Properties

Label 2352.4.a.cj
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.58461.1
Defining polynomial: \(x^{3} - x^{2} - 65 x - 126\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( 4 + \beta_{1} ) q^{5} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( 4 + \beta_{1} ) q^{5} + 9 q^{9} + ( 6 - \beta_{1} + \beta_{2} ) q^{11} + ( 7 - \beta_{1} + 3 \beta_{2} ) q^{13} + ( 12 + 3 \beta_{1} ) q^{15} + ( 34 - 2 \beta_{1} - \beta_{2} ) q^{17} + ( 69 + 5 \beta_{1} + 8 \beta_{2} ) q^{19} + ( 94 + 2 \beta_{1} - 7 \beta_{2} ) q^{23} + ( 67 + 15 \beta_{1} + 7 \beta_{2} ) q^{25} + 27 q^{27} + ( -28 - 11 \beta_{1} - 5 \beta_{2} ) q^{29} + ( -39 + 14 \beta_{1} - \beta_{2} ) q^{31} + ( 18 - 3 \beta_{1} + 3 \beta_{2} ) q^{33} + ( -113 - 13 \beta_{1} - 3 \beta_{2} ) q^{37} + ( 21 - 3 \beta_{1} + 9 \beta_{2} ) q^{39} + ( 166 - 18 \beta_{1} ) q^{41} + ( -13 - 3 \beta_{1} + 20 \beta_{2} ) q^{43} + ( 36 + 9 \beta_{1} ) q^{45} + ( -36 + 18 \beta_{1} - 11 \beta_{2} ) q^{47} + ( 102 - 6 \beta_{1} - 3 \beta_{2} ) q^{51} + ( -134 - 17 \beta_{1} - 4 \beta_{2} ) q^{53} + ( -204 - \beta_{1} - 11 \beta_{2} ) q^{55} + ( 207 + 15 \beta_{1} + 24 \beta_{2} ) q^{57} + ( 94 - 3 \beta_{1} - 16 \beta_{2} ) q^{59} + ( 22 + 32 \beta_{1} + 20 \beta_{2} ) q^{61} + ( -304 + 8 \beta_{1} - 19 \beta_{2} ) q^{65} + ( 25 - 25 \beta_{1} - 6 \beta_{2} ) q^{67} + ( 282 + 6 \beta_{1} - 21 \beta_{2} ) q^{69} + ( -16 - 14 \beta_{1} - 51 \beta_{2} ) q^{71} + ( 47 + 33 \beta_{1} - 39 \beta_{2} ) q^{73} + ( 201 + 45 \beta_{1} + 21 \beta_{2} ) q^{75} + ( 821 + 15 \beta_{2} ) q^{79} + 81 q^{81} + ( -32 + 19 \beta_{1} + 40 \beta_{2} ) q^{83} + ( -164 + 8 \beta_{1} - 10 \beta_{2} ) q^{85} + ( -84 - 33 \beta_{1} - 15 \beta_{2} ) q^{87} + ( 60 + 70 \beta_{1} - 24 \beta_{2} ) q^{89} + ( -117 + 42 \beta_{1} - 3 \beta_{2} ) q^{93} + ( 740 + 156 \beta_{1} + 3 \beta_{2} ) q^{95} + ( 982 + 5 \beta_{1} - 13 \beta_{2} ) q^{97} + ( 54 - 9 \beta_{1} + 9 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 9q^{3} + 11q^{5} + 27q^{9} + O(q^{10}) \) \( 3q + 9q^{3} + 11q^{5} + 27q^{9} + 19q^{11} + 22q^{13} + 33q^{15} + 104q^{17} + 202q^{19} + 280q^{23} + 186q^{25} + 81q^{27} - 73q^{29} - 131q^{31} + 57q^{33} - 326q^{37} + 66q^{39} + 516q^{41} - 36q^{43} + 99q^{45} - 126q^{47} + 312q^{51} - 385q^{53} - 611q^{55} + 606q^{57} + 285q^{59} + 34q^{61} - 920q^{65} + 100q^{67} + 840q^{69} - 34q^{71} + 108q^{73} + 558q^{75} + 2463q^{79} + 243q^{81} - 115q^{83} - 500q^{85} - 219q^{87} + 110q^{89} - 393q^{93} + 2064q^{95} + 2941q^{97} + 171q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 65 x - 126\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{2} + \nu - 45 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{2} + 10 \nu + 84 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{2} + 10 \beta_{1} + 178\)\()/4\)

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
−2.16736
−6.20369
9.37106
0 3.00000 0 −10.1566 0 0 0 9.00000 0
1.2 0 3.00000 0 −0.239289 0 0 0 9.00000 0
1.3 0 3.00000 0 21.3959 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.cj 3
4.b odd 2 1 1176.4.a.x 3
7.b odd 2 1 2352.4.a.ch 3
7.d odd 6 2 336.4.q.l 6
28.d even 2 1 1176.4.a.y 3
28.f even 6 2 168.4.q.e 6
84.j odd 6 2 504.4.s.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.e 6 28.f even 6 2
336.4.q.l 6 7.d odd 6 2
504.4.s.g 6 84.j odd 6 2
1176.4.a.x 3 4.b odd 2 1
1176.4.a.y 3 28.d even 2 1
2352.4.a.ch 3 7.b odd 2 1
2352.4.a.cj 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{3} - 11 T_{5}^{2} - 220 T_{5} - 52 \)
\( T_{11}^{3} - 19 T_{11}^{2} - 624 T_{11} - 3276 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -3 + T )^{3} \)
$5$ \( -52 - 220 T - 11 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( -3276 - 624 T - 19 T^{2} + T^{3} \)
$13$ \( -26976 - 3495 T - 22 T^{2} + T^{3} \)
$17$ \( 4032 + 2560 T - 104 T^{2} + T^{3} \)
$19$ \( 2225968 - 7243 T - 202 T^{2} + T^{3} \)
$23$ \( 1532736 + 6976 T - 280 T^{2} + T^{3} \)
$29$ \( 970992 - 29024 T + 73 T^{2} + T^{3} \)
$31$ \( -4156607 - 47869 T + 131 T^{2} + T^{3} \)
$37$ \( -17796 - 5247 T + 326 T^{2} + T^{3} \)
$41$ \( 15002144 + 4404 T - 516 T^{2} + T^{3} \)
$43$ \( -7204222 - 141111 T + 36 T^{2} + T^{3} \)
$47$ \( 11682152 - 149940 T + 126 T^{2} + T^{3} \)
$53$ \( 178128 - 20132 T + 385 T^{2} + T^{3} \)
$59$ \( 1793232 - 50532 T - 285 T^{2} + T^{3} \)
$61$ \( -22240152 - 293396 T - 34 T^{2} + T^{3} \)
$67$ \( 27246966 - 147039 T - 100 T^{2} + T^{3} \)
$71$ \( -207049704 - 779124 T + 34 T^{2} + T^{3} \)
$73$ \( 409658074 - 978339 T - 108 T^{2} + T^{3} \)
$79$ \( -492780761 + 1949223 T - 2463 T^{2} + T^{3} \)
$83$ \( 111709668 - 486372 T + 115 T^{2} + T^{3} \)
$89$ \( 347278464 - 1727024 T - 110 T^{2} + T^{3} \)
$97$ \( -866695284 + 2811496 T - 2941 T^{2} + T^{3} \)
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