Properties

Label 1600.2.a.bc
Level $1600$
Weight $2$
Character orbit 1600.a
Self dual yes
Analytic conductor $12.776$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{7} + 5 q^{9} +O(q^{10})\) \( q + \beta q^{3} + \beta q^{7} + 5 q^{9} + 2 \beta q^{11} -2 q^{13} -2 q^{17} + 8 q^{21} -\beta q^{23} + 2 \beta q^{27} -6 q^{29} + 2 \beta q^{31} + 16 q^{33} -10 q^{37} -2 \beta q^{39} + 2 q^{41} -3 \beta q^{43} + \beta q^{47} + q^{49} -2 \beta q^{51} + 6 q^{53} -4 \beta q^{59} + 2 q^{61} + 5 \beta q^{63} -\beta q^{67} -8 q^{69} + 2 \beta q^{71} + 6 q^{73} + 16 q^{77} + 4 \beta q^{79} + q^{81} + \beta q^{83} -6 \beta q^{87} + 10 q^{89} -2 \beta q^{91} + 16 q^{93} -2 q^{97} + 10 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{9} + O(q^{10}) \) \( 2q + 10q^{9} - 4q^{13} - 4q^{17} + 16q^{21} - 12q^{29} + 32q^{33} - 20q^{37} + 4q^{41} + 2q^{49} + 12q^{53} + 4q^{61} - 16q^{69} + 12q^{73} + 32q^{77} + 2q^{81} + 20q^{89} + 32q^{93} - 4q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 −2.82843 0 0 0 −2.82843 0 5.00000 0
1.2 0 2.82843 0 0 0 2.82843 0 5.00000 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.bc 2
4.b odd 2 1 inner 1600.2.a.bc 2
5.b even 2 1 320.2.a.g 2
5.c odd 4 2 1600.2.c.n 4
8.b even 2 1 800.2.a.m 2
8.d odd 2 1 800.2.a.m 2
15.d odd 2 1 2880.2.a.bk 2
20.d odd 2 1 320.2.a.g 2
20.e even 4 2 1600.2.c.n 4
24.f even 2 1 7200.2.a.cm 2
24.h odd 2 1 7200.2.a.cm 2
40.e odd 2 1 160.2.a.c 2
40.f even 2 1 160.2.a.c 2
40.i odd 4 2 800.2.c.f 4
40.k even 4 2 800.2.c.f 4
60.h even 2 1 2880.2.a.bk 2
80.k odd 4 2 1280.2.d.l 4
80.q even 4 2 1280.2.d.l 4
120.i odd 2 1 1440.2.a.o 2
120.m even 2 1 1440.2.a.o 2
120.q odd 4 2 7200.2.f.bh 4
120.w even 4 2 7200.2.f.bh 4
280.c odd 2 1 7840.2.a.bf 2
280.n even 2 1 7840.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 40.e odd 2 1
160.2.a.c 2 40.f even 2 1
320.2.a.g 2 5.b even 2 1
320.2.a.g 2 20.d odd 2 1
800.2.a.m 2 8.b even 2 1
800.2.a.m 2 8.d odd 2 1
800.2.c.f 4 40.i odd 4 2
800.2.c.f 4 40.k even 4 2
1280.2.d.l 4 80.k odd 4 2
1280.2.d.l 4 80.q even 4 2
1440.2.a.o 2 120.i odd 2 1
1440.2.a.o 2 120.m even 2 1
1600.2.a.bc 2 1.a even 1 1 trivial
1600.2.a.bc 2 4.b odd 2 1 inner
1600.2.c.n 4 5.c odd 4 2
1600.2.c.n 4 20.e even 4 2
2880.2.a.bk 2 15.d odd 2 1
2880.2.a.bk 2 60.h even 2 1
7200.2.a.cm 2 24.f even 2 1
7200.2.a.cm 2 24.h odd 2 1
7200.2.f.bh 4 120.q odd 4 2
7200.2.f.bh 4 120.w even 4 2
7840.2.a.bf 2 280.c odd 2 1
7840.2.a.bf 2 280.n even 2 1

Hecke kernels

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

\( T_{3}^{2} - 8 \)
\( T_{7}^{2} - 8 \)
\( T_{11}^{2} - 32 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -8 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -8 + T^{2} \)
$11$ \( -32 + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( -8 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -32 + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( -72 + T^{2} \)
$47$ \( -8 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( -128 + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( -8 + T^{2} \)
$71$ \( -32 + T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( -128 + T^{2} \)
$83$ \( -8 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
show more
show less