Properties

Label 543.2.a.d
Level 543543
Weight 22
Character orbit 543.a
Self dual yes
Analytic conductor 4.3364.336
Analytic rank 00
Dimension 88
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [543,2,Mod(1,543)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(543, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("543.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 543=3181 543 = 3 \cdot 181
Weight: k k == 2 2
Character orbit: [χ][\chi] == 543.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: 4.335876829764.33587682976
Analytic rank: 00
Dimension: 88
Coefficient field: Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x83x76x6+21x5+5x435x3+10x2+4x1 x^{8} - 3x^{7} - 6x^{6} + 21x^{5} + 5x^{4} - 35x^{3} + 10x^{2} + 4x - 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2q3+(β2+β1)q4+(β7+β4+β2+1)q5β1q6+(β6β5+β4)q7+(β3+β2+β1)q8++(β7β6+β1)q99+O(q100) q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{7} + \beta_{4} + \beta_{2} + 1) q^{5} - \beta_1 q^{6} + ( - \beta_{6} - \beta_{5} + \beta_{4}) q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8}+ \cdots + (\beta_{7} - \beta_{6} + \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+3q28q3+5q4+5q53q63q7+6q8+8q9+2q10+4q115q12+13q13+5q145q15+3q16+27q17+3q1810q19+21q20++4q99+O(q100) 8 q + 3 q^{2} - 8 q^{3} + 5 q^{4} + 5 q^{5} - 3 q^{6} - 3 q^{7} + 6 q^{8} + 8 q^{9} + 2 q^{10} + 4 q^{11} - 5 q^{12} + 13 q^{13} + 5 q^{14} - 5 q^{15} + 3 q^{16} + 27 q^{17} + 3 q^{18} - 10 q^{19} + 21 q^{20}+ \cdots + 4 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x83x76x6+21x5+5x435x3+10x2+4x1 x^{8} - 3x^{7} - 6x^{6} + 21x^{5} + 5x^{4} - 35x^{3} + 10x^{2} + 4x - 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2ν2 \nu^{2} - \nu - 2 Copy content Toggle raw display
β3\beta_{3}== ν3ν24ν+2 \nu^{3} - \nu^{2} - 4\nu + 2 Copy content Toggle raw display
β4\beta_{4}== ν73ν66ν5+21ν4+5ν335ν2+10ν+3 \nu^{7} - 3\nu^{6} - 6\nu^{5} + 21\nu^{4} + 5\nu^{3} - 35\nu^{2} + 10\nu + 3 Copy content Toggle raw display
β5\beta_{5}== ν7+3ν6+7ν521ν413ν3+34ν2+4ν3 -\nu^{7} + 3\nu^{6} + 7\nu^{5} - 21\nu^{4} - 13\nu^{3} + 34\nu^{2} + 4\nu - 3 Copy content Toggle raw display
β6\beta_{6}== ν73ν67ν5+22ν4+12ν339ν2ν+5 \nu^{7} - 3\nu^{6} - 7\nu^{5} + 22\nu^{4} + 12\nu^{3} - 39\nu^{2} - \nu + 5 Copy content Toggle raw display
β7\beta_{7}== 2ν75ν614ν5+35ν4+23ν358ν2+6 2\nu^{7} - 5\nu^{6} - 14\nu^{5} + 35\nu^{4} + 23\nu^{3} - 58\nu^{2} + 6 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+β1+2 \beta_{2} + \beta _1 + 2 Copy content Toggle raw display
ν3\nu^{3}== β3+β2+5β1 \beta_{3} + \beta_{2} + 5\beta_1 Copy content Toggle raw display
ν4\nu^{4}== β6+β5+β3+6β2+7β1+8 \beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + 7\beta _1 + 8 Copy content Toggle raw display
ν5\nu^{5}== β5+β4+8β3+9β2+27β1+2 \beta_{5} + \beta_{4} + 8\beta_{3} + 9\beta_{2} + 27\beta _1 + 2 Copy content Toggle raw display
ν6\nu^{6}== β7+7β6+9β5+10β3+35β2+46β1+36 \beta_{7} + 7\beta_{6} + 9\beta_{5} + 10\beta_{3} + 35\beta_{2} + 46\beta _1 + 36 Copy content Toggle raw display
ν7\nu^{7}== 3β7+12β5+7β4+52β3+63β2+153β1+19 3\beta_{7} + 12\beta_{5} + 7\beta_{4} + 52\beta_{3} + 63\beta_{2} + 153\beta _1 + 19 Copy content Toggle raw display

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
1.1
−2.09416
−1.57409
−0.330181
0.216541
0.495668
1.49117
2.31010
2.48495
−2.09416 −1.00000 2.38551 1.57030 2.09416 −3.68416 −0.807326 1.00000 −3.28847
1.2 −1.57409 −1.00000 0.477759 2.61079 1.57409 3.43627 2.39614 1.00000 −4.10962
1.3 −0.330181 −1.00000 −1.89098 −3.90200 0.330181 −4.54089 1.28473 1.00000 1.28837
1.4 0.216541 −1.00000 −1.95311 −1.13532 −0.216541 1.21084 −0.856012 1.00000 −0.245843
1.5 0.495668 −1.00000 −1.75431 3.58208 −0.495668 −1.17967 −1.86089 1.00000 1.77552
1.6 1.49117 −1.00000 0.223595 −1.54064 −1.49117 2.68646 −2.64893 1.00000 −2.29736
1.7 2.31010 −1.00000 3.33658 3.44398 −2.31010 1.03443 3.08764 1.00000 7.95596
1.8 2.48495 −1.00000 4.17495 0.370811 −2.48495 −1.96328 5.40464 1.00000 0.921444
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 +1 +1
181181 1 -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 543.2.a.d 8
3.b odd 2 1 1629.2.a.e 8
4.b odd 2 1 8688.2.a.bf 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
543.2.a.d 8 1.a even 1 1 trivial
1629.2.a.e 8 3.b odd 2 1
8688.2.a.bf 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T283T276T26+21T25+5T2435T23+10T22+4T21 T_{2}^{8} - 3T_{2}^{7} - 6T_{2}^{6} + 21T_{2}^{5} + 5T_{2}^{4} - 35T_{2}^{3} + 10T_{2}^{2} + 4T_{2} - 1 acting on S2new(Γ0(543))S_{2}^{\mathrm{new}}(\Gamma_0(543)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T83T7+1 T^{8} - 3 T^{7} + \cdots - 1 Copy content Toggle raw display
33 (T+1)8 (T + 1)^{8} Copy content Toggle raw display
55 T85T7+128 T^{8} - 5 T^{7} + \cdots - 128 Copy content Toggle raw display
77 T8+3T7++448 T^{8} + 3 T^{7} + \cdots + 448 Copy content Toggle raw display
1111 T84T7++64 T^{8} - 4 T^{7} + \cdots + 64 Copy content Toggle raw display
1313 T813T7+6298 T^{8} - 13 T^{7} + \cdots - 6298 Copy content Toggle raw display
1717 T827T7++640 T^{8} - 27 T^{7} + \cdots + 640 Copy content Toggle raw display
1919 T8+10T7+188416 T^{8} + 10 T^{7} + \cdots - 188416 Copy content Toggle raw display
2323 T83T7++3872 T^{8} - 3 T^{7} + \cdots + 3872 Copy content Toggle raw display
2929 T816T7+16928 T^{8} - 16 T^{7} + \cdots - 16928 Copy content Toggle raw display
3131 T8+T7+11200 T^{8} + T^{7} + \cdots - 11200 Copy content Toggle raw display
3737 T83T7++218 T^{8} - 3 T^{7} + \cdots + 218 Copy content Toggle raw display
4141 T820T7+1030 T^{8} - 20 T^{7} + \cdots - 1030 Copy content Toggle raw display
4343 T8+4T7+8348 T^{8} + 4 T^{7} + \cdots - 8348 Copy content Toggle raw display
4747 T812T7+78848 T^{8} - 12 T^{7} + \cdots - 78848 Copy content Toggle raw display
5353 T823T7+98 T^{8} - 23 T^{7} + \cdots - 98 Copy content Toggle raw display
5959 T87T7++591104 T^{8} - 7 T^{7} + \cdots + 591104 Copy content Toggle raw display
6161 T8T7++1262368 T^{8} - T^{7} + \cdots + 1262368 Copy content Toggle raw display
6767 T8+25T7++189700 T^{8} + 25 T^{7} + \cdots + 189700 Copy content Toggle raw display
7171 T811T7+111136 T^{8} - 11 T^{7} + \cdots - 111136 Copy content Toggle raw display
7373 T823T7+1191058 T^{8} - 23 T^{7} + \cdots - 1191058 Copy content Toggle raw display
7979 T8+26T7+126104 T^{8} + 26 T^{7} + \cdots - 126104 Copy content Toggle raw display
8383 T816T7+208076 T^{8} - 16 T^{7} + \cdots - 208076 Copy content Toggle raw display
8989 T814T7+15046 T^{8} - 14 T^{7} + \cdots - 15046 Copy content Toggle raw display
9797 T828T7++420704 T^{8} - 28 T^{7} + \cdots + 420704 Copy content Toggle raw display
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