Properties

Label 8100.2.a.t
Level $8100$
Weight $2$
Character orbit 8100.a
Self dual yes
Analytic conductor $64.679$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.6788256372\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1620)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{7} +O(q^{10})\) \( q + ( 1 + \beta ) q^{7} + \beta q^{11} + ( -2 - 2 \beta ) q^{13} + ( 3 + \beta ) q^{17} + ( -1 - 2 \beta ) q^{19} -2 \beta q^{23} + ( -6 - \beta ) q^{29} + ( -1 + 4 \beta ) q^{31} + ( 1 + 3 \beta ) q^{37} + ( -6 - 3 \beta ) q^{41} + ( -5 + \beta ) q^{43} + ( 3 - \beta ) q^{47} + ( -3 + 2 \beta ) q^{49} + ( 9 - \beta ) q^{53} + ( -6 - \beta ) q^{59} -4 q^{61} + ( 4 - 6 \beta ) q^{67} + ( -6 - 3 \beta ) q^{71} + ( -5 + 3 \beta ) q^{73} + ( 3 + \beta ) q^{77} + ( -4 - 6 \beta ) q^{79} + ( 3 + 7 \beta ) q^{83} + 3 \beta q^{89} + ( -8 - 4 \beta ) q^{91} + ( 1 - \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{7} - 4 q^{13} + 6 q^{17} - 2 q^{19} - 12 q^{29} - 2 q^{31} + 2 q^{37} - 12 q^{41} - 10 q^{43} + 6 q^{47} - 6 q^{49} + 18 q^{53} - 12 q^{59} - 8 q^{61} + 8 q^{67} - 12 q^{71} - 10 q^{73} + 6 q^{77} - 8 q^{79} + 6 q^{83} - 16 q^{91} + 2 q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
0 0 0 0 0 −0.732051 0 0 0
1.2 0 0 0 0 0 2.73205 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8100.2.a.t 2
3.b odd 2 1 8100.2.a.s 2
5.b even 2 1 1620.2.a.g 2
5.c odd 4 2 8100.2.d.m 4
15.d odd 2 1 1620.2.a.h yes 2
15.e even 4 2 8100.2.d.l 4
20.d odd 2 1 6480.2.a.bh 2
45.h odd 6 2 1620.2.i.m 4
45.j even 6 2 1620.2.i.n 4
60.h even 2 1 6480.2.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.g 2 5.b even 2 1
1620.2.a.h yes 2 15.d odd 2 1
1620.2.i.m 4 45.h odd 6 2
1620.2.i.n 4 45.j even 6 2
6480.2.a.bh 2 20.d odd 2 1
6480.2.a.bp 2 60.h even 2 1
8100.2.a.s 2 3.b odd 2 1
8100.2.a.t 2 1.a even 1 1 trivial
8100.2.d.l 4 15.e even 4 2
8100.2.d.m 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8100))\):

\( T_{7}^{2} - 2 T_{7} - 2 \)
\( T_{11}^{2} - 3 \)
\( T_{17}^{2} - 6 T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -2 - 2 T + T^{2} \)
$11$ \( -3 + T^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( 6 - 6 T + T^{2} \)
$19$ \( -11 + 2 T + T^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( 33 + 12 T + T^{2} \)
$31$ \( -47 + 2 T + T^{2} \)
$37$ \( -26 - 2 T + T^{2} \)
$41$ \( 9 + 12 T + T^{2} \)
$43$ \( 22 + 10 T + T^{2} \)
$47$ \( 6 - 6 T + T^{2} \)
$53$ \( 78 - 18 T + T^{2} \)
$59$ \( 33 + 12 T + T^{2} \)
$61$ \( ( 4 + T )^{2} \)
$67$ \( -92 - 8 T + T^{2} \)
$71$ \( 9 + 12 T + T^{2} \)
$73$ \( -2 + 10 T + T^{2} \)
$79$ \( -92 + 8 T + T^{2} \)
$83$ \( -138 - 6 T + T^{2} \)
$89$ \( -27 + T^{2} \)
$97$ \( -2 - 2 T + T^{2} \)
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