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: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + (\beta_{2} + 2) q^{9} + ( - 3 \beta_{2} + 3) q^{21} + (\beta_{3} - 5 \beta_1) q^{23} + ( - \beta_{3} - 2 \beta_1) q^{27} - 4 \beta_{2} q^{29} + 2 \beta_{2} q^{41} + (3 \beta_{3} + 3 \beta_1) q^{43} + (\beta_{3} - 5 \beta_1) q^{47} - 11 q^{49} - 8 q^{61} + (3 \beta_{3} - 3 \beta_1) q^{63} + ( - \beta_{3} - \beta_1) q^{67} + (3 \beta_{2} + 15) q^{69} + (4 \beta_{2} - 1) q^{81} + (\beta_{3} - 5 \beta_1) q^{83} + 4 \beta_{3} q^{87} + 8 \beta_{2} q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{9} + 12 q^{21} - 44 q^{49} - 32 q^{61} + 60 q^{69} - 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_1 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 18 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{23}^{2} - 90 \) Copy content Toggle raw display

Hecke characteristic polynomials

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