Properties

Label 1600.2.j.b
Level $1600$
Weight $2$
Character orbit 1600.j
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.j (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{3} + ( \zeta_{24} + 3 \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{7} + ( 1 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{3} + ( \zeta_{24} + 3 \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{7} + ( 1 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{9} + ( -3 + 3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{11} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{13} -\zeta_{24}^{3} q^{17} + ( 2 - 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{19} + ( -1 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{21} + ( \zeta_{24} + 3 \zeta_{24}^{3} - \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{23} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{27} + ( -4 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{29} + ( 3 - 6 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{31} + ( -6 \zeta_{24} + 3 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{33} + ( -5 \zeta_{24} - 5 \zeta_{24}^{3} + 7 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{37} + ( 2 - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{39} + ( -2 + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{41} + ( -\zeta_{24} - \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{43} + ( 2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{47} + ( -6 + 12 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{49} + ( \zeta_{24}^{2} - \zeta_{24}^{4} ) q^{51} + ( 7 \zeta_{24} - 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} ) q^{53} + ( 5 \zeta_{24} - 2 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{57} + ( -7 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 7 \zeta_{24}^{6} ) q^{59} + ( 5 - 6 \zeta_{24}^{2} - 6 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{61} + ( 4 \zeta_{24} + 6 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{63} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{7} ) q^{67} + ( 5 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{69} + ( 6 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{71} + ( 2 \zeta_{24} + 9 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{73} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{77} + ( -3 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{79} + ( -2 + 10 \zeta_{24}^{2} - 5 \zeta_{24}^{6} ) q^{81} + ( -9 \zeta_{24}^{5} - 9 \zeta_{24}^{7} ) q^{83} + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{87} + ( -12 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{89} + ( 12 \zeta_{24}^{2} + 12 \zeta_{24}^{4} ) q^{91} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{93} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{97} + ( 3 + 3 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{9} - 12q^{11} + 4q^{19} - 24q^{29} - 4q^{51} - 48q^{59} + 16q^{61} + 32q^{69} + 48q^{71} - 24q^{79} - 16q^{81} - 96q^{89} + 48q^{91} + 24q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(\zeta_{24}^{3}\) \(-\zeta_{24}^{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
0.258819 + 0.965926i
0 1.93185i 0 0 0 −0.896575 + 0.896575i 0 −0.732051 0
143.2 0 0.517638i 0 0 0 3.34607 3.34607i 0 2.73205 0
143.3 0 0.517638i 0 0 0 −3.34607 + 3.34607i 0 2.73205 0
143.4 0 1.93185i 0 0 0 0.896575 0.896575i 0 −0.732051 0
1007.1 0 1.93185i 0 0 0 0.896575 + 0.896575i 0 −0.732051 0
1007.2 0 0.517638i 0 0 0 −3.34607 3.34607i 0 2.73205 0
1007.3 0 0.517638i 0 0 0 3.34607 + 3.34607i 0 2.73205 0
1007.4 0 1.93185i 0 0 0 −0.896575 0.896575i 0 −0.732051 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1007.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
80.j even 4 1 inner
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.j.b 8
4.b odd 2 1 400.2.j.b 8
5.b even 2 1 inner 1600.2.j.b 8
5.c odd 4 2 1600.2.s.b 8
16.e even 4 1 400.2.s.b yes 8
16.f odd 4 1 1600.2.s.b 8
20.d odd 2 1 400.2.j.b 8
20.e even 4 2 400.2.s.b yes 8
80.i odd 4 1 400.2.j.b 8
80.j even 4 1 inner 1600.2.j.b 8
80.k odd 4 1 1600.2.s.b 8
80.q even 4 1 400.2.s.b yes 8
80.s even 4 1 inner 1600.2.j.b 8
80.t odd 4 1 400.2.j.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.j.b 8 4.b odd 2 1
400.2.j.b 8 20.d odd 2 1
400.2.j.b 8 80.i odd 4 1
400.2.j.b 8 80.t odd 4 1
400.2.s.b yes 8 16.e even 4 1
400.2.s.b yes 8 20.e even 4 2
400.2.s.b yes 8 80.q even 4 1
1600.2.j.b 8 1.a even 1 1 trivial
1600.2.j.b 8 5.b even 2 1 inner
1600.2.j.b 8 80.j even 4 1 inner
1600.2.j.b 8 80.s even 4 1 inner
1600.2.s.b 8 5.c odd 4 2
1600.2.s.b 8 16.f odd 4 1
1600.2.s.b 8 80.k odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + 4 T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( 1296 + 504 T^{4} + T^{8} \)
$11$ \( ( 81 - 54 T + 18 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$13$ \( ( -24 + T^{2} )^{4} \)
$17$ \( ( 1 + T^{4} )^{2} \)
$19$ \( ( 169 + 26 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$23$ \( 456976 + 1784 T^{4} + T^{8} \)
$29$ \( ( 144 + 144 T + 72 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$31$ \( ( 676 + 56 T^{2} + T^{4} )^{2} \)
$37$ \( ( 4356 - 156 T^{2} + T^{4} )^{2} \)
$41$ \( ( 9 + 42 T^{2} + T^{4} )^{2} \)
$43$ \( ( 36 - 84 T^{2} + T^{4} )^{2} \)
$47$ \( ( 16 + T^{4} )^{2} \)
$53$ \( ( 98 + T^{2} )^{4} \)
$59$ \( ( 4356 + 1584 T + 288 T^{2} + 24 T^{3} + T^{4} )^{2} \)
$61$ \( ( 2116 + 368 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$67$ \( ( 81 - 36 T^{2} + T^{4} )^{2} \)
$71$ \( ( -12 - 12 T + T^{2} )^{4} \)
$73$ \( 22667121 + 25074 T^{4} + T^{8} \)
$79$ \( ( -18 + 6 T + T^{2} )^{4} \)
$83$ \( ( 6561 + 324 T^{2} + T^{4} )^{2} \)
$89$ \( ( 141 + 24 T + T^{2} )^{4} \)
$97$ \( 26873856 + 72576 T^{4} + T^{8} \)
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