Properties

Label 6400.2.a.by
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $2$
CM discriminant -8
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.1042572936\)
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 128)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \beta q^{3} + 5 q^{9} +O(q^{10})\) \( q + 2 \beta q^{3} + 5 q^{9} + 2 \beta q^{11} -6 q^{17} + 6 \beta q^{19} + 4 \beta q^{27} + 8 q^{33} + 6 q^{41} + 6 \beta q^{43} -7 q^{49} -12 \beta q^{51} + 24 q^{57} + 10 \beta q^{59} -6 \beta q^{67} -2 q^{73} + q^{81} + 2 \beta q^{83} + 18 q^{89} + 10 q^{97} + 10 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{9} + O(q^{10}) \) \( 2q + 10q^{9} - 12q^{17} + 16q^{33} + 12q^{41} - 14q^{49} + 48q^{57} - 4q^{73} + 2q^{81} + 36q^{89} + 20q^{97} + O(q^{100}) \)

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 0 0 5.00000 0
1.2 0 2.82843 0 0 0 0 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
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 6400.2.a.by 2
4.b odd 2 1 inner 6400.2.a.by 2
5.b even 2 1 256.2.a.e 2
8.b even 2 1 inner 6400.2.a.by 2
8.d odd 2 1 CM 6400.2.a.by 2
15.d odd 2 1 2304.2.a.t 2
16.e even 4 2 3200.2.d.c 2
16.f odd 4 2 3200.2.d.c 2
20.d odd 2 1 256.2.a.e 2
40.e odd 2 1 256.2.a.e 2
40.f even 2 1 256.2.a.e 2
60.h even 2 1 2304.2.a.t 2
80.i odd 4 2 3200.2.f.o 4
80.j even 4 2 3200.2.f.o 4
80.k odd 4 2 128.2.b.a 2
80.q even 4 2 128.2.b.a 2
80.s even 4 2 3200.2.f.o 4
80.t odd 4 2 3200.2.f.o 4
120.i odd 2 1 2304.2.a.t 2
120.m even 2 1 2304.2.a.t 2
160.y odd 8 2 1024.2.e.a 2
160.y odd 8 2 1024.2.e.f 2
160.z even 8 2 1024.2.e.a 2
160.z even 8 2 1024.2.e.f 2
240.t even 4 2 1152.2.d.c 2
240.bm odd 4 2 1152.2.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 80.k odd 4 2
128.2.b.a 2 80.q even 4 2
256.2.a.e 2 5.b even 2 1
256.2.a.e 2 20.d odd 2 1
256.2.a.e 2 40.e odd 2 1
256.2.a.e 2 40.f even 2 1
1024.2.e.a 2 160.y odd 8 2
1024.2.e.a 2 160.z even 8 2
1024.2.e.f 2 160.y odd 8 2
1024.2.e.f 2 160.z even 8 2
1152.2.d.c 2 240.t even 4 2
1152.2.d.c 2 240.bm odd 4 2
2304.2.a.t 2 15.d odd 2 1
2304.2.a.t 2 60.h even 2 1
2304.2.a.t 2 120.i odd 2 1
2304.2.a.t 2 120.m even 2 1
3200.2.d.c 2 16.e even 4 2
3200.2.d.c 2 16.f odd 4 2
3200.2.f.o 4 80.i odd 4 2
3200.2.f.o 4 80.j even 4 2
3200.2.f.o 4 80.s even 4 2
3200.2.f.o 4 80.t odd 4 2
6400.2.a.by 2 1.a even 1 1 trivial
6400.2.a.by 2 4.b odd 2 1 inner
6400.2.a.by 2 8.b even 2 1 inner
6400.2.a.by 2 8.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\):

\( T_{3}^{2} - 8 \)
\( T_{7} \)
\( T_{11}^{2} - 8 \)
\( T_{13} \)
\( T_{17} + 6 \)
\( T_{29} \)
\( T_{31} \)