Properties

Label 1600.2.a.bd
Level $1600$
Weight $2$
Character orbit 1600.a
Self dual yes
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
CM discriminant -20
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( -3 + \beta ) q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( -3 + \beta ) q^{7} + ( 3 + 2 \beta ) q^{9} + ( 2 - 2 \beta ) q^{21} + ( -1 + 3 \beta ) q^{23} + ( 10 + 2 \beta ) q^{27} + 6 q^{29} -2 \beta q^{41} + ( 9 + \beta ) q^{43} + ( -7 - 3 \beta ) q^{47} + ( 7 - 6 \beta ) q^{49} + 6 \beta q^{61} + ( 1 - 3 \beta ) q^{63} + ( 3 - 5 \beta ) q^{67} + ( 14 + 2 \beta ) q^{69} + ( 11 + 6 \beta ) q^{81} + ( -11 - 3 \beta ) q^{83} + ( 6 + 6 \beta ) q^{87} + 6 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 6q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 6q^{7} + 6q^{9} + 4q^{21} - 2q^{23} + 20q^{27} + 12q^{29} + 18q^{43} - 14q^{47} + 14q^{49} + 2q^{63} + 6q^{67} + 28q^{69} + 22q^{81} - 22q^{83} + 12q^{87} + 12q^{89} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
0 −1.23607 0 0 0 −5.23607 0 −1.47214 0
1.2 0 3.23607 0 0 0 −0.763932 0 7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.a.bd 2
4.b odd 2 1 1600.2.a.z 2
5.b even 2 1 1600.2.a.z 2
5.c odd 4 2 320.2.c.d 4
8.b even 2 1 800.2.a.j 2
8.d odd 2 1 800.2.a.n 2
15.e even 4 2 2880.2.f.w 4
20.d odd 2 1 CM 1600.2.a.bd 2
20.e even 4 2 320.2.c.d 4
24.f even 2 1 7200.2.a.cr 2
24.h odd 2 1 7200.2.a.cb 2
40.e odd 2 1 800.2.a.j 2
40.f even 2 1 800.2.a.n 2
40.i odd 4 2 160.2.c.b 4
40.k even 4 2 160.2.c.b 4
60.l odd 4 2 2880.2.f.w 4
80.i odd 4 2 1280.2.f.g 4
80.j even 4 2 1280.2.f.g 4
80.s even 4 2 1280.2.f.h 4
80.t odd 4 2 1280.2.f.h 4
120.i odd 2 1 7200.2.a.cr 2
120.m even 2 1 7200.2.a.cb 2
120.q odd 4 2 1440.2.f.i 4
120.w even 4 2 1440.2.f.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.b 4 40.i odd 4 2
160.2.c.b 4 40.k even 4 2
320.2.c.d 4 5.c odd 4 2
320.2.c.d 4 20.e even 4 2
800.2.a.j 2 8.b even 2 1
800.2.a.j 2 40.e odd 2 1
800.2.a.n 2 8.d odd 2 1
800.2.a.n 2 40.f even 2 1
1280.2.f.g 4 80.i odd 4 2
1280.2.f.g 4 80.j even 4 2
1280.2.f.h 4 80.s even 4 2
1280.2.f.h 4 80.t odd 4 2
1440.2.f.i 4 120.q odd 4 2
1440.2.f.i 4 120.w even 4 2
1600.2.a.z 2 4.b odd 2 1
1600.2.a.z 2 5.b even 2 1
1600.2.a.bd 2 1.a even 1 1 trivial
1600.2.a.bd 2 20.d odd 2 1 CM
2880.2.f.w 4 15.e even 4 2
2880.2.f.w 4 60.l odd 4 2
7200.2.a.cb 2 24.h odd 2 1
7200.2.a.cb 2 120.m even 2 1
7200.2.a.cr 2 24.f even 2 1
7200.2.a.cr 2 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1600))\):

\( T_{3}^{2} - 2 T_{3} - 4 \)
\( T_{7}^{2} + 6 T_{7} + 4 \)
\( T_{11} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 4 + 6 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( -44 + 2 T + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( -20 + T^{2} \)
$43$ \( 76 - 18 T + T^{2} \)
$47$ \( 4 + 14 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( -180 + T^{2} \)
$67$ \( -116 - 6 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( 76 + 22 T + T^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( T^{2} \)
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