Properties

Label 1200.3.bg.d
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{7} -3 \beta_{2} q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} + ( -4 + 4 \beta_{1} - 4 \beta_{2} ) q^{7} -3 \beta_{2} q^{9} + ( 4 + 4 \beta_{1} - 4 \beta_{3} ) q^{11} + ( -3 + 3 \beta_{2} + 8 \beta_{3} ) q^{13} + ( -11 - 4 \beta_{1} - 11 \beta_{2} ) q^{17} + ( -4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 12 - 4 \beta_{1} + 4 \beta_{3} ) q^{21} + ( -4 + 4 \beta_{2} - 12 \beta_{3} ) q^{23} -3 \beta_{1} q^{27} + ( -4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} ) q^{29} + ( 20 + 8 \beta_{1} - 8 \beta_{3} ) q^{31} + ( 12 - 12 \beta_{2} - 4 \beta_{3} ) q^{33} + ( 27 + 27 \beta_{2} ) q^{37} + ( 3 \beta_{1} + 24 \beta_{2} + 3 \beta_{3} ) q^{39} + ( 8 + 4 \beta_{1} - 4 \beta_{3} ) q^{41} + ( 12 - 12 \beta_{2} + 20 \beta_{3} ) q^{43} + ( 24 - 12 \beta_{1} + 24 \beta_{2} ) q^{47} + ( -32 \beta_{1} + 31 \beta_{2} - 32 \beta_{3} ) q^{49} + ( -12 - 11 \beta_{1} + 11 \beta_{3} ) q^{51} + ( -25 + 25 \beta_{2} - 36 \beta_{3} ) q^{53} + ( -12 - 16 \beta_{1} - 12 \beta_{2} ) q^{57} -20 \beta_{2} q^{59} + ( -24 + 16 \beta_{1} - 16 \beta_{3} ) q^{61} + ( -12 + 12 \beta_{2} - 12 \beta_{3} ) q^{63} + ( -4 - 36 \beta_{1} - 4 \beta_{2} ) q^{67} + ( 4 \beta_{1} - 36 \beta_{2} + 4 \beta_{3} ) q^{69} + ( -16 + 4 \beta_{1} - 4 \beta_{3} ) q^{71} + ( 47 - 47 \beta_{2} + 8 \beta_{3} ) q^{73} + ( 32 - 16 \beta_{1} + 32 \beta_{2} ) q^{77} + ( -52 \beta_{1} + 12 \beta_{2} - 52 \beta_{3} ) q^{79} -9 q^{81} + ( 48 - 48 \beta_{2} - 28 \beta_{3} ) q^{83} + ( -12 - 16 \beta_{1} - 12 \beta_{2} ) q^{87} + ( 12 \beta_{1} + 88 \beta_{2} + 12 \beta_{3} ) q^{89} + ( -72 + 20 \beta_{1} - 20 \beta_{3} ) q^{91} + ( 24 - 24 \beta_{2} - 20 \beta_{3} ) q^{93} + ( -33 - 40 \beta_{1} - 33 \beta_{2} ) q^{97} + ( -12 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 16q^{7} + O(q^{10}) \) \( 4q - 16q^{7} + 16q^{11} - 12q^{13} - 44q^{17} + 48q^{21} - 16q^{23} + 80q^{31} + 48q^{33} + 108q^{37} + 32q^{41} + 48q^{43} + 96q^{47} - 48q^{51} - 100q^{53} - 48q^{57} - 96q^{61} - 48q^{63} - 16q^{67} - 64q^{71} + 188q^{73} + 128q^{77} - 36q^{81} + 192q^{83} - 48q^{87} - 288q^{91} + 96q^{93} - 132q^{97} + O(q^{100}) \)

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 −8.89898 + 8.89898i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 0.898979 0.898979i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −8.89898 8.89898i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 0.898979 + 0.898979i 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.d 4
4.b odd 2 1 150.3.f.b 4
5.b even 2 1 240.3.bg.b 4
5.c odd 4 1 240.3.bg.b 4
5.c odd 4 1 inner 1200.3.bg.d 4
12.b even 2 1 450.3.g.j 4
15.d odd 2 1 720.3.bh.i 4
15.e even 4 1 720.3.bh.i 4
20.d odd 2 1 30.3.f.a 4
20.e even 4 1 30.3.f.a 4
20.e even 4 1 150.3.f.b 4
40.e odd 2 1 960.3.bg.e 4
40.f even 2 1 960.3.bg.g 4
40.i odd 4 1 960.3.bg.g 4
40.k even 4 1 960.3.bg.e 4
60.h even 2 1 90.3.g.d 4
60.l odd 4 1 90.3.g.d 4
60.l odd 4 1 450.3.g.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.f.a 4 20.d odd 2 1
30.3.f.a 4 20.e even 4 1
90.3.g.d 4 60.h even 2 1
90.3.g.d 4 60.l odd 4 1
150.3.f.b 4 4.b odd 2 1
150.3.f.b 4 20.e even 4 1
240.3.bg.b 4 5.b even 2 1
240.3.bg.b 4 5.c odd 4 1
450.3.g.j 4 12.b even 2 1
450.3.g.j 4 60.l odd 4 1
720.3.bh.i 4 15.d odd 2 1
720.3.bh.i 4 15.e even 4 1
960.3.bg.e 4 40.e odd 2 1
960.3.bg.e 4 40.k even 4 1
960.3.bg.g 4 40.f even 2 1
960.3.bg.g 4 40.i odd 4 1
1200.3.bg.d 4 1.a even 1 1 trivial
1200.3.bg.d 4 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 16 T_{7}^{3} + 128 T_{7}^{2} - 256 T_{7} + 256 \) acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 256 - 256 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$11$ \( ( -80 - 8 T + T^{2} )^{2} \)
$13$ \( 30276 - 2088 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$17$ \( 37636 + 8536 T + 968 T^{2} + 44 T^{3} + T^{4} \)
$19$ \( 25600 + 704 T^{2} + T^{4} \)
$23$ \( 160000 - 6400 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$29$ \( 25600 + 704 T^{2} + T^{4} \)
$31$ \( ( 16 - 40 T + T^{2} )^{2} \)
$37$ \( ( 1458 - 54 T + T^{2} )^{2} \)
$41$ \( ( -32 - 16 T + T^{2} )^{2} \)
$43$ \( 831744 + 43776 T + 1152 T^{2} - 48 T^{3} + T^{4} \)
$47$ \( 518400 - 69120 T + 4608 T^{2} - 96 T^{3} + T^{4} \)
$53$ \( 6959044 - 263800 T + 5000 T^{2} + 100 T^{3} + T^{4} \)
$59$ \( ( 400 + T^{2} )^{2} \)
$61$ \( ( -960 + 48 T + T^{2} )^{2} \)
$67$ \( 14868736 - 61696 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$71$ \( ( 160 + 32 T + T^{2} )^{2} \)
$73$ \( 17859076 - 794488 T + 17672 T^{2} - 188 T^{3} + T^{4} \)
$79$ \( 258566400 + 32736 T^{2} + T^{4} \)
$83$ \( 5089536 - 433152 T + 18432 T^{2} - 192 T^{3} + T^{4} \)
$89$ \( 47334400 + 17216 T^{2} + T^{4} \)
$97$ \( 6874884 - 346104 T + 8712 T^{2} + 132 T^{3} + T^{4} \)
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