Properties

Label 1200.2.o.h
Level $1200$
Weight $2$
Character orbit 1200.o
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_1 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_1 q^{7} + 3 q^{9} - \beta_{3} q^{13} - \beta_{2} q^{19} + 3 q^{21} + 3 \beta_1 q^{27} + \beta_{2} q^{31} + 2 \beta_{3} q^{37} - \beta_{2} q^{39} + 7 \beta_1 q^{43} - 4 q^{49} - 3 \beta_{3} q^{57} - 13 q^{61} + 3 \beta_1 q^{63} + 9 \beta_1 q^{67} + 2 \beta_{3} q^{73} - 2 \beta_{2} q^{79} + 9 q^{81} - \beta_{2} q^{91} + 3 \beta_{3} q^{93} + \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} + 12 q^{21} - 16 q^{49} - 52 q^{61} + 36 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 10\zeta_{12}^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 5\beta_1 ) / 10 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 5 ) / 10 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 5 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1199.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 −1.73205 0 0 0 −1.73205 0 3.00000 0
1199.2 0 −1.73205 0 0 0 −1.73205 0 3.00000 0
1199.3 0 1.73205 0 0 0 1.73205 0 3.00000 0
1199.4 0 1.73205 0 0 0 1.73205 0 3.00000 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 \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.o.h 4
3.b odd 2 1 CM 1200.2.o.h 4
4.b odd 2 1 inner 1200.2.o.h 4
5.b even 2 1 inner 1200.2.o.h 4
5.c odd 4 1 1200.2.h.d 2
5.c odd 4 1 1200.2.h.f yes 2
12.b even 2 1 inner 1200.2.o.h 4
15.d odd 2 1 inner 1200.2.o.h 4
15.e even 4 1 1200.2.h.d 2
15.e even 4 1 1200.2.h.f yes 2
20.d odd 2 1 inner 1200.2.o.h 4
20.e even 4 1 1200.2.h.d 2
20.e even 4 1 1200.2.h.f yes 2
60.h even 2 1 inner 1200.2.o.h 4
60.l odd 4 1 1200.2.h.d 2
60.l odd 4 1 1200.2.h.f yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.2.h.d 2 5.c odd 4 1
1200.2.h.d 2 15.e even 4 1
1200.2.h.d 2 20.e even 4 1
1200.2.h.d 2 60.l odd 4 1
1200.2.h.f yes 2 5.c odd 4 1
1200.2.h.f yes 2 15.e even 4 1
1200.2.h.f yes 2 20.e even 4 1
1200.2.h.f yes 2 60.l odd 4 1
1200.2.o.h 4 1.a even 1 1 trivial
1200.2.o.h 4 3.b odd 2 1 CM
1200.2.o.h 4 4.b odd 2 1 inner
1200.2.o.h 4 5.b even 2 1 inner
1200.2.o.h 4 12.b even 2 1 inner
1200.2.o.h 4 15.d odd 2 1 inner
1200.2.o.h 4 20.d odd 2 1 inner
1200.2.o.h 4 60.h even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} - 3 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T + 13)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 243)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
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