Properties

Label 1856.4.a.s
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.19816.1
Defining polynomial: \(x^{3} - x^{2} - 42 x - 54\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{5} + ( -8 + 4 \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{5} + ( -8 + 4 \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{9} + ( 3 + \beta_{1} - 2 \beta_{2} ) q^{11} + ( 4 - 8 \beta_{1} - 11 \beta_{2} ) q^{13} + ( 39 + 13 \beta_{1} + 2 \beta_{2} ) q^{15} + ( -18 - 12 \beta_{1} - 16 \beta_{2} ) q^{17} + ( -52 - 8 \beta_{1} + 10 \beta_{2} ) q^{19} + ( 28 + 4 \beta_{1} + 16 \beta_{2} ) q^{21} + ( 66 + 6 \beta_{1} - 12 \beta_{2} ) q^{23} + ( 13 + 40 \beta_{1} + 11 \beta_{2} ) q^{25} + ( -53 + 17 \beta_{1} + 6 \beta_{2} ) q^{27} -29 q^{29} + ( 25 + 11 \beta_{1} + 36 \beta_{2} ) q^{31} + ( -42 - 4 \beta_{1} - 11 \beta_{2} ) q^{33} + ( 24 \beta_{1} + 12 \beta_{2} ) q^{35} + ( 10 + 12 \beta_{1} - 38 \beta_{2} ) q^{37} + ( 121 + 31 \beta_{1} - 20 \beta_{2} ) q^{39} + ( 186 + 4 \beta_{1} - 6 \beta_{2} ) q^{41} + ( 11 - 15 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -132 - 26 \beta_{1} - 4 \beta_{2} ) q^{45} + ( -207 - 33 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 201 - 64 \beta_{1} - 80 \beta_{2} ) q^{49} + ( 162 + 70 \beta_{1} - 28 \beta_{2} ) q^{51} + ( -276 + 116 \beta_{1} + 27 \beta_{2} ) q^{53} + ( -39 - 25 \beta_{1} - 8 \beta_{2} ) q^{55} + ( 254 + 66 \beta_{1} + 64 \beta_{2} ) q^{57} + ( 90 - 86 \beta_{1} + 4 \beta_{2} ) q^{59} + ( -134 + 80 \beta_{1} - 40 \beta_{2} ) q^{61} + ( 280 - 56 \beta_{1} - 56 \beta_{2} ) q^{63} + ( 468 + 90 \beta_{1} + 9 \beta_{2} ) q^{65} + ( -88 + 36 \beta_{1} - 72 \beta_{2} ) q^{67} + ( -204 - 72 \beta_{1} - 66 \beta_{2} ) q^{69} + ( 18 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -178 + 40 \beta_{1} + 30 \beta_{2} ) q^{73} + ( -968 - 144 \beta_{1} - 76 \beta_{2} ) q^{75} + ( -300 + 28 \beta_{1} + 24 \beta_{2} ) q^{77} + ( 691 + 37 \beta_{1} + 38 \beta_{2} ) q^{79} + ( -485 - 58 \beta_{1} - 108 \beta_{2} ) q^{81} + ( -150 + 162 \beta_{1} + 12 \beta_{2} ) q^{83} + ( 840 + 184 \beta_{1} + 38 \beta_{2} ) q^{85} + ( -29 + 29 \beta_{1} ) q^{87} + ( 282 + 68 \beta_{1} + 154 \beta_{2} ) q^{89} + ( -1064 + 208 \beta_{1} + 244 \beta_{2} ) q^{91} + ( 52 - 94 \beta_{1} + 111 \beta_{2} ) q^{93} + ( 552 + 244 \beta_{1} + 86 \beta_{2} ) q^{95} + ( 14 + 176 \beta_{1} - 34 \beta_{2} ) q^{97} + ( -114 + 38 \beta_{1} + 22 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9} + O(q^{10}) \) \( 3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9} + 10 q^{11} + 4 q^{13} + 130 q^{15} - 66 q^{17} - 164 q^{19} + 88 q^{21} + 204 q^{23} + 79 q^{25} - 142 q^{27} - 87 q^{29} + 86 q^{31} - 130 q^{33} + 24 q^{35} + 42 q^{37} + 394 q^{39} + 562 q^{41} + 18 q^{43} - 422 q^{45} - 654 q^{47} + 539 q^{49} + 556 q^{51} - 712 q^{53} - 142 q^{55} + 828 q^{57} + 184 q^{59} - 322 q^{61} + 784 q^{63} + 1494 q^{65} - 228 q^{67} - 684 q^{69} + 52 q^{71} - 494 q^{73} - 3048 q^{75} - 872 q^{77} + 2110 q^{79} - 1513 q^{81} - 288 q^{83} + 2704 q^{85} - 58 q^{87} + 914 q^{89} - 2984 q^{91} + 62 q^{93} + 1900 q^{95} + 218 q^{97} - 304 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 42 x - 54\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 4 \nu - 27 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2} + 4 \beta_{1} + 27\)

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
7.53003
−1.39712
−5.13291
0 −6.53003 0 −20.9205 0 −8.55839 0 15.6413 0
1.2 0 2.39712 0 3.28077 0 −33.9461 0 −21.2538 0
1.3 0 6.13291 0 −2.36031 0 18.5045 0 10.6126 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(1\)

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 1856.4.a.s 3
4.b odd 2 1 1856.4.a.r 3
8.b even 2 1 464.4.a.i 3
8.d odd 2 1 58.4.a.d 3
24.f even 2 1 522.4.a.k 3
40.e odd 2 1 1450.4.a.h 3
232.b odd 2 1 1682.4.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.d 3 8.d odd 2 1
464.4.a.i 3 8.b even 2 1
522.4.a.k 3 24.f even 2 1
1450.4.a.h 3 40.e odd 2 1
1682.4.a.d 3 232.b odd 2 1
1856.4.a.r 3 4.b odd 2 1
1856.4.a.s 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{3} - 2 T_{3}^{2} - 41 T_{3} + 96 \)
\( T_{5}^{3} + 20 T_{5}^{2} - 27 T_{5} - 162 \)
\( T_{7}^{3} + 24 T_{7}^{2} - 496 T_{7} - 5376 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 96 - 41 T - 2 T^{2} + T^{3} \)
$5$ \( -162 - 27 T + 20 T^{2} + T^{3} \)
$7$ \( -5376 - 496 T + 24 T^{2} + T^{3} \)
$11$ \( 2424 - 233 T - 10 T^{2} + T^{3} \)
$13$ \( -131706 - 5619 T - 4 T^{2} + T^{3} \)
$17$ \( -679368 - 10660 T + 66 T^{2} + T^{3} \)
$19$ \( -664448 - 124 T + 164 T^{2} + T^{3} \)
$23$ \( 677376 + 4284 T - 204 T^{2} + T^{3} \)
$29$ \( ( 29 + T )^{3} \)
$31$ \( 4766172 - 48089 T - 86 T^{2} + T^{3} \)
$37$ \( 7684896 - 79456 T - 42 T^{2} + T^{3} \)
$41$ \( -5982048 + 102432 T - 562 T^{2} + T^{3} \)
$43$ \( 196488 - 10369 T - 18 T^{2} + T^{3} \)
$47$ \( 3425124 + 98015 T + 654 T^{2} + T^{3} \)
$53$ \( -252120546 - 350571 T + 712 T^{2} + T^{3} \)
$59$ \( 57362928 - 311444 T - 184 T^{2} + T^{3} \)
$61$ \( 5254424 - 388372 T + 322 T^{2} + T^{3} \)
$67$ \( 47608192 - 327840 T + 228 T^{2} + T^{3} \)
$71$ \( 672 - 164 T - 52 T^{2} + T^{3} \)
$73$ \( -9410208 + 6112 T + 494 T^{2} + T^{3} \)
$79$ \( -285187172 + 1400543 T - 2110 T^{2} + T^{3} \)
$83$ \( -437606064 - 1038996 T + 288 T^{2} + T^{3} \)
$89$ \( 598011552 - 664800 T - 914 T^{2} + T^{3} \)
$97$ \( -17006112 - 1500768 T - 218 T^{2} + T^{3} \)
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