Properties

Label 900.3.l.f
Level $900$
Weight $3$
Character orbit 900.l
Analytic conductor $24.523$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{7} +O(q^{10})\) \( q + \beta_{1} q^{7} -\beta_{3} q^{13} + 26 \beta_{2} q^{19} + 46 q^{31} -5 \beta_{1} q^{37} + 6 \beta_{3} q^{43} + 143 \beta_{2} q^{49} + 74 q^{61} + 4 \beta_{1} q^{67} -10 \beta_{3} q^{73} + 142 \beta_{2} q^{79} + 192 q^{91} + 14 \beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 184q^{31} + 296q^{61} + 768q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 8 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( 8 \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/8\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)\(/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
0 0 0 0 0 −9.79796 9.79796i 0 0 0
757.2 0 0 0 0 0 9.79796 + 9.79796i 0 0 0
793.1 0 0 0 0 0 −9.79796 + 9.79796i 0 0 0
793.2 0 0 0 0 0 9.79796 9.79796i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.l.f 4
3.b odd 2 1 CM 900.3.l.f 4
5.b even 2 1 inner 900.3.l.f 4
5.c odd 4 2 inner 900.3.l.f 4
15.d odd 2 1 inner 900.3.l.f 4
15.e even 4 2 inner 900.3.l.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.l.f 4 1.a even 1 1 trivial
900.3.l.f 4 3.b odd 2 1 CM
900.3.l.f 4 5.b even 2 1 inner
900.3.l.f 4 5.c odd 4 2 inner
900.3.l.f 4 15.d odd 2 1 inner
900.3.l.f 4 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{4} + 36864 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 36864 + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 36864 + T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( 676 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( -46 + T )^{4} \)
$37$ \( 23040000 + T^{4} \)
$41$ \( T^{4} \)
$43$ \( 47775744 + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -74 + T )^{4} \)
$67$ \( 9437184 + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 368640000 + T^{4} \)
$79$ \( ( 20164 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 1416167424 + T^{4} \)
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