Properties

Label 1200.3.e.n
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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{7} -3 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{7} -3 q^{9} + ( 2 \beta_{1} - \beta_{2} ) q^{11} + ( 8 + \beta_{3} ) q^{13} + ( -12 - \beta_{3} ) q^{17} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{19} + ( 6 + \beta_{3} ) q^{21} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{23} + 3 \beta_{1} q^{27} + ( -12 + \beta_{3} ) q^{29} + ( 20 \beta_{1} - 2 \beta_{2} ) q^{31} + ( 6 - \beta_{3} ) q^{33} + ( -20 + 3 \beta_{3} ) q^{37} + ( -8 \beta_{1} - 3 \beta_{2} ) q^{39} -30 q^{41} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{43} + ( 32 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -23 - 4 \beta_{3} ) q^{49} + ( 12 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -24 - 3 \beta_{3} ) q^{53} + ( -12 - 2 \beta_{3} ) q^{57} + ( 2 \beta_{1} + 11 \beta_{2} ) q^{59} + ( 14 - 6 \beta_{3} ) q^{61} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{63} + ( 48 \beta_{1} - 6 \beta_{2} ) q^{67} + ( 12 + 4 \beta_{3} ) q^{69} + ( 68 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 2 + 2 \beta_{3} ) q^{73} + 48 q^{77} + ( -44 \beta_{1} + 2 \beta_{2} ) q^{79} + 9 q^{81} + ( 24 \beta_{1} - 6 \beta_{2} ) q^{83} + ( 12 \beta_{1} - 3 \beta_{2} ) q^{87} + ( 6 + 2 \beta_{3} ) q^{89} + ( 76 \beta_{1} + 14 \beta_{2} ) q^{91} + ( 60 - 2 \beta_{3} ) q^{93} + 14 q^{97} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + O(q^{10}) \) \( 4 q - 12 q^{9} + 32 q^{13} - 48 q^{17} + 24 q^{21} - 48 q^{29} + 24 q^{33} - 80 q^{37} - 120 q^{41} - 92 q^{49} - 96 q^{53} - 48 q^{57} + 56 q^{61} + 48 q^{69} + 8 q^{73} + 192 q^{77} + 36 q^{81} + 24 q^{89} + 240 q^{93} + 56 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu \)
\(\beta_{2}\)\(=\)\( 4 \nu^{3} - 4 \nu^{2} + 12 \nu + 2 \)
\(\beta_{3}\)\(=\)\( -6 \nu^{3} - 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} + 6 \beta_{1} + 6\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 3 \beta_{2} + 18 \beta_{1} - 18\)\()/24\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} - 12\)\()/6\)

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
−0.309017 0.535233i
0.809017 + 1.40126i
0.809017 1.40126i
−0.309017 + 0.535233i
0 1.73205i 0 0 0 4.28187i 0 −3.00000 0
751.2 0 1.73205i 0 0 0 11.2101i 0 −3.00000 0
751.3 0 1.73205i 0 0 0 11.2101i 0 −3.00000 0
751.4 0 1.73205i 0 0 0 4.28187i 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.n 4
3.b odd 2 1 3600.3.e.bh 4
4.b odd 2 1 inner 1200.3.e.n 4
5.b even 2 1 240.3.e.a 4
5.c odd 4 2 1200.3.j.f 8
12.b even 2 1 3600.3.e.bh 4
15.d odd 2 1 720.3.e.a 4
15.e even 4 2 3600.3.j.l 8
20.d odd 2 1 240.3.e.a 4
20.e even 4 2 1200.3.j.f 8
40.e odd 2 1 960.3.e.b 4
40.f even 2 1 960.3.e.b 4
60.h even 2 1 720.3.e.a 4
60.l odd 4 2 3600.3.j.l 8
120.i odd 2 1 2880.3.e.f 4
120.m even 2 1 2880.3.e.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.e.a 4 5.b even 2 1
240.3.e.a 4 20.d odd 2 1
720.3.e.a 4 15.d odd 2 1
720.3.e.a 4 60.h even 2 1
960.3.e.b 4 40.e odd 2 1
960.3.e.b 4 40.f even 2 1
1200.3.e.n 4 1.a even 1 1 trivial
1200.3.e.n 4 4.b odd 2 1 inner
1200.3.j.f 8 5.c odd 4 2
1200.3.j.f 8 20.e even 4 2
2880.3.e.f 4 120.i odd 2 1
2880.3.e.f 4 120.m even 2 1
3600.3.e.bh 4 3.b odd 2 1
3600.3.e.bh 4 12.b even 2 1
3600.3.j.l 8 15.e even 4 2
3600.3.j.l 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 \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{4} + 144 T_{7}^{2} + 2304 \)
\( T_{11}^{4} + 144 T_{11}^{2} + 2304 \)
\( T_{13}^{2} - 16 T_{13} - 116 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 2304 + 144 T^{2} + T^{4} \)
$11$ \( 2304 + 144 T^{2} + T^{4} \)
$13$ \( ( -116 - 16 T + T^{2} )^{2} \)
$17$ \( ( -36 + 24 T + T^{2} )^{2} \)
$19$ \( 36864 + 576 T^{2} + T^{4} \)
$23$ \( 831744 + 2016 T^{2} + T^{4} \)
$29$ \( ( -36 + 24 T + T^{2} )^{2} \)
$31$ \( 921600 + 2880 T^{2} + T^{4} \)
$37$ \( ( -1220 + 40 T + T^{2} )^{2} \)
$41$ \( ( 30 + T )^{4} \)
$43$ \( 831744 + 2016 T^{2} + T^{4} \)
$47$ \( 8020224 + 6624 T^{2} + T^{4} \)
$53$ \( ( -1044 + 48 T + T^{2} )^{2} \)
$59$ \( 52533504 + 14544 T^{2} + T^{4} \)
$61$ \( ( -6284 - 28 T + T^{2} )^{2} \)
$67$ \( 22581504 + 18144 T^{2} + T^{4} \)
$71$ \( 185831424 + 28224 T^{2} + T^{4} \)
$73$ \( ( -716 - 4 T + T^{2} )^{2} \)
$79$ \( 31002624 + 12096 T^{2} + T^{4} \)
$83$ \( 186624 + 7776 T^{2} + T^{4} \)
$89$ \( ( -684 - 12 T + T^{2} )^{2} \)
$97$ \( ( -14 + T )^{4} \)
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