Properties

Label 2352.3.m.k
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: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial: \(x^{4} - x^{3} - x^{2} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 336)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( 5 - \beta_{1} ) q^{5} -3 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( 5 - \beta_{1} ) q^{5} -3 q^{9} + ( 3 \beta_{2} + 5 \beta_{3} ) q^{11} + ( -8 + 2 \beta_{1} ) q^{13} + ( -5 \beta_{2} + 3 \beta_{3} ) q^{15} + ( -5 + \beta_{1} ) q^{17} -8 \beta_{3} q^{19} + ( -7 \beta_{2} + 11 \beta_{3} ) q^{23} + ( 21 - 10 \beta_{1} ) q^{25} + 3 \beta_{2} q^{27} + ( -14 + 4 \beta_{1} ) q^{29} -12 \beta_{2} q^{31} + ( 9 + 5 \beta_{1} ) q^{33} + ( -52 - 2 \beta_{1} ) q^{37} + ( 8 \beta_{2} - 6 \beta_{3} ) q^{39} + ( -23 + 7 \beta_{1} ) q^{41} + ( -8 \beta_{2} + 8 \beta_{3} ) q^{43} + ( -15 + 3 \beta_{1} ) q^{45} + ( -2 \beta_{2} - 6 \beta_{3} ) q^{47} + ( 5 \beta_{2} - 3 \beta_{3} ) q^{51} + ( 28 + 14 \beta_{1} ) q^{53} + ( -20 \beta_{2} + 16 \beta_{3} ) q^{55} -8 \beta_{1} q^{57} + ( -42 \beta_{2} - 10 \beta_{3} ) q^{59} + ( -42 - 8 \beta_{1} ) q^{61} + ( -82 + 18 \beta_{1} ) q^{65} + ( -54 \beta_{2} + 6 \beta_{3} ) q^{67} + ( -21 + 11 \beta_{1} ) q^{69} + ( -33 \beta_{2} + 5 \beta_{3} ) q^{71} + ( 8 + 10 \beta_{1} ) q^{73} + ( -21 \beta_{2} + 30 \beta_{3} ) q^{75} + ( -34 \beta_{2} - 6 \beta_{3} ) q^{79} + 9 q^{81} + ( 24 \beta_{2} - 28 \beta_{3} ) q^{83} + ( -46 + 10 \beta_{1} ) q^{85} + ( 14 \beta_{2} - 12 \beta_{3} ) q^{87} + ( -59 - 5 \beta_{1} ) q^{89} -36 q^{93} + ( 56 \beta_{2} - 40 \beta_{3} ) q^{95} + ( 76 - 10 \beta_{1} ) q^{97} + ( -9 \beta_{2} - 15 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 20q^{5} - 12q^{9} + O(q^{10}) \) \( 4q + 20q^{5} - 12q^{9} - 32q^{13} - 20q^{17} + 84q^{25} - 56q^{29} + 36q^{33} - 208q^{37} - 92q^{41} - 60q^{45} + 112q^{53} - 168q^{61} - 328q^{65} - 84q^{69} + 32q^{73} + 36q^{81} - 184q^{85} - 236q^{89} - 144q^{93} + 304q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} + \nu^{2} + 3 \nu + 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} + \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{3} - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 3 \beta_{2} + \beta_{1} + 3\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} + 5\)\()/2\)

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
1.39564 + 0.228425i
−0.895644 1.09445i
1.39564 0.228425i
−0.895644 + 1.09445i
0 1.73205i 0 0.417424 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 9.58258 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 0.417424 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 9.58258 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.k 4
4.b odd 2 1 inner 2352.3.m.k 4
7.b odd 2 1 336.3.m.a 4
21.c even 2 1 1008.3.m.f 4
28.d even 2 1 336.3.m.a 4
56.e even 2 1 1344.3.m.c 4
56.h odd 2 1 1344.3.m.c 4
84.h odd 2 1 1008.3.m.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.m.a 4 7.b odd 2 1
336.3.m.a 4 28.d even 2 1
1008.3.m.f 4 21.c even 2 1
1008.3.m.f 4 84.h odd 2 1
1344.3.m.c 4 56.e even 2 1
1344.3.m.c 4 56.h odd 2 1
2352.3.m.k 4 1.a even 1 1 trivial
2352.3.m.k 4 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 10 T_{5} + 4 \) acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( 4 - 10 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 21904 + 404 T^{2} + T^{4} \)
$13$ \( ( -20 + 16 T + T^{2} )^{2} \)
$17$ \( ( 4 + 10 T + T^{2} )^{2} \)
$19$ \( ( 448 + T^{2} )^{2} \)
$23$ \( 490000 + 1988 T^{2} + T^{4} \)
$29$ \( ( -140 + 28 T + T^{2} )^{2} \)
$31$ \( ( 432 + T^{2} )^{2} \)
$37$ \( ( 2620 + 104 T + T^{2} )^{2} \)
$41$ \( ( -500 + 46 T + T^{2} )^{2} \)
$43$ \( 65536 + 1280 T^{2} + T^{4} \)
$47$ \( 57600 + 528 T^{2} + T^{4} \)
$53$ \( ( -3332 - 56 T + T^{2} )^{2} \)
$59$ \( 21086464 + 11984 T^{2} + T^{4} \)
$61$ \( ( 420 + 84 T + T^{2} )^{2} \)
$67$ \( 72182016 + 18000 T^{2} + T^{4} \)
$71$ \( 9560464 + 6884 T^{2} + T^{4} \)
$73$ \( ( -2036 - 16 T + T^{2} )^{2} \)
$79$ \( 10342656 + 7440 T^{2} + T^{4} \)
$83$ \( 14137600 + 14432 T^{2} + T^{4} \)
$89$ \( ( 2956 + 118 T + T^{2} )^{2} \)
$97$ \( ( 3676 - 152 T + T^{2} )^{2} \)
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