Properties

Label 1519.1.c.a
Level $1519$
Weight $1$
Character orbit 1519.c
Self dual yes
Analytic conductor $0.758$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -31
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1519,1,Mod(1177,1519)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1519, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1519.1177");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1519 = 7^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1519.c (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: \(0.758079754190\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.31.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.329623.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{5} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{5} + q^{8} + q^{9} - q^{10} - q^{16} - q^{18} + q^{19} - q^{31} - q^{38} + q^{40} + q^{41} + q^{45} - 2 q^{47} + q^{59} + q^{62} + q^{64} + 2 q^{67} - q^{71} + q^{72} - q^{80} + q^{81} - q^{82} - q^{90} + 2 q^{94} + q^{95} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1519\mathbb{Z}\right)^\times\).

\(n\) \(344\) \(1179\)
\(\chi(n)\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1177.1
0
−1.00000 0 0 1.00000 0 0 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1519.1.c.a 1
7.b odd 2 1 31.1.b.a 1
7.c even 3 2 1519.1.n.a 2
7.d odd 6 2 1519.1.n.b 2
21.c even 2 1 279.1.d.b 1
28.d even 2 1 496.1.e.a 1
31.b odd 2 1 CM 1519.1.c.a 1
35.c odd 2 1 775.1.d.b 1
35.f even 4 2 775.1.c.a 2
56.e even 2 1 1984.1.e.b 1
56.h odd 2 1 1984.1.e.a 1
63.l odd 6 2 2511.1.m.e 2
63.o even 6 2 2511.1.m.a 2
77.b even 2 1 3751.1.d.b 1
77.j odd 10 4 3751.1.t.c 4
77.l even 10 4 3751.1.t.a 4
217.d even 2 1 31.1.b.a 1
217.m even 6 2 1519.1.n.b 2
217.n odd 6 2 1519.1.n.a 2
217.s even 6 2 961.1.e.a 2
217.u odd 6 2 961.1.e.a 2
217.v even 10 4 961.1.f.a 4
217.w odd 10 4 961.1.f.a 4
217.bd odd 30 8 961.1.h.a 8
217.be even 30 8 961.1.h.a 8
651.e odd 2 1 279.1.d.b 1
868.c odd 2 1 496.1.e.a 1
1085.b even 2 1 775.1.d.b 1
1085.o odd 4 2 775.1.c.a 2
1736.h even 2 1 1984.1.e.a 1
1736.n odd 2 1 1984.1.e.b 1
1953.cb even 6 2 2511.1.m.e 2
1953.cu odd 6 2 2511.1.m.a 2
2387.d odd 2 1 3751.1.d.b 1
2387.cb even 10 4 3751.1.t.c 4
2387.cv odd 10 4 3751.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 7.b odd 2 1
31.1.b.a 1 217.d even 2 1
279.1.d.b 1 21.c even 2 1
279.1.d.b 1 651.e odd 2 1
496.1.e.a 1 28.d even 2 1
496.1.e.a 1 868.c odd 2 1
775.1.c.a 2 35.f even 4 2
775.1.c.a 2 1085.o odd 4 2
775.1.d.b 1 35.c odd 2 1
775.1.d.b 1 1085.b even 2 1
961.1.e.a 2 217.s even 6 2
961.1.e.a 2 217.u odd 6 2
961.1.f.a 4 217.v even 10 4
961.1.f.a 4 217.w odd 10 4
961.1.h.a 8 217.bd odd 30 8
961.1.h.a 8 217.be even 30 8
1519.1.c.a 1 1.a even 1 1 trivial
1519.1.c.a 1 31.b odd 2 1 CM
1519.1.n.a 2 7.c even 3 2
1519.1.n.a 2 217.n odd 6 2
1519.1.n.b 2 7.d odd 6 2
1519.1.n.b 2 217.m even 6 2
1984.1.e.a 1 56.h odd 2 1
1984.1.e.a 1 1736.h even 2 1
1984.1.e.b 1 56.e even 2 1
1984.1.e.b 1 1736.n odd 2 1
2511.1.m.a 2 63.o even 6 2
2511.1.m.a 2 1953.cu odd 6 2
2511.1.m.e 2 63.l odd 6 2
2511.1.m.e 2 1953.cb even 6 2
3751.1.d.b 1 77.b even 2 1
3751.1.d.b 1 2387.d odd 2 1
3751.1.t.a 4 77.l even 10 4
3751.1.t.a 4 2387.cv odd 10 4
3751.1.t.c 4 77.j odd 10 4
3751.1.t.c 4 2387.cb even 10 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1519, [\chi])\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} - 1 \) 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
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 1 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 1 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 2 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T - 1 \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T + 1 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 1 \) Copy content Toggle raw display
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