Properties

Label 1200.2.h.l
Level $1200$
Weight $2$
Character orbit 1200.h
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM discriminant -20
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 240)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( \beta_{1} + \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( \beta_{1} + \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} + ( 3 - 3 \beta_{2} ) q^{21} + ( -5 \beta_{1} + \beta_{3} ) q^{23} + ( -2 \beta_{1} - \beta_{3} ) q^{27} -4 \beta_{2} q^{29} + 2 \beta_{2} q^{41} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{43} + ( -5 \beta_{1} + \beta_{3} ) q^{47} -11 q^{49} -8 q^{61} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{63} + ( -\beta_{1} - \beta_{3} ) q^{67} + ( 15 + 3 \beta_{2} ) q^{69} + ( -1 + 4 \beta_{2} ) q^{81} + ( -5 \beta_{1} + \beta_{3} ) q^{83} + 4 \beta_{3} q^{87} + 8 \beta_{2} q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{9} + O(q^{10}) \) \( 4q + 8q^{9} + 12q^{21} - 44q^{49} - 32q^{61} + 60q^{69} - 4q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

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

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
1.58114 + 0.707107i
1.58114 0.707107i
−1.58114 + 0.707107i
−1.58114 0.707107i
0 −1.58114 0.707107i 0 0 0 4.24264i 0 2.00000 + 2.23607i 0
1151.2 0 −1.58114 + 0.707107i 0 0 0 4.24264i 0 2.00000 2.23607i 0
1151.3 0 1.58114 0.707107i 0 0 0 4.24264i 0 2.00000 2.23607i 0
1151.4 0 1.58114 + 0.707107i 0 0 0 4.24264i 0 2.00000 + 2.23607i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.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.h.l 4
3.b odd 2 1 inner 1200.2.h.l 4
4.b odd 2 1 inner 1200.2.h.l 4
5.b even 2 1 inner 1200.2.h.l 4
5.c odd 4 2 240.2.o.a 4
12.b even 2 1 inner 1200.2.h.l 4
15.d odd 2 1 inner 1200.2.h.l 4
15.e even 4 2 240.2.o.a 4
20.d odd 2 1 CM 1200.2.h.l 4
20.e even 4 2 240.2.o.a 4
40.i odd 4 2 960.2.o.b 4
40.k even 4 2 960.2.o.b 4
60.h even 2 1 inner 1200.2.h.l 4
60.l odd 4 2 240.2.o.a 4
120.q odd 4 2 960.2.o.b 4
120.w even 4 2 960.2.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.o.a 4 5.c odd 4 2
240.2.o.a 4 15.e even 4 2
240.2.o.a 4 20.e even 4 2
240.2.o.a 4 60.l odd 4 2
960.2.o.b 4 40.i odd 4 2
960.2.o.b 4 40.k even 4 2
960.2.o.b 4 120.q odd 4 2
960.2.o.b 4 120.w even 4 2
1200.2.h.l 4 1.a even 1 1 trivial
1200.2.h.l 4 3.b odd 2 1 inner
1200.2.h.l 4 4.b odd 2 1 inner
1200.2.h.l 4 5.b even 2 1 inner
1200.2.h.l 4 12.b even 2 1 inner
1200.2.h.l 4 15.d odd 2 1 inner
1200.2.h.l 4 20.d odd 2 1 CM
1200.2.h.l 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 \(S_{2}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 18 \)
\( T_{11} \)
\( T_{13} \)
\( T_{23}^{2} - 90 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 4 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 18 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( -90 + T^{2} )^{2} \)
$29$ \( ( 80 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 20 + T^{2} )^{2} \)
$43$ \( ( 162 + T^{2} )^{2} \)
$47$ \( ( -90 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 8 + T )^{4} \)
$67$ \( ( 18 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( -90 + T^{2} )^{2} \)
$89$ \( ( 320 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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