Properties

Label 1200.2.h.j
Level $1200$
Weight $2$
Character orbit 1200.h
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -1 - 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( -1 - 2 \beta_{1} ) q^{9} + \beta_{3} q^{11} -\beta_{3} q^{13} + \beta_{2} q^{17} + \beta_{2} q^{19} + 6 q^{23} + ( 5 + \beta_{1} ) q^{27} + 2 \beta_{1} q^{29} + \beta_{2} q^{31} + ( 2 \beta_{2} - \beta_{3} ) q^{33} -\beta_{3} q^{37} + ( -2 \beta_{2} + \beta_{3} ) q^{39} -4 \beta_{1} q^{41} + 6 \beta_{1} q^{43} -6 q^{47} + 7 q^{49} + ( -\beta_{2} - \beta_{3} ) q^{51} + 3 \beta_{2} q^{53} + ( -\beta_{2} - \beta_{3} ) q^{57} -\beta_{3} q^{59} -2 q^{61} + 6 \beta_{1} q^{67} + ( -6 + 6 \beta_{1} ) q^{69} -2 \beta_{3} q^{71} -2 \beta_{3} q^{73} + 3 \beta_{2} q^{79} + ( -7 + 4 \beta_{1} ) q^{81} + 6 q^{83} + ( -4 - 2 \beta_{1} ) q^{87} -4 \beta_{1} q^{89} + ( -\beta_{2} - \beta_{3} ) q^{93} + ( -4 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{3} - 4 q^{9} + 24 q^{23} + 20 q^{27} - 24 q^{47} + 28 q^{49} - 8 q^{61} - 24 q^{69} - 28 q^{81} + 24 q^{83} - 16 q^{87} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

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
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
0 −1.00000 1.41421i 0 0 0 0 0 −1.00000 + 2.82843i 0
1151.2 0 −1.00000 1.41421i 0 0 0 0 0 −1.00000 + 2.82843i 0
1151.3 0 −1.00000 + 1.41421i 0 0 0 0 0 −1.00000 2.82843i 0
1151.4 0 −1.00000 + 1.41421i 0 0 0 0 0 −1.00000 2.82843i 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
15.d odd 2 1 inner
20.d odd 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.j 4
3.b odd 2 1 1200.2.h.n 4
4.b odd 2 1 1200.2.h.n 4
5.b even 2 1 1200.2.h.n 4
5.c odd 4 2 240.2.o.b 8
12.b even 2 1 inner 1200.2.h.j 4
15.d odd 2 1 inner 1200.2.h.j 4
15.e even 4 2 240.2.o.b 8
20.d odd 2 1 inner 1200.2.h.j 4
20.e even 4 2 240.2.o.b 8
40.i odd 4 2 960.2.o.d 8
40.k even 4 2 960.2.o.d 8
60.h even 2 1 1200.2.h.n 4
60.l odd 4 2 240.2.o.b 8
120.q odd 4 2 960.2.o.d 8
120.w even 4 2 960.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.o.b 8 5.c odd 4 2
240.2.o.b 8 15.e even 4 2
240.2.o.b 8 20.e even 4 2
240.2.o.b 8 60.l odd 4 2
960.2.o.d 8 40.i odd 4 2
960.2.o.d 8 40.k even 4 2
960.2.o.d 8 120.q odd 4 2
960.2.o.d 8 120.w even 4 2
1200.2.h.j 4 1.a even 1 1 trivial
1200.2.h.j 4 12.b even 2 1 inner
1200.2.h.j 4 15.d odd 2 1 inner
1200.2.h.j 4 20.d odd 2 1 inner
1200.2.h.n 4 3.b odd 2 1
1200.2.h.n 4 4.b odd 2 1
1200.2.h.n 4 5.b even 2 1
1200.2.h.n 4 60.h even 2 1

Hecke kernels

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

\( T_{7} \)
\( T_{11}^{2} - 24 \)
\( T_{13}^{2} - 24 \)
\( T_{23} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + 2 T + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -24 + T^{2} )^{2} \)
$13$ \( ( -24 + T^{2} )^{2} \)
$17$ \( ( 12 + T^{2} )^{2} \)
$19$ \( ( 12 + T^{2} )^{2} \)
$23$ \( ( -6 + T )^{4} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( 12 + T^{2} )^{2} \)
$37$ \( ( -24 + T^{2} )^{2} \)
$41$ \( ( 32 + T^{2} )^{2} \)
$43$ \( ( 72 + T^{2} )^{2} \)
$47$ \( ( 6 + T )^{4} \)
$53$ \( ( 108 + T^{2} )^{2} \)
$59$ \( ( -24 + T^{2} )^{2} \)
$61$ \( ( 2 + T )^{4} \)
$67$ \( ( 72 + T^{2} )^{2} \)
$71$ \( ( -96 + T^{2} )^{2} \)
$73$ \( ( -96 + T^{2} )^{2} \)
$79$ \( ( 108 + T^{2} )^{2} \)
$83$ \( ( -6 + T )^{4} \)
$89$ \( ( 32 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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