Properties

Label 1200.3.l.d
Level $1200$
Weight $3$
Character orbit 1200.l
Self dual yes
Analytic conductor $32.698$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 11q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 11q^{7} + 9q^{9} - q^{13} + 37q^{19} - 33q^{21} + 27q^{27} + 13q^{31} + 26q^{37} - 3q^{39} + 61q^{43} + 72q^{49} + 111q^{57} + 47q^{61} - 99q^{63} + 109q^{67} - 46q^{73} + 142q^{79} + 81q^{81} + 11q^{91} + 39q^{93} - 169q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
0
0 3.00000 0 0 0 −11.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.l.d 1
3.b odd 2 1 CM 1200.3.l.d 1
4.b odd 2 1 75.3.c.a 1
5.b even 2 1 1200.3.l.c 1
5.c odd 4 2 1200.3.c.a 2
12.b even 2 1 75.3.c.a 1
15.d odd 2 1 1200.3.l.c 1
15.e even 4 2 1200.3.c.a 2
20.d odd 2 1 75.3.c.b yes 1
20.e even 4 2 75.3.d.a 2
60.h even 2 1 75.3.c.b yes 1
60.l odd 4 2 75.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.a 1 4.b odd 2 1
75.3.c.a 1 12.b even 2 1
75.3.c.b yes 1 20.d odd 2 1
75.3.c.b yes 1 60.h even 2 1
75.3.d.a 2 20.e even 4 2
75.3.d.a 2 60.l odd 4 2
1200.3.c.a 2 5.c odd 4 2
1200.3.c.a 2 15.e even 4 2
1200.3.l.c 1 5.b even 2 1
1200.3.l.c 1 15.d odd 2 1
1200.3.l.d 1 1.a even 1 1 trivial
1200.3.l.d 1 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7} + 11 \)
\( T_{11} \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( T \)
$7$ \( 11 + T \)
$11$ \( T \)
$13$ \( 1 + T \)
$17$ \( T \)
$19$ \( -37 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( -13 + T \)
$37$ \( -26 + T \)
$41$ \( T \)
$43$ \( -61 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( -47 + T \)
$67$ \( -109 + T \)
$71$ \( T \)
$73$ \( 46 + T \)
$79$ \( -142 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( 169 + T \)
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