Properties

Label 1200.3.bg.o
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 60)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 5 - 5 \beta_{2} - 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 5 - 5 \beta_{2} - 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( 4 + 5 \beta_{1} - 5 \beta_{3} ) q^{11} -10 \beta_{1} q^{13} + ( 10 - 10 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -10 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{19} + ( -6 - 5 \beta_{1} + 5 \beta_{3} ) q^{21} + ( -10 + 6 \beta_{1} - 10 \beta_{2} ) q^{23} -3 \beta_{3} q^{27} + ( 5 \beta_{1} + 22 \beta_{2} + 5 \beta_{3} ) q^{29} + ( -4 + 10 \beta_{1} - 10 \beta_{3} ) q^{31} + ( -15 - 4 \beta_{1} - 15 \beta_{2} ) q^{33} + 6 \beta_{3} q^{37} + 30 \beta_{2} q^{39} + ( 50 - 10 \beta_{1} + 10 \beta_{3} ) q^{41} + ( 30 - 4 \beta_{1} + 30 \beta_{2} ) q^{43} -18 \beta_{3} q^{47} + ( -20 \beta_{1} - 13 \beta_{2} - 20 \beta_{3} ) q^{49} + ( 6 - 10 \beta_{1} + 10 \beta_{3} ) q^{51} + ( 50 - 12 \beta_{1} + 50 \beta_{2} ) q^{53} + ( -30 + 30 \beta_{2} - 10 \beta_{3} ) q^{57} + ( -15 \beta_{1} - 52 \beta_{2} - 15 \beta_{3} ) q^{59} + ( -78 - 10 \beta_{1} + 10 \beta_{3} ) q^{61} + ( 15 + 6 \beta_{1} + 15 \beta_{2} ) q^{63} + ( -10 + 10 \beta_{2} + 12 \beta_{3} ) q^{67} + ( 10 \beta_{1} - 18 \beta_{2} + 10 \beta_{3} ) q^{69} + ( 20 - 40 \beta_{1} + 40 \beta_{3} ) q^{71} + ( 5 + 32 \beta_{1} + 5 \beta_{2} ) q^{73} + ( 50 - 50 \beta_{2} - 58 \beta_{3} ) q^{77} + ( -10 \beta_{1} - 24 \beta_{2} - 10 \beta_{3} ) q^{79} -9 q^{81} + ( -60 - 34 \beta_{1} - 60 \beta_{2} ) q^{83} + ( 15 - 15 \beta_{2} - 22 \beta_{3} ) q^{87} + ( -60 \beta_{1} - 10 \beta_{2} - 60 \beta_{3} ) q^{89} + ( -60 - 50 \beta_{1} + 50 \beta_{3} ) q^{91} + ( -30 + 4 \beta_{1} - 30 \beta_{2} ) q^{93} + ( -75 + 75 \beta_{2} - 16 \beta_{3} ) q^{97} + ( 15 \beta_{1} + 12 \beta_{2} + 15 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 20q^{7} + O(q^{10}) \) \( 4q + 20q^{7} + 16q^{11} + 40q^{17} - 24q^{21} - 40q^{23} - 16q^{31} - 60q^{33} + 200q^{41} + 120q^{43} + 24q^{51} + 200q^{53} - 120q^{57} - 312q^{61} + 60q^{63} - 40q^{67} + 80q^{71} + 20q^{73} + 200q^{77} - 36q^{81} - 240q^{83} + 60q^{87} - 240q^{91} - 120q^{93} - 300q^{97} + O(q^{100}) \)

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 7.44949 7.44949i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 2.55051 2.55051i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 7.44949 + 7.44949i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 2.55051 + 2.55051i 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.o 4
4.b odd 2 1 300.3.k.a 4
5.b even 2 1 240.3.bg.d 4
5.c odd 4 1 240.3.bg.d 4
5.c odd 4 1 inner 1200.3.bg.o 4
12.b even 2 1 900.3.l.b 4
15.d odd 2 1 720.3.bh.f 4
15.e even 4 1 720.3.bh.f 4
20.d odd 2 1 60.3.k.a 4
20.e even 4 1 60.3.k.a 4
20.e even 4 1 300.3.k.a 4
40.e odd 2 1 960.3.bg.b 4
40.f even 2 1 960.3.bg.a 4
40.i odd 4 1 960.3.bg.a 4
40.k even 4 1 960.3.bg.b 4
60.h even 2 1 180.3.l.b 4
60.l odd 4 1 180.3.l.b 4
60.l odd 4 1 900.3.l.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.k.a 4 20.d odd 2 1
60.3.k.a 4 20.e even 4 1
180.3.l.b 4 60.h even 2 1
180.3.l.b 4 60.l odd 4 1
240.3.bg.d 4 5.b even 2 1
240.3.bg.d 4 5.c odd 4 1
300.3.k.a 4 4.b odd 2 1
300.3.k.a 4 20.e even 4 1
720.3.bh.f 4 15.d odd 2 1
720.3.bh.f 4 15.e even 4 1
900.3.l.b 4 12.b even 2 1
900.3.l.b 4 60.l odd 4 1
960.3.bg.a 4 40.f even 2 1
960.3.bg.a 4 40.i odd 4 1
960.3.bg.b 4 40.e odd 2 1
960.3.bg.b 4 40.k even 4 1
1200.3.bg.o 4 1.a even 1 1 trivial
1200.3.bg.o 4 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 20 T_{7}^{3} + 200 T_{7}^{2} - 760 T_{7} + 1444 \) acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1444 - 760 T + 200 T^{2} - 20 T^{3} + T^{4} \)
$11$ \( ( -134 - 8 T + T^{2} )^{2} \)
$13$ \( 90000 + T^{4} \)
$17$ \( 35344 - 7520 T + 800 T^{2} - 40 T^{3} + T^{4} \)
$19$ \( 250000 + 1400 T^{2} + T^{4} \)
$23$ \( 8464 + 3680 T + 800 T^{2} + 40 T^{3} + T^{4} \)
$29$ \( 111556 + 1268 T^{2} + T^{4} \)
$31$ \( ( -584 + 8 T + T^{2} )^{2} \)
$37$ \( 11664 + T^{4} \)
$41$ \( ( 1900 - 100 T + T^{2} )^{2} \)
$43$ \( 3069504 - 210240 T + 7200 T^{2} - 120 T^{3} + T^{4} \)
$47$ \( 944784 + T^{4} \)
$53$ \( 20866624 - 913600 T + 20000 T^{2} - 200 T^{3} + T^{4} \)
$59$ \( 1833316 + 8108 T^{2} + T^{4} \)
$61$ \( ( 5484 + 156 T + T^{2} )^{2} \)
$67$ \( 53824 - 9280 T + 800 T^{2} + 40 T^{3} + T^{4} \)
$71$ \( ( -9200 - 40 T + T^{2} )^{2} \)
$73$ \( 9132484 + 60440 T + 200 T^{2} - 20 T^{3} + T^{4} \)
$79$ \( 576 + 2352 T^{2} + T^{4} \)
$83$ \( 13927824 + 895680 T + 28800 T^{2} + 240 T^{3} + T^{4} \)
$89$ \( 462250000 + 43400 T^{2} + T^{4} \)
$97$ \( 109872324 + 3144600 T + 45000 T^{2} + 300 T^{3} + T^{4} \)
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