Properties

Label 315.3.e.b
Level $315$
Weight $3$
Character orbit 315.e
Self dual yes
Analytic conductor $8.583$
Analytic rank $0$
Dimension $1$
CM discriminant -35
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,3,Mod(244,315)]
 
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, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.244");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.e (of order \(2\), degree \(1\), minimal)

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: \(8.58312832735\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{4} + 5 q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{4} + 5 q^{5} + 7 q^{7} + 13 q^{11} - 19 q^{13} + 16 q^{16} - 29 q^{17} + 20 q^{20} + 25 q^{25} + 28 q^{28} - 23 q^{29} + 35 q^{35} + 52 q^{44} + 31 q^{47} + 49 q^{49} - 76 q^{52} + 65 q^{55} + 64 q^{64} - 95 q^{65} - 116 q^{68} - 2 q^{71} - 34 q^{73} + 91 q^{77} - 157 q^{79} + 80 q^{80} - 86 q^{83} - 145 q^{85} - 133 q^{91} + 149 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
0
0 0 4.00000 5.00000 0 7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.3.e.b 1
3.b odd 2 1 35.3.c.b yes 1
5.b even 2 1 315.3.e.a 1
7.b odd 2 1 315.3.e.a 1
12.b even 2 1 560.3.p.a 1
15.d odd 2 1 35.3.c.a 1
15.e even 4 2 175.3.d.e 2
21.c even 2 1 35.3.c.a 1
21.g even 6 2 245.3.i.b 2
21.h odd 6 2 245.3.i.a 2
35.c odd 2 1 CM 315.3.e.b 1
60.h even 2 1 560.3.p.b 1
84.h odd 2 1 560.3.p.b 1
105.g even 2 1 35.3.c.b yes 1
105.k odd 4 2 175.3.d.e 2
105.o odd 6 2 245.3.i.b 2
105.p even 6 2 245.3.i.a 2
420.o odd 2 1 560.3.p.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.c.a 1 15.d odd 2 1
35.3.c.a 1 21.c even 2 1
35.3.c.b yes 1 3.b odd 2 1
35.3.c.b yes 1 105.g even 2 1
175.3.d.e 2 15.e even 4 2
175.3.d.e 2 105.k odd 4 2
245.3.i.a 2 21.h odd 6 2
245.3.i.a 2 105.p even 6 2
245.3.i.b 2 21.g even 6 2
245.3.i.b 2 105.o odd 6 2
315.3.e.a 1 5.b even 2 1
315.3.e.a 1 7.b odd 2 1
315.3.e.b 1 1.a even 1 1 trivial
315.3.e.b 1 35.c odd 2 1 CM
560.3.p.a 1 12.b even 2 1
560.3.p.a 1 420.o odd 2 1
560.3.p.b 1 60.h even 2 1
560.3.p.b 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(315, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{13} + 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) 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
$11$ \( T - 13 \) Copy content Toggle raw display
$13$ \( T + 19 \) Copy content Toggle raw display
$17$ \( T + 29 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 23 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 31 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T + 34 \) Copy content Toggle raw display
$79$ \( T + 157 \) Copy content Toggle raw display
$83$ \( T + 86 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 149 \) Copy content Toggle raw display
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