Properties

Label 1200.2.h.n
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: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + ( - 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + ( - 2 \beta_1 - 1) q^{9} - \beta_{3} q^{11} - \beta_{3} q^{13} + \beta_{2} q^{17} - \beta_{2} q^{19} - 6 q^{23} + ( - \beta_1 - 5) q^{27} + 2 \beta_1 q^{29} - \beta_{2} q^{31} + ( - \beta_{3} + 2 \beta_{2}) q^{33} - \beta_{3} q^{37} + ( - \beta_{3} + 2 \beta_{2}) q^{39} - 4 \beta_1 q^{41} - 6 \beta_1 q^{43} + 6 q^{47} + 7 q^{49} + (\beta_{3} + \beta_{2}) q^{51} + 3 \beta_{2} q^{53} + ( - \beta_{3} - \beta_{2}) q^{57} + \beta_{3} q^{59} - 2 q^{61} - 6 \beta_1 q^{67} + (6 \beta_1 - 6) q^{69} + 2 \beta_{3} q^{71} - 2 \beta_{3} q^{73} - 3 \beta_{2} q^{79} + (4 \beta_1 - 7) q^{81} - 6 q^{83} + (2 \beta_1 + 4) q^{87} - 4 \beta_1 q^{89} + ( - \beta_{3} - \beta_{2}) q^{93} + (\beta_{3} + 4 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{9} - 24 q^{23} - 20 q^{27} + 24 q^{47} + 28 q^{49} - 8 q^{61} - 24 q^{69} - 28 q^{81} - 24 q^{83} + 16 q^{87}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) 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
1.22474 + 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.n 4
3.b odd 2 1 1200.2.h.j 4
4.b odd 2 1 1200.2.h.j 4
5.b even 2 1 1200.2.h.j 4
5.c odd 4 2 240.2.o.b 8
12.b even 2 1 inner 1200.2.h.n 4
15.d odd 2 1 inner 1200.2.h.n 4
15.e even 4 2 240.2.o.b 8
20.d odd 2 1 inner 1200.2.h.n 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.j 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 3.b odd 2 1
1200.2.h.j 4 4.b odd 2 1
1200.2.h.j 4 5.b even 2 1
1200.2.h.j 4 60.h even 2 1
1200.2.h.n 4 1.a even 1 1 trivial
1200.2.h.n 4 12.b even 2 1 inner
1200.2.h.n 4 15.d odd 2 1 inner
1200.2.h.n 4 20.d odd 2 1 inner

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} - 24 \) Copy content Toggle raw display
\( T_{13}^{2} - 24 \) Copy content Toggle raw display
\( T_{23} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

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