Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (of order \(4\), degree \(2\), 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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{3} + 2 \zeta_{24}^{6} q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{3} + 2 \zeta_{24}^{6} q^{9} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{11} -4 \zeta_{24}^{3} q^{13} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{17} + ( 3 - 6 \zeta_{24}^{4} ) q^{19} + ( -4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{23} -5 \zeta_{24}^{3} q^{27} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{29} + 4 \zeta_{24}^{6} q^{31} + ( \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{33} + ( 8 \zeta_{24} - 8 \zeta_{24}^{5} ) q^{37} + 4 q^{39} + 3 q^{41} + ( -8 \zeta_{24} + 8 \zeta_{24}^{5} ) q^{43} + ( 4 \zeta_{24}^{3} - 8 \zeta_{24}^{7} ) q^{47} + 7 \zeta_{24}^{6} q^{49} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{51} -12 \zeta_{24}^{3} q^{53} + ( 3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{57} + ( 6 - 12 \zeta_{24}^{4} ) q^{59} -7 \zeta_{24}^{3} q^{67} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{69} -12 \zeta_{24}^{6} q^{71} + ( 5 \zeta_{24}^{3} - 10 \zeta_{24}^{7} ) q^{73} + 8 q^{79} - q^{81} + ( -15 \zeta_{24} + 15 \zeta_{24}^{5} ) q^{83} + ( -4 \zeta_{24}^{3} + 8 \zeta_{24}^{7} ) q^{87} + 3 \zeta_{24}^{6} q^{89} -4 \zeta_{24}^{3} q^{93} + ( 2 - 4 \zeta_{24}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 32q^{39} + 24q^{41} + 64q^{79} - 8q^{81} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
543.1
0.965926 0.258819i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.258819 0.965926i
0.965926 + 0.258819i
−0.258819 0.965926i
−0.965926 0.258819i
0.258819 + 0.965926i
0 −0.707107 0.707107i 0 0 0 0 0 2.00000i 0
543.2 0 −0.707107 0.707107i 0 0 0 0 0 2.00000i 0
543.3 0 0.707107 + 0.707107i 0 0 0 0 0 2.00000i 0
543.4 0 0.707107 + 0.707107i 0 0 0 0 0 2.00000i 0
607.1 0 −0.707107 + 0.707107i 0 0 0 0 0 2.00000i 0
607.2 0 −0.707107 + 0.707107i 0 0 0 0 0 2.00000i 0
607.3 0 0.707107 0.707107i 0 0 0 0 0 2.00000i 0
607.4 0 0.707107 0.707107i 0 0 0 0 0 2.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
20.e even 4 2 inner
40.f even 2 1 inner
40.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.o.i yes 8
4.b odd 2 1 1600.2.o.h 8
5.b even 2 1 inner 1600.2.o.i yes 8
5.c odd 4 2 1600.2.o.h 8
8.b even 2 1 inner 1600.2.o.i yes 8
8.d odd 2 1 1600.2.o.h 8
20.d odd 2 1 1600.2.o.h 8
20.e even 4 2 inner 1600.2.o.i yes 8
40.e odd 2 1 1600.2.o.h 8
40.f even 2 1 inner 1600.2.o.i yes 8
40.i odd 4 2 1600.2.o.h 8
40.k even 4 2 inner 1600.2.o.i yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.o.h 8 4.b odd 2 1
1600.2.o.h 8 5.c odd 4 2
1600.2.o.h 8 8.d odd 2 1
1600.2.o.h 8 20.d odd 2 1
1600.2.o.h 8 40.e odd 2 1
1600.2.o.h 8 40.i odd 4 2
1600.2.o.i yes 8 1.a even 1 1 trivial
1600.2.o.i yes 8 5.b even 2 1 inner
1600.2.o.i yes 8 8.b even 2 1 inner
1600.2.o.i yes 8 20.e even 4 2 inner
1600.2.o.i yes 8 40.f even 2 1 inner
1600.2.o.i yes 8 40.k even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} + 1 \)
\( T_{7} \)
\( T_{79} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( -3 + T^{2} )^{4} \)
$13$ \( ( 256 + T^{4} )^{2} \)
$17$ \( ( 9 + T^{4} )^{2} \)
$19$ \( ( 27 + T^{2} )^{4} \)
$23$ \( ( 2304 + T^{4} )^{2} \)
$29$ \( ( -48 + T^{2} )^{4} \)
$31$ \( ( 16 + T^{2} )^{4} \)
$37$ \( ( 4096 + T^{4} )^{2} \)
$41$ \( ( -3 + T )^{8} \)
$43$ \( ( 4096 + T^{4} )^{2} \)
$47$ \( ( 2304 + T^{4} )^{2} \)
$53$ \( ( 20736 + T^{4} )^{2} \)
$59$ \( ( 108 + T^{2} )^{4} \)
$61$ \( T^{8} \)
$67$ \( ( 2401 + T^{4} )^{2} \)
$71$ \( ( 144 + T^{2} )^{4} \)
$73$ \( ( 5625 + T^{4} )^{2} \)
$79$ \( ( -8 + T )^{8} \)
$83$ \( ( 50625 + T^{4} )^{2} \)
$89$ \( ( 9 + T^{2} )^{4} \)
$97$ \( T^{8} \)
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