Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 3 - 6 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 3 - 6 \zeta_{6} ) q^{7} -3 q^{9} + 7 q^{13} + ( 3 - 6 \zeta_{6} ) q^{19} -9 q^{21} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -1 + 2 \zeta_{6} ) q^{31} -10 q^{37} + ( 7 - 14 \zeta_{6} ) q^{39} + ( -1 + 2 \zeta_{6} ) q^{43} -20 q^{49} -9 q^{57} - q^{61} + ( -9 + 18 \zeta_{6} ) q^{63} + ( 7 - 14 \zeta_{6} ) q^{67} + 10 q^{73} + ( -10 + 20 \zeta_{6} ) q^{79} + 9 q^{81} + ( 21 - 42 \zeta_{6} ) q^{91} + 3 q^{93} + 19 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{9} + O(q^{10}) \) \( 2q - 6q^{9} + 14q^{13} - 18q^{21} - 20q^{37} - 40q^{49} - 18q^{57} - 2q^{61} + 20q^{73} + 18q^{81} + 6q^{93} + 38q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 0 0 5.19615i 0 −3.00000 0
1151.2 0 1.73205i 0 0 0 5.19615i 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
12.b 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.h.g yes 2
3.b odd 2 1 CM 1200.2.h.g yes 2
4.b odd 2 1 inner 1200.2.h.g yes 2
5.b even 2 1 1200.2.h.c 2
5.c odd 4 2 1200.2.o.g 4
12.b even 2 1 inner 1200.2.h.g yes 2
15.d odd 2 1 1200.2.h.c 2
15.e even 4 2 1200.2.o.g 4
20.d odd 2 1 1200.2.h.c 2
20.e even 4 2 1200.2.o.g 4
60.h even 2 1 1200.2.h.c 2
60.l odd 4 2 1200.2.o.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.2.h.c 2 5.b even 2 1
1200.2.h.c 2 15.d odd 2 1
1200.2.h.c 2 20.d odd 2 1
1200.2.h.c 2 60.h even 2 1
1200.2.h.g yes 2 1.a even 1 1 trivial
1200.2.h.g yes 2 3.b odd 2 1 CM
1200.2.h.g yes 2 4.b odd 2 1 inner
1200.2.h.g yes 2 12.b even 2 1 inner
1200.2.o.g 4 5.c odd 4 2
1200.2.o.g 4 15.e even 4 2
1200.2.o.g 4 20.e even 4 2
1200.2.o.g 4 60.l odd 4 2

Hecke kernels

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

\( T_{7}^{2} + 27 \)
\( T_{11} \)
\( T_{13} - 7 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 27 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -7 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 27 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 3 + T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 3 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 1 + T )^{2} \)
$67$ \( 147 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( 300 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -19 + T )^{2} \)
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