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Properties

Label 152.3.s.a
Level $152$
Weight $3$
Character orbit 152.s
Analytic conductor $4.142$
Analytic rank $0$
Dimension $228$
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] = [152,3,Mod(13,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 152.s (of order \(18\), degree \(6\), 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: \(4.14170001828\)
Analytic rank: \(0\)
Dimension: \(228\)
Relative dimension: \(38\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 228 q - 6 q^{2} - 12 q^{4} + 12 q^{6} - 6 q^{7} - 9 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 228 q - 6 q^{2} - 12 q^{4} + 12 q^{6} - 6 q^{7} - 9 q^{8} - 12 q^{9} + 51 q^{10} - 9 q^{12} + 51 q^{14} - 12 q^{15} - 48 q^{16} - 12 q^{17} + 18 q^{20} - 18 q^{22} - 12 q^{23} - 84 q^{24} - 12 q^{25} + 69 q^{26} - 120 q^{28} + 264 q^{30} - 18 q^{31} - 51 q^{32} - 66 q^{33} + 60 q^{34} - 111 q^{36} - 42 q^{38} - 24 q^{39} + 60 q^{40} + 60 q^{41} - 231 q^{42} + 57 q^{44} - 414 q^{46} + 276 q^{47} + 219 q^{48} - 552 q^{49} - 360 q^{50} - 261 q^{52} - 63 q^{54} - 162 q^{55} - 12 q^{57} + 60 q^{58} + 216 q^{60} - 348 q^{62} + 282 q^{63} - 321 q^{64} - 18 q^{65} - 216 q^{66} - 240 q^{68} + 33 q^{70} - 12 q^{71} - 426 q^{72} - 252 q^{73} + 111 q^{74} + 66 q^{76} + 189 q^{78} - 12 q^{79} - 9 q^{80} - 126 q^{81} + 309 q^{82} + 963 q^{84} + 144 q^{86} - 6 q^{87} + 927 q^{88} - 12 q^{89} - 150 q^{90} + 336 q^{92} + 468 q^{95} + 702 q^{96} - 12 q^{97} - 309 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.99495 − 0.141980i 2.41350 + 0.878444i 3.95968 + 0.566486i −8.42500 + 1.48555i −4.69011 − 2.09512i 6.21724 − 10.7686i −7.81896 − 1.69231i −1.84106 − 1.54483i 17.0184 − 1.76743i
13.2 −1.96548 + 0.369979i −0.541850 − 0.197217i 3.72623 − 1.45437i 7.64702 − 1.34838i 1.13796 + 0.187154i −0.264608 + 0.458315i −6.78575 + 4.23717i −6.63969 − 5.57136i −14.5312 + 5.47944i
13.3 −1.95675 − 0.413654i 0.799849 + 0.291121i 3.65778 + 1.61884i −0.897321 + 0.158222i −1.44469 − 0.900514i −6.28139 + 10.8797i −6.48774 − 4.68073i −6.33939 − 5.31938i 1.82129 + 0.0615792i
13.4 −1.92972 − 0.525538i −4.03276 − 1.46780i 3.44762 + 2.02828i −5.51589 + 0.972601i 7.01070 + 4.95181i −1.32411 + 2.29343i −5.58700 − 5.72586i 7.21429 + 6.05351i 11.1553 + 1.02197i
13.5 −1.89927 + 0.626713i −4.09598 − 1.49081i 3.21446 − 2.38060i 0.517653 − 0.0912763i 8.71369 + 0.264457i 3.75490 − 6.50368i −4.61318 + 6.53594i 7.66011 + 6.42760i −0.925960 + 0.497779i
13.6 −1.82216 − 0.824465i 4.16216 + 1.51490i 2.64051 + 3.00461i 6.03602 − 1.06431i −6.33513 − 6.19195i 2.39719 − 4.15205i −2.33424 − 7.65188i 8.13427 + 6.82547i −11.8761 − 3.03714i
13.7 −1.69000 + 1.06953i 4.88537 + 1.77813i 1.71222 − 3.61501i 0.816358 − 0.143946i −10.1581 + 2.21999i −2.39152 + 4.14223i 0.972689 + 7.94065i 13.8107 + 11.5886i −1.22569 + 1.11639i
13.8 −1.67376 + 1.09477i 0.743650 + 0.270666i 1.60294 − 3.66478i −3.42840 + 0.604520i −1.54101 + 0.361099i 0.378578 − 0.655717i 1.32918 + 7.88881i −6.41465 − 5.38253i 5.07650 − 4.76515i
13.9 −1.49312 − 1.33064i −0.982131 − 0.357466i 0.458790 + 3.97360i −0.194535 + 0.0343018i 0.990776 + 1.84060i 1.72455 − 2.98701i 4.60241 − 6.54353i −6.05760 − 5.08293i 0.336107 + 0.207640i
13.10 −1.36569 + 1.46113i −2.95828 − 1.07673i −0.269808 − 3.99089i −6.19714 + 1.09272i 5.61332 − 2.85197i −4.21560 + 7.30163i 6.19969 + 5.05608i 0.697688 + 0.585430i 6.86673 − 10.5471i
13.11 −1.36485 − 1.46191i −4.79571 − 1.74549i −0.274380 + 3.99058i 8.51877 − 1.50209i 3.99365 + 9.39324i −4.18112 + 7.24191i 6.20837 − 5.04541i 13.0576 + 10.9567i −13.8228 − 10.4036i
13.12 −1.20270 − 1.59797i 4.79571 + 1.74549i −1.10703 + 3.84376i −8.51877 + 1.50209i −2.97855 − 9.76271i −4.18112 + 7.24191i 7.47364 − 2.85390i 13.0576 + 10.9567i 12.6458 + 11.8062i
13.13 −1.05115 − 1.70150i 0.982131 + 0.357466i −1.79017 + 3.57705i 0.194535 − 0.0343018i −0.424138 − 2.04684i 1.72455 − 2.98701i 7.96807 − 0.714039i −6.05760 − 5.08293i −0.262850 − 0.294945i
13.14 −0.973466 + 1.74710i 2.12018 + 0.771683i −2.10473 − 3.40149i 3.92989 − 0.692946i −3.41213 + 2.95297i 5.70315 − 9.87814i 7.99163 − 0.365936i −2.99472 − 2.51287i −2.61497 + 7.54048i
13.15 −0.925845 + 1.77280i −4.20352 − 1.52996i −2.28562 − 3.28267i 5.32019 − 0.938094i 6.60411 − 6.03549i −1.25733 + 2.17776i 7.93564 − 1.01271i 8.43442 + 7.07732i −3.26262 + 10.3002i
13.16 −0.495525 − 1.93764i −4.16216 − 1.51490i −3.50891 + 1.92030i −6.03602 + 1.06431i −0.872883 + 8.81545i 2.39719 − 4.15205i 5.45961 + 5.84745i 8.13427 + 6.82547i 5.05326 + 11.1682i
13.17 −0.398598 + 1.95988i 2.17810 + 0.792764i −3.68224 − 1.56241i −6.26573 + 1.10482i −2.42191 + 3.95282i −2.32018 + 4.01866i 4.52986 − 6.59396i −2.77875 − 2.33165i 0.332203 − 12.7204i
13.18 −0.182462 − 1.99166i 4.03276 + 1.46780i −3.93342 + 0.726803i 5.51589 − 0.972601i 2.18754 − 8.29970i −1.32411 + 2.29343i 2.16524 + 7.70141i 7.21429 + 6.05351i −2.94353 − 10.8083i
13.19 −0.0675830 − 1.99886i −0.799849 − 0.291121i −3.99087 + 0.270178i 0.897321 − 0.158222i −0.527854 + 1.61846i −6.28139 + 10.8797i 0.809762 + 7.95891i −6.33939 − 5.31938i −0.376907 − 1.78292i
13.20 0.0106968 + 1.99997i −2.12124 − 0.772068i −3.99977 + 0.0427865i −0.198451 + 0.0349922i 1.52142 − 4.25068i 2.46554 − 4.27044i −0.128356 − 7.99897i −2.99083 − 2.50961i −0.0721063 − 0.396522i
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
19.f odd 18 1 inner
152.s odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.3.s.a ✓ 228
8.b even 2 1 inner 152.3.s.a ✓ 228
19.f odd 18 1 inner 152.3.s.a ✓ 228
152.s odd 18 1 inner 152.3.s.a ✓ 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.s.a ✓ 228 1.a even 1 1 trivial
152.3.s.a ✓ 228 8.b even 2 1 inner
152.3.s.a ✓ 228 19.f odd 18 1 inner
152.3.s.a ✓ 228 152.s odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(152, [\chi])\).