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Properties

Label 243.4.e.b
Level $243$
Weight $4$
Character orbit 243.e
Analytic conductor $14.337$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [243,4,Mod(28,243)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("243.28");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 243.e (of order \(9\), degree \(6\), not 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: \(14.3374641314\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{2} + 3 q^{4} + 21 q^{5} + 3 q^{7} + 75 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{2} + 3 q^{4} + 21 q^{5} + 3 q^{7} + 75 q^{8} - 3 q^{10} + 159 q^{11} + 3 q^{13} + 336 q^{14} - 45 q^{16} + 207 q^{17} - 3 q^{19} - 681 q^{20} + 111 q^{22} - 33 q^{23} + 435 q^{25} - 1914 q^{26} - 12 q^{28} + 51 q^{29} + 111 q^{31} - 1647 q^{32} - 513 q^{34} + 1257 q^{35} - 3 q^{37} + 525 q^{38} - 6 q^{40} - 447 q^{41} + 516 q^{43} + 2211 q^{44} - 3 q^{46} - 2109 q^{47} - 591 q^{49} + 4938 q^{50} - 1350 q^{52} - 2736 q^{53} - 12 q^{55} + 7773 q^{56} - 888 q^{58} - 3048 q^{59} + 57 q^{61} + 2118 q^{62} - 195 q^{64} - 3297 q^{65} + 2082 q^{67} - 3573 q^{68} + 1524 q^{70} + 3105 q^{71} - 219 q^{73} - 9006 q^{74} - 1425 q^{76} + 8985 q^{77} - 1401 q^{79} - 9870 q^{80} - 12 q^{82} + 8511 q^{83} - 1827 q^{85} - 12507 q^{86} - 3693 q^{88} + 5202 q^{89} + 267 q^{91} - 5118 q^{92} - 2211 q^{94} - 5178 q^{95} + 1569 q^{97} + 4392 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1 −0.920010 − 5.21764i 0 −18.8598 + 6.86439i 8.05153 + 6.75603i 0 7.26495 + 2.64423i 31.9745 + 55.3815i 0 27.8430 − 48.2255i
28.2 −0.748840 − 4.24688i 0 −9.95772 + 3.62432i −14.2354 − 11.9449i 0 −17.6749 − 6.43313i 5.59920 + 9.69809i 0 −40.0687 + 69.4011i
28.3 −0.392076 − 2.22357i 0 2.72699 − 0.992543i 14.7308 + 12.3606i 0 −0.614463 − 0.223646i −12.3077 − 21.3175i 0 21.7091 − 37.6013i
28.4 −0.181650 − 1.03019i 0 6.48925 − 2.36189i −11.8235 − 9.92108i 0 22.1231 + 8.05215i −7.79628 − 13.5036i 0 −8.07284 + 13.9826i
28.5 −0.172250 − 0.976877i 0 6.59292 − 2.39963i −0.110676 − 0.0928685i 0 −23.8444 − 8.67865i −7.44756 − 12.8996i 0 −0.0716572 + 0.124114i
28.6 0.336951 + 1.91095i 0 3.97937 − 1.44837i 2.07510 + 1.74122i 0 11.1989 + 4.07607i 11.8703 + 20.5600i 0 −2.62817 + 4.55212i
28.7 0.652567 + 3.70089i 0 −5.75323 + 2.09401i −4.67730 − 3.92472i 0 −16.2160 − 5.90215i 3.52788 + 6.11047i 0 11.4727 − 19.8713i
28.8 0.751659 + 4.26287i 0 −10.0895 + 3.67228i 3.28965 + 2.76034i 0 19.2025 + 6.98913i −5.92381 − 10.2603i 0 −9.29428 + 16.0982i
55.1 −4.39879 + 1.60103i 0 10.6577 − 8.94287i 0.544255 + 3.08662i 0 −23.3667 − 19.6070i −13.8388 + 23.9695i 0 −7.33583 − 12.7060i
55.2 −3.91280 + 1.42414i 0 7.15349 − 6.00249i −2.58756 − 14.6748i 0 3.34538 + 2.80711i −2.78612 + 4.82571i 0 31.0237 + 53.7346i
55.3 −1.71186 + 0.623068i 0 −3.58609 + 3.00909i 2.34910 + 13.3224i 0 17.5105 + 14.6931i 11.5509 − 20.0068i 0 −12.3221 − 21.3425i
55.4 −1.08402 + 0.394551i 0 −5.10893 + 4.28690i −1.33360 − 7.56321i 0 9.93710 + 8.33822i 8.46114 − 14.6551i 0 4.42972 + 7.67249i
55.5 0.790355 − 0.287666i 0 −5.58645 + 4.68758i 1.87827 + 10.6522i 0 −20.4470 − 17.1571i −6.43113 + 11.1390i 0 4.54878 + 7.87873i
55.6 2.77394 − 1.00963i 0 0.547010 − 0.458996i −2.95956 − 16.7845i 0 −9.82851 − 8.24710i −10.7539 + 18.6263i 0 −25.1558 − 43.5711i
55.7 2.99548 − 1.09027i 0 1.65587 − 1.38944i 1.24508 + 7.06121i 0 3.72090 + 3.12220i −9.30563 + 16.1178i 0 11.4282 + 19.7942i
55.8 4.98740 − 1.81526i 0 15.4506 − 12.9646i −0.957594 − 5.43079i 0 18.8623 + 15.8274i 32.2942 − 55.9352i 0 −14.6342 − 25.3472i
109.1 −3.99118 + 3.34900i 0 3.32456 − 18.8545i 13.1992 − 4.80413i 0 −1.78859 − 10.1436i 29.0343 + 50.2889i 0 −36.5916 + 63.3784i
109.2 −2.82567 + 2.37102i 0 0.973494 − 5.52096i −1.51818 + 0.552573i 0 5.89800 + 33.4492i −4.41508 − 7.64714i 0 2.97972 − 5.16102i
109.3 −2.44296 + 2.04989i 0 0.376839 − 2.13716i −13.4713 + 4.90315i 0 −5.59176 − 31.7124i −9.29592 − 16.1010i 0 22.8590 − 39.5929i
109.4 −0.493691 + 0.414256i 0 −1.31706 + 7.46943i −0.483256 + 0.175891i 0 −1.66756 − 9.45719i −5.02191 − 8.69820i 0 0.165716 − 0.287028i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.4.e.b 48
3.b odd 2 1 243.4.e.c 48
9.c even 3 1 81.4.e.a 48
9.c even 3 1 243.4.e.a 48
9.d odd 6 1 27.4.e.a ✓ 48
9.d odd 6 1 243.4.e.d 48
27.e even 9 1 81.4.e.a 48
27.e even 9 1 243.4.e.a 48
27.e even 9 1 inner 243.4.e.b 48
27.e even 9 1 729.4.a.c 24
27.f odd 18 1 27.4.e.a ✓ 48
27.f odd 18 1 243.4.e.c 48
27.f odd 18 1 243.4.e.d 48
27.f odd 18 1 729.4.a.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.e.a ✓ 48 9.d odd 6 1
27.4.e.a ✓ 48 27.f odd 18 1
81.4.e.a 48 9.c even 3 1
81.4.e.a 48 27.e even 9 1
243.4.e.a 48 9.c even 3 1
243.4.e.a 48 27.e even 9 1
243.4.e.b 48 1.a even 1 1 trivial
243.4.e.b 48 27.e even 9 1 inner
243.4.e.c 48 3.b odd 2 1
243.4.e.c 48 27.f odd 18 1
243.4.e.d 48 9.d odd 6 1
243.4.e.d 48 27.f odd 18 1
729.4.a.c 24 27.e even 9 1
729.4.a.d 24 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + 3 T_{2}^{47} + 3 T_{2}^{46} - 57 T_{2}^{45} - 126 T_{2}^{44} + 702 T_{2}^{43} + \cdots + 18\!\cdots\!16 \) acting on \(S_{4}^{\mathrm{new}}(243, [\chi])\). Copy content Toggle raw display