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Properties

Label 2200.1.dd.b
Level $2200$
Weight $1$
Character orbit 2200.dd
Analytic conductor $1.098$
Analytic rank $0$
Dimension $16$
Projective image $D_{15}$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,1,Mod(499,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 5, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2200.499");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2200.dd (of order \(10\), degree \(4\), not 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: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{60})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{60}^{9} q^{2} + (\zeta_{60}^{23} - \zeta_{60}) q^{3} + \zeta_{60}^{18} q^{4} + ( - \zeta_{60}^{10} - \zeta_{60}^{2}) q^{6} + \zeta_{60}^{27} q^{8} + ( - \zeta_{60}^{24} + \cdots + \zeta_{60}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{60}^{9} q^{2} + (\zeta_{60}^{23} - \zeta_{60}) q^{3} + \zeta_{60}^{18} q^{4} + ( - \zeta_{60}^{10} - \zeta_{60}^{2}) q^{6} + \zeta_{60}^{27} q^{8} + ( - \zeta_{60}^{24} + \cdots + \zeta_{60}^{2}) q^{9}+ \cdots + ( - \zeta_{60}^{24} + \cdots - \zeta_{60}^{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 6 q^{6} + 2 q^{11} - 4 q^{16} + 6 q^{19} + 6 q^{24} - 4 q^{34} - 10 q^{36} + 4 q^{41} + 8 q^{44} + 4 q^{49} - 14 q^{51} + 4 q^{54} - 12 q^{59} + 4 q^{64} - 2 q^{66} + 4 q^{76} - 6 q^{81} + 12 q^{86} - 4 q^{89} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(177\) \(551\) \(1101\) \(1201\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{60}^{18}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
0.406737 − 0.913545i
−0.994522 + 0.104528i
0.994522 − 0.104528i
−0.406737 + 0.913545i
0.406737 + 0.913545i
−0.994522 − 0.104528i
0.994522 + 0.104528i
−0.406737 − 0.913545i
−0.207912 − 0.978148i
−0.743145 + 0.669131i
0.743145 − 0.669131i
0.207912 + 0.978148i
−0.207912 + 0.978148i
−0.743145 − 0.669131i
0.743145 + 0.669131i
0.207912 − 0.978148i
−0.587785 + 0.809017i −0.198825 − 0.0646021i −0.309017 − 0.951057i 0 0.169131 − 0.122881i 0 0.951057 + 0.309017i −0.773659 − 0.562096i 0
499.2 −0.587785 + 0.809017i 1.73767 + 0.564602i −0.309017 − 0.951057i 0 −1.47815 + 1.07394i 0 0.951057 + 0.309017i 1.89169 + 1.37440i 0
499.3 0.587785 − 0.809017i −1.73767 − 0.564602i −0.309017 − 0.951057i 0 −1.47815 + 1.07394i 0 −0.951057 − 0.309017i 1.89169 + 1.37440i 0
499.4 0.587785 − 0.809017i 0.198825 + 0.0646021i −0.309017 − 0.951057i 0 0.169131 − 0.122881i 0 −0.951057 − 0.309017i −0.773659 − 0.562096i 0
1499.1 −0.587785 − 0.809017i −0.198825 + 0.0646021i −0.309017 + 0.951057i 0 0.169131 + 0.122881i 0 0.951057 − 0.309017i −0.773659 + 0.562096i 0
1499.2 −0.587785 − 0.809017i 1.73767 − 0.564602i −0.309017 + 0.951057i 0 −1.47815 − 1.07394i 0 0.951057 − 0.309017i 1.89169 − 1.37440i 0
1499.3 0.587785 + 0.809017i −1.73767 + 0.564602i −0.309017 + 0.951057i 0 −1.47815 − 1.07394i 0 −0.951057 + 0.309017i 1.89169 − 1.37440i 0
1499.4 0.587785 + 0.809017i 0.198825 − 0.0646021i −0.309017 + 0.951057i 0 0.169131 + 0.122881i 0 −0.951057 + 0.309017i −0.773659 + 0.562096i 0
1699.1 −0.951057 + 0.309017i −0.786610 + 1.08268i 0.809017 − 0.587785i 0 0.413545 − 1.27276i 0 −0.587785 + 0.809017i −0.244415 − 0.752232i 0
1699.2 −0.951057 + 0.309017i 1.14988 − 1.58268i 0.809017 − 0.587785i 0 −0.604528 + 1.86055i 0 −0.587785 + 0.809017i −0.873619 − 2.68872i 0
1699.3 0.951057 − 0.309017i −1.14988 + 1.58268i 0.809017 − 0.587785i 0 −0.604528 + 1.86055i 0 0.587785 − 0.809017i −0.873619 − 2.68872i 0
1699.4 0.951057 − 0.309017i 0.786610 − 1.08268i 0.809017 − 0.587785i 0 0.413545 − 1.27276i 0 0.587785 − 0.809017i −0.244415 − 0.752232i 0
2099.1 −0.951057 − 0.309017i −0.786610 − 1.08268i 0.809017 + 0.587785i 0 0.413545 + 1.27276i 0 −0.587785 − 0.809017i −0.244415 + 0.752232i 0
2099.2 −0.951057 − 0.309017i 1.14988 + 1.58268i 0.809017 + 0.587785i 0 −0.604528 − 1.86055i 0 −0.587785 − 0.809017i −0.873619 + 2.68872i 0
2099.3 0.951057 + 0.309017i −1.14988 − 1.58268i 0.809017 + 0.587785i 0 −0.604528 − 1.86055i 0 0.587785 + 0.809017i −0.873619 + 2.68872i 0
2099.4 0.951057 + 0.309017i 0.786610 + 1.08268i 0.809017 + 0.587785i 0 0.413545 + 1.27276i 0 0.587785 + 0.809017i −0.244415 + 0.752232i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 499.4
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 2T_{3}^{14} + 13T_{3}^{12} - 49T_{3}^{10} + 150T_{3}^{8} + T_{3}^{6} + 523T_{3}^{4} - 37T_{3}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{8} - 3 T^{7} + 8 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( (T^{8} - 2 T^{7} + 3 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3 T^{2} + 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{8} + 9 T^{6} + 26 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} - 7 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{4} + T^{3} - 4 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + T^{6} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
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