Properties

Label 315.4.a.c
Level $315$
Weight $4$
Character orbit 315.a
Self dual yes
Analytic conductor $18.586$
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] = [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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - 7 q^{4} + 5 q^{5} + 7 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 7 q^{4} + 5 q^{5} + 7 q^{7} + 15 q^{8} - 5 q^{10} - 12 q^{11} - 78 q^{13} - 7 q^{14} + 41 q^{16} + 94 q^{17} + 40 q^{19} - 35 q^{20} + 12 q^{22} - 32 q^{23} + 25 q^{25} + 78 q^{26} - 49 q^{28} + 50 q^{29} - 248 q^{31} - 161 q^{32} - 94 q^{34} + 35 q^{35} - 434 q^{37} - 40 q^{38} + 75 q^{40} - 402 q^{41} - 68 q^{43} + 84 q^{44} + 32 q^{46} - 536 q^{47} + 49 q^{49} - 25 q^{50} + 546 q^{52} - 22 q^{53} - 60 q^{55} + 105 q^{56} - 50 q^{58} + 560 q^{59} - 278 q^{61} + 248 q^{62} - 167 q^{64} - 390 q^{65} - 164 q^{67} - 658 q^{68} - 35 q^{70} - 672 q^{71} + 82 q^{73} + 434 q^{74} - 280 q^{76} - 84 q^{77} - 1000 q^{79} + 205 q^{80} + 402 q^{82} + 448 q^{83} + 470 q^{85} + 68 q^{86} - 180 q^{88} + 870 q^{89} - 546 q^{91} + 224 q^{92} + 536 q^{94} + 200 q^{95} + 1026 q^{97} - 49 q^{98}+O(q^{100}) \) 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 7.00000 15.0000 0 −5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.c 1
3.b odd 2 1 35.4.a.a 1
5.b even 2 1 1575.4.a.g 1
7.b odd 2 1 2205.4.a.i 1
12.b even 2 1 560.4.a.p 1
15.d odd 2 1 175.4.a.a 1
15.e even 4 2 175.4.b.a 2
21.c even 2 1 245.4.a.d 1
21.g even 6 2 245.4.e.b 2
21.h odd 6 2 245.4.e.e 2
24.f even 2 1 2240.4.a.b 1
24.h odd 2 1 2240.4.a.bk 1
105.g even 2 1 1225.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.a 1 3.b odd 2 1
175.4.a.a 1 15.d odd 2 1
175.4.b.a 2 15.e even 4 2
245.4.a.d 1 21.c even 2 1
245.4.e.b 2 21.g even 6 2
245.4.e.e 2 21.h odd 6 2
315.4.a.c 1 1.a even 1 1 trivial
560.4.a.p 1 12.b even 2 1
1225.4.a.e 1 105.g even 2 1
1575.4.a.g 1 5.b even 2 1
2205.4.a.i 1 7.b odd 2 1
2240.4.a.b 1 24.f even 2 1
2240.4.a.bk 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\). 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
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 12 \) Copy content Toggle raw display
$13$ \( T + 78 \) Copy content Toggle raw display
$17$ \( T - 94 \) Copy content Toggle raw display
$19$ \( T - 40 \) Copy content Toggle raw display
$23$ \( T + 32 \) Copy content Toggle raw display
$29$ \( T - 50 \) Copy content Toggle raw display
$31$ \( T + 248 \) Copy content Toggle raw display
$37$ \( T + 434 \) Copy content Toggle raw display
$41$ \( T + 402 \) Copy content Toggle raw display
$43$ \( T + 68 \) Copy content Toggle raw display
$47$ \( T + 536 \) Copy content Toggle raw display
$53$ \( T + 22 \) Copy content Toggle raw display
$59$ \( T - 560 \) Copy content Toggle raw display
$61$ \( T + 278 \) Copy content Toggle raw display
$67$ \( T + 164 \) Copy content Toggle raw display
$71$ \( T + 672 \) Copy content Toggle raw display
$73$ \( T - 82 \) Copy content Toggle raw display
$79$ \( T + 1000 \) Copy content Toggle raw display
$83$ \( T - 448 \) Copy content Toggle raw display
$89$ \( T - 870 \) Copy content Toggle raw display
$97$ \( T - 1026 \) Copy content Toggle raw display
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