Properties

Label 8100.2.d.k
Level $8100$
Weight $2$
Character orbit 8100.d
Analytic conductor $64.679$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{7} + (\beta_{3} - 2) q^{11} + ( - \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{2} + 4 \beta_1) q^{17} - 2 q^{19} + (\beta_{2} + 2 \beta_1) q^{23} + ( - 2 \beta_{3} + 4) q^{29} + (2 \beta_{3} - 5) q^{31} + ( - \beta_{2} - 3 \beta_1) q^{37} - 3 \beta_{3} q^{41} + ( - \beta_{2} + 2 \beta_1) q^{43} + ( - 2 \beta_{2} - \beta_1) q^{47} + (\beta_{3} - 5) q^{49} + ( - \beta_{2} - 2 \beta_1) q^{53} + (\beta_{3} - 5) q^{59} + (\beta_{3} - 3) q^{61} - 7 \beta_1 q^{67} + 9 q^{71} + ( - 2 \beta_{2} + 3 \beta_1) q^{73} + (2 \beta_{2} - 11 \beta_1) q^{77} + ( - 2 \beta_{3} - 4) q^{79} + ( - 2 \beta_{2} - \beta_1) q^{83} + 12 q^{89} + (3 \beta_{3} - 14) q^{91} - 10 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{11} - 8 q^{19} + 12 q^{29} - 16 q^{31} - 6 q^{41} - 18 q^{49} - 18 q^{59} - 10 q^{61} + 36 q^{71} - 20 q^{79} + 48 q^{89} - 50 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{2} + 5\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times\).

\(n\) \(4051\) \(6401\) \(7777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.61803i
0.618034i
0.618034i
1.61803i
0 0 0 0 0 3.85410i 0 0 0
649.2 0 0 0 0 0 2.85410i 0 0 0
649.3 0 0 0 0 0 2.85410i 0 0 0
649.4 0 0 0 0 0 3.85410i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8100.2.d.k 4
3.b odd 2 1 8100.2.d.n 4
5.b even 2 1 inner 8100.2.d.k 4
5.c odd 4 1 8100.2.a.o 2
5.c odd 4 1 8100.2.a.q yes 2
15.d odd 2 1 8100.2.d.n 4
15.e even 4 1 8100.2.a.p yes 2
15.e even 4 1 8100.2.a.r yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8100.2.a.o 2 5.c odd 4 1
8100.2.a.p yes 2 15.e even 4 1
8100.2.a.q yes 2 5.c odd 4 1
8100.2.a.r yes 2 15.e even 4 1
8100.2.d.k 4 1.a even 1 1 trivial
8100.2.d.k 4 5.b even 2 1 inner
8100.2.d.n 4 3.b odd 2 1
8100.2.d.n 4 15.d odd 2 1

Hecke kernels

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

\( T_{7}^{4} + 23T_{7}^{2} + 121 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 9 \) Copy content Toggle raw display
\( T_{29}^{2} - 6T_{29} - 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 23T^{2} + 121 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T - 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 35T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{4} + 63T^{2} + 81 \) Copy content Toggle raw display
$19$ \( (T + 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T - 29)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 35T^{2} + 25 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T - 99)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 35T^{2} + 25 \) Copy content Toggle raw display
$47$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$59$ \( (T^{2} + 9 T + 9)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T - 5)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$71$ \( (T - 9)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 122T^{2} + 841 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T - 20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$89$ \( (T - 12)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
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