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Properties

Label 143.4.s.a
Level $143$
Weight $4$
Character orbit 143.s
Analytic conductor $8.437$
Analytic rank $0$
Dimension $320$
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] = [143,4,Mod(8,143)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(143, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([6, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("143.8");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 143 = 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 143.s (of order \(20\), degree \(8\), 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: \(8.43727313082\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(40\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 10 q^{2} - 12 q^{3} - 6 q^{5} - 10 q^{6} - 10 q^{7} - 10 q^{8} - 660 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 10 q^{2} - 12 q^{3} - 6 q^{5} - 10 q^{6} - 10 q^{7} - 10 q^{8} - 660 q^{9} - 124 q^{11} + 30 q^{13} - 300 q^{14} + 270 q^{15} + 1292 q^{16} - 10 q^{18} - 10 q^{19} - 840 q^{20} + 404 q^{22} + 970 q^{24} + 498 q^{26} + 828 q^{27} - 10 q^{28} + 680 q^{29} - 218 q^{31} + 730 q^{33} + 2092 q^{34} - 2620 q^{35} + 330 q^{37} + 1830 q^{39} - 1300 q^{40} - 2230 q^{41} - 1328 q^{42} + 634 q^{44} + 1128 q^{45} + 790 q^{46} - 910 q^{47} - 2120 q^{48} - 1260 q^{50} - 1460 q^{52} + 7404 q^{53} - 1672 q^{55} - 10 q^{57} + 1242 q^{58} - 946 q^{59} - 4324 q^{60} - 500 q^{61} + 4980 q^{63} - 18236 q^{66} + 2552 q^{67} + 220 q^{68} - 1940 q^{70} - 6354 q^{71} - 4570 q^{72} + 1910 q^{73} - 20 q^{74} - 2028 q^{78} + 9340 q^{79} - 13186 q^{80} + 6620 q^{81} + 1410 q^{83} + 9340 q^{84} - 170 q^{85} - 962 q^{86} - 5544 q^{89} - 3142 q^{91} + 1424 q^{92} + 9226 q^{93} - 20220 q^{94} + 21600 q^{96} - 2694 q^{97} + 3840 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −0.841299 + 5.31175i −1.40092 − 4.31158i −19.8985 − 6.46541i −0.0401213 − 0.253316i 24.0806 − 3.81399i −22.3819 + 11.4042i 31.5508 − 61.9220i 5.21634 − 3.78989i 1.37930
8.2 −0.824790 + 5.20752i −0.252948 − 0.778495i −18.8295 − 6.11808i −0.607785 − 3.83740i 4.26266 − 0.675139i 19.1517 − 9.75826i 28.2414 − 55.4268i 21.3014 − 15.4764i 20.4846
8.3 −0.797941 + 5.03800i 2.29368 + 7.05922i −17.1363 − 5.56792i −2.87862 − 18.1749i −37.3945 + 5.92271i −7.49320 + 3.81798i 23.1992 − 45.5310i −22.7281 + 16.5129i 93.8620
8.4 −0.750034 + 4.73553i 1.76440 + 5.43028i −14.2542 − 4.63148i 1.31373 + 8.29457i −27.0386 + 4.28249i −7.71950 + 3.93328i 15.2102 − 29.8518i −4.53133 + 3.29220i −40.2645
8.5 −0.714408 + 4.51059i −3.03834 − 9.35105i −12.2266 − 3.97267i 2.74904 + 17.3567i 44.3494 − 7.02426i 15.9403 − 8.12197i 10.0675 − 19.7587i −56.3672 + 40.9532i −80.2531
8.6 −0.642359 + 4.05569i −2.52417 − 7.76858i −8.42758 − 2.73829i −2.51799 − 15.8980i 33.1284 − 5.24703i −1.75499 + 0.894212i 1.60560 − 3.15117i −32.1360 + 23.3482i 66.0948
8.7 −0.635090 + 4.00980i −0.876764 − 2.69840i −8.06669 − 2.62103i 2.28503 + 14.4271i 11.3769 − 1.80192i −8.69165 + 4.42862i 0.888036 − 1.74287i 15.3308 − 11.1385i −59.3010
8.8 −0.615811 + 3.88808i 2.21250 + 6.80939i −7.12949 − 2.31651i 1.65081 + 10.4228i −27.8379 + 4.40909i 2.55147 − 1.30004i −0.900036 + 1.76642i −19.6291 + 14.2614i −41.5413
8.9 −0.598684 + 3.77994i −0.225426 − 0.693791i −6.32110 − 2.05385i 0.0516297 + 0.325977i 2.75745 − 0.436737i 31.0728 − 15.8324i −2.35181 + 4.61569i 21.4129 − 15.5574i −1.26308
8.10 −0.513728 + 3.24355i −0.643975 − 1.98195i −2.64827 − 0.860475i −1.20875 − 7.63172i 6.75939 − 1.07058i −18.6182 + 9.48645i −7.77569 + 15.2607i 18.3300 − 13.3175i 25.3749
8.11 −0.476058 + 3.00571i 0.821200 + 2.52739i −1.19922 − 0.389652i −3.14490 − 19.8561i −7.98755 + 1.26510i 8.03509 − 4.09408i −9.31051 + 18.2729i 16.1301 − 11.7192i 61.1789
8.12 −0.402348 + 2.54032i 2.85028 + 8.77224i 1.31710 + 0.427951i −0.837956 − 5.29065i −23.4311 + 3.71113i 15.9750 − 8.13967i −10.9583 + 21.5069i −46.9847 + 34.1364i 13.7771
8.13 −0.337026 + 2.12790i −2.18899 − 6.73702i 3.19410 + 1.03783i 0.874242 + 5.51975i 15.0734 − 2.38740i −9.13306 + 4.65353i −11.1096 + 21.8038i −18.7523 + 13.6243i −12.0401
8.14 −0.284645 + 1.79718i 0.773816 + 2.38156i 4.45963 + 1.44902i 3.21261 + 20.2836i −4.50034 + 0.712785i −10.3995 + 5.29881i −10.4821 + 20.5723i 16.7704 − 12.1844i −37.3677
8.15 −0.279073 + 1.76200i 1.07862 + 3.31966i 4.58171 + 1.48869i −0.785506 − 4.95949i −6.15023 + 0.974101i −20.6173 + 10.5050i −10.3809 + 20.3736i 11.9868 − 8.70890i 8.95781
8.16 −0.209587 + 1.32328i −2.43022 − 7.47945i 5.90131 + 1.91745i −1.35488 − 8.55439i 10.4067 − 1.64826i 17.1460 − 8.73634i −8.64012 + 16.9572i −28.1927 + 20.4832i 11.6038
8.17 −0.176074 + 1.11169i −0.887573 − 2.73167i 6.40360 + 2.08066i 1.91203 + 12.0721i 3.19304 − 0.505728i 15.8579 − 8.08001i −7.52845 + 14.7754i 15.1692 − 11.0211i −13.7570
8.18 −0.115668 + 0.730297i 2.95158 + 9.08402i 7.08850 + 2.30319i −0.461797 − 2.91567i −6.97544 + 1.10480i −27.0976 + 13.8069i −5.18737 + 10.1808i −51.9642 + 37.7542i 2.18272
8.19 −0.114387 + 0.722212i 1.63618 + 5.03566i 7.09995 + 2.30691i 1.50908 + 9.52793i −3.82397 + 0.605658i 24.4471 − 12.4564i −5.13394 + 10.0759i −0.837294 + 0.608330i −7.05381
8.20 0.00594438 − 0.0375313i −1.71697 − 5.28429i 7.60708 + 2.47169i −1.50254 − 9.48665i −0.208533 + 0.0330284i 12.9900 − 6.61872i 0.275995 − 0.541671i −3.13227 + 2.27573i −0.364978
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
13.d odd 4 1 inner
143.s even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.s.a ✓ 320
11.d odd 10 1 inner 143.4.s.a ✓ 320
13.d odd 4 1 inner 143.4.s.a ✓ 320
143.s even 20 1 inner 143.4.s.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.s.a ✓ 320 1.a even 1 1 trivial
143.4.s.a ✓ 320 11.d odd 10 1 inner
143.4.s.a ✓ 320 13.d odd 4 1 inner
143.4.s.a ✓ 320 143.s even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(143, [\chi])\).