Properties

Label 1200.2.a.r
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 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 4q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + 4q^{7} + q^{9} + 6q^{13} + 2q^{17} - 4q^{19} + 4q^{21} - 8q^{23} + q^{27} - 6q^{29} + 6q^{37} + 6q^{39} + 10q^{41} - 4q^{43} + 8q^{47} + 9q^{49} + 2q^{51} - 10q^{53} - 4q^{57} + 6q^{61} + 4q^{63} - 4q^{67} - 8q^{69} + 14q^{73} - 16q^{79} + q^{81} + 12q^{83} - 6q^{87} + 2q^{89} + 24q^{91} - 2q^{97} + O(q^{100}) \)

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 1.00000 0 0 0 4.00000 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.r 1
3.b odd 2 1 3600.2.a.bo 1
4.b odd 2 1 600.2.a.a 1
5.b even 2 1 240.2.a.a 1
5.c odd 4 2 1200.2.f.f 2
8.b even 2 1 4800.2.a.bh 1
8.d odd 2 1 4800.2.a.bl 1
12.b even 2 1 1800.2.a.c 1
15.d odd 2 1 720.2.a.f 1
15.e even 4 2 3600.2.f.l 2
20.d odd 2 1 120.2.a.a 1
20.e even 4 2 600.2.f.c 2
40.e odd 2 1 960.2.a.g 1
40.f even 2 1 960.2.a.n 1
40.i odd 4 2 4800.2.f.n 2
40.k even 4 2 4800.2.f.u 2
60.h even 2 1 360.2.a.e 1
60.l odd 4 2 1800.2.f.g 2
80.k odd 4 2 3840.2.k.a 2
80.q even 4 2 3840.2.k.z 2
120.i odd 2 1 2880.2.a.b 1
120.m even 2 1 2880.2.a.r 1
140.c even 2 1 5880.2.a.p 1
180.n even 6 2 3240.2.q.a 2
180.p odd 6 2 3240.2.q.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 20.d odd 2 1
240.2.a.a 1 5.b even 2 1
360.2.a.e 1 60.h even 2 1
600.2.a.a 1 4.b odd 2 1
600.2.f.c 2 20.e even 4 2
720.2.a.f 1 15.d odd 2 1
960.2.a.g 1 40.e odd 2 1
960.2.a.n 1 40.f even 2 1
1200.2.a.r 1 1.a even 1 1 trivial
1200.2.f.f 2 5.c odd 4 2
1800.2.a.c 1 12.b even 2 1
1800.2.f.g 2 60.l odd 4 2
2880.2.a.b 1 120.i odd 2 1
2880.2.a.r 1 120.m even 2 1
3240.2.q.a 2 180.n even 6 2
3240.2.q.m 2 180.p odd 6 2
3600.2.a.bo 1 3.b odd 2 1
3600.2.f.l 2 15.e even 4 2
3840.2.k.a 2 80.k odd 4 2
3840.2.k.z 2 80.q even 4 2
4800.2.a.bh 1 8.b even 2 1
4800.2.a.bl 1 8.d odd 2 1
4800.2.f.n 2 40.i odd 4 2
4800.2.f.u 2 40.k even 4 2
5880.2.a.p 1 140.c even 2 1

Hecke kernels

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

\( T_{7} - 4 \)
\( T_{11} \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T \)
$5$ 1
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 - 6 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 8 T + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 - 10 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 - 8 T + 47 T^{2} \)
$53$ \( 1 + 10 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 - 6 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 14 T + 73 T^{2} \)
$79$ \( 1 + 16 T + 79 T^{2} \)
$83$ \( 1 - 12 T + 83 T^{2} \)
$89$ \( 1 - 2 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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