Properties

Label 900.6.a.o
Level $900$
Weight $6$
Character orbit 900.a
Self dual yes
Analytic conductor $144.345$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{61}) \)
Defining polynomial: \( x^{2} - x - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 \(\beta = 2\sqrt{61}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 8 \beta q^{7} - 80 q^{11} + 24 \beta q^{13} + 35 \beta q^{17} - 12 q^{19} - 130 \beta q^{23} + 4560 q^{29} - 344 q^{31} + 280 \beta q^{37} - 14240 q^{41} + 1216 \beta q^{43} + 1570 \beta q^{47} - 1191 q^{49} - 1755 \beta q^{53} + 38000 q^{59} - 8206 q^{61} + 848 \beta q^{67} - 48480 q^{71} - 2720 \beta q^{73} + 640 \beta q^{77} - 9264 q^{79} - 2140 \beta q^{83} - 24320 q^{89} - 46848 q^{91} - 8752 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 160 q^{11} - 24 q^{19} + 9120 q^{29} - 688 q^{31} - 28480 q^{41} - 2382 q^{49} + 76000 q^{59} - 16412 q^{61} - 96960 q^{71} - 18528 q^{79} - 48640 q^{89} - 93696 q^{91}+O(q^{100}) \) 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
4.40512
−3.40512
0 0 0 0 0 −124.964 0 0 0
1.2 0 0 0 0 0 124.964 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
5.b even 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.o 2
3.b odd 2 1 900.6.a.p 2
5.b even 2 1 inner 900.6.a.o 2
5.c odd 4 2 180.6.d.a 2
15.d odd 2 1 900.6.a.p 2
15.e even 4 2 180.6.d.c yes 2
20.e even 4 2 720.6.f.b 2
60.l odd 4 2 720.6.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.6.d.a 2 5.c odd 4 2
180.6.d.c yes 2 15.e even 4 2
720.6.f.b 2 20.e even 4 2
720.6.f.e 2 60.l odd 4 2
900.6.a.o 2 1.a even 1 1 trivial
900.6.a.o 2 5.b even 2 1 inner
900.6.a.p 2 3.b odd 2 1
900.6.a.p 2 15.d odd 2 1

Hecke kernels

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

\( T_{7}^{2} - 15616 \) Copy content Toggle raw display
\( T_{11} + 80 \) 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
$7$ \( T^{2} - 15616 \) Copy content Toggle raw display
$11$ \( (T + 80)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 140544 \) Copy content Toggle raw display
$17$ \( T^{2} - 298900 \) Copy content Toggle raw display
$19$ \( (T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 4123600 \) Copy content Toggle raw display
$29$ \( (T - 4560)^{2} \) Copy content Toggle raw display
$31$ \( (T + 344)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 19129600 \) Copy content Toggle raw display
$41$ \( (T + 14240)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 360792064 \) Copy content Toggle raw display
$47$ \( T^{2} - 601435600 \) Copy content Toggle raw display
$53$ \( T^{2} - 751526100 \) Copy content Toggle raw display
$59$ \( (T - 38000)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8206)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 175461376 \) Copy content Toggle raw display
$71$ \( (T + 48480)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1805209600 \) Copy content Toggle raw display
$79$ \( (T + 9264)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 1117422400 \) Copy content Toggle raw display
$89$ \( (T + 24320)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 18689790976 \) Copy content Toggle raw display
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