Properties

Label 1200.1.z.b
Level $1200$
Weight $1$
Character orbit 1200.z
Analytic conductor $0.599$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -15
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.z (of order \(4\), degree \(2\), 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: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.153600.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{2} - i q^{3} + q^{4} - i q^{6} + q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - i q^{3} + q^{4} - i q^{6} + q^{8} - q^{9} - i q^{12} + q^{16} + ( - i + 1) q^{17} - q^{18} + (i - 1) q^{19} + ( - i - 1) q^{23} - i q^{24} + i q^{27} + i q^{31} + q^{32} + ( - i + 1) q^{34} - q^{36} + (i - 1) q^{38} + ( - i - 1) q^{46} + ( - i - 1) q^{47} - i q^{48} - i q^{49} + ( - i - 1) q^{51} + i q^{53} + i q^{54} + (i + 1) q^{57} + (i - 1) q^{61} + 2 i q^{62} + q^{64} + ( - i + 1) q^{68} + (i - 1) q^{69} - q^{72} + (i - 1) q^{76} + q^{81} + ( - i - 1) q^{92} + 2 q^{93} + ( - i - 1) q^{94} - i q^{96} - i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 2 q^{19} - 2 q^{23} + 2 q^{32} + 2 q^{34} - 2 q^{36} - 2 q^{38} - 2 q^{46} - 2 q^{47} - 2 q^{51} + 2 q^{57} - 2 q^{61} + 2 q^{64} + 2 q^{68} - 2 q^{69} - 2 q^{72} - 2 q^{76} + 2 q^{81} - 2 q^{92} + 4 q^{93} - 2 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000 0 1.00000i 0 1.00000 −1.00000 0
1043.1 1.00000 1.00000i 1.00000 0 1.00000i 0 1.00000 −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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
80.j even 4 1 inner
240.z odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.1.z.b yes 2
3.b odd 2 1 1200.1.z.a 2
5.b even 2 1 1200.1.z.a 2
5.c odd 4 1 1200.1.bd.a yes 2
5.c odd 4 1 1200.1.bd.b yes 2
15.d odd 2 1 CM 1200.1.z.b yes 2
15.e even 4 1 1200.1.bd.a yes 2
15.e even 4 1 1200.1.bd.b yes 2
16.f odd 4 1 1200.1.bd.a yes 2
48.k even 4 1 1200.1.bd.b yes 2
80.j even 4 1 inner 1200.1.z.b yes 2
80.k odd 4 1 1200.1.bd.b yes 2
80.s even 4 1 1200.1.z.a 2
240.t even 4 1 1200.1.bd.a yes 2
240.z odd 4 1 inner 1200.1.z.b yes 2
240.bd odd 4 1 1200.1.z.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.1.z.a 2 3.b odd 2 1
1200.1.z.a 2 5.b even 2 1
1200.1.z.a 2 80.s even 4 1
1200.1.z.a 2 240.bd odd 4 1
1200.1.z.b yes 2 1.a even 1 1 trivial
1200.1.z.b yes 2 15.d odd 2 1 CM
1200.1.z.b yes 2 80.j even 4 1 inner
1200.1.z.b yes 2 240.z odd 4 1 inner
1200.1.bd.a yes 2 5.c odd 4 1
1200.1.bd.a yes 2 15.e even 4 1
1200.1.bd.a yes 2 16.f odd 4 1
1200.1.bd.a yes 2 240.t even 4 1
1200.1.bd.b yes 2 5.c odd 4 1
1200.1.bd.b yes 2 15.e even 4 1
1200.1.bd.b yes 2 48.k even 4 1
1200.1.bd.b yes 2 80.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} - 2T_{17} + 2 \) acting on \(S_{1}^{\mathrm{new}}(1200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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