Properties

Label 9900.2.a.h
Level $9900$
Weight $2$
Character orbit 9900.a
Self dual yes
Analytic conductor $79.052$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9900.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.0518980011\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{7} + q^{11} + 4 q^{13} + 6 q^{17} + 8 q^{19} - 3 q^{23} + 5 q^{31} + q^{37} + 10 q^{43} - 3 q^{49} - 6 q^{53} - 3 q^{59} - 4 q^{61} + q^{67} - 15 q^{71} + 4 q^{73} - 2 q^{77} + 2 q^{79} + 6 q^{83} + 9 q^{89} - 8 q^{91} + 7 q^{97}+O(q^{100}) \) 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 −2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)

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 9900.2.a.h 1
3.b odd 2 1 1100.2.a.b 1
5.b even 2 1 396.2.a.c 1
5.c odd 4 2 9900.2.c.g 2
12.b even 2 1 4400.2.a.v 1
15.d odd 2 1 44.2.a.a 1
15.e even 4 2 1100.2.b.c 2
20.d odd 2 1 1584.2.a.p 1
40.e odd 2 1 6336.2.a.i 1
40.f even 2 1 6336.2.a.j 1
45.h odd 6 2 3564.2.i.j 2
45.j even 6 2 3564.2.i.a 2
55.d odd 2 1 4356.2.a.j 1
60.h even 2 1 176.2.a.a 1
60.l odd 4 2 4400.2.b.k 2
105.g even 2 1 2156.2.a.a 1
105.o odd 6 2 2156.2.i.b 2
105.p even 6 2 2156.2.i.c 2
120.i odd 2 1 704.2.a.f 1
120.m even 2 1 704.2.a.i 1
165.d even 2 1 484.2.a.a 1
165.o odd 10 4 484.2.e.a 4
165.r even 10 4 484.2.e.b 4
195.e odd 2 1 7436.2.a.d 1
240.t even 4 2 2816.2.c.k 2
240.bm odd 4 2 2816.2.c.e 2
420.o odd 2 1 8624.2.a.w 1
660.g odd 2 1 1936.2.a.c 1
1320.b odd 2 1 7744.2.a.bc 1
1320.u even 2 1 7744.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 15.d odd 2 1
176.2.a.a 1 60.h even 2 1
396.2.a.c 1 5.b even 2 1
484.2.a.a 1 165.d even 2 1
484.2.e.a 4 165.o odd 10 4
484.2.e.b 4 165.r even 10 4
704.2.a.f 1 120.i odd 2 1
704.2.a.i 1 120.m even 2 1
1100.2.a.b 1 3.b odd 2 1
1100.2.b.c 2 15.e even 4 2
1584.2.a.p 1 20.d odd 2 1
1936.2.a.c 1 660.g odd 2 1
2156.2.a.a 1 105.g even 2 1
2156.2.i.b 2 105.o odd 6 2
2156.2.i.c 2 105.p even 6 2
2816.2.c.e 2 240.bm odd 4 2
2816.2.c.k 2 240.t even 4 2
3564.2.i.a 2 45.j even 6 2
3564.2.i.j 2 45.h odd 6 2
4356.2.a.j 1 55.d odd 2 1
4400.2.a.v 1 12.b even 2 1
4400.2.b.k 2 60.l odd 4 2
6336.2.a.i 1 40.e odd 2 1
6336.2.a.j 1 40.f even 2 1
7436.2.a.d 1 195.e odd 2 1
7744.2.a.m 1 1320.u even 2 1
7744.2.a.bc 1 1320.b odd 2 1
8624.2.a.w 1 420.o odd 2 1
9900.2.a.h 1 1.a even 1 1 trivial
9900.2.c.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9900))\):

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17} - 6 \) 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
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 8 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 5 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 3 \) Copy content Toggle raw display
$61$ \( T + 4 \) Copy content Toggle raw display
$67$ \( T - 1 \) Copy content Toggle raw display
$71$ \( T + 15 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T - 2 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T - 9 \) Copy content Toggle raw display
$97$ \( T - 7 \) Copy content Toggle raw display
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