Properties

Label 1200.3.c.g
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: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{7} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{7} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{11} + 15 \zeta_{8}^{2} q^{13} + ( 14 \zeta_{8} - 14 \zeta_{8}^{3} ) q^{17} -23 q^{19} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{21} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{23} + ( -10 \zeta_{8} + 23 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{27} + ( -18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{29} -33 q^{31} + ( 6 \zeta_{8} + 24 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{33} -66 \zeta_{8}^{2} q^{37} + ( -15 - 30 \zeta_{8} - 30 \zeta_{8}^{3} ) q^{39} + ( 26 \zeta_{8} + 26 \zeta_{8}^{3} ) q^{41} + 7 \zeta_{8}^{2} q^{43} + ( 32 \zeta_{8} - 32 \zeta_{8}^{3} ) q^{47} + 48 q^{49} + ( -56 + 14 \zeta_{8} + 14 \zeta_{8}^{3} ) q^{51} + ( -26 \zeta_{8} + 26 \zeta_{8}^{3} ) q^{53} + ( 46 \zeta_{8} - 23 \zeta_{8}^{2} - 46 \zeta_{8}^{3} ) q^{57} + ( -72 \zeta_{8} - 72 \zeta_{8}^{3} ) q^{59} + 39 q^{61} + ( 4 \zeta_{8} + 7 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{63} + 113 \zeta_{8}^{2} q^{67} + ( -8 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{69} + ( -18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{71} + 58 \zeta_{8}^{2} q^{73} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{77} + 70 q^{79} + ( 17 - 56 \zeta_{8} - 56 \zeta_{8}^{3} ) q^{81} + ( 108 \zeta_{8} - 108 \zeta_{8}^{3} ) q^{83} + ( 18 \zeta_{8} + 72 \zeta_{8}^{2} - 18 \zeta_{8}^{3} ) q^{87} + ( -64 \zeta_{8} - 64 \zeta_{8}^{3} ) q^{89} -15 q^{91} + ( 66 \zeta_{8} - 33 \zeta_{8}^{2} - 66 \zeta_{8}^{3} ) q^{93} + \zeta_{8}^{2} q^{97} + ( -48 - 42 \zeta_{8} - 42 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 28q^{9} + O(q^{10}) \) \( 4q + 28q^{9} - 92q^{19} - 4q^{21} - 132q^{31} - 60q^{39} + 192q^{49} - 224q^{51} + 156q^{61} - 32q^{69} + 280q^{79} + 68q^{81} - 60q^{91} - 192q^{99} + O(q^{100}) \)

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
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 −2.82843 1.00000i 0 0 0 1.00000i 0 7.00000 + 5.65685i 0
449.2 0 −2.82843 + 1.00000i 0 0 0 1.00000i 0 7.00000 5.65685i 0
449.3 0 2.82843 1.00000i 0 0 0 1.00000i 0 7.00000 5.65685i 0
449.4 0 2.82843 + 1.00000i 0 0 0 1.00000i 0 7.00000 + 5.65685i 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.g 4
3.b odd 2 1 inner 1200.3.c.g 4
4.b odd 2 1 600.3.c.b 4
5.b even 2 1 inner 1200.3.c.g 4
5.c odd 4 1 1200.3.l.j 2
5.c odd 4 1 1200.3.l.o 2
12.b even 2 1 600.3.c.b 4
15.d odd 2 1 inner 1200.3.c.g 4
15.e even 4 1 1200.3.l.j 2
15.e even 4 1 1200.3.l.o 2
20.d odd 2 1 600.3.c.b 4
20.e even 4 1 600.3.l.a 2
20.e even 4 1 600.3.l.c yes 2
60.h even 2 1 600.3.c.b 4
60.l odd 4 1 600.3.l.a 2
60.l odd 4 1 600.3.l.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.c.b 4 4.b odd 2 1
600.3.c.b 4 12.b even 2 1
600.3.c.b 4 20.d odd 2 1
600.3.c.b 4 60.h even 2 1
600.3.l.a 2 20.e even 4 1
600.3.l.a 2 60.l odd 4 1
600.3.l.c yes 2 20.e even 4 1
600.3.l.c yes 2 60.l odd 4 1
1200.3.c.g 4 1.a even 1 1 trivial
1200.3.c.g 4 3.b odd 2 1 inner
1200.3.c.g 4 5.b even 2 1 inner
1200.3.c.g 4 15.d odd 2 1 inner
1200.3.l.j 2 5.c odd 4 1
1200.3.l.j 2 15.e even 4 1
1200.3.l.o 2 5.c odd 4 1
1200.3.l.o 2 15.e even 4 1

Hecke kernels

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

\( T_{7}^{2} + 1 \)
\( T_{11}^{2} + 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 14 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 72 + T^{2} )^{2} \)
$13$ \( ( 225 + T^{2} )^{2} \)
$17$ \( ( -392 + T^{2} )^{2} \)
$19$ \( ( 23 + T )^{4} \)
$23$ \( ( -8 + T^{2} )^{2} \)
$29$ \( ( 648 + T^{2} )^{2} \)
$31$ \( ( 33 + T )^{4} \)
$37$ \( ( 4356 + T^{2} )^{2} \)
$41$ \( ( 1352 + T^{2} )^{2} \)
$43$ \( ( 49 + T^{2} )^{2} \)
$47$ \( ( -2048 + T^{2} )^{2} \)
$53$ \( ( -1352 + T^{2} )^{2} \)
$59$ \( ( 10368 + T^{2} )^{2} \)
$61$ \( ( -39 + T )^{4} \)
$67$ \( ( 12769 + T^{2} )^{2} \)
$71$ \( ( 648 + T^{2} )^{2} \)
$73$ \( ( 3364 + T^{2} )^{2} \)
$79$ \( ( -70 + T )^{4} \)
$83$ \( ( -23328 + T^{2} )^{2} \)
$89$ \( ( 8192 + T^{2} )^{2} \)
$97$ \( ( 1 + T^{2} )^{2} \)
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