Properties

Label 1200.2.h.e
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: no (minimal twist has level 48)
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.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 2 - 4 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 2 - 4 \zeta_{6} ) q^{7} -3 q^{9} + 2 q^{13} + ( 2 - 4 \zeta_{6} ) q^{19} + 6 q^{21} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 6 - 12 \zeta_{6} ) q^{31} + 10 q^{37} + ( -2 + 4 \zeta_{6} ) q^{39} + ( 6 - 12 \zeta_{6} ) q^{43} -5 q^{49} + 6 q^{57} + 14 q^{61} + ( -6 + 12 \zeta_{6} ) q^{63} + ( -2 + 4 \zeta_{6} ) q^{67} -10 q^{73} + ( -10 + 20 \zeta_{6} ) q^{79} + 9 q^{81} + ( 4 - 8 \zeta_{6} ) q^{91} + 18 q^{93} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9} + O(q^{10}) \) \( 2 q - 6 q^{9} + 4 q^{13} + 12 q^{21} + 20 q^{37} - 10 q^{49} + 12 q^{57} + 28 q^{61} - 20 q^{73} + 18 q^{81} + 36 q^{93} + 28 q^{97} + O(q^{100}) \)

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 3.46410i 0 −3.00000 0
1151.2 0 1.73205i 0 0 0 3.46410i 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.e 2
3.b odd 2 1 CM 1200.2.h.e 2
4.b odd 2 1 inner 1200.2.h.e 2
5.b even 2 1 48.2.c.a 2
5.c odd 4 2 1200.2.o.i 4
12.b even 2 1 inner 1200.2.h.e 2
15.d odd 2 1 48.2.c.a 2
15.e even 4 2 1200.2.o.i 4
20.d odd 2 1 48.2.c.a 2
20.e even 4 2 1200.2.o.i 4
35.c odd 2 1 2352.2.h.c 2
40.e odd 2 1 192.2.c.a 2
40.f even 2 1 192.2.c.a 2
45.h odd 6 1 1296.2.s.b 2
45.h odd 6 1 1296.2.s.e 2
45.j even 6 1 1296.2.s.b 2
45.j even 6 1 1296.2.s.e 2
60.h even 2 1 48.2.c.a 2
60.l odd 4 2 1200.2.o.i 4
80.k odd 4 2 768.2.f.d 4
80.q even 4 2 768.2.f.d 4
105.g even 2 1 2352.2.h.c 2
120.i odd 2 1 192.2.c.a 2
120.m even 2 1 192.2.c.a 2
140.c even 2 1 2352.2.h.c 2
180.n even 6 1 1296.2.s.b 2
180.n even 6 1 1296.2.s.e 2
180.p odd 6 1 1296.2.s.b 2
180.p odd 6 1 1296.2.s.e 2
240.t even 4 2 768.2.f.d 4
240.bm odd 4 2 768.2.f.d 4
420.o odd 2 1 2352.2.h.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.c.a 2 5.b even 2 1
48.2.c.a 2 15.d odd 2 1
48.2.c.a 2 20.d odd 2 1
48.2.c.a 2 60.h even 2 1
192.2.c.a 2 40.e odd 2 1
192.2.c.a 2 40.f even 2 1
192.2.c.a 2 120.i odd 2 1
192.2.c.a 2 120.m even 2 1
768.2.f.d 4 80.k odd 4 2
768.2.f.d 4 80.q even 4 2
768.2.f.d 4 240.t even 4 2
768.2.f.d 4 240.bm odd 4 2
1200.2.h.e 2 1.a even 1 1 trivial
1200.2.h.e 2 3.b odd 2 1 CM
1200.2.h.e 2 4.b odd 2 1 inner
1200.2.h.e 2 12.b even 2 1 inner
1200.2.o.i 4 5.c odd 4 2
1200.2.o.i 4 15.e even 4 2
1200.2.o.i 4 20.e even 4 2
1200.2.o.i 4 60.l odd 4 2
1296.2.s.b 2 45.h odd 6 1
1296.2.s.b 2 45.j even 6 1
1296.2.s.b 2 180.n even 6 1
1296.2.s.b 2 180.p odd 6 1
1296.2.s.e 2 45.h odd 6 1
1296.2.s.e 2 45.j even 6 1
1296.2.s.e 2 180.n even 6 1
1296.2.s.e 2 180.p odd 6 1
2352.2.h.c 2 35.c odd 2 1
2352.2.h.c 2 105.g even 2 1
2352.2.h.c 2 140.c even 2 1
2352.2.h.c 2 420.o odd 2 1

Hecke kernels

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

\( T_{7}^{2} + 12 \)
\( T_{11} \)
\( T_{13} - 2 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 12 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -2 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 12 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 108 + T^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 108 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -14 + T )^{2} \)
$67$ \( 12 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( 300 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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