Properties

Label 2352.3.m.j
Level $2352$
Weight $3$
Character orbit 2352.m
Analytic conductor $64.087$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{37})\)
Defining polynomial: \(x^{4} - x^{3} + 10 x^{2} + 9 x + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( 3 + \beta_{2} ) q^{5} -3 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( 3 + \beta_{2} ) q^{5} -3 q^{9} + ( \beta_{1} + \beta_{3} ) q^{11} + ( -3 \beta_{1} + \beta_{3} ) q^{15} + ( -9 + \beta_{2} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{19} + ( 11 \beta_{1} - \beta_{3} ) q^{23} + ( 21 + 6 \beta_{2} ) q^{25} + 3 \beta_{1} q^{27} + ( 12 + 6 \beta_{2} ) q^{29} + ( 10 \beta_{1} - 2 \beta_{3} ) q^{31} + ( 3 - 3 \beta_{2} ) q^{33} + ( 16 - 6 \beta_{2} ) q^{37} + ( 33 - \beta_{2} ) q^{41} -16 \beta_{1} q^{43} + ( -9 - 3 \beta_{2} ) q^{45} + ( -12 \beta_{1} + 4 \beta_{3} ) q^{47} + ( 9 \beta_{1} + \beta_{3} ) q^{51} + ( 48 + 6 \beta_{2} ) q^{53} + ( -34 \beta_{1} + 2 \beta_{3} ) q^{55} + ( -6 + 6 \beta_{2} ) q^{57} + 4 \beta_{3} q^{59} + ( 66 + 6 \beta_{2} ) q^{61} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{67} + ( 33 + 3 \beta_{2} ) q^{69} + ( -7 \beta_{1} + 5 \beta_{3} ) q^{71} + ( -6 + 6 \beta_{2} ) q^{73} + ( -21 \beta_{1} + 6 \beta_{3} ) q^{75} + ( 50 \beta_{1} - 6 \beta_{3} ) q^{79} + 9 q^{81} + ( -24 \beta_{1} - 4 \beta_{3} ) q^{83} + ( 10 - 6 \beta_{2} ) q^{85} + ( -12 \beta_{1} + 6 \beta_{3} ) q^{87} + ( 69 - 13 \beta_{2} ) q^{89} + ( 30 + 6 \beta_{2} ) q^{93} + ( 68 \beta_{1} - 4 \beta_{3} ) q^{95} + ( 150 - 6 \beta_{2} ) q^{97} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{5} - 12q^{9} + O(q^{10}) \) \( 4q + 12q^{5} - 12q^{9} - 36q^{17} + 84q^{25} + 48q^{29} + 12q^{33} + 64q^{37} + 132q^{41} - 36q^{45} + 192q^{53} - 24q^{57} + 264q^{61} + 132q^{69} - 24q^{73} + 36q^{81} + 40q^{85} + 276q^{89} + 120q^{93} + 600q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 10 x^{2} + 9 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 10 \nu^{2} - 10 \nu + 36 \)\()/45\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 14 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} - 2 \nu^{2} + 38 \nu + 9 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 19 \beta_{1} - 19\)\()/4\)
\(\nu^{3}\)\(=\)\(5 \beta_{2} - 14\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
1.77069 + 3.06693i
−1.27069 2.20090i
1.77069 3.06693i
−1.27069 + 2.20090i
0 1.73205i 0 −3.08276 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 9.08276 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 −3.08276 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 9.08276 0 0 0 −3.00000 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.3.m.j yes 4
4.b odd 2 1 inner 2352.3.m.j yes 4
7.b odd 2 1 2352.3.m.e 4
28.d even 2 1 2352.3.m.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.3.m.e 4 7.b odd 2 1
2352.3.m.e 4 28.d even 2 1
2352.3.m.j yes 4 1.a even 1 1 trivial
2352.3.m.j yes 4 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6 T_{5} - 28 \) acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( -28 - 6 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 11664 + 228 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 44 + 18 T + T^{2} )^{2} \)
$19$ \( 186624 + 912 T^{2} + T^{4} \)
$23$ \( 63504 + 948 T^{2} + T^{4} \)
$29$ \( ( -1188 - 24 T + T^{2} )^{2} \)
$31$ \( 20736 + 1488 T^{2} + T^{4} \)
$37$ \( ( -1076 - 32 T + T^{2} )^{2} \)
$41$ \( ( 1052 - 66 T + T^{2} )^{2} \)
$43$ \( ( 768 + T^{2} )^{2} \)
$47$ \( 1806336 + 4416 T^{2} + T^{4} \)
$53$ \( ( 972 - 96 T + T^{2} )^{2} \)
$59$ \( ( 1776 + T^{2} )^{2} \)
$61$ \( ( 3024 - 132 T + T^{2} )^{2} \)
$67$ \( 15116544 + 8208 T^{2} + T^{4} \)
$71$ \( 6906384 + 5844 T^{2} + T^{4} \)
$73$ \( ( -1296 + 12 T + T^{2} )^{2} \)
$79$ \( 12278016 + 22992 T^{2} + T^{4} \)
$83$ \( 2304 + 7008 T^{2} + T^{4} \)
$89$ \( ( -1492 - 138 T + T^{2} )^{2} \)
$97$ \( ( 21168 - 300 T + T^{2} )^{2} \)
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