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Properties

Label 900.1.bb.a
Level $900$
Weight $1$
Character orbit 900.bb
Analytic conductor $0.449$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
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(19,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.19");
 
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.bb (of order \(10\), degree \(4\), 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.47460937500000000.8

$q$-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.

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

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{20}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.587785 − 0.809017i
0.587785 + 0.809017i
−0.587785 + 0.809017i
0.587785 − 0.809017i
0.951057 − 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 − 0.309017i
−0.951057 + 0.309017i 0 0.809017 − 0.587785i 0.587785 − 0.809017i 0 0 −0.587785 + 0.809017i 0 −0.309017 + 0.951057i
19.2 0.951057 − 0.309017i 0 0.809017 − 0.587785i −0.587785 + 0.809017i 0 0 0.587785 − 0.809017i 0 −0.309017 + 0.951057i
379.1 −0.951057 − 0.309017i 0 0.809017 + 0.587785i 0.587785 + 0.809017i 0 0 −0.587785 − 0.809017i 0 −0.309017 − 0.951057i
379.2 0.951057 + 0.309017i 0 0.809017 + 0.587785i −0.587785 − 0.809017i 0 0 0.587785 + 0.809017i 0 −0.309017 − 0.951057i
559.1 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i −0.951057 − 0.309017i 0 0 0.951057 + 0.309017i 0 0.809017 − 0.587785i
559.2 0.587785 − 0.809017i 0 −0.309017 − 0.951057i 0.951057 + 0.309017i 0 0 −0.951057 − 0.309017i 0 0.809017 − 0.587785i
739.1 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i −0.951057 + 0.309017i 0 0 0.951057 − 0.309017i 0 0.809017 + 0.587785i
739.2 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 0.951057 − 0.309017i 0 0 −0.951057 + 0.309017i 0 0.809017 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
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.e even 10 1 inner
75.h odd 10 1 inner
100.h odd 10 1 inner
300.r even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.1.bb.a ✓ 8
3.b odd 2 1 inner 900.1.bb.a ✓ 8
4.b odd 2 1 CM 900.1.bb.a ✓ 8
12.b even 2 1 inner 900.1.bb.a ✓ 8
25.e even 10 1 inner 900.1.bb.a ✓ 8
75.h odd 10 1 inner 900.1.bb.a ✓ 8
100.h odd 10 1 inner 900.1.bb.a ✓ 8
300.r even 10 1 inner 900.1.bb.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.1.bb.a ✓ 8 1.a even 1 1 trivial
900.1.bb.a ✓ 8 3.b odd 2 1 inner
900.1.bb.a ✓ 8 4.b odd 2 1 CM
900.1.bb.a ✓ 8 12.b even 2 1 inner
900.1.bb.a ✓ 8 25.e even 10 1 inner
900.1.bb.a ✓ 8 75.h odd 10 1 inner
900.1.bb.a ✓ 8 100.h odd 10 1 inner
900.1.bb.a ✓ 8 300.r even 10 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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