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Properties

Label 900.1.bh.a
Level $900$
Weight $1$
Character orbit 900.bh
Analytic conductor $0.449$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,1,Mod(287,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.287");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 900.bh (of order \(20\), degree \(8\), 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: \(0.449158511370\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{40}^{11} q^{2} - \zeta_{40}^{2} q^{4} + \zeta_{40}^{3} q^{5} + \zeta_{40}^{13} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{40}^{11} q^{2} - \zeta_{40}^{2} q^{4} + \zeta_{40}^{3} q^{5} + \zeta_{40}^{13} q^{8} - \zeta_{40}^{14} q^{10} + ( - \zeta_{40}^{16} + \zeta_{40}^{2}) q^{13} + \zeta_{40}^{4} q^{16} + (\zeta_{40}^{17} - \zeta_{40}^{9}) q^{17} - \zeta_{40}^{5} q^{20} + \zeta_{40}^{6} q^{25} + ( - \zeta_{40}^{13} - \zeta_{40}^{7}) q^{26} + ( - \zeta_{40}^{3} + \zeta_{40}) q^{29} - \zeta_{40}^{15} q^{32} + (\zeta_{40}^{8} - 1) q^{34} + ( - \zeta_{40}^{10} - \zeta_{40}^{8}) q^{37} + \zeta_{40}^{16} q^{40} + (\zeta_{40}^{7} - \zeta_{40}) q^{41} + \zeta_{40}^{10} q^{49} - \zeta_{40}^{17} q^{50} + (\zeta_{40}^{18} - \zeta_{40}^{4}) q^{52} + (\zeta_{40}^{19} + \zeta_{40}^{15}) q^{53} + (\zeta_{40}^{14} - \zeta_{40}^{12}) q^{58} + (\zeta_{40}^{18} + \zeta_{40}^{14}) q^{61} - \zeta_{40}^{6} q^{64} + ( - \zeta_{40}^{19} + \zeta_{40}^{5}) q^{65} + ( - \zeta_{40}^{19} + \zeta_{40}^{11}) q^{68} + ( - \zeta_{40}^{18} - \zeta_{40}^{4}) q^{73} + (\zeta_{40}^{19} - \zeta_{40}) q^{74} + \zeta_{40}^{7} q^{80} + ( - \zeta_{40}^{18} + \zeta_{40}^{12}) q^{82} + ( - \zeta_{40}^{12} - 1) q^{85} + ( - \zeta_{40}^{17} - \zeta_{40}^{15}) q^{89} + (\zeta_{40}^{8} - \zeta_{40}^{6}) q^{97} + \zeta_{40} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{13} + 4 q^{16} - 20 q^{34} + 4 q^{37} - 4 q^{40} - 4 q^{52} - 4 q^{58} - 4 q^{73} + 4 q^{82} - 20 q^{85} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{40}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−0.156434 − 0.987688i
0.156434 + 0.987688i
−0.156434 + 0.987688i
0.156434 − 0.987688i
0.453990 − 0.891007i
−0.453990 + 0.891007i
0.453990 + 0.891007i
−0.453990 − 0.891007i
0.891007 − 0.453990i
−0.891007 + 0.453990i
0.891007 + 0.453990i
−0.891007 − 0.453990i
−0.987688 − 0.156434i
0.987688 + 0.156434i
−0.987688 + 0.156434i
0.987688 − 0.156434i
−0.987688 + 0.156434i 0 0.951057 − 0.309017i 0.453990 + 0.891007i 0 0 −0.891007 + 0.453990i 0 −0.587785 − 0.809017i
287.2 0.987688 − 0.156434i 0 0.951057 − 0.309017i −0.453990 − 0.891007i 0 0 0.891007 − 0.453990i 0 −0.587785 − 0.809017i
323.1 −0.987688 − 0.156434i 0 0.951057 + 0.309017i 0.453990 − 0.891007i 0 0 −0.891007 − 0.453990i 0 −0.587785 + 0.809017i
323.2 0.987688 + 0.156434i 0 0.951057 + 0.309017i −0.453990 + 0.891007i 0 0 0.891007 + 0.453990i 0 −0.587785 + 0.809017i
467.1 −0.891007 − 0.453990i 0 0.587785 + 0.809017i −0.987688 + 0.156434i 0 0 −0.156434 − 0.987688i 0 0.951057 + 0.309017i
467.2 0.891007 + 0.453990i 0 0.587785 + 0.809017i 0.987688 − 0.156434i 0 0 0.156434 + 0.987688i 0 0.951057 + 0.309017i
503.1 −0.891007 + 0.453990i 0 0.587785 − 0.809017i −0.987688 − 0.156434i 0 0 −0.156434 + 0.987688i 0 0.951057 − 0.309017i
503.2 0.891007 − 0.453990i 0 0.587785 − 0.809017i 0.987688 + 0.156434i 0 0 0.156434 − 0.987688i 0 0.951057 − 0.309017i
647.1 −0.453990 − 0.891007i 0 −0.587785 + 0.809017i 0.156434 − 0.987688i 0 0 0.987688 + 0.156434i 0 −0.951057 + 0.309017i
647.2 0.453990 + 0.891007i 0 −0.587785 + 0.809017i −0.156434 + 0.987688i 0 0 −0.987688 − 0.156434i 0 −0.951057 + 0.309017i
683.1 −0.453990 + 0.891007i 0 −0.587785 − 0.809017i 0.156434 + 0.987688i 0 0 0.987688 − 0.156434i 0 −0.951057 − 0.309017i
683.2 0.453990 − 0.891007i 0 −0.587785 − 0.809017i −0.156434 − 0.987688i 0 0 −0.987688 + 0.156434i 0 −0.951057 − 0.309017i
827.1 −0.156434 + 0.987688i 0 −0.951057 − 0.309017i −0.891007 − 0.453990i 0 0 0.453990 − 0.891007i 0 0.587785 − 0.809017i
827.2 0.156434 − 0.987688i 0 −0.951057 − 0.309017i 0.891007 + 0.453990i 0 0 −0.453990 + 0.891007i 0 0.587785 − 0.809017i
863.1 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i −0.891007 + 0.453990i 0 0 0.453990 + 0.891007i 0 0.587785 + 0.809017i
863.2 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 0.891007 − 0.453990i 0 0 −0.453990 − 0.891007i 0 0.587785 + 0.809017i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner
100.l even 20 1 inner
300.u odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.1.bh.a ✓ 16
3.b odd 2 1 inner 900.1.bh.a ✓ 16
4.b odd 2 1 CM 900.1.bh.a ✓ 16
12.b even 2 1 inner 900.1.bh.a ✓ 16
25.f odd 20 1 inner 900.1.bh.a ✓ 16
75.l even 20 1 inner 900.1.bh.a ✓ 16
100.l even 20 1 inner 900.1.bh.a ✓ 16
300.u odd 20 1 inner 900.1.bh.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.1.bh.a ✓ 16 1.a even 1 1 trivial
900.1.bh.a ✓ 16 3.b odd 2 1 inner
900.1.bh.a ✓ 16 4.b odd 2 1 CM
900.1.bh.a ✓ 16 12.b even 2 1 inner
900.1.bh.a ✓ 16 25.f odd 20 1 inner
900.1.bh.a ✓ 16 75.l even 20 1 inner
900.1.bh.a ✓ 16 100.l even 20 1 inner
900.1.bh.a ✓ 16 300.u odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 20 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{8} - 2 T^{7} + 7 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + 5 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} + 10 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
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