Properties

Label 1200.3.bg.h
Level $1200$
Weight $3$
Character orbit 1200.bg
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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 300)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + 2 \beta_{1} q^{7} -3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + 2 \beta_{1} q^{7} -3 \beta_{2} q^{9} -6 q^{11} -10 \beta_{3} q^{13} -12 \beta_{1} q^{17} + 10 \beta_{2} q^{19} -6 q^{21} + 24 \beta_{3} q^{23} + 3 \beta_{1} q^{27} + 48 \beta_{2} q^{29} + 26 q^{31} -6 \beta_{3} q^{33} -26 \beta_{1} q^{37} + 30 \beta_{2} q^{39} + 30 q^{41} + 24 \beta_{3} q^{43} -12 \beta_{1} q^{47} -37 \beta_{2} q^{49} + 36 q^{51} + 12 \beta_{3} q^{53} -10 \beta_{1} q^{57} -78 \beta_{2} q^{59} + 2 q^{61} -6 \beta_{3} q^{63} -52 \beta_{1} q^{67} -72 \beta_{2} q^{69} -120 q^{71} + 68 \beta_{3} q^{73} -12 \beta_{1} q^{77} + 74 \beta_{2} q^{79} -9 q^{81} -36 \beta_{3} q^{83} -48 \beta_{1} q^{87} + 150 \beta_{2} q^{89} + 60 q^{91} + 26 \beta_{3} q^{93} -4 \beta_{1} q^{97} + 18 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 24q^{11} - 24q^{21} + 104q^{31} + 120q^{41} + 144q^{51} + 8q^{61} - 480q^{71} - 36q^{81} + 240q^{91} + O(q^{100}) \)

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 2.44949 2.44949i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −2.44949 + 2.44949i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 2.44949 + 2.44949i 0 3.00000i 0
1057.2 0 1.22474 1.22474i 0 0 0 −2.44949 2.44949i 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.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.bg.h 4
4.b odd 2 1 300.3.k.b 4
5.b even 2 1 inner 1200.3.bg.h 4
5.c odd 4 2 inner 1200.3.bg.h 4
12.b even 2 1 900.3.l.c 4
20.d odd 2 1 300.3.k.b 4
20.e even 4 2 300.3.k.b 4
60.h even 2 1 900.3.l.c 4
60.l odd 4 2 900.3.l.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.k.b 4 4.b odd 2 1
300.3.k.b 4 20.d odd 2 1
300.3.k.b 4 20.e even 4 2
900.3.l.c 4 12.b even 2 1
900.3.l.c 4 60.h even 2 1
900.3.l.c 4 60.l odd 4 2
1200.3.bg.h 4 1.a even 1 1 trivial
1200.3.bg.h 4 5.b even 2 1 inner
1200.3.bg.h 4 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 144 \) acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 144 + T^{4} \)
$11$ \( ( 6 + T )^{4} \)
$13$ \( 90000 + T^{4} \)
$17$ \( 186624 + T^{4} \)
$19$ \( ( 100 + T^{2} )^{2} \)
$23$ \( 2985984 + T^{4} \)
$29$ \( ( 2304 + T^{2} )^{2} \)
$31$ \( ( -26 + T )^{4} \)
$37$ \( 4112784 + T^{4} \)
$41$ \( ( -30 + T )^{4} \)
$43$ \( 2985984 + T^{4} \)
$47$ \( 186624 + T^{4} \)
$53$ \( 186624 + T^{4} \)
$59$ \( ( 6084 + T^{2} )^{2} \)
$61$ \( ( -2 + T )^{4} \)
$67$ \( 65804544 + T^{4} \)
$71$ \( ( 120 + T )^{4} \)
$73$ \( 192432384 + T^{4} \)
$79$ \( ( 5476 + T^{2} )^{2} \)
$83$ \( 15116544 + T^{4} \)
$89$ \( ( 22500 + T^{2} )^{2} \)
$97$ \( 2304 + T^{4} \)
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