Properties

Label 1200.3.bg.p
Level $1200$
Weight $3$
Character orbit 1200.bg
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.bg (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 6 - 6 \beta_{2} - \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 6 - 6 \beta_{2} - \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( -6 + 6 \beta_{1} - 6 \beta_{3} ) q^{11} + ( 12 + 3 \beta_{1} + 12 \beta_{2} ) q^{13} + ( -6 + 6 \beta_{2} + 6 \beta_{3} ) q^{17} + ( 6 \beta_{1} - 19 \beta_{2} + 6 \beta_{3} ) q^{19} + ( -3 - 6 \beta_{1} + 6 \beta_{3} ) q^{21} + ( 6 + 18 \beta_{1} + 6 \beta_{2} ) q^{23} -3 \beta_{3} q^{27} + ( 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{29} + ( 5 - 18 \beta_{1} + 18 \beta_{3} ) q^{31} + ( -18 + 6 \beta_{1} - 18 \beta_{2} ) q^{33} + ( 12 - 12 \beta_{2} + 20 \beta_{3} ) q^{37} + ( -12 \beta_{1} - 9 \beta_{2} - 12 \beta_{3} ) q^{39} + ( 48 + 6 \beta_{1} - 6 \beta_{3} ) q^{41} + ( -18 + 5 \beta_{1} - 18 \beta_{2} ) q^{43} + ( -36 + 36 \beta_{2} + 18 \beta_{3} ) q^{47} + ( -12 \beta_{1} - 26 \beta_{2} - 12 \beta_{3} ) q^{49} + ( 18 + 6 \beta_{1} - 6 \beta_{3} ) q^{51} + ( -30 + 24 \beta_{1} - 30 \beta_{2} ) q^{53} + ( 18 - 18 \beta_{2} + 19 \beta_{3} ) q^{57} -30 \beta_{2} q^{59} + ( 11 + 24 \beta_{1} - 24 \beta_{3} ) q^{61} + ( 18 + 3 \beta_{1} + 18 \beta_{2} ) q^{63} + ( 6 - 6 \beta_{2} + 9 \beta_{3} ) q^{67} + ( -6 \beta_{1} - 54 \beta_{2} - 6 \beta_{3} ) q^{69} + ( 24 + 6 \beta_{1} - 6 \beta_{3} ) q^{71} + ( 12 + 28 \beta_{1} + 12 \beta_{2} ) q^{73} + ( -18 + 18 \beta_{2} - 66 \beta_{3} ) q^{77} + ( -12 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} ) q^{79} -9 q^{81} + ( -12 + 42 \beta_{1} - 12 \beta_{2} ) q^{83} + ( 18 - 18 \beta_{2} - 6 \beta_{3} ) q^{87} + ( 12 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} ) q^{89} + ( 153 + 30 \beta_{1} - 30 \beta_{3} ) q^{91} + ( 54 - 5 \beta_{1} + 54 \beta_{2} ) q^{93} + ( -48 + 48 \beta_{2} + 5 \beta_{3} ) q^{97} + ( 18 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 24q^{7} + O(q^{10}) \) \( 4q + 24q^{7} - 24q^{11} + 48q^{13} - 24q^{17} - 12q^{21} + 24q^{23} + 20q^{31} - 72q^{33} + 48q^{37} + 192q^{41} - 72q^{43} - 144q^{47} + 72q^{51} - 120q^{53} + 72q^{57} + 44q^{61} + 72q^{63} + 24q^{67} + 96q^{71} + 48q^{73} - 72q^{77} - 36q^{81} - 48q^{83} + 72q^{87} + 612q^{91} + 216q^{93} - 192q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 1.22474i
−1.22474 + 1.22474i
0 −1.22474 1.22474i 0 0 0 7.22474 7.22474i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 4.77526 4.77526i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 7.22474 + 7.22474i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 4.77526 + 4.77526i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.bg.p 4
4.b odd 2 1 150.3.f.a 4
5.b even 2 1 1200.3.bg.a 4
5.c odd 4 1 1200.3.bg.a 4
5.c odd 4 1 inner 1200.3.bg.p 4
12.b even 2 1 450.3.g.h 4
20.d odd 2 1 150.3.f.c yes 4
20.e even 4 1 150.3.f.a 4
20.e even 4 1 150.3.f.c yes 4
60.h even 2 1 450.3.g.g 4
60.l odd 4 1 450.3.g.g 4
60.l odd 4 1 450.3.g.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.f.a 4 4.b odd 2 1
150.3.f.a 4 20.e even 4 1
150.3.f.c yes 4 20.d odd 2 1
150.3.f.c yes 4 20.e even 4 1
450.3.g.g 4 60.h even 2 1
450.3.g.g 4 60.l odd 4 1
450.3.g.h 4 12.b even 2 1
450.3.g.h 4 60.l odd 4 1
1200.3.bg.a 4 5.b even 2 1
1200.3.bg.a 4 5.c odd 4 1
1200.3.bg.p 4 1.a even 1 1 trivial
1200.3.bg.p 4 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 24 T_{7}^{3} + 288 T_{7}^{2} - 1656 T_{7} + 4761 \) acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 4761 - 1656 T + 288 T^{2} - 24 T^{3} + T^{4} \)
$11$ \( ( -180 + 12 T + T^{2} )^{2} \)
$13$ \( 68121 - 12528 T + 1152 T^{2} - 48 T^{3} + T^{4} \)
$17$ \( 1296 - 864 T + 288 T^{2} + 24 T^{3} + T^{4} \)
$19$ \( 21025 + 1154 T^{2} + T^{4} \)
$23$ \( 810000 + 21600 T + 288 T^{2} - 24 T^{3} + T^{4} \)
$29$ \( 32400 + 504 T^{2} + T^{4} \)
$31$ \( ( -1919 - 10 T + T^{2} )^{2} \)
$37$ \( 831744 + 43776 T + 1152 T^{2} - 48 T^{3} + T^{4} \)
$41$ \( ( 2088 - 96 T + T^{2} )^{2} \)
$43$ \( 328329 + 41256 T + 2592 T^{2} + 72 T^{3} + T^{4} \)
$47$ \( 2624400 + 233280 T + 10368 T^{2} + 144 T^{3} + T^{4} \)
$53$ \( 5184 + 8640 T + 7200 T^{2} + 120 T^{3} + T^{4} \)
$59$ \( ( 900 + T^{2} )^{2} \)
$61$ \( ( -3335 - 22 T + T^{2} )^{2} \)
$67$ \( 29241 + 4104 T + 288 T^{2} - 24 T^{3} + T^{4} \)
$71$ \( ( 360 - 48 T + T^{2} )^{2} \)
$73$ \( 4260096 + 99072 T + 1152 T^{2} - 48 T^{3} + T^{4} \)
$79$ \( 739600 + 1736 T^{2} + T^{4} \)
$83$ \( 25040016 - 240192 T + 1152 T^{2} + 48 T^{3} + T^{4} \)
$89$ \( 518400 + 2016 T^{2} + T^{4} \)
$97$ \( 20548089 + 870336 T + 18432 T^{2} + 192 T^{3} + T^{4} \)
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