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Properties

Label 2904.1.n.a
Level $2904$
Weight $1$
Character orbit 2904.n
Self dual yes
Analytic conductor $1.449$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -24
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2904,1,Mod(485,2904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2904, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2904.485");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2904 = 2^{3} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2904.n (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: \(1.44928479669\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 264)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.8433216.2

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + ( - \beta + 1) q^{5} + q^{6} - \beta q^{7} - q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + ( - \beta + 1) q^{5} + q^{6} - \beta q^{7} - q^{8} + q^{9} + (\beta - 1) q^{10} - q^{12} + \beta q^{14} + (\beta - 1) q^{15} + q^{16} - q^{18} + ( - \beta + 1) q^{20} + \beta q^{21} + q^{24} + ( - \beta + 1) q^{25} - q^{27} - \beta q^{28} + ( - \beta + 1) q^{29} + ( - \beta + 1) q^{30} - \beta q^{31} - q^{32} + q^{35} + q^{36} + (\beta - 1) q^{40} - \beta q^{42} + ( - \beta + 1) q^{45} - q^{48} + \beta q^{49} + (\beta - 1) q^{50} + \beta q^{53} + q^{54} + \beta q^{56} + (\beta - 1) q^{58} + \beta q^{59} + (\beta - 1) q^{60} + \beta q^{62} - \beta q^{63} + q^{64} - q^{70} - q^{72} + (\beta - 1) q^{73} + (\beta - 1) q^{75} + (\beta - 1) q^{79} + ( - \beta + 1) q^{80} + q^{81} + ( - \beta + 1) q^{83} + \beta q^{84} + (\beta - 1) q^{87} + (\beta - 1) q^{90} + \beta q^{93} + q^{96} + (\beta - 1) q^{97} - \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - q^{10} - 2 q^{12} + q^{14} - q^{15} + 2 q^{16} - 2 q^{18} + q^{20} + q^{21} + 2 q^{24} + q^{25} - 2 q^{27} - q^{28} + q^{29} + q^{30} - q^{31} - 2 q^{32} + 2 q^{35} + 2 q^{36} - q^{40} - q^{42} + q^{45} - 2 q^{48} + q^{49} - q^{50} + q^{53} + 2 q^{54} + q^{56} - q^{58} + q^{59} - q^{60} + q^{62} - q^{63} + 2 q^{64} - 2 q^{70} - 2 q^{72} - q^{73} - q^{75} - q^{79} + q^{80} + 2 q^{81} + q^{83} + q^{84} - q^{87} - q^{90} + q^{93} + 2 q^{96} - q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2904\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1453\) \(1937\) \(2785\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
1.61803
−0.618034
−1.00000 −1.00000 1.00000 −0.618034 1.00000 −1.61803 −1.00000 1.00000 0.618034
485.2 −1.00000 −1.00000 1.00000 1.61803 1.00000 0.618034 −1.00000 1.00000 −1.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2904.1.n.a 2
3.b odd 2 1 2904.1.n.d 2
8.b even 2 1 2904.1.n.d 2
11.b odd 2 1 2904.1.n.c 2
11.c even 5 2 264.1.t.b yes 4
11.c even 5 2 2904.1.t.f 4
11.d odd 10 2 2904.1.t.b 4
11.d odd 10 2 2904.1.t.c 4
24.h odd 2 1 CM 2904.1.n.a 2
33.d even 2 1 2904.1.n.b 2
33.f even 10 2 2904.1.t.d 4
33.f even 10 2 2904.1.t.e 4
33.h odd 10 2 264.1.t.a ✓ 4
33.h odd 10 2 2904.1.t.a 4
44.h odd 10 2 1056.1.bj.a 4
88.b odd 2 1 2904.1.n.b 2
88.l odd 10 2 1056.1.bj.b 4
88.o even 10 2 264.1.t.a ✓ 4
88.o even 10 2 2904.1.t.a 4
88.p odd 10 2 2904.1.t.d 4
88.p odd 10 2 2904.1.t.e 4
132.o even 10 2 1056.1.bj.b 4
264.m even 2 1 2904.1.n.c 2
264.t odd 10 2 264.1.t.b yes 4
264.t odd 10 2 2904.1.t.f 4
264.u even 10 2 2904.1.t.b 4
264.u even 10 2 2904.1.t.c 4
264.w even 10 2 1056.1.bj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.1.t.a ✓ 4 33.h odd 10 2
264.1.t.a ✓ 4 88.o even 10 2
264.1.t.b yes 4 11.c even 5 2
264.1.t.b yes 4 264.t odd 10 2
1056.1.bj.a 4 44.h odd 10 2
1056.1.bj.a 4 264.w even 10 2
1056.1.bj.b 4 88.l odd 10 2
1056.1.bj.b 4 132.o even 10 2
2904.1.n.a 2 1.a even 1 1 trivial
2904.1.n.a 2 24.h odd 2 1 CM
2904.1.n.b 2 33.d even 2 1
2904.1.n.b 2 88.b odd 2 1
2904.1.n.c 2 11.b odd 2 1
2904.1.n.c 2 264.m even 2 1
2904.1.n.d 2 3.b odd 2 1
2904.1.n.d 2 8.b even 2 1
2904.1.t.a 4 33.h odd 10 2
2904.1.t.a 4 88.o even 10 2
2904.1.t.b 4 11.d odd 10 2
2904.1.t.b 4 264.u even 10 2
2904.1.t.c 4 11.d odd 10 2
2904.1.t.c 4 264.u even 10 2
2904.1.t.d 4 33.f even 10 2
2904.1.t.d 4 88.p odd 10 2
2904.1.t.e 4 33.f even 10 2
2904.1.t.e 4 88.p odd 10 2
2904.1.t.f 4 11.c even 5 2
2904.1.t.f 4 264.t odd 10 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2904, [\chi])\):

\( T_{5}^{2} - T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 1 \) 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
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) 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
$59$ \( T^{2} - T - 1 \) 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
$79$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$83$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + T - 1 \) Copy content Toggle raw display
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