Properties

Label 1200.2.h.k
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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9} + (2 \beta_{3} - 2 \beta_1) q^{11} + 2 q^{13} + 2 \beta_{2} q^{17} + ( - 2 \beta_{3} - 2 \beta_1) q^{19} + (\beta_{2} - 3) q^{21} + (\beta_{3} - \beta_1) q^{23} + 3 \beta_{3} q^{27} + (4 \beta_{3} + 4 \beta_1) q^{31} + ( - 2 \beta_{2} - 6) q^{33} - 2 q^{37} + 2 \beta_1 q^{39} - 2 \beta_{2} q^{41} + ( - \beta_{3} - \beta_1) q^{43} + (5 \beta_{3} - 5 \beta_1) q^{47} + q^{49} + 6 \beta_{3} q^{51} + 2 \beta_{2} q^{53} + ( - 2 \beta_{2} + 6) q^{57} + ( - 4 \beta_{3} + 4 \beta_1) q^{59} + 8 q^{61} + (3 \beta_{3} - 3 \beta_1) q^{63} + (3 \beta_{3} + 3 \beta_1) q^{67} + ( - \beta_{2} - 3) q^{69} + (2 \beta_{3} - 2 \beta_1) q^{71} + 14 q^{73} - 4 \beta_{2} q^{77} + (2 \beta_{3} + 2 \beta_1) q^{79} - 9 q^{81} + ( - 3 \beta_{3} + 3 \beta_1) q^{83} - 4 \beta_{2} q^{89} + (2 \beta_{3} + 2 \beta_1) q^{91} + (4 \beta_{2} - 12) q^{93} - 10 q^{97} + ( - 6 \beta_{3} - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{13} - 12 q^{21} - 24 q^{33} - 8 q^{37} + 4 q^{49} + 24 q^{57} + 32 q^{61} - 12 q^{69} + 56 q^{73} - 36 q^{81} - 48 q^{93} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) 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 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
1.22474 + 1.22474i
0 −1.22474 1.22474i 0 0 0 2.44949i 0 3.00000i 0
1151.2 0 −1.22474 + 1.22474i 0 0 0 2.44949i 0 3.00000i 0
1151.3 0 1.22474 1.22474i 0 0 0 2.44949i 0 3.00000i 0
1151.4 0 1.22474 + 1.22474i 0 0 0 2.44949i 0 3.00000i 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 inner
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.k 4
3.b odd 2 1 inner 1200.2.h.k 4
4.b odd 2 1 inner 1200.2.h.k 4
5.b even 2 1 240.2.h.a 4
5.c odd 4 1 1200.2.o.e 4
5.c odd 4 1 1200.2.o.f 4
12.b even 2 1 inner 1200.2.h.k 4
15.d odd 2 1 240.2.h.a 4
15.e even 4 1 1200.2.o.e 4
15.e even 4 1 1200.2.o.f 4
20.d odd 2 1 240.2.h.a 4
20.e even 4 1 1200.2.o.e 4
20.e even 4 1 1200.2.o.f 4
40.e odd 2 1 960.2.h.c 4
40.f even 2 1 960.2.h.c 4
60.h even 2 1 240.2.h.a 4
60.l odd 4 1 1200.2.o.e 4
60.l odd 4 1 1200.2.o.f 4
120.i odd 2 1 960.2.h.c 4
120.m even 2 1 960.2.h.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.h.a 4 5.b even 2 1
240.2.h.a 4 15.d odd 2 1
240.2.h.a 4 20.d odd 2 1
240.2.h.a 4 60.h even 2 1
960.2.h.c 4 40.e odd 2 1
960.2.h.c 4 40.f even 2 1
960.2.h.c 4 120.i odd 2 1
960.2.h.c 4 120.m even 2 1
1200.2.h.k 4 1.a even 1 1 trivial
1200.2.h.k 4 3.b odd 2 1 inner
1200.2.h.k 4 4.b odd 2 1 inner
1200.2.h.k 4 12.b even 2 1 inner
1200.2.o.e 4 5.c odd 4 1
1200.2.o.e 4 15.e even 4 1
1200.2.o.e 4 20.e even 4 1
1200.2.o.e 4 60.l odd 4 1
1200.2.o.f 4 5.c odd 4 1
1200.2.o.f 4 15.e even 4 1
1200.2.o.f 4 20.e even 4 1
1200.2.o.f 4 60.l odd 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 150)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$73$ \( (T - 14)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{4} \) Copy content Toggle raw display
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