Properties

Label 2352.4.a.bn
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: \( 7 \)
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 \(\beta = 7\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} + ( -6 + 7 \beta ) q^{5} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( -6 + 7 \beta ) q^{5} + 9 q^{9} + ( 2 + 42 \beta ) q^{11} + ( -24 - 21 \beta ) q^{13} + ( 18 - 21 \beta ) q^{15} + ( -42 + 7 \beta ) q^{17} + ( -36 - 14 \beta ) q^{19} + ( 154 - 42 \beta ) q^{23} + ( 9 - 84 \beta ) q^{25} -27 q^{27} + ( -40 + 126 \beta ) q^{29} + ( 192 - 42 \beta ) q^{31} + ( -6 - 126 \beta ) q^{33} + ( 268 - 84 \beta ) q^{37} + ( 72 + 63 \beta ) q^{39} + ( -378 - 35 \beta ) q^{41} + ( -200 - 168 \beta ) q^{43} + ( -54 + 63 \beta ) q^{45} + ( 156 - 70 \beta ) q^{47} + ( 126 - 21 \beta ) q^{51} + ( -26 - 168 \beta ) q^{53} + ( 576 - 238 \beta ) q^{55} + ( 108 + 42 \beta ) q^{57} + ( 432 + 14 \beta ) q^{59} + ( -708 + 91 \beta ) q^{61} + ( -150 - 42 \beta ) q^{65} + ( -72 - 168 \beta ) q^{67} + ( -462 + 126 \beta ) q^{69} + ( 762 + 210 \beta ) q^{71} + ( -372 - 581 \beta ) q^{73} + ( -27 + 252 \beta ) q^{75} + ( -488 - 588 \beta ) q^{79} + 81 q^{81} + ( -156 + 952 \beta ) q^{83} + ( 350 - 336 \beta ) q^{85} + ( 120 - 378 \beta ) q^{87} + ( -54 + 203 \beta ) q^{89} + ( -576 + 126 \beta ) q^{93} + ( 20 - 168 \beta ) q^{95} + ( 372 + 875 \beta ) q^{97} + ( 18 + 378 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} - 12q^{5} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} - 12q^{5} + 18q^{9} + 4q^{11} - 48q^{13} + 36q^{15} - 84q^{17} - 72q^{19} + 308q^{23} + 18q^{25} - 54q^{27} - 80q^{29} + 384q^{31} - 12q^{33} + 536q^{37} + 144q^{39} - 756q^{41} - 400q^{43} - 108q^{45} + 312q^{47} + 252q^{51} - 52q^{53} + 1152q^{55} + 216q^{57} + 864q^{59} - 1416q^{61} - 300q^{65} - 144q^{67} - 924q^{69} + 1524q^{71} - 744q^{73} - 54q^{75} - 976q^{79} + 162q^{81} - 312q^{83} + 700q^{85} + 240q^{87} - 108q^{89} - 1152q^{93} + 40q^{95} + 744q^{97} + 36q^{99} + O(q^{100}) \)

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 −15.8995 0 0 0 9.00000 0
1.2 0 −3.00000 0 3.89949 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.bn 2
4.b odd 2 1 294.4.a.k yes 2
7.b odd 2 1 2352.4.a.cd 2
12.b even 2 1 882.4.a.bi 2
28.d even 2 1 294.4.a.j 2
28.f even 6 2 294.4.e.o 4
28.g odd 6 2 294.4.e.n 4
84.h odd 2 1 882.4.a.bc 2
84.j odd 6 2 882.4.g.bd 4
84.n even 6 2 882.4.g.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 28.d even 2 1
294.4.a.k yes 2 4.b odd 2 1
294.4.e.n 4 28.g odd 6 2
294.4.e.o 4 28.f even 6 2
882.4.a.bc 2 84.h odd 2 1
882.4.a.bi 2 12.b even 2 1
882.4.g.y 4 84.n even 6 2
882.4.g.bd 4 84.j odd 6 2
2352.4.a.bn 2 1.a even 1 1 trivial
2352.4.a.cd 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{2} + 12 T_{5} - 62 \)
\( T_{11}^{2} - 4 T_{11} - 3524 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( -62 + 12 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -3524 - 4 T + T^{2} \)
$13$ \( -306 + 48 T + T^{2} \)
$17$ \( 1666 + 84 T + T^{2} \)
$19$ \( 904 + 72 T + T^{2} \)
$23$ \( 20188 - 308 T + T^{2} \)
$29$ \( -30152 + 80 T + T^{2} \)
$31$ \( 33336 - 384 T + T^{2} \)
$37$ \( 57712 - 536 T + T^{2} \)
$41$ \( 140434 + 756 T + T^{2} \)
$43$ \( -16448 + 400 T + T^{2} \)
$47$ \( 14536 - 312 T + T^{2} \)
$53$ \( -55772 + 52 T + T^{2} \)
$59$ \( 186232 - 864 T + T^{2} \)
$61$ \( 484702 + 1416 T + T^{2} \)
$67$ \( -51264 + 144 T + T^{2} \)
$71$ \( 492444 - 1524 T + T^{2} \)
$73$ \( -536738 + 744 T + T^{2} \)
$79$ \( -453344 + 976 T + T^{2} \)
$83$ \( -1788272 + 312 T + T^{2} \)
$89$ \( -79502 + 108 T + T^{2} \)
$97$ \( -1392866 - 744 T + T^{2} \)
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