Properties

Label 1519.1.n.b
Level $1519$
Weight $1$
Character orbit 1519.n
Analytic conductor $0.758$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -31
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1519,1,Mod(30,1519)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1519, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1519.30");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1519 = 7^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1519.n (of order \(6\), degree \(2\), not 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: no
Analytic conductor: \(0.758079754190\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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: $C_3\times S_3$
Artin field: Galois closure of 6.0.71528191.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{5} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{5} + q^{8} + \zeta_{6}^{2} q^{9} - \zeta_{6} q^{10} - \zeta_{6}^{2} q^{16} + \zeta_{6} q^{18} - \zeta_{6}^{2} q^{19} - \zeta_{6} q^{31} - \zeta_{6} q^{38} - \zeta_{6}^{2} q^{40} - q^{41} + \zeta_{6} q^{45} + \zeta_{6}^{2} q^{47} + \zeta_{6} q^{59} - q^{62} + q^{64} - \zeta_{6} q^{67} - q^{71} + \zeta_{6}^{2} q^{72} - \zeta_{6} q^{80} - \zeta_{6} q^{81} + \zeta_{6}^{2} q^{82} + q^{90} + 2 \zeta_{6} q^{94} - \zeta_{6} q^{95} - q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{5} + 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{5} + 2 q^{8} - q^{9} - q^{10} + q^{16} + q^{18} + q^{19} - q^{31} - q^{38} + q^{40} - 2 q^{41} + q^{45} - 2 q^{47} + q^{59} - 2 q^{62} + 2 q^{64} - 2 q^{67} - 2 q^{71} - q^{72} - q^{80} - q^{81} - q^{82} + 2 q^{90} + 2 q^{94} - q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(344\) \(1179\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0 0 0.500000 0.866025i 0 0 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
557.1 0.500000 + 0.866025i 0 0 0.500000 + 0.866025i 0 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
\(n\): 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 \(\Q(\sqrt{-31}) \)
7.c even 3 1 inner
217.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1519.1.n.b 2
7.b odd 2 1 1519.1.n.a 2
7.c even 3 1 31.1.b.a 1
7.c even 3 1 inner 1519.1.n.b 2
7.d odd 6 1 1519.1.c.a 1
7.d odd 6 1 1519.1.n.a 2
21.h odd 6 1 279.1.d.b 1
28.g odd 6 1 496.1.e.a 1
31.b odd 2 1 CM 1519.1.n.b 2
35.j even 6 1 775.1.d.b 1
35.l odd 12 2 775.1.c.a 2
56.k odd 6 1 1984.1.e.b 1
56.p even 6 1 1984.1.e.a 1
63.g even 3 1 2511.1.m.e 2
63.h even 3 1 2511.1.m.e 2
63.j odd 6 1 2511.1.m.a 2
63.n odd 6 1 2511.1.m.a 2
77.h odd 6 1 3751.1.d.b 1
77.m even 15 4 3751.1.t.c 4
77.o odd 30 4 3751.1.t.a 4
217.d even 2 1 1519.1.n.a 2
217.e even 3 1 961.1.e.a 2
217.g even 3 1 961.1.e.a 2
217.k odd 6 1 961.1.e.a 2
217.m even 6 1 1519.1.c.a 1
217.m even 6 1 1519.1.n.a 2
217.n odd 6 1 31.1.b.a 1
217.n odd 6 1 inner 1519.1.n.b 2
217.o odd 6 1 961.1.e.a 2
217.z even 15 4 961.1.h.a 8
217.ba even 15 4 961.1.h.a 8
217.bb even 15 4 961.1.f.a 4
217.bg odd 30 4 961.1.f.a 4
217.bh odd 30 4 961.1.h.a 8
217.bn odd 30 4 961.1.h.a 8
651.bg even 6 1 279.1.d.b 1
868.r even 6 1 496.1.e.a 1
1085.bh odd 6 1 775.1.d.b 1
1085.cf even 12 2 775.1.c.a 2
1736.br odd 6 1 1984.1.e.a 1
1736.ct even 6 1 1984.1.e.b 1
1953.bf even 6 1 2511.1.m.a 2
1953.ck odd 6 1 2511.1.m.e 2
1953.cs even 6 1 2511.1.m.a 2
1953.eb odd 6 1 2511.1.m.e 2
2387.bm even 6 1 3751.1.d.b 1
2387.gi even 30 4 3751.1.t.a 4
2387.il odd 30 4 3751.1.t.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 7.c even 3 1
31.1.b.a 1 217.n odd 6 1
279.1.d.b 1 21.h odd 6 1
279.1.d.b 1 651.bg even 6 1
496.1.e.a 1 28.g odd 6 1
496.1.e.a 1 868.r even 6 1
775.1.c.a 2 35.l odd 12 2
775.1.c.a 2 1085.cf even 12 2
775.1.d.b 1 35.j even 6 1
775.1.d.b 1 1085.bh odd 6 1
961.1.e.a 2 217.e even 3 1
961.1.e.a 2 217.g even 3 1
961.1.e.a 2 217.k odd 6 1
961.1.e.a 2 217.o odd 6 1
961.1.f.a 4 217.bb even 15 4
961.1.f.a 4 217.bg odd 30 4
961.1.h.a 8 217.z even 15 4
961.1.h.a 8 217.ba even 15 4
961.1.h.a 8 217.bh odd 30 4
961.1.h.a 8 217.bn odd 30 4
1519.1.c.a 1 7.d odd 6 1
1519.1.c.a 1 217.m even 6 1
1519.1.n.a 2 7.b odd 2 1
1519.1.n.a 2 7.d odd 6 1
1519.1.n.a 2 217.d even 2 1
1519.1.n.a 2 217.m even 6 1
1519.1.n.b 2 1.a even 1 1 trivial
1519.1.n.b 2 7.c even 3 1 inner
1519.1.n.b 2 31.b odd 2 1 CM
1519.1.n.b 2 217.n odd 6 1 inner
1984.1.e.a 1 56.p even 6 1
1984.1.e.a 1 1736.br odd 6 1
1984.1.e.b 1 56.k odd 6 1
1984.1.e.b 1 1736.ct even 6 1
2511.1.m.a 2 63.j odd 6 1
2511.1.m.a 2 63.n odd 6 1
2511.1.m.a 2 1953.bf even 6 1
2511.1.m.a 2 1953.cs even 6 1
2511.1.m.e 2 63.g even 3 1
2511.1.m.e 2 63.h even 3 1
2511.1.m.e 2 1953.ck odd 6 1
2511.1.m.e 2 1953.eb odd 6 1
3751.1.d.b 1 77.h odd 6 1
3751.1.d.b 1 2387.bm even 6 1
3751.1.t.a 4 77.o odd 30 4
3751.1.t.a 4 2387.gi even 30 4
3751.1.t.c 4 77.m even 15 4
3751.1.t.c 4 2387.il odd 30 4

Hecke kernels

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

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$71$ \( (T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 1)^{2} \) Copy content Toggle raw display
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