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Properties

Label 72.5.p.b
Level $72$
Weight $5$
Character orbit 72.p
Analytic conductor $7.443$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,5,Mod(43,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.43");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 72.p (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(7.44263734204\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 7 q^{2} - 18 q^{3} + 31 q^{4} - 9 q^{6} - 386 q^{8} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 7 q^{2} - 18 q^{3} + 31 q^{4} - 9 q^{6} - 386 q^{8} + 30 q^{9} - 36 q^{10} + 44 q^{11} + 612 q^{12} - 318 q^{14} + 511 q^{16} - 1156 q^{17} + 234 q^{18} + 860 q^{19} - 1164 q^{20} + 215 q^{22} - 561 q^{24} + 5998 q^{25} - 5232 q^{26} + 504 q^{27} - 516 q^{28} + 2046 q^{30} + 5227 q^{32} - 3204 q^{33} + 1877 q^{34} - 2508 q^{35} - 5877 q^{36} - 4393 q^{38} + 2274 q^{40} + 2348 q^{41} - 8166 q^{42} + 3500 q^{43} + 2234 q^{44} + 3576 q^{46} - 1641 q^{48} + 16462 q^{49} + 5785 q^{50} - 12378 q^{51} - 1440 q^{52} - 7467 q^{54} + 1146 q^{56} - 6522 q^{57} - 4128 q^{58} - 3508 q^{59} + 11934 q^{60} + 2376 q^{62} - 5510 q^{64} - 2502 q^{65} + 834 q^{66} + 5132 q^{67} + 10169 q^{68} + 1134 q^{70} + 9663 q^{72} + 19004 q^{73} - 2946 q^{74} - 39858 q^{75} - 5875 q^{76} - 3330 q^{78} + 8592 q^{80} - 11346 q^{81} - 18454 q^{82} + 17090 q^{83} + 12360 q^{84} + 18581 q^{86} - 3253 q^{88} - 8272 q^{89} - 9408 q^{90} - 9612 q^{91} + 9318 q^{92} - 7590 q^{94} + 52650 q^{96} + 9980 q^{97} + 6826 q^{98} - 11742 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −3.95986 + 0.565272i −0.248502 − 8.99657i 15.3609 − 4.47679i −33.3324 + 19.2445i 6.06954 + 35.4847i 74.8398 + 43.2088i −58.2965 + 26.4105i −80.8765 + 4.47133i 121.113 − 95.0473i
43.2 −3.93517 − 0.717268i −7.94378 + 4.23041i 14.9711 + 5.64514i 0.827340 − 0.477665i 34.2944 − 10.9495i 48.6307 + 28.0769i −54.8645 − 32.9528i 45.2073 − 67.2109i −3.59833 + 1.28627i
43.3 −3.89968 − 0.890213i −7.14400 − 5.47387i 14.4150 + 6.94310i −7.87087 + 4.54425i 22.9864 + 27.7060i −59.3416 − 34.2609i −50.0332 − 39.9083i 21.0736 + 78.2106i 34.7393 − 10.7144i
43.4 −3.85922 + 1.05185i 8.92834 + 1.13350i 13.7872 − 8.11868i −26.1495 + 15.0974i −35.6487 + 5.01688i −39.5599 − 22.8399i −44.6682 + 45.8339i 78.4304 + 20.2405i 85.0364 − 85.7697i
43.5 −3.85475 + 1.06812i 5.47144 − 7.14586i 13.7182 − 8.23470i 34.3706 − 19.8439i −13.4584 + 33.3897i 4.61706 + 2.66566i −44.0847 + 46.3955i −21.1267 − 78.1963i −111.294 + 113.205i
43.6 −3.81882 + 1.19022i −2.65445 + 8.59965i 13.1667 − 9.09048i 17.4312 − 10.0639i −0.0986349 − 35.9999i −49.3138 − 28.4714i −39.4617 + 50.3862i −66.9078 − 45.6546i −54.5881 + 59.1791i
43.7 −3.79885 − 1.25250i 7.40748 + 5.11167i 12.8625 + 9.51614i 19.4388 − 11.2230i −21.7375 − 28.6964i 16.3361 + 9.43164i −36.9435 − 52.2606i 28.7416 + 75.7293i −87.9018 + 18.2873i
43.8 −3.42315 + 2.06931i −7.80714 − 4.47756i 7.43587 − 14.1671i 14.7009 − 8.48755i 35.9905 − 0.828075i −16.5269 − 9.54179i 3.86217 + 63.8834i 40.9029 + 69.9139i −32.7598 + 59.4749i
43.9 −3.33332 − 2.21111i 1.03663 + 8.94010i 6.22200 + 14.7406i −38.0026 + 21.9408i 16.3121 − 32.0923i −8.29349 − 4.78825i 11.8532 − 62.8928i −78.8508 + 18.5351i 175.188 + 10.8922i
43.10 −3.30101 − 2.25906i 4.44945 − 7.82320i 5.79331 + 14.9143i −2.53773 + 1.46516i −32.3607 + 15.7729i −39.8426 − 23.0032i 14.5686 − 62.3198i −41.4048 − 69.6178i 11.6869 + 0.896378i
43.11 −3.00451 + 2.64063i 3.93069 + 8.09627i 2.05417 − 15.8676i 0.239806 − 0.138452i −33.1891 − 13.9458i 73.5661 + 42.4734i 35.7286 + 53.0986i −50.0993 + 63.6479i −0.354899 + 1.04922i
43.12 −2.80039 − 2.85619i −5.44488 − 7.16612i −0.315605 + 15.9969i 39.0021 − 22.5179i −5.21999 + 35.6195i 62.8918 + 36.3106i 46.5739 − 43.8961i −21.7066 + 78.0373i −173.536 − 48.3384i
43.13 −2.74080 + 2.91342i −7.87501 + 4.35709i −0.976044 − 15.9702i −38.7055 + 22.3466i 8.88980 − 34.8851i −16.8840 − 9.74799i 49.2031 + 40.9275i 43.0316 − 68.6242i 40.9788 − 174.013i
43.14 −2.17989 − 3.35382i −1.94675 + 8.78693i −6.49617 + 14.6219i 21.6394 − 12.4935i 33.7135 − 12.6255i −24.5598 − 14.1796i 63.2001 − 10.0871i −73.4203 − 34.2119i −89.0724 − 45.3401i
43.15 −2.11362 − 3.39597i 8.77973 − 1.97899i −7.06524 + 14.3556i −19.3544 + 11.1743i −25.2776 − 25.6329i 66.9185 + 38.6354i 63.6843 − 6.34884i 73.1672 − 34.7500i 78.8552 + 42.1088i
43.16 −1.79643 − 3.57391i −8.96951 + 0.740137i −9.54568 + 12.8406i −15.9578 + 9.21327i 18.7583 + 30.7266i −11.6291 − 6.71408i 63.0392 + 11.0482i 79.9044 − 13.2773i 61.5946 + 40.4809i
43.17 −1.15270 + 3.83031i −7.87501 + 4.35709i −13.3426 − 8.83038i 38.7055 − 22.3466i −7.61149 − 35.1862i 16.8840 + 9.74799i 49.2031 − 40.9275i 43.0316 − 68.6242i 40.9788 + 174.013i
43.18 −0.942169 − 3.88746i 8.90483 + 1.30534i −14.2246 + 7.32528i 23.2381 − 13.4165i −3.31539 − 35.8470i −52.2165 − 30.1472i 41.8787 + 48.3960i 77.5922 + 23.2478i −74.0503 − 77.6964i
43.19 −0.784595 + 3.92230i 3.93069 + 8.09627i −14.7688 − 6.15483i −0.239806 + 0.138452i −34.8400 + 9.06504i −73.5661 − 42.4734i 35.7286 − 53.0986i −50.0993 + 63.6479i −0.354899 − 1.04922i
43.20 −0.269184 − 3.99093i −1.80425 − 8.81729i −15.8551 + 2.14859i −11.7270 + 6.77058i −34.7035 + 9.57411i −3.17774 − 1.83467i 12.8428 + 62.6982i −74.4894 + 31.8172i 30.1777 + 44.9791i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.5.p.b ✓ 88
3.b odd 2 1 216.5.p.b 88
4.b odd 2 1 288.5.t.b 88
8.b even 2 1 288.5.t.b 88
8.d odd 2 1 inner 72.5.p.b ✓ 88
9.c even 3 1 inner 72.5.p.b ✓ 88
9.d odd 6 1 216.5.p.b 88
12.b even 2 1 864.5.t.b 88
24.f even 2 1 216.5.p.b 88
24.h odd 2 1 864.5.t.b 88
36.f odd 6 1 288.5.t.b 88
36.h even 6 1 864.5.t.b 88
72.j odd 6 1 864.5.t.b 88
72.l even 6 1 216.5.p.b 88
72.n even 6 1 288.5.t.b 88
72.p odd 6 1 inner 72.5.p.b ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.5.p.b ✓ 88 1.a even 1 1 trivial
72.5.p.b ✓ 88 8.d odd 2 1 inner
72.5.p.b ✓ 88 9.c even 3 1 inner
72.5.p.b ✓ 88 72.p odd 6 1 inner
216.5.p.b 88 3.b odd 2 1
216.5.p.b 88 9.d odd 6 1
216.5.p.b 88 24.f even 2 1
216.5.p.b 88 72.l even 6 1
288.5.t.b 88 4.b odd 2 1
288.5.t.b 88 8.b even 2 1
288.5.t.b 88 36.f odd 6 1
288.5.t.b 88 72.n even 6 1
864.5.t.b 88 12.b even 2 1
864.5.t.b 88 24.h odd 2 1
864.5.t.b 88 36.h even 6 1
864.5.t.b 88 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{88} - 16749 T_{5}^{86} + 151466376 T_{5}^{84} - 946630365075 T_{5}^{82} + \cdots + 11\!\cdots\!76 \) acting on \(S_{5}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display