Properties

Label 1200.1.c.a
Level $1200$
Weight $1$
Character orbit 1200.c
Analytic conductor $0.599$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.300.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{3} - i q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} - i q^{7} - q^{9} - i q^{13} - q^{19} - q^{21} + i q^{27} + q^{31} - i q^{37} - q^{39} + i q^{43} + i q^{57} - q^{61} + i q^{63} - i q^{67} + i q^{73} + q^{79} + q^{81} - q^{91} - i q^{93} + i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} - 2 q^{19} - 2 q^{21} + 2 q^{31} - 2 q^{39} - 2 q^{61} + 4 q^{79} + 2 q^{81} - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000i 0 −1.00000 0
449.2 0 1.00000i 0 0 0 1.00000i 0 −1.00000 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 \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.1.c.a 2
3.b odd 2 1 CM 1200.1.c.a 2
4.b odd 2 1 300.1.b.a 2
5.b even 2 1 inner 1200.1.c.a 2
5.c odd 4 1 1200.1.l.a 1
5.c odd 4 1 1200.1.l.b 1
12.b even 2 1 300.1.b.a 2
15.d odd 2 1 inner 1200.1.c.a 2
15.e even 4 1 1200.1.l.a 1
15.e even 4 1 1200.1.l.b 1
20.d odd 2 1 300.1.b.a 2
20.e even 4 1 300.1.g.a 1
20.e even 4 1 300.1.g.b yes 1
60.h even 2 1 300.1.b.a 2
60.l odd 4 1 300.1.g.a 1
60.l odd 4 1 300.1.g.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.1.b.a 2 4.b odd 2 1
300.1.b.a 2 12.b even 2 1
300.1.b.a 2 20.d odd 2 1
300.1.b.a 2 60.h even 2 1
300.1.g.a 1 20.e even 4 1
300.1.g.a 1 60.l odd 4 1
300.1.g.b yes 1 20.e even 4 1
300.1.g.b yes 1 60.l odd 4 1
1200.1.c.a 2 1.a even 1 1 trivial
1200.1.c.a 2 3.b odd 2 1 CM
1200.1.c.a 2 5.b even 2 1 inner
1200.1.c.a 2 15.d odd 2 1 inner
1200.1.l.a 1 5.c odd 4 1
1200.1.l.a 1 15.e even 4 1
1200.1.l.b 1 5.c odd 4 1
1200.1.l.b 1 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1200, [\chi])\).

Hecke characteristic polynomials

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