Properties

Label 135.4.p.a
Level $135$
Weight $4$
Character orbit 135.p
Analytic conductor $7.965$
Analytic rank $0$
Dimension $312$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [135,4,Mod(4,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([2, 9]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.p (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: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(52\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 12 q^{4} - 18 q^{5} - 24 q^{6} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 312 q - 12 q^{4} - 18 q^{5} - 24 q^{6} + 36 q^{9} - 3 q^{10} - 24 q^{11} - 144 q^{14} + 24 q^{15} - 60 q^{16} - 6 q^{19} - 183 q^{20} + 132 q^{21} - 174 q^{24} - 222 q^{25} - 1896 q^{26} - 138 q^{29} + 417 q^{30} + 96 q^{31} - 60 q^{34} - 417 q^{35} + 3564 q^{36} + 120 q^{39} - 189 q^{40} - 1632 q^{41} - 1758 q^{44} + 18 q^{45} - 6 q^{46} + 582 q^{49} - 4668 q^{50} + 3186 q^{51} - 2808 q^{54} - 12 q^{55} + 3558 q^{56} - 672 q^{59} + 3630 q^{60} + 42 q^{61} + 7290 q^{64} + 2970 q^{65} - 1266 q^{66} + 3480 q^{69} - 1059 q^{70} + 5958 q^{71} + 4458 q^{74} - 1716 q^{75} + 372 q^{76} - 2820 q^{79} - 2994 q^{80} - 540 q^{81} - 14814 q^{84} - 381 q^{85} - 10422 q^{86} - 6654 q^{89} + 6246 q^{90} - 6 q^{91} + 3660 q^{94} + 4191 q^{95} - 10038 q^{96} + 5544 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.89118 + 5.19596i −0.0220314 5.19611i −17.2931 14.5107i −9.65682 5.63435i 27.0404 + 9.71227i 13.7470 + 16.3830i 69.7922 40.2945i −26.9990 + 0.228955i 47.5386 39.5209i
4.2 −1.85666 + 5.10114i −2.41243 + 4.60219i −16.4461 13.7999i 4.19064 + 10.3653i −18.9974 20.8509i −9.50379 11.3262i 63.3201 36.5579i −15.3604 22.2049i −60.6552 + 2.13223i
4.3 −1.76130 + 4.83912i 5.11954 0.888993i −14.1866 11.9040i 10.8512 2.69293i −4.71508 + 26.3399i −7.10582 8.46839i 46.9135 27.0855i 25.4194 9.10247i −6.08071 + 57.2533i
4.4 −1.65382 + 4.54382i 4.52413 + 2.55583i −11.7829 9.88700i −5.35022 + 9.81709i −19.0953 + 16.3299i 22.7072 + 27.0614i 30.9106 17.8462i 13.9354 + 23.1258i −35.7588 40.5461i
4.5 −1.63162 + 4.48285i 2.53614 + 4.53520i −11.3054 9.48636i −0.966721 11.1385i −24.4686 + 3.96938i −4.14355 4.93809i 27.9208 16.1201i −14.1360 + 23.0038i 51.5094 + 13.8401i
4.6 −1.62821 + 4.47348i −4.69383 + 2.22890i −11.2326 9.42528i −2.71524 10.8456i −2.32838 24.6269i 0.578750 + 0.689728i 27.4707 15.8602i 17.0640 20.9241i 52.9387 + 5.51241i
4.7 −1.53007 + 4.20384i −0.242589 5.19049i −9.20281 7.72208i 4.66423 + 10.1610i 22.1912 + 6.92202i −7.23087 8.61742i 15.5491 8.97729i −26.8823 + 2.51831i −49.8517 + 4.06069i
4.8 −1.49986 + 4.12082i −4.97861 1.48778i −8.60322 7.21896i 10.8774 + 2.58511i 13.5981 18.2845i 20.6597 + 24.6212i 12.2695 7.08378i 22.5730 + 14.8141i −26.9672 + 40.9464i
4.9 −1.47218 + 4.04479i −4.79721 1.99668i −8.06463 6.76703i −9.33180 + 6.15772i 15.1385 16.4642i −11.5687 13.7870i 9.42223 5.43993i 19.0265 + 19.1570i −11.1686 46.8104i
4.10 −1.37958 + 3.79035i 4.90110 1.72604i −6.33518 5.31585i −11.0884 1.43121i −0.219118 + 20.9581i −13.3794 15.9450i 0.943144 0.544524i 21.0415 16.9190i 20.7220 40.0543i
4.11 −1.23285 + 3.38722i 1.73354 4.89845i −3.82500 3.20956i 7.45058 8.33600i 14.4550 + 11.9109i 1.63240 + 1.94541i −9.38631 + 5.41919i −20.9897 16.9833i 19.0505 + 35.5138i
4.12 −1.15191 + 3.16486i 0.627092 + 5.15817i −2.56107 2.14899i 10.8432 2.72480i −17.0472 3.95712i 6.28747 + 7.49311i −13.5826 + 7.84192i −26.2135 + 6.46930i −3.86685 + 37.4560i
4.13 −1.11091 + 3.05220i −2.06434 + 4.76849i −1.95347 1.63916i −10.7613 + 3.03229i −12.2611 11.5982i 8.11002 + 9.66515i −15.3302 + 8.85092i −18.4770 19.6876i 2.69965 36.2142i
4.14 −1.00430 + 2.75929i −2.84251 4.34973i −0.476691 0.399991i 1.21373 11.1143i 14.8569 3.47489i −10.7265 12.7834i −18.7613 + 10.8319i −10.8402 + 24.7283i 29.4485 + 14.5111i
4.15 −0.944176 + 2.59410i 3.52324 + 3.81927i 0.290454 + 0.243719i 0.662990 + 11.1607i −13.2341 + 5.53358i −19.8929 23.7074i −20.0324 + 11.5657i −2.17360 + 26.9124i −29.5779 8.81777i
4.16 −0.935901 + 2.57137i 4.16251 3.11022i 0.392344 + 0.329216i 3.40677 + 10.6487i 4.10182 + 13.6142i 10.1992 + 12.1549i −20.1720 + 11.6463i 7.65302 25.8927i −30.5700 1.20603i
4.17 −0.842463 + 2.31465i −4.78143 + 2.03418i 1.48050 + 1.24229i 10.7613 3.03227i −0.680240 12.7811i −19.1625 22.8370i −21.1883 + 12.2330i 18.7242 19.4526i −2.04734 + 27.4632i
4.18 −0.716355 + 1.96817i 4.76638 + 2.06920i 2.76783 + 2.32248i −4.23077 10.3489i −7.48696 + 7.89877i 8.31506 + 9.90951i −21.0648 + 12.1618i 18.4368 + 19.7252i 23.3992 0.913357i
4.19 −0.627147 + 1.72307i −4.15670 3.11798i 3.55269 + 2.98106i −8.04407 7.76486i 7.97938 5.20687i 15.6915 + 18.7004i −20.0686 + 11.5866i 7.55637 + 25.9211i 18.4242 8.99080i
4.20 −0.617815 + 1.69743i 1.85019 4.85559i 3.62877 + 3.04490i −10.9792 + 2.11101i 7.09897 + 6.14044i 10.3906 + 12.3830i −19.9253 + 11.5039i −20.1536 17.9676i 3.19985 19.9407i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.52
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
27.e even 9 1 inner
135.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.4.p.a 312
5.b even 2 1 inner 135.4.p.a 312
27.e even 9 1 inner 135.4.p.a 312
135.p even 18 1 inner 135.4.p.a 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.p.a 312 1.a even 1 1 trivial
135.4.p.a 312 5.b even 2 1 inner
135.4.p.a 312 27.e even 9 1 inner
135.4.p.a 312 135.p even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(135, [\chi])\).