Properties

Label 1200.3.c.e
Level $1200$
Weight $3$
Character orbit 1200.c
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{3} ) q^{3} + \beta_{1} q^{7} + ( 1 + 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{3} ) q^{3} + \beta_{1} q^{7} + ( 1 + 2 \beta_{2} ) q^{9} -3 \beta_{2} q^{11} + 4 \beta_{1} q^{13} + 6 \beta_{3} q^{17} -34 q^{19} + ( 4 - \beta_{2} ) q^{21} + 18 \beta_{3} q^{23} + ( -11 \beta_{1} + 7 \beta_{3} ) q^{27} + 9 \beta_{2} q^{29} -14 q^{31} + ( 15 \beta_{1} - 12 \beta_{3} ) q^{33} -28 \beta_{1} q^{37} + ( 16 - 4 \beta_{2} ) q^{39} -6 \beta_{2} q^{41} -4 \beta_{1} q^{43} + 18 \beta_{3} q^{47} + 45 q^{49} + ( -30 - 6 \beta_{2} ) q^{51} + 18 \beta_{3} q^{53} + ( 34 \beta_{1} + 34 \beta_{3} ) q^{57} + 3 \beta_{2} q^{59} -46 q^{61} + ( \beta_{1} - 8 \beta_{3} ) q^{63} + 16 \beta_{1} q^{67} + ( -90 - 18 \beta_{2} ) q^{69} -12 \beta_{2} q^{71} -53 \beta_{1} q^{73} + 12 \beta_{3} q^{77} -22 q^{79} + ( -79 + 4 \beta_{2} ) q^{81} -54 \beta_{3} q^{83} + ( -45 \beta_{1} + 36 \beta_{3} ) q^{87} -24 \beta_{2} q^{89} -16 q^{91} + ( 14 \beta_{1} + 14 \beta_{3} ) q^{93} -61 \beta_{1} q^{97} + ( 120 - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} - 136q^{19} + 16q^{21} - 56q^{31} + 64q^{39} + 180q^{49} - 120q^{51} - 184q^{61} - 360q^{69} - 88q^{79} - 316q^{81} - 64q^{91} + 480q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 8 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.618034i
0.618034i
1.61803i
1.61803i
0 −2.23607 2.00000i 0 0 0 2.00000i 0 1.00000 + 8.94427i 0
449.2 0 −2.23607 + 2.00000i 0 0 0 2.00000i 0 1.00000 8.94427i 0
449.3 0 2.23607 2.00000i 0 0 0 2.00000i 0 1.00000 8.94427i 0
449.4 0 2.23607 + 2.00000i 0 0 0 2.00000i 0 1.00000 + 8.94427i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.c.e 4
3.b odd 2 1 inner 1200.3.c.e 4
4.b odd 2 1 300.3.b.c 4
5.b even 2 1 inner 1200.3.c.e 4
5.c odd 4 1 240.3.l.a 2
5.c odd 4 1 1200.3.l.r 2
12.b even 2 1 300.3.b.c 4
15.d odd 2 1 inner 1200.3.c.e 4
15.e even 4 1 240.3.l.a 2
15.e even 4 1 1200.3.l.r 2
20.d odd 2 1 300.3.b.c 4
20.e even 4 1 60.3.g.a 2
20.e even 4 1 300.3.g.d 2
40.i odd 4 1 960.3.l.d 2
40.k even 4 1 960.3.l.a 2
60.h even 2 1 300.3.b.c 4
60.l odd 4 1 60.3.g.a 2
60.l odd 4 1 300.3.g.d 2
120.q odd 4 1 960.3.l.a 2
120.w even 4 1 960.3.l.d 2
180.v odd 12 2 1620.3.o.b 4
180.x even 12 2 1620.3.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.g.a 2 20.e even 4 1
60.3.g.a 2 60.l odd 4 1
240.3.l.a 2 5.c odd 4 1
240.3.l.a 2 15.e even 4 1
300.3.b.c 4 4.b odd 2 1
300.3.b.c 4 12.b even 2 1
300.3.b.c 4 20.d odd 2 1
300.3.b.c 4 60.h even 2 1
300.3.g.d 2 20.e even 4 1
300.3.g.d 2 60.l odd 4 1
960.3.l.a 2 40.k even 4 1
960.3.l.a 2 120.q odd 4 1
960.3.l.d 2 40.i odd 4 1
960.3.l.d 2 120.w even 4 1
1200.3.c.e 4 1.a even 1 1 trivial
1200.3.c.e 4 3.b odd 2 1 inner
1200.3.c.e 4 5.b even 2 1 inner
1200.3.c.e 4 15.d odd 2 1 inner
1200.3.l.r 2 5.c odd 4 1
1200.3.l.r 2 15.e even 4 1
1620.3.o.b 4 180.v odd 12 2
1620.3.o.b 4 180.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11}^{2} + 180 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 2 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 4 + T^{2} )^{2} \)
$11$ \( ( 180 + T^{2} )^{2} \)
$13$ \( ( 64 + T^{2} )^{2} \)
$17$ \( ( -180 + T^{2} )^{2} \)
$19$ \( ( 34 + T )^{4} \)
$23$ \( ( -1620 + T^{2} )^{2} \)
$29$ \( ( 1620 + T^{2} )^{2} \)
$31$ \( ( 14 + T )^{4} \)
$37$ \( ( 3136 + T^{2} )^{2} \)
$41$ \( ( 720 + T^{2} )^{2} \)
$43$ \( ( 64 + T^{2} )^{2} \)
$47$ \( ( -1620 + T^{2} )^{2} \)
$53$ \( ( -1620 + T^{2} )^{2} \)
$59$ \( ( 180 + T^{2} )^{2} \)
$61$ \( ( 46 + T )^{4} \)
$67$ \( ( 1024 + T^{2} )^{2} \)
$71$ \( ( 2880 + T^{2} )^{2} \)
$73$ \( ( 11236 + T^{2} )^{2} \)
$79$ \( ( 22 + T )^{4} \)
$83$ \( ( -14580 + T^{2} )^{2} \)
$89$ \( ( 11520 + T^{2} )^{2} \)
$97$ \( ( 14884 + T^{2} )^{2} \)
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