Properties

Label 2352.4.a.bx
Level $2352$
Weight $4$
Character orbit 2352.a
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Defining polynomial: \(x^{2} - x - 48\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{193})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( -5 - \beta ) q^{5} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( -5 - \beta ) q^{5} + 9 q^{9} + ( -1 + 7 \beta ) q^{11} -5 \beta q^{13} + ( -15 - 3 \beta ) q^{15} + ( -52 + 4 \beta ) q^{17} + ( 32 + 3 \beta ) q^{19} + ( -28 - 20 \beta ) q^{23} + ( -52 + 11 \beta ) q^{25} + 27 q^{27} + ( 143 - 11 \beta ) q^{29} + ( 171 + 20 \beta ) q^{31} + ( -3 + 21 \beta ) q^{33} + ( 20 - 45 \beta ) q^{37} -15 \beta q^{39} + ( 72 + 18 \beta ) q^{41} + ( -362 + 3 \beta ) q^{43} + ( -45 - 9 \beta ) q^{45} + ( -126 + 36 \beta ) q^{47} + ( -156 + 12 \beta ) q^{51} + ( 243 + 9 \beta ) q^{53} + ( -331 - 41 \beta ) q^{55} + ( 96 + 9 \beta ) q^{57} + ( 113 - 53 \beta ) q^{59} + ( 246 + 40 \beta ) q^{61} + ( 240 + 30 \beta ) q^{65} + ( -94 + 77 \beta ) q^{67} + ( -84 - 60 \beta ) q^{69} + ( -778 - 44 \beta ) q^{71} + ( -634 + 53 \beta ) q^{73} + ( -156 + 33 \beta ) q^{75} + ( -699 - 62 \beta ) q^{79} + 81 q^{81} + ( -755 + 101 \beta ) q^{83} + ( 68 + 28 \beta ) q^{85} + ( 429 - 33 \beta ) q^{87} + ( 966 + 42 \beta ) q^{89} + ( 513 + 60 \beta ) q^{93} + ( -304 - 50 \beta ) q^{95} + ( -295 + 29 \beta ) q^{97} + ( -9 + 63 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} - 11q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} - 11q^{5} + 18q^{9} + 5q^{11} - 5q^{13} - 33q^{15} - 100q^{17} + 67q^{19} - 76q^{23} - 93q^{25} + 54q^{27} + 275q^{29} + 362q^{31} + 15q^{33} - 5q^{37} - 15q^{39} + 162q^{41} - 721q^{43} - 99q^{45} - 216q^{47} - 300q^{51} + 495q^{53} - 703q^{55} + 201q^{57} + 173q^{59} + 532q^{61} + 510q^{65} - 111q^{67} - 228q^{69} - 1600q^{71} - 1215q^{73} - 279q^{75} - 1460q^{79} + 162q^{81} - 1409q^{83} + 164q^{85} + 825q^{87} + 1974q^{89} + 1086q^{93} - 658q^{95} - 561q^{97} + 45q^{99} + O(q^{100}) \)

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
7.44622
−6.44622
0 3.00000 0 −12.4462 0 0 0 9.00000 0
1.2 0 3.00000 0 1.44622 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.bx 2
4.b odd 2 1 588.4.a.f 2
7.b odd 2 1 2352.4.a.bt 2
7.d odd 6 2 336.4.q.i 4
12.b even 2 1 1764.4.a.y 2
28.d even 2 1 588.4.a.i 2
28.f even 6 2 84.4.i.a 4
28.g odd 6 2 588.4.i.j 4
84.h odd 2 1 1764.4.a.o 2
84.j odd 6 2 252.4.k.f 4
84.n even 6 2 1764.4.k.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.a 4 28.f even 6 2
252.4.k.f 4 84.j odd 6 2
336.4.q.i 4 7.d odd 6 2
588.4.a.f 2 4.b odd 2 1
588.4.a.i 2 28.d even 2 1
588.4.i.j 4 28.g odd 6 2
1764.4.a.o 2 84.h odd 2 1
1764.4.a.y 2 12.b even 2 1
1764.4.k.q 4 84.n even 6 2
2352.4.a.bt 2 7.b odd 2 1
2352.4.a.bx 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{2} + 11 T_{5} - 18 \)
\( T_{11}^{2} - 5 T_{11} - 2358 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( -18 + 11 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -2358 - 5 T + T^{2} \)
$13$ \( -1200 + 5 T + T^{2} \)
$17$ \( 1728 + 100 T + T^{2} \)
$19$ \( 688 - 67 T + T^{2} \)
$23$ \( -17856 + 76 T + T^{2} \)
$29$ \( 13068 - 275 T + T^{2} \)
$31$ \( 13461 - 362 T + T^{2} \)
$37$ \( -97700 + 5 T + T^{2} \)
$41$ \( -9072 - 162 T + T^{2} \)
$43$ \( 129526 + 721 T + T^{2} \)
$47$ \( -50868 + 216 T + T^{2} \)
$53$ \( 57348 - 495 T + T^{2} \)
$59$ \( -128052 - 173 T + T^{2} \)
$61$ \( -6444 - 532 T + T^{2} \)
$67$ \( -282994 + 111 T + T^{2} \)
$71$ \( 546588 + 1600 T + T^{2} \)
$73$ \( 233522 + 1215 T + T^{2} \)
$79$ \( 347427 + 1460 T + T^{2} \)
$83$ \( 4122 + 1409 T + T^{2} \)
$89$ \( 889056 - 1974 T + T^{2} \)
$97$ \( 38102 + 561 T + T^{2} \)
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