Properties

Label 243.4.e.c
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

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 −0.751659 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
28.2 −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
28.3 −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
28.4 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
28.5 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
28.6 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
28.7 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
28.8 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
55.1 −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
55.2 −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
55.3 −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
55.4 −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
55.5 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
55.6 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
55.7 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
55.8 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
109.1 −3.85878 + 3.23790i 0 3.01699 17.1102i −14.6375 + 5.32761i 0 0.978906 + 5.55165i 23.6100 + 40.8938i 0 39.2325 67.9527i
109.2 −2.61428 + 2.19364i 0 0.633205 3.59108i 10.3200 3.75618i 0 −0.765086 4.33902i −7.42861 12.8667i 0 −18.7397 + 32.4581i
109.3 −1.78548 + 1.49819i 0 −0.445841 + 2.52849i 5.86493 2.13466i 0 −0.469354 2.66184i −12.3152 21.3306i 0 −7.27356 + 12.5982i
109.4 −0.228936 + 0.192100i 0 −1.37368 + 7.79050i −18.8424 + 6.85806i 0 3.73178 + 21.1640i −2.37749 4.11794i 0 2.99626 5.18968i
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.c 48
3.b odd 2 1 243.4.e.b 48
9.c even 3 1 27.4.e.a 48
9.c even 3 1 243.4.e.d 48
9.d odd 6 1 81.4.e.a 48
9.d odd 6 1 243.4.e.a 48
27.e even 9 1 27.4.e.a 48
27.e even 9 1 inner 243.4.e.c 48
27.e even 9 1 243.4.e.d 48
27.e even 9 1 729.4.a.d 24
27.f odd 18 1 81.4.e.a 48
27.f odd 18 1 243.4.e.a 48
27.f odd 18 1 243.4.e.b 48
27.f odd 18 1 729.4.a.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.e.a 48 9.c even 3 1
27.4.e.a 48 27.e even 9 1
81.4.e.a 48 9.d odd 6 1
81.4.e.a 48 27.f odd 18 1
243.4.e.a 48 9.d odd 6 1
243.4.e.a 48 27.f odd 18 1
243.4.e.b 48 3.b odd 2 1
243.4.e.b 48 27.f odd 18 1
243.4.e.c 48 1.a even 1 1 trivial
243.4.e.c 48 27.e even 9 1 inner
243.4.e.d 48 9.c even 3 1
243.4.e.d 48 27.e even 9 1
729.4.a.c 24 27.f odd 18 1
729.4.a.d 24 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 3 T_{2}^{47} + 3 T_{2}^{46} + 57 T_{2}^{45} - 126 T_{2}^{44} - 702 T_{2}^{43} + 23901 T_{2}^{42} - 48852 T_{2}^{41} - 38196 T_{2}^{40} + 23553 T_{2}^{39} + 2340873 T_{2}^{38} - 16419240 T_{2}^{37} + \cdots + 18\!\cdots\!16 \) acting on \(S_{4}^{\mathrm{new}}(243, [\chi])\). Copy content Toggle raw display