Properties

Label 2352.3.m.d
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( -6 + \beta_{1} ) q^{5} -3 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( -6 + \beta_{1} ) q^{5} -3 q^{9} + ( 6 \beta_{2} + 4 \beta_{3} ) q^{11} + ( -12 + 7 \beta_{1} ) q^{13} + ( 6 \beta_{2} + \beta_{3} ) q^{15} + ( -6 + 5 \beta_{1} ) q^{17} -6 \beta_{3} q^{19} + ( 18 \beta_{2} + 4 \beta_{3} ) q^{23} + ( 13 - 12 \beta_{1} ) q^{25} + 3 \beta_{2} q^{27} + ( 24 - 6 \beta_{1} ) q^{29} + ( 4 \beta_{2} + 18 \beta_{3} ) q^{31} + ( 18 - 12 \beta_{1} ) q^{33} + 24 \beta_{1} q^{37} + ( 12 \beta_{2} + 7 \beta_{3} ) q^{39} + ( 18 + 7 \beta_{1} ) q^{41} + ( -12 \beta_{2} - 20 \beta_{3} ) q^{43} + ( 18 - 3 \beta_{1} ) q^{45} + ( -8 \beta_{2} - 6 \beta_{3} ) q^{47} + ( 6 \beta_{2} + 5 \beta_{3} ) q^{51} + ( -22 + 36 \beta_{1} ) q^{53} + ( -44 \beta_{2} - 30 \beta_{3} ) q^{55} + 18 \beta_{1} q^{57} + ( -28 \beta_{2} + 18 \beta_{3} ) q^{59} + ( -24 - 41 \beta_{1} ) q^{61} + ( 86 - 54 \beta_{1} ) q^{65} + ( -12 \beta_{2} - 4 \beta_{3} ) q^{67} + ( 54 - 12 \beta_{1} ) q^{69} + ( 18 \beta_{2} - 24 \beta_{3} ) q^{71} + ( -12 + 35 \beta_{1} ) q^{73} + ( -13 \beta_{2} - 12 \beta_{3} ) q^{75} + ( 24 \beta_{2} + 20 \beta_{3} ) q^{79} + 9 q^{81} + ( -20 \beta_{2} + 12 \beta_{3} ) q^{83} + ( 46 - 36 \beta_{1} ) q^{85} + ( -24 \beta_{2} - 6 \beta_{3} ) q^{87} + ( 30 - 19 \beta_{1} ) q^{89} + ( 12 - 54 \beta_{1} ) q^{93} + ( 12 \beta_{2} + 36 \beta_{3} ) q^{95} + ( -108 + 11 \beta_{1} ) q^{97} + ( -18 \beta_{2} - 12 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 24q^{5} - 12q^{9} + O(q^{10}) \) \( 4q - 24q^{5} - 12q^{9} - 48q^{13} - 24q^{17} + 52q^{25} + 96q^{29} + 72q^{33} + 72q^{41} + 72q^{45} - 88q^{53} - 96q^{61} + 344q^{65} + 216q^{69} - 48q^{73} + 36q^{81} + 184q^{85} + 120q^{89} + 48q^{93} - 432q^{97} + O(q^{100}) \)

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

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

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
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
0 1.73205i 0 −7.41421 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 −4.58579 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 −7.41421 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 −4.58579 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.d 4
4.b odd 2 1 inner 2352.3.m.d 4
7.b odd 2 1 2352.3.m.l yes 4
28.d even 2 1 2352.3.m.l yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.3.m.d 4 1.a even 1 1 trivial
2352.3.m.d 4 4.b odd 2 1 inner
2352.3.m.l yes 4 7.b odd 2 1
2352.3.m.l yes 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 12 T_{5} + 34 \) acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( 34 + 12 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 144 + 408 T^{2} + T^{4} \)
$13$ \( ( 46 + 24 T + T^{2} )^{2} \)
$17$ \( ( -14 + 12 T + T^{2} )^{2} \)
$19$ \( ( 216 + T^{2} )^{2} \)
$23$ \( 767376 + 2136 T^{2} + T^{4} \)
$29$ \( ( 504 - 48 T + T^{2} )^{2} \)
$31$ \( 3594816 + 3984 T^{2} + T^{4} \)
$37$ \( ( -1152 + T^{2} )^{2} \)
$41$ \( ( 226 - 36 T + T^{2} )^{2} \)
$43$ \( 3873024 + 5664 T^{2} + T^{4} \)
$47$ \( 576 + 816 T^{2} + T^{4} \)
$53$ \( ( -2108 + 44 T + T^{2} )^{2} \)
$59$ \( 166464 + 8592 T^{2} + T^{4} \)
$61$ \( ( -2786 + 48 T + T^{2} )^{2} \)
$67$ \( 112896 + 1056 T^{2} + T^{4} \)
$71$ \( 6170256 + 8856 T^{2} + T^{4} \)
$73$ \( ( -2306 + 24 T + T^{2} )^{2} \)
$79$ \( 451584 + 8256 T^{2} + T^{4} \)
$83$ \( 112896 + 4128 T^{2} + T^{4} \)
$89$ \( ( 178 - 60 T + T^{2} )^{2} \)
$97$ \( ( 11422 + 216 T + T^{2} )^{2} \)
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