Properties

Label 900.6.a.t
Level $900$
Weight $6$
Character orbit 900.a
Self dual yes
Analytic conductor $144.345$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 180)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{241}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 40) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 40) q^{7} + ( - 5 \beta - 120) q^{11} + (8 \beta + 340) q^{13} + ( - 20 \beta + 270) q^{17} + (20 \beta + 548) q^{19} + (10 \beta + 1740) q^{23} + ( - 40 \beta - 4710) q^{29} + ( - 70 \beta + 1496) q^{31} + ( - 60 \beta + 3760) q^{37} + (60 \beta - 13560) q^{41} + ( - 88 \beta - 2360) q^{43} + (110 \beta + 19140) q^{47} + ( - 80 \beta - 6531) q^{49} + ( - 40 \beta + 26790) q^{53} + (275 \beta - 23400) q^{59} + (120 \beta + 10754) q^{61} + ( - 54 \beta + 5440) q^{67} + ( - 330 \beta - 22920) q^{71} + (240 \beta - 4790) q^{73} + ( - 80 \beta + 38580) q^{77} + (150 \beta + 45536) q^{79} + ( - 170 \beta - 7080) q^{83} + ( - 420 \beta - 17580) q^{89} + ( - 20 \beta - 55808) q^{91} + (1096 \beta + 49390) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 80 q^{7} - 240 q^{11} + 680 q^{13} + 540 q^{17} + 1096 q^{19} + 3480 q^{23} - 9420 q^{29} + 2992 q^{31} + 7520 q^{37} - 27120 q^{41} - 4720 q^{43} + 38280 q^{47} - 13062 q^{49} + 53580 q^{53} - 46800 q^{59} + 21508 q^{61} + 10880 q^{67} - 45840 q^{71} - 9580 q^{73} + 77160 q^{77} + 91072 q^{79} - 14160 q^{83} - 35160 q^{89} - 111616 q^{91} + 98780 q^{97}+O(q^{100}) \) 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
8.26209
−7.26209
0 0 0 0 0 −53.1450 0 0 0
1.2 0 0 0 0 0 133.145 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.t 2
3.b odd 2 1 900.6.a.u 2
5.b even 2 1 180.6.a.g yes 2
5.c odd 4 2 900.6.d.k 4
15.d odd 2 1 180.6.a.f 2
15.e even 4 2 900.6.d.n 4
20.d odd 2 1 720.6.a.bg 2
60.h even 2 1 720.6.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.6.a.f 2 15.d odd 2 1
180.6.a.g yes 2 5.b even 2 1
720.6.a.bc 2 60.h even 2 1
720.6.a.bg 2 20.d odd 2 1
900.6.a.t 2 1.a even 1 1 trivial
900.6.a.u 2 3.b odd 2 1
900.6.d.k 4 5.c odd 4 2
900.6.d.n 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(900))\):

\( T_{7}^{2} - 80T_{7} - 7076 \) Copy content Toggle raw display
\( T_{11}^{2} + 240T_{11} - 202500 \) 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
$7$ \( T^{2} - 80T - 7076 \) Copy content Toggle raw display
$11$ \( T^{2} + 240T - 202500 \) Copy content Toggle raw display
$13$ \( T^{2} - 680T - 439664 \) Copy content Toggle raw display
$17$ \( T^{2} - 540 T - 3397500 \) Copy content Toggle raw display
$19$ \( T^{2} - 1096 T - 3170096 \) Copy content Toggle raw display
$23$ \( T^{2} - 3480 T + 2160000 \) Copy content Toggle raw display
$29$ \( T^{2} + 9420 T + 8302500 \) Copy content Toggle raw display
$31$ \( T^{2} - 2992 T - 40274384 \) Copy content Toggle raw display
$37$ \( T^{2} - 7520 T - 17096000 \) Copy content Toggle raw display
$41$ \( T^{2} + 27120 T + 152640000 \) Copy content Toggle raw display
$43$ \( T^{2} + 4720 T - 61617344 \) Copy content Toggle raw display
$47$ \( T^{2} - 38280 T + 261360000 \) Copy content Toggle raw display
$53$ \( T^{2} - 53580 T + 703822500 \) Copy content Toggle raw display
$59$ \( T^{2} + 46800 T - 108562500 \) Copy content Toggle raw display
$61$ \( T^{2} - 21508 T - 9285884 \) Copy content Toggle raw display
$67$ \( T^{2} - 10880 T + 4294384 \) Copy content Toggle raw display
$71$ \( T^{2} + 45840 T - 419490000 \) Copy content Toggle raw display
$73$ \( T^{2} + 9580 T - 476793500 \) Copy content Toggle raw display
$79$ \( T^{2} - 91072 T + 1878317296 \) Copy content Toggle raw display
$83$ \( T^{2} + 14160 T - 200610000 \) Copy content Toggle raw display
$89$ \( T^{2} + 35160 T - 1221390000 \) Copy content Toggle raw display
$97$ \( T^{2} - 98780 T - 7982377916 \) Copy content Toggle raw display
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