Properties

Label 1600.2.d.g
Level $1600$
Weight $2$
Character orbit 1600.d
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM discriminant -40
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 320)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{7} + 3 q^{9} +O(q^{10})\) \( q -\beta_{3} q^{7} + 3 q^{9} -\beta_{1} q^{11} + \beta_{2} q^{13} -3 \beta_{1} q^{19} + \beta_{3} q^{23} -\beta_{2} q^{37} + 2 q^{41} + 3 \beta_{3} q^{47} + 13 q^{49} -3 \beta_{2} q^{53} -7 \beta_{1} q^{59} -3 \beta_{3} q^{63} + 2 \beta_{2} q^{77} + 9 q^{81} + 14 q^{89} -10 \beta_{1} q^{91} -3 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 8q^{41} + 52q^{49} + 36q^{81} + 56q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 8 \nu \)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1}\)\()/2\)

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
0.618034i
0.618034i
1.61803i
1.61803i
0 0 0 0 0 −4.47214 0 3.00000 0
801.2 0 0 0 0 0 −4.47214 0 3.00000 0
801.3 0 0 0 0 0 4.47214 0 3.00000 0
801.4 0 0 0 0 0 4.47214 0 3.00000 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
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.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.g 4
4.b odd 2 1 inner 1600.2.d.g 4
5.b even 2 1 inner 1600.2.d.g 4
5.c odd 4 2 320.2.f.a 4
8.b even 2 1 inner 1600.2.d.g 4
8.d odd 2 1 inner 1600.2.d.g 4
15.e even 4 2 2880.2.d.e 4
16.e even 4 1 6400.2.a.bi 2
16.e even 4 1 6400.2.a.bj 2
16.f odd 4 1 6400.2.a.bi 2
16.f odd 4 1 6400.2.a.bj 2
20.d odd 2 1 inner 1600.2.d.g 4
20.e even 4 2 320.2.f.a 4
40.e odd 2 1 CM 1600.2.d.g 4
40.f even 2 1 inner 1600.2.d.g 4
40.i odd 4 2 320.2.f.a 4
40.k even 4 2 320.2.f.a 4
60.l odd 4 2 2880.2.d.e 4
80.i odd 4 1 1280.2.c.b 2
80.i odd 4 1 1280.2.c.c 2
80.j even 4 1 1280.2.c.b 2
80.j even 4 1 1280.2.c.c 2
80.k odd 4 1 6400.2.a.bi 2
80.k odd 4 1 6400.2.a.bj 2
80.q even 4 1 6400.2.a.bi 2
80.q even 4 1 6400.2.a.bj 2
80.s even 4 1 1280.2.c.b 2
80.s even 4 1 1280.2.c.c 2
80.t odd 4 1 1280.2.c.b 2
80.t odd 4 1 1280.2.c.c 2
120.q odd 4 2 2880.2.d.e 4
120.w even 4 2 2880.2.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.a 4 5.c odd 4 2
320.2.f.a 4 20.e even 4 2
320.2.f.a 4 40.i odd 4 2
320.2.f.a 4 40.k even 4 2
1280.2.c.b 2 80.i odd 4 1
1280.2.c.b 2 80.j even 4 1
1280.2.c.b 2 80.s even 4 1
1280.2.c.b 2 80.t odd 4 1
1280.2.c.c 2 80.i odd 4 1
1280.2.c.c 2 80.j even 4 1
1280.2.c.c 2 80.s even 4 1
1280.2.c.c 2 80.t odd 4 1
1600.2.d.g 4 1.a even 1 1 trivial
1600.2.d.g 4 4.b odd 2 1 inner
1600.2.d.g 4 5.b even 2 1 inner
1600.2.d.g 4 8.b even 2 1 inner
1600.2.d.g 4 8.d odd 2 1 inner
1600.2.d.g 4 20.d odd 2 1 inner
1600.2.d.g 4 40.e odd 2 1 CM
1600.2.d.g 4 40.f even 2 1 inner
2880.2.d.e 4 15.e even 4 2
2880.2.d.e 4 60.l odd 4 2
2880.2.d.e 4 120.q odd 4 2
2880.2.d.e 4 120.w even 4 2
6400.2.a.bi 2 16.e even 4 1
6400.2.a.bi 2 16.f odd 4 1
6400.2.a.bi 2 80.k odd 4 1
6400.2.a.bi 2 80.q even 4 1
6400.2.a.bj 2 16.e even 4 1
6400.2.a.bj 2 16.f odd 4 1
6400.2.a.bj 2 80.k odd 4 1
6400.2.a.bj 2 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 \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3} \)
\( T_{7}^{2} - 20 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -20 + T^{2} )^{2} \)
$11$ \( ( 4 + T^{2} )^{2} \)
$13$ \( ( 20 + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 36 + T^{2} )^{2} \)
$23$ \( ( -20 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 20 + T^{2} )^{2} \)
$41$ \( ( -2 + T )^{4} \)
$43$ \( T^{4} \)
$47$ \( ( -180 + T^{2} )^{2} \)
$53$ \( ( 180 + T^{2} )^{2} \)
$59$ \( ( 196 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( -14 + T )^{4} \)
$97$ \( T^{4} \)
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