Properties

Label 1200.2.h.h
Level $1200$
Weight $2$
Character orbit 1200.h
Analytic conductor $9.582$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{6} ) q^{3} + 3 \zeta_{6} q^{9} -3 q^{11} -2 q^{13} + ( -3 + 6 \zeta_{6} ) q^{17} + ( -3 + 6 \zeta_{6} ) q^{19} + 6 q^{23} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -6 + 12 \zeta_{6} ) q^{29} + ( 2 - 4 \zeta_{6} ) q^{31} + ( -3 - 3 \zeta_{6} ) q^{33} + 8 q^{37} + ( -2 - 2 \zeta_{6} ) q^{39} + ( -3 + 6 \zeta_{6} ) q^{41} + ( 2 - 4 \zeta_{6} ) q^{43} -6 q^{47} + 7 q^{49} + ( -9 + 9 \zeta_{6} ) q^{51} + ( 6 - 12 \zeta_{6} ) q^{53} + ( -9 + 9 \zeta_{6} ) q^{57} -12 q^{59} + 8 q^{61} + ( 7 - 14 \zeta_{6} ) q^{67} + ( 6 + 6 \zeta_{6} ) q^{69} + 6 q^{71} + q^{73} + ( -4 + 8 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -9 q^{83} + ( -18 + 18 \zeta_{6} ) q^{87} + ( -3 + 6 \zeta_{6} ) q^{89} + ( 6 - 6 \zeta_{6} ) q^{93} + 10 q^{97} -9 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 3q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 3q^{9} - 6q^{11} - 4q^{13} + 12q^{23} - 9q^{33} + 16q^{37} - 6q^{39} - 12q^{47} + 14q^{49} - 9q^{51} - 9q^{57} - 24q^{59} + 16q^{61} + 18q^{69} + 12q^{71} + 2q^{73} - 9q^{81} - 18q^{83} - 18q^{87} + 6q^{93} + 20q^{97} - 9q^{99} + O(q^{100}) \)

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 1.50000 0.866025i 0 0 0 0 0 1.50000 2.59808i 0
1151.2 0 1.50000 + 0.866025i 0 0 0 0 0 1.50000 + 2.59808i 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
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.h yes 2
3.b odd 2 1 1200.2.h.b yes 2
4.b odd 2 1 1200.2.h.b yes 2
5.b even 2 1 1200.2.h.a 2
5.c odd 4 2 1200.2.o.c 4
12.b even 2 1 inner 1200.2.h.h yes 2
15.d odd 2 1 1200.2.h.i yes 2
15.e even 4 2 1200.2.o.d 4
20.d odd 2 1 1200.2.h.i yes 2
20.e even 4 2 1200.2.o.d 4
60.h even 2 1 1200.2.h.a 2
60.l odd 4 2 1200.2.o.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.2.h.a 2 5.b even 2 1
1200.2.h.a 2 60.h even 2 1
1200.2.h.b yes 2 3.b odd 2 1
1200.2.h.b yes 2 4.b odd 2 1
1200.2.h.h yes 2 1.a even 1 1 trivial
1200.2.h.h yes 2 12.b even 2 1 inner
1200.2.h.i yes 2 15.d odd 2 1
1200.2.h.i yes 2 20.d odd 2 1
1200.2.o.c 4 5.c odd 4 2
1200.2.o.c 4 60.l odd 4 2
1200.2.o.d 4 15.e even 4 2
1200.2.o.d 4 20.e even 4 2

Hecke kernels

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

\( T_{7} \)
\( T_{11} + 3 \)
\( T_{13} + 2 \)
\( T_{23} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 - 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( 27 + T^{2} \)
$19$ \( 27 + T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( 108 + T^{2} \)
$31$ \( 12 + T^{2} \)
$37$ \( ( -8 + T )^{2} \)
$41$ \( 27 + T^{2} \)
$43$ \( 12 + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( 108 + T^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( 147 + T^{2} \)
$71$ \( ( -6 + T )^{2} \)
$73$ \( ( -1 + T )^{2} \)
$79$ \( 48 + T^{2} \)
$83$ \( ( 9 + T )^{2} \)
$89$ \( 27 + T^{2} \)
$97$ \( ( -10 + T )^{2} \)
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