Properties

Label 6400.2.a.cf
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $2$
CM no
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1600)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -\beta q^{7} -2 q^{9} +O(q^{10})\) \( q + q^{3} -\beta q^{7} -2 q^{9} -3 q^{11} + \beta q^{13} -3 q^{17} + q^{19} -\beta q^{21} -5 q^{27} -3 \beta q^{29} + 2 \beta q^{31} -3 q^{33} -3 \beta q^{37} + \beta q^{39} + 9 q^{41} + 4 q^{43} + 3 \beta q^{47} + 5 q^{49} -3 q^{51} + q^{57} + 12 q^{59} + \beta q^{61} + 2 \beta q^{63} + 11 q^{67} -3 \beta q^{71} + 7 q^{73} + 3 \beta q^{77} -3 \beta q^{79} + q^{81} + 15 q^{83} -3 \beta q^{87} + 3 q^{89} -12 q^{91} + 2 \beta q^{93} -14 q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{9} - 6q^{11} - 6q^{17} + 2q^{19} - 10q^{27} - 6q^{33} + 18q^{41} + 8q^{43} + 10q^{49} - 6q^{51} + 2q^{57} + 24q^{59} + 22q^{67} + 14q^{73} + 2q^{81} + 30q^{83} + 6q^{89} - 24q^{91} - 28q^{97} + 12q^{99} + O(q^{100}) \)

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
1.73205
−1.73205
0 1.00000 0 0 0 −3.46410 0 −2.00000 0
1.2 0 1.00000 0 0 0 3.46410 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
8.d odd 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.cf 2
4.b odd 2 1 6400.2.a.bb 2
5.b even 2 1 6400.2.a.ba 2
8.b even 2 1 6400.2.a.bb 2
8.d odd 2 1 inner 6400.2.a.cf 2
16.e even 4 2 1600.2.d.e 4
16.f odd 4 2 1600.2.d.e 4
20.d odd 2 1 6400.2.a.cg 2
40.e odd 2 1 6400.2.a.ba 2
40.f even 2 1 6400.2.a.cg 2
80.i odd 4 2 1600.2.f.c 4
80.j even 4 2 1600.2.f.c 4
80.k odd 4 2 1600.2.d.f yes 4
80.q even 4 2 1600.2.d.f yes 4
80.s even 4 2 1600.2.f.g 4
80.t odd 4 2 1600.2.f.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.e 4 16.e even 4 2
1600.2.d.e 4 16.f odd 4 2
1600.2.d.f yes 4 80.k odd 4 2
1600.2.d.f yes 4 80.q even 4 2
1600.2.f.c 4 80.i odd 4 2
1600.2.f.c 4 80.j even 4 2
1600.2.f.g 4 80.s even 4 2
1600.2.f.g 4 80.t odd 4 2
6400.2.a.ba 2 5.b even 2 1
6400.2.a.ba 2 40.e odd 2 1
6400.2.a.bb 2 4.b odd 2 1
6400.2.a.bb 2 8.b even 2 1
6400.2.a.cf 2 1.a even 1 1 trivial
6400.2.a.cf 2 8.d odd 2 1 inner
6400.2.a.cg 2 20.d odd 2 1
6400.2.a.cg 2 40.f even 2 1

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{7}^{2} - 12 \)
\( T_{11} + 3 \)
\( T_{13}^{2} - 12 \)
\( T_{17} + 3 \)
\( T_{29}^{2} - 108 \)
\( T_{31}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( -12 + T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( -12 + T^{2} \)
$17$ \( ( 3 + T )^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( -108 + T^{2} \)
$31$ \( -48 + T^{2} \)
$37$ \( -108 + T^{2} \)
$41$ \( ( -9 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -108 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( -12 + T^{2} \)
$67$ \( ( -11 + T )^{2} \)
$71$ \( -108 + T^{2} \)
$73$ \( ( -7 + T )^{2} \)
$79$ \( -108 + T^{2} \)
$83$ \( ( -15 + T )^{2} \)
$89$ \( ( -3 + T )^{2} \)
$97$ \( ( 14 + T )^{2} \)
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