Properties

Label 1200.3.e.m
Level $1200$
Weight $3$
Character orbit 1200.e
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{11})\)
Defining polynomial: \(x^{4} + 11 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( \beta_{1} + \beta_{2} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( \beta_{1} + \beta_{2} ) q^{7} -3 q^{9} + \beta_{1} q^{11} + q^{13} + ( -12 - \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{2} ) q^{19} + ( -3 - \beta_{3} ) q^{21} + ( \beta_{1} + 12 \beta_{2} ) q^{23} -3 \beta_{2} q^{27} + ( -24 + \beta_{3} ) q^{29} + ( -\beta_{1} - 15 \beta_{2} ) q^{31} -\beta_{3} q^{33} + ( -22 + 2 \beta_{3} ) q^{37} + \beta_{2} q^{39} + ( -6 + 3 \beta_{3} ) q^{41} + ( -\beta_{1} + 21 \beta_{2} ) q^{43} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{47} + ( -86 - 2 \beta_{3} ) q^{49} + ( -3 \beta_{1} - 12 \beta_{2} ) q^{51} + ( 30 + 3 \beta_{3} ) q^{53} + ( 3 + \beta_{3} ) q^{57} + ( 4 \beta_{1} - 30 \beta_{2} ) q^{59} + ( -37 - 2 \beta_{3} ) q^{61} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{63} + ( -3 \beta_{1} - 43 \beta_{2} ) q^{67} + ( -36 - \beta_{3} ) q^{69} + ( \beta_{1} - 6 \beta_{2} ) q^{71} + ( 70 - 2 \beta_{3} ) q^{73} + ( -132 - \beta_{3} ) q^{77} + ( 4 \beta_{1} - 28 \beta_{2} ) q^{79} + 9 q^{81} + 42 \beta_{2} q^{83} + ( 3 \beta_{1} - 24 \beta_{2} ) q^{87} -96 q^{89} + ( \beta_{1} + \beta_{2} ) q^{91} + ( 45 + \beta_{3} ) q^{93} + ( -35 - 4 \beta_{3} ) q^{97} -3 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + O(q^{10}) \) \( 4 q - 12 q^{9} + 4 q^{13} - 48 q^{17} - 12 q^{21} - 96 q^{29} - 88 q^{37} - 24 q^{41} - 344 q^{49} + 120 q^{53} + 12 q^{57} - 148 q^{61} - 144 q^{69} + 280 q^{73} - 528 q^{77} + 36 q^{81} - 384 q^{89} + 180 q^{93} - 140 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 11 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} + 44 \nu \)\()/11\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{2} + 11 \)\()/11\)
\(\beta_{3}\)\(=\)\((\)\( -6 \nu^{3} \)\()/11\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(11 \beta_{2} - 11\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-11 \beta_{3}\)\()/6\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
1.65831 2.87228i
−1.65831 + 2.87228i
−1.65831 2.87228i
1.65831 + 2.87228i
0 1.73205i 0 0 0 13.2212i 0 −3.00000 0
751.2 0 1.73205i 0 0 0 9.75707i 0 −3.00000 0
751.3 0 1.73205i 0 0 0 9.75707i 0 −3.00000 0
751.4 0 1.73205i 0 0 0 13.2212i 0 −3.00000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.e.m yes 4
3.b odd 2 1 3600.3.e.bf 4
4.b odd 2 1 inner 1200.3.e.m yes 4
5.b even 2 1 1200.3.e.k 4
5.c odd 4 2 1200.3.j.d 8
12.b even 2 1 3600.3.e.bf 4
15.d odd 2 1 3600.3.e.bc 4
15.e even 4 2 3600.3.j.j 8
20.d odd 2 1 1200.3.e.k 4
20.e even 4 2 1200.3.j.d 8
60.h even 2 1 3600.3.e.bc 4
60.l odd 4 2 3600.3.j.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1200.3.e.k 4 5.b even 2 1
1200.3.e.k 4 20.d odd 2 1
1200.3.e.m yes 4 1.a even 1 1 trivial
1200.3.e.m yes 4 4.b odd 2 1 inner
1200.3.j.d 8 5.c odd 4 2
1200.3.j.d 8 20.e even 4 2
3600.3.e.bc 4 15.d odd 2 1
3600.3.e.bc 4 60.h even 2 1
3600.3.e.bf 4 3.b odd 2 1
3600.3.e.bf 4 12.b even 2 1
3600.3.j.j 8 15.e even 4 2
3600.3.j.j 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{4} + 270 T_{7}^{2} + 16641 \)
\( T_{11}^{2} + 132 \)
\( T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 16641 + 270 T^{2} + T^{4} \)
$11$ \( ( 132 + T^{2} )^{2} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( ( -252 + 24 T + T^{2} )^{2} \)
$19$ \( 16641 + 270 T^{2} + T^{4} \)
$23$ \( 90000 + 1128 T^{2} + T^{4} \)
$29$ \( ( 180 + 48 T + T^{2} )^{2} \)
$31$ \( 294849 + 1614 T^{2} + T^{4} \)
$37$ \( ( -1100 + 44 T + T^{2} )^{2} \)
$41$ \( ( -3528 + 12 T + T^{2} )^{2} \)
$43$ \( 1418481 + 2910 T^{2} + T^{4} \)
$47$ \( 4016016 + 4440 T^{2} + T^{4} \)
$53$ \( ( -2664 - 60 T + T^{2} )^{2} \)
$59$ \( 345744 + 9624 T^{2} + T^{4} \)
$61$ \( ( -215 + 74 T + T^{2} )^{2} \)
$67$ \( 19000881 + 13470 T^{2} + T^{4} \)
$71$ \( 576 + 480 T^{2} + T^{4} \)
$73$ \( ( 3316 - 140 T + T^{2} )^{2} \)
$79$ \( 57600 + 8928 T^{2} + T^{4} \)
$83$ \( ( 5292 + T^{2} )^{2} \)
$89$ \( ( 96 + T )^{4} \)
$97$ \( ( -5111 + 70 T + T^{2} )^{2} \)
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