Properties

Label 2200.1.dd.a
Level $2200$
Weight $1$
Character orbit 2200.dd
Analytic conductor $1.098$
Analytic rank $0$
Dimension $8$
Projective image $D_{5}$
CM discriminant -8
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

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

Self dual: no
Analytic conductor: \(1.09794302779\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.937024.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{20}^{3} q^{2} + ( \zeta_{20} - \zeta_{20}^{7} ) q^{3} + \zeta_{20}^{6} q^{4} + ( 1 + \zeta_{20}^{4} ) q^{6} + \zeta_{20}^{9} q^{8} + ( \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{8} ) q^{9} +O(q^{10})\) \( q + \zeta_{20}^{3} q^{2} + ( \zeta_{20} - \zeta_{20}^{7} ) q^{3} + \zeta_{20}^{6} q^{4} + ( 1 + \zeta_{20}^{4} ) q^{6} + \zeta_{20}^{9} q^{8} + ( \zeta_{20}^{2} - \zeta_{20}^{4} - \zeta_{20}^{8} ) q^{9} + \zeta_{20}^{4} q^{11} + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{12} -\zeta_{20}^{2} q^{16} + ( -\zeta_{20} + \zeta_{20}^{3} ) q^{17} + ( \zeta_{20} + \zeta_{20}^{5} - \zeta_{20}^{7} ) q^{18} + ( -1 - \zeta_{20}^{8} ) q^{19} + \zeta_{20}^{7} q^{22} + ( -1 + \zeta_{20}^{6} ) q^{24} + ( -\zeta_{20} + \zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{9} ) q^{27} -\zeta_{20}^{5} q^{32} + ( \zeta_{20} + \zeta_{20}^{5} ) q^{33} + ( -\zeta_{20}^{4} + \zeta_{20}^{6} ) q^{34} + ( 1 + \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{36} + ( \zeta_{20} - \zeta_{20}^{3} ) q^{38} + ( -\zeta_{20}^{2} - \zeta_{20}^{6} ) q^{41} + ( -\zeta_{20}^{3} - \zeta_{20}^{7} ) q^{43} - q^{44} + ( -\zeta_{20}^{3} + \zeta_{20}^{9} ) q^{48} + \zeta_{20}^{2} q^{49} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{51} + ( \zeta_{20}^{2} - \zeta_{20}^{4} + \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{54} + ( -\zeta_{20} - \zeta_{20}^{5} + \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{57} + ( -1 + \zeta_{20}^{2} ) q^{59} -\zeta_{20}^{8} q^{64} + ( \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{66} + ( -\zeta_{20} - \zeta_{20}^{9} ) q^{67} + ( -\zeta_{20}^{7} + \zeta_{20}^{9} ) q^{68} + ( -\zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{72} + ( -\zeta_{20}^{3} + \zeta_{20}^{9} ) q^{73} + ( \zeta_{20}^{4} - \zeta_{20}^{6} ) q^{76} + ( 1 - \zeta_{20}^{2} + \zeta_{20}^{4} - \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{81} + ( -\zeta_{20}^{5} - \zeta_{20}^{9} ) q^{82} + ( \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{83} + ( 1 - \zeta_{20}^{6} ) q^{86} -\zeta_{20}^{3} q^{88} + ( \zeta_{20}^{2} - \zeta_{20}^{8} ) q^{89} + ( -\zeta_{20}^{2} - \zeta_{20}^{6} ) q^{96} + ( -\zeta_{20} - \zeta_{20}^{5} ) q^{97} + \zeta_{20}^{5} q^{98} + ( \zeta_{20}^{2} + \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{4} + 6q^{6} + 6q^{9} + O(q^{10}) \) \( 8q + 2q^{4} + 6q^{6} + 6q^{9} - 2q^{11} - 2q^{16} - 6q^{19} - 6q^{24} + 4q^{34} + 4q^{36} - 4q^{41} - 8q^{44} + 2q^{49} + 2q^{51} + 8q^{54} - 6q^{59} + 2q^{64} - 4q^{66} - 4q^{76} + 6q^{86} + 4q^{89} - 4q^{96} + 6q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(177\) \(551\) \(1101\) \(1201\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{20}^{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
−0.951057 + 0.309017i
0.951057 0.309017i
−0.951057 0.309017i
0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
0.587785 0.809017i
−0.587785 + 0.809017i
−0.587785 + 0.809017i −1.53884 0.500000i −0.309017 0.951057i 0 1.30902 0.951057i 0 0.951057 + 0.309017i 1.30902 + 0.951057i 0
499.2 0.587785 0.809017i 1.53884 + 0.500000i −0.309017 0.951057i 0 1.30902 0.951057i 0 −0.951057 0.309017i 1.30902 + 0.951057i 0
1499.1 −0.587785 0.809017i −1.53884 + 0.500000i −0.309017 + 0.951057i 0 1.30902 + 0.951057i 0 0.951057 0.309017i 1.30902 0.951057i 0
1499.2 0.587785 + 0.809017i 1.53884 0.500000i −0.309017 + 0.951057i 0 1.30902 + 0.951057i 0 −0.951057 + 0.309017i 1.30902 0.951057i 0
1699.1 −0.951057 + 0.309017i −0.363271 + 0.500000i 0.809017 0.587785i 0 0.190983 0.587785i 0 −0.587785 + 0.809017i 0.190983 + 0.587785i 0
1699.2 0.951057 0.309017i 0.363271 0.500000i 0.809017 0.587785i 0 0.190983 0.587785i 0 0.587785 0.809017i 0.190983 + 0.587785i 0
2099.1 −0.951057 0.309017i −0.363271 0.500000i 0.809017 + 0.587785i 0 0.190983 + 0.587785i 0 −0.587785 0.809017i 0.190983 0.587785i 0
2099.2 0.951057 + 0.309017i 0.363271 + 0.500000i 0.809017 + 0.587785i 0 0.190983 + 0.587785i 0 0.587785 + 0.809017i 0.190983 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2099.2
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.a 8
5.b even 2 1 inner 2200.1.dd.a 8
5.c odd 4 1 88.1.l.a 4
5.c odd 4 1 2200.1.cl.a 4
8.d odd 2 1 CM 2200.1.dd.a 8
11.c even 5 1 inner 2200.1.dd.a 8
15.e even 4 1 792.1.bu.a 4
20.e even 4 1 352.1.t.a 4
40.e odd 2 1 inner 2200.1.dd.a 8
40.i odd 4 1 352.1.t.a 4
40.k even 4 1 88.1.l.a 4
40.k even 4 1 2200.1.cl.a 4
55.e even 4 1 968.1.l.b 4
55.j even 10 1 inner 2200.1.dd.a 8
55.k odd 20 1 88.1.l.a 4
55.k odd 20 1 968.1.f.b 2
55.k odd 20 2 968.1.l.a 4
55.k odd 20 1 2200.1.cl.a 4
55.l even 20 1 968.1.f.a 2
55.l even 20 1 968.1.l.b 4
55.l even 20 2 968.1.l.c 4
60.l odd 4 1 3168.1.ck.a 4
80.i odd 4 1 2816.1.v.c 8
80.j even 4 1 2816.1.v.c 8
80.s even 4 1 2816.1.v.c 8
80.t odd 4 1 2816.1.v.c 8
88.l odd 10 1 inner 2200.1.dd.a 8
120.q odd 4 1 792.1.bu.a 4
120.w even 4 1 3168.1.ck.a 4
165.v even 20 1 792.1.bu.a 4
220.i odd 4 1 3872.1.t.c 4
220.v even 20 1 352.1.t.a 4
220.v even 20 1 3872.1.f.a 2
220.v even 20 2 3872.1.t.b 4
220.w odd 20 1 3872.1.f.b 2
220.w odd 20 2 3872.1.t.a 4
220.w odd 20 1 3872.1.t.c 4
440.t even 4 1 3872.1.t.c 4
440.w odd 4 1 968.1.l.b 4
440.bh odd 10 1 inner 2200.1.dd.a 8
440.bp odd 20 1 352.1.t.a 4
440.bp odd 20 1 3872.1.f.a 2
440.bp odd 20 2 3872.1.t.b 4
440.br odd 20 1 968.1.f.a 2
440.br odd 20 1 968.1.l.b 4
440.br odd 20 2 968.1.l.c 4
440.bs even 20 1 88.1.l.a 4
440.bs even 20 1 968.1.f.b 2
440.bs even 20 2 968.1.l.a 4
440.bs even 20 1 2200.1.cl.a 4
440.bu even 20 1 3872.1.f.b 2
440.bu even 20 2 3872.1.t.a 4
440.bu even 20 1 3872.1.t.c 4
660.bp odd 20 1 3168.1.ck.a 4
880.ce odd 20 1 2816.1.v.c 8
880.ch even 20 1 2816.1.v.c 8
880.cz even 20 1 2816.1.v.c 8
880.da odd 20 1 2816.1.v.c 8
1320.dh even 20 1 3168.1.ck.a 4
1320.dr odd 20 1 792.1.bu.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.1.l.a 4 5.c odd 4 1
88.1.l.a 4 40.k even 4 1
88.1.l.a 4 55.k odd 20 1
88.1.l.a 4 440.bs even 20 1
352.1.t.a 4 20.e even 4 1
352.1.t.a 4 40.i odd 4 1
352.1.t.a 4 220.v even 20 1
352.1.t.a 4 440.bp odd 20 1
792.1.bu.a 4 15.e even 4 1
792.1.bu.a 4 120.q odd 4 1
792.1.bu.a 4 165.v even 20 1
792.1.bu.a 4 1320.dr odd 20 1
968.1.f.a 2 55.l even 20 1
968.1.f.a 2 440.br odd 20 1
968.1.f.b 2 55.k odd 20 1
968.1.f.b 2 440.bs even 20 1
968.1.l.a 4 55.k odd 20 2
968.1.l.a 4 440.bs even 20 2
968.1.l.b 4 55.e even 4 1
968.1.l.b 4 55.l even 20 1
968.1.l.b 4 440.w odd 4 1
968.1.l.b 4 440.br odd 20 1
968.1.l.c 4 55.l even 20 2
968.1.l.c 4 440.br odd 20 2
2200.1.cl.a 4 5.c odd 4 1
2200.1.cl.a 4 40.k even 4 1
2200.1.cl.a 4 55.k odd 20 1
2200.1.cl.a 4 440.bs even 20 1
2200.1.dd.a 8 1.a even 1 1 trivial
2200.1.dd.a 8 5.b even 2 1 inner
2200.1.dd.a 8 8.d odd 2 1 CM
2200.1.dd.a 8 11.c even 5 1 inner
2200.1.dd.a 8 40.e odd 2 1 inner
2200.1.dd.a 8 55.j even 10 1 inner
2200.1.dd.a 8 88.l odd 10 1 inner
2200.1.dd.a 8 440.bh odd 10 1 inner
2816.1.v.c 8 80.i odd 4 1
2816.1.v.c 8 80.j even 4 1
2816.1.v.c 8 80.s even 4 1
2816.1.v.c 8 80.t odd 4 1
2816.1.v.c 8 880.ce odd 20 1
2816.1.v.c 8 880.ch even 20 1
2816.1.v.c 8 880.cz even 20 1
2816.1.v.c 8 880.da odd 20 1
3168.1.ck.a 4 60.l odd 4 1
3168.1.ck.a 4 120.w even 4 1
3168.1.ck.a 4 660.bp odd 20 1
3168.1.ck.a 4 1320.dh even 20 1
3872.1.f.a 2 220.v even 20 1
3872.1.f.a 2 440.bp odd 20 1
3872.1.f.b 2 220.w odd 20 1
3872.1.f.b 2 440.bu even 20 1
3872.1.t.a 4 220.w odd 20 2
3872.1.t.a 4 440.bu even 20 2
3872.1.t.b 4 220.v even 20 2
3872.1.t.b 4 440.bp odd 20 2
3872.1.t.c 4 220.i odd 4 1
3872.1.t.c 4 220.w odd 20 1
3872.1.t.c 4 440.t even 4 1
3872.1.t.c 4 440.bu even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 4 T_{3}^{6} + 6 T_{3}^{4} + T_{3}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$3$ \( 1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( 1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8} \)
$19$ \( ( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( ( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$43$ \( ( 1 + 3 T^{2} + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( ( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( ( 1 + 3 T^{2} + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( 1 + T^{2} + 6 T^{4} - 4 T^{6} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( 1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8} \)
$89$ \( ( -1 - T + T^{2} )^{4} \)
$97$ \( 1 - 4 T^{2} + 6 T^{4} + T^{6} + T^{8} \)
show more
show less