Properties

Label 147.4.e.l
Level 147147
Weight 44
Character orbit 147.e
Analytic conductor 8.6738.673
Analytic rank 00
Dimension 44
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] = [147,4,Mod(67,147)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("147.67"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: N N == 147=372 147 = 3 \cdot 7^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 147.e (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,3,-6,-17,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 8.673280770848.67328077084
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(3,19)\Q(\sqrt{-3}, \sqrt{-19})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4x34x25x+25 x^{4} - x^{3} - 4x^{2} - 5x + 25 Copy content Toggle raw display
Coefficient ring: Z[a1,,a4]\Z[a_1, \ldots, a_{4}]
Coefficient ring index: 3 3
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β3+β1+1)q2+3β1q3+(3β33β2+7β1)q4+(2β32β12)q5+(3β23)q6+(5β241)q8++(90β272)q99+O(q100) q + (\beta_{3} + \beta_1 + 1) q^{2} + 3 \beta_1 q^{3} + (3 \beta_{3} - 3 \beta_{2} + 7 \beta_1) q^{4} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{5} + ( - 3 \beta_{2} - 3) q^{6} + ( - 5 \beta_{2} - 41) q^{8}+ \cdots + (90 \beta_{2} - 72) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+3q26q317q46q518q6174q818q9+66q10+6q1151q12+32q13+36q15137q16+6q17+27q1864q19+444q20552q22+108q99+O(q100) 4 q + 3 q^{2} - 6 q^{3} - 17 q^{4} - 6 q^{5} - 18 q^{6} - 174 q^{8} - 18 q^{9} + 66 q^{10} + 6 q^{11} - 51 q^{12} + 32 q^{13} + 36 q^{15} - 137 q^{16} + 6 q^{17} + 27 q^{18} - 64 q^{19} + 444 q^{20} - 552 q^{22}+ \cdots - 108 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4x34x25x+25 x^{4} - x^{3} - 4x^{2} - 5x + 25 : Copy content Toggle raw display

β1\beta_{1}== (ν3+4ν24ν25)/20 ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 Copy content Toggle raw display
β2\beta_{2}== (ν3+ν2+9ν+5)/5 ( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5 Copy content Toggle raw display
β3\beta_{3}== (3ν3+2ν2+8ν25)/10 ( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10 Copy content Toggle raw display
ν\nu== (β3+β22β11)/3 ( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3 Copy content Toggle raw display
ν2\nu^{2}== (β3+2β2+14β1+13)/3 ( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3 Copy content Toggle raw display
ν3\nu^{3}== (8β34β24β1+19)/3 ( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/147Z)×\left(\mathbb{Z}/147\mathbb{Z}\right)^\times.

nn 5050 5252
χ(n)\chi(n) 11 1β1-1 - \beta_{1}

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}
67.1
−1.63746 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 0.656712i
−1.13746 1.97014i −1.50000 + 2.59808i 1.41238 2.44631i 2.27492 + 3.94027i 6.82475 0 −24.6254 −4.50000 7.79423i 5.17525 8.96379i
67.2 2.63746 + 4.56821i −1.50000 + 2.59808i −9.91238 + 17.1687i −5.27492 9.13642i −15.8248 0 −62.3746 −4.50000 7.79423i 27.8248 48.1939i
79.1 −1.13746 + 1.97014i −1.50000 2.59808i 1.41238 + 2.44631i 2.27492 3.94027i 6.82475 0 −24.6254 −4.50000 + 7.79423i 5.17525 + 8.96379i
79.2 2.63746 4.56821i −1.50000 2.59808i −9.91238 17.1687i −5.27492 + 9.13642i −15.8248 0 −62.3746 −4.50000 + 7.79423i 27.8248 + 48.1939i
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.4.e.l 4
3.b odd 2 1 441.4.e.q 4
7.b odd 2 1 147.4.e.m 4
7.c even 3 1 21.4.a.c 2
7.c even 3 1 inner 147.4.e.l 4
7.d odd 6 1 147.4.a.i 2
7.d odd 6 1 147.4.e.m 4
21.c even 2 1 441.4.e.p 4
21.g even 6 1 441.4.a.r 2
21.g even 6 1 441.4.e.p 4
21.h odd 6 1 63.4.a.e 2
21.h odd 6 1 441.4.e.q 4
28.f even 6 1 2352.4.a.bz 2
28.g odd 6 1 336.4.a.m 2
35.j even 6 1 525.4.a.n 2
35.l odd 12 2 525.4.d.g 4
56.k odd 6 1 1344.4.a.bo 2
56.p even 6 1 1344.4.a.bg 2
84.n even 6 1 1008.4.a.ba 2
105.o odd 6 1 1575.4.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 7.c even 3 1
63.4.a.e 2 21.h odd 6 1
147.4.a.i 2 7.d odd 6 1
147.4.e.l 4 1.a even 1 1 trivial
147.4.e.l 4 7.c even 3 1 inner
147.4.e.m 4 7.b odd 2 1
147.4.e.m 4 7.d odd 6 1
336.4.a.m 2 28.g odd 6 1
441.4.a.r 2 21.g even 6 1
441.4.e.p 4 21.c even 2 1
441.4.e.p 4 21.g even 6 1
441.4.e.q 4 3.b odd 2 1
441.4.e.q 4 21.h odd 6 1
525.4.a.n 2 35.j even 6 1
525.4.d.g 4 35.l odd 12 2
1008.4.a.ba 2 84.n even 6 1
1344.4.a.bg 2 56.p even 6 1
1344.4.a.bo 2 56.k odd 6 1
1575.4.a.p 2 105.o odd 6 1
2352.4.a.bz 2 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(147,[χ])S_{4}^{\mathrm{new}}(147, [\chi]):

T243T23+21T22+36T2+144 T_{2}^{4} - 3T_{2}^{3} + 21T_{2}^{2} + 36T_{2} + 144 Copy content Toggle raw display
T54+6T53+84T52288T5+2304 T_{5}^{4} + 6T_{5}^{3} + 84T_{5}^{2} - 288T_{5} + 2304 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T43T3++144 T^{4} - 3 T^{3} + \cdots + 144 Copy content Toggle raw display
33 (T2+3T+9)2 (T^{2} + 3 T + 9)^{2} Copy content Toggle raw display
55 T4+6T3++2304 T^{4} + 6 T^{3} + \cdots + 2304 Copy content Toggle raw display
77 T4 T^{4} Copy content Toggle raw display
1111 T46T3++2005056 T^{4} - 6 T^{3} + \cdots + 2005056 Copy content Toggle raw display
1313 (T216T1988)2 (T^{2} - 16 T - 1988)^{2} Copy content Toggle raw display
1717 T46T3++2304 T^{4} - 6 T^{3} + \cdots + 2304 Copy content Toggle raw display
1919 T4+64T3++51609856 T^{4} + 64 T^{3} + \cdots + 51609856 Copy content Toggle raw display
2323 T4+6T3++271063296 T^{4} + 6 T^{3} + \cdots + 271063296 Copy content Toggle raw display
2929 (T2+252T+7668)2 (T^{2} + 252 T + 7668)^{2} Copy content Toggle raw display
3131 T4++5398134784 T^{4} + \cdots + 5398134784 Copy content Toggle raw display
3737 T4248T3++9560464 T^{4} - 248 T^{3} + \cdots + 9560464 Copy content Toggle raw display
4141 (T2+450T+37800)2 (T^{2} + 450 T + 37800)^{2} Copy content Toggle raw display
4343 (T2376T+2512)2 (T^{2} - 376 T + 2512)^{2} Copy content Toggle raw display
4747 T4++4337012736 T^{4} + \cdots + 4337012736 Copy content Toggle raw display
5353 T4++92705634576 T^{4} + \cdots + 92705634576 Copy content Toggle raw display
5959 T4+804T3++908660736 T^{4} + 804 T^{3} + \cdots + 908660736 Copy content Toggle raw display
6161 T4428T3++788261776 T^{4} - 428 T^{3} + \cdots + 788261776 Copy content Toggle raw display
6767 T4++25836061696 T^{4} + \cdots + 25836061696 Copy content Toggle raw display
7171 (T2954T+214704)2 (T^{2} - 954 T + 214704)^{2} Copy content Toggle raw display
7373 T4++81364139536 T^{4} + \cdots + 81364139536 Copy content Toggle raw display
7979 T4++7126061056 T^{4} + \cdots + 7126061056 Copy content Toggle raw display
8383 (T21944T+813456)2 (T^{2} - 1944 T + 813456)^{2} Copy content Toggle raw display
8989 T4++64438807104 T^{4} + \cdots + 64438807104 Copy content Toggle raw display
9797 (T2808T922292)2 (T^{2} - 808 T - 922292)^{2} Copy content Toggle raw display
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