Properties

Label 2352.4.a.ce
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: \(\Q(\sqrt{137}) \)
Defining polynomial: \(x^{2} - x - 34\)
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 \(\beta = \sqrt{137}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( 9 - \beta ) q^{5} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} + ( 9 - \beta ) q^{5} + 9 q^{9} + ( -29 - 3 \beta ) q^{11} + ( 24 + 4 \beta ) q^{13} + ( 27 - 3 \beta ) q^{15} + ( -3 - \beta ) q^{17} + ( -42 - 10 \beta ) q^{19} + ( -3 + 15 \beta ) q^{23} + ( 93 - 18 \beta ) q^{25} + 27 q^{27} + ( -52 + 6 \beta ) q^{29} + ( -126 - 2 \beta ) q^{31} + ( -87 - 9 \beta ) q^{33} + ( 112 - 6 \beta ) q^{37} + ( 72 + 12 \beta ) q^{39} + ( 219 - 23 \beta ) q^{41} + 388 q^{43} + ( 81 - 9 \beta ) q^{45} + ( 120 + 32 \beta ) q^{47} + ( -9 - 3 \beta ) q^{51} + ( 540 - 6 \beta ) q^{53} + ( 150 + 2 \beta ) q^{55} + ( -126 - 30 \beta ) q^{57} + ( -156 - 16 \beta ) q^{59} + ( -330 + 22 \beta ) q^{61} + ( -332 + 12 \beta ) q^{65} + ( 198 - 42 \beta ) q^{67} + ( -9 + 45 \beta ) q^{69} + ( -33 - 51 \beta ) q^{71} + ( 486 + 18 \beta ) q^{73} + ( 279 - 54 \beta ) q^{75} + ( 430 + 18 \beta ) q^{79} + 81 q^{81} + ( -1260 + 16 \beta ) q^{83} + ( 110 - 6 \beta ) q^{85} + ( -156 + 18 \beta ) q^{87} + ( 843 - 15 \beta ) q^{89} + ( -378 - 6 \beta ) q^{93} + ( 992 - 48 \beta ) q^{95} + ( 1050 + 22 \beta ) q^{97} + ( -261 - 27 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 18q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 18q^{5} + 18q^{9} - 58q^{11} + 48q^{13} + 54q^{15} - 6q^{17} - 84q^{19} - 6q^{23} + 186q^{25} + 54q^{27} - 104q^{29} - 252q^{31} - 174q^{33} + 224q^{37} + 144q^{39} + 438q^{41} + 776q^{43} + 162q^{45} + 240q^{47} - 18q^{51} + 1080q^{53} + 300q^{55} - 252q^{57} - 312q^{59} - 660q^{61} - 664q^{65} + 396q^{67} - 18q^{69} - 66q^{71} + 972q^{73} + 558q^{75} + 860q^{79} + 162q^{81} - 2520q^{83} + 220q^{85} - 312q^{87} + 1686q^{89} - 756q^{93} + 1984q^{95} + 2100q^{97} - 522q^{99} + O(q^{100}) \)

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
6.35235
−5.35235
0 3.00000 0 −2.70470 0 0 0 9.00000 0
1.2 0 3.00000 0 20.7047 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.ce 2
4.b odd 2 1 1176.4.a.s 2
7.b odd 2 1 2352.4.a.bm 2
28.d even 2 1 1176.4.a.t yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.4.a.s 2 4.b odd 2 1
1176.4.a.t yes 2 28.d even 2 1
2352.4.a.bm 2 7.b odd 2 1
2352.4.a.ce 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):

\( T_{5}^{2} - 18 T_{5} - 56 \)
\( T_{11}^{2} + 58 T_{11} - 392 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( -56 - 18 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -392 + 58 T + T^{2} \)
$13$ \( -1616 - 48 T + T^{2} \)
$17$ \( -128 + 6 T + T^{2} \)
$19$ \( -11936 + 84 T + T^{2} \)
$23$ \( -30816 + 6 T + T^{2} \)
$29$ \( -2228 + 104 T + T^{2} \)
$31$ \( 15328 + 252 T + T^{2} \)
$37$ \( 7612 - 224 T + T^{2} \)
$41$ \( -24512 - 438 T + T^{2} \)
$43$ \( ( -388 + T )^{2} \)
$47$ \( -125888 - 240 T + T^{2} \)
$53$ \( 286668 - 1080 T + T^{2} \)
$59$ \( -10736 + 312 T + T^{2} \)
$61$ \( 42592 + 660 T + T^{2} \)
$67$ \( -202464 - 396 T + T^{2} \)
$71$ \( -355248 + 66 T + T^{2} \)
$73$ \( 191808 - 972 T + T^{2} \)
$79$ \( 140512 - 860 T + T^{2} \)
$83$ \( 1552528 + 2520 T + T^{2} \)
$89$ \( 679824 - 1686 T + T^{2} \)
$97$ \( 1036192 - 2100 T + T^{2} \)
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