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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1856,4,Mod(1,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1856.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.19816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 42x - 54 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{2} - 2 \beta_1 - 6) q^{5} + (4 \beta_{2} - 8) q^{7} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + ( - \beta_{2} - 2 \beta_1 - 6) q^{5} + (4 \beta_{2} - 8) q^{7} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9} + ( - 2 \beta_{2} + \beta_1 + 3) q^{11} + ( - 11 \beta_{2} - 8 \beta_1 + 4) q^{13} + (2 \beta_{2} + 13 \beta_1 + 39) q^{15} + ( - 16 \beta_{2} - 12 \beta_1 - 18) q^{17} + (10 \beta_{2} - 8 \beta_1 - 52) q^{19} + (16 \beta_{2} + 4 \beta_1 + 28) q^{21} + ( - 12 \beta_{2} + 6 \beta_1 + 66) q^{23} + (11 \beta_{2} + 40 \beta_1 + 13) q^{25} + (6 \beta_{2} + 17 \beta_1 - 53) q^{27} - 29 q^{29} + (36 \beta_{2} + 11 \beta_1 + 25) q^{31} + ( - 11 \beta_{2} - 4 \beta_1 - 42) q^{33} + (12 \beta_{2} + 24 \beta_1) q^{35} + ( - 38 \beta_{2} + 12 \beta_1 + 10) q^{37} + ( - 20 \beta_{2} + 31 \beta_1 + 121) q^{39} + ( - 6 \beta_{2} + 4 \beta_1 + 186) q^{41} + (2 \beta_{2} - 15 \beta_1 + 11) q^{43} + ( - 4 \beta_{2} - 26 \beta_1 - 132) q^{45} + ( - 2 \beta_{2} - 33 \beta_1 - 207) q^{47} + ( - 80 \beta_{2} - 64 \beta_1 + 201) q^{49} + ( - 28 \beta_{2} + 70 \beta_1 + 162) q^{51} + (27 \beta_{2} + 116 \beta_1 - 276) q^{53} + ( - 8 \beta_{2} - 25 \beta_1 - 39) q^{55} + (64 \beta_{2} + 66 \beta_1 + 254) q^{57} + (4 \beta_{2} - 86 \beta_1 + 90) q^{59} + ( - 40 \beta_{2} + 80 \beta_1 - 134) q^{61} + ( - 56 \beta_{2} - 56 \beta_1 + 280) q^{63} + (9 \beta_{2} + 90 \beta_1 + 468) q^{65} + ( - 72 \beta_{2} + 36 \beta_1 - 88) q^{67} + ( - 66 \beta_{2} - 72 \beta_1 - 204) q^{69} + (4 \beta_{2} - 2 \beta_1 + 18) q^{71} + (30 \beta_{2} + 40 \beta_1 - 178) q^{73} + ( - 76 \beta_{2} - 144 \beta_1 - 968) q^{75} + (24 \beta_{2} + 28 \beta_1 - 300) q^{77} + (38 \beta_{2} + 37 \beta_1 + 691) q^{79} + ( - 108 \beta_{2} - 58 \beta_1 - 485) q^{81} + (12 \beta_{2} + 162 \beta_1 - 150) q^{83} + (38 \beta_{2} + 184 \beta_1 + 840) q^{85} + (29 \beta_1 - 29) q^{87} + (154 \beta_{2} + 68 \beta_1 + 282) q^{89} + (244 \beta_{2} + 208 \beta_1 - 1064) q^{91} + (111 \beta_{2} - 94 \beta_1 + 52) q^{93} + (86 \beta_{2} + 244 \beta_1 + 552) q^{95} + ( - 34 \beta_{2} + 176 \beta_1 + 14) q^{97} + (22 \beta_{2} + 38 \beta_1 - 114) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9} + 10 q^{11} + 4 q^{13} + 130 q^{15} - 66 q^{17} - 164 q^{19} + 88 q^{21} + 204 q^{23} + 79 q^{25} - 142 q^{27} - 87 q^{29} + 86 q^{31} - 130 q^{33} + 24 q^{35} + 42 q^{37} + 394 q^{39} + 562 q^{41} + 18 q^{43} - 422 q^{45} - 654 q^{47} + 539 q^{49} + 556 q^{51} - 712 q^{53} - 142 q^{55} + 828 q^{57} + 184 q^{59} - 322 q^{61} + 784 q^{63} + 1494 q^{65} - 228 q^{67} - 684 q^{69} + 52 q^{71} - 494 q^{73} - 3048 q^{75} - 872 q^{77} + 2110 q^{79} - 1513 q^{81} - 288 q^{83} + 2704 q^{85} - 58 q^{87} + 914 q^{89} - 2984 q^{91} + 62 q^{93} + 1900 q^{95} + 218 q^{97} - 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 4\nu - 27 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} + 4\beta _1 + 27 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

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

Hecke characteristic polynomials

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