Properties

Label 2352.4.a.cr
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.145408.2
Defining polynomial: \(x^{4} - 24 x^{2} + 142\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1176)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( 2 + \beta_{2} + 3 \beta_{3} ) q^{5} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( 2 + \beta_{2} + 3 \beta_{3} ) q^{5} + 9 q^{9} + ( 10 + 3 \beta_{1} - \beta_{2} + 4 \beta_{3} ) q^{11} + ( 12 - 3 \beta_{1} - 23 \beta_{3} ) q^{13} + ( 6 + 3 \beta_{2} + 9 \beta_{3} ) q^{15} + ( -38 - 5 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} ) q^{17} + ( -56 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{19} + ( 2 + 5 \beta_{1} - 13 \beta_{2} - 16 \beta_{3} ) q^{23} + ( -7 + 6 \beta_{1} + 4 \beta_{2} + 20 \beta_{3} ) q^{25} + 27 q^{27} + ( -36 + 3 \beta_{1} + 12 \beta_{2} + 18 \beta_{3} ) q^{29} + ( -100 - \beta_{1} - 6 \beta_{2} - 6 \beta_{3} ) q^{31} + ( 30 + 9 \beta_{1} - 3 \beta_{2} + 12 \beta_{3} ) q^{33} + ( -76 + 10 \beta_{1} - 14 \beta_{2} + 16 \beta_{3} ) q^{37} + ( 36 - 9 \beta_{1} - 69 \beta_{3} ) q^{39} + ( -38 - \beta_{1} + 13 \beta_{2} - 23 \beta_{3} ) q^{41} + ( -40 + 6 \beta_{1} - 30 \beta_{2} - 36 \beta_{3} ) q^{43} + ( 18 + 9 \beta_{2} + 27 \beta_{3} ) q^{45} + ( -136 - 5 \beta_{1} - 10 \beta_{2} - 170 \beta_{3} ) q^{47} + ( -114 - 15 \beta_{1} - 9 \beta_{2} + 21 \beta_{3} ) q^{51} + ( -330 - 18 \beta_{1} - 20 \beta_{2} + 128 \beta_{3} ) q^{53} + ( -4 + 7 \beta_{1} + 26 \beta_{2} + 318 \beta_{3} ) q^{55} + ( -168 - 9 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{57} + ( -260 - 23 \beta_{1} + 6 \beta_{2} - 126 \beta_{3} ) q^{59} + ( 224 + 5 \beta_{1} - 6 \beta_{2} + 37 \beta_{3} ) q^{61} + ( -162 - 29 \beta_{1} - 6 \beta_{2} - 298 \beta_{3} ) q^{65} + ( 104 + 6 \beta_{1} - 6 \beta_{2} - 252 \beta_{3} ) q^{67} + ( 6 + 15 \beta_{1} - 39 \beta_{2} - 48 \beta_{3} ) q^{69} + ( -62 - 27 \beta_{1} + 35 \beta_{2} + 28 \beta_{3} ) q^{71} + ( -188 - 26 \beta_{1} - 30 \beta_{2} + 57 \beta_{3} ) q^{73} + ( -21 + 18 \beta_{1} + 12 \beta_{2} + 60 \beta_{3} ) q^{75} + ( -216 + 12 \beta_{1} - 20 \beta_{2} + 260 \beta_{3} ) q^{79} + 81 q^{81} + ( -364 + 10 \beta_{1} - 64 \beta_{2} + 16 \beta_{3} ) q^{83} + ( -402 - 12 \beta_{1} - 74 \beta_{2} - 604 \beta_{3} ) q^{85} + ( -108 + 9 \beta_{1} + 36 \beta_{2} + 54 \beta_{3} ) q^{87} + ( 734 - 5 \beta_{1} + 23 \beta_{2} - 369 \beta_{3} ) q^{89} + ( -300 - 3 \beta_{1} - 18 \beta_{2} - 18 \beta_{3} ) q^{93} + ( 20 - 2 \beta_{1} - 70 \beta_{2} - 444 \beta_{3} ) q^{95} + ( 36 + 24 \beta_{1} + 54 \beta_{2} + 225 \beta_{3} ) q^{97} + ( 90 + 27 \beta_{1} - 9 \beta_{2} + 36 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} + 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} + 8q^{5} + 36q^{9} + 40q^{11} + 48q^{13} + 24q^{15} - 152q^{17} - 224q^{19} + 8q^{23} - 28q^{25} + 108q^{27} - 144q^{29} - 400q^{31} + 120q^{33} - 304q^{37} + 144q^{39} - 152q^{41} - 160q^{43} + 72q^{45} - 544q^{47} - 456q^{51} - 1320q^{53} - 16q^{55} - 672q^{57} - 1040q^{59} + 896q^{61} - 648q^{65} + 416q^{67} + 24q^{69} - 248q^{71} - 752q^{73} - 84q^{75} - 864q^{79} + 324q^{81} - 1456q^{83} - 1608q^{85} - 432q^{87} + 2936q^{89} - 1200q^{93} + 80q^{95} + 144q^{97} + 360q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 24 x^{2} + 142\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} - 24 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{2} - 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/4\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 12\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 6 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.25358
−3.66254
−3.25358
3.66254
0 3.00000 0 −11.4452 0 0 0 9.00000 0
1.2 0 3.00000 0 −4.11659 0 0 0 9.00000 0
1.3 0 3.00000 0 6.95987 0 0 0 9.00000 0
1.4 0 3.00000 0 16.6019 0 0 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.4.a.cr 4
4.b odd 2 1 1176.4.a.bb 4
7.b odd 2 1 2352.4.a.ck 4
28.d even 2 1 1176.4.a.bc yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.4.a.bb 4 4.b odd 2 1
1176.4.a.bc yes 4 28.d even 2 1
2352.4.a.ck 4 7.b odd 2 1
2352.4.a.cr 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{4} - 8 T_{5}^{3} - 204 T_{5}^{2} + 688 T_{5} + 5444 \)
\( T_{11}^{4} - 40 T_{11}^{3} - 2920 T_{11}^{2} + 79968 T_{11} + 2039184 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T )^{4} \)
$5$ \( 5444 + 688 T - 204 T^{2} - 8 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 2039184 + 79968 T - 2920 T^{2} - 40 T^{3} + T^{4} \)
$13$ \( -692156 + 153312 T - 4708 T^{2} - 48 T^{3} + T^{4} \)
$17$ \( -8992508 - 917008 T - 3820 T^{2} + 152 T^{3} + T^{4} \)
$19$ \( -1381824 + 254976 T + 14960 T^{2} + 224 T^{3} + T^{4} \)
$23$ \( 112149392 - 1217568 T - 38888 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( 37205568 - 3494016 T - 26928 T^{2} + 144 T^{3} + T^{4} \)
$31$ \( 38706752 + 2505088 T + 52176 T^{2} + 400 T^{3} + T^{4} \)
$37$ \( -163223296 - 5559552 T - 33440 T^{2} + 304 T^{3} + T^{4} \)
$41$ \( 165653252 - 2580880 T - 25452 T^{2} + 152 T^{3} + T^{4} \)
$43$ \( 5188983808 - 21880832 T - 170688 T^{2} + 160 T^{3} + T^{4} \)
$47$ \( 1123620416 - 15397376 T - 36624 T^{2} + 544 T^{3} + T^{4} \)
$53$ \( -51605020912 - 126878816 T + 363608 T^{2} + 1320 T^{3} + T^{4} \)
$59$ \( -13864209856 - 85236352 T + 140880 T^{2} + 1040 T^{3} + T^{4} \)
$61$ \( 1143730116 - 34465920 T + 280988 T^{2} - 896 T^{3} + T^{4} \)
$67$ \( 12240034816 + 39976960 T - 207552 T^{2} - 416 T^{3} + T^{4} \)
$71$ \( 259707024 - 19978272 T - 434728 T^{2} + 248 T^{3} + T^{4} \)
$73$ \( -31874858428 - 236159968 T - 283236 T^{2} + 752 T^{3} + T^{4} \)
$79$ \( 4954883072 - 42533888 T - 107200 T^{2} + 864 T^{3} + T^{4} \)
$83$ \( 55227560192 - 367160576 T + 10080 T^{2} + 1456 T^{3} + T^{4} \)
$89$ \( 26681619204 - 681366576 T + 2584084 T^{2} - 2936 T^{3} + T^{4} \)
$97$ \( -32096683836 - 429875424 T - 1058724 T^{2} - 144 T^{3} + T^{4} \)
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