Properties

Label 2352.4.a.by
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{113}) \)
Defining polynomial: \(x^{2} - x - 28\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{113}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( -3 - \beta ) q^{5} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( -3 - \beta ) q^{5} + 9 q^{9} + ( -1 - 3 \beta ) q^{11} -4 \beta q^{13} + ( -9 - 3 \beta ) q^{15} + ( -15 - \beta ) q^{17} + ( 6 - 2 \beta ) q^{19} + ( -39 - 9 \beta ) q^{23} + ( -3 + 6 \beta ) q^{25} + 27 q^{27} + ( 28 - 18 \beta ) q^{29} + ( 66 - 10 \beta ) q^{31} + ( -3 - 9 \beta ) q^{33} + ( 160 + 18 \beta ) q^{37} -12 \beta q^{39} + ( -9 + 25 \beta ) q^{41} -188 q^{43} + ( -27 - 9 \beta ) q^{45} + ( -120 - 16 \beta ) q^{47} + ( -45 - 3 \beta ) q^{51} + ( -36 - 30 \beta ) q^{53} + ( 342 + 10 \beta ) q^{55} + ( 18 - 6 \beta ) q^{57} + ( 276 + 32 \beta ) q^{59} + ( 246 + 14 \beta ) q^{61} + ( 452 + 12 \beta ) q^{65} + ( 270 - 42 \beta ) q^{67} + ( -117 - 27 \beta ) q^{69} + ( -429 - 27 \beta ) q^{71} + ( -450 + 18 \beta ) q^{73} + ( -9 + 18 \beta ) q^{75} + ( 310 + 66 \beta ) q^{79} + 81 q^{81} + ( 612 - 32 \beta ) q^{83} + ( 158 + 18 \beta ) q^{85} + ( 84 - 54 \beta ) q^{87} + ( 711 + 33 \beta ) q^{89} + ( 198 - 30 \beta ) q^{93} + 208 q^{95} + ( -558 - 58 \beta ) q^{97} + ( -9 - 27 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} - 6q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} - 6q^{5} + 18q^{9} - 2q^{11} - 18q^{15} - 30q^{17} + 12q^{19} - 78q^{23} - 6q^{25} + 54q^{27} + 56q^{29} + 132q^{31} - 6q^{33} + 320q^{37} - 18q^{41} - 376q^{43} - 54q^{45} - 240q^{47} - 90q^{51} - 72q^{53} + 684q^{55} + 36q^{57} + 552q^{59} + 492q^{61} + 904q^{65} + 540q^{67} - 234q^{69} - 858q^{71} - 900q^{73} - 18q^{75} + 620q^{79} + 162q^{81} + 1224q^{83} + 316q^{85} + 168q^{87} + 1422q^{89} + 396q^{93} + 416q^{95} - 1116q^{97} - 18q^{99} + O(q^{100}) \)

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
5.81507
−4.81507
0 3.00000 0 −13.6301 0 0 0 9.00000 0
1.2 0 3.00000 0 7.63015 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.by 2
4.b odd 2 1 1176.4.a.q 2
7.b odd 2 1 2352.4.a.bs 2
28.d even 2 1 1176.4.a.v yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.4.a.q 2 4.b odd 2 1
1176.4.a.v yes 2 28.d even 2 1
2352.4.a.bs 2 7.b odd 2 1
2352.4.a.by 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{2} + 6 T_{5} - 104 \)
\( T_{11}^{2} + 2 T_{11} - 1016 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( -104 + 6 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -1016 + 2 T + T^{2} \)
$13$ \( -1808 + T^{2} \)
$17$ \( 112 + 30 T + T^{2} \)
$19$ \( -416 - 12 T + T^{2} \)
$23$ \( -7632 + 78 T + T^{2} \)
$29$ \( -35828 - 56 T + T^{2} \)
$31$ \( -6944 - 132 T + T^{2} \)
$37$ \( -11012 - 320 T + T^{2} \)
$41$ \( -70544 + 18 T + T^{2} \)
$43$ \( ( 188 + T )^{2} \)
$47$ \( -14528 + 240 T + T^{2} \)
$53$ \( -100404 + 72 T + T^{2} \)
$59$ \( -39536 - 552 T + T^{2} \)
$61$ \( 38368 - 492 T + T^{2} \)
$67$ \( -126432 - 540 T + T^{2} \)
$71$ \( 101664 + 858 T + T^{2} \)
$73$ \( 165888 + 900 T + T^{2} \)
$79$ \( -396128 - 620 T + T^{2} \)
$83$ \( 258832 - 1224 T + T^{2} \)
$89$ \( 382464 - 1422 T + T^{2} \)
$97$ \( -68768 + 1116 T + T^{2} \)
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