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: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 2) q^{3} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{5} + ( - 2 \beta_{4} + \beta_{2} + 8) q^{7} + ( - 6 \beta_{3} + \beta_{2} + 5 \beta_1 + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 2) q^{3} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{5} + ( - 2 \beta_{4} + \beta_{2} + 8) q^{7} + ( - 6 \beta_{3} + \beta_{2} + 5 \beta_1 + 8) q^{9} + ( - 7 \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{11} + (10 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} - \beta_1 - 1) q^{13} + ( - \beta_{4} + 12 \beta_{3} + 2 \beta_{2} - 13 \beta_1 - 17) q^{15} + (3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 7 \beta_1 + 13) q^{17} + ( - 3 \beta_{4} - \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 43) q^{19} + (7 \beta_{4} + 9 \beta_{3} + 7 \beta_1 - 5) q^{21} + (4 \beta_{4} + 20 \beta_{3} - 7 \beta_{2} - 6 \beta_1 + 26) q^{23} + (15 \beta_{4} - 17 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 49) q^{25} + (3 \beta_{4} + 24 \beta_{3} + 11 \beta_{2} - 35 \beta_1 - 75) q^{27} + 29 q^{29} + ( - 8 \beta_{4} + 13 \beta_{3} - 13 \beta_{2} - 6 \beta_1 + 80) q^{31} + (11 \beta_{4} + 3 \beta_{3} - 20 \beta_{2} - 2 \beta_1 - 116) q^{33} + (30 \beta_{4} - 13 \beta_{2} + 12 \beta_1 + 8) q^{35} + ( - 12 \beta_{4} + 40 \beta_{3} - 22 \beta_{2} - 18 \beta_1 - 88) q^{37} + ( - 8 \beta_{4} - 3 \beta_{3} + 11 \beta_{2} - 8 \beta_1 - 72) q^{39} + (19 \beta_{4} + 25 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 241) q^{41} + (7 \beta_{4} + 4 \beta_{3} + 27 \beta_{2} - 29 \beta_1 + 45) q^{43} + (35 \beta_{4} - 91 \beta_{3} + 3 \beta_{2} + 43 \beta_1 + 329) q^{45} + (9 \beta_{4} - 16 \beta_{3} + 58 \beta_{2} - 25 \beta_1 + 61) q^{47} + ( - 2 \beta_{4} + 6 \beta_{3} + 32 \beta_{2} - 30 \beta_1 - 53) q^{49} + ( - 12 \beta_{4} + 52 \beta_{3} + 17 \beta_{2} - 42 \beta_1 - 58) q^{51} + ( - 2 \beta_{4} - 8 \beta_{3} - 26 \beta_{2} - 15 \beta_1 + 117) q^{53} + (86 \beta_{4} + 15 \beta_{3} - 27 \beta_{2} + 26 \beta_1 + 98) q^{55} + ( - 9 \beta_{4} - 19 \beta_{3} - 5 \beta_{2} - 33 \beta_1 + 23) q^{57} + (26 \beta_{4} - 10 \beta_{3} - 11 \beta_{2} - 20 \beta_1 - 80) q^{59} + ( - 9 \beta_{4} - 3 \beta_{3} + 40 \beta_{2} - 57 \beta_1 - 111) q^{61} + (40 \beta_{4} - 20 \beta_{3} + 10 \beta_{2} + 24 \beta_1 + 164) q^{63} + ( - 90 \beta_{4} - 56 \beta_{3} + 33 \beta_{2} + 41 \beta_1 - 317) q^{65} + (48 \beta_{4} - 48 \beta_{3} + 22 \beta_{2} + 92 \beta_1 - 232) q^{67} + ( - 29 \beta_{4} - 75 \beta_{3} - 8 \beta_{2} + 73 \beta_1 + 409) q^{69} + (16 \beta_{4} - 10 \beta_{3} + 32 \beta_{2} - 112 \beta_1 - 112) q^{71} + ( - 30 \beta_{4} - 2 \beta_{3} - 34 \beta_{2} - 36 \beta_1 - 382) q^{73} + ( - 24 \beta_{4} + 84 \beta_{3} - 10 \beta_{2} - 82 \beta_1 - 632) q^{75} + ( - 15 \beta_{4} - 33 \beta_{3} + 16 \beta_{2} - 143 \beta_1 + 365) q^{77} + ( - 127 \beta_{4} - 50 \beta_{3} - 38 \beta_{2} - 13 \beta_1 + 77) q^{79} + (27 \beta_{4} - 163 \beta_{3} - 83 \beta_{2} + 107 \beta_1 + 404) q^{81} + ( - 2 \beta_{4} - 58 \beta_{3} + 55 \beta_{2} + 44 \beta_1 - 88) q^{83} + ( - 45 \beta_{4} - 91 \beta_{3} + 8 \beta_{2} + 25 \beta_1 + 357) q^{85} + (29 \beta_{3} - 58) q^{87} + (121 \beta_{4} + 31 \beta_{3} + 28 \beta_{2} - 21 \beta_1 + 165) q^{89} + ( - 58 \beta_{4} - 36 \beta_{3} - 3 \beta_{2} + 72 \beta_1 - 516) q^{91} + ( - 23 \beta_{4} + 25 \beta_{3} - 39 \beta_{2} + 20 \beta_1 - 6) q^{93} + (47 \beta_{4} + 17 \beta_{3} + 35 \beta_{2} - 85 \beta_1 + 459) q^{95} + ( - 31 \beta_{4} - \beta_{3} - 22 \beta_{2} + 79 \beta_1 + 297) q^{97} + (107 \beta_{4} - 73 \beta_{3} - 5 \beta_{2} - 11 \beta_1 + 79) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} + 2\nu - 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + \nu^{2} - 8\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + \nu^{3} - 10\nu^{2} - 2\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - \nu^{3} - 14\nu^{2} + 18\nu + 16 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{4} - 5\beta_{3} + 7\beta_{2} + 3\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{4} + 9\beta_{3} - 8\beta_{2} + 4\beta _1 + 55 \) Copy content Toggle raw display

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 8 T^{4} - 52 T^{3} + \cdots + 1042 \) Copy content Toggle raw display
$5$ \( T^{5} + 10 T^{4} - 366 T^{3} + \cdots + 55534 \) Copy content Toggle raw display
$7$ \( T^{5} - 40 T^{4} + 84 T^{3} + \cdots + 243968 \) Copy content Toggle raw display
$11$ \( T^{5} + 12 T^{4} - 4892 T^{3} + \cdots - 30997958 \) Copy content Toggle raw display
$13$ \( T^{5} + 14 T^{4} - 7558 T^{3} + \cdots + 13078418 \) Copy content Toggle raw display
$17$ \( T^{5} - 66 T^{4} - 2444 T^{3} + \cdots + 19935872 \) Copy content Toggle raw display
$19$ \( T^{5} + 214 T^{4} + \cdots - 19441152 \) Copy content Toggle raw display
$23$ \( T^{5} - 164 T^{4} + \cdots - 7938109184 \) Copy content Toggle raw display
$29$ \( (T - 29)^{5} \) Copy content Toggle raw display
$31$ \( T^{5} - 420 T^{4} + 45552 T^{3} + \cdots - 2094346 \) Copy content Toggle raw display
$37$ \( T^{5} + 378 T^{4} + \cdots - 23564115968 \) Copy content Toggle raw display
$41$ \( T^{5} + 1158 T^{4} + \cdots + 59613728000 \) Copy content Toggle raw display
$43$ \( T^{5} - 204 T^{4} + \cdots - 198643410886 \) Copy content Toggle raw display
$47$ \( T^{5} - 248 T^{4} + \cdots - 203435244846 \) Copy content Toggle raw display
$53$ \( T^{5} - 554 T^{4} + \cdots + 786854101018 \) Copy content Toggle raw display
$59$ \( T^{5} + 440 T^{4} + \cdots - 109032704000 \) Copy content Toggle raw display
$61$ \( T^{5} + 618 T^{4} + \cdots - 2140697762176 \) Copy content Toggle raw display
$67$ \( T^{5} + 1164 T^{4} + \cdots + 39308070146048 \) Copy content Toggle raw display
$71$ \( T^{5} + 692 T^{4} + \cdots + 98341318953856 \) Copy content Toggle raw display
$73$ \( T^{5} + 1950 T^{4} + \cdots + 7201878016 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 240961986300538 \) Copy content Toggle raw display
$83$ \( T^{5} + 512 T^{4} + \cdots - 6057622580224 \) Copy content Toggle raw display
$89$ \( T^{5} - 866 T^{4} + \cdots + 21549994365568 \) Copy content Toggle raw display
$97$ \( T^{5} - 1562 T^{4} + \cdots - 20480102175488 \) Copy content Toggle raw display
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