Properties

Label 1200.3.c.j
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: \(\Q(i, \sqrt{35})\)
Defining polynomial: \(x^{4} - 17 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -8 \beta_{2} q^{7} + ( 9 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -8 \beta_{2} q^{7} + ( 9 + \beta_{3} ) q^{9} + ( 3 + 6 \beta_{3} ) q^{11} -2 \beta_{2} q^{13} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{17} + 11 q^{19} -8 \beta_{3} q^{21} + ( 12 \beta_{1} - 6 \beta_{2} ) q^{23} + ( 8 \beta_{1} + 9 \beta_{2} ) q^{27} + ( 6 + 12 \beta_{3} ) q^{29} + 46 q^{31} + ( -3 \beta_{1} + 54 \beta_{2} ) q^{33} -16 \beta_{2} q^{37} -2 \beta_{3} q^{39} + ( -9 - 18 \beta_{3} ) q^{41} + 62 \beta_{2} q^{43} + ( 12 \beta_{1} - 6 \beta_{2} ) q^{47} -15 q^{49} + ( -54 - 3 \beta_{3} ) q^{51} + ( 12 \beta_{1} - 6 \beta_{2} ) q^{53} + 11 \beta_{1} q^{57} + ( 12 + 24 \beta_{3} ) q^{59} -16 q^{61} + ( 8 \beta_{1} - 72 \beta_{2} ) q^{63} -113 \beta_{2} q^{67} + ( 108 + 6 \beta_{3} ) q^{69} + ( -18 - 36 \beta_{3} ) q^{71} -101 \beta_{2} q^{73} + ( 48 \beta_{1} - 24 \beta_{2} ) q^{77} + 68 q^{79} + ( 72 + 17 \beta_{3} ) q^{81} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -6 \beta_{1} + 108 \beta_{2} ) q^{87} + ( 9 + 18 \beta_{3} ) q^{89} -16 q^{91} + 46 \beta_{1} q^{93} -22 \beta_{2} q^{97} + ( -27 + 51 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 34q^{9} + O(q^{10}) \) \( 4q + 34q^{9} + 44q^{19} + 16q^{21} + 184q^{31} + 4q^{39} - 60q^{49} - 210q^{51} - 64q^{61} + 420q^{69} + 272q^{79} + 254q^{81} - 64q^{91} - 210q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 17 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 8 \nu \)\()/9\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 9\)
\(\nu^{3}\)\(=\)\(9 \beta_{2} + 8 \beta_{1}\)

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
−2.95804 0.500000i
−2.95804 + 0.500000i
2.95804 0.500000i
2.95804 + 0.500000i
0 −2.95804 0.500000i 0 0 0 8.00000i 0 8.50000 + 2.95804i 0
449.2 0 −2.95804 + 0.500000i 0 0 0 8.00000i 0 8.50000 2.95804i 0
449.3 0 2.95804 0.500000i 0 0 0 8.00000i 0 8.50000 2.95804i 0
449.4 0 2.95804 + 0.500000i 0 0 0 8.00000i 0 8.50000 + 2.95804i 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.j 4
3.b odd 2 1 inner 1200.3.c.j 4
4.b odd 2 1 300.3.b.d 4
5.b even 2 1 inner 1200.3.c.j 4
5.c odd 4 1 1200.3.l.k 2
5.c odd 4 1 1200.3.l.m 2
12.b even 2 1 300.3.b.d 4
15.d odd 2 1 inner 1200.3.c.j 4
15.e even 4 1 1200.3.l.k 2
15.e even 4 1 1200.3.l.m 2
20.d odd 2 1 300.3.b.d 4
20.e even 4 1 300.3.g.f 2
20.e even 4 1 300.3.g.g yes 2
60.h even 2 1 300.3.b.d 4
60.l odd 4 1 300.3.g.f 2
60.l odd 4 1 300.3.g.g yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.b.d 4 4.b odd 2 1
300.3.b.d 4 12.b even 2 1
300.3.b.d 4 20.d odd 2 1
300.3.b.d 4 60.h even 2 1
300.3.g.f 2 20.e even 4 1
300.3.g.f 2 60.l odd 4 1
300.3.g.g yes 2 20.e even 4 1
300.3.g.g yes 2 60.l odd 4 1
1200.3.c.j 4 1.a even 1 1 trivial
1200.3.c.j 4 3.b odd 2 1 inner
1200.3.c.j 4 5.b even 2 1 inner
1200.3.c.j 4 15.d odd 2 1 inner
1200.3.l.k 2 5.c odd 4 1
1200.3.l.k 2 15.e even 4 1
1200.3.l.m 2 5.c odd 4 1
1200.3.l.m 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 \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 64 \)
\( T_{11}^{2} + 315 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 17 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 64 + T^{2} )^{2} \)
$11$ \( ( 315 + T^{2} )^{2} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( ( -315 + T^{2} )^{2} \)
$19$ \( ( -11 + T )^{4} \)
$23$ \( ( -1260 + T^{2} )^{2} \)
$29$ \( ( 1260 + T^{2} )^{2} \)
$31$ \( ( -46 + T )^{4} \)
$37$ \( ( 256 + T^{2} )^{2} \)
$41$ \( ( 2835 + T^{2} )^{2} \)
$43$ \( ( 3844 + T^{2} )^{2} \)
$47$ \( ( -1260 + T^{2} )^{2} \)
$53$ \( ( -1260 + T^{2} )^{2} \)
$59$ \( ( 5040 + T^{2} )^{2} \)
$61$ \( ( 16 + T )^{4} \)
$67$ \( ( 12769 + T^{2} )^{2} \)
$71$ \( ( 11340 + T^{2} )^{2} \)
$73$ \( ( 10201 + T^{2} )^{2} \)
$79$ \( ( -68 + T )^{4} \)
$83$ \( ( -315 + T^{2} )^{2} \)
$89$ \( ( 2835 + T^{2} )^{2} \)
$97$ \( ( 484 + T^{2} )^{2} \)
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