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Properties

Label 141.4.g.a
Level $141$
Weight $4$
Character orbit 141.g
Analytic conductor $8.319$
Analytic rank $0$
Dimension $1012$
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] = [141,4,Mod(5,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, base_ring=CyclotomicField(46))
 
chi = DirichletCharacter(H, H._module([23, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 141.g (of order \(46\), degree \(22\), 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: \(8.31926931081\)
Analytic rank: \(0\)
Dimension: \(1012\)
Relative dimension: \(46\) over \(\Q(\zeta_{46})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{46}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1012 q - 25 q^{3} + 122 q^{4} - 75 q^{6} - 42 q^{7} - 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1012 q - 25 q^{3} + 122 q^{4} - 75 q^{6} - 42 q^{7} - 49 q^{9} - 46 q^{10} - 52 q^{12} - 46 q^{13} - 23 q^{15} - 558 q^{16} + 184 q^{18} - 46 q^{19} + 297 q^{21} - 46 q^{22} + 498 q^{24} - 1280 q^{25} + 839 q^{27} + 218 q^{28} - 23 q^{30} - 46 q^{31} - 23 q^{33} - 1430 q^{34} + 6825 q^{36} + 30 q^{37} + 667 q^{39} - 46 q^{40} - 3786 q^{42} - 46 q^{43} - 5083 q^{45} - 10226 q^{48} - 652 q^{49} - 1903 q^{51} - 46 q^{52} - 362 q^{54} - 358 q^{55} + 4393 q^{57} - 46 q^{58} + 12489 q^{60} + 1110 q^{61} - 113 q^{63} + 3874 q^{64} - 23 q^{66} - 46 q^{67} - 23 q^{69} - 46 q^{70} - 1791 q^{72} - 46 q^{73} + 2651 q^{75} + 9752 q^{76} - 23 q^{78} + 7098 q^{79} + 335 q^{81} + 16100 q^{82} - 7894 q^{84} - 3910 q^{85} - 23 q^{87} - 17434 q^{88} - 23 q^{90} - 32062 q^{91} - 18018 q^{94} - 8475 q^{96} - 7494 q^{97} - 23 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −5.19236 + 1.84537i 0.0386084 − 5.19601i 17.3496 − 14.1149i −0.245124 + 3.58358i 9.38807 + 27.0508i −6.02960 + 8.54199i −41.1325 + 67.6395i −26.9970 − 0.401219i −5.34024 − 19.0596i
5.2 −4.73504 + 1.68283i −3.61919 + 3.72847i 13.3830 − 10.8878i −0.546710 + 7.99262i 10.8626 − 23.7449i −6.85501 + 9.71134i −24.1583 + 39.7267i −0.802991 − 26.9881i −10.8615 − 38.7654i
5.3 −4.67844 + 1.66272i 0.586377 + 5.16296i 12.9174 − 10.5091i 0.858451 − 12.5501i −11.3279 − 23.1796i 5.82555 − 8.25292i −22.3214 + 36.7060i −26.3123 + 6.05488i 16.8511 + 60.1422i
5.4 −4.57888 + 1.62733i −4.73999 − 2.12896i 12.1122 − 9.85403i 1.27588 − 18.6527i 25.1684 + 2.03472i 11.1977 − 15.8636i −19.2255 + 31.6149i 17.9350 + 20.1825i 24.5120 + 87.4845i
5.5 −4.56224 + 1.62142i 3.91870 + 3.41230i 11.9793 − 9.74592i −1.48837 + 21.7592i −23.4108 − 9.21388i 0.898833 − 1.27336i −18.7246 + 30.7914i 3.71241 + 26.7436i −28.4905 − 101.684i
5.6 −4.54697 + 1.61599i 4.92197 − 1.66559i 11.8578 − 9.64703i 0.0815354 − 1.19201i −19.6885 + 15.5272i 15.6866 − 22.2228i −18.2691 + 30.0422i 21.4516 − 16.3960i 1.55553 + 5.55177i
5.7 −4.29625 + 1.52689i 5.19307 − 0.179081i 9.92066 − 8.07105i 0.797393 − 11.6575i −22.0373 + 8.69859i −20.6574 + 29.2648i −11.3456 + 18.6571i 26.9359 − 1.85995i 14.3738 + 51.3009i
5.8 −4.14469 + 1.47302i −5.15491 − 0.653341i 8.80295 − 7.16173i −0.258997 + 3.78640i 22.3279 − 4.88541i −10.4497 + 14.8038i −7.65229 + 12.5837i 26.1463 + 6.73583i −4.50399 − 16.0750i
5.9 −3.43029 + 1.21912i −0.244763 − 5.19038i 4.07492 − 3.31519i −0.583073 + 8.52422i 7.16733 + 17.5061i 4.97708 − 7.05091i 5.19579 − 8.54411i −26.8802 + 2.54083i −8.39198 − 29.9514i
5.10 −3.35294 + 1.19164i −4.16338 − 3.10907i 3.61654 − 2.94227i −0.826296 + 12.0800i 17.6644 + 5.46329i 7.72918 − 10.9498i 6.17118 − 10.1481i 7.66742 + 25.8884i −11.6245 − 41.4882i
5.11 −2.95070 + 1.04868i 1.60841 − 4.94095i 1.40121 − 1.13997i 1.06059 − 15.5052i 0.435547 + 16.2660i −0.214172 + 0.303412i 10.0776 − 16.5719i −21.8261 − 15.8941i 13.1305 + 46.8634i
5.12 −2.90305 + 1.03174i 4.06373 + 3.23823i 1.15749 − 0.941686i 0.272284 − 3.98065i −15.1382 − 5.20800i 6.34906 − 8.99457i 10.4178 − 17.1313i 6.02778 + 26.3185i 3.31655 + 11.8369i
5.13 −2.87253 + 1.02090i 0.576098 + 5.16412i 1.00350 − 0.816408i 0.102127 − 1.49304i −6.92689 − 14.2459i −13.1413 + 18.6169i 10.6227 − 17.4683i −26.3362 + 5.95008i 1.23088 + 4.39308i
5.14 −2.86263 + 1.01738i −3.31476 + 4.00155i 0.953907 − 0.776061i −0.592676 + 8.66462i 5.41783 − 14.8273i 20.1293 − 28.5168i 10.6870 − 17.5741i −5.02479 − 26.5283i −7.11859 − 25.4066i
5.15 −2.77958 + 0.987861i 3.93969 − 3.38805i 0.544481 − 0.442968i −1.31951 + 19.2905i −7.60374 + 13.3092i −15.7793 + 22.3542i 11.1859 − 18.3945i 4.04228 − 26.6957i −15.3887 − 54.9230i
5.16 −2.26702 + 0.805700i −4.70239 + 2.21077i −1.71546 + 1.39563i 0.945205 − 13.8184i 8.87921 − 8.80058i −4.69936 + 6.65748i 12.7652 − 20.9915i 17.2250 − 20.7918i 8.99068 + 32.0882i
5.17 −1.47552 + 0.524401i 4.72513 + 2.16176i −4.30352 + 3.50117i −0.290775 + 4.25098i −8.10567 − 0.711861i −2.54914 + 3.61131i 11.0230 − 18.1266i 17.6536 + 20.4291i −1.80017 − 6.42490i
5.18 −1.39022 + 0.494085i 4.76986 − 2.06117i −4.51709 + 3.67493i −0.270227 + 3.95058i −5.61277 + 5.22221i 8.02472 − 11.3684i 10.5968 − 17.4258i 18.5031 − 19.6630i −1.57625 − 5.62571i
5.19 −1.17850 + 0.418839i −3.25683 − 4.04883i −4.99226 + 4.06150i 0.194763 − 2.84734i 5.53398 + 3.40746i −12.9644 + 18.3664i 9.38106 − 15.4265i −5.78613 + 26.3727i 0.963046 + 3.43716i
5.20 −0.875116 + 0.311016i −1.85664 + 4.85313i −5.53659 + 4.50435i −1.01449 + 14.8313i 0.115372 − 4.82450i −14.4398 + 20.4566i 7.30471 − 12.0121i −20.1058 − 18.0210i −3.72499 − 13.2946i
See next 80 embeddings (of 1012 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
47.d odd 46 1 inner
141.g even 46 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 141.4.g.a ✓ 1012
3.b odd 2 1 inner 141.4.g.a ✓ 1012
47.d odd 46 1 inner 141.4.g.a ✓ 1012
141.g even 46 1 inner 141.4.g.a ✓ 1012
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.4.g.a ✓ 1012 1.a even 1 1 trivial
141.4.g.a ✓ 1012 3.b odd 2 1 inner
141.4.g.a ✓ 1012 47.d odd 46 1 inner
141.4.g.a ✓ 1012 141.g even 46 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(141, [\chi])\).