Properties

Label 1600.2.d.i
Level $1600$
Weight $2$
Character orbit 1600.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{3} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{3} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{7} -3 q^{9} -2 \zeta_{24}^{6} q^{11} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{13} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{17} + 6 \zeta_{24}^{6} q^{19} + ( 2 - 4 \zeta_{24}^{4} ) q^{21} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{23} + ( 4 - 8 \zeta_{24}^{4} ) q^{29} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{31} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{33} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{37} + ( -16 \zeta_{24}^{2} + 8 \zeta_{24}^{6} ) q^{39} -4 q^{41} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{43} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{47} -5 q^{49} + 12 \zeta_{24}^{6} q^{51} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{57} -2 \zeta_{24}^{6} q^{59} + ( 2 - 4 \zeta_{24}^{4} ) q^{61} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{63} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{67} + ( 10 - 20 \zeta_{24}^{4} ) q^{69} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{71} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{73} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{77} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{79} -9 q^{81} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} + 10 \zeta_{24}^{7} ) q^{83} + ( -12 \zeta_{24} - 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} ) q^{87} + 2 q^{89} -8 \zeta_{24}^{6} q^{91} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} ) q^{93} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{97} + 6 \zeta_{24}^{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} - 32q^{41} - 40q^{49} - 72q^{81} + 16q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
801.1
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
0 2.44949i 0 0 0 −1.41421 0 −3.00000 0
801.2 0 2.44949i 0 0 0 −1.41421 0 −3.00000 0
801.3 0 2.44949i 0 0 0 1.41421 0 −3.00000 0
801.4 0 2.44949i 0 0 0 1.41421 0 −3.00000 0
801.5 0 2.44949i 0 0 0 −1.41421 0 −3.00000 0
801.6 0 2.44949i 0 0 0 −1.41421 0 −3.00000 0
801.7 0 2.44949i 0 0 0 1.41421 0 −3.00000 0
801.8 0 2.44949i 0 0 0 1.41421 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 801.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.d.i 8
4.b odd 2 1 inner 1600.2.d.i 8
5.b even 2 1 inner 1600.2.d.i 8
5.c odd 4 2 320.2.f.b 8
8.b even 2 1 inner 1600.2.d.i 8
8.d odd 2 1 inner 1600.2.d.i 8
15.e even 4 2 2880.2.d.g 8
16.e even 4 1 6400.2.a.ct 4
16.e even 4 1 6400.2.a.cu 4
16.f odd 4 1 6400.2.a.ct 4
16.f odd 4 1 6400.2.a.cu 4
20.d odd 2 1 inner 1600.2.d.i 8
20.e even 4 2 320.2.f.b 8
40.e odd 2 1 inner 1600.2.d.i 8
40.f even 2 1 inner 1600.2.d.i 8
40.i odd 4 2 320.2.f.b 8
40.k even 4 2 320.2.f.b 8
60.l odd 4 2 2880.2.d.g 8
80.i odd 4 1 1280.2.c.g 4
80.i odd 4 1 1280.2.c.h 4
80.j even 4 1 1280.2.c.g 4
80.j even 4 1 1280.2.c.h 4
80.k odd 4 1 6400.2.a.ct 4
80.k odd 4 1 6400.2.a.cu 4
80.q even 4 1 6400.2.a.ct 4
80.q even 4 1 6400.2.a.cu 4
80.s even 4 1 1280.2.c.g 4
80.s even 4 1 1280.2.c.h 4
80.t odd 4 1 1280.2.c.g 4
80.t odd 4 1 1280.2.c.h 4
120.q odd 4 2 2880.2.d.g 8
120.w even 4 2 2880.2.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.b 8 5.c odd 4 2
320.2.f.b 8 20.e even 4 2
320.2.f.b 8 40.i odd 4 2
320.2.f.b 8 40.k even 4 2
1280.2.c.g 4 80.i odd 4 1
1280.2.c.g 4 80.j even 4 1
1280.2.c.g 4 80.s even 4 1
1280.2.c.g 4 80.t odd 4 1
1280.2.c.h 4 80.i odd 4 1
1280.2.c.h 4 80.j even 4 1
1280.2.c.h 4 80.s even 4 1
1280.2.c.h 4 80.t odd 4 1
1600.2.d.i 8 1.a even 1 1 trivial
1600.2.d.i 8 4.b odd 2 1 inner
1600.2.d.i 8 5.b even 2 1 inner
1600.2.d.i 8 8.b even 2 1 inner
1600.2.d.i 8 8.d odd 2 1 inner
1600.2.d.i 8 20.d odd 2 1 inner
1600.2.d.i 8 40.e odd 2 1 inner
1600.2.d.i 8 40.f even 2 1 inner
2880.2.d.g 8 15.e even 4 2
2880.2.d.g 8 60.l odd 4 2
2880.2.d.g 8 120.q odd 4 2
2880.2.d.g 8 120.w even 4 2
6400.2.a.ct 4 16.e even 4 1
6400.2.a.ct 4 16.f odd 4 1
6400.2.a.ct 4 80.k odd 4 1
6400.2.a.ct 4 80.q even 4 1
6400.2.a.cu 4 16.e even 4 1
6400.2.a.cu 4 16.f odd 4 1
6400.2.a.cu 4 80.k odd 4 1
6400.2.a.cu 4 80.q even 4 1

Hecke kernels

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

\( T_{3}^{2} + 6 \)
\( T_{7}^{2} - 2 \)
\( T_{17}^{2} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 6 + T^{2} )^{4} \)
$5$ \( T^{8} \)
$7$ \( ( -2 + T^{2} )^{4} \)
$11$ \( ( 4 + T^{2} )^{4} \)
$13$ \( ( 32 + T^{2} )^{4} \)
$17$ \( ( -24 + T^{2} )^{4} \)
$19$ \( ( 36 + T^{2} )^{4} \)
$23$ \( ( -50 + T^{2} )^{4} \)
$29$ \( ( 48 + T^{2} )^{4} \)
$31$ \( ( -48 + T^{2} )^{4} \)
$37$ \( ( 8 + T^{2} )^{4} \)
$41$ \( ( 4 + T )^{8} \)
$43$ \( ( 6 + T^{2} )^{4} \)
$47$ \( ( -18 + T^{2} )^{4} \)
$53$ \( T^{8} \)
$59$ \( ( 4 + T^{2} )^{4} \)
$61$ \( ( 12 + T^{2} )^{4} \)
$67$ \( ( 6 + T^{2} )^{4} \)
$71$ \( ( -48 + T^{2} )^{4} \)
$73$ \( ( -24 + T^{2} )^{4} \)
$79$ \( ( -48 + T^{2} )^{4} \)
$83$ \( ( 150 + T^{2} )^{4} \)
$89$ \( ( -2 + T )^{8} \)
$97$ \( ( -216 + T^{2} )^{4} \)
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