Properties

Label 2352.4.a.bu
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 294)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -3 q^{3} + ( 6 + \beta ) q^{5} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( 6 + \beta ) q^{5} + 9 q^{9} + ( -2 - 6 \beta ) q^{11} + ( 24 - 15 \beta ) q^{13} + ( -18 - 3 \beta ) q^{15} + ( 66 - 11 \beta ) q^{17} + ( -60 - 46 \beta ) q^{19} + ( 38 + 102 \beta ) q^{23} + ( -87 + 12 \beta ) q^{25} -27 q^{27} + ( -56 + 150 \beta ) q^{29} + ( -216 + 54 \beta ) q^{31} + ( 6 + 18 \beta ) q^{33} + ( -140 - 180 \beta ) q^{37} + ( -72 + 45 \beta ) q^{39} + ( 18 + 127 \beta ) q^{41} + ( 64 - 288 \beta ) q^{43} + ( 54 + 9 \beta ) q^{45} + ( 132 + 338 \beta ) q^{47} + ( -198 + 33 \beta ) q^{51} + ( 134 - 192 \beta ) q^{53} + ( -24 - 38 \beta ) q^{55} + ( 180 + 138 \beta ) q^{57} + ( -168 - 298 \beta ) q^{59} + ( -252 + 353 \beta ) q^{61} + ( 114 - 66 \beta ) q^{65} + ( 192 + 144 \beta ) q^{67} + ( -114 - 306 \beta ) q^{69} + ( 198 - 342 \beta ) q^{71} + ( -156 - 595 \beta ) q^{73} + ( 261 - 36 \beta ) q^{75} + ( 424 + 300 \beta ) q^{79} + 81 q^{81} + ( 324 - 80 \beta ) q^{83} + 374 q^{85} + ( 168 - 450 \beta ) q^{87} + ( -306 - 175 \beta ) q^{89} + ( 648 - 162 \beta ) q^{93} + ( -452 - 336 \beta ) q^{95} + ( -1092 + 133 \beta ) q^{97} + ( -18 - 54 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 12q^{5} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 12q^{5} + 18q^{9} - 4q^{11} + 48q^{13} - 36q^{15} + 132q^{17} - 120q^{19} + 76q^{23} - 174q^{25} - 54q^{27} - 112q^{29} - 432q^{31} + 12q^{33} - 280q^{37} - 144q^{39} + 36q^{41} + 128q^{43} + 108q^{45} + 264q^{47} - 396q^{51} + 268q^{53} - 48q^{55} + 360q^{57} - 336q^{59} - 504q^{61} + 228q^{65} + 384q^{67} - 228q^{69} + 396q^{71} - 312q^{73} + 522q^{75} + 848q^{79} + 162q^{81} + 648q^{83} + 748q^{85} + 336q^{87} - 612q^{89} + 1296q^{93} - 904q^{95} - 2184q^{97} - 36q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 −3.00000 0 4.58579 0 0 0 9.00000 0
1.2 0 −3.00000 0 7.41421 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.bu 2
4.b odd 2 1 294.4.a.o yes 2
7.b odd 2 1 2352.4.a.bw 2
12.b even 2 1 882.4.a.t 2
28.d even 2 1 294.4.a.l 2
28.f even 6 2 294.4.e.m 4
28.g odd 6 2 294.4.e.k 4
84.h odd 2 1 882.4.a.bb 2
84.j odd 6 2 882.4.g.be 4
84.n even 6 2 882.4.g.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.l 2 28.d even 2 1
294.4.a.o yes 2 4.b odd 2 1
294.4.e.k 4 28.g odd 6 2
294.4.e.m 4 28.f even 6 2
882.4.a.t 2 12.b even 2 1
882.4.a.bb 2 84.h odd 2 1
882.4.g.be 4 84.j odd 6 2
882.4.g.bk 4 84.n even 6 2
2352.4.a.bu 2 1.a even 1 1 trivial
2352.4.a.bw 2 7.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} - 12 T_{5} + 34 \)
\( T_{11}^{2} + 4 T_{11} - 68 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( 34 - 12 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -68 + 4 T + T^{2} \)
$13$ \( 126 - 48 T + T^{2} \)
$17$ \( 4114 - 132 T + T^{2} \)
$19$ \( -632 + 120 T + T^{2} \)
$23$ \( -19364 - 76 T + T^{2} \)
$29$ \( -41864 + 112 T + T^{2} \)
$31$ \( 40824 + 432 T + T^{2} \)
$37$ \( -45200 + 280 T + T^{2} \)
$41$ \( -31934 - 36 T + T^{2} \)
$43$ \( -161792 - 128 T + T^{2} \)
$47$ \( -211064 - 264 T + T^{2} \)
$53$ \( -55772 - 268 T + T^{2} \)
$59$ \( -149384 + 336 T + T^{2} \)
$61$ \( -185714 + 504 T + T^{2} \)
$67$ \( -4608 - 384 T + T^{2} \)
$71$ \( -194724 - 396 T + T^{2} \)
$73$ \( -683714 + 312 T + T^{2} \)
$79$ \( -224 - 848 T + T^{2} \)
$83$ \( 92176 - 648 T + T^{2} \)
$89$ \( 32386 + 612 T + T^{2} \)
$97$ \( 1157086 + 2184 T + T^{2} \)
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