Properties

Label 1856.4.a.u
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 232)
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.

\(f(q)\) \(=\) \( q + ( - 3 \beta_{2} + 1) q^{3} + (4 \beta_{2} - 3 \beta_1 - 1) q^{5} + (12 \beta_{2} - 2 \beta_1 - 2) q^{7} + ( - 6 \beta_{2} - 9 \beta_1 + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_{2} + 1) q^{3} + (4 \beta_{2} - 3 \beta_1 - 1) q^{5} + (12 \beta_{2} - 2 \beta_1 - 2) q^{7} + ( - 6 \beta_{2} - 9 \beta_1 + 10) q^{9} + ( - 5 \beta_{2} + 4 \beta_1 - 1) q^{11} + ( - 16 \beta_{2} - 13 \beta_1 - 19) q^{13} + ( - 11 \beta_{2} - 49) q^{15} + (10 \beta_{2} + 8 \beta_1 - 16) q^{17} + ( - 16 \beta_{2} - 26 \beta_1 - 36) q^{19} + (6 \beta_{2} + 28 \beta_1 - 146) q^{21} + (10 \beta_{2} + 10 \beta_1 - 52) q^{23} + (22 \beta_{2} + 23 \beta_1 + 12) q^{25} + ( - 9 \beta_{2} - 54 \beta_1 + 55) q^{27} + 29 q^{29} + (7 \beta_{2} - 74 \beta_1 + 41) q^{31} + (22 \beta_{2} + \beta_1 + 59) q^{33} + (56 \beta_{2} + 10 \beta_1 + 242) q^{35} + ( - 76 \beta_{2} - 34 \beta_1 - 184) q^{37} + ( - 37 \beta_{2} - 100 \beta_1 + 173) q^{39} + (42 \beta_{2} + 2 \beta_1 + 402) q^{41} + ( - 77 \beta_{2} - 52 \beta_1 - 33) q^{43} + (28 \beta_{2} + 48 \beta_1 + 110) q^{45} + (59 \beta_{2} - 14 \beta_1 + 261) q^{47} + (40 \beta_{2} - 84 \beta_1 + 269) q^{49} + (106 \beta_{2} + 62 \beta_1 - 136) q^{51} + (112 \beta_{2} + 53 \beta_1 - 49) q^{53} + ( - 37 \beta_{2} - 24 \beta_1 - 175) q^{55} + ( - 64 \beta_{2} - 152 \beta_1 + 156) q^{57} + (62 \beta_{2} - 10 \beta_1 + 32) q^{59} + ( - 126 \beta_{2} - 180 \beta_1 - 120) q^{61} + (288 \beta_{2} + 184 \beta_1 - 164) q^{63} + ( - 130 \beta_{2} + 177 \beta_1 + 75) q^{65} + (12 \beta_{2} - 124 \beta_1 - 76) q^{67} + (226 \beta_{2} + 70 \beta_1 - 172) q^{69} + ( - 82 \beta_{2} + 16 \beta_1 + 378) q^{71} + ( - 172 \beta_{2} + 102 \beta_1 - 396) q^{73} + (124 \beta_{2} + 158 \beta_1 - 252) q^{75} + ( - 102 \beta_{2} - 12 \beta_1 - 302) q^{77} + (217 \beta_{2} + 14 \beta_1 + 831) q^{79} + ( - 336 \beta_{2} - 107) q^{81} + (350 \beta_{2} - 166 \beta_1 - 540) q^{83} + ( - 30 \beta_{2} - 26 \beta_1 - 16) q^{85} + ( - 87 \beta_{2} + 29) q^{87} + ( - 458 \beta_{2} + 74 \beta_1 + 398) q^{89} + (406 \beta_1 - 522) q^{91} + ( - 560 \beta_{2} - 275 \beta_1 - 43) q^{93} + ( - 172 \beta_{2} + 332 \beta_1 + 404) q^{95} + (162 \beta_{2} - 394 \beta_1 + 406) q^{97} + ( - 14 \beta_{2} - 38 \beta_1 - 178) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9} - 2 q^{11} - 28 q^{13} - 136 q^{15} - 66 q^{17} - 66 q^{19} - 472 q^{21} - 176 q^{23} - 9 q^{25} + 228 q^{27} + 87 q^{29} + 190 q^{31} + 154 q^{33} + 660 q^{35} - 442 q^{37} + 656 q^{39} + 1162 q^{41} + 30 q^{43} + 254 q^{45} + 738 q^{47} + 851 q^{49} - 576 q^{51} - 312 q^{53} - 464 q^{55} + 684 q^{57} + 44 q^{59} - 54 q^{61} - 964 q^{63} + 178 q^{65} - 116 q^{67} - 812 q^{69} + 1200 q^{71} - 1118 q^{73} - 1038 q^{75} - 792 q^{77} + 2262 q^{79} + 15 q^{81} - 1804 q^{83} + 8 q^{85} + 174 q^{87} + 1578 q^{89} - 1972 q^{91} + 706 q^{93} + 1052 q^{95} + 1450 q^{97} - 482 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + \beta _1 + 6 ) / 2 \) 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
−1.86081
2.11491
−0.254102
0 −5.97021 0 12.4882 0 28.6773 0 8.64344 0
1.2 0 2.92622 0 −14.3315 0 −16.8804 0 −18.4372 0
1.3 0 9.04399 0 −2.15672 0 −27.7969 0 54.7938 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.u 3
4.b odd 2 1 1856.4.a.p 3
8.b even 2 1 464.4.a.g 3
8.d odd 2 1 232.4.a.b 3
24.f even 2 1 2088.4.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.a.b 3 8.d odd 2 1
464.4.a.g 3 8.b even 2 1
1856.4.a.p 3 4.b odd 2 1
1856.4.a.u 3 1.a even 1 1 trivial
2088.4.a.b 3 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{3} - 6T_{3}^{2} - 45T_{3} + 158 \) Copy content Toggle raw display
\( T_{5}^{3} + 4T_{5}^{2} - 175T_{5} - 386 \) Copy content Toggle raw display
\( T_{7}^{3} + 16T_{7}^{2} - 812T_{7} - 13456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 6 T^{2} - 45 T + 158 \) Copy content Toggle raw display
$5$ \( T^{3} + 4 T^{2} - 175 T - 386 \) Copy content Toggle raw display
$7$ \( T^{3} + 16 T^{2} - 812 T - 13456 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 301 T - 106 \) Copy content Toggle raw display
$13$ \( T^{3} + 28 T^{2} - 3999 T - 137518 \) Copy content Toggle raw display
$17$ \( T^{3} + 66 T^{2} - 184 T - 1696 \) Copy content Toggle raw display
$19$ \( T^{3} + 66 T^{2} - 9616 T - 393632 \) Copy content Toggle raw display
$23$ \( T^{3} + 176 T^{2} + 8192 T + 106688 \) Copy content Toggle raw display
$29$ \( (T - 29)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 190 T^{2} - 54433 T + 9505226 \) Copy content Toggle raw display
$37$ \( T^{3} + 442 T^{2} + \cdots - 10323328 \) Copy content Toggle raw display
$41$ \( T^{3} - 1162 T^{2} + \cdots - 53735864 \) Copy content Toggle raw display
$43$ \( T^{3} - 30 T^{2} - 81277 T - 8035162 \) Copy content Toggle raw display
$47$ \( T^{3} - 738 T^{2} + \cdots - 10647458 \) Copy content Toggle raw display
$53$ \( T^{3} + 312 T^{2} - 97471 T + 4921082 \) Copy content Toggle raw display
$59$ \( T^{3} - 44 T^{2} - 23280 T - 849664 \) Copy content Toggle raw display
$61$ \( T^{3} + 54 T^{2} - 559656 T - 87366816 \) Copy content Toggle raw display
$67$ \( T^{3} + 116 T^{2} + \cdots + 19023808 \) Copy content Toggle raw display
$71$ \( T^{3} - 1200 T^{2} + \cdots - 44065376 \) Copy content Toggle raw display
$73$ \( T^{3} + 1118 T^{2} + \cdots - 19620512 \) Copy content Toggle raw display
$79$ \( T^{3} - 2262 T^{2} + \cdots - 199598902 \) Copy content Toggle raw display
$83$ \( T^{3} + 1804 T^{2} + \cdots - 652294144 \) Copy content Toggle raw display
$89$ \( T^{3} - 1578 T^{2} + \cdots + 1024346504 \) Copy content Toggle raw display
$97$ \( T^{3} - 1450 T^{2} + \cdots + 1820417096 \) Copy content Toggle raw display
show more
show less