Properties

Label 1600.1.bd.a
Level $1600$
Weight $1$
Character orbit 1600.bd
Analytic conductor $0.799$
Analytic rank $0$
Dimension $8$
Projective image $A_{5}$
CM/RM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1600.bd (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.798504020213\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{5}\)
Projective field: Galois closure of 5.1.25000000.3

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\zeta_{20}^{6} - 1) q^{3} - \zeta_{20}^{3} q^{5} + \zeta_{20}^{5} q^{7} + ( - \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{6} - 1) q^{3} - \zeta_{20}^{3} q^{5} + \zeta_{20}^{5} q^{7} + ( - \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{9} - \zeta_{20}^{4} q^{11} + \zeta_{20} q^{13} + ( - \zeta_{20}^{9} + \zeta_{20}^{3}) q^{15} + ( - \zeta_{20}^{4} - 1) q^{17} + (\zeta_{20}^{8} - \zeta_{20}^{6}) q^{19} + ( - \zeta_{20}^{5} - \zeta_{20}) q^{21} - \zeta_{20}^{9} q^{23} + \zeta_{20}^{6} q^{25} + ( - \zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{2} - 1) q^{27} + \zeta_{20}^{3} q^{29} - \zeta_{20}^{7} q^{31} + (\zeta_{20}^{4} + 1) q^{33} - \zeta_{20}^{8} q^{35} + ( - \zeta_{20}^{7} + \zeta_{20}^{5}) q^{37} + (\zeta_{20}^{7} - \zeta_{20}) q^{39} - \zeta_{20}^{6} q^{41} + q^{43} + (\zeta_{20}^{9} + \zeta_{20}^{5} - \zeta_{20}^{3}) q^{45} + \zeta_{20}^{3} q^{47} + ( - \zeta_{20}^{6} + \zeta_{20}^{4} + 1) q^{51} + \zeta_{20}^{7} q^{55} + ( - \zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{57} + ( - \zeta_{20}^{7} + \zeta_{20}) q^{61} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}) q^{63} - \zeta_{20}^{4} q^{65} + \zeta_{20}^{2} q^{67} + (\zeta_{20}^{9} + \zeta_{20}^{5}) q^{69} - \zeta_{20}^{3} q^{71} + ( - \zeta_{20}^{6} - \zeta_{20}^{2}) q^{75} - \zeta_{20}^{9} q^{77} + (\zeta_{20}^{8} - \zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{2} + 1) q^{81} - \zeta_{20}^{2} q^{83} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{85} + (\zeta_{20}^{9} - \zeta_{20}^{3}) q^{87} + \zeta_{20}^{6} q^{91} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{93} + (\zeta_{20}^{9} + \zeta_{20}) q^{95} + (\zeta_{20}^{4} - \zeta_{20}^{2}) q^{97} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} + 4 q^{9} + 2 q^{11} - 6 q^{17} - 4 q^{19} + 2 q^{25} - 2 q^{27} + 6 q^{33} + 2 q^{35} - 2 q^{41} + 8 q^{43} + 12 q^{51} + 8 q^{57} + 2 q^{65} + 2 q^{67} - 4 q^{75} - 2 q^{83} + 2 q^{91} - 4 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(\zeta_{20}^{8}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
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 −1.30902 + 0.951057i 0 −0.587785 0.809017i 0 1.00000i 0 0.500000 1.53884i 0
31.2 0 −1.30902 + 0.951057i 0 0.587785 + 0.809017i 0 1.00000i 0 0.500000 1.53884i 0
671.1 0 −1.30902 0.951057i 0 −0.587785 + 0.809017i 0 1.00000i 0 0.500000 + 1.53884i 0
671.2 0 −1.30902 0.951057i 0 0.587785 0.809017i 0 1.00000i 0 0.500000 + 1.53884i 0
991.1 0 −0.190983 0.587785i 0 −0.951057 + 0.309017i 0 1.00000i 0 0.500000 0.363271i 0
991.2 0 −0.190983 0.587785i 0 0.951057 0.309017i 0 1.00000i 0 0.500000 0.363271i 0
1311.1 0 −0.190983 + 0.587785i 0 −0.951057 0.309017i 0 1.00000i 0 0.500000 + 0.363271i 0
1311.2 0 −0.190983 + 0.587785i 0 0.951057 + 0.309017i 0 1.00000i 0 0.500000 + 0.363271i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1311.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
25.d even 5 1 inner
200.n odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.1.bd.a 8
4.b odd 2 1 1600.1.bd.b yes 8
8.b even 2 1 1600.1.bd.b yes 8
8.d odd 2 1 inner 1600.1.bd.a 8
25.d even 5 1 inner 1600.1.bd.a 8
100.j odd 10 1 1600.1.bd.b yes 8
200.n odd 10 1 inner 1600.1.bd.a 8
200.t even 10 1 1600.1.bd.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.1.bd.a 8 1.a even 1 1 trivial
1600.1.bd.a 8 8.d odd 2 1 inner
1600.1.bd.a 8 25.d even 5 1 inner
1600.1.bd.a 8 200.n odd 10 1 inner
1600.1.bd.b yes 8 4.b odd 2 1
1600.1.bd.b yes 8 8.b even 2 1
1600.1.bd.b yes 8 100.j odd 10 1
1600.1.bd.b yes 8 200.t even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{3} + 4T_{3}^{2} + 2T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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