Properties

Label 900.6.a.l
Level $900$
Weight $6$
Character orbit 900.a
Self dual yes
Analytic conductor $144.345$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(144.345437832\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{94}) \)
Defining polynomial: \( x^{2} - 94 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 60\sqrt{94}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 71 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 71 q^{7} + \beta q^{11} - 137 q^{13} + \beta q^{17} - 1087 q^{19} + 7 \beta q^{23} - 7 \beta q^{29} - 2269 q^{31} + 6010 q^{37} + 3 \beta q^{41} - 4283 q^{43} + 2 \beta q^{47} - 11766 q^{49} - 43 \beta q^{53} + 50 \beta q^{59} + 12719 q^{61} - 6899 q^{67} - 69 \beta q^{71} + 24010 q^{73} - 71 \beta q^{77} + 12236 q^{79} + 46 \beta q^{83} - 96 \beta q^{89} + 9727 q^{91} + 19651 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 142 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 142 q^{7} - 274 q^{13} - 2174 q^{19} - 4538 q^{31} + 12020 q^{37} - 8566 q^{43} - 23532 q^{49} + 25438 q^{61} - 13798 q^{67} + 48020 q^{73} + 24472 q^{79} + 19454 q^{91} + 39302 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
−9.69536
9.69536
0 0 0 0 0 −71.0000 0 0 0
1.2 0 0 0 0 0 −71.0000 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

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.6.a.l 2
3.b odd 2 1 inner 900.6.a.l 2
5.b even 2 1 900.6.a.v yes 2
5.c odd 4 2 900.6.d.l 4
15.d odd 2 1 900.6.a.v yes 2
15.e even 4 2 900.6.d.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.6.a.l 2 1.a even 1 1 trivial
900.6.a.l 2 3.b odd 2 1 inner
900.6.a.v yes 2 5.b even 2 1
900.6.a.v yes 2 15.d odd 2 1
900.6.d.l 4 5.c odd 4 2
900.6.d.l 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(900))\):

\( T_{7} + 71 \) Copy content Toggle raw display
\( T_{11}^{2} - 338400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 71)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 338400 \) Copy content Toggle raw display
$13$ \( (T + 137)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 338400 \) Copy content Toggle raw display
$19$ \( (T + 1087)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 16581600 \) Copy content Toggle raw display
$29$ \( T^{2} - 16581600 \) Copy content Toggle raw display
$31$ \( (T + 2269)^{2} \) Copy content Toggle raw display
$37$ \( (T - 6010)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 3045600 \) Copy content Toggle raw display
$43$ \( (T + 4283)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 1353600 \) Copy content Toggle raw display
$53$ \( T^{2} - 625701600 \) Copy content Toggle raw display
$59$ \( T^{2} - 846000000 \) Copy content Toggle raw display
$61$ \( (T - 12719)^{2} \) Copy content Toggle raw display
$67$ \( (T + 6899)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 1611122400 \) Copy content Toggle raw display
$73$ \( (T - 24010)^{2} \) Copy content Toggle raw display
$79$ \( (T - 12236)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 716054400 \) Copy content Toggle raw display
$89$ \( T^{2} - 3118694400 \) Copy content Toggle raw display
$97$ \( (T - 19651)^{2} \) Copy content Toggle raw display
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