Properties

Label 900.6.a.h
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 4)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 88 q^{7} + O(q^{10}) \) \( q + 88 q^{7} - 540 q^{11} + 418 q^{13} + 594 q^{17} + 836 q^{19} - 4104 q^{23} + 594 q^{29} + 4256 q^{31} + 298 q^{37} - 17226 q^{41} + 12100 q^{43} - 1296 q^{47} - 9063 q^{49} + 19494 q^{53} + 7668 q^{59} - 34738 q^{61} - 21812 q^{67} + 46872 q^{71} - 67562 q^{73} - 47520 q^{77} - 76912 q^{79} + 67716 q^{83} - 29754 q^{89} + 36784 q^{91} + 122398 q^{97} + O(q^{100}) \)

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 88.0000 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.h 1
3.b odd 2 1 100.6.a.b 1
5.b even 2 1 36.6.a.a 1
5.c odd 4 2 900.6.d.a 2
12.b even 2 1 400.6.a.d 1
15.d odd 2 1 4.6.a.a 1
15.e even 4 2 100.6.c.b 2
20.d odd 2 1 144.6.a.c 1
40.e odd 2 1 576.6.a.bd 1
40.f even 2 1 576.6.a.bc 1
45.h odd 6 2 324.6.e.a 2
45.j even 6 2 324.6.e.d 2
60.h even 2 1 16.6.a.b 1
60.l odd 4 2 400.6.c.f 2
105.g even 2 1 196.6.a.e 1
105.o odd 6 2 196.6.e.g 2
105.p even 6 2 196.6.e.d 2
120.i odd 2 1 64.6.a.f 1
120.m even 2 1 64.6.a.b 1
165.d even 2 1 484.6.a.a 1
195.e odd 2 1 676.6.a.a 1
195.n even 4 2 676.6.d.a 2
240.t even 4 2 256.6.b.c 2
240.bm odd 4 2 256.6.b.g 2
420.o odd 2 1 784.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 15.d odd 2 1
16.6.a.b 1 60.h even 2 1
36.6.a.a 1 5.b even 2 1
64.6.a.b 1 120.m even 2 1
64.6.a.f 1 120.i odd 2 1
100.6.a.b 1 3.b odd 2 1
100.6.c.b 2 15.e even 4 2
144.6.a.c 1 20.d odd 2 1
196.6.a.e 1 105.g even 2 1
196.6.e.d 2 105.p even 6 2
196.6.e.g 2 105.o odd 6 2
256.6.b.c 2 240.t even 4 2
256.6.b.g 2 240.bm odd 4 2
324.6.e.a 2 45.h odd 6 2
324.6.e.d 2 45.j even 6 2
400.6.a.d 1 12.b even 2 1
400.6.c.f 2 60.l odd 4 2
484.6.a.a 1 165.d even 2 1
576.6.a.bc 1 40.f even 2 1
576.6.a.bd 1 40.e odd 2 1
676.6.a.a 1 195.e odd 2 1
676.6.d.a 2 195.n even 4 2
784.6.a.d 1 420.o odd 2 1
900.6.a.h 1 1.a even 1 1 trivial
900.6.d.a 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(900))\):

\( T_{7} - 88 \)
\( T_{11} + 540 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -88 + T \)
$11$ \( 540 + T \)
$13$ \( -418 + T \)
$17$ \( -594 + T \)
$19$ \( -836 + T \)
$23$ \( 4104 + T \)
$29$ \( -594 + T \)
$31$ \( -4256 + T \)
$37$ \( -298 + T \)
$41$ \( 17226 + T \)
$43$ \( -12100 + T \)
$47$ \( 1296 + T \)
$53$ \( -19494 + T \)
$59$ \( -7668 + T \)
$61$ \( 34738 + T \)
$67$ \( 21812 + T \)
$71$ \( -46872 + T \)
$73$ \( 67562 + T \)
$79$ \( 76912 + T \)
$83$ \( -67716 + T \)
$89$ \( 29754 + T \)
$97$ \( -122398 + T \)
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