Properties

Label 1200.3.bg.m
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 600)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 4 - 4 \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 4 - 4 \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( 2 - 4 \beta_{1} + 4 \beta_{3} ) q^{11} -2 \beta_{1} q^{13} + ( -4 + 4 \beta_{2} + 4 \beta_{3} ) q^{17} + ( -8 \beta_{1} + 14 \beta_{2} - 8 \beta_{3} ) q^{19} + ( -6 + 4 \beta_{1} - 4 \beta_{3} ) q^{21} + ( 12 - 8 \beta_{1} + 12 \beta_{2} ) q^{23} + 3 \beta_{3} q^{27} + ( 8 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} ) q^{29} + ( -14 - 16 \beta_{1} + 16 \beta_{3} ) q^{31} + ( -12 + 2 \beta_{1} - 12 \beta_{2} ) q^{33} + ( -8 + 8 \beta_{2} + 14 \beta_{3} ) q^{37} -6 \beta_{2} q^{39} + ( -2 + 24 \beta_{1} - 24 \beta_{3} ) q^{41} + ( -32 - 8 \beta_{1} - 32 \beta_{2} ) q^{43} + ( -20 + 20 \beta_{2} + 20 \beta_{3} ) q^{47} + ( 16 \beta_{1} + 5 \beta_{2} + 16 \beta_{3} ) q^{49} + ( -12 - 4 \beta_{1} + 4 \beta_{3} ) q^{51} + ( 8 - 36 \beta_{1} + 8 \beta_{2} ) q^{53} + ( 24 - 24 \beta_{2} + 14 \beta_{3} ) q^{57} + ( -20 \beta_{1} + 38 \beta_{2} - 20 \beta_{3} ) q^{59} + ( -30 + 32 \beta_{1} - 32 \beta_{3} ) q^{61} + ( 12 - 6 \beta_{1} + 12 \beta_{2} ) q^{63} + ( 24 - 24 \beta_{2} + 44 \beta_{3} ) q^{67} + ( 12 \beta_{1} - 24 \beta_{2} + 12 \beta_{3} ) q^{69} + ( 8 - 24 \beta_{1} + 24 \beta_{3} ) q^{71} + ( 64 - 12 \beta_{1} + 64 \beta_{2} ) q^{73} + ( 32 - 32 \beta_{2} + 36 \beta_{3} ) q^{77} -66 \beta_{2} q^{79} -9 q^{81} + ( 40 - 36 \beta_{1} + 40 \beta_{2} ) q^{83} + ( -24 + 24 \beta_{2} + 16 \beta_{3} ) q^{87} + ( 8 \beta_{1} + 10 \beta_{2} + 8 \beta_{3} ) q^{89} + ( 12 - 8 \beta_{1} + 8 \beta_{3} ) q^{91} + ( -48 - 14 \beta_{1} - 48 \beta_{2} ) q^{93} + ( -40 + 40 \beta_{2} + 76 \beta_{3} ) q^{97} + ( -12 \beta_{1} + 6 \beta_{2} - 12 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{7} + O(q^{10}) \) \( 4 q + 16 q^{7} + 8 q^{11} - 16 q^{17} - 24 q^{21} + 48 q^{23} - 56 q^{31} - 48 q^{33} - 32 q^{37} - 8 q^{41} - 128 q^{43} - 80 q^{47} - 48 q^{51} + 32 q^{53} + 96 q^{57} - 120 q^{61} + 48 q^{63} + 96 q^{67} + 32 q^{71} + 256 q^{73} + 128 q^{77} - 36 q^{81} + 160 q^{83} - 96 q^{87} + 48 q^{91} - 192 q^{93} - 160 q^{97} + O(q^{100}) \)

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 6.44949 6.44949i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 1.55051 1.55051i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 6.44949 + 6.44949i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 1.55051 + 1.55051i 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.m 4
4.b odd 2 1 600.3.u.a 4
5.b even 2 1 1200.3.bg.b 4
5.c odd 4 1 1200.3.bg.b 4
5.c odd 4 1 inner 1200.3.bg.m 4
12.b even 2 1 1800.3.v.j 4
20.d odd 2 1 600.3.u.f yes 4
20.e even 4 1 600.3.u.a 4
20.e even 4 1 600.3.u.f yes 4
60.h even 2 1 1800.3.v.q 4
60.l odd 4 1 1800.3.v.j 4
60.l odd 4 1 1800.3.v.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.u.a 4 4.b odd 2 1
600.3.u.a 4 20.e even 4 1
600.3.u.f yes 4 20.d odd 2 1
600.3.u.f yes 4 20.e even 4 1
1200.3.bg.b 4 5.b even 2 1
1200.3.bg.b 4 5.c odd 4 1
1200.3.bg.m 4 1.a even 1 1 trivial
1200.3.bg.m 4 5.c odd 4 1 inner
1800.3.v.j 4 12.b even 2 1
1800.3.v.j 4 60.l odd 4 1
1800.3.v.q 4 60.h even 2 1
1800.3.v.q 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 16 T_{7}^{3} + 128 T_{7}^{2} - 320 T_{7} + 400 \) acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 400 - 320 T + 128 T^{2} - 16 T^{3} + T^{4} \)
$11$ \( ( -92 - 4 T + T^{2} )^{2} \)
$13$ \( 144 + T^{4} \)
$17$ \( 256 - 256 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$19$ \( 35344 + 1160 T^{2} + T^{4} \)
$23$ \( 9216 - 4608 T + 1152 T^{2} - 48 T^{3} + T^{4} \)
$29$ \( 16384 + 1280 T^{2} + T^{4} \)
$31$ \( ( -1340 + 28 T + T^{2} )^{2} \)
$37$ \( 211600 - 14720 T + 512 T^{2} + 32 T^{3} + T^{4} \)
$41$ \( ( -3452 + 4 T + T^{2} )^{2} \)
$43$ \( 3444736 + 237568 T + 8192 T^{2} + 128 T^{3} + T^{4} \)
$47$ \( 160000 - 32000 T + 3200 T^{2} + 80 T^{3} + T^{4} \)
$53$ \( 14137600 + 120320 T + 512 T^{2} - 32 T^{3} + T^{4} \)
$59$ \( 913936 + 7688 T^{2} + T^{4} \)
$61$ \( ( -5244 + 60 T + T^{2} )^{2} \)
$67$ \( 21678336 + 446976 T + 4608 T^{2} - 96 T^{3} + T^{4} \)
$71$ \( ( -3392 - 16 T + T^{2} )^{2} \)
$73$ \( 60217600 - 1986560 T + 32768 T^{2} - 256 T^{3} + T^{4} \)
$79$ \( ( 4356 + T^{2} )^{2} \)
$83$ \( 473344 + 110080 T + 12800 T^{2} - 160 T^{3} + T^{4} \)
$89$ \( 80656 + 968 T^{2} + T^{4} \)
$97$ \( 199600384 - 2260480 T + 12800 T^{2} + 160 T^{3} + T^{4} \)
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