \( T + 380 \) Copy content Toggle raw display
$61$ \( T - 718 \) Copy content Toggle raw display
$67$ \( T - 412 \) Copy content Toggle raw display
$71$ \( T - 960 \) Copy content Toggle raw display
$73$ \( T - 1066 \) Copy content Toggle raw display
$79$ \( T - 896 \) Copy content Toggle raw display
$83$ \( T + 436 \) Copy content Toggle raw display
$89$ \( T - 1038 \) Copy content Toggle raw display
$97$ \( T + 702 \) Copy content Toggle raw display
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