Properties

Label 1600.2.d.d
Level $1600$
Weight $2$
Character orbit 1600.d
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( -4 - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( -4 - \beta_{3} ) q^{9} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( 3 - \beta_{3} ) q^{17} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{19} + ( -5 \beta_{1} - 3 \beta_{2} ) q^{27} + ( 9 + 2 \beta_{3} ) q^{33} + ( 3 - 2 \beta_{3} ) q^{41} -5 \beta_{1} q^{43} -7 q^{49} + ( -5 \beta_{1} + \beta_{2} ) q^{51} + ( 17 + \beta_{3} ) q^{57} + 3 \beta_{1} q^{59} + ( -5 \beta_{1} + 3 \beta_{2} ) q^{67} + ( -1 - 3 \beta_{3} ) q^{73} + ( 19 + 5 \beta_{3} ) q^{81} + ( 5 \beta_{1} - \beta_{2} ) q^{83} + ( -9 + \beta_{3} ) q^{89} + 10 q^{97} + ( 7 \beta_{1} + 10 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{9} + O(q^{10}) \) \( 4q - 16q^{9} + 12q^{17} + 36q^{33} + 12q^{41} - 28q^{49} + 68q^{57} - 4q^{73} + 76q^{81} - 36q^{89} + 40q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + \nu^{2} + 3 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{3} + 6 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)\(/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 6 \beta_{2} - 3 \beta_{1}\)\()/4\)

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
1.22474 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
0 3.44949i 0 0 0 0 0 −8.89898 0
801.2 0 1.44949i 0 0 0 0 0 0.898979 0
801.3 0 1.44949i 0 0 0 0 0 0.898979 0
801.4 0 3.44949i 0 0 0 0 0 −8.89898 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b 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.d yes 4
4.b odd 2 1 inner 1600.2.d.d yes 4
5.b even 2 1 1600.2.d.c 4
5.c odd 4 1 1600.2.f.f 4
5.c odd 4 1 1600.2.f.j 4
8.b even 2 1 inner 1600.2.d.d yes 4
8.d odd 2 1 CM 1600.2.d.d yes 4
16.e even 4 1 6400.2.a.bc 2
16.e even 4 1 6400.2.a.ci 2
16.f odd 4 1 6400.2.a.bc 2
16.f odd 4 1 6400.2.a.ci 2
20.d odd 2 1 1600.2.d.c 4
20.e even 4 1 1600.2.f.f 4
20.e even 4 1 1600.2.f.j 4
40.e odd 2 1 1600.2.d.c 4
40.f even 2 1 1600.2.d.c 4
40.i odd 4 1 1600.2.f.f 4
40.i odd 4 1 1600.2.f.j 4
40.k even 4 1 1600.2.f.f 4
40.k even 4 1 1600.2.f.j 4
80.k odd 4 1 6400.2.a.bd 2
80.k odd 4 1 6400.2.a.ch 2
80.q even 4 1 6400.2.a.bd 2
80.q even 4 1 6400.2.a.ch 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.c 4 5.b even 2 1
1600.2.d.c 4 20.d odd 2 1
1600.2.d.c 4 40.e odd 2 1
1600.2.d.c 4 40.f even 2 1
1600.2.d.d yes 4 1.a even 1 1 trivial
1600.2.d.d yes 4 4.b odd 2 1 inner
1600.2.d.d yes 4 8.b even 2 1 inner
1600.2.d.d yes 4 8.d odd 2 1 CM
1600.2.f.f 4 5.c odd 4 1
1600.2.f.f 4 20.e even 4 1
1600.2.f.f 4 40.i odd 4 1
1600.2.f.f 4 40.k even 4 1
1600.2.f.j 4 5.c odd 4 1
1600.2.f.j 4 20.e even 4 1
1600.2.f.j 4 40.i odd 4 1
1600.2.f.j 4 40.k even 4 1
6400.2.a.bc 2 16.e even 4 1
6400.2.a.bc 2 16.f odd 4 1
6400.2.a.bd 2 80.k odd 4 1
6400.2.a.bd 2 80.q even 4 1
6400.2.a.ch 2 80.k odd 4 1
6400.2.a.ch 2 80.q even 4 1
6400.2.a.ci 2 16.e even 4 1
6400.2.a.ci 2 16.f odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 14 T_{3}^{2} + 25 \)
\( T_{7} \)
\( T_{17}^{2} - 6 T_{17} - 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 25 + 14 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 9 + 30 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -15 - 6 T + T^{2} )^{2} \)
$19$ \( 2809 + 110 T^{2} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( -87 - 6 T + T^{2} )^{2} \)
$43$ \( ( 100 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 36 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( 25 + 206 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -215 + 2 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( 5625 + 174 T^{2} + T^{4} \)
$89$ \( ( 57 + 18 T + T^{2} )^{2} \)
$97$ \( ( -10 + T )^{4} \)
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