Properties

Label 6400.2.a.ca
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: \(\Q(\sqrt{10}) \)
Defining polynomial: \(x^{2} - 10\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + \beta q^{7} + 7 q^{9} +O(q^{10})\) \( q + \beta q^{3} + \beta q^{7} + 7 q^{9} + 6 q^{13} + 2 q^{17} -2 \beta q^{19} + 10 q^{21} + \beta q^{23} + 4 \beta q^{27} + 4 q^{29} -2 \beta q^{31} + 2 q^{37} + 6 \beta q^{39} -\beta q^{43} -3 \beta q^{47} + 3 q^{49} + 2 \beta q^{51} + 6 q^{53} -20 q^{57} + 2 \beta q^{59} -2 q^{61} + 7 \beta q^{63} -3 \beta q^{67} + 10 q^{69} -2 \beta q^{71} -14 q^{73} + 4 \beta q^{79} + 19 q^{81} -\beta q^{83} + 4 \beta q^{87} -10 q^{89} + 6 \beta q^{91} -20 q^{93} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{9} + O(q^{10}) \) \( 2q + 14q^{9} + 12q^{13} + 4q^{17} + 20q^{21} + 8q^{29} + 4q^{37} + 6q^{49} + 12q^{53} - 40q^{57} - 4q^{61} + 20q^{69} - 28q^{73} + 38q^{81} - 20q^{89} - 40q^{93} - 4q^{97} + O(q^{100}) \)

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
−3.16228
3.16228
0 −3.16228 0 0 0 −3.16228 0 7.00000 0
1.2 0 3.16228 0 0 0 3.16228 0 7.00000 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 6400.2.a.ca 2
4.b odd 2 1 inner 6400.2.a.ca 2
5.b even 2 1 1280.2.a.h 2
8.b even 2 1 6400.2.a.bz 2
8.d odd 2 1 6400.2.a.bz 2
16.e even 4 2 3200.2.d.j 4
16.f odd 4 2 3200.2.d.j 4
20.d odd 2 1 1280.2.a.h 2
40.e odd 2 1 1280.2.a.l 2
40.f even 2 1 1280.2.a.l 2
80.i odd 4 2 3200.2.f.p 4
80.j even 4 2 3200.2.f.q 4
80.k odd 4 2 640.2.d.a 4
80.q even 4 2 640.2.d.a 4
80.s even 4 2 3200.2.f.p 4
80.t odd 4 2 3200.2.f.q 4
240.t even 4 2 5760.2.k.t 4
240.bm odd 4 2 5760.2.k.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.a 4 80.k odd 4 2
640.2.d.a 4 80.q even 4 2
1280.2.a.h 2 5.b even 2 1
1280.2.a.h 2 20.d odd 2 1
1280.2.a.l 2 40.e odd 2 1
1280.2.a.l 2 40.f even 2 1
3200.2.d.j 4 16.e even 4 2
3200.2.d.j 4 16.f odd 4 2
3200.2.f.p 4 80.i odd 4 2
3200.2.f.p 4 80.s even 4 2
3200.2.f.q 4 80.j even 4 2
3200.2.f.q 4 80.t odd 4 2
5760.2.k.t 4 240.t even 4 2
5760.2.k.t 4 240.bm odd 4 2
6400.2.a.bz 2 8.b even 2 1
6400.2.a.bz 2 8.d odd 2 1
6400.2.a.ca 2 1.a even 1 1 trivial
6400.2.a.ca 2 4.b odd 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 10 \)
\( T_{7}^{2} - 10 \)
\( T_{11} \)
\( T_{13} - 6 \)
\( T_{17} - 2 \)
\( T_{29} - 4 \)
\( T_{31}^{2} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -10 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -10 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -6 + T )^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( -40 + T^{2} \)
$23$ \( -10 + T^{2} \)
$29$ \( ( -4 + T )^{2} \)
$31$ \( -40 + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( -10 + T^{2} \)
$47$ \( -90 + T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( -40 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -90 + T^{2} \)
$71$ \( -40 + T^{2} \)
$73$ \( ( 14 + T )^{2} \)
$79$ \( -160 + T^{2} \)
$83$ \( -10 + T^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( ( 2 + T )^{2} \)
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