Properties

Label 1200.2.a.j
Level $1200$
Weight $2$
Character orbit 1200.a
Self dual yes
Analytic conductor $9.582$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 5 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - 5 q^{7} + q^{9} + 6 q^{11} + 3 q^{13} + 2 q^{17} - q^{19} - 5 q^{21} - 2 q^{23} + q^{27} + 6 q^{29} - 3 q^{31} + 6 q^{33} + 6 q^{37} + 3 q^{39} + 4 q^{41} + 11 q^{43} - 10 q^{47} + 18 q^{49} + 2 q^{51} + 8 q^{53} - q^{57} + 6 q^{59} + 3 q^{61} - 5 q^{63} - q^{67} - 2 q^{69} + 12 q^{71} - 10 q^{73} - 30 q^{77} + 8 q^{79} + q^{81} - 6 q^{83} + 6 q^{87} - 16 q^{89} - 15 q^{91} - 3 q^{93} + 7 q^{97} + 6 q^{99}+O(q^{100}) \) 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 1.00000 0 0 0 −5.00000 0 1.00000 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 1200.2.a.j 1
3.b odd 2 1 3600.2.a.a 1
4.b odd 2 1 600.2.a.e 1
5.b even 2 1 1200.2.a.i 1
5.c odd 4 2 1200.2.f.i 2
8.b even 2 1 4800.2.a.a 1
8.d odd 2 1 4800.2.a.ct 1
12.b even 2 1 1800.2.a.x 1
15.d odd 2 1 3600.2.a.bq 1
15.e even 4 2 3600.2.f.b 2
20.d odd 2 1 600.2.a.f yes 1
20.e even 4 2 600.2.f.a 2
40.e odd 2 1 4800.2.a.b 1
40.f even 2 1 4800.2.a.cs 1
40.i odd 4 2 4800.2.f.a 2
40.k even 4 2 4800.2.f.bj 2
60.h even 2 1 1800.2.a.a 1
60.l odd 4 2 1800.2.f.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.e 1 4.b odd 2 1
600.2.a.f yes 1 20.d odd 2 1
600.2.f.a 2 20.e even 4 2
1200.2.a.i 1 5.b even 2 1
1200.2.a.j 1 1.a even 1 1 trivial
1200.2.f.i 2 5.c odd 4 2
1800.2.a.a 1 60.h even 2 1
1800.2.a.x 1 12.b even 2 1
1800.2.f.k 2 60.l odd 4 2
3600.2.a.a 1 3.b odd 2 1
3600.2.a.bq 1 15.d odd 2 1
3600.2.f.b 2 15.e even 4 2
4800.2.a.a 1 8.b even 2 1
4800.2.a.b 1 40.e odd 2 1
4800.2.a.cs 1 40.f even 2 1
4800.2.a.ct 1 8.d odd 2 1
4800.2.f.a 2 40.i odd 4 2
4800.2.f.bj 2 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7} + 5 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 5 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T - 3 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T + 2 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 3 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T - 4 \) Copy content Toggle raw display
$43$ \( T - 11 \) Copy content Toggle raw display
$47$ \( T + 10 \) Copy content Toggle raw display
$53$ \( T - 8 \) Copy content Toggle raw display
$59$ \( T - 6 \) Copy content Toggle raw display
$61$ \( T - 3 \) Copy content Toggle raw display
$67$ \( T + 1 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 16 \) Copy content Toggle raw display
$97$ \( T - 7 \) Copy content Toggle raw display
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