Properties

Label 3600.2.a.z
Level $3600$
Weight $2$
Character orbit 3600.a
Self dual yes
Analytic conductor $28.746$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.7461447277\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{7} + O(q^{10}) \) \( q + q^{7} + 6q^{11} + 5q^{13} + 6q^{17} - 5q^{19} - 6q^{23} + 6q^{29} + q^{31} + 2q^{37} + q^{43} + 6q^{47} - 6q^{49} + 12q^{53} - 6q^{59} - 13q^{61} - 11q^{67} + 2q^{73} + 6q^{77} - 8q^{79} - 6q^{83} + 5q^{91} - 7q^{97} + O(q^{100}) \)

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 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3600.2.a.z 1
3.b odd 2 1 1200.2.a.f 1
4.b odd 2 1 900.2.a.c 1
5.b even 2 1 3600.2.a.s 1
5.c odd 4 2 3600.2.f.v 2
12.b even 2 1 300.2.a.c yes 1
15.d odd 2 1 1200.2.a.n 1
15.e even 4 2 1200.2.f.a 2
20.d odd 2 1 900.2.a.e 1
20.e even 4 2 900.2.d.a 2
24.f even 2 1 4800.2.a.o 1
24.h odd 2 1 4800.2.a.cf 1
60.h even 2 1 300.2.a.b 1
60.l odd 4 2 300.2.d.a 2
120.i odd 2 1 4800.2.a.p 1
120.m even 2 1 4800.2.a.ce 1
120.q odd 4 2 4800.2.f.b 2
120.w even 4 2 4800.2.f.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.a.b 1 60.h even 2 1
300.2.a.c yes 1 12.b even 2 1
300.2.d.a 2 60.l odd 4 2
900.2.a.c 1 4.b odd 2 1
900.2.a.e 1 20.d odd 2 1
900.2.d.a 2 20.e even 4 2
1200.2.a.f 1 3.b odd 2 1
1200.2.a.n 1 15.d odd 2 1
1200.2.f.a 2 15.e even 4 2
3600.2.a.s 1 5.b even 2 1
3600.2.a.z 1 1.a even 1 1 trivial
3600.2.f.v 2 5.c odd 4 2
4800.2.a.o 1 24.f even 2 1
4800.2.a.p 1 120.i odd 2 1
4800.2.a.ce 1 120.m even 2 1
4800.2.a.cf 1 24.h odd 2 1
4800.2.f.b 2 120.q odd 4 2
4800.2.f.bi 2 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3600))\):

\( T_{7} - 1 \)
\( T_{11} - 6 \)
\( T_{13} - 5 \)
\( T_{17} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( -6 + T \)
$13$ \( -5 + T \)
$17$ \( -6 + T \)
$19$ \( 5 + T \)
$23$ \( 6 + T \)
$29$ \( -6 + T \)
$31$ \( -1 + T \)
$37$ \( -2 + T \)
$41$ \( T \)
$43$ \( -1 + T \)
$47$ \( -6 + T \)
$53$ \( -12 + T \)
$59$ \( 6 + T \)
$61$ \( 13 + T \)
$67$ \( 11 + T \)
$71$ \( T \)
$73$ \( -2 + T \)
$79$ \( 8 + T \)
$83$ \( 6 + T \)
$89$ \( T \)
$97$ \( 7 + T \)
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