Properties

Label 8100.2.a.be
Level $8100$
Weight $2$
Character orbit 8100.a
Self dual yes
Analytic conductor $64.679$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.28356903014400.1
Defining polynomial: \(x^{8} - 37 x^{6} + 399 x^{4} - 1195 x^{2} + 400\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{7} +O(q^{10})\) \( q + \beta_{4} q^{7} + ( -\beta_{2} + \beta_{7} ) q^{11} -\beta_{6} q^{13} + ( -\beta_{1} - \beta_{3} ) q^{17} + ( 2 + \beta_{5} ) q^{19} + 2 \beta_{1} q^{23} -3 \beta_{2} q^{29} + \beta_{5} q^{31} + ( \beta_{4} - \beta_{6} ) q^{37} + ( \beta_{2} + \beta_{7} ) q^{41} + \beta_{4} q^{43} + ( -2 \beta_{1} + \beta_{3} ) q^{47} + ( 5 - 2 \beta_{5} ) q^{49} -\beta_{3} q^{53} + ( 3 \beta_{2} - \beta_{7} ) q^{59} + ( 3 - \beta_{5} ) q^{61} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{67} + ( \beta_{2} + 3 \beta_{7} ) q^{71} + ( \beta_{4} + \beta_{6} ) q^{73} + ( -2 \beta_{1} + \beta_{3} ) q^{77} + 6 q^{79} + ( 2 \beta_{1} + \beta_{3} ) q^{83} + ( 3 \beta_{2} - 2 \beta_{7} ) q^{89} + ( 2 - 2 \beta_{5} ) q^{91} + ( -\beta_{4} - 2 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 12q^{19} - 4q^{31} + 48q^{49} + 28q^{61} + 48q^{79} + 24q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 37 x^{6} + 399 x^{4} - 1195 x^{2} + 400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} - 42 \nu^{4} + 429 \nu^{2} - 460 \)\()/180\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{7} + 69 \nu^{5} - 738 \nu^{3} + 2795 \nu \)\()/900\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 12 \nu^{4} - 261 \nu^{2} + 1640 \)\()/180\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 42 \nu^{5} - 519 \nu^{3} + 1630 \nu \)\()/180\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - 30 \nu^{4} + 225 \nu^{2} - 304 \)\()/36\)
\(\beta_{6}\)\(=\)\((\)\( 2 \nu^{7} - 69 \nu^{5} + 648 \nu^{3} - 1085 \nu \)\()/180\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{7} + 69 \nu^{5} - 648 \nu^{3} + 1445 \nu \)\()/180\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 2 \beta_{3} - 3 \beta_{1} + 19\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(19 \beta_{7} + 15 \beta_{6} - 20 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(23 \beta_{5} - 34 \beta_{3} - 81 \beta_{1} + 297\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(349 \beta_{7} + 269 \beta_{6} + 48 \beta_{4} - 520 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(537 \beta_{5} - 570 \beta_{3} - 1755 \beta_{1} + 5243\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(6427 \beta_{7} + 5143 \beta_{6} + 1656 \beta_{4} - 11460 \beta_{2}\)\()/2\)

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
−0.617787
3.52696
−4.46736
−2.05466
2.05466
4.46736
−3.52696
0.617787
0 0 0 0 0 −4.93536 0 0 0
1.2 0 0 0 0 0 −4.93536 0 0 0
1.3 0 0 0 0 0 −1.28148 0 0 0
1.4 0 0 0 0 0 −1.28148 0 0 0
1.5 0 0 0 0 0 1.28148 0 0 0
1.6 0 0 0 0 0 1.28148 0 0 0
1.7 0 0 0 0 0 4.93536 0 0 0
1.8 0 0 0 0 0 4.93536 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8100.2.a.be 8
3.b odd 2 1 inner 8100.2.a.be 8
5.b even 2 1 inner 8100.2.a.be 8
5.c odd 4 2 1620.2.d.e 8
15.d odd 2 1 inner 8100.2.a.be 8
15.e even 4 2 1620.2.d.e 8
45.k odd 12 4 1620.2.r.h 16
45.l even 12 4 1620.2.r.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.d.e 8 5.c odd 4 2
1620.2.d.e 8 15.e even 4 2
1620.2.r.h 16 45.k odd 12 4
1620.2.r.h 16 45.l even 12 4
8100.2.a.be 8 1.a even 1 1 trivial
8100.2.a.be 8 3.b odd 2 1 inner
8100.2.a.be 8 5.b even 2 1 inner
8100.2.a.be 8 15.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8100))\):

\( T_{7}^{4} - 26 T_{7}^{2} + 40 \)
\( T_{11}^{4} - 23 T_{11}^{2} + 100 \)
\( T_{17}^{4} - 83 T_{17}^{2} + 1690 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 40 - 26 T^{2} + T^{4} )^{2} \)
$11$ \( ( 100 - 23 T^{2} + T^{4} )^{2} \)
$13$ \( ( 360 - 51 T^{2} + T^{4} )^{2} \)
$17$ \( ( 1690 - 83 T^{2} + T^{4} )^{2} \)
$19$ \( ( -30 - 3 T + T^{2} )^{4} \)
$23$ \( ( 640 - 68 T^{2} + T^{4} )^{2} \)
$29$ \( ( -27 + T^{2} )^{4} \)
$31$ \( ( -32 + T + T^{2} )^{4} \)
$37$ \( ( 1690 - 89 T^{2} + T^{4} )^{2} \)
$41$ \( ( 16 - 35 T^{2} + T^{4} )^{2} \)
$43$ \( ( 40 - 26 T^{2} + T^{4} )^{2} \)
$47$ \( ( 4000 - 170 T^{2} + T^{4} )^{2} \)
$53$ \( ( 360 - 78 T^{2} + T^{4} )^{2} \)
$59$ \( ( 64 - 59 T^{2} + T^{4} )^{2} \)
$61$ \( ( -20 - 7 T + T^{2} )^{4} \)
$67$ \( ( 4000 - 260 T^{2} + T^{4} )^{2} \)
$71$ \( ( 6084 - 231 T^{2} + T^{4} )^{2} \)
$73$ \( ( 250 - 65 T^{2} + T^{4} )^{2} \)
$79$ \( ( -6 + T )^{8} \)
$83$ \( ( 2560 - 122 T^{2} + T^{4} )^{2} \)
$89$ \( ( 961 - 110 T^{2} + T^{4} )^{2} \)
$97$ \( ( 160 - 206 T^{2} + T^{4} )^{2} \)
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