Properties

Label 1200.2.o.d
Level $1200$
Weight $2$
Character orbit 1200.o
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.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + (3 \zeta_{12}^{2} - 3) q^{9} + 3 q^{11} + 2 \zeta_{12}^{3} q^{13} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{17} + ( - 6 \zeta_{12}^{2} + 3) q^{19} + 6 \zeta_{12}^{3} q^{23} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + (12 \zeta_{12}^{2} - 6) q^{29} + ( - 4 \zeta_{12}^{2} + 2) q^{31} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{33} + 8 \zeta_{12}^{3} q^{37} + (2 \zeta_{12}^{2} - 4) q^{39} + ( - 6 \zeta_{12}^{2} + 3) q^{41} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{43} + 6 \zeta_{12}^{3} q^{47} - 7 q^{49} + 9 \zeta_{12}^{2} q^{51} + ( - 6 \zeta_{12}^{3} + 12 \zeta_{12}) q^{53} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{57} - 12 q^{59} + 8 q^{61} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12}) q^{67} + (6 \zeta_{12}^{2} - 12) q^{69} - 6 q^{71} - \zeta_{12}^{3} q^{73} + ( - 8 \zeta_{12}^{2} + 4) q^{79} - 9 \zeta_{12}^{2} q^{81} - 9 \zeta_{12}^{3} q^{83} + (18 \zeta_{12}^{3} - 18 \zeta_{12}) q^{87} + (6 \zeta_{12}^{2} - 3) q^{89} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{93} + 10 \zeta_{12}^{3} q^{97} + (9 \zeta_{12}^{2} - 9) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{9} + 12 q^{11} - 12 q^{39} - 28 q^{49} + 18 q^{51} - 48 q^{59} + 32 q^{61} - 36 q^{69} - 24 q^{71} - 18 q^{81} - 18 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1199.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0 0 0 0 −1.50000 + 2.59808i 0
1199.2 0 −0.866025 + 1.50000i 0 0 0 0 0 −1.50000 2.59808i 0
1199.3 0 0.866025 1.50000i 0 0 0 0 0 −1.50000 2.59808i 0
1199.4 0 0.866025 + 1.50000i 0 0 0 0 0 −1.50000 + 2.59808i 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
5.b even 2 1 inner
12.b even 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.o.d 4
3.b odd 2 1 1200.2.o.c 4
4.b odd 2 1 1200.2.o.c 4
5.b even 2 1 inner 1200.2.o.d 4
5.c odd 4 1 1200.2.h.b yes 2
5.c odd 4 1 1200.2.h.i yes 2
12.b even 2 1 inner 1200.2.o.d 4
15.d odd 2 1 1200.2.o.c 4
15.e even 4 1 1200.2.h.a 2
15.e even 4 1 1200.2.h.h yes 2
20.d odd 2 1 1200.2.o.c 4
20.e even 4 1 1200.2.h.a 2
20.e even 4 1 1200.2.h.h yes 2
60.h even 2 1 inner 1200.2.o.d 4
60.l odd 4 1 1200.2.h.b yes 2
60.l odd 4 1 1200.2.h.i yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.2.h.a 2 15.e even 4 1
1200.2.h.a 2 20.e even 4 1
1200.2.h.b yes 2 5.c odd 4 1
1200.2.h.b yes 2 60.l odd 4 1
1200.2.h.h yes 2 15.e even 4 1
1200.2.h.h yes 2 20.e even 4 1
1200.2.h.i yes 2 5.c odd 4 1
1200.2.h.i yes 2 60.l odd 4 1
1200.2.o.c 4 3.b odd 2 1
1200.2.o.c 4 4.b odd 2 1
1200.2.o.c 4 15.d odd 2 1
1200.2.o.c 4 20.d odd 2 1
1200.2.o.d 4 1.a even 1 1 trivial
1200.2.o.d 4 5.b even 2 1 inner
1200.2.o.d 4 12.b even 2 1 inner
1200.2.o.d 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 \(S_{2}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T - 3)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$59$ \( (T + 12)^{4} \) Copy content Toggle raw display
$61$ \( (T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
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