Properties

Label 6400.2.a.ci
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 $2$

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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: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1600)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + (2 \beta + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + (2 \beta + 4) q^{9} + (\beta + 3) q^{11} + ( - 2 \beta + 3) q^{17} + ( - 3 \beta + 1) q^{19} + (3 \beta + 13) q^{27} + (4 \beta + 9) q^{33} + (4 \beta - 3) q^{41} + 10 q^{43} - 7 q^{49} + (\beta - 9) q^{51} + ( - 2 \beta - 17) q^{57} - 6 q^{59} + (3 \beta - 7) q^{67} + (6 \beta + 1) q^{73} + (10 \beta + 19) q^{81} + ( - \beta + 9) q^{83} + ( - 2 \beta + 9) q^{89} + 10 q^{97} + (10 \beta + 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 8 q^{9} + 6 q^{11} + 6 q^{17} + 2 q^{19} + 26 q^{27} + 18 q^{33} - 6 q^{41} + 20 q^{43} - 14 q^{49} - 18 q^{51} - 34 q^{57} - 12 q^{59} - 14 q^{67} + 2 q^{73} + 38 q^{81} + 18 q^{83} + 18 q^{89} + 20 q^{97} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−2.44949
2.44949
0 −1.44949 0 0 0 0 0 −0.898979 0
1.2 0 3.44949 0 0 0 0 0 8.89898 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 \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.ci 2
4.b odd 2 1 6400.2.a.bc 2
5.b even 2 1 6400.2.a.bd 2
8.b even 2 1 6400.2.a.bc 2
8.d odd 2 1 CM 6400.2.a.ci 2
16.e even 4 2 1600.2.d.d yes 4
16.f odd 4 2 1600.2.d.d yes 4
20.d odd 2 1 6400.2.a.ch 2
40.e odd 2 1 6400.2.a.bd 2
40.f even 2 1 6400.2.a.ch 2
80.i odd 4 2 1600.2.f.f 4
80.j even 4 2 1600.2.f.f 4
80.k odd 4 2 1600.2.d.c 4
80.q even 4 2 1600.2.d.c 4
80.s even 4 2 1600.2.f.j 4
80.t odd 4 2 1600.2.f.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.c 4 80.k odd 4 2
1600.2.d.c 4 80.q even 4 2
1600.2.d.d yes 4 16.e even 4 2
1600.2.d.d yes 4 16.f odd 4 2
1600.2.f.f 4 80.i odd 4 2
1600.2.f.f 4 80.j even 4 2
1600.2.f.j 4 80.s even 4 2
1600.2.f.j 4 80.t odd 4 2
6400.2.a.bc 2 4.b odd 2 1
6400.2.a.bc 2 8.b even 2 1
6400.2.a.bd 2 5.b even 2 1
6400.2.a.bd 2 40.e odd 2 1
6400.2.a.ch 2 20.d odd 2 1
6400.2.a.ch 2 40.f even 2 1
6400.2.a.ci 2 1.a even 1 1 trivial
6400.2.a.ci 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\):

\( T_{3}^{2} - 2T_{3} - 5 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 3 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} - 15 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 5 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 3 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T - 15 \) Copy content Toggle raw display
$19$ \( T^{2} - 2T - 53 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 87 \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 14T - 5 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 215 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 18T + 75 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 57 \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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