Properties

Label 324.3.d.e
Level 324324
Weight 33
Character orbit 324.d
Analytic conductor 8.8288.828
Analytic rank 00
Dimension 66
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [324,3,Mod(163,324)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("324.163"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: N N == 324=2234 324 = 2^{2} \cdot 3^{4}
Weight: k k == 3 3
Character orbit: [χ][\chi] == 324.d (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 8.828360565278.82836056527
Analytic rank: 00
Dimension: 66
Coefficient field: 6.0.3636603.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x6+8x4+12x2+3 x^{6} + 8x^{4} + 12x^{2} + 3 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 263 2^{6}\cdot 3
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ3q2β1q4+(β3+β2β1)q5+(β4β1)q7+(β4+β3β2+1)q8+(β5β4+2)q10++(2β5+2β4+5β3++76)q98+O(q100) q - \beta_{3} q^{2} - \beta_1 q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{5} + (\beta_{4} - \beta_1) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{8} + ( - \beta_{5} - \beta_{4} + 2) q^{10}+ \cdots + (2 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + \cdots + 76) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6qq23q42q57q8+9q10+6q1315q16+10q17+67q20+48q22+73q2648q28+22q2931q32+81q34+54q37168q3881q40++407q98+O(q100) 6 q - q^{2} - 3 q^{4} - 2 q^{5} - 7 q^{8} + 9 q^{10} + 6 q^{13} - 15 q^{16} + 10 q^{17} + 67 q^{20} + 48 q^{22} + 73 q^{26} - 48 q^{28} + 22 q^{29} - 31 q^{32} + 81 q^{34} + 54 q^{37} - 168 q^{38} - 81 q^{40}+ \cdots + 407 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6+8x4+12x2+3 x^{6} + 8x^{4} + 12x^{2} + 3 : Copy content Toggle raw display

β1\beta_{1}== (ν5+ν4+9ν3+5ν2+17ν+1)/2 ( \nu^{5} + \nu^{4} + 9\nu^{3} + 5\nu^{2} + 17\nu + 1 ) / 2 Copy content Toggle raw display
β2\beta_{2}== ν5+8ν3+ν2+12ν+3 \nu^{5} + 8\nu^{3} + \nu^{2} + 12\nu + 3 Copy content Toggle raw display
β3\beta_{3}== (ν5ν4+7ν37ν2+7ν5)/2 ( \nu^{5} - \nu^{4} + 7\nu^{3} - 7\nu^{2} + 7\nu - 5 ) / 2 Copy content Toggle raw display
β4\beta_{4}== (5ν5+ν4+37ν3+5ν2+33ν+1)/2 ( 5\nu^{5} + \nu^{4} + 37\nu^{3} + 5\nu^{2} + 33\nu + 1 ) / 2 Copy content Toggle raw display
β5\beta_{5}== ν52ν46ν313ν22ν8 -\nu^{5} - 2\nu^{4} - 6\nu^{3} - 13\nu^{2} - 2\nu - 8 Copy content Toggle raw display
ν\nu== (β52β4+4β3+β2+2β11)/12 ( -\beta_{5} - 2\beta_{4} + 4\beta_{3} + \beta_{2} + 2\beta _1 - 1 ) / 12 Copy content Toggle raw display
ν2\nu^{2}== (β3+β2β15)/2 ( -\beta_{3} + \beta_{2} - \beta _1 - 5 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (4β5+5β413β3β22β1+1)/6 ( 4\beta_{5} + 5\beta_{4} - 13\beta_{3} - \beta_{2} - 2\beta _1 + 1 ) / 6 Copy content Toggle raw display
ν4\nu^{4}== (β5+10β313β2+14β1+49)/4 ( -\beta_{5} + 10\beta_{3} - 13\beta_{2} + 14\beta _1 + 49 ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (26β528β4+83β3+5β2+7β15)/6 ( -26\beta_{5} - 28\beta_{4} + 83\beta_{3} + 5\beta_{2} + 7\beta _1 - 5 ) / 6 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/324Z)×\left(\mathbb{Z}/324\mathbb{Z}\right)^\times.

nn 163163 245245
χ(n)\chi(n) 1-1 11

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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}
163.1
1.25235i
1.25235i
2.47367i
2.47367i
0.559107i
0.559107i
−1.75941 0.951043i 0 2.19104 + 3.34655i 1.86325 0 11.3183i −0.672219 7.97171i 0 −3.27823 1.77203i
163.2 −1.75941 + 0.951043i 0 2.19104 3.34655i 1.86325 0 11.3183i −0.672219 + 7.97171i 0 −3.27823 + 1.77203i
163.3 −0.195350 1.99044i 0 −3.92368 + 0.777662i −7.23805 0 6.88025i 2.31438 + 7.65792i 0 1.41395 + 14.4069i
163.4 −0.195350 + 1.99044i 0 −3.92368 0.777662i −7.23805 0 6.88025i 2.31438 7.65792i 0 1.41395 14.4069i
163.5 1.45476 1.37247i 0 0.232642 3.99323i 4.37480 0 2.13525i −5.14216 6.12848i 0 6.36427 6.00429i
163.6 1.45476 + 1.37247i 0 0.232642 + 3.99323i 4.37480 0 2.13525i −5.14216 + 6.12848i 0 6.36427 + 6.00429i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.d.e 6
3.b odd 2 1 324.3.d.f yes 6
4.b odd 2 1 inner 324.3.d.e 6
9.c even 3 2 324.3.f.r 12
9.d odd 6 2 324.3.f.q 12
12.b even 2 1 324.3.d.f yes 6
36.f odd 6 2 324.3.f.r 12
36.h even 6 2 324.3.f.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.d.e 6 1.a even 1 1 trivial
324.3.d.e 6 4.b odd 2 1 inner
324.3.d.f yes 6 3.b odd 2 1
324.3.d.f yes 6 12.b even 2 1
324.3.f.q 12 9.d odd 6 2
324.3.f.q 12 36.h even 6 2
324.3.f.r 12 9.c even 3 2
324.3.f.r 12 36.f odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T53+T5237T5+59 T_{5}^{3} + T_{5}^{2} - 37T_{5} + 59 acting on S3new(324,[χ])S_{3}^{\mathrm{new}}(324, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6+T5++64 T^{6} + T^{5} + \cdots + 64 Copy content Toggle raw display
33 T6 T^{6} Copy content Toggle raw display
55 (T3+T237T+59)2 (T^{3} + T^{2} - 37 T + 59)^{2} Copy content Toggle raw display
77 T6+180T4++27648 T^{6} + 180 T^{4} + \cdots + 27648 Copy content Toggle raw display
1111 T6+516T4++1769472 T^{6} + 516 T^{4} + \cdots + 1769472 Copy content Toggle raw display
1313 (T33T2++991)2 (T^{3} - 3 T^{2} + \cdots + 991)^{2} Copy content Toggle raw display
1717 (T35T2++233)2 (T^{3} - 5 T^{2} + \cdots + 233)^{2} Copy content Toggle raw display
1919 T6+1476T4++20155392 T^{6} + 1476 T^{4} + \cdots + 20155392 Copy content Toggle raw display
2323 T6+2868T4++425115648 T^{6} + 2868 T^{4} + \cdots + 425115648 Copy content Toggle raw display
2929 (T311T2++32519)2 (T^{3} - 11 T^{2} + \cdots + 32519)^{2} Copy content Toggle raw display
3131 T6+5376T4++7077888 T^{6} + 5376 T^{4} + \cdots + 7077888 Copy content Toggle raw display
3737 (T327T2++144607)2 (T^{3} - 27 T^{2} + \cdots + 144607)^{2} Copy content Toggle raw display
4141 (T3+46T2+21976)2 (T^{3} + 46 T^{2} + \cdots - 21976)^{2} Copy content Toggle raw display
4343 T6+6372T4++204484608 T^{6} + 6372 T^{4} + \cdots + 204484608 Copy content Toggle raw display
4747 T6+8208T4++743620608 T^{6} + 8208 T^{4} + \cdots + 743620608 Copy content Toggle raw display
5353 (T3+58T2+5128)2 (T^{3} + 58 T^{2} + \cdots - 5128)^{2} Copy content Toggle raw display
5959 T6++15635054592 T^{6} + \cdots + 15635054592 Copy content Toggle raw display
6161 (T327T2++101191)2 (T^{3} - 27 T^{2} + \cdots + 101191)^{2} Copy content Toggle raw display
6767 T6++12897487872 T^{6} + \cdots + 12897487872 Copy content Toggle raw display
7171 T6+8532T4++808455168 T^{6} + 8532 T^{4} + \cdots + 808455168 Copy content Toggle raw display
7373 (T3+39T2+447851)2 (T^{3} + 39 T^{2} + \cdots - 447851)^{2} Copy content Toggle raw display
7979 T6+7764T4++113246208 T^{6} + 7764 T^{4} + \cdots + 113246208 Copy content Toggle raw display
8383 T6++900806787072 T^{6} + \cdots + 900806787072 Copy content Toggle raw display
8989 (T3125T2++107777)2 (T^{3} - 125 T^{2} + \cdots + 107777)^{2} Copy content Toggle raw display
9797 (T3+102T2+369272)2 (T^{3} + 102 T^{2} + \cdots - 369272)^{2} Copy content Toggle raw display
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