Properties

Label 900.6.a.b
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 20)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 218 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 218 q^{7} + 480 q^{11} + 622 q^{13} + 186 q^{17} - 1204 q^{19} - 3186 q^{23} - 5526 q^{29} + 9356 q^{31} - 5618 q^{37} + 14394 q^{41} + 370 q^{43} + 16146 q^{47} + 30717 q^{49} - 4374 q^{53} + 11748 q^{59} + 13202 q^{61} + 11542 q^{67} + 29532 q^{71} - 33698 q^{73} - 104640 q^{77} + 31208 q^{79} - 38466 q^{83} - 119514 q^{89} - 135596 q^{91} - 94658 q^{97}+O(q^{100}) \) 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 −218.000 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.b 1
3.b odd 2 1 100.6.a.a 1
5.b even 2 1 180.6.a.e 1
5.c odd 4 2 900.6.d.h 2
12.b even 2 1 400.6.a.m 1
15.d odd 2 1 20.6.a.a 1
15.e even 4 2 100.6.c.a 2
20.d odd 2 1 720.6.a.l 1
60.h even 2 1 80.6.a.b 1
60.l odd 4 2 400.6.c.c 2
105.g even 2 1 980.6.a.b 1
120.i odd 2 1 320.6.a.c 1
120.m even 2 1 320.6.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 15.d odd 2 1
80.6.a.b 1 60.h even 2 1
100.6.a.a 1 3.b odd 2 1
100.6.c.a 2 15.e even 4 2
180.6.a.e 1 5.b even 2 1
320.6.a.c 1 120.i odd 2 1
320.6.a.n 1 120.m even 2 1
400.6.a.m 1 12.b even 2 1
400.6.c.c 2 60.l odd 4 2
720.6.a.l 1 20.d odd 2 1
900.6.a.b 1 1.a even 1 1 trivial
900.6.d.h 2 5.c odd 4 2
980.6.a.b 1 105.g even 2 1

Hecke kernels

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

\( T_{7} + 218 \) Copy content Toggle raw display
\( T_{11} - 480 \) 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
$7$ \( T + 218 \) Copy content Toggle raw display
$11$ \( T - 480 \) Copy content Toggle raw display
$13$ \( T - 622 \) Copy content Toggle raw display
$17$ \( T - 186 \) Copy content Toggle raw display
$19$ \( T + 1204 \) Copy content Toggle raw display
$23$ \( T + 3186 \) Copy content Toggle raw display
$29$ \( T + 5526 \) Copy content Toggle raw display
$31$ \( T - 9356 \) Copy content Toggle raw display
$37$ \( T + 5618 \) Copy content Toggle raw display
$41$ \( T - 14394 \) Copy content Toggle raw display
$43$ \( T - 370 \) Copy content Toggle raw display
$47$ \( T - 16146 \) Copy content Toggle raw display
$53$ \( T + 4374 \) Copy content Toggle raw display
$59$ \( T - 11748 \) Copy content Toggle raw display
$61$ \( T - 13202 \) Copy content Toggle raw display
$67$ \( T - 11542 \) Copy content Toggle raw display
$71$ \( T - 29532 \) Copy content Toggle raw display
$73$ \( T + 33698 \) Copy content Toggle raw display
$79$ \( T - 31208 \) Copy content Toggle raw display
$83$ \( T + 38466 \) Copy content Toggle raw display
$89$ \( T + 119514 \) Copy content Toggle raw display
$97$ \( T + 94658 \) Copy content Toggle raw display
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