Properties

Label 1287.1.g.a
Level $1287$
Weight $1$
Character orbit 1287.g
Self dual yes
Analytic conductor $0.642$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -143
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1287,1,Mod(1000,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1287.1000");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1287.g (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: \(0.642296671259\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 143)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.20449.1
Artin image: $D_{10}$
Artin field: Galois closure of 10.0.1320972497559.1

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + \beta q^{4} + ( - \beta + 1) q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + \beta q^{4} + ( - \beta + 1) q^{7} - q^{8} + q^{11} - q^{13} + q^{14} + \beta q^{19} - \beta q^{22} + ( - \beta + 1) q^{23} + q^{25} + \beta q^{26} - q^{28} + q^{32} + ( - \beta - 1) q^{38} + (\beta - 1) q^{41} + \beta q^{44} + q^{46} + ( - \beta + 1) q^{49} - \beta q^{50} - \beta q^{52} + \beta q^{53} + (\beta - 1) q^{56} - \beta q^{64} + \beta q^{73} + (\beta + 1) q^{76} + ( - \beta + 1) q^{77} - q^{82} + (\beta - 1) q^{83} - q^{88} + (\beta - 1) q^{91} - q^{92} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{4} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{4} + q^{7} - 2 q^{8} + 2 q^{11} - 2 q^{13} + 2 q^{14} + q^{19} - q^{22} + q^{23} + 2 q^{25} + q^{26} - 2 q^{28} + 2 q^{32} - 3 q^{38} - q^{41} + q^{44} + 2 q^{46} + q^{49} - q^{50} - q^{52} + q^{53} - q^{56} - q^{64} + q^{73} + 3 q^{76} + q^{77} - 2 q^{82} - q^{83} - 2 q^{88} - q^{91} - 2 q^{92} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(937\) \(1145\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1000.1
1.61803
−0.618034
−1.61803 0 1.61803 0 0 −0.618034 −1.00000 0 0
1000.2 0.618034 0 −0.618034 0 0 1.61803 −1.00000 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
143.d odd 2 1 CM by \(\Q(\sqrt{-143}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1287.1.g.a 2
3.b odd 2 1 143.1.d.b yes 2
11.b odd 2 1 1287.1.g.b 2
12.b even 2 1 2288.1.m.a 2
13.b even 2 1 1287.1.g.b 2
15.d odd 2 1 3575.1.h.e 2
15.e even 4 2 3575.1.c.c 4
33.d even 2 1 143.1.d.a 2
33.f even 10 2 1573.1.l.b 4
33.f even 10 2 1573.1.l.d 4
33.h odd 10 2 1573.1.l.a 4
33.h odd 10 2 1573.1.l.c 4
39.d odd 2 1 143.1.d.a 2
39.f even 4 2 1859.1.c.c 4
39.h odd 6 2 1859.1.i.b 4
39.i odd 6 2 1859.1.i.a 4
39.k even 12 4 1859.1.k.c 8
132.d odd 2 1 2288.1.m.b 2
143.d odd 2 1 CM 1287.1.g.a 2
156.h even 2 1 2288.1.m.b 2
165.d even 2 1 3575.1.h.f 2
165.l odd 4 2 3575.1.c.d 4
195.e odd 2 1 3575.1.h.f 2
195.s even 4 2 3575.1.c.d 4
429.e even 2 1 143.1.d.b yes 2
429.l odd 4 2 1859.1.c.c 4
429.p even 6 2 1859.1.i.b 4
429.t even 6 2 1859.1.i.a 4
429.v odd 10 2 1573.1.l.b 4
429.v odd 10 2 1573.1.l.d 4
429.y even 10 2 1573.1.l.a 4
429.y even 10 2 1573.1.l.c 4
429.bc odd 12 4 1859.1.k.c 8
1716.m odd 2 1 2288.1.m.a 2
2145.k even 2 1 3575.1.h.e 2
2145.bf odd 4 2 3575.1.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.1.d.a 2 33.d even 2 1
143.1.d.a 2 39.d odd 2 1
143.1.d.b yes 2 3.b odd 2 1
143.1.d.b yes 2 429.e even 2 1
1287.1.g.a 2 1.a even 1 1 trivial
1287.1.g.a 2 143.d odd 2 1 CM
1287.1.g.b 2 11.b odd 2 1
1287.1.g.b 2 13.b even 2 1
1573.1.l.a 4 33.h odd 10 2
1573.1.l.a 4 429.y even 10 2
1573.1.l.b 4 33.f even 10 2
1573.1.l.b 4 429.v odd 10 2
1573.1.l.c 4 33.h odd 10 2
1573.1.l.c 4 429.y even 10 2
1573.1.l.d 4 33.f even 10 2
1573.1.l.d 4 429.v odd 10 2
1859.1.c.c 4 39.f even 4 2
1859.1.c.c 4 429.l odd 4 2
1859.1.i.a 4 39.i odd 6 2
1859.1.i.a 4 429.t even 6 2
1859.1.i.b 4 39.h odd 6 2
1859.1.i.b 4 429.p even 6 2
1859.1.k.c 8 39.k even 12 4
1859.1.k.c 8 429.bc odd 12 4
2288.1.m.a 2 12.b even 2 1
2288.1.m.a 2 1716.m odd 2 1
2288.1.m.b 2 132.d odd 2 1
2288.1.m.b 2 156.h even 2 1
3575.1.c.c 4 15.e even 4 2
3575.1.c.c 4 2145.bf odd 4 2
3575.1.c.d 4 165.l odd 4 2
3575.1.c.d 4 195.s even 4 2
3575.1.h.e 2 15.d odd 2 1
3575.1.h.e 2 2145.k even 2 1
3575.1.h.f 2 165.d even 2 1
3575.1.h.f 2 195.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1287, [\chi])\). 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$23$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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