Properties

Label 900.6.a.i
Level $900$
Weight $6$
Character orbit 900.a
Self dual yes
Analytic conductor $144.345$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 91 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 91 q^{7} + 174 q^{11} - 785 q^{13} + 1794 q^{17} - 925 q^{19} - 2346 q^{23} + 726 q^{29} - 811 q^{31} - 7922 q^{37} + 360 q^{41} + 4951 q^{43} + 9906 q^{47} - 8526 q^{49} - 8292 q^{53} - 7014 q^{59} - 51433 q^{61} - 581 q^{67} + 56520 q^{71} + 42478 q^{73} + 15834 q^{77} - 28912 q^{79} - 104586 q^{83} + 118080 q^{89} - 71435 q^{91} - 110273 q^{97}+O(q^{100}) \) 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
0
0 0 0 0 0 91.0000 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

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 900.6.a.i 1
3.b odd 2 1 300.6.a.f yes 1
5.b even 2 1 900.6.a.d 1
5.c odd 4 2 900.6.d.g 2
15.d odd 2 1 300.6.a.a 1
15.e even 4 2 300.6.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.6.a.a 1 15.d odd 2 1
300.6.a.f yes 1 3.b odd 2 1
300.6.d.b 2 15.e even 4 2
900.6.a.d 1 5.b even 2 1
900.6.a.i 1 1.a even 1 1 trivial
900.6.d.g 2 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(900))\):

\( T_{7} - 91 \) Copy content Toggle raw display
\( T_{11} - 174 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 91 \) Copy content Toggle raw display
$11$ \( T - 174 \) Copy content Toggle raw display
$13$ \( T + 785 \) Copy content Toggle raw display
$17$ \( T - 1794 \) Copy content Toggle raw display
$19$ \( T + 925 \) Copy content Toggle raw display
$23$ \( T + 2346 \) Copy content Toggle raw display
$29$ \( T - 726 \) Copy content Toggle raw display
$31$ \( T + 811 \) Copy content Toggle raw display
$37$ \( T + 7922 \) Copy content Toggle raw display
$41$ \( T - 360 \) Copy content Toggle raw display
$43$ \( T - 4951 \) Copy content Toggle raw display
$47$ \( T - 9906 \) Copy content Toggle raw display
$53$ \( T + 8292 \) Copy content Toggle raw display
$59$ \( T + 7014 \) Copy content Toggle raw display
$61$ \( T + 51433 \) Copy content Toggle raw display
$67$ \( T + 581 \) Copy content Toggle raw display
$71$ \( T - 56520 \) Copy content Toggle raw display
$73$ \( T - 42478 \) Copy content Toggle raw display
$79$ \( T + 28912 \) Copy content Toggle raw display
$83$ \( T + 104586 \) Copy content Toggle raw display
$89$ \( T - 118080 \) Copy content Toggle raw display
$97$ \( T + 110273 \) Copy content Toggle raw display
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