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Properties

Label 2200.1.fr.b
Level $2200$
Weight $1$
Character orbit 2200.fr
Analytic conductor $1.098$
Analytic rank $0$
Dimension $16$
Projective image $D_{10}$
CM discriminant -40
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,1,Mod(107,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 5, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2200.107");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2200.fr (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: \(1.09794302779\)
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: \(D_{10}\)
Projective field: Galois closure of 10.0.30181730444800000.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{40}^{19} q^{2} - \zeta_{40}^{18} q^{4} + ( - \zeta_{40}^{15} + \zeta_{40}^{11}) q^{7} - \zeta_{40}^{17} q^{8} + \zeta_{40}^{14} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{40}^{19} q^{2} - \zeta_{40}^{18} q^{4} + ( - \zeta_{40}^{15} + \zeta_{40}^{11}) q^{7} - \zeta_{40}^{17} q^{8} + \zeta_{40}^{14} q^{9} - \zeta_{40}^{8} q^{11} + (\zeta_{40}^{13} + \zeta_{40}^{5}) q^{13} + ( - \zeta_{40}^{14} + \zeta_{40}^{10}) q^{14} - \zeta_{40}^{16} q^{16} + \zeta_{40}^{13} q^{18} + (\zeta_{40}^{18} + \zeta_{40}^{6}) q^{19} - \zeta_{40}^{7} q^{22} + ( - \zeta_{40}^{17} - \zeta_{40}^{13}) q^{23} + (\zeta_{40}^{12} + \zeta_{40}^{4}) q^{26} + ( - \zeta_{40}^{13} + \zeta_{40}^{9}) q^{28} - \zeta_{40}^{15} q^{32} + \zeta_{40}^{12} q^{36} + ( - \zeta_{40}^{19} - \zeta_{40}^{7}) q^{37} + (\zeta_{40}^{17} + \zeta_{40}^{5}) q^{38} + ( - \zeta_{40}^{4} - 1) q^{41} - \zeta_{40}^{6} q^{44} + ( - \zeta_{40}^{16} - \zeta_{40}^{12}) q^{46} + ( - \zeta_{40}^{11} + \zeta_{40}^{3}) q^{47} + ( - \zeta_{40}^{10} + \cdots - \zeta_{40}^{2}) q^{49} + \cdots + \zeta_{40}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{11} + 4 q^{16} + 8 q^{26} + 4 q^{36} - 20 q^{41} - 8 q^{56} + 4 q^{81} + 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(551\) \(1101\) \(1201\)
\(\chi(n)\) \(-\zeta_{40}^{10}\) \(-1\) \(-1\) \(-\zeta_{40}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.987688 + 0.156434i
0.987688 − 0.156434i
−0.987688 − 0.156434i
0.987688 + 0.156434i
−0.156434 + 0.987688i
0.156434 − 0.987688i
−0.453990 + 0.891007i
0.453990 − 0.891007i
−0.891007 − 0.453990i
0.891007 + 0.453990i
−0.891007 + 0.453990i
0.891007 − 0.453990i
−0.156434 − 0.987688i
0.156434 + 0.987688i
−0.453990 − 0.891007i
0.453990 + 0.891007i
−0.987688 − 0.156434i 0 0.951057 + 0.309017i 0 0 −0.550672 + 0.280582i −0.891007 − 0.453990i −0.587785 − 0.809017i 0
107.2 0.987688 + 0.156434i 0 0.951057 + 0.309017i 0 0 0.550672 − 0.280582i 0.891007 + 0.453990i −0.587785 − 0.809017i 0
843.1 −0.987688 + 0.156434i 0 0.951057 − 0.309017i 0 0 −0.550672 − 0.280582i −0.891007 + 0.453990i −0.587785 + 0.809017i 0
843.2 0.987688 − 0.156434i 0 0.951057 − 0.309017i 0 0 0.550672 + 0.280582i 0.891007 − 0.453990i −0.587785 + 0.809017i 0
1107.1 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i 0 0 0.280582 − 0.550672i 0.453990 + 0.891007i 0.587785 − 0.809017i 0
1107.2 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 0 0 −0.280582 + 0.550672i −0.453990 − 0.891007i 0.587785 − 0.809017i 0
1443.1 −0.453990 − 0.891007i 0 −0.587785 + 0.809017i 0 0 −1.59811 + 0.253116i 0.987688 + 0.156434i −0.951057 − 0.309017i 0
1443.2 0.453990 + 0.891007i 0 −0.587785 + 0.809017i 0 0 1.59811 − 0.253116i −0.987688 − 0.156434i −0.951057 − 0.309017i 0
1707.1 −0.891007 + 0.453990i 0 0.587785 − 0.809017i 0 0 0.253116 + 1.59811i −0.156434 + 0.987688i 0.951057 + 0.309017i 0
1707.2 0.891007 − 0.453990i 0 0.587785 − 0.809017i 0 0 −0.253116 − 1.59811i 0.156434 − 0.987688i 0.951057 + 0.309017i 0
1843.1 −0.891007 − 0.453990i 0 0.587785 + 0.809017i 0 0 0.253116 − 1.59811i −0.156434 − 0.987688i 0.951057 − 0.309017i 0
1843.2 0.891007 + 0.453990i 0 0.587785 + 0.809017i 0 0 −0.253116 + 1.59811i 0.156434 + 0.987688i 0.951057 − 0.309017i 0
2043.1 −0.156434 + 0.987688i 0 −0.951057 − 0.309017i 0 0 0.280582 + 0.550672i 0.453990 − 0.891007i 0.587785 + 0.809017i 0
2043.2 0.156434 − 0.987688i 0 −0.951057 − 0.309017i 0 0 −0.280582 − 0.550672i −0.453990 + 0.891007i 0.587785 + 0.809017i 0
2107.1 −0.453990 + 0.891007i 0 −0.587785 − 0.809017i 0 0 −1.59811 − 0.253116i 0.987688 − 0.156434i −0.951057 + 0.309017i 0
2107.2 0.453990 − 0.891007i 0 −0.587785 − 0.809017i 0 0 1.59811 + 0.253116i −0.987688 + 0.156434i −0.951057 + 0.309017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
11.d odd 10 1 inner
40.k even 4 2 inner
55.h odd 10 1 inner
55.l even 20 2 inner
88.k even 10 1 inner
440.bm even 10 1 inner
440.br odd 20 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2200.1.fr.b ✓ 16
5.b even 2 1 inner 2200.1.fr.b ✓ 16
5.c odd 4 2 inner 2200.1.fr.b ✓ 16
8.d odd 2 1 inner 2200.1.fr.b ✓ 16
11.d odd 10 1 inner 2200.1.fr.b ✓ 16
40.e odd 2 1 CM 2200.1.fr.b ✓ 16
40.k even 4 2 inner 2200.1.fr.b ✓ 16
55.h odd 10 1 inner 2200.1.fr.b ✓ 16
55.l even 20 2 inner 2200.1.fr.b ✓ 16
88.k even 10 1 inner 2200.1.fr.b ✓ 16
440.bm even 10 1 inner 2200.1.fr.b ✓ 16
440.br odd 20 2 inner 2200.1.fr.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.1.fr.b ✓ 16 1.a even 1 1 trivial
2200.1.fr.b ✓ 16 5.b even 2 1 inner
2200.1.fr.b ✓ 16 5.c odd 4 2 inner
2200.1.fr.b ✓ 16 8.d odd 2 1 inner
2200.1.fr.b ✓ 16 11.d odd 10 1 inner
2200.1.fr.b ✓ 16 40.e odd 2 1 CM
2200.1.fr.b ✓ 16 40.k even 4 2 inner
2200.1.fr.b ✓ 16 55.h odd 10 1 inner
2200.1.fr.b ✓ 16 55.l even 20 2 inner
2200.1.fr.b ✓ 16 88.k even 10 1 inner
2200.1.fr.b ✓ 16 440.bm even 10 1 inner
2200.1.fr.b ✓ 16 440.br odd 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{1}^{\mathrm{new}}(2200, [\chi])\). Copy content Toggle raw display

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
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 11 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} - 11 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} + 10 T^{4} + \cdots + 25)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 15 T^{4} + 25)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} - 20 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$41$ \( (T^{4} + 5 T^{3} + 10 T^{2} + \cdots + 5)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} - 20 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$53$ \( T^{16} - 20 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$59$ \( (T^{8} - 4 T^{6} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( (T^{4} + 3 T^{2} + 1)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} \) Copy content Toggle raw display
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