Properties

Label 1600.2.a.bb
Level $1600$
Weight $2$
Character orbit 1600.a
Self dual yes
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
CM no
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 800)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q -\beta q^{3} + 2 \beta q^{7} + 2 q^{9} +O(q^{10})\) \( q -\beta q^{3} + 2 \beta q^{7} + 2 q^{9} + \beta q^{11} + 4 q^{13} + 7 q^{17} -3 \beta q^{19} -10 q^{21} -2 \beta q^{23} + \beta q^{27} -2 \beta q^{31} -5 q^{33} + 2 q^{37} -4 \beta q^{39} + 5 q^{41} -4 \beta q^{47} + 13 q^{49} -7 \beta q^{51} + 6 q^{53} + 15 q^{57} + 4 \beta q^{59} -10 q^{61} + 4 \beta q^{63} + \beta q^{67} + 10 q^{69} + 4 \beta q^{71} + 9 q^{73} + 10 q^{77} + 2 \beta q^{79} -11 q^{81} + 5 \beta q^{83} -5 q^{89} + 8 \beta q^{91} + 10 q^{93} -2 q^{97} + 2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} + 8q^{13} + 14q^{17} - 20q^{21} - 10q^{33} + 4q^{37} + 10q^{41} + 26q^{49} + 12q^{53} + 30q^{57} - 20q^{61} + 20q^{69} + 18q^{73} + 20q^{77} - 22q^{81} - 10q^{89} + 20q^{93} - 4q^{97} + O(q^{100}) \)

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
1.61803
−0.618034
0 −2.23607 0 0 0 4.47214 0 2.00000 0
1.2 0 2.23607 0 0 0 −4.47214 0 2.00000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.a.bb 2
4.b odd 2 1 inner 1600.2.a.bb 2
5.b even 2 1 1600.2.a.ba 2
5.c odd 4 2 1600.2.c.o 4
8.b even 2 1 800.2.a.k 2
8.d odd 2 1 800.2.a.k 2
20.d odd 2 1 1600.2.a.ba 2
20.e even 4 2 1600.2.c.o 4
24.f even 2 1 7200.2.a.cf 2
24.h odd 2 1 7200.2.a.cf 2
40.e odd 2 1 800.2.a.l yes 2
40.f even 2 1 800.2.a.l yes 2
40.i odd 4 2 800.2.c.g 4
40.k even 4 2 800.2.c.g 4
120.i odd 2 1 7200.2.a.cn 2
120.m even 2 1 7200.2.a.cn 2
120.q odd 4 2 7200.2.f.bg 4
120.w even 4 2 7200.2.f.bg 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.k 2 8.b even 2 1
800.2.a.k 2 8.d odd 2 1
800.2.a.l yes 2 40.e odd 2 1
800.2.a.l yes 2 40.f even 2 1
800.2.c.g 4 40.i odd 4 2
800.2.c.g 4 40.k even 4 2
1600.2.a.ba 2 5.b even 2 1
1600.2.a.ba 2 20.d odd 2 1
1600.2.a.bb 2 1.a even 1 1 trivial
1600.2.a.bb 2 4.b odd 2 1 inner
1600.2.c.o 4 5.c odd 4 2
1600.2.c.o 4 20.e even 4 2
7200.2.a.cf 2 24.f even 2 1
7200.2.a.cf 2 24.h odd 2 1
7200.2.a.cn 2 120.i odd 2 1
7200.2.a.cn 2 120.m even 2 1
7200.2.f.bg 4 120.q odd 4 2
7200.2.f.bg 4 120.w even 4 2

Hecke kernels

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

\( T_{3}^{2} - 5 \)
\( T_{7}^{2} - 20 \)
\( T_{11}^{2} - 5 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -5 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -20 + T^{2} \)
$11$ \( -5 + T^{2} \)
$13$ \( ( -4 + T )^{2} \)
$17$ \( ( -7 + T )^{2} \)
$19$ \( -45 + T^{2} \)
$23$ \( -20 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( -20 + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( ( -5 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( -80 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( -80 + T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( -5 + T^{2} \)
$71$ \( -80 + T^{2} \)
$73$ \( ( -9 + T )^{2} \)
$79$ \( -20 + T^{2} \)
$83$ \( -125 + T^{2} \)
$89$ \( ( 5 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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