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Properties

Label 900.3.c.b
Level $900$
Weight $3$
Character orbit 900.c
Self dual yes
Analytic conductor $24.523$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,3,Mod(451,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.451"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,0,4,0,0,0,-8,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} - 8 q^{8} + 10 q^{13} + 16 q^{16} - 16 q^{17} - 20 q^{26} + 40 q^{29} - 32 q^{32} + 32 q^{34} + 70 q^{37} - 80 q^{41} + 49 q^{49} + 40 q^{52} + 56 q^{53} - 80 q^{58} - 22 q^{61} + 64 q^{64}+ \cdots - 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0
−2.00000 0 4.00000 0 0 0 −8.00000 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.c.b 1
3.b odd 2 1 900.3.c.c 1
4.b odd 2 1 CM 900.3.c.b 1
5.b even 2 1 36.3.d.b yes 1
5.c odd 4 2 900.3.f.a 2
12.b even 2 1 900.3.c.c 1
15.d odd 2 1 36.3.d.a ✓ 1
15.e even 4 2 900.3.f.b 2
20.d odd 2 1 36.3.d.b yes 1
20.e even 4 2 900.3.f.a 2
40.e odd 2 1 576.3.g.c 1
40.f even 2 1 576.3.g.c 1
45.h odd 6 2 324.3.f.f 2
45.j even 6 2 324.3.f.e 2
60.h even 2 1 36.3.d.a ✓ 1
60.l odd 4 2 900.3.f.b 2
80.k odd 4 2 2304.3.b.e 2
80.q even 4 2 2304.3.b.e 2
120.i odd 2 1 576.3.g.a 1
120.m even 2 1 576.3.g.a 1
180.n even 6 2 324.3.f.f 2
180.p odd 6 2 324.3.f.e 2
240.t even 4 2 2304.3.b.d 2
240.bm odd 4 2 2304.3.b.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.d.a ✓ 1 15.d odd 2 1
36.3.d.a ✓ 1 60.h even 2 1
36.3.d.b yes 1 5.b even 2 1
36.3.d.b yes 1 20.d odd 2 1
324.3.f.e 2 45.j even 6 2
324.3.f.e 2 180.p odd 6 2
324.3.f.f 2 45.h odd 6 2
324.3.f.f 2 180.n even 6 2
576.3.g.a 1 120.i odd 2 1
576.3.g.a 1 120.m even 2 1
576.3.g.c 1 40.e odd 2 1
576.3.g.c 1 40.f even 2 1
900.3.c.b 1 1.a even 1 1 trivial
900.3.c.b 1 4.b odd 2 1 CM
900.3.c.c 1 3.b odd 2 1
900.3.c.c 1 12.b even 2 1
900.3.f.a 2 5.c odd 4 2
900.3.f.a 2 20.e even 4 2
900.3.f.b 2 15.e even 4 2
900.3.f.b 2 60.l odd 4 2
2304.3.b.d 2 240.t even 4 2
2304.3.b.d 2 240.bm odd 4 2
2304.3.b.e 2 80.k odd 4 2
2304.3.b.e 2 80.q even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{13} - 10 \) Copy content Toggle raw display
\( T_{17} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 10 \) Copy content Toggle raw display
$17$ \( T + 16 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 40 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 70 \) Copy content Toggle raw display
$41$ \( T + 80 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 56 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 22 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 110 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 160 \) Copy content Toggle raw display
$97$ \( T - 130 \) Copy content Toggle raw display
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