Properties

Label 1856.4.a.y
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.13458092.1
Defining polynomial: \(x^{5} - x^{4} - 14 x^{3} + 18 x^{2} + 20 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{3} ) q^{3} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( 8 + \beta_{2} - 2 \beta_{4} ) q^{7} + ( 8 + 5 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -2 + \beta_{3} ) q^{3} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( 8 + \beta_{2} - 2 \beta_{4} ) q^{7} + ( 8 + 5 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{9} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - 7 \beta_{4} ) q^{11} + ( -1 - \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} ) q^{13} + ( -17 - 13 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} - \beta_{4} ) q^{15} + ( 13 + 7 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{17} + ( -43 + 3 \beta_{1} - 5 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{19} + ( -5 + 7 \beta_{1} + 9 \beta_{3} + 7 \beta_{4} ) q^{21} + ( 26 - 6 \beta_{1} - 7 \beta_{2} + 20 \beta_{3} + 4 \beta_{4} ) q^{23} + ( 49 - 4 \beta_{1} + 2 \beta_{2} - 17 \beta_{3} + 15 \beta_{4} ) q^{25} + ( -75 - 35 \beta_{1} + 11 \beta_{2} + 24 \beta_{3} + 3 \beta_{4} ) q^{27} + 29 q^{29} + ( 80 - 6 \beta_{1} - 13 \beta_{2} + 13 \beta_{3} - 8 \beta_{4} ) q^{31} + ( -116 - 2 \beta_{1} - 20 \beta_{2} + 3 \beta_{3} + 11 \beta_{4} ) q^{33} + ( 8 + 12 \beta_{1} - 13 \beta_{2} + 30 \beta_{4} ) q^{35} + ( -88 - 18 \beta_{1} - 22 \beta_{2} + 40 \beta_{3} - 12 \beta_{4} ) q^{37} + ( -72 - 8 \beta_{1} + 11 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} ) q^{39} + ( -241 - 3 \beta_{1} - 2 \beta_{2} + 25 \beta_{3} + 19 \beta_{4} ) q^{41} + ( 45 - 29 \beta_{1} + 27 \beta_{2} + 4 \beta_{3} + 7 \beta_{4} ) q^{43} + ( 329 + 43 \beta_{1} + 3 \beta_{2} - 91 \beta_{3} + 35 \beta_{4} ) q^{45} + ( 61 - 25 \beta_{1} + 58 \beta_{2} - 16 \beta_{3} + 9 \beta_{4} ) q^{47} + ( -53 - 30 \beta_{1} + 32 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} ) q^{49} + ( -58 - 42 \beta_{1} + 17 \beta_{2} + 52 \beta_{3} - 12 \beta_{4} ) q^{51} + ( 117 - 15 \beta_{1} - 26 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} ) q^{53} + ( 98 + 26 \beta_{1} - 27 \beta_{2} + 15 \beta_{3} + 86 \beta_{4} ) q^{55} + ( 23 - 33 \beta_{1} - 5 \beta_{2} - 19 \beta_{3} - 9 \beta_{4} ) q^{57} + ( -80 - 20 \beta_{1} - 11 \beta_{2} - 10 \beta_{3} + 26 \beta_{4} ) q^{59} + ( -111 - 57 \beta_{1} + 40 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} ) q^{61} + ( 164 + 24 \beta_{1} + 10 \beta_{2} - 20 \beta_{3} + 40 \beta_{4} ) q^{63} + ( -317 + 41 \beta_{1} + 33 \beta_{2} - 56 \beta_{3} - 90 \beta_{4} ) q^{65} + ( -232 + 92 \beta_{1} + 22 \beta_{2} - 48 \beta_{3} + 48 \beta_{4} ) q^{67} + ( 409 + 73 \beta_{1} - 8 \beta_{2} - 75 \beta_{3} - 29 \beta_{4} ) q^{69} + ( -112 - 112 \beta_{1} + 32 \beta_{2} - 10 \beta_{3} + 16 \beta_{4} ) q^{71} + ( -382 - 36 \beta_{1} - 34 \beta_{2} - 2 \beta_{3} - 30 \beta_{4} ) q^{73} + ( -632 - 82 \beta_{1} - 10 \beta_{2} + 84 \beta_{3} - 24 \beta_{4} ) q^{75} + ( 365 - 143 \beta_{1} + 16 \beta_{2} - 33 \beta_{3} - 15 \beta_{4} ) q^{77} + ( 77 - 13 \beta_{1} - 38 \beta_{2} - 50 \beta_{3} - 127 \beta_{4} ) q^{79} + ( 404 + 107 \beta_{1} - 83 \beta_{2} - 163 \beta_{3} + 27 \beta_{4} ) q^{81} + ( -88 + 44 \beta_{1} + 55 \beta_{2} - 58 \beta_{3} - 2 \beta_{4} ) q^{83} + ( 357 + 25 \beta_{1} + 8 \beta_{2} - 91 \beta_{3} - 45 \beta_{4} ) q^{85} + ( -58 + 29 \beta_{3} ) q^{87} + ( 165 - 21 \beta_{1} + 28 \beta_{2} + 31 \beta_{3} + 121 \beta_{4} ) q^{89} + ( -516 + 72 \beta_{1} - 3 \beta_{2} - 36 \beta_{3} - 58 \beta_{4} ) q^{91} + ( -6 + 20 \beta_{1} - 39 \beta_{2} + 25 \beta_{3} - 23 \beta_{4} ) q^{93} + ( 459 - 85 \beta_{1} + 35 \beta_{2} + 17 \beta_{3} + 47 \beta_{4} ) q^{95} + ( 297 + 79 \beta_{1} - 22 \beta_{2} - \beta_{3} - 31 \beta_{4} ) q^{97} + ( 79 - 11 \beta_{1} - 5 \beta_{2} - 73 \beta_{3} + 107 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} + O(q^{10}) \) \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - x^{4} - 14 x^{3} + 18 x^{2} + 20 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + 2 \nu - 6 \)
\(\beta_{2}\)\(=\)\( \nu^{3} + \nu^{2} - 8 \nu - 2 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + \nu^{3} - 10 \nu^{2} - 2 \nu + 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 14 \nu^{2} + 18 \nu + 16 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 11\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{4} - 5 \beta_{3} + 7 \beta_{2} + 3 \beta_{1} - 3\)\()/2\)
\(\nu^{4}\)\(=\)\(-7 \beta_{4} + 9 \beta_{3} - 8 \beta_{2} + 4 \beta_{1} + 55\)

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
2.27399
−0.957567
0.328194
−3.68360
3.03898
0 −9.87991 0 16.8209 0 5.21997 0 70.6126 0
1.2 0 −4.64574 0 −12.8729 0 26.0540 0 −5.41713 0
1.3 0 −1.84328 0 −18.3339 0 −16.8583 0 −23.6023 0
1.4 0 1.90549 0 6.52855 0 5.22706 0 −23.3691 0
1.5 0 6.46343 0 −2.14270 0 20.3573 0 14.7760 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.y 5
4.b odd 2 1 1856.4.a.bb 5
8.b even 2 1 29.4.a.b 5
8.d odd 2 1 464.4.a.l 5
24.h odd 2 1 261.4.a.f 5
40.f even 2 1 725.4.a.c 5
56.h odd 2 1 1421.4.a.e 5
232.g even 2 1 841.4.a.b 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.b 5 8.b even 2 1
261.4.a.f 5 24.h odd 2 1
464.4.a.l 5 8.d odd 2 1
725.4.a.c 5 40.f even 2 1
841.4.a.b 5 232.g even 2 1
1421.4.a.e 5 56.h odd 2 1
1856.4.a.y 5 1.a even 1 1 trivial
1856.4.a.bb 5 4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{5} + 8 T_{3}^{4} - 52 T_{3}^{3} - 322 T_{3}^{2} + 187 T_{3} + 1042 \)
\( T_{5}^{5} + 10 T_{5}^{4} - 366 T_{5}^{3} - 2904 T_{5}^{2} + 21453 T_{5} + 55534 \)
\( T_{7}^{5} - 40 T_{7}^{4} + 84 T_{7}^{3} + 10768 T_{7}^{2} - 100288 T_{7} + 243968 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( 1042 + 187 T - 322 T^{2} - 52 T^{3} + 8 T^{4} + T^{5} \)
$5$ \( 55534 + 21453 T - 2904 T^{2} - 366 T^{3} + 10 T^{4} + T^{5} \)
$7$ \( 243968 - 100288 T + 10768 T^{2} + 84 T^{3} - 40 T^{4} + T^{5} \)
$11$ \( -30997958 + 4398787 T - 50174 T^{2} - 4892 T^{3} + 12 T^{4} + T^{5} \)
$13$ \( 13078418 + 1294565 T - 133312 T^{2} - 7558 T^{3} + 14 T^{4} + T^{5} \)
$17$ \( 19935872 - 3694112 T + 205448 T^{2} - 2444 T^{3} - 66 T^{4} + T^{5} \)
$19$ \( -19441152 - 1091328 T + 342272 T^{2} + 15136 T^{3} + 214 T^{4} + T^{5} \)
$23$ \( -7938109184 + 24181632 T + 3316000 T^{2} - 18812 T^{3} - 164 T^{4} + T^{5} \)
$29$ \( ( -29 + T )^{5} \)
$31$ \( -2094346 - 5574361 T - 1363354 T^{2} + 45552 T^{3} - 420 T^{4} + T^{5} \)
$37$ \( -23564115968 - 2452994048 T - 30918912 T^{2} - 69792 T^{3} + 378 T^{4} + T^{5} \)
$41$ \( 59613728000 + 4409752000 T + 74423880 T^{2} + 462908 T^{3} + 1158 T^{4} + T^{5} \)
$43$ \( -198643410886 + 1855476667 T + 11715386 T^{2} - 94388 T^{3} - 204 T^{4} + T^{5} \)
$47$ \( -203435244846 + 8179300863 T + 76509294 T^{2} - 332696 T^{3} - 248 T^{4} + T^{5} \)
$53$ \( 786854101018 - 14144920995 T + 72740336 T^{2} - 38334 T^{3} - 554 T^{4} + T^{5} \)
$59$ \( -109032704000 - 4107227200 T - 42988400 T^{2} - 95868 T^{3} + 440 T^{4} + T^{5} \)
$61$ \( -2140697762176 - 41066569056 T - 206442728 T^{2} - 204156 T^{3} + 618 T^{4} + T^{5} \)
$67$ \( 39308070146048 + 25268520960 T - 457331392 T^{2} - 251984 T^{3} + 1164 T^{4} + T^{5} \)
$71$ \( 98341318953856 + 153248941872 T - 572202480 T^{2} - 876848 T^{3} + 692 T^{4} + T^{5} \)
$73$ \( 7201878016 + 19306358272 T + 267870432 T^{2} + 1168032 T^{3} + 1950 T^{4} + T^{5} \)
$79$ \( -240961986300538 + 543174633815 T + 545111474 T^{2} - 1497888 T^{3} - 272 T^{4} + T^{5} \)
$83$ \( -6057622580224 + 85821473600 T - 118532752 T^{2} - 520156 T^{3} + 512 T^{4} + T^{5} \)
$89$ \( 21549994365568 + 138056266176 T + 69193352 T^{2} - 852420 T^{3} - 866 T^{4} + T^{5} \)
$97$ \( -20480102175488 - 81033100480 T + 290423688 T^{2} + 412988 T^{3} - 1562 T^{4} + T^{5} \)
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